THE
METHOD of FLUXIONS
AND
INFINITE SERIES;
WITH ITS
Application to the Geometry of CURVE-LINES.
By the INVENTOR
Sir I S A A C NEWTON,^
Late Prefident of the Royal Society.
^ranjlated from the AUTHOR'* LATIN ORIGINAL
not yet made publick.
To which is fubjoin'd,
A PERPETUAL COMMENT upon the whole Work,
Confiding of
ANN OTATIONS, ILLU STRATION s, and SUPPLEMENTS,
In order to make this Treatife
Acomplcat Inftitution for the ufe o/' LEARNERS.
By JOHN CO L SON, M. A. andF.R.S.
Mafter of Sir Jofeph fFilliamfon's free Mathematical-School at Rochejter.
LONDON: Printed by HENRY WOODFALLJ
And Sold by JOHN NOURSE, at the Lamb without Temple-Bar.
M.DCC.XXXVI.
'
T O
William Jones Efq; F.R S. SIR,
[T was a laudable cuftom among the ancient Geometers, and very worthy to be imitated by their SuccefTors, to addrefs their Mathematical labours, not fo much to Men of eminent rank and {ration in the world, as to Perfons of diftinguidi'd merit and proficience in the fame Studies. For they knew very well, that fuch only could be competent Judges of their Works, and would receive them with ''the efteem. they might deferve. So far at leaft I can copy after thofe great Originals, as to chufe a Patron for thefe Speculations, whofe known skill and abilities in fuch matters will enable him to judge, and whofe known candor will incline him to judge favourably, of the fhare I have had in the prefent performance. For as to the fundamental part of the Work, of which I am only the Interpreter, I know it cannot but pleafe you ; it will need no protection, nor ean it receive a greater recommendation, than to bear the name of its illuftrious Author. However, it very naturally applies itfelf to you, who had the honour (for I am fure you think it fo) of the Author's friendship and familiarity in his life-time ; who had his own confent to publifli nil elegant edition of fome of his pieces, of a nature not very different from this ; and who have fo juft an efteem for, as well as knowledge of, his other moft fublime, moil admirable, andjuftly celebrated Works.
A 2 But
iv DEDICATION.
\
But befides thefe motives of a publick nature, I had others that more nearly concern myfelf. The many per- fonal obligations I have received from you, and your ge- nerous manner of conferring them, require all the tefti- monies of gratitude in my power. Among the reft, give me leave to mention one, (tho' it be a privilege I have enjoy 'd in common with many others, who have the hap- pinefs of your acquaintance,) which is, the free accefs you have always allow'd me, to -your copious Collection of whatever is choice and excellent in the Mathernaticks. Your judgment and induftry, .in collecting -thofe. valuable ?tg{^t»fcu«., are not more conspicuous, than the freedom and readinefs with which you communicate them, to all fuch who you know will apply them to their proper ufe, that is, to the general improvement of Science.
Before I take my leave, permit me, good Sir, to join my wiOies to thofe of the publick, that your own ufeful Lu- cubrations may fee the light, with all convenie-nt ipeed ; which, if I rightly conceive of them, will be an excellent methodical Introduction, not only to the mathematical Sciences in general, but alfo to thefe, as well as to the other curious and abftrufe Speculations of our great Author. You are very well apprized, as all other good Judges muft be, that to illuftrate him is to cultivate real Science, and to make his Difcoveries eafy and familiar, will be no fmall improvement in Mathernaticks and Philofophy.
That you will receive this addrefs with your ufual can- dor, and with that favour and friendship I have fo long ind often experienced, is the earneil requeft of,
S I R,
Your moft obedient humble Servant^
J. C OLSON.
(*)
THE
PREFACE.
Cannot but very much congratulate with my Mathe- matical Readers, and think it one of the moft for- tunate ciicumftances of my Life, that I have it in my power to prefent the publick with a moft valuable
Anecdote, of the greatefl Ma fter in Mathematical and
Philofophical Knowledge, that ever appear 'd in the World. And fo much the more, becaufe this Anecdote is of an element ry nature, preparatory and introductory to his other moft arduous and fubh'me Speculations, and intended by himfelf for the instruction of Novices and Learners. I therefore gladly embraced the opportunity that was put into my hands, of publishing this pofthumous Work, be- caufe I found it had been compofed with that view and defign. And that my own Country-men might firft enjoy the benefit of this publication, I refolved upon giving it in an Englijh Translation, •with fome additional Remarks of my own. I thought it highly injurious to the memory and reputation of the great Author, as well as invidious to the glory of our own Nation, that fo curious and uleful a piece fhould be any longer fupprels'd, and confined to a few private hands, which ought to be communicated to all the learned World for general Inftruction. And more efpecially at a time when the Principles of the Method here taught have been fcrupuloufly fifted and examin'd, have been vigorouily .oppofed and (we may fay) ignominioufly rejected as infufficient, by fome Mathe- matical Gentlemen, who feem not to have derived their knowledge of them from their only true Source, that is, from cur Author's own Treatife wrote exprefsly to explain them. And on the other hand, the Principles of this Method have been zealouily and com- mendably defended by other Mathematical Gentlemen, who yet
a feem
x lie PREFACE.
fern to have been as little acquainted with this Work, (or at leaft to have over-look'd it,) the only genuine and original Fountain of this kind of knowledge. For what has been elfewhere deliver'd by our Author, concerning this Method, was only accidental and oc- calional, and far from that copioufnefs with which he treats of it here, and illuftrates it with a great variety of choice Examples.
The learned and ingenious Dr. Pemberton, as he acquaints us in his View of Sir Tfaac Newton's Philofophy, had once a defign of publishing this Work, with the confent" and under the infpectkm of the Author himfelf; which if he had then accomplim'd, he would certainly have deferved and received the thanks of all lovers of Science, The Work would have then appear'd with a double advantage, as receiving the la ft Emendations of its great Author, and likewife in faffing through the hands of fo able an Editor. And among the other good effects of this publication, poffibly it might have prevent- ed all or a great part of thofe Difputes, which have fince been raifed, and which have been fo ftrenuoufly and warmly pnrfued on both fides, concerning the validity of the Principles of this Method. They would doubtlefs have been placed in fo good a light, as would have cleared them from any imputation of being in any wife defective, or not fufficiently demonstrated. But fince the Author's Death, as the Doctor informs us, prevented the execution of that defign, and fince he has not thought fit to refume it hitherto, it became needful that this publication fhould be undertook by another, tho' a much in- ferior hand.
For it was now become highly necefTary, that at laft the great Sir Ijaac himfelf fhould interpofe, fhould produce his genuine Me- thod of Fluxions, and bring it to the teft of all impartial and con- fiderate Mathematicians ; to mew its evidence and Simplicity, to maintain and defend it in his own way, to convince his Opponents, and to teach his Difciples and Followers upon what grounds they mould proceed in vindication of the Truth and Himfelf. And that this might be done the more eafily and readily, I refolved to accom- pany it with an ample Commentary, according to the beft of my fkill, and (I believe) according to the mind and intention of the Au- thor, wherever I thought it needful ; and particularly with an Eye to the fore-mention'd Controverfy. In which I have endeavoui'd to obviate the difficulties that have been raifed, and to explain every thing in fo full a manner, as to remove all the objections of any force, that have been any where made, at leaft fuch as have occtu'd to my obfervation. If what is here advanced, as there is good rea-
fon
PREFACE. xi
fon to hope, fhall prove to the fatisfadtion of thofe Gentlemen, who ikfl darted thefe objections, and who (I am willing to fuppofe) had only the caufe of Truth at heart; I fhall be very glad to have con- tributed any thing, towards the removing of their Scruples. But if it fhall happen otherwife, and what is here offer'd fhould not appear to be furricient evidence, conviction, and demonflration to them ; yet I am perfuaded it will be fuch to moil other thinking Readers, who fhall apply themfelves to it with unprejudiced and impartial minds; and then I mall not think my labour ill beflow'd. It fhould however be well confider'd by thofe Gentlemen, that the great num- ber of Examples they will find here, to which the Method of Fluxions is fuccefsfuUy apply'd, are fo many vouchers for the truth of the Principles, on which that Method is founded. For the Deductions are always conformable to what has been derived from other uncon- troverted Principles, and therefore mufl be acknowledg'd us true. This argument mould have its due weight, even with fuch as can- not, as well as with fuch as will not, enter into the proof of the Principles themfelves. And the hypothefn that has been advanced to evade this conclufion, of one error in reafoning being ilill corrected by another equal and contrary to it, and that fo regularly, conftantly, and frequently, as it mufl be fiippos'd to do here ; this bvpothe/is, I fay, ought not to be ferioufly refuted, becaufe I can hardly think it is ferioufly propofed.
The chief Principle, upon which the Method of Fluxions is here built, is this very fimple one, taken from the Rational Mechanicks ; which is, That Mathematical Quantity, particularly Extenlion, may be conceived as generated by continued local Motion; and that all Quan- tities whatever, at leaflby analogy and accommodation, may be con- ceived as generated after a like manner. Confequently there mufl be comparativeVelocitiesofincreafeanddecreafe, during fuch generations, whole Relations are fixt and determinable, and may therefore /pro- blematically) be propofed to be found. This Problem our Author here folves by the hjip of another Principle, not lefs evident ; which fuppofes that Qnimity is infinitely divifible, or that it may (men- tally at leaft) fo far continually diminifh, as at lafl, before it is totally extinguifh'd, to arrive at Quantities that may be call'd vanilhing Quantities, or whk.li are infinitely little, and lefs than any afTign- able Quantity. Or it funnolcs that we may form a Notion, not indeed of abioiute, but of relative and comparative infinity. 'Tis a very jufl exception to the Method of Indivifibles, as aifo to the foreign infiniteiimal Method, that they have rccourfe at once to
a 2 infinitely
The PREFACE.
infinitely little Quantities, and infinite orders and gradations of thefe, not relatively but absolutely fuch. They affume thefe Quantities finnd & Jewel, without any ceremony, as Quantities that actually and obvioufly exift, and make Computations with them accordingly ; tlie refult of which muft needs be as precarious, as the abfblute ex- iftence of the Quantities they afiume. And fome late Geometricians have carry 'd thefe Speculations, about real and abfolute Infinity, ftill much farther, and have raifed imaginary Syftems of infinitely great and infinitely little Quantities, and their feveral orders and properties j which, to all fober Inquirers into mathematical Truths, muft cer- tainly appear very notional and vifionary.
Thefe will be the inconveniencies that will arife, if we do not rightly diftinguifh between abfolute and relative Infinity. Abfolute Infinity, as fuch, can hardly be the object either of our Conceptions or Calculations, but relative Infinity may, under a proper regulation. Our Author obferves this diftinction very ftrictly, and introduces none but infinitely little Quantities that are relatively fo ; which he arrives at by beginning with finite Quantities, and proceeding by a gradual and neceffary progrefs of diminution. His Computations always commence by finite and intelligible Quantities ; and then at laft he inquires what will be the refult in certain circumftances, when fuch or fuch Quantities are diminim'd in infinitum. This is a con- ftant practice even in common Algebra and Geometry, and is no more than defcending from a general Propofition, to a particular Cafe which is certainly included in it. And from thefe eafy Principles, managed with a vaft deal of fkill and fagacity, he deduces his Me- thod of Fluxions j which if we confider only fo far as he himfelf has carry'd it, together with the application he has made of it, either here or elfewhere, directly or indiredly, exprefly or tacitely, to the moft curious Difcoveries in Art and Nature, and to the fublimeft Theories : We may defervedly efteem it as the greateft Work of Genius, and as the nobleft Effort that ever was made by the Hun an Mind. Indeed it muft be own'd, that many uftful Improvement?, and new Applications, have been fince made by others, and proba- bly will be ftill made every day. For it is no mean excellence of this Method, that it is doubtlefs ftill capable of a greater degree of perfection ; and will always afford an inexhauftible fund of curious matter, to reward the pains of the ingenious and iuduftrious Analyft.
As I am defirous to make this as fatisfactory as poffible, efptcially to the very learned and ingenious Author of the Difcourle call'd The Analyjl, whofe eminent Talents I acknowledge myfelf to have a
J great
The PREFACE. xlii
great veneration for ; I fhall here endeavour to obviate fome of his principal Objections to the Method of Fluxions, particularly fuch as I have not touch'd upon in my Comment, which is foon to follow.
He thinks cur Author has not proceeded in a demonftrative and fcientifical matter, in his Princip. lib. 2. km. 2. where he deduces the Moment of a Rectangle, whole Sides are fuppofed to be variable Lines. I fhall reprefent the matter Analytically thus, agreeably (I think) to the mind of the Author.
Let X and Y be two variable Lines, or Quantities, which at dif- ferent periods of time acquire different values, by flowing or increa- fing continually, either equably or alike inequably. For inflance, let there be three periods of time, at which X becomes A — fa, A, and A -+- 7 a ; and Y becomes B — f3, B, and B -+- f b fuccefiively and reflectively ; where A, a, B, b, are any quantities that may be aiTumed at pleafure. Then at the fame periods of time the variable Produ<ft or Rectangle XY will become A" — fa x B — f4, AB, and A •+- f * x B -+- ±h, that is, AB — T<?B — fM. -f- ±ab, AB, and AB -+- f^B -f- 7$ A -f- ^ab. Now in the interval from the firft period of time to the fecond, in which X from being A — fa is become A, and in which Y from being B — 7^ is become B, the Product XY from being AB — f^B — i£A -f- ^ab becomes AB -, that is, by Sub- traction, its whole Increment during that interval is f#B -+- f£A — ^ab. And in the interval from the fecond period of time to the third, in which X from being A becomes A-f-ftZ, and in which Y frcm being B becomes B -hf^, the Product XY from being AB becomes AB-f- ffiB -f f 4A -+- -^ab ; that is, by Subtraction, its whole Increment during that interval is 7,76 + 7^A -+- ^ab. _ Add thefe two Increirents together, and we fhall have <?B -+- bA. for the compleat Increment of the Product XY, during the whole interval of time, while X fk w'd from the value A — \a to A -f- ftf , or Y flow'd from the value B — f£ to B +7''. Or U might have been found by tne Operation, thus: While X f.ows from A — \a to A, and therce to A -f- ft?, or Y flows f-om B — f3 to B, and thence to B -i- f A, the Product XY will flow fiom AB — f<?B— f3A -f- ±ab to AB, ?nd thence to AB -+- f^B + -J'k -f- ^ab •> therefore by Sub- traction the whole Increment during that interval of time will be tfB-4-M. Q^E. D.
This may eafily be illuftrated by Numbers thus: Make A,rf,B,/, equal to 9, 4, i 5, 6, refpeclively; (or any other Numbers to be af- fumed at pleafure.) Then the three fucceffive values of X will be 7, 9, ii, and the three fucceffive values of Y will be 12, 15, 18,
reipcciivcly.
xiv The PREFACE.
refpeftively. Alfo the three fucceflive values of the Produd XY will be 84, 135, 198. But rtB-f-M = 4xic-f- 6x9= 114 = 198_84. Q.E. O.
Thus the Lemma will be true of any conceivable finite Incre- ments whatever; and therefore by way of Corollary, it will be true of infinitely little Increments, which are call'd Moments, and which was the thing the Author principally intended here to demonflrate. 15ut in the cafe of Moments it is to be confider'd, that X, or defi- nitely A — ftf, A, and A -+- ±a, are to be taken indifferently for the fame Quantity ; as alfo Y, and definitely B — f/;, B, B -+- ~b. And the want of this Confutation has occafion'd not a few per- plexities.
Now from hence the reft of our Author's Conclufions, in the fame Lemma, may be thus derived fomething more explicitely. The Moment of the Reclangle AB being found to be Ab -+- ^B, when the contemporary Moments of A and B are reprelented by a and b refpedtively ; make B = A, and therefore b = a, and then the Moment of A x A, .or A*, will be Aa -+- aA, or 2aA. Again, make B = Aa, and therefore b-=. zaA, and then the Moment of AxA*, or A', will be 2rfA4-f- aA1, or 3^A*. Again, make B = A5, and therefore l> = ^aAs-, and then the Moment of A xA*, or A4, will be 3<?A3 -4-rfA3, or 4#A3. Again, make B==A-», and therefore ^ = 4^A3, and then the Moment of Ax A4, or A', will be 4<?A4 -i-tfA4, or 5<zA4. And fo on in infinitum. Therefore in general, afluming m to reprefent any integer affirmative Number, the Moment of A* will be maA™"1.
Now becaufe A* x A^ra= i, (where m is any integer affirmative •Number,) and becaufe the Moment of Unity, or any other conftant quantity, is = p ; we (hall have A* x Mom. A~m -f- A~m x Mom. A"= o, or Mom. A~"= — A-110 x Mom. A" . But Mom. A" = maAm~*, as found before ; therefore Mom. A"* = — A~iw x ma A"-' = — maA-"-' . Therefore the Moment of Am will be maAm~I, when m is any integer Number, whether affirmative or negative.
•
And univerfally, if we put A" =B, or A"=. B" , where m and n may be any integer Numbers, affirmative or negative ; then we
mall have ma A"-* = ;.^B"^' , or b= mgA<° = -aA»— i, which
is the Moment of B, or of A" . So that the Moment of A" will
be
The P E E F A C E. xv
be rtill wtfA"*"1, whether ;;/ be affirmative or negative, integer or fraction.
The Moment of AB being M -+- aB, and the Moment of CD being </C •+- cD ; fuppofe D = AB, and therefore d-=. b& •+- aB, and then by Subftitution the Moment of ABC will be bA •+- aB xC -f- c AB = MC -+- rfBC -h r AB. And likewife the Moment of A*B" will be «/>B"-'A" -f- maA.m~lBn. And fo of any others.
Now there is fo near a connexion between the Method of Mo- ments and the Method of Fluxions, that it will be very eafy to pafs from the one to the other. For the Fluxions or Velocities of in- creafe, are always proportional to the contemporary Moments. Thus if for A, B, C, &c. we write x, y, z, &c. for a, b, c, &c. we may write x, y, z, &c. Then the Fluxion of xy will be xy -f- xy, the Fluxion of xm will be rnxx*-* , whether m be integer or fraction, affiimative or negative; the Fluxion of xyz will be xyz -f- xyz -f- xjz, and the Fluxion of xmyn will be mxxm-*y» -J- nxmyy"~s . And fo of the reft.
Or the former Inquiry may be placed in another view, thus : Let A and A-f- a be two fucceflive values of the variable Quantity X, as alfo B and B -+- b be two fucceflive and contemporary values of Y ; then will AB and AB -f- aB-\~ bA+ab be two fucceflive and contemporary values of the variable Product XY. And while X, by increafing perpetually, flows from its value A to A -f- a, or Y flows from B to B -f- b ; XY at the fame time will flow from AB to AB •+- aB -+- bA. -f- abt during which time its whole Increment, as appears by Subtraction, will become aB -h bh. -+- ab. Or in Numbers thus: Let A, a, B, b, be equal to 7, 4, 12, 6, refpectively ; then will the two fucceflive values of X be 7, 1 1 , and the two fuc- ceflive values of Y will be 12, 18. Alib the two fucceflive values of the Product XY will be 84, 198. But the Increment aB -+- t>A -J- ah- — • 48 -f- 42 -+- 24= 1 14= 198 — 84, as before.
And thus it will be as to all finite Increments : But when the In- crements become Moments, that is, when a and b are fo far dirni- nifh'd, as to become infinitely lefs than A and B ; at the fame time ab will become infinitely lefs than either aB or ^A, (for aB. ab :: B. b, and bA. ab :: A. ay) and therefore it will vanifh in refpect of them. In which cafe the Moment of the Product or Rectangle will be aB -+- bA, as before. This perhaps is the more obvious and direct way of proceeding, in the t relent Inquiry ; but, as there was room for choice, our Author thought fit to chufe the former way,,
as
xvi The PREFACE.
as the more elegant, and in which he was under no neceflity of hav- ing recourfe to that Principle, that quantities arifing in an Equation, which are infinitely lefs than the others, may be neglected or ex- punged in companion of thofe others. Now to avoid the ufe of this Principle, tho' otherwife a true one, was all the Artifice ufed on this occaiion, which certainly was a very fair and justifiable one.
I fhall conclude my Obfervations with confidering and obviating the Objections that have been made, to the ufual Method of finding the Increment, Moment, or Fluxion of any indefinite power x» of the variable quantity x, by giving that Inveftigation in fuch a man- ner, as to leave (I think) no room for any juft exceptions to it. And the rather becaufe this is a leading point, and has been ftrangely perverted and mifreprefented.
In order to find the Increment of the variable quantity or power x», (or rather its relation to the Increment of x} confider'd as given ; becaufe Increments and Moments can be known only by comparifon with other Increments and Moments, as alfo Fluxions by comparifon with other Fluxions ;) let us make x"=y, and let X and Y be any fynchronous Augments of x and y. Then by the hypothefis we fhall have the Equation x-fc-X\* =y -+- Y ; for in any Equation the variable Quantities may always be increafed by their fynchronous Augments, and yet the Equation will flill hold good. Then by our Author's famous Binomial Theorejn we fhall have y -f- Y = xn
-+- nx"~'X -+- n x ^=-^—*X * + n x *~ x '-^-V^X 3 , &c. or re - moving the equal Quantities y and x", it will be Y = nxn~lX •+- ny. ^-x"--X * -+- n x ?-^- x ^^x'-'^X 3 , &c. So that when X deT
notes the given Increment of the variable quantity A,-, Y will here denote the fynchronous Increment of the indefinite power y or x" ; whofe value therefore, in all cafes, may be had from this Series. Now that we may be fure we proceed regularly, we will verify this thus far, by a particular .and familiar instance or two. Suppofe n = 2, then Y = 2xX -+- X l . That is, while x flows or increafes to x •+- X, .v* in the fame time, by its Increment Y = 2xX -+-X1, will increafe to .v1 4- 2xX -j- X1, which we otherwife know to be true. Again, fuppofe fl = 3, then Y = 3*1X -+- 3*Xa H- X3. Or while x in*. creafes to x r+- X, x"> by its Increment Y = 3^aX -h 3^XJ + X3 will increafe to x* -f- 3*1X -+- ^xX1 -+- X3. And fo in all ,other particular cafes, whereby we may plainly perceive, that this general Conclufion mud be certain and indubitable.
This
Tie PREFACE. xvii
This Series therefore will be always true, let the Augments X and Y be ever fo great, or ever fo little ; for the truth docs not at all de- pend on the circumftance of their magnitude. Nay, when they are infinitely little, or when they become Moments, it muft be true alfo, by virtue of the general Conclufion. But when X and Y are di- minifh'd in infinitum, fo as to become at laft infinitely little, the greater powers of X muft needs vanifli firft, as being relatively of an infinitely lefs vali e than the fmaller powers. So that when they are all expunged, we ihall neceflarily obtain the Equation Y=znx*~'X ; where the remaining Terms are likewife infinitely little, and confe- quently would vanifh, if there were other Terms in the Equation, which were (relatively) infinitely greater than themfelves. But as .there are not, we may fecurely retain this Equation, as having an undoubted right fo to do; and efpecially as it gives us anufeful piece of information, that X and Y, tho' themfelves infinitely little, or vanifhing quantities, yet they vanifli in proportion to each other as j to nx"~f. We have therefore learn 'd at laft, that the Moment by which x increafes, or X, is to the contemporary Moment by which xa increafes, or Y, as i is to nx"~s. And their Fluxions, or Velo- cities of increafe, being in the fame proportion as their fynchronous Moments, we fhall have nx*-'x for the Fluxion of X", when the Fluxion of x is denoted by x.
I cannot conceive there can be any pretence to infinuate here, that any unfair artifices, any leger-de-main tricks, or any Ihifting of the hypothefis, that have been fo feverely complain'd of, are at all made ufe of in this Inveftigation. We have legitimately derived this general Conclufion in finite Quantities, that in all cafes the re- lation of the Increments will be Y = nx"~lX + « x ~~x*'-1X*, &c. of which one particular cafe is, when X and Y are fuppofed conti- nually to decreafe, till they finally terminate in nothing. But by thus continually decreafing, they approach nearer and nearer to the Ratio of i to nx"~\ which they attain to at ihe very inftant of the'r vanifhing, and not before. This therefore is their ultimate Ratio, the Ratio of their Moments, Fluxions, or Velocities, by which x and xn continually increafe or decreafe. Now to argue from a general Theorem to a particular cafe contain'd under it, is certainly tine of the moft legitimate and logical, as well as one of the mofl ufual and ufeful ways of arguing, in the whole compafs of the Mathemc- ticks. To object here, that after we have made X and Y to ftand for fome quantity, we are not at liberty to make them nothing, or no quantity, or vanishing quantities, is not an Objection againft the
b Method
XVlll
Tte PREFACE.
Method of Fluxions, but againft the common Analyticks. This Method only adopts this way of arguing, as a conftant practice in the vulgar Algebra, and refers us thither for the proof of it. If we have an Equation any how compos'd of the general Numbers a, b, c, &c. it has always been taught, that we may interpret thefe by any particular Numbers at pleafure, or even by o, provided that the Equation, or the Conditions of the Queftion, do not exprefsly re- quire the contrary. For general Numbers, as fuch, may ftand for any definite Numbers in the whole Numerical Scale ; which Scale (I think) may be thus commodioufly reprefented, &c. — 3, — 2> — i, o, i, 2, 3,4, &c. where all poffible fractional Numbers, inter- mediate to thefe here exprefs'd, are to be conceived as interpolated. But in this Scale the Term o is as much a Term or Number as any other, and has its analogous properties in common with the refK We are likewife told, that we may not give fuch values to general Symbols afterwards, as they could not receive at firft ; which if ad- mitted is, I think, nothing to the prefent purpofe. It is always moft eafy and natural, as well as moll regular, inftruclive, and ele- gant, to make our Inquiries as much in general Terms as may be, and to defcend to particular cafes by degrees, when the Problem is nearly brought to a conclufion. But this is a point of convenience only, and not a point of neceffity. Thus in the prefent cafe, in- flead of defcending from finite Increments to infinitely little Mo- ments, or vanifhing Quantities, we might begin our Computation with thofe Moments themfelves, and yet we mould arrive at the fame Conclufions. As a proof of which we may confult our Au- thor's ownDemonftration of hisMethod, in oag. 24. of this Treatife. In fhort, to require this is jufl the famexthing as to infift, that a Problem, which naturally belongs to Algebra, mould be folved by common Arithmetick ; which tho' poflible to be done, by purluing backwards all the fleps of the general procefs, yet would be very troubkfome and operofe, and not fo inflrudtive, or according to the true Rules of Art
But I am apt to fufpedr, that all our doubts and fcruples about Mathematical Inferences and Argumentations, especially when we are fatisfied that they have been juftly and legitimately conducted, may be ultimately refolved into a fpecies of infidelity and diftruft. Not in refpecl of any implicite faith we ought to repofe on meer human authority, tho' ever fo great, (for that, in Mathematicks, we mould utterly difclaim,) but in refpedl of the Science itfelf. We are hardly brought to believe, that the Science is fo perfectly regular and uni- form,
72* PREFACE. xix
form, fo infinitely confident, conftant, and accurate, as we mall re&lly find it to be, when after long experience and reflexion we (hall have overcome this prejudice, and {hall learn to purfue it rightly. We do not readily admit, or eafily comprehend, that Quantities have an infinite number of curious and fubtile properties, fome near and ob- vious, others remote and abftrufe, which are all link'd together by a neceffary connexion, or by a perpetual chain, and are then only difcoverable when regularly and clofely purfued ; and require our . truft and confidence in the Science, as well as our induftry, appli- cation, and obftinate perfeverance, our fagacity and penetration, in order to their being brought into full light. That Nature is ever confiftent with herfelf, and never proceeds in thefe Speculations per faltum, or at random, but is infinitely fcrupulous and felicitous, as we may fay, in adhering to Rule and Analogy. That whenever we make any regular Portions, and purfue them through ever fo great a variety of Operations, according to the ftricT: Rules of Art ; we fhall always proceed through a feries of regular and well- connected tranlmutations, (if we would but attend to 'em,) till at laft we arrive at regular and juft Conclufions. That no properties of Quantity are intirely deftructible, or are totally loft and abolim'd, even tho' profecuted to infinity itfelf j for if we fuppofe fome Quantities to be- come infinitely great, or infinitely little, or nothing, or lefs than nothing, yet other Quantities that have a certain relation to them will only undergo proportional, and often finite alterations, will fym- pathize with them, and conform to 'em in all their changes ; and will always preferve their analogical nature, form, or magnitude, which will be faithfully exhibited and difcover'd by the refult. This we may colledl from a great variety of Mathematical Speculations, and more particularly when we adapt Geometry to Analyticks, and Curve-lines to Algebraical Equations. That when we purfue gene- ral Inquiries, Nature is infinitely prolifick in particulars that will refult from them, whether in a direct rubordination, or whether they branch out collaterally ; or even in particular Problems, we may often perceive that thefe are only certain cafes of fomething more general, and may afford good hints and afiiftances to a fagacious Analyft, for afcending gradually to higher and higher Difquilitions, which may be profecuted more univerfally than was at firft expe<5ted or intended. Thefe are fome of thofe Mathematical Principles, of a higher order, which we find a difficulty to admit, and which we {hall never be fully convinced of, or know the whole ufe of, but from much prac- tice and attentive confideration ; but more efpecially by a diligent
b 2 peruial,
xx The P R E F A C E.
peruial, and clofe examination, of this and the other Works of our illuftrious Author. He abounded in thefe fublime views and in- quiries, had acquired an accurate and habitual knowledge of all thefe, and of many more general Laws, or Mathematical Principles of a fuperior kind, which may not improperly be call'd The Philofophy of Quantity ; and which, aflifted by his great Genius and Sagacity, to- gether with his great natural application, enabled him to become fo compleat a Matter in the higher Geometry, and particularly in the Art of Invention. This Art, which he poflefl in the greateft per- fection imaginable, is indeed the fublimeft, as well as the moft diffi- cult of all Arts, if it properly may be call'd fuch ; as not being redu- cible to any certain Rules, nor can be deliver'd by any Precepts, but is wholly owing to a happy fagacity, or rather to a kind of divine Enthufiafm. To improve Inventions already made, to carry them on, when begun, to farther perfection, is certainly a very ufeful and excellent Talent ; but however is far inferior to the Art of Difcovery, as haying a TIV e^u, or certain data to proceed upon, and where juft method, clofe reasoning, ftrict attention, and the Rules of Analogy, may do very much. But to ftrike out new lights, to adventure where no footfteps had ever been fet before, nullius ante trita folo ; this is the nobleft Endowment that a human Mind is capable of, is referved for the chofen few quos Jupiter tequus amavit, and was the peculiar and diftinguifhing Character of our great Mathematical Philofopher. He had acquired a compleat knowledge of the Philofophy of Quan- tity, or of its moft eflential and moft general Laws ; had confider'd it in all views, had purfued it through all its difguifes, and had traced it through all its Labyrinths and Recefles j in a word, it may be faid of him not improperly, that he tortured and tormented Quantities all poflible ways, to make them confefs their Secrets, and difcover their Properties.
The Method of Fluxions, as it is here deliver'd in this Treatife, is a very pregnant and remarkable inftance of all thefe particulars. To take a cuifory view of which, we may conveniently enough divide it into thefe three parts. The firft will be the Introduction, or the Method of infinite Series. The fecond is the Method of Fluxions, properly fo culi'd. The third is the application of both thefe Methods to fome very general and curious Speculations, chiefly in the Geometry of Curve-lines.
As to the firft, which is the Method of infinite Series, in this the Author opens a new kind of Arithrnetick, (new at leaft at the time of his writing this,) or rather he vaftly improves the old. For
he
The PREFACE. xxi
he extends the received Notation, making it compleatly universal, and fhews, that as our common Arithmetick of Integers received a great Improvement by the introduction of decimal Fractions ; fo the common Algebra or Analyticks, as an univerfal Arithmetick, will receive a like Improvement by the admiffion of his Doctrine of in- finite Series, by which the fame analogy will be ftill carry'd on, and farther advanced towards perfection. Then he fhews how all com- plicate Algebraical Expreffions may be reduced to fuch Series, as will continually converge to the true values of thofe complex quantities, or their Roots, and may therefore be ufed in their ftead : whether thofe quantities are Fractions having multinomial Denominators, which are therefore to be refolved into fimple Terms by a perpetual Divi- fion ; or whether they are Roots of pure Powers, or of affected Equa- tions, which are therefore to be refolved by a perpetual Extraction. And by the way, he teaches us a very general and commodious Me- thod for extracting the Roots of affected Equations in Numbers. And this is chiefly the fubftance of his Method of infinite Series.
The Method of Fluxions comes next to be deliver'd, which in- deed is principally intended, and to which the other is only preparatory and fubfervient. Here the Author difplays his whole fkill, and fhews the great extent of his Genius. The chief difficulties of this he re- duces to the Solution of two Problems, belonging to the abftract or Rational Mechanicks. For the direct Method of Fluxions, as it is now call'd, amounts to this Mechanical Problem, tte length of the Space defer ibed being continually given, to find the Velocity of the Mo- tion at any time propofcd. Aifo the inverfe Method of Fluxions has, for a foundation, the Reverfe of this Problem, which is, The Velocity of the Motion being continually given, to find the Space defer ibed at any time propofcd. So that upon the compleat Analytical or Geometri- cal Solution of thefe two Problems, in all their varieties, he builds his whole Method.
His firft Problem, which is, The relation 6J the f owing Quantities being given, to determine the relation of their Fhixiom, he difpatches very generally. He does not propofe this, as is ufualiy done, A flow- ing Quantity being given, to find its Fluxion ; for this gives us too lax and vague an Idea of the thing, and does not fufficiently fhew that Comparifon, which is here always to be understood. Fluents and Fluxions are things of a relative n.iture, and fuppofe two at leafr, whofe relation or relations mould always be exprefs'd bv Equations. He requires therefore that all fhould be reduced to Equations, by which the relation of the flowing Quantities will be exhibited, and their
comparative
xxii f/jg PREFACE.
comparative magnitudes will be more eafily eftimated ; as alfo the comparative magnitudes of their Fluxions. And befides, by this means he has an opportunity of refolving the Problem much more generally than is commonly done. For in the ufual way of taking Fluxions,- we are confined to. the Indices of the Powers, which are to be made Coefficients ; whereas the Problem in its full extent will allow us to take any Arithmetical Progreflions whatever. By this means we may have an infinite variety of Solutions, which tho' dif- ferent in form, will yet all agree in the main ; and we may always chufe the fimpleft, or that which will beft ferve the prefent purpofe. He (hews alfo how the given Equation may comprehend feveral va- riable Quantities, and by that' means the Fluxional Equation maybe found, notwithstanding any furd quantities that may occur, or even any other quantities that are irreducible, or Geometrically irrational. And all this is derived and demonitrated from the properties of Mo- ments. He does not here proceed to fecond, or higher Orders of Fluxions, for a reafon which will be affign'd in another place.
His next Problem is, An Equation being propofed exhibiting the re- lation of the Fluxions of Quantities, to find the relation of thofe Quan- tities, or Fluents, to one another ; which is the diredt Converfe of the foregoing Problem. This indeed is an operofe and difficult Problem, taking it in its full extent, and, requires all our Author's fkill and ad- dreis ; which yet hefolyes very generally, chiefly by the affiftance of his Method of infinite Series. He firfl teaches how we may return from the Fluxional Equation given, to its correfponding finite Fluential or Algebraical Equation, when that can be done. But when it cannot be .done, or when there is no fuch finiie Algebraical Equation, as is moft commonly the cafe, yet however he finds the Root of that Equation by an infinite converging Series, which anfwers the fame purpofe. And often he mews how to find the Root, or Fluent required, by an infinite number of fuch Series. His proceffes for extracting thefe Roots are peculiar to himfelf, and always contrived with much fub- tilty and ingenuity.
The reft of his Problems are an application or an exemplification of the foregoing. As when he determines the Maxima and Minima of quantities in all cafes. When he mews the Method of drawing Tangents to Curves, whether Geometrical or Mechanical ; or how- ever the nature of the Curve may be defined, or refer'd to right Lines or other Curves. Then he {hews how to find the Center or Radius of Curvature, of any Curve whatever, and that in a fimple but general manner ; which he illuftrates by many curious Examples,
and
fbe PREFACE. xxiii
and purfues many other ingenious Problems, that offer themfelves by the way. After which he difcufTes another very fubtile and intirely new Problem about Curves, which is, to determine the quality of the Curvity of any Curve, or how its Curvature varies in its progrefs through the different parts, in refpect of equability or inequability.
He then applies himfelf to confider the Areas of Curves, and fhews us how we may find as many Quadrable Curves as we pleafe, or fuch whole Areas may be compared with thofe of right-lined Figures. Then he teaches us to find as many Curves as we pleafe, whofe Areas may be compared with that of the Circle, or of the Hyper- bola, or of any other Curve that (hall be affign'd ; which he extends to Mechanical as well as Geometrical Curves. He then determines the Area in general of any Curve that may be propofed, chiefly by the help of infinite Series ; and gives many ufeful Rules for afcer- taining the Limits of fuch Areas. And by the way he fquares the Circle and Hyperbola, and applies the Quadrature of this to the con- ftructing of a Canon of Logarithms. But chiefly he collects very- general and ufeful Tables of Quadratures, for readily finding the Areas of Curves, or for comparing them with the Areas of the Conic Sections; which Tables are the fame as. thofe he has publifh'd him- felf, in his Treatife of Quadratures. The ufe and application of thefe he (hews in an ample manner, and derives from them many curious Geometrical Conftructions, with their Demonftrations.
Laftly, he applies himfelf to the Rectification of Curves, and mews us how we may find as many Curves as we pleafe,. whofe Curve- lines are capable of Rectification ; or whofe Curve-lines, as to length, may be compared with the Curve-lines of any Curves that fha.ll be affign'd. And concludes in general, with rectifying any Curve-lines that may be propofed, either by the aflifbncc of his Tables of Quadra- tures, when that can be done, or however. by infinite Series. And this is chiefly the fubflance of the prefent Work. As to ,the account that perhaps" may be expected, of what I have added in my Anno- tations ; I {hall refer the inquifitive Reader to the PrefacCj which will go before that part of the Work.
THE
;• -
THE
CONTENTS.
CT^HE Introduction, or the Method of refolding complex Quantities into infinite Series of Jimple Terms. pag. i
Prob. i. From the given Fluents to find the Fluxions. p. 21
Prob. 2. From the given Fluxions to find the Fluents. — — p. 25
Prob. 3. To determine the Maxima and Minima of Quantities, p. 44
Prob. 4. To draw Tangents to Curves. p. 46
Prob. 5. To find the Quantity of Curvature in any Curve. P- 59
Prob. 6. To find the Quality cf Curvature in any Curve. p. 75
Prob. 7. To find any number of Quadrable Curves. p. 80
Prob. 8. To find Curves whofe Areas may be compared to thofe of the Conic SecJions. p. 8 1
Prob. 9. To find the Quadrature of any Curve ajjigrid. p. 86
Prob. 10. To find any number of rettifiable Curves. p. 124
Prob. 1 1. To find Curves whofe Lines may be compared with any Curve- lines ajfigrid. p. 129
Prob. 12. To rectify any Curve-lines ajpgn'd. •— p. 134
THE
METHOD of FLUXIONS,
AND
INFINITE SERIES.
INTRODUCTION : Or, the Refolution of Equations
by Infinite Series.
IAVING obferved that moft of our modern Geome-- tricians, neglecting the Synthetical Method of the Ancients; have apply'd themfelves chiefly to the cultivating of the Analytical Art ; by the affiftance of which they have been able to overcome fo many and fo great difficulties, that they feem to have exhaufted all the Speculations of Geometry, excepting the Quadrature of Curves, and Ibme other matters of a like nature, not yet intirely difcufs'd : I thought it not amifs, for the fake of young Students in this Science, to compofe the following Treatife, in which I have endeavour'd to enlarge the Boundaries of Analyticks, and to improve the Doctrine of Curve-lines.
2. Since there is a great conformity between the Operations in Species, and the fame Operations in common Numbers; nor do they feem to differ, except in the Characters by which they are re-
B prefented,.
'The Method of FLUXIONS,
prefented, the firft being general and indefinite, and the other defi- nite and particular : I cannot but wonder that no body has thought of accommodating the lately-difcover'd Doctrine of Decimal Frac- tions in like manner to Species, (unlels you will except the Qua- drature of the Hyberbola by Mr. Nicolas Mercator ;) efpecially fince it might have open'd a way to more abftrufe Discoveries. But iince this Doctrine of Species, has the fame relation to Algebra, as the Doctrine of Decimal Numbers has to common Arithme- tick ; the Operations of Addition, Subtraction, Multiplication, Di- vifion, and Extraction of Roots, may eafily be learned from thence,, if the Learner be but fk.ill'd in Decimal Arithmetick, and the Vulgar Algebra, and obferves the correfpondence that obtains be- tween Decimal Fractions and Algebraick Terms infinitely continued. For as in Numbers, the Places towards the right-hand continually decreafe in a Decimal or Subdecuple Proportion ; fo it is in Species refpedtively, when the Terms are difpofed, (as is often enjoin 'd in what follows,) in an uniform Progreflion infinitely continued, ac- cording to the Order of the Dimenfions of any Numerator or De- nominator. And as the convenience of Decimals is this, that all vulgar Fractions and Radicals, being reduced to them, in fome mea- fure acquire the nature of Integers, and may be managed as fuch ; fo it is a convenience attending infinite Series in Species, that all kinds of complicate Terms, ( fuch as Fractions whofe Denomina- tors are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Clafs of fimple Quantities ; that is, to an infinite Series of Fractions, whofe Numerators and Denominators are fimple Terms ; which will no longer labour under thofe difficulties, that in the other form feem'd almoft infuperable. Firft therefore I mail fhew how thefe Re- ductions are to be perform'd, or how any compound Quantities may be reduced to fuch fimple Terms, efpecially when the Methods of computing are not obvious. Then I fhall apply this Analyfis to the Solution of Problems.
