THE

METHOD of FLUXIONS

AND

INFINITE SERIES;

WITH ITS

Application to the Geometry of 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

ANN OTATIONS, ILLU STRATION s, and SUPPLEMENTS,

In order to make this Treatife

Acomplcat Inftitution for the ufe o/' LEARNERS.

By JOHN CO L SON, M. A. andF.R.S.

Mafter of Sir Jofeph fFilliamfon's free Mathematical-School at Rochejter.

LONDON: Printed by HENRY WOODFALLJ

And Sold by JOHN NOURSE, at the 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 merit and proficience in the fame Studies. For they knew very well, that fuch only could be competent Judges of their Works, and would receive them with ''the efteem. they might deferve. So far at leaft I can copy after thofe great Originals, as to chufe a Patron for thefe Speculations, whofe known skill and abilities in fuch matters will enable him to judge, and whofe known candor will incline him to judge favourably, of the fhare I have had in the prefent performance. For as to the fundamental part of the Work, of which I am only the Interpreter, I know it cannot but pleafe you ; it will need no protection, nor ean it receive a greater recommendation, than to bear the name of its illuftrious Author. However, it very naturally applies itfelf to you, who had the honour (for I am fure you think it fo) of the Author's friendship and familiarity in his life-time ; who had his own confent to publifli nil elegant edition of fome of his pieces, of a nature not very different from this ; and who have fo juft an efteem 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- monies of gratitude in 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- pinefs of your acquaintance,) which is, the free 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{^t»fcu«., 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, that is, to the general improvement of Science.

Before I take my leave, permit me, good Sir, to join my wiOies to thofe of the publick, that your own ufeful Lu- cubrations may fee the light, with all convenie-nt 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 are very well apprized, as all other 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 can- dor, 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, preparatory and introductory to his other moft arduous and fubh'me Speculations, and intended by himfelf for the instruction of Novices and Learners. I therefore gladly embraced the opportunity that was put into my hands, of publishing this pofthumous Work, be- caufe I found it had been compofed with that view and defign. And that my own Country-men might firft enjoy the benefit of this publication, I refolved upon giving it in an Englijh Translation, •with fome additional Remarks of my own. I thought it highly injurious to the memory and reputation of 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 (we may fay) ignominioufly rejected as infufficient, by fome Mathe- matical Gentlemen, who feem not to have derived their knowledge of them from their only true Source, that is, from cur 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 to have been as little acquainted with this Work, (or at leaft to have over-look'd it,) the 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- calional, and far from that copioufnefs with which he treats of it here, and illuftrates it with a great variety of choice Examples.

The learned and ingenious Dr. Pemberton, as he acquaints us in his View of Sir Tfaac Newton's Philofophy, had once a defign of publishing this Work, with the 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 receiving the la ft Emendations of its great Author, and likewife in faffing through the hands of fo able an Editor. And among the other good effects of this publication, poffibly it might have prevent- ed all or a great part of thofe Difputes, which have fince been raifed, and which have been fo ftrenuoufly and warmly pnrfued on both fides, concerning the validity of the Principles of this Method. They would doubtlefs have been placed in fo good a 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 in- ferior 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 Fluxions, and bring it to the teft of all impartial and con- fiderate Mathematicians ; to mew its evidence and Simplicity, to maintain and defend it in his own way, to convince his Opponents, and to teach his Difciples and Followers upon what grounds they mould proceed in vindication of the Truth and Himfelf. And that this might be done the more eafily and readily, I refolved to accom- pany it with an ample Commentary, according to the beft of my fkill, and (I believe) according to the mind and intention of the Au- thor, wherever I thought it needful ; and particularly with an Eye to the fore-mention'd Controverfy. In which I have endeavoui'd to obviate the difficulties that have been raifed, and to explain every thing in fo full a 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 rea-

fon

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 only the caufe of Truth at heart; I fhall be very glad to have con- tributed any thing, towards the removing of their Scruples. But if it fhall happen otherwife, and what is here offer'd fhould not appear to be furricient evidence, conviction, and demonflration to them ; yet I am perfuaded it will be fuch to moil other 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 num- ber of Examples they will find here, to which the Method of Fluxions is fuccefsfuUy apply'd, are fo many vouchers for the truth of the Principles, on which that Method is founded. For the Deductions 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 can- not, as well as with fuch as will not, enter into the proof of the Principles themfelves. And the hypothefn that has been advanced to evade this conclufion, of one error in reafoning being ilill corrected by another equal and contrary to it, and that fo regularly, conftantly, and frequently, as it mufl be fiippos'd to do here ; this bvpothe/is, I fay, ought not to be ferioufly refuted, becaufe I can hardly think it is ferioufly propofed.

The chief Principle, upon which the Method of Fluxions is here built, is this very fimple one, taken from the Rational 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 con- ceived as generated after a like manner. Confequently there mufl be comparativeVelocitiesofincreafeanddecreafe, during fuch generations, whole Relations are fixt and determinable, and may therefore /pro- blematically) be propofed to be found. This Problem our Author here folves by the hjip of another Principle, not lefs evident ; which fuppofes that Qnimity is infinitely divifible, or that it may (men- tally at leaft) fo far continually diminifh, as at lafl, before it is totally extinguifh'd, to arrive at Quantities that may be call'd vanilhing Quantities, or whk.li are infinitely little, and lefs than any afTign- able Quantity. Or it funnolcs that we may form a Notion, not indeed of abioiute, but of relative and comparative infinity. 'Tis a very jufl exception to the Method of Indivifibles, as aifo to the foreign infiniteiimal Method, that they have rccourfe at once to

a 2 infinitely

The PREFACE.

infinitely little Quantities, and infinite orders and gradations of thefe, not relatively but absolutely fuch. They affume thefe Quantities finnd & Jewel, without any ceremony, as Quantities that actually and obvioufly exift, and make Computations with them accordingly ; tlie refult of which muft needs be as precarious, as the abfblute ex- iftence of the Quantities they afiume. And fome late Geometricians have carry 'd thefe Speculations, about real and abfolute Infinity, ftill much farther, and have raifed imaginary Syftems of infinitely great and infinitely little Quantities, and their feveral orders and properties j which, to all fober Inquirers into mathematical Truths, muft cer- tainly 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 Infinity, as fuch, can hardly be 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 none but infinitely little Quantities that are relatively fo ; which he arrives at by beginning with finite Quantities, and proceeding by a gradual and neceffary progrefs of diminution. His Computations always commence by finite 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 con- ftant practice even in common Algebra and Geometry, and is no more than defcending from a general Propofition, to a particular Cafe which is certainly included in it. And from thefe eafy Principles, managed with a vaft deal of fkill and fagacity, he deduces his Me- thod of Fluxions j which if we confider only fo far as he himfelf has carry'd it, together with the application he has made of it, either here or elfewhere, directly or indiredly, exprefly or 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?, and new Applications, have been fince 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 perfection ; and will always afford an inexhauftible fund of curious matter, to reward the pains of the ingenious and iuduftrious Analyft.

As I am defirous to make this as fatisfactory as 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

great veneration for ; I fhall here endeavour to obviate fome of his principal Objections to the Method of Fluxions, particularly fuch 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 Lines. I fhall reprefent the matter Analytically thus, agreeably (I think) to the mind of the Author.

Let X and Y be two variable Lines, or Quantities, which at dif- ferent periods of time acquire different values, by flowing or increa- fing continually, either equably or alike inequably. For inflance, let there be three periods of time, at which X becomes A — fa, A, and A -+- 7 a ; and Y becomes B — f3, B, and B -+- f b fuccefiively and reflectively ; where A, a, B, b, are any quantities that may be aiTumed at pleafure. Then at the fame periods of time the variable Produ<ft or Rectangle XY will become A" — fa x B — f4, AB, and A •+- f * x B -+- ±h, that is, AB — T<?B — fM. -f- ±ab, AB, and AB -+- f^B -f- 7$ A -f- ^ab. Now in the interval from the firft period of time to the fecond, in which X from being A — fa is become A, and in which Y from being B — 7^ is become B, the Product XY from being AB — f^B — i£A -f- ^ab becomes AB -, that is, by Sub- traction, its whole Increment during that interval is f#B -+- f£A — ^ab. And in the interval from the fecond period of time to the third, in which X from being A becomes A-f-ftZ, and in which Y frcm being B becomes B -hf^, the Product XY from being AB becomes AB-f- ffiB -f f 4A -+- -^ab ; that is, by Subtraction, its whole Increment during that interval is 7,76 + 7^A -+- ^ab. _ Add thefe two Increirents together, and we fhall have <?B -+- bA. for the compleat Increment of the Product XY, during the whole interval of time, while X fk w'd from the value A — \a to A -f- ftf , or Y flow'd from the value B — f£ to B +7''. Or U might have been found by tne Operation, thus: While X f.ows from A — \a to A, and therce to A -f- ft?, or Y flows f-om B — f3 to B, and thence to B -i- f A, the Product XY will flow fiom AB — f<?B— f3A -f- ±ab to AB, ?nd thence to AB -+- f^B + -J'k -f- ^ab •> therefore by Sub- traction the whole Increment during that interval of time will be tfB-4-M. Q^E. D.

This may eafily be illuftrated by Numbers thus: Make A,rf,B,/, equal to 9, 4, i 5, 6, refpeclively; (or any other Numbers to be af- fumed at pleafure.) Then the three fucceffive values of X will be 7, 9, ii, and the three fucceffive values of Y will be 12, 15, 18,

reipcciivcly.

xiv The PREFACE.

refpeftively. Alfo the three fucceflive values of the Produd XY will be 84, 135, 198. But rtB-f-M = 4xic-f- 6x9= 114 = 198_84. Q.E. O.

Thus the Lemma will be true of any conceivable finite Incre- ments whatever; and therefore by way of Corollary, it will be true of infinitely little Increments, which are call'd Moments, and which was the thing the Author principally intended here to demonflrate. 15ut in the cafe of Moments it is to be confider'd, that X, or defi- nitely A — ftf, A, and A -+- ±a, are to be taken indifferently for the fame Quantity ; as alfo Y, and definitely B — f/;, B, B -+- ~b. And the want of this Confutation has occafion'd not a few per- plexities.

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, make B = Aa, and therefore b-=. zaA, and then the Moment of AxA*, or A', will be 2rfA4-f- aA1, or 3^A*. Again, make B = A5, and therefore l> = ^aAs-, and then the Moment of A xA*, or A4, will be 3<?A3 -4-rfA3, or 4#A3. Again, make B==A-», and therefore ^ = 4^A3, and then the Moment of Ax A4, or A', will be 4<?A4 -i-tfA4, or 5<zA4. And fo on in infinitum. Therefore in general, afluming m to reprefent any integer affirmative Number, the Moment of A* will be maA™"1.

Now becaufe A* x A^ra= i, (where m is any integer affirmative •Number,) and becaufe the Moment of Unity, or any other conftant quantity, is = p ; we (hall have A* x Mom. A~m -f- A~m x Mom. A"= o, or Mom. A~"= — A-110 x Mom. A" . But Mom. A" = maAm~*, as found before ; therefore Mom. A"* = — A~iw x ma A"-' = — maA-"-' . Therefore the Moment of Am will be maAm~I, when m is any integer Number, whether affirmative or negative.

•

And univerfally, if we put A" =B, or A"=. B" , where m and n may be any integer Numbers, affirmative or negative ; then we

mall have ma A"-* = ;.^B"^' , or b= mgA<° = -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

be rtill wtfA"*"1, whether ;;/ be affirmative or negative, integer or fraction.

The Moment of AB being M -+- aB, and the Moment of CD being </C •+- cD ; fuppofe D = AB, and therefore d-=. b& •+- aB, and then by Subftitution the Moment of ABC will be bA •+- aB xC -f- c AB = MC -+- rfBC -h r AB. And likewife the Moment of A*B" will be «/>B"-'A" -f- maA.m~lBn. 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 in- creafe, 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 write x, y, z, &c. Then the Fluxion of xy will be xy -f- xy, the Fluxion of xm will be rnxx*-* , whether m be integer or fraction, affiimative or negative; the Fluxion of xyz will be xyz -f- xyz -f- xjz, and the Fluxion of xmyn will be mxxm-*y» -J- nxmyy"~s . And fo of the reft.

Or the former Inquiry may be placed in 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 of Y ; then will AB and AB -f- aB-\~ bA+ab be two fucceflive 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 Numbers thus: Let A, a, B, b, be equal to 7, 4, 12, 6, refpectively ; then will the two fucceflive values of X be 7, 1 1 , and the two fuc- ceflive 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- nifh'd, as to become infinitely lefs than A and B ; at the fame time ab will become infinitely lefs than either aB or ^A, (for aB. ab :: B. b, and bA. ab :: A. ay) and therefore it will vanifh in 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 way of proceeding, in the t relent Inquiry ; but, as there was 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- ing recourfe to that Principle, that quantities arifing in an Equation, which are infinitely lefs than the others, may be neglected or ex- punged 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- ner, as to leave (I think) no room for any juft exceptions to it. 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 x», (or rather its relation to the Increment of x} confider'd as given ; becaufe Increments and Moments can be known only by comparifon with other Increments and Moments, as alfo Fluxions by comparifon with other Fluxions ;) let us make x"=y, and let X and Y be any fynchronous Augments of x and y. Then by the hypothefis we fhall have the Equation x-fc-X\* =y -+- Y ; for in 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 x ^=-^—*X * + n x *~ x '-^-V^X 3 , &c. or re - moving the equal Quantities y and x", it will be Y = nxn~lX •+- ny. ^-x"--X * -+- n x ?-^- x ^^x'-'^X 3 , &c. So that when X deT

notes the given Increment of the variable quantity A,-, Y will here denote the fynchronous Increment of the indefinite power y or x" ; 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, then Y = 2xX -+- X l . That is, while x flows or increafes to x •+- X, .v* in the fame time, by its Increment Y = 2xX -+-X1, will increafe to .v1 4- 2xX -j- X1, which we otherwife know to be true. Again, fuppofe fl = 3, then Y = 3*1X -+- 3*Xa H- X3. Or while x in*. creafes to x r+- X, x"> by its Increment Y = 3^aX -h 3^XJ + X3 will increafe to x* -f- 3*1X -+- ^xX1 -+- X3. 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 Y be ever fo great, or ever fo little ; for the truth docs not at all de- pend on the circumftance of their magnitude. Nay, when they are infinitely little, or when they become Moments, it muft be true alfo, by virtue of the general Conclufion. But when X and Y are di- minifh'd in infinitum, fo as to become at laft infinitely little, the greater powers of X muft needs vanifli firft, as being relatively of an infinitely lefs vali e than the fmaller powers. So that when they are all expunged, we ihall neceflarily obtain the Equation Y=znx*~'X ; where the remaining Terms are likewife infinitely little, and confe- quently would vanifh, if there were other Terms in the Equation, which were (relatively) infinitely greater than themfelves. But as .there are not, we may fecurely retain this Equation, as having an undoubted right fo to do; and efpecially as it gives us anufeful piece of information, that X and Y, tho' themfelves infinitely little, or vanifhing quantities, yet they vanifli in proportion to each other as j to nx"~f. We have therefore learn 'd at laft, that the Moment by which x increafes, or X, is to the contemporary Moment by which xa increafes, or Y, as i is to nx"~s. And their Fluxions, or Velo- 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, that any unfair artifices, any leger-de-main tricks, or any Ihifting of the hypothefis, that have been fo feverely complain'd of, are at all made ufe of in this Inveftigation. We have legitimately derived this general Conclufion in finite Quantities, that in all cafes the re- lation of the Increments will be Y = nx"~lX + « x ~~x*'-1X*, &c. of which one particular cafe is, when X and Y are fuppofed conti- nually to decreafe, till they finally terminate in nothing. But by thus continually decreafing, they approach nearer and nearer to the Ratio of i to nx"~\ which they attain to at ihe very inftant of the'r vanifhing, and not before. This therefore is their ultimate Ratio, the Ratio of their Moments, Fluxions, or Velocities, by which x and xn continually increafe or decreafe. Now to argue from a general Theorem to a particular cafe contain'd under it, is certainly tine of the moft legitimate and logical, as well as one of the mofl ufual and ufeful ways of arguing, in the whole compafs of the Mathemc- ticks. To object here, that after we have made X and Y to ftand for fome quantity, we are not at liberty to make them nothing, or no quantity, or vanishing quantities, is not an Objection againft the

b Method

XVlll

Tte PREFACE.

Method of Fluxions, but againft the common Analyticks. This Method only adopts this way of arguing, as a conftant practice in the vulgar Algebra, and refers us thither for the proof of it. If we have an Equation any how compos'd of the general Numbers a, b, c, &c. it has always been taught, that we may interpret thefe by any particular Numbers at pleafure, or even by o, provided that the Equation, or the Conditions of the Queftion, do not exprefsly re- quire the contrary. For general Numbers, as fuch, may ftand for any definite Numbers in the whole Numerical Scale ; which Scale (I think) may be thus commodioufly reprefented, &c. — 3, — 2> — i, o, i, 2, 3,4, &c. where all poffible fractional Numbers, inter- mediate to thefe here exprefs'd, are to be 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 We are likewife told, that we may not give fuch values to general Symbols afterwards, as they could not receive at firft ; which if ad- mitted is, I think, nothing to the prefent purpofe. It is always moft eafy and natural, as well as moll regular, inftruclive, and ele- gant, to make our Inquiries as much in 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 only, and not a point of neceffity. Thus in the prefent cafe, in- flead 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 oag. 24. of this Treatife. In fhort, to require this is jufl the famexthing as to infift, that a Problem, which naturally belongs to Algebra, mould be folved by common Arithmetick ; which tho' poflible to be done, by purluing backwards all the fleps of the general procefs, yet would be very troubkfome and operofe, and not fo inflrudtive, or according to the true Rules of Art

But I am apt to fufpedr, that all our doubts and fcruples about Mathematical Inferences and Argumentations, especially when we are fatisfied that they have been juftly and legitimately conducted, may be ultimately refolved into 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 utterly difclaim,) but in refpedl of the Science itfelf. We are hardly brought to believe, that the Science is fo perfectly regular and uni- form,

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 do not readily admit, or eafily comprehend, that Quantities have an infinite number of curious and fubtile properties, fome near and ob- vious, others remote and abftrufe, which are all link'd together by a neceffary connexion, or by a perpetual chain, and are then only difcoverable when regularly and clofely purfued ; and require our . truft and confidence in the Science, as well as our induftry, appli- cation, 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 we may fay, in adhering to Rule and Analogy. That whenever we make any regular Portions, and purfue them through ever fo great a variety of Operations, according to the ftricT: Rules of 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' profecuted to infinity itfelf j for if we fuppofe fome Quantities to be- 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- pathize with them, and conform to 'em in all their changes ; and will always preferve their analogical nature, form, or magnitude, which will be 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 gene- ral Inquiries, Nature is infinitely prolifick in particulars that will refult from them, whether in a direct rubordination, or whether they branch out collaterally ; or even in particular Problems, we may often perceive that thefe are 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. Thefe are fome of 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 prac- tice and attentive confideration ; but more efpecially by a diligent

b 2 peruial,

xx The P R E F A C E.

peruial, and clofe examination, of this and the other Works of our illuftrious Author. He abounded in thefe fublime views and in- quiries, had acquired an accurate and habitual knowledge of all thefe, and of many more general Laws, or Mathematical Principles of a fuperior kind, which may not improperly be call'd The Philofophy of Quantity ; and which, aflifted by his great Genius and Sagacity, to- gether 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 per- fection imaginable, is indeed the fublimeft, as well as the moft diffi- cult of all Arts, if it properly may be call'd fuch ; as not being redu- 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 excellent Talent ; but however is far inferior to the Art of 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, may do very much. But to ftrike out new lights, to adventure where no footfteps had ever been fet before, nullius ante 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- tity, or of its moft eflential and moft general Laws ; had confider'd it in all views, had purfued it through all its difguifes, and had traced it through all its Labyrinths and Recefles j in a word, it may be faid of him not improperly, that he tortured and tormented Quantities all poflible ways, to make them confefs their 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 time of his 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 great Improvement by the introduction of decimal Fractions ; fo the common Algebra or Analyticks, as an univerfal Arithmetick, will receive a like Improvement by the admiffion of his Doctrine of in- finite Series, by which the fame analogy will be ftill carry'd on, and farther advanced towards perfection. Then he fhews how all com- plicate Algebraical Expreffions may be reduced to fuch Series, as will continually converge to the true 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 they are Roots of pure Powers, or of affected 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 in- deed 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 re- duces 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 Space defer ibed being continually given, to find the Velocity of the Mo- tion at any time propofcd. Aifo the inverfe Method of Fluxions has, for a foundation, the Reverfe of this Problem, which is, The Velocity of the Motion being continually given, to find the Space defer ibed at any time propofcd. So that upon the compleat Analytical or Geometri- cal Solution of thefe two Problems, in all their varieties, he builds his whole Method.

His firft Problem, which is, The relation 6J the f owing Quantities being given, to determine the relation of their Fhixiom, he difpatches very generally. He does not propofe this, as is ufualiy done, A flow- ing Quantity being given, to find its Fluxion ; for this gives us too lax and vague an Idea of the thing, and does not fufficiently fhew that Comparifon, which is here always to be understood. Fluents and Fluxions are things of a relative n.iture, and fuppofe two at leafr, whofe relation or relations mould always be exprefs'd bv Equations. He requires therefore that all fhould be reduced to Equations, by which the relation of the flowing Quantities will be exhibited, and their

comparative

xxii f/jg PREFACE.

comparative magnitudes will be more eafily eftimated ; as alfo the comparative magnitudes of their Fluxions. And befides, by this means he has an opportunity of refolving the Problem much more generally than is commonly done. For in the ufual way of taking Fluxions,- we are confined to. 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 may have an infinite variety of Solutions, which tho' dif- ferent in form, will yet all agree in the main ; and we may always chufe the fimpleft, or that which will beft ferve the prefent purpofe. He (hews alfo how the given Equation may comprehend feveral va- riable Quantities, and by that' means the Fluxional Equation maybe found, notwithstanding any furd quantities that may occur, or even any other quantities that are irreducible, or Geometrically irrational. And all this is derived and demonitrated from the properties of Mo- ments. He does not here proceed to fecond, or higher Orders of Fluxions, for a reafon which will be affign'd in another place.

His next Problem is, An Equation being propofed exhibiting the re- lation of the Fluxions of Quantities, to find the relation of thofe Quan- tities, or Fluents, to one another ; which is the diredt Converfe of the foregoing Problem. This indeed is an operofe and difficult Problem, taking it in its 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 .done, or when there is no fuch finiie Algebraical Equation, as is moft commonly the cafe, yet however he finds the Root of that Equation by an infinite converging Series, which anfwers the fame purpofe. And often he mews how to find the Root, or Fluent required, by an infinite number of fuch Series. His proceffes for extracting thefe Roots are peculiar to himfelf, and always contrived with much fub- tilty and ingenuity.

The reft of his Problems are an application or an exemplification of the foregoing. As when he determines the Maxima and Minima of quantities in all cafes. When he mews the Method of drawing Tangents to Curves, whether Geometrical or Mechanical ; or how- ever the nature of the Curve may be defined, or refer'd to right Lines or other Curves. Then he {hews how to find the Center or Radius of Curvature, of any Curve whatever, and that in a fimple but general manner ; which he illuftrates by many curious Examples,

and

fbe PREFACE. xxiii

and purfues many other ingenious Problems, that offer themfelves by the way. After which he difcufTes another very fubtile and intirely new Problem about Curves, which is, to determine the quality of the Curvity of any Curve, or how its Curvature varies in its progrefs through the different parts, in refpect of equability or inequability.

He then applies himfelf to confider the Areas of Curves, and fhews us how we may find as many Quadrable Curves as we pleafe, or 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- bola, or of any other Curve that (hall be affign'd ; which he extends to Mechanical as well as Geometrical Curves. He then determines the Area in general of any Curve that may be propofed, chiefly by the help of infinite Series ; and gives many ufeful Rules for afcer- taining the Limits of fuch Areas. And by the way he fquares the Circle and Hyperbola, and applies the Quadrature of this to the con- ftructing of a Canon of Logarithms. But chiefly he collects very- general and ufeful Tables of Quadratures, for 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- felf, in his Treatife of Quadratures. The ufe and application of thefe 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 us how we may find as many Curves as we pleafe,. whofe Curve- lines are capable of Rectification ; or whofe Curve-lines, as to length, may be compared with the Curve-lines of any Curves that fha.ll be affign'd. And concludes in general, with rectifying any Curve-lines that may be propofed, either by the aflifbncc of his Tables of 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 that perhaps" may be expected, of what I have added in my Anno- tations ; I {hall refer the inquifitive Reader to the PrefacCj which will go before that part of the Work.

THE

;• -

THE

CONTENTS.

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

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

Prob. 2. From the given Fluxions to find the Fluents. — — p. 25

Prob. 3. To determine the Maxima and Minima of Quantities, p. 44

Prob. 4. To draw Tangents to Curves. p. 46

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

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

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

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

Prob. 9. To find the Quadrature of any Curve ajjigrid. p. 86

Prob. 10. To find any number of rettifiable Curves. p. 124

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

Prob. 12. To rectify any Curve-lines 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 cultivating of the Analytical Art ; by the affiftance of which they have been able to overcome fo many and fo great difficulties, that they feem to have exhaufted all the Speculations of Geometry, excepting the Quadrature of Curves, and 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 Species, and the fame Operations in common Numbers; nor do they feem to differ, except in the Characters by which they are re-

B prefented,.

'The Method of FLUXIONS,

prefented, the firft being general and indefinite, and the other defi- 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- drature of the Hyberbola by Mr. Nicolas Mercator ;) efpecially fince 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- tick ; the Operations of Addition, Subtraction, Multiplication, Di- vifion, and Extraction of 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 be- tween Decimal Fractions and Algebraick Terms infinitely continued. For as in Numbers, the Places towards the right-hand continually decreafe in a Decimal or Subdecuple Proportion ; fo it is in Species refpedtively, when the Terms are difpofed, (as is often enjoin 'd in what follows,) in an uniform Progreflion infinitely continued, ac- cording 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- fure acquire the nature of Integers, and may be managed as fuch ; fo it is a convenience attending infinite Series in Species, that all kinds of complicate Terms, ( fuch as Fractions whofe Denomina- tors are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Clafs of fimple Quantities ; that is, to an infinite Series of Fractions, whofe Numerators and Denominators are fimple Terms ; which will no longer labour under thofe difficulties, that in the other form 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.

3. Reduction by Divifion and Extraction of 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

• . ..ift Av

Examples of Reduttion by Dhifwn. IjfM/l^^ '* /•

.4. The Fraction ^™ being propofed, divide aa by b + x in the following manner :

faa aax aax1 a a x* aax* .

» " .

aax

aax O— --7 — -f-O

aax*

o -+-

o - +o

flt *» ** Jf*

~ ;•.

-rr^i_ *-\ " v i r * ^^ tf*^1 a* x* a* x* . a* X+ ~

The Quotient therefore is T_-JT-+-T_ . — rr+T7-, &c. which Series, being infinitely continued, will be equivalent to £j^. Or making x the firft Term of the Divifor, in this manner,

x + toaa + o (the Quotient will be - - ?4 4. 1^« —V &c~ e , , % r~ _ _ * **n*» AV

found as by the foregoing Procefs.

5. In like manner the Fraction ~- will be reduced to I — #• -{- x4 — ' A:* H- x8, &c. or to x-* — #-* _f. ^-« — ^-8

2* "

9 v

6. And the Fraction r will be reduced to 2x^ — 2x

i s i+x*— 3*

•+• yx1 — 13** -j- 34xT, &c.

7. Here it will be proper to obferve, that I make ufe of x-', x-', x-', x-*, &c. for i, ;r 7,' -• &c. of xs, xi, x^, xl, A4, &c.

for v/x, v/*S \/x*> vx , ^xl, &c. and of x'^, x-f. x-i &c for , i j_^ ' * **** 1Ui

^ x ^?>' y-^.' &c. And this by the Rule of Analogy, as may be apprehended from fuch Geometrical Progreflions as thefe ; x», x*, x«> (or i,) a"*,*-',*'*, *•», &c.

B 2 8.

x,

ffie Method of FLUXIONS,

er for ', &c.

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

q. And thus inftead of^/aa — xx may be wrote aa — 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- mative and Negative, Integral and Fractional.

Examples of Reduction by Extraction of Roots.

11. The Quantity aa -+- xx being propofed, you may thus ex- tract its Square-Root.

-„ _i_ Vv (a -4- — — — 4- — — — 5 x - 4- J— • — — — - — ' c*

aa-+- XX ^" 2a Sfl3 r i6«* 128«7 2560*

aa

xx

4. a*

x*

~*

a 4 64 ««

X*

sT*

64 a«

~

64^8 " z$6a'^

i; x

5*

64^

_ 256 *

64 a 6 I z8rt8

+

_- 7^ _ 2^1, &c.

1 i7R/3» n-- /7lt>

7'1

+

,__i!_lll, &c.'

Jo that the Root is found to be a~\--^-— ^ 4- ^T,&C. Where it may be obferved, that towards the end of the Operation I neg- lect all thofe Terms, whofe Dimenfions would exceed the Dimenfions of the laft Term, to which I intend only to continue the Root,

fuppofe to *—' ,2.

and INFINITE SERIES. 5

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

a a

10 A* iz« A- »

13. Thus the Root of aa — xx is « — ^ — -Jj -- ^7

14. The Root of x — xx is #'" — i** — 4-.v* — T'r**, 8cc.

. . £ AT A.' A' b*X* g

15. Of «« -+- «f — ## is a -f- — — — -- ^ , Sec.

. i + <z *• A- . i 4- '- « * * — i a * A- 4 + ,'_ n 3 x- 6. &c- j

1 6. And v/r^rr, « .Ii*«»--».«4-. ;,,»««. .c. and more-

over by adually dividing, it becomes

i -|- -i/^r + |^^4 -+- ^frx6, &c. -4- T^ -f- T^ H- rV^x

17. But thefe Operations, by due preparation, may very often 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 </ 1 — bxx, which would

y^i -f-rt*1— - ab x *

have produced — & and the reft of the work might

I — b x x

have been performed by extracting the Root of the Numerator only, and then dividing by the Denominator.

1 8. From hence I imagine it will fufficiently appear, by what means any other Roots may be extracted, and how any compound Quantities, however entangled with Radicals or Denominators, (fuch

Vx — \fi — xx Vxi!2xt — xi v

as x"> -}- — — — •; _. j may be reduced to

^/axx -\- A- 3 * x-{-xx — " 2X — x.1 '

infinite Series confifting of iimple Terms.

Of the ReduStion of offered Equations.

19. As to aftedled Equations, we mufl be fomething more par- ticular in explaining how their Roots are to be reduced to fuch Se- ries as thefe ; becaufe their Doctrine in Numbers, as hitherto de- liver'd by Mathematicians, is very perplexed, and incumber'd with fuperfluous Operations, fo as not to afford proper Specimens for per- forming the Work in Species. I fhall therefore firfl (hew how the

Refolu-

Method of FLUXIONS,

Refolutidn of affected Equations may be compendioufly perform'd in Numbers, and then I fhall apply the fame to Species.

20. Let this Equation _yl — zy — 5 = 0 be propofed to be re- folved, 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 produced a new Equation p> 4- 6pl 4- iop — i =o, whofe Root is to be fought for, that it may be added to the Quote. Thus rejecting />> 4- 6//1 becaufe of its 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 1,23^ 4- 0,06 1 =o is near the truth, or ^= — 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 prin- cipal 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 before. And thus I continue the Operation as far as I pleafe, in the manner of the following Diagram :

y~' — zy — 5 =o	+ 2, IOOOOOOO
+ 2,09455148, &c. =y
Z+p=J>. + 7 * — 27	— 4— zp
The Sum	-i + iop+6p* + p->
+ i°/	+ o3ooi+ 0,035 +o, 5 5 2 + 2* + o, 06 + i32 + 6, + 1, + 10,
1 he 6um	o, 061 -|- 1 1) 23 i + 6, 3 q * + 2*
— o,oo54 + r= q. <ji + II,2?? + 0,06 1	— o, oooooo i f74^+ o,ooo0#7-4&V — 0, 0tfai » +)•' + 0,00018370^ 0,06804: +^;? — 0,060642 +11,23 + o, 061
The Sum	+ 0,0005416 +II,l62r
— 0,000048^2 + * = r.

21.

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- gram, as there remain places to be fill'd up in 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 reject all that are after fo many fewer places. And fo on, by pro- ceeding 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 I fubftituted 0,0054 -f- r for q, where four 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 them ; and indeed I might alfo have omitted the firft Term r J, 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 Quote to the Period required. Then fubtracting the negative part of the Quote from the affirmative part, there arifes 2,09455148 for the Root of the propofed Equation.

22. It may likewife be obferved, that at the beginning of the Work, if I had doubted whether o, i -f-/> was a fufficient Ap- proximation to the Root, inftead of iof> — i = o, I might have fuppos'd that o/** -f- i op — i = o, and fo have wrote the firft Figure of its 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 Figure of the Quote, when in 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 the laft Term multiply'd into the Coefficient of the antepenulti- mate Term. And indeed you will often fave fome pains, efpecially in Equations of many Dimensions, if you feek for all the Figures

to-

8 Tie Method of FLUXION'S,

to be added to the Quote in this manner ; that is, if you extract 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 either is, or may be fuppos'd to be, much the leaft or greateft of all, or neareft 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 to the Root required ; as may appear from what is faid before of the Species x, in the Examples of Reduction by Divifion and Ex- traction of Roots. And for this Species, in what follows, I mall generally make ufe of A: or z ; as alfo I fliall ufe y, p, q, r, s, &c. for the Radical Species to be extracted.

25. Secondly, when any complex Fractions, or 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

y* -+- j— 1>'1 — x"= = o,. multiply by b — x, and from the Pro- duct by* Kyi'-l-fry* — bx^ -+• x*-= o extract the Root y. Or

we might fuppofe y x b — x=v, and then writing ^~x for yt we mould have i;J -+- &*v* — fax* -\- 3/5*** — ^hx' -+. x6 = o,. whence extracting the Root vr we might divide the Quote by b — x,, in order to obtain y. Affo if the Equation j3 — xy* -f- x$ = o were propofed, we might put y?= v, and xj = z, and fo wri- ting vv for y, and z* for x, there will arife v6 — z=v -f- z* = o ; which Equation being refolved, y and x may be reftored. For the Root will befound^=2-f-s3_|_5~s55cc.andrei1:onngjyandA;, we have y* = x^ -f- x -+- 6x^ &c. dien fquaring, y =x^-+- 2XJ ~f- 13*", &c..

26. After the fame manner if there mould be found negative Di- menfions ofx and jy, they may be removed by multiplying by the fame x andjy. As if we had the Equation x*-}-T>x*-y~I—'2.x~I — i6y-3=o, multiply by x and j3, and there would arife x*y* -+- 3#3jy1 — 2_v5

A J -r 1 -r-v • aa 2ai i 1 a 4»

O. And U tjie Equation were x = — — ~ + ?—r

y\. by;

and INFINITE SERIES.

by multiplying into jy} there would arife xy*-=.a'iy*— And fo of others.

27. Thirdly, when the Equation is thus prepared, the work be^ gins by finding the firfr. Term of the Quote ; concerning which, as alfo for finding the following Terms, we have this general Rule, when the indefinite Species (x or 2) is fuppofed to be fmall ; to which Caie the other two Cafes are reducible.

28. Of all the Terms, in which the Radical 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 which that Radical Species is found, fuch as that the Progreflion of the Dimenfions of each of the fore-mentioned Species, being con- tinued from the Term fir ft afTumed to this Term, may defcend as much as may be, or afcend as little as may be. And if there are any other Terms, whofe Dimenfions may fall in with this Progreflion continued at pleafure, they muft be taken in 1 ike- wife. Laftly, from thefe Terms thus felected, and made equal to nothing, find the Value of the faid Radical Species, and write it in the Quote.

29. But that this Rule may be more clearly apprehended, I fhall explain it farther by help of the following Diagram. Making a right Angle BAC, divide its fides AB, AC, into equal parts, and raifing Perpendiculars, diftribute the Angular Space into equal Squares or Parallelograms, which you may conceive to be denominated from the Dimenfions of the Species x and y,

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-

lograms fo mark'd, of which let one

be the loweft in the left-hand Column at AB, the other touching

the Ruler towards the right-hand ; and let all the reft, not 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.

30. Thus to extract the Root y out of the Equation y6 — 5xys-+-

— •)'* — ja*x1y1+6aix*-\-&1x4=o, I mark the Parallelograms belong-

C

B

A 4	ft	Xlj*	*4;5	.1-4:4
A3	*3	X3£	A? 3	A 5 4
X*	A'*	x*y*	**. 3
X	xy	*!*	A -;	v,4
1	y	}*	s1	4

ing

10

The Method of FLUXIONS,

B

A

*

C

ing to the Terms of this Equation 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 lower to the upper, till it begins in like manner to touch another, or perhaps more, of the Parallelograms that are mark'd ; and I fee that the places fo touch'd belong to x3, x*-y*y and_y5. Therefore from the Terms y6 — 7azx*-y<L-}-6a*x*, as if equal to nothing, (and moreover, if you pleafe, reduced to v6 — 7^*4- 6= o, by making $=rv'\fitxt) I feek the Value of y, and find it to be four- fold, -\-</ax, — </ax, -+-</2ax, and — ^/2ax, of which I may take any one for the initial Term of the Quote, according as I defign to extract this or that Root of the given Equation.

31. Thus having the Equation y* — 6y*-i-()&x* — x3=o, I chufe the Terms — by- -\-gbx*-, and thence I obtain 4-3* for the initial Term of the Quote.

32. And having y">-i-axy-{-aay — x* — 2rt3=o, I make choice of y'-i-a^y — 2<23, and its Root -\-a I write in the Quote.

33. Alfo having x*ys—— ^c^xy1 — cI.va4-£7=o, I felect vViyf4-<r7J

which gives — ^/c— for the firft Term of the Quote. And the

like of others.

34. But when this Term is found, if its Power fhould happen to be negative, I deprefs the Equation by the fame Power of the indefinite Species, that there may be no need of depreffing it in the Refolution ; and befides, that the Rule hereafter delivei'd, for the fuppreffion of fuperfluous Terms, may be conveniently apply'd. Thus the Equation 8z;6_)i34-^25>'a — 27^5=0 being propofed, whofe

Root is to begin by the Term ^ I deprefs by s% that it may be- come Sz+yt-^azy — 2ja!>z~1=o, before I attempt the Refolu- tion.

3 5. The fubfequent Terms of the Quotes are derived by the fame Method, in the Progrefs of the Work, from their feveral fecondary Equations, but commonly with lefs trouble. For the whole affair is perform'd by dividing the loweft of the Terms affected with the indefinitely fmall Species, (x, x1, x3, &c.) without the Radical Spe- (/>, q, r} &c.) by the Quantity with which that radical Species

i of

and INFINITE SERIES, n

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

Example, the Terms -> ~} - ~> &c. are produced by dividing

alx, TrW", TTT-v3, &c. by ^aa.

36. Thefe things being premifed, it remains now to exhibit the Praxis of Refolution. Therefore let the Equation y*-{-azy-\-axy — za* — xz=o be propofed to be refolved. And from its Terms y=-\-a*y — 2«3=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 a+p=y, and inftead of y, in the Terms of the Equation written in the Margin, I fubftitute a-\-p, and the Terms refulting (/>3-{- 3rf/1-f-,?,v/>, &c.) I again write in the Margin ; from which again, according to the third of the Premifes, I felect the Terms -+-^p -H2l.v=o for a fictitious Equation, which giving p= — ^x, I write — ~x in the Quote. Then becaufe — ^.v is not the accurate Value of p, I put — ±x-\-q=p, and in the marginal Terms for p I fubftitute — ^x-t-q, and the refulting Terms (j3 — -^x^+^a^, &c.) I again write in the Margin, out of which, according to the fore- going Rule, I again feledl the Terms 4^ — _I3-drx*=o for a ficti- tious Equation, which giving £=^> I write -^ in the Quote. Again, fince ^ is not the accurate Value of g, I make -^--{-r=qt and inftead of a I fubftitute ~--\-r in the marginal Terms. And

&4« '

thus I continue the Procefs at pleafare, as the following Diagram exhibits to view.

12

Method of FLUXIONS,

•X3

•2a'

• axp

; 643

— ±axq

*-

- X*

T '

•a*-x

*

*

'31** 509*4

37. If it were required to continue the Quote only to a certain Period, that x, for inilance, in the laft Term {hould not afcend beyond a given Dimenfion ; as I fubftitute the Terms, I 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- tity, fo many Terms are to be added to the right-hand, as 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 the Species .v in the Quote,) mould afcend no higher than to four Dimenfions, I omit all the Terms after A-*, and put only one after x=.

Therefore

and INFINITE SERIES. 13

Therefore the Terms after the Mark * are to be conceived to be expunged. And thus the Work being continued till at laft we come

to the Terms -^— -^--H-rfV— ±axr,'m which />, q, r, or

reprefenting the Supplement of the Root to be extracted, are only of one Dimenfion ; we may find fo many Terms by Divifion,

131*3 _, 509*4 \ as we fl^n £e wantjng to compleat the Quote.

16384(13 /

5121.

'SI*'

509*4

... XX 13 1.*' kuyAT _

So that at laft we {hall have y=a — 7*-f"6^-t-^l~*- r^I; icc-

39. For the fake of farther Illustration, I mail propofe another Example to be refolved. From the Equation -L_y< — .Ly4_f_iy3 — iy=. _^_y — z=o, let the Quote be found only to the fifth Dimenfion, and the fuperfluous Terms be rejected after the Mark,

_!_£5j &c.

+ ^5, &c.

-L;S4 Z'p, &C.

6cc.

2;

s, &c. % &c.

40. And thus if we propofe the Equation T4-rjrJ' '+TT|-T )'' + -rTT;'7-t-TW'J-i-r.)'3+y — £=o, to be refolved only to the ninth Di- menfion of the Quote ; before the Work begins we may reject the Term -^^y" ; then as we operate we may reject all the Terms beyond 2', beyond s7 we may admit but one, and two only after

Y4 The Method of FLUXIONS,

zf ; becaufe we may obferve, that the Quote ought always to afcerrd by the Interval of two Units, in this manner, z, .sj, zs , &c. Then at laft we fliall have ;'=c— fs3_j__|_.s»_ T_5__2;^_J_^_'T^_.39)&C. 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, exceeds the greateft Dimenfion required in the Quote ; or from, 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 injini- tum. And fo in this Equation

8 -f-31 4S4-f-92lS l6«8, &C.

) — j'1 in z* — s4-}- z6 — z*y &c.

that the Cubick Root may be extracted only to four Dimenfions of z, I omit all the Terms in infinitum beyond -f-j5 in z,1 — J.-4_|_.L2«> and all beyond — y- in z1 — a4-(-.c6, and all beyond -+-y in .c1 — 2z4, and beyond — S-}-;stt — 424. And therefore I aflurr.e this Equation only to be refolved, -^z6y* — ±z*y* -{-?•*•• ;> — s6^1-}-^4^1 — z^y* — 2z*y -i-z'-y — 4s4_j_si — 8=0. Becaufe?. ',(*''- ~^{' Term of the Quote,) being fubflituted inflead of y in the reft of the Equation deprefs'd by z^y gives every where more than four Dimenfions.

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

.r1 A* A 4 -

h-r-f--; &c.

as far as the Period xf, I omit all the Terms in infinititm., beyond — y in <?_[-*•+— ' and affume only this Equation, j* — ay — xy —

2" \ 4

-y+ —=0. This I refolve either in the ufual manner, by making

& 4-*-*

and IN FINITE SERIES.

j-^; or more expedition fly by the Method of affected Equations deliver'd before, by which we fhall have _}'=•— 3 — — #> where the laft Term required vanifhes, or

becomes equal to nothing.

43. Now after that Roots are extracted to a convenient Period, they may fometimes be continued at pleafure, only by oblerving the Analogy of the Series. So you may for ever continue this z-t-i-z* ^_^.25_j__'_2;4_{_Ti_2;sj &c. (which is the Root of the infinite Equa- tion 5r==)'-f-^i_j_^5_|_±y4j foe.) by dividing the laft Term by thefe Numbers in order 2, 3, 4, 5, 6, &c. And this, z — f^-H-rlo-^' — ' yj lTB.27-f_TrT'TTy2;9j &c. may be continued by dividing by thefe Num-

bers 2x3, 4x5, 6x7, 8x9, &c. Again, the Series

"-'g ,» &c. may be continued at pleafure, by multiplying the Terms refpectively by thefe Fractions, f } — 7, — £, — -£, — TV, &c> And fo of others.

44. But in difcovering the firft Term of the Quote, and fome- times 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 known. As in the Example y*-\-axy-{-ii*-y — x3 — 2a>=o : If the Root of this Equation y^^-a'-y — 2«5=o, had been furd, or un- known, I mould have put any Letter b for it, and then have per- form'd the Refolution as follows, fuppofe the Quote found only to the third Dimenfion.

i6

fbe Method of FLUXIONS,

y s -\-aay-\r£txy — 2 a 3 — ; , tf^A- «4£jCft	^=0. Make a--\-T,b1=c2, then ii | (v*r*
	rTTv* .8 ,8 ,10 . •
AT3	— \-b~i -f-?^i^-j-2^/:1-f-/)J
~" w :;	«5;'3A3 — ' — j — &C. A'3
6<?£1A.-^ C43.V1	«3i3,;S /,.4iA* ~X* 3 3.%3
iz / 1 4	~* + t« ( ,« h^ r8

45. Here writing £ in the Quote, I fuppofe b-±-p=y, and then for y I fubftitute as you fee. Whence proceeds p'^-^bp1, &c. re- jecting the Terms b'-^a'-b — 2tf3, as being equal to nothing : For b is fuppos'd to be a Root of this Equation jy3_j_fl*y — 2<?3=o. Then

the Terms ^p-^-a^p-^-abx give '/^V* :1 to be fet in the Quote,, and

to be fubflituted for p.

46. But for brevity's fake I write a- for aa-^-^l>l>, yet with this caution, that aa-\-^bb may be reflored, whenever I perceive that the Terms may be abbreviated by it. When the Work is finim'd, I aflume fome Number for a, and refolve this Equation y*-\-?.'-\' — 2^;=o, as is fhewn above concerning Numeral Equations ; and I fubftitute for b any one of its Roots, if it has three Roots. Or rather, I deliver fuch Equations from Species, as far as I can, efpe- cially from the indefinite Species, and that after the manner before insinuated. And for the reft only, if any remain that cannot be expunged, I put Numbers. Thus y'-^-a^y — 2^5=o will be freed from a, by dividing the Root by a, and it will become y*+)' — 2=0, whofe Root being found, and multiply'd by a, muft be fubftituted

inftead of b.

47-

and INFINITE SERIES, 17

47. Hitherto I have fuppos'd the indefinite Species to be little. But if it be fuppos'd to approach nearly to a given Quantity, for that indefinitely fmall difference I put fome Species, and that being fubftituted, I folve the Equation as before. Thus in the Equation •f}-' — ^y* -+- ^yl — ±y* -t-y -\-a — x = o, it being known or fup- pos'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 arife ±y — ±y* -f- jj5 — ±y* -{-y + z=o, which is to be folved as before.

48. But if that Species be fuppos'd to be indefinitely great, for its Reciprocal, which will therefore be indefinitely little, I put fome Species, which being fubflituted, I proceed in the Refolution as before. Thus having y* -+-\l -f-jv — x> =o, where x is known or fuppos'd to be very great, for the reciprocally little Quantity

- I put z, and fobflituting - for .v, there will arife y> -f-.)'1 •+• y — ~ =o, whofe Root is .y = ^ — •- — ^z + £z* -f- ^2', &c. where x being reflored. if you pleafe, it will be y=:x — - H- — H — —

J •* 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 in the Equation y* — x^y1 -+- xy* -f- Z)1 — 2y -+- i = o, whereas the firft Term ought to be obtain'd from the Suppofition that jy-4_j_2yt — 2y + 1 = 0, which yet admits of no poffible Root; you may try what can be done another way. As you may fuppofe that x is but little different from •+• 2, or that 2-{-z-=x. Then fubftituting 2-{-z inftead of A*, there will arife y* — z'-y* — -\zy* — 2y -f- 1 = 0, and the Quote will begin from -j- i. Or if you

fuppole x to be indefinitely great, or l- = z, you will have ^4—

>* y1

•--{-- -+-2y* — 2y H- i = o, and -f- z for the initial Term of the Quote. ,

50. And thus by proceeding according to feveral Suppofitions, you may extract and exprefs Roots after various ways.

51. If you mould delire 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 fake, will happen in the Equation y* -}-axy-+-aly — x> — 203 = o,

D by

4

1 8 The Method of FLUXIONS,

by fubftituting -f-rf, or — a, or — za, or — 2«}|T, &c. inftead of .v. And thus you may conveniently fuppofe the 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. And farther ftill., by fubftituting any fictitious Values for the inde- finite Species, fuch as az + bz1, •£-> ~n^> &c. and then proceeding as before in the Equations that will refult.

52. But now that the truth of thefe Conclufions may be mani- feft ; 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 whofe Roots are p, y, r, s, &c. and that as they vanifh, the Roots p, q, r, s, &c. that is, the differences between the 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- ftroy one' another, you may conclude^ that the Quote fo far ex- tracted is the perfect Root of the Equation. But if it be other- wife, you will fee however, that the Terms in which the indefi- nitely fhiall Species is of few Dimenfions, that is, the greate ft 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.

53. Laftly, altho' the Species, which for the fake of perfpieuity I have hitherto fuppos'd to be indefinitely little, fhould however be fuppos'd to be as great as you pleafe, yet the Quotes will ftill 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

when there Is the greateft or leaft difference between the Sums of the affirmative Terms, and of the negative Terms ; and is limited when the indefinite Quantity, (which therefore not improperly I fuppos'd to be fmall,) cannot be taken greater, but that the Mag- nitude of the Root will immediately become infinite, that is, will 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

— Sax -f- f- S^x 4- — Sax> &c- that is> when * = ** i but

la " ga* V i6a> V

it will be terminated when x — a. For if we take x greater than

at the Sum of all the Terms — ^ Sax — s7» Vax — TbTs *Sax> &c. will be infinite. There is another Limit alfo, when x = o, by reafon of the impoffibility of the Radical S — ax ; to which Terms or Limits, the Limits of the Semicircle A, B, and D, are cor^ refpondent.

Tranfttion to the METHOD OF FLUXIONS.

55. And thus much for the Methods of Computation, of which I mall make frequent ufe in what follows. Now it remains, that , for an Illuftration of the Analytick Art, I mould give fome Speci- mens of Problems, efpecially fuch as the nature of Curves will fup- ply. But firft it may be obferved, that all the difficulties of thefe x may be reduced to thefe two Problems only, which I mall propofe concerning a Space defcribed by local Motion, any how accelerated ' or retarded. ~

56. I. The Length of the Space defcribed being continually ( that -*"*£ ?V, at fill Times) given; to find the Velocity of the Motion at any ffo^

Tune propofed. / SJLJ tt

57. II. The Velocity of the Motion being continually given ; to find JbotA.*** if* the Length of the Space defcribed at any Time propofed.

58. Thus in the Equation xx=y, if y reprefents the Length of the Space «t any time defcribed, which (time) another Space x,

by increafing with an uniform Celerity #, mea/ures and exhibits as

D 2 defcribed :

20 ?%e Method of FLUXIONS,

defcribed : Then zxx will reprefent the Celerity by which the Space y, at the fame moment of Time, proceeds to be defcribed ; 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 Fluxion, to which the reft may be referr'd, as it 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 under- ftood 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 2 indefinitely increafing, I fhall hereafter call Fluents, or Flowing

Quantities, and fhall reprefent them by the final Letters of the f £ Alphabet v, x, y, and z ; that I may diftinguifh them from other

Quantities, which in Equations are to be confider'd as known and. T H > %f& f'df** determinate, and which therefore are reprefented by the initial U» _' .i V*> i*i~- Letters a, b, c, &c. And the Velocities 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 pointed thus -y, x, y., and z. That is, for the Celerity of K t4 JO the Quantity v I fhall put v, and fo for the Celerities of the other id tti Quantities x, y, and z, I fhall put x, y, and z refpeftively.

J '(/ 6 1. Thefe 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- prefs'd, according to the 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 all the Products equal to nothing,, aad you will have the Equation required.

2. EXAMPLE i. If the Relation of the flowing Quantities A; and y be X' — ax*--{- axy — ^3=o; firft difpofe the Terms according to x, and then according to y, and multiply them in the follow- ing, manner.

Mult.

by

makes %xx* — zaxx -{- axy * — zyy* -f- ayx *

• • • • *

The Sum of the Produdls is -jx** — zaxx -k- axy — W*-f- ayx=zo,

i . •

which Equation gives the Relation between the Fluxions x and y.

For if you take x at pleafure, the Equation .v3 — ax1 -{-axy — yt = o will give y. Which being determined, it will be x : y :: 7v* — ax : yx^—zax -{- ay.

3.. Ex. 2. If the Relation of the Quantities x, y,. and zr be ex- preis'd by the Equation 2j3 -f- x*y — zcyz •+- yz* — z'' = QJ

*»	— ax*	+ ffxy-	-r	— >':	•JT axy.	—ax1
3*	2x	X		iy .	_v
—	-^ •	V •	o		~- *	O
X	X	x		3	y

Mult. 2j3 -i-xxxy — z*	yx* -+- zy*	— z* -fc- 3_>-21 — zcyz •+• x'y
— zcz ~f"~ 32;*	' — zcyz	-h zy3
ay y	2X	; 2~ ±
DV *"* • O . "•	— . o .	— . — . - o.
'• y y	x	z z z
makes 4^-* % 4-'~	zxxy %	-2zz*+6zzy-zcZy .

Where-

22 *The Method of FLUXIONS,

Wherefore the Relation of the Celerities of Flowing, or of the Fluxions ,v, v, and z, is tyy* -\- +• 2xxy — $zzl -f- 6zzy — zczy

.

4. But fince there are here three flowing Quantities, .v, y, and z, another Equation ought alfo to be given, by which the Relation among them, as alfo among their Fluxions, may be intirely deter- mined. 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- tions, by expunging any one of the three Quantities, and alfo any one of the Fluxions, and then you will obtain an Equation which will intirely determine the Relation of the reft.

5. In the Equation propos'd, whenever there are complex Frac- tions, or furd Quantities, I put fo many Letters for each, and fup- pofing them to reprefent flowing Quantities, I work as before. Af- terwards I fupprefs and exterminate the afTumed Letters, as you fee done here.

6. Ex. 3. If the Relation of the Quantities .v and y be yy — aa

— x\/aa — ## = o; for x</aa — xx I write z, and thence I have the two Equations^' — aa — %,•=.&., and a3-*1 — x4 — 2* i — . o, of which the firfl will give zyy — z = o, as before, for the Relation of the Celerities y and z, and the latter will give 2<j*xx

o, or a*xx~ **** = z, for the Relation of the

Celerities x and z. Now z being expunged, it will be zyy -

= o, and then reftoring x^aa — xx for z, we fhall have zyy

-./»** 4- g*.>* __ 0> for the Relation between x and y, as was re-

^ aa — XX

quired.

7. Ex. 4. If .v3 — ay* 4- j4r — XX \fay -+- xx = o, expreffes

the Relation that is between AT and v : I make ^^ = 5;, and

^x \/~ay-+-xx=v, from whence I fhall Lave the three Equations x- — ay* + & — -u = o, az-\-yz — ^3=o, and ax*y •+• x6 — 1^=0. The firft gives 3**' — zayy •+• z — -0=0, the fecond gives az •+• Zy^-yz — 3^& = o, and the third gives 4.axx>y-+-6xx'-i-a}>x* — 2W= o, for the Relations of the Velocities -y, .v, y, and «. But

the

and INF i NIT E SERIES. 23

the Values of & and i', found by the fecond and third Equations, iSj ££? for z and

/. /. 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>

. XX \/ ay -+- xx, there will arife the Equation fought ^xx*-—2ayy

— 6*- A- 3 — awMf ... . _ . . r ,

— = o. by which the Relation or the

•>

. aa -f- 2^ + yy 2

Velocities x and y will be exprefs'd.

8. After what manner the Operation is to be performed in other Cafes, I believe is manifefl from hence j as when in the Equation propos'd there are found furd Denominators, Cubick Radicals, Ra-

dicals within Radicals, as v ax -+- \/ 'aa — xx} or any other com- plicate Terms of the like kind.

9. Furthermore, altho' in the Equation propofed there fhould be Quantities involved, which cannot be determined or exprefs'd by any Geometrical Method, fuch as Curvilinear Areas or the Lengths of Curve-lines ; yet the Relations of their Fluxions may be 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 by an Equation. Let AB be called A;, and the Area of the Curve ADB, apply 'd to Unity, be call'd z. Then erect the Perpendicular AC equal to Unity, and thro' C draw CE parallel to AB, and meeting BD in E. Then conceiving thefe two Superficies ADB and ACEB to be generated by the Motion of the right Line BED ; it is manifeft that their Fluxions, (that is,, the Fluxions of the Quantities i x zt. and i x v, or of the Quantities s and x,) are to each other as the generating Lines BD and BE. Therefore « : x :: BD : BE or i, and therefore z = * x BD.

1 1. And hence it is, that z may be involved in any Equation, expre fling the Relation between .v and any other flowing'Quantityjv ; and yet the Relation of the Fluxions x and y may however be dif- cover'd, 12.

24 <fhe Method <J/" FLUXION s,

12. Ex. 5. As if the Equation zz -\-axz — _y*=r=o were pro- pos'd to exprefs the Relation between x and;1, as alfo \/ax—xx = BD, for determining a Curve, which therefore will be a Circle. The Equation zz-^-axz — j^=o, as before, will give 2zz-i- azX -f- axz — 4_y_y» = o, for the Relation of the Celerities x,y, and z. And therefore fince it is z = x x BD or • — -x \/ax — xxt iubftitute this Value inftead of it, and there will 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.

13. The Moments of flowing Quantities, (that is, their indefi- nitely fmall Parts, by the acceffjon of which, in indefinitely fmall portions of Time, they are continually increafed,) are as the Ve- locities of their Flowing or Increafing.

14. Wherefore if the Moment of any one, as x, be reprefented t>y the Product of its Celerity x into an indefinitely fmall Quantity o (that is, by xo,} the Moments of the others <y, y, z, will be reprefented by vot yo, zo ; becaufe voy xo, yo, and zo, are to each other as v, x, y, and x.

,. p. , 15. Now fince the Moments, as xo and yo, are the indefinitely

/fc«, »// natti** cttA uttie ^cceflions of the flowing Quantities .v and y, by which thofe

any

And therefore the Equation, which at all times indifferently exprefles the Relation of the flowing Quantities, will as well exprefs the Relation between x -3- xo and y-+-yo, as between x and y: So that x -+- xo and y -f- yo may be fubftituted in the fame Equation for thofe Quantities, inftead of x and y.

1 6. Therefore let any Equation #' — ax* -+- axy — ^' = 0 be given, and fubftitute x~\-xo for x} and y -j- yo for y, and there will arife

•+• $x*oox -f- x*o''

ax1 — 2axox — ax*oo

• • axy •+- axoy -h ayox -h axyoo

y: —lyoy- ~ yfooy —

and INFINITE SERIES. 25

17. Now by Suppofition x3 — ax°--3raxy — _}'3=o, which there- fore being expunged, and the remaining Terms being divided by o, there will remain ^xx* -f- ^ox -+- x>oo — zaxx — ax1o -f- axy -f- ayx _f_ axyo — 3_vy* — 3y*oy — y*oo = o. But whereas o is fuppofed to be infinitely little, that it may reprefent the Moments of Qiian- tities ; the Terms that are multiply'd by it will be nothing in relbedl 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 being divided by o, will always acquire the form that they ought to have by the foregoing Rule : Which was the thing to be proved.

19. And this being now fhewn, the other things included in the Rule will eafily follow. As that in the propos'd Equation feveral flowing Quantities may be involved ; and that the Terms may be multiply'd, not only by the Number of the Dimenlions of the flow- ing Quantities, but alfo by any other Arithmetical 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 Examp. 3, 4, and 5, will be plain enough of itfelf.

P R O B. II.

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

A PARTICULAR SOLUTION.

i. As this Problem is the Converfe of the foregoing, it muft be folved by proceeding in a contrary manner. That is, the Terms multiply'd by x being difpofed according to the Dimenfions of x ;

they muft be divided by *x , and then by the number of their Di- menfions, or perhaps by fome other Arithmetical Progreffion. Then the fame work muft be repeated with the 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

by - • Quot. 3A:5 — 2ax* -\-ayx

Divide by 3 . 2 i.

Quote A;5 — ax1 -{-ayx

Divide —

by ^. Quot. —3

Divide by 3

Quote — _y5

* -f- ayx

* 4- axy

2 . i.

* 4- axy

Therefore the Sum #3 — ax* -f- axy — y* = o, will be the required Relation of the Quantities x and y. Where it is to be 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 Sagacity of the Artift -, for it would be needlefs to dwell 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,

after I have found the Equation x> ax1- -{- 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- axy — i,yy* -f- ayx= o. Whence it is plain, that the Equation AT3 • -ax*-+-axy — _y3 = o is rightly found. But if the Equation xx — xy -\- ay = o were propofed, by the prefcribed Method I fhould obtain this ^x* — xy + ay = o, for the Relation between x and y ; which Conclufion would be erroneous: Since by Prob. i. 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.

5. Firft it mufl be obferved, that in the propofed Equation the Symbols of the Fluxions, (fince they are Quantities of a diffe- rent kind from the Quantities of which they are the Fluxions,) ought to afcend in every Term to the fame number of Dimenfions :• And when it happens otherwife, another Fluxion of fome flowing Quantity mufl be underflood to be 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 the Equation x -+• x'yx — axx = o were propofed, the Fluxion z of fome third flowing Quantity z mufl be underilood to be Unity, by which the firfl Term x mufl be multiply'd once, and the lafl axx twice, that the Fluxions in them may afcend to as 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 ima- gine x to be Unity, by which the Term yy is multiply'd.

6. Now Equations, in which there are only two flowing Quan- tities, which every where arife to the fame number of Dimenfions, may always be reduced to fuch a form, as that on one fide may be

had the Ratio of the Fluxions, (as 4 , or - , or ~ ,&c.) and on the

\ x . y x

other fide the Value of that Ratio, exprefs'd by fimple Algebraic

*

Terms ; as you may fee here, 4- = 2 -h 2X — y. And when the

foregoing particular Solution will not take place, it is required that you fhould bring the Equations to this form.

7. Wherefore when in the Value of that Ratio any Term is de- nominated-by a Compound quantity, or is Radical, or if that Ratio be the Root of an affected Equation ; the Reduction mufl be per- form'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- pofed j firfl by Reduction this becomes T-=i-f--^-, or -==

x a—x y

a—v+y' And in the firfl Cafe, if I reduce the Term ^£^., deno- minated by the compound Quantity a — x, to an infinite Series of

E 2 fimple

28 The Method of FLUXIONS,

fimple Terms j -f- - -f- ~ -+- ^ &c. by dividing the Numerator y by the Denominator a — x, I mall have - — — i •+- - -f- ^ -f.

^ -f- 7; &c. by the help of which the Relation between x and y is to be determined.

9. So the Equation _y_y = xy -j- .XVY.V A: being given, or ^- = 4,

A-* x

•i- xx, and by a farther Reduction 4=4 +V/T -+- A-* : I extract

AT —

the fquare Root out of the Terms -J -f- xr, and obtain the infinite Series f -{-x* — x* -f- 2X6 — 5*" -f- 14*'°, &c. which if I fubfti-

tute for \/t H- xx, I (hall have - = i -f- x* — x* -f- 2x6

X

&c. or. ~ = — x^-ir-x* — 2X6 -+- 5*8, &c. according as

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"> — ~

2x*a>=o were propofed, or '— -f- ax— -f- a1- >v3 — 2rf3 = o

A:5 A: x

I extract the Root of the affected Cubick Equation, and there.

•/- V X XX 111*5 COQi'4 0

anfes ~ =a ^-—_|_ ^_ _ 4. » ^ &c. as may be feen

x 4 640 5i2«a 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 com- pounded only in refpect of given Quantities. For they may be re- duced to fimple Quantities by luppofing them equal to other givea

Quantities. Thus I eonfider the Quantities " -•> "-TT, — rr-

^ — - ^^' c a*4- b' ax-\~bx >

1 4 — — — — —

~^,L,xi > v/tfA- H- bx, &c. as fimple Quantities, becaufe they may may all be reduced to the fimple Quantities —^ i, -^-, — , \/ex (or

£x*} &cc. by fuppofing a -f- b =r= e.

12. Moreover, that the flowing Quantities may the more 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

flowing Quantities may be diftinguifli'd by the fame Names refpec- tively. And for the better understanding of what follows, you may conceive, that the Correlate Quantity is Time, or rather any other Quantity that flows equably, by which Time is expounded and 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 of Time, the Space defcribed in the whole Time may be deter- mined.

13. But in refpedt of this Problem Equations may be diftinguifli'd

into three Orders.

14. Firft: In which two Fluxions of Quantities, and only one of their flowing Quantities are involved.

15. Second: In which the two flowing Quantities are involved, together with their Fluxions.

1 6. Third: In which the Fluxions of more than two Quantities are involved.

17. With thefe Premifes I {hall attempt the Solution of the Problem, according to thefe three Cafes.

SOLUTION OF CASE I.

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

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

reduced, we mall have •- = i -f- x1 — .v4 -f- 2X&, &c. Now I mul-

tiply the Value of — into x, and there arifes .v-f-AT3' — xf -{- 2X\

&c. which Terms I divide feverally by their number cf Dimenfions, and the Refult x •+- fv' — fv'-f-fv1, &c. I put =y. And by

this

30 77je Method ^/"FLUXIONS,

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

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

x 4 6-}<z 5i2«*

will arife y = ax — y -+- ~ j- -^ ' &c. for determining the

' y ZM —OJ.oi.t~ o

Relation between A; and y.

21. And thus the Equation — = _i_ -, •, — x* -t- #*,

v-J *.! I — • I

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

*; Jf^ X X ,

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

22. After the fame manner the Equation -. =5-7=== 4- -^— -+-

\/ f S7- 1. A •

\- cy, gives A- = — ^_ -}- — H- - v/^)'3 -i- cy~> . For the Value of - being multiply'd by j, there arifes ~ -^ — *— _j_

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

23. And fo =? =z\ gives y = $z*. And -1 =- 4 , gives r= , ~ * «7

3f^L3. But the Equation ^ = ; , gives 7 = f . For f multiply'd

into A: makes a, which being divided by the number of Dimen- fions, which is o, there arifes ~ , an infinite 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 only ; inftead of the Correlate Quantity, fubftitute the Sum or the Difference between the fame and fome other given Quantity to be affumed at pleafure. For there will be the fame Relation of Flowing, of the Fluents in the Equation fo. produced, as of the Equation at firft propofed j and the infinite Relate Quan-

tity

and INFINITE SERIES. 31

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

25. Therefore the Equation 4 = - being propofed, if for x I write ^4- x, affuming the Quantity b at pleafure, there will arife

v 11 T^« • /* v fl a^ ax^ ax^ c At

•- = , — : and by Divifion 4 = T — rr 4- 77 — -rr &c- And

u-^r~X * v O & £ b +

now the Rule aforegoing will give_}'= j — - ^ 4- 3~£p — ~j^ &c. for the Relation between x and y.

26. So if you have the Equation - = - 4-3 — xx; becaufe

X X

of the Term ~x-> if you write i -f- x for x, there will arife 4 . — _f (_ 2 — 2X —xx. Then reducing the Term ~-^ into an in- finite Series 4-2 — 2x4- 2xl — 2Ar3 4- 2x% &c. you will have 4 ,

X

— ^ — 4* _{_ x* — 2x3 4- 2x4, &c. And then according to the Rule y = 4.x — ax1 4- fx3 — |x4 4- ^xs, 6cc. for the Relation of x

and y.

