METHOD FLUXIONS

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(1)THE. METHOD of FLUXIONS AND. INFINITE SERIES; WITH ITS of CURVE-LINES. Application to the Geometry the. By. Sir. I. NEWTON,^. AA C. S. INVENTOR. Late Prefident of the Royal Society.. ^ranjlated from the. AUTHOR'* LATIN ORIGINAL. not yet. made. To which. is. publick.. fubjoin'd,. A PERPETUAL COMMENT. upon the whole Work,. Confiding of. ANN OTATIONS,. ILLU STRATION In order to. make. Acomplcat Inftitution for By Mafter of. JOHN. and. SUPPLEMENTS,. this Treatife. the ufe o/'. CO L SON,. Sir Jofeph fFilliamfon's free. s,. M.. LEARNERS.. A. andF.R.S.. Mathematical-School at Rochejter.. LONDON: Printed by. And. Sold by. HENRY WOODFALLJ. JOHN NOURSE,. at the. Lamb. M.DCC.XXXVI.. without Temple-Bar..

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(3) 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 of labours, not fo much to. Men. Mathematical eminent rank. the world, as to Perfons of diftinguidi'd For they knew merit and proficience in the fame Studies. fuch only could be competent Judges of very well, that. and. {ration. in. Works, and would receive them with ''the efteem. So far at leaft I can copy after thofe they might deferve. their. as to chufe a Patron for thefe Speculations, great Originals, whofe known skill and abilities in fuch matters will enable. and whofe known candor will incline him I have had in the prefent judge favourably, of the fhare. him to. to judge,. performance.. For. Work, of which. as. to. am. I. the fundamental. only the Interpreter,. part I. of the. know. it. it will need no cannot but pleafe you protection, nor ean it receive a greater recommendation, than to bear the name of its illuftrious Author. However, it very naturally ;. I am fure you, who had the honour (for you think it fo) of the Author's friendship and familiarity in his life-time ; who had his own confent to publifli nil. applies. itfelf to. of fome of his pieces, of a nature not very elegant edition. and who have. an efteem for, as well as knowledge of, his other moft fublime, moil admirable, andjuftly celebrated Works. But A 2. different. from. this. ;. fo. juft.

(4) DEDICATION.. iv. \. But befides thefe motives of a publick nature, I had The many perothers that more nearly concern myfelf. fonal obligations I have received from you, and your generous manner of conferring them, require all the teftimonies 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 hapof your acquaintance,) which is, the free accefs you pinefs 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. 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, are not. ?tg{^tfcu.,. that. is,. improvement of Science. leave, permit me, good Sir,. to the general. Before. I. take. my. to join. my. wiOies to thofe of the publick, that your own ufeful Lucubrations 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 candor, and with that favour and friendship I have fo long ind often experienced,. S. I. is. the earneil requeft of,. R, Your moft obedient humble Servant^ J.. C. OLSON..

(5) (*). THE. PREFACE. Cannot but very much congratulate with my Mathematical Readers, and think it one of the moft forLife, that I have it in to the prefent publick with a moft valuable power fter in Mathematical and Anecdote, of the greatefl. tunate ciicumftances of. my. my. Ma. the World. And Philofophical Knowledge, of this Anecdote is an element becaufe the much fo more, ry nature, to his other moft arduous and fubh'me and introductory preparatory for the instruction of Novices Speculations, and intended by himfelf and Learners. I therefore gladly embraced the opportunity that was put into my hands, of publishing this pofthumous Work, bethat ever appear 'd in. 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, I thought it highly with fome additional Remarks of my own. and of the great Author, as reputation injurious to the memory 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 caufe. I. found. it. World for general Inftruction. And more efpecially at a when the Principles of the Method here taught have been. learned. time. fcrupuloufly fifted and examin'd, have been vigorouily .oppofed and. (we may fay) ignominioufly rejected as infufficient, by fome Mathematical Gentlemen, who feem not to have derived their knowledge of them from their only true Source, that is, from cur Author's. And on the other Treatife wrote exprefsly to explain them. of Method have this been hand, the Principles zealouily and commendably defended by other Mathematical Gentlemen, who yet own. a. feem.

(6) x. lie. fern. have been as. PREFACE.. acquainted with this Work, (or at leaft 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 octo. little. to have over-look'd. and far from that copioufnefs with which he treats of it and illuftrates it with a here, 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 this with the confent" and under the Work, publishing 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, calional,. The Work would. have then appear'd with a double advantage, as Emendations of its great Author, and likewife in And among the faffing through the hands of fo able an Editor. other good effects of this publication, it poffibly might have prevented 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 inferior hand. receiving the. la ft. was now become highly necefTary, that at laft the great himfelf fhould interpofe, fhould produce his genuine MeIjaac thod of Fluxions, and bring it to the teft of all impartial and confiderate 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 accomit with an pany ample Commentary, according to the beft of For. it. Sir. my. and (I believe) according to the mind and intention of the Author, wherever I thought it needful ; and particularly with an Eye fkill,. to the fore-mention'd In which I have endeavoui'd to Controverfy. 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.

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

(8) The. PREFACE.. little Quantities, and infinite orders and gradations of thefe, thefe Quantities affume not relatively but absolutely fuch. They without as that Quantities finnd any ceremony, Jewel, actually and. infinitely. &. and make Computations with them accordingly ; tlie refult of which muft needs be as precarious, as the abfblute exiftence 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 certainly appear very notional and vifionary. Thefe will be the inconveniencies that will arife, if we do not Abfolute rightly diftinguifh between abfolute and relative Infinity. can be as the either of our fuch, Infinity, hardly object 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 His Computations gradual and neceffary progrefs of diminution. finite and intelligible commence by Quantities ; and then at always laft he inquires what will be the refult in certain circumftances, when fuch or fuch Quantities are diminim'd in infinitum. This is a conftant practice even in common Algebra and Geometry, and is no more than defcending from a general Propofition, to a particular Cafe which is certainly included in it. And from thefe eafy Principles, managed with a vaft deal of fkill and fagacity, he deduces his Method 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 obvioufly exift,. here or elfewhere, directly or indiredly, exprefly or tacitely, to the moft curious Difcoveries in Art and Nature, and to the fublimeft Theories may defervedly efteem it as the greateft Work of nobleft Effort that ever was made by the Hun an and as the Genius, Mind. Indeed it muft be own'd, that many uftful Improvement?, and new Applications, have been fince made by others, and probaFor it is no mean excellence of bly will be ftill made every day.. We. :. Method, that it is doubtlefs ftill capable of a greater degree of and will always afford an inexhauftible fund of curious perfection 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. this. ;. J. great.

