Isaac Newton.

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

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of all thefe orders of Fluxions and Fluents, and that they differ from
each other by mere relation only, and in the manner of conceiving.
And in general, what has been obferved from this Example, may
be eafily accommodated to any other cafes whatfoever.

Or thefe different orders of Fluents and Fluxions may be thus ex-
plain'd abftractedly and Analytically, without the afliftance of Curve-
lines, by the following general Example. Let any conflant and
known quantity be denoted by a, and let a" be any given Power
or Root of the lame. And let x n be the like Power or Root of
the variable and indefinite quantity x. Make a m : x m : : a : y, or


y = ^ = a l ~ m x m . Here y alfo will be an indefinite quantity,


which will become known as foon as the value of x is affign'd.
Then taking the Fluxions, it will be y = ma l ~ m xx m ~ 1 ; and fup-
pofing x to flow or increafe uniformly, and making its constant
Velocity or Fluxion x = a, it will be y = ma* m x m -*. Here if

for a 1 n x m we write its value y, it will be y = , that is, x :

ma : : y : y. So that y will be alfo a known and affignable Quan-


tity, whenever x (and therefore y) is affign'd. Then taking the
Fluxions again, we mall have^=wxw ia* "xx"- 1 - = ;;; x
; irtS"""^*"" 1 ; or for ma"-~ m x m ~ l writing its value y, it will be

y = ~~ x ta - v , that is, x : m la : : y : y. So that y will be-
come a known quantity, when x (and therefore y and y) is affign'd.
Then taking the Fluxions again, we fhall have y = m x m i x
m ?.a*- m \ m -*, or y=.-^~ , that is, x : m za :: y : y

where alfo y will be known, when x is given. And taking the
Fluxions again, we fhall have y = rnx m i x m 2 x m

= - - ; that is, x : m 30 :: y : /. So that y will alfo be
known, whenever x is given. And from this Inductipn we may
conclude in general, that if the order of Fluxions be denoted by any
integer number ?/, or if n be put for the number of points over the

^_____ n ll-l-i

Letter y t it will always be x : m na : : y : y ; or from the
Fluxion of any order being given, the Fluxion of the next imme-
diate order may be hence found.

_______ "+t n

Or we may thus invert the proportion m na : x : : y : y }
and then from the Fluxion given, we fhall find its next immedi-

ate Fluent. As if = 2, 'tis m za : A; : : y : y. If n - i
'tis m \a : x :: y : y. If 72 = 0, 'tis ma : x : : y : y. And
obferving the fame analogy, if n== i, 'tis m - ia : x :: y :

y ; where y is put for the Fluent of; 1 , or for y with a negative point.
And here becaufe y=.a l - m x m , it will be m 4- la : x :: a 1 -" 1 *" :

' I ~V+ l v" 1 -*- 1

y, or y = _ = ^ : which alfo may thus appear. Be-

m-\-\a m-\-\a

caufe y = {a*-<"x*> = __Zj__T =) il , taking the Fluents, (fee the

/ m-f,,

next Problem,) it will be y = ^ - . Again, if we make = 2,


. - I II I +...

tism-{-2a : x :: y ; y } or y = .. v .. = * - . For


M m becaufe


The Method of FLUXIONS,

becaufe y = .f - x - ==


, taking the Fluents it will be


-. _ m+ i . Again, if we make = 3, 'tis m -|-

And fo for

-t-3 m + l X m -\- 2 x j+3a l "~'~*

all other fuperior orders of Fluents.

And this may fuffice in general, to mew the comparative nature
and properties of thefe feveral orders of Fluxions and Fluents, and
to teach the operations by which they are produced, or to find their
refpeftive fluxional Equations. As to the ufes they may be apply 'd
to, when found, that will come more properly to be confider'd in
another place.

