FLUXIONS. 
479 
its proper units ; although therefore there is no relation 
between the area and either of its Tides, yet it is expreffed 
in terms of the Tides. And if a be conftant and x varia¬ 
ble, the fluxion of the area will be ax by Prop. 3 ; if 
therefore (x) the fluxion of the abfcifta x be 1 inch in 
length , the correfponding fluxion of the area will be a 
fquare inches ; if x be 2 inches in length , the fluxion of 
the area will be 2 a fquare inches. And in general, when 
we confider any two quantities which are not homoge¬ 
neous, although their fluxions, which are exprefted by 
their increments uniformly generated in a given time, can 
have no relation to each other, if we carry our ideas no 
further than the increments themfelves ; yet when we 
confider the numerical values of thefe fluxions, the ana¬ 
lytical expreflion for one may be comprifed in terms of 
the other without any impropriety, and our conclufions 
will be perfectly juft and correct, in the fenfe in which 
the units of the refpedtive quantities are underftood, not- 
withftanding the fluxions themfelves may be heteroge¬ 
neous. Sir Ifaac Newton, in his Quadrature of Curves,' 
in finding the area of a curve, defer ibes a parallelogram 
on the abfeida ( x ), the other fide ( a) of which is con¬ 
ftant; and then he compares the fluxion of the area of 
this parallelogram with the fluxion of the area of the 
curve, they being homogeneous quantities; and the 
fluxion of the area of the parallelogram being ax, he 
gets the fluxion of the area of the curve. From what 
lias been faid above, when we reduce thefe matters to 
calculation, there appears to be no abfolute neceffity for 
this ; but it is more fcientific to make the comparifon 
between homogeneous quantities, than, between thofe 
which are not homogeneous, and therefore the former 
method is always to be preferred in cafes where it can be 
applied, notwithfianding the conclufions which are other- 
wife deduced are perfectly true and fatisfaffory. 
20. The ingenious and juftly-celebrated author of the 
Analyft has endeavoured to fliow, that the principles of 
fluxions, as delivered by its author, are not founded upon 
reafoning ftriftly logical and concltifive. He lays this 
down as a lemma : “ If you make any fuppofition, and 
in virtue thereof deduce any confequence; if you deftroy 
that fuppofition, every confequence before deduced mud 
be deftroyed and rejefted, fo as from thenceforward to 
be no more fupplied or applied in the demonftration.” 
This, he thinks, is fo plain as to need no proof. It may 
perhaps be admitted to be true, when we want to deduce 
the abfolute value of a quantity which is to be obtained in 
virtue of a fuppofition ; but it is not true when we want 
to obtain the relative values of quantities. He feems not 
to have attended to the connedfion which there mull ne- 
celfarily be between the two terms which conftitute a 
ratio, and the two terms which exprefs the ratio to 
•which they approach as their limit, when you diminifh 
them fine limite, called the limit of the ratio. It is 
agreed, that by diminifhing the increments you approach 
to the ratio of the velocities which they had at the 
points from whence they began to be generated, and that 
by rpaking them become indefinitely fmall, you arrive 
at a ratio indefinitely near to that of the velocities at 
thofe points. Now the two terms exprefling the limit of 
a ratio mud depend upon the terms themfelves of the 
ratio, and the terms are obtained upon the fuppofition of 
the exiftence of an increment; the limit therefore is ob¬ 
tained upon the fuppofition of the exiftence of an incre¬ 
ment; but the limit is a certain determinate invariable 
ratio, totally independent of the magnitude of the terms 
of the ratio, and confequendy of the increment, as appears 
by Art. 8. When we therefore deduce the limit by making 
the increments vanilh, the ejfeEl of the prior exiftence of 
the terms of the ratio (fill remains in the terms which 
exprefs the limit of the ratio. If the exigence of the terms 
which exprefs the limit of the ratio, depended upon the 
exijltnce of the terms themfelves of the ratio, the fuppofi¬ 
tion which makes the latter vaniftt would necellarily make 
the former alfo vanilh, and then no conclufion could be 
deduced by making the terms of the ratio vanilh; but 
as that is not the cafe, the limit, which is obtained by 
making the terms become equal to nothing, contains an 
effect, after the increments are actually vanifhed, which 
depends upon their having exifted. The lemma, there¬ 
fore of the author, however true it may be under fome 
circumftances, cannot be applied againft the reafoning 
upon which the Principles of Fluxions are founded. The 
author admits the conclufions to be true. He favs, “ I 
have no controverfy about your conclufions, but only 
about your logic ; and it muft: be remembered that I am 
not concerned about the truth of your theorems, but 
only about the way of coming at them.” The above 
obfervations (how, not only that our conclufions are true, 
but that they are deduced by fteps which are perfectly 
fatisfadtory, and ftridtly logical. 
