THE QUADRATURE OF THE PARABOLA 87 
e greater over 
itself be made 
ed the solution 
by means of 
3 Method; the 
of mechanics 
ps. 1-17), and 
means of pure 
18-24). 
3 properties of 
he base, and P 
oint of contact 
P V through P 
e tangent at Q 
vith the funda- 
dinedes uses to 
eeting QT, QP 
in E, F, R, 0 
(Props. 4, 5.) 
aced in relation 
at the diameter 
ical, and let the 
in E. Produce 
at 0 X , 0 2 ... 0 n , 
)E, i.e. vertical 
E 2 , ..., and the 
curve in R 13 R 2 , .... Join QR l3 and produce it to meet OE in 
F, QR 2 meeting 0 1 E 1 in F 1 , and so on. 
A O Ht H 2 H 3 Ht, q, 
Now Archimedes has proved in a series of propositions 
(6-13) that, if a trapezium such as 0 l E l E 2 0 2 is suspended 
from H l Pf 2 , and an area P suspended at A balances 0 1 E 1 E 2 0 2 
so suspended, it will take a greater area than P suspended at 
A to balance the same trapezium suspended from H 2 and 
a less area than P to balance the same trapezium suspended 
from If 1 . A similar proposition holds with regard to a triangle 
such as E n H n Q suspended where it is and suspended at Q and 
H n respectively. 
Suppose (Props. 14, 15) the triangle QqE suspended where 
it is from OQ, and suppose that the trapezium E0 1 , suspended 
where it is, is balanced by an area suspended at A, the 
trapezium E 1 0 2> suspended where it is, is balanced by P 2 
suspended at A, and so on, and finally the triangle E n O n Q, 
suspended where it is, is balanced by P n+1 suspended at A ; 
then P + P 9 + ... + P„ ... at A balances the whole triangle, so that 
P 1 + P 2 +... + P n+l = ^AEqQ > 
since the whole triangle may be regarded as suspended from 
the point on OQ vertically above its centre of gravity. 
Now AO : 0H 1 = QO : 0H l 
= Qq ■ q0 1 
— E l 0 1 :0 1 R 1 , by Prop. 5, 
= (trapezium EOJ: (trapezium FO a ),