Isaac Newton.

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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
x a 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"~ l X + x ~~x*'- 1 X*, &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 x n 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 fame x thing as to infift, that a
Problem, which naturally belongs to Algebra, mould be folved by
common Arithmetick ; which tho' poflible to be done, by purluing
backwards all the fleps of the general procefs, yet would be very
troubkfome and operofe, and not fo inflrudtive, or according to the
true Rules of Art

But I am apt to fufpedr, that all our doubts and fcruples about
Mathematical Inferences and Argumentations, especially when we are
fatisfied that they have been juftly and legitimately conducted, may
be ultimately refolved into a fpecies of infidelity and diftruft. Not
in refpecl of any implicite faith we ought to repofe on meer human
authority, tho' ever fo great, (for that, in Mathematicks, we mould
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
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
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-
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 aax 1 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 # -{- x 4 ' A:* H- x 8 , &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*

+ yx 1 13** -j- 34x T , &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, A 4, &c.

for v/x, v/*S \/ x *> vx , ^x l , &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

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_ V v (a -4- 4- 5 x - 4- J - ' c *

aa-+- XX ^" 2a Sfl3 r i6* 1287 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 z8rt 8

+

_- 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