APPROXIMATIONS TO THE VALUE OF U 233 
as given is greater than the true value, and the higher limit is 
greater than the earlier upper limit 3^. Slight corrections by 
Tannery (fx y aooof3 for fx y acooe and /x^cott/S for /x^cottt]) give 
better figures, namely 
195882 211872 
62351 > ^ > "67441 
or 3-1416016 > 7T > 3-1415904.... 
<r ‘T 
Another suggestion 1 is to correct ¡x^v/xa into ¡x y £v[x8 and 
l9 l0 
ixfairf] into fX^COTTT], giving 
195888 211875 
> 7T > 
62351 67444 
or 3-141697... > 7T >3-141495.... 
If either suggestion represents the true reading, the mean 
between the two limits gives the same remarkably close 
approximation 3-141596. 
Ptolemy 2 gives a value for the ratio of the circumference 
of a circle to its diameter expressed thus in sexagesimal 
fractions, y rj A, i.e. 3+~ + ^ or 3-1416. He observes 
60 60 z 
that this is almost exactly the mean between the Archimedean 
limits 3^- and 3^. It is, however, more exact than this mean, 
and Ptolemy no doubt obtained his value independently. He 
had the basis of the calculation ready to hand in his Table 
of Chorda. This Table gives the lengths of the chords of 
a circle subtended by arcs of |°, 1°, 1^°, and so on by half 
degrees. The chords are expressed in terms of 120th parts 
of the length of the diameter. If one such part be denoted 
by l p , the chord subtended by an arc of 1° is given by the 
Table in terms of this unit and sexagesimal fractions of it 
thus, l p 2'50". Since an angle of 1° at the centre subtends 
a side of the regular polygon of 360 sides inscribed in the 
circle, the perimeter of this polygon is 360 times l p 2 / 50" 
or, since l p — 1/120th of the diameter, the perimeter of the 
polygon expressed in terms of the diameter is 3 times 1 2' 50", 
that is 3 8 r 30", which is Ptolemy’s figure for n. 
3 J. L. Heiben in Nor disk Tidsskrift for Filologi, 3 e Ser. xx. Fasc. 1-2. 
2 Ptolemy, Syntaxis, vi. 7, p. 513. 1-5, Heib.