(9) The. PREFACE.. xlii. for ; I fhall here endeavour to obviate fome of his great veneration to the Method of Fluxions, particularly fuch as principal Objections I have not touch'd upon in Comment, which is foon to follow.. my. He. thinks cur Author has not proceeded in a demonftrative and fcientifical matter, in his Princip. lib. 2. km. 2. where he deduces the Moment of a Rectangle, whole Sides are fuppofed to be variable I fhall reprefent the matter Analytically thus, Lines. agreeably (I. mind of the Author. Y be two variable Lines,. think) to the. X. and. or Quantities, which at different periods of time acquire different values, by flowing or increaor alike inequably. For inflance, let fing continually, either equably. Let. becomes A fa, A, b and B -+f3, B, f fuccefiively b, are any quantities that may be the fame periods of time the variable. there be three periods of time, at. and and. A -+- 7 a. ;. reflectively. aiTumed. ;. Y. which. X. becomes B where A, a, B,. and. at pleafure.. Then. XY. Produ<ft or Rectangle +- f * x B -+- h, that. A. AB -+- f^B -f- 7$ A -f- âb.. at. will. become A". AB Now. T <?B. is,. in. fa x B. fM.. f4,. -f-. ab,. the interval from the. AB, and AB, and. firft. period. of time to the fecond, in which X from being A fa is become A, the is become Product XY and in which Y from being B B, 7^ AB that becomes from being AB -f- âb is, f^B by Subis f#B -+traction, its whole Increment during that interval of time to the in the fecond interval from the And âb. period in which Y from and X A in becomes which A-f-ftZ, third, being frcm being B becomes B -hf^, the Product XY from being AB becomes AB-f- ffiB -f f 4A -+- -âb that is, by Subtraction, its whole Increment during that interval is 7,76 7Â -+- âb. _ 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 thus: While X f.ows from A tne \a to A, and by Operation, to therce to A -f- ft?, or Y flows f-om B B, and thence to f3 -ithe will Product flow fiom XY AB B f<?B f3A -f- ab f A, to AB, ?nd thence to AB -+- f^B -J'k -f- âb > therefore by Subtraction the whole Increment during that interval of time will be tfB-4-M. QÊ. D. This may eafily be illuftrated by Numbers thus: Make A,rf,B,/, equal to 9, 4, i 5, 6, refpeclively; (or any other Numbers to be affumed at pleafure.) Then the three fucceffive values of X will be the three fucceffive values of Y will be 12, 15, 18, 7, 9, ii, and. iA. -,. fA. ;. +. ,. +. reipcciivcly..

(10) PREFACE.. The. xiv. Produd XY = 4xic-f- 6x9= =. the three fucceflive values of the. Alfo refpeftively. will be 84, 135, 198. 19. 8_8 4. Thus. But rtB-f-M. 114. Q.E. O.. .. the. Lemma. will be. true of any conceivable finite Increments 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. nitely. Moments. the cafe of. 15ut in. A. A, and. ftf,. to be confider'd, that X, or defito be taken indifferently for and definitely B f/;, B, B -+- ~b.. it is. A -+-. a, are. the fame Quantity ; as alfo Y, the want of this Confutation has occafion'd not a few per-. And. 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 A, and therefore b a, and then the refpedtively ; make B A .or of x will be Moment Aa -+- aA, or 2aA. Again, A, A*, A a and therefore b-=. zaA, and then the Moment of make B. =. =. =. =. ,. AxA*, or A', will be 2rfA 4 -f- aA 1 , or 3Â*. Again, make B s 5 and therefore l> , âA -, and then the Moment of xA*, or. A A. =. A. 3 3 3 3<?A -4-rfA , or 4#A Again, make 3 therefore ^ and of then the Moment , 4Â be 4<?A 4 -i-tfA 4 , or 5<zA 4 And fo on in infinitum.. 4. ,. will be. =. .. Ax A. B==A-, 4. .. m to. general, afluming reprefent any integer affirmative 1 Moment of A* will be .. Number,. maA". Now. and. or A', will , Therefore in. the. =. i, (where m is any integer affirmative of Unity, or any other conftant and the becaufe Moment Number,) A* we (hall have x Mom. A~m -f- A~m x Mom. p; quantity, is 110 A- x Mom. A" But Mom. A" o, or Mom. maA m ~*, as found before ; therefore Mom. A"* A~ iw x maA-"-' ma A"-' Therefore the Moment of Am will be ~ m maA , when m is any integer Number, whether affirmative or. becaufe. A*. =. A"=. =. x. A^. ra. A~"=. =. =. .. .. I. negative.. n. And may. or A"=. B" we put A" where m and be any integer Numbers, affirmative or negative ; then we. mall have. is. the. =B,. univerfally, if. ma A"-*. Moment. =. ;.^B"^'. of B, or of. ,. A". or. .. b=. mgA<. So that the. ,. =. -aA. Moment. i,. of. which. A". will. be.

(11) P E E F A C. The be. wtfA"*". rtill. 1. whether. ,. ;;/. fraction.. The Moment being </C +- cD. of. AB. fuppofe. ;. be affirmative or negative, integer or. = MAB,. aB, and the Moment of CD and therefore d-=. b& +- aB,. -+-. being. D. xv. E.. Moment of ABC will be bA +- aB xC r AB. And likewife the Moment of ~ m n B maA. And fo of any others. connexion between the Method of Mo-. and then by Subftitution the -f- c. AB. =. MC. -+-. rfBC -h. l />B"-'A" -fthere is fo near a ments and the Method of Fluxions, that it will be very eafy to pafs from the one to the other. For the Fluxions or Velocities of increafe, are always proportional to the contemporary Moments. Thus if for A, B, C, &c. we write x, y, z, &c. for a, b, c, &c. we may Then the Fluxion of xy will be xy -f- xy, the write x, y, z, &c. m whether m be integer or fraction, will be rnxx*-* Fluxion of x affiimative or negative; the Fluxion of xyz will be xyz -f- xyz -f-. A*B". will be. .. Now. ,. x my n. xjz, and the Fluxion of. will be. mxx m -*y. -J-. fo of the reft. Or the former Inquiry. nx myy"~ s. .. And. may be placed in another view, thus 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 And while X, contemporary values of the variable Product XY. from A -f- a, or Y flows value to its A by increafing perpetually, flows from B to B -f- b ; XY at the fame time will flow from AB to AB +- aB -+- bA. -f- ab t during which time its whole Increment, Or in as appears by Subtraction, will become aB -h bh. -+- ab. 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 fucLet. :. A and. ceflive values. of. the Product. XY. ah-. 42. 48. And. -f-. thus. it. Y. Alib the two fucceflive values of But the Increment aB -+- t>A -J-. will be 12, 18. will be 84, 198.. 24= 14=. 1 -+198 84, as before. will be as to all finite Increments : But when the In-. crements become Moments, that. is,. when a and b are fo far dirniA and B at the fame time aB or Â, (for aB. ab ::. nifh'd, as to become infinitely lefs than ab will become infinitely lefs than either. and bA. ab. A. a. and therefore. ;. will vanifh in refpect of them. In which cafe the Moment of the Product or Rectangle This perhaps is the more obvious and will be aB -+- bA, as before. in the t relent of direct way proceeding, Inquiry but, as there was room for choice, our Author thought fit to chufe the former way,,. B.. b,. ::. y. ). it. ;. as.

