Isaac Newton.

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

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ax 1

grerTion Mjs " 7, ~7~7' "^' ^ w *^ P ro( ^ uce t ^ ie Terms

* H- H- iaxy-\-nx~>yy-* nax l yy~ l . Then collecting the Terms,
we ilia 11 have m -\- 3. vx 1 ; -i- arfxv H- m-\-.iaxy my"'.\x~ i

w ~f- 3.X) 1 * * 4-^-t- irftfy -f- nx*yy~~ l nax*yy* = o, for the
Fluxional Equation required. Or the ratio of the. Fluxions will be

m -4- 3*" js -(- z* -j- m -j- I ay m)$x * . . 1 . . .

- = - - - ^ - - - : - : ; which might have been

n -j- D * * ;? -j- I ax nx'j r -f- nax^y l

found immediately from the given Equation, by the foregoing Rule.

Here the general Numbers m and n may be determined pro lubitu,
by which means we may obtain as many .Fluxional Equations as we
pleafe, which will all belong to the given Equation. And thus we
may always find the fimpleft Expreffion, or that which is beft ac-
commodated to the prefent exigence. Thus if we make m = o,
and ;; == o, we mall have 4 = *'*-*"* + "> , as found before. Or if

X 3j a ax

n 11 l y 4* a

we make ; = i, and n= i , we ihall have - = : ^

x *-

as before. Or if we make m=- i, and n = i, we fhall have

- =. - ax +_> A - _

x fy- zax x*j '-\~axij '

i n 11 l. V

and n =. -r- 7, we ihall have - =


before. Or if we make m = ?,


of Qthers _ Now th j s var i ety of Solutions

y -(- 3^4 ^axi J

will beget no ambiguity in the Conclusion, as poffibly might have
been fufpected; for it is no other than what ought neceffarily to
arife, from the different forms the given Equation may acquire, as
will appear afterwards. If we confine ourfelves to the Progremon of
the Indices, it will bring the Solution to the common Method of
taking Fluxions, which our Author has taught elfewhere, and which,
becaufe it is eafy and expeditious, and requires no certain order of
the Terms, I mall here fubjoin.

For every Term of the given Equation, fo many Terms mufr. be
form'd in the Fluxional Equation, as there are flowing Quantities in
that Term. And this muft be done, (i.) by multiplying the Term
by the Index of each flowing Quantity contain'd in it. (2.) By
dividing it by the quantity itfelf j and, (3.) by multiplying by its
Fluxion. Thus in the foregoing Equation x> ax* -f- ayx y 3

= o, the Fluxion belonging to the Term .v 3 is 3 , or ^x^x.



The Fluxion belonging to ax 1 is - - , or zaxx. The

. avxv ayxx

Fluxion belonging to ayx is 1- , or axy -f- ayx. And the

Fluxion belonging to / 3 is , or y-y. So that the

Fluxion of the whole Equation, or the whole Fluxional Equation,
is 3-v a A- zaxx -f- ayx -f- ayx 3_>' 1 _y=o. Thus the Equation
x m =}', will give mxx m -* =.y ; and the Equation x m z," y, will
give mxx m l z" -f- nx m zz"~ t = y for its Fluxional Equation. And
the like of other Examples.

If we take the Author's funple Example, in pag. 19, or the Equa-
tion y = xx, or rather ay x* = o, that is ayx x l y = o,
in order to find its moft general Fluxional Equation ; it may be per-
form'd by the Rule before given, fuppofing the Index of x to be
encreas'd by m, and the Index of y by ;;. For then we {hall have
diredtly ? = "-'-g+-'* _ For the firft Term of the given

x nx z y ' n -|- \a

Equation being ayx, this multiply'd by the Index of x increas'd by
7/7, that is by ;;z, and divided by x, will give mayx~ l for the firlt
Term of the Numerator. Alfo the fecond Term being x*y, this
multiply'd by the Index of A- increas'd by m, that is by w-f- 2, and
divided by ,v, will give m -h 2X for the fecond Term of the Nu-
merator. Again, the firft Term of the given Equation may be now

