SSiSil 
integers of 
those of the 
rnyrna after 
nonnt for the 
ued fractions 
an approxi- 
an approxi- 
ibed polygon, 
itli centre 0. 
right angle, 
by OE, the 
Produce GA 
GOH is then 
ght angle, so 
polygon with 
(1) 
‘(2) 
wmm 
MEASUREMENT OF A CIRCLE 53 
And, since OE bisects the angle CO A, 
CO :0A = CD: DA, 
so that {G0 + OA) : OA = CA : DA, 
or (CO + OA) : CA = OA : AD. 
Hence OA : AD > 571 :153, by (1) and (2). 
And OD 2 : AD 2 = (OA 2 + AD 2 ) : AD 2 
> (571 2 + 153 2 ) : 153 2 
> 349450:23409. 
Therefore, says Archimedes, 
OD : DA > 591|: 153. 
Next, just as we have found the limit of OD: AD 
from OG: CA and the limit of OA : AG, we find the limits 
of OA : AE and OE: AE from the limits of OD: DA and 
OA : AD, and so on. This gives ultimately the limit of 
OA : AG. 
Dealing with the inscribed polygon, Archimedes gets a 
similar series of approximations. ABC being a semicircle, the 
angle BA G is made equal to one-third of a right angle. Then, 
if the angle BAG 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. 
i 1 
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