328 
HERON OF ALEXANDRIA 
Heron. As a matter of fact, however, C (| + /0) = -■$- exactly, 
and only the Metrica gives the more accurate calculation. 
The regular heptagon. 
Heron assumes (chap. 20) that, if a be the side and r the 
radius of the circumscribing circle, a = |r, being approxi 
mately equal to the perpendicular from the centre of the 
circle to the side of the regular hexagon inscribed in it (for | 
is the approximate value of | a/3). This theorem is quoted by 
Jordanus Nemorarius (d. 1237) as an ‘Indian rule’; he pro 
bably obtained it from Abul Wafa (940-98). The Metrica 
shows that it is of Greek origin, and, if Archimedes really 
wrote a book on the heptagon in a circle, it may be due to 
him. If then p is the perpendicular from the centre of the 
circle on the side (a) of the inscribed heptagon, r/(%a) = 8/3-| 
or 16/7, whence p*/(^a) 2 = ?-£■£-, and p/\a = (approxi 
mately) 14|/7 or 43/2*1, Consequently the area of the 
heptagon = 7 . \pa = 7 . a 2 = a 1 . 
The regular octagon, decagon and dodecagon. 
In these cases (chaps. 21, 23, 25) Heron finds p by drawing 
the perpendicular OC from 0, the centre of the 
circumscribed circle, on a side AB, and then making 
the angle OAD equal to the angle ADD. 
d\ For the octagon, 
or \a . ft approximately. 
For the decagon, 
A ADC = f R, and AD : DC =5:4 nearly (see preceding page); 
hence AD : AG =5:3, and p = \a (| + f) = %a. 
For the dodecagon, 
Z ADC = IR, and p = \a (2 + V3) = \a (2 +1) = ^-a 
approximately. 
Accordingly A 8 = -% 9 -a 2 , A w = ^-a 2 , A 12 = -\ 5 -a 2 , where a is 
the side in each case. 
The regular enneagon and hendecagon. 
In these cases (chaps. 22, 24) the Table of Chords (i.e.