SECT. III. Tfte Geometrical and Mechanical Elements

of Fluxions,

THE foregoing- Principles of the Doftrine of Fluxions being
chiefly abftradted and Analytical I mail here endeavour, af-
ter a general manner, to (hew fomething analogous to them in Geo-
metry a.nd Mechanicks ; by which they may become, not only the
objeft of the Underftanding, and of the Imagination, (which will only
prove their poffible exiftence,) but even of Senfe too, by making
them adually to exift in a vifible and fenfible form. For jt is now
become neceffary to exhibit them all manner of ways, in order to
give a fatisfaclpry proof, thai they have indeed any real exiftence at

And fir ft, by way of prepara-
tion, it will be convenient to con-
fider Uniform and equable motions,
as alfo fuch as are alike inequable.

Let the right Line AB be defcribed

by the equable motion of a point,

which is now at E, and will pre-

fently be at G. Alfo let the Line

CD, parallel to the former, be de-

fcribed by the equable motion of a point, which is in H and K, at

the farne times as the former is in E and G. Then will EG and

HK be contemporaneous Lines, and therefore will be proportional to




the Velocity of each moving point refpedlively. Draw the indefi-
nite Lines EH and GK, meeting in L ; then becaufe of like Tri-^
angles ELG and HLK, the Velocities of the points E and H, which
were before as EG and HK, will be now as EL and HL. Let
the defcribing points G and K be conceived to move back, again,
with the fame Velocities, towards A and C, and before they ap-
proach to E and H let them be found in g and ^, at any fmall
diftance from E and H, and draw gk, which will pafs through L ;
then ftill their Velocities will be in the ratio of Eg and H/, be thofe
Lines ever fo little, that is, in the ratio of EL and HL. Let
the moving points g and k continue to move till they coincide with
E and H ; in which cafe the decreeing Lines Eg and H will pafs
through all polYible magnitudes that are lefs and lefs, and will finally
become vanishing Lines. For they muft intirely vanifh at the fame
moment, when the points g and k mall coincide with E and H.
In all which ftates and circumftances they will ftill retain the ratio
of EL to HL, with which at laft they will finally vaniih. Let
thofe points ftill continue to move, after they have coincided with
E and H, and let them be found again at the fame time in y and
K, at any diftance beyond E and H, Still the Velocities, which are
now as Ey and H*, and may be efteemed negative, will be as EL
and HL, whether thofe Lines Ey and Hx are of any finite magni-
tude, or are only nafcent Lines ; that is, if the Line yx.L, by its
angular motion, be but juft beginning to emerge and divaricate from
EHL. And thus it will be when both thefe motions are equable
motions, as alfo when they are alike inequable ; in both which
cafes the common interfedlion of all the Lines EHL, GKL, gkL, &c.
-will be the fixt point L. But when either or both thefe motions
are fappos'd to be inequable motions, or to be any how continually
accelerated or retarded, thefe Symptoms will be fomething different ;
for then the point L, which will ftill be the common interfeclion
of thofe Lines when they firft begin to coincide, or to divaricate,
will no longer be a fixt but a moveable point, and an account muft
be had of its motion. For this purpofe we may have recourfe to
the following Lemma.

Let AB be an indefinite and fixt right Line, along which anothe:
indefinite but moveable right Line DE may be conceived to move or
roll in fuch a manner, as to have both a progreflive motion, as alfo aa
angular motion about a moveable Center C. That is, the common
interfection C of the two Lines AB and DE may be fuppofed to
move with any progreffive motion from A towards B, while at the

M m 2 fame

26S The Method of FLUXIONS,

tame time the moveable Line DE revolves about
the lame point C, with any angular motion. Then
as the Angle ACD continually decreafes, and at
laft vanifhes when the two Lines ACB and DCE
coincide ; yet even then the point of interfection
C, (as it may be ftill call'd,) will not be loft and
annihilated, but will appear again, as foon as the
Lines begin to divaricate, or to feparate from each
other. That is, if C be the point of interfeclion
before the coincidence, and c the point of interfec-
tion after the coincidence, when the Line dee {hall
again emerge out of AB ; there will be fome inter-
mediate point L, in which C and c were united in
the fame point, at the moment of coincidence. This
point, for diftin&ion-fake, may be call'd the Node,
or the point of no divarication. Now to apply this
to inequable Motions :

