476 FLUX 
ral method of deriving their afFe£Vions from their geneiis, 
vve conceive the quantities to be increafed and diminifhed, 
or to be wholly generated by motion, or by a continual 
flux analogous to it. The quantity that is thus generated 
is fetid to [flow, and is hence called a fluent. 
METHOD of FLUXIONS. 
Definitions. —i. Every quantity is here confidered 
as generated by motion ; a line by the motion of a point; 
a furface by the motion of a line; a folid by the motion 
of a furface.—Sir Ifaac Newton, in the Introduction to 
his Quadrature of Curves, obferves, that “ thefe genefes 
really take place in the nature of things, and are daily 
feen in the motion of bodies. And after this manner, the 
ancients, by drawing moveable right lines along immove¬ 
able right lines, taught the geneiis of rectangles.” 
2. The quantity thus generated is called the fluent, or 
flowing quantity. 
3. T he velocities with which flowing quantities increafe 
or decreafe at any point of time, are called the fluxions of 
thofe quantities at that inftant. 
Cor. 1. As the velocities are in proportion to the in¬ 
crements or decrements uniformly generated in a given 
time, fuch increments or decrements will reprefent the 
fluxions.—This is agreeable to fir Ifaac Newton’s ideas 
on the fubjecl. He fays, “ I fought a method of deter¬ 
mining quantities from the velocities of the motions or 
increments with which they are generated; and calling 
thefe velocities of the motions or increments, fluxions, and 
the generated quantities fluents, 1 fell by degrees upon 
the method of fluxions.” 
Cor. 2. ITence, as any given time may be aflumed, the 
fluxion is not an abfolute but a relative quantity. When 
we have feveral cotemporary fluxions, we may alfume one 
fluxdon what we pleafe, and thence determine the values 
of the others. Thus, if x and y increafe uniformly, and 
if x increafe by p in the time that y increafes by q, then 
the cotemporary increments of xandj will be p and q, 
2 p and jj, 3 p and 3 q, &c. hence, if p be affirmed the 
fluxion of x, the fluxion of y will be y; if the former 
fluxion be 2 p, the latter will be 2 q, See. Sec. 
Cor. 3. A conflant quantity has no fluxion. 
4. The fir ft letters, a, b, c, See. of the alphabet are 
ufually put for conftant quantities, and the laft v, zo, x, 
y, z, for variable ones; and they are to be thus under- 
ftood, unlefs the contrary be exprefled. 
5. The fluxion of a limple quantity as x, is exprefled 
by placing a point over it, thus x. 
To find the FLUXIONS of QUANTITIES. 
Prop. I.— If two quantities increafe or decreafe uniformly, 
the increments or decrements generated in a given time will be 
as their fluxions. 
6. This appears from Art. 3. Cor. 1. 
Prop. II.— If one quantity increafe uniformly, and another 
of the fame kind increafe with an accelerated or retarded velocity, 
and two increments be affumed which are generated in the fame 
time ; if thofe increments be diminifhed till they vanifh, that ratio 
to which they approach as their limit is the ratio of the fluxions 
of thofe quantities. 
7. Let the line FK be deferibed with an uniform velo¬ 
city, and AZ with an accelerated velocity, and let the 
increments Gr, Pm, be generated in the fame time ; let 
alfo Pv be the increment that would have been generated 
A___ P ffl Z 
in the fame time, if the velocity at P had been continued 
uniform; then by Prop. 1. the fluxions of FK, AZ at 
the points G and P will be reprefented by Gs and Py. 
