A history of Greek mathematics - Wilbourhall.org

MEASUREMENT OF A CIRCLE
53
And, since OD bisects the angle COA,
CO:OA = CD: DA,
so that
or
(CO + OA):OA = CA: DA,
(CO + OA) :CA = OA: AD.
Hence OA :
AD
> 571 :
153,
by (1) and (2).
And OD 2 : AD = 2 (OA 2 + AD 2 ) AD :
Therefore, says Archimedes,
2
> (571 2 +153 2 ): 153 2
> 349450: 23409.
OD.DA > 591$: 153.
Next, just as we have found the limit of OD :
from OC : CA and the limit of OA : AC, we find the limits
of OA-.AE and OE-.AE from the limits of 0D:DA and
AD
Oil : ^.D, and so on. This gives ultimately the limit of
0A:AG.
Dealing with the inscribed polygon, Archimedes gets a
similar series of approximations.
ABC being a semicircle, the
Then,
angle BAC is made equal to one-third of a right angle.
if the angle BAC is bisected by AD, the angle BAD by AE,
the angle BAE by AF, and the angle BAF by AG, the
straight line BG is the side of an inscribed polygon with
96 sides. r