FLUXIONS. 
4S9 
tude, yet the fluents are not at al! affected by varying. x, 
tb.e fluents of thefe quantities x and nx' : 1 ix being x and 
x H , whatever be the value of x. Hence, of whatever 
magnitude we a flume the fluxion of any quantity, the 
fluent will always give the quantity generated. In the 
following problems, therefore, the fluxion of the area, 
folid, curve line or furface, may be aflumed of any mag¬ 
nitude, and the fluent, cerrefted if neceflary, will give 
the quantity which has been generated. 
To find the AREAS of CURVES. 
Prop. XIX .—To find the area ABC rf any curve, whofe 
ordinate BC is perpendicular to the abfeiffa AB. 
49. Let ABC be any curvilinear area generated by the 
uniform motion of the ordinate BC; on AB, BC, deferibe 
the parallelogram ABCD, and conceive this to have been 
generated by the fame 
uniform motion of a 
line equal and parallel 
to AD j draw bm pa¬ 
rallel to BC,and com¬ 
plete the parallelo¬ 
gram B bmn, and pro¬ 
duce DC to c. Then 
AD being conftant 
whilft BC varies, the 
next increment of the parallelogram is BCcb, and the co¬ 
temporary increment of the area ABC is BC mb-, hence, 
the ratio of the increment BCci of the parallelogram to 
the cotemporary increment BC mb of the area ABC, is 
always nearer to a ratio of equality, than BCr£: B nmb, 
cr nearer than BC : bm ; now let bm move up to and 
coincide with BC, in order to obtain the limiting ratio of 
the increments, and we get the limiting ratio of BC : bm, 
a ratio of equality ; hence, a fortiori, the limiting ratio of 
the increment B Ccb of the parallelogram, to the cotem¬ 
porary increment B Cmb of the area ABC, is a ratio of 
equality; therefore by Prop. 2. Cor. 1. the fluxion of 
the parallelogram ABCD is equal to the fluxion of the 
area ABC ; but BCc£ being the increment of the paral¬ 
lelogram uniformly generated, will reprefent its fluxion, 
by Prop. 1. hence, the fluxion of the area of the curve 
ABC will be reprefented by BC cb, the cotemporary 
fluxion of the abfeiffa AB being B b. If therefore 
AB=x, B C—y, B bz=.x, and A — the area ABC, then 
will A=BCc 5 =y.r; the fluent of which, corrected if 
neceflary, gives A. 
Cor. Hence, the fluxion of any area, generated by the 
motion of a ftraight line in a direction perpendicular to 
itfelf, is as the length of the generating line and its velo¬ 
city conjointly. And as a curve line, moving in a direc¬ 
tion perpendicular to itfelf, mud deferibe the fame area 
as a ftraight line of the fame length moving with the 
fame velocity, the fluxion of the furface generated by a 
curve line, fo moving, mud be as its length and velocity 
conjointly, 
Ex. 1. Let AC be any parabola; to find its area. 
_ v,y n 1 y 
Here ax—y -; hence, ax—ny n 1 y, and 
Ex. 2. To find the area c.f a circle, whofe radius is 
unity. Let A be the center 
of the circle ; draw BC, AP, 
perpendicular to QR, and 
join AC. Put AC=t, AB 
—x,BC =y; thenx--j-)*=r, 
—*— x2 A ' 4 
(Art.34); 
16 12S 
A—yxz 
— &c. the 
fluxion of the area BAPC whofe fluent is A=x— : 
x s x 7 $ x ° 
40 112 115 2 
C=o ; hence, A—x- 
+ &c. C; now when x~ 0, A=o, 
3 x 5 
x‘ 
112 
5 * 
Jt 
yx—-^- = A, whofe fluent (Art. 37.) A: 
a 
n/'+i 
, — Sec. 
6 40 112 1152 
Now if the arc PC=30°, *•—£ ; and the area ABCP= 
0,5 — 0,0208333 — 0,000781 2 —0,0000698 — 0,0000085 — 
0,0000012 — &c. =20,4783055. But as a-=£, y—Vi > 
therefore the area of the triangle ACB=^x^| — 
0,2165063, which fubtrafiled from 0,4783055 leaves 
0,2617992 the area of the fedtor ACP ; which multiplied 
by 12 gives t 3,14159 Sec. — the area of the whole circle. 
Cor. If r 2= radius of any circle, <2— its area ; then, 
fince circles vary as the fquares of their radii, i 2 3 : r 2 :: 
'3,14159 See. : *2=3,14159 Sec. X r*' If £=the diame- 
d . d , # d* 
ter, then r—~ , and r 2 =— ; hence, a— 3,14159 &c. X — 
24 4 
—•0,78539 & c - X d 2 . 
Ex. 3. To find the area of an hyperbola between the 
afymptotes AP, AM, and the curve MP. Put AB=.v, 
1162227; then y— 
1 P \P 
—, and the flux- 
x" 
ion of the area 
APCB= v.v = — 
x” 
— x ” x — A, 
whofe fluent is A 
=— + c. 
i —n 
Cafe 1. If n be 
lefs than unity, 
when A=o, x—o, 
x'~~" _ 
i —n ’ 
hence, C = o > 
therefore the area 
• X ^ " ,f 
APCB (infinite in extent) is-a finite quantity when 
x is finite. 
Cafe 2. If n be greater than unity, tb.e irdex 1 —11 
being negative, x mult come into the denominator, and 
”+iX« t j ie ft uen t will become A 
-j-C (C being the correction if neceflary) 
n y” 
- ■ - ^ y 
n- j-1 a 
+ C: 
C = o; hence, At 
now when A=o, =0, 
11—1 X *"' 
n -j-i 
X xy. 
1 — n x x n 1 
+ C ; now when A=o, x=o, confequently 
i» infinite, becaufe the denominator 
If n—i, it becomes the common parabola, and the 
If =, the figure becomes a triangle, and 
n — iX*"" * 
becomes —0 ; therefore the area APCB=- 
2 
area —-xy. 
3 , , . • 
the area =i.vy. If n— 1, ax—y, and x :y:: 1 : a, that is, in a 
conftant ratio, which is the cafe when AC is a ftraight line, 
becaufe the triangle ABC continues always fimilar to itfelf. 
Vol. VII. No. 443- 
t —n X x* 
-!-C is infinite. Whenever there is a negative index, the 
quantity riuift always be transferred from the numerator 
to the denominator, or the contrary, before its value, in 
any particular cafe, can be found. 
61 Ca/e 
