Isaac Newton.

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

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in general, whatever other flowing .quantities, as well as Lines and

I Spaces,

2 54 "*fl je Method of FLUXIONS,

Spaces, arc reprefented by A-, y, z, -v, &c. as o may (land for a.
-vanishing quantity of the fame kind, and as x, y, z, v, &c. may
ftand for their Velocities of increafe or decreafe, (or, if you pleafe,
fpr Numbers proportional to thofe Velocities,) then may xo, yo,
zo, i-o, &c. always denote their refpedive fynchronal Moments,
.or momentary accefiions, and may be admitted into Computations
.accordingly. And this we corne now to apply.

15. We muft now have recourfe to a very notable, ufeful, and
extenfive property, belonging to. all Equations that involve flowing
Quantities. Which property is, that in the progrefs of flowing,
the Fluents will continually acquire new values, .by the accefilon of
contemporary parts of thofe Fluents, and yet the Equation will be
equally true in all thcfe, cafes. This is a neceffary refult from the Na-
ture and Definition of variable Quantities. Confequently thefe Fluents
.rnay be any .how increafed or diminifh'd by their contemporary
Increments or Decrements ; which Fluents, fo increafed or dimi-
niihed, may be fubflituted for the others in the Equation. As if
an Equation mould involve the Fluents x and _y, together with any
given quantities, and X and Y are fuppofed to be any of their con-
temporary Augments reflectively. Then in the given Equation we
may fubflitute x -f- X for x, and y -+- Y for -y, and yet the Equa-
tion will be .good, or .the equality of the Terms will be prefer ved.
.So if X and Y were contemporary Decrements, inflead of x and
y we might fubflitute x X and y Y reflectively. And as this
inuft hold good of all contemporary Increments or Decrements what-
ever, whether finitely great or infinitely little, it will be true like-
wife of contemporary Moments. That is, in flea d of .r and y in
any Equation, we may fubflitute .v-f- xo and y -t-jo, and yet we
ihall flill have a good Equation. The tendency of this will appear
from what immediately follows.

16. The Author's fingle Example is a kind of Induction, and the
proof of this may ferve for all cafes. Let the Equation x s a.\*
+ a xy _>' 5 =o be given as before, including the variable quan-
tities x and r, inftead of which we may fubflitute thefe quan-
tities increas'd by their contemporary Moments, or x - xo and
y -i-yo respectively. Tlien we ihall have the Equation x -+- xo | 3
a x x + AO i a -f- a x x -|- xo x y -Jo~ T+"}f > = o. Thefe
Terms .being expanded, and reduced to three orders or columns,
according as the vanifhing quantity o is of none, one, or of more
/limenfions, will ftand as in the Margin.


17, 18. Here the Terms of the fir ft * 3 + ?* w * +3 A *"* r 1
order, or column, remove or deftroy one _ fl . v i_ 2f ,^ ox _ a ll'l \
another, as being absolutely equal to no- +a.rj> + a.\iy -\-axjs- )>=o,
thing by the given Equation. They be- _ )3 ^ _,;>, |
ing therefore expunged, the remaining _ "j. j

Terms may all be divided by the com-

mon Multiplier <?, whatever it is. This being- done, all the Terms
of the third order will ftiil be affecled by o, of one or more dimen-
fions, and may therefore be expunged, as infinitely lels than the
others. Laftly, there will only remain thofe of the fecond order or
column, that is 3.vA.' i zaxx -+- axy 4- ayx Tjy- = o, which
will be the Fluxional Equation required. Q^. E. D.

The fame Conclufions may be thus derived, in fomething a dif-
ferent manner. Let X and Y be any fynchronal Augments of the
variable quantities A* and y, as befoie, the relation of which quan-
tities is exhibited by any Equation. Then may tf-J-X and y 4- Y
be fubfKtuted for x and y in that Equation. Suppofe for inftance
that x> ax* 4- axy - _y 3 = o ; then by fubftitution we flwll
have x 4- X | 3 a x .v 4- X | a 4-#x.v4-Xx/4-Y y 4- V | 3
= o ; or in termini* expanfis .v 5 -f- 3X 1 X -f- 3xX z -+- X 3 ax 1
2rfxX aX* -t- axy -\- <?.vY4- aXy -f- ^XY j 3 3j a Y 3;'Y 4