,Y*J, which multiply'd by the Index of y increas'd by n, that
is by ;;, and divided by r, will give (changing the fign) nx l y~ l for
the firft Term of the Denominator. Alib the fecond Term will
then be cyx, which multiply'd by the Index of/ increas'd by ;/,
that is by n -f- i, and divided by y, will give (changing the Sign)

n -|- \a for die fecond Term of the Denominator, as found above.
Now from this general relation of the Fluxions, we may deduce as
many particular ones as we pleaie. Thus if we make ///= o, and

7/r=o, we fhall have - r - , or ay = 2xx, agreeable to our
Author's Solution in the place before cited. Or if we make;= 2,

1 n II 1 2TA 2tfl> 1 . .,

aiid ;z= r, we lhall have - = -7^7 = -77 Or if we make
//v = o, and ;/ = i, we (hall have - = - ^ = - . Or if

V X j ' A"

we make n = o. and m-=. 2, we fhall have - = -^-^ . -1-,

v a m , v

as before. All which, and innumerable other cafes, may be eafilv*
proved by a fubftitution of equivalents. Or we may prove it c:


246 e tf >e Method of FLUXIONS,

rally thus. Becaufe by the given Equation it is y=x i a~ I , in the


/-i V mayx m-\-zx c r , ,,. . , ,

value of the ratio 4 = gA& -. _7^ri7 f r 7 mbmtute its value, and

/, 11 V V OT * - + 2X 2X

we fhall have ~- = - = = as above.

x na n -+- i a a

3. The Equation of the fecond Example is 2j 3 -f- x*y 2cysi
-4- ^z* Z' = o, in which there are three flowing quantities y, x,
and z, and therefore there muft be three operations, or three Tables
mufl be form'd. Firft difpofe the Terms according to y, thus ;
2j3 _j_ oja _{_ x*y z~>y= o, and multiply by the Terms of the Pro-

- 2CZ

greffion 2 xjj"" 1 , ixj/y" 1 , oxj/y"" 1 , i xj//- 1 , relpeclively, (where
the Coefficients are form'd by diminishing the Indices of y by the com-
mon Number :,) and the refulting Terms will be qyy* * * -f- &yy*.
Secondly ; difpofe theTerms according to x, thus-> yx* - }-ox-t-2y">x=o 3 .


and multiply by the Terms of the ProgreiTion 2xxx~\ i xxv~ r , .
oy.xx~ l , (\vhere the Coefficients are the fame as the Indices of x,)
and the only refulting Term here is -+- 2yxx * *. Laftly, difpofe
the Terms according to z, thus ; z= -+-^y^ 2cyz - x*yz=o J

-4- 2}"

and multiply by the Progreffion 3xs~ I , 2xzz~' f , fx.zz~ ! , oxzz*,
(where the Coefficients are alfo the fame as the Indices of z,} and
the Terms will be ^zz* -h 6yzz-~-2cyz * . Then collecting all
thefe Terms together, we fhall have the Fluxional Equation fyrj 1 +
~3yy i _|_ av,v.v yzz* -+- 6yzz 2cyz =. o.

Here we have a notable inftance of our Author's dexterity, at

finding expedients for abbreviating. For in every one of thefe Ope-

rations fuch a Progreffion is chofe, as by multiplication will make

the greateft deftrudtion of the Terms. By which means he arrives

at the fhorteft Expreffion, that the nature of the Problem will allow.

It we mould feck the Fluxions of this Equation by the ufaal me-

thod, which is taught above, that is, if we always a flu me the Pro-

oreffions of the Indices, we fhall have 6yy* -+ 2xxy -\- xy 2cyz

zcyz -+- ~}yz* ~r- dyzz 3'zz* = o ; which has two Terms

more' than the other form. And if the Progreffions of the Indices

t (-j increas'd, in each cafe, by any common general Numbers, we

may form the moil: general Expreilion for the Fluxional Equation,

that the Problem will admit of.