(12) The. xvi. PREFACE.. elegant, and in which he was under no neceflity of having recourfe to that Principle, that quantities arifing in an Equation, which are infinitely lefs than the others, may be neglected or exas the. more. companion of. in. punged. Now. thofe others.. avoid the ufe of. to. tho' otherwife a true one, was all the Artifice ufed which certainly was a very fair and justifiable one.. this Principle, this occaiion,. on. 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 manner, 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 have the Equation x-fc-X\*. fhall. =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 -+-. nx"~'X. ^=-^*X. n x. -+-. *. we. fhall. have y. + n x *~ x '-^-V^X. moving the equal Quantities y and. x",. it. -f-. 3 ,. =. Y. =. &c. or. nx n ~. xn re -. X +that when X deT. Y. will be. l. So n x ?-^- x ^^x'-'^X 3 , &c. Y will here denote notes the given Increment of the variable quantity 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. 2, Suppofe n then Y 2xX -+- X l That is, while x flows or increafes to x +- X, .v* in the fame time, 2xX -+-X 1 will increafe by its Increment Y ny.. *. ^-x"--X. -+-. A,-,. to. .v. 1. 4-. fuppofe. =. = 2xX = fl. creafes to. = =. .. X. ,. which we otherwife know to be true. Again, -ja 1 3 Or while x in*. 3, then Y 3* X -+- 3*X H- X J a x r+- X, x"> by its Increment Y X X3 -h 3^ 3^X. will increafe. 1. ,. to x* -f-. =. 3*. 1. X -+- ^xX. 1. -+- X. 3 .. .. +. And. whereby we may plainly perceive, Conclufion mud be certain and indubitable.. particular cafes,. fo. in all ,other. that. this general. This.

(13) Tie. PREFACE.. xvii. X. and Series therefore will be always true, let the Augments docs not at all defo little for the truth or ever fo ; ever be great, of their when circumftance the on magnitude. Nay, they are. This. Y. pend. they become Moments, it muft be true alfo, But when and Y are diby virtue of the general Conclufion. minifh'd in infinitum, fo as to become at laft infinitely little, the muft needs vanifli firft, as being relatively of an of or infinitely little,. when. X. X. greater 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 confeif there were other Terms in the Equation, quently would vanifh, 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 vanifli in proportion to each other as vanifhing quantities, yet they f have therefore learn 'd at laft, that the Moment by j to nx"~ vali e than the fmaller powers. infinitely lefs. .. We. which x increafes, or X, is to the contemporary Moment by which And their Fluxions, or Velox a increafes, or Y, as i is to nx"~ in the fame cities of increafe, being proportion as their fynchronous s. .. 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 have legitimately derived made ufe of in this Inveftigation. this general Conclufion in finite Quantities, that in all cafes the re-. Moments, we. fhall. We. lation. of the Increments will be. of which one particular cafe nually to decreafe, till they. Y. is,. =. nx"~. when. X. l. X+. and. Y. x ~~x*'-1X*, &c. are fuppofed conti-. But by finally terminate in nothing. 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 This therefore is their ultimate Ratio, vanifhing, and not before. the Ratio of their. and x n continually general tine. Moments, Fluxions, increafe. or decreafe.. or Velocities, by which x to argue from a. Now. Theorem. of the moft. to a particular cafe contain'd under it, is certainly legitimate and logical, as well as one of the mofl ufual. in the whole compafs of the Mathemcto ftand we have made and after that object here, for fome quantity, we are not at liberty to make them nothing, or no is not an Objection againft the quantity, or vanishing quantities,. and. ufeful. ticks.. ways of arguing,. X. To. b. Y. Method.

(14) Tte. XVlll. 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 require the contrary. definite. any. For general Numbers, as fuch, may ftand for in the whole Numerical Scale which Scale. Numbers. ;. be thus commodioufly 2> reprefented, &c. 3, &c. where all i, o, i, 2, 3,4, poffible fractional Numbers, intermediate 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 (I. think). We. may. are likewife told, that. we may. not give fuch values to general as they could not receive at firft ; which if adafterwards, Symbols mitted is, I think, nothing to the prefent purpofe. It is always moft eafy and natural, as well as moll regular, inftruclive, and elegant, to. make our. Inquiries as. much. and. to defcend to particular cafes nearly brought to a conclufion.. by degrees, But this is a point of convenience. only, and not a point of neceffity. flead of defcending. from. finite. ments, or vanifhing Quantities,. Terms as may be, when the Problem is. in general. Thus. Increments. in the prefent cafe, into infinitely little. Mo-. we might. begin our Computation with thofe Moments themfelves, and yet we mould arrive at the As a proof of which we may confult our Aufame Conclufions. thor's ownDemonftration of hisMethod, in oag. 24. of this Treatife. x In fhort, to require this is jufl the fame thing 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 are hardly utterly difclaim,) but in refpedl of the Science itfelf. to that fo Science is the believe, brought perfectly regular and uni-. We. form,.

(15) 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. do not readily admit, or eafily comprehend, that Quantities have an infinite number of curious and fubtile properties, fome near and obvious, others remote and abftrufe, which are all link'd together by a neceffary connexion, or by a perpetual chain, and are then only difcoverable when regularly and clofely purfued ; and require our truft and confidence in the Science, as well as our induftry, application, and obftinate perfeverance, our fagacity and penetration, in That Nature is ever order to their being brought into full light.. We. .. with. confiftent. herfelf,. and never proceeds. faltum, or at random, but. is. in thefe Speculations. infinitely fcrupulous. and. per. felicitous,. as. adhering to Rule and Analogy. That whenever we 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 That no properties of Quantity at regular and juft Conclufions. are intirely deftructible, or are totally loft and abolim'd, even tho' profecuted to infinity itfelf j for if we fuppofe fome Quantities to become infinitely great, or infinitely little, or nothing, or lefs than nothing, yet other Quantities that have a certain relation to them. we may. fay,. in. make any. and often finite alterations, will fymand with conform to 'em in all their changes ; and them, pathize their will always preferve 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 will only undergo proportional,. Curve-lines to Algebraical Equations. ral. That when we purfue gene-. Nature is infinitely prolifick in particulars that will from them, whether in a direct rubordination, or whether they. Inquiries,. refult. collaterally ; or even in particular Problems, we may often that thefe are only certain cafes of fomething more perceive general, and may afford good hints and afiiftances to a fagacious Analyft, for. branch out. 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. and which we {hall never be or know the whole ufe of, but from much pracof, attentive confideration ; but more efpecially by a diligent b 2 peruial, find a difficulty to admit,. fully convinced tice. and.

