Isaac Newton.

The method of fluxions and infinite series : with its application to the geometry of curve-lines online

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that have been raifed againft it : But with what fuccefs I muft leave
to the judgment of others.

I then treat concerning Fluxions of fuperior orders, and give the
Method of deriving their Equations, with its Demonftration. For
tho' our Author, in this Treatife, does not expreffly mention thefe
orders of Fluxions, yet he has fometimes recourfe to them, tho' ta-
citely and indirectly. I have here ("hewn, that they are a necelTary
refult from the nature and notion of nrft Fluxions ; and that
all thefe feveral orders differ from each other, not abfolutely
and effentially, but only relatively and by way of companion.
And this I prove as well from Geometry as from Anaiyticks ;
and I actually exhibit and make fenfible thefe feveral orders ot

But more efpecially in what I call the Geometrical and Mechani-
cal Elements of Fluxions, I lay open a general Method, by the help
of Curve-lines and their Tangents, to reprefent and exhibit Fluxions
and Fluents in all cafes, with all their concomitant Symptoms and


334 f ^ 3e Method of FLUXIONS,

Aiic&ions, after a plain and familiar manner, and that even to ocular
view and infpedlion. And thus I make them the Objects of Senfe,
by which not only their exiitence is proved beyond all poflible con-
tradiftion, but alfo the Method of deriving them is at the fame time
fully evinced, verified, and illuftrated.

Then follow my Annotations upon our Author's fecond Problem,
or the Relation of the Fluxions being given, to determine the Re-
lation of the flowing Quantities or Fluents ; which is the fame thing
as the Inverfe Method of Fluxions. And firft I explain (what out-
Author calls) a particular Solution of this Problem, becaufe it cannot
be generally apply 'd, but takes place only in fuch Fluxional Equa-
tions as have been, or at leaft might have been, previoufly derived
from fome finite Algebraical or Fluential Equations. Whereas the
Fluxional Equations that ufually occur, and whofe Fluents or Roots
are required, are commonly fuch as, by reafon of Terms either re-
dundant or deficient, cannot be refolved by this particular Solution ;
but muft be refer'd to the following general Solution, which is here
distributed into thefe three Cafes of Equations.

The firft Cafe of Equations is, when the Ratio of the Fluxions
of the Relate and Correlate Quantities, (which Terms are here ex-
plain'd,) can be exprefs'd by the Terms of the Correlate Quantity
alone ; in which Cafe the Root will be obtain'd by an eafy pro-
cefs : In finite Terms, when it may be done, or at leaft by an
infinite Series. And here a ufeful Rule is explain'd, by which
an infinite Expreffion may be always avoided in the Conclufion,
which otherwife would often occur, and render the Solution inexpli-

The fecond Cafe of Equations comprehends fuch Fluxional Equa-
tions, wherein the Powers of the Relate and Correlate Quantities,
with their Fluxions, are any how involved. Tho' this Cale is much
more operofe than the former, yet it is folved by a variety of eafy
and fimple Analyfts, (more fimple and expeditious, I think, than
thofe of our Author,) and is illuftrated by a numerous collection of

The third and laft Cafe of Fluxional Equations is, when there are
more than two Fluents and their Fluxions involved j which Cafe,
without much trouble, is reduced to the two former. But here are
alfo explain'd fome other matters, farther to illuftrate this Dodlrinej
as the Author's Demonftration of the Inverfe Method of Fluxions,
the Rationale of the Tranfmutation of the Origin of Fluents to other

i places


places at pleafure, the way of finding the contemporaneous Incre-
ments of Fluents, and fuch like.

Then to conclude the Method of Fluxions, a very convenient and
general Method is propofed and explain'd, for the Refolution of all
kinds of Equations, Algebraical or Fluxional, by having recourfe
to fuperior orders of Fluxions. This Method indeed is not con-
tain'd in our Author's prelent Work, but is contrived in purfu-
ance of a notable hint he gives us, in another part of his Writings.
And this Method is exemplify 'd by feveral curious and ufeful Pro-

Laftly, by way of Supplement or Appendix, fome Terms in the
Mathematical Language arc farther explain'd, which frequently oc-
cur in the foregoing work, and which it is very neceflary to appre-
hend rightly. And a fort of Analytical Praxis is adjoin'd to this
Explanation, to make it the more plain and intelligible ; in which is
exhibited a more direct and methodical way of refolving fuch Alge-
braical or Geometrical Problems as are ufually propofed ; or an at-
tempt is made, to teach us to argue more cloiely, dhtinctly, and Ana-