3 4-


4. On occafion of the laft Example, in which are three Fluents
and their Fluxions, our Author makes an ufeful Obfervation, for
the Reduction and compleat Determination of fuJi Equations, tho'
it be derived from the Rules of the vulgar Algebra ; which matter
may be confider'd thus. Every Equation, conlilling of two flowing
or variable Quantities, is what correfponds to an indetcrmin'd Pro-
blem, admitting of an infinite number of Anfwcrs. Therefore one
of thofe quantities being afiumed at pleafure, or a particular value
being affign'd to it, the other will alfb be compleatly determined.
And in the Fluxional Equation derived from thence, thofe particular
values being fubftituted, the Ratio of the Fluxions will be given in
Numbers, in any particular cafe. And one of the Fluxions being
taken for Unity, or of any determinate value, the value of the other
may be exhibited by a Number, which will be a compleat Determi-

But if the given Equation involve three flowing or indeterminate
Quantities, two of them muft be a/Turned to determine the third ;
or, which is the fame thing, fome other Equation muft be either
given or aflumed, involving fome or all the Fluents, in order to a
compleat Determination. For then, by means of the two Equa-
tions, one of the Fluents may be eliminated, which will bring this
to the former cafe. Alfo two Fluxional Equations may be derived,
involving the three Fluxions, by means of which one of them may be
eliminated. And fo if the given Equation mould involve four Fluents,
two other Equations fliould be either given or afTumed, in order to
a compleat Determination. This will be fufficiently explain 'd by the
two following Examples, which will alfo teach us how compli-
cate Terms, fuch as compound Fractions and Surds, are to be ma-
naged in this Method.

5, 6. Let the given Equation be y* a* x*/ a* x- = o,
of which we are to take the Fluxions. To the two Fluents y and
x we may introduce a third ;c, if we aflume another Equation.
Let that be z = x\/a*- x~, and we mall have the two Equations
y- a- & = o, and a'-x 1 x* z* r= o. Then by the fore-
going Solution their Fluxional Equations (at leaft in one cafe) will
be 2jy z = o, and a*xx zxx> zz = o. Thefe two Fluen-
tial Equations, and their Fluxional Equations, may be reduced
to one Fluential and one Fluxional Equation, by the ufual methods
of Reduction : that is, we may eliminate z and z by fubftituting
their values yy a a and zyy. Then we fhall havej 1 a 1 x\/ a 1 .v 1

248 fix Method of FLUXIONS,

" "

- ! Q J and 2yy " "__ = == o. Or by taking away the furds, ,

'tis a"x z ^ 4 y* 4- 2a l y z rt 4 = o, and then a*xx 2xx=.
- za* = o.

7. Or if the given Equation be x 5 ay* -f- - x^^/ay -\-x*-

= o, to find its corresponding Fluxional Equation ; to the two 1
flowing quantities ,v and y we may introduce two others .z and i',
and thereby remove the Fraction and the Radical, if we affume the

two Equations -~ = z, and x*~i/ay-t-xx=zv. Then we (hall

T. +_>' ^

have the three Equations x= ay 1 -\- z i;=o, az-\-yz
by* r o, and ayx* -f- x 6 i<-~ = o, which will give the three
Fluxional Equations ^xx* zayy -+- z V = o, az +- yz -+- yz
"^byy* = o, and ay'x* -+- ^.ayxx' -f- 6xx s 2vv= o. Thefeby,-
known Methods of the common Algebra may be reduced to on&
Fluential and one Fluxional Equation, iavolving x and y y and their
Fluxions, as is required.

8. And by the fame Method we may take the Fluxions of Bino-
mial or other Radicals, of any kind, any how involved or compli-
cated with one another. As for inflance, if we were to find the

Fluxion oF-Vwf -\-*/aa xx, put it equal to y, or make ax-i~
xx=yy. Alfo make </ aa xx = s$. Then we fhall

have the two Fluential Equations ax-\-z y 1 = o, and a* AT*;
z 1 = o, from whence we mall have the two Fluxional Equations
ax-}- z 2j/y = o, and 2xx 2zz = o, or xx -f- zz = o.'
This laft Equation, if for z and z we fubftitute their values^ ax~
and zyy ax, will become xx -f- 2yy* zaxyy axy* -{- a^xx-

o ; whence y = ~ " A ' ~ A - . And here if for y we fubfti -

' 2\i - 2HX1 *

tute its value vax-+-\/aa xx, we mall have the Fluxion re-

ax -J an A Jf xx , , 1 T^

quired y = - - - - - - : ..., - . And many other Exam -

7.1/fta xx x yax + y aa xx

pies of a like- kind will be found in the fequel of this Work.