On the MAXIMA and MINIMA of QUANTITIES. 
Prop. IX. — To determine the value of a quantity when it 
becomes a maximum or minimum. 
21. If a quantity fir ft increafe and then decreafe, at the 
end of its increafe it becomes a maximum ; and if it firft 
decreafe and then increafe, at the end of its decreafe 
it becomes a minimum. And as the fluxion of a quan¬ 
tity is the rate of its increafe or decreafe (Art. 3.) 
when it becomes a maximum or minimum its fluxion 
muft; be —o, the quantity having, at that point of time, 
no further increafe or decreafe. If any quantity be a 
maximum or minimum, any power or root of that quan¬ 
tity muft then, evidently, be a maximum or minimum. 
For the power or root of a quantity will increafe or de¬ 
creafe as long as the quantity ilfelf increafes ordecreafes, 
and no longer. Any conftant multiple, or part of a quan¬ 
tity which is a maximum or minimum, muft alfo be a 
maximum or minimum. For the multiple, or part of a 
quantity will increafe or decreafe as long as the quantity 
itfelf increafes or decreafes, and no longer; therefore 
when its fluxion is made =0, the conftant multiplier may 
be neglefted. 
Ex. 1. To divide a given number a into two parts, x,y, 
fo that x w y n may be a maximum. Since x-\-y=za, and x m y"zzz 
max. the fluxion of each —o, the former, becaufe it is 
conftant, and the latter, becaufe it is a maximum ; 
x-\-y—o, and my n x m — 1 x-\-nx"’y —*/— 0 > hence, x— — -j, 
nx m v"— i V nxy . nxy 
-—- therefore —J — - — 
and x — - 
my"x' 
my—nx, and m : 
x : y. 
my 
Now/; 
confequendy xz 
and y 
H)= 
mfn 
my 
nx 
. x-\ - — a. 
m 
If m—n. the 
m-\-u 
two parts are equal. 
Cor." Hence, to divide a quantity a into three parts, 
x, y, z, fo that xyz may be a max. the parts muft be 
equal. For fuppofe x to remain conftant, and y, z, to 
vary ; the product yz, and confequendy xyz, will be 
greateft when/==*. Or, if y remain conftant, the pro- 
du6t xz, and confequeadyjxx, will be greateft when x—2. 
Thus it appears that the parts muft be equal. And in 
like manner it may be fliown, that whatever be the num¬ 
ber of parts, they will be equal. 
Ex. 2. Given x-\-y-\-z=a, and xy 2 z 3 a maximum, to find 
*> y, *• 
As x,y, z, muft have fome certain determinate values 
to anfwer thefe conditions, let us fuppofe fuch a value of 
y to remain conftant, wliilft x and z vary till diey anfwer 
the conditions, and then a+a—o and z 3 x + 3x2'■ £zzo ; 
1 • • 3X2^ 3*Z -T 
hence, x——»= — -—— =-’ z—$x. Now 
z 3 z 
let us fuppofe the value of 2: to remain conftant, and x 
and / to vary, fo as to fatisfy the conditions; then 
x-\-yzzz o, y^x+ 2xyjz=o ; hence, x ——^ — 
**y. 
y 
values 
•y— 
of y 
y i 
2x5 fabftitute in the given equation, thefe 
and z in terms of x, and x-^-zx-fsx=za, or 