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

pofed j becaufe I here obferve the Term x l (or ~j to be found, I tranfmute x, by fubftituting I — • x for it, and there arifes 4 — . _' _L _•_ - - — v/ 1 — A;". Now the Term - l—x produces i _{_ x _|_ x1 4- x3, &c. and the Term \/i — x is equivalent to

j, .i# — 4-x1 • — —V^S an(^ therefore or •i_±v_JL;(.a ^ • is

the fame as i 4- -i-x 4- 4-x1 4- |-x3 , &c. So that when thefe Values are fubftituted, I fhall have 4 = i ~f- 2x 4- 4xi4-4-^-x3,6cc. And

X

then by the Rule y •=. x 4- x1 4- 4-x* 4- ri*4, &c- An<i ^ oi others.

28. Alfo in other Cafes the Equation may fometimes be con- veniently reduced, by fuch a Tranfmutation of the flowing Quantity.

As if this Equation were propofed 4 = -^ ^^.c^_xi • inflead

•52 ^ Method of FLUXIONS,

O i/

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

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

SOLUTION OF CASE II.

29". PREPARATION. And fo much for Equations that involve only one Fluent. But when each of them are found in the 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.

30. And befides, if in the Equations fo reduced there be any Fractions denominated by the flowing Quantity, they muft be freed from thofe Denominators, by the above-mentioned Tranfmutation of the flowing Quantity.

31. So the Equation yax — xxy — aax = o being propofed, or

i_l _{_ f . becaufe of the Term -, I afiume b at pleafure, and

x a x *

for x I either write b -+- x, or b — x, or x — - b. As if I fhould write b -+- x, it will become 4 = - -f- rrr. . And then the Term

being converted byDivifion into an infinite Series, we mall have

-1—-1 , - - < — — , &C.

72. And after the fame manner the Equation £••= 37 — 2x +

•J X

X 2v

- .. being propofed; if, by reafon of the Terms - and^.,

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

X

_ oV -4- 2 x -f- ^-=-^ -4- — 2-v~.2 r . But the Term '-— ^ by

3/ 1 y I ZX -\- X* 1 y J

infinite Divjfion gives i — x -+-y — xy -f-_ya — xy* -J-_y3 — xy*t &c. and the Term -t _^2~+ xx by a like Divifion gives 2_y — 2 -i- ^xy — ^x _f- 6x*-y — . 6xa 4- S*3^ — 8x5 + iox*y — IOAT*, &c. There- fore r-= — 3^-i- 3^J -f->'a' — xy* -{- y3 — ^y5, &c. -i- 6^^ — • 6x*

X

33-

and INFINITE SERIES. 33

33. RULE. The Equation being thus prepared, when need re- quires, difpofe the Terms according to the Dimenfions of the flow- ing Quantities, by fetting down fir ft thofe that are not affected by the Relate Quantity, then thofe that are affected by its lead Dimen- fion, and fo 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 to- wards the right hand, and the reft in a defcending Series in the left- hand Column, as the following Diagrams indicate. The work be- ing 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 man- ner as you obtain'd the firft. And by repeating the Operation you may continue the Quote as far as you pleafe. But this will appear plainer by an Example or two.

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

be propofed, whofe Terms i — T.V -+- A'1, which are not affected by the Relate Quantity _v, you fee difpos'd collaterally in the up-

	-h I T,X -\- XX
+'*,	* -+- A' X,Y-f-l.,V3 ^.x-4_|__'_,v	r,&c. J_ ^ V ' "5""^"	s,&c
The Sum	I ' 2.V "--I-"- &X * — V ^ - 1 * v4i T ^_ \s	, &c.
y	A—A-X -»4*I - >4 + ^,__Vx6^c.

permoft Row, and the reft ' y -and .vy in the left-hand Column. And rirft I multiply the initial Term i into the Correlate Quantity .v, .ind it makes x, which being divided by the number of Dimen- fions i, I place it in the Quote under-written. Then fubftkuting rhis Term inftead of y in the marginal Terms -f- y and -f- .vy, I have -\-x and -+- xx, which I write over againft them to the right hand. Then from the reft I take the loweft Terms — ?.v and -±-x, whofe aggregate — zx multiply'd into x becomes — 2.v.v, and

F being

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 being likewifc afiumed to compleat the Value of the Marginals -{-y and -+- xv, there will arife alfo — xx and — x5, to be 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 -|-.ix;, to be put in the Value of y. Again, taking this Term -i-x3 into the Values of the marginal Terms, from the next loweft -f-y#3 and — x3 added together, I obtain — ^-x4 for the fourth Term of the Value of y. And fo on in infinitum.

35. Ex AMP. 2. In like manner if it were required to determine

the Relation of x and y in this Equation, y- -=. I -f- - -f- --v -f- — r'-f-

< ^ a &* &*

- , &c. which Series is fuppofed to proceed ad infinitum ; I put I

in the beginning, and the other Terms in the left-hand Column, and then purfue the work according to the following Diagram.

	-hi
	A" A* *3 .X 4	-.j
+ ~		h —, , &c.
XV	A"a v 3 A 4	A *
4- £	a1 2^3 2^4	h z~ . &C-
Xs" V	_1_ 'v3 i A'4	, . 5
4- ~		h — , &c.
-4- ~	* * * * -+- — -	h S ' &c-
4-*-?	* * * * * -	h-J , &c.
a*
Sum	.V 3** 2\= CAT4 T _l_ *_ 1 — . 1 1 * I i ^ — r — i — "^~" —	3.V5 c h 4y , &c.
	a ^ai a= z.;4
y ==	* + Ta-+- ili + £ + ^ -	^6 o h — j , &c.

36. As I here propofed to extradl: the Value of y as far as fix Dimenfions of x only ; for that reafon I omit all the Terms in the Operation which I forefee will contribute nothing to my pur- pofe, as is intimated by the Mark, &c. which I have fubjoin'd to the Series that are cut off.

3 37-

and INFINITE SERIES. 35

37. EXAMP. 3. In like manner if this Equation were propofed

• = — 3,v -+- i*y -4-;* — Xj* -t-j3 — .vy3 -4-;-« — A^

— 6..Y1 -f- SA-J_V - — 8.v3 4- \oxy* — IOA-*, &c. and it is intended to extract the Value ot y as far as 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 beyond the degree of x~* ,) I fubjoin the fecond and third Powers of the Value of y, fo far gradually produced, that when they are fubftitu- ted by degrees to the right-hand, in the Values of the Marginals

	_ 3.v _ 6X> — 8*3 — IO.V^ — I2A- — M£ ,&CC.
+ 3*7	9v, 2"	— 6x*	b zo ' "
-+- 6x*y	* * *	— gx*	— I2.V — ^V ,&C.
-f- 8*7	* * *	*	I2AT* l6x6,fxc.
-f- IOA:^	* * *	#	* ^[J^6 j&C-
&c.
+-;•*	* * #	^|*4	-f- 6xs -{-~^7x6 ,&;c.
— xy*	* * *	*	4 * '
&C.
H-.v;	* * *	*	* — ~--xs ,6cc.
Sum	— 3 A- — 6x* — ^f.v	3 9' 4	— -^-'v' — -Z.v-6 li-r- ^ •* — .X ,tXC. h S '
3 2S	qi	111 6 ^"
y= -A1 2X> -*<	20	"16^ "77"r > C '
^ A '°7 * " 4"^ 8	«, &C.
y; — — — x6, 6cc.

to the left, Terms may arife of fo many Dimenfions rs I obferve to'be required for the following Operation. And by this Method

there arifes at length y= — ^x1 — 6.x13 — ^^+, &c. which is the

F 2 Equation

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 in- creafes. 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 Equation, - = iy — ^y- -+- zyx* — -J.v1 -f- 77* -f- 2_y;, in

	H— 5-7 * — 4-y1 -+• jy1 •+• 2>'3
I	* * +)'* * — 2_)'3-|-4}'T — 2_y4, &c. * * * * * * — ~y4y&tc.
Sum	+±y #_3r_f_7/ . +4/— 44-VS&C.
ATT=±= •+ 4_y — y1 -+- 2y* ' — _)•* , &c. A;*= -V74> ^c-

which ,v in the Term a^'-x11 (or zy^/x) is affected with the Frac- tional Dimenlion -i- From the Value of x I derive by degrees the Value of A?% (that is, by extracting its fquafe-Root,) as may be obferved in the lower part of this Diagram ; that it may be in- ferted and transfer'd gradually into the Value of the marginal Term 2yx'f. And fo at laft I fliall have the Equation x = ±.yl — y* _|_ 2_y^ -(- ^ — TVo^'f> &c- by which x is exprefs'd indefinitely in re- ipect of y. And thus you may operate in any other cafe what- foever.

39. I foid before, that thefe Solutions may be perform'd by an infinite variety of ways. T'his may 'be done if you afiiime at pleafure not only the initial quantity of the upper Series, but any other given quantity for the firft Term of the Quote, and then you may proceed as before. Thus in the firft of the preceding Exam- ples, if you affume i for the firft Term of the Value of 7, and fubftitute it for y in the marginal Terms -h_y and -t-xy, and pur- fue the reft of the Operation as before, (of which I have here given a

and INFINITE SERIES.

37

	-f- I 3x4- XV
4-*V	-4- i 4- 2x * 4- AT3 4- .ix4, 6cc. * -t- X 4- 2Arl * 4- X4, &C.
Sum	4-2 * 4- 3** 4- A;3 4-4-A"4, &c.
y - — i -f- 2.v * 4- x"' -\- ix4 4-^-A'5, 6cc.

Specimen,) another Value of y will arife, i -f- 2x-\- x* -h i*4, 6cc. And thus another and another Value may be produced, by afTum- ing 2, or 3, or any other number for its firfl Term. Or if you make ufe of any Symbol, as a, to reprefent the firft Term inde- finitely, by the fame method of Operation, (which I fhall here fet down,") you will find y = a -+- x -+- ax — xx -f- axx -+- ~x*+±ax*, &c. which being found, for a you may fubfHtute i, 2, o, 4-, or any other Number, and thereby obtain the Relation between x and y an infinite variety of ways.

	4- i — 3 x 4- A* AT
+y	_|_ fl _|_ x .v.V -	H yX3 , &c.
	4™ #^" 4~ ^ATX -	f- -i^.v3, 6cc.
4-#y	* -f. tf.v 4- AT1 -	- *s , &c.
	-(- ^ZAT1 -	f- ax* , &c.
Sum	4-1 2X 4- AT1 -	— AAr5 , &C.
	4-^4- 2^-4- 2«x»-	-f-l^x3, &c.
j = a 4- A; — x1	-h y-V3 ^-.V4 , &C.
4- ax 4- fl.v1 -	f- j.tfJfJ + _V^V45 &C.

40. And it is to be obferved, that when the Quantity to be ex- trailed is affected with a Fractional Dimenfion, (as you fee in the fourth of the preceding Examples,) then it is convenient to take Unity, or fome other proper Number, for its firft Term. And in- deed this is neceflliry, when to obtain the Value of that fractional Dimenfion, the Root cannot otherwife be extracted, becaufe oi the negative Sign ; as alib when there are no Terms to be diJpofcd in the firft or capital Clafs, from which that initial Term may be deduced. 41.

38 tte Method of FLUXIONS,

41. And thus at laft I have compleated this moft troublefo'me and of all others moft difficult 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 there- fore the Equation mould be reduced to another form. For thus

the Equation y = - — xx being propofed, where y is of one ne- gative Dimenfion, I might indeed reduce it to another Form, as by writing i -f- y for y ; but the Refolution will be more expe- dite as you have it in the following Diagram.

	#	* XX
I y Sum	i i	— V* -•-! — • ^ V JK* ^CC
y	4- .V "'i-YAT -f- |-.V3, &C. - — x-t-^xx, 5cc.

43. Here affuming i for the initial Term of the Value of y., . I extract the reft of the Terms as befoie, and in the mean time

I deduce from thence, by degrees, the Value of - by Divifion, and infert it in the Value of the marginal Term.

44. II. Neither is it neceffary that the Dimenfions of the other flowins Quantity fhould be always affirmative. For from the Equa- tion y = 3 -\- zy — '- , without the prefcribed Reduction of the

Term }~ , there will arife_y = 3 A; — ±xx -f- 2XJ, &c.

4^. And from the Equation y = — }'-+--. — ~x > the Value

of y will be found y ==• ^, if the Operation be perform 'd after the Manner of the following Specimen.

i

XX

and INFINITE SERIES,

3.9

	I	.
	*A:	.V
		I
— V	*	" .V
Sum	i	o
	ATA:
y =
* X

46. Here we may obferve by the way, that among the infinite manners by which any Equation may be refolved, it often happens that there are fome, that terminate at a finite Value of the Quan- tity to be extracted, as in the foregoing Example, And thefe are not difficult to find, if fome Symbol be aflumed for the firft Term. For when the Refolution is perform'd, then fome proper Value may -be given to that Symbol, which may render the whole finite.

47. III. Again, if the Value of y is to be extracted from this

Equation y = ^. -+- i — zx -f- ±xxy it may be done conveniently

enough, without any Reduction of the Term ~ , by fuppofing

(after the manner of Analyfts,) that to be given which is required. Thus for the firit Term of the Value of y I put zcx, taking 2<? for the numeral Coefficient which is yet unknown. And fubltituting 2.cx inftead of y, in the marginal Term, there ariies e, which I write on the right-hand ; and the Sum i -f- e will give x -f- ex for the fame firft Term of the Value of yt which I had firfi repre- fented by the Term zcx. Therefore I make 2cx = x-}-ex, and thence I deduce e =•. i. So that the firfl Term zex of the Value of y is 2.x. After the fame manner I make ufe of the fidlitious Term 2/x* to reprefent the fecond Term of the Value of r, and thence at laft I derive — ^ for the Value of y, and therefore that fe- cond Term is — ±xx. And fo the fictitious Coefficient g in the third Term will give TV, and b in the fourth Term will be o. Wherefore iince there are no other Terms remaining, I conclude the work is finiOi'd, and that the Value of y is exadtl-y zx — ±xl -if-^X', See the Operation in the following Diagram.

i

The Method ^FLUXIONS,

	I ~2X +iXX
y
	? 4~ /A* | - cfxx [ /yv'
Zx	6
Sum	4"~i ~~~ 2 A" 4~ •£ XX
Hvpothetically r= zex-{- 2fx*-\- 2gx* 4- 2&c+ II II 1l II
Confequentially y= 4->v — A* 4- ^x* 4- ^6^«
Real Value j'= 2 A* — l^1 4- ^-A-'

48. Much after the fame manner, if it were y = ^- ; fuppoie

y=.exs, where e denotes the unknown Coefficient, and s the num- ber of Dimeniions, which is alfo unknown. And ex' being fub-

ftituted for y, there will arife y •=. -— , and thence again 7 =

*— . Compare thefe two Values of y, and you will find ^ = e, and therefore s = •£•, and e will be indefinite. Therefore afTuming

e at pleafure, you will have y = ex*.

49. IV. Sometimes alfo the Operation may be begun from the higheft Dimenfion of the equable Quantity, and continually pro- ceed to the lower Powers. As if this Equation were given, ^=: 2.1.1 _i_T_i_2;r — -, and we would begin from the higheft

xx ~ XX ,. 3 * . °

Term zx, by difpofing the capital Series in an order contraiy to the foregoing ; there will arife at laft y = xx -f- 4.* — - , &c. as may be feen in the form of working here fet down.

	4 '
+.i	* H- i 4-^ *	i i e — - -h — > &C. -v * ^A *r
Sum	i	— • rr •+• ^7* ' ^cc>
_j> = A'1 4- 4.v * — ;	+ 1^ SIT > &c-

50.

and INFINITE SERIES, 41

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

the Terms 4** and — - , for the intermediate Term that is 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- menfion, to be equal to any third Fluent ; and then by ftibftitutii g that Quantity, as alfo its Fluxion, ariling from that fictitious Equation, inftead of the Relate Quantity and its Fluxion.

52. As if the Equation y= 3*7* -\- y were propofed, where the Relate Quantity is affected with the fractional Index .1 of its Dimen- fion; a Fluent z being afTumed at pleafure, fuppofe y^ = z, or y = z'> ; the Relation of the Fluxions, by Prob. i. will be y = 32Z1. Therefore fubftituting ^zz* for v, as alfo z* for y, and z* for y$, there will arife yzz1 = ^xz*- -+- z3, or z = x -\-^z, where z performs the office of the Relate Quantity. But after the

Value of z is extracted, as z = ±x* -f- — -f- ^ -J- -^-Q , &c. in- ftead of z reftore y\ and you will have the defired Relation be- tween x and v; that is, y? = i.v1 + -V^3H- T-nr*4; &c- an(^ ^7 Cubing each fide, y •=.^x6-\- T'_.v7 -+- TYTXS> ^c-

53. In like manner if the Equation y = </^y -+- </xy were given, or_y = 2^^ -J- xM ; I make z =)'^ or zz=y, and thence by Prob. i. 2zz = y, and by confequence 2zz = 2z -f- x*z, or z = i -+- {-x^. Therefore by the firft Cafe of this 'tis z = x -f-

-i-v1", or y'1 = Ar-f- -i.v1, then by fquaring each fide, v=y>; -+- -|Jf^ -i- -i-x5. But if you mould defire to have the Value of y exhibited an infinite number of ways, make z =. c -f- x -f- -ytf , aiTuming any initial Term c, and it will be ss, that is y, = c* -{- zcx + ^cx* •+• -v1 -+- -i-x1* -t- ^v3. But perhaps I may feem too minute, in treat- ing 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 may be found from thence ; fo that either of them, together with its Fluxion, may be exterminated. For which reafon if there are found the Fluxions of three Quantities, only one Equation need to be affumedj two if there be 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.

55. Let the Equation propofed be zx — z -f- yx = o ; that I may obtain the Relation of the Quantities x, y, and z, whofe Fluxions x, y, and z are contained in the Equation ; I form a Relation at pleafure between any two of them, as x and y, fuppofing that x=y, or 2y = a -+- z, or x=yy, &c. But fuppofe at prefent x=yy, and thence x = 2yy. Therefore writing zyy for x, and yy for x, the Equation propofed will be transform'd into this : q.yy — z-^-yy* = o. And thence the Relation between y and z will arife, 2yy-{-

^y= =.z. In which if x be written for yy, and x* for y~>, we mall have 2X -f- ~x^ = z. So that among the infinite ways in which x, y, and z, may be related to each other, one of them is here found, which is reprefented by thefe Equations, .v =yy, 2y* •+- ±y* = z, and 2X -+- ^x* = z.

DEMONSTRATION.

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

57. So if the Equation propofed were y = x, the derived Equa- tion will be y={xl; and on the contrary, by Prob. i. we have y — xx, that is, y=.x, becaufe x is fuppofed Unity. And thus

from

and INFINITE SERIES. 4.3

from y = I — 3* -+-y -f- xx -+- xy is derived _y = tf — x* -f- Lx1 — ^v+ -+- ^o x! — -4T'vS> &c- And thence by Prob. i. y = i — 2x ^-x1 — %x> -+- ^-x* • — -Vx!) &c. Which two Values of y agree with each other, as appears by fubftituting x — xx+^x> — -^x* ->-J-xs, <5cc. inftead of^ in the firft Value.

.,8. But in the Reduction of Equations I made ufe of an Opera- tion, of which alfo it will be convenient to give fome account. And that is, the Tranfmutation of a flowing Quantity by its connexion 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

point in AE is eftimated ; fo that — 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 are refer'd ; then o 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- traclion of the given Quantities AB and ac ; but they are not changed as to the Celerity of their Motions, and the mutual refpect of their Fluxion. For the 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 : For the only Scope of this Problem is, to determine the contempo- raneous Parts, or the contemporary Differences of the abfolute Quan- tities f, x, _>', or z, defcribed with a given Rate of Flowing. And 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.

59. The reaibn of this matter may alfo be thus explain'd Al- gebraically. Let the Equation y=xxy be propofed, and fup-

G 2 pole

44. 77je Method of FLUXIONS,

pofe x= i -+-Z- Then by Prob. i. x = z. So that for y =-. xxy , may be wrote y •=. xy -h xzy. Now fince ,v=s, it is plain,, that though the Quantities x and z be not of the fame length, yet that they flow alike in refpecl: of y, and that they have equal contem- poraneous parts. Why therefore may I not reprefent by the fame Symbols Quantities that agree in their Rate of Flowing,; and to de- termine, 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 parts may be found, from an Equation involving flowing Quantities.

Thus if y = ~ -+- x be the Equation, when # = 2, then _y = 24. But when x = 3, then y =. 3.1. Therefore while x flows from 2 to 3, y will flow from 2-i to 3.1. So that the 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.

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

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- fently be greater than it is. And the contrary if it flows backwards, or decreafes. Wherefore find its Fluxion, by Prob. i. and fuppofe it to be nothing.

2. Ex AMP. i. If in the Equation x> — ax1 + axy — jy3 = o the greatefl Value of, x be required ; find the Relation of the Fluxions of x and y, and you will have 3X.va — 2axx -f- axy — %yyl -i-ayx = o. Then making x = o, there will remain — yyy1 -\- ayx=o, or 3j* = ax. By the help of this you may exterminate either x or y out of the primary Equation, and by the refulting Equation you may determine the other, and then both of them by — 3^* -f- ax = o.

3. This Operation is the fame, as if you had multiply 'd 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

famous Rule of Huddenius, that, in order to obtain the greateft or leaft Relate Quantity, the Equation mufl be difpofed according to the Dimenfions of the Correlate Quantity, and then the Terms are to be multiply 'd by any Arithmetical ProgrelTion. But fince neither this Rule, nor any other that I know yet publiihed, extends to Equa- tions affected with iiird Quantities, without a previous Reduction j I fhall give the following Example for that purpofe.

4. EXAMP. 2. If the greatest Quantity y in the Equation x* —

ay~ + 7+ -- xx ^ ay ~+" xx= ° be to be determin'd, feek the .Fluxions of xand^y, and there will arife the Equation 3^^* — zayy-{-

^«^v)1 + 2^n5 Aaxxy-\-6x\* + atx2 A j r \ r r •

I __ - — _ -— = 0. And fince by fuppofition y = o, ,

a1 -\- zay +j* 2 ^ ay -\- xx

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 — ^- "*"-'** = o. When the Re-

a"xx

duction is made, there will arife ^ay-\- %xx = o, by help of which you may exterminate either of the quantities x or y out of the pro- pos'd Equation, and then from the refulting Equation, which will, be Cubical, you may extract the Value of the other.

5. From this Problem may be had the Solution of thefe fol- lowing.

I. In a given .Triangle, or in a Segment of any given Curve, ft> ir.fcribe the greatejl Reft angle.

II. To draw the greatejl or the leafl right Line, 'which can lie: between a given Point, and a Curve given in pofition. Or, to draw. a Perpendicular to a Curve from a given Point.

III. To draw the greatejl or the leajl right Lines, which pajjin?.- through a given Point, can lie bet-ween two others, either right Lines or Curves.

IV. From a given Point within a Parabola, to draw a rivbt Line, which Jhall cut the Parabola more obliquely than any other. And to do the fame in other Curves.

V. To determine the Vertices of Curves, their greatejl or lealT Breadths, the Points in which revolving parts cut each other, 6cc.

VI. To find the Points in Curves, where they hcrce the great ejT or leajl Curvature.

VII. To find the Icaft Angle in a given EHi£/is, in which the. Ordinates can cut their Diameters.

VIII..

4.6 The Method of FLUXIONS,

VIII. Of EHipfes that pafs through four given Points, to deter- mine the greateft, or that which approaches neareft to a Circle.

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.

And many other Problems of a like nature may more eafily be propofed than refolved, becaufe of the labour of Computation.

P R O B. IV.

To draw Tangents to Curves.

Firft Manner.

1. Tangents may be varioufly drawn, according to the various 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- fcifs, and terminated at the Curve ED. Let this Ordinate move through an inde- finitely finall Space to the place bd, fo that it may be increafed by the Moment cd, while AB is increafed by the Moment — ^ A Bb, to which DC is equal and parallel. Let Da1 be produced till it meets with AB in T, and this Line will touch the Curve in D or 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 Equation, by which the nature of the Curve is determined ; feek for 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.

3. Ex. i. Calling AB = x, and BD =jy, let their Relation be x-, — ax* -h axy — _y3 = o. And the Relation of the Fluxions will be 3xx-i — 2axx-i-axy — ^yy* -+- ayx-=. o. So that y : x :: ^xx — 2ax -4- ay : ^ —ax :: BD (;-) : BT. Therefore BT = ... w* ~~ f!X~ — • 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

4. But this Method of Operation may be thusconcinnated. Make the Terms of the propofed Equation equal to nothing : multiply by the proper number of the Dimenfions 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

5. Thus the Equation*3 — ax* -f- axy — y*=o, being multi-

3 z 10

ply'd by the upper Numbers, gives axy — 3_y3 for the Numerator j and multiply 'd by the lower Numbers, and then divided by x, gives 3-x-1 — zax -+- ay for the Denominator of the Value of BT.

6. Thus the Equation jy3 — by* — cdy -f- bed -\-dxy = o, (which denotes a Parabola of the fecond kind, by help of which Des Cartes confirufted Equations of fix Dimenfions ; fee his Geometry, p. 42. Amfterd. Ed. An. 1659.) by Infpeftion gives ^--"fr+'^v ^ Qr

7. And thus a1 — r-x* — y1 = o, (which denotes an Ellipfis whofe Center is A,) gives —^ , or ^ = BT. And fo in others.

- — X 1

1

8. And you may take notice, that it matters not of what quantity the Angle of Ordination ABD may be.

9. But as this Rule does not extend to Equations afFefted by furd Quantities, or to mechanical Curves ; in thefe Cafes we mufl have recourfe to the fundamental Method.

10. Ex. 2. Let A;S — ay1 -+- j-£ xx \/'ay -+- xx = o be the

Equation exprefling the Relation between AB and BD ; and by Prob. i. the Relation of the Fluxions will be 3*** — zayy -f. *"*"* + 2V

=0. Therefore it will be <ixx

*/,.,,,

4 v ~

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

fay — p ^^

II.

TJoe Method of FLUXIONS,

48

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

'GA = £, LD = c, AB=.v, andBD=;>. And becaufe of fimi- lar Triangles DEL and DMG, it will be LB : BD : : DM : MG ; that is, v/ 'cc — yy : y : : x : b -+- y, and therefore b-\-y ^/cc — yy =yx. Having got this Equation, I fuppofe V cc — yy = z, and thus I fliall have two Equations bz ~\-yz =yx, andzz = cc — yy. By the help of thefe I find the Fluxions of the Quantities x, y, and z, by Prob. i. From the firft arifes 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

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

(y : x ::) BD : BT. But as BD is y, therefore BT= «— .3- That is, — BT = AL -f- - — ~ -; where the Sign

BL

iff !-• J_l (_J

prefixt to BT denotes, that the Point T mufl be taken contrary to the Point A.

12. SCHOLIUM. And hence it appears by the bye, how that point of the Conchoid may be found, which Separates the concave from the convex part. For when AT is the lea ft poffible, D will be that point. Therefore make AT = v ; and fmce BT • — - z

• x

then v = — z -+- 2K -+-

by -\- yv

Here to morten

the work, for x fubftitute - ^l!5 > which Value is derived from what is before, and it will be - ? -f. z -+- - - = v. Whence the Fluxions v, y, and z being found by Prob. i. and fuppofing ^=0

and INFINITE SERIES. 49

.,, ... iy, )K ' iy-l-zyy Azy-4-zvy

bvProb. -3. there will anfe --- ~-t-z + • -- °--=i; = o.

J J y jy z za

Laflly, fubftituting in this : - for z, and cc — yy for zz, (which

values of z and zz are had from what goes before,) and making a due Reduction, you will have y'- -+- ^by* — -2.be* = o. By the Con- ftrudlion of which Equation y or AM, will be given. Then thro' M drawing MD parallel to AB, it will fall upon the Point D of contrary Flexure.

13. Now if the Curve be Mechanical whofe Tangent is to be 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.

14. Ex. 4. Let AC and AD be two Curves, which are cut in 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 (by Prob. i.

Preparat. to Examp. 5.) it will be z = x ~T> ^ ^ B~ xBC.

15. Now let AC be a Circle, or any known Curve ; and to deter- mine the other Curve AD, let any Equation be propofed, in which z is involved, as zz •+- axz =_y4. 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 -{- az : 4jyJ :: (y : x ::) BD : BT. So that if the nature of the Curve AC be given, the Ordinate BC, and the Area 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

Curve AD ; 'twill be (3.3) 3^ x BC = zy. So that 3BC : 2 :: (y : x ::) BD : BT. And fo in others.

17. Ex. 5. Let AB=,v, BD =y, as before, and let the length of any Curve AC be z. And drawing a Tangent to it, as Cl, 'twill

x x C/

be Bt : Ct :: x : z, or 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- volved, fuppofe z ==)'. Then it will be z=y, fo that Ct : Bf '•'• (y : x : :} : BD : BT. But the Point T being found, the Tan- gent DT may be drawn.

H 19-

The Method of FLUXIONS,

19. Thus fuppofmg xzsssyy, 'twill be KZ + zx = zyj >, and for z writing ^ there will 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 Tangent is Ct, and let AD be any

other Curve whofe Tangent DT is to be drawn, and let it be defin'd by afTuming AB = to the Arch AC ; and (CE, BD being Ordinates to AB in a given Angle,) let the Relation of BD to CE or AE be exprels'd by any Equation.

21. Therefore call AB or AC = x, BD =y, AE=z, and CE = v. And it is plain that v, x, and z, the Fluxions of CE, AC, and AE, are^to each other as CE, Ct, and Et. Therefore *x C7 = i>, and .v x ^ = z.

22. Now let any Equation be given to define the Curve AD,

as y = «. Then y = z ; and therefore Et : Ct :: (v • x ••) BD : BT. K "'

23. Or let the Equation be y—z+v—x, and it will be

• . r~>T? I TT- . y-.

And therefore CE -4- Et

t. T

— Ct : Ct :: (y : x ::) BD : BT.

24. Or finally, let the Equation be ayy = v*y and it will be zayy = (3^ =) 3*1;' x— . So that 31;* x CE : 2 ay x Ct :: BD : BT.

25. Ex. 7. Let FC be a Circle, which is touched by CS in C; and let FD be a Curve, which is de- fined by affuming any Relation of the

Ordinate DB to 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

CF = /; and it will be tz=(t^=)

K B

T ,S

• . . ^..

v, and — tv = (/x -^ =) z. Here I put z negatively, becaufe AE is dirninifh'd while EC is increafed. And befides AE : EC ::

AB :

and INFINITE SERIES. 51

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

• — • vx -f- xv. Then exterminating v, z, and v, 'tis 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- pofe let ^=_y, (an Equation to the firft Quadratrix,) and by Prob. i. it will be / = y, fo that yx — yy* — yx* = xy. Whence y : xx — x :: (y : _ x : :) BD(;') : BT. Therefore BT = x*

ADa

- — x; and AT = xx+yy = ^/.

27. After the fame manner, if it is // = ly, there will arife = 6r, and thence AT= - x~ . And fo of others.

z/ /» r

28. Ex. 8. Now if AD be taken equal to the Arch FC, the Curve ADH being then the Spiral of Archimedes ; the fame names of the Lines ftill remaining as were put

afore : Becaufe of the right Angle ABD 'tis xx -{-yy=tf) and therefore (by Prob. i.) xx +yy = //. Tis alfo AD : AC : : DB : CE, fo that tv=ytznd thence (by Prob. i.) tv -4- vf =y. Laftly, the Fluxion of the Arch FC is to the Fluxion of the right Line CE, as AC to AE, or as AD to AB, that is, t : v : : t : x, and thence ix = vf. Compare the Equations now found, and you will fee

'tis tv -+-ix=y, and thence xx -\-yy = (tt =) ^^ . And there- fore compleating the Parallelogram ABDQ^_, if you make QD : QP_ :: (BD : BT :: y : —x ::) X : y — ^ ; that is, if you

take AP = ; ! > PD will be perpendicular to the Spiral.

29. And from hence (I imagine) it will be fufficiently manifeft, by what methods the Tangents of all fcrts of Curves are to be 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- fer'd to right Lines, after any other manner whatever : So that hav- ing the choice of feveral Methods, the eafieft and moil fimple may always be ufed.

H 2 Second

$2 The Method of FLUXIONS,

Second Manner.

30. Let D be a point in 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

are diminifh'd while D is transfer'd to 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

Dcdk and DBTF will be like; and therefore DB : DF :: DC : Dk.

31. Since then the Relation of BD to GD is exhibited by the Equation for determining the Curve ; find the 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.

32. Ex. i. Call GD = x, and BD =_>', and let their Relation be x~, — ax1 -f- axy — y"= = o. Then the Relation of the Fluxions will be ^xx1 — 2axx •+- axy -f- ayx — ^yy- = o. Therefore ^xx — zax -h ay : ^yy — ax :: (y : x : :) DB (y) : DF. So that

.' V — axy, — . Then any Point D in the Curve being given,

~ 1 — « •

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.

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

you

and IN FINITE SERIES,

53

you may obferve, that it is no matter how far diftant the Point G is from the Abfcifs AB, or if it be at all diftant, nor what is the Angle of Ordination ABD.

34. Let the Equation be as before x* — ax* -f- axy — J3 = o ; it gives immediately axy — 3>'3 for the Numerator, and 3** — 2ax -+- ay for the Denominator of the Value of DF.

35. Let alfo a -+- -x—~y=o, (which Equation is to a Conick Sedtion,) it gives — y for the Numerator, and •• for the Denomi-

fly

nator of the Value of DF, which therefore will be — 7 •

36. And thus in the Conchoid, (wherein thefe things will be perform'd more expeditioufly than before,) putting GA = b,

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

is, x — <r=DF. Therefore prolong GD to F, fo that DF = LG, and at F raife the perpendicular FT meeting the Alymptote AB in T, and DT being drawn will touch the Conchoid.

37. But when compound or furd Quantities are found in the Equation, you mufl have recourfe to the general Method, except you fliould chufe rather to reduce the Equation.

38. Ex. 2. If the Equation

xv/cr — yy =zyx, were gven

for the Relation between GD and BD ; (fee the foregoing Figure, p. 52.) find the Relation of the Fluxions by Prob. i. As fuppoiing v/ff — )')' = z) you will have the Equations bz -+- yz = yx, and cc — yy=.zz, and thence the Relation of the Fluxions bz-\-yx

= yx -f- yx, and — 2yy=2Z,z. And now z, and z being i exter-

T&e Method of FLUXIONS, exterminated, there will arife v \/ cc — yy — 'JjlvU — \x = xy.

Therefore y : ^/cc — yy — — J2^ — .v :: (y : ,v ::) BD (ji1) : DF.

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 D to flow on through an infinitely little Space Del in the Curve ; and in AD and BD take Ak = Ad, and Bc = Bc/; and then kD and cD will be contempora- neous 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 perpendicu- lar 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- tion be a -f- ej — y = o. This Equation is to the Ellipfes of

the fecond Order, whofe Properties for Refracting of Light are fhewn by Des Cartes, in the fecond Book of his Geometry. Then the

Relation of the Fluxions will be e- — 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.

43. COROL. i. Hence if d-=.e, (in which cafe the Curve be- 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 the Tangent. v -K A

44.

and INFINITE SERIES. 55

44. COROL. 2. And hence alfo thofe things will be manifeft of themfelves, which are demonstrated, in a very prolix manner, by Des Cartes concerning the Refraction of thcfe Curves. For as much as DF and DB, (which are in the given Ratio of d to e,) in refpect of the Radius DT, are the Sines of the Angles DTF and DTB, that is, of the Ray of Incidence AD upon the Surface of the Curve, and of its Reflexion or Refraction DB. And there is a like reafon- ing concerning the Refractions of the Conick Sections, fuppofing that either of the Points A or B be conceived to be at an infinite diftance.

45. It would be eafy to modify this Rule in the manner of 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 com- modioufly 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 be- ing exprefs'd by any E- quation ; draw BF pa- rallel 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.

47. But if the Point 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.

48. Or again, if the Equation exprefles the Relation between AC and CD; let AC and FT meet in 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 con- ceived to move any how, fo as to defcribe the Spiral ADE. And fuppofe ~Dd to be an in- finitely little part of the Curve thro' 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 BG. Therefore draw Af parallel to 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 exprefs'd between BG and AD ; find 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 A:3 — ax1 -f- axy — jy5 = o, and by Prob. i. 3^* — zax-\- ay : 3^* — ax : : (y : x : :) AD : At. The Point / being thus found, draw Gt, and DT parallel to it, which will touch the Curve.

52. Ex. 2. If 'tis y =y> (which is the Equation to the Spiral

of Archimedes,} 'twill be j = y, and therefore a : b : : (y : x : :)

AD : At. Wherefore by the way, if TA be produced to P, that it may be AP : AB :: a : by PD will be perpendicular to the Curve.

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

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 whatever : Conceive the Point D to move in the Curve for an infinite- ly {mail Interval to d, and the Pa- rallelogram dhHk being compleat- Jf ed, produce Ad to c, fo that

Ac = AD ; then Gg and D/' will be contemporaneous Moments of 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 the Perpendicular TF on DcF. Then the Trapezia Dkdc and 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- fore ex aquo, 'tis D£ : G^ : : DH : Gf, that is, as the Moments or Fluxions of the Lines DH and BG.

55. Therefore by the Equation which exprefies the Relation of BG to DH, find the Relation of the Fluxions (by Prob. i.) and in- that Ratio take Gf, the Tangent of the Circle BG, to DH. Draw DF parallel to Gf, which may meet A/* produced in F. And at F creel the perpendicular FT, meeting AB in T; and the right Line DT being drawn, will touch the Quadratrix.

56. Ex. i. Making EG = x, and DH=;', let it be xx = fy; then (by Prob. i.)2xx = by. Therefore 2.x : b :: (y : x ::) DH : 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 ATD, 'tis AD x DF= AT x DH, and therefore AT : AD : : (DF or

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

57. Ex.2. Let x=y, (which is the Equation to the Quadratrix of the Ancients,) then #=v. Therefore AB : AD :: AD : AT.

I 8.

58 *fhe Method ^FLUXIONS,

58. Ex. 3. Let axx=y*, then zaxx=sMy*. Therefore make 3;-* : zax : : (x : y : :) AB : AL. Then AL : AD : : AD : AT. And

thus you may determine 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 right Line Bt ; and a 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,

by taking (in the Tangent of this ^— ^ <f-

Curve,) BT in the fame Ratio to

BD, as the Fluxion of the Curve AB hath to the Fluxion of the

right Line BD.

60. Ex. i. Calling AB ==x, and BD =y-t let it be ax==yy, and therefore ax = zyy. Then a : zy : : (y : x : :) BD : BT.

6j. Ex.2. Let ^#==7, (the Equation to the Trochoid, if ABF be a Circle,) then fX=yt and a : b :: BD : BT.

62. And with the fame eafe may Tangents be drawn, when the Relation of BD to AC, or toBC, is exprefs'd by any Equation; or when the Curves are refer 'd to right Lines, or to any other Curves, after any other manner whatever.

63. There are alfo many other Problems, whofe Solutions are to be derived from the fame Principles ; fuch as thefe following.

I. To find a Point of a Curve, where the Tangent is parallel to the Abfcife, or to any other right Line given in pofition ; or is perpendicular to it, or inclined to it in any given Angle.

II. To find the Point where the Tangent is moft or leajl inclined to the Abfcifs, or to any other right Line given in 'pofition. That is, to find the confine of contrary Flexure. Of this I have already given a Spe- cimen, in the Conchoid.

III. From any given Point without the Perimeter of a Curve, to draw a right Line, which with the Perimeter may make an Angle of

Contact.

and IN FINITE SERIES. 59

Contaft, or a right Angle, or any other given Angle, that is, from a given Point, to draw 'Tangents, or Perpendiculars^ or right Lines that Jhall have any other Inclination to a Curve-line.

IV. From any given Point within a Parabola, to draw a right Line, which may make with the Perimeter the greateji or leaft Angle poj/ible. And Jb of all Curves whatever.

V. To draw a right Line which may touch two Curves given in pojition, or the fame Curve in two Points, when that can be done.

VI. To draw any Curve with given Conditions, which may touch another Curve given in pojition, in a 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 not be fo difficult, abating the tedioufnefs of Computation, as 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 this, or that give a greater Infight into their nature. In order to cits 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.

3. II. If a Circle touches any Curve on its concave fide, in any given Point, and if it be of fuch magnitude, that no other tangent Circle can be interleribed in the Angles of Contact near that Point ; 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.

4. III. Therefore the Center of Curvature to any Point of a Curve, is the Center of a Circle equally curved. And thus the Ra- dius or Semidiameter of Curvature is part of the Perpendicular to the Curve, which is terminated at that Center.

5. IV. And the proportion of Curvature at different Points will 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.

7. Imagine therefore that at three Points of the Curve <f , D, and d, Peipendkulars are drawn, of which thofe that are

at D and ^ meet in H, and thofe that are at D and d meet in h : And the Point D being in the / middle, if there is a greater Curyity at the part Dj^ than at DJ, then DH will be lefs than db. But by how much the Perpendiculars /H and dh are nearer the intermediate Perpendicular, fo much the lefs will the diftance be of the Points H and h : And at laft when the Perpendiculars meet, thofe Points will coincide. Let them coincide in the Point C, then will C be the Center of Curvature, at the Point D of the Curve, on which the Perpendicu- lars ftand ; which is manifeft of itfelf.

8. But there are feveral Symptoms or 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 fide, are feparated and divided by it ; fo that thofe which are on the more curved fide D,f fooner meet at H, and thofe which are on the other iefs curved fide -Dd meet more remotely at h.

11. III. If DC be conceived to move, while it infifts perpendi- cularly on the Curve, that point of it C, (if you except the motion of approaching to or 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 di- ftance DC, no other Circle can be defcribed, that can lie between at the Contact.

and INFINITE SERIES.

61

n. V. Laftly, if the Center II or b of any other touching Circle approaches by degrees to C the Center of this, till at la it it co- incides with 'it ; then any of the points in which that Circle mall cut the Curve, will coincide with the point of Contact D.

14. And each of thefe Properties may fupply the means of folving the Problem different ways : But we fliall here make choice of the firlt, as being the moit fimple.

15. At any Point D of the Curve let DT be a Tangent, DC a Perpendicular, and C the Center of Curvature, as before. And let AB be the Abfcifs, to which let DB be apply 'd at right Angles, and which DC meets in P. Draw

DG parallel to AB, and CG per- pendicular to it, in which take Cg of any given Magnitude, and draw gb perpendicular to it, which meets DC in <T. Then it will be Cg : gf : : (TB : BD : :) the Fluxion 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 drawing de perpendicular to 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 contemporaneous Momentum of the right Line g£. Therefore DF —-De^.^t . Having therefore the Ratio's of thefe Moments, or,

LJC ' *

which is the fame thing, of their generating Fluxions, you will have the Ratio of CG to the given Line C^, (which is the fame as that of DF to Sf,) and thence the Point C will be determined.

16. Therefore let AB = x, BD =y, Cg- = i, and g£ = z ;

then it will be i : z : : x : y, or z = r- . Now let the Mo-

X

mentum S-f of z be zxo, (that is, the Product of the Velocity

and of an infinitely fmall Quantity o,} and therefore the Momenta

Dt'==xxo, de=yx.o, and thence DF = .\o -f- — . Therefore

X

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

xx \y

J7-

62 7%e Method of FLUXIONS,

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

J ' '

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

may be DH = - - , and draw HC parallel to AB, and meet-

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

PT TM-T PT Tk/-> DP

make DH== ' or

z

19. Ex. i. Thus the Equation ax^-hx* — y1 =;o being pro- pofed, (which is an Equation to the Hyperbola whofe Latus redtum

is a, and Tranfverfum 2;) there will arife (by Prob. i.) a •+. zbx — 2zy • — o, (writing l for x, and z for y in the refulting Equation, which otherwife would have been ax -+• 2&xx — zyy = o ;) and hence again there arifes zb — 2zz — 2zy = o, (i and z being again

wrote for ,v and y.) By the firft we have z = CL±^L } an(j by tne

i ^^ latter z = — — • Therefore any Point D of the Curve being given,

and confequently xand y, from thence z and z will be given, which being known, make ••• 7 = GC or DH, and draw HC.

Z

20. As if definitely you make 0 = 3, and b=i, fo that 3#-f- xx=yy may be the condition of the Hyperbola. And if you aliume x=i, ^11^ = 2, z=±, z= — T9T, and DH= — gL. li being found, raife the Perpendicular HC meeting the Perpendi-

cular

and IN FINITE SERIES. 63

cular DC before drawn ; or, which is the fame thing, make HD : HC :: (i : z ::) i : £. Then draw DC the Radius of Curva- ture.

21. When you think the Computation will not be too perplex, you

may fabfHtute the indefinite Values of z and z into - , the

Value of CG. Thus in the prefent Example, by a due Reduction you will have DH =y -j- 4'S^r* . Yet the Value of DH by

Calculation conies out negative, as may be feen in the numeral Ex- ample. But this only fhews, that DH mufl be taken towards B ; for if it had come out affirmative, it ought to have been drawn the contrary way.

22. COROL. Hence let the Sign prefixt to the Symbol -\-b be changed, that it may be ax — -bxx — yy=zo, (an Equation to the

Ellipfis,) then DH=;--f- ilLll^: .

23. But fuppofing b=. o, that the Equation may become ax — yy —-- o, (an Equation to the Parabola,) then DH = y -f- ~ ; and

thence DG = \a -f- 2X.

24. From thefe feveral Exprefilons it may eafily be concluded, that the Radius of Curvature of any Conick Seftion is always

aa

25. Ex. 2. If x*=ay* — xy- be propofed, (which is the Equa- tion to the CiiToid of Diodes,") by Prob. i. it will be firft T>xl=.2azy

— zxzy — y-t and then 6x = 2azy-+-2azz — -2zy — zxzy — 2xzz

„ 1 3*x -4- yy , • T.X — a%z -4- 2cv+ *~~ n-.!

— 2Z\ : So that z= - — 3-^. and z= - - ^ ••••• — . There-

J zay — 2.vy' ay — xj

fore any Point of the Ciflbid being given, and thence .v and y, there will be given alfo & and z, ; which being known, make -

K

= CG. _ _

26. Ex. 3. If b-jf-y^/cc — yy =.vy were given, (which is the Equation to the Conchoid, inpag.48;) make \/cc — y\=zv, and there will arife hi) -+- yv = xy. Now the firft of thele, (cc — _vv = vv,) will give (by Prob. i.) — 2yz = 2vv, (writing z for v ;) and the latter will give l>v -+-yv + zv =y -{- xz. And from thefe Equations rightly difpofed v and z will be determined. But that z may alfo be found; out of the laft Equation exterminate the Fluxion

i>, by fubilituting — ^ , and there will arife — —7 — — -I- ~"^

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. But when z and z are known, make ' + zz =± CG.

27. If we had divided the laft Equation but one by z, then by Prob. i . we mould have had — - -f- ^ — — -f- --- -f. -i; =

2 — ^, ; which would have been a more fimple Equation than the

former, for determining z.

28. I have given this Example, that it may appear, how the ope- ration is to be perform'd in furd Equations: But the Curvature of the Conchoid may be thus found a fhorter way. The parts of the Equation b -\-y ^/cc — v\' = xy being fquared, and divided by yy, there arifes ~ -f. — *" ^ — 2by — y* = x*, and thence by Prob. i.

or

x

...

And hence again by Prob. i. ^^ -f- ~ — z— 1 — ™ m By

*^ J y4 y/9 z, zz

the firft refult z is determined, and z by the latter.

29. Ex. 4. Let ADF be a Trochoid [or Cycloid] belonging to the Circle ALE, whofe Diameter is AE j and making the Ordinate 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 of the

Bafe or Abfcifs AB will be to the Fluxion of the Arch AL, as BL

to

and INFINITE SERIES. 65

to PL ; that is, A* or I : / : : v : ~a. And therefore ^ = /. Then from the nature of the Circle ax — xx = -y-y, and therefore by Prob. i. a — 2X = 2-yy, or -~~* = v.

30. Moreover from the nature of the Trochoid, 'tis LD= Arch AL, and therefore -y -M =y. And thence (by Prob. i ) v -h / =z. Laftly, inftead of the Fluxions v and / let their Values be lubfti- tuted, and there will arife a-^ =z. Whence (by Prob. i.) is de- rived — - -f- — — - = z. And thefe being found, make —

*ut/ w *v z,

== — DH, and raife the perpendicular HC.

31. COR. i. Now it follows from hence, that DH = 2BL, and CH — 2BE, or that EF bifeds the radius of Curvature CO in N. And this will appear by fubftituting the values of z and z now found, in the Equation '• . **= DH, and by a proper reduction of

the refult.

32. COR, 2. Hence the Curve FCK, defcribed indefinitely by the 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 the Circle FA, equal and alike pofited to ALE, be defcribed, and let C/3 be drawn parallel to EF, meeting the Circle in A : Then will Arch FA = (Arch EL= NF =) CA.

33. COR. 3. The right Line CD, which is at right Angles to the 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 fuppos'd to apply itfelf to the parts of the Trcchoid KF and KI, which refift it on each fide, that it may not be extended into a right Line, but compel it (as it departs from the 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 perpen- dicular to it.

35. COR. 5. Therefore the 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,

36. COR. 6. Since the Thread by its ofcillating Motion revolves about the moveable Point C, as a Center ; the Superficies through which the whole Line CD continually pafles, will be to the Super- ficies through whichjthe part CN above the right Line IF pafles at the fame time, as CD* to CN*, that is, as 4 to i. Therefore the Area CFN is a fourth part of the Area CFD ; and the Area KCNE is a fourth part of the Area AKCD.

37. COR. 7. Alfo fince the fubtenfe EL is equal and parallel to CN, and is converted about the immoveable Center E, juft as CN moves about the moveable Center C ; the Superficies will be equal through which they pafs in the fame time, that is, the Area CFN, and the Segment of the Circle EL. And thence the Area NFD will be the triple of that Segment, and the whole area EADF will be the triple of the Semicircle.

38. COR. 8. When the Weight D arrives at the point F, the 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 Circle ; and makes an Angle of Contact, with the Tangent /3F produ- ced, infinitely greater than a Circle can make with a right Line.

39. But there are Angles of Contact that are infinitely greater than Trochoidal ones, and others infinitely greater than thefe, and fo on in infinitum ; and yet the greateft of them all are infinitely lefs than right-lined Angles. Thus xx = ay, x3 = £y», x* ==ry5, x* = dy+, &cc. denote a Series of Curves, of which every fucceeding one makes an Angle of Contact with its Abfciis, which is infinitely greater than the preceding can make with the fame Abfcifs. And the Angle of Contact which the firft xx=ay makes, is of the fame kind with Circular ones; and that which the fecond x*-=byz makes, is of the fame kind with Trochoidals. And tho' the Angles of the fucceed- in° Curves do always infinitely exceed the Angles of the preceding, yet they can never arrive at the magnitude of a right-lined Angle.

40. After the fame manner x ==y, xx=ay, x*=l>1y, x4 = 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 be found ad infwitum, that mall infinitely exceed each other.

41. Now it appears, that Angles of Contact of one kind are in- finitely greater than thofe of another kind ; fince a Curve of one kind, however great it may be, cannot, at the Point of 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 necefTarily contain an Angle of Contact of another kind, as the whole contains a part. Thus the Angle of Contaft of the Curve x* = cy*, or the Angle which it makes with its Abfcifs, neceflarfly includes the Angle of Contacl of the Curve x~' =^yi, and can never be contain'd 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

Equation becomes zx — zyl — zx*

= y ; and thence, by Prob. i. zx

— zy* — zx* -f- zx — zzxx — zzyy = ym Then reducing, and

again writing i for x and z for y, there arifes z —

x—xx—jy

J, ——

But z and & being found, make ' T ** =— DH, and draw HC as

above.

44. If you defire a Conftrudtion of the Problem, you will find it very mort. 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 -_^_.. into zy-\-x-=. — z- into — AP=2. Moreover it is i-4-zz =

AE x BTy

"PT* T> P\ TAT1 . I nrfr T3T

r 1 /i f. BlJq U I a \ j i r 1 -j- ** r 1

:= i-{- rrTT =-T-:TI ,) and tnereiore — : — = —

Bl? BI? " 2- —

BT

= DH. Laftly, it is BT : BD :: DH : CH==^^. Here

the negative Value only mews, that CH mufl be taken the fame way as AB from DH.

45. In the fame manner the Curvature of Spirals, or of any other Curves whatever, may be determined by a very mort Calculation.

K 2 46.

68 7&e Method of FLUXIONS,

46. Furthermore, to determine the Curvature without any pre- vious reduction, when the Curves are refer'd to right Lines in any other manner, this Method might have been apply'd, as has beer* done already for drawing Tangents. But as all Geometrical Curves, as alfo Mechanical, (efpecially when the defining conditions are re- duced to infinite Equations, as I mail mew hereafter,) may be re- fer'd to rectangular Ordinates, I think I have done enough in this matter. He that defires more, may eafily fupply it by his own in- duftry ; efpecially if for a farther illuflration I mall add the Method for Spirals.

A its Center, and B a given Point in

47. Let BK be its Circumference.

a Circle, 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-

.

ceive the Point D to move in the Spiral for an infinitely little Spree Drf', and then through rfdraw the Semidiameter A/£, and Cg parallel and equal to it, draw gf perpendicular to gC, fo that G/ cuts gf in/ and GF in P; produce GF to <p, fo that G£p=<§/, and draw de perpendicular to AK, and produce it till it meets CD at I. Then the contemporaneous Moments of BK, AD, and G<p, will be Kk, De and Fa, which therefore may be call'd xo, yo, and zo.

48. Now it is AK : Ae (AD) :: kK : Je=yo, where I aflurne x=i, as above. Alfo CG : GF :: de : eD = oyz, and there- fore yz — yf Befides CG : CF : : de : dD = oy x CF : : dD : d\ = oy x CF?. Moreover, becaufe Z_PC<p (=Z-GG?) = LDAd, and /.CPp (= LCdl = £- eSQ -f- Red.) = L. ADJ, the Triangles CP<p and AD</ are fimilar, and thence AD : Dd :: CP (CF) : P<p = o x CFq. From whence take F<pt and there will remain PF = oxCF^ — ex z. Laftly, letting fall CH perpendicular to AD} 'tis PF : dl :: CG : eH or DH = LlHf . Or fubftituting i+zz

CFy—x

for CFa, 'twill be DH =

y -ya!g

Here it may be obferved,

that

and IN FINITE SERIES. 69

that in this kind of Computations, I take thofe Quantities (AD and Ae) for equal, the Ratio of which differs but infinitely little from the Ratio of Equality.

49. Now from hence arifes the following Rule. The Relation of x and y being exhibited by any Equation, find the Relation of the Fluxions x and y, (by Prob. i.) and fubftitute i for x, and yz for y. Then from the refulting Equation find again, (by Prob. i.) the Relation between x, y, and z, and again fubftitute i for x. The firft refult by due reduction will give y and z, and the latter will eive z ; which being known, make — — =—• = DH, and raife

1 -f- Z.X.—Z.

the Perpendicular HC, meeting the Perpendicular to the 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.

50. Ex. i. If the Equation be ax=y, (which will belong to

the Spiral at Archimedes,) then (by Prob. i.) ax=yy or (writing i for x, and yz for_y,)7^ =yz. And hence again (by Prob i.) o = yz+y'z. Wherefore any Point D of the Spiral being given,, and thence the length AD or y, there will be given z = - , and 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. Produce AB to Q, fo that AB : Arch BK :: Arch BK : BC^, and make AB -+- AQ^: AQj: DA : DH :: a : HC.

51. Ex. 2. If ax1 =_)" be the Equation that determines the Re- lation between BK and AD; (by Prob. i.) you will have 2axx=. 3Jy,-*, or 2ax= 3«y». Thence again 2a'x= ^zys -+- gsiyy*. 'Tis therefore z = ^7 , and z = 'a~9~z'- . Thefe being known, make

i-\-zz — K : i-t-zz ••• DA : DH. Or, the work being reduced to a better form, make gxx1 -f- 10 : gxx -f- 4 :: DA : DH.

52. Ex. 3. After the lame manner, if ax* — bxy=yi determines

the Relation of BK to AD ; there will arife I"* ~ '• = z,, and

bxy -f- $)*.

.g*~;*7~^;~9*'-8 = g. From which DH/ and thence the. Point C, is determined as before.

5q i-

yo I'he Method of FLUXIONS,

53. And thus you will eafily determine the Curvature of any- other Spirals ; or invent Rules for any other kinds of Curves, in imitation of thefe already given.

£4. And now I have finim'd the Problem ; but having made ufe 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 illuflra- tion 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 be defcribed, which may cut any Curve in feveral Points ; if that Circle be fuppos'd to be contracted, or enlarged, till two of the Points of interfeclion coincide, it will there touch the Curve. And befides, if its Center be fuppos'd to approach towards, or recede 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 be cequicurved with the Curve in that Point of Contadt : In like man- ner 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.

55. Therefore with Center C, and Radius CD, let a Circle be defcribed, that cuts the Curve in the Points d, D, and <f ; and letting fall the Perpendi- culars DB, db, <T/3, and CF, to the Abfcifs AB ; call AB = x, BD = y, AF = v, FC=/,andDC=J. Then BF=v—x, and DB-f-FC =_>>-{-/. The fum of the Squares of thefe is equal to the Square of DC ; that is, -D1—

2VX -+- X* -f- )"• -h 2yt -+- /»

=ss. If you would abbrevi- ate this, make v* -f-/1 — s1 =f, (any Symbol at pleafure,) and it becomes x1 — 2vx -f-jy1 -f- zfy -+- q1 = o. After you have found

/, «y, and q*, you will have s-=\/rv1 -+- 1* — q*.

56. Now let any Equation be propofed for defining the Curve, the quantity of whofe Curvature is to be found. By the help of this Equation you may exterminate either of the Quantities x or y,

and

and INFINITE SERIES. 71

and there will arife an Equation, the Roots of which, (db, DB, <f/g, &c. if y°u exterminate x ; or A/>, AB, A/3, &c. if you exterminate _y,) are "at the Points of interfedtion d, D, J\ &c. Wherefore fince "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 of Dimenfions, which has three equal Roots ; as Des Cartes has fhew'd. Or more expeditioufly by multiplying its Terms twice by an Arithmetical Progreflion.

57. EXAMPLE. Let the Equation be ax =yy, (which is an Equation to the Parabola,) and exterminating x, (that is, fubftitu-

ting its Value -- in the forego- ing Equation,) there will arife £ * — ^~y*_ -+• zty -f- ?a = o. Three of whofe Roots ^ are to be _j_ yi made equal. And for this purpofe 4*2 I o

I multiply the Terms twice by an * i o i

Arithmetical Progrellion, as you — —

fee done here j and there arifes — — -J1 + 2JX = °-

Or «u = — + \a. Whence it is eafily infer'd, that BF = 2x -{-

\a, as before.

58. Wherefore any Point D of the Parabola being given, draw the Perpendicular DP to the Curve, and in the Axis take PF = 2AB, and erect FC Perpendicular to FA, meeting DP in C; then will C be the Center of Curvity defired.

59. The fame may be perform'd in the Ellipfis and 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 may be perform'd ; fuch 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 -+- ^.ax, which muft be 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 inelegant, may be derived from this Problem. Which is, the longer the Radius of Flexure is, fo much the lefs the Angle DCJ (Fig.pag.6i.) becomes, and alfo the Moment <F/j fo that the Fluxion of the Quantity z is diminim'd along with it, and by the Infinitude of that Radius, altogether vanimes. Therefore find the Fluxion z, and fuppofe it to become nothing.

63. As if we would determine the Limit of contrary Flexure in the Parabola of the fecond kind, by the help of which Cartefius con- ftructed Equations of fix Dimenfions ; the Equation to that Curve is AT3 — bx* — cdx -+- bed 4- dxy = o. And hence (by Prob. i .) arifes 3*** — 2bxx — - cdx -4- dxy -f- dxy = o. Now writing i for xt and z for y, it becomes 3-va — zbx — cd-{- dy -f- dxz=.o ; whence again (by Prob. i,) 6xx — zbx -+• dy + dxz •+- dxz = o. Here again writing i for x, & for y, and o for z, it becomes (>x — zb -+- zdz = o. And exterminating z, by putting b — 3* for dz in the Equation 3^,v — zbx — cd -+- dy -f- dxz = o, there will arife — bx — cd-$-dy = o) ory=c-{-^; this being fubftituted in the room

of y in the Equation of the Curve, we fhall have 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 between parts of contrary Flexure. As if the Equation x* — 4<w3 -}- ba^x* — b>y = o ex- prefs'd the nature of a Curve ; you have 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

III. To find the Point of infinite Flexure.

65. Find the Radius of Curvature, and fuppofe it to be nothing. Thus to the Parabola of the fecond kind, whole Equation is A;* =

<7ya, that Radius will be CD = 4"6aq* \/q.ax-\- gxx , which be- comes nothing when x = o.

IV. To determine the Point of the greatefl or leaft Flexure.

66. At thefe Points the Radius of Curvature becomes either the greateft or leaft. Wherefore the Center of Curvature, at that mo- ment of Time, neither moves towards the point of Contact, nor the contrary way, but is intirely at reft. Therefore let the Fluxion of the Radius CD be found; or more ex-

peditioufly, let the Fluxion of either of the Lines BH or AK be found, and let it be made equal to nothing.

67. As if the Queftion were propofed con- cerning the Parabola of the fecond kind xl = o*y ; firft to determine the Center of

Curvature you will find DH = aa , 9X->

• ox

and therefore BH = 6^'?AV; make BH

Hence (by Prob. i.) — "- - _j_ ^y==t}. But now fuppofe -y, or the Fluxion of BH, to be nothing ; and belides, lince by Hypothecs A- "' = rf1.y, and thence (by Prob. i.) yxx1 =<?*.}', putting x= i, fub- ftitute ^ for v, and there will arife 4.5x4=0+. Take therefore

^

AB ==a y'^j- =<7 x45| , and raifrng the perpendicular BD, it will" 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 reprefented by the Equation xyl = «3, will be moft inflected in the points D and d, which you may determine by taking in the Abfcifs AQ== r, and erecting the Perpendicular QP_=z=v/5, and Q^/> equal to it on the other fide. Then draw- ing 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 de- fcribe the Curve, in which tbaf* Center is always found,

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- courje of the Rays that are ref rafted at any of its 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 refrac- ted 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, and the rectangular Ordinate y, and thence (by Prob. i.) the rela- tion between the Fluxions x and y ; the Value yy, if you fubftitute r for x into it, and make y = o, will be the Radius of Curva- ture.

72. Thus in the Ellipfis ax — £xX=yy, it is -* — "•— = yy ;

which Value of yy, if we fuppofe^=o, and confequently x = />, ^writing i for x, becomes ±a for the Radius of Curvature. And fo 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 zbx — xx = yy, the Value of yyt (found by

zicc + cc ~T — bb

Prob. i.) will be ^ ""* IT — ^ "~~ *• Now 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 or lefs inequable, or as it is varied 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 demand- ed, 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 uni- form 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 to its Logarithm : I fay, if it be 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, ac- cording to the variation of their Curvature.

2. Therefore the Inequability or Variation of Curvature is re- quired at any Point of a Curve. Concerning which it may be ob- ferved,

3. I. That at Points placed alike in like 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.

5. III. And therefore, that where thofe Fluxions are not propor- tional, 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 greater. And therefore that ratio of the Fluxions may not impro- perly be call'd the Index of the Inequability or of the Variation 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</ being the Moment of the Curve, Cc will be the contemporaneous Moment

of the Radius of Curvature, and -^ will be the Index of the Inequability of Curvature. For the Inequability may be call'd fuch and fo great, as the quan- tity of that ratio 7^ mews it to be :

j ±Ja

Or the Curvature may be faid to be fo much the more unlike to the uniform Curvature of a Circle.

7. Now letting fall the perpendicular Ordinates DB and dbt to any line AB meeting DC in P j make AB = #, BD = y\

and thence B& = xo, it will be Cc = vo; and -1 — T^ = — , making x = i.

Wherefore

£>

II

and IN FINITE SERIES. 77

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

Fluxion <y of that Radius, (by Prob. i.) the Index '^ of the Inequabi- lity of Curvature will be given alfo.

8. Ex. i. Let the Equation to the Parabola tax = vy be given ; then (by Prob. 4.) BP = a, and therefore DP= ^a-\-\y=^t. Alfo (by Prob. 5.) BF = a -+- 2X, and BP : DP :: BF : "i)C =

- =1;. Now the Equations 2ax =}'}', aa-\-yy=tt, -and

t-~ =v, (by Prob. i.) give 2ax = 2jvy, and zyy = ztt, and at + Zfx + 2fx __ ^ Which being reduced to order, and putting

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

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

of the Inequability of Curvature.

9. As if in Numbers it were determin'd, that^=ja or 2#==n>,

and x= 4 ; then y (==

+ 7 + "= 3v/2. So that

j^= 3, which therefore is the Index of Inequability.

10. But if it were determin'd, that A: =2, then y = 2, ^'=T> / = v/5, f = </±, and -17 = 3^/5. So that ^-=) 6 will be here

the Index of Inequability.

11. Wherefore the Inequability of Curvature at the Point of the Curve, from whence an Ordinate, equal to the Latus reftum of the Parabola, being drawn perpendicular to the Axis, will-be double to the Inequability at that Point, from whence the Ordinate fo drawn is half the Latus rectum ; that is, the Curvature at the firft Point is as unlike a- gain to the Curvature of the Circle, as the Curvature at the fecond Point.

12. Ex. 2. Let the Equation be zax — bxx-=.yy, and (by Prob. 4.) it will be a — &v=BP, and thence tf=(aa — 2a6x-lrb

=) na — byy -±- yy. Alfo (by Prob. 5.) it is 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 U1

the Equations zax—bxx^yy, aa — byy-\-y\-=^t!, and

give

7 8 77je Method of FLUXIONS,

•

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

being found, the Index ^ of the Inequability of Curvature, will aJib be known.

13. Thus in the Ellipfis 2X — 3 ATA: =}'}', where it is a = r, and b=-.^ ; if we make x=-, then r-— L v —

* * S " a 3 x ~~ ~"

A

b P

Jl

o

V

and therefore ;v=|, which is the In-

dex of the Inequability of Curvature.

Hence it appears, that the Curvature of

this Ellipfis, at the Point D here af-

fign'd, is by two times left inequable,

(or 'by two times more like to the Cur-

vature of the Circle,) than the Curva-

ture of the Parabola, at that Point of

its Curve, from whence an Ordinate let fall upon the Axis is equaj

to half the Latus rectum.

14. If we have a mind to compare the Conclufions derived in

thefe Examples, in the Parabola 2ax=yy arifes (~ =>)^vfor the

V ' s a

Index of Inequability j and in the Ellipfis zax — bxx=yy, arifes (^7- =J - - x BP j and fo in the Hyperbola 2ax -+- bxx =yy, the analogy being obferved, there arifes the Index ("2- — ^ y+3b

\. 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.

15. Now as the Parabola is 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 Ellipfis 2X — $xx=yy, at that Point of the Perimeter which is determined by affuming x = ±: Becaufe its Index is 4., as before, it might be anfwer'd, that it is like the Curvature of the Parabola

6.v

and IN FINITE SERIES.

79

6.v =)')', at that Point of the Curve, between which and the Axis the perpendicular Odinate is equal to |.

17. Thus, as the Fluxion of the Spiral ADE is to the Fluxion of the Subtenfe AD, in a certain given Ratio, fuppofe as d to e; on its concave fide erect

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

bility. So that this Spiral has every where 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

Trochoid, (fee Fig. in Art. 29. pag. 64.) is found to be — . 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 -^ •

19. And from thefe Confiderations the Senfe of the 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 things above deliver'd, to furnifh himfelf with more Examples, and to contrive many other Methods of operation, as occafion may re- quire. So that he will be able to manage Problems of a like nature, (where he is not difcouraged by tedious and perplex Calculations,) with little or no difficulty. Such are thefe following ;

I. To find the Point of any Curve, where there is either no Inequabi- lity of Curvature, or infinite, or tie grcatej?, or the leajl.