Y 5 = o. But the Terms ,v 3 ax* -+- axy _y 3 = o will va-
niHi out of the Equation, and leave 3# 1 X 4- 3xX a 4-X 3 2axX

aX* 4- axV 4- aXy 4- XY y* Y 3/i 7 * Y- == o, for
the relation of the contemporary Augments, let their magnitude be
what it will. Or refolving this Equation into an Analogy, the ratio

,- , ,. A , ,. Y ?r*-|- ^rX-L. X 1 2 ..* /7X -Lv

of thele Augments may be this, =. -

X a* ..v _|- -j* - r 3.., + l *

Now to find the ultimate rc.tio of thefe Augments, or their ratio
when they become Moments, fuppofe X and V to diminil'h till they
become vanishing quantities, and then they may be expunged out
of this value of the ratio. Or in thofe circumftances it will be

, which is now the ratio of the Moments. And


P = ~^ -

- y ax

this is the fame ratio as that of the Fluxions, or it will be

.V 1 2f>x - ai . . -

or 3_)' a axy = $x-x zaxx 4- ayx, as wss

found before.

In this way of arguing there is no aflumption made, but what is
iuflifiable by the received Methods both of the ancient and modern
Geometricians. We only defend from a general Proportion, which
is undeniable, to a particular cafe which is certainly included in ir.


256 The Method of FLUXIONS,

That is, having the relation of the variable Quantities, we thence
da-eddy deduce the relation or ratio of their contemporary Aug-
ments ; and having this, we directly deduce the relation or ratio of
thofc contemporary Augments when they are nafcent or evanefcent,
juft beginning or juft ceafing to be ; in a word, when they are Mo-
ments, or vanilliing Quantities. To evade this realbning, it ought
to be proved, that no Quantities can be conceived lefs than afiign-
able Quantities; that the Mind has not the privilege of conceiving
Quantity as perpetually diminiiLingy/w^w ; that the Conception of
a .vanishing Quantity, a Moment, an Infinitefimal, &c. includes a
contradiction : In fhort, that Quantity is not (even mentally) divifi-
ble ad infinitum ; for to that the Controverfy mufb be reduced at
laft. But I believe it will be a very difficult matter to extort this
Principle from the Mathematicians of our days, who have been fo
long in quiet poiTefTion of it, who are indubitably convinced of the
evidence and. certainty of it, who continually and fuccefslully ap-
ply it, arid who- are ready to acknowledge the extreme fertility and
ufefulnefs of it, upon fo many important occalions.

19. Nothing remains, I think, but to account for thefe two cir-
.cumilances, belonging to the Method of Fluxions, which our Au-
thor briefly mentions here. Firft that the given Equation, whofe
Fluxional Equation is to be found, may involve any number of
flowing quantities. This has been fufficiently proved already, and
we have feen feveral Examples of it. Secondly, that in taking
Fluxions we need not always confine ourfelves to the progreffion of
the Indices, but may affume infinite other Arithmetical Progreflions,
as conveniency may require. This will deferve a little farther illu-
ftration, tho' it is no other than what muft neceiTarily refult from
the different forms, which any given Equation may afTume, in an
infinite variety. Thus the Equation x 3 ax 1 -4- axy j 3 = o,
being multiply'd by the general quantity x m y", will become #"+*>'
-r- ax m -$- 1 y" -h ax m+l y"'t' 1 x m y"~^^ = o, which is virtually the fame
Equation as it was before, tho' it may aiTume infinite forms, accor-
ding as we pleafe to interpret m and n. And if we take the
Fluxions of this Equation, in the ufual way, we mall have
m + i y* -j- nx^rty}*- 1 m -+- zaxx m ^y" nax m ^yy n ~^ -f-

l -f- n + irf.Y" ! 'j/)- B mxx m ~ I y a ''* n
.5= o. Now if we divide this again by x"}", we mail have m
4- nx*j>y~* m -f- 2axx nax*yy~~ l -+- m -+- laxy 4- n-\- \axy
/xx~*y* n -f- 3j/y a ?= o, which is the fame general Equation as
was derived before. And the like may be underftood of all other
Examples. SECT.


SECT. II. Concerning Fluxions of fuperior orders^ and
the method of deriving their Equations.