And this is chiefly the fubftance of my Comment upon this part
of our Author's work, in which my conduct has always been, to
endeavour to digeft and explain every thing in the moft direct and
natural order, and to derive it from the moft immediate and genuine
Principles. I have always put myfelf in the place of a Learner, and
have endeavour'd to make fuch Explanations, or to form this into fuch
an Inftitution of Fluxions and infinite Series, as I imagined would
have been ufeful and acceptable to myfelf, at the time when I fidl
enter'd upon thefe Speculations. Matters of a trite and eafy nature
I have pafs'd over with a flight animadverfion : But in things of more
novelty, or greater difficulty, I have always thought myfelf obliged
to be more copious and explicite ; and am conlcious to myfelf, that
I have every where proceeded cumjincero ammo docendi. Wherever
I have fallen fhort of this defign, it fliould not be imputed to any
want of care or good intentions, but rather to the want of fkill, or
to the abftrufe nature of the lubject. I (hall be glad to fee my de-
fects fupply'd by abler hands, and (hall always be willing and thank-
ful to be better instructed.

What perhaps will give the greateft difficulty, and may furnifli
mod matter of objection, as I apprehend, will be the Explanations
before given, of Moments, -vanifiing quantities, infinitely little quan-


236 The Method of FLUXIONS,

fjfies, and the like, which our Author makes ufe of in this Treatife,
and elfe where, for deducing and demonftrating hisMethod of Fluxions.
I fhall therefore here add a word or two to my foregoing Explana-
tions, in hopes farther to clear up this matter. And this feems to
be the more necefTary, becaufe many difficulties have been already
ftarted about the abftracl nature of theie quantities, and by what
name they ought to be call'd. It has even been pretended, that they
are utterly impoffible, inconceiveable, and unintelligible, and it may
therefore be thought to follow, that the Conclu lions derived by their
means muft be precarious at leaft, if not erroneous and impoflible.

Now to remove this difficulty it mould be obferved, that the only
Symbol made ufe of by our Author to denote thefe quantities, is the
letter o, either by itfelf, or affected by fome Coefficient. But this
Symbol o at firft reprefents a finite and ordinary quantity, which
mu ft be understood to diminim continually, and as it were by local
Motion ; till after fome certain time it is quite exhaufted, and termi-
nates in mere nothing. This is furely a very intelligible Notion.
But to go on. In its approach towards nothing, and juft before it
becomes abfolute nothing, or is quite exhaufted, it muft neceflarily
pafs through vanifhing quantities of all proportions. For it cannot
pafs from being an affignable quantity to nothing at once ; that were
to proceed per fa/turn, and not continually, which is contrary to the
Suppofition. While it is an affignable quantity, tho' ever fo little,
it is not yet the exact truth, in geometrical rigor, but only an Ap-
proximation to it ; and to be accurately true, it muft be lefs than
any affignable quantity whatfoever, that is, it muft be a vanifhing
quantity. Therefore the Conception of a Moment, or vanishing
quantity, muft be admitted as a rational Notion.

But it has been pretended, that the Mind cannot conceive quan-
tity to be fo far diminifh'd, and fuch quantities as thefe are repre-
fented as impoffible. Now I cannot perceive, even if this impofli-
bility were granted, that the Argumentation would be at all affected
by it, or that the Concluiions would be the lefs certain. The im-
poffibility of Conception may arife from the narrownefs and imper-
fection of our Faculties, and not from any inconfiftency in the na-
ture of the thing. So that we need not be very folicitious about
the pofitive nature of thefe quantities, which are fo volatile, fub-
tile, and fugitive, as to efcape our Imagination ; nor need we be
much in pain, by what name they are to be call'd j but we may
confine ourfelves wholly to the ufe of them, and to difcover their



properties. They are not introduced for their own fakes, but only
as fo many intermediate fteps, to bring us to the knowledge of other
quantities, which are real, intelligible, and required to be known.
It is fufficient that we arrive at them by a regular progrefs of di-
minution, and by a juft and neceflary way of reafoning ; and that
they are afterwards duly eliminated, and leave us intelligible and
indubitable Conclusions. For this will always be the confequence,
let the media of ratiocination be what they will, when we argue
according to the ftriet Rules of Art. And it is a very common
thing in Geometry, to make impoffib'e and nbfurd Suppofitions,
which is the fame thing as to introduce iinpoffible quantities, and
by their means to difcover truth.