3. Reduction by Divifion and Extraction of Roots will be plain from the following Examples, when you compare like Methods of Operation in Decimal and in Specious Arithmetick.
Examples
and INFINITE SERIES, 3
• . ..ift Av
Examples of Reduttion by Dhifwn. IjfM/l^^ '* /•
.4. The Fraction ^™ being propofed, divide aa by b + x in the following manner :
faa aax aax1 a a x* aax* .
» " .
aax
aax O— --7 — -f-O
aax*
o -+-
o - +o
flt *» ** Jf*
~ ;•.
-rr^i_ *-\ " v i r * ^^ tf*^1 a* x* a* x* . a* X+ ~
The Quotient therefore is T_-JT-+-T_ . — rr+T7-, &c. which Series, being infinitely continued, will be equivalent to £j^. Or making x the firft Term of the Divifor, in this manner,
x + toaa + o (the Quotient will be - - ?4 4. 1^« —V &c~ e , , % r~ _ _ * **n*» AV
found as by the foregoing Procefs.
5. In like manner the Fraction ~- will be reduced to I — #• -{- x4 — ' A:* H- x8, &c. or to x-* — #-* _f. ^-« — ^-8
2* "
9 v
6. And the Fraction r will be reduced to 2x^ — 2x
i s i+x*— 3*
•+• yx1 — 13** -j- 34xT, &c.
7. Here it will be proper to obferve, that I make ufe of x-', x-', x-', x-*, &c. for i, ;r 7,' -• &c. of xs, xi, x^, xl, A4, &c.
for v/x, v/*S \/x*> vx , ^xl, &c. and of x'^, x-f. x-i &c for , i j_^ ' * **** 1Ui
^ x ^?>' y-^.' &c. And this by the Rule of Analogy, as may be apprehended from fuch Geometrical Progreflions as thefe ; x», x*, x«> (or i,) a"*,*-',*'*, *•», &c.
B 2 8.
x,
ffie Method of FLUXIONS,
er for ', &c.
8. In the fame manner for -- — 1^ + 1^!, &c. may be wrote
q. And thus inftead of^/aa — xx may be wrote aa — xxl^ > .and aa — xv|* inftead of the Square of aa — xx; and
3
inftead of v/
10. So that we may not improperly diftinguim Powers into Affir- mative and Negative, Integral and Fractional.
Examples of Reduction by Extraction of Roots.
11. The Quantity aa -+- xx being propofed, you may thus ex- tract its Square-Root.
-„ _i_ Vv (a -4- — — — 4- — — — 5 x - 4- J— • — — — - — ' c*
aa-+- XX ^" 2a Sfl3 r i6«* 128«7 2560*
aa
xx
4. a*
x*
~*
a 4 64 ««
X*
sT*
64 a«
~
64^8 " z$6a'^
i; x
5*
64^
_ 256 *
64 a 6 I z8rt8
+
_- 7^ _ 2^1, &c.
1 i7R/3» n-- /7lt>
7'1
+
,__i!_lll, &c.'
Jo that the Root is found to be a~\--^-— ^ 4- ^T,&C. Where it may be obferved, that towards the end of the Operation I neg- lect all thofe Terms, whofe Dimenfions would exceed the Dimenfions of the laft Term, to which I intend only to continue the Root,
fuppofe to *—' ,2.
and INFINITE SERIES. 5
iz. Alfo the Order of the Terms may be inverted in this man- ner xx •+- aa, in which cafe the Root will be found to be
a a
10 A* iz« A- »
13. Thus the Root of aa — xx is « — ^ — -Jj -- ^7
14. The Root of x — xx is #'" — i** — 4-.v* — T'r**, 8cc.
. . £ AT A.' A' b*X* g
15. Of «« -+- «f — ## is a -f- — — — -- ^ , Sec.
. i + <z *• A- . i 4- '- « * * — i a * A- 4 + ,'_ n 3 x- 6. &c- j
1 6. And v/r^rr, « .Ii*«»--».«4-. ;,,»««. .c. and more-
over by adually dividing, it becomes
i -|- -i/^r + |^^4 -+- ^frx6, &c. -4- T^ -f- T^ H- rV^x
17. But thefe Operations, by due preparation, may very often be abbreviated; as in the foregoing Example to find \/;_***' if the Form of the Numerator and Denominator had not been the fame, I might have multiply'd each by </ 1 — bxx, which would
y^i -f-rt*1— - ab x *
have produced — & and the reft of the work might
I — b x x
have been performed by extracting the Root of the Numerator only, and then dividing by the Denominator.
1 8. From hence I imagine it will fufficiently appear, by what means any other Roots may be extracted, and how any compound Quantities, however entangled with Radicals or Denominators, (fuch
Vx — \fi — xx Vxi!2xt — xi v
as x"> -}- — — — •; _. j may be reduced to
^/axx -\- A- 3 * x-{-xx — " 2X — x.1 '
infinite Series confifting of iimple Terms.
Of the ReduStion of offered Equations.
19. As to aftedled Equations, we mufl be fomething more par- ticular in explaining how their Roots are to be reduced to fuch Se- ries as thefe ; becaufe their Doctrine in Numbers, as hitherto de- liver'd by Mathematicians, is very perplexed, and incumber'd with fuperfluous Operations, fo as not to afford proper Specimens for per- forming the Work in Species. I fhall therefore firfl (hew how the
Refolu-
Method of FLUXIONS,
Refolutidn of affected Equations may be compendioufly perform'd in Numbers, and then I fhall apply the fame to Species.
20. Let this Equation _yl — zy — 5 = 0 be propofed to be re- folved, and let 2 be a Number (any how found) which differs from the true Root lefs than by a tenth part of itfelf. Then I make 2 -\-p =y, and fubftitute 2 4-/> for y in the given Equation, by which is produced a new Equation p> 4- 6pl 4- iop — i =o, whofe Root is to be fought for, that it may be added to the Quote. Thus rejecting />> 4- 6//1 becaufe of its fmallnefs, the remaining Equation io/> — i = o, or/>=o,i, will approach very near to the truth. Therefore I write this in the Quote, and fuppofe o, i 4- ^ =/>, and fubftitute this fictitious Value of p as before, which produces q* 4- 6,3^ 4- 1 1,23? 4- 0,06 1 =o. And fince 1 1,23^ 4- 0,06 1 =o is near the truth, or ^= — 0,0054 nearly, (that is, dividing 0,06 1 by 11,23, ^ *° many Figures arife as there are places between the firft Figures of this, and of the prin- cipal QmDte exclufively, as here there are two places between 2 and 0,005) I write — 0,0054 in the lower part of the Quote, as being negative; and fuppofing — 0,0054 4- r=sg, I fubftitute this as before. And thus I continue the Operation as far as I pleafe, in the manner of the following Diagram :
y~' — zy — 5 =o
+ 2, IOOOOOOO
+ 2,09455148, &c. =y
Z+p=J>. + 7 *
— 27
— 4— zp
The Sum
-i + iop+6p* + p->
+ i°/
+ o3ooi+ 0,035 +o, 5 5 2 + 2* + o, 06 + i32 + 6,
+ 1, + 10,
1 he 6um
o, 061 -|- 1 1) 23 i + 6, 3 q * + 2*
— o,oo54 + r= q. <ji
+ II,2??
+ 0,06 1
— o, oooooo i f74^+ o,ooo0#7-4&V — 0, 0tfai » +)•'
+ 0,00018370^ 0,06804: +^;? — 0,060642 +11,23
+ o, 061
The Sum
+ 0,0005416 +II,l62r
— 0,000048^2 + * = r.
21.
and INFINITE SERIES. 7
21. But the Work may be much abbreviated towards the end by this Method, efpecially in Equations of many Dimenfions. Having firft determin'd how far you intend to extract the Root, count fo many places after the firft Figure of the Coefficient of the laft Term but one, of the Equations that refult on the right fide of the Dia- gram, as there remain places to be fill'd up in the Quote, and reject the Decimals that follow. But in the laft Term the Decimals may be neglected, after fo many more places as are the decimal places that are fill'd up in the Quote. And in the antepenultimate Term reject all that are after fo many fewer places. And fo on, by pro- ceeding Arithmetically, according to that Interval of places: Or, which is the fame thing, you may cut off every where fo many Figures as in the penultimate Term, fo that their loweft places may be in Arithmetical Progreffion, according to the Series of the Terms, or are to be fuppos'd to be fupply'd with Cyphers, when it happens otherwife. Thus in the prefent Example, if I defired to continue the Quote no farther than to the eighth place of Decimals, when I fubftituted 0,0054 -f- r for q, where four decimal places are compleated in the Quote, and as many remain to be compleated, I might have omitted the Figures in the five inferior places, which therefore I have mark'd or cancell'd by little Lines drawn through them ; and indeed I might alfo have omitted the firft Term r J, although its Coefficient be 0,99999, Thofe Figures therefore being expunged, for the following Operation there arifes the Sum 0,0005416 -f- 1 1,1 62?% which by Divifion, continued as far as the Term prefcribed, gives — 0,00004852 for r, which compleats the Quote to the Period required. Then fubtracting the negative part of the Quote from the affirmative part, there arifes 2,09455148 for the Root of the propofed Equation.
22. It may likewife be obferved, that at the beginning of the Work, if I had doubted whether o, i -f-/> was a fufficient Ap- proximation to the Root, inftead of iof> — i = o, I might have fuppos'd that o/** -f- i op — i = o, and fo have wrote the firft Figure of its Root in the Quote, as being nearer to nothing. And in this manner it may be convenient to find the fecond, or even the third Figure of the Quote, when in the fecondarjr Equation, about which you are converfant, the Square of the Coefficient of the penultimate Term is not ten times greater than the Product of the laft Term multiply'd into the Coefficient of the antepenulti- mate Term. And indeed you will often fave fome pains, efpecially in Equations of many Dimensions, if you feek for all the Figures
to-
8 Tie Method of FLUXION'S,
to be added to the Quote in this manner ; that is, if you extract the lefier Root out of the three lafl Terms of its fecondary Equation : For thus you will obtain, at every time, as many Figures again in the Quote.
23. And now from the Refolution of numeral Equations, I mall proceed to explain the like Operations in Species; concerning which, it is neceflary to obferve what follows.
24. Firft, that fome one of the fpecious or literal Coefficients, if there are more than one, fliould be diftinguifh'd from the reft, which either is, or may be fuppos'd to be, much the leaft or greateft of all, or neareft to a given Quantity. The reafon of which is, that becaufe of its Dimeniions continually increafing in the Numerators, or the Denominators of the Terms of the Quote, thofe Terms may grow lefs and lefs, and therefore the Qtipte may conftantly approach to the Root required ; as may appear from what is faid before of the Species x, in the Examples of Reduction by Divifion and Ex- traction of Roots. And for this Species, in what follows, I mall generally make ufe of A: or z ; as alfo I fliall ufe y, p, q, r, s, &c. for the Radical Species to be extracted.
25. Secondly, when any complex Fractions, or furd Quantities, happen to occur in the propofed Equation, or to arife afterwards in the Procefs, they ought to be removed by fuch Methods as are fufficiently known to Analyfts. As if we mould have
y* -+- j— 1>'1 — x"= = o,. multiply by b — x, and from the Pro- duct by* Kyi'-l-fry* — bx^ -+• x*-= o extract the Root y. Or
we might fuppofe y x b — x=v, and then writing ^~x for yt we mould have i;J -+- &*v* — fax* -\- 3/5*** — ^hx' -+. x6 = o,. whence extracting the Root vr we might divide the Quote by b — x,, in order to obtain y. Affo if the Equation j3 — xy* -f- x$ = o were propofed, we might put y?= v, and xj = z, and fo wri- ting vv for y, and z* for x, there will arife v6 — z=v -f- z* = o ; which Equation being refolved, y and x may be reftored. For the Root will befound^=2-f-s3_|_5~s55cc.andrei1:onngjyandA;, we have y* = x^ -f- x -+- 6x^ &c. dien fquaring, y =x^-+- 2XJ ~f- 13*", &c..
26. After the fame manner if there mould be found negative Di- menfions ofx and jy, they may be removed by multiplying by the fame x andjy. As if we had the Equation x*-}-T>x*-y~I—'2.x~I — i6y-3=o, multiply by x and j3, and there would arife x*y* -+- 3#3jy1 — 2_v5
A J -r 1 -r-v • aa 2ai i 1 a 4»
O. And U tjie Equation were x = — — ~ + ?—r
y\. by;
and INFINITE SERIES.
by multiplying into jy} there would arife xy*-=.a'iy*— And fo of others.
27. Thirdly, when the Equation is thus prepared, the work be^ gins by finding the firfr. Term of the Quote ; concerning which, as alfo for finding the following Terms, we have this general Rule, when the indefinite Species (x or 2) is fuppofed to be fmall ; to which Caie the other two Cafes are reducible.
28. Of all the Terms, in which the Radical Species (y,/>, q, or r, &c.) is not found, chufe the loweft in refpect of the Dimenlions of the indefinite Species (x or z, &c.) then chufe another Term in which that Radical Species is found, fuch as that the Progreflion of the Dimenfions of each of the fore-mentioned Species, being con- tinued from the Term fir ft afTumed to this Term, may defcend as much as may be, or afcend as little as may be. And if there are any other Terms, whofe Dimenfions may fall in with this Progreflion continued at pleafure, they muft be taken in 1 ike- wife. Laftly, from thefe Terms thus felected, and made equal to nothing, find the Value of the faid Radical Species, and write it in the Quote.
29. But that this Rule may be more clearly apprehended, I fhall explain it farther by help of the following Diagram. Making a right Angle BAC, divide its fides AB, AC, into equal parts, and raifing Perpendiculars, diftribute the Angular Space into equal Squares or Parallelograms, which you may conceive to be denominated from the Dimenfions of the Species x and y,
as they are here infcribed. Then, when
any Equation is propofed, mark fuch of
the Parallelograms as correfpond to all
its Terms, and let a Ruler be apply'd
to two, or perhaps more, of the Paralle-
lograms fo mark'd, of which let one
be the loweft in the left-hand Column at AB, the other touching
the Ruler towards the right-hand ; and let all the reft, not touching
the Ruler, lie above it. Then felecl: thofe Terms of the Equation
which are reprefented by the Parallelograms that touch the Ruler,
and from them find the Quantity to be put in the Quote.
30. Thus to extract the Root y out of the Equation y6 — 5xys-+-
— •)'* — ja*x1y1+6aix*-\-&1x4=o, I mark the Parallelograms belong-
C
B
A 4
ft
Xlj*
*4;5
.1-4:4
A3
*3
X3£
A? 3
A 5 4
X*
A'*
x*y*
**. 3
X
xy
*!*
A -;
v,4
1
y
}*
s1
4
ing
10
The Method of FLUXIONS,
B
A
*
C
ing to the Terms of this Equation with the Mark #, as you fee here done. Then I apply the Ruler DE to the lower of the Parallelo- grams mark'd in the left-hand Column, and I make it turn round towards the right-hand from the lower to the upper, till it begins in like manner to touch another, or perhaps more, of the Parallelograms that are mark'd ; and I fee that the places fo touch'd belong to x3, x*-y*y and_y5. Therefore from the Terms y6 — 7azx*-y<L-}-6a*x*, as if equal to nothing, (and moreover, if you pleafe, reduced to v6 — 7^*4- 6= o, by making $=rv'\fitxt) I feek the Value of y, and find it to be four- fold, -\-</ax, — </ax, -+-</2ax, and — ^/2ax, of which I may take any one for the initial Term of the Quote, according as I defign to extract this or that Root of the given Equation.
31. Thus having the Equation y* — 6y*-i-()&x* — x3=o, I chufe the Terms — by- -\-gbx*-, and thence I obtain 4-3* for the initial Term of the Quote.
32. And having y">-i-axy-{-aay — x* — 2rt3=o, I make choice of y'-i-a^y — 2<23, and its Root -\-a I write in the Quote.
33. Alfo having x*ys—— ^c^xy1 — cI.va4-£7=o, I felect vViyf4-<r7J
which gives — ^/c— for the firft Term of the Quote. And the
like of others.
34. But when this Term is found, if its Power fhould happen to be negative, I deprefs the Equation by the fame Power of the indefinite Species, that there may be no need of depreffing it in the Refolution ; and befides, that the Rule hereafter delivei'd, for the fuppreffion of fuperfluous Terms, may be conveniently apply'd. Thus the Equation 8z;6_)i34-^25>'a — 27^5=0 being propofed, whofe
Root is to begin by the Term ^ I deprefs by s% that it may be- come Sz+yt-^azy — 2ja!>z~1=o, before I attempt the Refolu- tion.
3 5. The fubfequent Terms of the Quotes are derived by the fame Method, in the Progrefs of the Work, from their feveral fecondary Equations, but commonly with lefs trouble. For the whole affair is perform'd by dividing the loweft of the Terms affected with the indefinitely fmall Species, (x, x1, x3, &c.) without the Radical Spe- (/>, q, r} &c.) by the Quantity with which that radical Species
i of
and INFINITE SERIES, n
of one Dimenfion only is affected, without the other indefinite Spe- cies, and by writing the Refult in the Quote. So in the following
Example, the Terms -> ~} - ~> &c. are produced by dividing
alx, TrW", TTT-v3, &c. by ^aa.
36. Thefe things being premifed, it remains now to exhibit the Praxis of Refolution. Therefore let the Equation y*-{-azy-\-axy — za* — xz=o be propofed to be refolved. And from its Terms y=-\-a*y — 2«3=o, being a fictitious Equation, by the third of the foregoing Premifes, I obtain y — a=o, and jtherefore I write -{-a in the Quote. Then becaufe -\~a is not the compleat Value ofy, I put a+p=y, and inftead of y, in the Terms of the Equation written in the Margin, I fubftitute a-\-p, and the Terms refulting (/>3-{- 3rf/1-f-,?,v/>, &c.) I again write in the Margin ; from which again, according to the third of the Premifes, I felect the Terms -+-^p -H2l.v=o for a fictitious Equation, which giving p= — ^x, I write — ~x in the Quote. Then becaufe — ^.v is not the accurate Value of p, I put — ±x-\-q=p, and in the marginal Terms for p I fubftitute — ^x-t-q, and the refulting Terms (j3 — -^x^+^a^, &c.) I again write in the Margin, out of which, according to the fore- going Rule, I again feledl the Terms 4^ — _I3-drx*=o for a ficti- tious Equation, which giving £=^> I write -^ in the Quote. Again, fince ^ is not the accurate Value of g, I make -^--{-r=qt and inftead of a I fubftitute ~--\-r in the marginal Terms. And
&4« '
thus I continue the Procefs at pleafare, as the following Diagram exhibits to view.
12
Method of FLUXIONS,
•X3
•2a'
• axp
; 643
— ±axq
*-
- X*
T '
•a*-x
*
*
'31** 509*4
37. If it were required to continue the Quote only to a certain Period, that x, for inilance, in the laft Term {hould not afcend beyond a given Dimenfion ; as I fubftitute the Terms, I omit fuch as I forefee will be of no ufe. For which this is the Rule, that after the firft Term refulting in the collateral Margin from every Quan- tity, fo many Terms are to be added to the right-hand, as the In- dex of the higheft Power required in the Quote exceeds the Index of that firft refulting Term.
38. As in the prefent Example, if I defired that the Quote, (or the Species .v in the Quote,) mould afcend no higher than to four Dimenfions, I omit all the Terms after A-*, and put only one after x=.
Therefore
and INFINITE SERIES. 13
Therefore the Terms after the Mark * are to be conceived to be expunged. And thus the Work being continued till at laft we come
to the Terms -^— -^--H-rfV— ±axr,'m which />, q, r, or
reprefenting the Supplement of the Root to be extracted, are only of one Dimenfion ; we may find fo many Terms by Divifion,
131*3 _, 509*4 \ as we fl^n £e wantjng to compleat the Quote.
16384(13 /
5121.
'SI*'
509*4
... XX 13 1.*' kuyAT _
So that at laft we {hall have y=a — 7*-f"6^-t-^l~*- r^I; icc-
39. For the fake of farther Illustration, I mail propofe another Example to be refolved. From the Equation -L_y< — .Ly4_f_iy3 — iy=. _^_y — z=o, let the Quote be found only to the fifth Dimenfion, and the fuperfluous Terms be rejected after the Mark,
_!_£5j &c.
+ ^5, &c.
-L;S4 Z'p, &C.
6cc.
2;
s, &c. % &c.
40. And thus if we propofe the Equation T4-rjrJ' '+TT|-T )'' + -rTT;'7-t-TW'J-i-r.)'3+y — £=o, to be refolved only to the ninth Di- menfion of the Quote ; before the Work begins we may reject the Term -^^y" ; then as we operate we may reject all the Terms beyond 2', beyond s7 we may admit but one, and two only after
Y4 The Method of FLUXIONS,
zf ; becaufe we may obferve, that the Quote ought always to afcerrd by the Interval of two Units, in this manner, z, .sj, zs , &c. Then at laft we fliall have ;'=c— fs3_j__|_.s»_ T_5__2;^_J_^_'T^_.39)&C. 41. And hence an Artifice is difcover'd, by which Equations, tho' affected hi injinitum, and confiding of an infinite number of Terms, may however be refolved. And that is, before the Work begins all the Terms are to be rejected, in which the Dimenfion of the indefinitely fmall Species, not affected by the radical Species, exceeds the greateft Dimenfion required in the Quote ; or from, which, by fubftituting inftead of the radical Species, the firfl Term, of the Quote found by the Parallelogram as before, none but fuch exceeding Terms can arife. Thus in the laft Example I mould have omitted all the Terms beyond y>, though they went on ad injini- tum. And fo in this Equation
8 -f-31 4S4-f-92lS l6«8, &C.
) — j'1 in z* — s4-}- z6 — z*y &c.
that the Cubick Root may be extracted only to four Dimenfions of z, I omit all the Terms in infinitum beyond -f-j5 in z,1 — J.-4_|_.L2«> and all beyond — y- in z1 — a4-(-.c6, and all beyond -+-y in .c1 — 2z4, and beyond — S-}-;stt — 424. And therefore I aflurr.e this Equation only to be refolved, -^z6y* — ±z*y* -{-?•*•• ;> — s6^1-}-^4^1 — z^y* — 2z*y -i-z'-y — 4s4_j_si — 8=0. Becaufe?. ',(*''- ~^{' Term of the Quote,) being fubflituted inflead of y in the reft of the Equation deprefs'd by z^y gives every where more than four Dimenfions.
42. What I have faid of higher Equations may alib be apply'd to Qi\adraticks. As if I defired the Root of this Equation
r
.r1 A* A 4 -
h-r-f--; &c.
as far as the Period xf, I omit all the Terms in infinititm., beyond — y in <?_[-*•+— ' and affume only this Equation, j* — ay — xy —
2" \ 4
-y+ —=0. This I refolve either in the ufual manner, by making
& 4-*-*
and IN FINITE SERIES.
j-^; or more expedition fly by the Method of affected Equations deliver'd before, by which we fhall have _}'=•— 3 — — #> where the laft Term required vanifhes, or
becomes equal to nothing.
43. Now after that Roots are extracted to a convenient Period, they may fometimes be continued at pleafure, only by oblerving the Analogy of the Series. So you may for ever continue this z-t-i-z* ^_^.25_j__'_2;4_{_Ti_2;sj &c. (which is the Root of the infinite Equa- tion 5r==)'-f-^i_j_^5_|_±y4j foe.) by dividing the laft Term by thefe Numbers in order 2, 3, 4, 5, 6, &c. And this, z — f^-H-rlo-^' — ' yj lTB.27-f_TrT'TTy2;9j &c. may be continued by dividing by thefe Num-
bers 2x3, 4x5, 6x7, 8x9, &c. Again, the Series
"-'g ,» &c. may be continued at pleafure, by multiplying the Terms refpectively by thefe Fractions, f } — 7, — £, — -£, — TV, &c> And fo of others.
44. But in difcovering the firft Term of the Quote, and fome- times of the fecond or third, there may ftill remain a difficulty to be overcome. For its Value, fought for as before, may happen to be furd, or the inextricable Root of an high affected Equation. Which when it happens, provided it be not alfo impoffible, you may reprefent it by fome Letter, and then proceed as if it were known. As in the Example y*-\-axy-{-ii*-y — x3 — 2a>=o : If the Root of this Equation y^^-a'-y — 2«5=o, had been furd, or un- known, I mould have put any Letter b for it, and then have per- form'd the Refolution as follows, fuppofe the Quote found only to the third Dimenfion.
i6
fbe Method of FLUXIONS,
y s -\-aay-\r£txy — 2 a 3 — ;
, tf^A- «4£jCft
^=0. Make a--\-T,b1=c2, then
ii | (v*r*
rTTv* .8 ,8 ,10 . •
AT3
— \-b~i -f-?^i^-j-2^/:1-f-/)J
~" w :;
«5;'3A3
— ' — j — &C.
A'3
6<?£1A.-^ C43.V1
«3i3,;S /,.4iA* ~X* 3 3.%3
iz / 1 4
~* + t« ( ,« h^ r8
45. Here writing £ in the Quote, I fuppofe b-±-p=y, and then for y I fubftitute as you fee. Whence proceeds p'^-^bp1, &c. re- jecting the Terms b'-^a'-b — 2tf3, as being equal to nothing : For b is fuppos'd to be a Root of this Equation jy3_j_fl*y — 2<?3=o. Then
the Terms ^p-^-a^p-^-abx give '/^V* :1 to be fet in the Quote,, and
to be fubflituted for p.
46. But for brevity's fake I write a- for aa-^-^l>l>, yet with this caution, that aa-\-^bb may be reflored, whenever I perceive that the Terms may be abbreviated by it. When the Work is finim'd, I aflume fome Number for a, and refolve this Equation y*-\-?.'-\' — 2^;=o, as is fhewn above concerning Numeral Equations ; and I fubftitute for b any one of its Roots, if it has three Roots. Or rather, I deliver fuch Equations from Species, as far as I can, efpe- cially from the indefinite Species, and that after the manner before insinuated. And for the reft only, if any remain that cannot be expunged, I put Numbers. Thus y'-^-a^y — 2^5=o will be freed from a, by dividing the Root by a, and it will become y*+)' — 2=0, whofe Root being found, and multiply'd by a, muft be fubftituted
inftead of b.
47-
and INFINITE SERIES, 17
47. Hitherto I have fuppos'd the indefinite Species to be little. But if it be fuppos'd to approach nearly to a given Quantity, for that indefinitely fmall difference I put fome Species, and that being fubftituted, I folve the Equation as before. Thus in the Equation •f}-' — ^y* -+- ^yl — ±y* -t-y -\-a — x = o, it being known or fup- pos'd that x is nearly of the fame Quantity as a, I fuppofe z to be their difference; and then writing a-\-z or a — z for x, there will arife ±y — ±y* -f- jj5 — ±y* -{-y + z=o, which is to be folved as before.
48. But if that Species be fuppos'd to be indefinitely great, for its Reciprocal, which will therefore be indefinitely little, I put fome Species, which being fubflituted, I proceed in the Refolution as before. Thus having y* -+-\l -f-jv — x> =o, where x is known or fuppos'd to be very great, for the reciprocally little Quantity
- I put z, and fobflituting - for .v, there will arife y> -f-.)'1 •+• y — ~ =o, whofe Root is .y = ^ — •- — ^z + £z* -f- ^2', &c. where x being reflored. if you pleafe, it will be y=:x — - H- — H — —
J •* 3 9* 8 i**
&c'
49. If it fhould happen that none of thefe Expedients mould fucceed to your defire, you may have recourfe to another. Thus in the Equation y* — x^y1 -+- xy* -f- Z)1 — 2y -+- i = o, whereas the firft Term ought to be obtain'd from the Suppofition that jy-4_j_2yt — 2y + 1 = 0, which yet admits of no poffible Root; you may try what can be done another way. As you may fuppofe that x is but little different from •+• 2, or that 2-{-z-=x. Then fubftituting 2-{-z inftead of A*, there will arife y* — z'-y* — -\zy* — 2y -f- 1 = 0, and the Quote will begin from -j- i. Or if you
fuppole x to be indefinitely great, or l- = z, you will have ^4—
>* y1
•--{-- -+-2y* — 2y H- i = o, and -f- z for the initial Term of the Quote. ,
50. And thus by proceeding according to feveral Suppofitions, you may extract and exprefs Roots after various ways.
51. If you mould delire to find after how many ways this may be done, you mufl try what Quantities, when fubfHtuted for the indefinite Species in the propofed Equation, will make it divifible by_y, -f-or — • fome Quantity, or by^ alone. Which, for Example fake, will happen in the Equation y* -}-axy-+-aly — x> — 203 = o,
D by
4
1 8 The Method of FLUXIONS,
by fubftituting -f-rf, or — a, or — za, or — 2«}|T, &c. inftead of .v. And thus you may conveniently fuppofe the Quantity x to differ little from -j-tf, or — a, or — 2a, or — za*l^, and thence you may extract the Root of the Equation propofed after fo many ways. And perhaps alfo after fo many other ways, by fup- poling thofe differences to be indefinitely great. Befides, if you take for the indefinite Quantity this or that of the Species which exprefs the Root, you may perhaps obtain your defire after other ways. And farther ftill., by fubftituting any fictitious Values for the inde- finite Species, fuch as az + bz1, •£-> ~n^> &c. and then proceeding as before in the Equations that will refult.
52. But now that the truth of thefe Conclufions may be mani- feft ; that is, that the Quotes thus extracted, and produced ad libi-* turn, approach fb near to the Root of the Equation, as at laft to differ from it by lefs than any afilgnable Quantity, and therefore when infinitely continued, do not at all differ from it : You are to confider, that the Quantities in the left-hand Column of the right- hand fide of the Diagrams, are the laft Terms of the Equations whofe Roots are p, y, r, s, &c. and that as they vanifh, the Roots p, q, r, s, &c. that is, the differences between the Quote and the Root fought, vanifh at the fame time. So that the Quote will not then differ from the true Root. Wherefore at the beginning of the Work, if you fee that the Terms in the faid Column will all de- ftroy one' another, you may conclude^ that the Quote fo far ex- tracted is the perfect Root of the Equation. But if it be other- wife, you will fee however, that the Terms in which the indefi- nitely fhiall Species is of few Dimenfions, that is, the greate ft Terms, are continually taken out of that Column, and that at laft none will remain there, unlefs fuch as are lefs than any given Quantity, and therefore not greater than nothing when the Work is continued ad infinitum. So that the Quote, when infinitely extracted, will at laft be the true Root.
53. Laftly, altho' the Species, which for the fake of perfpieuity I have hitherto fuppos'd to be indefinitely little, fhould however be fuppos'd to be as great as you pleafe, yet the Quotes will ftill be true, though they may not converge fo faft to the true Root. This is manifeft from the Anal'ogy of the thing. But here the Limits of the Roots, or the greateft and leaft Quantities, come to be confider'd. For thefe Properties are in common both to finite and infinite Equations. The Root in thefe is then greateft or leaft,.
when
and INF INITE SERIES. 19
when there Is the greateft or leaft difference between the Sums of the affirmative Terms, and of the negative Terms ; and is limited when the indefinite Quantity, (which therefore not improperly I fuppos'd to be fmall,) cannot be taken greater, but that the Mag- nitude of the Root will immediately become infinite, that is, will become impoffible.
54. To illuftrate this, let AC D be a Semicircle defcribed on the Diameter AD, and BC be an Ordinate. MakeAB = ^,BC=7,AD = ^. Then
— xx
as before.
Therefore BC, or y, then becomes greateft when iax moft exceeds all the Terms
— Sax -f- f- S^x 4- — Sax> &c- that is> when * = ** i but
la " ga* V i6a> V
it will be terminated when x — a. For if we take x greater than
at the Sum of all the Terms — ^ Sax — s7» Vax — TbTs *Sax> &c. will be infinite. There is another Limit alfo, when x = o, by reafon of the impoffibility of the Radical S — ax ; to which Terms or Limits, the Limits of the Semicircle A, B, and D, are cor^ refpondent.
Tranfttion to the METHOD OF FLUXIONS.
55. And thus much for the Methods of Computation, of which I mall make frequent ufe in what follows. Now it remains, that , for an Illuftration of the Analytick Art, I mould give fome Speci- mens of Problems, efpecially fuch as the nature of Curves will fup- ply. But firft it may be obferved, that all the difficulties of thefe x may be reduced to thefe two Problems only, which I mall propofe concerning a Space defcribed by local Motion, any how accelerated ' or retarded. ~
56. I. The Length of the Space defcribed being continually ( that -*"*£ ?V, at fill Times) given; to find the Velocity of the Motion at any ffo^
Tune propofed. / SJLJ tt
57. II. The Velocity of the Motion being continually given ; to find JbotA.*** if* the Length of the Space defcribed at any Time propofed.
58. Thus in the Equation xx=y, if y reprefents the Length of the Space «t any time defcribed, which (time) another Space x,
by increafing with an uniform Celerity #, mea/ures and exhibits as
D 2 defcribed :
20 ?%e Method of FLUXIONS,
defcribed : Then zxx will reprefent the Celerity by which the Space y, at the fame moment of Time, proceeds to be defcribed ; and contrary-wife. And hence it is, that in what follows, I confider Quantities as if they were generated by continual Increafe, after the manner of a Space, which a Body or Thing in Motion defcribes.
59. But whereas we need not confider the Time here, any farther than as it is expounded and meafured by an equable local Motion ; and befides, whereas only Quantities of the fame kind can be compared together, and alfo their Velocities of Increafe and Decreafe : Therefore in what follows I fhall have no regard to Time formally conficter'd,, but I fhall fiippofe fome one of the Quantities propofed, being of the fame kind, to be increafed by an equable Fluxion, to which the reft may be referr'd, as it were to Time j and therefore, by way of Analogy, it may not improperly receive the name of Time. Whenever therefore the word Time occurs in what follows, (which for the fake of perfpicuity and diftindlion I have fometimes ufed,) by that Word I would not have it under- ftood as if I meant Time in its formal Acceptation, but only that other Quantity, by the equable Increafe or Fluxion whereof, Time is expounded and meafured.
'60. Now thofe Quantities which I confider as gradually and 2 indefinitely increafing, I fhall hereafter call Fluents, or Flowing
Quantities, and fhall reprefent them by the final Letters of the f £ Alphabet v, x, y, and z ; that I may diftinguifh them from other
Quantities, which in Equations are to be confider'd as known and. T H > %f& f'df** determinate, and which therefore are reprefented by the initial U» _' .i V*> i*i~- Letters a, b, c, &c. And the Velocities by which every Fluent
is increafed by its generating Motion, (which I may call Fluxions,
( oi V* ffm***4t*'Qr fimply Velocities or Celerities,) I fhall reprefent by the fame
Letters pointed thus -y, x, y., and z. That is, for the Celerity of K t4 JO the Quantity v I fhall put v, and fo for the Celerities of the other id tti Quantities x, y, and z, I fhall put x, y, and z refpeftively.
J '(/ 6 1. Thefe things being premifed, I mall now forthwith proceed
to the matter in hand } and firft I fhall give the Solution of the: two Problems juft now propofed.
PROF,
and INFINITE SERIES.
21
P R O B. I.
The Relation of the Flowing Quantities to one another being given, to determine the Relation of their Fluxions.
SOLUTION.
1. Difpofe the Equation, by which the given Relation is ex- prefs'd, according to the Dimenftons of fome one of its flowing Quantities, fuppofe x, and multiply its Terms by any Arithmetical
Progreflion, and then by - . And perform this Operation feparately
for every one of the flowing Quantities. Then make the Sum of all the Products equal to nothing,, aad you will have the Equation required.
2. EXAMPLE i. If the Relation of the flowing Quantities A; and y be X' — ax*--{- axy — ^3=o; firft difpofe the Terms according to x, and then according to y, and multiply them in the follow- ing, manner.
Mult.
by
makes %xx* — zaxx -{- axy * — zyy* -f- ayx *
• • • • *
The Sum of the Produdls is -jx** — zaxx -k- axy — W*-f- ayx=zo,
i . •
which Equation gives the Relation between the Fluxions x and y.
For if you take x at pleafure, the Equation .v3 — ax1 -{-axy — yt = o will give y. Which being determined, it will be x : y :: 7v* — ax : yx^—zax -{- ay.
3.. Ex. 2. If the Relation of the Quantities x, y,. and zr be ex- preis'd by the Equation 2j3 -f- x*y — zcyz •+- yz* — z'' = QJ
*»
— ax*
+ ffxy-
-r
— >':
•JT axy.
—ax1
3*
2x
X
iy .
_v
—
-^ •
V •
o
~- *
O
X
X
x
3
y
Mult. 2j3 -i-xxxy — z*
yx* -+- zy*
— z* -fc- 3_>-21 — zcyz •+• x'y
— zcz ~f"~ 32;*
' — zcyz
-h zy3
ay y
2X
; 2~ ±
DV *"* • O . "•
— . o .
— . — . - o.
'• y y
x
z z z
makes 4^-* % 4-'~
zxxy %
-2zz*+6zzy-zcZy .
Where-
22 *The Method of FLUXIONS,
Wherefore the Relation of the Celerities of Flowing, or of the Fluxions ,v, v, and z, is tyy* -\- +• 2xxy — $zzl -f- 6zzy — zczy
.
4. But fince there are here three flowing Quantities, .v, y, and z, another Equation ought alfo to be given, by which the Relation among them, as alfo among their Fluxions, may be intirely deter- mined. As if it were fuppofed that x -\-y — 2 = 0. From whence another Relation among the Fluxions AT-HV — z = o would be found by this Rule. Now compare thefe with the foregoing Equa- tions, by expunging any one of the three Quantities, and alfo any one of the Fluxions, and then you will obtain an Equation which will intirely determine the Relation of the reft.
5. In the Equation propos'd, whenever there are complex Frac- tions, or furd Quantities, I put fo many Letters for each, and fup- pofing them to reprefent flowing Quantities, I work as before. Af- terwards I fupprefs and exterminate the afTumed Letters, as you fee done here.
6. Ex. 3. If the Relation of the Quantities .v and y be yy — aa
— x\/aa — ## = o; for x</aa — xx I write z, and thence I have the two Equations^' — aa — %,•=.&., and a3-*1 — x4 — 2* i — . o, of which the firfl will give zyy — z = o, as before, for the Relation of the Celerities y and z, and the latter will give 2<j*xx
o, or a*xx~ **** = z, for the Relation of the
Celerities x and z. Now z being expunged, it will be zyy -
= o, and then reftoring x^aa — xx for z, we fhall have zyy
-./»** 4- g*.>* __ 0> for the Relation between x and y, as was re-
^ aa — XX
quired.
7. Ex. 4. If .v3 — ay* 4- j4r — XX \fay -+- xx = o, expreffes
the Relation that is between AT and v : I make ^^ = 5;, and
^x \/~ay-+-xx=v, from whence I fhall Lave the three Equations x- — ay* + & — -u = o, az-\-yz — ^3=o, and ax*y •+• x6 — 1^=0. The firft gives 3**' — zayy •+• z — -0=0, the fecond gives az •+• Zy^-yz — 3^& = o, and the third gives 4.axx>y-+-6xx'-i-a}>x* — 2W= o, for the Relations of the Velocities -y, .v, y, and «. But
the
and INF i NIT E SERIES. 23
the Values of & and i', found by the fecond and third Equations, iSj ££? for z and
/. /. v-. . 11 . ,. • • 7n — vz
nrft Equation, and there anies %xx* — 2a)y-^-~^T. —
= o. Then inflead of z and v refloring their Values — f— and
a>
. XX \/ ay -+- xx, there will arife the Equation fought ^xx*-—2ayy
— 6*- A- 3 — awMf ... . _ . . r ,
— = o. by which the Relation or the
•>
. aa -f- 2^ + yy 2
Velocities x and y will be exprefs'd.
8. After what manner the Operation is to be performed in other Cafes, I believe is manifefl from hence j as when in the Equation propos'd there are found furd Denominators, Cubick Radicals, Ra-
dicals within Radicals, as v ax -+- \/ 'aa — xx} or any other com- plicate Terms of the like kind.
9. Furthermore, altho' in the Equation propofed there fhould be Quantities involved, which cannot be determined or exprefs'd by any Geometrical Method, fuch as Curvilinear Areas or the Lengths of Curve-lines ; yet the Relations of their Fluxions may be found, as will appear from the following Example.
Preparation for EXAMPLE 5*
10. Suppofe BD to be an Ordinate at right Angles to AB, ancL that ADH be any Curve, which is defined by the Relation between AB and BD exhibited by an Equation. Let AB be called A;, and the Area of the Curve ADB, apply 'd to Unity, be call'd z. Then erect the Perpendicular AC equal to Unity, and thro' C draw CE parallel to AB, and meeting BD in E. Then conceiving thefe two Superficies ADB and ACEB to be generated by the Motion of the right Line BED ; it is manifeft that their Fluxions, (that is,, the Fluxions of the Quantities i x zt. and i x v, or of the Quantities s and x,) are to each other as the generating Lines BD and BE. Therefore « : x :: BD : BE or i, and therefore z = * x BD.
1 1. And hence it is, that z may be involved in any Equation, expre fling the Relation between .v and any other flowing'Quantityjv ; and yet the Relation of the Fluxions x and y may however be dif- cover'd, 12.
24 <fhe Method <J/" FLUXION s,
12. Ex. 5. As if the Equation zz -\-axz — _y*=r=o were pro- pos'd to exprefs the Relation between x and;1, as alfo \/ax—xx = BD, for determining a Curve, which therefore will be a Circle. The Equation zz-^-axz — j^=o, as before, will give 2zz-i- azX -f- axz — 4_y_y» = o, for the Relation of the Celerities x,y, and z. And therefore fince it is z = x x BD or • — -x \/ax — xxt iubftitute this Value inftead of it, and there will arife the Equation
2xz -t- axx \/ax-r— xx 4- axz — qyy* = o, which determines the Relation of the Celerities x and y.
DEMONSTRATION of the Solution.