Let the Line AB be defcribed by the continually accelerated mo-
tion of a point, which is now in E, and will be prefently found inr
G. Alfo let the Line CD, parallel to the former, be defcribed by
the equable mo-
tion of a point,
which is found
in H and K, at
the fame times as
the other point
is in E and G.
Then willEG and
HK be contem-
poraneous Lines ;
and producing
EH and GK till
they meet in I,
thofe contempo-
raneous Lines will be as El and HI refpedlively. Let the defcribing
points G and K be conceived to move back again towards A and C,
each with the fame degrees of Velocity, in every point of their mo-
tion, as they had before acquired ; and let them arrive at the fame
time at g and k, at fome fmall diftance from E and H, and draw
gki meeting EH in /. Then Eg and Hk, being contemporary Lines
alfo, and very little by fuppofuion, they will be nearly as the Ve-



locities at g and k, that is, at E and H ; which contemporary
Lines will be now as E/' and H/'. Let the points g and k continue
their motion till they coincide with E and H, or let the Line GKI
or gki continue its progremve and angular motion in this manner,
till it coincides with EHL, and let L be the Node, or point of no
divarication, as in the foregoing Lemma. Then will the laft ratio
of the vanifhing Lines Eg- and lik, which is the ratio of the Velo-
' cities at E and H, be as EL and HL refpe&ively.

Hence we have this Corollary. If the point E (in the foregoing
figure,) be fuppos'd to move from A towards B, with a Velocity
any how accelerated, and at the fame time the point H moves from
C towards D with an equable Velocity, (or inequable, if you pleafe ; )
thofe Velocities in E and H will be refpectively as the Lines EL and
HL, which point L is to be found, by fuppofmg the contemporary
Lines EG and HK continually to dkninim, and finally to vanim.
Or by fuppofmg the moveable indefinite Line GKI to move with a
progreffive and angular motion, in fuch manner, as that EG and
HK fhall always be contemporary Lines, till at laft GKI mall co-
incide with the Line EHL, at which time it will determine the Node
L, or the point of no divarication. So that if the Lines AE and
CH reprefent two Fluents, any how related, their Velocities of de-
fcription at E and H, or their refpe&ive Fluxions, will be in the
ratio of EL and HL.

And hence it will fol- ^

low alfo, that the Lo-
cus of the moveable
point or Node L-, that
is, of all the points of C
no divarication, will be
fome Curve-line L/, to
which the Lines EHL
and GK/ will always be
Tangents in L and /.
And the nature of this
Curve L/ may be deter-
mined by the given re-
lation of the Fluents or Lines AE and CH ; and vice versa. Or
however the relation of its intercepted Tangents EL and HL may
be determined in all cafes ; that is, the ratio of the Fluxions of the
given Fluents.




270 tte Method of FLUXLONS,

For illuftration-fake, let us apply this to an Example. Make the
Fluents AE= y and CH ;= x, and let the relation of thefe be always
exprefs'd by this Equation y = x". Make the contemporary Lines
EG = Y and HKs=X.; and becaufe AE and CH are contempo-
rary by fuppofition, we fhall have the whole Lines AG and CK
contemporary alfo, and thence the Equation y -f-Y= x -j-X | . This
by our Author's Binomial Theorem will produce y -+- Y = x" +

