A history of Greek mathematics - Wilbourhall.org

88 ARCHIMEDES
that is, it takes the trapezium F0 Y
suspended at A to balance
the trapezium E0 1
suspended at H v And P 2
balances E0 X
where it is.
Therefore (FO,) > P 2
.
Similarly (^1^2) > ^2' anc^ so on -
Again AO :0H 1
= E^: X
R
X
= (trapezium E :
1 2 )
(trapezium P
2 2),
that is, {R x 2 ) at A will balance (E 1 2 ) suspended at H 1 ,
while P 2
at A balances (Efi^) suspended where it is,
whence P 2
> R Y 2
.
Therefore (F x
0^ > P 2
> (PjOJ,
(P 2 3 ) > P 6
> P 2 3
, and so on;
and finally,
&H n °nQ > P n+i > & R n °nQ'
By addition,
(R 1 2
)+(R 2
O i ) + ...+(AR n O n
Q)