(16) xx. The. P. R E. F. A C. E.. and clofe examination, of this and the other Works of our He abounded in thefe fublime views and inAuthor. had acquired an accurate and habitual knowledge of all thefe, quiries, and of many more general Laws, or Mathematical Principles of a not improperly be call'd The Philofophy of fuperior kind, which may aflifted and which, Quantity ; by his great Genius and Sagacity, together with his great natural application, enabled him to become fo compleat a Matter in the higher Geometry, and particularly in the Art of Invention. This Art, which he poflefl in the greateft perfection imaginable, is indeed the fublimeft, as well as the moft diffiperuial,. illuftrious. cult of all Arts, if it properly may be call'd fuch ; as not being reducible to any certain Rules, nor can be deliver'd by any Precepts, but. wholly owing to a happy fagacity, or rather to a kind of divine Enthufiafm. To improve Inventions already made, to carry them. is. 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 eû, 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. the nobleft. had ever been. Endowment. fet before,. that a. nullius ante trita folo. human Mind. is. capable. of,. is. this. ;. is. referved. and was the peculiar and diftinguifhing Character of our great Mathematical Philofopher. He had acquired a compleat knowledge of the Philofophy of Quanor of its moft eflential and moft general Laws ; had confider'd it tity, 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 make them confefs their Secrets, and difcover all poflible ways, to for the chofen. few. quos Jupiter tequus amavit,. 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 The firft will be the Introduction, into thefe three parts. it or the Method of infinite Series. The fecond is the Method of The third is the application of both Fluxions, properly fo culi'd. thefe. Methods. to. fome very general and curious Speculations,. Geometry of Curve-lines. As to the firft, which is the Method of infinite the Author opens a new kind of Arithrnetick, (new. chiefly. in the. time of his writing. this,). Series, in this at leaft at the. or rather he vaftly improves the old.. For he.

(17) 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 infinite Series,. by which the fame analogy. farther advanced towards perfection.. will be. ftill. carry'd on,. Then he fhews how. all. and. com-. be reduced to fuch Series, as will plicate Algebraical Expreffions may 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-. whether they are Roots of pure Powers, or of affected Equawhich are therefore to be refolved by a perpetual Extraction. And by the way, he teaches us a very general and commodious Method for extracting the Roots of affected Equations in Numbers. fion. ;. or. tions,. And this is chiefly the fubftance of The Method of Fluxions comes. his. Method of. infinite Series.. next to be deliver'd, which indeed is principally intended, and to which the other is only preparatory and fubfervient. Here the Author difplays his whole fkill, and fhews the great extent of his Genius. The chief difficulties of this he reduces to the Solution of two Problems, belonging to the abftract or Rational Mechanicks. For the direct Method of Fluxions, as it is now call'd, amounts to this Mechanical Problem, tte length of the ibed being continually given, to find the Velocity of the Modefer Aifo the inverfe Method of Fluxions has, tion at any time propofcd. 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. Space. So that upon the compleat Analytical or Geometritime propofcd. cal Solution of thefe two Problems, in all their varieties, he builds. whole Method. His firft Problem, which. his. being given,. to. f. The relation 6J the owing Quantities is, the determine relation of their Fhixiom, he difpatches. A. He does not propofe this, as is ufualiy done, flowvery generally. ing Quantity being given, to find its Fluxion ; for this gives us too lax and vague an Idea of the thing, and does not fhew fufficiently. that Comparifon, which is here always to be understood. Fluents and Fluxions are things of a relative n.iture, and two at leafr,. fuppofe. whofe. mould always be exprefs'd bv Equations. He all fhould be reduced to Equations, by which. relation or relations. requires therefore that the relation of the flowing Quantities will be. exhibited, and their. comparative.

(18) f/jg. xxii. PREFACE.. comparative magnitudes will be more eafily eftimated ; as alfo the And befides, by this comparative magnitudes of their Fluxions. means he has an opportunity of refolving the Problem much more For in the ufual way of generally than is commonly done. taking we are confined to. the Indices of the Powers, which are Fluxions,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' different 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. the given Equation may comprehend feveral variable 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.. He. (hews. And. how. alfo. derived and demonitrated from the properties of Modoes 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 relation of the Fluxions of Quantities, to the relation of find thofe Quantities, or Fluents, to one another ; which is the diredt Converfe of the This indeed is an operofe and difficult foregoing Problem. all this is. He. ments.. Problem,. taking dreis. ;. it. in. its full. which. extent, and, requires all our Author's fkill and adyet hefolyes very generally, chiefly by the affiftance of his. Series. He firfl teaches how we may return from the Fluxional Equation given, to its correfponding finite Fluential or when be that can done. But when it cannot be Algebraical Equation, or when there is no finiie fuch .done, Algebraical Equation, as is moft commonly the cafe, yet however he finds the Root of that. Method of infinite. Equation. by an. infinite. converging. Series,. which anfwers the fame purpofe. the Root, or Fluent required, by. often he mews how to find an infinite number of fuch Series. His proceffes for extracting thefe Roots are peculiar to himfelf, and always contrived with much fubtilty 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 to whether Geometrical or Mechanical ; or howCurves, Tangents 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. and.

(19) fbe. PREFACE.. xxiii. and purfues many other ingenious Problems, that offer themfelves by After which he difcufTes another very fubtile and intirely the way. new Problem about Curves, which is, to determine the quality of the Curvity of any Curve, or how its Curvature varies in its progrefs or inequability. through the different parts, in refpect of equability He then applies himfelf to confider the Areas of Curves, and fhews. 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 Hyperus. bola, or of any other Curve that (hall be affign'd to Mechanical as well as Geometrical Curves.. ;. which he extends. 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 afcerAnd by the way he fquares the Areas. taining the Limits of fuch the and and Circle Quadrature of this to the conapplies Hyperbola, of Logarithms. But chiefly he collects veryftructing of a Canon of Quadratures, for readily finding the general and ufeful Tables 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 himThe ufe and application of thefe felf, in his Treatife of Quadratures. 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 Curvelines 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 And concludes in general, with rectifying any Curve-lines affign'd. that may be propofed, either by the aflifbncc of his Tables of QuadraAnd tures, when that can be done, or however. by infinite Series. this. is. chiefly the fubflance of the prefent. that perhaps". may. be expected, of what. I. Work.. As. to ,the account. have added in. my Anno-. the inquifitive Reader to the PrefacCj tations ; will go before that part of the Work. I. {hall. refer. which. THE.

(20) -. ;. 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. p.. 44. p.. 46. Maxima and Minima. Prob. 3.. To determine the. Prob. 4.. To draw Tangents. 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.. To find Curves whofe Areas may be compared. 8.. of Quantities,. to Curves.. Conic SecJions.. Prob. 9.. Prob.. 1 1.. thofe. of the p. 8. To find the Quadrature of any Curve. Prob. 10. To find any number of. to. ajjigrid.. rettifiable Curves.. p. p.. 1. 86. 124. To find Curves whofe Lines may be compared with any Curvelines ajfigrid.. Prob. 12. To rectify any Curve-lines ajpgn'd.. p.. 129. p.. 134.

(21) THE. METHOD. of. FLUXIONS,. AND. INFINITE SERIES. INTRODUCTION. :. Or, the Refolution of Equations. by Infinite Series.. IAVING. obferved that moft of our modern Geome-neglecting the Synthetical Method of the Ancients; have apply'd themfelves chiefly to the the affiftance cultivating of the Analytical Art ; by. tricians,. of which they have been able to overcome fo many and fo great difficulties, that they feem to have exhaufted all the of Curves, and Speculations of Geometry, excepting the Quadrature 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 and the fame Operations in common Numbers; nor do they Species, :. feem to. differ,. except in the Characters by which they. B. are re-. prefented,..