IN this Treatifc our Author confiders only fir ft Fluxions, and has
not thought fit to extend his Method to fuperior orders, as not di-
rectly foiling within his prefent purpofe. For tho' he here purfues
Speculations which require the ufe of fecond Fluxions, or higher
orders, yet he has very artfully contrived to reduce them to firft
Fluxions, and to avoid the necefTity of introducing Fluxions of fu-
perior orders. In his other excellent Works of this kind, which
have been publifh'd by himfelf, he makes exprefs mention of them,
he difcovers their nature and properties, and gives Rules for deriving
their Equations. Therefore that this Work may be the more fer-
viceable to Learners, and may fulfil the defign of being an Inftitu-
tion, I mall here make fome inquiry into the nature of fuperior
Fluxions, and give fome Rules for finding their Equations. And
afterwards, in its proper place, I mail endeavour to (hew fomething
of their application and ufe.

Now as the Fluxions of quantities which have been hitherto con-
fider'd, or their comparative Velocities of increafe and decreafe, are
themfelves, and of their own nature, variable and flowing quantities
alfo, and as fuch are themfelves capable of perpetual increafe and de-
crea&, or of perpetual acceleration and retardation ; they may be
treated as other flowing quantities, and the relation of their Fluxions
may be inquired and difcover'd. In order to which we will adopt
our Author's Notation already publifh'd, in which we are to con-
ceive, that as x, y, z, &c. have their Fluxions #, j, z., &c. fo thefe
likewife have their Fluxions x, /, z,&c.which are the fecond Fluxions
of x, v, z, &c. And thefe again, being ftill variable quantities, have


their Fluxions denoted by x, y, z, &c. which are the third Fluxions
of x, y, z, &c. And thefe again, being ftill flowing quantities,

have their Fluxions x, /, z, &c. which are the fourth Fluxions of
x, y, z, &c. And fo we may proceed to fuperior orders, as far as
there mall be occafion. Then, when any Equation is propofed, con-
futing of variable quantities, as the relation of its Fluxions may be
found by what has been taught before ; fo by repeating only the fame
operation, and confidering the Fluxions as flowing Quantities^ the

L 1 relation

258 The Method of FLUXIONS,

relation of the fecond Fluxions may be found. And the like for all
higher orders of Fluxions.

Thus if we have the Equation y* ax = o, in which are the
two Fluents y and x, we fhall have the firft Fluxional Equation zyy

ax - o. And here, as we have the three Fluents j>, y, and x,
if we take the Fluxions again, we fhall have the fecond Fluxional
Equation zyy -+- zy* ax= o. And here, as there are four Fluents
y, y, y, and x, if we take the Fluxions again, we fhall have the

.. .

third Fluxional Equation zyy + zyy -f- ^.yy ax = o, or zyy 4-

bjy ax = o. And here, as there are five Fluents y, y, y, y, and x,
if we take the Fluxions again, we fhall have the fourth Fluxional

Equation zyy + zyy -f- 6yy -+- 6y l ax = o, or zyy -+- Syy -f- 6y*

ax = o. And here, as there are fix Fluents y, y, y, y, y, and x y
if we take the Fluxions again, we fhall have zyy + zyy -f- 8yy -{-

fyy _j_ i zyy ax = o, or zyy +- i oyy -f- zoyy ax = o, for the
fifth Fluxional Equation. And fo on to the fixth, feventh, 6cc.

Now the Demonftration of this will proceed much after the man-
ner as our Author's Demonftration of firft Fluxions, and is indeed
virtually included in it. For in the given Equation^* ax = o }
if we fuppofe y and x to become at the fame time y -f- yo and x-)- xo,
(that is, if we fuppofe yo and xo to denote the fynchronal Moments
of the Fluents y and x,) then by fubftitution we fhall have ~y +yo\ *

a x x -f- xo = o, or in termini* expanjis, y 1 -f- zyyo -+-y*o* ax

axo = o. Where expunging y 1 ax = o, andj/ 1 ^ 1 , and divi-
ding the reft by o, it will be zyy ax = o for the firft fluxional
Equation. Now in this Equation, if we fuppofe the fynchronal
Moments of the Fluents y, y, and x, to beyo } yo t and xo refpedively ;
for thofe Fluents we may fubftitute y -f-jj/o, y -+-yo, and x+ xo in
the kft Equation, and it will become zy-t-zyoxy-l-yo axx + xo
r. o, or expanding, zyy -f- zyyo +- zyyo -+- zy'yoo ax axo = o.
Here becaufe zyy ax= o by the given Equation, and becaufe
zy'yoo vanishes ; divide the reft by o, and we fhall have zy* + zyy

ax = o for the fecond fluxional Equation. Again in this Equa-
tion, if we fuppofe the Synchronal Moments of the Fluents y, y t

y, and x t to be yo, yo, yo, and xo refpedively ; for thofe Fluents



we may fubftitute y+yo, y + yo, y-t-yo, a^id x +- xo in the lad

.. j a

Equation, and it will become 2x7 -\-yo \ +- zy -+- 2yo x y -f- yo
a x x _j_ xo o, or expanding and collecting, 2j* + 6yyo -t- 2y*o l