20. Thus at the Vertices of the Conic Sections, there is no In- equability of Curvature; at the Cuf] id of the 1 rcchoid it is infi- nite ; and it is greatefl at thofe Points of the Ellif.fis, where the Rectangle BD x BP is greatefl, that is, where the Diagor.al-Lines of the circumfcribed Parallelogram cut the Elliriis, whofe Sides touch it in their principal Vertices.

II. 1o determine a Curve of fame definite Species, l'nfprje a Cr.n:c Section, liioje Curvature at any Point may be cqiu:l and Jiitiilar to the 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 the fecond 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 find as many Curves as you pleafey ivbofe Areas may be exhibited by finite Equations.

I. Let AB be the Abfcifs of a Curve, at whofe Vertex A let the perpendicular AC = i be raifed, and let CE be D

drawn parallel to AB. Let alfo DB be a rectan- gular 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 always as 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.

2. Now if any Equation be a/Turned at pleafure, for determining the relation of z and x, from thence, (by Prob. i.) may z be de- rived. And thus there will be two Equations, the 'latter of which will determine the Curve, and the former its Area.

EXAMPLES.

3. Aflume ##:=:£, and thence (by Prob. i.) 2xx=s} or 2x=c:, becaufe x=, i.

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

5. A flume ax* =zz, or a'fx*=z, and there will arife \a^x'£=^^, or ^(?x = zz, an Equation again to the Parabola.

i 6.

and INFINITE SERIES. 81

6. Affume a6x~1=zz,or a*x-' =z, and there arifes — a*xf* = z, or a'' -j-2xx = o. Here the negative Value of z only infinuates, that BD is to be taken the contrary way from BE.

7. Again if you affume c'-a1 -+- c^x* = z1, you will have zc*x

= 2zz ; and z being eliminated, there will arife

8. Or if you affume

aa-^-xx

aa -J-.VA-

'Z.

\/aa -+- xx = z, make

- -- , <z -}- ATA-

= 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 if you exterminate <u, it will become 3-j^- = z = j- \/ aa-^-xx.

9. Laftly, if you affume 8 — 3^2 -f- ^&=. zz, you will obtain — 32; — 3x2; -f- $z = 2Z&. Wherefore by the affumed Equation firflieek the Area z, and then the Ordinate z by the reiulting Equa- tion.

10. And thus from the Areas, however they may be feign'd, you may always determine the Ordinates to which they belong.

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,

/V

their Abfciffes or Bafes AB and AC. Then the Increments or Fluxions of the Areas which they defcribe, will be as thofe Ordinates drawn

M into

82 fhe Method of FLUXIONS,

into their Velocities of moving, that is, into the Fluxions of their Abfcifles. Therefore make AB = x, BD = v, AC = z, and CE =y, the Area AFDB = j, and the Area AGEC = /, and let 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

zy = t, and thence - =y.

2. Therefore let any two Equations be affumed ; one of which may exprefs the relation of the Areas s and t, and the other the relation of their Abfciffes x and z, and thence, (by Prob. i.) let the

Fluxions t and z be found, and then make - =>'.

3. Ex. i. 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-

pothefis s=:f, and thence s = f, and y = - =^-. It remains

to determine z, by afluming fome relation between the Abfciffes x and z.

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

(\/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.

5. After the fame manner if you fuppofe xx =. z, there will

ariie 2x =s, and thence _)'= (—==] ~; whence -j and x being

; — I exterminated, it will be y=- 7"-'

2Z2-

6. Or if you fuppofe cc = xz, there arifes o = z + xz, and

T-V (5 /

thence -- = y = -- , v az — cc.

2 2^3

7. Again, fuppofing ax •+- '- = z, (by Prob. i.) \t'isa + s=:z, and thence -^- — y — —?—> which denotes a mechanical Curve.

8. Ex. 2. Let the Circle ax — xx = w be given again, and let Curves be fought, whofe Areas may have any other aflumed relation to the Area of the Circle. As if you afliime cx + s = t, and fup-

pofe alfo ax = ZZ. (By Prob. i.) 'tis c + s = t, and a =

Therefore

and INFINITE SERIES. 83

Therefore y = ~ =2~~; and fubftituting ^ ax — xx for j, and 5f for x, 'tis;'= ™ 4- ^ v'^

9. But if you affume j — — =/, and x = z, you will have

s — ^! =/, and i = z. Therefore y—- - =j — 2!2L Oi

« K fl '

= i; — — — . Now for exterminating v, the Equation ax — xx = iJ'u, (by Prob. i.) gives ^ — 2x= 2vv, and therefore 'tis y=. — . Where if you expunge v and A; by fubftituting their values

\/ ax — xx and 2;, there will arile _)•=-" \/tf;s —

10. But if you affume ss = f, and x = zz, there will arife 2w=r^, and i = 2zz; and therefore _y = V = 4^- Anc

5 and x fubftituting \/ ax — xx and &z, it will become y = \Sa-zz;, which is an Equation to a mechanical Curve.

1 1. Ex. 3. After the fame manner Figures may be found, which have an aflumed relation to any other given Figure. Let the Hyper- bola cc -{- xx = wu be given ; then if you affume s = /, and xx=cz, you will have s = f and 2X = cz; and thence _)' =

.r= -. Then fubftituting v/cc -+- xx for j, and C-z^ for x, it

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 y= - = ~. But now (by Prob. i.) cc-\-xx

% ***

= 1?^ gives x=^-ui;, and 'tis y = ^ Then fubftituting \/i<,-t-xx for -u, and c*z* for x, it becomes y •=. , ^ c~ ^

13. Ex. 4. Moreover if the Ciffoid ^-^^_- - =1; were given, to which other related Figures are to be found, and for that purpofe you affume - ^/ax — xx •+- - s = t ; fuppofe - */ ax — xx = h,

and its Fluxion /' -, therefore h •+- - s = /. But the Equation *** -

M 2 =M

84 7%e Method of FLUXIONS,

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

And bcfides fuice it is - s — -

3 3

— xx,

= t. Now to determine z and z, afTume \/ aa — ax = z ; then (by Prob. i .) — a = 2zz, or z = — - •

V.. Jax—xx 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.

14. And thus if you had aflumed V ">/ ax — xx -h ~ s = fy

and x = z, there would have been derived y-=.\/as> — .22-, an Equation again to the Circle.

15. In like manner if 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 Bafe or Abfcifs ; and that the Area which compleats the Paralle- logram be fought, by fuppofing its Fluxion to be equivalent to the Abfcifs, drawn into the Fluxion of the Ordinate.

1 6. Ex. 5. Thus the Trochoid ADF being propofed, I refer it 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

GD, and therefore its Fluxion to be equivalent to the Line GD drawn into the Velocity of the Motion ; that is, x*v. Now where- as 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

that is, as r to -j. So that <u = — and therefore xv == BL.

A h

Therefore the Area ADG is described by the Fluxion BL ; fince therefore the circular Area ALB is defcribed by the fame Fluxion, they will be equal.

17. In like manner if you conceive ADF to be a Figure of Arches, or of verfed Sines, that is, whole Ordinate BD is equal to the Arch AL ; lince the Fluxion of the Arch AL is to the Fluxion of the Abfcifs AB, as PL to BL, that is, v : i :: ±a : \/ ax — .v.v, then -y = --/— . Then vx, the Fluxion of the Area ADG,

2 v ax — xx

will be — 7=^=. Wherefore if a right Line equal to — --.'

2V<,* — 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 Circle, of an Hyperbola, or of any other Curve, to the Sines right or verfed of thole Arches, or to any other 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 AG, cutting the right Line AF in G, and the Spiral in D ; fince that Arch, as a Line moving upon the

Abfcifs AG, delcribes the Area of the Spiral AHDG, fo that the Fluxion of that Area is to the Fluxion of the Rectangle i x AG, as the Arch GD to i ; if you raife the perpen- dicular 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 = | xGL = |AGx GD = Sector AGDj therefore the complernental Segments AL/ and ADH will alfo be 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, or (I believe) can be found out, is in fome manner contain'd herein, and is here determined for the moil part with lefs trouble, and with- out the ufual perplexities.

21. But the chief ufe of this and the foregoing Problem is, that nffuming the Conic Sections, or any other Curves of a known mag- nitude, 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 Centers of Gravity, their Solids generated by their rotation, the Su- perficies of thofe Solids, and to the finding of any other 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 may be found, (as in Prob. 2.) And firft, if the right Line BD, by the motion of which the Area required AFDB

is defcribed, move upright upon an Abfcifs AB given in pofition, conceive (as before) the Paral- lelogram ABEC to be defcribed in the mean time on the other fide AB, by a line equal to unity. And BE being fuppos'd the Fluxion of the Pa- rallelogram, BD will be the Fluxion of the Area required.

2. Therefore make AB = x, and then alfo ABEC=i \x=x, and BE = x. Call alfo the Area AFDB = z, and it will be

BD=z, as alfo =~, becaufe x=i. Therefore by the Equa-

X

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

IS

and INFINITE SERIES. 87

is exprefs'd, and thence (by Prob. 2. Cafe i.) may be found the relation of the flowing quantities x and z.

3. Ex. i. When BD, or z, is equal to fome fimple quantity.

4. Let there be given ~ = z, °r — , (the Equation to the Pa-

rabola,) and (Prob. 2.) there will arife -a = z. Therefore ^> or -L AB x BD, = Area of the Parabola AFDB.

c. Let there be eiven — = z, fan Equation to a Parabola of

J ^ aa *

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

6. Let there be given — — z

XX ~ '

or a^x— 1 = x-:, (an Equation to an Hyperbola of the fecond kind,) and there will arife — a 3 x—1

• z

or — 7 = z. That is, AB x BD „

= Area HDBH, of an infinite length, lying on the other fide of

the Ordinate BD, as its negative value insinuates.

j. And thus if there were given ^ = z, there would arife

2XX

Z.

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

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

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

1 1. Let ax* = z~> ; then f«V = z, or i AB xBD = AFDH. And fo in others.

12. Ex. 2. Where z is equ.il to an Aggregate of fuch Quantities.

13. LetAT-H^—ij then^-h *-£ = z>

, J

14. Let <z -{- ^ = ~ . then ax — \ = &>

15. Let 3*i — £ — ^ — z ; then 2x^ +-x — 4** = 2r-

1 6. Ex. 3. Where a previous reduction by Divifion is required.

17. Let there be given j~, =.& (an Equation to the Apollonian Hyperbola,) and the divifion being performed in injinittun, it will be

l%e Method of FLUXION s

,

•x __ «« _ ^ 4. ?f£ — ^l5, &c. And thence, (by Prob. 2.) as

•11 1. • a*x ^^

in the fecond Set of Examples, you will obtain z= -y • --^

/. "xa U~A ^ «

5^/3 A/4 *

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

^ 1 ~J XX

~=i — x*-{-x* — x6, &c. or elfe s= -^ — l- -f- -., &c. And

"° X1 X4 A.0'

thence (by Prob. 2.) 2 = x — ^3-f-^r — 1#7, &c. =AFDBi or 2 = — -i H- ^- -5, &c. =HDBH.

X 3X* SA.''

L A

19. Let there be given , ™~!LiX =z, and by divifion it will be z = 2x^ — 2X + 7^ — I3AT1 -f- 34*% 6cc. And thence (by

Prob. 2.) z = $x* — x' -f- yx* — 'T3x3 •+• V8 A<^ &c-

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 Hyperbola,) and the Root being extracted to an infinite multitude

of terms, it will be z=i a 4- * „ ~\ — 7- — - — r, &c. whence

* ft a Qf,9.*if.f,l I I -7 fit*

. r . X X x ^x „

as in the foregoing ss = ax+ 6~ — — , -h 77^ — TT^ &c-

22. In the fame manner if the Equation z = \/aa — xx were given, (which is to the Circle,) there would be produced z=ax —

b« ^Oi.J ii2as

23. And fo if there were given z-=\/x — xx, (an Equation alfo to the Circle,) by extracting the Root there would arife z = x'f — x* — 4-** — -r'-g-x^, 6cc. And therefore z = .ix*

1

Vz

TT .

___ _

24. Thus s === v//z<* -^- AV — xx, (an Equation again to the Cir-

bx

x .\

cle,) by extraction of the Root it gives z=a-\- — - • - — gjsj occ.

, ^* Jf3 /'*v3 -

whence 2; = <7Ar -f- -- -, --- , &c.

4« 6<» 24^ I

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

z=i-+- T^-V* -h 43^4, &c. then 2 = AT -f- ^3 -f- TV^ S &c. H-irf -f±^ _l_^ +TV^

— T™ ' - Vo^

26.

and INFINITE SERIES. 89

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

*=** + -~, — gfr •+• T£> &c. = AFDB. Or elfe *=

C'' A 1 1 **

And thence * = 7

&c. = HDBH.

567*

27. Ex. 5. Where a previous reduction is required, by the refo- lution of an affected Equation.

28. If a Curve be defined by this Equation z> •+• a*z

— 2a"' — x3 = o, extradl the Root, and there will arife z = a — x. j_ : : _j_ !4^-. &c. whence will be obtain'd as before z-=ax —

64.2 5 i zaa

„

29. But if z~' — cz* — 2x*z — c *z -f- 2x? -+- c* = o were the Equation to the Curve, the refolution will afford a three-fold Root; either z = c + .v— f? + Jl, &c. or S = c — .v-f- !i' -,

V 32'1 <-

or s = — c — A- — f -- (_ ±- &c. And hence will arife the

2£" 2rc At T

values of tb,e three correfponding Areas, z = ex + ±x* — — -f- T^t, &c. 2; = r^ — i.v1 + ^ — ^0, &e. and x = — ex — •

A 5 X4 .,S

~ - -- !— - flrr 6c 8.1 24^' CCC>

30. I add nothing here concerning mechanical Carves, becaufe their reduction to the form of geometrical Curves will be taught af- terwards.

31. But whereas the values of z thus found belong to Areas which are fituate, fometimes to a finite part AB of the Abfcifs, fometimes to a part BH produced infinitely towards H, and fome- times to both parts, according to their different terms: That the due value of the Area may be alTign'd, adjacent to any portion of the Abfcifs, that Area is always to be made equal to the difference of the values of z, which belong to the 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} _l_ 4_,vS &c. Now that I may de- termine the quantity of the Area MDll, adjacent to the part of the •Abfcifs /'B; from the value of z, which arifes by putting AB = x, I take the value of z, which arifes by putting Ab=x, and there remains x — -Lx* + ^-x', &c. — x + ±x> — -J-x', &c. the value of that Area WDB. Whence if A*, or x, be put equal to nothing, jqere will be had the whole Area AFDB = x — £x' -+- -^x', &c.

33. To the fame Curve there is alfo found z, •==. — - -+• —

L, &c. Whence again, according to what is before, the Area

5**

I 1 1 ^ I I &/~f* *"T * ppr^TOt'f1

1 ]V\T\ •__ t _.— I ,_ -- oCC '- " — T- 1 ' ' ^""1 — -) OCC. J. ijCl CIUI C

if AB, or x, be fuppofed infinite, the adjoining Area bdH toward H, which is alfo infinitely long, will be equivalent to - — ^ -f- — . &c. For the latter Series — - -f- — • ~, &c. will

CA ^-35

vanifh, becaufe of its infinite denominators.

34. To the Curve reprefented by the Equation a-\- — = Z,^ it

: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- pofed nothing, or x infinite ; and therefore 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 value, when the figns of all 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 points b and B, fuppofe in E, inftead of the Area will be had the difference &/E*— BDE of the Areas at the diffe- rent parts of the Abfcifs ; to which if there be added the Rectangle

he Area dEDG will be obtain'd. t

G

and INFINITE SERIES.

36. But it is chiefly to be regarded, that when in the value of & any term is divided by x of only one dimension ; the Area corre- fponding to that term belongs to the Conical Hyperbola ; and there- fore is to be exhibited by it felf, in an infinite Series : As is done in what follows.

77. Let fl3~glA'= z, be an Equation to a Curve ; and by divifion

J • ax -f- xx J

it becomes z = - — 2a •+- 2X — —_ — h^ &c. and thence

aa y

2X> l X*

Z = l^ j — 2ax -f- x1 — ^T ' To* &c. And the Area &/DB

— £,&*.—

aa

zax

xx

2*5

— ,

I denote the little Areas belonging

Where by the Marks — and

1-1 1

- aa aa

to the Terms — and —

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

therefore,

ing to £B, that is, j -

A4

or -

x

-* ' '

. and therefore the whole Area required

WDB = —

X 2A3

21 3 .

— xx H -, &c.

39. After the fame manner, AB, or x, might have been ufed for

a definite Line, and then it would have been

40. Moreover, if Z>B be bifefted in C, and AC be affumed to be of a definite length, and Cb and CB indefinite ; then making AC

= i>, and C£ or CB =_)', 'twill be bd= -^ s=™-\- '— -)- —~

-{- — _i- ^-^-' &c. and therefore the Hyperbolical Area adjacent

N 2 to

'A Mt&od of FLUXIONS,

to the Part of the Abfcifs &C will be

a V

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 Sum of thefe

f 2fl Jf1 4' * 5'1

Areas 7- -\~ ~r ~r, &c. will be equivalent to -|

41. Thus in the Equation a3 -f- z,* ~$- z — x~= =o, denoting the •nature of a Curve, its Root will be z = ,v — y

&c. Whence there arifes z, =-. Lxx — -x —

6cc. And the Area

' Y» TA

7

ox

8 J A X 8 I A i

_ _1 _ _!_»

/^r^ Six' KC'

T, &c. that is,=:|.v

.'X— ^' TA Six

&c. _ - ^

&c. - i - -

42. But this Hyperbolical term, for the moft pnrt, may be very commodioufly avoided, by altering the beginning of the Abfcifs, that is, by increafing or diminiihing it by fome gi\ en quantity. As

in the former Example, where ?v +*v v = z was the Equation to the Curve, if I fhould make b to be the beginning of the Ablcifs* and fuppofmg Al> to be of any determinate length 4/7, for the re- mainder of the Abfcifs £B, I fliall now write x : Thst is, if I dimi- nifti the Abfcifs by ±a, by writing x -f- ±a inftead of x, it will

-become ^~^,. = ~> and

£_!£±! &Ci whence arifes s = \ax — ' 4^z 4--^' -' &c. = 273 j bia

Area .

43. And thus by affuming another and another point for the be- ginning of the Abfcifs, the Area of any Curve may be exr-ivib'd an infinite variety of ways.

44. Alfo the Equation rj-p£ = z might have been refolved into the two infinite Series z, — - -- "— -+- "-^ &c. — a -f .v

.V2 X1 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 infinitely in the numerators of the one, and in the denominators 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.

45. Hardly any thing difficult can occur to any one, who is to un- dertake fuch a computation in Numbers, after the value of the Area is obtain'd in Species. Yet for the more compleat illufhation of the foregoing Doctrine, I mall add an Example or two.

46. Let the Hyperbola AD be propofed, whofe Equation is \/x-+-xx=z; its Vertex be- ing at A, and each of its Axes is equal to Unity. From what goes before, its Area ADB=-i.v>

-+- j'^ — A*'1" -+- T'T*? — T^P*'"' &c' that is x* into Lx -+- ±x* — T'T.v * + y'T.v 4 — T4T-V s > &c. which Series may be infinitely produced by multiplying thelaft term continually by the fucceeding terms of this

Proereffion i-J#. — 5.v ^^r ~~'"qx ^^'v &c. That is

2 S 47 6-9A- 8.-nXi 10.15*. »

the firft term ^..v1 x I_3 x makes the fecond term -L.v* : Which

2-5 ' multiply 'd by " l-~x makes the third term — TV-vl : Which mul-

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

°-°^3333333> &c- This into '- ^— makeso.oo625 the fecond term. This into ~ ' v 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 difpole in two Tables; the affirmative terms in one, and the nega- tive 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

4954581 38676

190948 1663

7963 75 352 ±_

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- gatives, and there remains 0.0893284166257043 for the quantity of the Hyperbolic Area ADB ; which was to be found.

47. Now let the Circle AdF be propofed, which isexpreffed by the equation \/x — xx = z > that is, whofe Diameter is unity, and from what goes before its Area AdB will be -!#* — .£.#* — T'Txi — -fT^i &c> In 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 Signs -4- and — ; nothing elfe remains to be done, than to conned: the fame numeral terms with other fignsj that is, by fubtracting the connected fums- of both the afore -mention'd tables, 0.08989 3 560 503 6 1 93 from the firft term doubled 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- tion, yet each of them is dilcover'd by the fame Arithmetical com- putation.

48. The portion of the circle A^/B being found, from thence the whole Area may be derived. For the Radius dC being drawn, multiply Ed, or -^v/S? Ulto -^C, or i, and half of the product

•s-Vs/3' or °-°5412^58773^5275 w'" ^e ^e va^ue °f the 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.

50. To thefe we mail add the calculation of the Area compre- hended between the Hyperbola dfD and its Afymptote CA. Let C be the Center of the Hyperbola, and putting

'twill be -^— =BD, and -^— -=.bd; whence

a+x

"

the Area AFDB = bx — - - 4- 4 — -*, &c. and the Area

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

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

O.2OOOOOOOOOOOOOOO

6666666666666

40000000000 285714286

2222222 l8l82

The fum 0.200670695462151 1= Area bdDB. 51. If the parts of this Area Ad and AD be defired feparately, fubtract the lefler BA from the greater dA, and there will remain

•3-+ -^4- - — h --» &c. Where if i be wrote for a and b,

and -jig. for x, the terms being reduced to decimals will iland thus;

O.O IOOOOOOOOOOOOOO

500000000000

3333333333

25000000

2OOOOO 1667

The fum o.

= A^— AD,

52-

96 The Method of FLUXIO N s,

52. Now if this difference of 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 d> = o.gg, if 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 o 020000(5667066(195 =

Sum 0.0001000050003333 AJ — AD.

==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.

55. In the fame manner (if CA and AF= i) putting AB and 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</= 0.0202027073 175194, and AD = 0.0 19802 627296 1797, or AW=o.oo2oo2 andAp = o.ooi

56. From thefe Areas thus found it will be eafy to derive others,

I f \ 2.

by addition and fubtradtion alone. For as it is — ' into -^ = 2, the fum of the Areas 0.693 I47I^°5599453 belonging to the Ratio's ^|and ^-2, (that is, infifting upon the parts of the Abfcifs 1.2 — o 8 and 1.2 — o.9,)will be the Area AFcPjS, C/3 being = 2, as is known. Again, fince —^ into 2 = 3, the fum 1.0986122886681097 of the

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

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

- =499 ; it is plain, that the Area AF^/3 may be found by the compofition of the Areas found before, when C/3 = i oo j i 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 which the Logarithms may ealily be derived,) correfponding to fo 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 ?

57. Firfl therefore having affumed o for the Logarithm of the number i, and i for the Logarithm of the number 10, as is gene- rally done, the Logarithms of the Prime numbers 2, 3, 5, 7, 1 1, 13, 17, 37, are to be inveftigated, by dividing the Hyperbolical Areas now found by 2.3025850929940457, which is the Area cor- refponding to the number 10: Or which is the fame thing, by mul- tiplying by its reciprocal 0.4342944819032518. Thus for 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.

58. Then the Logarithms of all the numbers in the 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 places are to be interpolated afterwards, by the help of this Theorem.

59. Let « be a Number to which a Logarithm is to be adapted, A- 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- rithm of the Number n will be obtain'd by adding d-\- £ •+- gr^,

&c. to the Logarithm of the leffer number. For if the numbers are expounded by C/>, C/3, and CP, the rectangle CBD or C,&T=i, as before, and the Ordinates pq and PQ^being raifed ; if n be wrote

for C/3, and x for £p or /3P, the Area pgQP or ^ -+- ~} + ~, &c. will be to the Area pq}$ or *- •+- ^ -f- ^, &c. as the diffe- rence between the Logarithms of the extream 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* 0

- -+- — -f- — &C.

one: which therefore will be —. , that is, when the

x A'3 A"* a

- •+- — -4- — &c. divifion is perform'd, d-\- — -4- -— &c.

* 2n i Zfjs

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 tho' they were to be produced to fourteen or fifteen figures; pro- vided the number, whofe Logarithm is to be found, be not lefs than 1000. And this can give little trouble in the calculation, be- caufe x is generally an unit, or the number 2. Yet it is not necef- fary 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- lion, the void places may be more expeditioufly fupply'd ; the fore- going Rule being to be apply'd only, when the continuation of fome full places is wanted, in order to obtain thofe 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 eafily difcover'd, for the conftruction of the tables 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 are exprefs'd by Equations confirming of complicate terms ; and that by means of their reduction to Equations, which confift of an infi- nite number of fimple terms. But whereas fuch Curves may fome- times be fquared by finite Equations alfo, or however may be com- pared with other Curves, whofe Areas in a manner may be confi- der'd as known ; of which kind are the Conic Sections : For this reafon I thought fit to adjoin the two following catalogues or tables of Theorems, according to 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 fquared ; and the fecond contains fuch Curves, whole Areas may be 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 integer or fractional, affirmative or negative. As if »=3, then

JZ1ZZZ23, zl"=zs, z-«=z-~> or-'3, &+' = z*, and z*-' =z*.

65. Moreover in the values of the Areas, for the fake of brevity, is written R inftead of this Radical \Se-{-f&t or </e-t-fzi-\-gz**>, and/ inflead of </b-t-iz*t by which the value of the Ordinate^ is affected.

10O

"fhe Method of FLUXIONS,

, t

I I

i

•s

1

I

a

CO

rt

ii

'5

CO

U

3

Curve

u

+

n

en

e*

H«» N

•-1 N CO

*

•« 1 v,

N 1 S1

' T *~

1 *J-

•» ~

V ^

and INFINITE SERIES.

101

T-» •»»*

II II

a

CO

bo

• t

II

U

—

a

o u

f ^

^ \«

O ol \O

iM

""* I I

1 I

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

N

II II

T 1

v

t

*•

i

H-

T*»

II II

?r

-f'

c

•f

M

s

OJ

M M

x

o

G

* *•*

X

01

X"

i

X X

s CO s:

x"

IO2 ejff>e Method -o^ FLUXIONS,

67. Other things of the fame kind might have been added ; but I fhall now pafs on to another fort .of Curves, which may be com- pared with the Conic Sections. And in this Table or Catalogue you have the propofed Curve reprefented by the Line QE^R, the beginning of 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- monly either 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 per- pendicular a^/.

68. After the fame manner you have the Conic Sedlion repre- fented by the Line PDG, whofe Center is 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.

69. Therefore retaining the letters before defined, it will be AC = z, CE=y, a.%EC = t, AB or aB = x, BD = i;, and ABDP or aGDB=j. And befides, when two Conic Sections are required, for the determination of any Area, the Area of the latter mall be call'd <r, the Abfcifs |, and the Ordinate T. Put p for

and INFINITE

103

S

S

u

o

en

_2 3

rt

-y

CO

5 U

o

U.

oa

a O Q

V

V

Tl »

BL, O

Q O

14

o

c

.

+ i?

.«V3 >*,

Q O

rt 2

ea Q O

rt >s| =

O

eg

Q

Q O

M

OH

Q

pa

CO

Q O

**

I

4-

--V-

104

Method of FLUXIONS,

OJ

m

(LI

3

Q

O a c c

h

O

Q

Pi O C

O

rt O

^1^

a

o

Q O

o .5

o Q O

2

O Q

rt O B

dina

•ed

C

3

- 3

U

I

o

fe

4

I

a ji

^«

H

-

4-

.<v fa.

cj

o o

n

X

15 ^

V

4-

fr

SJ

t

3| •

s

"T

II II

+

+

I

K +

%

v

-V"

u

13

O

U

S)

u

O

and INFINITE SERIES.

U

<J-.

o

1= o

$

N

4- H

Mj

P s-

^ ^

It tt

*«-

<

Q

CO

Q O

4

i

'

.

I.Hf

Tf-

1

+ 1

5

+ 1

V

1 +

<5

+ b<j

+

««)

I?

4-

E -V

H-|

^i t.

X

io6

Method of FLUXIONS,

s

J*

U

«J

to

_u

o

I

3 U

t/>

o

fa

+

X

•v.

i

s

+

x

and INFINITE SERIES, 107

71. Before I go on to illuftrate by Examples the Theorems that are deliver'd in thefe claffes of Curves, I think it proper to obferve,

72. I. That whereas in the Equations reprefenting Curves, I have all along fuppofed all the figns of the quantities d, e, f\ g, />, and i to be affirmative ; whenever it fhall happen that they are negative, they muft be changed in the fubfequent values of the Abfcifs and Or- ninate of the Conic Section, and alfo of the Area required.

73. II. Alfo the figns of the numeral Symbols » and 0, when they are negative, muft be changed in the values of the Areas. More- over their Signs being changed, the Theorems themfclvcs may ac- quire a new form. Thus in the 4th Form of Table 2, the Sign ot «

d '

being changed, the 3d Theorem becomes -;_iv,.-j-I ~~~' ,-^ -—}> ~~^

— x, &c. that is, 7=^=— =}', *"==*,

' cz -f-/a

into 2.w — 3^===^. And the fame is to be obferved in others.

74. III. The feries of each order, excepting the 2d of the ift Ta- ble, may be continued each way ad infinitum. For in the Series of the -;d and 4th Order cf Table i, the numeral co-efficients of the initial terms, (2, — 4, 16, — 96, 768, Sec.) are fonn'd by multi- plying the numbers — 2, — 4, — 6, — 8, — ro, &c. continually into each other ; and the co-efficients of the fubfequcnt terms are de- rived from the initials in the 3d Order, by multiplying gradually by — 1> — A, — £, — £, — -Li, &rc. or in the 4th Order by multi- plying by * — i, — 4-, — f, — T> — -rV. &C. But the co-efficients of' the denominators i, 3, 15, 105, &c. a rife by multiplying the numbers i, 3, 5, 7, 9, &c. gradually into each other.

75. But in the ad Table, the Series of the ift, 2d, 3'', 4h, c;1", and ioth Orders are produced in infinitum by diviiion alone. Thus having

= v, in the ift Order, if you perform the diviiion to a con- venient period, there will arifo j~ — ~z ^ 'j7 ^

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

t/x

.4--1-'

Table i, and the fourth term belongs to the ift Species cf this Order.

d 3n Jc --4 <•:• ~ n

Whence it appears, that the Area is 7^- - ~ 1^fz + r?r ~

__ _il s.} putting s for the Area of the Conic Section, whofe Abfcifi

*' d

is x=r» , and Ordinate v = g-:r- -.

P 2

io8 7&e Method of FLUXIONS,

76. But the Series of the ^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 far- ther continued, fuppofe 6 = — 4>j, and the ift Theorem of the

jth Order of Table i. wll become — 8»fts~4l|~~1.— 5«/b~3>1~1 into =. -^-=^f. But according to the 4th Theorem of

this Series to be produced, writing — —^ for </, it is — ~ f%>

<x=v, and 'Qfr'-'S/*' __ t

ize

So that fubtrafting the former values of / and /, there will remain

— 4»J— ' / J- 1 10/1/3 Ii;/?} RS a _,, - , .

qnez v/^-h/2 =/> I2e ft Thefe being mul-

ij j

tiplied by — - ; and, (if you pleafe) for -~ writing xv*, there will arife a 5th Theorem of the Series to be produced,'

, 1 — ! s -

= v, and -r- — = f.

77. IV. Some of thefe Orders may alfo be otherwife 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, but that they may be of fome ufe, tho' not altogether necefftry. Yet I have omitted fome Orders, which I 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.

78. V. If the defining Equation of any Curve is compounded of feveral Equations of different Orders, or of different Species of the fame Order, its Area mufl be compounded of the correlponding A- reas ; taking care however, that they may be rightly connected with their proper Signs. For we mufl not always add or fubtra<fl at the fame time Ordinates to or -from Ordinates, or correfponding Areas to or from correfponding Areas ; but fometimes the fum of thefe, and the difference of thofe, is to be taken for a new Ordinate, or to conftitute a correfponding Area. And this muft be done, when the constituent Areas are pofited on the contrary fide of the Ordinate. Huf that the cautious Geometrician may the more readily avoid this

in-

and INFINITE SERIES. 109

inconveniency, I have prefix' d their proper Signs to the feveral Va- lues of the Areas, tho' ibmetimes negative, as is done in the jth and yth Order of Table 2.

70. VI. It is farther to be obferved, about the Signs of the Areas, that -f- * denotes, either that the Area of the Conic Section, adjoin- ing to the Abfcifs, is to be added to the other quantities in the value of t •, ( fee the ifl Example following ;) or that the Area on the other 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 is to be added, as it may feem convenient. Alfo the Value of f, if it comes out affirmative, denotes the Area of the Curve propoled ad- joining to its Abfcifs : And contrariwife, if it be negative, it repre- fents the Area on the other fide of the Ordinate.

80. VII. But that this Area may be more certainly defined, we mull enquire after its Limits. And as to its Limit at the Abfcifs, at the Ordinate, and at the Perimeter of the Curve, there can be no un- certainty: But its initial Limit, or the beginning from whence its de- fcription commences, may obtain various pofitions. In the 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 may be placed elfewhere. And wherever it is, it may be found, by ieeking that length of the Abfcifs, at which the value of f becomes nothing, and there erecting an Ordinate. For the Ordinate fo raifed will be the Limit required.

8 1. VIII. If any part of the Area is pofited below the 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 .v, i;, and /, (hall afcend too high, or defcend too low, they may be .reduced to a juft degree, by dividing or multiplying fo often by any

given quantity, which may be fuppos'd to perform the office of Uni- ty, as often as thole dimenfions mail be either too high or too low.

83. X. Befides the foregoing Catalogues, or Tables, we might allb conftrucT: Tables of Curves related_tp_ other Curves, which may be the

moftfimple intheirkind; as to <Ja-\-fx* =v, ortox</e-t-fx* =v, or to ^/e-\-Jx* =<y, &c. So that we might at all times derive the Area of any propoled Curve from the fimpleft original, and know to what Curves it llands related. But now let us illuitrute by Ex- amples. what has been already delivered.

84-

no

The Method ^FLUXIONS,

84. EXAMPLE I. Let QER be a Conchoidal of fuch a kind, that the Q Semicircle QH A 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-

pendicular HE be let fall to 1C ; then the Point E will defcribe a Curve, whole Area ACEQJs fought.

^.Therefore make AQ^==a, AC=z, CE=y, and becaufe of the

continual Proportionals AI, AQ^, AH, EC, 'twill be 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 Equation of the ift Species of the ad Order of Table 2,

a -\-x,

and the Terms being compared, it will be^ = rf3, e = a*, and f= I j .fo that 4/ .J'' i

«/ v ii — T-£<

x,

3 — tf1.*;1 = -u, and xv — 2s

t.

87. Now that the values found of x and v may be reduced to a 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 in the value of v, a> may be divided once, and ^x1 twice. And by

this means you will obtain s/"^niTr =^,^/al — .v1 =1', and xv — 2s, — t: of which the conllradion is thus.

88. Center A, and Radius AQ^_ defcribe the Qigadrahtal Arch QDP ; in AC take AB = AH ; raiie the perpendicular BD meeting that Arch in D, and draw AD. Then the double of the Scclof

ADP will be equal to the Area fought ACEQ^ For

' — AB.?=) BD, or-y ; and .vj — 2s= 2 A ADB — 2 or = 2*A ADB'-f- aBDP, that 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.

89. The folutions 'of Problems thus found may fometimes be made more elegant. Thus in the prefent cafe, drawing RH the le-

midiameter

and INFINITE SERIES.

in

midiameter of the Circle QH A, becaufe of equal Arches QH and DP, the Sector QRH is half the Sector DAP, and therefore a fourth part of the Surface ACEQ^

90. EXAMPLE II. Let AGE be a Curve, which is 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, flides upon the right Line AF gi- ven in pofition. Let fall EH per- pendicular to AF, and compleat the Parallelogram AHEC ; and calling AC = z, CE =_y, and EF = rf, becaufe of HF, HE, HA continual Proportionals, it will be

HAor y=

r,

91. Now that the Area AGEC may be known, fuppofe »» = £*,

t«r-i or 2 = », and thence it will be j== =}'• Here fl"ce z in the

' a •~z^1

numerator is of a fraded dimenfion, deprefs the value of/ by di-

~V)~I viding by z&, and it will be 7=7= = S> an Equation of the

y a ~ * — i

ad Species of the ;th Order of Table 2. And the terms being com- pared, it is </= i, e= — i, and /= a*. So that z1 = /- ' __ N A.ijV/^i _ .v1 — -u, and 5 — xv = /. Therefore fince

\*~« ) _ ;

*• and z are equal, and fince ^a-—x* = v is an Equation to a Circle whofe Diameter is a : with the Center A, and diftancq a or EF let the Circle PDQ^be defcribed, which CE meets in D, and let the' Parallelogram ACDI be compleated ; then will AC = ^, CD=<u, and the Area fought AGEC = ^ — xv = ACDP

92. Ex-

The Method of FLUXIONS,

112

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

:, and dividing by z, 'tis

X

y = / ~~ • Therefore zr~l

' az — I

==^, or — i = »,and thence

y =

V aai-i

an Equation or

the 3d Species of the 4th Order of Table 2. The Terms therefore being compared, 'tis d-=. I, e = — • i, and f=a. Therefore

% — — = x, </ax — xx = v, and 3^ — 2x1; = /. Wherefore

it is *AC = x, CD = v, and thence ACDH = s ; fo that 3ACDH — 4AADC = 3* — 2xv = t = Area of the Ciflbid ACEGA. Or, which is the fame thing, 3 Segments ADHA = Area ADEGA, or 4 Segments ADHA = Area AHDEGA.

93. EXAMPLE IV. Let 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 Ap . — • c ; becaufe of the Pro- portionals A C : CE — AL : : GC : CE, it will be CE or y

04. * Now that its Area PEC may be found from hence, the paits'of the Ordinate CE are to be confider'd feparately. And if

the Ordinate CE is fo divided in D, that it is CD = v/^— «»,

and

and INFINITE SERIES.

and DE = *\/V — ^ ; CD will be the Ordinate of a Circle de-

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

pofe 2 = w, and it becomes -^/e* — z* = DE, an Equation of the ift Species of the 3d Order of Table 2. The terms therefore being compared, itisd=t>, f = ct, and/= — i; and therefore

1 — . j — = x, \/ — i -+- c* x1 = v, and zbcls -- • - = t.

1

Z Z.

95. Thefe things being found, reduce them to a juft number of dimenfions, by multiplying the terms that are too deprefs'd, and dividing thofe that are too high, by fome given Quantity. If this be done by c, there will arife ~ = x, </ — c * -t- x% = v, and

— -- — = t : The Conflruclion of which is in this manner.

c ex

96. With the Center A, principal Vertex P, and Parameter 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.

97. EXAMPLE 5. Let the Norma GFE fo revolve about the Pole G, as that its angular point F may continually flide upon the right Line AF given in pofition ; then conceive the Curve PE to be de- 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 per- pendicular 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 </cc — zz is the Ordinate

of a Circle defcribed with the Semidiameter c ; about the Center A

let

*fhe Method of FLUXIONS, let fuch a Circle PDQ_be defcribed, which CE produced meets ia

D ; then it will be DE = ^=rS : B? the helP of which EqUa~ tion there remains the Area PDEP or DERQ^to be determin'd.

Suppofe therefore »=:2, and G=^3 and it will be DE=— • — i^~ >

V ft — ^

sn Equation of the ift Species of the 4th Order of Table i. And the Terms being compared, it will be b-= d, cc =e, and — j ==/;

fo that — bV cc — zz = — l>R=f.

98. Now as the value of t is negative, and therefore the Area reprefented by / lies beyond the Line DE ; that its initial Limit may be found, feek for that length of z, at which t becomes no- thing, and you will find it to be c. Therefore continue 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.

99. If you fhould define to know the quantity of the Area PDE, pofited at the Abfcifs AC, and co-extended with it, without knowing the Limit QR, you may thus determine it.

100. From the Value which / obtains at the length of the Ab- fcifs AC, fubtract its value at the beginning of the Abfcifs ; 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 in M ; and the Parallelogram PKML will be equal to the 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 by divifion, nor by any other means ; it muft be transform'd into other Equations of Curves related to it, in the manner 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- pofed cannot be 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

103. EXAMPLE 6. Let it be propofed to determine the Area of the Figure of the Arches of any Conic Section, when they aie 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 A EC is fought, or com- pleating the 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=_>';

then it will be 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 AC, as PD to CD ; if the fluxion of the Abfcifs be fup- pos'd i, the Fluxion of the Arch QD, or of the Ordinate CE,

**+"-*~~ will be i/4 — . Draw this into FE, or z, and there

for the fluxion of the Area AEF.

will arife z »/

If therefore in the Ordinate CD you take CG — -

-zz

V

-zz

, the Area AGC, which is defcribed by CG

moving upon AC, will be equal to the Area AEF, and the Curve

AG

<•

i;

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 &*-* \/^-j-, j-^ = CG, an 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 \/ ^bb ~] — zz=x. \/ — — — -f- • xx —r, i>. and

a * •* a ' Afl a /

~s = t. That is, CD = x, DP = v, and 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 the comprehended Area HIKQ^will be to the Area fought AEF, as b to a, or as PC to AC.

106. Here obferve, that if you change the fign of b, the Conic Section, to whofe Arch the right Line CE is equal, will become an Ellipfis; and befides, if you make b = — «, the Ellipfis becomes- a Circle. And in this cafe the line KI becomes a right line parallel

107. After the Area of any Curve has been thus found and con- ftrucled, we fhould confider about the demonftration of the con- ftruction ; that laying afide all Algebraical calculation, as much as may be, the Theorem may be adorn'd, and 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 follow- ing Examples.

Demonftration of the Conjlruflion in Example 5.

1 08. In the Arch PQ^take a point d indefinitely near to D, (Figure p. 113.) and draw de and dm parallel to DE and DM, meeting DM and AP in p and /. Then will DE^/ be the mo- ment of the Area PDEP, and LM/»/ will be the moment of the Area LMKP. Draw the femidiameter AD, and conceive the inde- finitely fmall arch ~Dd to be as it were a right line, and the tri- angles -D/^/ and ALD will be like, and therefore D/> : pd:: AL : LD. But it is HF : EH :: AG : AF ; that is, AL : LD :: ML : DE; and therefore Dj> : pd : : ML : DE. Wherefore Dp x DE = pd x ML

That

and IN FINITE SERIES,

117

That is, the moment DEed is equal to the moment LM;;//. 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 are equal to each other. C^JE. D.

Demonftration of the ConftruSfion in Example 3.

109. Let DEed be the momentum of the fuperficies AHDE, and A</DA be the contemporary moment of the Segment ADH. Draw the femidiameter DK, and let de meet AK in c -, and it is Cc : Dd :: CD : DK. 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 mo- ment of the periphery Dd produced, that is, to the tan- gent of the Circle, let fall the perpendicular AI, and AI will be equal to AC. So that zDd x AC = zDd x AI = 4

Triangles AD</. So that 4 Triangles AD^/=C^xDE= moment DE^/. Therefore every moment of the fpace AHDE is quadruple of the contemporary moment of the Segment ADH, and therefore that whole fpace is quadruple of the whole Segment. Q^E. D.

Bemvnftratwn

iiS

"The Method of FLUXIONS,

Demonftration of the ConftruRion in Example 4.

no. Draw ce parallel to CE, and at an indefinitely fmall diflance from it, and the tangent of the Hyperbola ckt 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? : GLq :: AC?: LE? (or APV') :: AP? : AM? ; and divlfim* AG/ : AL? (DE?) ::.AP?: AM? — AP?(MK?) ; And invent, AG: AP :: DE : MK. But the little Area DEed is to the Tri- angle CKr, as 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*/ parallel and infinitely near to CD, (Fig. in p. 115-) meeting the Curve AE in e, and draw hi and fe meeting DCJ in p and q. Then by the Hypothefis ~Dd= Eg, and from the fimi- litude of the Triangles Ddp and DCP, it will be D/> : (Dd) Eq :: ( P : (PD) HI, fo that Dp x HI = Eg xCPj and thence Dp x HI (the moment HI/'/.)): Eg x AC (the moment EF/e) :: E?xCP : EyxAC :: CP : AC. Wherefore fince 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 them- felves will be in the fame ratio. Q-^E. D.

112. In this kind of demonilrations it is to be obferved, that I affume fuch quantities for equal, whofe ratio is that of equality : And that is to be efteem'd a ratio of equality, which differs lefs from equality than by any unequal ratio that can be affign'd. Thus in the laft demon ftration I fuppos'd the rectangle E^xAC, or FE?/, to be equal to the fpace FEt/j becaufe (by realon of the difference Eqe infinitely lefs than them, or nothing in comparifon of them,)

they

and INFINITE SERIES.

119

they have not a ratio of inequality. And for the fame reafon I made DP x HI = HI//6 ; and fo in others.

1 13. I have here made ufe of this method of proving the Areas of Curves to be equal, or to have a given ratio, by the equality, or by the given ratio, of their moments ; becaufe it has an affinity to the ufual methods in thefe matters. But that feems more natural which depends upon the generation of Superficies, by Motion or Fluxion. Thus if the Confbuclion in Example 2. was to be de- monftrated : From the nature of the Circle, the fluxion of the right line ID (Fig. p.i 1 1.) is to the fluxion of the right line IP, as AI to ID ; and it is AI : ID : : ID : CE, from the nature of the Curve

AGE ; and therefore CE x ID = ID x IP. But CE x ID = to the fluxion of the Area PDI. And therefore thofe Areas, being ge- nerated by equal fluxion, muft be equal. Q^E. D.

1 14. For the fake of farther illustration, I fliall add the demon- flration of the Confrruc~r.ion, by which the Area of the Ciffoid is determin'd, in Example 3. Let the 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 DQ __ is perpendicular at the end of AD, revolving about

A ; and i AD x QD = to the fluxion generating the Area

its quadruple alfo ED x CQ^== fluxion generating the Ciffoidal Area QREDO. Wherefore that Area QREDO infinitely long, is gene- rated 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 quantities of any other 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 unity ; if fo be that the fluxion generating that quantity be to an unit of its kind, as the fluxion generating the Area is to an unit of its kind alfo ; that is, as the right Line 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 expounded by the Area defcribed by fuch Ordinate ; or if the Fluxion be expounded by the fame Algebraic terms as the Ordinate, the generated quantity will be expounded by the fame as the de- 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 equal to y, and that it may be reduced to the form of the Equations in the Tables, fubftitute z* for z, and it will be z~ ' </ 1 -+- — z«

43

7—y, an Equation of the firft Species of the 3d Order of Table i. And comparing the terms, it will be </= i, e=i,f=2.>

8a + i8z ,~ gz -id -p. _, _

and thence — - — \S i •+- -a== — R> =/. Therefore it is the quantity Z^~ 1/1 -4- which is generated by the Fluxion

4"

3 17, And thus if v'l -f- J^l- reprefents a Fluxion, by a due re-

9«7

duftion, (or by extracting & out of the 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

terms, it is d=. i, e = —, and/= i. So that x7 = - = *•*•,

'

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

«7

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

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

1 1 8. Thus fuppofing \/i 4- l~ to reprefent a linear Fluxion, I

9«T

imagine t no longer to fignify a Superficies, but a Line ; that Line, for inftance, which is to a linear unit, as the Area: which (accord- ing to the Tables) is reprefented by t, is to a fuperficial unit, or that which is produced by applying that Area to a linear unit. On which account, if that linear unit be made e, the length generated by the foregoing fluxion will be ~ . And upon this foundation

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,

119. The method is this, that the values of two or more right- lined Figures may be fo compounded together, that they may very nearly conftitute the value of the Curvilinear Area required.

120. Thus for the Circle AFD which is denoted by the Equa- tion .v — xx =rzz} having found the value of

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 any different letters, that ftand for numbers indefinitely, and then

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* — TV*% &c. there arifes £+/=-!, and— i^= — 4., or e = £, and /= % — e = T*r • So that ^-BD x AB -f- T4TAD x AB = Area AFDB very nearly. For ^-BD x AB -f. T*TAD x AB is equivalent to .!#* — 4.** — _^.v* -— _L.,v*, &c. which being fub- tracted 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 1024 r

the rectangle AD x AB, gives 8DE + zAD into AB = Area AFDB, the error being only

J-x* -\ -- —x* &c. which is always lefs than 560 5760

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

II. The Area being g hen, to determine the 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

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

124. Thus as to the Ellipfis which is exprefs'd by the Equation ax — -xx = zz, after the Area is found z = ^a?x* — a%x* —

1 i I £ ,x ,

^!^ — Hf_, &c. write i;' for — , and / for x*, and it becomes = t* — ^ — — — -i-j, &c. and extracting the root /=

&c. is equal to x. And this value being fubflituted inftead of x in the Equation ax — a-xx = zz, and the root being extracted, there arifes * = **«—. ^L3 — 38«*«' __ 4Q7^7 5cc> So that from

5<: '7Sf* 225018

z, the given Area, and thence v or ./"I, the Abfcifs # will be

f za*

given, and the Ordinate z. All which things may be accommo- dated to the Hyperbola, if only the flgn of 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 following pofitions prepare the way for the foltirion of 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.

3. II. If 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- pofe more curved towards <T, and lefs toward A; that Center will continually change its place, approaching nearer to the parts more curved, as in K, and going farther off at the parts lefs curved, as in. kt and by that means will defcribe fome line, as KG£.

5. IV. The right Line DC will continually touch the line de- fcribed by the Center of Curvature. For if the Point D of this line moves towards ^, its point G, which in the mean time pafTes to K, and is fituate on the fame fide of the Center C, will move the fame way, by pofition 2. Again, if the fame point D moves towards A, the point g, which in the mean time paffes to k, and k fituate on the contrary fide of the Center C, will move the con- trary way, that is, the fame way that G moved in the former cafe, while it pafs'd to K. Wherefore K and k lie on the fame fide of the right Line DC. But as K and k are taken indefinitely f :>r any

points,

and INFINITE SERIES. 125

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

6. Here it is fuppos'd, that the line <rDA is continually more curved towards <T, and lefs towards A ; for if its greateft or leaft Curvature is in D, then the right line DC will cut the Curve KC ; but yet in an angle that is lefs than any right-lined angle, which is the fame thing as if it were faid to touch it. Nay, the point C in this cafe is the Limit, or Cufpid, at which the two parts of the Curve, finishing in the moft oblique concourfe, touch each other ; and therefore may more juftly be faid to be touch'd, than to be cut, by the right line DC, which divides the Angle of contact.

7. V. The right Line CG is equal to the Curve CK. For con- ceive all the points r, 2r, 3;-, ^.r, &c. of that right Line to defcribe the arches of Curves rs, 2r2s, 3^3;, &c. in the mean time that they approach to the Curve CK, by the motion of that right line ; and fmce thofe arches, (by polition i.) are perpendicular to the right lines that touch the Curve CK, (by pofition 4.) it follows that they will be alfo perpendicular to that Curve. Wherefore the parts of the line CK, intercepted between thofe arches, which by reafon of their infinite fmallnefs may be confider'd as right lines, are equal to the intervals of the fame arches ; that is, (by polition i.) are equal to fo many parts of the right line CG. And equals being added to equals, the whole Line CK will be equal to the whole Line CG.

8. The fame thing would appear by conceiving, that every part of the right Line CG, as it moves along, will apply itfelf fuccef- fively to every part of the Curve CK, and thereby will meafure them ; juft as the Circumference of a wheel, as it moves forward by revolving upon a Plain, will meafure the diflance that the point of ContacT; continually defcribes.

9. And hence it appears, that the Problem may be refolved, by afiuming any Curve at pleaflue A/'DA, and thence by determining the other Curve KC£, in which the Center of Curvature of the aftumed Curve is always found. Therefore letting fall the perpen- diculars DB and CL, to a right Line AB given in pofition, and in AB taking any point A, and calling AB = .v and BD = v ; to define the Curve AD let any relation be affumed between x and v, and then by Prob 5. the point C may be found, by which may be determined both the Curve KC, and its Length GC.

10.

Method of FLUXIONS,

126

10. EXAMPLE. Let ax =yy be the Equation to the Curve, which therefore will be the Apollonian Parabola. And, by Prob. 5. will be found AL=|« ^ , and DC = 2±if

* a

-+. ax.

Which being obtain'd, the Curve KC is determin'd by AL and LC, and its Length by DC. For as we are at liberty to aflume the points K and C anf where in the Curve KC, let us fuppofe K to be the Center of Cur- vature of the Parabola at its Vertex ; and putting therefore AB and BD, or x and y, to be nothing, it will be 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.

11. Now if you defire to know what Curve this is, and what is its Length, without any relation to the Parabola ; call KL = zt and LC = v, and it will be &•==. AL — \a = 3 x, or ^z = AT, and - = ax =yy. Therefore 4v/- = S! = CL = v, or — ' ==

2 •''' 27 £t aa 2 7 #

•u* j which fhews the Curve KC to be a Parabola of the fecond kind. And for its Length there arifes ll±il ^/^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 be- tween 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 the place Cpd} and in CD and Cd ta- king CA and CeT both of the fame given length, fuppofe = r, and to CL let fall the perpendiculars A^ and fyy of which Ag, (which call =z) may meet Cd inf. Then compleat the Parallelogram gyfe, and making x,y, and z the fluxions of the quantities ,v, y, and x, as before

it

and IN FINITE SERIES. 127

it will be Ae : A/ :t A?P '• All* " Q"P : CA]1 :: TT ' And A/: P/> :: CA : C P. Then «? a>quot Ae:Pp:: ^11 : CP. But P/> is the moment of the Abfcifs AP, by the acceiTion of which it becomes Ap ; and Ae is the contemporaneous moment of the per- pendicular Ag-, by the decreafe of which it becomes fy. There- fore Ae and Pp are as the fluxions of the lines Ag (z) and AP (x),

that is, as z and x. Wherefore 2, : x :: ~- : CP. And fmce it

is Cgl * = CAI a — AgT = i — &&, and CA = i ; it will be

CP_= * ~*z . Moreover fmce we may aflume any one of the

•

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

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

—~z

j — Zz. Laftly, drawing /^parallel to the infinitely fmall

X

Arch D</, or perpendicular to DC, P^- will be the momentum of DP, by the acceflion of which it becomes dp, at the fame time that AP becomes A/>. Therefore Pp and Pg are as the fluxions of AP (x) and PD (;'), that is, as i and y. Therefore becaufe of fimilar triangles Ppq and CAg, fmce CA and Ag, or i and z, are in the fame ratio, it will be y = «. Whence we have this folution of the Problem.

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

being found make ^21= CP, z x CP = PL, and CP x v/ 1 — yy

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 tangents, drawn perpendicularly to the Curve \)d from the points C and K,

I28 7%e Method of FLUXIONS,

15. Ex. Let ax=yy be the Equation which exprefles the rela- tion between AP and PD ; and (by Trob. i.) it will be firft ax= 2yy, or a = 2yz. Then zyz -f- zyz = o, or

— = z. Thence it is CP =

y

I —yy £l_J

— 4-vv

aa.

c

And from CP and PL taking away y and x. there remains CD = — — ,

aa

and AL = ?a — ~ . Now I take

away y and x, becaufe when CP and PL have affirmative values, they fall on the fide of the point P to- wards 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 affir- mative quantities PD and AP.

1 6. Now to know the Length of the Curve, in which the point C is found, between any two of its points K and C ; we rauft ieek 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 in the Abicifs itfelf AP ; write i for z in 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 Tangent at the point K, or of DG ; the difference between which and the foregoing indefinite value of CD, is — -- -i#> that

is GC, to which the part of the Curve KC is equal.

17. Now that it may appear what Curve this is, from AL (hav- ing 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 — \/^at = v ; or — = 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- tion ^x-^-^y — _}'3=o were affumed; from hence (by Prob. i.) firft there arifes a1 4-^*2 — y*z = o, then aaz — zyyz — y*z=o,

and therefore it is z = , and z = — — . Whence are

yy — aa ' aa — yy

given PC = •••""'- , and PL = 2rxPC, by which the point C is

determined, which is in the Curve. And the length of the Curve, between two fuch points, will be known by the difference of the two correfponding Tangents, DC or PC — y.

19. For Example, if we make a= i, and in order to determine fome point C of the Curve, we take y = 2 ; then AP or x becomes

.y»— 3"'.v_ _ . -_.« - * PC 2 and PI -

Zaa T' z T> z T> 1V"- 2> ana rLl ?•

Then to determine another point, if we take ^' = 3, it will be AP=6, «=i, z = — >ir, PC=— 84, andPL=— ioi. Which being had, if y be taken from PC, there will remain — 4 in the firil cafe, and — 87 in the fecond, for the 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 conti- nued between the two points C and c, or between K and C, with- out that Term or Limit, which we call'd its Cufpid. For when one or more fuch terms come between thofe points, (which terms are found by the determination of the greateft or leaft PC or DC,) the lengths of each of the parts of the Curve, between them and the points C or K, muft be feparately found, and then added together.

PROB. XI.

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

i. It is performed by involving the Length, or the Area of the •propofed Curve, in the Equation which is 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,

Prob. i.) the fluxion of the Length, or of 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 = s, NR=/, and QR='i>, and their Fluxions s, /, and <u refpeclively ; con- ceive the Line NR to move into the place nr infinitely near the former, and letting _ ^ fall RJ perpendicular to nr, then RJ, sr, M" v N"

and Rr will be the contemporaneous moments of the lines MN, NR, and QR, by the accetfion of which they become M«, nr, and And as thefe are to each other as the fluxions of the fame

lines, and becaufe of the right Angle Rsr, it will be >/R/ -f-Tr* = Rr, or \/V -f- f- = <v.

3. But to determine the fluxions s and t there are two Equations- required; one of which is to define the relation between MN and NR,. or s and /, from whence the relation between the fluxions s and t- is to be derived ; and another which may define the relation be- tween MN or NR in the given Figure, and of AP or x in that re- quired, from whence the relation of the fluxion s or t to the fluxion x or i may be difcover'd.

4. Then <u being found, the fluxions y and z are to be fought by a third aflumed Equation, by which the length PD or y may be

defined. Then we are to take PC = '-^, PL =y x PC, and

DC = PC — y, as in the foregoing Problem.

5. Ex. i. Let as — ss=tt be an Equation to the given Curve QR, which will be a Circle; xx = as the relation between the lines AP and MN, and Lv=.y, the relation between the length of the Curve given QR, and the right Line PD. By the firft it will

be as — 2ss = 2tt, or a ~ 2's=i. And thence - =v s*-i-t*==:v.

zt zt

By the fecond it is 2X = as, and therefore -t •=. v. And by the

third £u=y, that is, ^ = z} and hence ^ — ^'=2;. Which

being

and INFINITE SERIES. 131

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

==PC — y, or PC — £QR- Where it appears, that the length of the given Curve QR cannot be found, but at the fame time 'the length of the right Line DC muft be known, and from thence the length of the Curve, in which the point C is found ; and fo on the contrary.

6. Ex.2. The Equation as — ss = ff remaining, make # = j,

and irv — ^ax-=.^ay. And by the firft there will be found — ^ = -y, as above. But by the fecond i = s, and therefore ^ = v. And by the third 2iw — 4^ = 407, or (eliminating -y) ^ — i = z.

Then from hence "— — 3L == z,

j. Ex. 3. Let there be fuppos'd three Equations, aa = st, a •+• *s = x, and A: -f- v =}'• Then by the firft, which denotes an

Hyperbola, it is o=rf+/i, or— 7 = ', and therefore '-V" 4- "

— V/M -f- tf = v. By the fecond it is 3* = i, and therefore - v/w -+- « = v. And by the third it is i + -u == yt or i +

3'

— </ss-4-tt=:z; then it is from hence w =s, that is, putting w 3'

for the Fluxion of the radical -^ </" -t- ^, which if it be made

equal to iv, or | -f- ~ = 7C'i£;, there will arife from thence ^ — ^ = 2W7i;. And firft fubftituting — ~ for /', then 1. for s, and

dividing by aw, there will arife P^3 = iv = z. Now _>' and z

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 let fall on MN, and a Curve is to be found whofe length may be known from the length which arifes by applying the Area QRNV

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

which is produced by fuch application be call'd <y, and its fluxion v. And fince the fluxion of the Area QRNV is to the Fluxion of the Area of a reiTtangular parallelogram made upon VN, with the height E, as the Ordinate or moving line NR = t, by which this is dc- fcribed, to the moving Line E, by which the other is deicribcd in

S 2 the

132 tte Method of FLUXION s,

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

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

by this Rule the value of v is to be inquired, and the reft to be perform'd as in the Examples aforegoing.

9. Ex. 4. Let QR be an Hyperbola which is defined by this

Equation, aa -+• — = // ; and thence arifes (by Prob. I.) — =tf, or — = t. Then if for the other two Equations are aflumed x=s and y = v ; the firft will give i = j, whence v = ^ = £ } and the latter will give y = v, or z = -g, then from hence z= ^ , 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 fuffi- ciently fliewn before.

10. There is alfo another method, by which the Problem may be refolved ; and that is by finding Curves whofe fluxions are either 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 Curves ; of which thing there is a remarkable Example in fpiral lines.

1 1. Let AB be a right Line given in pofition, BD an Arch mov-< ing upon AB as an Abfcifs, and yet re- taining A as its Center, AD^ a Spiral, at

which that arch is continually terminated, bd an arch indefinitely near it, or the place into which the arch BD by its motion next arrives, DC a perpendicular to the arch bdt dG the difference of the arches, AH an- other Curve equal to the Spiral AD, BH a right Line moving perpendicularly upon

AB, and terminated at the Curve AH, bh the ^ ~B~<T

next place into which that right lane moves, andHK perpendicular to

bb.

and INFINITE SERIES. 133

bb. And in the infinitely little triangles DG/ and HK£, lince DC and HK are equal to the fame third Line Bb, and therefore equal to each other, and Dd and Hh (by hypothecs) are correfpondent parts of equal Curves, and therefore equal, as alfo the angles at G and K are right angles ; the third fides dC and hK will be equal alfb. Moreover fince it is AB : BD :: Ab : bC :: hb — AB (Qb) : bC •— BD (CG) j therefore - A*B - = CG. If this be taken away from dG, there will remain dG — • • *& • = dC = /6K. Call therefore AB=*, BD=-y, andBH=>', and their fluxions z, v, and y refpedtively, fince B£, dG, and /jK are the contempora- neous moments of the fame, by the acceflion pf which they become A£, bdt and bb, and therefore are to each other as the fluxions. Therefore for the moments in the lafl Equation let the fluxions be fubftituted, as alfo the letters for the Lines, and there will arife-y— .

^==-.y. Now of thefe fluxions, if z be fuppos'd equable, or the

~ " *'

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

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

i?. Ex. i. If the Equation jzrzr-u were given, which is to the

Spiral of Archimedes, thence (by Prob. i.) -2-^ = v. From hence take - , or - , and there will remain - =y, and thence (by Prob. 2.) 2?_-r. Which fhews the Curve AH, to which the Spiral AD i

2U

equal, to be the Parabola of Apollonius, whofe Latus reclum is 2??; or whole Ordinate BH is always equal to half the Arch BD.

14. Ex. 2. If the Spiral be propofed which is defined by the

5 }_

Equation a3 =a'v1, or v =;^ , there arifes (by Prob. i.) — =-r,

«T 2^T

_l_ I

from which if you take ^, or ~- , there will remain — , = v, ano

flx 2iT

i thence (by Prob. 2.) will be produced ^l = v. That i.;; -BD nrr

3^

EU, AH being a Parabola of the fecond kind, t >

is

134 tte Method of FLUXIONS,

15. Ex. 3. If the Equation to the Spiral be z</"—^ =-y, thence (by Prob. i.) -a, . ?.~- = v ; from whence if you take away ""- or

' 2 V ac -\- cz K

^/- ?, there will remain , ~.. - = y. Now fince the 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- lium to Prob. 9. I reduce it to the form of the Equations in the firft column of the Tables, by fubftituting z* for z, ; then it 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, andf=c, fo that -~2- ^ 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 Equation -- -f- - '- =_y ; making the Abfcifs AB = s, and the

moving Ordinate DB =y. Then Jr

from the Equation will be had,

(by Prob. i.) 3— — — = y, the ^ J v-

\ s ' aa 12Z.S. -/'

fluxion of z being i, and y being

the fluxion of y. Then adding the X~

fquares of the fluxions, the fum

v/ill be — -h |-f- -^ == it, and extracting the root, —

and INFINITE SERIES. 135

= t, and thence (by Prob. 2.) ^ — ^ =— : t . Here / ftands for the

fluxion of the Curve, and / for its Length.

3. Therefore if the length </D of any portion of this Curve were required, from the points d and D let fall the perpendiculars db and DB to AB, and in the value of t fubftitute the quantities Ab and AB feverally for z, and the difference of the refults will be JD the Length required. As if Ab === ?a, and AB = a, writing La for #,

it becomes t = — — ; then writing a for #, it becomes / = — from whence if the firfl value be taken away, there will remain ^ for the length </D. Or if only h.b be determin'd to be ^a, and

AB be look'd upon as indefinite, there will remain — -— — _i_ -1

aa 1 2ft 24

for the value of

4. If you would know the portion of the Curve which is repre- fented by /, fuppofe the value of / to be equal to nothing, and there

arifes z* = — , or z= -£- .. Therefore if you take AB=-^- > 12 V*z y,2

and eredT: the perpendicular bdt the length of the Arch ^D will be

t or — — — • And the fame is to be underflood of all Curves

11%

aa

in general.

5. After the fame manner by which we have determin'd the

length of this Curve, if the Equation ^ -f- -^L =y be propofed, for defining the nature of another Curve ; there will be deduced

^ . _lL -=.t\ or if this Equation be propofed, — — La*y?—~*. «» 3"1 „* "*

2_

there will arife ^ -f-i^5'= t. Or in general, if it is cz* -{-

*• _ .- =_>', where 6 is u fed for reprefenting any number, either

,-—8"

Integer or Fraction, we (hall have cz* — — = /.

o 4&Qi — od<r

6. Ex.2. Let the Curve be propofed which is defined by this Equation ••"" + ^ \/ #a -t- £•£ =t^,V; then1 (by Prob. i.) will be had _y = ^^-r ^f*-* + 4* ^ or exterminating yt y= '-'</~aa-{- zz. To the fquare of which add i. and the fum will be i -J- ~ 4- 4-4 .

aa a*

and

136 ttt Method of FLUXIONS,

and its Root i -f- *— = t. Hence (by Prob. 2.) will be ob-

aa * * /

tain'd 2 + — ^

A«

7. Ex. 3. Let a Parabola of the fecond kind be propofed, whofe Equation is z* = ay1, or ~ =_y, and thence .by Prob. i. is derived

r==y. Therefore < 1 -+- 2f: = ~ i -+- yy s±s . Now fmce the

2aa 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-

ing to it, I have / = ' v/ 1 -t- — . And thus you may find

the lengths of thefe Parabolas Z1 = ay*, 2? r= ay*, z> = ay*, &c.

8. Ex. 4. Let the Parabola be propofed, whofe Equation is «*

4 *

= rfy3, or ^=:^; and thence (by Prob. i.) will arife 1^ = _y.

"

Therefore v/ 1 -f- i^ = </yy -+- i = t. This being found, I

ga7

confult the Tables according to the aforefaid Scholium, and by com- paring with the 2d Theorem of the 5th Order of Table 2, I have

sF = x, v/i -f- 1—^ = v, and |j=?. Where x denotes the Ab-

9«7

fcifs, y the Ordinate, and s the Area of the Hyperbola, and / the length which arifes by applying the Area %s to linear unity.

9. After the fame manner the lengths of the Parabolas z6 =ay', z* :z=«y7, z'° =ay', &c. may alfo be reduced to the Area of the Hyperbola.

jo. Ex. 5. Let the CuToid of the Ancients be propofed, whole

Equation is ^T^jL" __;. and thence (by Prob. i.)

V az. — 2.Z. '

22,*

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

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

i /: 20;' 4</c • i3

= 'u, and 6; — ___.l_into s=f.

AT My 2iA?

And

and INFINITE SERIES.