(22) 'The. Method of FLUXIONS,. firft being general and indefinite, prefented, the I cannot but wonder that nite and particular. and the other defino body has thought of accommodating the lately-difcover'd Doctrine of Decimal Fractions in like manner to Species, (unlels you will except the Quadrature 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 Arithmetick ; the Operations of Addition, Subtraction, Multiplication, Divifion, and Extraction of Roots, may eafily be learned from thence,, and the if the Learner be but fk.ill'd in Decimal Arithmetick, and obferves the that obtains beVulgar Algebra, correfpondence and Decimal Fractions Terms continued. tween Algebraick infinitely For as in Numbers, the Places towards the right-hand continually decreafe in a Decimal or Subdecuple Proportion ; fo it is in Species :. when. the Terms are difpofed, (as is often enjoin 'd in in an uniform Progreflion infinitely continued, acwhat follows,) cording to the Order of the Dimenfions of any Numerator or De-. refpedtively,. And as the convenience of Decimals is this, that all and Radicals, being reduced to them, in fome meaFractions vulgar fure acquire the nature of Integers, and may be managed as fuch ; nominator.. it is a convenience attending infinite Series in Species, that all kinds of complicate Terms, ( fuch as Fractions whofe Denominators are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Clafs. fo. of fimple Quantities ; that is, to an infinite Series of Fractions, whofe Numerators and Denominators are fimple Terms ; which will no under thofe difficulties, that in the other form feem'd longer labour. almoft infuperable. Firft therefore I mail fhew how thefe Reductions are to be perform'd, or how any compound Quantities may be reduced to fuch fimple Terms, efpecially when the Methods of computing are not obvious. Then I fhall apply this Analyfis to the Solution of Problems. Divifion and Extraction of Roots will be 3. Reduction by plain from the following Examples, when you compare like Methods of Operation in Decimal and in Specious Arithmetick.. Examples.

(23) and INFINITE SERIES,. 3 .. ..ift. Examples of Reduttion by Dhifwn.. The. .4.. Fraction. being propofed, divide. ^. following manner. aa by b. +. IjfM/l^^ x. in the. :. aax 1. aax. faa. aax*. a a x*. .. " .. aax aax. O. --7. -f-O. aax*. o. -+-. -. o. ~. +o **. *. flt. Jf*. ;. v *-\ The Quotient ". -rrî_. which. i r therefore. being. Series,. Or making x. j^.. infinitely. the. tf*^ a* x* T _-JT-+ -T_. ^^. *. is. - - ?4 **. + toaa + o e. (the Quotient will be * r~ _ _ , % found as by the foregoing Procefs. ,. #. I. 6.. +. manner the Fraction. In like. 5.. -{-. x4. And. '. A:*. H- x. -j-. 34x. T ,. ~-. x-*. will. r. i+x*. s. 13**. Here. or to , &c. 9 v " 2 *. the Fraction. i1. yx. 8. be. equivalent. of the Divifor, in. x. 4. n. this. ~. will. be. reduced. to. manner, ~. 1^ *. #-* _f. ^-. be. a* X+. .. rr + T7-, &c.. .. will. continued,. Term. firft. a* x*. 1. V &c AV reduced. to. ^-8. to. 2x^. 2x. 3*. &c.. will be. proper to obferve, that I make ufe of x-', - &c. of for &c. i, ;r 7,' x-', x-', x-*, xs, xi, x^, xl, A4, &c. for v/x, v/*S \/ x *> vx , ^x l , &c. and of x'^, x-f. x i &c for **** 1Ui ' * i 7.. it. -. ,. ^x. j_^. ^ ? >' y-^.'. &c.. And. this. by the Rule of Analogy,. as. apprehended from fuch Geometrical Progreflions as thefe x> (or i,) a"*,*-',*'*, x, *, &c.. B. 2. may. be. x,. x*,. ;. 8.. '* /. Av.

(24) ffie. Method of FLUXIONS,. er for -In the fame manner. q.. And. 1^ + 1^!,. 8.. .and. aa. xv|*. inftead. may. be wrote. &c.. ',. thus inftead. &c.. xx may be wrote aa of the Square of aa xx; and. >. of^/aa. xxl^. 3. inftead of v/. So that we may not improperly diftinguim Powers into Affirmative and Negative, Integral and Fractional. 10.. 11.. tract. Examples of Reduction by Extraction of Roots. The Quantity aa -+- xx being propofed, you may thus exits. Square-Root.. - _i_ XX V v (a aa-+^". -4-. 4r. Sfl3. 2a. 5. i6*. x -. 1287. 4- J. -. '. c*. 2560*. aa. xx 4.. a*. x*. ~*. a 4. 64 64 a. X* sT*. 64^8. ~ i;. z$6a'^. _. x. 64^ 5* 64 a. ". 6. + 2^1, &c 7^ _ n-i7R/3 7' + I. _-1. 256 *. z8rt 8. .. /7lt>. 1. ,__i!_lll, &c.'. found to be. ^. 4a~\--^^T,&C. Where it may be obferved, that towards the end of the Operation I neglect all thofe Terms, whofe Dimenfions would exceed the Dimenfions of the laft Term, to which I intend only to continue the Root,. Jo that the Root. fuppofe to. is. *'. ,2..

(25) and INFINITE SERIES.. Order of the Terms may be inverted in this manbe in which cafe the Root will be found to. iz. Alfo the. xx. ner. 5. +- aa,. aa 10 A*. Thus. 13.. the. 15.. . i. 1. of. xx. .. <z. *. A-. .. AT. *. '-. *. A.'. A'. g. i a. *. A-. 4. +. ,'_. n. 3 x- 6. + |^^4. T^. -f-. T^. ^frx H- rV^ x -+-. 6 ,. But thefe Operations, by due preparation,. 17.. Sec.. ,. .. &c-. .c. j. and more -. .. becomes. it. -|- -i/^r. -4-. **, 8cc.. 4-.v*. .Ii*--.4-. ;,,.. over by adually dividing, i. 4-. i. ^ -Jj --^7 Tr --b*X* ^ '. i**. .. f. +. is. xx is #'" ## is a -f-. x. And v/r^rr,. 6.. A-. Root of aa. The Root Of -+-. 14.. iz. &c.. may. very often. the foregoing. Example to find \/;_***' if the Form of the Numerator and Denominator had not been the bxx, which would fame, I might have multiply'd each by </ 1 be abbreviated;. as. in. 1. -. yî -f-rt*. have produced. ab x. b. I. *. and the. &. of the work might. reft. xx. 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 as. x">. \fi. V x i! 2x t. xx. -}-. ;. ^/axx. -\-. A-. *. 3. infinite Series confifting. Of 19.. As. xi. " x-{-xx. v. _. 2X. j. x.1. may. be reduced to. '. of iimple Terms.. the ReduStion of offered Equations.. to aftedled Equations, we mufl be fomething more parhow their Roots are to be reduced to fuch Se-. ticular in explaining ries as thefe ; becaufe. their Doctrine. in. Numbers, as hitherto devery perplexed, and incumber'd with fuperfluous Operations, fo as not to afford proper Specimens for performing the Work in Species. I fhall therefore firfl (hew how the Refoluliver'd. by Mathematicians,. is.