We have an inftance fimilar to this, in another fpecies of Quan-
tities, which, though as inconceiveable and as impofTible as thefe
can be, yet when they arife in Computations, they do not affect
the Conclufion with their impoffibility, except when they ought
fo to do; but when they are duly eliminated, by juft Methods of
Reduction, the Conclufion always remains found and good. Thefe.
Quantities are thofe Quadratick Surds, which are diftinguifh'd by
the name of impoffible and imaginary Quantities ; fuch as ^/ i,
^/a, v/ 3, v/ 4, &c. For they import, that a quantity or
number is to be found, which multiply'd by itfelf mall produce a -
negative quantity ; which is manifeftly impoffible. And yet thefe
quantities have all varieties of proportion to one another, as thofe
aforegoing are proportional to the poffible and intelligible numbers
I, ^/2, v/3, 2, 8cc. respectively -,. and when they arife in Compu-
tations, and are regularly eliminated and excluded, they always leave
a juft and good Conclufion.

Thus, for Example, if we had the Cubick Equation x~> lax"
-J-4IX 42 =o, from whence we were to extract the Root x ;
by proceeding according to Rule, we mould have this fiird Ex-

preffion for the Root, x = 4 -f- y'3 4- v/ -"fr-f- ^J^ - ,/ -^,
in which the impoffible quantity ^/ -~ is involved ; and
yet this Expreffion ought not to be rejected as abfurd and ufelefs,
becaufe, by a due Reduction, we may derive the true Roots of
the Equation from it. For when the Cubick Root of the firft inn-
culum is rightly extracted, it will be found to be the impoffible
Number i -+- ^/ , as may appear by cubing ; and when the
Cubick Root of the fecond vinculum is extracted, it will be found
to be i \/- j- Then by collecting thefe Numbers, the

X.x im-

338 77je Method of FLUXIONS,

impoffibie Number </ will be eliminated, and the Root of
the Equation will be found x = 4 i i = 2.

Or the Cubick Root of the firft vinculum will alfo be A -f- y/ T ' T)
as may likewife appear by Involution ; and of the fecond vincu-
lum it will be | </ _' T . So that another of the Roots of
the given Equation will be x = 4 -f- 1 -f- A = 7. Or the Cu-
bick Root of the fame firft vincuhtm will be \ v/ i| J
and of the fecond will be i H- ^/ .11. So that the third
Root of the given Equation will be x = 4 - 4 T = 3- And
in like manner in all other Cubick Equations, when the furd vin-
cula include an impoffible quantity, by extracting the Cubick
Roots, and then by collecting, the impoffible parts will be exclu-
ded, and the three Roots of the Equation will be found, which
will always be poffible. But when the aforefaid furd vincula do not
include an impoffible quantity, then by Extraction one poffible
Root only will be found, and an impoffibility will affect the other
two Roots, or will remain (as it ought) in the Conclufion.
And a like judgment may be made of higher degrees of Equa-

So that thefe impoffible quantities, in all thefe and many other
inftances that might be produced, are fo far from infecting or de-
ftroying the truth of thefe Conclufions, that they are the neceflary
means and helps of difcovering it. And why may we not conclude
the fame of that other fpecies of impoffible quantities, if they muft
needs be thought and call'd fo ? Surely it may be allow'd, that
if thefe Moments and infinitely little Quantities are to be elteem'd
a kind of impoffible Quantities, yet neverthelefs they may be made
ufeful, they may affift us, by a juft way of Argumentation, in find-
ing the Relations of Velocities, or Fluxions, or other poffible Quan-
tities required. And finally, being themfelves duly eliminated and
excluded, they may leave us finite, poffible, and intelligible Equa-
tions, or Relations of Quantities.

Therefore the admitting and retaining thefe Quantities, how-
ever impoffible they may feem to be, the investigating their Pro-
perties with our utmoft induftry, and applying thofe Properties to
ufe whenever occafion offers, is only keeping within the Rules of
Reafon and Analogy; and is alfo following the Example of our
fagacious aud illuftrious Author, who of all others has the greateffc
right to be our Precedent in thefe matters. 'Tis enlarging the num-
ber of general Principles and Methods, which will always greatly

i con-

[ '43 ]



CONTENTS of the following Comment.

I /JNnotations on the Introduction ; or the Refolution of
*-* Equations by Infinite Series. pag. 143

Sedt. I. Of the nature and conjlruttion of infinite or converg-
ing Series. P-H3

Sedt. II. The Refolution offimple Equations, or of pure Powers,
by infinite Series. = ~ p. 1 59

Sedt. III. The Refolution of Numeral Affected Equations, p. 1 8 6

Sedt. IV. The Refolution of Specious Equations by infinite Se-
ries ; andfirjifor determining the forms of the Series, ami
their initial Approximations. P- 1 9 1

Sedt. V. The Refolution ofAJfe&ed fpecious Equations proje-
cuted by various Methods of Analyjis. . . p. 209

Sedt. VI. Tranfition to the Method of Fluxions. P-235

II. Annotations on P rob. i. or, the Relation of the flow-
ing Quantities being given-t to determine the Relation
of their Fluxions. p. 241

Sedt. I. Concerning Fluxions of the firft Order , and to find
their Equations. p.24i

Sedt. II. Concerning Fluxions of fuperior Orders, and the
method of deriving their Equations. > ]

Seft. III. The Geometrical and Mechanical Elements of
Fluxions, i p.266

[T] III.