37

= v, and — — —

a ax

And taking a for Unity, by the Multiplication or Divifion Of which, thefe Quantities may be reduced to a juft number of Di-

menfions, it becomes az = xx, < — f : Which are thus conftructed.

1 1. The Ciflbid being VD, AV the Diameter of the Circle to which it is adapted, AF its Afymptotc, and DB perpendicular to AV, cutting the Curve in D ; with the Semiaxis AF = AV, and the Semipara- meter AG = jAV, let the Hyperbola YkK be defcribed ; and taking AC a mean Proportional between AB and AV, at C and V let CA and VK drawn perpendi- cular to AV, <:ut the Hyperbola in £, and K, and let right Lines kt and KT touch it in thofe points, and cut AV in /and T; and at AV let the Rectangle AVNM be defcribed, equal to the Space TK&. Then the length of the Ciflbid VD will be fextuple of the Altitude VN.

d

12. Ex. 6. Suppofing Ad to be an Ellipfis, which the Equation i/az — 2zz =y reprefents ; let the mechani- cal Curve AD be propofed of fuch a nature, that v'' if B</, or_)', be produced till it meets this Curve at D, let BD be equal to the Elliptical Arch &d. Now that the length of this may be deter- min'd, the Equation \/ az — 2.zz=. y will give

=y, to the fquare of which if i be added, there ariies

— , the fquare of the fluxion of the arch A.J. To which

zy az • aa — 4

02— Szz

if i be added again, there will arife -^ ^ ^- , whofe 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 Species of the 4th Order of

'

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

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

and -1 + ,= into,

J38

Method of FLUXION s.

i-i. The Conftruaion of which is thus; that the right line </G being drawn to the center of the Ellipfis, a parallelogram may be made upon AC, equal to the fedlor AC/, and the double of its height will be the length of the Curve AD.

14. Ex. 7. Making A/3= tp, (Fig, i.) and CL£ being an Hyper- bola, whofe Equation is v/— a -+• % = $&, and its tangent <TT being drawn ; let the Curve WD be propofed, whofe

Abfcifs is — , and its per- pendicular Ordinate is the 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<rTa, when AB flows uniformly,

and I find it to be -^

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

'tis AT = £ = £ </z, and its fluxion is -rV , whofe half drawn

t>p o za v z

into the altitude /3<^, or v/— a •+• - , is the fluxion of the Area

, defcribed by the Tangent <TT. Therefore that fluxion is -p v/ ' b — az, and this apply'd to unity becomes the fluxion of the Ordinate BD. To the fquare of this ~^~ add i, the fquare of

the fluxion BD, and there arifes ^~fl^+al6^ta , whofe root -^

</a*b — a>z-\- ibfrz*- is the fluxion of the Curve VD. But this is a fluxion of the ift Species of the 7th Order of Table 2 : and

the terms being collated, there will be - = </, aab=e, — a*=f,

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

and */i6bb— - a*% + a&t-i = Y) (an Equation to another Conic

Section,

and INFINITE SERIES. 139

Section, (hppofe ML (Fig. 3.) whofe Area IKLM is <r, where IK

i T/"T *w* \ T /XT 2aftbb^f — fl5^Y*—tf4y — Aaabb? — T.2abbs

• — g and Kl_/= TiJ L,aitiy — " 2 — /\

15. Wherefore that the length of any portion DJ of the Curve VD may be known, let fall db perpendicular to AB, and make Kb = z ; and thence, by what is now found, feek the value of t. Then make AB=,s, and thence alfo feek for /. And the diffe- rence of thefe two values of / will be the length Dd required.

16. Ex. 8. Let the Hyperbola be propos'd, whofe Equation is

=)', and thence, (by Prob. i.) will be had^ = - ( Or To the fquare of this add i, and the root of the fum = /. Now as this fluxion is not to be found

aa -\- bz.z. + bits.

\/aa 4- tzz

will be ^/

in the Tables, I 'reduce it to an infinite Series ; and firft by divifion

•" y i / 3 y 4 / 1

it becomes t ;= </ 1 -f-jaS1 — ^2:4H-r«2'5 — 7*z* > &c- a«d extracting

the root, t ==

— a-

c A , z&, &c. And

•*

hence (by Prob. 2.) may be had the length of the Hyperbolical Arch

17. If the Ellipfis \/aa — bz,z=.y were propofed, the Sign of b ought to be every where changed, and there will be had z 4-

—& _f- - — ^— *-z' -^ 1— i— ^t_s7, &c. for the length of its

Arch. And likewife putting Unity for b, it will be z -+- -^ -f- 3ii_4_ Jil , &c. for the length of the Circular Arch. Now the

10«4 I I 2V.'' > O

numeral coefficients of this feries may be found adinfinitumt by mul- tiplying continually the terms of this Progreflion j— , —— , •^- >

S x 9 ' 10 x i i '

18. Ex. 9. Laftly, let the Quadratrix VDE be propofed, whole 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 to AE } call AV =.a, AG = c;, VK = x, and BD = y, and it

T 2 will

The Method of FLUXIONS.

will be as in the foregoing Example, x =.z 4- -r~ 4- j—; 4- - &c. Extract the root js, and there will arife z= x — ^ 4-

, — *7 • , &c. whofe Square fubtract from AKq. or als and the

s°4°" a 4

root of the remainder # — — 4- -^ —, ; , &c. will be GK.

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 there will arife y = a — ^ — —^ — -^--, , &c. And thence, (by Prob. i.) y = - ^ — ^.— ^ , &c. to the fquare of which add i , and the root of the fum will be i 4- ^ -f- —

'-J il il

_6o4^« &c^ __ • \vhence (by Prob. 2.) / may be obtain'd,

1Z/S~SU

or the Arch of the Quadratrix ; viz. YD = x 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 mo- dern 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 contrary he has very nauch 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 ex- amples of the Synthetical Method likewife, which are very mafterly and elegant. Almoft all that remains 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 :-tbe 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- ner, till after a .fcries of argumentation, in which the -feveral fteps have a. neceftary. connexion wjth each other, it arrives at the know- ledge of the propofition required, by comparing it with fomething really known or given. This therefore being the Art of Invention, it certainly deferves to be cultivated with the utmoft induftry. Many of our modern Geometricians have been perfuaded, by confidering the intricate and labour'd Demonftrations of the 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 demon- ftrated Synthetically. But however this may be, the lofs of that Analyfis, if any fuch there were, is amply compenfated, I think, by our prefent Arithmetical or 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 from thence ; as 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 con- fiderable 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- hend fomething abftrufe and difficult in infinite Series ; whereas in- deed 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 Principles, of our common Scale of Decimal Arithmetick. Secondly what is the nature of other particular Scales, which have been, or

may

and INFINITE SERIES. 145

may be, occasionally introduced. Thirdly, what is the nature of a general Scale, which lays the foundation for the Doctrine of infinite Series. Laftly, I ihall add a word or two concerning that Scale ot Arithmetick in which the Root is unknown, and thcrefoi-e propofcd to be found ; which gives occafion to the Doctrine of Affected Equa- tions.

Firft then as to the common Scale of Decimal Arithmetick, it is that ingenious Artifice of expreffing, in a regular manner, all con- ceivable Numbers, whether Integers or Fractions, Rational or Surd, by the feveral Powers of 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 Root and Numeral Indexes, (X1 = 10, X1 = ico, X3 = 1000, X4 = 10000, &c.) as is ufual ; then by ailuming the Coefficients o, i, 2, 3, 4, 5, 6, 7, 8, 9, as occafion (hall require, we may form or exprefs any Number in this Scale. Thus for inflance 5X4-f- jX3 -f- 4X1 + 8X1 -rf- 3X° will be a particular Number exprefs'd by this 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 Notation only in this, that here the feveral Powers of X (or Ten) are fupprefs'd, together with the Sign of Addition -f-, and are left to be fupply'd by the Underftanding. For as thofe Powers afcend regularly from the place of Units, (in which is always X°, or i, muhiply'd by its Coefficient, which here is 3,) the feveral Powers will ealily be understood, and may therefore be omitted, and the Coefficients only need to be fet down in their proper order. Thus the Number 7906538 will (land for yX6 -+- gX5 -f- oX* -+-6X3 -f- ^X* -f-3X' -f-3X°, when you fupply all that is underftood. And the Number 1736 (by fuppreffing what may be ealiiy -underftood,) will be equivalent to X3 -+- 7X1 -f- 3X -f- 6 ; and the like of all other 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

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

U becomes

Method <?/* FLUXIONS,

becomes 5X4 •+• 2X> -f- 6X° -f- 3X~' -f- 8X"1 -f- 4X-» ; and the infinite or interminate Decimal Fraction 0,9999999, &c. ftands for 9X^' -f- gX-1 -4- 9X~3 H- 9X~4-f- 9X~5 -+- yX~& , &c. which infi- 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 fracled, rational or irrational ; at leaft by admitting of a 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 make frequent life of other Scales ; and therefore in 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 Sixty, and certain numeral Coefficients not exceeding fifty- nine. Thefe Coefficients, for want of peculiar Characters to repre- fent 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 53^ -f- 9^' -+- 34!°, or in the Sexagenary Scale 53", 9*, 34°, which is equivalent to 191374° in the Decimal Scale. Again, the Sexagefimal Fraclion 53°, 9', 34", will be the fame as 53^= -f- 9|f+ 34£~z, which in Decimal Numbers will be 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 pro- greffion 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 com- mon 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 popular Scale of Arithmetick ; as not being fo lig but that its Multiples, and all below it, might be eafily committed to memory ; and it admits of a greater variety of Divifors than any Number not much greater than itfelf. Befides, it is not fo fmall, 'but that Numbers exprefs'd hereby would fufficiently converge, or by a few figures would arrive near enough to the Number required; the contrary of which is an 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 by it, and to find considerable advantages in it. But this can never be a convenient Scale to manage and exprefs large Numbers by, be- caufe 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 25 -f- i x 2* -h o x23 •+• 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 common affairs we have 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 Root of the Scale is 1000. So when we reckon by Millions, Billions, Trillions, &c. a Million may be conceived as the Root of our Arith- metick. Alfo when we divide a Number into pairs of figures, for the Extraction of the Square-root ; into ternaries of figures for the Extraction of the Cube-root ; &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 exprefs'd or computed by that Scale, admitting only of integral and affirmative Coefficients, whofe number (including the Cypher c) need not be greater than the Root. Thus in (Quinary Arithmetick, in which the Scale is compofed of the Powers of the Root 5, the Coefficients need be only the five Numbers o, i, 2, 3, 4, and yet all Numbers whatever are 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 2827,92 in this Arithmetick would be 4 x 54 -+- 2 x 5' -|- 3 x 5* -\~ ox5IH-2x5°-f-4x5~IH-3x 5~s ; or if we may fupply the feveral Powers of 5 by the Imagination only, as we do thofe of 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 expreffing of Numbers by a particular Scale, or making the Deno- minator of the Fraction to be the Root of a new Scale. Thus ± is in effect o x 3° + 2 x^"1 ; and 8-f- is the fame as 8 x 5° '-f- 3 x j-'j and 25-5- reduced to this Notation will be 25x9° + 4x 9—' , or ra- ther 2x9' -4- 7x9° -4-4X9""1. And fo of all other Fractions and mixt Numbers.

A Number computed by any one of thefe Scales is eafily reduced to any other Scale affign'd, by fubftituting inftead of the Root in one Scale, what is equivalent to it exprefs'd by the Root of the other Scale. Thus to reduce Sexagenary Numbers to Decimals, becaufe 60 = 6x10, or|=6X, and therefore |s = 3 6X1, ^=2i6X3, &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 affirmative integer Numbers lefs than the Root, (tho' they mould be fuch if we would have the Scale to be regular,) but as occafion may require they may be any Numbers whatever, affirmative or negative, integers or fractions. And indeed they generally come out promif- cuoully in the Solution of Problems. Nor is it neceflary that the Indices of the Powers mould be always integral Numbers, but may be any regular Arithmetical Progreffion whatever, and the Powers themielves either rational or irrational. And thus (thirdly) we are come by degrees to the Notion of what is call'd an univerfal Series, or an indefinite or infinite Series. For fuppofing the Root of the Scale to be indefinite, or a general Number, which may therefore be reprefcnted by x, or y, &c. and affuming the general Coefficients a, b, c, d, &c. which are Integers or Fractions, affirmative or nega- tive, as it may happen ; we may form fuch a Series as this, ax* -f- lx* _j_ ex* -f- dxl -f- ex°, which will reprefent fome certain Number, exprefs'd by the Scale whofe Root is x. If fuch a Number pro- ceeds in hfif.itum, then it is truly and properly call'd an Infinite Series, or a Converging Series, x being then fuppos'd greater than Unity. Such for example is x + \x~ '-\-^.x—'--+ ^*~3, &c. where 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 for its Indices, as xm — \xm~l -+- ^v*—1 -+-"*.. \—s, Gfc.

And thus we have been led by proper gradations, (that is, by arguing from what is well known and commonly received, to what before appear'd to be difficult and obfcure,) to the knowledge of 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 t-hefe Series ; viz. That the firft Term of every regular Series is al- ways the mo ft coniiderable, or that which approaches nearer to the Number intended, (denoted by the Aggregate of the Series,) than any other lingle Term : That the fecond is next in value, and fo on : That therefore the Terms of the Series ought always to be difpoled in this regular defcending order, as is often inculcated by our 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 be a fufficient Approximation to the whole ; and that thefe 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 bigger the Root of the Scale is, by fo much the fafter, cafen'.i paribus, the Series will converge ; for then the Reciprocals of the Powers will be fo much the lefs, and therefore may the more fafely be neglected : That if a Series coir e Tos by increafing Powers, fuch as ax -^ bx* -+- ex* -|-</.v4, &c. the Root x of the Scale mull be un- derftood to be a proper Fraction, the lefler the better. Yet when- ever a Series can be made to conveige by the Reciprocals of Ten, or its Compounds, it will be more convenient than a Series that converges fafter j becaufe it will more eafily acquire the form of the Decimal Scale, to which, in particular Cafes, all Series are to be ul- timately reduced. LafHy, from fuch general Series as thefe, which are commonly the refill t in the higher Problems, we muft pafs (by fubftitution) to particular Scales c; Series, and thofe are finally to be reduced to the Decimal Scale. And the Art of finding fuch general Series, and then their Reduction to -particular Scales, and laft •©£ all to the common Scale of Decimal Numbers, is ulmoll the whole of

j abrtiull-r pares of Amly ticks, as may be fecn in a good meaiiire'by the prefent TrcuUic.

I

Method of FLUXIONS,

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 the other Inflances of particular Scales, and even of general ones, the Root of the Scale, the Coefficients, and the Indices, are all fiip- pos'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- quired. Thus in the affected Equation $x* -j- 3*2 -f- ox* -+- 7*- — • 53070, the Aggregate of the Series is given, viz. the Number 53070, to find x the Root of the Scale. This is 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- derftood in the Aggregate, thus 5X4 -+- 3X3 -f-oX1 +7X' -4-oX0 = 53070. Whence by companion 'tis x = X=io. But this will not be fo eafily perceived in other instances. As if I had the Equation 4^+4- ax3 -f- 3** -f-ox" -f- 2x° -f- ^x~f -f- ^x~1 = 2827,92 I Ihould not 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 form, 4x5* -+- 2x $3 + 3*5* + 0x5' -f- 2 x 5° H- 4x5— * -+- 3 x 5~~;= 2827,92, when by comparifon it would preiently ap- pear, that the Root fought muft be 5. So that finding the 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 given in the common Scale ; which is a Problem of great ufe 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 con- ceivable Numbers whatever may be exprefs'd by any one of theie Scales, or by help of an Aggregate or Scries of Powers derived frcm any Root ; fo likewife any Number whatever may be 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. that is, the feveral integral Powers of 10 and their Reciprocals, may be exprefs'd by the fingle Powers of X or 10, viz. X° , X' , X1, Xs, or X°, X-1, X~% X--% &c. refpectively, but alfo all the inter- mediate

and INFINITE SERIES. 151

mediate Numbers, as 2, 3, 4, Gff. u, 12, 13, Gfr. may be exprefs'd by fuch fingle Powers of X or 10, if we aflame proper Indices.

Thus 2 = X°'JOI03> &C- , 3 =X0'477",&c. 4=__ Xo/o-.o«, &e. g^ Qr jj

_.X''°4'3!>.&C- i2===X'>°7i"8'&e> 456 = X*.«s89s,&c. And the like of all other Numbers. Thefe Indices are ufually call'd the Logarithms of the Numbers (or Powers) to which they belong, and are fo many Ordinal Numbers, declaring what Power (in order or fucceflion) any given Number is, of any Root aflign'd : And different Scales of Lo- garithms will be form'd, by afluming different Roots 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.

It is eafy to obferve from the Arenariiu of Archimedes, that he had fully confider'd and difcufs'd this Subject of Arithmetical Scales, in a particular Treatife which he there quotes, by the name of his a'^^tl, or Principles ; in which (as it there appears) he had laid the 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 appears likewife, that he had acquired a very general notion of the Dodtrine and Ufe of Indices alfo. But how far he had accommo- dated an Algorithm, or Method of Operation, to thofe his Princi- ples, muft remain uncertain till that Book can be recover'd, which is a thing more to be wim'd than expedled. However it may be fairly concluded from his great Genius and Capacity, that 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- ration with infinite Series, it may be expedient to enlarge a little farther upon their nature and formation, and to 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.

Let the quantity a -+- bx -{-ex1 -+- <A'3 -+- ex4, 6cc. be aflumed as a Multiplier, confming either of a finite or an infinite number of

Terms ; and let alfo - -+- x = o be fuch a Multiplier, as will give the Root x= — - . If thefe two are multiply'd together, they

will produce 3 + 2£Xf?* + 2±f_V + "1^5^ + *i £V, &c.

* a a — a n

152 The Method of FLUXIONS,

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

ap fy+"<! f tp+bq f- dp + cq /3 ' ef+t/f p*

wi 1 become - — TTT — - x - -f- x — —-*- x -. -f- -^-^-? x - >

q q q if 11* 9 j*

&c. = o ; or if we divide by -, and tranfpofe, it will be •• "*" aq — .

tp + bg p dj> + eg /* ep + Jq t* ....

— — x y + —j— x ^ — x - , &c. = ,7 : which Series,

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 were continued to infinity,) mufl always be equal to a, whatever may be fuppofed to be the values of p, q, a, by c, d} &c. For

- , the firft part of the firflTerm, will always be removed or deflroy'd by its equal with a contrary Sign, in the fecond part of the feeond

Term. And — x- , the firfl 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 Term, and fo on : So as finally to leave -- , or a, for 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 .the integrity of the Equation. But if the Series be converging, or if the Terms continually decreafe in any compound Ratio, and there- fore finally vanifh or approach to nothing ; the Supplement may be fafely neglected, as vanishing alfb, and any number of Terms may- be taken, the more the better, as an Approximation to the Qium- tity a. And thus from a due confederation of this fictitious Series, the nature of all converging or diverging Series may eafily be appre- hended. 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 along with the Terms of the Series, and finally becomes lefs than any aflignable Quantity.

The. lame Quantity may often be exhibited or exprefs'd by feveral converging Scries ; but that Series is to be mod edeem'd that has the greateft Rate of Convergency. The foregoing Series will converge fo much the fader, cteteris paribus, as p is lefs than qy or as the

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 the fader, as p is 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 / re- prefent two Terms indefinitely, which immediately fucceed each other in the progrefTion of the Coefficients of the Multiplier a -+- bx -if ex* -f-^x3, &c. and let the number n reprefent the order or place of k. Then any Term of the Series indefinitely may be repre-

fented by -f- l—'-Jf»-~*- where the Sign mud be -+- or — , accor-

?" ding as n is an odd or an even Number. Thus if «== i, then

k = a, 1 = 1', and the firft Term will be -f- *_LlL^Z . ]f «==2j

then & = />, l = c, and the fecond Term will be — c^—~p. And fo of the red. Alib if m be the next Teim in the aforefaid pro-

grefTion after /, then -f- -^~lp"~l -f- ^ — 7/." will be any two fuc-

?" ?"

cefiive Terms in the fame Series. Now in order to a due Conver- gency, the former Term abfolutely confider'd, that is fetting afide the Signs, mould be as much greater than the fucceeding Term, as

conveniently may be. Let us fuppoie therefore that JL^—Jp»-i js

i"

greater than ' —^p", or ( dividing all by the common factor c" } \

r" ~^ ' t" '

that ^ + /f? is greater than — ^ - , or ( multiplying both by pq, )

that Ipq -f- krf is greater than nip* •+- Ipq, or (taking away the com- mon IpqJ that kf is greater than //.y,1, or (by a farther Diviiion,)

that - x — is greater than unity ; and as much greater as may be. fl X This

7%e Method of FLUXIONS,

This will take effeft on a double account ; firft, the greater k is in refpecl: of ;;;, and fecondly, the greater 5* is in refpect of p\ Now in the Multiplier a -\-bx -f- ex* -\-dx>, &c. if the Coefficients a, b, r, &c. are in any decreafing ProgreiTion, then k will be greater than /, which is greater than m ; fo that a fortiori k will be greater than m. Alfo if q be greater than p, and therefore (in a duplicate ratio) j* will be greater than /*. So that (cater is faribus) the degree of 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 be fuppos'd to be greater than />.

— / n

The fame things obtaining as before, the Term .j_ A will be

»

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 all the reft of the Terms in itifinitutn, we may introduce this Sup- plement, and then we fhall have the accurate value of a, inftead 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

n= i, and confequently k=a, and /= <£, we fhall have —

— *£ == a. Or if « == 2, and /= c, then bl±X — et±ll x t + q ill

c\i „ . f 7 j .i bb-^-a-j ff->rf-a p <{$ -4- cq

L---a. Or if n = 3, /= a, then J-I—f — _L_L_I x - -4- - f i 1 i q

x ^ — — •=.$. And fo on. Here the taking in of the Supple- ment always compleats the value of a, and makes it perfect, whether the Series be converging or diverging ; which will always be the beft way of proceeding, when that Supplement can readily be known. But as this rarely happens, in fuch infinite Series as ge- nerally occur, we muft have recourfe to infinite converging Series, wherein this Supplement, as well as the Terms of the Series, are infinitely diminifh'd ; and therefore after a competent number of them are collected, the reft may be all neglected in infinitum.

From this general Series, the better to aflift the Imagination, we will defcend to a few particular Inftances of converging Series in pure Numbers. Let the Coefficients a, />, c-, d, &c. be expounded by

,, • , |; < , to, refpectively ; then *±* _ »±* x ^ + ^ x

^ ^c—! orL^_f£±l^x^H_-2±^x^_^-+5ix/4, &C. 5»'(XC<— J' 27 r.x;? 7 3x4? f 4x55. 53'

'. — r. That the Series hence arifmg may converge, make/ lefs

than

a?:d IN FINITE SERIES. 155

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

A — |.x|H-4^x^ — TV x -J., &c. = i. That is, this Series of Fractions, which is computed by Binary Arithmetick, or by the Reciprocals of the Powers of Two, if infinitely continued will finally be equal to Unity. Or if 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 j x Ty, or TV, we Hull have 4 — | -+- ££- — T^ -f- T'o- = i, as is eafy to prove. Or let the fame Coemdents be ex- pounded by i, — |, -i, — i, -f, &c. then it will be - - -+-

f 4iz^ £ 1f=4f /• & Thu Series m ehhei.

1 3X47 J* 4X5? i3

be continued infinitely, or may be fum'd after any number of Terms

i, _ n

exprefs'd by ;?, by introducing the Supplement ; ~ — infteadof all

H-IXJ*

the reft. Or more particularly, if we make (jr= $p, then -2 _f. 7-^-. -+- -- — - -f ( ^— . &c. = i, v/hich is a Number

6x5! liXjS 20X^4 30X;;!'

exprefs'd by Quinary Arithmetick. And this is eafily reduced to the Decimal Scale, by writing ~ for -f, and reducing the Coefficients ; for then it will become 0,99999, &c. = i. Now if we take thefe five Terms, together with the Supplement, we mall have exadly

— -f- r11- + -12- -f- — - + -~ 4- ^-, = i. Again, if

2x5 6x,i 12x5} 20x54 30x5' 6x;«

we make here 77= ioo/^, we fhall have the Series

JJ •"

^^-6 >c -i- + 40°~9 x 9 -f- <co-': x 27 - x 3 iccoo 3 X4 i oooooo 4X 5 locoocooo

which converges very fa ft. And if we would reduce this to the re- gular Decimal Scale of Arithmetick, (which is always fuppos'd to be done, before any particular Problem can be faid to 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 they will ftand as in the Margin. Here the Ag- gregate of the firfc five Terms is 0,99999999595, 0,985 which is a near Approximation to the Amount of the whole infinite Series, or to Unity. And if, for proof-

lake, we add to this the Supplement _+/' = 1L ,- —

„ + , ,/' °' 5 '" |OJ

= 0,00000000405, the wh< . be Unity exaclly.

X 2 There

Tf6 The Method of FLUXIONS,

3 *f

There are alfo other Methods of forming converging Series, whe- ther general or particular, which fhall approximate to a known quan- tity, and therefore will be very proper to explain the nature of Con- vergency, and to mew how the Supplement is to be introduced, when it can be done, in order to make the Series finite ; which of late has been call'd the Summing of a Series. Let A, B, C, D, E, &c. and a, />, c, d, e, &c. be any two Progrcffions of Terms, of 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.

Then is A = a -}-/>, or p = ~a . As this is the whole Supple- ment, in order to form a Series, I fhall only take fuch a part of it as is denominated by the Fraction - , and put q for the fecond Sup- plement. That is, I will afiimie • - = (p=) - -XTJ -\-q, or

/A — a b \ A — a E — b .. .... .,

q — f xi — R=7 ~~B~ x ' Again, as ™1S 1S the whole

value of the Supplement q> I fhall only aflume fuch a part of it as is de- nominated by the Fradion £> and for the next Supplement put r.

/A— a

orr= (-§- x Now as this is the whole value of the Supplement r, I only afTume fuch a part of it as is denominated by the Fraction - , and for the next Supplement put s. That is, — ~

B—l> C—c A— a B— /; C—c, A— a

x -7— x — — = ( r = ) — — - x — - x -rr-a -+- s, or s = -77- x

^ I ^ ' D ^ U Ij

B— /; C—c 7 A— •a B — '•> C—c T>—d A j /- c

— — x x i — TJ — — r- x — r- x —77— x . And lo on as far

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

A — a B — b A— a E — l> C—c ,

inent q •==• —g— x — TT-C -}- r, where r = — g— x —^~ x -]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,

_,. • r 11 A A— a. A— a B — /; A— a E—b C— c ,

That is finally A = a -+- —b .+- — x —^-c -\ — x -7— x -jj-«

A— a 7,— b C—c D— d c \ -a r^ TT\ -O Of*

-\- — — x —TV- x -jj- x -J7- e, Kc. where A, B, C, D, E, ere. and ay

b, r, d, e, 6cc. may be any two Progreffions of Numbers whatever, whether regular or defultory, afcending or defcending. And when

it

. . —

— = (?=) -g- x

x — rr- x

and INFINITE SERIES. 157

it happens in thefe Progreffions, that either A = a, or B=^, or £___£• 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- proximates indefinitely to the value of A. But in the cafe of an infinite Approximation, the faid Progreffions ought to proceed re- "ularlv, according to feme Hated Law. Here it will be eafy to ob- fcrvc," that if 1C and k are put to reprefent any two Terms indefi- nitely in the aforefaid Progreffions, whofe places are denoted by the number ;/, and if L and / are the Terms immediately following ; then the Term in the Series denoted by n -f- i will be form'd from

(v /-

the preceding Term, by multiplying it by -^— /. As if n = i, K = A, k = a, L = B, l=b, and the fecond Term will

« i /t j ', A T f-* 17 L « t"1~ipn TC - -L- "R k —-- " u

DC ** 1 r> " H * '

A— a, B — £ A — a B — f>

I —z c, and the third Term will be — jr-^* ~7cTr ==~tT x~Tr~r; and fo of the reft. And whenever it fhall happen that L =/, then the Series will ftop at this Term, and proceed no farther. And the Series approximates fo much the fafter, catcris paribus, as the Numbers A, B, C, D, &c. and a, b, c, d, &c. approach nearer to each other refpedively.

Now to give fome Examples in pure Numbers. Let A, B,C, D, &c. = 2, 2, 2, 2, &c. and a, b, c, d, &c. = i, i, i, i, &c- then we fhall have 2 = i -h 1 H- £ + T •+* -V> &c- And fo always, when the given Progreffions are Ranks of equals, the Series will be a G<~ metrical Progrefnon. If we would have this Progieffion ftop at the next Term, we may either fuppofe the firft given Progreilion to be 2, 2, 2, 2, 2, i, or the fecond to be i, i, i, i, i, 2, 'tis all one which. For in either cafe we mall have L= /, that is F ==/,

TC— ^ p*

and therefore the laft Term muft be multiply'dby - — , or — — = i.

Then the Progreffion or Series becomes 2 = I +T-r-ir~+"T + Tci •+-TT- Again, 'if A, B, C, D, &c. = 5, 5, 5, 5, &c. and a, b, c, d, &c. =4, 4> 4, 4, &c- then 5 = 4 H- T + T+T + TTT -H *-TT, &c- or ^. = ± H- T'T -i- -4T + T!T> &c« Or if A, B, C, D, &c. = 4, 4, 4, 4, 6cc. and </, *, f , d, &c. = 5, 5, 5, 5, &c. then 4=5 — i -4- -fV - - *-ST H- ^Tr, &c. If A, B, C, D, &c. = 5, 5, 5, 5, &c. and tf, ^, c, d, &c. = 6, 7, 8, 9, &c. then 5 = 6 — T7-f-4-Xy8 — ^-xf x-f-y -h -^- x-fx AX ±10, &c. If we would have the Series ftop here, or if we v/oiud find one more Term, or Supplement, which fhculd be equivalent to all the reft ad inftnitumy (which in- deed

Method of FLUXIONS,

deed might be deiirable here, and in fuch cafes as this, becaufe of thc- llow Convergency, or rather Divergency of the Series,) fuppofe F==/j

and therefore ~— - = ""7^ ^ — T mu^ be niultiply'd by the la ft

Term. So that the Series becomes 5 = 6 — 1.7 -f- .1. x -^-S - - f x .§.

^ 3 n I * v * ^ •' v 4 TO ' v '* v 3' v •* f Tf A R CD for -->

XT9 • TXT XT XT10 T X T X>T x TJ1 r **> ^ ^> ^> (XU 2>

3, 4, 5, &c. and <--, b, c, d, &c. = i, 2, 3, 4, &c. then 2 = 1-4- T-+^x^3 +|x^xi4-|-|x^xix^5, &c. If A,B,C,D,6cc. =^ i, 2, 3, 4, &c. and ^, b, c, d, &c. = 2, 3,4, 5, &c. then i =

- 13 + T x|4 — T XT;<i5 H- T *TXTXT6> &c- And from this general Series may infinite other particular Series be eafily de- rived, which fliali perpetually converge to given Quantities ; the chief ufe of which Speculation, I think, will be, to iliew us the nature of Convergency in general.

There are many other fuch like general Series that may be 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, toge- ther with a Rank of Fractions, both negative and affirmative, as here follows.

'* - - - &c I--"""""'"'°

-h	a ' A	b	c - c	e £r
A-\-a	-+•	Ab — Ba	-J-	ti —^/> L — De-Ed c_
A	AB	BC DE ' C* ]

Then proceeding obliquely, I collect the Terms of each Series toge- ther, by adding the two nrit, then the two fecond, and fo on. So that' the whole Series thus conftrudled muft neceflarily be equal to Unity ; which alfo is manifeft by a bare Infpeclion of the Series. From this Series it is eafy to defcend to any number of particular Cafes. As if we make A, B, C, D, &c. = 2, 3, 4, 5, 6cc. and a, b, i, &c. then A— J- — ^ __l___i_6,

&c. And fo in all

.= . , .

other Cafes. The Series will flop at a finite number of Terms, whenfoever you omit to take in the firft part of the Numerator of any Term. As here | — -JL ? _ -1- — ^ — -1--.^ = ,.

Laftly, to conftru6t one more Series of this kind, which mail converge to Unity ; I fet down i, with a Rank of Fractions along

with

and INFINITE SERIES, 159

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

i 4-	a f A ~*~ a	ab	«£(T	"+" A BCD ~ «/>«:</	abcJe /, hrvrr
AB a!>	"t" ABC abt	abctle „ N~C
A	AB	AB^	' ABCDE' C' *
A— a	l-b	?4	_«~- L^_	D — rf ,	E — c , , „ I .._ , fjhrii ATP T
A			•*" AB^DE^™' ~C> J<

This Series may be made to flop at any finite number of Terms, if you omit to take in the latter part of the Binomial in any Term. Or you may derive particular Series from it, which fhall have any Rate of Convergency.

For an Example of this Series, make A, B, C, D, &c. = 3, 3, 3, 3,6cc. and a, b, c,dy &c. = i, i, i, i, &c. then y4-f -+-TV + TT> &c. = i, or JL 4- £ 4- -\ 4- TV, &c. = ±. And whenever A, B, C, &c. and a, b, f, &c. are Ranks of Equals, the Series will be a Geometrical PiogrefTion.

Again, make A, B, C, D, &c. = 2, 3, 4, 5, &c. and a, b, c, d, &c.

= i, i, i, i, &c. then i-4- 7^ 4- 7777; + rx 3x4x5 + 2x3x4x5x6° &c. = i. Or in a finite number of Terms T + T+ 77^ + 2X 3xS

_i I = i. And the like may be obferved of others in an

2x3x4x5

infinite variety.

And thus having prepared the way for what follows, by explain- ing the nature of infinite Series in general, by difcovering their origin and manner of convergency, and by fhewing their connexion with cur common Arithmetick ; I mall now return to our Author's Me- thods of Oj , or to the Reduction of compound Quantities to fuch infinite Series.

SECT. II. The Resolution of fimph Equations, or pure Powers, by I?ifihi.'d Szries.

3, 4. ' | ^HE Author begins his Reduction of compound (

tit; -, to an equivalent infinite Series of fmiple Tc-ms, fir ft by fhevr: j; how the Piocefs may be peiform'd in Divifion. Now in his Example the manner of the Operation is thus, in imi-

taton

j6o *fi>e Method of FLUXIONS,

tation of the ufual praxis of Divifion in Numbers. In order to ob- tain the Quotient of aa divided by b -f- x, or to relblve the com- pound Fraction T|T- into a Series of fimple Terms, firft find the Quotient of aa divided by l>} the firft Term of the Divifor. This is ^ , which write in the Quote. Then multiply the Divifor by

this Term, and fet the Product aa -h ^ under the Dividend, from

whence it muft be fubtracted, and will leave the Remainder — ~ .

Then to find the next Term (or Figure) of the Quotient, divide 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 fet orderly under the laft Remainder ; from whence it muft be fub- tracted, to find the new Remainder -h "-^- . Then to find the

bo

next Term of the Quotient, you are to proceed with th-is new Remainder as with the former ; and fo on in infimtum. The Qup-

r . a* K* c c*x* a*x3 c , - «* •

tient therefore is j — — -+- — ^- , &c. (or -j into i —

? .+- ^ — ^ , 6cc.) So that by this Operation the Number or

Quantity ^— , (or a1 x^-t-*!"1) is reduced from that Scale in Arithmetick whofe Root is b -+• x, to an equivalent Number, the Root of whofe Scale, (or whofe converging quantity) is £ . And this Number, or infinite Series thus found, will converge fo much the fafter to the truth, as b is greater than x.

To- apply this, by way of illustration, to an inftance or two in common Numbers. Suppofe we had the Fraction |, and would jeduce it from the feptenary Scale, in which it now appears, to an equivalent Series, that mall converge by the Powers of 6. Then

, we (hall have j = ^ ^ ; and therefore in the foregoing general

\ Fraction -^- , make a-=. i, b = 6, and #==1, and the Series

b -"j~ x

will become f — ~ + ^ — ^, &c. which will be equivalent to Y. Or if we would reduce it to a Series converging by the Powers of 8, becaufe f= ~ , make a= i, ^=8, and .v = — i,

then

and IN FINITE SERIES. j6r

then ~ = T •+• ~* -+- & -+- ^ > &-c- which Series will converge fafter than the former. Or if we would reduce it to the common Denary (or Decimal) Scale, becaufe f — -~r- , niake a= i, l> = 10, and

x= — 3 ; then 7 = -rV -4- -4-0- -+- Wo-o- -f- -o-Vo-o- + TO-O-^O-S-J <*c' = 0,1428, &c. as may be eafily collected. And hence we may obferve, that this or any other Fraction maybe reduced a great va- riety of ways to infinite Series ; but that Series will converge iafteft 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 mould here refolve 7 into the parts 3 -f- 4, or 2+5, or i -f- 6, &c. inftead of converging we mould have diverging Series, or :fuch as require a Supplement to be taken in.

And we may here farther obferve, that as in .Divifion of com- mon Numbers, we may flop the procefs of Divifion whenever we pleafe, and inftead of all the reft of the Figures (or Terms) ad in- finituniy we may write the Remainder as a Numerator, and the 'Divifor as the Denominator of a Fraction, which Fraction will be the Supplement to the Quotient : fo the fame will obtain in the Divifion of Species. Thus in the prefent Example, if we will flop

at the firft Term of the Quotient, we mall have -^- = "~ — a^L. .

^•— ' b •+• X o b X /; — |- x

Or if we will ft op at the fecond Term, then -£-r\. = j — "-~r -f- Or if we will flop at the third Term, then ^- = ^ — _ ^-x . And fo in the fucceeding Terms, in which thefe Supplements may always be introduced, to make the Quotient compleat. This Obfervation will be found of good ufe in fome of the following Speculations, when a complicate Fraction is not to be intirely refolved, but only to be deprefs'd, or to be reduced to a fimpler and more commodious form.

Or we may hence change Divifion into Multiplication. For hav- ing found the firft Term of the Quotient, and its Supplement, or

aa £ta aax i • i *'• -i K /lit

the Equation ^— = - — -^x -, multiplying it by ? , we fhall have -^- = — — T~^- , fo that fubftituting this value of

IldVC i * 3-a '

ant Ml aa aa aaX

_ffL_ in the firft Equation, it will become ^ = y -^ -f-

.a>A'*- where the two firft Terms of the Quotient are now known.

Y Multiply

162 The. Method of FLUXIONS*

Multiply this by ^ , and it will become

*L- , which being fubfthuted in the laft Equation, it will become

aa ra aav fi^.v1 a*** a'ix*' i .1 c r- n

— r— =. - ---- -4 — • -- - -- 1- -. — r- . where the four nrlt

t-^-x b b* b* I* iS+i*X '

Terms of the Quotient are now known. Again, multiply this

,-, . , A.4 rf5.v4 fl-.v4 a*x* a*x6'

Equation by ^ , and it will become ^7^x = —* --- JT+ ~

-p — r- -, r£8- , which being fubftituted in the laft Equations

... , aa a* az.v a*x* «7.<3 a**4 a1x!

it will become - — - = — — 4 — — — - — 1 — —f p- 4-

i 6 17 i V8

fyi- — ^- -4 5T- , where eight of the firft Terms are now

hi b° t9-^-6°x

known. And fo every fucceeding Operation will double the num- ber of Terms, that were before found in the Quotient.

This method of Reduction may be thus very conveniently imi- tated in Numbers, or we may thus change Divifion into Multipli- cation. Suppofe (for inftance). I would find the Reciprocal of the Prime Number 29, or the value of the Fraction T'T. m Decimal 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- comes a fingle Figure, and then I afliime the Supplement to 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 give ^ = 0,275844^., or rather ^.==0,27586^. I fubftitute this initead of the Fraction in the firft Equation, and I (hall have ^=1:0,0344827586^. Again, I multiply this Equation by 6, and it will give T*7 = o, 2068965517^, and then by Subftitution T'7== 0,03448275862068965517^. Again, I multiply this Equation by 7, anditbecomesT7?=o,24i3793io3448275862oi|-,andthenbySubfti-

where every Operation will at leaft double the number of Figures found by the preceding Operation. And this will be an eafy Expe- dient for converting Divifion into Multiplication in all Cafes. For the Reciprocal of the Divifor being thus found, it may be multi- ply'd into the Dividend to produce the Quotient.

. . , c , , aa aa n*x «-** «**S

Now as it is here found, that j— =7 — 77 -+• ~jr Z7~>

&c. which Series will converge when b is greater than A* ; fo when it happens to be otherwife, or when x is greater than b, that the Powers of x may be in the Denominators we muft have recourfe to

the

and INFINITE SERIES, 163

the other Cafe of Divifion, in which we fhall find -^-^ = ^ — £i _j_ a^- — "^ , &c. and where the Divifion is perform 'd as

before.

5, 6. In thefe Examples of our Author, the Procefs of Divifion (for the exercife of the Learner) may be thus exhibited :

o — xi+o

— AT1 — .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

Unity, we muft invert the Terms, and then we fhall have — l—

XX "^ 1

i i

1 I I I c «*•

= ^ — ^ + 7* — »•» &c-

ii

*•/•*

7, 8, 9, io. This Notation of Powers and Roots 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- ther difcuffion of it here, as not properly belonging to this place, or fubject, but rather to the vulgar Algebra.

1 1. The Author proceeds to the Extraction of the Roots of pure Equations, which he thus performs, in imitation of the ufual Pro^ cefs in Numbers. To extract the Square-root of aa •+- xx ; firft the Root of aa is a, which muft be 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. Multiply the Divifor thus compleated, or -za -J- x~ , by the fecond

Term of the Root, and the Produft xx + — muft be fubtrafted from the Refolvend. This will leave — — , for a new Refolvend,

4-"

which being divided by the firft Term of the double Root, or 2tf,

. A

will give j for the third Term of the Root. Twice the Root

before found, with this Term added to it, or 2a -+- ^ — -^ , be- 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,

for finding the next Term of the Root ; and fo on as far as you pleafe. So that we (hall have \/ ' aa -+-xx = a+ '- _ £-' _i_ ~

1 T.a oa* io»*

It is eafy to obferve from hence, that in 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 ex- traded, as many more may be found by Divifion only. Thus hav- ing; found the three firft Terms of the Root a -f- — , by

2a fcu3 " J

v^ v4

their double -za -\ — , dividing the third Remainder or Re- folvend -\- 7^—: , the three firft Terms of the Quotient —

in. 4 04*. ° ^*— l6&^

c*8 7xl°

— ; H — '—,- will be the three fucceeding Terms of the Root.

1 2 Oil ' 2 COrt* fj

The Series a -f- ^i H — TT* •> ^c< t^us f°untl f°r the fquare-

root of the irrational quantity aa -f- xx, is to be understood in the following manner. In order to a due convergency a is to 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- root required. But as this is too little, it is enereafed by the fmall

quantity — , which now makes it too big. Then by the next

Operation

and INFINITE SERIES. 165

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

minifli'd by the next Term. And thus it proceeds in infinitum, the Augmentations and Diminutions continually correcting one another, till at lalt ihey become inconfiderable, and till the Series (fo far con- tinued) is a lufficiemly near Approximation to the Root required.

12. Wh-ii a is Ids than x, the order of the Terms muft be in- verted, 01 ihe fquare-root of xx -+- aa muft be extracted as before;

in which cafe it will be x -+- — — -f-. , &c. And in this Series

5 '

2X

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 of the quantity aa -+- xx -, for each of the two Series will exhibit thofe two Roots, by only changing the Signs. But they are 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 yv=.cni-\- xx, where we may fi'-d the value of the Root y by the f 11 ,wir ;.j Proccfy ; yy = aa -+- XX= (\f)' = rf-f-/) aa^-zap -\-pp-,

or zap -+- pp = xx = (If p = — •+• q} xx •+• zaq -{- ~ -±-qq; or 2rf?-J- ^ -H^=— —- = (if ?===_^

+ r > or

rr = '-, . -- 6 -— (if r = •— + j) &c. which Procefs may

oi,'» O-|t.° 1 VU* J

be thus explain 'd in wo-ds.

In order to find V ua -±-xx, or the Root y of this Equation yy-=aa-\-xx, iuppofc1 y = ^-f-/', wheie a is to be undeiftood as a pretty near Approxii: arion to the value of _y, (the nearer the bet- ter,) and p is the lnv.,11 Supplement to that, or the quantity which makes it compleat. Then by Subftitution is deiivcd the fir It Sun- plementiil Lqu^i'oa zap -+-//; = xx, whole Root/; is to bt fou:,d. INOW as 2uJ> is n:iich bigger than ff, (lor za is bigger than the Sup-

plement/,) v;c fh;.!l have nearly p — - , or at leaft ve (hall have

exactly ;- = : ; -f- -', fuppofmg q to reprefent the fecoiid 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 fecond Supplement. Therefore —q will be a little quantity, and qq much lefs, fo -that we mall have nearly q= — g--3, or accurately q =. — £^ -f- r, if r be made the third Supplement to the Root. 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 there- fore will make nearer and nearer Approaches to the Root y, to which they always converge. For y =5= a -{-/>, where p is the Root of this

Equation zap-±- pp-=xx. Or y =: a~\- — -+-g, where q is the

Root of this Equation zaq -\ — -q-\-qq-=z -- —^ . Or y ; — a -f- *— — £-. -f- r. where r is the Root of this Equation zar -f- — r—

Ztt oa> a

~ I rr-=. -~ — ~. And fo on. The "Refolution of any one

of thefe Quadratick Equations, in the ordinary way, will give the refpeclive Supplement, which will compleat the value 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 carry on the Analogy, we mall hereafter find a like Comparifon of Fluents ; where one Fluent, (fuppofe, for inftance, a 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 anticipation, that I might mew the conftant uniformity and har- mony of Nature, in thefe Speculations, when they are duly and re- gularly purfued.

But I mall here give, ex abundanti, another Method for this, and fuch kind of Extractions, tho' perhaps it may more properly be- long to the Refolution of Affected Equations, which is foon to fol- low ; however it may ferve as an Introduction to their Solution.

j The

and INFINITE SERIES. 167

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

Bccaufe />= -^-, it will be by Divilion p = - — -{ -f- ^ —

' za + ty 3 " za Aa* »«»

** ! x*tA

,-^7 •+• -^ , &c. Divide all the Terms of this Series (except the fir ft) by p, and then multiply them by the whole Series, or by the value of />, and you will have p = - — — + — ' — 3-^ -f-

ia 8*' 8*4 3Z«»

^ -g , 6cc. where the two firft Terms are clear'd of />. Divide all the

Terms of this Series, except the two firft, by />, and multiply them by the value of />, or by the firft Series, and you will have a Series for p in which the three firft Terms are clear'd of p. And by re- peating the Operation, you may clear as many Terms of p as you

pleafe. So that at laft you will have p = •£ — ~ -+- £, — 7^

-+- ^~, &c. which will give the fame value of y as before.

13, 14, 15, 16, 17, 18. The feveral Roots of thefe Examples, and 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- ral Theorems computed for that purpofe. And as there will be fre- quent occalion, in the enfuing Treatiie, for certain general Opera- tions to be perform'd with infinite Series, fuch as Multiplication, Divilion, railing of Powers, and extracting of Roots ; 1 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 a_l_£_j_^_{_j\.4_g) &c. reprefent the Terms of three feveral Series refpedlively, and let A-|-B-{-C-f-D-|-E, &c. into P+Q-t-R-f-S+T, &c. = a, -\- /B -{- y -i- <f~ -\- e, &c. Then by the known Rules of Multip'ication, by which every Term of one Factor is to be multi- ply'd into every Term of the other, it will be « = AP, /3 = AQ^-j- BP, 7=AR-i-BQ^-CP, ^z^AS-i-BR-i-LQH-DP, g=AT-f- BS + CR-t-DQ^-4- E'P ; and fo on. Then by Subftitution it will be

. 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 infinite Series into each other 5 as in the following Example.

(A) (B) (C) (D) (E) (P) (QJ (R) (S) (T)

X* *J A'4 ,, . x1

afr£*+ £ + & + &> &c- mto*-fx-f- - &>+X^+i*?,rb£ +~, $cc, =^+t^+tf^1

X* A 4

JL/rv _ v*a — -P.

•=— , t* A ^^TT1** V-1 — • '- _

• 9# \2a"

+**'+£ +7|?

— i! *4_

7 a 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

will ferve commodioufly for the Divifion of one infinite Series by another. Here for conveniency-fake the Capitals A, B, C, D, &c. are retained in the Theorem, to denote the firft, fecond, third, fourth, &c. Terms of the Series refpedively.

M (0) Thus, for Example, if we would divide the Series #* _f. £.ax -+.

(>) (/) (t) (pJ (QJ W (S) (ij

iix* _}_ ^-ii^-_{_ .2' " z , &c. by the Series a+^x-i- — -f- ~^. — , &c.

the Quotient will . be a -f- -a*"~T* -f- ±fx*

, &c. Or reftoring the Values of A, B, C, D, &c. which reprefent the feveral Terms as they /land in order, the Quotient will become a — f # + — — — _i_ .11 &r

5« 7«z *^ ga3 ' Ut^'

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 Thus if we would know the Reciprocal of the Series a-\- f.v-{-

<R) (S) (T)

£ _|_ ^ 4- ^ , &c. we fhall have by Subftitution I — t_i _

&c. And reftoring the Values of A, B, C, D, &c. it will be *- — — --- - -- h — — ^~- > &c. for the Reciprocal required.

la1- 12^ 8«4 720«*'

^- 2. l,.x + f..-A<..{iff. = i + f.v + i*»+i<», &c. And fa of others.

IV. In the firft Theorem if we make P=A, Q^==B, R = C, S=D, &c. that is, if we make both to be the fame Series ; we mail have

A+B+C+D+E+F+G7&^ I * tf= A»+ zAB + zAC+ zAD + 2AE + zAF + zAG.tff.

+ B1 + zBC + 2BD + zBE + aBF + L* + zCD+ zCE + D*

which will be a Theorem for finding the Square of any infinite Series.

Fv i -'_

-_— -— .

aa Sa'^lba5 iz8a7 256^ 4^* ga^iea* I zSafl"1" 25!., .'<>

64«8 S i 2as

i t1x'L txl A-4

•»-• *^c x1 bx* o

Ex. 3. --- H -- &c.

J 2« ^a ^3 '

u «„« i TTTI

.7/4 4 64*4

Ex.4. _lH_H_l^-Ii^-, I *_ ii._fil , » 30."

2 J 8 2434" -9<?4 — >

v.

Method of FLUXIONS, V. In this laft Theorem, if we make A*= P, aAB = Qv, 2 AC

_f- B1 =R, 2AD -+- 2BC = S, 2 AE -+- 2BD -f- C1 = T, &c. we

O R .-— R *• S "* BC

fhallhave A = P^ B = ^-, C==-^- , D = -^- , E== T~2BD~C- , &c. Or p + Q + K-hS+TH-U, &c. | ^ = pi

iA

Q R— Bz S— 2RC T— 2BD— C^ U— 2BE — ^CD

-4- — r -4~ 1 -4- -4- -• &c

zA ^^ 2A zA 2A 2A ' <xu

By this Theorem the Square-root of any infinite Series may eafily be extracted. Here A, B, C, D, &c. will reprelent the feveral Terms of the Series as they are in fucceffion.

^1 ^i ~ i^- _i_fli a4

Ex 2^1-— o

"'~~

VI. Becaufeit is by the fourth Theorem a-{-@-i--y-\-<f<-t-t,&tc. |* = «,a 4-2a/3-f- 2a^ + 2a^H- 2ae, &c. in the third Theorem for

P, Q^ R, S, T, &c. write a1, 2a/3, 2«> + j8S 2a«^ -f- 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 A*, 2AB, 2AC + B4, 2AD -H 2BC, &c. refpedively, (that is A+B+C+D,&c.|13byTheor.4.)wemallhaveA+B+C+D+E+F36cc.|5. = As -i- 3A*B + sAB1 -h sA*D -j- 3AC1 -f- 360, &c.

6ABC+ 36^0 + 3B»D B' + 6ABD-f- 6ACD -- 6ABE

t

which will readily give the Cube of any infinite Series. "

v9 A-11

r. 13 ^ X'

*' *'* ^ yjf^ ^^

* *" " » "T~ 2*11 " " «15 3

Ex.

and INFINITE SERIES, 171

Ex.2. t*1-i~

VIII. In the laft Theorem, if we make A3 =P, 3A*B • — O , '+.3A'C = R, B'-f-6ABC-|-3A1D = S, &c. then A=PT,

Q_ „ R — 3AB* _ S-6ABC — Bi

B = p: , C = ?Ax , U = - j^ - , fisc. that is , Sec. I i^K + +l + ^

root of any infinite Series may be extracted. Here alfo A, B, C, D, &c. will reprefent the Terms as they ftand in order.

T? x'1 8*15 7*"* 1 7 _ _*» xs ;** IPX'* ^

-~"I"I~ - z' I ~ ^+8i«8 243a"'

Ex. 2. f*4 -h T'7A;7 H-T|Tx8, 6cc. l^ =t**-t-r'T** H-Trr^4, &c.

IX. Becaufe it is by the feventh Theorem a •+• £ -f- y -\- £, &c. j J

a* + 3ai/3 -f- 3 a/31 -f- /35, &c. in the third Theorem for P,

R, S, T, &c. write «', 3««j8, 3«/Si -f-Sa1^ /3}-f- 6a/3>-f- 3«'fr &c. refpeflively ; then

This Theorem will give the Reciprocal of the Cube of any infinite Series ; where A, B, C, D, &c. ftand for the Terms in order.

X. Laftly, in the firft Theorem if we make P=A;, Q4==3A1B,

> &c. we {hall have

A+B-f-C-i-D, &c. I 4 =A^H-4AsB-{-6A1B1-|-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- cals, or any fractional Powers compounded of thefe ; all which will 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,

Or we may have recourfe to a kind of Mechanical Artifice, by which all the 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 Multiplicand, and the other the Multiplier. Write dawn upon your Paper the Terms of the Multiplicand, with their Signs, in a defcend- ing order, fo that 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 the Terms of the Multiplier, with their proper Signs, in an afcend- ing 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 place, for the firft Term of the Product required. Move your move- abie Paper a ftep lower, fo that two of the firft Terms of the Mul- tiplier may ftand over-againft two of the firft Terms of the Multi- plicand. Find the two Produces, by multiplying each pair of the Terms together, that ftand over-againft one another ; abbreviate them if it may be done, and- fet down the Refult for the fecond Term of the Product required. Move your moveable Paper a ftep lower, fo that three of the firft Terms of the Multiplier may ftand over-againft three of the firft Terms of the Multiplicand. Find the three Products, by multiplying each pair of the Terms together that ftand over-againft one another j abbreviate them, and fet 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 from the common Scale of Denary 01 Decimal Arithmetick ; which will equally explain the Procefs in all other infinite Series whatever.

Let the Numbers to be multiply 'd be 37,528936, &c. and 528,73041, &c. which, by fupplying X or 10 where it is under- ftood, will become the Series 3X -\- jX° -+- jX-'-f- aX-'-f- 8X-* _j_ 9X-44- 3X-5H- 6X-« &c. and 5X* -f- aX 4- 8X° -j-7X- + 3X~l-t- oX-J-+-4X-4-f- iX-s, &c. and call the firft the Multipli- cand, and the fecohd the Multiplier. Thefe being difpofed as is prefcribed, will ftand as follows.

Multiplier,

and INFINITE SERIES.

Multiplier,

-H4X-+

-f-oX-'

8X

Multiplicand •?X	Product TrY3	iX* - - - oX3
3A	*5A	8Xl
^X-1	-i- 6^^f	. - 4.X
2X-»		T**
- 8X— s	- •- -1 . T 1 1 Y— x	AV-i
?X~5	i i-8X— *	-8X— s
-i- 6X-5	1 1JO^\. ~* ^ ^ ~* ~ ' t" 2 O I ^\. ^ ^ r^j /*	©c.

Now the firft Term of the rrioveable Paper, or Multiplier, being apply'd to the firft Term of the Multiplicand, will give jX1 x 3X = i5X3 for the firft Term of the Product. Then the' two firft Terms of each being apply'd together, they will give jXa xyX0 -f- 2X x 3X = 4-iX1 for the fecond Term of the Product. Then the three firft Terms of each being apply'd together, they will give 5X1 x5X-'-t-2X x7X° -f- 8X° x 3X = 63X for the third Term of the Product. And fo on. So that the Product required will be ,5X» + 4IX1 -H 63X H- oyX0 -f- i42X—-f- 133%.-*+. I38X-3 -i-2OiX~4, &c. Now this will be a Number in the Decimal Scale of Arithmetick, becaufe X = 10. But in that Scale, when it is re- gular, the Coefficients muft always be affirmative Integers, lefs than the Root 10 j and therefore to reduce thefe to fuel), fet them orderly under one another, as is done here, and beginning at the loweft, col- lect them as they ftand, by adding up each Column. The reafon of which is this. Becaufe aoiX~4== aoX" *-f- iX~4, we muft fet down iX~4, and add 2oX~5 to the line above,- Then becaufe 2oX~3 H- i38X~s= i58X-*=i5X-ir4-8X-*) we muft fet down and add i ^X.~- to the line above. Then becaufe i^X~^-f- i = i48X-*=-i4X-'+ 8X~l, we muft let down 8X-S, and add i4X~' to the line above. And fo we muft' proceed through the whole Number. So that at lift we (hall find the Product to be iX4 _!_ 9X 3 H- 8X * -+- 4X -f- 2X° -f- 6X— + 8X-1 -f- 8X-J , &c. Or by fuppreffing X, or 10, and leaving it to be fuppiv'a by the Ima- gination, the Product required wil' be 19842,688, &c.

When one ihfinite Series is to be divided by another, wiite down the Terms of the Dividend, wkh , eir \ >^ er Signs, in a defccnd- ing order, fo that the Tunis may be at equal diftances, and juil nn-

dcr

174 t^}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 changed except the firft, fo that the 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 Dividend, together with the Product of the firft Term of the Quo- tient now found, multiply'd by the Terms over-againft it in the left- hand Paper ; thefe divided by the firft Term of the Divifor will be the fecond Term of the Quotient required. Move your moveable Paper a ftep lower, fo that three of the firft Terms of the 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- ing multiply'd into the Term over-againft it, in the left-hand 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- tient required. And fo on to find the fifth, and the fucceeding Terms.

For an Example let it be propofed to divide the infinite Series

I2IA-5 28|X4 ., , 1C' 1

.a* + tax 4- —x1 H- ^ + 7^1 , &c. by the Series a 4- f x _]_ — -j_ — L -+- -^ , &c. Thefe being difpofed as is prefcribed, will ftand as here follows.

Divifor,

and INFINITE SERIES.

175

Divifor, ; £«fr		Dividend tf»	^___	Quotient
X4		-4- itfA:	-f-y/z.V — — ,7V — ~<7V	*z 1 v
5*3 X* ~4^ **		-f- tt-v1 ( I 2ix3	•4-tt** + f ^ — f x1 = -f- f x* 121*3 AT* *3 ^J Aj	• J'V xl -*-?= ^3
3« _ * V		izbca 281*4	*~I26oa lOa ""ga 43 7,1 f 28l.v4 ( A4 A4 *4 A4 . A4	7*» *4
»•* 0		' I2boa' &C.	*~126o«*~1 I4«l ,;«* 1 ,2s1 ra' ~t~9a* &c.	+ 9.. &c.

Here if we apply the firft Term of the Divifor a, to the firft Term of the Dividend a1, by Divifion we fhall have a for the firft Term of the Quotient. Then applying the two firft Terms of the.- Divifor to the two firft Terms of the JDividend, we fhall have ^ax to be colledled with the Produdl a x — f AT, or — ±ax, which will make — -^ax -, and this divided by a, the firft Term of the Divifor, will give — ±x for the fecond Term of the Quotient. And fo of the other Terms ; and in like manner for all other Examples.

When an infinite Series is to be raifed to any Power, or when any Root of it is to be extradled, it may be perform'd in all cafes by a like Artifice. Prepare your fixt or right-hand Paper, by wri- ting down the natural Numbers o, i, 2, 3, 4, &c. juft under one an- other at equal diftances, referving places to the right-hand for the feveral Terms of the Power or Root, as they fhall be found. The firft Term of which Series may be immediately known from the firft Term of the given Series, and from the given Index of the Power or Root, whether that Index be an Integer or a Fraclion, affirmative or negative ; and that Term therefore may be fet down in its place, . over-againft the firft Number o. Prepare your moveable or left- hand Paper, by writing down, towards the edge of the Paper at the right-hand, all the Terms of the given Series, except the firft, over one another in order, at the fame diftances as the Numbers in the other Paper. After which, nearer the edge of the Paper, write juft over one another, fiift the Index of the Power or Root to be found, then its double, then its triple, and fo the reft of its multiples, with the negative Sign after each, as far as the Terms of the Series extend. And alfo the firft Term of the given Series may be wrote below. Thus will the moveable Paper be prepared. Thefe multi- ples, together with the following negative Signs, and the Numbers •

7^^ Method of FLUXIONS,

°> ij 2' 3-4> ^c- on tne otner Paper, when they meet together, will compkat the numeral Coefficients. Apply therefore the fecond Term .of the move-able Paper to the uppertnoft Term of the fixt Paper, ;ind the Product made by the continual Multiplication of 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,] di- vided 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- -againft one another, (in which, and elfewhere, care muft be 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 re- quired, 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 Multiples of the Index, as m, zm, pn, 40;, &c. with the negative Sign — after them. Likewife a vinculum may be undei flood to be placed over them, to connect them with the other 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 feparated from the reft by a line, to denote its being a Divifor, or the Denominator of a Fraction. And thus is the moveable Paper prepared.

To prepare the fixt or right-hand Paper, write down the natu- ral Numbers o, i, 2, 3, 4, &c. under one another, at the fame equal diftances as the Terms in the other Paper, with a Point after them as a Mark of Multiplication ; and over-againft the firft 1 erm o

write

and INFINITE SERIES.

write a*"zm for the firft Term of the Series required. The reft ot the Terms are to be wrote down orderly under this, as they (hall be 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

will then exhibit this Fraction *-*• m~~ °- " z , which being reduced

as,. I

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

az. 2

which being reduced will become mum-lc-{- mx "L—Lam--b* xzm+'~,

to be put down for the third Term of the Series required. Bring down the moveable Paper a ftep lower, and you will have the Fraction -f- dz,*. yn — o. amzn .+- cz*.

bz?. m

ma

*c -+- m x

Lam-lb3-

az. 3

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

Moveable Paper, &c.

*. m

az.

Fixt Paper

o. i.

2.

- a^i-o* -f- mam~*c x z"

-3 . m x - — -x- — -am—*l>>+mx.'- — •' am—16c+Mam—ldxz"

J 7. 1 T

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 place. And this therefore will fufficiently prove the truth of the 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 jit the fame time to prepare the way for the Solution of Affected and Fluxional Equations; we will here make a kind of retrofpect, and refume our Author's Examples of fimple Extractions, beginning with Divifion 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 fimple Terms ; make r^—=y, or by -f- xy —- , Now to find the quantity y difpofe the Terms of this Equation after this manner + *-J J = a1, and proceed in the Refolution as you fee is done here.

I a*x a*.*1 a^J a*xt

=** — -T-*--T* 7T •+• -77- > &C..

+ xy\ h-r TT + -75 — ,

C

IT + -JT- > OCC.

Here by the difpofition of the Terms a*- is made the firft Term of the Series belonging (or equivalent) to by, and therefore dividing by b, — will be the firfl Term of the Series 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 down over-againft by, but with a contrary Sign, to be the fecond

Term of that Series. Then will — a~ be the fecond Term of yt

to be fet down in its place, which will give — a-^- for the fe- cond Term of -f- xy ; and this with a contrary Sign muit be fet down for the third Term of by. Then will + ~- be the third Term of

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

Now the Rationale of this Procefs, and of all that will here fol- low 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- gregate of which muft (till be equal to the fame known Term aa ; (or perhaps Terms.) Therefore all the fubfidiary and adventitious Terms, which are introduced into the Equation to aflift the Solution, (or the Supplemental Terms,) muft mutually deftroy one another. Or we may refolve the fame Equation in the following manner :

a* la* k*a* Ha* .

y = — -- - -4 -- - -- , &c.

•^ A" .V X» A;4 •

Here a1 is made the firft Term of -+- xv, and therefore — muft

•" x

be put down for the firft Term of y. This will give + — for the firft Term of by, which with a contrary Sign muft be the fecond Term of -+- xy, and therefore -- ~ muft be put down for the fe-

cond Term of y. Then will — — ^ be the fecond Term of byy which with a contrary Sign will be 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"

• — vt or y -+- xly=. i, the Refolution of which Equation is little rrxpre than writing down the Terms, in the manner following :

y = i— x+x— xx,&cc. y 7 ---- (-x-1— *-4-|_x- «

. +x*y 3 = i— x-'-+x-±— x~&,

, ccc. +x*y 3 = i— x-'-+x-±— x~&, &c.

Here in the firft Paradigm, as i is made the firft Term of y, fo will x1 be the firft Term of x*y, and therefore — x*- will be the fecond Term of y, and therefore — x* will be the fecond Term of x*y, and therefore -+- x* will be third Term of y ; &c. Alfo in 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 Term of x*y, or — x~* will be the fecond Term of y ; &c.

A a 2 To

180 tte Method of FLUXIONS,

i 3.

To refolve the compound Fraction . zx ~* - into fimple Terms, i i

2.ya •£* 13 i

make — 7 =y, or 2** — #v = y 4- AT^ — ^xy, which E-

I+*1— 3*

quation may be thus refolved :

= 2A^ * X^

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

Place the Terms of the Equation, in which the unknown quan- tity 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 4- x*y, which mufl be wrote with a contrary Sign for the fecond Term of y. Then will the fecond 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 of y, had not the Term • — x* been above, which reduces it to 4- jxJ* for the third Term of y. Then will 4- yx* be the third Term of 4- x*y, and 4- 6x* will be the fecond Term of — 3fly, which being collected with a contrary Sign, will make — 1 3** for the fourth Term of y ; and fo on, as in the Paradigm.

If we would refolve this Fraction, or this Equation, fo as to ac- commodate it to the other cafe of convergency, we may invert the Terms, and proceed thus :

O W f — •— V1 *- -i-

3-v / x

y = f X' -f- 7 — 4f *

1, &c.

— ft*-1' , 6cc.

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

firfl

and INFINITE SERIES. 181

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

•will glve — ?j-x~* f°r the third Term of y. Then will the third Term of -+- xh be — if, and the fecond Term of y being -+- -%, thefe two collected with a contrary Sign will make -f- if for the fourth Term of — T,xy, and therefore — TT*""1 will be l^e fourth Term of y -, and fo on.

And thus much for Divifion ; now to go on to the Author's pure or fimple Extractions.

To find the Square-root of aa -f- xx, or to extract the Root y of this Equation yy = aa-{- xx ; make y = a -+-/>, then we fhall have by Subftitution zap -f- pp = xx, of which affected Quadratick Equa- tion we may thus extract the Root p. Difpofe the Terms in this manner zap-^= xx, the unknown Terms in a defcending order oa

H-/AJ one fide, and the known Term or Terms on the other fide of the

Equation, and proceed in the Extraction as is here directed.

-) *4 *« 5x8 7*'°

**/==* - — - + s74 — 5£i + H53.

+ -\.__ +^ _^i +^!_^:,

"J J 4«* 8*4 640* 12Sa8'

x* A4 *6 CA9 7*10 .,

* -I- —, h -t-f-t » &C.

•f za 8al \6a! izSa1 25O«'

By this Difpofition of the Terms, x1 is made the firft Term of

x

the Series belonging to zap ; then we fhall have — for the firft 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

place over-againft p1. 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 of/> to be — - ^ . Having there- fore

77jt2 Method of F L n v i o !-; s,-

«/

fore the two firft Terms of P = *- — ~, we fhall liave, (by any of the foregoing Methods for finding the Square of an infinite Se- ries,) the two firft Terms of p1 = ~ — — . which la ft Term

AfCi3- #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 p*

* ' L

zap ---- -f- zax — za* -+-

- * — * ,&c.