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

(27) and INFINITE SERIES. 21. But the. Work may. this Method, efpecially in far firft determin'd. be. much. 7. abbreviated towards the end by. Equations of. many. Dimenfions.. Having. how. you intend to extract the Root, count fo after the firft Figure of the Coefficient of the laft Term many places but one, of the Equations that refult on the right fide of the Diagram, as there remain places to be fill'd up in the Quote, and reject But in the laft Term the Decimals may the Decimals that follow. be neglected, after that are reject. fill'd. all. up. many more places as are the decimal places Quote. And in the antepenultimate Term And fo on, by proafter fo many fewer places. fo. in the. that are. ceeding Arithmetically, according to that Interval of places: Or, is the fame thing, you may cut off every where fo many in as the penultimate Term, fo that their loweft places may Figures 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 , Thofe Figures therefore although its Coefficient be 0,99999, for the being expunged, following Operation there arifes the Sum 1 which by Divifion, continued as far as 0,0005416 -f- 1,1 62?%. which. the Term prefcribed, gives 0,00004852 for r, which compleats Then fubtracting the negative the Quote to the Period required. from of the the affirmative Quote part, there arifes 2,09455148 part for the Root of the propofed Equation. 22. It may likewife be obferved, that at the beginning of the. had doubted whether o, i -f-/> was a fufficient Apto the Root, inftead of i o, I might have iof> proximation i o, and fo have wrote the firft fuppos'd that o/** -f- i op of its Root in the as nearer to And Quote, Figure. Work,. if I. =. =. being nothing. 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 antepenultimate Term. And indeed you will often fave fome in this. manner. in Equations of. it. may. many. Dimensions,. if. you feek. pains, efpecially for all the Figures. to-.

(28) Tie Method of FLUXION'S,. 8. to be added to the. this. manner. lefier. lafl. Terms of. Quote in Root out of the three. that. ;. its. is,. if. you. extract the. 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, 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 it is. much the leaft or greateft of The reafon of which is, that. is, or may be fuppos'd to be, or neareft to a given Quantity.. either all,. 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-. And. traction of Roots.. make. for this Species, in what follows, I z ; as alfo I fliall ufe y, p, q, r, s,. mall. &c. generally extracted. Radical to be for the Species or furd Quantities, 25. Secondly, when any complex Fractions, or to in to arife afterwards occur the happen propofed Equation, in. are. ufe of. or. A:. the Procefs, they ought to be removed by known to Analyfts. As if fufficiently. y* -+- j. 1. duct by*. =. x"=. 1 >'. o,.. Kyi'-l-fry*. fuppofe y x b have i; J -+- &*v* whence extracting the Root v r in order to obtain y.. have. x, and from the Promultiply by b Or bx^ -+ x*-= o extract the Root y.. x=v,. we might we mould. fuch Methods as. we mould. and then writing. ^~x. for. =y = =. ^hx' -+. x we might divide the Quote by b Affo if the Equation j 3 xy* -f- x$. fax*. 6. -\- 3/5***. xj. =. were propofed, we might put v, and will arife v 6 for and z* vv for there x, ting y, which Equation being refolved, y and x may be. y?=. t. o,. x,,. o. and fo wrio; -f- z* reftored. For the Root will befound^=2-f-s3_|_5~s 5cc.andrei1:onngjyandA;, we have x^ -f- x -+- 6x^ &c. dien fquaring, y =x^-+- 2X J ~f- 13*", &c.. y* 26. After the fame manner if there mould be found negative Dimenfions ofx and jy, they may be removed by multiplying by the fame '2.x~ x andjy. As if we had the Equation x*-}-T x*-y~ i6y- =o, 1 3 5 and and x arife there would x*y* -+- 3# jy multiply by j , 2_v. =. z,. z=v. 5. I. I. 3. >. A. O.. J. -r. And U. 1. tjie. -r-v. Equation were. x. =. 3. aa. ~. 2ai. a + ?r y i. 4. 1. \.. by;.

(29) and INFINITE SERIES. by multiplying. And. fo. into. }. there. jy. would. arife. i. xy*-=.a' y*. of others.. when. the Equation is thus prepared, the work be^ of the Quote ; concerning which, as gins by finding the firfr. Term alfo for finding the following Terms, we have this general Rule, when the indefinite Species (x or 2) is fuppofed to be fmall ; to 27. Thirdly,. which Caie the other two Cafes are reducible. 28. Of all the Terms, in which the Radical. Species. (y,/>, q,. or. not found, chufe the loweft in refpect of the Dimenlions &c.) 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 continued from the Term fir ft afTumed to this Term, may defcend as is. r,. much. as. may. or. be,. afcend. as. little. as. may. be.. And. if. there. any other Terms, whofe Dimenfions may fall in with this muft be taken in 1 ikeProgreflion continued at pleafure, they thus felected, and made equal to wife. Laftly, from thefe Terms are. and write nothing, find the Value of the faid Radical Species, the Quote.. may be more. 29. But that this Rule. explain right. it. farther. Angle BAC,. its. fides. in. clearly apprehended, I fhall. by help of the following Diagram. divide. it. AB, AC,. into. Making. equal parts,. a. and. Angular Space into equal Squares or Parallelograms, which you may conceive to be denominated from the Dimenfions of the Species x and y, raifing Perpendiculars, diftribute the. Then, when mark fuch of propofed,. as they are here infcribed.. any Equation. is. B. A4. the Parallelograms as correfpond to all and let a Ruler be apply'd its Terms, to two, or perhaps more, of the Parallelet one lograms fo mark'd, of which 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 Then felecl: thofe Terms of the Equation the Ruler, lie above it. which are reprefented by the Parallelograms that touch the Ruler, and from them find the Quantity to be put in the Quote. s out of the Equation y 6 30. Thus to extract the Root y 5xy -+1. )'*. ja*x y. 1. +6a. i. x*-\-&. 1. x4=o, C. I. mark. the Parallelograms belong-.

(30) The Method of. 10. ing to the Terms of this Equation with the Mark #, as you fee here Then I apply the Ruler done. to the lower of the Parallelo-. FLUXIONS,. B *. DE. mark'd in the left-hand Column, and I make it turn round towards the right-hand from the grams. C. A. 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. 5 that the places fo touch'd belong to x 3 , x*-y* y Therefore and_y z 6 as if from the Terms y to equal 7a x*-y -}-6a*x*, nothing, (and .. <L. moreover,. if. you. 6=. reduced to v 6. o, by making 7^*4of and find it to be four- fold, $=rv'\fitxt ) y, -\-</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. x =o, I chufe 31. Thus having the Equation y* 6y*-i-()&x* the Terms thence and I obtain -\-gbx*-, by4-3* for the initial Term of the Quote. x* 2rt =o, I make choice of 32. And having y">-i-axy-{-aay 2<2 3 and its Root -\-a I write in the Quote. y'-i-a^y I. pleafe,. feek the Value. 3. 3. ,. 33. Alfo having x*y. 1. s. ^c^xy. c. which like. ^/. gives. for. the. firft. c I .v a 4-. Term. 7. =o,. I felect. vV. i. f. y 4-<r. And. of the Quote.. 7 J. the. of others.. Term. is found, if its Power fhould happen the to be negative, I deprefs 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. But when. 34.. this. fuppreffion of fuperfluous Terms, Thus the Equation 8z; 6_)i3 4-^2 5>' a. Root. is. to begin. by the Term. come Sz+yt-âzy. 2ja. !>. z~. 1. ^ =o,. may. be. 27^5=0 I deprefs. before. conveniently apply'd. being propofed, whofe. by s% I. that. attempt. it. may. be-. the Refolu-. tion. 3 5.. The. fubfequent. Terms. of the Quotes are derived by the fame. Work, from their feveral fecondary For the whole affair but lefs trouble. with commonly Equations, the of the loweft Terms affected with the is perform'd by dividing 1 3 Radical Speindefinitely fmall Species, (x, x , x , &c.) without the the radical r with which that } &c.) Species by Quantity (/>, q, Method,. in the Progrefs of the. i. of.

