ANTIPHON AND BRYSON 
223 
perimeter of the inscribed polygon of 96 sides, which is 
constructed in Antiphon’s manner from an inscribed equilateral 
triangle. The same construction starting from a square was 
likewise the basis of Yieta’s expression for 2/7r, namely 
2 it it tv 
- = cos v. cos -. cos — ... 
tv 4 8 16 
= 4'^( 1 + \ / |).n/5( 1 + ^(11^))... {ad inf.) 
Bryson, who came a generation later than Antiphon, being 
a pupil of Socrates or of Euclid of Megara, was the author 
of another attempted quadrature which is criticized by 
Aristotle as ‘ sophistic ’ and ‘ eristic ’ on the ground that it 
was based on principles not special to geometry but applicable 
equally to other subjects. 1 The commentators give accounts 
of Bryson’s argument which are substantially the same, except 
that Alexander speaks of squares inscribed and circumscribed 
to a circle 2 3 , while Themistius and Philoponus speak of any 
polygons. 1 * According to Alexander, Bryson inscribed a square 
in a circle and circumscribed another about it, while he also 
took a square intermediate between them (Alexander does not 
say how constructed); then he argued that, as the intermediate 
square is less than the outer and greater than the inner, while 
the circle is also less than the outer square and greater than 
the inner, and as things which are greater and less than the 
same things respectively are equal, it follows that the circle is 
equal to the intermediate square: upon which Alexander 
remarks that not only is the thing assumed applicable to 
other things besides geometrical magnitudes, e.g. to numbers, 
times, depths of colour, degrees of heat or cold, &c., but it 
is also false because (for instance) 8 and 9 are both less than 
10 and greater than 7 and yet they are not equal. As regards 
the intermediate square (or polygon), some have assumed that 
it was the arithmetic mean between the inscribed and circum 
scribed figures, and others that it was the geometric mean. 
Both assumptions seem to be due to misunderstanding 4 ; for 
1 Post 1, 9 75 b 40 
2 Alexander on Soph. El., p. 90. 10-21, Wallies, 806 b 24 sq., Brandis. 
3 Them, on An. Post., p. 19. 11-20, Wallies, 211 b 19, Brandis; Philop. 
on An. Post., p. 111. 20-114. 17 W., 211 b 30, Brandis. 
4 Psellus (11th cent, a.d.) says, ‘there are different opinions as to the