9, 10, 1 1, 12. In Examp. 5. the propofed Equation is zz -{-
a xz .)' 4 => m which there are three variable quantities x, y, and
z, and therefore the relation of the Fluxions will be 2zz -|- axz
_j_ ax ~ 4j/j-3 === o. But as there wants another Fluential Equa-
tion, and thence another Fluxional Equation, to make a compleat
determination ; if only another Fluxional Equation were given or <
afTurned, we mould have the required relation of the Fluxions x and y,..



Suppofe this Fluxional Equation were i=.vv/^-v xx ; then by
fubftitution we mould have the Equation zz -f- ax x x^/ax xx
-f- axz 4)7 5 = o, or the Analogy x :y :: 4_>' 3 : 2Z -4- ax x
v/rftf .vx -f- rf;s, which can be reduced no farther, (or & cannot
be eliminated,) till we have the Fluential Equation, from which the
Fluxional Equation z=x\/ax xx is fuppos'd to be derived.
And thus we may have the relation of the Fluxions, even in fuch
cafes as \re have not, or perhaps cannot have, the relation of the

But tho' this Reduction may not perhaps be conveniently per-
forni'd Analytically, or by Calculation, yet it may poffibly be per-
form'd Geometrically, as it were, and by the Quadrature of Curves ;
as we may learn from our Author's preparatory Proportion, and
from the following general Conliderations. Let the right Line AC,
perpendicular to the right Line AB, be conceived to move always
parallel to itfelf, fo as that its extremity A may defcribe the line AB.
Let the point C be fixt, or always at the fame diftance from A, and
let another point move from A towards C, with a velocity any how
accelerated or retarded. The parallel motion of the line AC does
not at all affect the progreffive motion of the point moving from
A towards C, but from a combination of thefe two independent
morions, it will defcribe the Curve ADH ;
while at the fame time the fixt point C will
defcribe the right line CE, parallel to AB.
Let the line AC be conceived to move thus,
till it comes into the place BE, or BD. Then
the line AC is conftant, and remains the fame,
while the indefinite or flowing line becomes
BD. Alfo the Areas defcribed at the fame time, ACEB and ADB,
are likewife flowing quantities, and their velocities of defcription,
or their Fluxions, muft neceflarily be as their refpeclive defcribing
lines, or Ordinates, BE and BD. Let AC or BE be Linear Unity,
or a conftant known right line, to which all the other lines are to
be compared or refer'd ; juft as in Numbers, r.M other Numbers
are tacitely refer'd to i, or to Numeral Unity, as being the fim-
pleft of all Numbers. And let the Area ADB be fuppos'd to be
apply 'd to BE, or Linear Unity, by which it will be reduced from
the order of Surfaces to that of Lines j ami let the refulting line
be call'd z. That is, make the Area ADB = z x BE ; and if AB
be call'd x, then is the Area ACEB = x x BE. Therefore the

K k Fluxions

25 o" 1 Ibe Method of FLUXIONS,

Fluxions of thefe Areas will be z x BE and x x BE, which are as z
and x. But the Fluxions of the Areas were found before to be as
BD to BE. So that it is z : x : : ED : BE = i, or z = x x BD.
Consequently in any Curve, the Fluxion of the Area will be as the
Ordinate of the Curve, drawn into the Fluxion of the Abfcifs.

Now to apply this to the prefent cafe. In the Fluxional Equa-
tion before affumed z=x</ax xx, if x reprefents the Abfcifs
of a Curve, and \/ ax xx be the Ordinate ; then will this Curve
be a Circle, and z will reprefent the corresponding Area. So that
we fee from hence, whether the Area of a Circle can be exhibited
or no, or, in general Terms, tho' in the Equation proppfed there
fhould be quantities involved, which cannot be determined or ex-
prefs'd by any Geometrical Method, luch as the Areas or Lengths
of Curve-lines ; yet the relation of their Fluxions may neverthelefs

be found.