(by the aforefaid Methods) will be -~} which is to be wrote with a contrary Sign, as the fourth Term of zap. Then the fourth

8

Term of p will be — -||_, and therefore the fourth Term of/* is """ 7IsI« ' which is to be wrote with a contrary Sign for the fifth Term of zap. This will give 2^—- for the fifth Term of p •. and fo

2^O«'

we may proceed in the Extraction as far as we pleafe.

Or we may difpofe the Terms of the Supplemental Equation thus :

>J a* c

~x * — ^ ' &c'

— , &c.

A X 3 y

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

Extraction) the third Term of / will be -- . This will make the .third Term of zap to be — , which makes the fourth Term of/4

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

I zap

and INFINITE SERIES, 183

zap will be — — . , which will make the fixth Term of />* to be

f 4*5

.ll ; and therefore o will be the fixth Term of p, &c.

Here the Terms will be alternately deficient ; fo that in the given Equation yy = aa -\- xx, the Root will be y = a -f- x — a -f- "-•>

&c. that is y = x -f- °— — ^-} -h -^-s , &c. which is the fame as 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 Root y of the Equation yy = aa — xx ; make y = a -f- p, as be- fore ; then zap -f-/*x = — x*-, which may be refolved as in the fol- lowing Paradigm :

•J _ .v4 X6 ^.V8

•f I" 4flz 8a4 64,1°

f *4 X'6 <;*8

+ t>1\ 1- — ; H- ^~; -f- f — -„ -{- r J 4« 8«4 6^.«6

^ "4 JC CX "7 X c

J* — — . ,^_ __ k~^_ • i . . ^^^^ - ^^ ^ J, cSCC*

Here if we mould attempt to make — x1 the firft Term of -J-/1, we mould have ^/ — x1, or x^/ — i, for the fi rfl Term of/ ; which being rnpoflible, fliews no Series can be form'd from that Suppofi- tion.

To find the Square-root of # — xx, or the Root y in this Equa- tion yy = x — xx, make y = x^ + p, then x -+- zx^p -f- p1 = 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 ~\- px3 will be the firft Term of /a, which is alfo to be wrote with a contrary Sign for the fecoiid Term of 2x'-p, which will give — f A-* for the lecond Term of p. Then (by fquaring) the fecond Term of ^ will be i^4, which will give — - i*4 for the fecond Term of

184 ffi? Method of FLUXIONS,

zx^p, and therefore — -V^ for the third Term of p ; and fq op. Therefore in this Equation it will be y=z x'* — f A-'" — f x* — rV*''"* &c.

So to extract the Root y of this Equation yy =.aa-\-bx — xxt make y = a-{-p} then zap -+- p* = bx — xx, which may be thus refolved.

=fa—X* +*fi, &C. L'-x'1

tx x* Ixl

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

20.

of p. Therefore the firft Term of p1 will be -+- b—^ , which is alfo to be wrote with a contrary Sign, fo that the fecond Term of

zap will be -<— x* ^- , which will make the fecond Term of

p to be — *— — gjr • ' Then by fquaring, the fecond Term of/*

will be — —7 *-* -g^ 5 which muft be wrote with a 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

Quantity a^ -f- bx — xx will be a -+• •£ — ^ ~ -f. -^ _£,

Alfo if we would extract the Square-root of •< _ a* , we may ex-

tract the Roots of the Numerator, and likewife of the Denomi- nator, and then divide one Series by the other, as before ; but more

dire.ctly thus. Make ••_**! = yy, or i -{- ax1 = yy — b*x*y*.

Suppofe y = i -f- p, then ax* = zp -+-pz — bx*- — zbx*-p— bx*-p*, which Suppplemental Equation may be thus refolved.

zp

and INFINITE SERIES,

185

— bx^p* _

—ab —^ab1, &c.

~a^bt &c.

+TT*\

, &c.

Make ax1 -f- bx* the firfl Term of 2/>, then will frf.vl -f- f &v» "be the firfl Term of /. Therefore — abx* — b*x* will be the firfl Term of — 2bx*p, and ^a*x* -f- -^abx* -f- -^bx* will be the firfl Term of/*. Thefe being collected, and their Signs changed, muil 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 •will be — -^ab^x6 — ±l>*x6 -f- •^aibx6> and the fecond Term of p* (by fquaring) will be found f albx6 •+- \ab*x6 — j-a''X6 -{- $frx6, and the firfl Term of — bx*pl will be — ^a^bx6 — -^ab'-x6 — f^'AT4 ; which being collected and the Signs changed, will make the third Term of 2p, half which will be the third Term of p ; and fo on as far as you pleafe.

And thus if we were to extract the Cube-root of a* 4- x*, or the Root y of this Equation 7' = a 3 4- tf3 j make y = a -f-/, then by Subflitution a3 -f- 3d1/ -f- ^ap1 -+- p> = & -+- x*, or 3«i/ -f = A:S, which fupplemental Equation may be thus refolved.

243«l

B b

The

Method of FLUXIONS,

The Terms being difpos'd in order, the firft Term of the Series <ia*p will be #', which will make the firft Term ofp to be *—. Thiss

will make the firft Term of/1 to be —^ . And this will make the

firfl Term of ^ap* to be ^ , which with a contrary Sign muft be the fecond Term of 3/z*/>, and therefore the fecond Term ofp will be — — r . Then (by fquaring) the fecond Term of ^ap1 will be

. ff! and (by cubing) the firft Term of *"= win be — fi . Thefe

Oa* 27<:<6

r y9

being collected make — — , which with a contrary Sign muft be the third Term of ^a^p, and therefore the third Term of p will be _j_ ill . Then by fquaring, the third Term of ^ap* will be -

__ .

and by cubing, the fecond Term of/3 will be — ^—^, which being collected will make y^-j > anc^ therefore the fourth Term of-^^p will be — ^—T, and the fourth Term of p will be -- •°*11- . And

8j<i' ' 243a

fo on;'

Arid thus may the Roots of all pare Equations- be extracted, but in a more direct and fimple manner by the foregoing Theorems. All that is here intended, is, to prepare the way for the Refolution of affected Equations, both in Numbers and Species, as alfo of Fluxional Equations, in- which this Method will be found to be of very extenfive ufe. And firfl we mall proceed with our Author to the Solution of numerical affected Equations.

SECT. II L The Refolution of Nttmeral AffeSted Equations*

W as to the Refolution of affected Equations, and firft in Numbers ; our Author very juftly complains, that be- 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- ever was extremely perplex'd and operofe. So that he had good rea- fon to reject their Methods, efpecially as he has fubftituted a much better in their room. They -affected too great accuracy in purfuing

exact

and INFINITE SERIES. 187

exact Roots, which led them into tedious perplexities ; but he knew very well, that legitimate Approximations would proceed much more regularly and expeditioufly, and would anfwer the fame intention much better.

20, 21, 22. His Method may be 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 takes as near an Approximation to the Root as 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, together with this Supplement, are to be fubftituted in the given. Equation inftead of the Root. By this means, (expunging what will be fuperfluous,) a Supplemental Equation will be form'd, whole Root is now p, which will confift of the Powers of the affumed Approxima- tion orderly defcending, involved with the Powers of the Supplement regularly afcending, on both which accounts the Terms will be con- tinually decreafmg, in a decuple ratio or falter, if 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 nearly exhauft the Supplement p, it will be fufficient to retain only the two firft Terms of this Equation, and to feek the Value ofp from the refulting fimple Equation. [Or fometimes the 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- mation, together with a new Supplement g, muft be fuhftituted in- itead of p in this laft fupplemental Equation, in order to form a fecond, whofe Root will be q. And the fame things may be obferved of this fecond fupplemental Equation as of the firft; and its Root, or an Approximation to it, may be difcover'd after the fame manner. And 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 prefent Example jy3 — 2y — 5 = o, 'tis eaiy to perceive, that y = 2 fere ; for 2x2x2 — 2x2 = 4, which mould make 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, fere, or p •=. Tx_. fere ; or affuming a fecond Supplement q, 'tis p = T'o- -h ? accurately. This being fubftituted for p in the laft Equation, it becomes o, 6 1 -+- 1 1,237 + 6>3?* 4- <f = o, which is a new Supplemental Equation, in which all the Terms are farther deprefs'd, and in which the Supplement q will be much lefs than the former Supplement p. Therefore it is 0,61 -f- 1 1,23.^ = o, ym?,.

or q= f^e, or q-=. — 0,0054-)^ accurate, by afluming

r for the third Supplement. This being fubftituted will give 0,000541554- 11,162;-, &c. =o, and therefore r-= — °'°°^^-^

= — 0,00004852, &c. So that at laft/=2 -{-^> = &c. or_y = 2,09455148, &c.

And thus our Author's Method proceeds, for finding the Roots of affedted Equations in Numbers. Long after this was wrote, Mr. Rapb- Jon publifh'd his Analyfis Mquationum imiverjalis, containing a Me- thod for the Solution of Numeral Equations, not very much diffe- rent from this of our Author, as may appear by the following Com— parifon.

To find the Root of the Equation y* — zy = 5, Mr. Rapbfon would proceed thus. His firft Approximation he calls g, which he takes as near the true Root as he can, and makes the Supplement x, fo that he has_y==g-+Ar. Then by Subftitution <g3-f-3^1Ar+3^xa-f-x3=5,

•— 2£— 2

or if g=2, 'tis iOAr-f-6.v* -4- x* = i, to determine the Supple- ment x. This being fuppofed fmall, its Powers may be rejected, and therefore iox= i, or ,v = o, i nearly. This added tog or 2, makes a new £ = 2,1, and x being ftill the Supplement, 'tis y = 2,1 +x, which being fubftituted in the original Equation _y5 — zy = 5, produces 11,23^-4- 6,3** + x3 = — 0,6 1, to determine the, new Supplement x. He rejects the Powers of x, and thence derives ^___o£j __ — 0,0.054, and confequently y = 2,0946, which

I 1 ,25

not being exaft, becaufe the Powers of x were rejected, he makes the Supplement again to be x, fo that y= 2,0946 -f- x, which be- ing fubftituted in the Original Equation, gives 11,162^-+- &c. = — 0,00054155. Therefore to find the third Supplement x, he has

.v =•" 0,'°°106524'5S = — 0,00004852, fo that y =.2,0946 + *= 2,09455148, &c. and To on.

By

and IN FINITE SERIES. 189

By this Procefs we may fee how nearly thefe two Methods agree, and wherein they differ. For the difference is only this, that our Author conftantly profecutes the Refidual or Supplemental Equations, to find the firft, fecond, third, &c. Supplements to the Root : But Mr. Raphjbn continually corrects the Root itfelf from the fame fup- plementaf Equations, which are formed by fubftituting the corrected Roots in the Original Equation. And the Rate of Convergency will1 be the fame in both.

In imitation of thefe Methods, we may thus profecute this In- quiry after a very general manner. Let the given Equation to be refolved be in this form aym -+- by"-* -4- cy™-* -J- dym~* , &c. = o, in which fuppofe P to be any near Approximation to the Root y, and the little Supplement to be p. Then is y = P -4-/>. Now from what is (hewn before, concerning the raifing of Powers and extrac- ting Roots, it will follow that ym = P -h/> I m = P* -f- wPm-'/>, &c. or that thefe will be the two firft Terms of ym ; and all the reft, , being multiply'd into the Powers of />, may be rejected. And for the fame reafon ym~l = Pm~I -h m — iPm~lp, &c. ym~l = Pm~- -+• m — 2P"-=p, &c. and fo of all the reft. Therefore thefe being fub- ftituted into the Equation, it will be a]>>» .4- niaPn-lp , &c."l ~ -, &c.

m — 2c¥n~*p, &c. >= o ; Or dividing by P" , m — 7 JP "-"<•/>, &c.

&c.

'- -j-^/P-s , &c.

-\-m — ^dP~*p, &c. = o. From whence taking the Value of />, we mail have/ = -- « + *P-' + cP-» + rfp-* . ar,. _ and

ma?-1 +« — lbV~' -{.m — z^~3 + m — J^P-4 , Jjff .

confequently r=

^ J

,.

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

P—=A-'B, P-I=A-iB% &c. which being fubftituted, and alfo multiplying the Numerator and Denominator by A"7, it will be

~ ~ ~ 'B +" A."-"-B*+ =4rfA"-?Bi. ^c-. will

be a nearer Approach to the Rootjy, than jp or P, and fo much

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. Let th,e general Equation aym -^- by"1—1 -+• cym~~- -f- d)m~s, &c.

=: o be propofed to be folved ; if the Fraction - be affumed as near the Root y as conveniently may be, the Fraction

. iAAm— 131 +7«— zcAm-S F 3 4. »z— 3n/A«— 4B4,'feff .

nearer Approximation to the .Root. And this Fraction, when com- puted, may,be,ufed inflead of the Fraction •- , by which means a

Bearer Approximation may again -be had ; and fo on, till we ap- proach 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-

A1 -4- rBz

tion y1 •+• by === c, it will be y = , ,"7D p , fere. In the Cubick

if if * f2t\ — J— DO X D

. . .... 2A

Equation y* + ty + cy = d, it will be y == 3<i. y^r^. In the Biquadratick Equation y* -{- by* -\- cy1 -+-dy=ze, it

irt* -+- 2/)AB-4- iB~ x A1 -f- rB4 _ . 111-1 c i • l_

be ^ == >/^' And the llke of hlSher

Equations.

;For an 'Example of the Solution of a Quadratick Equation, let it.be propofed to extract the Square-root of 12, or let us find the value of_y in this Equation y1 #= 1.2. Then by comparing with the general formula, we fliall have b •=. o, and <: = 12. And

/I

taking 3 for the firft approach to the Root, or making g =T> that is, Az=3 and B;= i, we fliall have by Substitution y ^==. ^~ =4-, fora nearer Approximation. Again, making A = 7

and B = 2, we fliall have y = 12l == || for a nearer Approxi-

14 X 2

mation. Again, making A = 97 and B = 28, we fliall have _y=± 97j — + i:! x 28i . __ l£il7 for a nearer Approximation. Aeain,

'94* 2» S452 _ _ _

making A= 18817 and B=543 2> we ^a11 have y= '

7o8ic8o77 /- A • • i -r • i

== — — 1|^ ior a nearer Approximation. And if we go on in the

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 Equation in the fhorteft and fimpleft manner. That is, it will al- 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 _yj * — 2? = 5, and therefore by Companion b = o, €•=. — 2, and d==. 5. And taking 2 for the firft Approach to the

Root, or making ^- = 4., that is, A = 2 and B=i, we mall

have by Subftitution y ==- = 44 f°r a nearer Approach to

the Root. Again, make A = 21 and B = 10, and then we mall have y = 9- 1 + 2500 __ Hj-L for a nearer Approximation.

6615 — 1000 5615

Again, make A= 11761 and 6 = 5615, and we mall have

~ — .

y = =

3x11761 1 ^561 5— 2x5615 1 3 J 9759573 16495

proximation. And fo we might proceed to find as near an Approxi' mation as we think fit. And when we have computed the Root near enough in a Vulgar Fraction, we may then (if we pleafe) re- duce 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 firft for determining the forms of the Series^ and their initial Approximations.

23, 24. TTT^ROM the Refolution of numeral affected Equations, J/ our Author proceeds to find the 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 they are to be exprefs'd by Numbers belonging to a general Arithme- tical Scale, as has been explain'd before, of which the Root is de- noted by .v or z. The affigning or chufing this Root is what he means here, by diftinguiming one of the literal Coefficients from the reft, if there are feverul. And this is 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 converge as faft as may be. If it be the leaft, or a Fraction lefs than Unity, its afcending Powers muft be in the Numerators of the Terms. If it be the greateft quantity, then its afcending Powers muft be in the Denominators, to make the Series duly converge. If it be very near a given quantity, then that quantity may be con- veniently made the firft Approximation, and that fmall difference, or Supplement, may be made the Root of the Scale, or the con- verging quantity. The Examples will make this plain.

25, 26. The Equation to be refolved, for conveniency-fake, iliould always be reduced to the fimpleft form it can be, before its Refo- Jution be attempted ; for this will always give the leaft trouble. But all the Reductions mention'd by the Author, and of which he gives us Examples, are not always neceflary, tho' they may be often con- venient. The Method is general, and will find the Roots of 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- guifhing or affigning a proper quantity, by which its Root is to con- verge, the Author before has made three cafes or varieties ; all which, for the fake of uniformity, he here reduces to one. For becaufe the Series mull neceffarily converge, that quantity muft be as fmall -,as poffible, in refpect of the other -quantities, that its afcending Powers may continually diminim. If it be thought proper to chufe the greateil quantity, inftead of that its Reciprocal muft be intro- duced, which will bring it to the foregoing cafe. And if it approach near to a given quantity, then their fmall difference may be intro- duced into the Equation, which again will bring it to the firft cafe. So that we need only purfue that cale, becaufe the Equation is al- ways 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 con- fider the .nature of thofe Numbers, or Expreffions, to which thefe literal Equations are reduced, whofe Roots are required ; and in this Inquiry we ihall be much aiTifted by what has been already difcourfed of Arithmetical Scales. In affected Equations that were purely nume- ral, the Solution of which was juft now taught, the feveral Powers of the Root were orderly difpoied, according to a fingle or limple Arithmetical Scale, which proceeded only in longum, and was there

fufficient

#nd INFINITE SERIES.

fafficient for their Solution. 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 Scale, or of the converging quantity, by which the Series for the Root of the Equation is to be form'd ; on account of each of which circumftances the Terms of the Equation are to be regularly difpofed, and therefore are to conftitute a double or combined Arithmetical Scale, which muft proceed both ways, in latum as well as in longum, as it were in a Table. For the Powers of the Root to be extraded, fuppofe y, are to be 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 pf the Root, by which the Series is to converge, fuppofe x, are to be difpofed in latum, fo as to conftitute an Arithmetical Progreffion, and the vacancies may likewife be fill'd up by the fame Mark *, when it £hall be thought neceffary. And both thefe together will make a combined or double Arithmetical Scale. Thus if the Equa- tion ys— $xy • 4- i!y4 — 7* •#•/» 4- -6a* x* 4- fax* =a=-o, were given, to find the Root y, the Terms may be thus difpofed :

y6 y* V4 yS y* yl yo

= 0;

Alfo the Equation vf — by* 4- gbx\ — x* =o fhould be thus dif- jpofed, in order to its Solution :

y' * * — by* *

x°	y *	*	#	#	# *
X	* —-5*7*	*	*	*	* *
X*	* *	*	* "~"7	ga^i.a	* *
		** 4		*
X*	* *		*	*	* +6a*x*
X*	* *	•	*	»	* 4~^*#4

: 1-

4- tfx* r

And the Equation y* 4- axy -J- a*y — #» — 2<z3 = o thus :

jKJ * + a*y — za

za*-*

* f = °> *' J

And the Equation x'y* — y*xy* — c'x* 4-^ = 0 thus :

* • *

* * * •—y*xy* * * ^ sss 0.

x*y' * * And the like of all other Equations.

C c When

Method of FLUXIONS,

When the Terms of the Equation are thus regularly difpos'd, ft is then ready for Solution ; to which the 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- fully inquired into : The reafon and foundation of which may be thus generally explained from the following Table, of which the Construction is thus.

— za-\-zb

—za+b

—za — b

— za — zb

+4*

— b

—zl

a^-bt,

a — b

a—zb

2-+J*

— b

3 a

3 a — zb

40+4^

\a-\-zb

« — b

50 — b

;«— 3

ba+bb

ba—b

ba—zb

•ja-\-bb

711+3^

70+ zJ

7—3*

In a Pfor.e draw any number of Lines, parallel and equidiftant, and c»thers_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- ceffively",' and the Terms — b, — zb, — 3^, &c. underneath. And thefe Ave ma^f call primary Terms. Now to infert its proper. Term in any other afitgiVd. Parallelogram, add the two primary 'Terms together.., that' ftand over-againft if each- way, and write the Sum in the given Parallelogram. And-* thus all the Parallelograms be- ing fill's, as-far as there is oecafion every way, the whole. Space

will

and INFINITE SERIES. 195

-will become a Table, which may be called a combined Arithmetical ProgreJJion in piano, compofed of the two general Numbers a and t\ of which thefe following will be the chief properties.

Any Row of Terms, parallel to the primary Series o, a, za, ^a, &c. will be an Arithmetical Progreflion, whofe common Difference is a ; and it may be any fuch Progreflion at pleafure. Any Row or Column parallel to the primary Series o, £, zb, 3^, &c. will be an 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- grams whatever ; all the Terms of the Parallelograms, whofe Cen- ters mail at the fame time touch the Edge of the Ruler, will conftitute an Arithmetical Progreflion, whofe common difference will coniiit of two parts, the firfl of which will be fome Multiple of a, and the other a Multiple of b. If this Progreflion be fuppos'd to proceed injeriora. verjus, or from the upper Term or Parallelogram towards the lower ; each part of the common difference may be feparately found, by fub- tracling the primary Term belonging to the lower, from the primary Term belonging to the upper Parallelogram. If this common diffe- rence, when found, be made equal to nothing, and thereby the Re- lation of a and b be determined ; the Progreflion degenerates into a Hank of Equals, or (if you pleafe) it becomes an Arithmetical Progref- fion, whofe common difference is infinitely little. In which cafe, if the Ruler be moved by a parallel motion, all the Terms of the Parallelo- grams, whofe Centers mall at the fame time be found to touch the 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- flinguifh'd from. the reft, or aflign'd promifcuoufly and at pleafure, through whofe Centers, as before, the Edge of the Ruler ihall fuc- ceflively pafs in its parallel motion, beginning from any two (or more) initial or external Parallelograms, :whofe Terms are 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 Table, or of a combined Arithmetical Progreflion in 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- promifcuoufly, with other known quantities, of which x is to be the Root of the Scale, (or Series,) as explain'd before ; fuch a value of y is to be found, as when fubftituted in the Equation inftead of y, the whole (hall be deftroy'd, and become equal to nothing. And firft the initial Term of the Series^or the firfl Approximation, is to be found, wtyich in all cafes may be Analytically reprefented by Ax*1 ; or we may always put y = Axm , &c. So that we mail have y1 = A*x™, &c. _>'3 = A*x*m, 6cc. _}>4 = A4*4*, &c. And fo of other Powers or Roots. Thefe when fubftituted in the Equation, and by that means compounded with the feveral Powers of 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 Table j but the vacancies may always be conceived to be fupply'd, and then it will have the properties before mention'd. That is, the Ruler being apply'd to two (or perhaps more) initial or external Terms, (for if they were not external, they could not be at the be- 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 by increafing. But the higheft may be chofen, and then a Series 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 alfo, and an Arithmetical Progreffion may be found', in which they are all comprehended, and confequently the form 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 rightly placed at equal diftances j otherwife the Ruler cannot be ac- tually apply'd, to difcover the Progreffions of the Indices, as 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- cular cafe, in which a-=.m and b=. i -, which will th.n appear thus :

- 2M+0

— zm-if 3

— 2m — 3

_,, +5

— w+4

- AO-f- 2

— m-\- 1

— m— 3

+ 5

+ 4 + 3

2

— 3

m+6

"-•+5

CT+2

OT+I

<— 3

2W+4

2OT-4-2

2OT+ 1

JOT+6

>— 3

4'"+ 4

4W+2

— 3

5 M-6

5^—2

— 3

601 — 3

"'"+5

7m+4

Now the chief properties of this Table, fubfervient to the prefent purpofe, will be thefe. If any Parallelogram be feledted, and an- other any how below it towards the right hand, and if their included Numbers be made equal, by determining the general Number m, which in this cafe will always be affirmative ; alfo if the Edge of the Ruler be apply 'd to the Centers of thefe two Parallelograms ; all the Numbers of the other Parallelograms, whofe Centers at the fame time touch the Ruler, will likewife be equal to each other. Thus if the Parallelogram denoted by m -+- 4 be feleded, as alfo the Parallelo- gram 377* -f- 2 ; and if we make m -t- 4 = ^m -H 2, we mall have m=i. Alfo the Parallelograms — ;;; -h 6, m -f- 4, 3^7 -|- 2, $m, jm — 2, &c. will at the fame time be found to touch the Edge of 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 in- cluded 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 the Parallelograms 6w+6, 5^4-4, 4^ + 2, 3?;;, zm — 2, 6cc.

will

198 7%e Method of FLUXIONS,

will be found at the fame time to 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 equal diftances will form an afcending Arithmetical Progeffion. Thus if the two firft felected Parallelograms be zm — 1 = 5;;; — 3, whence m=.±, the Numbers in all the correfponding Parallelograms will be -j. Then if the Ruler moves towards the 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 4/7; -{-3, 7/» •+- i, &c. which will each be 5^. If it moves forward again to the fame diftance, it will arrive at yn -f- 5, %m -f- 3, &c. which will each be 8f. And fo on. But the Numbers f, 3, 52., 8y, &c. are in an Arithmetical Progreffion whofe common diffe- rence is 2-i. And the like, mutatis mutandis, in other circum- fiances.

And hence it will follow <? contra, that if from the firft fituation of the Ruler, it moves towards the left-hand by a parallel motion, it will continually arrive at lefler and leifer Numbers, which at equal diftances will form a decreafing Arithmetical Progreffion.

But in the other fituation of the Ruler, in which it inclines down- wards towards the left-hand, if it be moved towards the right-hand by a parallel motion, it will continually arrive at greater and greater Numbers, which at equal diftances will form an increafing Arith- metical Progreffion. Thus if the two firft feleded Numbers or Pa- rallelograms be 8m + i = $m — i, whence m = ~ ~} and the Numbers in all the correfponding Parallelograms will be — 4!.. If the Ruler moves upwards into the parallel fituation 5^-4-2, 2;;;, 8fc. thefe Numbers will each be — i f. If it move on at the fame diftance, it will arrive at 2m + 3, — m-+- i, 6cc. which will each be i-i. If it move forward again to the fame diftance, it will arrive at — m -f- 4, — 4/;z -+- 2, &c. which will each be 4^. And fo on. But the Num- bers — 4,1, — i|, i-i, 4.1, &c. or — .Ll, — i, |, -L±, &c. are in an in- creafing Arithmetical Progreffion, whofe common difference is ±, or 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 conti- nually 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 num- ber of Parallelograms, in their proper places, with their refpeclive

Num-

and INFINITE SERIES,

199

Numbers included, neglecYmg all the reft ; we mould form fome cer- tain Figure, fuch as this, of which thefe would be the 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, that is, to two of the external Parallelograms, and their Numbers being made equal, by determining the general Number m ; if the Ruler paffes over all the reft of the Parallelograms by a parallel mo- tion, 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 diftan- ces ; or atleaft-they may be reduced to fuch, by fupplyingany Terms 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 ^m ~}_ 5, and if they be made equal, we mall have m = o, fo that each of thefe Numbers will be 5. The next Numbers that the Ruler will arrive at will be m -f- 3, 4;;; +3, 6/« -f- 3, of which each will be 3. The la ft are zm -f- i, 5>«-f- i, of which each is i. So that here #2 = 0, and the Numbers arifing are 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 Progreffion, or at leaft it might have been interpolated with other Terms, fo as to comprehend them all, however promifcuoufly and irregularly they might have been taken.

Thus fecondly, if the Ruler be apply'd to the two external Pa- rallelograms 5/72+ 5 and 6m-}- 3, and if thefe Numbers be made equal, we mail have m = 2, and the Numbers themfelves will be each ic. The three next Numbers which the Ruler .will arrive at

will

20O The Method of FLUXIONS,

will be each 11, and the two laft will be ^ach 5. But the Num- bers 15, n> 5. will be comprehended in the decreafing Arithmetical Progreffion 15, 13, 1 1, 9, 7, 5, whofe common difference is 2.

Thirdly, if the Ruler be apply'd to the two external Parallelograms 6m -f- 3 and 5*0-4-1, and if thefe Numbers be made equal, we fhall have tn = — 2, and the Numbers will be each — 9. The two next 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 comprehended in the afcending Arithmetical 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

Parallelograms 2m-\-i and 5/77 -+- i, and if they be made equal, we fhall have again m = o, fo that each of thefe Numbers will be i . The next three Numbers that the Ruler will approach to, will each be 3, and the laft 5. But the Numbers i, 3, 5, will be compre- hended in an afcending Arithmetical Progreffion, whofe common difference is 2.

Fifthly, if the Ruler 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 5. The three next Numbers that the Ruler will approach to will each be 1 1, and the two next will be each 15. But the Numbers 5, 1 1, 15, will be comprehended in the afcending Arithmetical Progreffion 5, 7, 9, II, 13, 15, of which the common difference is 2.

Laftly, if the Ruler be apply'd to the two external Parallelograms pn -f- 5 and m-\- 3, and if thefe 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 defcending Arithmetical 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- fulting Terms fhall be comprehended ; find their differences, and the greateft common Divifor of thofe differences fhall be the common difference of the Progreffion. Thus in the fifth cafe before, the refulting Numbers were 5, 1 1,15, whofe differences are 6, 4, and their greateft common Divifor is 2. Therefore 2 will be the common difference of the Arithmetical Progreffion, which will include all the refulting Numbers 5, n, 15, without any fuperfluous Terms. But the .ap- plication of all this will be beft apprehended from the Examples that are to follow. 30

and INFINITE SERIES. 201

30. We have before given the form of this Equation, y< — $xy*

_j_ I!y4 — ja*x1)* •+- 6<?3.Y5 4-^Ar* = o, when the Terms are dif-

pofed according to a double or combined Arithmetical Scale, in or- der to its Solution. Or obferving the fame difpofition of the Terms, they may be inferted in their refpedive Parallelograms, as the Table requires. Or rather, it may be fufficient to tabulate the feveral In- dices of A; only, when they are derived as follows. Let Ax" repre- fent the firft Term of the Series to be form'd for y, as before, or let y=;Ax'", &c. Then by fubftituting this for y in the given Equa- tion, we fhall have A6.\-6m — $Asx$m+l -+- -^xv+s — 7«*Aa.vtIB+« -f.

6fl3xJ -f-^.x'4, &c. = o. Thefe Indices of AT, when felected from the general Table, with their refpective Parallelograms, will ftand thus:

4
3				4w-h3
		2/W-J-2
					5/W-f- I
						6m

Here if we would have an afcending Series for the Root yy we may apply the Ruler to the three external Terms 3, 2/;;-f- 2, 6/w, which being made equal to each other, will give ;« = -£-> and each of the Numbers will be 3. The Ruler in its parallel motion will next arrive at $m -+- i, or 37; then at 4; then at 4^-1-3, or 5; which Numbers will be comprehended in the Arithmetical Progref- fion 3, 37, 4, 47, 5, whofe common difference is f. This there- fore will be the common difference of the Progreffion of the Indices, in the Series to be derived for y. So that now we intirely know the form of the Series, which will refuk from this Cafe. For if A, B, C, D, &c. be put to reprefent the feveral Coefficients of the Series in order, and as the firft Index m is found to be 7, and the common difference of the afcending Series is allo 7, we ihall have here j =

A^ H- Bx -{- CV-H- DAT% &c.

As to the Value of the firft Coefficient A, this is found by putting the initial or external Terms of the Parallelogram equal to nothing.

D d This

202 Tfo Method of FLUXIONS,

This here will give the Equation A6 — 7rt*A* -j- 6<z5 = o, which, has thefe fix Roots, A = ± ,/tf, A — ±^/2a, A=±v/ — 3*7,. of which the two laft are impoffible, and to be rejected. Of the others any one may be taken for A, according as we would profecute this or that Root of the Equation.

Now that this is a legitimate Method for rinding the firft Ap- proximation Axm , may appear from confidering, that when the Terms of the Equation are thus ranged, according to a double Arith- metical Scale, the initial or external Terms, (each Cafe in its turn,) become the moil confiderable of the Series, and the reft continually decreafe, or become of lefs and lefs value, according as they recede more and more from thofe initial Terms. Confequently they may be all rejected, as leaft confiderable, which will make thofe initial or external Terms to be (nearly) equal to nothing ; which Suppofi- tion gives the Value of A, or of Axn , for the fir ft Approximation, And this Suppofition is afterwards regularly purfued in the fubfe- quent Operations, and proper Supplements are found, by means of which the remaining Terms of the Root are extracted.

We may try here likewife, if we can obtain a defcending Series for the Root y, by applying the Ruler to the two external Terms ^m _j_ y and 6m ; which being made equal to each other, will give m =T> and hence each of the Numbers will be 9. The Ruler in its motion will next arrive at $m-\- i, or 8f. Then at zm -f- 2, or 5. Then at 4. And laftly at 3. But thefe Numbers 9, 8f, 5, 4, 3, will be comprehended in an Arithmetical Progreffion, of which the common difference is i. So that the form of the Series here •will be y =A.v* -f- Ex -+- Cx^ -f- D^°, &c. But if we put the two external Terms equal to nothing, in order to obtain the firft Ap-

A4 I

proximation, we mail have A6 •+• — =o, or A1 -f- - = o, which

will afford none but impoffible Roots. So that we can have no ini- tial Approximation from this fuppofition, and confequently no Series.

But laftly, to try the third and laft cafe of external Parallelograms, we may apply the Ruler to 4 and 4^2-4-3, which being made equal, will give m = -£, and each of the Numbers will be 4. The next Number will be 3 ; the next 2m -\- 2, or 2| ; the next 50* -{- i, or 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 6cc. may reprefent the form of this Series, if the 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.

3 i. The Equation ys — by1 -+- qbx* — #; =o, when the Terms are difpofed according to a double Arithmetical Scale, will have the form as was (hewn before ; from whence it may be 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 Ax1" for the firft Term of the Series y, this Equation will become by Subftitution A'A.-?" — M1*1" -f- gbx* — x*, 6cc. = o. So 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 Series for y, we may apply the Ruler to the two external Parallelograms 2 and 2W, which therefore being made equal, will give m — - i, and each of the Numbers will be 2. The Ruler then in its parallel

progreis will firft come to 3, and then to yn, or 5. But the Num- bers 2, 3, 5, are all contain'd in an afcending Arithmetical Progrefiion, whofe common difference is i . Therefore the form of the Series will here be ;' = AA; -f- B*1 -f-(*', &c. And to determine the firft Coefficient A, we fhall have the Equation — bfcx1 -f- qbx* — - - o, or Aa= 9, that is A = + 3. So that either 4-3*, or — 3^ may be the initial Approximation, according as we intend to extract the affirmative or the negative Root.

We mall have another cafe of external Terms, and perhaps an- other afcending Series for_y, by applying the Ruler to the Parallelo- grams 2;« and 5;^, which Numbers being made equal, will g;ive m =zo. (For by the way, when we put 2;»= 5/77, we are not at liberty to argue by Diviiion, that 2=5, becaufe this would bring us to an absurdity. And the laws of Argumentation require, that no Abfurdities muft be admitted, but when they are inevitable, and when they are of ufe to mew the falfity of fome Supposition. We fliould therefore here argue by Subtraction, thus: Becanfe cm — ^t>it then 5//f — 2:>i = o, or pn = o, and therefore m = o. This Cau- 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- plies the Principles of his Art.) Having therefore 777. = o, we {hall alfo have the Numbers 2/77. = 577*' = o. The Ruler in its parallel motion will next arrive at 2 ; and then at 3. But the Numbers o, 2, 3, will 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* — bA* = o, or A3 = 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- bly may afford a defcending Series for y. For applying the Ruler to the Parallelograms 3 and 5777, and making thefe equal, we (hall have 7/7=4, and each of thefe Numbers will be 3. Then the Ruler will come to 2 ; and laftly 2777, or -§-• But the Numbers 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^ _f. BA"T -|- CA^ -f- D, &c. And the external Terms Ar.v3 — A:3 = 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 than Unity ; fo this cafe will afford us a Series when x is greater than Unity ; which will converge fo much the fafter, the 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 Ax* to reprefent the firft Approximation to the Root ;', we {hall have by fubftitution A'X=m -f- aAxm+l -+- a'-Ax" — A'3 — 2^3, Sec, = 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 Ruler will come to 777.4- i, or i ; and laftly to 3. But the Numbers o, i, 3, are contain'd in the Arithmetical Progreffion o, i, 2, 3, whofe common difference is i. Therefore the form of the Root is y=. A -+- Ex -{-Cx1 -+- Dx>, 6cc. Now if the Equation A3 + a1 A —<2a' =o, (which is derived from the

initial

and INFINITE SERIES.

205

initial Term?,) is divided by the factor A1 -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 may apply the Ruler to the external Parallelograms 3, yn, which being made equal to each other, will give m = i ; alio thefe Num- bers will each be 3. Then the Ruler will approach to m-\- i, or 2 ; then to //;, or i ; laftly to o. But the Numbers 3, 2, i, o, are a de- creafing Arithmetical Progreflion, of which the common difference is i. So that the form of the Series will here be y=Ax -+- B -+- CA,— ' -f- Dx~- , &c. And the Equation form'd by the external Terms will be A3x3 — .v3 = o, or A= i.

33. The form of the Equation x*)'s — y+X}1 — c'x3- -f- c7 = o, as exprefs'd by a combined Arithmetical Scale, we have already feen, which will eafily mew us all the varieties of external Terms, with their other Circumftances. But for farther illuftration, putting A,vra for the firft Term of the Root y, we ("hall have by fubftitution Atx^m+l — 36--»AI,vi"I+I — c'x* -+- c\ &c. =o. Thefe Indices of x being tabulated, will ftand thus.

Now to have an afcending Series, we mufl apply the Ruler to the two external Terms o and yn -\- 2, which being made equal, will give m — — .*-, and the two Numbers anting will be each o. The next Number that the Ruler arrives at is zm + r, or .J. ; and the la ft is 2. But the Numbers o, i, 2, will be found in an afcending Arithmeti- cal Progreffion, whofe common difference is -i-. Therefore y =. Ax~ _l_ B.v '> -f- C -f- D.x^, &c. will be the form of the Root. To deter- mine the firft Coefficient A, we fhall have from the exterior Terms A'-f-6-7 = o, which will give A = — y^c7 = — c'\ Therefore the firft Term or Approximation to the Root will be y ==. — J/-^ ,

&c.

We may try if we can obtain a defcending Series, by applying the Ruler to the two external Parallelograms, whofe Numbers are 2 and 5;»-f-2, which being made equal, will give ;;; = o, and thefe Numbers will each be 2. The Ruler will next arrive at 2///-J- i, or i ; and laftly at o. But the Numbers 2, i, o, form a de Icon cling ProgreiTion, whofe common difference is i. So that die form of the Series will here be y = A -f. B,v— + Cv-J , &c, And putting the

initial

2					;«+z
		ZOT-J- I
O

206 The Method of FLUXIONS,

initial Terms equal to nothing, as they ftand in the Equation, we ihall have A'*1 — c*x* = o, or A = <r, for the firft Approximation to the Root. And this Series will be accommodated to the cafe of Con- vergency, when x is greater than c -, as the other Series is accommo- dated to the other cafe, when x is lefs than c.

34. If the propofed Equation be 8z,6f> -\- a^y* — 27^ = o, it may be thus refolved without any preparation. When reduced to

our form, it will ftand thus, 8z6/3 -}-az6\* * * 1 ,,

y J f=o; and by

* — 27^9 3 *

putting_y=A£B',&:c.it willbecome 8A*z*"!+6+aA1z*'m+'s * * &c.7

* — 27^3 °*

The firft cafe of external Terms will give $A*z*m+s — 27.^' = o, whence 3/^-1-6 = 0, or m=s — 2. Thefe Indices or Numbers therefore will be each o ; and the other 2/»-f- 6 will be 2. But 0,2, will be in an afcending Arithmetical Progreffion, of which the com- mon difference is 2. So that the form of the Series will be y=. Az~~- -|- B -h Cs.1 -+- Dz*, &c. And bccaufe 8A' = 27^9, or 2A=3^3, it will be A = J-03. Therefore the firft Term or Approximation to

the Root will be 3-^-

2

2. *

But another cafe of external Terms will give aA*-z~-mJc6 — = o, whence 2w-f-6 = o, or /;; = — 3. Thefe Indices or Num- bers therefore will be each o j and the other yn -+- 6 will be — 3. But o, — 3, will be found in a defcending Arithmetical P/ogrefiion, whofe common difference is 3 . So that the form of the Series will be y = Az~* -f- Ez~6 -f- Cs-' , ccc. And becaufe ^A1 = 27^', Jtis A = + 3v/3 x^4> f°r ^ ^''^ Coefficient.

Laftly, there is another cafe of external Terms, which may pom"- bly afford us a defcending Series, by making SA*z3a+6 -f- aA*z"-m^~6 =: o ; whence m = o. And the Numbers will be each equal to 6 ; the other Number, or Index of z, is o. But 6, o, will be in a defcending Arithmetical Progreffion, of which the common difference is 6. Therefore the form of the Series will be _y= A -f- Ez~ 6 -f- Oc-11, &c. Alib becaufe 8A« -+- a A1 = o, it is A = — {a for the firft Coefficient.

I fhall produce one Example more, in order to fhew 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 prefent ftate of the Equation. Let us therefore affume this Equation,

1»,vl «3^a /.« ClI I6 fl\*

y* -- _ + xz -- _ + - _ _ _j_ _ _ _ .+. ^ = o, or

rather y3 — a~1yixl -\- x> — a>y~- x3- -+- a\)— 3 — a \y~z A.—1 -}- a6x~ s ~i -+- a= = o. Which if we make }' = A.\m, &c. and

difpofe

and INFINITE SERIES. 207

difpofe the Terms according to a combined Arithmetical Progref- fion, will appear thus :

* *

*** .*x"m+** *

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 eight cafes of 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- ing Powers of x ; and afterwards for the other four cafes, when the Series converges by the defcending Powers of x.

I. Apply the Ruler, or, (which is the fame thing,) afTume the Equation asA~=x~^ — a"' A-1*-1"1-* = o, which will give — 3/77

= — 2in — 2, or 7/7= 2; alfo A=^. The Number refulting

from thefe Indices is — 6. But the Pailer in its parallel motion will next come to the Index — 3 •. then to — zm-{- 2, or — 2 ; then to o ; then to zm — 2, or 2 ; then to 3 ; and laftly to 3/7; and 2/774- 2, or 6. But the Numbers — 6, — 3, — 2, o, 2, 3, 6, are in an af- cending Arithmetical Progrellion, of which the common difference is i ; and therefore the form of the Series will be y = Ax1- -±-Bx*

-f- C.v«, &c. and its firft Term will be - .

a

II. Affume the Equation a6x~l — a''A—tx-™-i==z o, which will give — 3 = — zm — 2, or m = f } alfo A = ± a*. The Num- ber refulting hence is — 3 ; the next will be — 37/7, or — iJL ; the next 2/72 — 2, or — i ; the next o ; the next — 2/>»-f- 2, or j ; the next 3/7;, or i± ; the two laft zm 4- 2 and 3, are 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 f ; and therefore 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,

III. Aflame the Equation a6x~* — a* A.1.*11""- = o, which will give — 3 = 2?/7 — 2, or;;;= — f; alfo A = + a*. The Num- ber refulting is — 3 ; the next 3;?;, or — if ; the next — 2m — 2, or — i ; the next o ; the next 2m -+- 2, or i ; the next — 3»z, or if; the two laft 3 and — 2m -f- 2, which are each 3. But the Numbers — 3, — if, — i, o, i, if, 3, will be all comprehended in an afcending Arithmetical Progreiiion, of which the common dif- ference is f ; and therefore the form of the Series will be y— - A.y~ ~h B -f- Cx* -f- Dx, &c. and the firft Term will be ± a*x~'f, or

±"v/;-

IV. Affume the Equation A3 A:3™ — ^'A1*-'*-2 = o, which will give 3« = 2;« — 2, or ;/z = — 2; alfo A = a*. The Number refulting is — 6 ; the next will be — 3 ; the next 2m -{-2, or — 2 ; the next o; the next — 2m — 2, or 2 ; the next 3 ; the two laft

— 3/tf and — 2#?4-2, each of which is 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.

I. Afllune the Equation Au3™ — a-'A1x"-m+l — o, which will

give 3;;z = 2/w -4- 2, or #2 = 2; alfo A = - . The Number re- fulting is 6 ; the next will be 3 ; the next 2m — 2, or 2 ; the next O; the next — 2;;z-f-2, or < — 25 the next — 3; the two laft

— 3/72 and — 2m — 2, each of which is — 6. But the Numbers 6, 3, 2, o, — 2, — 3, — 6, belong to a defcending Arithmetical Pro- greflion, of which the common difference is i. Therefore the form of the Series will be/ = Ax* -i- Ex ~f- C -f- D.*-1, &c. and the firft

XS,

Term will be — .

a

II. Affume the Equation x* — a~lAixim~Jri = o, which will give 2m -+- 2 = 3, or ;;:= f ; alfo A = + a*. The Number refulting is 3 ; the next wi: be 3;^, or if; the next — 2;/z-f-2, or i ; the next o ; the next ,/ — 2, or — i ; the next — yn, or — if; the two laft — 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 be_)' = Ax^-i-Ex°+Cx~^-{- DAT*', &c. and the firft Term will be + ^/ax.

III.

gve 3

and INFINITE SERIES. 209

III. Aflume the Equation x5 — <7»A-**— *»+» = o, which will' = — 2 w H- 2, or TW = — f ; alfo A = + a*'. The Num- ber refulting from hence is 3 ; the next will be — 3;;?, or if ; the next 2m -+-2, or i ; the next o ; the next — 2m — 2, or — i ; the next 3777, or — if ; the two laft — 3 and 2m — 2, each of whichare — 3. But the 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 bcy=Ax~*-t-Bx~'-i-Cx~~l-l-Dx- % &c- and the firft Term will be + a*x~* or + a - .

IV. Laftly, aflume the Equation a6A-ix~im — rf»A.-I which will give — 3;;; = — 2m -f- 2, or m = — 2 ; alfo A ==='#». The Number refulting is 6 ; the next will be 3 ; the next — zm — 2, or 2 ; the next o ; the next 2m -f- 2, or — 2 ; the next — 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 the Series will be/=A<x—1H-BA— 34-Cx— 4-t-Dx-5, &c. and the firft

rn • «'

Term is — .

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 pro- ceeds 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- mental Equations, in a regular fuccefTion alfo, the Roots of which are a continued Series of 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 compleat, is the Root of an inferior Equation> the Sup- plement of which is again the Root of an inferior Equation ; and fo on for ever. Or retaining that Supplement, we may flop where we pleafe. 36. The Author's Diagram, or his Procefs of Refolution, is very eafy to be underflood ; yet however it may be thus farther 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- mation to the Root to be a ; inflead of the Root y he fubflitutes its 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 left-hand Column ; of which rejecting all the Terms of too high a compofition, he retains only the two loweft Terms ^.aip-\-aix=.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 Terms, he retains only, the two loweft ^a*q — TIT-cxi = oi which

give a • — • — for the third Term of the Root. And fo on.

Or in imitation of a former Procefs, (which may be feen-, ,pag; 165.) the Refolution of this, and all fuch like Equations, may be thus perform'd. i)3_|_tf^y= 2fl'= (if y=-a-irp} a* + '$a*'p-\-T>ap!L-}-p* } Or collecting

-+.axy-h A?3 +a* +ay > andexpung-

J ing,

©r collecting and expunging,

I

X

By which Procefs the Root will be found _y = <z — 7* 4- ^, &c.

Or

and INFINITE SERIES. 211

Or in imitation of the Method before taught, (pag. 178, &c.) we may thus refolve the firft Supplemental Equation of this Example ; «w>. W-p -f- axp -f- 3^/>a -4-/3 = — a'-x -+• x* -, where the Terms muft be difpos'd in the following manner. But to avoid a great deal of unneceffary prolixity, it may be here obferved, that 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. infinuates, that -4- r— is the third Term of the Series y, with-

1 643 **

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 what- ever.

4*'/l ==— a*x * +*»

^ , &c.

40963

13 1x4 c

7T7T » &c-

To explain this Procefs, it may be obferved, that here — a*x is made the firft Term of the Series, into which ^alp is to be re- folved ; or 4t.a*p = — a*x, &c. and therefore p = — ±x, &c. which is fet down below, Then is -f- axp = — ^ax1, &c. and (by fquaring) _f-3^)a = -t-Tz-axl, &c. each of which are let down in their pro- per Places. Thefe Terms being collecled, will make — -V^S which with a contrary Sign muft be fet down for the fecond Term

of ^a*p ; or 4da/> = * + -r'^axl} &c. and therefore p = * -f- -?- >

&c. Then axp=.*-^-^— , &c. and (by fquaring) 3<?/>a = * — il. > &c. and (by cubing) />5 = — W^"3' ^c- T^efe being collected •will make — ^, to be wrote down with a contrary Sign; and this, together with A:3, one of the Terms of the given Equation,

will make ±a*p = * * -f- -- 'x*} &c. and therefore />= * * -f- l-^- ?

i 1*4 s'*"a

&c. Then axp= * * -J- ~~ , &c. and (by fquaring) 3^/1* = * *

E e 2 —

212 Ih* Method of FLUXIONSJ

_, 1&22 &c. and (by cubing) *» = * -f- -1^1 , &c. all which

4096.1 ' N ] ft! 1024* '

being collected with a contrary Sign, will make 4tf1/> = ***-f- 5£9i_* , &c. and therefore /—=*** -f- ^ ,' &c. And by the

40961* 163841 '

fame Method we may continue the Extraction as far 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- what a .different manner. The Terms of 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 fide, and t,he known Terms on the other fide ; any adventitious Terms may be introduced, fuch as will be neceffary for forming the feveral Series, provided they are made mutually to deftroy one an- other, that the integrity of the Equation may be thereby preferved. Thefe adventitious Terms will be fupply'd by a kind of Circulation, , which w^ill make the work eafy and pleafant enough ; and the ne- ceffary Terms of the fimple Powers or Roots, of fuch Series as com- pofe the Equation, muft be derived one by one, by any of the foregoing Theorems. , - -

Or if we are willing to avoid too many, and to0 high Powers - in thefe Extraction's, we may proceed' in the following manner. The Example mall be the fame Supplemental Equation as before, , which may be reduced to this form, 4a* -f- ax -+- ^ap -4- pp -x.p =s

— a*x * 4-#3j of which the Refolution may be thus : ,

— " ' .'

-•'! '• •'• 4rfaH- ax

__•, 3. . .- $ - .

5 i 2a

-4-/a - h TV** — 77^ , &c:

X* I3I*S

64« 5i2«i 16384^3^

The Terms 4^* -f- ax-\- ^p-^-fp I call the aggregate Factor, of which I place the known part or parts 4<2* -{- ax .above, and the unknown, parts ^ap -f- pp in a Column to the left-hand, fa as that their refpeclive Series, as they come to be known, may be placed regularly over-againft them. Under thefe a Line is drawn, to receive

2, the

and INFINITE SERIES. 2*3

the aggregate Series beneath it, which is -form'd by the Terms of the aggregate Factor, as they become known. Under this aggregate Se- ries comes the fimple Factor />, or the fymbol of the Root to be extracted, as its Terms become known alfo. Laftly, under all are the known Terms of the Equation in their 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.

Firft, the initial Term 4^ (of the aggregate Series) is brought down into its place, as having no other Term to be collected with it. Then becaufe this Term, multiply 'd by the firft Term of />, fuppofe q, is equal to the firft Term of the Product, that is, ^.a'-q = — a*x, it will be q = — ~x, cr p = — -L.v, &c. to be put down in its place. Thence we (hall have T>ap-=. — %ax, &c. which to- gether with -}-ax above, will make -^^'ax for the fecond Term; of the aggregate Series. Now if we fuppofe r to reprefent the le-- cond Term of p, and to be wrote in its 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*'

may now be fet down in its place. And hence yap = * -f- -^x3-, &c. and p* = ^x*, &c. which being collected will make ^~xl, 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 by our Theorem for Multiplication of infinite Series,) q.a1; 4-

— • — ^ = *3 ; (for x3 is the third Term of the Product.) There- 256 256

fore s = l- , to be fet down in its place. Then -lap = * * -4- 512^*

&c. and i1 = * — A- , &c. which together will make

• zea

5 I 2a

_j_ ^2l3 for the fourth Term of the aggregate Series. Then putting /.to reprefent the fourth Term of p, by multiplication we fliall have

— ^ = o, whence / = -^L to be

'

2048^ 4096*

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- ries 3«/, and the third Term of p*-, in order to find the fifth Term of the aggregate Series. And thus we may eafily and furely carry on the Root to what degree of accuracy we pleafe, without any danger of 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- plemental Equation of the fame Example, ^cfp •+- axp -f- ^ap* -f- p= = — a*x-{-x*, (and all others in imitation of this,) may be

reduced to this form, /\.a*- -+- ax-+- ^a -{- p x/> x/> = — a*x •+• x>, which may be thus refolved.

4#* -f- ax

The Terms being difpofed as in this Paradigm, bring down 4^* for the firft Term of the aggregate Series, as it may ftill be call'd, and fuppofe q to reprefent the firft Term of the Series p. Then will 4^*5' = — a*x, or^= — ~x, which is to be wrote every where for the firft Term of p. Multiply •+- 3*2 by — ±x for the firft Term of 3#-f-/>x/>, with which product — ±ax collect the Term above, or -+- ax ; the Refult ~#x will be the fecond Term of the aggregate Series. Then let r reprefent the fecond Term of />, and we fhal'l

have by Multiplication q.a'-r — -r-s-^x1 = o, or r = ^— , to be wrote every where for the fecond Term of^>. Then as above, by crofs- multiplication we fhall have 3^ x ~~a -I- -rV-v1 = ^V^1 ^or tne third Term of the aggregate Series. Again, fuppofing s to reprefent the third Term of p, we ftiall have by Multiplication, (fee the Theorem,

for that purpofe,) A.a1s + —, — '—, =xi. that is, s=^^ . to

' 256 256 5iz«a '

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 Term -of p to be

And fo on.

Among all this variety of Methods for thefe Extractions, we muft not omit to ftipply the Learner with one more, which is com-

mon

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- ready fhewn how to determine the form and number of fuch Se- ries, in any cafe propofed. This Method confifts in the aflumption of a general Series for the Root, fuch as may conveniently repre- fent it, by the fubftitution of which in the given Equation, the ge- neral Coefficients may be determined. Thus in the prefent Equa- tion y= 4- axy 4- aay — A'5 — 2a 3 = 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

A3 4~ 3 AlijX 4~ ^n.Dlx1 4~ -D'AJ"3 4~ 3 ^"*> *^c* 4^ 3A*C 4- 6ABC4- 36^ } 4- 6ABD

aA.x 4- aBx1- 4- aCx* -f- aDx*, &c.

o.

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- termined to congruous Values ; and by this means the Root^ will be known. Thus, ( i.) A3 4- a1 A — 2a~> = o, which will give A=/r,

as before. (2.) 3AaB -+- aA -+- a*-B = o, or B==

(4.) B3-4-6ABC-j-3A*D4-rfC-H«aD— i = o, orD'=^_> (5.) 3AO + 3B-C + 6ABD 4- 3 A*E 4- aD 4- ^E = o, or £=»

_^2_ . And. fo on, to determine F, G, H, &c. Then by fubfti-

163^4^^

tuting thefe Values of A, B, C, D, &c. in the aflumed Root, we

(hall have the former Series y =a—±x + ^4- '•—+• ;%^> &c. Or laftly, we may conveniently enough refolve this Equation, or any other of the fame kind, by applying it to the general Theorem, raCT. 1 90. for extracting the Roots of any affedted Equations in Num- bers. For this Equation being reduced to this form ;i3 * 4- a1 4-^x

2. x/

2lb The Method of FLUXIONS,

•x« — 2rt3 -+-A:' x.yc = o, we fliall have there #2 = 3. And inftead of the firft, fecond, third, fourth, fifth,- &c. Coefficients of the Powers of y in the Theorem, if we write 1,0, aa -f- ax, — 2#5 — x?, o,

.&c. refpectively ; and if we make the firft Approximation - = - •> or A= a and B = i ; we. fliall have 4" , , A* for a nearer Approxi-

4«a -f- «*

mation to the Root. Again, if we make A= 4^' -f- x*, and B — — 4^ -f- ax, by Subftitution we fliall have the Fraction

.t5 -(- 48a*.v4 -f- i Zfi4 ,* -f- zq«Sjt« .* * +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-

king B = i ,) we fliall have y = -^ ^ ** , &c. = a — ~x, &cc. .And if again we make this the fecond Approximation, or A • — a — t*, (making B = i,) we fliall have y =

4«a — ax ~T -i * 4« 5 1 z«4

if again we make this the third Approximation, or A=:a

_ &c. (making 'B== J,) we fliall have the Value of 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.

To proceed ftill with the fame Equation ; we have found before, pag. 205, that we might likewife have a defcending Series in this form, v = AA'H-B -j-Cx-1, &c. for the Root y, which we fliall extract two or three ways, for the more abundant exemplification of this Doctrine. It has been already found, that A= i, or that x is the firft Approximation to the Root. Make therefore y =. x •and fubftitute this in the given Equation jy3 -f- axy -f- any — x> *a= = o, which will then become ^p -f- axp -f- a?p -\- ^xf -4- ax1 -f- a^x — 2«3 = o. This may be reduced to this form

-^ _t_ ax -+- a* -f- 3.v/> -{-/* x/> = — ax"- — a*x + 2a*, and may be refolved as follows. OAT*

and INFINITE SERIES. 217

3A'1 -f- aX -f- rt*

ax _ a* _+_ IE1 t Sec.

4_ p* . _ + ^ _+_ ±_; s &c>

3-v* * -f- trfl •+- ^ , &c.

/,.. _i_^ — -U_ *""* _i- 64"4 c--

y ' 3 " 3-v ~~ 81^^ ~

The Terms of the aggregate Factor, as alib the known Terms of the Equation, being difpofed as in the Paradigm, bring down ^xl for lire firft Term of the aggregate Series ; and fuppofing q to repre- lent the 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 Term of 3^ to be put down in its place. This will make the fecond Term of the aggregate Series to be nothing ; fo that if ;• re- prefent the fecond Term of p, we fliall have by multiplication 3«vV

= — a1*;, or /• = — "—_ for the fecond Term of p, to be put down

in its place. Then will — a1 be the fecond Term of $xp, as alfo •^d"~ will be the firft Term of/1, to be fet down each in their places. The Refult of this Column will be -^z1, which is to be made the third Term of the aggregate Series. Then putting s for the third Term of/, we mall have by Multiplication ^x^s — -»Vrt3 == 2(l' •> or s= $52- . And thus by the next Operation we fhall have / =

1 J

and fo on.

"Or if we would refolve this reildual Equation by one of the fore- going; Methods, by which the railing of Powers was avoided, and wherein the whole was performed by Multiplication alone ; we may

reduce it to this form, 3*1 •+• fix -f- a1 -f- 3* -j-/» x/> x/ = — ^.v1 — d^X _j_ 2n* , the Refolution of which will be thus :

F f j.v-

2I8 tte Method of FLUXIONS,

3** -f- ax -h a* * --+ 3* — T* ~

a „

fa* + — , &c.

--- •—.»

3*- 8ix* 243*3'

The Terms being difpos'd as in the Example, bring down 3*'* for the firft Term of the aggregate Series, and fuppofing q to reprefent the firft Term of the Series p, it will be yx^q = — ax*, or q = — La. Put down -+- 3* in its proper place, and under it (as alfo after it) put down the firft Term of/, or — La, which being multiply'd, and collected with -j- ax above, will make o for the fecond Term of the aggregate Series. If the fecond Term of p is now reprefented

by r, we fhall have ix^r * = — a'-x, or r = — — , to 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 putting s for the third Term of p, we fhall have by Multiplication

3Ar*j — T'Trf3 =2d3, or j= |^ . And fo on as far as we pleafe.

Laftly, inftead of the Supplemental Equation, we may refolve the given Equation itfelf in the following manner :

*

28*4 — ax1 — %a*x -f- \a* - — , Sec.

• ----- f- ax1 — La*x — La* •+- , &c.

y=x—'a —

243^5

Here becaufe it is y~> =x*, &c. it will be y = x, &c. and therefore _t_ «xy =-f- «A-I, &c. which muft be fet down in its place. Then it muft be wrote again with a contrary fign, that it may be y= == * — rfx*, &c. and therefore (extracting the cube-root,) /= * — ±a &c. Then -+- a*y = 4- a*x, &c. and •+• axy = * — , j-^^, &c.

which

and INFINITE SERIES. 219

which being collected with a contrary fign, will makers = * * — JLa*x, &c. and (by Extraction) y = * * — — , &c. Hence -f- aly

= # — frt3, &c. and -f- ^v>'= * * — ±a*, &c. which being col- lected with a contrary fign, and united with -f- 20 J above, will

make y"' = * * * f^5, &c. whence (by Extraction) y = * * * 4-^ >

&c. Then -+- a 7 = * * — j*, &c. and -f- axy = * * * + ^7' &c. which being collected with a contrary fign, will make y* = * * * * — 177 ' &c' and l^en (by Extraction)^' ==* * * * +

&c. And fo on.

37, 38. I think I need not trouble the Learner, or myfelf, with giving any particular Explication (or Application) of the Author's Rules, for continuing the Quote only to fuch a certain period as {hall be before determined, and for preventing the computation of fuper- fluous Terms ; becaufe mod of the Methods of Analyfis here deli- ver'd require no Rules at all, nor is there the leaft danger of making any unneceflary Computations.

39. When we are to find the Root y of fuch an Equation as

this, y — t)'1 -+- fj3 — tv4 + t>"> &c- = *> tllis is ufually call'd the Reverfion of a Series. For as here the Aggregate z is 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 (lands, fuppofes z (or the Aggregate of the Series) to be unknown, and that we are to approximate to it indefinitely, by means of the known Number y and its Powers. Or otherwife ; the unknown Number z is equivalent to an infinite Series of decreafing Terms, exprefs'd by an Arithmetical Scale, of which the 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 fol- low, that z, alfo will be much lefs than Unity. This is ufually cal- led a Logarithmick Series, becaufe in certain circumftanp.es it ex- preffes the Relation between the 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 be exprefs'd by a Series of Terms compos'd of the known Num- ber z and its Powers. The Author's Method for reverting this Se- ries will be very obvious from the confideration of his Diagram ; and we mall meet with another Method hereafter, in another part of his Works. It will be fuffiqient therefore in this place, to perform it after the manner of fome of the foregoing Extractions.

F f 2 y

Method of FLUXIONS,

y 1 = a + |~a -+- f:i3 + TV-4 4- T5o-3}> &c-

f V- > * h f *5 -4- 'f^4 -H AS',' &c.

fS4 _. 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. and therefore — fj* = — fa1, Sec. which is to be fet down in its place, and alfo with a contrary fign, fo that _}'= * -f- f £% &c- And therefore (fquaring) — f^1 = * — f 2', Sec. and (cubing) -h fy3 =4- fa3, Sec. which Terms collected with a contrary fign, make y= * * -f- -.^s, Sec. And therefore (fquaring) — f_y* = * * — rV24» &c- and (cubing) -4- f_)'3 = * -|- fa4, Sec. and — f/4 =: — f.?.4, Sec. which Terms collected with a contrary fign, make y = ***-{- -j1-?-54' ^c- Therefore — y_ya = * # * — f.sf, &c.

H- f^5 = -{- f^% &c. which Terms collected with a contrary fign, make y= #- ***-{- -4-as, Sec. And fo of the reft.

40. Thus if we were to revert the Series y -f- f/3 -f- ^V>'5 •+• TT-T^ -f- T.fyTy' -h TTTS->''S ^c- = ^, (where the Aggregate of the Se- ries, or the unknown Number a, will reprefent the Arch of a Circle, whole Radius is i, if its right Sine is reprefented by the known Number y,) or if we were to find the value of r, confider'd as un- known, to be exprefsd by the Powers of a,, now confider'd as known ; we may proceed thus :

Lo*3 ,^1^ f ]_ o* 5 ± „ ____'__ *?" 1 -.{ ^9 o^C

+ 3 5 i>>9 ATr*

T"T"3""a" 3 vVv»

Sec. j

The Terms being difpofed as you fee here, we mall have jy==a, Sec. and therefore (cubing) fj5 = fa3, Sec. which makes y = * — fs3, Sec. fo that (cubing) we lhall have + f_>'3 = * — -rV^'j Sec. and alfo -^y1 =-5?5-a!r, Sec. and collecting with a contrary fign,

y

and INFINITE SERIES, 221

r==* * -+-TT.T-'. &c- Hence ±\<-> = * * T'TV«7> &c. and ^y ' — * — TV~7, &c- and TTT.v7 = TTT~7> &<--. and collecting with a contrary tign, v = * * * — WT^"' &c< Anct lo on'

It" we fhould defire to perform this Extraction by another of the foregoing Methods, that is, by fuppoiing; the Equation to be reduced

to this- form i -+- j-_v* 4- -rV4 •+- TTT.'6 + TTTT^'^ &<•'• x ;•==£;, it may be fufHcient to let down the Praxis, as here follows.

I	* * * *
-f- f y*	h i"5-'1 — —V"-'4 4~ TTT^ "~~ TT'po-'2 > ^c-
1 3 V4	1 I A i t ft &
	4°^ _j_ s°^« "s0-8' &c
FT^s	'11" 3 5*. ,9* ^
1 ' . T * .*	1 . i 5 l >

41. The afFedted Cubick Equation, which the Author here affumes to be folved, has infinite Series for the Coefficients of the Powers of y ; and therefore its Terms being difpoled (as is taught before) accor- ding to a double Arithmetical Scale, the Roots of each of which are V and z,, it will ftand as is reprefented here below. Or taking As" for the fir ft Approximation to the Root y, and lubftituting it in the firft Table, it will appear as is here let down in the fecond Table.

« . * — S 1 . * * —

A-.;,-'"-H- 4. Aim+z+

J2-V. C-V. 45V. &f. J

Now the only cafe of external Terms, to be difcover'd by apply- ing the Ruler, will give the Equation A3£*m+» — 8 = o, whence j;«-|-2=o, or w= — .1, and the Coefficient A = 2. The next Number or Index, to which the Ruler in its parallel motion will apply itfelf, will be 2m H- 2, or .1 ; the next will be m -f- 2., or ± ; and fo on. Which afcending Arithmetical Progreflion o, |, i, 6cc. will have ~ for its common difference. Therefore y—_ A.g~ -f- B +Cs^+ D^J -{- E^, &c. will be the form of the Root in this Equation. It may be refolved by any of the foregoing Methods,

bat

The Method of FLUXIONS.

222

but perhaps moft readily by fubftituting the Value of y now found in the given Equation, and thence determining the general Coeffi- cients as before. By which the Root will be found to be _)' =

or J - I- — — —Z^ I 3 '_ *»;7 i * *_2-» I » 9 9 fy7 I 6 i i i ~3 f, _ Z.O 3 gt** TT» T i ~ TTTT'6 T^ T JTT^rT > *•*-*-.

42. To refolve this affected Quadratick Equation, in which one of the Coefficients is an infinite Series ; if we fuppofe y •=. Axm, &c. we (hall have (by Subftitution) the Equation as it ftands here below.

Then by applying the Ruler, we {hall have — aAxm -+- — 4 =o,

whence m = 4, and A = ~ . The next Index, that the Ruler

in its parallel motion will arrive at, is m -+- I, or 5; the next is m-\-2, or 6; &c. fo that the common difference of the Progrel- fion is i, and the Root may be reprefented by y = Ax* -{-Ex* -f- Cx6, &c. which may be extracted as here follows.

x"	« *• — aAx	&e. "
*	. m-l-i — A*	*
	A M-J-S
#	— ~ X a	*
	A_ «+?	*
*	^^~ j_X
	_ A VH	AT4
*		«•?

-ay

— xy x*

X4

-7^

6T*.

-fy

4-2

A

,777 . &c-

Here becaufe it is — ay = — —. . &c. it will be y = fl

4« J 4^5 >

Therefore — xy = — ^ , &c, which wrote with a contrary Sign will make — ay = *-+-—,, and therefore y = * ^ &c

4" ' 4^.4 '

Then — xy = * -4- £-4 , &c. and — 77 = — — 4 , &c. which

t" 4«~

collected will deftroy each other, and therefore — ay = * * -f- o, &c. and confequently y = * * -j- o, &c. &c.

