198 THE ELEMENTS DOWN TO PLATO’S TIME 
‘ Therefore 
segment on GI [ = 2(segmt. on GH) + 6 (segmt. on ABj] 
= (segmts. on GH, HI) + {all segmts. in 
inner circle). 
[‘ Add to each side the area bounded by GH, HI and the 
arc 6r/;] 
therefore (A G HI) = (lune G HI) + {all segmts. in inner circle). 
Adding to both sides the hexagon in the inner circle, we have 
(A G HI) + (inner hexagon) = (lune G HI) + (inner circle). 
‘ Since, then, the sum of the two rectilineal figures can be 
squared, so can the sum of the circle and the lune in question.’ 
Simplicius adds the following observations : 
‘ Now, so far as Hippocrates is concerned, we must allow 
that Eudemus was in a better position to know the facts, since 
he was nearer the times, being a pupil of Aristotle. But, as 
regards the “squaring of the circle by means of segments” 
which Aristotle reflected on as containing a fallacy, there are 
three possibilities, (1) that it indicates the squaring by means 
of lunes (Alexander was quite right in expressing the doubt 
implied by his words, “if it is the same as the squaring by 
means of lunes”), (2) that it refers, not to the proofs of 
Hippocrates, but some others, one of which Alexander actually 
reproduced, or (3) that it is intended to reflect on the squaring 
by Hippocrates of the circle plus the lune, which Hippocrates 
did in fact prove “ by means of segments ”, namely the three 
(in the greater circle) and those in the lesser circle. . . . On