13. We now come to the Author's Demonftration of his Solutions
or to the proof of the Principles of the Method of Fluxions, here laid
down, which certainly deferves to engage our mcft ferious attention.
And more efpecially, becaufe thefe Principles have been lately drawn
into debate, without being well confider'd or imderftoqd ; polfibly be T
caufe this Treatife of our Author's, expreffly wrote on the fubjed, had
not yet feen the light. As thefe Principles therefore have been treated
as precarious at leaft, if not wholly inefficient to fupport the Doo
trine derived from them ; I Shall endeavour to examine into every
the moll: minute circumflance of this Demonstration, and that with
the utmoft circumipeclion and impartiality.

We have here in the firft place a Definition and a Theorem to-r
gether, Moments are defined to be the indefinitely jmall parts offoiv-
itig quantities, by the acceflion of which, in indefinitely fmall portions
of time, tboj'e quantities are continually increajed. The word Moment
(momentum^ movimentum, a mevcoj by analogy feems to have been
borrow'd from Time. For as Time is conceived to be in continual
flux, or motion, and as a greater and a greater Time is generated
by the acceffion of more and more Moments, which are conceived
as the fmalleit particles of Time : So all other flowing Quantities
may be underitood, as perpetually, increafing, by the accellion of
their fmallefr, particles, which therefore may not improperly be call'd
their Moments. But what are here call'd their jmalleft particles,
are not to be underftood as if they were Atoms, or of any definite
and determinate magnitude, as in the Method of Indivisibles.} but
to be indefinitely fmall, or continually decreafing, till they are lefs



than any afiignable quantities, and yet may then retain all poffible
varieties of proportion to one another. That thefe Moments are
not chimerical, vifionary, or merely imaginary things, but have an
existence Jut generis, at leaft Mathematically and in the Underftand-
ing, is a neceflary confequence from the infinite Divifibility of Quan-
tity, which I think hardly any body now contefts *. For all con-
tinued quantity whatever, tho' not indeed actually, yet mentally
may be conceived to be divided in infinitutn, Perhaps this may be
beft illuftrated by a comparative gradation or progrefs of Magnitudes.
Every finite and limited Quantity may be conceived as divided into
any finite number of fmaller parts. This Divifion may proceed,
and thofc parts may be conceived to be farther divided in very lit-
tle, but flill finite parts, or particles, which yet are not Moments.
But when thefe particles are farther conceived to be divided, not
actually but mentally, fo far as to become of a magnitude Ids than
any afiignable, (and what can flop the progrefs of the Mind ?) then
are they properly the Moments which are to be understood here. As
this gradation of diminution certainly includes no abfurdity or con-
tradiction, the Mind has the privilege of forming a Conception of
thefe Moments, a poffible Notion at leaft, though perhaps not an
adequate one ; and then Mathematicians have a right of applying
them to ufe, and of making fuch Inferences from them, as by any
flrict way of reafoning may be derived.

It is objected, that we cannot form an intelligible and adequate
Notion of thefe Moments, becaufe fo obfcure and incomprehenfible
an Idea, as that of Infinity is, muft needs enter that Notion ; and
therefore they ought to be excluded from all Geometrical Difquifi-
tions. It may indeed be allowed, that we have not an adequate
Notion of them on that account, fuch as exhatifts the whole nature
of the thing, neither is it at all neceflary ; for a partial Notion,
which is that of their Divifibility fine Jine, without any regard to
their magnitude, is fufficient in the preient cafe. There are many
other Speculations in the Mathematicks, in which a Notion of In-
finity is a neceflary ingredient, which however are admitted by all
Geometricians, as ufeful and dcmonftrable Truths. The Doctrine
of commenfurable and incommenfurable magnitudes includes a No-
tion of Infinity, and yet is received as a very demonftrablc Doctrine.
We have a perfect Idea of a Square and its Diagonal, and yet we