13. The Moments of flowing Quantities, (that is, their indefi- nitely fmall Parts, by the acceffjon of which, in indefinitely fmall portions of Time, they are continually increafed,) are as the Ve- locities of their Flowing or Increafing.
14. Wherefore if the Moment of any one, as x, be reprefented t>y the Product of its Celerity x into an indefinitely fmall Quantity o (that is, by xo,} the Moments of the others <y, y, z, will be reprefented by vot yo, zo ; becaufe voy xo, yo, and zo, are to each other as v, x, y, and x.
,. p. , 15. Now fince the Moments, as xo and yo, are the indefinitely
/fc«, »// natti** cttA uttie ^cceflions of the flowing Quantities .v and y, by which thofe
any
And therefore the Equation, which at all times indifferently exprefles the Relation of the flowing Quantities, will as well exprefs the Relation between x -3- xo and y-+-yo, as between x and y: So that x -+- xo and y -f- yo may be fubftituted in the fame Equation for thofe Quantities, inftead of x and y.
1 6. Therefore let any Equation #' — ax* -+- axy — ^' = 0 be given, and fubftitute x~\-xo for x} and y -j- yo for y, and there will arife
•+• $x*oox -f- x*o''
ax1 — 2axox — ax*oo
• • axy •+- axoy -h ayox -h axyoo
y: —lyoy- ~ yfooy —
and INFINITE SERIES. 25
17. Now by Suppofition x3 — ax°--3raxy — _}'3=o, which there- fore being expunged, and the remaining Terms being divided by o, there will remain ^xx* -f- ^ox -+- x>oo — zaxx — ax1o -f- axy -f- ayx _f_ axyo — 3_vy* — 3y*oy — y*oo = o. But whereas o is fuppofed to be infinitely little, that it may reprefent the Moments of Qiian- tities ; the Terms that are multiply'd by it will be nothing in relbedl of the reft. Therefore I reject them, and there remains $xx* — zaxx -f- axy -+- ayx — 3_yj*= o, as above in Examp. i.
1 8. Here we may obferve, that the Terms that are not multiply'd by o will always vaniih, as alfo thole Terms that are multiply'd by o of more than one Dimenfion. And that the reft of the Terms being divided by o, will always acquire the form that they ought to have by the foregoing Rule : Which was the thing to be proved.
19. And this being now fhewn, the other things included in the Rule will eafily follow. As that in the propos'd Equation feveral flowing Quantities may be involved ; and that the Terms may be multiply'd, not only by the Number of the Dimenlions of the flow- ing Quantities, but alfo by any other Arithmetical Progreilions ; fo that in the Operation there may be the lame difference of the Terms according to any of the flowing Quantities, and the ProgrefTion be difpos'd according to the fame order of the Dimenlions of each of them. And thele things being allow'd, what is taught belides in Examp. 3, 4, and 5, will be plain enough of itfelf.
P R O B. II.
An Equation being propofed, including the Fluxions of O^uantitieS) to find the Relations of tbofe Quantities to one another.
A PARTICULAR SOLUTION.
i. As this Problem is the Converfe of the foregoing, it muft be folved by proceeding in a contrary manner. That is, the Terms multiply'd by x being difpofed according to the Dimenfions of x ;
they muft be divided by *x , and then by the number of their Di- menfions, or perhaps by fome other Arithmetical Progreffion. Then the fame work muft be repeated with the Terms multiply'd by v, y,
E or
26 The Method of FLUXIONS,
or z, and the Sum refulting muft be made equal to nothing, re- jeding the Terms that are redundant.
2. EXAMPLE. Let the Equation propofed be ^xx* — 2axx 4- axy 4- ayx = o. The Operation will be after this manner :
Divide 3 ATA?* — 2axx-i-axy
by - • Quot. 3A:5 — 2ax* -\-ayx
Divide by 3 . 2 i.
Quote A;5 — ax1 -{-ayx
Divide —
by ^. Quot. —3
Divide by 3
Quote — _y5
* -f- ayx
* 4- axy
2 . i.
* 4- axy
Therefore the Sum #3 — ax* -f- axy — y* = o, will be the required Relation of the Quantities x and y. Where it is to be obferved, that tho' the Term axy occurs twice, yet I do not put it twice in the Sum x'> — ax* -+- axy — y* •=. o, but I rejed the redundant Term. And fo whenever any Term recurs twice, (or oftener when there are feveral flowing Quantities concern'd,) it muft be wrote only once in the Sum of the Terms.
3. There are other Circumftances to be obferved, which I mall/ leave to the Sagacity of the Artift -, for it would be needlefs to dwell too long upon this matter, becaufe the Problem cannot always be folved by this Artifice. I mail add however, that after the Rela- tion of the Fluents is obtain'd by this Method, if we can return, by Prob. i. to the propofed Equation involving the Fluxions, then the work is right, otherwife not. Thus in the Example propofed,
after I have found the Equation x> ax1- -{- axy — y* = o, if from
thence I feek the Relation of the Fluxions x and y by the firft Problem, I mall arrive at the propofed Equation ^xx* — 2axx 4- axy — i,yy* -f- ayx= o. Whence it is plain, that the Equation AT3 • -ax*-+-axy — _y3 = o is rightly found. But if the Equation xx — xy -\- ay = o were propofed, by the prefcribed Method I fhould obtain this ^x* — xy + ay = o, for the Relation between x and y ; which Conclufion would be erroneous: Since by Prob. i. the Equation xx — xy — yx -+- ay = o would be produced, which is different from the former Equation.
4. .Having therefore premiled this in a perfundory manner, I lhall now undertake the general Solution.
A
and IN FINITE SERIES. 27
A PREPARATION FOR THE GENERAL SOLUTION.
5. Firft it mufl be obferved, that in the propofed Equation the Symbols of the Fluxions, (fince they are Quantities of a diffe- rent kind from the Quantities of which they are the Fluxions,) ought to afcend in every Term to the fame number of Dimenfions :• And when it happens otherwife, another Fluxion of fome flowing Quantity mufl be underflood to be Unity, by which the lower Terms are fo often to be multiply'd, till the Symbols of the Fluxions arife to the fame number of Dimenfions in all the Terms. As if the Equation x -+• x'yx — axx = o were propofed, the Fluxion z of fome third flowing Quantity z mufl be underilood to be Unity, by which the firfl Term x mufl be multiply'd once, and the lafl axx twice, that the Fluxions in them may afcend to as many Di- menfions as in the fecond Term xyx : As if the propofed Equation had been derived from this xz -{-xyx- — azzx*- = o, by putting z = i. And thus in the Equation yx =}')'-, you ought to ima- gine x to be Unity, by which the Term yy is multiply'd.
6. Now Equations, in which there are only two flowing Quan- tities, which every where arife to the fame number of Dimenfions, may always be reduced to fuch a form, as that on one fide may be
had the Ratio of the Fluxions, (as 4 , or - , or ~ ,&c.) and on the
\ x . y x
other fide the Value of that Ratio, exprefs'd by fimple Algebraic
*
Terms ; as you may fee here, 4- = 2 -h 2X — y. And when the
foregoing particular Solution will not take place, it is required that you fhould bring the Equations to this form.
7. Wherefore when in the Value of that Ratio any Term is de- nominated-by a Compound quantity, or is Radical, or if that Ratio be the Root of an affected Equation ; the Reduction mufl be per- form'd either by Divifion, or by Extraction of Roots, or by the Refolution of an affected Equation, as has been before fhewn.
8. As if the Equation ya — yx — xa -+- xx — xy = o were pro- pofed j firfl by Reduction this becomes T-=i-f--^-, or -==
x a—x y
a—v+y' And in the firfl Cafe, if I reduce the Term ^£^., deno- minated by the compound Quantity a — x, to an infinite Series of
E 2 fimple
28 The Method of FLUXIONS,
fimple Terms j -f- - -f- ~ -+- ^ &c. by dividing the Numerator y by the Denominator a — x, I mall have - — — i •+- - -f- ^ -f.
^ -f- 7; &c. by the help of which the Relation between x and y is to be determined.
9. So the Equation _y_y = xy -j- .XVY.V A: being given, or ^- = 4,
A-* x
•i- xx, and by a farther Reduction 4=4 +V/T -+- A-* : I extract
AT —
the fquare Root out of the Terms -J -f- xr, and obtain the infinite Series f -{-x* — x* -f- 2X6 — 5*" -f- 14*'°, &c. which if I fubfti-
tute for \/t H- xx, I (hall have - = i -f- x* — x* -f- 2x6
X
&c. or. ~ = — x^-ir-x* — 2X6 -+- 5*8, &c. according as
is either added to -I, or fubtracled from it.
10. And thus if the Equation y* -j- axx*y -f- a'-x^y — x*x"> — ~
2x*a>=o were propofed, or '— -f- ax— -f- a1- >v3 — 2rf3 = o
A:5 A: x
I extract the Root of the affected Cubick Equation, and there.
•/- V X XX 111*5 COQi'4 0
anfes ~ =a ^-—_|_ ^_ _ 4. » ^ &c. as may be feen
x 4 640 5i2«a 16384^3 ^
before.
11. But here it may be obferved, that I look upon thofc Terms only as compounded, which are compounded in refpect of flowing Quantities. For I efteem thofe as fimple Quantities which are com- pounded only in refpect of given Quantities. For they may be re- duced to fimple Quantities by luppofing them equal to other givea
Quantities. Thus I eonfider the Quantities " -•> "-TT, — rr-
^ — - ^^' c a*4- b' ax-\~bx >
1 4 — — — — —
~^,L,xi > v/tfA- H- bx, &c. as fimple Quantities, becaufe they may may all be reduced to the fimple Quantities —^ i, -^-, — , \/ex (or
£x*} &cc. by fuppofing a -f- b =r= e.
12. Moreover, that the flowing Quantities may the more eafily be diflinguifh'd from one another, the Fluxion that is put in the Numerator of the Ratio, or the Antecedent of the Ratio, may not improperly be call'd the Relate Quantify, and the other in the De- nominator, to which it is compared, the Correlate : Alfo the
flowing
and INFINITE SERIES. 29
flowing Quantities may be diftinguifli'd by the fame Names refpec- tively. And for the better understanding of what follows, you may conceive, that the Correlate Quantity is Time, or rather any other Quantity that flows equably, by which Time is expounded and meafured. And that the other, or the Relate Quantity, is Space, which the moving Thing, or Point, any how accelerated or retarded, defcribes in that Time. And that it is the Intention of the Problem, that from the Velocity of the Motion, being given at every Inftant of Time, the Space defcribed in the whole Time may be deter- mined.
13. But in refpedt of this Problem Equations may be diftinguifli'd
into three Orders.
14. Firft: In which two Fluxions of Quantities, and only one of their flowing Quantities are involved.
15. Second: In which the two flowing Quantities are involved, together with their Fluxions.
1 6. Third: In which the Fluxions of more than two Quantities are involved.
17. With thefe Premifes I {hall attempt the Solution of the Problem, according to thefe three Cafes.
SOLUTION OF CASE I.
1 8. Suppofe the flowing Quantity, which alone is contain 'd in the Equation, to be the Correlate, and the Equation being accord- ingly difpos'd, (that is, by making on one fide to be only the Ratio of the Fluxion of the other to the Fluxion of this, and on the other fide to be the Value of this Ratio in fimple Terms,) mul- tiply the Value of the Ratio of the Fluxions by the Correlate Quan- tity, then divide each of its Terms by the number of Dimenfions with which that Quantity is there afTeded, and what arifes will be equivalent to the other flowing Quantity.
19. So propofing the Equation yy = xy -+- xxxx ; I fuppofe x to be the Correlate Quantity, and the Equation being accordingly
reduced, we mall have •- = i -f- x1 — .v4 -f- 2X&, &c. Now I mul-
tiply the Value of — into x, and there arifes .v-f-AT3' — xf -{- 2X\
&c. which Terms I divide feverally by their number cf Dimenfions, and the Refult x •+- fv' — fv'-f-fv1, &c. I put =y. And by
this
30 77je Method ^/"FLUXIONS,
this Equation will be defined the Relation between x and y, as was • required.
20. Let the Equation be -- = a — - -4- — -f- '3'*3 &c. there
x 4 6-}<z 5i2«*
will arife y = ax — y -+- ~ j- -^ ' &c. for determining the
' y ZM —OJ.oi.t~ o
Relation between A; and y.
21. And thus the Equation — = _i_ -, •, — x* -t- #*,
v-J *.! I — • I
gives y = — ^ -f- ^ . + 2^ — |.x*+ £** . For multiply the Value of - into A;, and it becomes — — - -f. ax^ - . x* -*- v*
*; Jf^ X X ,
or A:-1 — x'1 -\- ax*— x^-i-x^, which Terms being divided by the number of Dimenfions, the Value of y will arife as be- fore.
22. After the fame manner the Equation -. =5-7=== 4- -^— -+-
\/ f S7- 1. A •
\- cy, gives A- = — ^_ -}- — H- - v/^)'3 -i- cy~> . For the Value of - being multiply'd by j, there arifes ~ -^ — *— _j_
-{-n'3 or 2^^-y* -h -~i ;'3 + v/^ •+• c %y*. And thence -the Value of x refults, by dividing by the number of the Dimen- lions of each Term.
23. And fo =? =z\ gives y = $z*. And -1 =- 4 , gives r= , ~ * «7
3f^L3. But the Equation ^ = ; , gives 7 = f . For f multiply'd
into A: makes a, which being divided by the number of Dimen- fions, which is o, there arifes ~ , an infinite Quantity for the Value
_
24. Wherefore, whenever a like Term mail occur in the Value
of •-. , whofe Denominator involves the Correlate Quantity of one
Dimenfion only ; inftead of the Correlate Quantity, fubftitute the Sum or the Difference between the fame and fome other given Quantity to be affumed at pleafure. For there will be the fame Relation of Flowing, of the Fluents in the Equation fo. produced, as of the Equation at firft propofed j and the infinite Relate Quan-
tity
and INFINITE SERIES. 31
tity by this means will be diminifh'd by an infinite part of itfelf, and will become finite, but yet confifting of Terms infinite in number.
25. Therefore the Equation 4 = - being propofed, if for x I write ^4- x, affuming the Quantity b at pleafure, there will arife
v 11 T^« • /* v fl a^ ax^ ax^ c At
•- = , — : and by Divifion 4 = T — rr 4- 77 — -rr &c- And
u-^r~X * v O & £ b +
now the Rule aforegoing will give_}'= j — - ^ 4- 3~£p — ~j^ &c. for the Relation between x and y.
26. So if you have the Equation - = - 4-3 — xx; becaufe
X X
of the Term ~x-> if you write i -f- x for x, there will arife 4 . — _f (_ 2 — 2X —xx. Then reducing the Term ~-^ into an in- finite Series 4-2 — 2x4- 2xl — 2Ar3 4- 2x% &c. you will have 4 ,
X
— ^ — 4* _{_ x* — 2x3 4- 2x4, &c. And then according to the Rule y = 4.x — ax1 4- fx3 — |x4 4- ^xs, 6cc. for the Relation of x
and y.
27. And thus if the Equation -.-•=x'^-i-x-1 — AT* were pro-
pofed j becaufe I here obferve the Term x l (or ~j to be found, I tranfmute x, by fubftituting I — • x for it, and there arifes 4 — . _' _L _•_ - - — v/ 1 — A;". Now the Term - l—x produces i _{_ x _|_ x1 4- x3, &c. and the Term \/i — x is equivalent to
j, .i# — 4-x1 • — —V^S an(^ therefore or •i_±v_JL;(.a ^ • is
the fame as i 4- -i-x 4- 4-x1 4- |-x3 , &c. So that when thefe Values are fubftituted, I fhall have 4 = i ~f- 2x 4- 4xi4-4-^-x3,6cc. And
X
then by the Rule y •=. x 4- x1 4- 4-x* 4- ri*4, &c- An<i ^ oi others.
28. Alfo in other Cafes the Equation may fometimes be con- veniently reduced, by fuch a Tranfmutation of the flowing Quantity.
As if this Equation were propofed 4 = -^ ^^.c^_xi • inflead
•52 ^ Method of FLUXIONS,
O i/
of .v I write c — AT, and then I mall have 4= — ^— or 75 — ~i>
and then by the Rule y = - — J ^ -f,. L. But the ufe of fuch Tranf- mutations will appear more plainly in what follows.
SOLUTION OF CASE II.
29". PREPARATION. And fo much for Equations that involve only one Fluent. But when each of them are found in the Equation, fiift it muft be reduced to the Form prefcribed, by making, that on one fide may be had the Ratio of the Fluxions, equal to an aggregate of fimple Terms on the other fide.
30. And befides, if in the Equations fo reduced there be any Fractions denominated by the flowing Quantity, they muft be freed from thofe Denominators, by the above-mentioned Tranfmutation of the flowing Quantity.
31. So the Equation yax — xxy — aax = o being propofed, or
i_l _{_ f . becaufe of the Term -, I afiume b at pleafure, and
x a x *
for x I either write b -+- x, or b — x, or x — - b. As if I fhould write b -+- x, it will become 4 = - -f- rrr. . And then the Term
being converted byDivifion into an infinite Series, we mall have
-1—-1 , - - < — — , &C.
72. And after the fame manner the Equation £••= 37 — 2x +
•J X
X 2v
- .. being propofed; if, by reafon of the Terms - and^.,
I write i — y for yy and i — x for x, there will arife — =
X
_ oV -4- 2 x -f- ^-=-^ -4- — 2-v~.2 r . But the Term '-— ^ by
3/ 1 y I ZX -\- X* 1 y J
infinite Divjfion gives i — x -+-y — xy -f-_ya — xy* -J-_y3 — xy*t &c. and the Term -t _^2~+ xx by a like Divifion gives 2_y — 2 -i- ^xy — ^x _f- 6x*-y — . 6xa 4- S*3^ — 8x5 + iox*y — IOAT*, &c. There- fore r-= — 3^-i- 3^J -f->'a' — xy* -{- y3 — ^y5, &c. -i- 6^^ — • 6x*
X
33-
and INFINITE SERIES. 33
33. RULE. The Equation being thus prepared, when need re- quires, difpofe the Terms according to the Dimenfions of the flow- ing Quantities, by fetting down fir ft thofe that are not affected by the Relate Quantity, then thofe that are affected by its lead Dimen- fion, and fo on. In like manner alfo diipofe the Terms in each of thefe Clafies according to the Dimenfions of the other Correlate Quantity, and thofe in the firft Clafs, (or fuch as are not affected by the Relate Quantity,) write in a collateral order, proceeding to- wards the right hand, and the reft in a defcending Series in the left- hand Column, as the following Diagrams indicate. The work be- ing thus prepared, multiply the firft or the loweft of the Terms in the firft Clafs by the Correlate Quantity, and divide by the number of Dimenfions, and put this in the Quote for the initial Term of the Value of the Relate Quantity. Then fubftitute this into the Terms of the Equation that are difpofed in the left-hand Column, inftead of the Relate Quantity, and from the next loweft Terms you will obtain the fecond Term of the Quote, after the fame man- ner as you obtain'd the firft. And by repeating the Operation you may continue the Quote as far as you pleafe. But this will appear plainer by an Example or two.
34. EXAMP. i. Let the Equation 4 = i — ^x-\-y-\- x*-{-.vy
be propofed, whofe Terms i — T.V -+- A'1, which are not affected by the Relate Quantity _v, you fee difpos'd collaterally in the up-
-h I T,X -\- XX
+'*,
* -+- A' X,Y-f-l.,V3 ^.x-4_|__'_,v
r,&c.
J_ ^ V
' "5""^"
s,&c
The Sum
I ' 2.V "--I-"- &X * — V ^ - 1 * v4i T ^_ \s
, &c.
y
A—A-X -»4*I - >4 + ^,__Vx6^c.
permoft Row, and the reft ' y -and .vy in the left-hand Column. And rirft I multiply the initial Term i into the Correlate Quantity .v, .ind it makes x, which being divided by the number of Dimen- fions i, I place it in the Quote under-written. Then fubftkuting rhis Term inftead of y in the marginal Terms -f- y and -f- .vy, I have -\-x and -+- xx, which I write over againft them to the right hand. Then from the reft I take the loweft Terms — ?.v and -±-x, whofe aggregate — zx multiply'd into x becomes — 2.v.v, and
F being
3-4
The Method of FLUXIONS,
beino; divid'-d by the number of Dimenfions 2, gives — xx for the fecund Term of the Value of y in the Quote. Then this Term being likewifc afiumed to compleat the Value of the Marginals -{-y and -+- xv, there will arife alfo — xx and — x5, to be added to the Terms -j-x and -{-xx that were before inferted. Which being done, I again a flume the next loweil Terms -f-xx, — xx, and -{-xx, which I collect into one Sum xx, and thence I derive (as before) the third Term -|-.ix;, to be put in the Value of y. Again, taking this Term -i-x3 into the Values of the marginal Terms, from the next loweft -f-y#3 and — x3 added together, I obtain — ^-x4 for the fourth Term of the Value of y. And fo on in infinitum.
35. Ex AMP. 2. In like manner if it were required to determine
the Relation of x and y in this Equation, y- -=. I -f- - -f- --v -f- — r'-f-
< ^ a &* &*
- , &c. which Series is fuppofed to proceed ad infinitum ; I put I
in the beginning, and the other Terms in the left-hand Column, and then purfue the work according to the following Diagram.
-hi
A" A* *3 .X 4
-.j
+ ~
h —, , &c.
XV
A"a v 3 A 4
A *
4- £
a1 2^3 2^4
h z~ . &C-
Xs" V
_1_ 'v3 i A'4
, . 5
4- ~
h — , &c.
-4- ~
* * * * -+- — -
h S ' &c-
4-*-?
* * * * * -
h-J , &c.
a*
Sum
.V 3** 2\= CAT4
T _l_ *_ 1 — . 1 1
* I i ^ — r — i — "^~" —
3.V5 c
h 4y , &c.
a ^ai a= z.;4
y ==
* + Ta-+- ili + £ + ^ -
^6 o
h — j , &c.
36. As I here propofed to extradl: the Value of y as far as fix Dimenfions of x only ; for that reafon I omit all the Terms in the Operation which I forefee will contribute nothing to my pur- pofe, as is intimated by the Mark, &c. which I have fubjoin'd to the Series that are cut off.
3 37-
and INFINITE SERIES. 35
37. EXAMP. 3. In like manner if this Equation were propofed
• = — 3,v -+- i*y -4-;* — Xj* -t-j3 — .vy3 -4-;-« — A^
— 6..Y1 -f- SA-J_V - — 8.v3 4- \oxy* — IOA-*, &c. and it is intended to extract the Value ot y as far as feven Dimensions of x. I place the Terms in order, according to the following Diagram, and I work as before, only with this exception, that iince in the left-hand Co- lumn y is not only of one, but alfo of two and three Dimensions; (or of more than three, if I intended to produce the Value of y beyond the degree of x~* ,) I fubjoin the fecond and third Powers of the Value of y, fo far gradually produced, that when they are fubftitu- ted by degrees to the right-hand, in the Values of the Marginals
_ 3.v _ 6X> — 8*3 — IO.V^ — I2A- — M£ ,&CC.
+ 3*7
9v,
2"
— 6x*
b zo ' "
-+- 6x*y
* * *
— gx*
— I2.V — ^V ,&C.
-f- 8*7
* * *
*
I2AT* l6x6,fxc.
-f- IOA:^
* * *
#
* ^[J^6 j&C-
&c.
+-;•*
* * #
^|*4
-f- 6xs -{-~^7x6 ,&;c.
— xy*
* * *
*
4 * '
&C.
H-.v;
* * *
*
* — ~--xs ,6cc.
Sum
— 3 A- — 6x* — ^f.v
3 9'
4
— -^-'v' — -Z.v-6 li-r-
^ •* — .X ,tXC.
h S '
3 2S
qi
111 6 ^"
y= -A1 2X> -*<
20
"16^ "77"r > C '
^ A '°7 * " 4"^ 8
«, &C.
y; — — — x6, 6cc.
to the left, Terms may arife of fo many Dimenfions rs I obferve to'be required for the following Operation. And by this Method
there arifes at length y= — ^x1 — 6.x13 — ^^+, &c. which is the
F 2 Equation
3 6 The Method of FLUXIONS,
Equation required. But whereas this Value is negative, it appears that one of the Quantities x or y decreafes, while the other in- creafes. And the fame thing is allb to be concluded, when one of the Fluxions is affirmative, and the other negative.
38. EXAMP. 4. You may proceed in like manner to refolve the Equation, when the Relate Quantity is affected with fractional Di- menfions. As if it were propofed to extract the Value of x from
this Equation, - = iy — ^y- -+- zyx* — -J.v1 -f- 77* -f- 2_y;, in
H— 5-7 * — 4-y1 -+• jy1 •+• 2>'3
I
* * +)'* * — 2_)'3-|-4}'T — 2_y4, &c. * * * * * * — ~y4y&tc.
Sum
+±y #_3r_f_7/ . +4/— 44-VS&C.
ATT=±= •+ 4_y — y1 -+- 2y* ' — _)•* , &c. A;*= -V74> ^c-
which ,v in the Term a^'-x11 (or zy^/x) is affected with the Frac- tional Dimenlion -i- From the Value of x I derive by degrees the Value of A?% (that is, by extracting its fquafe-Root,) as may be obferved in the lower part of this Diagram ; that it may be in- ferted and transfer'd gradually into the Value of the marginal Term 2yx'f. And fo at laft I fliall have the Equation x = ±.yl — y* _|_ 2_y^ -(- ^ — TVo^'f> &c- by which x is exprefs'd indefinitely in re- ipect of y. And thus you may operate in any other cafe what- foever.
39. I foid before, that thefe Solutions may be perform'd by an infinite variety of ways. T'his may 'be done if you afiiime at pleafure not only the initial quantity of the upper Series, but any other given quantity for the firft Term of the Quote, and then you may proceed as before. Thus in the firft of the preceding Exam- ples, if you affume i for the firft Term of the Value of 7, and fubftitute it for y in the marginal Terms -h_y and -t-xy, and pur- fue the reft of the Operation as before, (of which I have here given a
and INFINITE SERIES.
37
-f- I 3x4- XV
4-*V
-4- i 4- 2x * 4- AT3 4- .ix4, 6cc.
* -t- X 4- 2Arl * 4- X4, &C.
Sum
4-2 * 4- 3** 4- A;3 4-4-A"4, &c.
y - — i -f- 2.v * 4- x"' -\- ix4 4-^-A'5, 6cc.
Specimen,) another Value of y will arife, i -f- 2x-\- x* -h i*4, 6cc. And thus another and another Value may be produced, by afTum- ing 2, or 3, or any other number for its firfl Term. Or if you make ufe of any Symbol, as a, to reprefent the firft Term inde- finitely, by the fame method of Operation, (which I fhall here fet down,") you will find y = a -+- x -+- ax — xx -f- axx -+- ~x*+±ax*, &c. which being found, for a you may fubfHtute i, 2, o, 4-, or any other Number, and thereby obtain the Relation between x and y an infinite variety of ways.
4- i — 3 x 4- A* AT
+y
_|_ fl _|_ x .v.V -
H yX3 , &c.
4™ #^" 4~ ^ATX -
f- -i^.v3, 6cc.
4-#y
* -f. tf.v 4- AT1 -
- *s , &c.
-(- ^ZAT1 -
f- ax* , &c.
Sum
4-1 2X 4- AT1 -
— AAr5 , &C.
4-^4- 2^-4- 2«x»-
-f-l^x3, &c.
j = a 4- A; — x1
-h y-V3 ^-.V4 , &C.
4- ax 4- fl.v1 -
f- j.tfJfJ + _V^V45 &C.
40. And it is to be obferved, that when the Quantity to be ex- trailed is affected with a Fractional Dimenfion, (as you fee in the fourth of the preceding Examples,) then it is convenient to take Unity, or fome other proper Number, for its firft Term. And in- deed this is neceflliry, when to obtain the Value of that fractional Dimenfion, the Root cannot otherwife be extracted, becaufe oi the negative Sign ; as alib when there are no Terms to be diJpofcd in the firft or capital Clafs, from which that initial Term may be deduced. 41.
38 tte Method of FLUXIONS,
41. And thus at laft I have compleated this moft troublefo'me and of all others moft difficult Problem, when only two flowing Quantities, together with their Fluxions, are comprehended in an Equation. But befides this general Method, in which I have taken in all the Difficulties, there are others which are generally fhorter, by which the Work may often be eafed; to givefome Specimens of which, ex abundantly perhaps will not be diiagreeable to the Reader.
42. I. If it happen that the Quantity to be refolved has in fome places negative Dimenfions, it is not of ablblute necefllty that there- fore the Equation mould be reduced to another form. For thus
the Equation y = - — xx being propofed, where y is of one ne- gative Dimenfion, I might indeed reduce it to another Form, as by writing i -f- y for y ; but the Refolution will be more expe- dite as you have it in the following Diagram.
#
* XX
I
y Sum
i i
— V* -•-! — • ^ V JK* ^CC
y
4- .V "'i-YAT -f- |-.V3, &C.
- — x-t-^xx, 5cc.
43. Here affuming i for the initial Term of the Value of y., . I extract the reft of the Terms as befoie, and in the mean time
I deduce from thence, by degrees, the Value of - by Divifion, and infert it in the Value of the marginal Term.
44. II. Neither is it neceffary that the Dimenfions of the other flowins Quantity fhould be always affirmative. For from the Equa- tion y = 3 -\- zy — '- , without the prefcribed Reduction of the
Term }~ , there will arife_y = 3 A; — ±xx -f- 2XJ, &c.
4^. And from the Equation y = — }'-+--. — ~x > the Value
of y will be found y ==• ^, if the Operation be perform 'd after the Manner of the following Specimen.
i
XX
and INFINITE SERIES,
3.9
I
.
*A:
.V
I
— V
*
" .V
Sum
i
o
ATA:
y =
* X
46. Here we may obferve by the way, that among the infinite manners by which any Equation may be refolved, it often happens that there are fome, that terminate at a finite Value of the Quan- tity to be extracted, as in the foregoing Example, And thefe are not difficult to find, if fome Symbol be aflumed for the firft Term. For when the Refolution is perform'd, then fome proper Value may -be given to that Symbol, which may render the whole finite.
47. III. Again, if the Value of y is to be extracted from this
Equation y = ^. -+- i — zx -f- ±xxy it may be done conveniently
enough, without any Reduction of the Term ~ , by fuppofing
(after the manner of Analyfts,) that to be given which is required. Thus for the firit Term of the Value of y I put zcx, taking 2<? for the numeral Coefficient which is yet unknown. And fubltituting 2.cx inftead of y, in the marginal Term, there ariies e, which I write on the right-hand ; and the Sum i -f- e will give x -f- ex for the fame firft Term of the Value of yt which I had firfi repre- fented by the Term zcx. Therefore I make 2cx = x-}-ex, and thence I deduce e =•. i. So that the firfl Term zex of the Value of y is 2.x. After the fame manner I make ufe of the fidlitious Term 2/x* to reprefent the fecond Term of the Value of r, and thence at laft I derive — ^ for the Value of y, and therefore that fe- cond Term is — ±xx. And fo the fictitious Coefficient g in the third Term will give TV, and b in the fourth Term will be o. Wherefore iince there are no other Terms remaining, I conclude the work is finiOi'd, and that the Value of y is exadtl-y zx — ±xl -if-^X', See the Operation in the following Diagram.
i
The Method ^FLUXIONS,
I ~2X +iXX
y
? 4~ /A* | - cfxx [ /yv'
Zx
6
Sum
4"~i ~~~ 2 A" 4~ •£ XX
Hvpothetically r= zex-{- 2fx*-\- 2gx* 4- 2&c+
II II 1l II
Confequentially y= 4->v — A* 4- ^x* 4- ^6^«
Real Value j'= 2 A* — l^1 4- ^-A-'
48. Much after the fame manner, if it were y = ^- ; fuppoie
y=.exs, where e denotes the unknown Coefficient, and s the num- ber of Dimeniions, which is alfo unknown. And ex' being fub-
ftituted for y, there will arife y •=. -— , and thence again 7 =
*— . Compare thefe two Values of y, and you will find ^ = e, and therefore s = •£•, and e will be indefinite. Therefore afTuming
e at pleafure, you will have y = ex*.
49. IV. Sometimes alfo the Operation may be begun from the higheft Dimenfion of the equable Quantity, and continually pro- ceed to the lower Powers. As if this Equation were given, ^=: 2.1.1 _i_T_i_2;r — -, and we would begin from the higheft
xx ~ XX ,. 3 * . °
Term zx, by difpofing the capital Series in an order contraiy to the foregoing ; there will arife at laft y = xx -f- 4.* — - , &c. as may be feen in the form of working here fet down.
4 '
+.i
* H- i 4-^ *
i i e — - -h — > &C.
-v * ^A *r
Sum
i
— • rr •+• ^7* ' ^cc>
_j> = A'1 4- 4.v * — ;
+ 1^ SIT > &c-
50.
and INFINITE SERIES, 41
50. And here it may be obferved by the way, that as the Opera- tion proceeded, I might have inferted any given Quantity between
the Terms 4** and — - , for the intermediate Term that is deficient,
and fo the Value of y might have been exhibited an infinite variety of ways.
51. V. If there are befides any fractional Indices of the Dimen- fions of the Relate Quantity, they may be reduced to Integers by fuppofing that Quantity, which is affected by its fractional D- menfion, to be equal to any third Fluent ; and then by ftibftitutii g that Quantity, as alfo its Fluxion, ariling from that fictitious Equation, inftead of the Relate Quantity and its Fluxion.
52. As if the Equation y= 3*7* -\- y were propofed, where the Relate Quantity is affected with the fractional Index .1 of its Dimen- fion; a Fluent z being afTumed at pleafure, fuppofe y^ = z, or y = z'> ; the Relation of the Fluxions, by Prob. i. will be y = 32Z1. Therefore fubftituting ^zz* for v, as alfo z* for y, and z* for y$, there will arife yzz1 = ^xz*- -+- z3, or z = x -\-^z, where z performs the office of the Relate Quantity. But after the
Value of z is extracted, as z = ±x* -f- — -f- ^ -J- -^-Q , &c. in- ftead of z reftore y\ and you will have the defired Relation be- tween x and v; that is, y? = i.v1 + -V^3H- T-nr*4; &c- an(^ ^7 Cubing each fide, y •=.^x6-\- T'_.v7 -+- TYTXS> ^c-
53. In like manner if the Equation y = </^y -+- </xy were given, or_y = 2^^ -J- xM ; I make z =)'^ or zz=y, and thence by Prob. i. 2zz = y, and by confequence 2zz = 2z -f- x*z, or z = i -+- {-x^. Therefore by the firft Cafe of this 'tis z = x -f-
-i-v1", or y'1 = Ar-f- -i.v1, then by fquaring each fide, v=y>; -+- -|Jf^ -i- -i-x5. But if you mould defire to have the Value of y exhibited an infinite number of ways, make z =. c -f- x -f- -ytf , aiTuming any initial Term c, and it will be ss, that is y, = c* -{- zcx + ^cx* •+• -v1 -+- -i-x1* -t- ^v3. But perhaps I may feem too minute, in treat- ing of fuch things as will but feldom come into practice.
SOLUTION OF CASE III.
54. The Refolution of the Problem will foon be difpatch'd, when the Equation involves three or more Fluxions of Quantities. For
G between
42 ?$£ Method of FLUXIONS,
between any two of thofe Quantities any Relation may be afiumed, when it is not determined by the State of the Queftion, and the Re- lation of their Fluxions may be found from thence ; fo that either of them, together with its Fluxion, may be exterminated. For which reafon if there are found the Fluxions of three Quantities, only one Equation need to be affumedj two if there be four, and fo on j that the Equation propos'd may finally be transform'd into another Equation, in which only two Fluxions may be found. And then this Equation being refolved as before, the Relations of the other Quantities may be difcover'd.
55. Let the Equation propofed be zx — z -f- yx = o ; that I may obtain the Relation of the Quantities x, y, and z, whofe Fluxions x, y, and z are contained in the Equation ; I form a Relation at pleafure between any two of them, as x and y, fuppofing that x=y, or 2y = a -+- z, or x=yy, &c. But fuppofe at prefent x=yy, and thence x = 2yy. Therefore writing zyy for x, and yy for x, the Equation propofed will be transform'd into this : q.yy — z-^-yy* = o. And thence the Relation between y and z will arife, 2yy-{-
^y= =.z. In which if x be written for yy, and x* for y~>, we mall have 2X -f- ~x^ = z. So that among the infinite ways in which x, y, and z, may be related to each other, one of them is here found, which is reprefented by thefe Equations, .v =yy, 2y* •+- ±y* = z, and 2X -+- ^x* = z.
DEMONSTRATION.
56. And thus we have folved the Problem, but the Demonftra- tion is ftill behind. And in fo great a variety of matters, that we may not derive it fynthetically, and with too great perplexity, from its genuine foundations, it may be fufficient to point it out thus in fhort, by way of Analyfis. That is, when any Equation is propos'd, after you have finifh'd the work, you may try whether from the derived Equation you can return back to the Equation propos'd, by Prob. I. And therefore, the Relation of the Quantities in the de- rived Equation requires the Relation of the Fluxions in the propofed Equation, and contrary-wife : which was to be fhewn.
57. So if the Equation propofed were y = x, the derived Equa- tion will be y={xl; and on the contrary, by Prob. i. we have y — xx, that is, y=.x, becaufe x is fuppofed Unity. And thus
from
and INFINITE SERIES. 4.3
from y = I — 3* -+-y -f- xx -+- xy is derived _y = tf — x* -f- Lx1 — ^v+ -+- ^o x! — -4T'vS> &c- And thence by Prob. i. y = i — 2x ^-x1 — %x> -+- ^-x* • — -Vx!) &c. Which two Values of y agree with each other, as appears by fubftituting x — xx+^x> — -^x* ->-J-xs, <5cc. inftead of^ in the firft Value.
.,8. But in the Reduction of Equations I made ufe of an Opera- tion, of which alfo it will be convenient to give fome account. And that is, the Tranfmutation of a flowing Quantity by its connexion with a given Quantity. Let AE and ae be two Lines indefinitely extended each way, along which two moving Things or Points may pafs from afar, and at the fame time
may reach the places A and a, B and A E c p E
b, C and c, D and d, &c. and let B '
be the Point, by its diftance from which, -4 : — i £ ^ ?—
the Motion of the moving thing or
point in AE is eftimated ; fo that — BA, BC, BD, BE, fucceffively, may be the flowing Quantities, when the moving thing is in the places A, C, D, E. Likewife let b be a like point in the other Line. Then will — BA and — ba be contemporaneous Fluents, as alfo BC and be, BD andZv/, BE and be, 6cc. Now if inftead of the points B and b, be fubftituted A and c, to which, as at reft, the Motions are refer'd ; then o and — ca, AB and — cb, AC and o, AD and cd, AE and ce, will be contemporaneous flowing Quantities. There- fore the flowing Quantities are changed by the Addition and Sub- traclion of the given Quantities AB and ac ; but they are not changed as to the Celerity of their Motions, and the mutual refpect of their Fluxion. For the contemporaneous parts AB and ab, BC and be, CD and cd, DE and de, are of the fame length in both cafes. And thus in Equations in which thefe Quantities are reprefented, the contemporaneous parts of Quantities are not therefore changed, not- withftanding their ablblute magnitude maybe increafed or diminimed by fome given Quantity. Hence the thing propofed is manifeft : For the only Scope of this Problem is, to determine the contempo- raneous Parts, or the contemporary Differences of the abfolute Quan- tities f, x, _>', or z, defcribed with a given Rate of Flowing. And it is all one of what abfolute magnitude thofe Quantities are, fo that their contemporary or correfpondent Differences may agree with the prcpofed Relation of the Fluxions.
59. The reaibn of this matter may alfo be thus explain'd Al- gebraically. Let the Equation y=xxy be propofed, and fup-
G 2 pole
44. 77je Method of FLUXIONS,
pofe x= i -+-Z- Then by Prob. i. x = z. So that for y =-. xxy , may be wrote y •=. xy -h xzy. Now fince ,v=s, it is plain,, that though the Quantities x and z be not of the fame length, yet that they flow alike in refpecl: of y, and that they have equal contem- poraneous parts. Why therefore may I not reprefent by the fame Symbols Quantities that agree in their Rate of Flowing,; and to de- termine, their contemporaneous Differences, why may not I uie
v === xy •+•• xxy initead of y = xxy ?
60.. Lartly it appears plainly in what manner the contemporary parts may be found, from an Equation involving flowing Quantities.
Thus if y = ~ -+- x be the Equation, when # = 2, then _y = 24. But when x = 3, then y =. 3.1. Therefore while x flows from 2 to 3, y will flow from 2-i to 3.1. So that the parts defcribed in this time are 3 — 2 = i, and 3-^ — 2-i = f .
6 1. This Foundation being thus laid for what follows, I fhall now proceed to more particular Problems.
PROB. m.
A ltijt'1 ^° determine the Maxima and Minima of H^
1. When a Quantity is the greateft or the leaft that it can be, at that moment it neither flows backwards or forwards. For if it flows forwards, or increafes, that proves it was lefs, and will pre- fently be greater than it is. And the contrary if it flows backwards, or decreafes. Wherefore find its Fluxion, by Prob. i. and fuppofe it to be nothing.
2. Ex AMP. i. If in the Equation x> — ax1 + axy — jy3 = o the greatefl Value of, x be required ; find the Relation of the Fluxions of x and y, and you will have 3X.va — 2axx -f- axy — %yyl -i-ayx = o. Then making x = o, there will remain — yyy1 -\- ayx=o, or 3j* = ax. By the help of this you may exterminate either x or y out of the primary Equation, and by the refulting Equation you may determine the other, and then both of them by — 3^* -f- ax = o.