(31) n. and INFINITE SERIES,. of one Dimenfion only is affected, without the other indefinite SpeSo in the following cies, and by writing the Refult in the Quote. ->. ~. ~>. Example, the Terms a l x, TrW", TTT-v &c. by âa. 36. Thefe things being }. -. &c. are. produced. by dividing. 3. ,. premifed,. it. remains. now. to exhibit the. Praxis of Refolution. za* xz be. Therefore let the Equation y*-{-a zy-\-axy And from its Terms propofed to be refolved. 2 3 =o, being a fictitious Equation, by the third of the. =o. y=-\-a*y. a=o, and jtherefore I write -{-a in foregoing Premifes, I obtain y 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 Terms refulting (/> -{a-\-p, and the 1 from which again, I in the write ; 3rf/ -f-,?,v/>, &c.) Margin again Terms -+-^p the to felect the third of the I Premifes, according 3. p=. -H2 l .v=o. for a fictitious ^x, I Equation, which giving in the Quote. is not the becaufe Then accurate ^.v of p, I put in Terms for and the marginal x-\-q=p, p. ~x. write. Value. 3 -^x^+â^, &c.) ^x-t-q, and the refulting Terms (j I again write in the Margin, out of which, according to the fore_I3-drx*=o for a fictigoing Rule, I again feledl the Terms. I fubftitute. tious Equation,. Again, fince. ^. and inftead of a thus. I. continue. exhibits to view.. which giving is. I. 4^ =^> I write. not the accurate Value of fubftitute. g,. -^ I. in. the Quote.. make -^--{-r=q. ~--\-r in the marginal Terms. '. &4 the Procefs at pleafare,. as the following. t. And. Diagram.

(32) Method of FLUXIONS,. 12. X. -. 3. X*. 2a'. axp. T. '. a*-x. ;. 643. *. *. axq *-. '31**. 509*4. were required to continue the Quote only to a certain 37. If it that x, for inilance, in the laft Term {hould not afcend Period, a beyond given Dimenfion ; as I fubftitute the Terms, I omit fuch as For which this is the Rule, that after I forefee will be of no ufe. the firft Term refulting in the collateral Margin from every QuanTerms are to be added to the right-hand, as the Intity, fo many 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.

(33) and INFINITE SERIES. Therefore the Terms after the. And. expunged. to the. Terms. Work being. thus the. -^. Mark. 13. are to be conceived to. *. continued. till. axr,'m which. -^--H-rfV. at laft />, q,. r,. be. we come or. the Supplement of the Root to be extracted, are only of one Dimenfion ; we may find fo many Terms by Divifion, reprefenting. 509*4. 131*3 _, 5121.. So that. \. as. 16384(13 /. at laft. we. we. fl^n. ... {hall have. e. want n g. y=a. to compleat the Quote.. j. XX. 1.*' 'SI*' 13. kuyAT 509*4. 7*-f"6^-t-^l~*- rÎ;. _. icc -. farther Illustration, I mail propofe another 39. For the fake of From the Equation -L_y< .Ly4_f_iy3 iy=. to be refolved.. Example _^_y. z=o,. let. the. Quote be found only. and the fuperfluous Terms be. to. rejected after the. the fifth Dimenfion,. Mark,. _!_. +^. 5 ,. -L;S 4. 5j. &c.. &c. Z'p, &C. 6cc.. 2; s. ,. &c.. % &c.. +. 40. And thus if we propofe the Equation T 4-rjrJ' '+TT|-T )'' 7 J 3 to be refolved only to the ninth Di-rT T ;' -t-TW' -i-r.)' menfion of the Quote ; before the Work begins we may reject the Term -^^y" ; then as we operate we may reject all the Terms 7 beyond 2', beyond s we may admit but one, and two only after. +y. =o,.

(34) The Method of. Y4 z. becaufe. f. ;. we may. obferve,. FLUXIONS,. that the. to afcerrd. Quote ought always. two Units, in this manner, z, .s , z &c. Then 9 s have ;'=c T_5__ 2; ^_ _^_'T ^_. 3 & C . fs3_j__|_. is difcover'd, by which Equations, 41. And hence an Artifice tho' affected hi injinitum, and confiding of an infinite number of Terms, may however be refolved. And that is, before the Work of begins all the Terms are to be rejected, in which the Dimenfion not the radical affected by the indefinitely fmall Species, Species, Interval of. by the. we. at laft. _. fliall. ,. J. Dimenfion required. the greateft. exceeds. s. j. 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 Terms can arife. Thus in the laft Example I mould have exceeding omitted all the Terms beyond y>, though they went on ad And fo in this Equation tum. 8 -f-3 1 1 ). 4S 4 -f-92. lS. s 4 -}- z. 6. in z*. j'. 8. l6. injini-. &C.. ,. z*y &c.. Root may be extracted only to four Dimenfions of z, Terms in infinitum beyond -f-j 5 in z, 1 J.-4_|_.L 2 > 1 a 4 -(-.c 6 and all beyond -+-y in .c 1 2z 4 and all beyond y- in z 4 And therefore I aflurr.e this Equation and beyond S-}-;s 42 6 s 6^ 1 -}-^ 4^ 1 be to refolved, -^z y* 2z*y z^y* z*y* -{-?* ;> only. that the Cubick I omit all the. ,. ,. tt. .. '. 4s4_j_ s. -i-z'-y. 8=0.. i. of y. being fubflituted inflead. by. ',(*''-. in the reft. What. I. Qi\adraticks.. have. As. faid. if I. of the. 1. A*. A4. h-r-f--;. y 2". as the. -y+ &. in \. Equation deprefs'd. of higher Equations may alib be apply'd to the Root of this Equation. .r. far. of the Quote,). defired. r. as. Term. ~^ {. every where more than four Dimenfions.. z^y gives. 42.. Becaufe?.. Period x f , I omit. <?_[-*+. '. all. the. and affume only. Terms. this. -. &c.. in infinititm.,. Equation, j*. ay. beyond xy. 4. 4-*-*. =0.. This. I. refolve either in the ufual. manner, by making.