-nx"~ 1 X -+- n x"-^-x*~*X* , &c. which ( becaufe y = x" ) will be-
come Y= x"- I X-J- x ^-^Ar'-^X 1 , &c. or in an Analogy, X :

y :: i : nx n ~ l + x ^-^""^X, &c. which will be the general re-
lation of the contemporary Lines or Increments EG and HK. Now
let us fuppofe the indefinite Line GKI, which limits thefe contem-
porary Lines, to return back by a progrefiive and angular motion,
fo as always to intercept contemporary Lines EG and HK, and
finally to coincide with EHL, and by that means to determine the
Node L; that is, we may fuppofe EG = Y and HK = X, to di-
minifli hi i-nfinitum, and to become vanifhing Lines, in which cafe
we fhall have X : Y : : i : nx"~ l . But then it will be like wife X :
Y : : HK : EG :: HL : EL : : x : y, or i : nx"~' : : x :y, ory=nxx'.
And hence we may have an expedient for exhibiting Fluxions
and Fluents Geometrically and Mechanically, in all circumftances,
fo as to make them the objects of Senfe and ocular Demonftration.
Thus in the laft figure, let the two parallel lines AB and CD be de-
fcribed by the motion of two points E and H, of which E moves
any how inequably, and (if you pleafe) H may be fuppos'd to move
equably and uniformly ; and let the points H and K correfpond to
E and G. Alfo let the relation of the Fluents AE =r y and
CH = x be defined by any Equation whatever. Suppofe now the
defcribing points E and H to carry along with them the indefinite
Line EHL, in all their motion, by which means the point or Node
L will defcribe fome Curve L/, to which EL will always be a Tan-
gent in L. Or fuppofe EHL to be the Edge of a Ruler, of an in-
definite length, which moves with a progreffive and angular mo-
tion thus combined together ; the moveable point or Node L in this
Line, which will have the leaft angular motion, and which is always
the point of no divarication, will defcribe the Curve, and the Line
or Edge itfelf will be a Tangent to it in L. Then will the feg-
ments EL and HL be proportional to the Velocity of the points
E and H refpeclively ; or will exhibit the ratio of the Fluxions y
a-nd x, belonging to the Fluents AE=y and CF = x.

i Or


Or if we fuppofe the Curve L/to be given, or already conftmcled,
we may conceive the indefinite Line EHIL to revolve or roll about
it, and by continually applying itfelf to it, as a Tangent, to move
from the fituation EHIL to GK.ll. Then will AE and CH be the
Fluents, the fenfible velocities of the defcribing points E and H will
be their Fluxions, and the intercepted Tangents EL and HL will
be the redlilinear meafures of thofe Fluxions or Velocities. Or it
may be reprefented thus : If L/ be any rigid obftacle in form of a
Curve, about which a flexible Line, or Thread, is conceived to be
wound, part of which is ftretch'd out into a right Line LE, which
will therefore touch the Curve in L ; if the Thread be conceived to
be farther wound about the Curve, till it comes into the fituation
L/KG ; by this motion it will exhibit, even to the Eye, the fame
increafing Fluents as before, their Velocities of increafe, or their
Fluxions, as alfo the Tangents or rectilinear reprefentatives of thofe
Fluxions. And the fame may be done by unwinding the Thread,
in the manner of an Evolute. Or inftead of the Thread we may
make ufe of a Ruler, by applying its Edge continually to the
curved Obftacle L/, and making it any how revolve about the move-
able point of Contadl L or /. In all which manners the Fluents,
Fluxions, and their rectilinear meafures, will be fenfibly and mecha-
nically exhibited, and therefore they muft be allowed to have a place
in rernm naturd. And if they are in nature, even tho' they were but
barely pofiible and conceiveable, much more if they are fenfible
and vifible, it is the province of the Mathematicks, by fome me-
thod or other, to investigate and determine their properties and pro-

Or as by one Thread EHL, perpetually winding about the curved
obftacle L/, of a due figure, we mall fee the Fluents AE and CH
continually to increafe or decreafe, at any rate aflign'd, by the mo-
tion of the Thread EHL either backwards or forwards ; and as we
(hall thereby fee the comparative Velocities of the points E and H,
that is, the Fluxions of the Fluents AE and CH, and alfo the Lines
EL and HL, whofe variable ratio is always the rectilinear meafure of
thofe Fluxions : So by the help of another Thread GK/L, wind-
ing about the obftacle in its part /L, and then ftretching out into a
right Line or Tangent /KG, and made to move backwards or for-
wards, as before ; if the firft Thread be at reft in any given fitua-
tion EHL, we may fee the fecond Thread defcribe the contempo-
porary Lines or Increments EG and HK, by which the Fluents
AE and CH are continually increafed ; and if GK/ is made to ap-

