In the Redangle triangle A l> 3 , A B = a .. X] g q 
— S, A D = t = a — $ $ then is ADq = ABq-f 
B D q : or a a a a S' +. SS = a a — 2 a x + x x + S S.. 
2 3 X - XX 
which being reduced, gives S = ■ 
2 cl 
Let la be a right line fuppofed infinitely near and pa- 
rallel to L A, and interfeding A B in C. Becaufe of like 
triangles A C a, A B D 5 A B ; B D : ; A C : C a that is 
a— x : S C = - a . x ”~ . — ; ; x ; y . therefore y = 
2 a x «— x x . 
x . 
2 a a — .-a ax 
2 ax — x x . 
s • 
a — x 
Multiply each by a a, and ’tis 2 a j 
The Plowing quantity of a a y is 2 a y 
2 21 X r X X 
and the flowing quantity of x is theHyperbolie 
3. 3C 
Area I L M f as is (hewn before. J Thefe two Area's be- 
ginning together at F and I, and having every where 
equal Fluxions fir Augments, are therefore themfelves every 
where equal. 1 
N, The Quadrature of the Trilinear Figure I L M 
bei,ng thus found, any other Area bounded with 
the Curve line I L. and any other Right Lines is alfo 
given. 
)n;: Jo r- ' /y , .... 4 
FV. Suppofing the fame things as in the precedent Vro- 
pofition, I Ay, that the Area of the Quadratrix F a b H F 
is equal to half the fquare of F g, wanting the Cube of 
F g divided by fix F H, or F a hH F ~ ■— - - f ^Fhe 
x 6 a 
Fluxion of this Area is the Redangle GabB — a — x » j 
= a 
2 ax —xx. 
— xx x = x x 
2 a a — a x 
x xv~ . 
— X " 
3 a 
The 
flowing quantity or x % is i x x : And the flowing 1 quin- 
sy 
