A history of Greek mathematics - Wilbourhall.org

THE REGULAR POLYGONS 329
presumably Hipparehus'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 ^ as the respective approximate values of the
ratio AB /AC. The angles subtended at the centre by the
side AB are 40° and 32 X
8
T °
respectively,
and Ptolemy's Table gives,
as the chords subtended by angles of
40° and 33° respectively, 41? 2' 33"
and 34P 4' 55" (expressed in
120th
parts of the diameter) ; Heron's
figures correspond to 40^ and 33^
36' respectively. For the enneagon
AG = 9AB 2 2 , whence BC 2 =, SAB 2
or approximately 2££-AB 2 ,
and
BC = --§-a\ therefore (area of
enneagon)^ . AABC^-%
1 ^. For
the hendecagon AC = 2 %5
-AB 2 and BC = 2 ^f-AB 2 , so that
BC = %r a >
and area of hendecagon = ^
2
. AABC = --f-a
.
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
— n—. Now the ratio na n /d n tends to it as the
o
number (
n) of sides increases, and the formula indicates a time
when it was generally taken as = 3.
(e)
The Circle.
Coming to the circle (Metrica I. 26) Heron uses Archimedes's
value for 7r, namely - 2 2
T-, making the circumference of
a circle *f-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 77- 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 toVt^ 71 < "
-w&stT' an