But there is another cafe of external Terms, which will be dif- cover'd by the Ruler, and which will give A*x"-m — a Axm - — o, whence m = o, and A =a. Here the Progrefiion of the Indices will be o, i, 2, &c. fo that ;> == A -j- B* -h Cx1, &c. will be the form of the Series. And if this Root be profecuted by any of the

Methods

and INFINITE SERIES.

223

Methods taught before, it will be found y = a •+• x -f-— _,_

g, &c.

Now in the given Equation, becaufe the infinite Series a -\- x -+-

1 * vj v4

— -4- ji 4- ^r y &c. is a Geometrical ProgrcfTion, and therefore is

a

equal to — - , as may be proved by Divifion ; if we fubftitute this,

the Equation will become _>•* — -£— y -f- ^ — - o> And if we ex- tract 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 ufe- ful and elegant. This may commonly be difcover'd in the Compu- tation, by attending to the 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 Series, for instance, z=y — {y1 -f- .lys — ±y4, &c. the Law of Confecution is very obvious, fo that any Term, tho' ever fo remote, may eafily be aflign'd at pleafure. For if we put T to reprefent any Term indefinitely, whcfe order in the Series is exprefs'd by the na- tural Number «, then will T = + -j", where the Sign muft be

•4- or — according as m is an odd or an even Number. So that the hundredth Term is — —L-yl°°, the next is -j-_J_^101, &c. In the Reverie of this Series, or y = z,~\- f-s* •+- fa;3 -+- -y^a4 -4- TTo-^S &c. the Law of Continuation is thus. Let T reprefent any Term indefinitely, whofe order in the Series is exprefs'd by m ; then is

m

T= — — — ^p- , which Series in the Denominator mud: be con- 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=

_ <ym

m

Again, in the Series y = z — fz"> -+- TTO-S' — ToV^27 + _rTITT_r'> &c. (by which the Relation between the Circular Arch a*nd its light Sine is exprefs'd,) the Law of Continuation will be thus.

224 *?%* Method of FLUXIONS,

If T be any Term of the Series, whofe order is exprefs'd by w, and

_ _im— I

if c be the Coefficient immediately before ; then T = f ". — ,

zm — I x zm — 2

And in the Reverie of this Series, or z. = y -f- ^v3 -f- -£?)•' -f- TAT y~ _f-_|^-Tr9, £cc. the Law of Confecution will be thus. If T repre- fents any Term, the Index of whole place in the Series is ;//, and if

c be the preceding Coefficient j then T = "" . . — • — ,,i»-i_ 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 Equation be v*_ -}- a\v -\- rf.vv — 2^—x-=o, _z<jj + fl»r , + j5 the Terms of which Ihould be difpofed as » + «*,•„ * Q in the Margin. Suppofe y = l> -}- p, where _ *; * * » y /; is to be conceived as a near Approximation

to the Root y, and p as its fmall Supplement. When this is fubfti- tuted, the Equation will ftand as it

does here. Now becaufe .v and f> — -^ ? + a~P I + ^'f" + f=~\ are both fmall quantities, the moil _i_ '/3 ^ " f* confiderable quantities are at the be- + «/* + «*/. a s ^ =

ginning of the Equation, from _ x* '

whence they proceed gradually di-

minifliing, both downwards and towards the right-hand ; as oug;ht always to be fuppos'd, when the Terms of an Equation are dilpos'd according to a double Arithmetical Scale. And becaufe inftead of one unknown quantity _v, we have here introduced two, If and />, we may determine one of them b, as the neceffity of the Relblution iliall require. To remove therefore the moft confiderable Quantities out of the Equation, and to leave only a Supplemental Equation, whofe Root is/>; we may put 6* -+- a*b — 2^ = o, which Equa- tion will determine b, and which therefore henceforward we are to look upon as known. And for brevity fake, if we put a1 -+- 3^* • — c, we mail have the Equation in the Margin.

Now here, becaufe the two initialTerms

-f- cp -+- abx are the moft con fiderable of \±. 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- prefs'd lower ; therefore make p = — — _f- q, and by fubftitution we

-flwll have this Equation following.

Or

and INFINITE SERIES. 225

Or in this Equation, if • + J7 . ^+ 3*?*

e make ^ — = ^,

3

*

.e.

-+- i =/; itwillaffume a^ > + — **? * »

' 1 V

this form.

<3

— x*

Here becaufe the Terms to be next removed are-f- cq -j-^-x^we may put y = — -xl •+- r, and by Sub- * +<•? + 3V -f?3l ftitution we fhall have another • +«•? _ 1^-^* » | Supplemental Equation, which ?«^»

will be farther deprefs'd, and fo on as far as we pleafe. Therefore *

we mall have the Root y = <£ — a-x -- x* Sec. where b will be

c c

the Root of this Equation b* -+. a*b — za* = o, c = a1 -f- 3^*

Or by another Method of Solution, if in this Equation we affumc (as before) y = A -\-Bx -}- Cx* •+- Dx3, &c. and fubflitute this in the Equation, to determine the general Coefficients, we fhall have

. a\ e c -ja' — a c , • . • ,

y = A -- -x -t- —x*-\ -- — - — ,.- - x"'t &c. wherein A is the

Root of the Equation AS -f. a* A — art5 = o, and c1 = 3 A* + a*.

47. All Equations cannot be thus immediately refolved, or their Roots cannot always be exhibited by an Arithmetical Scale, whofe Root is one of the Quantities in the given Equation. But to per- form 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 converg- ing 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, x = fi-\-y — 4/1 -Hy}'3 — ^_}'4, &c. it would be in vain to 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 fome fmall Number lefs than Unity ; but x and a are 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-

226 Ibe Method of FLUXIONS,

form'd of its Powers may converge to y. Now it is plain, that x ami rf, tho' ever fo great, muft always be near each other, becaufe their difference y — ±y*} &.c. is a (mall quantity. Aflame therefore 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- ^y* — -ly4, &c. whofe Root being extracted will be y = z->t-^^ -j-y.23 4-TVs4> ^cc> as before.

48. Thus if we had the Equation _>i3 -f-j* -h_y — x3 = o, to find the Root y ; we might have a Series for y compofed of the afcending Powers of x, which would converge if x were a fmall quantity, lels than Unity, but would diverge in contrary Circumftances. Suppo- fing then that x was known to be a large Quantity ; in this cafe the Author's Expedient is this. Making & the Reciprocal of x, or fup-

pofing the Equation x= l- , inftead of x he introduces z into the

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, to keep within the Cafe he propofed to himfelf ; but in the Method here purfued, there is no occafion to have recourfe to this Expedient, it being an indifferent matter, whether the Powers of the converg- ing quantity afcend in the Numerators or the Denominators. Thus in the given Equation y> 4-j5- 4- jy * ?

° '

king y = Axm , 6cc.) A**'" 4- A1*"" 4- Axm * , &c.?

j v* .3

by applying the Ruler we mall have the exterior Terms A3 A*™ — A'5 = o, or m=. i, and A = i. Alfo the refulting Number or Index is 3. The next Term to which the Ruler approaches will 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 y = Ax 4- B 4- Cx"1 4- DAT"* , &c. which we may thus extract.

+y '

Becaufe >)'3=Al},&c.it will be _y=x,&c.and therefore _y1=A-1, &c. which will make y* = * — #*, &c. and (by Extraction)/ = * — -^ &c. Then (by fquaring)^* = * — ~x} &c. which with A* below, and changing the Sign, makes j3 = * * — ~x} &c. and therefore

y

and INFINITE SERIES. 227

v = * * — }*~", &c. Then ;•* = * * — .1, &c. and y = * — ±, &c. which together, changing the Sign, make y> = * * * -4- ±, &c. and ;'=*** + TV*~S &c. Then y- — * * » -f- 44*-', Sec. and _>' = * * — £*""'> &c. and therefore 75 — _ ^ # ,. ^ _^_ ^.^-^ &c. and _v = * * * * -f- -pTx~3 > &c-

Now as this Series is accommodated to the cafe of convergency when x is a large Quantity, fo we may derive another Series from hence, which will be accommodated to the cafe when .v is a fmall quantity. For the Ruler will direct us to the external Terms Ax* — x5 = o, whence m= 3, and A= i ; and the refulting Num- ber is 3. The next Term will give 2m, or 6 ; and the lair, is 3*77, or 9. But 3, 6, 9 will form an afcending Progremon, of which the common difference is 3. Therefore v =Ax'' -+- Ex6 -t-Cy9, &c. will be the form of the Series in this cafe, which may be thus derived.

y -} = x> — x6 -+- #» * — 4*" -f- 14*' 8, &c. h xs — 2X> •+- 3AT11 — 2x" — 7A-8, &c.

Here becaufe_)' = Ar3, &c. it will bej>*=x6, &c. and therefore v = * — x6, &c. Then y- = * — 2x», &c. and ^5=^9, &c. and therefore y = * * H- .V, &c. Then y- •= * * -j- 3^'*, 6cc. and y3 = * — S-^11) &c. and therefore7= * * * -f. o, &c.

The Expedient of the Ruler will indicate a third cafe of external Terms, which may be try'd alfo. For we may put A=x=m -{- A*x*'K -f- Axm = o, whence m = o, and the Number refulting from the other Term is 3. Therefore 3 will be the common difference of the Progrelfion, and the form of the Root will be _y= A -{- Bx' -{- Cx6, &c. But the Equation A5 -f- Aa + A = o, will give A = o, which will reduce this to the former Series. And the other two Roots of the Equation will be impofftble.

If the Equation of this Example jy3 -f- y* -{- r — x"' — • o be multiply'd by the factor y — i, we mall have the Equation y* — y

— X~'y -f- x' = o, or r+ * # — y * •) ...

- t C=o, which when re-

- A'_) -- AT J

'•folved, will only afford the fame Series for the Root y as before.

49. This Equation \* — x\yl -h xy1 + 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,

Now the finl Cafe of external _>•» * +2."- — 2>-f i Terms, fhewn by the Ruler, in _ *- L » *

order for an afcending Series, will Or making y _ Axm> fc

make A'.**"1 _|- 2A^i" — 2 A*" M^m # zAljc« -+- i = o, or ;;/ = o ; where the tmjri

refulting; Number is alfo o. The \«4-s

i • 11-1 — A1* ^

fecond is zm -h i, or i ; the third zm-}- 2, or 2. Therefore the Arithmetical Progreffion will be o, i, 2, whofe common difference is i ; and confequently it will be v == A -f- Bx -+• Cx1 -+- Dx*, &c. But the Equation A« •+- zA* "_ zA -H i — o, which mould give the Value of the firft Coeffi- cient, will fupply us with none but impoffible Roots ; fo that y, the Root of this Equation, cannot be exprefs'd by an Arithmetical Scale whofe Root is x, or by an afcending Series that converges by the Powers of x, when x is a fmall quantity.

As for defcending Series, there are two cafes to be try'd ; firft the Ruler will give us A**** — A1ATim+l = o, whence ^m = zm -f- 2, or fff — i} and A=+ i. The Number arifing is 4; the next will be zm -f- i, or 3 ; the next 2w, or 2 ; the next m, or i ; the laft o. But the Arithmetical Progreffion 4, 3, 2, i, o, has^ i for its common difference, and therefore the form of the Series will be y = Ax -+• B 4- CAT"', 8cc. But to extract this Series by our ufual Method, it will be beft to reduce the Equation to this form, / — x* 4- x -+- z _ _ 2 y~l 4- y~* = o, and then to proceed thus :

-_ xi x — 2 -f- ZX~* |A:~1> &c-

h A;~I, &C.

97' 77 c

Becaufe jy» = x* — x — 2, &c. 'tis therefore (by Extradlion) y _— x — JL — %x~l , &c. Then (by Divifion) — zy~* = — zx-*, &c. fo that y = * * * -f- 2*-1, &c. and (by Extradion) y = * * * _j_ -VAT-% &c. Then — zy~l = * -f- -i-^"1? &c- and y~* == *"*, &c.' which being united with a contrary fign, make^1=* * * * — TA'~I> &c> ant^ therefore by Extraction y = ****—-• 4-i-s-v~ 3»

.

In the other cafe of a defcending Series we mall have the Equation

, A1*""^ -f- i =o, whence zm •+- z = o, or m = — i, and

A — ± i . The Number hence arifing is o ; the next will be zm + r,

Cjf

and INFINITE SERIES. 229

or — i ; the next 2//v, or — 2 -, and the laft 4w, or — 4. But 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

Here becaufe it is y~- = x1, &c. it will be by Extraction of the Square-root y~l =x, &c. and by finding the Reciprocal, y = x~', &c. Then becaufe — zy~l = — 2X, &c. this with a contrary Sign, and collected with — x above, will make_y—1 = * -{- x, &c. which (by Extraction) makes y~ I = * -+• i, &c. and by taking the Reci- procal, /== * — ^^~i, 6cc. Then becaufe — zy~* = * — i, &c. this with a contrary fign, and collected with — 2 above, will make y~* = * * — i, 8ec. and therefore (by Extraction) y~l = * * — 4*"" , &c. and (by Divifion) jy = * * -f- ^x~3y &cc. Then becaufe — 2y~' = * * -}- -^"S ^ wiH be y— l = # * * — %x~*y &c. and j-1 = * * * — 4*1"1, &c- and >'=*** — -V*~J> &c- Then becaufe — 2y~' = * * * -f- -fA;-1, &c. and /* = x~~-, &c. thefe

collected with a contrary fign will make y~z = * * * * — V-v~% &cc. and y~' = * * * * — -V*~S &c- an^ 7 = ** ** -f- rlT*"4* &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 we rmy aflame the Equation .v — 2 = &. 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- '

ries, we muft put A+z*? — zAz'" ~ ?*' > = l>

+ 1=0, whence m = o, and Or k.

* _ T-l__ -KT 1 1

A4.,4

A = i. The Number hence A 4,»

arifing is o ; the next is 2/«H-i, or i ; and the laft 2m -f- 2, or 2. But o, 1,2, are in an afcending Progreffion, whole common difference is i. Therefore the form of the Series is y = A -f- B;s -f- Csa -+- D;s3, 6cc. And if the Root y be extracted by any of the foregoing Methods, it will be found y =. i -+--iz — -^s1, 6cc. Alfo we may hence find two defcending Se- ries, which would converge by the Root of the Scale z, if it were a large quantity.

50, 51. Our Author has here opened a large field for the Solution of thefe Equations, by Shewing, that the indeterminate quantity, or what we call the Root of the Scale, or the converging quantity, may be changed a great variety of ways, and thence new Series 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 an- other that (hall converge falter. But that we may not be left to uncertain interpretations of the indeterminate quantity, or be obliged to make Suppositions at random j he gives us this Rule for 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 fub- ftituted 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 diffe- rence 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 _y3 -f- 2aiy — 3^*, which are divisible by y — a, the Quotient be- ing y* -h ay -f- 3*2*. Therefore we may fuppofe, by the 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 become y* -f- 2a*y — azy -\- y-z — 3"£* -t- z= — 2#5 = o. A

Series

and INF j NITE SERIES, 231

Series derived from hence, compofcd of the afcending Powers of z9 mull converge faft, crtfcris parifats, becaule the Root of the Scale z is a (mail quantity.

Or in the fame Equation, if for x we fubftitute — a, we fliall have the Terms \* — a3, which are divifible by y — a, the Quo- tient being y* -4- ay -f- a*. Therefore we may fuppofe the diffe- rence between — a and .v to be but little, or that — -a — x = z is a fmall quantity, and therefore in (lead of .v we may fubftitute its- equal — a — z in the given Equation. This will then become r3 — azy -f- T,alz -f- 303* — a* = o, where the Root y will con- verge by the Powers of the fmall quantity z.

Or if for x we fubftitute — za, we lhall have the Terms _>'3 — a*? -4- 6^3, which are divilible by y-\- za, the Quotient being _)•*

— zay-i- 3rtx. Wherefore we may fuppofe there is but a fmall dif- ference between — za and x, or that — za — x =z is a fmall quantity ; and therefore infread of x we may introduce its equal

— za — z into the Equation, which will then become jv* — a'-y — azy -4- 6a> -f- iza*z -f- 6az* -f-s} = o.

Laftly, if for x we fubftitute — z~*a, we fliall have the Terms jy3 — z^a'-y -4-tf*y, which are divifible by y, the Radical Species alone. Wherefore we may fuppofe there is but a fmall difference between

— z^'a and x, or that — 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

-\- 3^2 x az1 -f- Z' = o, wherein the Series for 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 the Equation y~> -\- axy -f- a*y — x3 — za* =o, the indeterminate quantity x, of its own nature, muft be fufceptible of all poffible Values ; at leaft, if it had any limitations, they would be fhew'd 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* •+- za*y — 30* = o, 7; — a1 = o, y* — a1y -4- 6a* = o, _y3 — 2^a*y -f- a'-y •=. o, &cc. refpedtively. Now as thefe Equations admit of jull Roots, as appears by their being divifible by y -f- or — an- other 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 pne of thofe Values, muft deprefs the Equation ; and z itfelf mull 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- ples, which mall be this, _y3 — azy -f- y-z -4- ^az* • — a* = o, or

Here becaufe _>'5 = #;, &c. it will be y = a, &c. Then — azy — — — fl»2r, 6cc. which muft be wrote again with a contrary fign, and united with — 3^*2 above, to make y* = * — 2a*z, &c. and therefore y •==. * — -f-*' &c- Then — «s>' = * -f- ^az1, 6cc. and

V = * * — — , &c. Then —

•* •" 3a

ssz # * -f- .Is3, &c. and _)'* = *** — J-.ss, &c. and y = * * * —

217Z5 <>

, OCC.

The Author hints at many other ways of deriving a variety of Series from the fame Equation ; as when we fuppofe the afore-men- tion'd difference z to be indefinitely great, and from that Suppofition we find Series, in which the Powers of z (hall afcend in the Deno- minators. This Cafe we have all along purfued indifcriminately with the other Cafe, in which the Powers of the converging quantity afcend in the Numerators, and therefore we need add nothing here about it. Another Expedient is, to affume for the converging quantity fome other quantity of the Equation, which then may be confider'd as indeterminate. So here, for inftance, we may change a into x, and x into a. Or laftly, to affume any Relation at pleafure, (fup- pofe x = az -f- bz\ x = ~- , x —3 J±^ 5 &c.) between the in- determinate quantity of the Equation x, and the quantity z we would introduce into its room, by which new equivalent Equations may be form'd, and then their Roots may be extracted. And after- wards the value of z may be exprefs'd by x} by means of the af- fumed Equation.

52. The

and INFINITE SERIES. 233

52, The Author here, in a fummary way, gives us a Rationale of his whole Method of Extractions, proving a priori, that the Series thus form'd, and continued in infinitum, will then be the juft Roots of the propofed Equation. And if they are only continued to a competent number of 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- proach is made to the Root, iuppofe y, as can be had in a lingle Term, compofed of the quantities given by the Equation ; and be*. 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 is made to /», as can be done by a lingle Term, and a 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 will permit, fo as to leave the leaft 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 figures. And the fame way of Argumentation, that proves our Au- thor's Method of Extraction, may ealily be apply'd to the other ways of Analylis that are here found.

53, 54. Here it is feafonably obferved, that tho' the indefinite Quantity fhould not be taken fo fmall, as to make the Series con- verge very faft, yet it would however converge to the true Root, tho' by more fteps and flower degrees. And this would obtain in proportion, even if it were taken never fo large, provided we do not exceed the due Limits of 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 ob- ferve, 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 would vaniih in refpedt of — xx, which would give the Value ofjyy impoffible on this Supposition. Nor can x be negative; for then the Value of yy would be negative, and therefore the Value of_y would again become impoffible. If x = o, then is^ = o allb ; which is one Limitation of both quantities. As yy is the difference between ax and xx, when that difference is greateft, then will yy, and con- fequently^, be greateft alfo. But this happens when x = ±a, as alfo y = ftf, as may appear from the following Prob. 3. And in general, when y is exprefs'd by any number of Terms, whether finite or infinite, it will then come to its Limit when the difference is greateft between the affirmative and negative Terms j as may ap- pear from the fame Problem. This laft will be a Limitation for yt but not for x. Laftly, when x = a, then_y = o; which will limit both x and y. For if we fuppofe x to be greater than a, the ne- gative Term will prevail over the affirmative, and give the Value of yy negative, which will make the Value of y impoffible. So that 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 infinite Series, we may ftill difcover the fame Limitations from thence. For from the Equation yy = ax — xx, by extracting the

3. _5.

fquare-root, as before, we fhall have y =. a^ — —L — ~ —

za* Sa1

X1 ' i • X ** *3 O TT

— - , c. that is, y == d*x* into i — - — — — — , &c. Here

i6az

x cannot be negative ; for then x? would be an impoffible quantity. Nor can x be greater than a ; for then the converging quantity ~ »

or the Root of the Scale by which the Series is exprefs'd, would be greater than Unity, and confequently the Series would diverge,, and not converge as it ought to do. The Limit between converging and diverging will be found, by putting x=a, and therefore y = o ; in which cafe we fhall have the identical Numeral Series i = i _l_ ^ -if. _'r, &c. of the fame nature with fome of thofe, which we 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 Series. But the nearer it is taken to the Limit o, fo much fafter the Series will converge to the true Root ; and the nearer it is taken to the Limit a, it will converge fo much the flower. But it will

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- tity, or but juft beginning to be pofitive, all the Terms but the rirft may be negledted, and y will be a mean proportional between a and x. Alfo y = o, when the affirmative Term is equal to all the negative

Terms.or when i= - -f- - — u- -?— , &c. that is, when x = a.

z« 8a* ib«3 '

For then i = ± -f. 4. _f_ _frj &c. as above. Laftly, y will be a Maximum when the difference between the affirmative Term and all the negative Terms is greateft, which by Prob. 3. will be found 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 = _y will be exprefs'd by the fore- going Series. But it is plain from the na- ture of the Circle, that the Abfcifs AB cannot be extended back- wards, fo as to become negative ; neither can it be continued for- 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. 'Trqnfitton fo the Method of Fluxions.

55. ' | "^HE learned and fagacious Author having thus accom- 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 Method of Fluxions, which is 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 con- ceived 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. A right line, or a curve line, is defcribed by the motion of a point, a fmface by the motion of a line, a folid by the motion of a fur- face, an angle by the rotation of a radius ; all which motions we may conceive to be performed according to any ftated law, as occa- fion (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- ducing their properties from fuch adhial defcriptions. And by ana- logy, all other quantities, as well as thefe continued geometrical quantities, may be conceived as generated by a 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 gene- rations 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 concep- tion, of their being now generated by local Motion, is a very fertile notion, and an exceeding ufeful artifice for discovering their pro- perties, and a great help to the Mind for a clear, diftincl:, and me- thodical perception of them. For local Motion fuppofes a notion of time, and time implies a fucceffion of Ideas. We eafily diflin- guifh it into what was, what is, and what will be, in thefe ge- nerations of quantities ; and fo we commodioufly confider thofe things by parts, which would be too much for our faculties, and ex- tream difficult for the Mind to take in the whole together, without fuch artificial partitions and distributions.

Our Author therefore makes this eafy fuppofition, that a Line may be conceived as now defcribing by a Point, which moves either equably or inequably, either with an uniform motion, or elfe accor- ding to any rate of continual Acceleration or Retardation. Velocity is a Mathematical Quantity, and like all fuch, it is fufceptible of infinite gradations, may be intended or remitted, may be 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 to each other, and muft mutually determine each other ; fo that one of them being affign'd, the other by neceflary inference may be derived from it. And therefore this is ftrictly a Geometrical Pro- blem, and capable of a full Determination. And all Geometrical Propoluions once demonftrated,1 or 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 Space or Velocity is affign'd, at any given time, in order to find the other. Arid this has given occasion to that diftin<5lion which has lince obtain'd, of the dirctt and irrcerje Method of Fluxions, each of which we fhall now confider apart.

56. In the direct Method the Problem is thus abftractedly pro- pofed. From the Space defer i bed, being continually given, or affumed, or being known at any point of Time ajjigrid ; to find the Velocity of the 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 defcription conjunclly ; or the Velocity is directly as the Spice de- 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- tions ; and flill the nearer, the fmaller thole are taken. But if we may fuppofe the Times and Spaces to be indefinitely fmall, or if they are nafcent or evanefcent quantities, then we fhall have the Ve- locity in any infinitely little Space, as that Space directly, and as the tempufculum inverlely. This property therefore of all inequable Mo- tions being thus deduced, will afford us a medium for folving the prefent Problem, as will be fhewn afterwards. So that the Space defcribed being thus continually given, and the whole time of its defcription, the Velocity at the end of that time will be thence de- terminable.

57. The general abflract Mechanical Problem, which 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- termine the Space defcribed, at any point of Time 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- nitely little Spare defcribed in any indefinitely little Time, will be in a compound ratio of the Time and the Velocity ; or thejpafiolum will 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 may be known alfo ; as will more fully appear from what follows. ThisTroblem, in all its cafes, will be capable of a juft determina- tion ; tho' taking it in its full extent, we mult acknowledge it to be a very difficult and operofe Problem. So that our Author had good reafon for calling it moleftijfimum & omnium difficilltmum pro- blema.

58. To fix the Ideas of his Reader, our Author illuftrates his general Problems by a particular Example. If two Spaces x 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 being reprefented by the Symbol x ; and if the Space y increafes in- equably, but after fuch a rate, as that the Equation y •=. xx ihall always determine the relation between thofe Spaces j (or x being continually given, y will be thence known ;) then the velocity of the increafe of y fhall always be reprefented by 2xx. That is, if the fymbol y be put to reprefent the velocity of the increafe of 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 Spaces y and x, (that is, the Space and Time, or its 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- locity y, by which the Space y is defcribed, were continually known from the Time x being given, and its Velocity x •, and if 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 .general Problem, or of a particular Queftion of the inverfe Method of Fluxions. And in analogy to this defcription of Spaces by mov- ing points, our Author confiders all other quantities whatever as ge- nerated

and INFINITE SERIES. 239

nerated and produced by continual augmentation, or by the perpe- tual acceffion and accretion of new particles of the fame kind.

59. In fettling the Laws of his Calculus of Fluxions, our Author very fkilfully and judicioufly difengages himfelf from all confidera- tion 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, when they come to be compared or meafured, may feem neceflarily to include a notion of Time; yet Time, like all other quantities, may be reprefented by Lines and Symbols, as in the foregoing ex- ample, 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 pur- pofe 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 conceive to be generated by the 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 any Problem or Equation are fuppos'd to be fufceptible of continual increafe or decreafe, he calls Fluents, or flowing Quantities ; which are fometimes call'd variable or 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- creafe of fuch quantities are call'd 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 quantities. This diftindlion of quantities, when once made, is care- fully obferved through the whole Problem, and infinuated by proper Symbols. For the firft Letters of the Alphabet are generally appro- priated for denoting conftant quantities, and the laffc Letters com- monly lignify variable quantities, and the fame Letters, being pointed, repreient the Fluxions of thofe variable quantities or Fluents refpec- tivcly. This diftinction between thefe quantities is 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

24-O 7#* Method of FLUXIONS.

or variable quantities may be conceived as now generating by Motion, and the conftant or invariable quantities as fome how o other al- .ready generated. Thus in any given Circle or Parabola, the Diame- ter or Parameter are conftant lines, or already generated ; but the Abfcifs, Ordinate, Area, Curve-line, &c. are flowing and variable quantities, becaufe they are to be underftood as now defcribing by local Motion, while their properties are derived. Another diftinc- tion of thefe quantities may be this. A conftant or given Irne in any Problem is tinea qtitzdam^ but an indeterminate line is line a 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 Unit, of their own kind, is fuppos'd to be known ; but in variable quantities that ratio cannot be known, becaufe it is varying perpe- tually. 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 conveniency, it has a power of abolifhing it, and of converting de- terminate quantities into indeterminate, and vice versa, as occaiion may require ; of which we fhall fee Inftances in what follows. In a Problem, or Equation, theie may be any number of conftant quan- tities, but there muft be at leaft two that are flowing and indeter- minate ; for one cannot increafe or diminifh, while all the reft con- tinue 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 of the flowing Quantities being given, to determine the relation of their Fluxions.

SECT. I. Concerning Fluxions of the firft orcler^ and t(f Jlnd their Equations.

HE Author having thus propofed his fundamental Pro-' blemss in an abftra<ft and general manner, and gradually brought them down to the form mod convenient for* his Method ; he now proceeds to deliver the Precepts of Solution, which he illuftrates by a fufficient variety of Examples,! referving the Demonftration to be given afterwards, when his Rea- ders will be better prepared to apprehend the force of it, and when their notions will be better fettled and confirm'd. Theie Precepts of Solution, or the Rules for finding the Fluxions of any given' Equation, are very fliort, elegant, and compreheniive ; and appeal- to have but little affinity with the Rules ufually given for this pur- pofe : But that is owing to their great degree of univerfality. We are to form, as it were, fo many different Tables for the Equation, as there are flowing quantities in it, by difpofing the Terms accor- ding to the Powers of each quantity, fo as that their Indices may' form an Arithmetical Progreflion. Then the Terms are to be mul- tiply'd in each cafe, either by the Progreflion of the Indices, or by ' the Terms of any other Arithmetical Progreflion, (which yet mould .have the fame common difference with the Progreffion of the Indices ;) '

I i as

242 Tfo Method of FLUXIONS.

as alfo by the Fluxion of that Fluent, and then to be divided by the Fluent itfelf. La ft of all, thefe Terms are to be collected, accor- ding to their proper Signs, and to be made equal to nothing; which will be a new Equation, exhibiting the relation of the Fluxions. This procefs indeed is not fo fhort as the Method for taking Fluxions, (to be given p relent lyv) which he el fe where delivers, and which is commonly follow' d ; but it makes fufficient amends by the univer- lality of 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> — ax1 -{- axy — y"> = o, where the Terms are always brought over to one fide. Thefe Terms being difpofed according to the powers of the Fluent x, or being conlider'd as a Number ex- prefs'd by the Scale whofe Root is x, will iland thus x> - — ax1 -f- ayx* — y>x° = o; and affuming the Arithmetical Progrefiion 3, 2, ], o, which is here that of the Indices of x, and multiplying each Term by each refpedlively, we fhall have the Terms jx3 — zax- H- ayx * j which again multiply'd by i , or xx~l, according to the Rule, will make ^xx1 — 2axx -f- ayx. Then in the fame Equa- tion making the other Fluent/ the Root of the Scale, it will ftand thus, — _y5 -f- oy*-i- axy1 — ax*y° = o ; and 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

— 3_)'; * -+- axy *, which multiply'd by -- , or yy~*, will make

— 3i>'a -+• axJ- Tlien colle(^ing the Terms, the Equation yxx1 — zaxx + ayx — tyy* -f- axj = o will give the required relation of the Fluxions. For if we refolve this Equation into an Analogy, we fhall have x : y : : 3>'2 — ax i^x1— zax -h ay -, which, in all the values that x and y can affume, will give the ratio of their Fluxions, or the comparative velocity of their increafe or 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' by fewer 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- tion, difpos'd according to x, with their refpective figns ; each be- ing multiply'd by the Index of x in that Term, (increafed or di- minifh'd, if you pleafe, by any common Number,) as alib divided by .v. In the Denominator do the fame by the Terms, when dii- pofed according to y, only changing the figns. Thus in the pre- fent Equation x"' — ax1 -f- axy — ;'3 = o, we (lull 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 x1 — ax* -(- ayx1 — y*x° •=. o ; and fuppofing the Indices of x to be increas'd by an unit, or aifuming the Arithmetical Pro- greffion -j- , -~~t ^ , ~ , and multiplying the Terms refpectively,

we fhall have thefe Terms ^.xx* — ^axx -}- zayx — y-xx-1. Then ordering the Terms according to /, they will become — _)'3-f- oy1 -\-axyf-i- x*y° =.0; and fuppofing the Indices ofy to be diminifli'd — ax*

by an unit, or afluming the Arithmetical Progreffion ^ , L Si iJ,

.> y ' y ' y and multiplying the Terms refpecYively, we mall have thefe Terms

— 2yy* * * — x*yy-1 + ax*yy~*. So that collecting the Terms, we lhall have 4.v.v* — -^axx •+- 2 ayx — y>xx~l — zyy* — x'>yy-' -+- ax*yy~* = o, for the Fluxional Equation required. Or the ratio

c ^1 T>1 • -11 i y 4X* — Ta-f-f- 2ay — v'v * . . ,

of the Fluxions will be - = —, -. — 3_J — : _ . which ratio

x Z)2 * * -f-As, J — axl\< l

may be found immediately by applying the foregoing Rule.

Or contrary-wife, if we multiply the Equation in the fir ft form by the Progreffion ~ , ? } ~ , ^v , we flinll have the Terms zxx1

— axx * -\-ytx\-1. And if we multiply the Equation in the fc- cond form by -• , ^ , ly 5 y- , we fiiall have the Terms — 4^* *,

H- zcixy -+- x=j}~! — cx-yy~'. Therefore collecting 'tis a.v.v1 — ^v.v + rxx~> — ^v}*+ 2axy-i-x>j}->~fix1y}-'~o. Or the ratio of the Fluxions will be | = ^ ^ ~^:^^.,-r , which might l.avc been found at once by the foregoing Rule.

And in general, if the Equation x"> - -ax% -±- axy — y* — o, in the form x- — a\- -f- <?.yv -->;,v° = o, be multiply'd by the Terms of this Arithmetical Progreffion ";+ 3v "L+J. ;r w;n

O ) -v, .v „ JL \> 11

produce the Terms m -\-y.\-~ — m-+-2n>:x-{- m -\- icxt — mj'xx-'-,

I i 2 • and

244 e^)e Method of FLUXIONS,

and if the fame Equation, reduced to the form — y*-\-

_f- K\y°= o, b; multiply 'd by the Terms of this Arithmetical Pro-

— ax1

grerTion — Mjs " — 7, ~7~7' "^' ^ w*^ Pro(^uce t^ie Terms —

* H- » H- iaxy-\-nx~>yy-* — naxlyy~l. Then collecting the Terms, we ilia 11 have m -\- 3. vx1 — ;« -i- arfxv H- m-\-.iaxy — my"'.\x~i

— w~f- 3.X)1* * 4-^-t- irftfy -f- nx*yy~~l — nax*yy—* = o, for the Fluxional Equation required. Or the ratio of the. Fluxions will be

« m -4- 3*" — js -(- z«* -j- m -j- I ay — m)$x * . . 1 . . .

- = - - - ^ - — - - : - : — — ; which might have been

n -j- D * * — ;? -j- I ax — nx'j — r -f- nax^y l

found immediately from the given Equation, by the foregoing Rule.

Here the general Numbers m and n may be determined pro lubitu, by which means we may obtain as many .Fluxional Equations as we pleafe, which will all belong to the given Equation. And thus we may always find the fimpleft Expreffion, or that which is beft ac- commodated to the prefent exigence. Thus if we make m = o, and ;; == o, we mall have 4 = *'*-*"* + "> , as found before. Or if

X 3ja — ax

n 11 l y 4*a

we make »; = i, and n= — i , we ihall have - = :^

x *-

as before. Or if we make m=- — i, and n = i, we fhall have

- •=. - — ax +_> A - _

x fy- — zax — x*j '-\~axij '

i n 11 l. V

and n •=. -r- 7, we ihall have - =

'

before. Or if we make m = — ?,

JJ

of Qthers_ Now thjs variety of Solutions

y -(- 3^4 — ^axi J

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 by the Index of each flowing Quantity contain'd in it. (2.) By dividing it by the quantity itfelf j and, (3.) by multiplying by its Fluxion. Thus in the foregoing Equation x> — ax* -f- ayx — y3

= o, the Fluxion belonging to the Term .v3 is 3— , or ^x^x.

The

and INFINITE SERIES. 245

The Fluxion belonging to — ax1 is — - - , or — zaxx. The

. avxv ayxx

Fluxion belonging to ayx is 1- — , or axy -f- ayx. And the

Fluxion belonging to — /3 is — — , or — y-y. So that the

Fluxion of the whole Equation, or the whole Fluxional Equation, is 3-vaA- — zaxx -f- ayx -f- ayx — 3_>'1_y=o. Thus the Equation xm =}', will give mxxm-* =.y ; and the Equation xmz," • — y, will give mxxm—lz" -f- nxmzz"~t = y for its Fluxional Equation. And the like of other Examples.

If we take the Author's funple Example, in pag. 19, or the Equa- tion y = xx, or rather ay — x* = o, that is ayx° — xly° = o, in order to find its moft general Fluxional Equation ; it may be per- form'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 diredtly •? = ™"-'-g+-'* _ For the firft Term of the given

x nxzy ' — n -|- \a

Equation being ayx°, this multiply'd by the Index of x increas'd by 7/7, that is by ;;z, and divided by x, will give mayx~l for the firlt 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 divided by ,v, will give — m -h 2X for the fecond Term of the Nu- merator. Again, the firft Term of the given Equation may be now

— ,Y*J°, which multiply'd by the Index of y increas'd by n, that is by ;;, and divided by r, will give (changing the fign) nxly~l for the firft Term of the Denominator. Alib the fecond Term will then be cyx°, which multiply'd by the Index of/ increas'd by ;/, that is by n -f- i, and divided by y, will give (changing the Sign)

— n -|- \a for die fecond Term of the Denominator, as found above. Now from this general relation of the Fluxions, we may deduce as many particular ones as we pleaie. Thus if we make ///= o, and

7/r=o, we fhall have -- r— -- , or ay = 2xx, agreeable to our Author's Solution in the place before cited. Or if we make;«= — 2,

1 n II 1 2£TA 2tfl>1 . .,

aiid ;z= — r, we lhall have - = -7^7 = -77 • Or if we make /•/v = o, and ;/ = — i, we (hall have - = - ^ = - . Or if

•V X j ' A"

we make n = o. and m-=. — 2, we fhall have •- = -^-^ . -1-,

•v a m ,v

as before. All which, and innumerable other cafes, may be eafilv* proved by a fubftitution of equivalents. Or we may prove it c:

rally

246 etf>e Method of FLUXIONS,

rally thus. Becaufe by the given Equation it is y=xia~I, in the

~

/-i V mayx — m-\-zx c r , ,,. . , ,

value of the ratio 4 = gA& -. _7^ri7 f°r 7 mbmtute its value, and

/, 11 V V OT* - » + 2X 2X

we fhall have ~- = - =—— = — as above.

x na — n -+- i a a

3. The Equation of the fecond Example is 2j3 -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 mufl be form'd. Firft difpofe the Terms according to y, thus ; 2j3 _j_ oja _{_ x*y — z~>y°= o, and multiply by the Terms of the Pro-

- 2CZ

greffion 2 xjj""1, ixj/y"1, oxj/y""1, — i xj//-1, relpeclively, (where the Coefficients are form'd by diminishing the Indices of y by the com- mon Number :,) and the refulting Terms will be qyy* * * -f- &yy—*. Secondly; difpofe theTerms according to x, thus-> yx*--}-ox-t-2y">x°=o3 .

— 2cz

and multiply by the Terms of the ProgreiTion 2xxx~\ i xxv~r, . oy.xx~l, (\vhere the Coefficients are the fame as the Indices of x,) and the only refulting Term here is -+- 2yxx * *. Laftly, difpofe the Terms according to z, thus ; — z= -+-^y^ — 2cyz-±-x*yz°=oJ

-4- 2}"

and multiply by the Progreffion 3x£s~I, 2xzz~'f, fx.zz~!, 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 thefe Terms together, we fhall have the Fluxional Equation fyrj1 + ~3yy— i _|_ av,v.v — yzz* -+- 6yzz — 2cyz =. o.

Here we have a notable inftance of our Author's dexterity, at

finding expedients for abbreviating. For in every one of thefe Ope-

rations fuch a Progreffion is chofe, as by multiplication will make

the greateft deftrudtion of the Terms. By which means he 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-

thod, which is taught above, that is, if we always a flu me the Pro-

oreffions of the Indices, we fhall have 6yy* -+• 2xxy -\- xy — 2cyz

— zcyz -+- ~}yz* ~r- dyzz — 3'zz* = o ; which has two Terms

more' than the other form. And if the Progreffions of the Indices

t(-j increas'd, in each 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 being affign'd to it, the other will alfb be compleatly determined. And in the Fluxional Equation derived 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 Quantities, two of them muft be a/Turned to determine the third ; 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 compli- cate Terms, fuch as compound Fractions and Surds, are to be ma- naged in this Method.

5, 6. Let the given Equation be y* — a* — x*/ a* — x- = o, of which we are to take the Fluxions. To the two Fluents y and x we may introduce a third ;c, if we aflume another Equation. Let that be z = x\/a*- — x~, and we mall have the two Equations y- — a- — & = o, and a'-x1 — x* — z* r= o. Then by the fore- going Solution their 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 of Reduction : that is, we may eliminate z and z by fubftituting their values yy — a a and zyy. Then we fhall havej1 — a1 — x\/ a1 — .v1

248 fix Method of FLUXIONS,

" "

- — !QJ and 2yy — " "__ = == o. Or by taking away the furds, ,

'tis a"xz — ^4 — y* 4- 2alyz — rt4 = o, and then a*xx — 2xx=. -- za* = o.

7. Or if the given Equation be x5 — ay* -f- • - — x^^/ay -\-x*-

= o, to find its corresponding Fluxional Equation ; to the two1 flowing quantities ,v and y we may introduce two others .z and 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. « +_>' ^

have the three Equations x= — ay1 -\- z — i;=o, az-\-yz — by* —r o, and ayx* -f- x6 — i<-~ = o, which will give the three Fluxional Equations ^xx* — zayy -+- z — V = o, az •+- yz -+- yz — "^byy* = o, and ay'x* -+- ^.ayxx' -f- 6xxs — 2vv= o. Thefeby,- known Methods of the common Algebra may be reduced to on& Fluential and one Fluxional Equation, iavolving x and yy and their 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 compli- cated 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 </ aa — xx = s$. Then we fhall

have the two Fluential Equations ax-\-z — y1 = o, and a* — AT*; — z1 = o, from whence we mall have the two Fluxional Equations ax-}- z — 2j/y = o, and — 2xx — 2zz = o, or xx -f- zz = o.' This laft Equation, if for z and z we fubftitute their values^ — ax~ and zyy — ax, will become xx -f- 2yy* — zaxyy — axy* -{- a^xx-

• — o ; whence y = ~ " A' ~A- . And here if for y we fubfti--

•' 2\i - 2HX1 •*

tute its value vax-+-\/aa — xx, we mall have the Fluxion re-

ax -J an — A Jf — xx , , 1 T^

quired y = — ---------- - : .„.., - . And many other Exam--

7.1/fta — xx x yax + y aa — xx

pies of a like- kind will be found in the fequel of this Work.

9, 10, 1 1, 12. In Examp. 5. the propofed Equation is zz -{- axz — .)'4=°> m which there are three variable quantities x, y, and z, and therefore the relation of the Fluxions will be 2zz -|- axz • _j_ ax~ — 4j/j-3 === o. But as there wants another Fluential Equa- tion, and thence another Fluxional Equation, to make a compleat determination ; if only another Fluxional Equation were given or < afTurned, we mould have the required relation of the Fluxions x and y,..

Suppofe

and INFINITE SERIES. 249

Suppofe this Fluxional Equation were i=.vv/^-v — xx ; then by fubftitution we mould have the Equation zz -f- ax x x^/ax — xx -f- axz — 4)7 5 = o, or the Analogy x :y :: 4_>'3 : 2Z -4- ax x 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- forni'd Analytically, or by Calculation, yet it may poffibly be per- form'd Geometrically, as it were, and by the Quadrature of Curves ; as we may learn from our Author's preparatory Proportion, and from the following general Conliderations. Let the right Line AC, perpendicular to the right Line AB, be conceived to move always parallel to itfelf, fo as that its extremity A may defcribe the line AB. Let the point C be fixt, or always at the fame diftance from A, and let another point move from A towards C, with a velocity any how accelerated or retarded. The parallel motion of the line AC does not at all affect the progreffive motion of the point moving from A towards C, but from a combination of thefe two independent morions, it will defcribe the 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 flowing quantities, and their velocities of defcription, or their Fluxions, muft neceflarily be as their refpeclive defcribing lines, or Ordinates, BE and BD. Let AC or BE be Linear Unity, or a conftant known right line, to which all the other lines are to be compared or refer'd ; juft as in Numbers, r.M other Numbers are tacitely refer'd to i, or to Numeral Unity, as being the fim- pleft of all Numbers. And let the Area 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 j ami let the refulting line be call'd z. That is, make the Area ADB = z x BE ; and if AB be call'd x, then is the Area ACEB = x x BE. Therefore the

K k Fluxions

25 o"1 Ibe Method 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. Consequently in any Curve, the Fluxion of the Area will be as 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</ax — xx, if x reprefents the Abfcifs of a Curve, and \/ ax — xx be the Ordinate ; then will this Curve be a Circle, and z will reprefent the corresponding Area. So that we fee from hence, whether the Area of a Circle can be exhibited or no, or, in general Terms, tho' in the Equation proppfed there fhould be quantities involved, which cannot be determined or ex- prefs'd by any Geometrical Method, luch as the Areas or Lengths of Curve-lines ; yet the relation of their Fluxions may neverthelefs

be found.

13. We now come to the Author's Demonftration of his Solutions or to the proof of the Principles of the Method of Fluxions, here laid down, which certainly deferves to engage our mcft ferious attention. And more efpecially, becaufe thefe Principles have been lately drawn into debate, without being well confider'd or imderftoqd ; polfibly beT caufe this Treatife of our Author's, expreffly wrote on the fubjed, had not yet feen the light. As thefe Principles therefore have been treated as precarious at leaft, if not wholly inefficient to fupport the Doo trine derived from them ; I Shall endeavour to examine into every the moll: minute circumflance of this Demonstration, and that with the utmoft circumipeclion and impartiality.

We have here in the firft place a Definition and a Theorem to-r gether, Moments are defined to be the indefinitely jmall parts offoiv- itig quantities, by the acceflion of which, in indefinitely fmall portions of time, tboj'e quantities are continually increajed. The word Moment (momentum^ movimentum, a mevcoj by analogy feems to have been borrow'd from Time. For as Time is conceived to be in continual flux, or motion, and as a greater and a greater Time is generated by the acceffion of more and more Moments, which are conceived as the fmalleit particles of Time : So all other flowing Quantities may be underitood, as perpetually, increafing, by the accellion of their fmallefr, particles, which therefore may not improperly be call'd their Moments. But what are here call'd their jmalleft particles, are not to be underftood as if they were Atoms, or of any definite and determinate magnitude, as in the Method of Indivisibles.} but to be indefinitely fmall, or continually decreafing, till they are lefs

than

and INFINITE SERIES. 251

than any afiignable quantities, and yet may then retain all poffible varieties of proportion to one another. That thefe Moments are not chimerical, vifionary, or merely imaginary things, but have an existence Jut generis, at leaft Mathematically and in the Underftand- ing, is a neceflary confequence from the infinite Divifibility of Quan- tity, which I think hardly any body now contefts *. For all con- tinued quantity whatever, tho' not indeed actually, yet mentally may be conceived to be divided in infinitutn, Perhaps this may be beft illuftrated by a comparative gradation or progrefs of Magnitudes. Every finite and limited Quantity may be conceived as divided into any finite number of fmaller parts. This Divifion may proceed, and thofc parts may be conceived to be farther divided in very lit- tle, but flill finite parts, or particles, which yet are not Moments. But when thefe particles are farther conceived to be divided, not actually but mentally, fo far as to become of a magnitude Ids than any afiignable, (and what can flop the progrefs of the Mind ?) then are they properly the Moments which are to be understood here. As this gradation of diminution certainly includes no abfurdity or con- tradiction, the Mind has the privilege of forming a Conception of thefe Moments, a poffible Notion at leaft, though perhaps not an adequate one ; and then Mathematicians have a right of applying them to ufe, and of making fuch Inferences from them, as by any flrict way of reafoning may be derived.

It is objected, that we cannot form an intelligible and adequate Notion of thefe Moments, becaufe fo obfcure and incomprehenfible an Idea, as that of Infinity is, muft needs enter that Notion ; and therefore they ought to be excluded from all Geometrical Difquifi- tions. It may indeed be allowed, that we have not an adequate Notion of them on that account, fuch as exhatifts the whole nature of the thing, neither is it at all neceflary ; for a partial Notion, which is that of their Divifibility fine Jine, without any regard to their magnitude, is fufficient in the preient cafe. There are many other Speculations in the Mathematicks, in which a Notion of In- finity is a neceflary ingredient, which however are admitted by all Geometricians, as ufeful and dcmonftrable Truths. The Doctrine of commenfurable and incommenfurable magnitudes includes a No- tion of Infinity, and yet is received as a very demonftrablc Doctrine. We have a perfect Idea of a Square and its Diagonal, and yet we

K k 2 know

The Method of FLUXIONS,

know they will admit of no finite common meafure, or that their pro- portion cannot be exhibited in rational Numbers, tho' ever fo fmall, but may by a feries of decimal or other parts continued ad infini- tum. In common Arithmetick we know, that the vulgar Fraction 1., and the decimal Fraction 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- icribe a right line 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 remainder, and fo on. That is, I actually run over all thofe infinite divifions and fubdivifions, before I have com- pleated the Line, tho' I do not attend to them, or cannot diftin- guifh them. And by this I am indubitably certain, that this Series of Fractions i -f- JL _j_ -£.-}- _'r> &c. continued ad infinitum, is pre- cifely equal to 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 aflign- able 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 their nature and properties are hardly intelligible or difcoverable. And in general, many of the 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 of Infinity will be neeeflarily included. We may therefore as 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 of fairing quantities are as the Velocities of flowing or increafing ; that is, as their Fluxions. Now if this be proved of Lines, it will equally obtain in all flowing quantities whatever, which may always be adequately rcprefented and ex- pounded 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 infi- nitely little Spaces, which we call Moments. And even in Mo- tions continually accelerated or retarded, the Motions in infinite- ly 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 promifcu- oufly 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 appointed already, that when x, y, z, v, &c. ftand for variable or flowing quantities, then their Velocities of increafe, or their Fluxions, fhall be reprefented by x, y, z, -j, &c. which therefore will be pro- portional to the contemporary Moments. But as thefe are only Velocities, or magnitudes of another Species, they cannot be the Mo- ments themfelves, which we conceive as indefinitely little Spaces, or other analogous quantities. We may therefore here aptly intro- duce the Symbol o, not to ftand for abfolute nothing, as in Arith- rnetick, but a vanifhing Space or Qtiantity, which was juft now finite, but by continually decrealing, in order prefently to terminate in mere nothing, is now become lefs than any affignable Qinintify. And we have certainly a right fo to do. For if the notion is in- telligible, and implies no contradiction as was argued before, it may furely be infinuated by a Character appropriate to it. This is not aligning the quantity, which would be contrary to the hypothefis, but is only appointing a mark to reprefent it.- Then multiplying the' Fluxions by the vanishing quantity <?, we fhall have the fcve- ral quantities .\o, yo, zo, r?, £cc. which are vanifhing likewife, and pioportional to the Fluxions refpedlively. Thefe therefore may now reprefent the contemporary Moments- -of x, y, z, v, &c. And in general, whatever other flowing .quantities, as well as Lines and

I Spaces,

2 54 "*flje Method of FLUXIONS,

Spaces, arc reprefented by A-, y, z, -v, &c. as o may (land for a. -vanishing quantity of the fame kind, and as x, y, z, v, &c. may ftand for their Velocities of increafe or decreafe, (or, if you pleafe, fpr Numbers proportional to thofe Velocities,) then may xo, yo, zo, i-o, &c. always denote their refpedive fynchronal Moments, .or momentary accefiions, and may be admitted into Computations .accordingly. And this we corne now to apply.

15. We muft now have recourfe to a very notable, ufeful, and extenfive property, belonging to. all Equations that involve flowing Quantities. Which property is, that in the 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 .rnay be any .how increafed or diminifh'd by their contemporary Increments or Decrements ; which Fluents, fo increafed or dimi- niihed, may be fubflituted for the others in the Equation. As if an Equation mould involve the Fluents x and _y, together with any given quantities, and X and Y are fuppofed to be any of their con- temporary Augments reflectively. Then in the given Equation we may fubflitute x -f- X for x, and y -+- Y for -y, and yet the Equa- tion will be .good, or .the equality of the Terms will be prefer ved. .So if X and Y were contemporary Decrements, inflead of x and y we might fubflitute x — X and y — Y reflectively. And as this inuft hold good of all contemporary Increments or Decrements what- ever, whether finitely great or infinitely little, it will be true like- wife of contemporary Moments. That is, in flea d of .r and y in any Equation, we may fubflitute .v-f- xo 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 xs — a.\* + a xy — _>'5=o be given as before, including the variable quan- tities x and r, inftead of which we may fubflitute thefe quan- tities increas'd by their contemporary Moments, or x -±- xo and y -i-yo respectively. Tlien we ihall have the Equation x -+- xo | 3 — a x x + AO i a -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

17, 18. Here the Terms of the fir ft *3+ ?*w* +3A*"*r 1 order, or column, remove or deftroy one _fl.vi_ 2f,^ox _ all'l \ 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-

mon Multiplier <?, whatever it is. This being- done, all the Terms of the third order will ftiil be affecled by o, of one or more dimen- fions, and may therefore be expunged, as infinitely lels than the others. Laftly, there will only remain thofe of the fecond order or column, that is 3.vA.'i — zaxx -+- axy 4- ayx — Tjy- = o, which will be the Fluxional Equation required. Q^. E. D.

The fame Conclufions may be thus derived, in fomething a dif- ferent manner. Let X and Y be any fynchronal Augments of the variable quantities A* and y, as befoie, the relation of which quan- tities is exhibited by any Equation. Then may tf-J-X and y 4- Y be fubfKtuted for x and y in that Equation. Suppofe for inftance that x> — ax* 4- axy - — _y3 = o ; then by fubftitution we flwll have x 4- X | 3 — a x .v 4- X | a 4-#x.v4-Xx/4-Y — y 4- V | 3 = o ; or in termini* expanfis .v5 -f- 3X1X -f- 3xXz -+- X3 — ax1 — 2rfxX — aX* -t- axy -\- <?.vY4- aXy -f- ^XY — j3 — 3jaY — 3;'Y4

— Y5 = o. But the Terms ,v3 — ax* -+- axy • — _y3 = o will va- niHi out of the Equation, and leave 3#1X 4- 3xXa 4-X3 — 2axX

— aX* 4- axV 4- aXy 4- «XY — y* Y — 3/i7* — Y- == o, for the relation of the contemporary Augments, let their magnitude be what it will. Or refolving this Equation into an Analogy, the ratio

,- , ,. A , ,. Y ?r*-|- ^rX-L. X1 — 2 ..*— /7X -L«v

of thele Augments may be this, — =. - •

X — a* — ..v _|- -j* -r 3.., + l *

Now to find the ultimate rc.tio of thefe Augments, or their ratio when they become Moments, fuppofe X and V to diminil'h till they become vanishing quantities, and then they may be expunged out of this value of the ratio. Or in thofe circumftances it will be

, which is now the ratio of the Moments. And

-

P = — — ~^ - —

- y — ax

this is the fame ratio as that of the Fluxions, or it will be

.V1 — 2f>x--ai . . • • -

or 3_)'a — axy = $x-x — zaxx 4- ayx, as wss

found before.

In this way of arguing there is no aflumption made, but what is iuflifiable by the received Methods both of the ancient and modern Geometricians. We only defend from a general Proportion, which is undeniable, to a particular cafe which is certainly included in ir.

That

256 The Method of FLUXIONS,

That is, having the relation of the variable Quantities, we thence da-eddy deduce the relation or ratio of their contemporary Aug- ments ; and having this, we directly deduce the relation or ratio of thofc contemporary Augments when they are nafcent or evanefcent, juft beginning or juft ceafing to be ; in a word, when they are 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 Quantity as perpetually diminiiLingy/w^w ; that the Conception of a .vanishing Quantity, a Moment, an Infinitefimal, &c. includes a contradiction : In fhort, that Quantity is not (even mentally) divifi- ble ad infinitum ; for to that the Controverfy mufb be reduced at 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 long in quiet poiTefTion of it, who are indubitably convinced of the evidence and. certainty of it, who continually and fuccefslully ap- ply it, arid who- are ready to acknowledge the extreme fertility and ufefulnefs of it, upon fo many important occalions.

19. Nothing remains, I think, but to account for thefe two cir- .cumilances, belonging to the Method of Fluxions, which our Au- thor briefly mentions here. Firft that the given Equation, whofe Fluxional Equation is to be found, may involve any number of flowing quantities. This has been fufficiently proved already, and we have feen feveral Examples of it. Secondly, that in taking Fluxions we need not always confine ourfelves to the progreffion of the Indices, but may affume infinite other Arithmetical Progreflions, as conveniency may require. This will deferve a little farther illu- ftration, tho' it is no other than what muft neceiTarily refult from the different forms, which any given Equation may afTume, in an infinite variety. Thus the Equation x3 — ax1 -4- axy — j3 = o, being multiply'd by the general quantity xmy", will become #«"+*>'» -r- axm-$-1y" -h axm+ly"'t'1 — xmy"~^^ = o, which is virtually the fame Equation as it was before, tho' it may aiTume infinite forms, accor- ding as we pleafe to interpret m and n. And if we take the Fluxions of this Equation, in the ufual way, we mall have m+iy* -j- nx^rty}*-1 — m -+- zaxxm^y" — naxm^yyn~^ -f-

l -f- n •+• irf.Y"!'j/)-B — mxxm~Iya''* — n .5= o. Now if we divide this again by x"}", we mail have m 4- nx*j>y~* — m -f- 2axx — nax*yy~~l -+- m -+- laxy 4- n-\- \axy — »/xx~*y* — n -f- 3j/ya ?= o, which is the fame general Equation as was derived before. And the like may be underftood of all other Examples. SECT.

and INFINITE SERIES. 257

SECT. II. Concerning Fluxions of fuperior orders^ and the method of deriving their Equations.

IN this Treatifc our Author confiders only fir ft Fluxions, and has not thought fit to extend his Method to fuperior orders, as not di- rectly foiling within his prefent purpofe. For tho' he here purfues Speculations which require the ufe of fecond Fluxions, or higher orders, yet he has very artfully contrived to reduce them to firft Fluxions, and to avoid the necefTity of introducing Fluxions of fu- perior orders. In his other excellent Works of this kind, which have been publifh'd by himfelf, he makes exprefs mention of them, he difcovers their nature 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- tion, I mall here make fome inquiry into the 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- fider'd, or their comparative Velocities of increafe and decreafe, are themfelves, and of their own nature, variable and flowing quantities alfo, and as fuch are themfelves capable of perpetual increafe and de- crea&, or of perpetual acceleration and retardation ; they may be treated as other flowing quantities, and the relation of their Fluxions may be inquired and difcover'd. In order to which we will adopt our Author's Notation already publifh'd, in which we are to con- ceive, that as x, y, z, &c. have their Fluxions #, j, z., &c. fo thefe likewife have their Fluxions x, /, z,&c.which are the fecond Fluxions of x, v, z, &c. And thefe again, being ftill variable quantities, have

j.

their Fluxions denoted by x, y, z, &c. which are the third Fluxions of x, y, z, &c. And thefe again, being ftill flowing quantities,

have their Fluxions x, /, z, &c. which are the fourth Fluxions of x, y, z, &c. And fo we may proceed to fuperior orders, as far as there mall be occafion. Then, when any Equation is propofed, con- futing of variable quantities, as the relation of its Fluxions may be found by what has been taught before ; fo by repeating only the fame operation, and confidering the Fluxions as flowing Quantities^ the

L 1 relation

258 The Method of FLUXIONS,

relation of the fecond Fluxions may be found. And the like for all higher orders of Fluxions.

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

— ax - — o. And here, as we have the three Fluents j>, y, and x, if we take the Fluxions again, we fhall have the fecond Fluxional Equation zyy -+- zy* — ax= o. And here, as there are four Fluents y, y, y, and x, if we take the Fluxions again, we fhall have the

.. .»

third Fluxional Equation zyy •+• zyy -f- ^.yy — ax = o, or zyy 4-

bjy — ax = o. And here, as there are five Fluents y, y, y, y, and x, if we take the Fluxions again, we fhall have the fourth Fluxional

Equation zyy •+• zyy -f- 6yy -+- 6yl — ax = o, or zyy -+- Syy -f- 6y*

— ax = o. And here, as there are fix Fluents y, y, y, y, y, and xy if we take the Fluxions again, we fhall have zyy •+• zyy -f- 8yy -{-

fyy _j_ i zyy — ax = o, or zyy •+- i oyy -f- zoyy — ax = o, for the fifth Fluxional Equation. And fo on to the fixth, feventh, 6cc.

Now the Demonftration of this will proceed much after the man- ner as our Author's Demonftration of firft Fluxions, and is indeed virtually included in it. For in the given Equation^* — ax = o} if we fuppofe y and x to become at the fame time y -f- yo and x-)- xo, (that is, if we fuppofe yo and xo to denote the fynchronal Moments of the Fluents y and x,) then by fubftitution we fhall have ~y +yo\ *

— a x x -f- xo = o, or in termini* expanjis, y1 -f- zyyo -+-y*o* — ax

— axo = o. Where expunging y1 — ax = o, andj/1^1, and divi- ding the reft by o, it will be zyy — ax = o for the firft fluxional Equation. Now in this Equation, if we fuppofe the fynchronal Moments of the Fluents y, y, and x, to beyo}yot and xo refpedively ; for thofe Fluents we may fubftitute y -f-jj/o, y -+-yo, and x+ xo in the kft Equation, and it will become zy-t-zyoxy-l-yo — axx + xo •r. — o, or expanding, zyy -f- zyyo •+- zyyo -+- zy'yoo — ax — axo = o. Here becaufe zyy — ax= o by the given Equation, and becaufe zy'yoo vanishes ; divide the reft by o, and we fhall have zy* •+• zyy

— ax •=• o for the fecond fluxional Equation. Again in this Equa- tion, if we fuppofe the Synchronal Moments of the Fluents y, yt

y, and xt to be yo, yo, yo, and xo refpedively ; for thofe Fluents

we

and INFINITE SERIES. 259

we may fubftitute y+yo, y + yo, y-t-yo, a^id x •+- xo in the lad

.. j a

Equation, and it will become 2x7 -\-yo \ •+- zy -+- 2yo x y -f- yo — a x x _j_ xo — o, or expanding and collecting, 2j* + 6yyo -t- 2y*ol

_}_ 2yy -+- 2yyo -t- s;^1 — ax — axo = o. But here becaufe 2j's _l_ 2/_y — rfx = o by the laft Equation ; dividing the reft by o, and expunging all the Terms in which o will ftill be found, we fliall

have 6yy -+- 2yy — ax = o for the third fluxional 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 Example, let us take our Author's Equation #5 — ax3- -{-axy — y> = o, in which he has found the fimpleft relation of the Fluxions to be 3x^a — zaxx -h axy •+- axy — 3^/7* = o. Here we have the flowing quantities x, y, x, y ; and by the fame Rules the Fluxion of this Equation, when contracled, will be 3#wi + 6x*x — 2axx — zax* H- axy -+- 2axy -\- axy — 3vys — 6jf!Ly = o. And in this 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,

when contracted, ^xxl -f- iSxxx -{- 6x3 — 2axx — 6axx -f- axy -f-

%axy -+- T,a.\y -f. axy • — 3 yy* — i fyyy — 6ys = o. And as in this Equation there are found the flowing quantities x, y, x, yy x, y,

x, y, we might proceed in like manner to find the relations of the fourth Fluxions belonging to this Equation, and all 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 y -, and as the firft fluxional Equation exprefles the conftant re- lation of the variable (but finite i.nd 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 exprefs the conftant relation of the variable (but finite and aflig- nable) quantities x and yy which denote the comparative Velocity of 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,

comparative Velocity of the increafe or decreafe of "x and "y in the foregoing Equation. And fo on for ever. Here the Velocity of a Velocity, however uncouth it may found, will be no abfurd Idea when rightly conceived, but on the contrary will be a very rational and intelligible Notion. If there be fuch a thing as Motion any how continually accelerated, that continual Acceleration will be the Ve- locity of a Velocity ; and as that variation may be continually va- ried, that is, accelerated or retarded, there will 'be in nature, or at leafl in the Understanding, the Velocity of a Velocity of a Velocity. Or in other words, the Notion offecond, third, and higher Fluxions, muft be admitted as found and genuine. But to proceed :

We may much abbreviate the Equations now derived, by the known Laws of Analyticks. From the given Equation x* — ax1 -+- axy — y"' =0 ^ere is found a new Equation, wherein, becaufe of two new Symbols x and y introduced, we are at liberty to aflume another Equation, belides this now found, in order to a jufl De- termination. For fimplicity-fake we may make x Unity, or any other conftant quantity ; that is, we may fuppofe x to flow equably, and therefore its Velocity is uniform. Make therefore x = i} and

the firft fluxional Equation will become 3^* — 2ax -+- ay + axy

3j)/)'1 = o. So in the Equation 3x.va -f- 6x*x — 2axx — 2ax* -+. axy -i- zaxy -h axy — 3 vj* — 6y\y = o there are four new Sym- bols introduced, x, y, x, and r, and therefore we may afiume two other congruous Equations, which together with the two now found, will amount to a compleat Determination. Thus if for the fake of fimplicity we make one to be x = i, the other will' neceflarily be .v =o ; and thefe being fubftituted, will reduce the fecond fluxionaj Equation to this, 6x — 2.0. -f- iay -f- axy — ^yy- — 6y*y — o. And thus in the next Equation, wherein there are fix new Symbols

x, }', x, y, x, y ; befides the three Equations now found, we may take x= i, and thence 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 Equations are derived. Thus the 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 3** — zax + ay + axy — zyy* = o. And taking the Fluxions

again.

and INFINITE SERIES.

261

again, it will be 6x — 20. -f- zay •+- axy — 3 yy* — 6y*y = i o.

And taking the Fluxions again, it will be 6 -f- $d'y -+- axy — %yy* 6y*> = o. And taking the Fluxions again, it will be

axy — 3^4 — 24-yy'y — i%y1y — 3677* = o. And fo on, as 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 Figure, adapted to a iimple and a particular cafe. Let us allume the Equation y1 r=ax, otyzs=ia*x*, which will therefore belong to the Parabola ABC, whole Parameter is AP = tf, Abfcifs AD = x, and Ordinate LD =y ; where AP is a Tangent at the Vertex A. Then taking the Fluxions, we fhall have y = yaPsve~~*. And fup- pofing the Parabola to be defcribed by the equable motion of the Ordinate upon the Abfcifs, that equable Velocity may be expounded by the given Line or Parameter a, that is, we may put x = a. Then

\t\v]\ibey=(±a*x *= ~ • = "—?— = ) -•?- , which will give us

zxk 2X ' 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.

AE and AP ; for its Equation isjx= -11 ,

Or yy = £ .

Again, from the Equation y = "± , or 2*y = ay, by taking the Fluxions again, and putting x =a as before, we fhall have zay -{- 2xy=aj,or—y = J j where the negative fign {hews only,

that_y is to be confider'd rather as a 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) ; DI = y, and the Line DI will 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

Again, from the Equation — y = ^-v , or — 2xy = ay't

^by taking the Fluxions we mail have — 2ay • — zxy =: ay., or ~ y=~ , which will give us this Conftrudlion. Make x (AD) :

y (DI) :: \a (|AP) : DL=y, and the Line DL will therefore reprefent the Fluxion of DI, or of y, the fecond Fluxion of DG, or of y, and the third Fluxion of BD, or of^. And if this be done every where upon AE, a Curve LM will be conflructed, whofe Ordinates will always expound the third Fluxions of the correfpon- ding Ordinates of the Parabola ABC. This Curve will be an Hyper- bola, and its Equation will be — y=.— '=-§1 ; , or 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.

Fluxions, when given. As if we had y = —, = La*xx~*} or if the Curve GH were given ; by taking the Fluents, (as will be taught in the next Problem,) it would be y = (a^x*= ^-r = )

- , which will give us this Conftruction. Make \a (|AP) :

.v (AD) :: y (DG) : DB =± 2-J , and the Line DB will reprefent

the Fluent of DG, or of y. And if this 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

Parameter is the Line AP = a. For its Equation is y = a*x'*t or yy=ax.