272 ^ e Method of FLUXIONS,

proach towards EHL, we may fee thofe contemporary Lines conti-
imallv to diminim, and their ratio continually approaching towards
the ratio of EL to HL ; and continuing the motion, we may pre-
fently fee thofe two Lines actually to coincide, or to unite as one
Line, and then we may fee the contemporary Lines actually to va-
ntfh at the fame time, and their ultimate ratio actually to become
that of EL to HL. And if the motion be ftill continued, we mall
fee the Line GK/ to emerge again out of EHL, and begin to de-
fcribe other contemporary Lines, whofe nafcent proportion will be
that of EL to HL. And fo we may go on till the Fluents are ex-
haufted. All thefe particulars may be thus eafily made the objects
of fight, or of Ocular Demonftration.

This may ftill be added, that as we have here exhibited and re-
prefented firft Fluxions geometrically and mechanically, we may do
the fame thing, mutatis mutandis, by any higher orders of Fluxions.
Thus if we conceive a fecond figure, in which the Fluential Lines fhall
increafe after the rate of the ratio of the intercepted Tangents (or the
Fluxions) of the firft figure ; then its intercepted Tangents will ex-
pound the ratio of the fecond Fluxions of the Fluents in the firft
figure. Alfo if we conceive a third figure, in which the Fluential
Lines fhall increafe after the rate of the intercepted Tangents of
the fecond figure ; then its intercepted Tangents will expound the
third Fluxions of the Fluents in the firft figure. And fo on as far
as we pleafe. This is a neceflary confequence from the relative na-
ture of thefe feveral orders of Fluxions, which has been fhewn be-

And farther to mew the univerfality of this Speculation, and how
well it is accommodated to explain and reprefent all the circumftan-
ces of Fluxions and Fluents; we may here take notice, that it may
be alfo adapted to thofe cafes, in which there are more than two
Fluents, which have a mutual relation to each other, exprefs'd by
one or more Equations. For we need but introduce a third parallel
Line, and fuppofc it to be defcribed by a third point any how mov-
ing, and that any two of thefe defcribing points carry an indefinite
Line along with them, which by revolving as a Tangent, defcribes
the Curve whofe Tangents every where determine the Fluxions. As
alfo that any other two of thofe three points are connected by an-
other indefinite Line, which by revolving in like manner defcribes
another fuch Curve. And fo there may be four or more parallel
Lines. All but one of thefe Curves may be affumed at pleafure,
when they are not given by the ftate of the Queftion. Or Analy-

' ///. j //'//<'. i , i( //. uvuutm / v /( v Y fa /?////



tically, fo many Equations may be aflumed, except one, (if not
given by the Problem,) as is the number of the Fluents concern'd.