_}_ 2 yy -+- 2yyo -t- s;^ 1 ax axo = o. But here becaufe 2j' s
_l_ 2/_y rfx = o by the laft Equation ; dividing the reft by o, and
expunging all the Terms in which o will ftill be found, we fliall

have 6yy -+- 2yy ax = o for the third fluxional Equation. And
in like manner for all other orders of Fluxions, and for all other
Examples. Q^ E. D.

To illuftrate the method of rinding fuperior Fluxions by another
Example, let us take our Author's Equation # 5 ax 3 - -{-axy y>
= o, in which he has found the fimpleft relation of the Fluxions
to be 3x^ a zaxx -h axy +- axy 3^/7* = o. Here we have the
flowing quantities x, y, x, y ; and by the fame Rules the Fluxion of
this Equation, when contracled, will be 3#w i + 6x*x 2axx
zax* H- axy -+- 2axy -\- axy 3vy s 6jf !L y = o. And in this Equa-
tion we have the flowing quantities x, y, x,y, x, y, fo that taking
the Fluxions again by the fame Rules, we fhall have the Equation,

when contracted, ^xx l -f- iSxxx -{- 6x 3 2axx 6axx -f- axy -f-

%axy -+- T,a.\y -f. axy 3 yy* i fyyy 6y s = o. And as in this
Equation there are found the flowing quantities x, y, x, y y x, y,

x, y, we might proceed in like manner to find the relations of the
fourth Fluxions belonging to this Equation, and all the following
orders of Fluxions.

And here it may not be amifs to obferve, that as the propofed
Equation expreffes the conflant -elation of the variable quantities x
and y -, and as the firft fluxional Equation exprefles the conftant re-
lation of the variable (but finite i.nd alTignable) quantities x and y,
which denote the comparative Velocity of increafe or decreale of x
and y in the propcfed Equation : So the fecond fluxional Equation
will exprefs the conftant relation of the variable (but finite and aflig-
nable) quantities x and y y which denote the comparative Velocity of
the increafe or decreafe ot .v and_y in the foregoing Equation. And in
the third fluxional Equation we have the conftant relation ot the variable

(but finite and aflignable) quantities .v and r, which will denote the

L 1 2 com-

260 The Method of FLUXIONS,

comparative Velocity of the increafe or decreafe of "x and "y in the
foregoing Equation. And fo on for ever. Here the Velocity of a
Velocity, however uncouth it may found, will be no abfurd Idea
when rightly conceived, but on the contrary will be a very rational
and intelligible Notion. If there be fuch a thing as Motion any how
continually accelerated, that continual Acceleration will be the Ve-
locity of a Velocity ; and as that variation may be continually va-
ried, that is, accelerated or retarded, there will 'be in nature, or at
leafl in the Understanding, the Velocity of a Velocity of a Velocity.
Or in other words, the Notion offecond, third, and higher Fluxions,
muft be admitted as found and genuine. But to proceed :

We may much abbreviate the Equations now derived, by the
known Laws of Analyticks. From the given Equation x* ax 1 -+-
ax y y"' = ^ere is found a new Equation, wherein, becaufe of
two new Symbols x and y introduced, we are at liberty to aflume
another Equation, belides this now found, in order to a jufl De-
termination. For fimplicity-fake we may make x Unity, or any
other conftant quantity ; that is, we may fuppofe x to flow equably,
and therefore its Velocity is uniform. Make therefore x = i } and

the firft fluxional Equation will become 3^* 2ax -+- ay + axy

3j)/)' 1 = o. So in the Equation 3x.v a -f- 6x*x 2axx 2ax* -+.
axy -i- zaxy -h axy 3 vj* 6y\y = o there are four new Sym-
bols introduced, x, y, x, and r, and therefore we may afiume two
other congruous Equations, which together with the two now found,
will amount to a compleat Determination. Thus if for the fake of
fimplicity we make one to be x = i, the other will' neceflarily be
.v =o ; and thefe being fubftituted, will reduce the fecond fluxionaj
Equation to this, 6x 2.0. -f- iay -f- axy ^yy- 6y*y o. And
thus in the next Equation, wherein there are fix new Symbols

x, }', x, y, x, y ; befides the three Equations now found, we may
take x= i, and thence x=o, x= o, which will reduce it

to 6 -f- $ay -+- axy yy* i $yyy 6f> == o. And the like of
Equations of fucceeding orders.

