A history of Greek mathematics - Wilbourhall.org

a
MEASUREMENT OF A CIRCLE 55
be
The series of values found by Archimedes are shown in the
following table
:
> 2339J 1007 (< \/(1007 2 + 66 2 ) 66
n a b c
265 306 153 1351 1560 780
571 > |V(571 2 +153 2 )]
153 1 2911 < 591|
< 3013JJ
1162|>[V{(1162|) 2 +153 2 }] 153 2 f5924j|
780]*
>1172|
1823 ( < v / (1823 2 +240 2 ) 240/
1
9
< 1838 TT
2334i>[v / {(2334£) 2 +153 2 }] 153 3 3661 T 9 T ... 240 f
4673| 153 4
2016|
( < 1009J
< V{(2016^) 2 + 66 2 } 66
< 2017f
and, bearing in mind that in the first case the final ratio
a :
4
c is the ratio A : AG = 2 OA : OH, and in the second case
the final ratio 6 4
: 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
96X153 96x66
> 7T >
4673| 2017J
Archimedes simply infers from this that
As a matter of fact
3i >tt > 3if
.
96 x 153
4673J
667| 667
= 3 * _ i
and
"
4673|'
4672J
1
It is also to be observed that 3^£ = 3 -\ -, and it may
have been arrived at by a method equivalent to developing
the 6336
fraction in the form of a continued fraction.
2017j
It should be noted that, in the text as we have it,
the values
of b lf b 2
, 63, 6 4
are simply stated in their final form without
the intermediate step containing the radical except in the first
* t Here the ratios of a to c are in the first instance reduced to lower
terms.