3. This Operation is the fame, as if you had multiply 'd the Terms of the propofed Equation by the number of the Dimenfions of the other flowing Quantity.^. From whence we may .derive the
famous 2.
and INFINITE SERIES. 45
famous Rule of Huddenius, that, in order to obtain the greateft or leaft Relate Quantity, the Equation mufl be difpofed according to the Dimenfions of the Correlate Quantity, and then the Terms are to be multiply 'd by any Arithmetical ProgrelTion. But fince neither this Rule, nor any other that I know yet publiihed, extends to Equa- tions affected with iiird Quantities, without a previous Reduction j I fhall give the following Example for that purpofe.
4. EXAMP. 2. If the greatest Quantity y in the Equation x* —
ay~ + 7+ -- xx ^ ay ~+" xx= ° be to be determin'd, feek the .Fluxions of xand^y, and there will arife the Equation 3^^* — zayy-{-
^«^v)1 + 2^n5 Aaxxy-\-6x\* + atx2 A j r \ r r •
I __ - — _ -— = 0. And fince by fuppofition y = o, ,
a1 -\- zay +j* 2 ^ ay -\- xx
omit the Terms multiply'd by y, (which, to fhorten the labour, might have been done before, in the Operation,) and divide the reft
by xx, and there will remain %x — ^- "*"-'** = o. When the Re-
a"xx
duction is made, there will arife ^ay-\- %xx = o, by help of which you may exterminate either of the quantities x or y out of the pro- pos'd Equation, and then from the refulting Equation, which will, be Cubical, you may extract the Value of the other.
5. From this Problem may be had the Solution of thefe fol- lowing.
I. In a given .Triangle, or in a Segment of any given Curve, ft> ir.fcribe the greatejl Reft angle.
II. To draw the greatejl or the leafl right Line, 'which can lie: between a given Point, and a Curve given in pofition. Or, to draw. a Perpendicular to a Curve from a given Point.
III. To draw the greatejl or the leajl right Lines, which pajjin?.- through a given Point, can lie bet-ween two others, either right Lines or Curves.
IV. From a given Point within a Parabola, to draw a rivbt Line, which Jhall cut the Parabola more obliquely than any other. And to do the fame in other Curves.
V. To determine the Vertices of Curves, their greatejl or lealT Breadths, the Points in which revolving parts cut each other, 6cc.
VI. To find the Points in Curves, where they hcrce the great ejT or leajl Curvature.
VII. To find the Icaft Angle in a given EHi£/is, in which the. Ordinates can cut their Diameters.
VIII..
4.6 The Method of FLUXIONS,
VIII. Of EHipfes that pafs through four given Points, to deter- mine the greateft, or that which approaches neareft to a Circle.
IX. 70 determine fuch a part of a Spherical Superficies, which can be illuminated, in its farther part, by Light coming from a great dijlance, and which is refracted by the nearer Hemijphere.
And many other Problems of a like nature may more eafily be propofed than refolved, becaufe of the labour of Computation.
P R O B. IV.
To draw Tangents to Curves.
Firft Manner.
1. Tangents may be varioufly drawn, according to the various Relations of Curves to right Lines. And firft let BD be a right Line, or Ordinate, in a given Angle to
another right Line AB, as a Bafe or Ab- fcifs, and terminated at the Curve ED. Let this Ordinate move through an inde- finitely finall Space to the place bd, fo that it may be increafed by the Moment cd, while AB is increafed by the Moment — ^ A Bb, to which DC is equal and parallel. Let Da1 be produced till it meets with AB in T, and this Line will touch the Curve in D or d ; and the Triangles dcD, DBT will be fimilar. So that it is TB : BD : : DC (or B£) : cd.
2. Since therefore the Relation of BD to AB is exhibited by the Equation, by which the nature of the Curve is determined ; feek for the Relation of the Fluxions, by Prob. i. Then take TB to BD in the Ratio of the Fluxion of AB to the Fluxion of BD, and TD will touch the Curve in the Point D.
3. Ex. i. Calling AB = x, and BD =jy, let their Relation be x-, — ax* -h axy — _y3 = o. And the Relation of the Fluxions will be 3xx-i — 2axx-i-axy — ^yy* -+- ayx-=. o. So that y : x :: ^xx — 2ax -4- ay : ^ —ax :: BD (;-) : BT. Therefore BT = ... w* ~~ f!X~ — • Therefore the Point D being given, and thence DB and AB, or v and x, the length BT will be given, by which the Tan- gent TD is determined.
4-
and INFINITE SERIES. 47
4. But this Method of Operation may be thusconcinnated. Make the Terms of the propofed Equation equal to nothing : multiply by the proper number of the Dimenfions of the Ordinate, and put the Refult in the Numerator : Then multiply the Terms of the fame Equation by the proper number of the Dimenfions of the Abfcifs, and put the Produdl divided by the Abfcifs, in the Denominator of the Value of BT. Then take BT towards A, if its Value be affirmative, but the contrary way if that Value be negative.
o o 13
5. Thus the Equation*3 — ax* -f- axy — y*=o, being multi-
3 z 10
ply'd by the upper Numbers, gives axy — 3_y3 for the Numerator j and multiply 'd by the lower Numbers, and then divided by x, gives 3-x-1 — zax -+- ay for the Denominator of the Value of BT.
6. Thus the Equation jy3 — by* — cdy -f- bed -\-dxy = o, (which denotes a Parabola of the fecond kind, by help of which Des Cartes confirufted Equations of fix Dimenfions ; fee his Geometry, p. 42. Amfterd. Ed. An. 1659.) by Infpeftion gives ^--"fr+'^v ^ Qr
7. And thus a1 — r-x* — y1 = o, (which denotes an Ellipfis whofe Center is A,) gives —^ , or ^ = BT. And fo in others.
- — X 1
1
8. And you may take notice, that it matters not of what quantity the Angle of Ordination ABD may be.
9. But as this Rule does not extend to Equations afFefted by furd Quantities, or to mechanical Curves ; in thefe Cafes we mufl have recourfe to the fundamental Method.
10. Ex. 2. Let A;S — ay1 -+- j-£ xx \/'ay -+- xx = o be the
Equation exprefling the Relation between AB and BD ; and by Prob. i. the Relation of the Fluxions will be 3*** — zayy -f. *"*"* + 2V
=0. Therefore it will be <ixx
*/,.,,,
4 v ~
T^T- :: (y : x ::) BD : BT.
fay — p ^^
II.
TJoe Method of FLUXIONS,
48
ii. Ex. 3. Let ED be the Conchoid of Nicomedes, defcribed with the Pole G, the Afymptote AT, and the Diftance LD ; and let
'GA = £, LD = c, AB=.v, andBD=;>. And becaufe of fimi- lar Triangles DEL and DMG, it will be LB : BD : : DM : MG ; that is, v/ 'cc — yy : y : : x : b -+- y, and therefore b-\-y ^/cc — yy =yx. Having got this Equation, I fuppofe V cc — yy = z, and thus I fliall have two Equations bz ~\-yz =yx, andzz = cc — yy. By the help of thefe I find the Fluxions of the Quantities x, y, and z, by Prob. i. From the firft arifes bz -+-yz -\- yz =y'x -+- xy, and from the fecond 2zz = — 2yy, or zz -j- yy = o. Out of
thefe if we exterminate z, there will arife — — — -^ -i-yz =yx
-+• xy, which being refolved it will be y : z •- — x : :
(y : x ::) BD : BT. But as BD is y, therefore BT= «— .3- That is, — BT = AL -f- - — ~ -; where the Sign
BL
iff !-• J_l (_J
prefixt to BT denotes, that the Point T mufl be taken contrary to the Point A.
12. SCHOLIUM. And hence it appears by the bye, how that point of the Conchoid may be found, which Separates the concave from the convex part. For when AT is the lea ft poffible, D will be that point. Therefore make AT = v ; and fmce BT • — - z
• x
then v = — z -+- 2K -+-
by -\- yv
Here to morten
the work, for x fubftitute - ^l!5 > which Value is derived from what is before, and it will be - ? -f. z -+- - - = v. Whence the Fluxions v, y, and z being found by Prob. i. and fuppofing ^=0
and INFINITE SERIES. 49
.,, ... iy, )K ' iy-l-zyy Azy-4-zvy
bvProb. -3. there will anfe --- ~-t-z + • -- °--=i; = o.
J J y jy z za
Laflly, fubftituting in this : - for z, and cc — yy for zz, (which
values of z and zz are had from what goes before,) and making a due Reduction, you will have y'- -+- ^by* — -2.be* = o. By the Con- ftrudlion of which Equation y or AM, will be given. Then thro' M drawing MD parallel to AB, it will fall upon the Point D of contrary Flexure.
13. Now if the Curve be Mechanical whofe Tangent is to be drawn, the Fluxions of the Quantities are to be found, as in Examp.5. of Prob. i. and then the reft is to be perform'd as before.
14. Ex. 4. Let AC and AD be two Curves, which are cut in the Points C and D by the right Line
BCD, apply 'd to the Abfcifs AB in a given Angle. Let AB = x, BD = y,
and — - = z. Then (by Prob. i.
Preparat. to Examp. 5.) it will be z = x ~T> ^ ^ B~ xBC.
15. Now let AC be a Circle, or any known Curve ; and to deter- mine the other Curve AD, let any Equation be propofed, in which z is involved, as zz •+- axz =_y4. Then by Prob. i. 2zz •+- axz -+- axz = 4X7*. And writing x x BC for z, it will be zxz x BC -+- axx x BC H- axz = 4)7'. Therefore 2z x BC -+- ax x BC -{- az : 4jyJ :: (y : x ::) BD : BT. So that if the nature of the Curve AC be given, the Ordinate BC, and the Area ACB or z ; the Point T will be given, through which the Tangent DT will pafs.
1 6. After the fame manner, if 32 = zy be the Equation to the
Curve AD ; 'twill be (3.3) 3^ x BC = zy. So that 3BC : 2 :: (y : x ::) BD : BT. And fo in others.
17. Ex. 5. Let AB=,v, BD =y, as before, and let the length of any Curve AC be z. And drawing a Tangent to it, as Cl, 'twill
x x C/
be Bt : Ct :: x : z, or z = — ^-«
18. Now for determining the other Curve AD, whofe Tangent is to be drawn, let there be given any Equation in which z is in- volved, fuppofe z ==)'. Then it will be z=y, fo that Ct : Bf '•'• (y : x : :} : BD : BT. But the Point T being found, the Tan- gent DT may be drawn.
H 19-
The Method of FLUXIONS,
19. Thus fuppofmg xzsssyy, 'twill be KZ + zx = zyj >, and for z writing ^ there will arife xz -f- ^-^ = ayy. There-
y-> O/ •'•'
fore * -I- f~-' : 27 : : BD : DT.
20. Ex. 6. Let AC be a Circle, or any other known Curve, whofe Tangent is Ct, and let AD be any
other Curve whofe Tangent DT is to be drawn, and let it be defin'd by afTuming AB = to the Arch AC ; and (CE, BD being Ordinates to AB in a given Angle,) let the Relation of BD to CE or AE be exprels'd by any Equation.
21. Therefore call AB or AC = x, BD =y, AE=z, and CE = v. And it is plain that v, x, and z, the Fluxions of CE, AC, and AE, are^to each other as CE, Ct, and Et. Therefore *x C7 = i>, and .v x ^ = z.
22. Now let any Equation be given to define the Curve AD,
as y = «. Then y = z ; and therefore Et : Ct :: (v • x ••) BD : BT. K "'
23. Or let the Equation be y—z+v—x, and it will be
• . r~>T? I TT- . y-.
And therefore CE -4- Et
t. T
— Ct : Ct :: (y : x ::) BD : BT.
24. Or finally, let the Equation be ayy = v*y and it will be zayy = (3^ =) 3*1;' x— . So that 31;* x CE : 2 ay x Ct :: BD : BT.
25. Ex. 7. Let FC be a Circle, which is touched by CS in C; and let FD be a Curve, which is de- fined by affuming any Relation of the
Ordinate DB to the Arch FC, which is intercepted by DA drawn to the Center. Then letting fall CE, the Ordinate in the Circle, call AC or AF=i, AB
CF = /; and it will be tz=(t^=)
K B
T ,S
• . . ^..
v, and — tv = (/x -^ =) z. Here I put z negatively, becaufe AE is dirninifh'd while EC is increafed. And befides AE : EC ::
AB :
and INFINITE SERIES. 51
AB : BD, fo that zy = vx, and thence by Prob. i. zy -f- yx
• — • vx -f- xv. Then exterminating v, z, and v, 'tis yx — ty* — •
tx* = xy.
26. Now let the Curve DF be defined by any Equation, from
which the Value of t may be derived, to be fubftituted here. Sup- pofe let ^=_y, (an Equation to the firft Quadratrix,) and by Prob. i. it will be / = y, fo that yx — yy* — yx* = xy. Whence y : xx — x :: (y : _ x : :) BD(;') : BT. Therefore BT = x*
ADa
- — x; and AT = xx+yy = ^/.
27. After the fame manner, if it is // = ly, there will arife = 6r, and thence AT= - x~ . And fo of others.
z/ /» r
28. Ex. 8. Now if AD be taken equal to the Arch FC, the Curve ADH being then the Spiral of Archimedes ; the fame names of the Lines ftill remaining as were put
afore : Becaufe of the right Angle ABD 'tis xx -{-yy=tf) and therefore (by Prob. i.) xx +yy = //. Tis alfo AD : AC : : DB : CE, fo that tv=ytznd thence (by Prob. i.) tv -4- vf =y. Laftly, the Fluxion of the Arch FC is to the Fluxion of the right Line CE, as AC to AE, or as AD to AB, that is, t : v : : t : x, and thence ix = vf. Compare the Equations now found, and you will fee
'tis tv -+-ix=y, and thence xx -\-yy = (tt =) ^^ . And there- fore compleating the Parallelogram ABDQ^_, if you make QD : QP_ :: (BD : BT :: y : —x ::) X : y — ^ ; that is, if you
take AP = ; ! > PD will be perpendicular to the Spiral.
29. And from hence (I imagine) it will be fufficiently manifeft, by what methods the Tangents of all fcrts of Curves are to be drawn. However it may not be foreign from the purpofe, if I alfo fliew how the Problem may be perform'd, when the Curves are re- fer'd to right Lines, after any other manner whatever : So that hav- ing the choice of feveral Methods, the eafieft and moil fimple may always be ufed.
H 2 Second
$2 The Method of FLUXIONS,
Second Manner.
30. Let D be a point in the Curve, from which the Subtenfe DG is drawn to a given Point G, and let DB be anOrdinate in any given Angle to the Abfcifs AB. Now let the
Point D flow for an infinitely fmall fpace
D^/ in the Curve, and in GD let Gk be
taken equal to Gd, and let the Parallelo-
gram dcBl> be compleated. Then Dk
and DC will be the contemporary Mo- ---
ments of GD and BD, by which they
are diminifh'd while D is transfer'd to d. Now let the right Line
~Dd be produced, till it meets with AB in T, and from the Point T to
the Subtenfe GD let fall the perpendicular TF, and then the Trapezia
Dcdk and DBTF will be like; and therefore DB : DF :: DC : Dk.
31. Since then the Relation of BD to GD is exhibited by the Equation for determining the Curve ; find the Relation of the Fluxions, and take FD to DB in the Ratio of the Fluxion of GD to the Fluxion of BD. Then from F raife the perpendicular FT, which may meet with AB in T, and DT being drawn will touch the Curve in D. But DT muft be taken towards G, if it be affirmative, and the contrary way if negative.
32. Ex. i. Call GD = x, and BD =_>', and let their Relation be x~, — ax1 -f- axy — y"= = o. Then the Relation of the Fluxions will be ^xx1 — 2axx •+- axy -f- ayx — ^yy- = o. Therefore ^xx — zax -h ay : ^yy — ax :: (y : x : :) DB (y) : DF. So that
.' V — axy, — . Then any Point D in the Curve being given,
~ 1 — « •
and thence BD and GD or y and x, the Point F will be given alfo. From whence if the Perpendicular FT be raifed, from its concourfe T with the Abfcifs AB, the Tangent DT may be drawn.
3 3 . And hence it appears, that a Rule might be derived here, as well as in the former Cafe. For having difpofed all the Terms of the given Equation on one fide, multiply by the Dimensions of the Ordinatejy, and place the refult in the Numerator of a Fraction. Then multiply its Terms feverally by the Dimenfions of the Subtenfe x, and dividing the refult by that Subtenfe x, place the Quotient in the Deno- minator of the Value of DF. And take the fame Line DF to- wards G if it be affirmative, otherwile the contrary way.. Where
you
and IN FINITE SERIES,
53
you may obferve, that it is no matter how far diftant the Point G is from the Abfcifs AB, or if it be at all diftant, nor what is the Angle of Ordination ABD.
34. Let the Equation be as before x* — ax* -f- axy — J3 = o ; it gives immediately axy — 3>'3 for the Numerator, and 3** — 2ax -+- ay for the Denominator of the Value of DF.
35. Let alfo a -+- -x—~y=o, (which Equation is to a Conick Sedtion,) it gives — y for the Numerator, and •• for the Denomi-
fly
nator of the Value of DF, which therefore will be — 7 •
36. And thus in the Conchoid, (wherein thefe things will be perform'd more expeditioufly than before,) putting GA = b,
= c, GD=x, and BD=^, it will be BD (;•) : DL (c) :: G A (5) : GL (x — <:). Therefore xy — cy = cb, or xy — cy — cb = o. This Equation according to the Rule gives ^-^ - , that
is, x — <r=DF. Therefore prolong GD to F, fo that DF = LG, and at F raife the perpendicular FT meeting the Alymptote AB in T, and DT being drawn will touch the Conchoid.
37. But when compound or furd Quantities are found in the Equation, you mufl have recourfe to the general Method, except you fliould chufe rather to reduce the Equation.
38. Ex. 2. If the Equation
xv/cr — yy =zyx, were gven
for the Relation between GD and BD ; (fee the foregoing Figure, p. 52.) find the Relation of the Fluxions by Prob. i. As fuppoiing v/ff — )')' = z) you will have the Equations bz -+- yz = yx, and cc — yy=.zz, and thence the Relation of the Fluxions bz-\-yx
= yx -f- yx, and — 2yy=2Z,z. And now z, and z being i exter-
T&e Method of FLUXIONS, exterminated, there will arife v \/ cc — yy — 'JjlvU — \x = xy.
Therefore y : ^/cc — yy — — J2^ — .v :: (y : ,v ::) BD (ji1) : DF.
Third Manner.
39. Moreover, if the Curve be refer'd to two Subtenfes AD and BD, which being drawn from two given Points A and B, may meet at the Curve: Conceive that Point D to flow on through an infinitely little Space Del in the Curve ; and in AD and BD take Ak = Ad, and Bc = Bc/; and then kD and cD will be contempora- neous Moments of the Lines AD and - BD. Take therefore DF to BD in
the Ratio of the Moment D& to the /r
Moment DC, (that is, in the Ratio of the Fluxion of the Line AD to the Fluxion of the LineBD,) and draw BT, FT perpendicu- lar to BD, AD, meeting in T. Then the Trapezia DFTB and DM: will be fimilar, and therefore the Diagonal DT will touch the Curve.
40. Therefore from the Equation, by which the Relation is defined between AD and BD, find the Relation of the Fluxions by Prob. i. and take FD to BD in the fame Ratio.
41. Ex AMP. Suppofing AD = x, andBD=;', let their Rela- tion be a -f- ej — y = o. This Equation is to the Ellipfes of
the fecond Order, whofe Properties for Refracting of Light are fhewn by Des Cartes, in the fecond Book of his Geometry. Then the
Relation of the Fluxions will be e- — y ==o. 'Tis therefore e : d ::(>:# ::) BD : DF.
42. And for the fame reafon if a — ^ — y = o, 'twill be
e : _ d : : BD : DF. In the firft Cafe take DF towards A, and contrary-wife in the other cafe.
43. COROL. i. Hence if d-=.e, (in which cafe the Curve be- comes a Conick Section,) 'twill be
DF = DB. And therefore the Tri- angles DFT and DBT being equal, the Angle FDB will be bifected by the Tangent. v -K A
44.
and INFINITE SERIES. 55
44. COROL. 2. And hence alfo thofe things will be manifeft of themfelves, which are demonstrated, in a very prolix manner, by Des Cartes concerning the Refraction of thcfe Curves. For as much as DF and DB, (which are in the given Ratio of d to e,) in refpect of the Radius DT, are the Sines of the Angles DTF and DTB, that is, of the Ray of Incidence AD upon the Surface of the Curve, and of its Reflexion or Refraction DB. And there is a like reafon- ing concerning the Refractions of the Conick Sections, fuppofing that either of the Points A or B be conceived to be at an infinite diftance.
45. It would be eafy to modify this Rule in the manner of the foregoing, and to give more Examples of it : As alfo when Curves are refer'd to Right lines after any other manner, and cannot com- modioufly be reduced to the foregoing, it will be very eafy to find out other Methods in imitation of thefe, as occafion mall require.
Fourth Manner.
46. As if the right Line BCD mould revolve about a given Point B, and one of its Points D mould defcribe a Curve, and another Point C fhould be the
interfection of the right Line BCD, with another right Line AC given in pofition. Then the Re- lation of BC and BD be- ing exprefs'd by any E- quation ; draw BF pa- rallel to AC, fo as to meet DF, perpendicular to BD, in F. Alfo erect FT perpendicular to DF; and take FT in the fame Ratio to BC, that the Fluxion of BD has to the Fluxion of BC. Then DT being drawn will touch the Curve.
Fifth Manner.
47. But if the Point A being given, the Equation ihould exprefs the Relation between AC and BD } draw CG parallel to DF, and take FT in the fame Ratio to BG, that the Fluxion of BD has to the Fluxion of AC.
Sixth Manner.
48. Or again, if the Equation exprefles the Relation between AC and CD; let AC and FT meet in H ; and take HT in the fune Ratio to BG, that the Fluxion of CD has to the Fluxion of AC. A. id the like in others. Seventh
*fhe Method of FLUXION
Seventh Manner : For Spirals.
49. The Problem is not otherwise perform'd, when the Curves are refer'd, not to right Lines, but to other Curve-lines, as is ufiial in Mechanick Curves. Let BG be the Circumference of a Circle, in whole Semidiameter AG, while it revolves
about the Center A, let the Point D be con- ceived to move any how, fo as to defcribe the Spiral ADE. And fuppofe ~Dd to be an in- finitely little part of the Curve thro' which D flows, and in AD take Ac = Ad, then cD and Gg will be contemporaneous Moments of the right Line AD and of the Periphery BG. Therefore draw Af parallel to cd, that is, perpendicular to AD, and let the Tangent DT meet it in T ; then it will be cD : cd : : AD : AT. Alfo let Gt be parallel to the Tangent DT, and it will be cd : Gg :: (Ad or AD : AG ::) AT : At.
50. Therefore any Equation being propofed, by which the Re- lation is exprefs'd between BG and AD ; find the Relation of their Fluxions by Prob. i. and takeAi? in the fame Ratio to AD: And then Gt will be parallel to the Tangent.
51. Ex. i. Calling EG = x, and AD=^, let their Relation be A:3 — ax1 -f- axy — jy5 = o, and by Prob. i. 3^* — zax-\- ay : 3^* — ax : : (y : x : :) AD : At. The Point / being thus found, draw Gt, and DT parallel to it, which will touch the Curve.
52. Ex. 2. If 'tis y =y> (which is the Equation to the Spiral
of Archimedes,} 'twill be j = y, and therefore a : b : : (y : x : :)
AD : At. Wherefore by the way, if TA be produced to P, that it may be AP : AB :: a : by PD will be perpendicular to the Curve.
53. Ex. 3. If xx = by, then 2XX = by, and 2x : b :: AD : A£. And thus Tangents may be eafily drawn to any Spirals what- ever.
Eighth
and INFINITE SERIES. 57
Eighth Manner : For Quad ratr ices.
CA. Now if the Curve be fuch, that any Line AGD, being drawn from the Center A, may meet the Circular Arch inG, and the Curve in D; and if the Relation between the Arch BG, and the right Line DH, which is an Ordinate to the Bafe or Abfcifs AH in a given Angle, be determin'd by any Equation whatever : Conceive the Point D to move in the Curve for an infinite- ly {mail Interval to d, and the Pa- rallelogram dhHk being compleat- Jf ed, produce Ad to c, fo that
Ac = AD ; then Gg and D/' will be contemporaneous Moments of the Arch BG and of the Ordinate DH. Now produce Dd ftrait on to T, where it may meet with AB, and from thence let fall the Perpendicular TF on DcF. Then the Trapezia Dkdc and DHTF will be fimilar; and therefore D/fc : DC :: DH : DF. And befides if Gf be raifed perpendicular to AG, and meets AF in f; becaufe of the Parallels DF and Gf, it will be DC : Gg :: DF : Gf. There- fore ex aquo, 'tis D£ : G^ : : DH : Gf, that is, as the Moments or Fluxions of the Lines DH and BG.
55. Therefore by the Equation which exprefies the Relation of BG to DH, find the Relation of the Fluxions (by Prob. i.) and in- that Ratio take Gf, the Tangent of the Circle BG, to DH. Draw DF parallel to Gf, which may meet A/* produced in F. And at F creel the perpendicular FT, meeting AB in T; and the right Line DT being drawn, will touch the Quadratrix.
56. Ex. i. Making EG = x, and DH=;', let it be xx = fy; then (by Prob. i.)2xx = by. Therefore 2.x : b :: (y : x ::) DH : GJ; and the Pointy being found, the reft will be determin'd as above.
But perhaps this Rule may be thus made fomething neater : Make x :y :: AB : AL. Then AL : AD :: AD : AT, and then DT will touch the Curve. For becaufe of equal Triangles AFD and ATD, 'tis AD x DF= AT x DH, and therefore AT : AD : : (DF or
JB x Gf : DH or 1 G/::) AD : f- AG or) AL.
57. Ex.2. Let x=y, (which is the Equation to the Quadratrix of the Ancients,) then #=v. Therefore AB : AD :: AD : AT.
I 8.
58 *fhe Method ^FLUXIONS,
58. Ex. 3. Let axx=y*, then zaxx=sMy*. Therefore make 3;-* : zax : : (x : y : :) AB : AL. Then AL : AD : : AD : AT. And
thus you may determine expeditioufly the Tangents of any other Quadratrices, howfoever compounded.
Ninth Manner.
59. Laftly, if ABF be any given Curve, which is touch'd by the right Line Bt ; and a part BD of
the right Line BC, (being an Or- dinate in any given Angle to the Abfcifs AC,) intercepted between this and another Curve DE, has a Relation to the portion of the Curve AB, which is exprefs'd by any Equation: You may draw a Tangent DT to the other Curve,
by taking (in the Tangent of this ^— ^ <f-
Curve,) BT in the fame Ratio to
BD, as the Fluxion of the Curve AB hath to the Fluxion of the
right Line BD.
60. Ex. i. Calling AB ==x, and BD =y-t let it be ax==yy, and therefore ax = zyy. Then a : zy : : (y : x : :) BD : BT.
6j. Ex.2. Let ^#==7, (the Equation to the Trochoid, if ABF be a Circle,) then fX=yt and a : b :: BD : BT.
62. And with the fame eafe may Tangents be drawn, when the Relation of BD to AC, or toBC, is exprefs'd by any Equation; or when the Curves are refer 'd to right Lines, or to any other Curves, after any other manner whatever.
63. There are alfo many other Problems, whofe Solutions are to be derived from the fame Principles ; fuch as thefe following.
I. To find a Point of a Curve, where the Tangent is parallel to the Abfcife, or to any other right Line given in pofition ; or is perpendicular to it, or inclined to it in any given Angle.
II. To find the Point where the Tangent is moft or leajl inclined to the Abfcifs, or to any other right Line given in 'pofition. That is, to find the confine of contrary Flexure. Of this I have already given a Spe- cimen, in the Conchoid.
III. From any given Point without the Perimeter of a Curve, to draw a right Line, which with the Perimeter may make an Angle of
Contact.
and IN FINITE SERIES. 59
Contaft, or a right Angle, or any other given Angle, that is, from a given Point, to draw 'Tangents, or Perpendiculars^ or right Lines that Jhall have any other Inclination to a Curve-line.
IV. From any given Point within a Parabola, to draw a right Line, which may make with the Perimeter the greateji or leaft Angle poj/ible. And Jb of all Curves whatever.
V. To draw a right Line which may touch two Curves given in pojition, or the fame Curve in two Points, when that can be done.
VI. To draw any Curve with given Conditions, which may touch another Curve given in pojition, in a given Point.
VII. To determine the RefraSlion of any Ray of Light, that falls upon any Curve Superficies.
The Refolution of thefe, or of any other the like Problems, will not be fo difficult, abating the tedioufnefs of Computation, as that there is any occalion to dwell upon them here : And I imagine if may be more agreeable to Geometricians barely to have mention 'd them.
; : P R O B. V.
At any given Point of a given Curve^ to find the Quantity of Curvature.
1. There are few Problems concerning Curves more elegant than this, or that give a greater Infight into their nature. In order to cits Refolution, I mufl: premife thefe following general Confederations.
2. L The fame Circle has every where trie fame Curvature, and in different Circles it is reciprocally proportional to their Diameters. If the Diameter of any Circle is as little again as the Diameter of another, the Curvature of its Periphery will be as great again. If the Diameter be one-third of the other, the Curvature will be thrice as much, &c.
3. II. If a Circle touches any Curve on its concave fide, in any given Point, and if it be of fuch magnitude, that no other tangent Circle can be interleribed in the Angles of Contact near that Point ; that Circle will be of the lame Curvature as the Curve is of, in that Point of Contact. For the Circle that conies between the Curve and another Circle at the Point of Contact, varies lefs from the Curve, and makes a nearer approach to its Curvature, than that other Circle does. And therefore that Circle approaches nea'-eil to its
I 2 Curvature,
60 *fbe Method of FLUXIONS,
Curvature, between which and the Curve no other Circle can in- tervene.
4. III. Therefore the Center of Curvature to any Point of a Curve, is the Center of a Circle equally curved. And thus the Ra- dius or Semidiameter of Curvature is part of the Perpendicular to the Curve, which is terminated at that Center.
5. IV. And the proportion of Curvature at different Points will be known from the proportion of Curvature of aequi-curve Circles, or from the reciprocal proportion of the Radii of Curvature.
6. Therefore the Problem is reduced to this, that the Radius, or Center of Curvature may be found.
7. Imagine therefore that at three Points of the Curve <f , D, and d, Peipendkulars are drawn, of which thofe that are
at D and ^ meet in H, and thofe that are at D and d meet in h : And the Point D being in the / middle, if there is a greater Curyity at the part Dj^ than at DJ, then DH will be lefs than db. But by how much the Perpendiculars /H and dh are nearer the intermediate Perpendicular, fo much the lefs will the diftance be of the Points H and h : And at laft when the Perpendiculars meet, thofe Points will coincide. Let them coincide in the Point C, then will C be the Center of Curvature, at the Point D of the Curve, on which the Perpendicu- lars ftand ; which is manifeft of itfelf.
8. But there are feveral Symptoms or Properties of this Point C', which may be of ufe to its determination.
9. I. That it is the Concourfe of Perpendiculars that are on each lide at an infinitely little diftance from DC.
10. II. That the Interfeftions of Perpendiculars, at any little finite diftance on each fide, are feparated and divided by it ; fo that thofe which are on the more curved fide D,f fooner meet at H, and thofe which are on the other iefs curved fide -Dd meet more remotely at h.
11. III. If DC be conceived to move, while it infifts perpendi- cularly on the Curve, that point of it C, (if you except the motion of approaching to or receding from the Point of Influence C,) will be leaft moved, but will be as it were the Center of Motion.
12. IV. If a Circle be defcribed with the Center C, and the di- ftance DC, no other Circle can be defcribed, that can lie between at the Contact.
and INFINITE SERIES.
61
n. V. Laftly, if the Center II or b of any other touching Circle approaches by degrees to C the Center of this, till at la it it co- incides with 'it ; then any of the points in which that Circle mall cut the Curve, will coincide with the point of Contact D.
14. And each of thefe Properties may fupply the means of folving the Problem different ways : But we fliall here make choice of the firlt, as being the moit fimple.
15. At any Point D of the Curve let DT be a Tangent, DC a Perpendicular, and C the Center of Curvature, as before. And let AB be the Abfcifs, to which let DB be apply 'd at right Angles, and which DC meets in P. Draw
DG parallel to AB, and CG per- pendicular to it, in which take Cg of any given Magnitude, and draw gb perpendicular to it, which meets DC in <T. Then it will be Cg : gf : : (TB : BD : :) the Fluxion of the Ablcifs, to the Fluxion of the Ordinate. Likewife imagine the Point D to move in the Curve an infinitely little diftance Dd, and drawing de perpendicular to DG, and Cd perpendicular to the Curve, let Cd meet DG in F, and $g in/ Then will De be the Momen- tum of the Abfcifs, de the Momentum of the Ordinate, and J/ the contemporaneous Momentum of the right Line g£. Therefore DF —-De^.^t . Having therefore the Ratio's of thefe Moments, or,
LJC ' *
which is the fame thing, of their generating Fluxions, you will have the Ratio of CG to the given Line C^, (which is the fame as that of DF to Sf,) and thence the Point C will be determined.
16. Therefore let AB = x, BD =y, Cg- = i, and g£ = z ;
then it will be i : z : : x : y, or z = r- . Now let the Mo-
X
mentum S-f of z be zxo, (that is, the Product of the Velocity
and of an infinitely fmall Quantity o,} and therefore the Momenta
Dt'==xxo, de=yx.o, and thence DF = .\o -f- — . Therefore
X
'tisQ-(r) : CG :: (Jf : DF ::) zo : xo + ^ . That is, CG=
xx \y
J7-
62 7%e Method of FLUXIONS,
17. And whereas we are at liberty to afcribe whatever Velocity we pleafe to the Fluxion of the Abfcifs x, (to which, as to an equable Fluxion, the reft may be referr'd j) make x = i, and then y = z, and CG = '-±^ . And thence DG = z-±^. } and
J ' '
18. Therefore any Equation being propofed, in which the Rela- tion of BD to AB is exprefs'd for denning the Curve ; firft find the Relation betwixt x and yt by Prob. r. and at the fame time fub- ftitute i for ,v, and z for y. Then from the Equation that arifes, by the fame Prob. i. find the Relation between «#, y, and z, and at the fame time fubftitute i for x, and z for y, as before. And thus by the former operation you will obtain the Value of z, and by the latter you will have the Value of z ; which being obtain'd, pro- duce DB to H, towards the concave part of the Curve, that it
may be DH = - - , and draw HC parallel to AB, and meet-
ing the Perpendicular DC in C j then will C be the Center of Cur- vature at the Point D of the Curve. Or fince it is i -|- r.y. -7—
PT TM-T PT Tk/-> DP
make DH== ' or
z
19. Ex. i. Thus the Equation ax^-hx* — y1 =;o being pro- pofed, (which is an Equation to the Hyperbola whofe Latus redtum
is a, and Tranfverfum 2;) there will arife (by Prob. i.) a •+. zbx — 2zy • — o, (writing l for x, and z for y in the refulting Equation, which otherwife would have been ax -+• 2&xx — zyy = o ;) and hence again there arifes zb — 2zz — 2zy = o, (i and z being again
wrote for ,v and y.) By the firft we have z = CL±^L } an(j by tne
i ^^ latter z = — — • Therefore any Point D of the Curve being given,
and confequently xand y, from thence z and z will be given, which being known, make ••• 7 = GC or DH, and draw HC.
Z
20. As if definitely you make 0 = 3, and b=i, fo that 3#-f- xx=yy may be the condition of the Hyperbola. And if you aliume x=i, ^11^ = 2, z=±, z= — T9T, and DH= — gL. li being found, raife the Perpendicular HC meeting the Perpendi-
cular
and IN FINITE SERIES. 63
cular DC before drawn ; or, which is the fame thing, make HD : HC :: (i : z ::) i : £. Then draw DC the Radius of Curva- ture.
21. When you think the Computation will not be too perplex, you
may fabfHtute the indefinite Values of z and z into - , the
Value of CG. Thus in the prefent Example, by a due Reduction you will have DH =y -j- 4'S^r* . Yet the Value of DH by
Calculation conies out negative, as may be feen in the numeral Ex- ample. But this only fhews, that DH mufl be taken towards B ; for if it had come out affirmative, it ought to have been drawn the contrary way.
22. COROL. Hence let the Sign prefixt to the Symbol -\-b be changed, that it may be ax — -bxx — yy=zo, (an Equation to the
Ellipfis,) then DH=;--f- ilLll^: .
23. But fuppofing b=. o, that the Equation may become ax — yy —-- o, (an Equation to the Parabola,) then DH = y -f- ~ ; and
thence DG = \a -f- 2X.
24. From thefe feveral Exprefilons it may eafily be concluded, that the Radius of Curvature of any Conick Seftion is always
aa
25. Ex. 2. If x*=ay* — xy- be propofed, (which is the Equa- tion to the CiiToid of Diodes,") by Prob. i. it will be firft T>xl=.2azy
— zxzy — y-t and then 6x = 2azy-+-2azz — -2zy — zxzy — 2xzz
„ 1 3*x -4- yy , • T.X — a%z -4- 2cv+ *~~ n-.!
— 2Z\ : So that z= - — 3-^. and z= - - ^ ••••• — . There-
J zay — 2.vy' ay — xj
fore any Point of the Ciflbid being given, and thence .v and y, there will be given alfo & and z, ; which being known, make -
K
= CG. _ _
26. Ex. 3. If b-jf-y^/cc — yy =.vy were given, (which is the Equation to the Conchoid, inpag.48;) make \/cc — y\=zv, and there will arife hi) -+- yv = xy. Now the firft of thele, (cc — _vv = vv,) will give (by Prob. i.) — 2yz = 2vv, (writing z for v ;) and the latter will give l>v -+-yv + zv =y -{- xz. And from thefe Equations rightly difpofed v and z will be determined. But that z may alfo be found; out of the laft Equation exterminate the Fluxion
i>, by fubilituting — ^ , and there will arife — —7 — — -I- ~"^
Method of FLUXIONS,
= y -f- xz, an Equation that comprehends the flowing Quantities, without any of their Fluxions, as the Refolution of the firft Pro- blem requires. Hence therefore by Prob. i. we mall have —
^2* byz Ijzv 2)zs )•?£ \vzv
" +- ZV = 2Z •+- XZ.
This Equation being reduced, and difpofed in order, will give z. But when z and z are known, make ' + zz =± CG.
27. If we had divided the laft Equation but one by z, then by Prob. i . we mould have had — - -f- ^ — — -f- --- -f. -i; =
2 — ^, ; which would have been a more fimple Equation than the
former, for determining z.
28. I have given this Example, that it may appear, how the ope- ration is to be perform'd in furd Equations: But the Curvature of the Conchoid may be thus found a fhorter way. The parts of the Equation b -\-y ^/cc — v\' = xy being fquared, and divided by yy, there arifes ~ -f. — *" ^ — 2by — y* = x*, and thence by Prob. i.
or
x
...
And hence again by Prob. i. ^^ -f- ~ — z— 1 — ™ m By
*^ J y4 y/9 z, zz
the firft refult z is determined, and z by the latter.
29. Ex. 4. Let ADF be a Trochoid [or Cycloid] belonging to the Circle ALE, whofe Diameter is AE j and making the Ordinate BD to cut the Circle in L,
AB=x, BD
and the Arch AL=/, and the Fluxion of the fame Arch = /. And firfl (drawing the Semidia- meterPL,)the Fluxion of the
Bafe or Abfcifs AB will be to the Fluxion of the Arch AL, as BL
to
and INFINITE SERIES. 65
to PL ; that is, A* or I : / : : v : ~a. And therefore ^ = /. Then from the nature of the Circle ax — xx = -y-y, and therefore by Prob. i. a — 2X = 2-yy, or -~~* = v.
30. Moreover from the nature of the Trochoid, 'tis LD= Arch AL, and therefore -y -M =y. And thence (by Prob. i ) v -h / =z. Laftly, inftead of the Fluxions v and / let their Values be lubfti- tuted, and there will arife a-^ =z. Whence (by Prob. i.) is de- rived — - -f- — — - = z. And thefe being found, make —
*ut/ w *v z,
== — DH, and raife the perpendicular HC.
31. COR. i. Now it follows from hence, that DH = 2BL, and CH — 2BE, or that EF bifeds the radius of Curvature CO in N. And this will appear by fubftituting the values of z and z now found, in the Equation '• . **= DH, and by a proper reduction of
the refult.
32. COR, 2. Hence the Curve FCK, defcribed indefinitely by the Center of Curvature of ADF, is another Trochoid equal to this, whofe Vertices at I and F adjoin to the Cufpids of this. For let the Circle FA, equal and alike pofited to ALE, be defcribed, and let C/3 be drawn parallel to EF, meeting the Circle in A : Then will Arch FA = (Arch EL= NF =) CA.
33. COR. 3. The right Line CD, which is at right Angles to the Trochoid IAF, will touch the Trochoid IKF in the point C.
34. COR. 4. Hence (in the in verted Trochoids,) if at theCufpid K of the upper Trochoid, a Weight be hung by a Thread at the di- ilance KA or 2EA, and while the Weight vibrates, the Thread be fuppos'd to apply itfelf to the parts of the Trcchoid KF and KI, which refift it on each fide, that it may not be extended into a right Line, but compel it (as it departs from the Perpendicular) to be by degrees inflected above, into the Figure of the Trochoid, while the lower part CD, from the loweft Point of Contact, ftill remains a right Line : The Weight will move in the Perimeter of the lower Trochoid, becaufe the Thread CD will always be perpen- dicular to it.
35. COR. 5. Therefore the whole Length of the Thread KA is equal to the Perimeter of the Trochoid KCF, and its part CD is equal to the part of the Perimeter CF.
K 36.
66 The Method of FLUXIONS,
36. COR. 6. Since the Thread by its ofcillating Motion revolves about the moveable Point C, as a Center ; the Superficies through which the whole Line CD continually pafles, will be to the Super- ficies through whichjthe part CN above the right Line IF pafles at the fame time, as CD* to CN*, that is, as 4 to i. Therefore the Area CFN is a fourth part of the Area CFD ; and the Area KCNE is a fourth part of the Area AKCD.