But all thefe Reductions and Abbreviations will be beft made as
the Equations are derived. Thus the propofed Equation being x~>
ax* + axy y= = o, taking the Fluxions, and at the fame time

making x= i, (and confequently x, x, &c. =o,) we (hall have
3** zax + ay + axy zyy* = o. And taking the Fluxions




again, it will be 6x 20. -f- zay +- axy 3 yy* 6y*y = i o.

And taking the Fluxions again, it will be 6 -f- $d'y -+- axy %yy*
6y*> = o. And taking the Fluxions again, it will be

axy 3^ 4 2 4-yy'y i%y 1 y 3677* = o. And fo on, as
far as there is occafion.

But now for the clearer apprehenfion of thefe feveral orders of
Fluxions, I (hall endeavour to illuftrate them by a Geometrical
Figure, adapted to a iimple and a particular cafe. Let us allume
the Equation y 1 r=ax, otyzs=ia*x*, which will therefore belong to
the Parabola ABC, whole Parameter is AP = tf, Abfcifs AD = x,
and Ordinate LD =y ; where AP is a Tangent at the Vertex A.
Then taking the Fluxions, we fhall have y = yaPsve~~*. And fup-
pofing the Parabola to be defcribed by the equable motion of the
Ordinate upon the Abfcifs, that equable Velocity may be expounded
by the given Line or Parameter a, that is, we may put x = a. Then

\t\v]\ibey=(a*x *= ~ = "? = ) -?- , which will give us

zx k 2X ' 2X '

this Conftrudtion. Make x (AD) : y (BD) :: a (|AP) : DG =
= y, and the Line DG will therefore

zx J

reprefent the Fluxion of y or BD. And if
this be done every where upon AE, (or if
the Ordinate DG be fuppos'd to move upon
AE with a parallel motion,) a Curve GH
will be conftiucted or delcribed, whofe Ordi-
nates will every where expound the Fluxions
of the correfponding Ordinates of the Pa-
rabola ABC. This Curve will be one of
the Hyperbola's between the Afymptotes


AE and AP ; for its Equation isjx= -11 ,

Or yy = .

Again, from the Equation y = " , or 2 *y = ay, by taking
the Fluxions again, and putting x =a as before, we fhall have
zay -{- 2xy=aj,ory = J j where the negative fign {hews only,

that_y is to be confider'd rather as a retardation than an acceleration,
or an acceleration the contrary way. Now this will give us the


202 ?2* Method of FLUXIONS,

following Conftruaion. Make x (AD) : y (DG) : : \a (iAP) ;
DI = y, and the Line DI will therefore reprefent the Fluxion of
DG, or of j, and therefore the fecond Fluxion of BD, or of/.
And if this be done every where upon AE, a Curve IK will be
comlructed, whofe Ordinates will always expound the fecond Fluxions
of the correfponding Ordinates of the Parabola ABC. This Curve
likewife will be one of the Hyperbola's, for its Equation is y =

/Jy fl* G. *

a* ^ 1 6*5

Again, from the Equation y = ^- v , or 2xy = ay' t

^by taking the Fluxions we mail have 2ay zxy =: ay., or
~ y=~ , which will give us this Conftrudlion. Make x (AD) :

y (DI) :: \a (|AP) : DL=y, and the Line DL will therefore
reprefent the Fluxion of DI, or of y, the fecond Fluxion of DG,
or of y, and the third Fluxion of BD, or of^. And if this be
done every where upon AE, a Curve LM will be conflructed, whofe
Ordinates will always expound the third Fluxions of the correfpon-
ding Ordinates of the Parabola ABC. This Curve will be an Hyper-
bola, and its Equation will be y=. '=-1 ; , or yy= 64*"* "

And fo we might proceed to conftrucl Curves, the Ordinates of
which (in the prefent Example) would expound or reprefent the
fourth, fifth, and other orders of Fluxions.