So if we had the Parabola ABC, we might conceive its Ordinates to reprefent Fluxions, of which the 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 .. .:

of y, &c. (That is, let, &c. _/, y, y, /, j/, y, y, &c. be a Series of Terms proceeding both ways indefinitely, of which every fucceed- ing Term reprefents the Fluxion of the preceding, and vice versa ; according to a Notation of our Author's, deliver'd elfewhere.) Then

becaufe it is_y = (div*=<z^x^ =) ^r , taking the Fluents it

' .x ' — %• \

will be y = [—, = 2f!i! = ) ZJ2. ; which will give us this Con- W 3« y 3*

ftrudion. Make $a (|AP) : ft (AD) :: y (BD) : -^ =y = DQ^

and the Line DQ^will reprefent the Fluent of DB, or of y. 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

of which is^= — * , or yy = — .

3«^

Again, becaufe y = fzx~ == 3ilJL£ =\ 2^fl . taking the Flu-

\ $a? $al v. a J ->.a.'-

" / "* i 7

ents it will be y=( JfL.—sff! |x"= W , which will give us this

Conftruaion. Make |« (|AP) : x (AD) : : y ( DQ^J : — = _y

/

= DS, and the Line DS will reprefent the Fluent of DQ^, or 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 Equation is jy= 1^1 , or yy =

— ^-. . And fo we might go on as far as we pleafe,

Laftlv,

264 The Method of FLUXIONS,

Laftly, if we conceive DB, the common Ordinate of all thefe Curves, to be any where thus conftrucled upon AD, that is, to be thus divided in the points S, Q^ B, G, I, L, 6cc. from whence to AP are drawn Ss, Qtf, B^, Gg, I/, L/, 6cc. parallel to 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 afore- faid Parallels along with it in its motion : Then the points s, q, b,g, i, /, &c. will likewife move in fuch a manner, in the 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- prefented by Aq, the Velocity of q by A£, of b by Ag, of g by A/, of / by A/, &c. Or in other words, Aq will be the Fluxion of A.S ; Al> will be the Fluxion of Ag, or the fecond Fluxion of As ; Ag will be the Fluxion of Ab, or the fecond Fluxion of Aq, or the third. Fluxion of As ; Ai will be the Fluxion of Ag, or the fecond Fluxion of Ah, or the third Fluxion of Aq, or the fourth 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 beginning ; which fufficiently mews the relative nature of all thefe orders of Fluxions and Fluents, and that they differ from each other by mere relation only, and in the manner of conceiving. And in general, what has been obferved from this Example, may be eafily accommodated to any other cafes whatfoever.

Or thefe different orders of Fluents and Fluxions may be thus ex- plain'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 or Root of the lame. And let xn be the like Power or Root of the variable and indefinite quantity x. Make am : xm : : a : y, or

m

y = ^ = al~mxm . Here y alfo will be an indefinite quantity,

a

which will become known as foon as the value of x is affign'd. Then taking the Fluxions, it will be y = mal~mxxm~1 ; and fup- pofing x to flow or increafe uniformly, and making its constant Velocity or Fluxion x = a, it will be y = ma*— mxm-*. Here if

for a1— nxm we write its value y, it will be y = — , that is, x :

ma : : y : y. So that y will be alfo a known and affignable Quan- tity,

and I N FINITE SERIES. 265

tity, whenever x (and therefore y) is affign'd. Then taking the Fluxions again, we mall have^=wxw — ia*— "xx"-1- = ;;; x ;« — irtS"""^*""1 ; or for ma"-~mxm~l writing its value y, it will be

y = ~~xta-v , that is, x : m — la : : y : y. So that y will be- come a known quantity, when x (and therefore y and y) is affign'd. Then taking the Fluxions again, we fhall have y = m x m — i x m — ?.a*-m\m-*, or y=.-^~ , that is, x : m — za :: y : y •

where alfo y will be known, when x is given. And taking the Fluxions again, we fhall have y = rnx m — i x m — 2 x m —

= — - — - ; that is, x : m — 30 :: y : /. So that y will alfo be known, whenever x is given. And from this Inductipn we may conclude in general, that if the order of Fluxions be denoted by any integer number ?/, or if n be put for the number of points over the

^_____ n ll-l-i

Letter yt it will always be x : m — na : : y : y ; or from the Fluxion of any order being given, the Fluxion of the next imme- diate order may be hence found.

_______ "+t n

Or we may thus invert the proportion m — na : x : : y : y} and then from the Fluxion given, we fhall find its next immedi-

ate Fluent. As if « = 2, 'tis m — za : A; : : y : y. If n - — i 'tis m — \a : x :: y : y. If 72 = 0, 'tis ma : x : : y : y. And obferving the fame analogy, if n== — i, 'tis m-±- ia : x :: y :

y ; where y is put for the Fluent of;1, or for y with a negative point. And here becaufe y=.al-mxm, it will be m 4- la : x :: a1-"1*" :

' «I~V+l v"1-*-1

y, or y = _ = ^ : which alfo may thus appear. Be-

m-\-\a m-\-\a

caufe y = {a*-<"x*> = __Zj__T =) il , taking the Fluents, (fee the

/ m-f,,

next Problem,) it will be y = ^— - . Again, if we make «=— 2,

m

. -- I 0 II I „+...

tism-{-2a : x :: y ; y} or y = .. v .. = * - . For

a

M m becaufe

266

The Method of FLUXIONS,

becaufe y = .f — - x - =±=

m

, taking the Fluents it will be

w

-. _ m+i . Again, if we make « = — 3, 'tis m -|-

And fo for

«-t-3« m + l X m -\- 2 x j»+3al"~'~*

all other fuperior orders of Fluents.

And this may fuffice in general, to mew the comparative nature and properties of thefe feveral orders of Fluxions and Fluents, and to teach the operations by which they are produced, or to find their refpeftive fluxional Equations. As to the ufes they may be apply 'd to, when found, that will come more properly to be confider'd in another place.

SECT. III. Tfte Geometrical and Mechanical Elements

of Fluxions,

THE foregoing- Principles of the Doftrine of Fluxions being chiefly abftradted and Analytical I mail here endeavour, af- ter a general manner, to (hew fomething analogous to them in Geo- metry a.nd Mechanicks ; by which they may become, not only the objeft of the 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 con- fider Uniform and equable motions, as alfo fuch as are alike inequable.

Let the right Line AB be defcribed

by the equable motion of a point,

which is now at E, and will pre-

fently be at G. Alfo let the Line

CD, parallel to the former, be de-

fcribed by the 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

HK be contemporaneous Lines, and therefore will be proportional to

the

~

and INFINITE SERIES. 267

the Velocity of each moving point refpedlively. Draw the indefi- nite Lines EH and GK, meeting in L ; then becaufe of like Tri-^ angles ELG and HLK, the Velocities of the points E and H, which 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- proach to E and H let them be found in g and ^, at any fmall 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 the moving points g and k continue to move till they coincide with E and H ; in which cafe the decreeing Lines Eg and H£ will pafs through all polYible magnitudes that are lefs and lefs, and will finally become vanishing Lines. For they muft intirely vanifh at the fame moment, when the points g and k mall coincide with E and 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 K, at any diftance beyond E and H, Still the Velocities, which are 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- tude, or are only nafcent Lines ; that is, if the Line yx.L, by its angular motion, be but juft beginning to emerge and divaricate from EHL. And thus it will be when both thefe motions are equable motions, as alfo when they are alike inequable ; in both 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.

Let AB be an indefinite and fixt 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 angular motion about a moveable Center C. That is, the common interfection C of 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 ; yet even then the point of interfection C, (as it may be ftill call'd,) will not be loft and annihilated, but will appear again, as foon as the Lines begin to divaricate, or to feparate from each other. That is, if C be the point of interfeclion before the coincidence, and c the point of interfec- tion after the coincidence, when the Line dee {hall again emerge out of AB ; there will be fome inter- mediate 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 :

Let the Line AB be defcribed by the continually accelerated mo- tion of a point, which is now in E, and will be prefently found inr G. Alfo let the Line CD, parallel to the former, be defcribed by the equable mo- tion 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- poraneous Lines ; and producing EH and GK till they meet in I, thofe contempo- raneous Lines will be as El and HI refpedlively. Let the defcribing points G and K be conceived to move back again towards A and C, each with the fame degrees of Velocity, in every point of their mo- tion, as they had before 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 alfo, and very little by fuppofuion, they will be nearly as the Ve- locities

and INFINITE SERIES,

269

locities at g and k, that is, at E and H ; which 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 or gki continue its progremve and angular motion in this manner, till it coincides with EHL, and let 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.

Hence we have this Corollary. If the point E (in the foregoing figure,) be fuppos'd to move from A towards B, with a Velocity any how accelerated, and at the fame time the point H moves from C towards D with an equable Velocity, (or inequable, if 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 progreffive and angular motion, in fuch manner, as that EG and HK fhall always be contemporary Lines, till at laft GKI mall co- incide 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 CH reprefent two Fluents, any how related, their Velocities of de- fcription at E and H, or their refpe&ive Fluxions, will be in the ratio of EL and HL.

And hence it will fol- ^

low alfo, that the Lo- cus of the moveable point or Node L-, that is, of all the points of C— 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 deter- mined by the given re- lation of the Fluents or 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

IH

T>

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 exprefs'd by this Equation y = x". Make the contemporary Lines EG = Y and HKs=X.; and becaufe AE and CH are contempo- rary by fuppofition, we fhall have the whole Lines AG and CK contemporary alfo, and thence the Equation y -f-Y= x -j-X | «. This by our Author's Binomial Theorem will produce y -+- Y = x" •+•

-nx"~1X -+- n x"-^-x*~*X* , &c. which ( becaufe y = x" ) will be- come Y= «x"-IX-J-» x ^-^Ar'-^X1, &c. or in an Analogy, X :

y :: i : nxn~l •+• »x ^-^""^X, &c. which will be the general re- lation of the contemporary Lines or Increments EG and HK. Now let us fuppofe the indefinite Line GKI, which limits thefe contem- porary Lines, to return back by a progrefiive and angular motion, fo as always to intercept contemporary Lines EG and HK, and finally to coincide with EHL, and by that means to determine the Node L; that is, we may fuppofe EG = Y and HK = X, to di- minifli hi i-nfinitum, and to become vanifhing Lines, in which cafe we fhall have X : Y : : i : nx"~l. But then it will be like wife X : Y : : HK : EG :: HL : EL : : x : y, or i : nx"~' : : 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 de- fcribed by the motion of two points E and H, of which E moves any how inequably, and (if you pleafe) H may be fuppos'd to move equably and uniformly ; and let the points H and 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- gent in L. Or fuppofe EHL to be the Edge of a Ruler, of an in- definite length, which moves with a progreffive and angular mo- tion thus combined together ; the moveable point or Node L in this Line, which will have the leaft angular motion, and which is always the point of no divarication, will defcribe the Curve, and the Line or Edge itfelf will be a Tangent to it in L. Then will the feg- ments EL and HL be proportional to the Velocity of the points E and H refpeclively ; or will exhibit the ratio of the 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 it, and by continually applying itfelf to it, as a Tangent, to move 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 may be reprefented thus : If L/ be any rigid obftacle in form of a Curve, about which a flexible Line, or Thread, is conceived to be wound, part of which is ftretch'd out into a right Line LE, which 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 L/KG ; by this motion it will exhibit, even to the Eye, the fame increafing Fluents as before, their Velocities of increafe, or their Fluxions, as alfo the Tangents or rectilinear reprefentatives of thofe Fluxions. And the fame may be done by unwinding the Thread, in the manner of an Evolute. Or inftead of the Thread we may make ufe of a Ruler, by applying its Edge continually to the curved Obftacle L/, and making it any how revolve about the move- able point of Contadl L or /. In all which manners the Fluents, Fluxions, and their rectilinear meafures, will be fenfibly and mecha- nically exhibited, and therefore they muft be allowed to have a place in rernm naturd. And if they are in nature, even tho' they were but barely pofiible and conceiveable, much more if they are fenfible and vifible, it is the province of the Mathematicks, by fome me- thod or other, to investigate and determine their properties and pro- portions.

Or as by one Thread EHL, perpetually winding about the curved obftacle L/, of a due figure, we mall fee the Fluents AE and CH continually to increafe or decreafe, at any rate aflign'd, by the mo- tion of the Thread EHL either backwards or forwards ; and as we (hall thereby fee the comparative Velocities of the points E and H, that is, the Fluxions of the Fluents AE and CH, and alfo the Lines EL and HL, whofe variable ratio is always the rectilinear meafure of thofe Fluxions : So by the help of another Thread GK/L, wind- ing about the obftacle in its part /L, and then ftretching out into a right Line or Tangent /KG, and made to move backwards or for- wards, as before ; if the firft Thread be at reft in any given fitua- tion EHL, we may fee the fecond Thread defcribe the contempo- porary Lines or Increments EG and HK, by which the Fluents AE and CH are continually increafed ; and if GK/ is made to ap- proach

272 ^e Method of FLUXIONS,

proach towards EHL, we may fee thofe contemporary Lines conti- imallv to diminim, and their ratio continually approaching towards the ratio of EL to HL ; and continuing the motion, we may pre- fently fee thofe two Lines actually to coincide, or to unite as 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 de- fcribe other contemporary Lines, whofe nafcent proportion will be that of EL to HL. And fo we may go on till the Fluents are ex- haufted. 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- prefented firft Fluxions geometrically and mechanically, we may do 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 Fluxions) of the firft figure ; then its intercepted Tangents will ex- 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 the fecond figure ; then its intercepted Tangents will expound the 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 na- ture of thefe feveral orders of Fluxions, which has been fhewn be- fore.

And farther to mew the univerfality of this Speculation, and how well it is accommodated to explain and reprefent all the circumftan- ces 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 mov- ing, 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 an- other indefinite Line, which by revolving in like manner defcribes another fuch Curve. And fo there may be four or 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 Y fa /?////

2-3.

and INFINITE SERIES. 273

tically, fo many Equations may be aflumed, except one, (if not given by the Problem,) as is the number of the Fluents concern'd.

But laftly, I believe it 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 re- prefented in the Frontifpiece. Let us fuppofe the right Line AB to be parallel to the Horizon, or level with the Ground, in which a Bird is now flying at G, which was lately at F, and a little be- fore at E. And let this Bird be 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 I, and a little before at H ; juft at the fame points of time as 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 fly equably, or always to defcribe 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 point L, fo that the proportion (or ratio) of the two Lines EL and HL may be the fame as that of the Velocities of the two Birds, when they were at E and H refpeclively. And let us far- ther fuppole, that the Eye of a Fowler was at the fame time at the point L, and that he directed his Gun, or Fowling-piece, according to the right Line LHE, in hopes to moot both the Birds at once. But not thinking himfelf 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 I, in the fame right Line 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 Eye is now at N, and the Birds in the fame right Line with his Eye, at K and G. The Path of his Eye, delcribed by this

N a double

274 The Method of FLUXIONS,

double motion, (or compounded of a progreffive and angular mo- tion,) will be ibme Curve-line LMN, in the fame Plain as the reft of the figure, which will have this property, that the proportion of the diftances of his Eye from each Bird, will be the fame every where as that of their refpeftive Velocities. That is, when his 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 propor- tion 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 proceed farther to inflrudl, and to 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 known ; their relation may always be exprefs'd by an Equation, which will be compos'd of the variable quantities x andjy, together with any known quantities. And that Equation will have this pro- perty, becaufe of thofe variable quantities, that as FG and IK, EG and HK, and infinite others, are alfo contemporaneous Fluents; it •will indifferently exhibit the relation of thofe Lines alfo, as well as of EF and HI ; or they may be fubflituted in the 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. For making FG =Y,.and IK = X ; in the given Equation I may fubftitute y -}- Y inftead of ^y, and x -f- X inftead of x, by which I fhall obtain an Equation, which in all circumftances will exhibit the relation of thofe Quantities or Increments. Now it may be plainly perceived, that if the Line MIF is conceived continually to approach nearer and nearer to the Line NKG, (as jufl now, in 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 ap- proach 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 thofc Velocities, that is, in the 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 ; we mail thereby obtain the Ratio of GN to 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 relation may always be exprefs'd by this Equation x1 — ax* •+• axy — y* = o ; then making FG=Y, and IK = X, by fubftituting y -f- Y for j, and x -+- X for x, and reducing the Equation that will arife, (fee before, pag, 255.) we fhall have ^x"-X -f- 3#XZ -f- X3 -— zaxX — aX1 -f- axY -\- aXj -+- rfXY — 3y*Y — 3jyY* — Y! = o, which may be thus exprefs'd in an Analogy, Y : X :: 3** — 2 ax .+. ay •+- 3tfX -h X1 — aX : ^ — ax — aX -+- 37 Y -+- Y*. This Analogy, when Y and X are vanishing quantities, or their ultimate Ratio, will become Y : X : : 3** — ^ax -f- ay : 3^* — ax. And becaufe it is then Y : X :: GN : KN :: v : x, it will be y : x :: 3X1 — zax -+- ay : 3^* — ax. Which gives the proportion of the Fluxions. And the like in all other cafes. Q^. E. I.

We might alfo lay a foundation for thefe Speculations in the fol- lowing manner. Let ABCDEF, 6cc. be the Periphery of a Polygon, or any part of 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,

DEes, &c. meeting the parallels as in the figure, Now if we fup-

N n 2 pofe

276 7%e Method of FLUXIONS,

pofe two moving points, or bodies, to be at $ and b, and to move in the fame time to y and c, with any equable Velocities ; thofe Velocities will be to each other as @y and be, that is, becaufe of the parallels, as /3B and bE. Let them fet out again from y and c, and arrive at the fame time at ^ and d, with any equable Velocities ; thole Velocities will be as yfr and cd, that is, as yC and cC. Let them depart again from £ and d, and arrive in the fame time at g and e, with any equable Velocities ; thofe Velocities will be as S-t and de, that is, as J^D and dD. And it will be 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 magni- tude be diminim'd in infinitum, and then the Periphery of the Poly- gon will continually approach towards a Curve-line, to which the Lines AB^/3, ECcy, CDd£, &c. will become Tangents -, as alfo the 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- gents cTD and dD ; and the Lines /3^ and bd, intercepted by any two Tangents J>D and /SB, will be the contemporaneous Lines, or Fluents. Now from the nature of the Curve being given, or from the property of its Tangents, the contemporaneous Lines may be found, or the relation of the Fluents. And vice versa, from the Rate of flowing being given, the correfponding Curve may be found.

ANNO-

and INFINITE SERIES.

277

ANNOTATIONS on Prob.i-

O R,

The Relation of the Fluxions being given, to

o & 7

find the Relation of the Fluents.

SECT. I. A particular Solution ; 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 fe- cond 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 quantities of the Spaces defcribed ; or to find the Fluents from the given Fluxions. In difcuffing which important Problem, there will be occafion to expatiate fome thing more at 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- rectly, in computing their feveral qit(?fita> and not tentatively or by way of tryal. Such are Addition, Multiplication, Railing of Powers, and taking of Fluxions. But the Refolutative or Analytical Opera- tions, as Subtraction, Divifion, Extraction of Roots, and finding of Fluents, are forced to proceed indirectly and tentatively, by long deductions, to arrive at their feveral qutefita ; and fuppofe or require the contrary Synthetic Operations, to prove and confirm 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,

or infinite, will produce finite conclufions ; whereas very often in the Refolutative Operations, tho' the data are in finite Terms, yet the quafita cannot be obtain'd without an infinite Series of Terms. Of this we mall fee frequent Inftances in the fubfequent Operation, of returning to the Fluents from the Fluxions given.

The Author's particular Solution of this Problem extends to fuch <afes only, wherein the Fluxional Equation propofed either has been, or at leafl might have been, derived from fome finite Algebraical Equation, which is now required. Here all the necefTary Terms being prefent, and no more than what are neceflary, it will not be difficult, by a Procefs juft contrary to the former, to return back again to the original Equation, But it will moft commonly happen, either if we aflume a Fluxional Equation at pleafure, or if we arrive at one as the refult of fome Calculation, that fuch an Equation is to be refolved, as could not be derived from any previous finite Al- gebraical Equation, but will have Terms either redundant or defi- cient ; and confequently the Algebraic Equation required, or its Root, mufl be had by Approximation only, or by an infinite Series. In all which cafes we mult have recourfe to the general Solution of this Problem, which we fhall find afterwards.

The Precepts for this particular Solution are thefe. (i.) All fuch Terms of the given Equation as are multiply 'd (fuppofe) by x, muft be difpofed according to the Powers of x, or muft be made a Num- ber belonging to the Arithmetical Scale whofe Root is x. (2.) Then they muft be divided by A-, and multiply'd by x ; or x muft be changed into A', by expunging the point. (3.) And laftly, the Terms muft be feverally divided by the Progreilion of the Indices of the Powers of x, or by fome other Arithmetical ProgrerTion, as need mail require. And the fame things muft be repeated for every one of the flowing quantities in the given Equation.

Thus in the Equation $xx- — zaxx -f- axy — . ^yy- -f- ajx — -Q^ the Terms -^xx1 — zaxx -\-axy by expunging the points become ^x'' — zax*- -+- axy, which divided by the Progreffion of the Indi- ces 3, 2, I, reflectively, will give A'5 — ax* -+- axy. Alfo the Terms — 3.X)'a * -+- ayx by expunging the points become — 3j3 * -f- ayx, which divided by the Progreffion of the Indices 3, 2, i, refpectively, will give — y> * -+- ayx. The aggregate of thefe, neglecting the redundant Term ayx, is x* — ax* -\- axy — _}" = o, the Equation required. Where it muft be noted, that every Term, which occurs more than once, mult be accounted a redundant Term.

So

and INFINITE SERIES. 279

So if the propoied Equation were m -f- ^yxx* — m-\- 2(jyxx1 -f- ;// -+- i ay* xx — m} 4-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 Root is x, they will ftand thus : m -+- yyxx' — m -+- zayx*.1 -+• m-\-\ay*xx — my**-, or expunging the points they will become m -+- %yx+ — m -f- Ziivx* •+- m -+- \ay-x1 — m\*x. Thefe being di- vided refpedtively by the Arithmetical Progreffion m -f- 3, m-\-2,. m-\- i, m, will give the Terms yx+ — ayx1 -f- ay'-x1 — y+x. Alio the Terms in which y is found ; being reduced to the Scale whofc

Root isy, will ftand thus : — n -4- ^xyy* * •+- n -+• iax*jy-{- nx*y;

— nax"=y

or expunging the points they will become — n-\~^x^ * -^~n~^~ iaxi)" •+- nx+y. Thefe being divided reipeclively by the Arithmetical Pro-

grefTion ^-{-3, ?i-{- 2, ?z-|-i, ;;, will give the Terms — xy* -\- ax*}'1-)- x+y — ax*y. But thefe Terms, being the fame as the former, mull all be confider'd as redundant, and therefore are to be rejected. So that yx* — ayx* -f- ayixi — y^x=o) or dividing by yx, the Equation x* — ax1 -\-ayx — y* = o will arife as before.

Thus if we had this Fluxional Equation mayxx~l — m -+- 2xx — nx*yy~* -+- ;z-f- \ay •=. o, to find the Fluential Equation to which it belongs ; the Terms mayxx~I * — m -f- 2xx, by expunging the points, and dividing by the Terms of the Progreffion m, m-\- 1, w-t-2, will give the Terms ay — x*. Alfo the Terms — nx^yf1 -+- n-\-iay, by expunging the points, and dividing by n, n-\- i, will give the Terms — x1 -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 ay — x1 = o. And when the given Fluxional Equation is a gene- ral one, and adapted to all the forms of the Fluential Equation, as is the cafe of the two laft Examples ; then all the Terms ariling from the fecond Operation will be always redundant, fo that it will be fufficient to make only one Operation.

Thus if the given Equation were ^.yy1 -f- z3yy~J -f- 2yxx — 3:32* H- 6}'z.z — 2cyz = o, in which there are found three flowing quan- tities j the only Term in which x is found is 2yxx, in which ex- punging the point, and then dividing by the Index 2, it will be- come^*1. Then the Terms in which y is found are 4^*4- z*yy~~l t which expunging the points become ^ # * 4-s3, and dividing

by,

280 72k Method of FLUXIONS,

by the Progreffion 2, i, o, — i, give the Terms aj5 — s;. Laftly the Terms in which z is found are — yzz* -J- 6yzz — zcyz, which expunging the points become — 32;"' -f- 6yz* — 29-2, and dividing by the Progreffion 3, 2, i, give the Terms — & •+• T.yz1 — zcyz. Now if we collect thefe Terms, and omit the redundant Term — z*, we mall have yxz -+- 2y> — z"' -f- yz1 — 2cyz = o for the Equa- tion required.

3, 4. But thefe deductions are not 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- pofitive Operation. So if the Fluxional Equation xx — <xy — xy-\- ny'= o were given, to return to the Equation involving the Fluents ; by the foregoing Rule we fliall firft have the Terms xx — xy, which by expunging the points will become x* — .vy, and dividing by the Progreffion 2, i, will give the Terms ^x1 — xy. Alfo the Terms, or rather Term, — xy -+- ay, by expunging the points will become — xy. -+- ay, which are only to be divided by Unity. So that leaving out the redundant Term — xy, we fhall have the Fluential Equation ±xl — xy -+- ay •== o. Now if we take the Fluxions of this Equation, we iliall find by the foregoing Problem xx — xy — xy -+- ay =o, which being the fame as the Equation given, we are to conclude our work is •true. But if either of the Fluxional Equations xx — xy -f- ay =o, or xx — xy -f- ay = o had been propofed, tho' by purfuing the foregoing method we fhould arrive at the Equation ±x* — xy-\-ay = o, for the relation of the Fluents ; yet as this conclulion would not fland the teft of this proof, we muft reject it as erroneous, and have recourfe to the following general Method ; which will give the value of y in either of thofe Equations by an infinite Series, and therefore for ufe and practice will be the moil commodious So- lution.

5. As Velocities can be compared only with Velocities, and all other quantities with others of the fame Species only ; therefore in every Term of an Equation, the Fluxions muft always afcend to the lame number of Dimenfions, that the homogeneity may not be de- ftroy'd. Whenever it happens otherwife, 'tis becaufe fome Fluxion; taken for Unity, is there underftood, and therefore muft be fupply'd when occafion requires. The Equation xz -+- xyx — az'-x* = o, by making z=i, may become -x -f- xyx — ax*==o> and like wile vice versa. And as this Equation virtually involves three variable

quantities,

and INFINITE SERIES. 281

quantities, it will require another Equation, either Fluential or Fluxionai, for a compleat determination, as has been already ob- ferved. So as the Equation yx = xyy, by putting x = i becomes yx=yy; in like manner this Equation requires and fuppofes the other.

6, 7, 8, 9, 10, II. Here we are taught fome ufeful Reductions, in 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 be extracted ; and the Confequent, for the greater fimplicity, may be made Unity. Thus the Equation zx •+- 2xx — yx — y = o is

reduced to this, y- = 2 -+• 2X — y, or making x=i, 'tis y = 2 _^_ 2x — y. So the Equation ya — yx — xa -f- xx — xy = o, ma- king x= i, will become y = (a~^+y = i -f- jdb = ) i + £ _!_ 2L _j_ f!? _|_ ^ , &c. by Divilion. But we may apply the par- ticular Solution to this Example, by which we mail have {x1 — xy __ ## _{_ tfy = o, and thence y •=."- ^~- . Thus the Equation yjr = xy-{-xxxx, making x=i, becomes yy =y -+- xx, and ex- tracting the fquare-root, 'tis y = -i ± \/± -j- xx = ~ ± the Series 2.-4-X1 — x*-{-2X6 — 5X8-f- I4x'°, &c. that is, either y = i -j- x* — x4-{-2x6 — jx8 -f- I4*10, &c. or y = — x1 -f- x4 — zx6 _j_ rx8 — I4-.V10, &c. Again, the Equation y> -+-axx*iy-{-a1x1y — X3x3 — 2x'tf? =o, putting x= i, becomes _y3 -\-axy -\-ay — x» — 2<73 - — • o. Now an affected Cubic Equation of this form has been refolved before, (pag. 1 2.) by which we mail have y = a — ^x •+•

xx iji*? ^°9^4 c,

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 he fuppofes to flow equably, which is given, or 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 very properly denote Time ; and its Velocity or Fluxion, being an 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,

ral are concern'd,) is that which is fuppos'd to flow inequably, with;

any degrees of acceleration or retardation ; and ts inequability may

be meafured, or reduced as it were to equability, by conihntly com-

paring it with its correfponding Correlate or equable Quantity. This

therefore is the Quantity to be found by the Proble'm, or whofe

Root is to be extracted from the given Equation. And it may be

conceived as a Space defcribed by the inequable Velocity of a Body

or Point in motion, while the equable Quantity, or the Correlate,

reprefents or meaiures the time of defcription. This may be illu-

ftrated by our common Mathematical Tables, of Logarithms, Sines,

Tangents, Secants, &c. In the Table of Logarithms, for inflance,

the Numbers are the Correlate Quantity, as proceeding equably, or

by equal differences, while their Logarithms, as a Relate Quantity,

proceed inequably and by unequal differences. And this refemblance

would more nearly obtain, if wre mould fuppofe infinite other Num-

bers and their Logarithms to be interpolated, (if that infinite Num-

ber be every where the fame,) fo as that in a manner they may be-

come continuous. So the Arches or Angles may be confider'd as

the Correlate Quantity, becaule they proceed by equal differences,

while the Sines, Tangents, Secants, &c. are as fo many Relate Quan-

tities, whofe rate of increafe is 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 their Fluxions, tho' it be not of abfolute neceflity for the Solution, may yet ferve to make it more expedite and methodical, and may fupply us with convenient places to reft at.

SECT. II . Solution of the Jirft Cafe of Equations.

18, 19, 20, 21, 22, 23. r~|~^HE firft Cafe of Equations is, wherr

-i. the Quantity ? , or what fupplies

its place, can always te found in Terms compofed of the Powers of x, and known Quantities or Numbers.. Thefc Terms are to be multiply'd by x, and to be divided by the Index of .v in each Term,, which will then exhibit the Value of jr. Thus in the Lquationj/a = .xi/

-+- xlx*, it has been found that ~ = i -t-x* — x* -f- 2xs — $x* -f-

I4*10, &cc. Therefore • - =^-4- x* — x'-f- 2,v7 — 5^' -t- 14*'*,

&c. and confequently y = x + jX* — -fx1 -f- ^x"1 — J-x9 -j-l^A"3, &c. as may ealily be proved by the direct Method.

But

and INFINITE SERIES. 283

But this, and the like Equations, may be refolved more 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 will bej)* =j/-l-.v*, which is thus refolved :

H

— y4J

= X* -+- X* 2XS -f- pC9, &C.

y*- $ .V4 -f- 2X& 5X9, &C.

Make — AT* the firft Term of y ; then will — x4 be the firft Term of — j/1, which is to be put with a contrary Sign for the fecond Term of y. Then by fquaring, -f- 2X6 will be the fecond Term of — j/», and — 2x* will be the third Term of y. Therefore

— 5#8 will be the third Term of — j/», and -f- 5*" will be the fourth Term of y ; and fo on. Therefore taking the Fluents, y =

— I..V5 -+- -fx* — ix7-f-4-x», &c. which will be one Root of the Equation. And if we fubtradt this from x, we (hall have y = x -+- ±x3 — ^.v* -f- -i-A;7 — AX', &e. for the other Root.

So if -v = a — 4-r -4- r — h -^— > &c- that is, if ^ = ax —

f 04^ 5 I 2«* A'

c , # I?IA.'4 .

6? li ' &c" then v=^->^4- — + ^, &c.

•* A jj-T ^

-^yii: yx i ya &c then Y j. v — s.

or

ex?

==fc*. If 4 = -, = , or ^--**.

* ex? x c

then _y=^f.

Laftly, if '-v = ~, or ' - = ^ = ^v° ; dividing by the In- dex o, it will be y = a- , or y is infinite. That this Expreffion, or value of y, mufl be infinite, is very plain. For as o is a vanim- ing quantity, or lefs than any affignable quantity, its Reciprocal -

or • muft be bigger than any affignable quantity, that is, in- finite.

O o 2

284 The Method of FLUXIONS,

Now that this quantity ought to be infinite, may be thus proved. In the Equation 4 = -x , let AB reprefent the conftant quantity a, and in CE let a point move equably from C towards E, and de- fcribe the Line CDE, of Avhich let any indefinite part CD be x, and its equable Velocity in D, (and every where elfe,) is reprefented

A o, E

c	c	— t — F 1	T> -p.
f		1 —--• — — -— - — — J. &

by x. Alfo let a point move from a diftant point c along the Line cde, with an inequable Velocity, and let the Line defcribed in the fame time, or the indefinite part of it cd, be call'd yy and let the

Velocity in d be call'd y. The Equation 4- = - muft always ob- tain, whatever the contemporaneous values of x and^ may be; or in the whole Motion the conftant Line AB (a) muft be to the variable Line CD (x), as the Velocity in d (y) is to the Velocity in D (x). But at the beginning of the Motion, or when CD (x) was indefi- nitely little, as the ratio of AB to CD was then greater than any

aflignable ratio, fo alfo was the ratio 4 of the Velocities, or the

Velocity y was infinitely greater than the Velocity x. But an infi- nite Velocity muft defcribe an infinite Space in a finite time, or the point c is at an infinite diftance from the point d, that is, y is an infinite quantity.

24, 25. But to avoid fuch infinite ExpreiTions, from whence we can conclude nothing ; we are at liberty to change the initial points of the Fluents, by which their Rate of flowing, (the only thing to be here regarded,) will not at all be affected. Thus in the foregoing Figure, we fuppofed the points D and d to be fuch, as limited the contemporaneous Fluents, or in which the two defcribing points were found at the fame time. Let F and f be any other two fuch points, and then the finite Line CF = b will be contemporaneous to, or will correspond with, the infinite Line cf=c ; and FD, which may be made the new .v, will correfpond to fdt which wiH

be the new y. So that in the given Equation - === - , inftead of

and INFINITE SERIES, 285

x we may write b •+- x, and we fhall have ~ = —£-- , and then by

•» r i • «• • i -r-v • r • • vx f ax \ ax "X1

Multiplication and Divifion it is -4- = ( •:—. — = J -r — — -f.

x V*-}-* / b tl

~ — -77 , &c. and therefore }'= ^- — "- \ -f- ~ — ~, 6cc.

2.6. So if ~ = - -J- 3 — ' xx, becaufe of the Term -' , which would give an infinite value for ^, we may write j -f- x inftead of X, and we fhall then have - = — ~ -4-2 — zx — xx, or y— =

X 1 | X X

-^ 1- 2X — 2X1 — x"', or by Divifion y-x- = 4x — 4x* -f- xj —

-f- zxs, &c. and therefore y=.^x — 2x* -+- ^.x3 — |x4 -f-

^xr, &c.

Or the Equation y~ = -^^ •+- z — zx — x1, that is y -f- xy = 4 — jx1 — xj, may be thus refolved :

y^ = 4 * — 3*11 — A:J

^" — 4X -J- 4X1 AT3 -j- 2X4, &C,

H- xyj h 4^ — 4x* 4- x3 — 2x4, 6cc.

y = 4 — 4.x -f- AT* — 2x3 -f- 2x4, &c.

T = 4.V— 2X"' -{- ^X3 _ iX4 _|_ £x*t &c.

Make 4 the firft Term of j, then 4x will be the firfl Term of xy, and confequently — 4* will be the fecond Term of j. Then — 4xa will be the fecond Term of .vy, and therefore -(- 4x* — 3xfc, or x*, will be the third Term of_y ; and fo on.

27. So if -. = AT~^ -f- x~J — x'~, becaufe of the Term x~' change x into i — x, then — == — -.'— -+- — s/ i — x. But

X y»i ,v I X

by the foregoing; Methods of Reduction 'tis — — = i -f- x -+- x*

* I — X

-+- x5, 6cc. and v/i — x = I — 4-^ — r-^4 — -rrx^ &c. a"d

Therefore collecting thefe according to their Signs, 'tis 4- — i 4-

2.v-|- ix1 -t- T^-x3, &c. that is-^ =x4-2x* -f- |x3 + ±^x4, &c. and therefore y = x 4- xa -f- ixs 4- ^x4, &c.

28. So if the given Equation were — == ~

X i. * ~^ ii^^C -!-• 3t"A* ~™ ~" X% "

. '^ - ; change the beginning of x. that is. inftead of x write

t A | '

x,

286 7$£ Method of FLUXIONS,

- y f3 — c*-X yx

c — x, then -- = Al = c">x~* — clx-1, or ^ = c*x~* — . c*x-1, and therefore _>' = — ^c=x-~ -\-c*x~l.

SECT. III. Solution of the fecond Cafe of Equations.

29> 3°- TT^Quations belonging to this fecond cafe are thofe, M^ wherein the two Fluents and their Fluxions, fuppofe x and y, x and j, or any Powers of them, are promifcuoufly in- volved. As our Author's Analyfes are very intelligible, and fee'm to want but little explication, I mall endeavour to refolve his Examples in fomething an eafier and fimpler manner, than is done here ; by applying to them his own artifice of the Parallelogram, when need- ful, or the properties of a combined Arithmetical Progreffion in piano, as explain'd before : As alfo the Methods before made ufe of, in the Solution of afTeclcd Equations.

31. The Equation yax — xxy — aax = o by a due Reduction

.becomes ~ = ~ -+- "- , in which, becaufe of the Term -• there is occafion for a Tranfmutation, or to change the beginning of the Correlate Quantity x. ArTurning therefore the conftant quantity b,

we may put 4- = ^ -f- -^— , whence by Divifion will be had

y v a ax axz ax* e i • i -.-, . ,

-j == -^ -I- y — £ -+- -ji 77 > &c' which Equation is then

prepared for the Author's Method of Solution.

But without this previous Reduction to 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- tion is now 4 = j- -|- -£— , or putting x = i, it is aby -f- axj — /y ~+- yx -f- a1, which may be thus refolved :

aby =

— xy

\	ab — a* 2/1* /7* .. V 1 ,	4. fc — at> ^ /34	-in*b — ab1 — 6,3	c__
	b • - 4- "*x -f-	ab — aa	babl v J	OCC. £^x.
>	b	a — I ,	2.U 3 J /?.6 — ^ ifi — h^	OCC. c.
	(IX -(—	ib * + " v»-J-	i}ab~ f a — I,	OtC.
)	a b — a z«* J-	b A "^ "*-"*•.., , I' +2.	•,V, — «^J — 6^3	OCC.
	b 4- h; ^ -f- 2, a b — a -iri' -L-	•i "•"		OCC. rt Arr«

2 Difpofing

and INFINITE SERIES. 287

Difpoiing the Terms as you fee is done here, make a1- the firft Term of aby, then ~ will be the firft Term of j, and thence -|x will be the firft Term of y. So that a—x will be the firft Term of

* b

axv, and — ax will be the firfl Term of — by. Thefe two to- •/ * i i /

gether, or -,x — ax = - —x, with a contrary Sign, mud be put down for the fecond 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-

fecond Term of .«.\y will be '-'-~a A*, and the fecond Term of

b"

a—b

— ly will be -^~x*, and the firft Term of — xy will be — yA*. Thefe three together make - ~ 2* _ ~ — A*, which with a contrary Sign muft be made the third Term of aby. Therefore the third Term of y will be — ~-^-'A* and the third Term of y will be

•r ZaL ' •*

t , / * _^_ 7

- A'3. And fo on. Here in' a particular cafe if we make.

b =. a, we mall have the fimple Series y •=. x * -+- ^ — —7 , &c.

Or if we would have a defcending Series for the Root y of this Equation, we may proceed as follows :

xy~\ =<!*• — a ~f- b x a**-1 -+- zu\ •+- zab -+- i x a*x-*, &c. [,y\ --- _f_ a*bx~l — a-i-l) y.a!-bx~-, &c,

~' -~t, &c.

'-, &c. 3 6cc.

JJ/=: rtaA~; - ^ -4-/>X 2rt*A-~3, &C.

Difpofe the Terms as you fee, and make a* the firft Term of the Series — xy, then will — - be the firft Term of y, and a*x~"- will

be the firft Term of y. Then will -f- a"-bx~l be the firft Term of — by, and a"'X~I will be the firft Term of axyy which together make a-\- b x alx~t ; this therefore with a contrary Sign muft be the fecond Term of — xy. Then the fecond Term of y will be a-\-by.a3ix~~) and the fecond Term ofj/ will be — a-\- l>-x.2a1x~3,

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 firft Term of aby will be a^bx~- ; which three together make — ^za1 -I- zab -+- b* xa1*.—*. ThiswithacontrarvSisnmuftbethethird

Term of — xy, which will give — 2a* -+- zab -+- b% x a2x~3 for the third Term of y ; and fo on. Here if we make b=.a, thenj=

a1 za* ca4 .

-- •+- — r — ^T 3 &C.

x *' x* '

And thefe are all the Series, by which the value of y can be ex- hibited 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 as Axm may reprefent the initial Term of the Root jr, in both thefe kinds of Equations, or becaufe it may be y = Axm, &c. fo in Fluxional Equations (making #=1, we mall have a\foy=mAxm~I) 6cc. or writing y for Axm, 6cc. 'tis y = myx~*t, &c. So that in 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 Quantity, fuppofe y ; according to which Reduction 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.

Thus in the prefent Equation aby -+- axy = by -f- yx •+- aa, the Terms mufl be inferted in the Parallelogram, as if yx~ ' were fub- ftituted inftead of y ; fo that the Indices will ftand as in the Margin, and the Ruler will give only two Cafes of exter- nal 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 Terms which have y of one Dimenfion, or what _ y i_ is equivalent to it. In the fecond Column is — a1, +axi ; J 2 C~ or y of no Dimenfions. Alfo in the firft Line is . — xy, or fuch Terms in which x is of one Dimenfion. In the,

fecond Line are the Terms — by~l , . ,

<f — a1, which have no Dimen-

{iqiis of .v, becaufe -j- axy is regarded as if it were ay. Laftly, in the third line is abyt or the Term in which x is of one negative

Dimenlion

— p-

2.V + Xj~

and INFINITE SERIES. 289

Dimension, becaufe -\-aly is confider'd as if it were -f- abx~~J)\ 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 oropofed Equation be — = -TV — 2 x -4- — — — or

O jy y xx >

making x= i, 'tis — y -f- 3_v — 2.v -f- xy~l — 2yx~1 = o ; the Solution of which we mall attempt without any preparation, or without any new interpretation of the Quantities. Firft, the Terms are to be difpos'd according to a double Arithmetical Scale, the Roots of which are y and .Y, and then they will Itand as in the Margin. The Method of doing this with certainty in all cafes is as follows. I obferve in the Equation there are three powers of 1 y, which are y1, y°, and 7-' ; there- * fore I place thefe in order at the top of the Table. I obferve likewife that there are four Powers of x, which are .v1, x°, A—I, and x~l, which I place in order in a Column at the right hand ; or it will be enough to conceive this to be done. Then I infert every Term of the Equation in its proper place, ac- cording to its Dimenfions of y and x in that Term ; filling up the vacancies with Aflerifms, to denote the abfence of the Terms be- longing to them. The Term — y I infert as if it were — _}'*""', as is explain'd before. Then we may perceive, that if we apply the Rukr to the exterior Terms, we mail have three cafes that may pro- duce Series ; for the fourth cafe, which is that of direft afcent or defcent, is always to be omitted, as never affording any Series. To begin with the defcending Series, which will arife from the two external Terms — 2x and -f- xy~s. The Terms are to bsdifpos'd, and the Analyfis to be performed, as here follows :

- 2JX-

-J44*-*, &c.

Make xy~l = 2X, 6cc. then y-1 = 2, &c. and by Divifion T=4, &c. Therefore 3>'=T, &c. and confequently A->— '=« " — 4, &c. or y-1 = # — -I*"1, &c. and by Divifion y = * -f- -f.*-1, &c. Therefore 2)' = *^x~1y &c- and confequently x\~l r—^ % * — T*"1) &c. So that y~l = * * — T*"~% ^c' anc^ ^7 ^'v^" fion y = * * -f- -rvfx~'^c . Then 3v = * # -j- 4rA~% ^c- anc^

P p —y

Method of FLUXIONS,

y == * -f- i*-"-, &c. and — zyx~* = — A— % 6cc. Thefe three- together make 4- r^x-i, and therefore xy~l = * * * — 44*"% &c. fo that y — * * * -f- V|T*~~J» &c- A"d fo on.

Another defcending Series will arife from the two external Terms -4- -y and — 2X, which may be thus extracted :

zx — f -f- 41*-' — i|*-3, &c. ' _ ^4x-*, &c.

+^X-», &C.

i-x-% &c.

Make 3/ = 2X, &c. then y = ^x, &c. and (by Divifion) y— * = ±x~*, &c. and x>'~1=|,&c. and — y=- — T> &c- There- fore 3_y = * — £, &c. and _y = * — TST) &c. and (by Divifion) xy~* = * -g-*"1, &c. and — _y= * o, &c. and — zyx~* = — •~x~\ &c. Therefore 3_y=* * 4- i-j..*-1, &c. and jy = * * 4- ^.i^— J, 6cc. &c.

The afcending Series in this Equation will arife from the two ex- ternal Terms — 2yx~* and xy~l ; or multiplying the whole Equa- tion by — y, (that one of the external Terms may be clear'd from y,) we mall have yy — 3^* 4- zxy — x 4- 2yix~1 = o, of which the Refolution is thus :

- v\ S v* -1 vl- _I_ 9 - v3 &r

M^~« t^V •# ^S~" ^^™* ^ . -• "T"*\ • LA/V *

•r »/ ^ *r '

Jl	2 4	6cc.
; :*	v 2	&c.
	^a J_ '35 A	&c.

y—

y __ 3 ^.I # 1^.^ 3^^,

Make aj1^-2 = ^, &c. then y* = 4-AT3, &c. and y = — x Here becaufe of the fractional Indices, and that the firft Term of 4- kxy, or 4 — —x%} may be afterwards admitted, we mufl take o

for the fecond Term of 2)-»A— % and therefore for the fecond Term

i of

and INFINITE SERIES. 291

of y. Then y'y = £**> &c. and confequently 2v*x~a = * * — A*1, 8cc. and y1 = * * — 4-v*, &c. and by extracting the fquare-roor,

Then yy = » -f- o, &c. and 2.vy = -4--V-'

&c. and therefore 2)'lx~t = « * *• — -^^S &c. and _>' = * * *

— |.v5, &c. &c.

33, 34. The Author's Procefs of Refolution, in this and the fol- lowing Examples, is very natural, fimple, and intelligible; it pro- ceeds 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 given Equation ; and then by the former 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 hap- pen'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.

The Equation of this Example i — 3^ -f- y •+• xl -+- xj — y = o being reduced to the form of a double Arithmetical Scale, will (land as here in the Margin ; and the v, v<)

Ruler will difcover two cafes to be try'd, of ~ which one may give us an afcending, and the xI0 other a defcending Series for the Root y. And »— • firft for the afcending Series.

The Terms being difpofed as you fee, makej/=i, &c. then y=x, &c. Therefore — y = — x, &c. the Sign of which Term being changed, it will bej/= * -{- x — 3 AT, &c. = * — 2.v, &c.

P p 2 and

292 77->e Method of FLUXIONS,

and therefore y = * — xx, &c. Then — y = * -+- #% &c. and — .vy = — *% &c. thefe deftroying each other, 'tis y = * * -+•*•*, &c. and therefore _y = * *-t-7.*3, &c. Then — _y=** — ^x*, &c. and — xy = * -f- x', &c. it will be j- = * * * — .I*5, &c. und therefore y = * * * — ^x*y &c. &c. The Analyfis in the fecond cafe will be thus :

h x — 4 ~t-

V =

AT

4

* -+• I2X~3, &C.

6*-1 * , &c.

f~2 1 2X~1, &C.

6*-"1 •+- 6.V~Z * I2AT~*, 6CC.

Make — xy = xl, &c. then ;' = — x, &c. Therefore — _y = x, &c. and changing the Sign, 'tis — xy-=. % — x — 3*, &c. = * — 4*, &c. and therefore y = * -h 4, &c. Then — jy= * — 4, &c. andj = — i, &c. and changing the Signs, 'tis — xy = * * H- 5 -f- i, 6cc. = * # -l- 6, &c. and y = * * — 6x~*, &cc. &c.

35, 36. If the given Equation were ^==:i-f-^-f.^._f_^-y H — ^ , &c. its Refolution may be thus perform'd :

zx'i

XV

a*

a X

A*

A1*

• £ » &c-

• — 4 > &c-

— , &c. £1, &c. - , &c.

A*

A4

f *" * 2^2 *^ ^^4 "T~ i/)3 "I" o>,4

Make y &c. and y

i, &.c. then y= x, &c. Therefore — 2 — • — f »

o a

* + ;, &c. and therefore _y =*•+-—, &c. Then

therefore

y =

* *

~i = — J7 j , &c. and j = * * + , &c. And fo on.

Now

and INFINITE SERIES. 293

Now in this Example, becaufe the Series | -+- ^ 4- ^ -+. — . &c. is equal to — =?— it will be y= — h I, or ay — . xy

«4 ' a — x ' / « — A; ' J

•=jy + tf — *", that is, jx -f- rfx — xx — ay -+- A j = o ; which Equation, by the particular Solution before deliver'd, will give the relation of the Fluents yx — ay -{-ax — I** = o. Hence y = a* — -_xx an(j , Divifion y= x -f- * — h — -, -f- — r , &c. as found

a — x • J za za~ 2a* '

above.

37. The Equation of this Example being tabulated, or reduced to a double Arithmeti- cal Scale, will ftand as here in the Margin. Where it may be ob- ferved, that becaufe of the Series proceeding both ways ad injinitum, there can be but one cafe of exterior Terms, of which the Solution here follows:

x-'	» —	>	» 1
x°	(	*	~H j1 *4" j* *4* ^* > ^f-
X*	— 3*; 4-	3 xv	— X)!1 — jyJ — Xj4,&c.
X1		6#1j'	* * *
xt	— 8*3 +	8*5,	* * *	'= O.
Jt4	— 1 ox 4 -f-	0*4y	* * *
X*	I 2X* -j-	z.'. '_y	» • »
&f.	— 14*« <	5^f.

= — , A; — 6xl

— iox+ — i2xs — i4.xs, &c.

-f- |

7 3

-

-f-

cs, &c.

— Y6 &C

•\ j CW .

— ±X4 — 6x-' — -4-1*6, &c.

s, &c.

X6, &C. X6, &C.

- 4- ¥*6> &c.

v . — — ±£i 2%l ,— X4— ^'Y' —— — Y15! . 3&7,V7 &C.

y*

&c.

Make y = — 3^-, &c. then y = — 4xa, &c. Then y = * — 6x*, &c. and_y == # — 2A'3, &c. Then — 3^ = -f- |x3, &c. and therefore j= * * — ±x3 — 8^3, &c. = * * — "V x3, 6cc. and _y = * * — VAT*, &c. And fo of the reft.

The Author here takes notice, that as the value of y is negative, and therefore contrary to that of x, it fhews that as x increaies, / muft decreafe, and on the contrary. For a negative Velocity is a Velocity backwarks, or whole direction is contrary to that which

was

294 Th* Method of FLUXIONS,

was fuppos'J to be an affirmative Velocity. This Remark mull take place hereafter, as often as there is occafion for it.

38. In this Example the Author puts x to reprefent the Relate Quantity, or the Root to be extracted, and y to reprefent the Cor- relate. Bat to prevent the confufion of Ideas, we mall here change .v into y, and / into A", fo that y (hall denote the Relate, and x the Correlate Quantity, as ufual. Let the given Equation there-fare be

- = ~x — 4** -\- 2xy'i — -f^1 -h 7#* -+- zx'} whofe Root y is to

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.

X1 X1

a *x

X1

At

. 1

* * * « + **5 J I _j_ ^i — 2J34-4*1

* » * '* _|-7A;i: If -t Z-

J = I*'1 * - A'H-ZX • + I*

** -

« * * * »

4..1

— x/ # * » *

* » * * *

* *

— y

j/=|A;, &c. then^ = -l-xa, &c. Now becaufe it is jx=s * o, &c. it will be alfo y= * o, &c. And whereas it is^ = |-x, &c. it will be — zxy^ = — x*, &c. and therefore y = * # -f- x*

— 4**, &c. = * » — 3**, &c. then _y= * * — >r3, &c. Now be- caufe it is y = * -f- o, &c. it will be alfo y^ == * -\- o, &c. and

— 2Ay5 = * -f- o, &c. and confequently y E= ***-{- 7^^, &c. and therefore y = ***-{_ 2**, &c. And fo on.

There are two other cafes of external Terms, which will fupply us with two other Series for the Root y, but they will run too much into Surds. This may be fufficient to (hew the univerfality of the Method, and how we are to proceed in like cafes.

39. The Author mews here, that the fame Fluxional Equation may often afford a great variety of Series for the Root, according as we fhall introduce any conftant quantity at pleafure. Thus the Equation of Art. .34. or j/=i — 3* -\-y -j- #• -f- xy, may be re- folved after the following general manner:

and INFINITE SERIES.

295

r^3*+ ** y = «+ * — *l

a 4. x -fza*1 + i<?*', ££<:. + ax+ax1

Ji — ax — ax1 — ^ — ax— x*— ,

axt, isV.

Here inftead of making ji/ = i, 6cc. we may make _y=o, &c. and therefore y = a, &c. becaufe then y = o, &c. then — y •— • — a, 6cc. and confequently y = * -f- <z -+- i, &c. and therefore^1 =; * -+- ax -±- x, 6cc. Then — y = * — ax — x, &c. and — xy == _- ##, &c. and therefore y = * * -f- zax -f- x — 3*, &c. = * * -f- 2ax — zx, &c. and then y = * * -f- ax* — x1, &c. There- fore — y = * * — ax1 -f- x1, 6cc. and — xy= * — ax* — AT*, &C. and confequently y = * * * -f. tax* -f- x*, 6cc. and y = a, * * -f- .iflx5 -f- -i-*3, &c. &c. Here if we make a = o, we fhall have the fame value of y as was extracted before. And by what- ever Number a is interpreted, fo many different Series we fhall obtain for y.

40. The Author here enumerates three cafes, when an arbitrary Number mould be affumed, if it can be done, for the firft Term of the Root. Firft, when in the given Equation the Root is affected with a Fractional Dimenfion, or when fome Root of it is to be ex- tracted ; for then it is convenient to have Unity for the firft Term, or fome other Number whofe Root may be extracted without aSurd, if fuch Number does not offer itfelf of its own accord. As in the fourth Example >tisA' = i}'1, &c. and therefore we may eafily have

x^ = -i->'> &c> Secondly, it muft be done, when by reafon of the fquare-root of a negative Quantity, we fhould otherwife fall upon impoflible Numbers. Laftly, we muft aflame fuch a Number, when otherwife there would be no initial Quantity, from whence to begin the computation of the Root ; that is, when the Relate 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- ples by the. foregoing general Method. As if the given Equation

werej=:- — x1, or y — /-' =— x4, the Refolu-tion might be thus :

y

296 The Method of FLUXIONS,

'y T = O * * < — . X* l<3-7x3, &c.

— f1 f — ' a.~I -f- a~*x — \a~$x*- -\- ±a~7x*, &c.

J — • 4-^~*A";

/7 _1_ * ** _I_ 'IL 1^ AT

« -t- - — j -4- ,77 ga, , see.

Make _y = o, &c. then afluming any conftant quantity a, it may be y= a, &c. 'Then by Divifion — y~l = > — a~l, &c. and therefore _y = * -f- a*1, &c. and confequently _y = * -4- a~lx, 6cc. Then by Divifion — y~l = * -{- «-3x, &c. and therefore y =

* * — a~ix, &c. and confequently _y = * * — 4«~3^S &c- Then again by Divifion — y-'1 = » » — 4^— s'x1, &c. and therefore y =

* * *H-|d~5A;1 — .vv&c.and confequently/ = * * * ±a-*x* — ^x', &c. And fo of the reft. Here if we make a = i, we ihall have y = i -f- x — IK* -f- £.x* — |-|-Ar4, &c.

Or the fame Equation may be thus refolved :

- y~ ' J == - A"1 -f- 2 AT" 3 -f- I4AT-8 -+- 2 l6x-J3, &C.

— 8'— 2i6x'~~I3, &c.

= AT"2 -f- 2AT~7 + l8x-JZ + 28ox-I7) &C,

Make — y~'= — A-S&c. or_y=^~z, &c. Thenj/= — 2Ar~3, 2fc.and therefore — y—l=z* -\-2x~3, &cc. and confequently by Divifion r=* -f-2 .v~7, &c. Then j/=* — i4x~8,&c. and therefore — y-1 = * *4-i4.v~8, &c. and by Divifion _)'= * *+i8^^12, &c. Then y = * * — 21 6jf— J33 &c. and therefore — y~l = * * * -f- 2 l6x~l^} &c. and by Divifion y = * * * + 28o^~17, &c. And fo on.

Another afcending Series may be had from this Equation, viz.

y=^/2x — \ X' -f- ** -f- ^— , &c. by multipying it by y, and

then making i the firft Term of yj.

44. The Equation y = 3 -+- 2y — x~Jy- may be thus refolved :

-4- ojc-1, &c.	r	y y°
-|- 3-x11, &c. ^1° — 9xa, &c. x~l l-{- ,?x3, 6cc.	*	+27+3 ?:=:0> 'a— • JV * J

Make

IN FINITE SERIES. 297

Ma ke y = 3 ,Scc. then y = ^x, 6cc. Therefore — zy = — 6x, &c. and x~Iyi = c)x, &c. and confequently_>' = * — 3*, 6cc. Therefore y •== * — IA*, &c. Then — a_>' = * -f- 3*1, 6cc. and x~Iyl=== * — 9**, 6cc. Therefore^ = * * -f- 6#4, &c. and / = * * -f- 2AT3, &c. &c.

Or the Refolution may be perform'd after thefe two following manners :

— zy 1=3— £*-' -f- IA— *,&c. ;'*-'\=

*

— _•? v~~l 1 — -* v~~ '—4— ' v~"~2 &"f*

_j _/ ^ ^^^r.-v 4 "F^ j ••A'^* •

T = 2A--j-i 3A-~' &C.

Make — zy-=. 3, &c. or_y =r — ?, 6cc. then j= o, &c. and x-iy-- =-j- %x-1, &c. Therefore — 2_y = * — %x~l, &c. or / = * -f- T*"",1} &c. and y = * — %x~z> &c- ancl by fquaring x~'j* = * — I1-'''"2, &c. and therefore — 2_>'=r* * -f- ^A*~2, 6cc. and y = * * — I-*"2} &c. And fo on.

Again, divide the whole Equation by y, and make x—Jy = 2, &c. thenj' = 2A;, &c. And becaufe j/=2, &c. and j—'^ni^1"1, &c. 'tis^"1 = Aj1""1, &c. and — 3>'~~1 = — I-*1""1} ^c- therefore yx~* = * H- T-^"1) ^C- an(l y = * H- T> &c- Then becaufe j^y"1 == * + o, &c. and — y I = * -+- -I-*"2, &c- 'tis jx~l = * * — T"v~4> &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 _!_*-

/;	)=A— '«— A'"1		y' y°
^ (	> -f-A" *	x°	— y * )
+^:	!- ' X~z	AT"1	— j -f-A-~'>
	*>	A—1	* — x~'j

Makejj/= — x~z, &c. then y =r A— ', &c. Confequently }'=*. * o, &c. and therefore _y= * o, &c. that is, y = x~I.

Again, make y^ix"1, &c. then y =. — A-~% 6cc. and confe- quently _>' = * + o, &c. that is, y = A—'.

That this fhould be fo, may appear by the direcl Method. For ify = x~l, 'tisj/ = — A-A-~2 ; a\foyx= .\x~'. Then adding tlieie two Equations together, 'tis yx -\-y =xx~ ' — xx~*, orj = — y

x~l — x~*. Thus may we form as many Fluxional Equations

as

298 The Method of FLUXIONS,

as we pleafe, of which the Fluents may be exprefs'd in finite Terms; but to return to thefe again may ibmetimes require particular Expe- dients. Thus if we aflume the Equation y = 2x — • ±x* -f- ^}, taking the Fluxions, and putting x= i, we {hall have jr = 2— •

|x -f- -for1, as alfo ~ = i — ^x -+- ~x*. Subtract this laft from the foregoing Equation, and we fhall have j/ — — = i — 2#-f-l*t,

ZX

the Solution of which here follows.

47. Let the propos'd Equation be y=— -f- i — , 2#-{- !#*, of which the Solution may be thus :

— \ — e —fx —gx* __ — ex'1 —fx

'

= o.

By tabulating the Terms of this Equation, as ufual, it may be obferved, that one of the external Terms — y -+- ^yx~l is a double Term, to which the other external Term i belongs in common.

Therefore to feparate thefe, afllime y = zex, &c. then — . i-

= — e, &c. and confequently y = i -f- e , &c. and therefore y ==x -f- ex, &c. That is, becaule 2ex = x -f- ex, or ze=. i -j-e, 'tis ez= i, or _y= 2x, &c. So if we make _y=* -f- 2/xz, «5cc.

then — i- = * — y^j &c. therefore y = * ~\~fx — 2x, &c. and

v = * -+- l/x1 — A-*, &c. that is, 2/= |/ — i, or /= — -i. So that _y == * — -f-A'% &c. So if we makejv' = * * •+- zgx*, &c. then

— — s== * * — gx1, &c. and therefore jj/ = * * -+-gx* -4- ix1, <Scc. and/ = * * -f- |£*-> -f- ^s &c. or 2g = -jg+±, or ^ -_ _«_.> fo that j =# * ^-.v3, &c. So if we make / = * *. * 2/^x4, &c. then

-=*#*-— /w3, &c. and therefore _y= * * * -f- /JA:3, &c.

and /= * * * -f- ^r^4, &c. Butbecaufe here 2^=^^, this Equa- tion would be ubfurd except £ = o. And fo all the fubfequent Terms will vanifh in infinitum, and this 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 we would demonftrate it fynthctically. Becaufe^ =ex*} as is here

found,

and INFINITE SERIES. 299

found, therefore y = ±e: v+~', or y = Iff! . Here in (lead of ex'(

fubftitute y, and we fhall have_y = ~x , as given at firft.

49, 50. The given Equation y =yx~- 4- x~- -f- 3 4- 2.V — 4*-' may be thus reiblved after a general manner.

y n = 2x 4- 3 — 4A-1 4- x-2 — .v-* 4- i-*-* , &c. / -f- i -f- 4-v— ' 4-rf.v~I — rtx~~3 4- ±ax~*

-~x~z)\ ----- * — 4A"~' — a*~* •+• *~3 — T*~* , &c. J -\-ax~* — ±ax~*

y= xl -f- 4.v + a — #-' •+- fx~z — ±.x~* ,.&c. — ^A-"~' -j- ±ax~z — fax"3.

Make_y = 2*1, &c. then;1 = x1, 6cc. Therefore — x~zy = — i , ficc. conlequently y = * -f- i -+- 3, &c. = * 4, 6cc. and therefore j = * -+- 4.x, &c. Then — x~zy = * — 4-v~", &c. and confe- quently y = * * -f- o, 6cc. and therefore afluming any conftant quantity a, it may be y = * * -+- a, &c. Then — #-*_y = * * — ^A,-"1, &c. and therefore , j = * * * -+- ax~* •+- x~z, &c. and y=. * * * — fix'1 — x"1, &c. And fo on. Here if we make

a = o, 'tis ^ = *1 + 4* * — ; H- ^ — ^3 , &c.

51, 52. The Equation of this Example is y=. ^xy* -\-y, which we fliall refolve by our ufual Method, without any other prepara- tion than dividing the whole by j*, that one of the Terms may be clear'd from the Relate Quantity ; which will reduce it yy~^ — ^ c= 3«r, of which the Refolution may be thus :

3x -f- f X-- -f- T'T*3 -|- ^rx* + -~-xs, &c.

±X* — -V^3 — TTTT^4 — TT4<T*'> &C'

y = f x6 -+• T'^7 -4- TTY*"> &c-

Make jJ;y"~^ = 3#, .&c. or taking the Fluents, %y~' = |jc% 6co.

y^ = f x1, &c. or y = fAr6, &c. And becaufe — y$ = — fx*, &c. it will be jj)T~7 ==*_{- f^1, &c. and therefore ^ = ^ 4- ^.x5, &c. and y^ = * -f- ^xl, &c. and by cubing y= * 4- TVx7> &c. Then becaufe — y"' = * — rV^'S &c- 'tis ji/y""^ = * * -f- -Vv?> &c.and therefore 37' = * # -+- T'T*4, &c. and ;-j = * * + TTr-v'4> &c. and by cubing ;•= * * 4- TTT-v8) &c- And *° on-

Qjl 2 53-

or

-oo 7%e Method of FLUXIONS,

53. Laftly, in the Equation y = zy^ -+- x ty*, orjj/y— i== zx -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 y*=c -+- x -f- -i-x'. Then by fquaring, _>• = c1 -+- 2cx -f- A-* -f.

-icx* + •!•** -+• f^3- Here the Root _y may receive as many diffe- rent values, while x remains the fame, as c can be interpreted diffe- rent ways. Make c = o, then y = x1 -+- -ix* -+- loc*.

The Author is pleas'd here to make an Excufe for his being fo minute and particular, in dilcuffing matters which, as he fays, will but feldom come into practice ; but I think any Apology of this kind is needlefs, and we cannot be too minute, when the perfec- tion 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 may happen, and for obviating every difficulty that may arife. The ufe of thefe Extractions is certainly very exteniive ; for there are no Problems in the inverfe 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.

54. TT* O R the more methodical Solution 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- tity occur in the given Equation, he has difpatch'd without much difficulty, by the affiftance 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 re- duced 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

termination, as has been 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 Queftion. And the reft of the work will only require the two former Cafes, with fome common Algebraic Reductions, as we fhall fee in the Author's Example.

55. Now to confider the Author's Example, belonging to this third Cafe of finding Fluents from their Fluxions given, or when there are more than two variable Quantities, and their Fluxions, ei- ther exprefs'd or underftood in the given Equation. This Example is zx — z 4- yx = o, in which becaufe there are three Fluxions A-, y, and z, (and therefore virtually three Fluents x, y, and z,) and but one Equation given ; I may affume (for inftance) x=y, whence x =JK, and by fubftitution zy — z -\-yy = o, and therefore zy — & •+• T)'* = °« Now as here are only two Equations x — y== o and zy — z-\-^yl =o, the Quantities x, y, and z are ftill variable Quantities, and 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 only derived from the other two, it brings no new limi- tation with it, but leaves the quantities ftill flowing and indetermi- nate quantities. Thus if I mould affume zy=a-\-z for the fc- cond Equation, then zy=z, and by fubftitution zx — zjr-k-yx=;o,

or y = j^ - = x -f- .Ixv -f- •^x'-x, &c. and therefore y = x -+- ix1

H-TT#SJ &c- which two Equations are a compleat Determination. Again, if we affume with the Author x=js, and thence x=Z)yt we mall have by fubftitution <\.yy — z -^-yy1 = o, and thence zy1 — z -+- ^ = o, which two Equations are a fufficient Determina- tion. We may indeed have a third, zx — z -+- ^x^ = o ; but as this is included in the other two, and introduces no new limitation, the quantities will ftill remain fluent. And thus an infinite variety of fecond Equations may be aflumed, tho1 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 zx — z = o to be 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 Fluxions, and their flowing Quantities •, but it would be needlefs here to multiply Examples. And thus our Author has com- pleatly folved this Cafe alfo, which at firft view might appear for- midable

302 7%e Method of FLUXIONS,

midable enough, by reducing all its difficulties to the two former Cafes.