But laftly, I believe it may not be difficult to give a pretty good
notion of Fluents and Fluxions, even to fuch Perlbns as are not
much verfed in Mathematical Speculations, if they are willing to be
iniorm'd, and have but a tolerable readinefs of apprehenfion. This
I {hall here attempt to perform, in a familiar way, by the inftance
of a Fowler, who is aiming to (lioot two Birds at once, as is re-
prefented in the Frontifpiece. Let us fuppofe the right Line AB
to be parallel to the Horizon, or level with the Ground, in which
a Bird is now flying at G, which was lately at F, and a little be-
fore at E. And let this Bird be conceived to fly, not with an equable
or uniform fwiftnefs, but with a fwiftnefs that always increafes, (or
with a Velocity that is continually accelerated,) according to fome
known rate. Let there alfo be another right Line CD, parallel to
the former, at the fame or any other convenient diftance from the
Ground, in which another Bird is now flying at K, which was lately
at I, and a little before at H ; juft at the fame points of time as the
firft Bird was at G, F, E, refpectively. But to fix our Ideas, and
to make our Conceptions the more fimple and eafy, let us imagine
this fecond Bird to fly equably, or always to defcribe equal parts of
the Line CD in equal times. Then may the equable Velocity of
this Bird be ufed as a known meafure, or ftandard, to which we
may always compare the inequable Velocity of the firft Bird. Let
us now fuppofe the right Line EH to be drawn, and continued to
the point L, fo that the proportion (or ratio) of the two Lines EL
and HL may be the fame as that of the Velocities of the two
Birds, when they were at E and H refpeclively. And let us far-
ther fuppole, that the Eye of a Fowler was at the fame time at the
point L, and that he directed his Gun, or Fowling-piece, according
to the right Line LHE, in hopes to moot both the Birds at once.
But not thinking himfelf then to be fufficiently near, he forbears
to difcharge his Piece, but ftill pointing it at the two Birds, he
continually advances towards them according to the direction of his
Piece, till his Eye is prefently at M, and the Birds at the fame time in F
and I, in the fame right Line FIM. And not being yet near enough,
we may fuppofe him to advance farther in the fame manner, his
Piece being always directed or level'd at the two Birds, while he
himfelf walks forward according to the direction of his Piece, till
his Eye is now at N, and the Birds in the fame right Line with
his Eye, at K and G. The Path of his Eye, delcribed by this

N a double

274 The Method of FLUXIONS,

double motion, (or compounded of a progreffive and angular mo-
tion,) will be ibme Curve-line LMN, in the fame Plain as the reft
of the figure, which will have this property, that the proportion of
the diftances of his Eye from each Bird, will be the fame every
where as that of their refpeftive Velocities. That is, when his Eye
was at L, and the Birds at E and H, their Velocities were then as
EL and HL, by the Conftruftion. And when his Eye was at M,
and the Birds at F and I, their Velocities were in the fame propor-
tion as the Lines FM and IM, by the nature of the Curve LMN".
And when his Eye is at N, and the Birds at G and K, their Velo-
cities are in the proportion of GN to KN, by the nature of the
fame Curve. And fo univerfally, of all other fituations. So that
the Ratio of thofe two Lines will always be the fenfible meafure of
the ratio of thofe two fenfible Velocities. Now if thefe Velocities,
or the fwiftneffes of the flight of the two Birds in this inflance, are
call'd Fluxions; then the Lines defcribed by the Birds in the fame
time, may be call'd their contemporaneous Fluents; and all inftances
whatever of Fluents and Fluxions, may be reduced to this Example,
and may be illuflrated by it.

And thus I would endeavour to give fome notion of Fluents and
Fluxions, to Perfons not much converfant in the Mathematicks j
but fuch as had acquired fome fkill in thefe Sciences, I would thus
proceed farther to inflrudl, and to apply what has been now deliver'd.
The contemporaneous Fluents being EF=_y, and Hl=.v, and
their rate of flowing or increafing. being fuppos'd to be given or
known ; their relation may always be exprefs'd by an Equation,
which will be compos'd of the variable quantities x andjy, together
with any known quantities. And that Equation will have this pro-
perty, becaufe of thofe variable quantities, that as FG and IK, EG
and HK, and infinite others, are alfo contemporaneous Fluents; it
will indifferently exhibit the relation of thofe Lines alfo, as well as
of EF and HI ; or they may be fubflituted in the Equation, inftead
of x and y. And hence we may derive a Method for determining
the Velocities themfelves, or for finding Lines proportional to them.
For making FG =Y,.and IK = X ; in the given Equation I may
fubftitute y -}- Y inftead of ^y, and x -f- X inftead of x, by which
I fhall obtain an Equation, which in all circumftances will exhibit
the relation of thofe Quantities or Increments. Now it may be plainly
perceived, that if the Line MIF is conceived continually to approach
nearer and nearer to the Line NKG, (as jufl now, in the inftance

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