We might likewife proceed in a retrograde order, to find the.
Curves whofe Ordinates mall reprefent the Fluents of any of thefe


Fluxions, when given. As if we had y = , = La*xx~* } or if
the Curve GH were given ; by taking the Fluents, (as will be
taught in the next Problem,) it would be y = (a^x*= ^-r = )

- , which will give us this Conftruction. Make \a (|AP) :

.v (AD) :: y (DG) : DB = 2 -J , and the Line DB will reprefent

the Fluent of DG, or of y. And if this be done every where upon
the Line AE, a Curve AB will be con ftru died, whofe Ordinates
will always expound the Fluents of the correfponding Ordinates of
the Curve GH. This Curve will be the common Parabola, whofe

i Parameter


Parameter is the Line AP = a. For its Equation is y = a*x'* t
or yy=ax.

So if we had the Parabola ABC, we might conceive its Ordinates
to reprefent Fluxions, of which the correfponding Ordinates
of fome other Curve, fiippofe QR, would reprefent the Fluents.

To find which Curve, put y for the Fluent of y, y for the Fluent

/ Iff n I .. .:

of y, &c. (That is, let, &c. _/, y, y, /, j/, y, y, &c. be a Series of
Terms proceeding both ways indefinitely, of which every fucceed-
ing Term reprefents the Fluxion of the preceding, and vice versa ;
according to a Notation of our Author's, deliver'd elfewhere.) Then

becaufe it is_y = (div*=<z^x^ =) ^r , taking the Fluents it

' .x ' % \

will be y = [, = 2f!i! = ) Z J2. ; which will give us this Con-
W 3 y 3*

ftrudion. Make $a (|AP) : ft (AD) :: y (BD) : -^ =y = DQ^

and the Line DQ^will reprefent the Fluent of DB, or of y. And
if the fame be done at every point of the Line AE, a Curve QR
will be form'd, the Ordinates of which will always expound the
Fluents of the correfponding Ordinates of the Parabola ABC. This
Curve alfo will be a Parabola, but of a higher order, the Equation

3. I I

of which is^= * , or yy = .


Again, becaufe y = f zx ~ == 3ilJL = \ 2 ^fl . taking the Flu-

\ $a? $a l v. a J ->.a.'-

" / "* i 7

ents it will be y=( JfL.sff! |x"= W , which will give us this

Conftruaion. Make | (|AP) : x (AD) : : y ( DQ^J : = _y


= DS, and the Line DS will reprefent the Fluent of DQ^, or of_y.
And if the fame be done at every point of the Line AE, a Curve
ST will thereby be form'd, the Ordinates of which will expound
the Fluents of the correfponding Ordinates of the Curve QR. This

// i ////

Curve will be a Parabola, whofe Equation is jy= 1^1 , or yy =

^-. . And fo we might go on as far as we pleafe,


264 The Method of FLUXIONS,

Laftly, if we conceive DB, the common Ordinate of all thefe
Curves, to be any where thus conftrucled upon AD, that is, to be
thus divided in the points S, Q^ B, G, I, L, 6cc. from whence to
AP are drawn Ss, Qtf, B^, Gg, I/, L/, 6cc. parallel to AE ; and
if this Ordinate be farther conceived to move either backwards or
forwards upon AE, with an equable Velocity, (reprefented by
AP = tf = x,) and as it defcribes thefe Curves, to carry the afore-
faid Parallels along with it in its motion : Then the points s, q, b,g,
i, /, &c. will likewife move in fuch a manner, in the Line AP, as
that the Velocity of each point will be reprefented by the diflance
of the next from the point A. Thus the Velocity of s will be re-
prefented by Aq, the Velocity of q by A, of b by Ag, of g by A/,
of / by A/, &c. Or in other words, Aq will be the Fluxion of A.S ;
Al> will be the Fluxion of Ag, or the fecond Fluxion of As ; Ag
will be the Fluxion of Ab, or the fecond Fluxion of Aq, or the third.
Fluxion of As ; Ai will be the Fluxion of Ag, or the fecond Fluxion
of Ah, or the third Fluxion of Aq, or the fourth Fluxion of As ;
and fo on. Now in this inftance the feveral orders of Fluxions, or
Velocities, are not only expounded by their Proxies and Reprefen-
tatives, but alfo are themfelves actually exhibited, as far as may be
done by Geometrical Figures. And the like obtains wherever elfe
we make a beginning ; which fufficiently mews the relative nature

Online LibraryIsaac NewtonThe method of fluxions and infinite series : with its application to the geometry of curve-lines → online text (page 23 of 30)