37. COR. 7. Alfo fince the fubtenfe EL is equal and parallel to CN, and is converted about the immoveable Center E, juft as CN moves about the moveable Center C ; the Superficies will be equal through which they pafs in the fame time, that is, the Area CFN, and the Segment of the Circle EL. And thence the Area NFD will be the triple of that Segment, and the whole area EADF will be the triple of the Semicircle.
38. COR. 8. When the Weight D arrives at the point F, the whole Thread will be wound about the Perimeter of the Trochoid KCF, and the Radius of Curvature will there be nothing. Where- fore the Trochoid IAF is more curved, at its Cufpid F, than any Circle ; and makes an Angle of Contact, with the Tangent /3F produ- ced, infinitely greater than a Circle can make with a right Line.
39. But there are Angles of Contact that are infinitely greater than Trochoidal ones, and others infinitely greater than thefe, and fo on in infinitum ; and yet the greateft of them all are infinitely lefs than right-lined Angles. Thus xx = ay, x3 = £y», x* ==ry5, x* = dy+, &cc. denote a Series of Curves, of which every fucceeding one makes an Angle of Contact with its Abfciis, which is infinitely greater than the preceding can make with the fame Abfcifs. And the Angle of Contact which the firft xx=ay makes, is of the fame kind with Circular ones; and that which the fecond x*-=byz makes, is of the fame kind with Trochoidals. And tho' the Angles of the fucceed- in° Curves do always infinitely exceed the Angles of the preceding, yet they can never arrive at the magnitude of a right-lined Angle.
40. After the fame manner x ==y, xx=ay, x*=l>1y, x4 = c*y, &c. denote a Series of Lines, of which the Angles of the fubfequents, made with their Abfcifs's at the Vertices, are always infinitely lefs than the Angles of the preceding. Moreover, between the Angles of Contact of any two of thefe kinds, other Angles of Contact may be found ad infwitum, that mall infinitely exceed each other.
41. Now it appears, that Angles of Contact of one kind are in- finitely greater than thofe of another kind ; fince a Curve of one kind, however great it may be, cannot, at the Point of Contact,
I he
and INFINITE SERIES. 67
lie between the Tangent and a Curve of another kind, however fmall that Curve may be. Or an Angle of Contacl of one kind cannot necefTarily contain an Angle of Contact of another kind, as the whole contains a part. Thus the Angle of Contaft of the Curve x* = cy*, or the Angle which it makes with its Abfcifs, neceflarfly includes the Angle of Contacl of the Curve x~' =^yi, and can never be contain'd by it. For Angles that can mutually exceed each other are of the fame kind, as it happens with the aforefaid Angles of the Trochoid, and of this Curve x> = by*.
42. And hence it appears, that Curves, in fome Points, may be infinitely more ftraight, or infinitely more curved, than any Circle, and yet not, on that account, lofe the form of Curve-lines. But all this by the way only.
43. Ex. 5. Let ED be the Quadratrix to the Circle, defcribed from Center A; and letting fall DB
perpendicular to AE, make AB = x, BD =y, and AE = i. Then 'twill
be yx — yy* — yx* =xy, as before.
Then writing i for x, and z for y, the
Equation becomes zx — zyl — zx*
= y ; and thence, by Prob. i. zx
— zy* — zx* -f- zx — zzxx — zzyy = ym Then reducing, and
again writing i for x and z for y, there arifes z —
x—xx—jy
J, ——
But z and & being found, make ' T ** =— DH, and draw HC as
above.
44. If you defire a Conftrudtion of the Problem, you will find it very mort. Thus draw DP perpendicular to DT, meeting AT in P,
and make aAP : AE :: PT : CH. For * =r
and zy = £g. =— -BP; and;ey + x = — AP, and -_^_.. into zy-\-x-=. — z- into — AP=2. Moreover it is i-4-zz =
AE x BTy
"PT* T> P\ TAT1 . I nrfr T3T
r 1 /i f. BlJq U I a \ j i r 1 -j- ** r 1
:= i-{- rrTT =-T-:TI ,) and tnereiore — : — = —
Bl? BI? " 2- —
BT
= DH. Laftly, it is BT : BD :: DH : CH==^^. Here
the negative Value only mews, that CH mufl be taken the fame way as AB from DH.
45. In the fame manner the Curvature of Spirals, or of any other Curves whatever, may be determined by a very mort Calculation.
K 2 46.
68 7&e Method of FLUXIONS,
46. Furthermore, to determine the Curvature without any pre- vious reduction, when the Curves are refer'd to right Lines in any other manner, this Method might have been apply'd, as has beer* done already for drawing Tangents. But as all Geometrical Curves, as alfo Mechanical, (efpecially when the defining conditions are re- duced to infinite Equations, as I mail mew hereafter,) may be re- fer'd to rectangular Ordinates, I think I have done enough in this matter. He that defires more, may eafily fupply it by his own in- duftry ; efpecially if for a farther illuflration I mall add the Method for Spirals.
A its Center, and B a given Point in
47. Let BK be its Circumference.
a Circle, Let ADd be a Spiral, DC its Perpen- dicular, and C the Center of Curvature at the Point D. Then drawing the right Line ADK, and CG parallel and equal to AK, as alfo the Per- pendicular GF meeting CD inF: Make AB or AK = i=CG, BK=#, AD==y, and GF = z. Then con-
.
ceive the Point D to move in the Spiral for an infinitely little Spree Drf', and then through rfdraw the Semidiameter A/£, and Cg parallel and equal to it, draw gf perpendicular to gC, fo that G/ cuts gf in/ and GF in P; produce GF to <p, fo that G£p=<§/, and draw de perpendicular to AK, and produce it till it meets CD at I. Then the contemporaneous Moments of BK, AD, and G<p, will be Kk, De and Fa, which therefore may be call'd xo, yo, and zo.
48. Now it is AK : Ae (AD) :: kK : Je=yo, where I aflurne x=i, as above. Alfo CG : GF :: de : eD = oyz, and there- fore yz — yf Befides CG : CF : : de : dD = oy x CF : : dD : d\ = oy x CF?. Moreover, becaufe Z_PC<p (=Z-GG?) = LDAd, and /.CPp (= LCdl = £- eSQ -f- Red.) = L. ADJ, the Triangles CP<p and AD</ are fimilar, and thence AD : Dd :: CP (CF) : P<p = o x CFq. From whence take F<pt and there will remain PF = oxCF^ — ex z. Laftly, letting fall CH perpendicular to AD} 'tis PF : dl :: CG : eH or DH = LlHf . Or fubftituting i+zz
CFy—x
for CFa, 'twill be DH =
y -ya!g
Here it may be obferved,
that
and IN FINITE SERIES. 69
that in this kind of Computations, I take thofe Quantities (AD and Ae) for equal, the Ratio of which differs but infinitely little from the Ratio of Equality.
49. Now from hence arifes the following Rule. The Relation of x and y being exhibited by any Equation, find the Relation of the Fluxions x and y, (by Prob. i.) and fubftitute i for x, and yz for y. Then from the refulting Equation find again, (by Prob. i.) the Relation between x, y, and z, and again fubftitute i for x. The firft refult by due reduction will give y and z, and the latter will eive z ; which being known, make — — =—• = DH, and raife
1 -f- Z.X.—Z.
the Perpendicular HC, meeting the Perpendicular to the Spiral DC before drawn in C, and C will be the Center of Curvature. Or which comes to the fame thing, take CH : HD :: z : i, and draw CD.
50. Ex. i. If the Equation be ax=y, (which will belong to
the Spiral at Archimedes,) then (by Prob. i.) ax=yy or (writing i for x, and yz for_y,)7^ =yz. And hence again (by Prob i.) o = yz+y'z. Wherefore any Point D of the Spiral being given,, and thence the length AD or y, there will be given z = - , and z=
( — 3- or) — — . Which being known, make i-t-zz-—z : H-iz :: DA (y) : DH. And i : z :: DH : CH.
And hence you will eafily deduce the following Conftrucftion. Produce AB to Q, fo that AB : Arch BK :: Arch BK : BC^, and make AB -+- AQ^: AQj: DA : DH :: a : HC.
51. Ex. 2. If ax1 =_)" be the Equation that determines the Re- lation between BK and AD; (by Prob. i.) you will have 2axx=. 3Jy,-*, or 2ax= 3«y». Thence again 2a'x= ^zys -+- gsiyy*. 'Tis therefore z = ^7 , and z = 'a~9~z'- . Thefe being known, make
i-\-zz — K : i-t-zz ••• DA : DH. Or, the work being reduced to a better form, make gxx1 -f- 10 : gxx -f- 4 :: DA : DH.
52. Ex. 3. After the lame manner, if ax* — bxy=yi determines
the Relation of BK to AD ; there will arife I"* ~ '• = z,, and
bxy -f- $)*.
.g*~;*7~^;~9*'-8 = g. From which DH/ and thence the. Point C, is determined as before.
5q i-
yo I'he Method of FLUXIONS,
53. And thus you will eafily determine the Curvature of any- other Spirals ; or invent Rules for any other kinds of Curves, in imitation of thefe already given.
£4. And now I have finim'd the Problem ; but having made ufe of a Method which is pretty different from the common ways of operation, and as the Problem itfelf is of the number of thofe which are not very frequent among Geometricians : For the illuflra- tion and confirmation of the Solution here given, I mall not think much to give a hint of another, which is more obvious, and has a nearer relation to the ufual Methods of drawing Tangents. Thus if from any Center, and with any Radius, a Circle be conceived to be defcribed, which may cut any Curve in feveral Points ; if that Circle be fuppos'd to be contracted, or enlarged, till two of the Points of interfeclion coincide, it will there touch the Curve. And befides, if its Center be fuppos'd to approach towards, or recede from, the Point of Contadt, till the third Point of interfedtion fhall meet with the former in the Point of Contadt ; then will that Circle be cequicurved with the Curve in that Point of Contadt : In like man- ner as I infmuated before, in the laft of the five Properties of the Center of Curvature, by the help of each of which I affirm'd the Problem might be folved in a different manner.
55. Therefore with Center C, and Radius CD, let a Circle be defcribed, that cuts the Curve in the Points d, D, and <f ; and letting fall the Perpendi- culars DB, db, <T/3, and CF, to the Abfcifs AB ; call AB = x, BD = y, AF = v, FC=/,andDC=J. Then BF=v—x, and DB-f-FC =_>>-{-/. The fum of the Squares of thefe is equal to the Square of DC ; that is, -D1—
2VX -+- X* -f- )"• -h 2yt -+- /»
=ss. If you would abbrevi- ate this, make v* -f-/1 — s1 =f, (any Symbol at pleafure,) and it becomes x1 — 2vx -f-jy1 -f- zfy -+- q1 = o. After you have found
/, «y, and q*, you will have s-=\/rv1 -+- 1* — q*.
56. Now let any Equation be propofed for defining the Curve, the quantity of whofe Curvature is to be found. By the help of this Equation you may exterminate either of the Quantities x or y,
and
and INFINITE SERIES. 71
and there will arife an Equation, the Roots of which, (db, DB, <f/g, &c. if y°u exterminate x ; or A/>, AB, A/3, &c. if you exterminate _y,) are "at the Points of interfedtion d, D, J\ &c. Wherefore fince "three of them become equal, the Circle both touches the Curve, and will alfo be of the fame degree of Curvature as the Curve, in the point of Contact But they will become equal by comparing the Equation with another fictitious Equation of the fame number of Dimenfions, which has three equal Roots ; as Des Cartes has fhew'd. Or more expeditioufly by multiplying its Terms twice by an Arithmetical Progreflion.
57. EXAMPLE. Let the Equation be ax =yy, (which is an Equation to the Parabola,) and exterminating x, (that is, fubftitu-
ting its Value -- in the forego- ing Equation,) there will arife £ * — ^~y*_ -+• zty -f- ?a = o. Three of whofe Roots ^ are to be _j_ yi made equal. And for this purpofe 4*2 I o
I multiply the Terms twice by an * i o i
Arithmetical Progrellion, as you — —
fee done here j and there arifes — — -J1 + 2JX = °-
Or «u = — + \a. Whence it is eafily infer'd, that BF = 2x -{-
\a, as before.
58. Wherefore any Point D of the Parabola being given, draw the Perpendicular DP to the Curve, and in the Axis take PF = 2AB, and erect FC Perpendicular to FA, meeting DP in C; then will C be the Center of Curvity defired.
59. The fame may be perform'd in the Ellipfis and Hyperbola, but the Calculation will be troublefome enough, and in other Curves generally very tedious.
Of ^uefiions that have fome Affinity to the preceding
Problem.
60. From the Refolution of the preceding Problem fome others may be perform'd ; fuch are,
I. To find the Point where the Curve has a given degree of Cur- vature.
6 1. Thus in the Parabola, ax=yy, if the Point be required whofe Radius of Curvature is of a given length f: From the Cen- ter of Curvature, found as before, you will determine die Radius
72 7%e Method of FLUXIONS,
to be -~^ \/aa -+- ^.ax, which muft be made equal to f. Then by reduction there arifes x = — ^a -f- 1/^aff. II. To find the Point of ReElitude.
62. I call that the Point of ReEiitude, in which the Radius of Flexure becomes infinite, or its Center at an infinite diftance : Such it is at the Vertex of the Parabola a*x=y*. And this fame Point is commonly the Limit of contrary Flexure, whole Determination I have exhibited before. But another Determination, and that not inelegant, may be derived from this Problem. Which is, the longer the Radius of Flexure is, fo much the lefs the Angle DCJ (Fig.pag.6i.) becomes, and alfo the Moment <F/j fo that the Fluxion of the Quantity z is diminim'd along with it, and by the Infinitude of that Radius, altogether vanimes. Therefore find the Fluxion z, and fuppofe it to become nothing.
63. As if we would determine the Limit of contrary Flexure in the Parabola of the fecond kind, by the help of which Cartefius con- ftructed Equations of fix Dimenfions ; the Equation to that Curve is AT3 — bx* — cdx -+- bed 4- dxy = o. And hence (by Prob. i .) arifes 3*** — 2bxx — - cdx -4- dxy -f- dxy = o. Now writing i for xt and z for y, it becomes 3-va — zbx — cd-{- dy -f- dxz=.o ; whence again (by Prob. i,) 6xx — zbx -+• dy + dxz •+- dxz = o. Here again writing i for x, & for y, and o for z, it becomes (>x — zb -+- zdz = o. And exterminating z, by putting b — 3* for dz in the Equation 3^,v — zbx — cd -+- dy -f- dxz = o, there will arife — bx — cd-$-dy = o) ory=c-{-^; this being fubftituted in the room
of y in the Equation of the Curve, we fhall have x* •+- bcd-=z. Q } which will determine the Confine of contrary Flexure.
64. By a like Method you may determine the Points of Rectitude, which do not come between parts of contrary Flexure. As if the Equation x* — 4<w3 -}- ba^x* — b>y = o ex- prefs'd the nature of a Curve ; you have firfl, (byProb. i.)4^3 — i2ax*-+- i2a*x — faz=o, and hence again 12X* — 24^7^ -f- 12^' — b*z «=o. Here fuppofe z = o, and by Reduc- tion there will arife x = a. Wherefore take
ABi=fl, and erect the perpendicular BDj this will meet Curve in the Point of Re&itude D, as was required.
III.
and IN FINITE SERIES. 73
III. To find the Point of infinite Flexure.
65. Find the Radius of Curvature, and fuppofe it to be nothing. Thus to the Parabola of the fecond kind, whole Equation is A;* =
<7ya, that Radius will be CD = 4"6aq* \/q.ax-\- gxx , which be- comes nothing when x = o.
IV. To determine the Point of the greatefl or leaft Flexure.
66. At thefe Points the Radius of Curvature becomes either the greateft or leaft. Wherefore the Center of Curvature, at that mo- ment of Time, neither moves towards the point of Contact, nor the contrary way, but is intirely at reft. Therefore let the Fluxion of the Radius CD be found; or more ex-
peditioufly, let the Fluxion of either of the Lines BH or AK be found, and let it be made equal to nothing.
67. As if the Queftion were propofed con- cerning the Parabola of the fecond kind xl = o*y ; firft to determine the Center of
Curvature you will find DH = aa , 9X->
• ox
and therefore BH = 6^'?AV; make BH
Hence (by Prob. i.) — "- - _j_ ^y==t}. But now fuppofe -y, or the Fluxion of BH, to be nothing ; and belides, lince by Hypothecs A- "' = rf1.y, and thence (by Prob. i.) yxx1 =<?*.}', putting x= i, fub- ftitute ^ for v, and there will arife 4.5x4=0+. Take therefore
^
AB ==a y'^j- =<7 x45| , and raifrng the perpendicular BD, it will" meet the Curve in the Point of the greateft Curvature. Or, which is the fame thing, make AB : BD : : 3^/5 : I.
68. After the fame manner the Hyperbola of the lecond kind reprefented by the Equation xyl = «3, will be moft inflected in the points D and d, which you may determine by taking in the Abfcifs AQ== r, and erecting the Perpendicular QP_=z=v/5, and Q^/> equal to it on the other fide. Then draw- ing AP and A/>, they will meet the Curve in the points D and d required.
V,
74 The Method of FLUXIONS,
V. To determine the Locus of the Center of Curvature, or to de- fcribe the Curve, in which tbaf* Center is always found,
69. We have already {hewn, that the Center of Curvature of the Trochoid is always found in another Trochoid. And thus the Cen- ter of Curvature of the Parabola is found in another Parabola of the fecond kind, reprefented by the Equation axx=y*, as will eafily appear from Calculation.
VI. Light falling upon any Curve, to find its Focus, or the Con- courje of the Rays that are ref rafted at any of its Points.
70. Find the Curvature at that Point of the Curve, and defcribe a Circle from the Center, and with-the Radius of Curvature. Then find the Concourfe of the Rays, when they are refracted by a Cir- cle about that Point : For the fame is the Concourfe of the refrac- ted Rays in the propofed Curve.
71. To thefe may be added a particular Invention of the Curva- ture at the Vertices of Curves, where they cut their Abfcifles at right Angles. For the Point in which the Perpendicular to the Curve, meeting with the Abfcifs, cuts it ultimately, is the Center of its Curvature. So that having the relation between the Abfcifs x, and the rectangular Ordinate y, and thence (by Prob. i.) the rela- tion between the Fluxions x and y ; the Value yy, if you fubftitute r for x into it, and make y = o, will be the Radius of Curva- ture.
72. Thus in the Ellipfis ax — £xX=yy, it is -* — "•— = yy ;
which Value of yy, if we fuppofe^=o, and confequently x = />, ^writing i for x, becomes ±a for the Radius of Curvature. And fo at the Vertices of the Hyperbola and Parabola, the Radius of Cur- vature will be always half of the Latus rectum.
73-
and INFINITE SERIES. 75
73. And in like manner for the Conchoid, defined by the Equation zbx — xx = yy, the Value of yyt (found by
zicc + cc ~T — bb
Prob. i.) will be ^ ""* IT — ^ "~~ *• Now fuppofing y = o,
and thence # = c or — f, we mail have — zb — c, or •
2(5 -f- f, for the Radius of Curvature. Therefore make AE : EG :: EG : EC, and he : eG :: eG : ec, and you will have the Centers of Curvature C and c, at the Vertices E and e of the Conjugate Conchoids.
PROB. VI.
To determine the Quality of the Curvature, at a given
Point of any Curve.
I. By the Quality of Curvature I mean its Form, as it is more or lefs inequable, or as it is varied more or lefs, in its progrefs thro' different parts of the Curve. So if it were demanded, what is the Quality of the Curvature of the Circle ? it might be anfwer'd, that it is uniform, or invariable. And thus if it were demand- ed, what is the Quality of the Curvature of the Spiral, which is described by the motion of the point D, proceeding from A in AD with an accelerated velocity, while the right Line AK moves with an uni- form rotation about the Cen- ter A ; the acceleration of
L 2 which
76 7&? Method of FLUXIONS,
which Velocity is fuch, that the right Line AD has the fame ratio to the Arch BK, defcribed from a given point B, as a Number has to its Logarithm : I fay, if it be afk'd, What is the Quality of the Curvature of this Spiral 1 It may be anfwer'd, that it is uniformly varied, or that it is equably inequable. And thus other Curves, in their feveral Points, may be denominated inequably inequable, ac- cording to the variation of their Curvature.
2. Therefore the Inequability or Variation of Curvature is re- quired at any Point of a Curve. Concerning which it may be ob- ferved,
3. I. That at Points placed alike in like Curves, there is a like Inequability or Variation of Curvature.
4. II. And that the Moments of the Radii of Curvature, at thofe Points, are proportional to the contemporaneous Moments of the Curves, and the Fluxions to the Fluxions.
5. III. And therefore, that where thofe Fluxions are not propor- tional, the Inequability of the Curvature will be unlike. For there will be a greater Inequability, where the Ratio of the Fluxion of the Radius of Curvature to the Fluxion of the Curve is greater. And therefore that ratio of the Fluxions may not impro- perly be call'd the Index of the Inequability or of the Variation of Curvature.
6. At the points D and d, infinitely near to each other, in the Curve AD^, let there be drawn the
Radii of Curvature DC and dc •, and D</ being the Moment of the Curve, Cc will be the contemporaneous Moment
of the Radius of Curvature, and -^ will be the Index of the Inequability of Curvature. For the Inequability may be call'd fuch and fo great, as the quan- tity of that ratio 7^ mews it to be :
j ±Ja
Or the Curvature may be faid to be fo much the more unlike to the uniform Curvature of a Circle.
7. Now letting fall the perpendicular Ordinates DB and dbt to any line AB meeting DC in P j make AB = #, BD = y\
and thence B& = xo, it will be Cc = vo; and -1 — T^ = — , making x = i.
Wherefore
£>
II
and IN FINITE SERIES. 77
Wherefore the relation between x and y being exhibited by any Equation, and thence, (according to Prob. 4. and 5.) the Perpendicu- lar DP or /, being found, and the Radius of Curvature i1, and the
Fluxion <y of that Radius, (by Prob. i.) the Index '^ of the Inequabi- lity of Curvature will be given alfo.
8. Ex. i. Let the Equation to the Parabola tax = vy be given ; then (by Prob. 4.) BP = a, and therefore DP= ^a-\-\y=^t. Alfo (by Prob. 5.) BF = a -+- 2X, and BP : DP :: BF : "i)C =
- =1;. Now the Equations 2ax =}'}', aa-\-yy=tt, -and
t-~ =v, (by Prob. i.) give 2ax = 2jvy, and zyy = ztt, and at + Zfx + 2fx __ ^ Which being reduced to order, and putting
.v = i, there will arife y = -, / = r^ = ) -f> an<^ v= - —
And thus y, t, and v being found, there will be had ^v the Index
of the Inequability of Curvature.
9. As if in Numbers it were determin'd, that^=ja or 2#==n>,
and x= 4 ; then y (==
+ 7 + "= 3v/2. So that
j^= 3, which therefore is the Index of Inequability.
10. But if it were determin'd, that A: =2, then y = 2, ^'=T> / = v/5, f = </±, and -17 = 3^/5. So that ^-=) 6 will be here
the Index of Inequability.
11. Wherefore the Inequability of Curvature at the Point of the Curve, from whence an Ordinate, equal to the Latus reftum of the Parabola, being drawn perpendicular to the Axis, will-be double to the Inequability at that Point, from whence the Ordinate fo drawn is half the Latus rectum ; that is, the Curvature at the firft Point is as unlike a- gain to the Curvature of the Circle, as the Curvature at the fecond Point.
12. Ex. 2. Let the Equation be zax — bxx-=.yy, and (by Prob. 4.) it will be a — &v=BP, and thence tf=(aa — 2a6x-lrb
=) na — byy -±- yy. Alfo (by Prob. 5.) it is DH =}' -{ where, if for yy — byy you fubftitute // — aa, there ariies DH = Tis alfo BD : DP :: DH : DC= - =v. Now (hv Prob.i.)
f.ll U1
the Equations zax—bxx^yy, aa — byy-\-y\-=^t!, and
give
7 8 77je Method of FLUXIONS,
•
give a — bx =}')', and yy — byy = /'/, and ~ = v. And thus v
being found, the Index ^ of the Inequability of Curvature, will aJib be known.
13. Thus in the Ellipfis 2X — 3 ATA: =}'}', where it is a = r, and b=-.^ ; if we make x=-, then r-— L v —
* * S " a 3 x ~~ ~"
A
b P
Jl
o
V
and therefore ;v=|, which is the In-
dex of the Inequability of Curvature.
Hence it appears, that the Curvature of
this Ellipfis, at the Point D here af-
fign'd, is by two times left inequable,
(or 'by two times more like to the Cur-
vature of the Circle,) than the Curva-
ture of the Parabola, at that Point of
its Curve, from whence an Ordinate let fall upon the Axis is equaj
to half the Latus rectum.
14. If we have a mind to compare the Conclufions derived in
thefe Examples, in the Parabola 2ax=yy arifes (~ =>)^vfor the
V ' s a
Index of Inequability j and in the Ellipfis zax — bxx=yy, arifes (^7- =J - - x BP j and fo in the Hyperbola 2ax -+- bxx =yy, the analogy being obferved, there arifes the Index ("2- — ^ y+3b
\. t J &&
x BP. Whence it is evident, that at the different Points of any Conic Section conn'der'd apart, the Inequability of Curvature is as the Rectangle BD x BP. And that, at the feveral Points of the Pa- raboh, it is as the Ordinate BD.
15. Now as the Parabola is the moft fimple Figure of thofe that are curved with inequable Curvature, and as the Inequability of its Curvature is fo eafily determined, (for its Index is 6x^ll^i,) there-
.. .
fore the Curvatures of other Curves may not improperly be compared to the Curvature of this.
1 6. As if it were inquired, what may be the Curvature of the Ellipfis 2X — $xx=yy, at that Point of the Perimeter which is determined by affuming x = ±: Becaufe its Index is 4., as before, it might be anfwer'd, that it is like the Curvature of the Parabola
6.v
and IN FINITE SERIES.
79
6.v =)')', at that Point of the Curve, between which and the Axis the perpendicular Odinate is equal to |.
17. Thus, as the Fluxion of the Spiral ADE is to the Fluxion of the Subtenfe AD, in a certain given Ratio, fuppofe as d to e; on its concave fide erect
AP = - x AD perpendicular to AD,
y dd — ee
and P will be the Center of Curvature, and
A P t
— or — — r=? will be the Index of Inequa-
«1J y a.i — ee
bility. So that this Spiral has every where its Curvature alike inequable, as the Parabola 6x = yy has in that Point of its Curve, from whence to its Abfcifs a perpendicular Ordi- nate is let fall, which is equal to the
1 8. And thus the Index of Inequability at any Point D of the
AB
Trochoid, (fee Fig. in Art. 29. pag. 64.) is found to be — . Where-
fore its Curvature at the fame Point D is as inequable, or as unlike to that of a Circle, as the Curvature of any Parabola ax - — yy is at
AB
the Point where the Ordinate is ^a x -^ •
19. And from thefe Confiderations the Senfe of the Problem, as I conceive, mufl be plain enough; which being well underftood, it will not be difficult for any one, who obferves the Series of the things above deliver'd, to furnifh himfelf with more Examples, and to contrive many other Methods of operation, as occafion may re- quire. So that he will be able to manage Problems of a like nature, (where he is not difcouraged by tedious and perplex Calculations,) with little or no difficulty. Such are thefe following ;
I. To find the Point of any Curve, where there is either no Inequabi- lity of Curvature, or infinite, or tie grcatej?, or the leajl.
20. Thus at the Vertices of the Conic Sections, there is no In- equability of Curvature; at the Cuf] id of the 1 rcchoid it is infi- nite ; and it is greatefl at thofe Points of the Ellif.fis, where the Rectangle BD x BP is greatefl, that is, where the Diagor.al-Lines of the circumfcribed Parallelogram cut the Elliriis, whofe Sides touch it in their principal Vertices.
II. 1o determine a Curve of fame definite Species, l'nfprje a Cr.n:c Section, liioje Curvature at any Point may be cqiu:l and Jiitiilar to the Curvature of any other Curve, at a given P./:./ of it.
8 o "The Method of FLUXIONS,
III. To iL-termine a Conk Sctfion, at any Point of which, the Cur- ri?//i7V and Pojition of the tangent, (in refpeSt of the AxisJ) may be like to the Curvature and Pofition of the Tangent, at a Point ajfigrid of any other Curir.
21. The ufe of which Problem is this, that inftead of Ellipfes of the fecond kind, whofe Properties of refradling Light are explain'd by Des Cartes in his Geometry, Conic Sections may be fubftituted, which mall perform the fame thing, very nearly, as to their Re- fractions. And the fame may be underfhood of other Curves.
P R O B. VII.
To find as many Curves as you pleafey ivbofe Areas may be exhibited by finite Equations.
I. Let AB be the Abfcifs of a Curve, at whofe Vertex A let the perpendicular AC = i be raifed, and let CE be D
drawn parallel to AB. Let alfo DB be a rectan- gular Ordinate, meeting the right Line CE in E, and the Curve AD in D. And conceive thefe Areas ACEB and ADB to be generated by the right Lines BE and BD, as they move along the Line AB, Then their Increments or Fluxions will
be always as the defcribing Lines BE and BD. Wherefore make the Parallelogram ACEB, or AB x i, =.v, and the Area of the Curve ADB call z. And the Fluxions x and z will be as BE and
BD; fo that making x = i = BE, then z = BD.
2. Now if any Equation be a/Turned at pleafure, for determining the relation of z and x, from thence, (by Prob. i.) may z be de- rived. And thus there will be two Equations, the 'latter of which will determine the Curve, and the former its Area.
EXAMPLES.
3. Aflume ##:=:£, and thence (by Prob. i.) 2xx=s} or 2x=c:, becaufe x=, i.
4. Aflame ^=z, and thence will arife — =;s? an Equation to the Parabola.
5. A flume ax* =zz, or a'fx*=z, and there will arife \a^x'£=^^, or ^(?x = zz, an Equation again to the Parabola.
i 6.
and INFINITE SERIES. 81
6. Affume a6x~1=zz,or a*x-' =z, and there arifes — a*xf* = z, or a'' -j-2xx = o. Here the negative Value of z only infinuates, that BD is to be taken the contrary way from BE.
7. Again if you affume c'-a1 -+- c^x* = z1, you will have zc*x
= 2zz ; and z being eliminated, there will arife
8. Or if you affume
aa-^-xx
aa -J-.VA-
'Z.
\/aa -+- xx = z, make
- -- , <z -}- ATA-
= v, and it will be ^ =s,and then (by Prob.i.) ^p — ^ Alfo
the Equation aa -f- xx = 011; gives 2X = zvv, by the help of which if you exterminate <u, it will become 3-j^- = z = j- \/ aa-^-xx.
9. Laftly, if you affume 8 — 3^2 -f- ^&=. zz, you will obtain — 32; — 3x2; -f- $z = 2Z&. Wherefore by the affumed Equation firflieek the Area z, and then the Ordinate z by the reiulting Equa- tion.
10. And thus from the Areas, however they may be feign'd, you may always determine the Ordinates to which they belong.
P R O B. VIII.
To fad as many Curves as you pleafe, -wbofe Areas fiall have a relation to the Area of any given Curve, a/fign- able by finite Equations.
i. Let FDH be a given Curve, and GEI the Curve required, and conceive their Ordinatss DB and EC to move at right Angles upon
A C
.11
G,
/V
their Abfciffes or Bafes AB and AC. Then the Increments or Fluxions of the Areas which they defcribe, will be as thofe Ordinates drawn
M into
82 fhe Method of FLUXIONS,
into their Velocities of moving, that is, into the Fluxions of their Abfcifles. Therefore make AB = x, BD = v, AC = z, and CE =y, the Area AFDB = j, and the Area AGEC = /, and let the Fluxions of the Areas be s and t : And it will be xv : zy : : s : t. Therefore if we fuppofe x = i, and v=s, as before; it will be
zy = t, and thence - =y.
2. Therefore let any two Equations be affumed ; one of which may exprefs the relation of the Areas s and t, and the other the relation of their Abfciffes x and z, and thence, (by Prob. i.) let the
Fluxions t and z be found, and then make - =>'.
3. Ex. i. Let the given Curve FDH be a Circle, exprefs'd by the Equation ax — xx = w, and let other Curves be fought, whofe Areas may be equal to that of the Circle. Therefore by the Hy-
pothefis s=:f, and thence s = f, and y = - =^-. It remains
to determine z, by afluming fome relation between the Abfciffes x and z.
4. As if you fuppofe ax=zz; then (by Prob. i.) a •=. 2zz: So that fubflituting :[ for z, then y = " = — . But it is v =
(\/ax — xx =) - \/ aa — s.s, therefore — \/ ' aa — zz = y is the
a aa *
Equation to the Curve, whofe Area is equal to that of the Circle.
5. After the fame manner if you fuppofe xx =. z, there will
ariie 2x =s, and thence _)'= (—==] ~; whence -j and x being
; — I exterminated, it will be y=- 7"-'
2Z2-
6. Or if you fuppofe cc = xz, there arifes o = z + xz, and
T-V (5 /
thence -- = y = -- , v az — cc.
2 2^3
7. Again, fuppofing ax •+- '- = z, (by Prob. i.) \t'isa + s=:z, and thence -^- — y — —?—> which denotes a mechanical Curve.
8. Ex. 2. Let the Circle ax — xx = w be given again, and let Curves be fought, whofe Areas may have any other aflumed relation to the Area of the Circle. As if you afliime cx + s = t, and fup-
pofe alfo ax = ZZ. (By Prob. i.) 'tis c + s = t, and a =
Therefore
and INFINITE SERIES. 83
Therefore y = ~ =2~~; and fubftituting ^ ax — xx for j, and 5f for x, 'tis;'= ™ 4- ^ v'^
9. But if you affume j — — =/, and x = z, you will have
s — ^! =/, and i = z. Therefore y—- - =j — 2!2L Oi
« K fl '
= i; — — — . Now for exterminating v, the Equation ax — xx = iJ'u, (by Prob. i.) gives ^ — 2x= 2vv, and therefore 'tis y=. — . Where if you expunge v and A; by fubftituting their values
\/ ax — xx and 2;, there will arile _)•=-" \/tf;s —
10. But if you affume ss = f, and x = zz, there will arife 2w=r^, and i = 2zz; and therefore _y = V = 4^- Anc
5 and x fubftituting \/ ax — xx and &z, it will become y = \Sa-zz;, which is an Equation to a mechanical Curve.
1 1. Ex. 3. After the fame manner Figures may be found, which have an aflumed relation to any other given Figure. Let the Hyper- bola cc -{- xx = wu be given ; then if you affume s = /, and xx=cz, you will have s = f and 2X = cz; and thence _)' =
.r= -. Then fubftituting v/cc -+- xx for j, and C-z^ for x, it
s;
will be y =: - i/cz -{- zz.
* 2Z
12. And thus if you affume xv — s=zf, and xx = cz, you will have v-^-vx — s=t, and 2X = cz. But v=.s, and thence
•vx = i. Therefore y= - = ~. But now (by Prob. i.) cc-\-xx
% ***
= 1?^ gives x=^-ui;, and 'tis y = ^ Then fubftituting \/i<,-t-xx for -u, and c*z* for x, it becomes y •=. , ^ c~ ^
13. Ex. 4. Moreover if the Ciffoid ^-^^_- - =1; were given, to which other related Figures are to be found, and for that purpofe you affume - ^/ax — xx •+- - s = t ; fuppofe - */ ax — xx = h,
and its Fluxion /' -, therefore h •+- - s = /. But the Equation *** -
M 2 =M
84 7%e Method of FLUXIONS,
=/j/j gives 3*A ^ .V. ==2.^, where if you exterminate /&, it will be
And bcfides fuice it is - s — -
3 3
— xx,
= t. Now to determine z and z, afTume \/ aa — ax = z ; then (by Prob. i .) — a = 2zz, or z = — - •
V.. Jax—xx a — A; '
v/tftf — £~. And as this Equation belongs to the Circle, we mall have the relation of the Areas of the Circle and of the Ciflbid.
14. And thus if you had aflumed V ">/ ax — xx -h ~ s = fy
and x = z, there would have been derived y-=.\/as> — .22-, an Equation again to the Circle.
15. In like manner if any mechanical Curve were given, other mechanical Curves related to it might be found. But to derive geometrical Curves, it will be convenient, that of right Lines de- pending Geometrically on each other, fome one may be taken for the Bafe or Abfcifs ; and that the Area which compleats the Paralle- logram be fought, by fuppofing its Fluxion to be equivalent to the Abfcifs, drawn into the Fluxion of the Ordinate.
1 6. Ex. 5. Thus the Trochoid ADF being propofed, I refer it to the Abfcifs
ABj and the Parallelogram ABDG being compleated, I leek for the complemen- tal Superficies ADG,byfup- pofing it to be defcribed by the Motion of the right Line
GD, and therefore its Fluxion to be equivalent to the Line GD drawn into the Velocity of the Motion ; that is, x*v. Now where- as AL is parallel to the Tangent DT, therefore AB will be to BL as the Fluxion of the fame AB to the Fluxion of the Ordinate BD,
that
and INFINITE SERIES. 85
that is, as r to -j. So that <u = — and therefore xv == BL.
A h
Therefore the Area ADG is described by the Fluxion BL ; fince therefore the circular Area ALB is defcribed by the fame Fluxion, they will be equal.
17. In like manner if you conceive ADF to be a Figure of Arches, or of verfed Sines, that is, whole Ordinate BD is equal to the Arch AL ; lince the Fluxion of the Arch AL is to the Fluxion of the Abfcifs AB, as PL to BL, that is, v : i :: ±a : \/ ax — .v.v, then -y = --/— . Then vx, the Fluxion of the Area ADG,
2 v ax — xx
will be — 7=^=. Wherefore if a right Line equal to — --.'
2V<,* — xx ly . .x — ATV
be conceived to be apply 'd as a rectangular Ordinate at B, a point of the Line AB, it will be terminated at a certain geometrical Curve, whole Area, adjoining to the Abfcifs AB, is equal to the Area ADG.
1 8. And thus geometrical Figures may be found equal to other Figures, made by the application (in any Angle) of Arches of a Circle, of an Hyperbola, or of any other Curve, to the Sines right or verfed of thole Arches, or to any other right Lines that may be Geometrically determin'd.
19. As to Spirals, the matter will be very fliort For from the Center of Rotation A, the Arch DG being defcribed, with any Radius AG, cutting the right Line AF in G, and the Spiral in D ; fince that Arch, as a Line moving upon the
Abfcifs AG, delcribes the Area of the Spiral AHDG, fo that the Fluxion of that Area is to the Fluxion of the Rectangle i x AG, as the Arch GD to i ; if you raife the perpen- dicular right Line GL equal to that Arch, by moving in like manner upon the fame Line AC, it will defcribe the Area A/LG equal to the Area of the Spiral AHDG : The Curve A/L being a geometrical Curve. And fartlirr, if the Subtenfe AL be drawn, then A ALG = | xGL = |AGx GD = Sector AGDj therefore the complernental Segments AL/ and ADH will alfo be equal. And this not only agrees to the Spiral of Archimedes^ (in which cafe A/L becomes the Parabola of Apoliomus,) but to any other whatever; fo that all of them may be converted into equal geometrical Curves with the fame eale.
20.
86 tte Method of FLUXIONS,
20. I might have produced more Specimens of the Conftruction of this Problem, but thefe may fuffice; as being fo general, that whatever as yet has been found out concerning the Areas of Curves, or (I believe) can be found out, is in fome manner contain'd herein, and is here determined for the moil part with lefs trouble, and with- out the ufual perplexities.
21. But the chief ufe of this and the foregoing Problem is, that nffuming the Conic Sections, or any other Curves of a known mag- nitude, other Curves may be found out that may be compared with thefe, and that their defining Equations may be difpofed orderly in a Catalogue or Table. And when fuch a Table is contracted, when the Area of any Curve is to be found, if its defining Equation be either immediately found in the Table, or may be transformed into another that is contain'd in the Table, then its Area may be known. Moreover fuch a Catalogue or Table may be apply'd to the determining of the Lengths of Curves, to the finding of their Centers of Gravity, their Solids generated by their rotation, the Su- perficies of thofe Solids, and to the finding of any other flowing quantity produced by a Fluxion analogous to it.
P R O B. IX.
To determine the Area of any Curve propofed.
1. The refolution of the Problem depends upon this, that from the relation of the Fluxions being given, the relation of the Fluents may be found, (as in Prob. 2.) And firft, if the right Line BD, by the motion of which the Area required AFDB
is defcribed, move upright upon an Abfcifs AB given in pofition, conceive (as before) the Paral- lelogram ABEC to be defcribed in the mean time on the other fide AB, by a line equal to unity. And BE being fuppos'd the Fluxion of the Pa- rallelogram, BD will be the Fluxion of the Area required.
2. Therefore make AB = x, and then alfo ABEC=i \x=x, and BE = x. Call alfo the Area AFDB = z, and it will be
BD=z, as alfo =~, becaufe x=i. Therefore by the Equa-
X
tion expreffing BD, at the fame time the ratio of the Fluions -
IS
and INFINITE SERIES. 87
is exprefs'd, and thence (by Prob. 2. Cafe i.) may be found the relation of the flowing quantities x and z.
3. Ex. i. When BD, or z, is equal to fome fimple quantity.
4. Let there be given ~ = z, °r — , (the Equation to the Pa-
rabola,) and (Prob. 2.) there will arife -a = z. Therefore ^> or -L AB x BD, = Area of the Parabola AFDB.
c. Let there be eiven — = z, fan Equation to a Parabola of
J ^ aa *
the fecond kind,) and there will arife -^ = z, that is, ~ AB x BD = Area AFDB.
6. Let there be given — — z
XX ~ '
or a^x— 1 = x-:, (an Equation to an Hyperbola of the fecond kind,) and there will arife — a 3 x—1
• z
or — 7 = z. That is, AB x BD „
= Area HDBH, of an infinite length, lying on the other fide of
the Ordinate BD, as its negative value insinuates.
j. And thus if there were given ^ = z, there would arife
2XX
Z.