56, 57. The Author's way of demonstrating the Inverfe Method of Fluxions is Short, but fatisfactory enough. We have argued elfe- where, that from the Fluents given to find the Fluxions, is a direct and fynthetical Operation ; and on the contrary, from the Fluxions given to find the Fluents, is indirect and analytical. And in the order of nature Synthefis mould always precede Analyfis, or Com- pofidon mould go before Refolution. But the Terms Synthefis and Analyfis are often ufed in a vague fenfe, and taken only relatively, as in this place. For the direct Method of Fluxions being already demonftrated fynthetically, the Author declines (for the reafons he gives) to demonstrate the Inverfe Method fynthetically alfo, that is, primarily, and independently of the direct Method. He contents himfelf to prove it analytically, that is, by fuppofing the direct Me- thod, as fufficiently demonstrated already, and Shewing the neceSTary connexion between this and the inverSe Method. And this will al- ways be a full proof of the truth of the conclufions, as Multiplica- tion is a good proof of Division. Thus in the firlt Example we found, that if the given Equation is y -f- xy — y=^x — x1 — I, we Shall have the Root y=x — x1 -j- f x3 — -£*« -f- ^.x* — -^r*6, £cc. To prove the truth of which conclufion, we may hence find, by the direct Method, _y = i — 2x -{-x1 — .i*3 -f-f#* — TTX', &c. and then fubStitute theSe two Series in the given Equation, as follows;

y ------- f_ X _ jf» 4- ]_X1 _ . ±X< _J_ _>_X, __ _^X6

_{_ Xy --------- 1_ X- __ . #3 _j_ £3.4 __ ^f _{_ _?_X65

— y -- r -f. 2X — A-* -{- ^ — ^ + -rX' — ^X6

— x1

Now by collecting thefe Series, we mall find the refult to pro- duce the given Equation, and therefore the preceding Operation will be fufticiently proved.

58. In this and the fubfequent paragraphs, our Author comes to open and explain fome of the chief 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 Lines, AE, aet as, &c. indefinitely extended both ways, along which a Body, or 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 point in the fame Line ; and any fuch Points may be chofen for this purpole, in each Line, fuppoie B, I), /3, in 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 nothing when in it, and as affirmative when they are got beyond it. In the rlrft Line AE, whole Fluent we denominate by x, we may luppofe the Body to move uniformly, or with any equable Velocity ; then may the Fluent x, or the Line which is continually defcribed,

A B C J> :E

a,

, * /? 9" c/^ 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 any Equation compos'd of x and y and known quantities j) then 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 Correlate x, the correfponding or contemporaneous value of 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 by y may always be known. The Origin (as we may call it) of the Fluent x is mark'd by the point B, and the Origin of the Fluent y by the point b. If the Bodies at the fame time are found in A and «, then will the contemporaneous Fluents be — BA and — ba. If at the fame time, as was fuppofed, they are found in their refpec- tive Origins B and £, then will each Fluent be nothing. If at the 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 begin to compute them, (for tho' they muft be conceived to be variable and indetermined in refpedt of one of their Limits, where the de- fcribing points are at prefent, yet they are 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 neceffity that they mould begin together, or at the fame Moment of Time. All that is neceflary is this, that the Motions may continue as before, or that they may obferve the fame rate 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 may remove them to what other points we pleafe. If 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 we fliall have -+- AB, inftead of -+- BC we fliall have -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 is, in the Equation which determines the general Law of flowing or increafing, we may always increafe or diminifh x, or yy or both, by any given quantity, as occafion may require, and yet the Equa- tion that arifes will ftill exprefs the rate of flowing ; which is all that is neceffary here. Of the ufe and conveniency of which Reduction we have feen feveral in fiances before.

If there be a third Line a.e, defcribed in like manner, whofe Fluent may be z, having its parts correfponding with the others, as a/3, &y, y£, &c- there muft be another Equation, either given or 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 two Equations will limit and determine the Law of flowing in each Line. And we may likewife remove the Origin of the Fluent z to what point we pleafe of the Line a£. And fo if there were more Lines, or more Fluents.

59. To exemplify what has been faid by an eafy inftance. Thus inftead of the Equation y=xxy, we may aflume y = xy -+- xxy, where the Origin of x is changed, or x is diminifli'd by Unity ; for j -J-- x is fubftituted inftead of x, The lawfulnefs of which Re- duction

and INFINITE SERIES.

305

duftion may be thus proved from the Principles of Analyticks. Make x — i -\-z, whence x=z, which (hews, that xand2 flow or increale alike. Subftitute thefe infteadof x 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 given at 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 give any of their contemporaneous parts ; for 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

given Equation were y = x -{- - , where x is fuppos'd to be a quan- tity equably increafmg or decreafing ; make x = o, i, 2, 3, 4, 5, &c. fucceifively, then y = infinite, 2, 2|, 3.1, 4^, 5-^-, &c. refpec- tively. And taking their differences, while x flows from o to i, from i to 2, from 2 to 3, &c. y will flow from infinite to 2, from 2 to 2-i-, from 2| to 3.1, &cc. that is, their contemporaneous parts will be i, i, i, i, &c. and infinite, i, £, -{.I, &c. refpeclively. Likewife, if we go backwards, or if we make x negative, we mall have x = o, — i, — 2, &c. which will make _y= infinite, — 2, — 2-i-, &c. fo that the contemporaneous differences will be as be- 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 be oppofite Hyperbola's ; bifed the An- gle GOK by the indefinite 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 pa- rallel to GOH. Now if OL is made to reprefent the indefinite and equable

quantity x in the Equation y = x -f- -'

then CR may reprefent y. For CL = ^ = l- , (fuppofing

= OL = x •, therefore CR =^ LR -4-

R r or

306 *The Method of FLUXIONS,

or y = x -f- ^ • Now the Origin of OL, or x, being in O j if

x = o, then CR, or y, will coincide with the Afymptote OG, and therefore will be infinite. If x= i = OQ^ then _y = BP=2. If x = 2 = OL, then y = CR = 2i. And fo of the reft. Alfo proceeding the contrary way, if 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 _y = Dv = — 2~, &c. And thus we may purfue, at leaft by Imagination, the correfpondent values of the flow- ing quantities x and_y, as alfo their contemporary differences, through all their poiTible varieties ; according to their relation to each other,

as exhibited by the Equation y = x •+- - . .

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) any of their contemporary parts or differences ; fo if thefe differences are taken very fmall, they will be nearly as the Velocities of the moving Bodies, or points, by which they are defcribed. For Mo- tions continually accelerated or retarded, when perform'd in very fmall fpaces, become nearly equable Motions. But if thofe diffe- rences are conceived to be dirninifhed in infiriitum, fo as from finite differences to become Moments, or vanifhing Quantities, the Mo- tions in them will be perfectly equable, and therefore the Velocities of their Defcription, or the Fluxions of the Fluents, will be accu- rately 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. and their refpedlive contemporary Moments to be op, oq, or, &c. where p, q, r, &c. will be the Exponents of the Proportions of the Moments, and o denotes a vanifhing quantity, as the nature of Moments requires. Then x, y, z, Sec. will be as op, oq, or, &c. that is, as p, g, r, &c. So that ,v, y, z, &c. may be ufed inflead of/>, ?> r-> ^c- ni the designation of the Moments. That is, the fyn- chronous Moments of x, y, z, &c. may be reprefented by ox, oy, oz, &c. Therefore in any Equation the Fluent x may be fuppofed to be increafed by its Moment ox, and the Fluent y by its Moment oy, &cc. or x -+- ox, y -{- oy, &c. may be fubftitnted in the Equation inflead of x, y, &c. and yet the Equation will flill be 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 neceflarily exhibit the relation of the Fluxions.

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- caufe y = x -+- z, will become oy = ox -f- oz, or y = x -f- z, winch

is the relation of the Fluxions. Here again, if we afllime z = - > or zx =• i, by increafing the 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

refpect of the others, 'tis zx -f- A z = o, or z === — -f = — — •

Now as the Fluxion of z conies out negative, 'tis an indication that as A- increafes z will decreafe, and the contrary. Therefore in the

Equation y = x -+- z, if z = - , or if the relation of the Fluents be y = x -+- - , then the relation of the Fluxions will be y = x

And as before, from the Equation y =: x -+- - we derived the contemporaneous parts, or differences of the Fluents ; fo from the 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.

For becaufe it is x : y : : I : i \ : : x1 : x1 — i ; when

# = o, or when the Fluent is but beginning to flow, (confequently when y is infinite,) it will be x : y :: o : — i. That is, the Ve- locity wherewith x is defcribed is infinitely little in comparifon of the velocity wherewith^ is defcribed; and moreover it is infinuated, (becaufe of — i,) that while x increafes by any finite quantity, tho' never fo little, y will decreafe by an infinite quantity at the fame time. This will appear from the infpeclion of the foregoing Figure. When x= i, (and confequently _)'= 2,) then x : y : : i : o. That is, x will then flow infinitely faflrer than y. The reafon of which is, that y is then at its Limit, or 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 x=2, (and therefore y = 27,) it will be A- : y :: 4 : 3. Or

R r 2 the

Method of FLUXIONS,

the Velocity of x is there greater than that of^, in the ratio of 4 to 3. When x=3, then x : y :: 9 : 8. And fo on. So that the Velocities or Fluxions conftantly tend towards equality, which

they do not attain till - (or CL) finally vanishing, x and y become

equal. And the like may be obferved of the negative values of x and y.

SECT. V. 77je Refolution of Equations^ whether Algebrai- cal or Fluxionalt by the ajfiftance of fuperior orders of Fluxions.

ALL the foregoing Extractions (according to a hint of our Au- thor's,) may be perform'd fomething more expeditioufly, and without the help of fubfidiary Operations, if we have recourfe to fuperior orders of Fluxions. To (hew this firft by an eafy Inftance.

Let it be required to extradl the Cube-root of the Binomial a> -f- x3, or to find the Root y of this Equation y1 = a3 -\- x"' ; or rather, for fimplicity-fake, let it be_)»3 =a* -+- z. Then y=.a, &c. or the initial Term of y will be a. Taking the Fluxions of this Equation, we fhall have -$yy°- = z = i, or y = -^y~l. But as it is y = a, &c. by fubftitution it will be y = jtf"1, 6cc. and taking the Fluents, 'tis y= * -f- \a~1z^ &c. Here a vacancy is 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/~* ; whence y = — %yy~* = — jy~5- Then becaufe y = a, &c. 'tis y = — ?&"*, &c. and taking the Fluents, 'tis y = * — $a~3z, &c. and taking the Fluents again, 'tis jy- = * * — ^a-^zl, 6cc. Here two vacancies are to be left for the two firfl Terms of y, which are already known. For the next Operation take the

Fluxions of the Equation y=. — ^y~s, that is, y = -f- -9-yy~6 =.

_l_^.£ji-8. Or becaufe _7=c,&c. 'tis _y=if(S-8, &c. Then taking the Fluents, 'tis y= * L^.a-*z, &c. y = * * -^a-^z1, 6cc. and y = * * * -vsTa~*z*, &c. Again, for another Operation take the

Fluxions of the Equation y =: ~~y—^ ; whence y = — -*f^~9 = — Iry"11- Or becaufe y = a, &c. 'tis "y = — IT^"", &c. Then taking the Fluents, y = * — ^a-1^, &c. jy = * * —

and INFINITE SERIES. 309

H-a-^z1, &c. y = * * * — _+._0_.rt— "z5, &c. and _)' = * * * * — • '^-?-a-"z+, &c. And fo we may go on as far as we pleafe. We

have therefore found at laft, that v = a -\ — -. — — , -f- ~g —

y.1 9«» 8i«B

^ , &c. or for z writing x*. 'tis fa + x- = « -+- ~ — ^(

C* 1* -

• I - - n — •" ~ « l\f\^»

' XI ,3» »^1«« I '

bi

Or univerfally, if we would refolve a -f- x \m into an equivalent infinite Series, make y = a -f- x \ ra, and we fhall have am for the

firfl Term of the Series y, or it will be y = am, &c. Then be- j_ i j

caufe y = a •+- x, taking the Fluxions we fhall have ~yym =

x = i, or y = my m. But becaufe it is y = am, &c. it will be y = t/mm~1, &c. and now taking the Fluents, 'tis y = * mam~txy

6cc. Again, becaufe it is y = my m, taking the Fluxions it will

be y = m — lyv *= m x « — ly m ; and becaufe y = am, Sec. 'tis/ = wxw — iam~'i, &c. And taking the Fluents, 'tis y = * ?« x /» — i«™~% 6cc. and therefore y = * * m x ^=-Idm~2x% &c.

_ j — 2

Again, becaufe it is y = m x //; — 17 "', taking the Fluxions it will be y = m - - i x ;// — 2j/y m = m x /« — i x

and becaufe 7 := am, &c. 'tis _y = m x And taking the Fluents, 'tis /== * m x w — i x w/ — y =**/;/ x '^^- x m — 2am-ix*, &c. and v = * x CT~y-3X3, 6cc. And fo we might proceed as far as we pleafe, if the Law of Continuation had not already been fufficiently ma- nifefl. So that we fhall have here a -+- x I » = a* -f- mam~lx -+-

ftt __ r m ^— 2 ni — * * m ~~" ^

m x !LUfl— »xl 4- w x -— - x - — am~^ -+- m x -—- x -

2 2 3

?^-V-4;c<, &c.

Vhis is a famous Theorem of our Author's, tho' difcover'd by

him after a very different manner of Investigation, or rather by •.,- lldufr

Indudion. It is commonly known by the name of his Binomial > '$ vntrrrU*t> *> v Theorem, becaufe by its amftance any Binomial, as a + x; may JujJf&rcL f be railed to any Power at pleafure, or any Root of it may be ex- -^fajfifa & tradted. And it is obvious, that when m is interpreted by any m- Cff1 ' .

**'—*•

r^ t£ r .3 i

^F ^ fr S ^v

(*K3 °>\y i'r NI ^ ^

I'l

w

A-i

310 7%e Method 0/* FLUXIONS,

teger affirmative Number, the Series will break off, and become finite, at a number of Terms denominated by m. But in all other cafes it will be an infinite Series, which will converge when x is lefs than a.

Indeed it can hardly be faid, that this, or any other that is de- rived from the Method of Fluxions, is a ftridl Inveftigation of this Theorem. Becaufe that Method itfelf is originally derived from the Method of railing Powers, at leaft integral Powers, and previoufly fuppofes the knowledge of the Unci<z, or the numeral Coefficients. However it may anfwer the intention, of being a proper Example of this Method of Extraction, which is all that is neceffary here.

There is another Theorem for this purpofe, which I found many years ago, and then communicated it to my ingenious Friend Mr. A. dc Moivre, who liked it fo well as to infert it in a Mathematical Treadle he was then publiming. I mall here give the Reader its Inveftigation, in the fame manner it was found.

Let us fuppofe a -f- x \ m = am -f- />, and that a -f- x = z, and therefore z = x = i. Now becaufe zm=am-i-f, it will be

m

p = niz"'—1 = ^- ; where for zm writing its value am -+- f, we

fH fnf, M

ma . mP . . ma x

fhall have/ = — H — 1 . Now if we make/>= —7- -J- q, it

m m

will be p = '—- — "^r -+- q> And comparing thefe two values

• ttn*1 ?%P

of p, we mall have q = —^ -f- ~ ; where if for p we write its

mamx m1amx m1

value as above, it will be q = -^- + ——^ •+• ~ , or q = m x

_ m .„„ _i_ w j

m ~\- i x °~ + ~l ; make q = m x " 2 - x ^~ -f- r; therefore

-m -a

- -I- r. From which

m a x

= m x m -+- i x — — m x m -+- i x

m 4 OT_

two values of <7 we fhall have r = m -x. m -}- i x — ^- ~1~ 7" . And

2 2,3 -

for y fubftituting its value, it will be r = m x m -h i x — » + ' ..- «OT*a

mr

i .

Or r = m

AT 1

Make r-=.m

4- j ; then, &c. So that we

mall

and INFINITE SERIES. 311

«»* "'+' *'"*=• ,

/lull have a + x I ra = a" -f- m x ^. -f- w x -7- ^7 ~

=^*=fX;0i.^

Now this Series will Hop of its own accord, at a finite number of Terms, when m is any integer and negative Number ; that is, when the Reciprocal of any Power of a Binomial is to be found. But in all other cafes we mall have an infinite converging Series for the Power or Root required, which will always converge when a and x have the fame Sign ; becaufe the Root of the Scale, or the

converging quantity, is • ^ , which is always lefs than Unity.

By comparing thefe two Series together, or by collecting from

• , : m _ a»i

each the common quantity - ^ — - , we fliall have the two

_ . TO — x — — „

equivalent Series - -+- -j- x - -f- -j-x — x - , &c. = — , 1±1 x x + 1±1 x "L±_2 x ==^=- , &c. from whence we

a + x |* 2 3 a-i-x I 5

might derive an infinite number of Numeral Converging Series, not inelegant, which would be proper to explain and illuftrate the na- ture of Convergency in general, as has been attempted in the for- mer 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. And then a and A; may be afterwards determined at pleafure.

As another Example of this Method, we fliall fhew (according to promife) how to derive Mr. de Moivre's elegant Theorem ; for raifing an Infinitinomial to any indeterminate Power, or for extract- ing 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 communi- cated and demonftrated by the Author, in the Philofbphicai Tranf- actions, N° 230. Yet for the dignity of the Problem, and the bet- ter to illuftrate the prefent Method of Extraction of Roots, I lhall deduce it here as follows.

Let us aflume the Equation a -+- b~ -+- cz* -f- itz* -4- ez*t &.c. | *•• = )', where the value of y is to be found by an infinite Series, of which the firfl: Term is already known to be am, or it is y • — a", &c. Make v = a -f- l>z -f- cz,1 -J- dz> -f- ez*, &c. and putting z = i, and taking the Fluxions, we mail have -y r^zi b -+- 2cz -+-

,I2 Tie Method of FLUXIONS,

7</jz» -f- 4«s*. &c- Then becaufe y = u", it is ^ = mviT-', where if we make -y = <*, &c. and V = t>, &c. we fliall have _y = ma'-'b, &c. and taking the Fluents, it will be _>• = * maF*kst

For another Operation, becaufe _y = mvv"-1, it is _y =

_l_ ;,; x ;» ii>i;"'~z. And becaufe <u = 2f -f- 6<fe + 12^2;*, &c. for

•y, v, and -yfubftituting their values a, &c. ^, &c. and 2c, 6cc. refpec- tively, we fliall have jy = zmca"1-1 -+- m x OT — i6*am-z, &c. and taking the Fluents 7 = * 2fncam-Jz -\-m-x. m — i£*tf*-**, 6cc. and taking the Fluents again, y = * * mcam-lz* -\-m~x. ^^•^am-2z1)

&c. ..

For another Operation, becaufe/ = mwm-1 -+- mxm — iv1vm~1t

'tis y • — miv"jm~l •+• yn x /» — ivm~z-vv-{-m\m — ixw — 2tvm—*v">. And becaufe -u = 6^+24^2;, &c. for v, v, v, v, fubftituting a,

6cc. b, &c. 2c, &c. 6^/, &c. we fliall have )"=, 6mdam~l -f- 6m x

w \bcam~'i -\- m x. m — i x m — 2l>3am~*, &c. And taking the

Fluents it will be y = * 6mdam~lz -+- 6m x m — \bcam-*z -+- m x w i Km — 2foam~*z, 8cc. l/ = * *

x ^^ x w — 2foam-iz1, &c. and/ = * * * mdam-lz* -+• m x

m x

. And fo on z'« /«-

finitum. We fliall therefore have tf

And if the whole be multiply'd by s", and continued to a due length, it will have the form of Mr. de Moivres Theorem.

The Roots of all Algebraical or Fluential Equations may be ex- tradled by this Method. For an Example let us take the Cubick Equation y* -\~axy -\-a*y — x"' — 2«3=o, fo often before refolved, in which y = a, &c. Then taking the Fluxions, and making x = i, we fliall have ^yy- -+- ay •+- axy -f- a*-y — 3xz = o. Here if for y we fubftitute a, &c. we fliall have ^y -\- a1 4- axy — 3**,

&c. =o, or>= -j:+^" = ^' &c- =.-- &C' A"d taking the Fluents, v = * — -i-x, &c. Then taking the Fluxions

j again

and INFINITE SERIES. 313

again of the lafl Equation, we fliall have 37}* -f- 6y1y 4- 2 ay 4- axy 4- aly — 6x= o. Where if we make 7 = a, &c. and 7 = — ^., &c. we fliall have y= ZZif+j''-^: __ _!_ &c. and therefore

' 43% CSV. 32a '

AT1 '""

y = * 4 , &c. and y= * * -h z ' » &c- Again, ?yy* -{- iSyy'y

* "\2.ii (3 -I.'? ^

_j_ 675 -f- 3^7 -f- axy -\- a*-j — 6 = 0. Make 7 = a, &c. 7 = — ^ &c. and 7 = — , 6cc. then 7 — ^ + ^~ ^-4"6 5j;C- __

' 32a 4a

^|j, , &c. and therefore 7 = * ^^ , &c. 7 ^ * * = ^ , 6cc.

and 7 = * * * - ^ 5 &c. Again, 377* -1-24777-1- 18^*74- 3677* 4- 4^7 4- ^7x7 4- a*y = o. Make 7 = a, &c. 7 = — ^, &c./ = - , &c. and 7 = - |^ , &c. then 7 = —

:, &c. and7=# •'- ^-. , &c. 7 = * *

, &c. and y = * * * * r68a» ' ^C' ^nc^ ^° on as ^ar as we Therefore the Root is y = a—i x + f- H-H^ + -|2|fl &c.

643 12^- ' i68«'J

The Series for the Root, when found by this Method, muft al- ways have its Powers afcending ; but if we defire likewife to find a Series with defcending Powers, it may be done by this eafy arti- fice. 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 quantity ; or rather, to prevent a confufion of Ideas, we may change a into x, and x into a, and then the Equation will be _y5 -f- axy -+• x*y — rt3 — ax3 = o. In this we mall have y = a, &c. and ta- king the Fluxions, 'tis 3j/y* -4- ay -f- axy -f- 2xy •+- x*y — 6x* = o,

— ay — 2X],"-r- 6-Vz n . i r o > • • "~" #& e>

or y= — 7-1 — H — r . But becauie y=a, &c. tis y= - , ficc. y +<*•*•+* r ^ 3««

== — i, &c. and therefore/ = * — -i-.v,&c. Again taking the Fluxions 'tis 3_)^4 -f- 6)/1/ 4- zay •+- axy -\-2y-\- ^.xy -+- x*y — I2x = o, or - -yj>-^-'j-4y+"* -6^-2^-2, Qr

/ yJ+flX-f-*1 y*

king _y = a, &c. and _y = — |, &c. 'tis y == ~^+f~2a , &c.

3^

=s — yrt , &c. and _y = « — ^ , &c. and ;'=**— ^ , &c.

Again it is 3vy* -t- i8;7/ -f- 6y5 -f- 3^} 4- axy -+. 67 H- 6x/ H- x*_y

S f —12

The Method of FLUXIONS,

_ 12 = O, Or y = -€rv-?--

v = at &c. y = — 4-' &c- and y = — -, &c.)~4 + v -T2 + 2 + 12, / • y 3«> / 3**

&c. = -^ , &c. Then taking the Fluents, 7 = * — , &c.

y = * * ^T , &c. and y = « * * |^ , £c. And fo on. There-

• 2"<Z Gift

fore we mall have y = <z — . ±x — 7^ + 7^7 , &c. Or now we may again change x into #, and a into x ; then it will be y = x

_ itf _ -. _i_ ii^j , &c. for the Root of the given Equation, as

3*

was found before, pag. 2 16, &c.

Alfo in the Solution of Fluxional Equations, we may proceed in the fame manner. As if the given Equation were ay — a*x -{- xy - — o,. (in which, if the Radius of a Circle be reprefented by a, and if y be any Arch of the fame, the corresponding Tangent will be reprefented by x 3) let it be required to extradl the Root y out of this Equation, or to exprefs it by a Series compofed of the Powers of a and x. Make x = i, then the Equation will be ay — a1 -J-

a

xa-yr=o. Here becaufe_y' = _, = i, &c. taking the Fluents it will be y = * x, &c. Then taking the Fluxions of this Equa- tion, we (hall have ay -f- 2xy -+- xy = o, or y= — -^T 1 . But becaufe we are to have a conftant quandty for the firft Term of y, we may fuppofe y='~^^i = o, &c. Then taking the Fluents 'tisj/= * o, &c. and y =. * * o, &c. Then taking the Fluxions

again, 'tis a'y -f- 27 + 4*7 -{- xy = o, or y = "J'^.'T* • Here if for y and_y we write their values i, &c. and o, &c. we mall have

}'= — ^ , &c. whence y — * — il , &c. y =•» * — ^ , £c. and 7 == * * * — ,71 > ^c- Taking the Fluxions again, 'tis

ay +6y + 6xy +xy = o, 01-7 — ~*?~^J = o, <5cc. There- fore y = * o, &c. _y= * * o., &c. j.'= * * * o, &c. y =. **#» Q, &-C. Again, ay -+- I2y 4- 8^,7 -f- A'4j = o, or y = ~ l2^!.''1-

and INFINITE SERIES.

= +•£ , &c. Then Jr^*^^, &c. ;< = **4- ^ , &c. _>>=:***-{- , &c. jy = * * * * 4- — , &c. and _}•=*****

— » 45

» - - - } &c. Again, a*y-\- zoy -f- loxy -f- xy = o, whence y=s

5"' r567?

***** o, &c. Again, a*j+ 307 -f- i2*y -+- A,"^ = o, or y=.

y~^=:- 30 x 24rf-9, &c. Then > =- . _ !±^2f , &c.

12x30** 0 4x301-3

j ==- * * -- ^ — , &c. y = * * * - —^— , &c. ;• = *** *

30*4 6*5 o • *6

-- -, &c. _y = # * * * * — -^ , c^c. j)/=******

A'7

£c. and _/ = ******* — — «, &c. And fo on. So that we

have here y = * # -H ox-1 — ^» + OA'4 + ^; > &c- that is, _y =• fl , . *! &c

* — ' P + 5^ /«' '

This Example is only to {hew the universality of this Method, and how we are to proceed in other like cafes ; for as to the Equa- tion itfelf, it might have been refolved much more fimply and ex-

peditioufly, in the following manner. Becaufe y = -^- — - } by Divifion it will be y = i — ~ + ^ — *-6 -f- *L , &c. And ta-

king the Fluents, y = x — ^ -+- — ; — — -6 4- ^ > &c.

In the fame Equation ^ — a*x 4- x1^ = o, if it were requir'd to exprefs x by y, (the Tangent by the Arch,) or if x were made the Relate, and y the Correlate, we might proceed thus. Make

jcx

y = i, then a* — a?x •+- x* = o, or si = 14 — - = i 5cc

•J G,

Then x = * y, &e. And taking the Fluxions, 'tis x = ^ •=s

~-^ , &c. = 04-^, &c. whence A- = * o, &c. and x = * * o,

&c. So that the Terms of this Series will be alternately deficient, and therefore we need not compute them. Taking the Fluxions

• . ,• * Zxx 2 n i-i->i r 2V

again, tis x = — -j- — =: — , &c. Therefore x = * -^ , &c.

A- = * * j-, , &c. and ^== # * * ^ , &c. Again, x=z -^ -l-^r >

S f 2 and

Method of FLUXIONS,

2XX

>X xx XX c ,n_. tf ,2

and again, A- = — -f- -^ — h — . Subitituting i, &c. and — >-

&c. for # and x} and alfo o, 6cc. for # and #, it will be x = 16 c t. i6y c gy*

— , &c. whence x •=. * — - , occ. * = ** — , &c. x = * # *

<j4 ** *•» '

8v' c • 2>'4 C J 21* £•

_, , &c. A- = * * * * ^ , &c. and . x ==-***»#—, &c..

6 •• •'• " J 7 •'• •• :: 5 6

. • 20xx+ iQAr«- 4" 2A';xr „ J „ • 2n*-I-4--;o.»;x4-i2.v.v-(-2.vA-

Again, x = - — - , and again, x = - — - — -- • Here for x, x, and x writing i, &c. ^ , &c. and l— , &c. re- fpeaively, 'tis x = 8±±^U.6 , &c. — ^ , Sec. Then x = » ^iv, &c. J = ..i4\ &c. x=***i^-J, &c. 2==.....

fi 3^

ig4 , &c. ^= * * * * * 7^ , &c. x= ****** JJpr, &c. and *, &c. That is, X=y+.^ + -£l. +

7.1 Cfl6

For another Example,, let us take the Equation ^*jj* — .ylj> — ,

/z*x* — o, (in which, if the Radius of a Circle be denoted by a, and if y be any Arch of the fame,, then the correfponding right Sine •will be denoted by x j) from which we are to extract the Root y.

Make x=i, then it will be a*y*~ — x^y1 = a*, or ;}* = -^^ M— J} &c. or j/= i, &c. and therefore y=. * x, &c. Taking the Fluxions we fhall have za*yy — zxy* — 2X1yy = o) or a*y—> Xy __ xy = 0, or "y = ^r~7i = °' ^^ ^nc^ ta^ing. the Fluxions

again, 'tis a*y — j/ — 3* — ^a/= o, or ^ = __. . y

Therefore } = * J } &c. ^ = * * — , &c. and _y = * * * 1- , &c. Then «a;y • — 4^ — 5^' — x*y = o, and again <7aj/ — 9_y — , x^=o, orj = i±^=§,&c. = J,&C. There-

fore y := * — , &c, y — ' * * ^ > &c. ^ = * * * — 4 , &c. ^5^

* • * *

INFINITE SERIES, 317

* * , . » j^-4 , 6cc. and r = ***#* -IL , &c. Taking the

6 :: S 6 7

Fluxions again, 'tis nly — i6y — g.y — x*y = o, and again, a*jr

2^Y + i i xv c

— z$y — 1 1*7 — x*y = o, or ;< = ~— -^ = -/, &c. = ^ , &c. Therefore j = * ^.v, &c. v = * * ^V, &c.

«« ' J a* J 2a6

y = * * * -2-3 xrt &c. _)' = * * * * •gJTx4> ^c> y === * * * * *

^-|xr, &c. y = ****** —r—6 , &c. and y = *****#* -^-. . &c. Or v = x -i- ~ 4- -^- 4- -^— „ , &c.

TI2a6 J •* t><il 40^.4 II2«*

If we were required to extraft the Root x out of the fame Equa- tion, aly* — x1)/1 — rt1*1 = o, (or to exprefs the Sine by the: Arch,) put y = i, then a1 — x1 — a**1 = o, or x* = i —

-*, and therefore x = i, &c. and x = * y, &c. Taking the-

Fluxions 'tis — axx — 2a*xx = o, or x=— — =: o, &c,- Therefore x = * o, &c. x =: * * o, &c. Taking the Fluxions

again, 'tis x = — ^ = — ^ , &c. Thence x= * — ^, &c. x = »« — — , &c. andx=* * * — £- , &c. Again, x = — ^>

2^1 O£i fi

and x = — ~ = -+- ^ , &c. Therefore x = « ^ , &c. x =

t1 )J j4

» « -^— , occ. x = * * * 7— , &c. x = * * * » — - , occ. and x

v* x x

=== » » » * * - — , &c. Again, x = , and x = — —

i ?r/24 * O * a fl

i 6 v ? y1

- — 6 , 6cc. Therefore x = » — - ^j , x = « * — ~« *

&c. x — » * « — -^-j , 6cc. x — » * » * — -1— g , &c. x = * * » * »

j.J V* » J

«_ -^-— t , &c. x = .»••««—. -±— f , &c. and x= « *»*»««

I 2Ofl* 72Ofl

*__ - -^ -6 , 6cc. And therefore x=<y>— ^ -f-

&c.

If it were required to extraft the Root y out of this Equation, *y» .— .x*y* •+• m*y* — w1^ = o, (where x =s i,) we might pro- ceed

3i 8 The Method of FLUXIONS,

ceed thus. Becaufe y~ •==. '" ^ ~^ -v- = ;«*, &c. 'tis ;/ = ;;;, See.

and_y= -# w*> &c. Taking the Fluxions, we fhall have 2a*yy — 2xy* — 2x*yy -bJzr$*yy = o, or ay — xy — xy -+- my • — o, or r= p— ^r= °j ^c- Therefore taking the Fluxions again, 'tis

3xy — x1/ = o, or y = -I — ^^_vjf^l; and making y = m, &c. 'tis y= OTX ' ~OT &c. and therefore y = * '" x ' "~ ;"'x, &c. y =

^ X /2A -^

i^LV, &c. and y = * * * OT x ' ~^\-', &c. Taking the

za'' * zx. ya

Fluxions again, 'tis a*y -+- m1 — 4 x_y — $xy — xy = o j and again, cfy — {- in* — 9 x y — — 7xy — x2i/ — • o, or y — Q "'' x->' ~t~ "•vv ___

,-«-x9-^ &c> Therefore y= * ;"x ' ~ g'lx 9~CT\y &c v

«4 «4 ' ^

This Series is equivalent to a Theorem of our Author's, which (in another place) he gives us for Angular Sections, For if A- be the Sine of any given Arch, to Radius a ; then will y be the Sine of an- other Arch, which is to the firft Arch in the given Ratio of m to i. Here if m be any odd Number, the Series will become finite j and in other cafes it will be a converging Series.

And thefe Examples may be lufficient to explain this Method of Extraction of Roots ; which, tho' it carries its own Demonftration along with it, yet for greater evidence may be thus farther illustrated. In Equations whofe Roots (for example) may be reprefented by the general Series y = A -+- Ex -f- Cx4 •+• Dx3, 6cc. (which by due 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

Equation, by Methods that have been abi ^antly explain'd already. Then making x= i, we flrall have have y = B -f- 2C.v -+- 30**, &c. where B likewife is a conftant quantity, or perhaps = o, and reprefents the firft Term of the Series y. This therefore is to be derived from the firft Fluxional Equation, either given or elfe to be found ; and then, becaufe it is y = B, &c. by taking the 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 the conftant quantity zC will reprefent the firft Term of y ; this is 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 Fluents it will be y = * zCx, &c. and again y = * * Cx1, £cc. 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 Term of the Series y ; this is to be derived from the

third Fluxional Equation. And then, becaufe it is y = 6D, &c. by taking the Fluents it will be v = * 6Dx, &c. y = * * 3D*'-, See. and _)'==.* * * D*3, &c. by which the 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 you ihould defire to know how the foregoing Trigonometri- cal Equations are derived from the Circle, it may be fhewn thus : on the Center A, with Radius AB = at let the Quadrantal Arch BC be defcribed, and draw the Radius AC. Draw the Tangent BK, and through any point of the Circum- 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 con- tinually to approach towards D, till it finally c<-:.ncides 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

c

320 The Method of FLUXIONS,

Moment of the Circumference, it will be K-! = ^4 x — = —

Da &.1 L)J ~ AB

x ^ . Make AB = a, the Tangent BK = x, and the Arch BD=y ; and inflead of the Moments Kk and Dd, fubftitute the proportional Fluxions x and y, and it will be - = " -+*- , or a2v

J y a* J

•+• x*y —— a*x = o.

From D to AB and de let fall the Perpendiculars DE and Dg-, which Dg meets de, parallel to DE, in g. Then the ultimate form of the Trilineum Ddg will be that of a right-lined Triangle fimi- lar to DAE. Whence "Dd : dg :: AD : AE = v//iJJ$F — 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 be y : x : : a : \/ a1 — AT*. Or^1 : x1 :: a* : a3- — x1, or a^y1 - — x^y* — a'-x1- = o.

Hence the Fluxion of an Arch, whofe right Sine is x, being

exprefs'd by f^_^ ; and likewife the Fluxion of an Arch, whofe

right Sine is y, being exprefs'd by i°?_ ,. ; if thefe Arches are to each other as i to m, their Fluxions will be in the fame proportion, and vice versa. Therefore "x , : °v x : : i : »;, or . ™x .

•J a — x */ a — y da — x

= -T37i } or putting #= i, 'tis a*yl —

== o ; the fame Equation as before refolved. ' 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 Equation zaxy* — x'-y'- — a^x* = o. Or if y were the Arch, and x the correfponding Secant, it would be x^y* — a*-x*yl — #4x* = o. Or inftead of the natural, we might derive Equations for the artificial Sines, Tangents, Secants, &c. But I fhall leave thefe Difquifitions, and many fuch others that might be propofed, to ex- ercile the Induftry and Sagacity of the Learner.

SECT,

and INFINITE SERIES. 321

'SECT. VI. An Analytical Appendix ', explaining fome Terms and Expreffions in the foregoing work.

BEcaufe mention has been frequently made of given Equations, and others a framed ad libitum, and the like ; I mall take oc- calion from hence, by way of Appendix, to attempt fome kind of explanation of this Mathematical Language, or of the Terms giver/, afligjfd, affiimed, and required Quantities or Equations, which may give light to fome things that may otherwife feem obfcure, and may remove fome doubts and fcruples, which are apt to arife in the Mind of a Learner. Now the origin of fuch kind of ExpreiTions in all probability feems to be this. The whole affair of purfuing Mathematical Inquiries, or of refolving Problems, is fuppofed (tho' tacitely) to be tranfacled between two Perfons, or Parties, the Pro- pofer and the Refolver of the Problem, or (if you pleafe) between the Mafter (or Inftruclor) and his Scholar. Hence this, and fuch like 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 Terms and Conditions as he thinks fit ; and the Scholar is 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 they involve any inconfiftency or contradiction, the Problem will be unfair, or will become ab- furd 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 expects to receive are the qii(?/ita. As it may fometimes happen, that the data may be more than are neceffary for determining the^Qiu ft'.on , and lo perhaps may interfere with one another, and the Problem (as now propofed) may become impolTible ; fo they may be 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 fupply what is wanting, and at his difcretioa miy a (Jit me fuch and fo 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- pleft, the eafieft, and neateft Solution that may be had, and yet in the moft general manner. For it is convenient the Problem fhouM be propofed as particular as may be, the better to fix the Imagina- tion; and .yet the Solution mould be made as 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- thing different from this, but which ultimately centers in it. As that is call'd a datum, when one quantity is not immediately given, but however is neceffirily infer'd from another,, which other perhaps is neceffarily infer'd from a third, and fo on in a continued Series, till it is neceffarily infer'd from a quantity, which is known or given in the fenfe before explained. This is the Notion of Euclid'?, data, and other Analytical Argumentations of that kind. Again, that is often call'd a given quantity, which always remains conftant and in- variable, while other quantities or circumftances vary ; becaufe fuch as thefe only can be the given quantities in, a Problem, when taken in the foregoing fenfe.

To make all this the more fenfible and intelligible, I /hall have, recourfe to a few pradlical inftances, by way of Dialogue, (which, was the old didadlic method,) between Mafter and Scholar; and this only in the common Algebra or 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- tion will confift in this, that as faft as the Mafter propofes the Con- ditions of his Queftion, the Scholar applies thole Conditions to ufe, argues from them Analytically, makes all the aeceffary deduc- tions, and derives fuch confequences from them, in the fame order they are propofed, as he apprehends will be mcft fubfervient to the Solution. And he that can do this, in all cafes, after the fureft, fim- pleft, and readieft manner, will be the beft ex-tempore Mathemati- cian. But this method will be beft explain'd from the following Examples.

I. M. A Gentleman being 'willing to diftribiite Abns S. Let

the Sum he intended to diftribute be reprefented by x. M. Among fbme poor people. S. Let the number of poor be _}>, then - would

have been the fhare of each. M. He wanted 3 fiillings S. Make

3 = rf, for the lake of univerfality, and let the pecuniary Unit be one Shilling ; then the Sum to be distributed would have been x-{-a,

and"

and INFINITE SERIES. 32"

and the fhnre of each would have been ^^- . M. So that each

y

might receive $ fallings. S. Make 5=^, then —^ = b, whence

x = by — a. M. "Therefore he gave every ot.e 4 fallings. S. Make

4=f, then the Money diftributed will be cy. M. And he has 10

fallings remaining. S. Make io = d, then cy -f- d was the Money

he intended at firft to diftribute; or cy -+- d = (x =) by- — a, or

y =5 ^jt-f . M. J^rf* w<2J the number of poor people ? S. The

number was y = 7 = 3"*"'° = 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? S. His Money was at firft 62 fhillings, and the number of poor people was 1 3. But if his Money had been 62-4-3 === ^5 ^ r3 x 5 fhillings, then each poor perfon might have received 5 millings. But as he gives to each 4 fhillings, that will be 13x4=52 fhillings diftributed in all, which will leave him a Remainder of 62 — 52 c= 10 fhillings.

II. M. A young Merchant, at his 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 himfclf the Jirjl year, he expended 100 pounds. S. Make the given number ioo = tf; then he had to trade with x — a. M. He traded with the reft, and at the end of the year had improved it by a third part. .S. For univerfality-fake I will aflume the general num- ber n, and will make ^ = n — i, (or n = ± ;) then the 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 MX — na. M. He did the fame thing the fecond year. S. 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 n — ix nx — na — a, or n'-x — n*a — nx -f- a for this year's Im- provement, which together make n'-x — n*a — na for his Eftate at die end of the fecond year. M. As aljo the third year. S. His whole Stock being now ;<ax — nxa — 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, or «Jx — 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

_ n1 a — na, or in a better form n"'x -+• "-^na. In like manner

T 2 if

324 Th* Method of FLUXIONS,

if he proceeded thus the fourth year, his Eftate being now n*x — nia _ 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 — if a — 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

year. And fo, by Induction, his Eftate will be found nsx -4- - — na at the end of the fifth year. And univerfally, if I aflume the ge-

m _

neral Number m. his Eftate will be n™x -f- ~l-na at the end of

i — «

any number of years denoted by m. M. But he made his Eftate double to -what it was at firft. S. Make 2 = £, then nmx -t-

m m

-l_^na = bx, or x = — — - ==-«#. M. At the end of 3 years.

n — i x n" — b

S. Then #2=3, a-=. 100, b = 2, n = %, and therefore x=s..

64

400 = 1480. M. j%!2<z/ <was his Eftate atjirji? S. It was 1480 pounds.

III. M. Two Bvdies A and B are at a given diftance from each other. S. As their diftance is faid to be given, though it is not fo. actually, I may therefore aflume it. Let the initial diftance of the Bodies be 59 = ey and let the Linear Unit be one Mile. M. And move equably towards one another. S. Let x reprefent the whole fpace defcribed by A before they meet ; then will e — x be the whole fpace defcribed by B. M. With given Velocities. S. I will aflume the Velocity of A to be fuch, that it will move 7 = c Miles in 2 =f Hours, the Unit of Time being one Hour. Then be-

caufe it is c : f : : x : - , A will move his whole fpace x in the

time Y • Alfo I will aflume the Velocity of B to be fuch, that it will move 8 = d Miles in 3 ==:g Hours. Then becaufe it is d : g :: e — x : —j-g> B will move his whole fpace e — x in the time '-7%. M. But A moves a given time - - - S. Let that time be i = b Hour. M. Before B begins to move. S. Then A"s time is equal to U's time added to the time hy or ^ = '^g -f- h.

M,'

and INFINITE SERIES, 325

M. Where will they meet., or what will be the fpace that each have defcribed ? S. From this Equation we fhall have x =

* s*i X7==1^X7== r x 7 = 35 Miles, which will

' ' J '

x2

oxz-r7x3'/ 37

be the whole fpace defcribed by A. Then e — x= 59 — 35 =

24 Miles will be the whole fpace defcribed by B. . IV. M. Jf 12 Oxen can be maintained by the Pafture 0/37 Acres of Meadow-ground for 4 weeks, S. Make 12 = #, 3! = ^, 4=cj then aiTuming the general Numbers e, f, h, to be determin'd after- wards as occafion Ihall require, we mall have by analogy

Oxen

If	«	a '
Then	(J j	fae
Alfo '	t_ Q .	ae T
And "	^"	ace T
Alfo	.1	ace
S-L, i or
\ • " . . r <-> ace Alf°J^J^J

> require <

'afture Time
" b '		c ~
be		c
e		c
	. during <	i
e		/
e		b

M, ^4«^ tf, becaufe of the continual growth of the Grafs after the four weeks, it be found that 2 1 Oxen can be maintain d by the pafture of \ o fuc h Acres for 9 weeks, S. Make 2i=J, e= io3 f== 9 ; then becaufe on this fuppofition, the Oxen d require the

pafture e during the time f; and in the former cafe the Oxen ~ required the fame pafture during the fame time : Therefore the growth of the Grafs of the quantity of pafture e, (commencing after 4 or c weeks, and continuing to the end of the Time f, or during the whole time f — c,) is fuch, as alone was fufficient to

maintain the difference of the Oxen, or the number d — ^ , for the whole time f. Then reciprocally that growth would be fuffi- cient to maintain the number of Oxen df — ' ' for the time i,

or the number of Oxen -£ — ~

h bb

ace J

for the time h. And becaufe this growth will be proportional to the time, and will maintain a greater number of Oxen in proportion as the time is greater j we ihall have

Time

3 26 7%e Method ^FLUXIONS,

-• ~* 1»

Time Oxen Time Oxen

,, df ace , h — c . df ace

t - c • r - Ti *•'• " - c • 7 - mtO T — ' TZ »

* bh f — c h bh '

which will be the number of Oxen that may be maintain'd by the growth only of the pafture e, during the whole time h. But it 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 together, or ^ _f_ -^-^ x - ~ ""• , will be the

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- tain d by 24 Acres of fitch pafture for 1 8 week s ? S. Suppofe x to be that number of Oxen, and make 24=^, and h= 18. Then by analogy

Oxen Pafture

If

Then ex requij.e J <g during the time b.

. , r i ex ace

And conlequently — = j-r

T J g bh

dft a^j ac /> — r

x f ~ jf = T +

21*9 1 2 x 4 • i 14 fi

I O J-j

V. M. If I have an Annuity S. Let x be the prefent value

of i pound to be received i year hence, then (by analogy) x* will be the prefent value of I pound to be received 2 years hence, &c. and in general, x" will be the prefent value of i pound to be re- ceived m years hence. Therefore, in the cafe of an Annuity, the 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 Annuity of i pound, to be continued for m years. But becaufe

»+« . r

-—- =x-{- x1 -f- x'' H-A'4, &c. continued to fo many Terms

I X

as there are Units in in, (as may appear by Divifion 3) therefore *~y will reprefent the Amount of an Annuity of i pound,

to be continued for ;;/ years. M. Of Pounds. S. Make

= a*

and INFINITE SERIES. 327

= a, then the Amount of this Annuity for m years will be

^— — a. M. To be continued for 5 years fuccejji'vely. S. Then m = 5. M. Which I Jell for pounds in ready Money. S. Make

= c, then

-a = c, or x™""1 --

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 year will be i — x, and therefore the Intereft of 100 (or K) pounds for i year will be b — bxy which will be 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. M. In the right Line AB I give you the ftuo points A and B. S. Then their 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

in mag-

given

nitude and fpe- cie ; and pro- ducing CA and CB towards d and <T, I can takeA</=AD, and Bf=KD. M. Aljb I give you the indefi- nite right Line EF in po/iticn, pajjing thro' the

green point D. S. Then the Angles ADE and BDF are given, to which (producing AB both ways, if need be, to e and fy) I can make the Angles h2e and B<f/~ equal refpedlively, and that will determine the points e and f, or the Lines Ae = a, and Ef=c. And becaufe de and <T/"are thereby known, I can continue de to G, fo that^/G. = Sj\ and make the given line eG c= b. Likewife I can draw CH

and

228 The Method of FLUXIONS,

and CK parallel to ed and f$ refpeftively, .meeting AB in H and K ; and becaufe the Triangle CHK will be given in magnitude snd fpecie, I will make CK = d, CH=e, and HK=/ M. Now let the given Angles CAD and C BD be conceived to revolve about the green points or Poles A .and B. S. Then the Lines AD and CA^ will move into another fituation AL and cAt, fo as that the Angles DAL, </A/, and CAc will be equal. Alfo the Lines BD and CB^ will obtain a new fituation BL and cBA, fo as that the Angles DBL, <fBAand CBc will be equal. M. And let D, the Inter feflion of the Lines AD and BD, always move in the right Line EF. S. Then the new point of In- terfedtion L is in EF; then the Triangles DAL and </A/, as alfo DBL andJ'BA, are equal and iimilar ; then^//= DL= cTA, and therefore G/==/A. M. What will be the nature of the Curve defer ibed by the other point of Inter feSt ion C ? S. From the new point of Interfection c to AB, I will draw the Lines ch and ck, parallel to CH and CK refpec- tively. Then will the Triangle chk be given in fpecie, though not in magnitude, for it will be Iimilar to ^CHK. Alfo the Triangle Bck will be fimilar to Btf. And the indefinite Line Bk=x may- be aflumed for an Abfcifs, and ck = y may be the correfponding Ordinate to the Curve Cc. Then becaufe it is Bk (x) : ck (y )

:: Bf(c) :/A = ; = G/. Subtraft this from Ge-=&, and there will remain le=.b — - . Then becaufe of the fimilar Triangles chk

and CHK, it will be CK (d) : CH (e) : : ck (y) : ch= 'j . And CK (/O : HK (/) :: ck ( y] : hk = -\ . Therefore A/J = AB— . Bk — hk = m — X—.£. But it is A/6 (m — x — 'f ) : cb 2) ::

(a) : le (b — c-v) . Therefore m — x — f x b — J = ^,

or

dc — ae — bft. xv — demy — bdx* + bdmx = o. In which Equation, becaufe the indeterminate quantities x and y arife only to two Dimenfions, it mews that the Curve defcribed by the point C is a Conic Section.

M. Ton have therefore folved the Problem in general, but you fionld now apply your Solution to the feveral fpecies of Conic Sections in par- ticular. S. That may eafily be done in the following manner : e + l'f—ctl __ 2pt an(j then the foregoing Equation will be-

come fcf •—- zpcxj> — demy — bdx'- 4- bdmx = o, and by ex-

trading

and INFINITE SERIES. 329

trading the Square - root it will be y = -.x -f. — -f-

V I'P ft I1*"1 ''d " </*»;* XT . . . , .

!Z + - x x* + — -. . _XA. + __. Now here it is plain, that if the Term j£ 4- ~ x XL were abfent, or if jj£ -4- ^ = o, or r = — £• ; that is, if the quantity -- (changing its fign) fhould

be equal to ^ , then the Curve would be a Parabola. But if the fame Term were prefent, and equal to fome affirmative quantity, that is, if •?• -f- - be affirmative, (which will always be when

• is affirmative, or 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

y> the Curve will be an Ellipfls or a Circle.

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 fully convinced by experience, that whoever fhall go through the reft of our Author's curious Problems, in the fame manner, (where- in, according to his ufual brevity, he has left many things to be fupply'd by the fagacity of his Reader,) or fuch other Queflions and Mathematical Diiquifitions, whether Arithmetical, Algebraical, Geometrical, &c. as may eafily be collected from Books treating on theie Subjects ; I fay, whoever fhall do this after the foregoing manner, will find it a very agreeable as well as profitable exercife : As being the proper means to acquire a habit of Investigation, or of arguing furely, methodically, and Analytically, even in other Sciences as well as fuch as 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,

SECT. VII. The Conclufion ; containing a Jhort recapitu- lation or review of the whole.

E are now arrived at a period, which may properly enough be call'd the conclujion of tie Method of Fluxions ami Infinite Series ; for the defign of this Method is to teach the nature of Series in general, and of Fluxions and Fluents, what they are, how they are derived, and what Operations they may undergo ; which defign (I think) may now be faid to be accompliili'd. As to the applica- tion 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 done, either by Series or by Fluxions, may eafily be reduced to the Refolution of Equations, either Algebraical or Fluxional, as it has been already deliver'd, and will be farther ap- ply'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 finding the matter to grow fo faft under 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 opportunity, if I mall find 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- mary account of it, in 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 recapitula- tion 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 pur- fued 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 other, both in their Notation and Operations ; the firft com- puting after a definite and particular manner, the latter after a ge- neral 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 Root ; fo in Algebra, or Analyticks, when the Terms are orderly

dilpos'd

and INFINITE SERIES. 331

difpos'd as is prefcribed, we reckon by any other Root and its Powers, or we may take any general Number for the Root of our Arithmetical Scale, by which to exprefs and compute any Numbers required. And as in common Arithmetick we approximate continually to the truth, by admitting Decimal Parts /;; infnititm, or by the ufe of Decimal Fractions, which are compofed of the reciprocal Powers of the Root Ten ; fo in our Author's improved Algebra, or in the Method of infinite converging Series, we may continually ap- proximate to the Number or Quantity required, by an orderly fuc- cefiion of Fractions, which are compofed of the reciprocal Powers of any Root in general. And the known Operations in common Arithmetick, having a due regard to Analogy, will generally afford us proper patterns and fpecimens, for performing the like 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 two principal circumftances of Convergency and Divergency; where- in I attempt to (hew, that in all fuch Series, whether converging or diverging, there is always a Supplement, which if not exprefs'd is however to be underftood ; which Supplement, when it can be ai- 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 may be neglected, becaufe it continually diminifhes with the Terms of the Series, and finally becomes lefs than any affignable quantity. And hence arifes the benefit and conveniency of infinite converging Scries ; that whereas that Supplement is commonly fo implicated and entangled with the Terms of the Series, as often to be impoiliblc to be extricated and exhibited ; in converging Series it may fafely be neg- lected, and yet we mall continually approximate to the quantity re- quired, And of this I produce a variety of Inftances, in 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- tion'd Supplement, by which Scries may be render'd finite, that is, may be compared with other quantities, which are confider'd as given. I then deduce feveral ufeful Theorems, or other Artifices,

Una for

332 tte Method of FLUXION s,

•

for the more expeditious Multiplication, Divifion, Involution, and Evolution of infinite Series, by which they may be eafily and rea- dily managed in all cafes. Then I fhew the ufe of thefe in pure Equations, or Extractions; from whence I take occasion to intro- duce a new praxis of Refolution, which I believe will be found to be very eafy, natural, and general, and which is afterwards ap- ply'd to all fpecies of Equations.

Then I go on with our Author to the Exegefis numerofa, or to the Solution of affefted Equations in Numbers ; where we mall find his Method to be the fame that has been publifh'd more than once in other of his pieces, to be very {hort, neat, and elegant, and was a great Improvement at the time of its firft 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- fected, may be computed with great exactnefs and facility.

From Numeral we pafs on to the Refolution of Literal or Speci- ous affected Equations by infinite Series ; in which the firfl 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 hap- pen 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 Arithmetical Progreffion in p/ano, as is there explain'd ; or 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 Progreffions of the Indices ; and therefore all the -forms of the fe- veral Series that may be derived for the Root, as alfo their initial Coefficients, Terms, or Approximations.

We then farther profecute the Refolution of Specious Equations, by diverfe Methods of Analyfis -, or we give a great variety of Pro- cefTes, by which the Series for the Roots are eafily produced to any number of Terms required. Thefe ProcefTes are generally very lim- ple, and depend chiefly upon the Theorems before 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 na- ture and foundation of that Method is explain'd at large. And fome general Observations are made, chiefly from the Science of Rational Mechanicks, by which the whole Method is 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 de- termine 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 fa- tisfactory 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- citely and indirectly. I have here ("hewn, that they are a necelTary refult from the nature and notion of nrft Fluxions ; and that all thefe feveral orders differ from each other, not abfolutely and effentially, but only relatively and by way of companion. And this I prove as well from Geometry as from Anaiyticks ; and I actually exhibit and make fenfible thefe feveral orders ot Fluxions.

But more efpecially in what I call the Geometrical and Mechani- cal Elements of Fluxions, I lay open a general Method, by the help of Curve-lines and their Tangents, to reprefent and exhibit Fluxions and Fluents in all cafes, with all their concomitant Symptoms and

AffecYions,

334 f^3e Method 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, by which not only their exiitence is proved beyond all poflible con- tradiftion, but alfo the 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 flowing Quantities or Fluents ; which is the fame thing as the Inverfe Method of Fluxions. And firft I explain (what out- Author calls) a particular Solution 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- dundant or deficient, cannot be refolved by this 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- plain'd,) can be exprefs'd by the Terms of the Correlate Quantity alone ; in which Cafe the Root will be obtain'd by an eafy pro- cefs : In finite Terms, when it may be done, or at leaft by an infinite Series. And here a ufeful Rule is explain'd, by which an infinite Expreffion may be always avoided in the Conclufion, which otherwife would often 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 more operofe than the former, yet it is folved by a 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.

The third and laft Cafe of Fluxional Equations is, when there are more than two Fluents and their Fluxions involved 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

places at pleafure, the way of finding the contemporaneous Incre- ments of Fluents, and fuch like.

Then to conclude the Method of Fluxions, a very convenient and general Method is propofed and explain'd, for the Refolution of all kinds of Equations, Algebraical or Fluxional, by having recourfe to fuperior orders of Fluxions. This Method indeed is not con- tain'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 oc- cur in the foregoing work, and which it is very neceflary to appre- hend rightly. And a fort of Analytical Praxis is adjoin'd to this Explanation, to make it the more plain and intelligible ; in which is exhibited a more direct and methodical way of refolving fuch Alge- braical or Geometrical Problems as are ufually propofed ; or an at- 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 Principles. I have always put myfelf in the place of a 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 I have pafs'd over with a flight animadverfion : But in things of more novelty, or greater difficulty, I have always thought myfelf obliged to be more copious and explicite ; and am conlcious to 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- fects fupply'd by abler hands, and (hall always be willing and thank- ful to be better instructed.

What perhaps will give the greateft difficulty, and 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,

fjfies, and the like, which our Author makes ufe of in this Treatife, and elfe where, for deducing and demonftrating hisMethod of Fluxions. I fhall therefore here add a word or two to my foregoing Explana- tions, in hopes farther to clear up this matter. And this feems to 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 are utterly impoffible, inconceiveable, and unintelligible, and it may therefore be thought to follow, that the Conclu lions derived by their means muft be precarious at leaft, if not erroneous and impoflible.

Now to remove this difficulty it mould be obferved, that the only Symbol made ufe of by our Author to denote thefe quantities, is the letter o, either by itfelf, or affected by fome Coefficient. But this Symbol o at firft reprefents a finite and ordinary quantity, which mu ft be understood to diminim continually, and as it were by local Motion ; till after fome certain time it is quite exhaufted, and termi- 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 pafs through vanifhing quantities of all proportions. For it cannot pafs from being an affignable quantity to nothing at once ; that were to proceed per fa/turn, and not continually, which is contrary to the Suppofition. While it is an affignable quantity, tho' ever fo little, it is not yet the exact truth, in geometrical rigor, but only an Ap- proximation to it ; and to be accurately true, it muft be lefs than any affignable quantity whatfoever, that is, it muft be a vanifhing quantity. Therefore the Conception of a Moment, or vanishing quantity, muft be admitted as a rational Notion.

But it has been pretended, that the Mind cannot conceive quan- tity to be fo far diminifh'd, and fuch quantities as thefe are repre- fented as impoffible. Now I cannot perceive, even if this impofli- bility were granted, that the Argumentation would be at all affected by it, or that the Concluiions would be the lefs certain. The im- poffibility of Conception may arife from the narrownefs and imper- fection of our Faculties, and not from any inconfiftency in the na- ture of the thing. So that we need not be very folicitious about the pofitive nature of thefe quantities, which are fo volatile, fub- tile, and fugitive, as to efcape our Imagination ; nor need we be much in pain, by what name they are to be call'd j but we may confine ourfelves wholly to the ufe of them, and to difcover their

properties,

and INFINITE SERIES. 337

properties. They are not introduced for their own fakes, but only as fo many intermediate fteps, to bring us to the knowledge of other quantities, which are real, intelligible, and required to be known. It is fufficient that we arrive at them by a regular progrefs of di- minution, and by a juft and neceflary way of reafoning ; and that they are afterwards duly eliminated, and leave us 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 the name of impoffible and imaginary Quantities ; fuch as ^/ — i, ^/—a, v/ — 3, v/ — 4, &c. For they import, that a quantity or number is to be found, which multiply'd by itfelf mall produce a - negative quantity ; which is manifeftly impoffible. And yet thefe quantities have all varieties of proportion to one another, as thofe aforegoing are proportional to the poffible and intelligible numbers I, ^/2, v/3, 2, 8cc. respectively -,. and when they arife in 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" -J-4IX — 42 =o, from whence we were to extract the Root x ; by proceeding according to Rule, we mould have this fiird Ex-

preffion for the Root, x = 4 -f- y'3 4- v/ — -"fr-f- ^J^ - ,/ — -^, in which the impoffible quantity ^/ — -~ is involved ; 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 may appear by cubing ; and when the Cubick Root of the fecond vinculum is extracted, it will be found to be — i — \/—- j- Then by collecting thefe Numbers, the

X.x im-

338 77je Method of FLUXIONS,

impoffibie Number </ — ± will be eliminated, and the Root of the Equation will be found x = 4 — i — i = 2.

Or the Cubick Root of the firft vinculum will alfo be A -f- y/ — T'T) as may likewife appear by Involution ; and of the fecond vincu- lum it will be | — </ — _'T. So that another of the Roots of the given Equation will be x = 4 -f- 1 -f- A = 7. Or the Cu- bick Root of the fame firft vincuhtm will be — \ — v/ — i| J and of the fecond will be — i H- ^/ — .11. So that the third Root of the given Equation will be x = 4 -— 4 — T = 3- And in like manner in all other Cubick Equations, when the furd vin- cula include an impoffible quantity, by extracting the Cubick Roots, and then by collecting, the impoffible parts will be exclu- ded, and the three Roots of the Equation will be found, which will always be poffible. But when the aforefaid furd vincula do not include an impoffible quantity, then by Extraction one poffible Root only will be found, and an impoffibility will affect the other two Roots, or will remain (as it ought) in the Conclufion. And a like judgment may be made of higher degrees of Equa- tions.

So that thefe impoffible quantities, in all thefe and many other inftances that might be produced, are fo far from infecting or de- ftroying the truth of thefe Conclufions, that they are the neceflary means and helps of difcovering it. And why may we not conclude the fame of that other fpecies of impoffible quantities, if they muft needs be thought and call'd fo ? Surely it may be allow'd, that if thefe Moments and infinitely little Quantities are to be elteem'd a kind of impoffible Quantities, yet neverthelefs they may be made ufeful, they may affift us, by a juft way of Argumentation, in find- ing the Relations of Velocities, or Fluxions, or other poffible Quan- tities required. And finally, being themfelves duly eliminated and excluded, they may leave us finite, poffible, and intelligible Equa- tions, or Relations of Quantities.

Therefore the admitting and retaining thefe Quantities, how- ever impoffible they may feem to be, the investigating their Pro- perties with our utmoft induftry, and applying thofe Properties to ufe whenever occafion offers, is only keeping within the Rules of Reafon and Analogy; and is alfo following the Example of our fagacious aud illuftrious Author, who of all others has the greateffc right to be our Precedent in thefe matters. 'Tis enlarging the num- ber of general Principles and Methods, which will always greatly

i con-

[ '43 ]

•••v

THE

CONTENTS of the following Comment.

I /JNnotations on the Introduction ; or the Refolution of •*-* Equations by Infinite Series. pag. 143

Sedt. I. Of the nature and conjlruttion of infinite or converg- ing Series. — — P-H3

Sedt. II. The Refolution offimple Equations, or of pure Powers, by infinite Series. = — ~ p. 1 59

Sedt. III. The Refolution of Numeral Affected Equations, p. 1 8 6

Sedt. IV. The Refolution of Specious Equations by infinite Se- ries ; andfirjifor determining the forms of the Series, ami their initial Approximations. P- 1 9 1

Sedt. V. The Refolution ofAJfe&ed fpecious Equations proje- cuted by various Methods of Analyjis. — . . p. 209

Sedt. VI. Tranfition to the Method of Fluxions. P-235

II. Annotations on P rob. i. or, the Relation of the flow- ing Quantities being given-t to determine the Relation of their Fluxions. p. 241

Sedt. I. Concerning Fluxions of the firft Order , and to find their Equations. p.24i

Sedt. II. Concerning Fluxions of fuperior Orders, and the method of deriving their Equations. • > ]

Seft. III. The Geometrical and Mechanical Elements of Fluxions, i p.266

[T] III.

CGNTENTa

,111. Annotations on Prob. 2. ory the Relation of the Fluxions being given, to determine the Relation of the Fluents.

p.277

Se£. I. A particular Solution ; 'with a preparation to the general Solution, by which it is dijlributed into three Cafes. p.a//

Sedl. II. Solution of the firft Cafe of Equations. - p. 282 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 Fluxional, by the afliftance of [uperior Orders of . •> diJ j j r j

r lux ions. - -— - p-3°o

Sedt. VI. An Analytical Appendix, explaining Jome Terms and ExpreJJiom in the foregoing Work. P-32I

,Se<5t. VII. The Conclufwn •, containing a j}:ort recapitulation or review of the whole. - - P-33°

THE Reader is defired to correfl the following Errors, which I hope will be thought but few, and fuch as in works of this kind are hardly to be avoided. 'Tis here ne- ceflary 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 two,_ who fupplyfd my frequent abfence from, the IVefs ; as alfo to the care of a diligent Printer.

ERRATA.

In tie Preface, pag. xiii< lirt, 3. read which is here fubjoin'd. Ibid. 1. 5. for matter read manner. Pag. xxiii. 1. alt. far Preface, &e. nWConclufton of this Work. P- 7. \.T,i.for ~{- read •=.. P. 15. 1.9. ready — !>**+ -&*'• —

-',4, &c. P'. 17. 1. 17. read — — . P. 32.

9*

l,'2j. read - . P. 35, 1. 3: for lOtfjr* read

loxty. P. 62. 1.4. read ~~r'~ . P. 63.!. 31.

firyreatl-y. Ibld:\.ult.for — y- read—y~-

P. 64. \.q. for 2 read z. Ibid. \. 30. read t. P. 82. 1. 17. read zzz. P. 87. 1. 22. read

+ 2A>. Ibid. 1.22,24. reaJAVDK. P.t \,2-[. read. *•• P.-gz. 1.5. read-\- .7 -,', . Ibid.

\.z\.for z read x. P. 104. 1.8. read 6;jt1. P. 109. 1. 33. dele as ofen. P. I 10. 1. 29. read

and v/^1 — xl '=. P. nj, 1 1 7. for Parabola

read Hyperbola. P. 1 19. I. 1 2. read CE x \Q = to the Fluxion of the' Area, ACEG);and

lDxIP =

P.I3J.1.8. readJf - . Ibid.

P.

\. 19, read. ! . P. 135. 1. 15. read

9"

13.8. I. 9. ^WAb&Jifs^AB. P. 145. \.fenult. read 7\~~3. P. 149. 1. 2O. read whkh irt; P. 157. I.i3./-f«^ ax. P.i68. l.j. retd^ax. P-I7I. \.\j.fir Reread $*. P. 1 77.' \.l$.rcait

. ....

P. 204. 1. 1 6. read to 2m, P. 213^ [.-j. far- Species read Series. P. 229, 1. 21* for x — 5 retu( x— 4. /i/V. 1. 24. for x— 4'readx— *. P. 234. 1. 2. ^or yy ready. P. 236. 1. 26. ;vW genera- ting. P. 243. 1. 29. read. — ax*yi—*. P. 284.

1. uit. read i . P. 289. 1. 17. fur right read

j^

left. P. 295. 1. i, 2.- read' — ', x 4 --^-J-ax*. P. 297. \.ig.forjx— ' read y*— '. P. 298. 1.14. read — y. 1\3O4. 1. 20, 21 . dil: -(- be. P^og.

1. 1 8. read am~t> . P. 3 1 7. 1. tilt, read a'-j1.

ADVERTISEMENT.

Lately publijtid by the Author,

THE BRITISH HEMISPHERE, or a Map of a new contrivance, proper for initiating young Minds in the firft 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 as ufe j and fufficient Inftructions are annex'd, to make it intelligible to every Capacity.

Sold by W. REDKNAP, at the Leg and Dial near the Sun Tavern in Fleet-jlreet ; and by ]. SEN EX, at the Globe near St. Dunftan's Church. Price, Haifa Guinea.

and INFINITE SERIES. 339

contribute to the Advancement of true Science. In fhort, it will enable us to make a much greater progrefs and profkience, than we othervvife can do, in cultivating and improving what I have elfe- where call'd The Philofopby of Quantity.

FINIS.

3T .

.. I

I