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