8. Moreover, let ax = zz, or a*x* = z, (an Equation again to the Parabola,) and there will arife ~a^x^ = z,, that is, i-AB x BD = Area AFDB.
9. Let ~=zz-t then — za*x± = s, or 2 AB x BD = AFDH.
10. Let £=zz', then — ^ f = s, or 2 AB x BD = HDBH.
1 1. Let ax* = z~> ; then f«V = z, or i AB xBD = AFDH. And fo in others.
12. Ex. 2. Where z is equ.il to an Aggregate of fuch Quantities.
13. LetAT-H^—ij then^-h *-£ = z>
, J
14. Let <z -{- ^ = ~ . then ax — \ = &>
15. Let 3*i — £ — ^ — z ; then 2x^ +-x — 4** = 2r-
1 6. Ex. 3. Where a previous reduction by Divifion is required.
17. Let there be given j~, =.& (an Equation to the Apollonian Hyperbola,) and the divifion being performed in injinittun, it will be
l%e Method of FLUXION s
,
•x __ «« _ ^ 4. ?f£ — ^l5, &c. And thence, (by Prob. 2.) as
•11 1. • a*x ^^
in the fecond Set of Examples, you will obtain z= -y • --^
/. "xa U~A ^ «
5^/3 A/4 *
1 8. Let there be given — ^— ==*, and by divifion it will be
^ 1 ~J XX
~=i — x*-{-x* — x6, &c. or elfe s= -^ — l- -f- -., &c. And
"° X1 X4 A.0'
thence (by Prob. 2.) 2 = x — ^3-f-^r — 1#7, &c. =AFDBi or 2 = — -i H- ^- -5, &c. =HDBH.
X 3X* SA.''
L A
19. Let there be given , ™~!LiX =z, and by divifion it will be z = 2x^ — 2X + 7^ — I3AT1 -f- 34*% 6cc. And thence (by
Prob. 2.) z = $x* — x' -f- yx* — 'T3x3 •+• V8 A<^ &c-
20. Ex. 4. Where a previous reduction is required by Extraction of Roots.
21. Let there be given z = \/ aa -\- xx, (an Equation to the Hyperbola,) and the Root being extracted to an infinite multitude
of terms, it will be z=i a 4- * „ ~\ — 7- — - — r, &c. whence
* ft a Qf,9.*if.f,l I I -7 fit*
. r . X X x ^x „
as in the foregoing ss = ax+ 6~ — — , -h 77^ — TT^ &c-
22. In the fame manner if the Equation z = \/aa — xx were given, (which is to the Circle,) there would be produced z=ax —
b« ^Oi.J ii2as
23. And fo if there were given z-=\/x — xx, (an Equation alfo to the Circle,) by extracting the Root there would arife z = x'f — x* — 4-** — -r'-g-x^, 6cc. And therefore z = .ix*
1
Vz
TT .
___ _
24. Thus s === v//z<* -^- AV — xx, (an Equation again to the Cir-
bx
x .\
cle,) by extraction of the Root it gives z=a-\- — - • - — gjsj occ.
, ^* Jf3 /'*v3 -
whence 2; = <7Ar -f- -- -, --- , &c.
4« 6<» 24^ I
25. And thus v^~ZT7~ = ^, by a due reduction gives
z=i-+- T^-V* -h 43^4, &c. then 2 = AT -f- ^3 -f- TV^ S &c. H-irf -f±^ _l_^ +TV^
— T™ ' - Vo^
26.
and INFINITE SERIES. 89
26. Thus finally z=l/a* -t-A'5, by the extraction of the Cubic Root, gives z=a -+- — — ~ -+- ~w &c. and then (by Prob. 2.)
*=** + -~, — gfr •+• T£> &c. = AFDB. Or elfe *=
C'' A 1 1 **
And thence * = 7
&c. = HDBH.
567*
27. Ex. 5. Where a previous reduction is required, by the refo- lution of an affected Equation.
28. If a Curve be defined by this Equation z> •+• a*z
— 2a"' — x3 = o, extradl the Root, and there will arife z = a — x. j_ : : _j_ !4^-. &c. whence will be obtain'd as before z-=ax —
64.2 5 i zaa
„
29. But if z~' — cz* — 2x*z — c *z -f- 2x? -+- c* = o were the Equation to the Curve, the refolution will afford a three-fold Root; either z = c + .v— f? + Jl, &c. or S = c — .v-f- !i' -,
V 32'1 <-
or s = — c — A- — f -- (_ ±- &c. And hence will arife the
2£" 2rc At T
values of tb,e three correfponding Areas, z = ex + ±x* — — -f- T^t, &c. 2; = r^ — i.v1 + ^ — ^0, &e. and x = — ex — •
A 5 X4 .,S
~ - -- !— - flrr 6c 8.1 24^' CCC>
30. I add nothing here concerning mechanical Carves, becaufe their reduction to the form of geometrical Curves will be taught af- terwards.
31. But whereas the values of z thus found belong to Areas which are fituate, fometimes to a finite part AB of the Abfcifs, fometimes to a part BH produced infinitely towards H, and fome- times to both parts, according to their different terms: That the due value of the Area may be alTign'd, adjacent to any portion of the Abfcifs, that Area is always to be made equal to the difference of the values of z, which belong to the parts of the Abfci/s, that are terminated at the beginning and end of the Area.
•32. For Inflance ; to the Curve exnrefs'd bv the Equation — —
i-^-'xx
•£ ' m^^ JTC-
fhe Method of FLUXIONS,
—— ~, it is found that z=x — ^x} _l_ 4_,vS &c. Now that I may de- termine the quantity of the Area MDll, adjacent to the part of the •Abfcifs /'B; from the value of z, which arifes by putting AB = x, I take the value of z, which arifes by putting Ab=x, and there remains x — -Lx* + ^-x', &c. — x + ±x> — -J-x', &c. the value of that Area WDB. Whence if A*, or x, be put equal to nothing, jqere will be had the whole Area AFDB = x — £x' -+- -^x', &c.
33. To the fame Curve there is alfo found z, •==. — - -+• —
L, &c. Whence again, according to what is before, the Area
5**
I 1 1 ^ I I &/~f* *"T * ppr^TOt'f1
1 ]V\T\ •__ t _.— I ,_ -- oCC '- " — T- 1 ' ' ^""1 — -) OCC. J. ijCl CIUI C
if AB, or x, be fuppofed infinite, the adjoining Area bdH toward H, which is alfo infinitely long, will be equivalent to - — ^ -f- — . &c. For the latter Series — - -f- — • ~, &c. will
CA ^-35
vanifh, becaufe of its infinite denominators.
34. To the Curve reprefented by the Equation a-\- — = Z,^ it
:s found, that z=.ax — -. Whence it is that «x — - — ax
i X X
-4- - = Area &/DB. But this becomes infinite, whether x be fup- pofed nothing, or x infinite ; and therefore each Area AFDB and &/H is infinitely great, and the intermediate parts alone, fuch as &/DB, can be exhibited. And this always happens when the Ab- fcifs x is found as well in the numerators of fome of the terms, as in the denominators of others, of the value of z. But when x is only found in the numerators, as in the firft Example, the value of z, belongs to the Area fituate at AB, on this fide the Ordinate. And when it is only in the denominators, as in the fecond Example, that value, when the figns of all the terms are changed, belongs to the whole Area infinitely produced beyond the Ordinate.
35. If at any time the Curve-line cuts the Abfcifs, between the points b and B, fuppofe in E, inftead of the Area will be had the difference &/E*— BDE of the Areas at the diffe- rent parts of the Abfcifs ; to which if there be added the Rectangle
he Area dEDG will be obtain'd. t
G
and INFINITE SERIES.
36. But it is chiefly to be regarded, that when in the value of & any term is divided by x of only one dimension ; the Area corre- fponding to that term belongs to the Conical Hyperbola ; and there- fore is to be exhibited by it felf, in an infinite Series : As is done in what follows.
77. Let fl3~glA'= z, be an Equation to a Curve ; and by divifion
J • ax -f- xx J
it becomes z = - — 2a •+- 2X — —_ — h^ &c. and thence
aa y
2X> l X*
Z = l^ j — 2ax -f- x1 — ^T ' To* &c. And the Area &/DB
— £,&*.—
aa
zax
xx
2*5
— ,
I denote the little Areas belonging
Where by the Marks — and
1-1 1
- aa aa
to the Terms — and —
38. Now that |^ and |j| may be found, I make Kb, or xy to be definite, and bE indefinite, or a flowing Line, which therefore I call ;' ; fo that it will be -^; = to that Hyperbolical Area adjoin- But by Divifion it will be - - = - J x~ \ y x
therefore,
ing to £B, that is, j -
A4
or -
x
-* ' '
. and therefore the whole Area required
WDB = —
X 2A3
21 3 .
— xx H -, &c.
39. After the fame manner, AB, or x, might have been ufed for
a definite Line, and then it would have been
40. Moreover, if Z>B be bifefted in C, and AC be affumed to be of a definite length, and Cb and CB indefinite ; then making AC
= i>, and C£ or CB =_)', 'twill be bd= -^ s=™-\- '— -)- —~
-{- — _i- ^-^-' &c. and therefore the Hyperbolical Area adjacent
N 2 to
'A Mt&od of FLUXIONS,
to the Part of the Abfcifs &C will be
a V
r I
.&c. Twill be alfo DB = -~ = ? - ~ + ~ - ^ + » &c. And therefore the Area adjacent to the other part of the Abfcifs CB
1 11 7* st^ • 4 Si"*- 1
= "• , + - -f- ' •' , &c. And the Sum of thefe
f 2fl Jf1 4' * 5'1
Areas 7- -\~ ~r ~r, &c. will be equivalent to -|
41. Thus in the Equation a3 -f- z,* ~$- z — x~= =o, denoting the •nature of a Curve, its Root will be z = ,v — y
&c. Whence there arifes z, =-. Lxx — -x —
6cc. And the Area
' Y» TA
7
ox
8 J A X 8 I A i
_ _1 _ _!_»
/^r^ Six' KC'
T, &c. that is,=:|.v
.'X— ^' TA Six
&c. _ - ^
&c. - i - -
42. But this Hyperbolical term, for the moft pnrt, may be very commodioufly avoided, by altering the beginning of the Abfcifs, that is, by increafing or diminiihing it by fome gi\ en quantity. As
in the former Example, where ?v +*v v = z was the Equation to the Curve, if I fhould make b to be the beginning of the Ablcifs* and fuppofmg Al> to be of any determinate length 4/7, for the re- mainder of the Abfcifs £B, I fliall now write x : Thst is, if I dimi- nifti the Abfcifs by ±a, by writing x -f- ±a inftead of x, it will
-become ^~^,. = ~> and
£_!£±! &Ci whence arifes s = \ax — ' 4^z 4--^' -' &c. = 273 j bia
Area .
43. And thus by affuming another and another point for the be- ginning of the Abfcifs, the Area of any Curve may be exr-ivib'd an infinite variety of ways.
44. Alfo the Equation rj-p£ = z might have been refolved into the two infinite Series z, — - -- "— -+- "-^ &c. — a -f .v
.V2 X1 X • }
—**--}-; &c. where there is found no Term divided b} the fir ft
2 Power
and INFINITE SERIES. 93
Power of x. But fuch kind of Series, where the Powers of A* afcend infinitely in the numerators of the one, and in the denominators of the other, are not fo proper to derive the value of z from, by Arithmetical computation, when the Species are to be changed in- to Numbers.
45. Hardly any thing difficult can occur to any one, who is to un- dertake fuch a computation in Numbers, after the value of the Area is obtain'd in Species. Yet for the more compleat illufhation of the foregoing Doctrine, I mall add an Example or two.
46. Let the Hyperbola AD be propofed, whofe Equation is \/x-+-xx=z; its Vertex be- ing at A, and each of its Axes is equal to Unity. From what goes before, its Area ADB=-i.v>
-+- j'^ — A*'1" -+- T'T*? — T^P*'"' &c' that is x* into Lx -+- ±x* — T'T.v * + y'T.v 4 — T4T-V s > &c. which Series may be infinitely produced by multiplying thelaft term continually by the fucceeding terms of this
Proereffion i-J#. — 5.v ^^r ~~'"qx ^^'v &c. That is
2 S 47 6-9A- 8.-nXi 10.15*. »
the firft term ^..v1 x I_3 x makes the fecond term -L.v* : Which
2-5 ' multiply 'd by " l-~x makes the third term — TV-vl : Which mul-
tiply'd by •— ^ x makes T'T.v? the fourth term; and fo ad hifini- tuin. Now let AB be affumed of any length, fuppofe ^, and writing this Number for .v, and its Root 4 for x*, and the firft term ^x^ or y x T> being reduced to a decimal Fraction, it becomes
°-°^3333333> &c- This into '- ^— makeso.oo625 the fecond term. This into ~ ' v makes — 0.0002790178, &c. the third term. And
4-7 4
fo on for ever. But the term?, which I thus deduce by degrees, I difpole in two Tables; the affirmative terms in one, and the nega- tive in another, and I add them up as you fee here.
-i-o.
94 "The Method of FLUXIONS,
•+ 0.0833333333333333 — 00002790178571429
62500000000000 34679066051
271267361111 834^65027
5135169396 26285354
144628917 961296
4954581 38676
190948 1663
7963 75 352 ±_
1 1 — 0.0002825719389575
-f- 0.0896109885646518
4- 0.0896109885640518 "0^3284166257043
Then from the fum of the Affirmatives I take the fum of the ne- gatives, and there remains 0.0893284166257043 for the quantity of the Hyperbolic Area ADB ; which was to be found.
47. Now let the Circle AdF be propofed, which isexpreffed by the equation \/x — xx = z > that is, whofe Diameter is unity, and from what goes before its Area AdB will be -!#* — .£.#* — T'Txi — -fT^i &c> In which Series, fince the terms do not differ from the terms of the Se- ries, which above exprefs'd the Hyperbolical Area, unlefs in the Signs -4- and — ; nothing elfe remains to be done, than to conned: the fame numeral terms with other fignsj that is, by fubtracting the connected fums- of both the afore -mention'd tables, 0.08989 3 560 503 6 1 93 from the firft term doubled 0.1666666666666, &c. and the remainder 0.0767731061630473 will be the portion A^B of the ciicular Area, fuppoiing AB to be a fourth part of the diameter. And hence we may obferve, that tho' the Areas of the Circle and Hyperbola are not compared in a Geometrical confidera- tion, yet each of them is dilcover'd by the fame Arithmetical com- putation.
48. The portion of the circle A^/B being found, from thence the whole Area may be derived. For the Radius dC being drawn, multiply Ed, or -^v/S? Ulto -^C, or i, and half of the product
•s-Vs/3' or °-°5412^58773^5275 w'" ^e ^e va^ue °f the Triangle cWB; which added to the Area AdB, there will be had the Sector ACd = 0.1308996938995747, the fextuple of which
whole Area.
49. And
and INFINITE SERIES.
95
49- And hence by the way the length of the Circumference will be 3.1415926535897928, by dividing the Area by a fourth part of the Diameter.
50. To thefe we mail add the calculation of the Area compre- hended between the Hyperbola dfD and its Afymptote CA. Let C be the Center of the Hyperbola, and putting
'twill be -^— =BD, and -^— -=.bd; whence
a+x
"
the Area AFDB = bx — - - 4- 4 — -*, &c. and the Area
4- — , &c. and the fum 0aL>&=. 2ox-\ — —? c £AB /'/.<
4- ~ 4- ^?, &c. Now let us fuppofe CA = AF=i, and Kb or AB = TL., Cb being 0.9, and CB = i.i ; and fubftituting thefe numbers for a, b, and x, the firft term of the Series becomes 0.2, the fecond 0.0006666666, &c. the third 0.000004 ; and fo on, as you fee in this Table.
O.2OOOOOOOOOOOOOOO
6666666666666
40000000000 285714286
2222222 l8l82
The fum 0.200670695462151 1= Area bdDB. 51. If the parts of this Area Ad and AD be defired feparately, fubtract the lefler BA from the greater dA, and there will remain
•3-+ -^4- - — h --» &c. Where if i be wrote for a and b,
and -jig. for x, the terms being reduced to decimals will iland thus;
O.O IOOOOOOOOOOOOOO
500000000000
3333333333
25000000
2OOOOO 1667
The fum o.
= A^— AD,
52-
96 The Method of FLUXIO N s,
52. Now if this difference of the Areas be added to, and fubtracted from,their fum before found, half the aggregate o. 1053605156578263 will be the greater Area hd, and half or the remainder 0.0953101798043248 will be the lefler Area AD.
53. By the fame tables thofe Areas AD and hd will be obtain'd alfo, when AB and Ab are fuppos'd T~, or CB=i.oi, and d> = o.gg, if the numbers are but duly transferr'd to lower places, as may be here feen.
O O2OOOOOOOOOOOOOC0 O.O30ICOOOOOOOO3OO
66666666666 50020000
4000000 3^
28
Sum o 020000(5667066(195 =
Sum 0.0001000050003333 AJ — AD.
==AD.
54. And fo putting AB andA£=-~o-> orCB=i.oor, and' 0^ = 0.999, there will be obtain'd Ad= 0.0010005003335835, and AD = o. 0009995003330835.
55. In the fame manner (if CA and AF= i) putting AB and A£ = o.2, or 0.02, or 0.002, the fe Areas will arife,
A^=o.223 1435513 142097, and ADz=o. 1823215567939546, or A</= 0.0202027073 175194, and AD = 0.0 19802 627296 1797, or AW=o.oo2oo2 andAp = o.ooi
56. From thefe Areas thus found it will be eafy to derive others,
I f \ 2.
by addition and fubtradtion alone. For as it is — ' into -^ = 2, the fum of the Areas 0.693 I47I^°5599453 belonging to the Ratio's ^|and ^-2, (that is, infifting upon the parts of the Abfcifs 1.2 — o 8 and 1.2 — o.9,)will be the Area AFcPjS, C/3 being = 2, as is known. Again, fince —^ into 2 = 3, the fum 1.0986122886681097 of the
Area's belonging to ^-| and 2, will be the Area AFcT/3, C/3 being 3.. Again, as it is ~ = 5, and 2 x5= 10, by a due addition of Areas will be obtain'd 1.6093379124341004 = AF^/3, when c/3=5; and 2.3025850929940457 =AF<T/3, when C/3 = 10. And thus, fince 10x10=100, and 10x100=1000, and ^5
x 10 xo.98 = 7, and lox i.i = n, and .'°°°x' °°' — — I^) and
- =499 ; it is plain, that the Area AF^/3 may be found by the compofition of the Areas found before, when C/3 = i oo j i ooo i
7>
and IN FIN ITE SERIES, 97
7; or any other of the above-mention'd numbers, AB = BF being llill unity. This I was willing to infinuate, that a method might be derived from hence, very proper for the conftrudtion of a Canon of Logarithms, which determines the Hyperbolical Areas, (from which the Logarithms may ealily be derived,) correfponding to fo many Prime numbers, as it were by two operations only, which are not very troublefome. But whereas that Canon feems to be deriva- ble from this fountain more commodioufly than from any other, what if I mould point out its contraction here, to compleat the whole ?
57. Firfl therefore having affumed o for the Logarithm of the number i, and i for the Logarithm of the number 10, as is gene- rally done, the Logarithms of the Prime numbers 2, 3, 5, 7, 1 1, 13, 17, 37, are to be inveftigated, by dividing the Hyperbolical Areas now found by 2.3025850929940457, which is the Area cor- refponding to the number 10: Or which is the fame thing, by mul- tiplying by its reciprocal 0.4342944819032518. Thus for Inftance, if 0.69314718, &c. the Area correfponding to the number 2, were multiply'd by 0.43429, &c. it makes 0.3010299956639812 the Lo- garithm of the number 2.
58. Then the Logarithms of all the numbers in the Canon, which are made by the multiplication of thefe, are to be found by the addition of their Logarithms, as is ufual. And the void places are to be interpolated afterwards, by the help of this Theorem.
59. Let « be a Number to which a Logarithm is to be adapted, A- the difference between that and the two neareft numbers equally diflant on each fide, whofe Logarithms are already found, and let d be half the difference of the Logarithms. Then the required Loga- rithm of the Number n will be obtain'd by adding d-\- £ •+- gr^,
&c. to the Logarithm of the leffer number. For if the numbers are expounded by C/>, C/3, and CP, the rectangle CBD or C,&T=i, as before, and the Ordinates pq and PQ^being raifed ; if n be wrote
for C/3, and x for £p or /3P, the Area pgQP or ^ -+- ~} + ~, &c. will be to the Area pq}$ or *- •+- ^ -f- ^, &c. as the diffe- rence between the Logarithms of the extream numbers or 2(i, to the difference between the Logarithms of the leffer and of the middle
O one;
g 8 Tie Method of FLUXIONS,
dx dx* dx* 0
- -+- — -f- — &C.
one: which therefore will be —. , that is, when the
x A'3 A"* a
- •+- — -4- — &c. divifion is perform'd, d-\- — -4- -— &c.
* 2n i Zfjs
60. The two firft terms of this Series d-\- — I think to be accu-
2n
rate enough for the construction of a Canon of Logarithms, even tho' they were to be produced to fourteen or fifteen figures; pro- vided the number, whofe Logarithm is to be found, be not lefs than 1000. And this can give little trouble in the calculation, be- caufe x is generally an unit, or the number 2. Yet it is not necef- fary to interpolate all the places by the help of this Rule. For the Logarithms of numbers which are produced by the multiplication or divifion of the number laft found, may be obtain'd by the numbers whofe Logarithms were had before, by the addition or fubtraction of their Logarithms. Moreover by the differences of the Loga- rithms, and by their fecond and third differences, if there be occa- lion, the void places may be more expeditioufly fupply'd ; the fore- going Rule being to be apply'd only, when the continuation of fome full places is wanted, in order to obtain thofe differences.
6 1. By the fame method rules may be found for the intercalation of Logarithms, when of three numbers the Logarithms of the leffer and of the middle number are given, or of the middle number and of the greater; and this although the numbers mould not be in Arithmetical progreffion.
62. Alfo by purfuing the fteps of this method, rules might be eafily difcover'd, for the conftruction of the tables of artificial Sines and Tangents, without the affiftance of the natural Tables. But of thefe things only by the bye.
63. Hitherto we have treated of the Quadrature of Curves, which are exprefs'd by Equations confirming of complicate terms ; and that by means of their reduction to Equations, which confift of an infi- nite number of fimple terms. But whereas fuch Curves may fome- times be fquared by finite Equations alfo, or however may be com- pared with other Curves, whofe Areas in a manner may be confi- der'd as known ; of which kind are the Conic Sections : For this reafon I thought fit to adjoin the two following catalogues or tables of Theorems, according to my promife, conflructed by the help of the jtb and Bth aforegoing Propofitions.
64.
and IN FINITE SERIES. 99
64. The firft of thefe exhibits the Areas of fuch Curves as can be fquared ; and the fecond contains fuch Curves, whole Areas may be compared with the Areas of the Conic Sections. In each of thefe, the letters d, e, f, g, and h, denote any given quantities, x and z the Abfcifles of Curves, v and y parallel Ordinares, and s and t Areas, as before. The letters » and 6, annex'd to the quantity z, denote the number of the dimenfions of the fame z, whether it be integer or fractional, affirmative or negative. As if »=3, then
JZ1ZZZ23, zl"=zs, z-«=z-~> or-'3, &+' = z*, and z*-' =z*.
65. Moreover in the values of the Areas, for the fake of brevity, is written R inftead of this Radical \Se-{-f&t or </e-t-fzi-\-gz**>, and/ inflead of </b-t-iz*t by which the value of the Ordinate^ is affected.
10O
"fhe Method of FLUXIONS,
, t
I I
i
•s
1
I
a
CO
rt
ii
'5
CO
U
3
Curve
u
+
n
en
e*
H«» N
•-1 N CO
*
•« 1 v,
N 1 S1
' T *~
1 *J-
•» ~
V ^
and INFINITE SERIES.
101
T-» •»»*
II II
a
CO
bo
• t
II
U
—
a
o u
f ^
^ \«
O ol \O
iM
""* I I
1 I
cno *?> oa'j j? N cr>
N
II II
T 1
v
t
*•
i
H-
T*»
II II
?r
-f'
c
•f
M
s
OJ
M M
x
o
G
* *•*
X
01
X"
i
X X
s CO s:
x"
IO2 ejff>e Method -o^ FLUXIONS,
67. Other things of the fame kind might have been added ; but I fhall now pafs on to another fort .of Curves, which may be com- pared with the Conic Sections. And in this Table or Catalogue you have the propofed Curve reprefented by the Line QE^R, the beginning of whole Abfcifs is A, the Abfcifs AC, the Ordinate CE, the beginning of the Area a^, and the Area
defcribed a^EC. But the beginning of this Area, or the initial term, (which com- monly either commences at the beginning of the Abfcifs A, or recedes to an infinite diftance,) is found by feeking the length of the Abfcifs Aa, when the value of the Area is nothing, and by eredling the per- pendicular a^/.
68. After the fame manner you have the Conic Sedlion repre- fented by the Line PDG, whofe Center is A, Vertex a, rectangular
Semidiameters Aa and AP, the beginning of the Abfcifs A, or a, or a, the Abfcifs AB, or aB, or aB, the Ordinate BD, the Tangent DT meeting AB in T, the Subtenfe aD, and the Re&angle infcribed or adfcribed ABDO.
69. Therefore retaining the letters before defined, it will be AC = z, CE=y, a.%EC = t, AB or aB = x, BD = i;, and ABDP or aGDB=j. And befides, when two Conic Sections are required, for the determination of any Area, the Area of the latter mall be call'd <r, the Abfcifs |, and the Ordinate T. Put p for
and INFINITE
103
S
S
u
o
en
_2 3
rt
-y
CO
5 U
o
U.
oa
a O Q
V
V
Tl »
BL, O
Q O
14
o
c
.
+ i?
.«V3 >*,
Q O
rt 2
ea Q O
rt >s| =
O
eg
Q
Q O
M
OH
Q
pa
CO
Q O
**
I
4-
--V-
104
Method of FLUXIONS,
OJ
m
(LI
3
Q
O a c c
h
O
Q
Pi O C
O
rt O
^1^
a
o
Q O
o .5
o Q O
2
O Q
rt O B
dina
•ed
C
3
- 3
U
I
o
fe
4
I
a ji
^«
H
-
4-
.<v fa.
cj
o o
n
X
15 ^
V
4-
fr
SJ
t
3| •
s
"T
II II
+
+
I
K +
%
v
-V"
u
13
O
U
S)
u
O
and INFINITE SERIES.
U
<J-.
o
1= o
$
N
4- H
Mj
P s-
^ ^
It tt
*«-
<
Q
CO
Q O
4
i
'
.
I.Hf
Tf-
1
+ 1
5
+ 1
V
1 +
<5
+ b<j
+
««)
I?
4-
E -V
H-|
^i t.
X
io6
Method of FLUXIONS,
s
J*
U
«J
to
_u
o
I
3 U
t/>
o
fa
+
X
•v.
i
s
+
x
and INFINITE SERIES, 107
71. Before I go on to illuftrate by Examples the Theorems that are deliver'd in thefe claffes of Curves, I think it proper to obferve,
72. I. That whereas in the Equations reprefenting Curves, I have all along fuppofed all the figns of the quantities d, e, f\ g, />, and i to be affirmative ; whenever it fhall happen that they are negative, they muft be changed in the fubfequent values of the Abfcifs and Or- ninate of the Conic Section, and alfo of the Area required.
73. II. Alfo the figns of the numeral Symbols » and 0, when they are negative, muft be changed in the values of the Areas. More- over their Signs being changed, the Theorems themfclvcs may ac- quire a new form. Thus in the 4th Form of Table 2, the Sign ot «
d '
being changed, the 3d Theorem becomes -;_iv,.-j-I ~~~' ,-^ -—}> ~~^
— x, &c. that is, 7=^=— =}', *"==*,
' cz -f-/a
into 2.w — 3^===^. And the fame is to be obferved in others.
74. III. The feries of each order, excepting the 2d of the ift Ta- ble, may be continued each way ad infinitum. For in the Series of the -;d and 4th Order cf Table i, the numeral co-efficients of the initial terms, (2, — 4, 16, — 96, 768, Sec.) are fonn'd by multi- plying the numbers — 2, — 4, — 6, — 8, — ro, &c. continually into each other ; and the co-efficients of the fubfequcnt terms are de- rived from the initials in the 3d Order, by multiplying gradually by — 1> — A, — £, — £, — -Li, &rc. or in the 4th Order by multi- plying by * — i, — 4-, — f, — T> — -rV. &C. But the co-efficients of' the denominators i, 3, 15, 105, &c. a rife by multiplying the numbers i, 3, 5, 7, 9, &c. gradually into each other.
75. But in the ad Table, the Series of the ift, 2d, 3'', 4h, c;1", and ioth Orders are produced in infinitum by diviiion alone. Thus having
= v, in the ift Order, if you perform the diviiion to a con- venient period, there will arifo j~ — ~z ^ 'j7 ^
==.)'. The firft three terms belong to the ift Order of
t/x
.4--1-'
Table i, and the fourth term belongs to the ift Species cf this Order.
d 3n Jc --4 <•:• ~ n
Whence it appears, that the Area is 7^- - ~ 1^fz + r?r ~
__ _il s.} putting s for the Area of the Conic Section, whofe Abfcifi
*' d
is x=r» , and Ordinate v = g-:r- -.
P 2
io8 7&e Method of FLUXIONS,
76. But the Series of the ^th and 6th Orders may be infinitely continued, by the help of the two Theorems in the 5th Order of Table i. by a due addition or fubtraction : As alib the 7th and 8th Scries, by means of the Theorems in the 6th Order of Table i. and the Series of the nth, by the Theorem in the roth Order of Table i.
For inftance, if the Series of the 3d Order of Table 2. beto be far- ther continued, fuppofe 6 = — 4>j, and the ift Theorem of the
jth Order of Table i. wll become — 8»fts~4l|~~1.— 5«/b~3>1~1 into =. -^-=^f. But according to the 4th Theorem of
this Series to be produced, writing — —^ for </, it is — ~ f%>
<x=v, and 'Qfr'-'S/*' __ t
ize
So that fubtrafting the former values of / and /, there will remain
— 4»J— ' / J- 1 10/1/3 Ii;/?} RS a _,, - , .
qnez v/^-h/2 =/> I2e ft Thefe being mul-
ij j
tiplied by — - ; and, (if you pleafe) for -~ writing xv*, there will arife a 5th Theorem of the Series to be produced,'
, 1 — ! s -
= v, and -r- — = f.
77. IV. Some of thefe Orders may alfo be otherwife derived from others. As in the 2d Table, the 5th, 6th, 7th, and nth, from the 8th; and the 9th from the loth : So that I might have omitted them, but that they may be of fome ufe, tho' not altogether necefftry. Yet I have omitted fome Orders, which I might have derived from the ifr, and 2d, as alfo from the 9th and loth, becaufe they were affected by Denominators that were more complicate, and therefore can hardly be of any ufe.
78. V. If the defining Equation of any Curve is compounded of feveral Equations of different Orders, or of different Species of the fame Order, its Area mufl be compounded of the correlponding A- reas ; taking care however, that they may be rightly connected with their proper Signs. For we mufl not always add or fubtra<fl at the fame time Ordinates to or -from Ordinates, or correfponding Areas to or from correfponding Areas ; but fometimes the fum of thefe, and the difference of thofe, is to be taken for a new Ordinate, or to conftitute a correfponding Area. And this muft be done, when the constituent Areas are pofited on the contrary fide of the Ordinate. Huf that the cautious Geometrician may the more readily avoid this
in-
and INFINITE SERIES. 109
inconveniency, I have prefix' d their proper Signs to the feveral Va- lues of the Areas, tho' ibmetimes negative, as is done in the jth and yth Order of Table 2.
70. VI. It is farther to be obferved, about the Signs of the Areas, that -f- * denotes, either that the Area of the Conic Section, adjoin- ing to the Abfcifs, is to be added to the other quantities in the value of t •, ( fee the ifl Example following ;) or that the Area on the other fide of the Ordinate is to be fubtracled. And on the contrary, — s denotes ambiguoufly, either that the Area adjacent to the Abfcifs is to be fubtradled, or that the Area on the other fide of the Ordinate is to be added, as it may feem convenient. Alfo the Value of f, if it comes out affirmative, denotes the Area of the Curve propoled ad- joining to its Abfcifs : And contrariwife, if it be negative, it repre- fents the Area on the other fide of the Ordinate.
80. VII. But that this Area may be more certainly defined, we mull enquire after its Limits. And as to its Limit at the Abfcifs, at the Ordinate, and at the Perimeter of the Curve, there can be no un- certainty: But its initial Limit, or the beginning from whence its de- fcription commences, may obtain various pofitions. In the following Examples it is either at the beginning of the Abfcifs, or at an infinite diftance, or in the concourfe of the Curve with its Abfcifs. But it may be placed elfewhere. And wherever it is, it may be found, by ieeking that length of the Abfcifs, at which the value of f becomes nothing, and there erecting an Ordinate. For the Ordinate fo raifed will be the Limit required.
8 1. VIII. If any part of the Area is pofited below the Abfcifs, / will denote the difference of that, and of the part above the Ab- fcifs.
82. IX. Whenever the dimenfions of the terms in the values of .v, i;, and /, (hall afcend too high, or defcend too low, they may be .reduced to a juft degree, by dividing or multiplying fo often by any
given quantity, which may be fuppos'd to perform the office of Uni- ty, as often as thole dimenfions mail be either too high or too low.
83. X. Befides the foregoing Catalogues, or Tables, we might allb conftrucT: Tables of Curves related_tp_ other Curves, which may be the
moftfimple intheirkind; as to <Ja-\-fx* =v, ortox</e-t-fx* =v, or to ^/e-\-Jx* =<y, &c. So that we might at all times derive the Area of any propoled Curve from the fimpleft original, and know to what Curves it llands related. But now let us illuitrute by Ex- amples. what has been already delivered.
84-
no
The Method ^FLUXIONS,
84. EXAMPLE I. Let QER be a Conchoidal of fuch a kind, that the Q Semicircle QH A being defcribed, and AC being creeled perpendicular to R the Diameter A Q^_ if the Parallelo- gram QACI be compleated, the Dia- gonal AI be drawn, meeting the Se- micircle in H, and from H the'per-
pendicular HE be let fall to 1C ; then the Point E will defcribe a Curve, whole Area ACEQJs fought.
^.Therefore make AQ^==a, AC=z, CE=y, and becaufe of the
continual Proportionals AI, AQ^, AH, EC, 'twill be ECor_>'= -—-^
86. Now that this may acquire the Form of the Equations in the Tables, make »=2, and for z~- in the denominator write z*, and a*z~-* * for
or
;]-' in the numerator, and there will arife_y = flf > an Equation of the ift Species of the ad Order of Table 2,
a -\-x,
and the Terms being compared, it will be^ = rf3, e = a*, and f= I j .fo that 4/ .J'' i
«/ v ii — T-£<
x,
3 — tf1.*;1 = -u, and xv — 2s
t.
87. Now that the values found of x and v may be reduced to a number of dimen lions, choofe any given quantity, as a, by
which, as unity, a* may be multiplied once in the value of x, and in the value of v, a> may be divided once, and ^x1 twice. And by
this means you will obtain s/"^niTr =^,^/al — .v1 =1', and xv — 2s, — t: of which the conllradion is thus.
88. Center A, and Radius AQ^_ defcribe the Qigadrahtal Arch QDP ; in AC take AB = AH ; raiie the perpendicular BD meeting that Arch in D, and draw AD. Then the double of the Scclof
ADP will be equal to the Area fought ACEQ^ For
' — AB.?=) BD, or-y ; and .vj — 2s= 2 A ADB — 2 or = 2*A ADB'-f- aBDP, that is, either = — aOAD, or=2DAP: Of which values the affirmative aDAP belongs to the Area ACEQ, on this fide EC, and the negative — aC^AD belongs to the Area RE R extended ad infi.ritum beyond EC.
89. The folutions 'of Problems thus found may fometimes be made more elegant. Thus in the prefent cafe, drawing RH the le-
midiameter
and INFINITE SERIES.
in
midiameter of the Circle QH A, becaufe of equal Arches QH and DP, the Sector QRH is half the Sector DAP, and therefore a fourth part of the Surface ACEQ^
90. EXAMPLE II. Let AGE be a Curve, which is defcribed by the Angular point E of the Norma AEF, whilft one of the Legs AE, being interminate, paffes continually through the given point A, and the other CE, of a given length, flides upon the right Line AF gi- ven in pofition. Let fall EH per- pendicular to AF, and compleat the Parallelogram AHEC ; and calling AC = z, CE =_y, and EF = rf, becaufe of HF, HE, HA continual Proportionals, it will be
HAor y=
r,
91. Now that the Area AGEC may be known, fuppofe »» = £*,
t«r-i or 2 = », and thence it will be j== =}'• Here fl"ce z in the
' a •~z^1
numerator is of a fraded dimenfion, deprefs the value of/ by di-
~V)~I viding by z&, and it will be 7=7= = S> an Equation of the
y a ~ * — i
ad Species of the ;th Order of Table 2. And the terms being com- pared, it is </= i, e= — i, and /= a*. So that z1 = /- ' __ N A.ijV/^i _ .v1 — -u, and 5 — xv = /. Therefore fince
\*~« ) _ ;
*• and z are equal, and fince ^a-—x* = v is an Equation to a Circle whofe Diameter is a : with the Center A, and diftancq a or EF let the Circle PDQ^be defcribed, which CE meets in D, and let the' Parallelogram ACDI be compleated ; then will AC = ^, CD=<u, and the Area fought AGEC = ^ — xv = ACDP
92. Ex-
The Method of FLUXIONS,
112
92. EXAMPLE III. Let AGE be the Ciflbid belonging to the Circle ADQj defcribed with the diameter AQ.. Let DCE be drawn perpendicular to the diameter, and meeting the Curves in D and E. And na- ming AC = zt CE =.y, and AQj== a ; becaufe of CD, CA, CE continual Proportio- nals, it will be CE or y =
:, and dividing by z, 'tis
X
y = / ~~ • Therefore zr~l
' az — I
==^, or — i = »,and thence
y =
V aai-i
an Equation or
the 3d Species of the 4th Order of Table 2. The Terms therefore being compared, 'tis d-=. I, e = — • i, and f=a. Therefore
% — — = x, </ax — xx = v, and 3^ — 2x1; = /. Wherefore
it is *AC = x, CD = v, and thence ACDH = s ; fo that 3ACDH — 4AADC = 3* — 2xv = t = Area of the Ciflbid ACEGA. Or, which is the fame thing, 3 Segments ADHA = Area ADEGA, or 4 Segments ADHA = Area AHDEGA.
93. EXAMPLE IV. Let PE be the firft Conchoid of the Ancients, defcribed from Center G, with the Afymptote AL,. and diftance LE. Draw its Axis GAP, and let fall the Or- dinate EC. Then calling AC =: z, CE =.y, GA = a, and Ap . — • c ; becaufe of the Pro- portionals A C : CE — AL : : GC : CE, it will be CE or y
04. * Now that its Area PEC may be found from hence, the paits'of the Ordinate CE are to be confider'd feparately. And if
the Ordinate CE is fo divided in D, that it is CD = v/^— «»,
and
and INFINITE SERIES.
and DE = *\/V — ^ ; CD will be the Ordinate of a Circle de-
fcribcd from Center A, and with the Radius AP. Therefore the part of the Area PDC is known, and there will remain the other part DPED to be found. Therefore fince DE, the part of the Or- dinate by which it is defcribed, is equivalent to -\/e* — z* ; fup-
pofe 2 = w, and it becomes -^/e* — z* = DE, an Equation of the ift Species of the 3d Order of Table 2. The terms therefore being compared, itisd=t>, f = ct, and/= — i; and therefore
1 — . j — = x, \/ — i -+- c* x1 = v, and zbcls -- • - = t.
1
Z Z.
95. Thefe things being found, reduce them to a juft number of dimenfions, by multiplying the terms that are too deprefs'd, and dividing thofe that are too high, by fome given Quantity. If this be done by c, there will arife ~ = x, </ — c * -t- x% = v, and
— -- — = t : The Conflruclion of which is in this manner.
c ex
96. With the Center A, principal Vertex P, and Parameter aAP, defcnbe the Hyperbola PK. Then from the point C draw the right Line CK, that may touch the Parabola in K : And it will be, as AP to 2AG, fo is the Area CKPC to the Area required DPED.
97. EXAMPLE 5. Let the Norma GFE fo revolve about the Pole G, as that its angular point F may continually flide upon the right Line AF given in pofition ; then conceive the Curve PE to be de- fcribed by any Point E in the
other Leg EF. Now that the Area of this Curve may be found, let fall GA and EH per- pendicular to the right Line AF, and compleating the Pa- rallelogram AHEC, call AC = 2, CE=j, AG = £, and EF=£; and becaufe of the Proportionals HF : EH : : AG : AF, we mall have AF =
, bz . Therefore CE or y
V a — zz b
But whereas </cc — zz is the Ordinate
of a Circle defcribed with the Semidiameter c ; about the Center A
let
*fhe Method of FLUXIONS, let fuch a Circle PDQ_be defcribed, which CE produced meets ia
D ; then it will be DE = ^=rS : B? the helP of which EqUa~ tion there remains the Area PDEP or DERQ^to be determin'd.
Suppofe therefore »=:2, and G=^3 and it will be DE=— • — i^~ >
V ft — ^
sn Equation of the ift Species of the 4th Order of Table i. And the Terms being compared, it will be b-= d, cc =e, and — j ==/;
fo that — bV cc — zz = — l>R=f.
