186 THE ELEMENTS DOWN TO PLATO’S TIME 
2. Next take three consecutive sides GE, EF, FB of a regular 
hexagon inscribed in a circle of diameter CD. Also take AB 
equal to the radius of the circle and therefore equal to each of 
the sides. 
On AB, GE, EF, FB as diameters describe semicircles (in 
the last three cases outwards with reference to the circle). 
Then, since 
GB 2 = 4 AB 2 = AB 2 + GE 2 + EF 2 + FB 2 , 
and circles are to one another as the squares on their 
diameters, 
semicircle GEFB) = 4 (semicircle ALB) 
— (sum of semicircles ALB, GGE, ERF, FKB). 
H 
Subtracting from each side the sum of the small segments 
on GE, EF, FB, we have 
(trapezium GEFB) = (sum of three lunes) + (semicircle ALB). 
The author goes on to say that, subtracting the rectilineal 
figure equal to the three lunes (‘ for a rectilineal figure was 
proved equal to a lune’), we get a rectilineal figure equal 
to the semicircle ALB, ‘ and so the circle will have been 
squared ’. 
This conclusion is obviously false, and, as Alexander says, 
the fallacy is in taking what was proved only of the lune on 
the side of the inscribed square, namely that it can be squared, 
to be true of the lunes on the sides of an inscribed regular 
hexagon. It is impossible that Hippocrates (one of the ablest 
of geometers) could have made such a blunder. We turn there 
fore to Eudemus’s account, which has every appearance of 
beginning at the beginning of Hippocrates’s work and pro 
ceeding in his order.