MEASUREMENT OF A CIRCLE 
55 
The series of values found by Archimedes are shown in the 
following table: 
a 
h 
c 
n 
a 
265 
306 
153 
0 
1351 
571 
> [ V(571 2 + 153 2 )] 
153 
i 
2911 
> 591| 
1162| 
>|V{(1162|) 2 + 153 2 }] 
153 
2 
59241 * 
>1172| 
1823 [ 
2334| 
>[V{(2334i) 2 + 153 2 }] 
153 
3 
i 
3661 X 9 T 
> 2339-J 
1007 [ 
4673i 
153 
4 
( 
2016|f- 
h 
1560 
c V(2911 2 + / 
< 3013i \ 
: V(1823 2 + 2 
< 1838 X 9 X 
-/(1007 2 + 6i 
< 1009| 
V {(2016^) 2 - 
< 20171 
c 
780 
780 
240 f 
66 
and, bearing in mind that in the first case the final ratio 
a 4 : c is the ratio OA : AG = 2 OA : GH, and in the second case 
the final ratio h i : c is the ratio AB: BG, while GH in the first 
figure and BG in the second are the sides of regular polygons 
of 96 sides circumscribed and inscribed respectively, we have 
finally 
96 X 153 96 x 66 
46731 > 77 > 20171 ‘ 
Archimedes simply infers from this that 
3i >7T > 3lf. 
As a matter of fact 
96 x 153 
4673i 
= 3 
667^ 
4673| 5 
and 
667 ^ _ i 
4672| 7 ‘ 
It is also to be observed that 3^ x = 3 + — ? and it may 
have been arrived at by a method equivalent to developing 
the fraction in the form of a continued fraction. 
2017i 
It should be noted that, in the text as we have it, the values 
of h x , 6 2 , h 3 , h i are simply stated in their final form without 
the intermediate step containing the radical except in the first 
* t Here the ratios of?« to c are in the first instance reduced to lower 
terms.