98. Now as the value of t is negative, and therefore the Area reprefented by / lies beyond the Line DE ; that its initial Limit may be found, feek for that length of z, at which t becomes no- thing, and you will find it to be c. Therefore continue AC to Q^> that it may be AQ==c, and erect the Ordinate QR.; and DQRED will be the Area whofe value now found is — b\/cc — zz.
99. If you fhould define to know the quantity of the Area PDE, pofited at the Abfcifs AC, and co-extended with it, without knowing the Limit QR, you may thus determine it.
100. From the Value which / obtains at the length of the Ab- fcifs AC, fubtract its value at the beginning of the Abfcifs ; that is, from — b\/ cc zz fubtract — &•, and there will arife the defired quantity A: — b\/ LC — zz. Therefore compleat the Parallelogram PAGK, and let fall DM perpendicular to AP, which meets GK in M ; and the Parallelogram PKML will be equal to the Area PDE.
101. Whenever the Equation defining the nature of the Curve cannot be found in the Tables, nor can be reduced to limpler terms by divifion, nor by any other means ; it muft be transform'd into other Equations of Curves related to it, in the manner fhewn in Prob. 8. till at laft one is produced, whofe Area may be known by the Tables. And when all endeavours are ufed, and yet no fuch can be found, it may be certainly concluded, that the Curve pro- pofed cannot be compared, either with rectilinear Figures, or with the Conic Sedions.
102. In the fame manner when mechanical Curves are concern'd, they muft fir ft be transform'd into equal Geometrical Figures, as is fhewn in the fame Prob. 8. and then the Areas of fuch Geometri- cal Curves are to be found from the Tables. Of this matter take the following Example.
103.
and IN FINITE SERIES. 115
103. EXAMPLE 6. Let it be propofed to determine the Area of the Figure of the Arches of any Conic Section, when they aie made Ordinates on their Right Sines. As let A be the Center of the Conic Section,
AQ_and AR the — ^ " V .' ^\
Semiaxes, CD the Ordinate to the Axis AR, and PD a Per- pendicular at the point D. Alfo let AE be the fa id mechanical Curve meeting CD in E; and from its nature before defined, CE will be equal to the Arch QD. There- fore the Area A EC is fought, or com- pleating the parallelogram ACEF, the excefs AEF is required. To which purpole let a be the Latus rectum of the Conic Section, and b its Latus tranfverfum, or 2AQ^_ Alfo let AC=z, and CD=_>';
then it will be V ^bb -f- -zz =y, an Equation to a Conic Section,
as is known. Alfo PC= -z, and thence PD = v/^H ~- zz.
104. Now fince the fluxion of the Arch QD is to the fluxion of the Abfcifs AC, as PD to CD ; if the fluxion of the Abfcifs be fup- pos'd i, the Fluxion of the Arch QD, or of the Ordinate CE,
**+"-*~~ will be i/4 — . Draw this into FE, or z, and there
for the fluxion of the Area AEF.
will arife z »/
If therefore in the Ordinate CD you take CG — -
-zz
V
-zz
, the Area AGC, which is defcribed by CG
moving upon AC, will be equal to the Area AEF, and the Curve
AG
<•
i;
n6 77je Method of FLUXIONS,
AG will be a Geometrical Curve. Therefore the Area AGC is fought. To this purpofe let z* be fubflituted for z* in the laft
Equation, and it becomes &*-* \/^-j-, j-^ = CG, an Equa.-
M
tion of the ad Species of the i ith Order of Table 2. And from a comparifon of terms it is d = i, e-=.i-bb =£,/= - ~ , and
$=— : fo that \/ ^bb ~] — zz=x. \/ — — — -f- • xx —r, i>. and
a * •* a ' Afl a /
~s = t. That is, CD = x, DP = v, and Jj = /. And this is the Conftruction of what is now found.
105. At Q^ erect QK perpendicular and equal to QA, and thro* the point D draw HI parallel to it, but equal to DP. And the Line KI, at which HI is terminated, will be a Conic Section, and the comprehended Area HIKQ^will be to the Area fought AEF, as b to a, or as PC to AC.
106. Here obferve, that if you change the fign of b, the Conic Section, to whofe Arch the right Line CE is equal, will become an Ellipfis; and befides, if you make b = — «, the Ellipfis becomes- a Circle. And in this cafe the line KI becomes a right line parallel
107. After the Area of any Curve has been thus found and con- ftrucled, we fhould confider about the demonftration of the con- ftruction ; that laying afide all Algebraical calculation, as much as may be, the Theorem may be adorn'd, and made elegant,, fo as to become fit for publick view. And there is a general method of de-- monftrating, which I mail endeavour to iiluftrate by the follow- ing Examples.
Demonftration of the Conjlruflion in Example 5.
1 08. In the Arch PQ^take a point d indefinitely near to D, (Figure p. 113.) and draw de and dm parallel to DE and DM, meeting DM and AP in p and /. Then will DE^/ be the mo- ment of the Area PDEP, and LM/»/ will be the moment of the Area LMKP. Draw the femidiameter AD, and conceive the inde- finitely fmall arch ~Dd to be as it were a right line, and the tri- angles -D/^/ and ALD will be like, and therefore D/> : pd:: AL : LD. But it is HF : EH :: AG : AF ; that is, AL : LD :: ML : DE; and therefore Dj> : pd : : ML : DE. Wherefore Dp x DE = pd x ML
That
and IN FINITE SERIES,
117
That is, the moment DEed is equal to the moment LM;;//. And fince this is demonflrated indeterminately of any contemporaneous moments whatever, it is plain, that all the moments of the Area PDEP are equal to all the contemporaneous moments of the Area PLMK, and therefore the whole Areas compofed of thofe moments are equal to each other. C^JE. D.
Demonftration of the ConftruSfion in Example 3.
109. Let DEed be the momentum of the fuperficies AHDE, and A</DA be the contemporary moment of the Segment ADH. Draw the femidiameter DK, and let de meet AK in c -, and it is Cc : Dd :: CD : DK. Befides it is DC : QA (aDK) : : AC : DE. And therefore Cc : 2Dd :: DC : aDK :: AC : DE, and Cc x DE = zDd-x. AC. Now to the mo- ment of the periphery Dd produced, that is, to the tan- gent of the Circle, let fall the perpendicular AI, and AI will be equal to AC. So that zDd x AC = zDd x AI = 4
Triangles AD</. So that 4 Triangles AD^/=C^xDE= moment DE^/. Therefore every moment of the fpace AHDE is quadruple of the contemporary moment of the Segment ADH, and therefore that whole fpace is quadruple of the whole Segment. Q^E. D.
Bemvnftratwn
iiS
"The Method of FLUXIONS,
Demonftration of the ConftruRion in Example 4.
no. Draw ce parallel to CE, and at an indefinitely fmall diflance from it, and the tangent of the Hyperbola ckt and let fall KM perpendicular to AP. Now from the nature of the Hyper- bola it will be AC : A? :: AP : AM, and therefore AC? : GLq :: AC?: LE? (or APV') :: AP? : AM? ; and divlfim* AG/ : AL? (DE?) ::.AP?: AM? — AP?(MK?) ; And invent, AG: AP :: DE : MK. But the little Area DEed is to the Tri- angle CKr, as the altitude DE is to half the altitude KM ; that is, as AG to -LAP. Wherefore all the moments of the Space PDE are to all the contemporaneous moments of the Space PKC, as AG to 4-AP. And therefore thofe whole Spaces are in the fame ratio.
Demonjlration of the Conjlruftion in Example 6.
in. Draw c*/ parallel and infinitely near to CD, (Fig. in p. 115-) meeting the Curve AE in e, and draw hi and fe meeting DCJ in p and q. Then by the Hypothefis ~Dd= Eg, and from the fimi- litude of the Triangles Ddp and DCP, it will be D/> : (Dd) Eq :: ( P : (PD) HI, fo that Dp x HI = Eg xCPj and thence Dp x HI (the moment HI/'/.)): Eg x AC (the moment EF/e) :: E?xCP : EyxAC :: CP : AC. Wherefore fince PC and AC are in the given ratio of the latus tranlverfum to the Jatus rectum of the Conic Section QD, and fince the moments HI//) and EFfe of the Areas HIKQ^and AEF are in that ratio, the Areas them- felves will be in the fame ratio. Q-^E. D.
112. In this kind of demonilrations it is to be obferved, that I affume fuch quantities for equal, whofe ratio is that of equality : And that is to be efteem'd a ratio of equality, which differs lefs from equality than by any unequal ratio that can be affign'd. Thus in the laft demon ftration I fuppos'd the rectangle E^xAC, or FE?/, to be equal to the fpace FEt/j becaufe (by realon of the difference Eqe infinitely lefs than them, or nothing in comparifon of them,)
they
and INFINITE SERIES.
119
they have not a ratio of inequality. And for the fame reafon I made DP x HI = HI//6 ; and fo in others.
1 13. I have here made ufe of this method of proving the Areas of Curves to be equal, or to have a given ratio, by the equality, or by the given ratio, of their moments ; becaufe it has an affinity to the ufual methods in thefe matters. But that feems more natural which depends upon the generation of Superficies, by Motion or Fluxion. Thus if the Confbuclion in Example 2. was to be de- monftrated : From the nature of the Circle, the fluxion of the right line ID (Fig. p.i 1 1.) is to the fluxion of the right line IP, as AI to ID ; and it is AI : ID : : ID : CE, from the nature of the Curve
AGE ; and therefore CE x ID = ID x IP. But CE x ID = to the fluxion of the Area PDI. And therefore thofe Areas, being ge- nerated by equal fluxion, muft be equal. Q^E. D.
1 14. For the fake of farther illustration, I fliall add the demon- flration of the Confrruc~r.ion, by which the Area of the Ciffoid is determin'd, in Example 3. Let the lines mark'd with points in the fcheme be expunged; draw the Chord DQ^ and the Afymptote QR of the Ciffoid. Then, from the nature of the Circle, it Is DQj- = AQ_x CQ^, and thence (by Prob. i.)
Fluxion of DQj= AQjcCQ.
And therefore AQ_:
2DQj CX^ Alfo from the nature of the Ciffoid it is ED : AD :: AQ^: DQ^ There-
fore ED : AD : :
and EDxCC^=ADx2DQ^,
or 4xiADxDQ^ Nowfmce DQ __ is perpendicular at the end of AD, revolving about
A ; and i AD x QD = to the fluxion generating the Area
its quadruple alfo ED x CQ^== fluxion generating the Ciffoidal Area QREDO. Wherefore that Area QREDO infinitely long, is gene- rated quadruple of the other ADOQ^ Q^E. D.
SCHOLIUM.
120 The Method of FLUXIONS,
SCHOLIUM.
115. By the foregoing Tables not only the Areas of Curves, but quantities of any other kind, that are generated by an analogous way of flowing, may be derived from their Fluxions, and that by the affiftance of this Theorem : That a quantity of any kind is to an unit of the lame kind, as the Area of a Curve is to a fuperficial unity ; if fo be that the fluxion generating that quantity be to an unit of its kind, as the fluxion generating the Area is to an unit of its kind alfo ; that is, as the right Line moving perpendicularly upon the Abfcifs (or the Ordinate) by which the Area is defcribed, to a linear Unit. Wherefore if any fluxion whatever is expounded by fuch a moving Ordinate, the quantity generated by that fluxion will be expounded by the Area defcribed by fuch Ordinate ; or if the Fluxion be expounded by the fame Algebraic terms as the Ordinate, the generated quantity will be expounded by the fame as the de- fcribed Area. Therefore the Equation, which exhibits a Fluxion of any kind, is to be fought for in the firft Column of the Tables, and the value of t in the laft Column will mow the generated Quan- tity. _
1 1 6. As if \/ 1 -h — exhibited a Fluxion of any kind, make it equal to y, and that it may be reduced to the form of the Equations in the Tables, fubftitute z* for z, and it will be z~ ' </ 1 -+- — z«
43
7—y, an Equation of the firft Species of the 3d Order of Table i. And comparing the terms, it will be </= i, e=i,f=2.>
8a + i8z ,~ gz -id -p. _, _
and thence — - — \S i •+- -a== — R> =/. Therefore it is the quantity Z^~ 1/1 -4- which is generated by the Fluxion
4"
3 17, And thus if v'l -f- J^l- reprefents a Fluxion, by a due re-
9«7
duftion, (or by extracting & out of the radical, and writing «_»» for 2~^) there will be had -or, */s&-±-—! =7, an Equation of
z ga*
the ad Species of the 5th Order of Table 2. Then comparing the
terms,
and INFINITE SERIES. 121
terms, it is d=. i, e = —, and/= i. So that x7 = - = *•*•,
'
_j_ '— ^ = -u, and 4 J = - * = A Which being found, the
«7
quantity generated by the fluxion v/ j + L^Z will be known, by
making it to be to an Unit of its own kind, as the Area j* is to fuperficial unity ; or which comes to the fame, by fuppofing the quantity t no longer to reprefent a Superficies, but a quantity of an- other kind, which is to an unit of its own kind, as that fuperficies k to fuperficial unity. _
1 1 8. Thus fuppofing \/i 4- l~ to reprefent a linear Fluxion, I
9«T
imagine t no longer to fignify a Superficies, but a Line ; that Line, for inftance, which is to a linear unit, as the Area: which (accord- ing to the Tables) is reprefented by t, is to a fuperficial unit, or that which is produced by applying that Area to a linear unit. On which account, if that linear unit be made e, the length generated by the foregoing fluxion will be ~ . And upon this foundation
thofe Tables may be apply'd to the determining the Lengths of Curve-lines, the Contents of their Solids, and any other quantities whatever, as well as the Areas of Curves.
Of ^uejlions that are related hereto.
I. To approximate to the Areas of Curves mechanically,
119. The method is this, that the values of two or more right- lined Figures may be fo compounded together, that they may very nearly conftitute the value of the Curvilinear Area required.
120. Thus for the Circle AFD which is denoted by the Equa- tion .v — xx =rzz} having found the value of
the Area AFDB, viz. £** — £#* — /,** — J-x*, &c. the values of fome Rectangles are to be fought, fuch is the value x\/x — xx, or x* — ±z* — T#* — TV#% &c- of the rectangle BD x AB, and x^/x, or #', the value of AD x AB. Then thefe values are to be multiply'd by any different letters, that ftand for numbers indefinitely, and then
R to
122 2^2 Method of FLUXIONS,
to be added together, and the terms of the fum are to be compared with the correfponding terms of the value of the Area AFDB, that as far as is poffible they may become equal. As if thofe Parallelo- grams were multiply'd by e and f, the fum would be ex* — \ex^
— {•$$, &c. the terms of which being compared with thefe terms ^x* — ,^x* — TV*% &c. there arifes £+/=-!, and— i^= — 4., or e = £, and /= % — e = T*r • So that ^-BD x AB -f- T4TAD x AB = Area AFDB very nearly. For ^-BD x AB -f. T*TAD x AB is equivalent to .!#* — 4.** — _^.v* -— _L.,v*, &c. which being fub- tracted from the Area AFDB, leaves the error only T'-#» -j- TV#*, &c.
121. Thus if AB were bifected in E, the value of the rectangle AB x DE will be x\/x — %xx, or x* — -^x* — •
-2-#* -- —x*, &c. And this compared with
128 1024 r
the rectangle AD x AB, gives 8DE + zAD into AB = Area AFDB, the error being only
J-x* -\ -- —x* &c. which is always lefs than 560 5760
•TJ^JTJ. part of the whole Area, even tho' AFDB were a quadrant of a Circle. But this Theorem may be thus pro- pounded. As 3 to 2, fo is the rectangle AB into DE, added to a fifth part of the difference between AD and DE, to the Area AFDB, very nearly.
122. And thus by compounding two rectangles ABxED and AB x BD, or all the three rectangles together, or by taking in ftill more rectangles, other Rules may be invented, which will be fo much the more exacT:, as there are more Rectangles made ufe of. And the fame is to be understood of the Area of the Hyperbola, or of any other Curves. Nay, by one only rectangle the Area may often be very commodioufly exhibited, as in the foregoing Circle, by taking BE to AB as v/io to 5, the rectangle AB x ED will be to the Area AFDB, as 3 to 2, the error being only TfTAT* -fr-
II. The Area being g hen, to determine the Abfcifs and Ordinate. 123. When the Area is exprefs'd by a finite Equation, there can be no difficulty : But when it is exprefs'd by an infinite Series, the affected root is to be extracted, which denotes the Abfcifs. So for
the
W^^ ** • w
and INFINITE SERIES. 123
the Hyperbola, defined by the Equation —^ = z, after we have found * = bx — -^ -+- -£ — — * , &c. that from the given Area the Abfcifs x may be known, extract the affedled Root, and there
will arife x = + ^ + £- 4- -JjjL , &c. And
moreover, if the Ordinate .5 were required, divide ab by /z 4- AT, that is, by a -f- } -+• -^ -f- ~s , &c. and there will arife z=l>—>
124. Thus as to the Ellipfis which is exprefs'd by the Equation ax — -xx = zz, after the Area is found z = ^a?x* — a%x* —
1 i I £ ,x ,
^!^ — Hf_, &c. write i;' for — , and / for x*, and it becomes = t* — ^ — — — -i-j, &c. and extracting the root /=
&c. is equal to x. And this value being fubflituted inftead of x in the Equation ax — a-xx = zz, and the root being extracted, there arifes * = **«—. ^L3 — 38«*«' __ 4Q7^7 5cc> So that from
5<: '7Sf* 225018
z, the given Area, and thence v or ./"I, the Abfcifs # will be
f za*
given, and the Ordinate z. All which things may be accommo- dated to the Hyperbola, if only the flgn of the quantity c be changed, wherever it is found of odd dimenfions.
R O B.
124-
*The Method of FLUXIONS,
P R O B. X.
1o find as many Curves as we pleafe, vohofe Lengths may be exprcfsd by finite Equations.
1. The following pofitions prepare the way for the foltirion of this Problem.
2. I. If the right Line DC, ftanding perpendicularly upon any. Curve AD, be conceived thus to move,
all its points G, g, r, &c. will defcribe other Curves, which are equidiftant, and perpendicular to that line : As GK, gk, rs, &c.
3. II. If that right Line is continued indefinitely each way, its extremities will move contrary ways, and therefore there will be a Point between, which will have no motion, but may therefore be call'd the Center of Motion. This Point will be the fame as the Center of Curvature, which the Curve AD hath at the point D, as is mention'd before. Let that point beC.
4. III. If we fuppofe the line AD not to be circular, but unequably curved, fup- pofe more curved towards <T, and lefs toward A; that Center will continually change its place, approaching nearer to the parts more curved, as in K, and going farther off at the parts lefs curved, as in. kt and by that means will defcribe fome line, as KG£.
5. IV. The right Line DC will continually touch the line de- fcribed by the Center of Curvature. For if the Point D of this line moves towards ^, its point G, which in the mean time pafTes to K, and is fituate on the fame fide of the Center C, will move the fame way, by pofition 2. Again, if the fame point D moves towards A, the point g, which in the mean time paffes to k, and k fituate on the contrary fide of the Center C, will move the con- trary way, that is, the fame way that G moved in the former cafe, while it pafs'd to K. Wherefore K and k lie on the fame fide of the right Line DC. But as K and k are taken indefinitely f :>r any
points,
and INFINITE SERIES. 125
points, it is plain that the whole Curve lies on the fame fide of the right line DC, and therefore is not cut, but only touch'd by it.
6. Here it is fuppos'd, that the line <rDA is continually more curved towards <T, and lefs towards A ; for if its greateft or leaft Curvature is in D, then the right line DC will cut the Curve KC ; but yet in an angle that is lefs than any right-lined angle, which is the fame thing as if it were faid to touch it. Nay, the point C in this cafe is the Limit, or Cufpid, at which the two parts of the Curve, finishing in the moft oblique concourfe, touch each other ; and therefore may more juftly be faid to be touch'd, than to be cut, by the right line DC, which divides the Angle of contact.
7. V. The right Line CG is equal to the Curve CK. For con- ceive all the points r, 2r, 3;-, ^.r, &c. of that right Line to defcribe the arches of Curves rs, 2r2s, 3^3;, &c. in the mean time that they approach to the Curve CK, by the motion of that right line ; and fmce thofe arches, (by polition i.) are perpendicular to the right lines that touch the Curve CK, (by pofition 4.) it follows that they will be alfo perpendicular to that Curve. Wherefore the parts of the line CK, intercepted between thofe arches, which by reafon of their infinite fmallnefs may be confider'd as right lines, are equal to the intervals of the fame arches ; that is, (by polition i.) are equal to fo many parts of the right line CG. And equals being added to equals, the whole Line CK will be equal to the whole Line CG.
8. The fame thing would appear by conceiving, that every part of the right Line CG, as it moves along, will apply itfelf fuccef- fively to every part of the Curve CK, and thereby will meafure them ; juft as the Circumference of a wheel, as it moves forward by revolving upon a Plain, will meafure the diflance that the point of ContacT; continually defcribes.
9. And hence it appears, that the Problem may be refolved, by afiuming any Curve at pleaflue A/'DA, and thence by determining the other Curve KC£, in which the Center of Curvature of the aftumed Curve is always found. Therefore letting fall the perpen- diculars DB and CL, to a right Line AB given in pofition, and in AB taking any point A, and calling AB = .v and BD = v ; to define the Curve AD let any relation be affumed between x and v, and then by Prob 5. the point C may be found, by which may be determined both the Curve KC, and its Length GC.
10.
Method of FLUXIONS,
126
10. EXAMPLE. Let ax =yy be the Equation to the Curve, which therefore will be the Apollonian Parabola. And, by Prob. 5. will be found AL=|« ^ , and DC = 2±if
* a
-+. ax.
Which being obtain'd, the Curve KC is determin'd by AL and LC, and its Length by DC. For as we are at liberty to aflume the points K and C anf where in the Curve KC, let us fuppofe K to be the Center of Cur- vature of the Parabola at its Vertex ; and putting therefore AB and BD, or x and y, to be nothing, it will be DC = -irf. And this is the Length AK, or DG, which being fubtracted from the former indefinite value of
DC, leaves GC or KC = -^- V ±aa +.ax — \a.
11. Now if you defire to know what Curve this is, and what is its Length, without any relation to the Parabola ; call KL = zt and LC = v, and it will be &•==. AL — \a = 3 x, or ^z = AT, and - = ax =yy. Therefore 4v/- = S! = CL = v, or — ' ==
2 •''' 27 £t aa 2 7 #
•u* j which fhews the Curve KC to be a Parabola of the fecond kind. And for its Length there arifes ll±il ^/^aa -f- ±az — ±a, by
writing ~z for >r in the value of CG.
12. The Problem alfo may be refolved by taking an Equation, which fhall exprefs the relation be- tween AP and PD, fuppofing P to
be the interfeclion of the Abfcifs and Perpendicular. For calling AP=,v, and PD =/, conceive CPD to move an infinitely fmall fpace, fuppofe to the place Cpd} and in CD and Cd ta- king CA and CeT both of the fame given length, fuppofe = r, and to CL let fall the perpendiculars A^ and fyy of which Ag, (which call =z) may meet Cd inf. Then compleat the Parallelogram gyfe, and making x,y, and z the fluxions of the quantities ,v, y, and x, as before
it
and IN FINITE SERIES. 127
it will be Ae : A/ :t A?P '• All* " Q"P : CA]1 :: TT ' And A/: P/> :: CA : C P. Then «? a>quot Ae:Pp:: ^11 : CP. But P/> is the moment of the Abfcifs AP, by the acceiTion of which it becomes Ap ; and Ae is the contemporaneous moment of the per- pendicular Ag-, by the decreafe of which it becomes fy. There- fore Ae and Pp are as the fluxions of the lines Ag (z) and AP (x),
that is, as z and x. Wherefore 2, : x :: ~- : CP. And fmce it
is Cgl * = CAI a — AgT = i — &&, and CA = i ; it will be
CP_= * ~*z . Moreover fmce we may aflume any one of the
•
three x,y, and z for an uniform fluxion, to which the reft are to be referr'd, if x be that fluxion, and its value is unity, then CP =
13. Befides it is CA (i) : Ag (z} :: CP : PL; alfo CA (i) : Cg — zz) : : CP : CL ; therefore it is PL = 2Z± , and CL =
—~z
j — Zz. Laftly, drawing /^parallel to the infinitely fmall
X
Arch D</, or perpendicular to DC, P^- will be the momentum of DP, by the acceflion of which it becomes dp, at the fame time that AP becomes A/>. Therefore Pp and Pg are as the fluxions of AP (x) and PD (;'), that is, as i and y. Therefore becaufe of fimilar triangles Ppq and CAg, fmce CA and Ag, or i and z, are in the fame ratio, it will be y = «. Whence we have this folution of the Problem.
14. From the propofed Equation, which exprefles the relation between x and^x, find the relation of the fluxions x and y, (by Prob. i.) and putting x = i, there will be had the value of _)-, to which z is equal. Then fubftituting z for/, by the help of the lafl Equa- tion find the relation of the Fluxions x,y, and z, (by Prob. i.) and again fubftituting i for x, there will be had the value of z. Thefe
being found make ^21= CP, z x CP = PL, and CP x v/ 1 — yy
Z
= CL; and C will be a Point in the Curve, any part of which KG is equal to the right Line CG, which is the difference of the tangents, drawn perpendicularly to the Curve \)d from the points C and K,
I28 7%e Method of FLUXIONS,
15. Ex. Let ax=yy be the Equation which exprefles the rela- tion between AP and PD ; and (by Trob. i.) it will be firft ax= 2yy, or a = 2yz. Then zyz -f- zyz = o, or
— = z. Thence it is CP =
y
I —yy £l_J
— 4-vv
aa.
c
And from CP and PL taking away y and x. there remains CD = — — ,
aa
and AL = ?a — ~ . Now I take
away y and x, becaufe when CP and PL have affirmative values, they fall on the fide of the point P to- wards D and A, and they ought to be diminiihed, by taking away the affirmative quantities PD and AP. But when they have negative values, they will fall on the contrary fide of the point P, and then they muft be encreafed, which is alfo done by taking away the affir- mative quantities PD and AP.
1 6. Now to know the Length of the Curve, in which the point C is found, between any two of its points K and C ; we rauft ieek the length of the Tangent at the point K, and fubtradt it from CD. As if K were the point, at which the Tangent is terminated, when CA and Ag, or i and z, are made equal, which therefore is fituate in the Abicifs itfelf AP ; write i for z in the Equation a= 2yz, whence a=2y. Therefore for y write ^a in the value of CD,
that is in — — , and it comes out — ±a. And this is the length of the Tangent at the point K, or of DG ; the difference between which and the foregoing indefinite value of CD, is — -- -i#> that
is GC, to which the part of the Curve KC is equal.
17. Now that it may appear what Curve this is, from AL (hav- ing firft changed its fign, that it may become affirmative,) take AK,
which will be ^a, and there will remain KL = — — %a, which call /, and in the value of the line CL, which call v, write — for
aa-> anc^ l^ere
l a"fe — \/^at = v ; or — = vv, which is an Equation to a Parabola of the fecond kind, as was found before.
i a,
and INFINITE SERIES. 129
1 8. When the relation between t and v cannot conveniently be reduced to an Equation, it may be fufficient only to find the lengths PC and PL. As if for the relation between AP and PD the Equa- tion ^x-^-^y — _}'3=o were affumed; from hence (by Prob. i.) firft there arifes a1 4-^*2 — y*z = o, then aaz — zyyz — y*z=o,
and therefore it is z = , and z = — — . Whence are
yy — aa ' aa — yy
given PC = •••""'- , and PL = 2rxPC, by which the point C is
determined, which is in the Curve. And the length of the Curve, between two fuch points, will be known by the difference of the two correfponding Tangents, DC or PC — y.
19. For Example, if we make a= i, and in order to determine fome point C of the Curve, we take y = 2 ; then AP or x becomes
.y»— 3"'.v_ _ . -_.« - * PC 2 and PI -
Zaa T' z T> z T> 1V"- 2> ana rLl ?•
Then to determine another point, if we take ^' = 3, it will be AP=6, «=i, z = — >ir, PC=— 84, andPL=— ioi. Which being had, if y be taken from PC, there will remain — 4 in the firil cafe, and — 87 in the fecond, for the lengths DC j the difference of which 83 is the length of the Curve, between the two points found C and c.
20. Thefe are to be thus underftood, when the Curve is conti- nued between the two points C and c, or between K and C, with- out that Term or Limit, which we call'd its Cufpid. For when one or more fuch terms come between thofe points, (which terms are found by the determination of the greateft or leaft PC or DC,) the lengths of each of the parts of the Curve, between them and the points C or K, muft be feparately found, and then added together.
PROB. XI.
To find as many Curves as you pie of e, whofe Lengths may be compared with the Length of any Curve propofed, or with its Area applied to a given Liney by the help of finite Equations.
i. It is performed by involving the Length, or the Area of the •propofed Curve, in the Equation which is affumed in the foregoing Problem, to determine the relation between AP and PD (Figure
Art. 12. pjg. 126.) Eut that z, and z may be thence derived, (by
S Prob.
130 7%4 Method of FtuxioNS,
Prob. i.) the fluxion of the Length, or of the Area, muft be firft difcowr'd.
2. The fluxion of the Length is determin'd by putting it" equal to the fquare-root of the fum of the fquares of the fluxion of the Ab- fcifs and of the Ordinate. For let RN be the perpendicular Ordi- nate, moving upon the Abfcifs MN, and let QR be the propofed Curve, at which RN is terminated. Then calling MN = s, NR=/, and QR='i>, and their Fluxions s, /, and <u refpeclively ; con- ceive the Line NR to move into the place nr infinitely near the former, and letting _ ^ fall RJ perpendicular to nr, then RJ, sr, M" v N"
and Rr will be the contemporaneous moments of the lines MN, NR, and QR, by the accetfion of which they become M«, nr, and And as thefe are to each other as the fluxions of the fame
lines, and becaufe of the right Angle Rsr, it will be >/R/ -f-Tr* = Rr, or \/V -f- f- = <v.
3. But to determine the fluxions s and t there are two Equations- required; one of which is to define the relation between MN and NR,. or s and /, from whence the relation between the fluxions s and t- is to be derived ; and another which may define the relation be- tween MN or NR in the given Figure, and of AP or x in that re- quired, from whence the relation of the fluxion s or t to the fluxion x or i may be difcover'd.
4. Then <u being found, the fluxions y and z are to be fought by a third aflumed Equation, by which the length PD or y may be
defined. Then we are to take PC = '-^, PL =y x PC, and
DC = PC — y, as in the foregoing Problem.
5. Ex. i. Let as — ss=tt be an Equation to the given Curve QR, which will be a Circle; xx = as the relation between the lines AP and MN, and Lv=.y, the relation between the length of the Curve given QR, and the right Line PD. By the firft it will
be as — 2ss = 2tt, or a ~ 2's=i. And thence - =v s*-i-t*==:v.
zt zt
By the fecond it is 2X = as, and therefore -t •=. v. And by the
third £u=y, that is, ^ = z} and hence ^ — ^'=2;. Which
being
and INFINITE SERIES. 131
being found, you muft take PC = 1-^. , PL=/x PC, and DC
==PC — y, or PC — £QR- Where it appears, that the length of the given Curve QR cannot be found, but at the fame time 'the length of the right Line DC muft be known, and from thence the length of the Curve, in which the point C is found ; and fo on the contrary.
6. Ex.2. The Equation as — ss = ff remaining, make # = j,
and irv — ^ax-=.^ay. And by the firft there will be found — ^ = -y, as above. But by the fecond i = s, and therefore ^ = v. And by the third 2iw — 4^ = 407, or (eliminating -y) ^ — i = z.
Then from hence "— — 3L == z,
j. Ex. 3. Let there be fuppos'd three Equations, aa = st, a •+• *s = x, and A: -f- v =}'• Then by the firft, which denotes an
Hyperbola, it is o=rf+/i, or— 7 = ', and therefore '-V" 4- "
— V/M -f- tf = v. By the fecond it is 3* = i, and therefore - v/w -+- « = v. And by the third it is i + -u == yt or i +
3'
— </ss-4-tt=:z; then it is from hence w =s, that is, putting w 3'
for the Fluxion of the radical -^ </" -t- ^, which if it be made
equal to iv, or | -f- ~ = 7C'i£;, there will arife from thence ^ — ^ = 2W7i;. And firft fubftituting — ~ for /', then 1. for s, and
dividing by aw, there will arife P^3 = iv = z. Now _>' and z
being found, the reft is perform'd as in the fivft Example.
8. Now if from any point Q_of a Curve, a perpendicular QV is let fall on MN, and a Curve is to be found whofe length may be known from the length which arifes by applying the Area QRNV
to any given Line ; let that given Line be call'd E, the length — —
which is produced by fuch application be call'd <y, and its fluxion v. And fince the fluxion of the Area QRNV is to the Fluxion of the Area of a reiTtangular parallelogram made upon VN, with the height E, as the Ordinate or moving line NR = t, by which this is dc- fcribed, to the moving Line E, by which the other is deicribcd in
S 2 the
132 tte Method of FLUXION s,
the fame time ; and the fluxions v and } of the lines v and MN, (or s,) or of the lengths which arife by applying thofe Areas to the
given Line E, are in the fame ratio ; it will be v= s~ . Therefore
by this Rule the value of v is to be inquired, and the reft to be perform'd as in the Examples aforegoing.
9. Ex. 4. Let QR be an Hyperbola which is defined by this
Equation, aa -+• — = // ; and thence arifes (by Prob. I.) — =tf, or — = t. Then if for the other two Equations are aflumed x=s and y = v ; the firft will give i = j, whence v = ^ = £ } and the latter will give y = v, or z = -g, then from hence z= ^ , and fubftituting — or — for t, it becomes z = ~ . Now y and z
° ct ft hit
being found, make -r~ === CP, and_y x CP =n PL, as beforehand
thence the Point C will be determin'd, and the Curve in which all fuch points are fituated : The length of which Curve will be known from the length DC, which is equivalent to CP — v, as is fuffi- ciently fliewn before.
10. There is alfo another method, by which the Problem may be refolved ; and that is by finding Curves whofe fluxions are either equal to the fluxion of the propofed Curve, or are compounded of the fluxion of that, and of other Lines. And this may fometimes be of ufe, in converting mechanical Curves into equable Geometri- cal Curves ; of which thing there is a remarkable Example in fpiral lines.
1 1. Let AB be a right Line given in pofition, BD an Arch mov-< ing upon AB as an Abfcifs, and yet re- taining A as its Center, AD^ a Spiral, at
which that arch is continually terminated, bd an arch indefinitely near it, or the place into which the arch BD by its motion next arrives, DC a perpendicular to the arch bdt dG the difference of the arches, AH an- other Curve equal to the Spiral AD, BH a right Line moving perpendicularly upon
AB, and terminated at the Curve AH, bh the ^ ~B~<T
next place into which that right lane moves, andHK perpendicular to
bb.
and INFINITE SERIES. 133
bb. And in the infinitely little triangles DG/ and HK£, lince DC and HK are equal to the fame third Line Bb, and therefore equal to each other, and Dd and Hh (by hypothecs) are correfpondent parts of equal Curves, and therefore equal, as alfo the angles at G and K are right angles ; the third fides dC and hK will be equal alfb. Moreover fince it is AB : BD :: Ab : bC :: hb — AB (Qb) : bC •— BD (CG) j therefore - A*B - = CG. If this be taken away from dG, there will remain dG — • • *& • = dC = /6K. Call therefore AB=*, BD=-y, andBH=>', and their fluxions z, v, and y refpedtively, fince B£, dG, and /jK are the contempora- neous moments of the fame, by the acceflion pf which they become A£, bdt and bb, and therefore are to each other as the fluxions. Therefore for the moments in the lafl Equation let the fluxions be fubftituted, as alfo the letters for the Lines, and there will arife-y— .
^==-.y. Now of thefe fluxions, if z be fuppos'd equable, or the
~ " *'
unit to which the reft are refer'd, the Equation will be i;— ^=)'-
12. Wherefore the relation between AB and BD, (or between z and v,) being given by any Equation, by which the Spiral is defined, the fluxion v will be given, (by Prob. i.) and thence alfo the fluxion ;', by putting it equal to v. — ^ . And (by Prob. 2.) this will give the line y, or BH, of which it is the fluxion.
i?. Ex. i. If the Equation jzrzr-u were given, which is to the
Spiral of Archimedes, thence (by Prob. i.) -2-^ = v. From hence take - , or - , and there will remain - =y, and thence (by Prob. 2.) 2?_-r. Which fhews the Curve AH, to which the Spiral AD i
2U
equal, to be the Parabola of Apollonius, whofe Latus reclum is 2??; or whole Ordinate BH is always equal to half the Arch BD.
14. Ex. 2. If the Spiral be propofed which is defined by the
5 }_
Equation a3 =a'v1, or v =;^ , there arifes (by Prob. i.) — =-r,
«T 2^T
_l_ I
from which if you take ^, or ~- , there will remain — , = v, ano
flx 2iT
i thence (by Prob. 2.) will be produced ^l = v. That i.;; -BD nrr
3^
EU, AH being a Parabola of the fecond kind, t >
is
134 tte Method of FLUXIONS,
15. Ex. 3. If the Equation to the Spiral be z</"—^ =-y, thence (by Prob. i.) -a, . ?.~- = v ; from whence if you take away ""- or
' 2 V ac -\- cz K
^/- ?, there will remain , ~.. - = y. Now fince the quantity
generated by this fluxion y cannot be found by Prob. 2. unlefs it be refolved into an infinite Series; according to the tenor of the Scho- lium to Prob. 9. I reduce it to the form of the Equations in the firft column of the Tables, by fubftituting z* for z, ; then it becomes
=.y, which Equation belongs to the 26. Species of the 4th
Orderof Table i. And by comparing the terms, it is d=±,e=:ac, andf=c, fo that -~2- ^ ac -f- cz == f=y. Which Equation
belongs to a Geometrical Curve AH, which is equal in length to the Spiral AD.
PROB. XII.
To determine the Lengths of Curves.
1. In the foregoing Problem we have fhewn, that the Fluxion of a Curve-line is equal to the fquare-root of the fum of the fquares of the Fluxions of the Abfcifs and of the perpendicular Ordinate. Wherefore if we take the Fluxion of the Abfcifs for an uniform and determinate meafure, or for an Unit to which the other Fluxions are to be refer'd, and alfo if from the Equation which defines the Curve, we find the Fluxion of the Ordinate, we mall have the Fluxion of the Curve-line, from whence (by Problem 2.) its Length may be deduced.
2. Ex. i. Let the Curve FDH be propofed, which is defined by
the Equation -- -f- - '- =_y ; making the Abfcifs AB = s, and the
moving Ordinate DB =y. Then Jr
from the Equation will be had,
(by Prob. i.) 3— — — = y, the ^ J v-
\ s ' aa 12Z.S. -/'
fluxion of z being i, and y being
the fluxion of y. Then adding the X~
fquares of the fluxions, the fum
v/ill be — -h |-f- -^ == it, and extracting the root, —
and INFINITE SERIES. 135
= t, and thence (by Prob. 2.) ^ — ^ =— : t . Here / ftands for the
fluxion of the Curve, and / for its Length.
3. Therefore if the length </D of any portion of this Curve were required, from the points d and D let fall the perpendiculars db and DB to AB, and in the value of t fubftitute the quantities Ab and AB feverally for z, and the difference of the refults will be JD the Length required. As if Ab === ?a, and AB = a, writing La for #,
it becomes t = — — ; then writing a for #, it becomes / = — from whence if the firfl value be taken away, there will remain ^ for the length </D. Or if only h.b be determin'd to be ^a, and
AB be look'd upon as indefinite, there will remain — -— — _i_ -1
aa 1 2ft 24
for the value of
4. If you would know the portion of the Curve which is repre- fented by /, fuppofe the value of / to be equal to nothing, and there
arifes z* = — , or z= -£- .. Therefore if you take AB=-^- > 12 V*z y,2
and eredT: the perpendicular bdt the length of the Arch ^D will be
t or — — — • And the fame is to be underflood of all Curves
11%
aa
in general.
5. After the fame manner by which we have determin'd the
length of this Curve, if the Equation ^ -f- -^L =y be propofed, for defining the nature of another Curve ; there will be deduced
^ . _lL -=.t\ or if this Equation be propofed, — — La*y?—~*. «» 3"1 „* "*
2_
there will arife ^ -f-i^5'= t. Or in general, if it is cz* -{-
*• _ .- =_>', where 6 is u fed for reprefenting any number, either
,-—8"
Integer or Fraction, we (hall have cz* — — = /.
o 4&Qi — od<r
6. Ex.2. Let the Curve be propofed which is defined by this Equation ••"" + ^ \/ #a -t- £•£ =t^,V; then1 (by Prob. i.) will be had _y = ^^-r ^f*-* + 4* ^ or exterminating yt y= '-'</~aa-{- zz. To the fquare of which add i. and the fum will be i -J- ~ 4- 4-4 .
aa a*
and
136 ttt Method of FLUXIONS,
and its Root i -f- *— = t. Hence (by Prob. 2.) will be ob-
aa * * /
tain'd 2 + — ^
A«
7. Ex. 3. Let a Parabola of the fecond kind be propofed, whofe Equation is z* = ay1, or ~ =_y, and thence .by Prob. i. is derived
r==y. Therefore < 1 -+- 2f: = ~ i -+- yy s±s . Now fmce the
2aa 4<*
length of the Curve generated by the Fluxion / cannot be found by Prob. 2. without a reduction to an infinite Series of fimple Terms, I confult the Tables in Prob. 9. and according to the Scholium belong-
ing to it, I have / = ' v/ 1 -t- — . And thus you may find
the lengths of thefe Parabolas Z1 = ay*, 2? r= ay*, z> = ay*, &c.
8. Ex. 4. Let the Parabola be propofed, whofe Equation is «*
4 *
= rfy3, or ^=:^; and thence (by Prob. i.) will arife 1^ = _y.