K k 2 know

The Method of FLUXIONS,

know they will admit of no finite common meafure, or that their pro-
portion cannot be exhibited in rational Numbers, tho' ever fo fmall,
but may by a feries of decimal or other parts continued ad infini-
tum. In common Arithmetick we know, that the vulgar Fraction
1., and the decimal Fraction 0,666666, &c. continued ad infinitum^
are one and the fame thing j and therefore if we have a fcientifick
notion of the one, we have likewife of the other. When I de-
icribe a right line with my Pen, fuppofe of an Inch long, I defcribe
firft one half of the line, then one half of the remainder, then one
half of the next remainder, and fo on. That is, I actually run
over all thofe infinite divifions and fubdivifions, before I have com-
pleated the Line, tho' I do not attend to them, or cannot diftin-
guifh them. And by this I am indubitably certain, that this Series
of Fractions i -f- JL _j_ -.-}- _' r> &c. continued ad infinitum, is pre-
cifely equal to Unity. Euclid has demonflrated in his Elements, ,
that the Circular Angle of Contact is lefs than any aflignable right-
lined Angle, or, which is the fame thing, is an infinitely little Angle
in comparifon with any finite Angle : And our Author fhews us
fHll greater My fteries, about the infinite gradations of Angles of Con-
tact. In Geometry we know, that Curves may continually approach
towards their Arymptotes, and yet will not a&ually meet with them;
till both are continued to an infinite diftance. We know likewife,
that many of their included Areas, or Solids, will be but of a finite
and determinable magnitude, even tho' their lengths mould be actually
continued ad infinitum. We know that fome Spirals make infinite
Circumvolutions about a Pole, or Center, and yet the whole Line,
thus infinitely involved, is but of a finite, determinable, and aflign-
able length. The Methods of computing Logarithms fuppofe, that
between any two given Numbers, an infinite number of mean Pro-
portionals maybe interpofedj and without fome Notion of Infinity
their nature and properties are hardly intelligible or difcoverable.
And in general, many of the moft fublime and ufeful parts of
knowledge muft be banifh'd out of the Mathematicks, if we are
fo fcrupulous as to admit of no Speculations, in which a Notion
of Infinity will be neeeflarily included. We may therefore as fafely
admit of Moments, and the Principles upon which the Method
of Fluxions is here built, . as any of the fore-mention'd Specula-

The nature and notion of Moments being thus eftablifli'd, we
may pafs on to the afore -mcnticn'd Theorem, which is this.


(contemporary) Moments of fairing quantities are as the Velocities of
flowing or increafing ; that is, as their Fluxions. Now if this be
proved of Lines, it will equally obtain in all flowing quantities
whatever, which may always be adequately rcprefented and ex-
pounded by Lines. But in equable Motions, the Times being given,
the Spaces defcribed will be as the Velocities of Defcription, as is
known in Mechanicks. And if this be true of any finite Spaces
whatever, or of all Spaces in general, it muft alfo obtain in infi-
nitely little Spaces, which we call Moments. And even in Mo-
tions continually accelerated or retarded, the Motions in infinite-
ly little Spaces, or Moments, muft degenerate into equability. So
that the Velocities of increafe or decreafe, or the Fluxions, will be
always as the contemporary Moments. Therefore the Ratio of
the Fluxions of Quantities, and the Ratio of their contemporary
Moments, will always be the fame, and may be ufed promifcu-
oufly for each other.

14. The next thing to be fettled is a convenient Notation for
thefe Moments, by which they may be diftinguifh'd, reprefented,
compared, and readily fuggefted to the Imagination. It has been
appointed already, that when x, y, z, v, &c. ftand for variable or
flowing quantities, then their Velocities of increafe, or their Fluxions,
fhall be reprefented by x, y, z, -j, &c. which therefore will be pro-
portional to the contemporary Moments. But as thefe are only
Velocities, or magnitudes of another Species, they cannot be the Mo-
ments themfelves, which we conceive as indefinitely little Spaces,
or other analogous quantities. We may therefore here aptly intro-
duce the Symbol o, not to ftand for abfolute nothing, as in Arith-
rnetick, but a vanifhing Space or Qtiantity, which was juft now
finite, but by continually decrealing, in order prefently to terminate
in mere nothing, is now become lefs than any affignable Qinintify.
And we have certainly a right fo to do. For if the notion is in-
telligible, and implies no contradiction as was argued before, it may
furely be infinuated by a Character appropriate to it. This is not
aligning the quantity, which would be contrary to the hypothefis,
but is only appointing a mark to reprefent it.- Then multiplying
the' Fluxions by the vanishing quantity <?, we fhall have the fcve-
ral quantities .\o, yo, zo, r?, cc. which are vanifhing likewife,
and pioportional to the Fluxions refpedlively. Thefe therefore may
now reprefent the contemporary Moments- -of x, y, z, v, &c. And

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 22 of 30)