52 
ARCHIMEDES 
and Zeuthen) is that the successive solutions in integers of 
the equations 
x 2 ~3y 2 =l I 
- 2 .3y 2 - -23 
x“ 
may have been found in a similar way to those of the 
equations ic 2 — 2 y 2 = +1 given by Theon of Smyrna after 
the Pythagoreans, The rest of the suggestions amount for the 
most part to the use of the method of continued fractions 
more or less disguised. 
or 
Applying the above formula, we easily find 
2-±>V3 >2-|, 
1>V3 >|. 
Next, clearing of fractions, we consider 5 as- an approxi 
mation to V3.3 2 or V27, and we have 
5 + x^>3v'3>5 + 1 2 T) 
whence ft > V 3 > xf • 
Clearing of fractions again, and taking 26 as an approxi- 
mation to 3.15 2 or V675, we have 
26-#2 > 15V3 > 26—^r, 
which reduces to 
13S1 ./q 265 
7 8 0' > V 0 > X5 3-. 
Archimedes first takes the case of the circumscribed polygon. 
Let CA be the tangent at A to a circular arc with centre 0. 
Make the angle AGO equal to one-third of a right angle. 
Bisect the angle AGO by OD, the angle AOD by OE, the 
angle AO A by GF, and the angle AGF by OG. Produce GA 
to AH, making AH equal to AG. The angle GOH is then 
equal to the angle FOA which is ^th of a right angle, so 
that GH is the side of a circumscribed regular polygon with 
96 sides. 
GA:AG[= V3 : 1] > 265:153, 
GC : CA = 2:1 = 306:153. 
(1) 
(2) 
And 
so that 
Henc 
And 
Thert 
Nex 
from ( 
of 0A 
G A: A 
G A: A 
Deal 
similar 
angle 
if the 
the ar 
straigl 
96 sidi