(35) and IN FINITE SERIES. j-^; or more expedition fly by the. Method of. have _}'=. affected Equations deliver'd before, #>. 3. where the. Term. laft. by which we. required. vanifhes,. fhall. 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_{_ T i_2;s &c. (which is the Root of the infinite Equaj. the laft Term by thefe 5r==)'-f-^ _j_^5_|_y4 j foe.) by dividing Numbers in order 2, 3, 4, 5, 6, &c. And this, z f^-H-rlo-^' yj TB.27-f_ TTy2;9 j &c. may be continued by dividing by thefe Numi. tion. '. l. '. TrT. bers. 2x3, 4x5, 6x7, 8x9, &c.. "-'g. ,. &c.. refpectively fo of others.. may by. Again, the Series. be continued at pleafure, by multiplying the. Terms. TV, &c>. And. thefe Fractions,. f}. 7,. ,. -,. 44. But in difcovering the firft Term of the Quote, and fometimes of the fecond or third, there may ftill remain a difficulty For its Value, fought for as before, may happen to be overcome. to be furd, or the inextricable Root of an high affected Equation. Which when it happens, provided it be not alfo impoffible, you. and then proceed as if it were known. As in the Example y*-\-axy-{-ii*-y x 3 2a>=o If the Root of this Equation y^^-a'-y 2 =o, had been furd, or unknown, I mould have put any Letter b for it, and then have perform'd the Refolution as follows, fuppofe the Quote found only to the third Dimenfion.. may. reprefent. it. by fome. Letter,. :. 5.

(36) fbe Method of. i6 s. y -\-aay-\rtxy ,. tfÂ-. 2a3 4jC ft. ;. FLUXIONS,.

(37) 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. Equation as before. Thus in the Equation x o, it being known or fupf}-' y* -t-y -\-a ^y* -+- ^y 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* -fwhich is to be folved y* -{-y jj fubftituted, I folve the. =. l. y. + z=o,. 5. as before.. 48. But if that Species be fuppos'd to be indefinitely great, for Reciprocal, which will therefore be indefinitely little, I put fome Species, which being fubflituted, I proceed in the Refolution as. its. Thus having. before.. y* -+-\. or fuppos'd to be very -. I. and. put z,. being reflored.. &c. great,. if. is. .y. =. =o,. x>. -f-jv. for the. - for fobflituting. ~ =o, whofe Root x. l. .v,. reciprocally. +. ^z. you pleafe, J. -f-. known. Quantity 1. -f-.)'. +. y. ^2', &c. where - H-. y=:x *. will be. it. z*. is. little. there will arife y>. ^. where x. 3. 9*. H. 8 i**. '. 49. If it fhould happen that none of thefe Expedients mould fucceed to your defire, you may have recourfe to another. Thus 1 1 in the Equation y* i whereas -+-fx^y o, xy* Z) 2y -+-. =. Term. be obtain'd from the Suppofition that ought 2 1 which 0, jy-4_j_2yt y 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 inftead of there will arife 2-{-z A*, fubftituting y* the. firft. +. =. to. +. 2y. -f-. 1. y. and the Quote. -+-2y*. 2y H-. i. =. o,. and. the Quote. 50. And thus by you may extract and 51. If. from. will begin l. 1. --{--. -\zy*. z'-y*. 0,. to be indefinitely great, or -. fuppole x >*. =. -f-. =. z. z,. -j- i.. if. you. will. have ^ 4. initial. Term. you. for the. Or. of. ,. proceeding according to feveral Suppofitions, exprefs Roots after various ways.. 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 fome Quantity, or by^ alone. Which, for Example by_y, -f-or l x> 20 fake, will happen in the Equation y* -}-axy-+-a y o, 3. D. 4. =. by.

(38) 1. The Method of. 8. by of. fubftituting -f-rf, or. And. .v.. from. differ little. you may. thus. -j-tf,. extract the. or. T. &c. inftead , the conveniently fuppofe Quantity x to 2a, or za*l^, and thence a, or a,. you may. FLUXIONS,. or. za,. or. 2. }. |. Root of the Equation propofed. after. fo. And perhaps many other ways, by fupbe differences to thofe Befides, if you take indefinitely great. poling for the indefinite Quantity this or that of the Species which exprefs many. ways.. your defire after other ways. farther ftill., by fubftituting any fictitious Values for the indebz 1 , -> ~n^> &c. and then proceeding Species, fuch as az. the Root, you. And finite. alfo after fo. may. perhaps obtain. +. as before in the Equations that will refult. 52. But now that the truth of thefe Conclufions. may be maniand feft ; is, that the Quotes thus extracted, 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 righthand 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 that. :. Work,. if. you. fee. that the. Terms. in the faid. Column. will. all. de-. you may conclude^ that the Quote fo far exBut if it be otherthe perfect Root of the Equation. will fee however, that the Terms in which the indefi-. one' another,. ftroy tracted. wife,. is. you. of few Dimenfions, that is, the greate ft Terms, out of that Column, and that at laft none unlefs fuch as are lefs than any given Quantity, will remain there, and therefore not greater than nothing when the Work is continued ad infinitum. So that the Quote, when infinitely extracted, will at laft be the true Root. fake of perfpieuity I 53. Laftly, altho' the Species, which for the 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 This true, though they may not converge fo faft to the true Root. here the Limits But is manifeft from the Anal'ogy of the thing. of the Roots, or the greateft and leaft Quantities, come to be For thefe Properties are in common both to finite and confider'd. The Root in thefe is then greateft or leaft,. infinite Equations. is. nitely fhiall Species are continually taken. when.

(39) and INF INITE SERIES.. 19. 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 but that the Magfuppos'd to be fmall,) cannot be taken greater, nitude of the Root will immediately become infinite, that is, will. when. there. become 54.. Is. impoffible.. To. Diameter. MakeAB. illuftrate this,. and. BC. AC D. be a Semicircle defcribed on the be an Ordinate.. let. =AD,^,BC=7,AD =. ^.. Then. xx before.. as. Therefore BC, or y, then becomes greateft. iax moft exceeds all the Terms " f- V S^x 4- i6a> VS ax &c that la Sax -f- ga*. when. -. >. it. will. when x. be terminated. a.. For. S ax. if. is >. when *. we. =. **. i. but. take x greater than. V. ax ax> TbTs *S s7 &c. will be infinite. There is another Limit alfo, when x o, ax Radical to which the of reafon of the ; by impoffibility Terms or Limits, the Limits of the Semicircle A, B, and D, are cor^ a t the. Sum. of. all. the. Terms. ^. =. S. refpondent. Tranfttion to the. 55.. And. thus. much. METHOD. for the. OF FLUXIONS.. Methods of Computation, of which. Now it remains, that mall make frequent ufe in what follows. I mould of an Illuftration the for give fome SpeciAnalytick Art, of the nature Curves will fupmens of Problems, efpecially fuch as But firft it may be obferved, that all the difficulties of thefe ply. be reduced to thefe two Problems only, which I mall propofe I. may. concerning a Space defcribed by local Motion, any. how. ,. x. '. accelerated. ~. or retarded.. The Length of the Space defcribed being continually ( that -*"* 56. at fill ?V, Times) given; to find the Velocity of the Motion at any ffo^ Tune propofed. JbotA.*** if* 57. II. The Velocity of the Motion being continually given ; to find I.. Length of the Space defcribed at any Time propofed. in the Equation xx=y, if y reprefents the Length of 58. Thus the Space t any time defcribed, which (time) another Space x, the. by increafing with an uniform Celerity #, mea/ures and. D. 2. exhibits as. defcribed. :. /. SJLJ tt.