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

^