Let v be the velocity with which Gs is deferibed, and V 
the velocity with which Pv is deferibed, and let the ve- 
I O N S. 
locity at m be V-pr; then vm is the increment which is 
deferibed in confequence of the increafe r of vel. city 
finee the deferibing point left P. Now let Pm be de¬ 
feribed with the uniform velocity V + ra in the fame time 
that Py and Pa;, are deferibed; then it is manifeft, that 
this uniform velocity muft be between the velocities at 
P and at m, that is, V-pw is greater than V and lefts than 
V+r, or zo is greater than 0 and lefs than r. Alfo, finee 
the fpaces deferibed in the fame time are as the veloci¬ 
ties, V : V-fa; :: Py : Pm: Now diminifli the times in 
which thefe increments are deferibed ; then as the points 
v and m approach to P, Py will continue to be deferibed 
with the uniform velocity V; but r will be diminifhed, 
and by diminifhing the time till it becomes indefinitely 
fmall, r will become indefinitely fmall ; but vm is de¬ 
feribed in confequence of this increafe r of velocity ; 
hence, when r becomes indefinitely fmall in refpedt to V, 
the fpace vm muft become indefinitely fmall in refpedt to 
Py; therefore the ratio Py : Pm is, in that ftate, indefi¬ 
nitely near to a ratio of equality ; but it is manifeft that 
it can never become accurately a ratio of equality, be- 
caufe vm will not vanifh until Py and Pm vanifh ; confe- 
quently the ratio of the adtual increments Gs : Pm can 
never actually exprefs the ratio of the fluxions, that ratio 
being exprefled by the ratio of Gs : Py. Let us then 
confider, what ratio Py : Pm approaches as its limit, when 
we make the time in which the increments are deferibed, 
and confequently the increments themfelves, vanifh. In 
every ftate of thefe increments, V : V-pzv :: Py : Pm ; and 
by continually diminifhing the time, and confequently 
the increments, we diminifli r, and confequently zo, but 
V remains conftant; it is manifeft, therefore, that the 
ratio of V : V-p zo, and confequently that or Py : Pm, 
continually approaches to a ratio of equality, agreeable 
to what we have already (hewn; and when the time, and 
confequently the increments, become actually — o, then 
r=o ; confequently wr=o : therefore the limit of the ra¬ 
tio of Py : Pm becomes that of V : V, a ratio of equality. 
Hence, the limit of the ratio of Gs : Pm is the fame as the 
limit of ratio of Gs : Py, or as Gs : Py, that ratio being 
conftant; that is, the limiting ratio of the increments is the 
ratio of the fluxions. 
The fame is manifeftly true for the limiting ratio of 
the decrements of two quantities ; for,, conceiving the 
deferibing points to move backwards, and to be retarded 
by the fame law, the decrements sG, mP, in this cafe be¬ 
come the fame as the increments in the other ; confe¬ 
quently their limiting ratio will exprefs the ratio of the 
fluxions at G and P, or the rate at which FG, AP, are, 
at that inftant, decreafing. 
Cor. t . Hence, the limiting ratio of the increments or 
decrements of two quantities which are both generated by 
variable velocities, will be the ratio of their fluxions. 
And as the velocities with which thefe two lines increafe 
or decreafe may be made to agree with the rate of in¬ 
creafe or decreafe of any two like quantities, the propo- 
fition muft be true for quantities of any kind. 
Cor. 2. As the limiting ratio of the increments is the 
ratio of the fluxions, it'’is manifeft that when the incre¬ 
ments are in an increafing or decreafing ftate," the fluxions 
will be increafing or decreafing. 
8. It has been laid, that when the increments are ac¬ 
tually vanifhed, it is abf'urd to talk of any ratio between 
them. It is true; but we fpeak not here of any ratio 
then exifting between the quantities, but of that ratio to 
which they have approached as their limit; and that ra¬ 
tio ftill remains. Thus, let the increments of two quan¬ 
tities be denoted by <*x 2 -pyzx and bx 2 -\-nx\ then the limit 
of their ratio, when x— o, is m : n ; for ax 1 -\-rnx : bx 2 flnx 
axfm : bx-\-n :: (when x=o) m : n. As the quantities 
therefore approach to nothing, the ratio approaches to 
that of m : n as its limit. Hence, if m—n , the limit of this 
ratio is a ratio of equality. We mult therefore be care¬ 
ful to diftinguifh between the ratio of two evanelcent 
quantities and the limit of their ratio j the former ratio 
^ never 