"
Therefore v/ 1 -f- i^ = </yy -+- i = t. This being found, I
ga7
confult the Tables according to the aforefaid Scholium, and by com- paring with the 2d Theorem of the 5th Order of Table 2, I have
sF = x, v/i -f- 1—^ = v, and |j=?. Where x denotes the Ab-
9«7
fcifs, y the Ordinate, and s the Area of the Hyperbola, and / the length which arifes by applying the Area %s to linear unity.
9. After the fame manner the lengths of the Parabolas z6 =ay', z* :z=«y7, z'° =ay', &c. may alfo be reduced to the Area of the Hyperbola.
jo. Ex. 5. Let the CuToid of the Ancients be propofed, whole
Equation is ^T^jL" __;. and thence (by Prob. i.)
V az. — 2.Z. '
22,*
v/ az — zz=y, and therefore -^ ^/"—^ = ^ yy -f- i = t ;
which by writing 2? for ^ or z~\ becomes ^ v/ ' az" -f- 3 = /, an Equation of the ift Species of the 3d Order of Table 2 ; then comparing the Terms, it is ^ = d, 3 =•. e, and ^ =^5 fo that
i /: 20;' 4</c • i3
= 'u, and 6; — ___.l_into s=f.
AT My 2iA?
And
and INFINITE SERIES.
37
= v, and — — —
a ax
And taking a for Unity, by the Multiplication or Divifion Of which, thefe Quantities may be reduced to a juft number of Di-
menfions, it becomes az = xx, < — f : Which are thus conftructed.
1 1. The Ciflbid being VD, AV the Diameter of the Circle to which it is adapted, AF its Afymptotc, and DB perpendicular to AV, cutting the Curve in D ; with the Semiaxis AF = AV, and the Semipara- meter AG = jAV, let the Hyperbola YkK be defcribed ; and taking AC a mean Proportional between AB and AV, at C and V let CA and VK drawn perpendi- cular to AV, <:ut the Hyperbola in £, and K, and let right Lines kt and KT touch it in thofe points, and cut AV in /and T; and at AV let the Rectangle AVNM be defcribed, equal to the Space TK&. Then the length of the Ciflbid VD will be fextuple of the Altitude VN.
d
12. Ex. 6. Suppofing Ad to be an Ellipfis, which the Equation i/az — 2zz =y reprefents ; let the mechani- cal Curve AD be propofed of fuch a nature, that v'' if B</, or_)', be produced till it meets this Curve at D, let BD be equal to the Elliptical Arch &d. Now that the length of this may be deter- min'd, the Equation \/ az — 2.zz=. y will give
=y, to the fquare of which if i be added, there ariies
— , the fquare of the fluxion of the arch A.J. To which
zy az • aa — 4
02— Szz
if i be added again, there will arife -^ ^ ^- , whofe fquare-root — =.* __ is the fluxion of the Curve-line AD. Where if z be ex-
2y/az — 2ZZ
tracted out of the radical, and for z ~ be written c", there will be " -- , a Fluxion of the ift Species of the 4th Order of
'
=rt; fo that z= — = x, \/ ux — _.v.v = <
Table 2. Therefore the terms being collated, there will arife d=.^a, e = —2,
and -1 + ,= into,
J38
Method of FLUXION s.
i-i. The Conftruaion of which is thus; that the right line </G being drawn to the center of the Ellipfis, a parallelogram may be made upon AC, equal to the fedlor AC/, and the double of its height will be the length of the Curve AD.
14. Ex. 7. Making A/3= tp, (Fig, i.) and CL£ being an Hyper- bola, whofe Equation is v/— a -+• % = $&, and its tangent <TT being drawn ; let the Curve WD be propofed, whofe
Abfcifs is — , and its per- pendicular Ordinate is the length BD, which arifes by applying the Area a^To. to linear unity. Now that the length of this Curve VD may be determin'd, I feek the fluxion of the Areaa<rTa, when AB flows uniformly,
and I find it to be -^
v/ ' b — ax, putting AB =«, and its fluxion unity. For
'tis AT = £ = £ </z, and its fluxion is -rV , whofe half drawn
t>p o za v z
into the altitude /3<^, or v/— a •+• - , is the fluxion of the Area
, defcribed by the Tangent <TT. Therefore that fluxion is -p v/ ' b — az, and this apply'd to unity becomes the fluxion of the Ordinate BD. To the fquare of this ~^~ add i, the fquare of
the fluxion BD, and there arifes ^~fl^+al6^ta , whofe root -^
</a*b — a>z-\- ibfrz*- is the fluxion of the Curve VD. But this is a fluxion of the ift Species of the 7th Order of Table 2 : and
the terms being collated, there will be - = </, aab=e, — a*=f,
=g, and therefore z = x, and \/alb — a*x -f- (an" Equation to one Conic Section, fuppofe HG, (Fig. 2.) whofe Area EFGH is j, where EF = #, and FG = v ;) alfo *- ==%,
and */i6bb— - a*% + a&t-i = Y) (an Equation to another Conic
Section,
and INFINITE SERIES. 139
Section, (hppofe ML (Fig. 3.) whofe Area IKLM is <r, where IK
i T/"T *w* \ T /XT 2aftbb^f — fl5^Y*—tf4y — Aaabb? — T.2abbs
• — g and Kl_/= TiJ L,aitiy — " 2 — /\
15. Wherefore that the length of any portion DJ of the Curve VD may be known, let fall db perpendicular to AB, and make Kb = z ; and thence, by what is now found, feek the value of t. Then make AB=,s, and thence alfo feek for /. And the diffe- rence of thefe two values of / will be the length Dd required.
16. Ex. 8. Let the Hyperbola be propos'd, whofe Equation is
=)', and thence, (by Prob. i.) will be had^ = - ( Or To the fquare of this add i, and the root of the fum = /. Now as this fluxion is not to be found
aa -\- bz.z. + bits.
\/aa 4- tzz
will be ^/
in the Tables, I 'reduce it to an infinite Series ; and firft by divifion
•" y i / 3 y 4 / 1
it becomes t ;= </ 1 -f-jaS1 — ^2:4H-r«2'5 — 7*z* > &c- a«d extracting
the root, t ==
— a-
c A , z&, &c. And
•*
hence (by Prob. 2.) may be had the length of the Hyperbolical Arch
17. If the Ellipfis \/aa — bz,z=.y were propofed, the Sign of b ought to be every where changed, and there will be had z 4-
—& _f- - — ^— *-z' -^ 1— i— ^t_s7, &c. for the length of its
Arch. And likewife putting Unity for b, it will be z -+- -^ -f- 3ii_4_ Jil , &c. for the length of the Circular Arch. Now the
10«4 I I 2V.'' > O
numeral coefficients of this feries may be found adinfinitumt by mul- tiplying continually the terms of this Progreflion j— , —— , •^- >
S x 9 ' 10 x i i '
18. Ex. 9. Laftly, let the Quadratrix VDE be propofed, whole Vertex is V, A being the Center, and AV the femidiameter of the interior Circle, to which it is adapted, and the Angle VAE being a right Angle. Now any right Line AKD being drawn through A, cutting the Circle in K, and the Quadratrix in D, and the perpendiculars KG, DB being let fall to AE } call AV =.a, AG = c;, VK = x, and BD = y, and it
T 2 will
The Method of FLUXIONS.
will be as in the foregoing Example, x =.z 4- -r~ 4- j—; 4- - &c. Extract the root js, and there will arife z= x — ^ 4-
, — *7 • , &c. whofe Square fubtract from AKq. or als and the
s°4°" a 4
root of the remainder # — — 4- -^ —, ; , &c. will be GK.
2^j 9Aa9 *7?r\/j9 *
Now whereas by the nature of the Quadratrix 'tis AB = VR = x, and fince it is AG : GK :: AB : BD (y), divide AB x GK by AG,
and there will arife y = a — ^ — —^ — -^--, , &c. And thence, (by Prob. i.) y = - ^ — ^.— ^ , &c. to the fquare of which add i , and the root of the fum will be i 4- ^ -f- —
'-J il il
_6o4^« &c^ __ • \vhence (by Prob. 2.) / may be obtain'd,
1Z/S~SU
or the Arch of the Quadratrix ; viz. YD = x 4- ^j -f
6°4'v7 &c.
895025
THE
THE
METHOD of FLUXIONS
AND
INFINITE SERIES;
O R,
A PERPETUAL COMMENT upon the foregoing TREATISE,
u
.
;
THE
METHOD of FLUXIONS
AND
INFINITE SERIES.
ANNOTATIONS on the Introduction :
OR,
The Refolution of Equations by INFINITE SERIES.
S E c T. I. Of the Nature and ConftruElion of Infinite
or Converging Series.
great Author of the foregoing Work begins it with a fhort Preface, in which he lays down his main defign very concifely. He is not to be here underftood, as if he would reproach the mo- dern Geometricians with deferting the Ancients, or with abandoning their Synthetical Method of Demonftration, much lefs that he intended to difparage the Analy- tical Art ; for on the contrary he has very nauch improved both Methods, and particularly in this Treatife he wholly applies himfelf to cultivate Analyticks, in which he has fucceeded to univerial ap- plaufe and admiration. Not but that we mail find here fome ex- amples of the Synthetical Method likewife, which are very mafterly and elegant. Almoft all that remains of the ancient Geometry is indeed Synthetical, and proceeds by way of demonftrating truths already known, by mewing their dependence upon the Axioms, and
other
144 :-tbe Method of FLUXIONS,
other fir ft Principles, either mediately or immediately. But the hiiinefs of Analyticks is to invcftiga'te fuch Mathematical Truths as really are, or may be fuppos'd at leaft to be unknown. It afiumes thofe Truths as granted, and argues from them in a general man- ner, till after a .fcries of argumentation, in which the -feveral fteps have a. neceftary. connexion wjth each other, it arrives at the know- ledge of the propofition required, by comparing it with fomething really known or given. This therefore being the Art of Invention, it certainly deferves to be cultivated with the utmoft induftry. Many of our modern Geometricians have been perfuaded, by confidering the intricate and labour'd Demonftrations of the Ancients, that they .were Mailers of an Analyfis purely Geometrical, which they ftudi- ouily conceal'd, and by the help of which they deduced, in a direct and fcientifical manner, thofe abftrufe Proportions we fo much ad- mire in tome of their writings, and which they afterwards demon- ftrated Synthetically. But however this may be, the lofs of that Analyfis, if any fuch there were, is amply compenfated, I think, by our prefent Arithmetical or Algebraical Analyfis, especially as it is now improved, I might fay perfected, by our fagacious Author in the Method before us. It is not only render 'd vaftly more univerfal, and exterriive than that other in all probability could ever be, but is likewife a moft compendious Analyiis for the more abftrufe Geome- trical Speculations, and for deriving Conftructions and Synthetical Demonftrations from thence ; as may abundantly appear from the enfuing Treatife.
2. The conformity or correfpondence, which our Author takes notice of here, between his new-invented Doctrine of infinite Series, and the commonly received Decimal Arithmetick, is a matter of con- fiderable importance, and well deferves, I think, to be let in 3. fuller Light, for the mutual illuftration of both ; which therefore I fhall here attempt to perform. For Novices in .this Doctrine, tJho' they inay already be well acquainted with the Vulgar Arithmetick, and with the Rudiments of the common Algebra, yet are apt to appre- hend fomething abftrufe and difficult in infinite Series ; whereas in- deed they have the fame general foundation as Decimal Arithmetick, efpecially Decimal Fractions, and the fame Notion or Notation is only tarry'd ftill farther, and rendered more univerfal. But to mew this in fome kind of order, I muft inquire into thefe following particulars. Firft I muft (hew what is the true Nature, and what are the genuine Principles, of our common Scale of Decimal Arithmetick. Secondly what is the nature of other particular Scales, which have been, or
may
and INFINITE SERIES. 145
may be, occasionally introduced. Thirdly, what is the nature of a general Scale, which lays the foundation for the Doctrine of infinite Series. Laftly, I ihall add a word or two concerning that Scale ot Arithmetick in which the Root is unknown, and thcrefoi-e propofcd to be found ; which gives occafion to the Doctrine of Affected Equa- tions.
Firft then as to the common Scale of Decimal Arithmetick, it is that ingenious Artifice of expreffing, in a regular manner, all con- ceivable Numbers, whether Integers or Fractions, Rational or Surd, by the feveral Powers of the number Ttv/, and their Reciprocals; with the affiftance of other fmall Integer Numbers, not exceeding Nine, which are the Coefficients of thofe Powers. So that Ten is here the Root of the Scale, which if we denote by the Character X, as in the Roman Notation and its feveral Powers by the help of this Root and Numeral Indexes, (X1 = 10, X1 = ico, X3 = 1000, X4 = 10000, &c.) as is ufual ; then by ailuming the Coefficients o, i, 2, 3, 4, 5, 6, 7, 8, 9, as occafion (hall require, we may form or exprefs any Number in this Scale. Thus for inflance 5X4-f- jX3 -f- 4X1 + 8X1 -rf- 3X° will be a particular Number exprefs'd by this Scale, and is the fame as 57483 in the common way of Notation. Where we may obferve, that this laft differs from the other way of Notation only in this, that here the feveral Powers of X (or Ten) are fupprefs'd, together with the Sign of Addition -f-, and are left to be fupply'd by the Underftanding. For as thofe Powers afcend regularly from the place of Units, (in which is always X°, or i, muhiply'd by its Coefficient, which here is 3,) the feveral Powers will ealily be understood, and may therefore be omitted, and the Coefficients only need to be fet down in their proper order. Thus the Number 7906538 will (land for yX6 -+- gX5 -f- oX* -+-6X3 -f- ^X* -f-3X' -f-3X°, when you fupply all that is underftood. And the Number 1736 (by fuppreffing what may be ealiiy -underftood,) will be equivalent to X3 -+- 7X1 -f- 3X -f- 6 ; and the like of all other Integer Numbers whatever, exprefs'd by this Scale, or with this Root X, or Ten.
The fame Artifice is uniformly carry'd on, for the expreffing of all Decimal Fractions, by means of the Reciprocals of the ll-vcral
Powers of Ten, fuch as ^ = o, i ; 5^1 = 0,0 1 ; ^ = 0,001 ; c.:c. which Reciprocals may be intimated by negative Indices. Thus the Decimal Fraction 0,3172 (lands for 3X~'-j- iX~~:-f-7X -{- 2\~~4 i and the mixt Number 526,384 (by {applying what is underfl ;
U becomes
Method <?/* FLUXIONS,
becomes 5X4 •+• 2X> -f- 6X° -f- 3X~' -f- 8X"1 -f- 4X-» ; and the infinite or interminate Decimal Fraction 0,9999999, &c. ftands for 9X^' -f- gX-1 -4- 9X~3 H- 9X~4-f- 9X~5 -+- yX~& , &c. which infi- nite Series is equivalent to Unity. So that by this Decimal Scale, (or by the feveral Powers of Ten and their Reciprocals, together with their Coefficients, which are all the whole Numbers below Ten,) all conceivable Numbers may be exprefs'd, whether they are integer or fracled, rational or irrational ; at leaft by admitting of a continual progrefs or approximation ad infinitum,
And the like may be done by any other Scale, as well as the Deci- mal Scale, or by admitting any other Number, befides Ten, to be the Root of our Arithmetick. For the Root Ten was an arbitrary Number, and was at firft aflumed by chance, without any previous confideration of the nature of the thing. Other Numbers perhaps may be affign'd, which would have been more convenient, and which have a better elaim for being the Root of the Vulgar Scale of Arith- metick. But however this may prevail in common affairs, Mathe- maticians make frequent life of other Scales ; and therefore in the fecond place I (hall mention fome other particular Scales, which have been occafionally introduced into Computations.
The moft remarkable of thefe is the Sexagenary or Sexagefimal Scale of Arithmetick, of frequent ufe among Aflronomers, which expreffes all poffible Numbers, Integers or Fractions, Rational or Surd, by the Powers of Sixty, and certain numeral Coefficients not exceeding fifty- nine. Thefe Coefficients, for want of peculiar Characters to repre- fent them, muit be exprefs'd in the ordinary Decimal Scale. Thus if £ ftands for 60, as in the Greek Notation, then one of the/e Num- bers will be 53^ -f- 9^' -+- 34!°, or in the Sexagenary Scale 53", 9*, 34°, which is equivalent to 191374° in the Decimal Scale. Again, the Sexagefimal Fraclion 53°, 9', 34", will be the fame as 53^= -f- 9|f+ 34£~z, which in Decimal Numbers will be 53,159444, &c. aa infinitum. Whence it appears by the way, that fome Numbers may be exprefs'd by a finite number of Terms in one Scale, which in another cannot be exprefs'd but by approximation, or by a pro- greffion of Terms in infinitum.
Another particular Scale that has been confider'd, and in fome meafure has been admitted into practice, is the Duodecimal Scale, which exprefles all Numbers by the Powers of Twelve. So in com- mon affairs we fay a Dozen, a Dozen of Dozens or a Grofs, a Dozen of GrofTes or a great Grofs, Off. And this perhaps would have been the mod convenient Root of all otherSj by the Powers of which
to
and IN FINITE SERIES. 147
to conftruct the popular Scale of Arithmetick ; as not being fo lig but that its Multiples, and all below it, might be eafily committed to memory ; and it admits of a greater variety of Divifors than any Number not much greater than itfelf. Befides, it is not fo fmall, 'but that Numbers exprefs'd hereby would fufficiently converge, or by a few figures would arrive near enough to the Number required; the contrary of which is an inconvenience, that muft neceflarily attend the taking too fmall a Number for the Root. And to admit this Scale into practice, only two fingle Characters would be wanting, to denote the Coefficients Ten and Eleven.
Some have confider'd the Binary Arithmetick, or that Scale in which TIDO is the Root, and have pretended to make Computations by it, and to find considerable advantages in it. But this can never be a convenient Scale to manage and exprefs large Numbers by, be- caufe the Root, and confequently its Powers, are fo very fmall, that they make no difpatch in Computations, or converge exceeding flowly. The only Coefficients that are here necelTary are o and i. Thus i x 25 -f- i x 2* -h o x23 •+• i x2* -f- i x 2' -f- 0x2° is one of thefe Numbers, (or compendioufly 110110,) which in the common No- tation is no more than 54. Mr. Leibnits imngin'd he had found great Myfteries in this Scale. See the Memoirs of the Royal Academy of Paris, Anno 1703.
In common affairs we have frequent recourfe, though tacitly, to Millenary Arithmetick, and other Scales, whofe Roots are certain Powers of Ten. As when a large Number, for the convenience of read- ing, is diftinguifli'd into Periods of three figures: As 382,735,628,490. Here 382, and 735, &c. may be confider'd as Coefficients, and the Root of the Scale is 1000. So when we reckon by Millions, Billions, Trillions, &c. a Million may be conceived as the Root of our Arith- metick. Alfo when we divide a Number into pairs of figures, for the Extraction of the Square-root ; into ternaries of figures for the Extraction of the Cube-root ; &c. we take new Scales in effect, whofe Roots are 100, 1000, &c.
Any Number whatever, whether Integer or Fraction, may be made the Root of a particular Scale, and all conceivable Numbers may be exprefs'd or computed by that Scale, admitting only of integral and affirmative Coefficients, whofe number (including the Cypher c) need not be greater than the Root. Thus in (Quinary Arithmetick, in which the Scale is compofed of the Powers of the Root 5, the Coefficients need be only the five Numbers o, i, 2, 3, 4, and yet all Numbers whatever are expreffible by this Scale, at leaft by approxi-
U 2 mation,
j^B 77oe Method of FLUXIONS,
mation, to v/hat accu-racy we pleafe. Thus the common Number 2827,92 in this Arithmetick would be 4 x 54 -+- 2 x 5' -|- 3 x 5* -\~ ox5IH-2x5°-f-4x5~IH-3x 5~s ; or if we may fupply the feveral Powers of 5 by the Imagination only, as we do thofe of Ten in the common Scale, this Number will be 42302,43 in Quinary Arithme- tick.
All vulgar Fractions and mixt Numbers are, in fome meafure, the expreffing of Numbers by a particular Scale, or making the Deno- minator of the Fraction to be the Root of a new Scale. Thus ± is in effect o x 3° + 2 x^"1 ; and 8-f- is the fame as 8 x 5° '-f- 3 x j-'j and 25-5- reduced to this Notation will be 25x9° + 4x 9—' , or ra- ther 2x9' -4- 7x9° -4-4X9""1. And fo of all other Fractions and mixt Numbers.
A Number computed by any one of thefe Scales is eafily reduced to any other Scale affign'd, by fubftituting inftead of the Root in one Scale, what is equivalent to it exprefs'd by the Root of the other Scale. Thus to reduce Sexagenary Numbers to Decimals, becaufe 60 = 6x10, or|=6X, and therefore |s = 3 6X1, ^=2i6X3, &c. by the fubilitution of thefe you will eafily find the equivalent Decimal Number. And the like in all other Scales.
The Coefficients in thefe Scales are not neceflarily confin'd to be affirmative integer Numbers lefs than the Root, (tho' they mould be fuch if we would have the Scale to be regular,) but as occafion may require they may be any Numbers whatever, affirmative or negative, integers or fractions. And indeed they generally come out promif- cuoully in the Solution of Problems. Nor is it neceflary that the Indices of the Powers mould be always integral Numbers, but may be any regular Arithmetical Progreffion whatever, and the Powers themielves either rational or irrational. And thus (thirdly) we are come by degrees to the Notion of what is call'd an univerfal Series, or an indefinite or infinite Series. For fuppofing the Root of the Scale to be indefinite, or a general Number, which may therefore be reprefcnted by x, or y, &c. and affuming the general Coefficients a, b, c, d, &c. which are Integers or Fractions, affirmative or nega- tive, as it may happen ; we may form fuch a Series as this, ax* -f- lx* _j_ ex* -f- dxl -f- ex°, which will reprefent fome certain Number, exprefs'd by the Scale whofe Root is x. If fuch a Number pro- ceeds in hfif.itum, then it is truly and properly call'd an Infinite Series, or a Converging Series, x being then fuppos'd greater than Unity. Such for example is x + \x~ '-\-^.x—'--+ ^*~3, &c. where the reft of the Terms are underftood ad in/initum, and are iniinuated
and INFINITE SERIES. 149
bv, oV. And it may have any dcfcending Arithmetical Progreffion for its Indices, as xm — \xm~l -+- ^v*—1 -+-"*.. \—s, Gfc.
And thus we have been led by proper gradations, (that is, by arguing from what is well known and commonly received, to what before appear'd to be difficult and obfcure,) to the knowledge of infinite Series, of which the Learner will find frequent Examples in the lequel of this Treatife. And from hence it will be eafy to make the following general Inferences, and others of a like nature, which will be of good ufe in the farther knowledge and practice of t-hefe Series ; viz. That the firft Term of every regular Series is al- ways the mo ft coniiderable, or that which approaches nearer to the Number intended, (denoted by the Aggregate of the Series,) than any other lingle Term : That the fecond is next in value, and fo on : That therefore the Terms of the Series ought always to be difpoled in this regular defcending order, as is often inculcated by our Author : That when there is a Progreflion of fuch Terms-/;? infinitum, a few of the firft Terms, or thofe at the beginning of the Series, are or fhould be a fufficient Approximation to the whole ; and that thefe may come as near to the truth as you pleafe, by taking in ftill more Terms : That the fame Number in which one Scale may be exprefs'd by a finite number of Terms, in another cannot be exprefs'd but by an infinite Series, or by approximation only, and vice versei : That the bigger the Root of the Scale is, by fo much the fafter, cafen'.i paribus, the Series will converge ; for then the Reciprocals of the Powers will be fo much the lefs, and therefore may the more fafely be neglected : That if a Series coir e Tos by increafing Powers, fuch as ax -^ bx* -+- ex* -|-</.v4, &c. the Root x of the Scale mull be un- derftood to be a proper Fraction, the lefler the better. Yet when- ever a Series can be made to conveige by the Reciprocals of Ten, or its Compounds, it will be more convenient than a Series that converges fafter j becaufe it will more eafily acquire the form of the Decimal Scale, to which, in particular Cafes, all Series are to be ul- timately reduced. LafHy, from fuch general Series as thefe, which are commonly the refill t in the higher Problems, we muft pafs (by fubftitution) to particular Scales c; Series, and thofe are finally to be reduced to the Decimal Scale. And the Art of finding fuch general Series, and then their Reduction to -particular Scales, and laft •©£ all to the common Scale of Decimal Numbers, is ulmoll the whole of
j abrtiull-r pares of Amly ticks, as may be fecn in a good meaiiire'by the prefent TrcuUic.
I
Method of FLUXIONS,
I took notice in the fourth place, that this Doctrine of Scales, and Series, gives us an eafy notion of the nature of affected Equations, or fhews us how they ftand related to fuch Scales of Numbers. In the other Inflances of particular Scales, and even of general ones, the Root of the Scale, the Coefficients, and the Indices, are all fiip- pos'd to be given, or known, in order to find the Aggregate of the Series, which is here the thing required. But in affected Equations, on the contrary, the Aggregate and the reft are known, and the Re ot of the Scale, by which the Number is computed, is unknown and re- quired. Thus in the affected Equation $x* -j- 3*2 -f- ox* -+- 7*- — • 53070, the Aggregate of the Series is given, viz. the Number 53070, to find x the Root of the Scale. This is eafily difcern'd to be 10, or to be a Number exprefs'd by the common Decimal Scale, efpecially if we fupply the feveral Powers of 10, where they are un- derftood in the Aggregate, thus 5X4 -+- 3X3 -f-oX1 +7X' -4-oX0 = 53070. Whence by companion 'tis x = X=io. But this will not be fo eafily perceived in other instances. As if I had the Equation 4^+4- ax3 -f- 3** -f-ox" -f- 2x° -f- ^x~f -f- ^x~1 = 2827,92 I Ihould not fo eafily perceive that the Root x was 5, or that this is a Number exprefs'd by Quinary Arithmetick, except I could reduce it to this form, 4x5* -+- 2x $3 + 3*5* + 0x5' -f- 2 x 5° H- 4x5— * -+- 3 x 5~~;= 2827,92, when by comparifon it would preiently ap- pear, that the Root fought muft be 5. So that finding the Root of an affected Equation is nothing elfe, but finding what Scale in Arith- jnetick that Number is computed by, whofe Refult or Aggregate is given in the common Scale ; which is a Problem of great ufe and extent in all parts of the Mathematicks. How this is to be done, either in Numeral, Algebraical, or Fluxional Equations, our Author will inflruct us in its due place.
Before I difmiis this copious and ufeful Subject of Arithmetical Scales, I fhall here make this farther Observation ; that as all con- ceivable Numbers whatever may be exprefs'd by any one of theie Scales, or by help of an Aggregate or Scries of Powers derived frcm any Root ; fo likewife any Number whatever may be exprefs'd by fome fingle Power of the fame Root, by affuming a proper Index, integer or fracted, affirmative or negative, as occafion fhall require. Thus in the Decimal Scale, the Root of which is 10, or X, not only the Numbers i, 10, 100, 1000, &c. or i, o.i, o.oi, o.ooi, &c. that is, the feveral integral Powers of 10 and their Reciprocals, may be exprefs'd by the fingle Powers of X or 10, viz. X° , X' , X1, Xs, or X°, X-1, X~% X--% &c. refpectively, but alfo all the inter- mediate
and INFINITE SERIES. 151
mediate Numbers, as 2, 3, 4, Gff. u, 12, 13, Gfr. may be exprefs'd by fuch fingle Powers of X or 10, if we aflame proper Indices.
Thus 2 = X°'JOI03> &C- , 3 =X0'477",&c. 4=__ Xo/o-.o«, &e. g^ Qr jj
_.X''°4'3!>.&C- i2===X'>°7i"8'&e> 456 = X*.«s89s,&c. And the like of all other Numbers. Thefe Indices are ufually call'd the Logarithms of the Numbers (or Powers) to which they belong, and are fo many Ordinal Numbers, declaring what Power (in order or fucceflion) any given Number is, of any Root aflign'd : And different Scales of Lo- garithms will be form'd, by afluming different Roots of thofe Scales. But how thefe Indices, Logarithms, or Ordinal Numbers may be conveniently found, our Author will likewife inform us hereafter. All that I intended here was to give a general Notion of them, and to mew their dependance on, and connexion with, the feveral Arith- metical Scales before defcribed.
It is eafy to obferve from the Arenariiu of Archimedes, that he had fully confider'd and difcufs'd this Subject of Arithmetical Scales, in a particular Treatife which he there quotes, by the name of his a'^^tl, or Principles ; in which (as it there appears) he had laid the foundation of an Arithmetick of a like nature, and of as large an extent, as any of the Scales now in ufe, even the moft univerlal. It appears likewife, that he had acquired a very general notion of the Dodtrine and Ufe of Indices alfo. But how far he had accommo- dated an Algorithm, or Method of Operation, to thofe his Princi- ples, muft remain uncertain till that Book can be recover'd, which is a thing more to be wim'd than expedled. However it may be fairly concluded from his great Genius and Capacity, that fince he thought fit to treat on this Subject, the progrefs he had made in it was very confiderable.
But before we proceed to explain cur Author's methods of Ope- ration with infinite Series, it may be expedient to enlarge a little farther upon their nature and formation, and to make fome general Reflexions on their Convergency, and other circumftances. Now their formation will be beft explain'd by continual Multiplication after the following manner.
Let the quantity a -+- bx -{-ex1 -+- <A'3 -+- ex4, 6cc. be aflumed as a Multiplier, confming either of a finite or an infinite number of
Terms ; and let alfo - -+- x = o be fuch a Multiplier, as will give the Root x= — - . If thefe two are multiply'd together, they
will produce 3 + 2£Xf?* + 2±f_V + "1^5^ + *i £V, &c.
* a a — a n
152 The Method of FLUXIONS,
. — o ; and if inftead of x we here fubflitute its value — - , the Series
ap fy+"<! f tp+bq f- dp + cq /3 ' ef+t/f p*
wi 1 become - — TTT — - x - -f- x — —-*- x -. -f- -^-^-? x - >
q q q if 11* 9 j*
&c. = o ; or if we divide by -, and tranfpofe, it will be •• "*" aq — .
tp + bg p dj> + eg /* ep + Jq t* ....
— — x y + —j— x ^ — x - , &c. = ,7 : which Series,
thus derived, may give us a good infight into the nature of infinite Series in general. For it is plain that this Series, (even though it were continued to infinity,) mufl always be equal to a, whatever may be fuppofed to be the values of p, q, a, by c, d} &c. For
- , the firft part of the firflTerm, will always be removed or deflroy'd by its equal with a contrary Sign, in the fecond part of the feeond
Term. And — x- , the firfl part of the fecond Term, will be re-
i i moved by its equal with a contrary Sign, in the fecond part of -the
third Term, and fo on : So as finally to leave -- , or a, for the
Aggregate of the whole Series. And here it is likewile to be obferv'd, that we may flop whenever we pleafe, and yet the Equation will be good, provided we take in the Supplement, or a due part of the next Term. And this will always obtain, whatever the nature of the Series may be, or whether it be converging or diverging. If the Series be diverging, or if the Terms continually increafe in value, then there is a neceflity of taking in that Supplement, to preferve .the integrity of the Equation. But if the Series be converging, or if the Terms continually decreafe in any compound Ratio, and there- fore finally vanifh or approach to nothing ; the Supplement may be fafely neglected, as vanishing alfb, and any number of Terms may- be taken, the more the better, as an Approximation to the Qium- tity a. And thus from a due confederation of this fictitious Series, the nature of all converging or diverging Series may eafily be appre- hended. Diverging Series indeed, unlefs when the afore-mention'd increafing Supplement can be affign'd and taken in, will be of no feivice. And this Supplement, in Series that commonly occur, will •be generally fo entangled and complicated with the Coefficients of the Terms of the Scries, that altho* it is always to be understood., neverthelef?, ii is often impoffible to be extricated and affign'd. But however, converging Series will always be of excellent ufe, as Affording a convenient Approximation to the quantity required, when it cannot be othei wile exhibited. In thefe the Supplement aforefaid,
tho'
and INFINIT E SERIES. 153
tho' generally inextricable and unnflignable, yet continually decreafes along with the Terms of the Series, and finally becomes lefs than any aflignable Quantity.
The. lame Quantity may often be exhibited or exprefs'd by feveral converging Scries ; but that Series is to be mod edeem'd that has the greateft Rate of Convergency. The foregoing Series will converge fo much the fader, cteteris paribus, as p is lefs than qy or as the
Fraction - is lefs than Unity. For if it be equal to, or greater than
Unity, it may become a diverging Series, and will diverge fo much the fader, as p is greater than q. The Coefficients will contribute little or nothing to this Convergency or Divergency, if they are fuppos'd to increafe or decreafe (as is generally the cafe) rather in a fimple and Arithmetical, than a compound and Geometrical Propor- tion. To make fome Edimate of the Rate of Convergency in this Series, and by analogy in any other of this kind, let k and / re- prefent two Terms indefinitely, which immediately fucceed each other in the progrefTion of the Coefficients of the Multiplier a -+- bx -if ex* -f-^x3, &c. and let the number n reprefent the order or place of k. Then any Term of the Series indefinitely may be repre-
fented by -f- l—'-Jf»-~*- where the Sign mud be -+- or — , accor-
?" ding as n is an odd or an even Number. Thus if «== i, then
k = a, 1 = 1', and the firft Term will be -f- *_LlL^Z . ]f «==2j
then & = />, l = c, and the fecond Term will be — c^—~p. And fo of the red. Alib if m be the next Teim in the aforefaid pro-
grefTion after /, then -f- -^~lp"~l -f- ^ — 7/." will be any two fuc-
?" ?"
cefiive Terms in the fame Series. Now in order to a due Conver- gency, the former Term abfolutely confider'd, that is fetting afide the Signs, mould be as much greater than the fucceeding Term, as
conveniently may be. Let us fuppoie therefore that JL^—Jp»-i js
i"
greater than ' —^p", or ( dividing all by the common factor c" } \
r" ~^ ' t" '
that ^ + /f? is greater than — ^ - , or ( multiplying both by pq, )
that Ipq -f- krf is greater than nip* •+- Ipq, or (taking away the com- mon IpqJ that kf is greater than //.y,1, or (by a farther Diviiion,)
that - x — is greater than unity ; and as much greater as may be. fl X This
7%e Method of FLUXIONS,
This will take effeft on a double account ; firft, the greater k is in refpecl: of ;;;, and fecondly, the greater 5* is in refpect of p\ Now in the Multiplier a -\-bx -f- ex* -\-dx>, &c. if the Coefficients a, b, r, &c. are in any decreafing ProgreiTion, then k will be greater than /, which is greater than m ; fo that a fortiori k will be greater than m. Alfo if q be greater than p, and therefore (in a duplicate ratio) j* will be greater than /*. So that (cater is faribus) the degree of Convergency is here to be eftimated, from, the Rate according to w hich the Coefficients a, b, c, &c. continually decreafe, compounded with the Ratio, (or rather its duplicate,) according to which q fhall be fuppos'd to be greater than />.
— / n
The fame things obtaining as before, the Term .j_ A will be
»
i
what was call'd the Supplement of the Series. For if the Series be continued to a number of Terms denominated by n, then inftead of all the reft of the Terms in itifinitutn, we may introduce this Sup- plement, and then we fhall have the accurate value of a, inftead of an approximation to that value. Here the firft Sign is to be taken if n is an odd number, and the other when it is even. Thus if
n= i, and confequently k=a, and /= <£, we fhall have —
— *£ == a. Or if « == 2, and /= c, then bl±X — et±ll x t + q ill
c\i „ . f 7 j .i bb-^-a-j ff->rf-a p <{$ -4- cq
L---a. Or if n = 3, /= a, then J-I—f — _L_L_I x - -4- - f i 1 i q
x ^ — — •=.$. And fo on. Here the taking in of the Supple- ment always compleats the value of a, and makes it perfect, whether the Series be converging or diverging ; which will always be the beft way of proceeding, when that Supplement can readily be known. But as this rarely happens, in fuch infinite Series as ge- nerally occur, we muft have recourfe to infinite converging Series, wherein this Supplement, as well as the Terms of the Series, are infinitely diminifh'd ; and therefore after a competent number of them are collected, the reft may be all neglected in infinitum.
From this general Series, the better to aflift the Imagination, we will defcend to a few particular Inftances of converging Series in pure Numbers. Let the Coefficients a, />, c-, d, &c. be expounded by
,, • , |; < , to, refpectively ; then *±* _ »±* x ^ + ^ x
^ ^c—! orL^_f£±l^x^H_-2±^x^_^-+5ix/4, &C. 5»'(XC<— J' 27 r.x;? 7 3x4? f 4x55. 53'
'. — r. That the Series hence arifmg may converge, make/ lefs
than
a?:d IN FINITE SERIES. 155
than q in any given ratio, fuppofe - = ~, or /> = i, q = 2, then
A — |.x|H-4^x^ — TV x -J., &c. = i. That is, this Series of Fractions, which is computed by Binary Arithmetick, or by the Reciprocals of the Powers of Two, if infinitely continued will finally be equal to Unity. Or if we defire to flop at thefe four Terms, and inftead of the reft ad infinitum if we would introduce •the Supplement which is equivalent to them, and which is here known to be j x Ty, or TV, we Hull have 4 — | -+- ££- — T^ -f- T'o- = i, as is eafy to prove. Or let the fame Coemdents be ex- pounded by i, — |, -i, — i, -f, &c. then it will be - - -+-
f 4iz^ £ 1f=4f /• & Thu Series m ehhei.
1 3X47 J* 4X5? i3
be continued infinitely, or may be fum'd after any number of Terms
i, _ n
exprefs'd by ;?, by introducing the Supplement ; ~ — infteadof all
H-IXJ*
the reft. Or more particularly, if we make (jr= $p, then -2 _f. 7-^-. -+- -- — - -f ( ^— . &c. = i, v/hich is a Number
6x5! liXjS 20X^4 30X;;!'
exprefs'd by Quinary Arithmetick. And this is eafily reduced to the Decimal Scale, by writing ~ for -f, and reducing the Coefficients ; for then it will become 0,99999, &c. = i. Now if we take thefe five Terms, together with the Supplement, we mall have exadly
— -f- r11- + -12- -f- — - + -~ 4- ^-, = i. Again, if
2x5 6x,i 12x5} 20x54 30x5' 6x;«
we make here 77= ioo/^, we fhall have the Series
JJ •"
^^-6 >c -i- + 40°~9 x 9 -f- <co-': x 27 - x 3 iccoo 3 X4 i oooooo 4X 5 locoocooo
which converges very fa ft. And if we would reduce this to the re- gular Decimal Scale of Arithmetick, (which is always fuppos'd to be done, before any particular Problem can be faid to be coinplcatly folved,) we muit let the Terms, when decimally reduced, orderly under one another, that their Amount or Aggregate may be tlifco- ver'd ; and then they will ftand as in the Margin. Here the Ag- gregate of the firfc five Terms is 0,99999999595, 0,985 which is a near Approximation to the Amount of the whole infinite Series, or to Unity. And if, for proof-
lake, we add to this the Supplement _+/' = 1L ,- —
„ + , ,/' °' 5 '" |OJ
= 0,00000000405, the wh< . be Unity exaclly.
X 2 There
Tf6 The Method of FLUXIONS,
3 *f
There are alfo other Methods of forming converging Series, whe- ther general or particular, which fhall approximate to a known quan- tity, and therefore will be very proper to explain the nature of Con- vergency, and to mew how the Supplement is to be introduced, when it can be done, in order to make the Series finite ; which of late has been call'd the Summing of a Series. Let A, B, C, D, E, &c. and a, />, c, d, e, &c. be any two Progrcffions of Terms, of which A is to be exprefs'd by a Series, either finite or infinite, compos'd of itfelf and the other Terms. Suppofe therefore the firft Term of the Series to be a, and that p is the fupplement to the value of a.
Then is A = a -}-/>, or p = ~a . As this is the whole Supple- ment, in order to form a Series, I fhall only take fuch a part of it as is denominated by the Fraction - , and put q for the fecond Sup- plement. That is, I will afiimie • - = (p=) - -XTJ -\-q, or
/A — a b \ A — a E — b .. .... .,
q — f xi — R=7 ~~B~ x ' Again, as ™1S 1S the whole
value of the Supplement q> I fhall only aflume fuch a part of it as is de- nominated by the Fradion £> and for the next Supplement put r.
/A— a
orr= (-§- x Now as this is the whole value of the Supplement r, I only afTume fuch a part of it as is denominated by the Fraction - , and for the next Supplement put s. That is, — ~
B—l> C—c A— a B— /; C—c, A— a
x -7— x — — = ( r = ) — — - x — - x -rr-a -+- s, or s = -77- x
^ I ^ ' D ^ U Ij
B— /; C—c 7 A— •a B — '•> C—c T>—d A j /- c
— — x x i — TJ — — r- x — r- x —77— x . And lo on as far
as we pleafe. So that at lafr. we have the value of A.'=a-\-p, where the Supplement p = - ~—l)-\-q, where the fecond Supple-
A — a B — b A— a E — l> C—c ,
inent q •==• —g— x — TT-C -}- r, where r = — g— x —^~ x -]y» 4- s,
A B— b C c D d
where s = '—^- x — -7- x -rr— x —r-e-\- 1. And fo on ad tnfinitum.
D (*. U H,
_,. • r 11 A A— a. A— a B — /; A— a E—b C— c ,
That is finally A = a -+- —b .+- — x —^-c -\ — x -7— x -jj-«
A— a 7,— b C—c D— d c \ -a r^ TT\ -O Of*
-\- — — x —TV- x -jj- x -J7- e, Kc. where A, B, C, D, E, ere. and ay
b, r, d, e, 6cc. may be any two Progreffions of Numbers whatever, whether regular or defultory, afcending or defcending. And when
it
. . —
— = (?=) -g- x
x — rr- x
and INFINITE SERIES. 157
it happens in thefe Progreffions, that either A = a, or B=^, or £___£• 5cc. then the Series terminates of itfelf, and exhibits the vilue of A in a finite number of Terms : But in other cafes it ap- proximates indefinitely to the value of A. But in the cafe of an infinite Approximation, the faid Progreffions ought to proceed re