THE SQUARING OF THE CIRCLE 
231 
of a town ; thè word reTpayoavos then really means ‘ with four 
(right) angles ’ (at the centre), and not { square ’, but the word 
conveys a laughing allusion to the problem of squaring all 
the same. 
We have already given an account of Hippocrates’s quadra 
tures of lunes. These formed a sort of prolusio, and clearly 
did not purport to be a solution of the problem ; Hippocrates 
was aware that ‘ plane ’ methods would not solve it, but, as 
a matter of interest, he wished to show that, if circles could 
not be squared by these methods, they could be employed 
to find the area of some figures bounded by arcs of circles, 
namely certain lunes, and even of the sum of a certain circle 
and a certain lune. 
Antiphon of Athens, the Sophist and a contemporary of 
Socrates, is the next person to claim attention. We owe 
to Aristotle and his # commentators our knowledge of Anti 
phon’s method. Aristotle observes that a geometer is only 
concerned to refute any fallacious arguments that may be 
propounded in his subject if they are based upon the admitted 
principles of geometry ; if they are not so based, he is not 
concerned to refute them : 
‘ thus it is the geometer’s business to refute the quadrature by 
means of segments, but it is not his business to refute that 
of Antiphon 
As we have seen, the quadrature ‘ by means of segments ’ is 
probably Hippocrates’s quad 
rature of lunes. Antiphon’s 
method is indicated by Themis- 
tius 2 and Simplicius. 3 Suppose 
there is any regular polygon 
inscribed in a circle, e.g. a square 
or an equilateral triangle. (Ac 
cording to Themistius, Antiphon 
began with an equilateral triangle, 
and this seems to be the authentic 
version; Simplicius says he in 
scribed some one of the regular polygons which can be inscribed 
1 Arisi. Phys. i. 2, 185 a 14-17. 
2 Them, in Phys., p. 4. 2 sq.. Schenkl. 
3 Simpl. in Phys., p. 54. 20-55. 24, Diels.