t 
THE REGULAR POLYGONS 
329 
presumably Hipparchus’s Table) is appealed to. If AB be the 
side (a) of an enneagon or hendecagon inscribed in a circle, AC 
the diameter through A, we are told that the Table of Chords 
gives | and W as the respective approximate values of the 
ratio AB/AC. The angles subtended at the centre 0 by the 
side AB are 40° and 32 T 8 T ° respec 
tively, and Ptolemy’s Table gives, 
as the chords subtended by angles of 
40° and 33° respectively, 4U 2' 33" 
and 344* 4' 55" (expressed in 120th 
parts of the diameter); Heron's 
figures correspond to 40^ and 33P 
36' respectively. For the enneagon 
AG 2 = 9 AB 2 , whence BG 2 = 8 AB 2 
or approximately 2££-AB 2 , and 
BG = ; therefore (area of 
enneagon) = f . A A BG = a 2 . For 
the hendecagon AG 2 = $££-AB 2 and BG 2 = -AB 2 , so that 
BG = -ij-a, and area of hendecagon = . A ABC = --^-a 2 . 
An ancient formula for the ratio between the side of any 
regular polygon and the diameter of the circumscribing circle 
is preserved in Geepon. 147 sq. (= Pseudo-Dioph. 23-41), 
namely d n = Now the ratio na n /d n tends to 77 as the 
number (n) of sides increases, and the formula indicates a time 
when 7T was generally taken as = 3. 
(e) The Circle. 
Coming to the circle (.Metrica I. 26) Heron uses Archi 
medes’s value for tt, namely - 2 T 2 -, making the circumference of 
a circle - 4 T 4 -r and the area \\d 2 , where r is the radius and d the 
diameter. It is here that he gives the more exact limits 
for tt which he says that Archimedes found in his work On 
Plinthides and Cylinders, but which are not convenient for 
calculations. The limits, as we have seen, are given in the 
text as WAY < 77 < WAY> and with Tannery’s alteration to 
WAY <■ tt < WAY are quite satisfactory. 1 
1 See vol. i, pp. 232-3.