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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{^tfcu., are not more conspicuous, than the freedom
and readinefs with which you communicate them, to all
fuch who you know will apply them to their proper ufe,
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 iA -f- ^ab becomes AB -, that is, by Sub-
traction, its whole Increment during that interval is f#B -+- fA
^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 =
19 8_8 4 . 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 = A a , and therefore b-=. zaA, and then the Moment of
AxA*, or A', will be 2rfA 4 -f- aA 1 , or 3^A*. Again, make B =
A 5 , and therefore l> = ^aA s -, and then the Moment of A xA*, or
A 4 , will be 3<?A 3 -4-rfA 3 , or 4#A 3 . Again, make B==A-, and
therefore ^ = 4^A 3 , and then the Moment of Ax A 4 , or A', will
be 4<?A 4 -i-tfA 4 , or 5<zA 4 . 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"
= maA m ~*, as found before ; therefore Mom. A"* = A~ iw x
ma A"-' = maA-"-' . Therefore the Moment of A m will be
maA m ~ 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< = - a A 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 ~ l B n . 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 x m 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 x m y n will be mxx m -*y -J- nx m yy"~ 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- ab t 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. a y ) 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 = x n

-+- 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 = nx n ~ l X +-
ny. ^-x" - X * -+- n x ?-^- x ^^x'-'^X 3 , &c. So that when X de T

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 -+-X 1 , will increafe
to .v 1 4- 2xX -j- X 1 , which we otherwife know to be true. Again,
fuppofe fl = 3, then Y = 3* 1 X -+- 3*X a H- X 3 . Or while x in*.
creafes to x r+- X, x"> by its Increment Y = 3^ a X -h 3^X J + X 3
will increafe to x* -f- 3* 1 X -+- ^xX 1 -+- X 3 . 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-



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