,111. Annotations on Prob. 2. or y the Relation of the Fluxions
being given, to determine the Relation of the Fluents.

p.2 77

Se. I. A particular Solution ; 'with a preparation to the
general Solution, by which it is dijlributed into three
Cafes. p.a//

Sedl. II. Solution of the firft Cafe of Equations. - p. 282
Sedt. III. Solution of the fecond Caje of Equations. -p.286

Seft. IV. Solution of the third Caje of Equations, with fome
neceffary Demonftrations. - . P-3OQ

Sedt V. The Refolution of Equations, whether Algebraical

or Fluxional, by the afliftance of [uperior Orders of
. > diJ j j r j

r lux ions. - - - p-3o

Sedt. VI. An Analytical Appendix, explaining Jome Terms
and ExpreJJiom in the foregoing Work. P-3 2 I

,Se<5t. VII. The Conclufwn , containing a j}:ort recapitulation
or review of the whole. - - P-33

THE Reader is defired to correfl the following Errors, which I hope will be thought
but few, and fuch as in works of this kind are hardly to be avoided. 'Tis here ne-
ceflary to take notice of even literal Miftakes, which in fubjefts of this nature are often very
material. That the Errors are fo few, is owing to the kind affillance of a flcilful Friend or
two,_ who fupplyfd my frequent abfence from, the IVefs ; as alfo to the care of a diligent Printer.


In tie Preface, pag. xiii< lirt, 3. read which
is here fubjoin'd. Ibid. 1. 5. for matter read
manner. Pag. xxiii. 1. alt. far Preface, &e.
nWConclufton of this Work. P- 7. \.T,i.for
~{- read =.. P. 15. 1.9. ready !>**+ -&*'

-',4, &c. P'. 17. 1. 17. read . P. 32.


l,'2j. read - . P. 35, 1. 3: for lOtfjr* read

loxty. P. 62. 1.4. read ~~r'~ . P. 63.!. 31.

firyreatl-y. Ibld:\.ult.for y - read y ~-

P. 64. \.q. for 2 read z. Ibid. \. 30. read t.
P. 82. 1. 17. read zzz. P. 87. 1. 22. read

+ 2 A>. Ibid. 1.22,24. reaJAVDK. P.t
\,2-[. read. * P.-gz. 1.5. read-\- .7 -,', . Ibid.

\.z\.for z read x. P. 104. 1.8. read 6;jt 1 .
P. 109. 1. 33. dele as ofen. P. I 10. 1. 29. read

and v / ^ 1 x l '=. P. nj, 1 1 7. for Parabola

read Hyperbola. P. 1 19. I. 1 2. read CE x \Q
= to the Fluxion of the' Area, ACEG ); and

lDxIP =

P.I3J.1.8. readJf - . Ibid.


\. 19, read. ! . P. 135. 1. 15. read


13.8. I. 9. ^WAb&Jifs^AB. P. 145. \.fenult.
read 7\~~ 3 . P. 149. 1. 2O. read whkh irt;
P. 157. I.i3./-f^ ax. P.i68. l.j. retd^ax.
P-I7I. \.\j.fir Reread $*. P. 1 77.' \.l$.rcait

. ....

P. 204. 1. 1 6. read to 2m, P. 213^ [.-j. far-
Species read Series. P. 229, 1. 21* for x 5 retu(
x 4. /i/V. 1. 24. for x 4'readx *. P. 234.
1. 2. ^or yy ready. P. 236. 1. 26. ;vW genera-
ting. P. 243. 1. 29. read. ax*yi*. P. 284.

1. uit. read i . P. 289. 1. 17. fur right read


left. P. 295. 1. i, 2.- read' ', x 4 - ^-J-ax*.
P. 297. \.ig.forjx ' read y* '. P. 298. 1.14.
read y. 1\3O4. 1. 20, 21 . dil: -(- be. P^og.

1. 1 8. read a m ~t> . P. 3 1 7. 1. tilt, read a'-j 1 .


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3T .

.. I

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 30 of 30)