324 
THE SQUARING OF THE CIRCLE 
the ancient commentators do not attribute to Bryson any such 
statement, and indeed, to judge by their discussions of difierent 
interpretations, it would seem that tradition was by no means 
clear as to what Bryson actually did say. But it seems 
important to note that Themistius states (1) that Bryson 
declared the circle to be greater than all inscribed, and less 
than all circumscribed, polygons, while he also says (2) that 
the assumed axiom is true, though not peculiar to geometry. 
This suggests a possible explanation of what otherwise seems 
to be an absurd argument. Bryson may have multiplied the 
number of the sides of both the inscribed and circumscribed 
regular polygons as Antiphon did with inscribed polygons; 
he may then have argued that, if we continue this process 
long enough, we shall have an inscribed and a circumscribed 
polygon differing so little in area that, if we can describe 
a polygon intermediate between them in area, the circle, which 
is also intermediate in area between the inscribed and circum 
scribed polygons, must be equal to the intermediate polygon. 1 
If this is the right explanation, Bryson’s name by no means 
deserves to be banished from histories of Greek mathematics; 
on the contrary, in so far as he suggested the necessity of 
considering circumscribed as well as inscribed polygons, he 
went a step further than Antiphon; and the importance of 
the idea is attested by the fact that, in the regular method 
of exhaustion as practised by Archiniedes, use is made of both 
inscribed and circumscribed figures, and this compression, as it 
were, of a circumscribed and an inscribed figure into one so 
that they ultimately coincide with one another, and with the 
proper method of finding the area of a circle, but that which has found 
the most favour is to take the geometric mean between the inscribed and 
circumscribed squares’. I am not aware that he quotes Bryson as the 
authority for this method, and it gives the inaccurate value n = ^/8 or 
2-8284272.... Isaac Argyrus (14th cent.) adds to his account of Bryson 
the following sentence: ‘ For the circumscribed square seems to exceed 
the circle by the same amount as the inscribed square is exceeded by the 
circle.’ 
1 It is true that, according to Philoponus, Proclus had before him an 
explanation of this kind, but rejected it on the ground that it would 
mean that the circle must actually be the intermediate polygon and not 
only be equal to it, in which case Bryson’s contention would be tanta 
mount to Antiphon’s, whereas according to Aristotle it was based on 
a quite different principle. But it is sufficient that the circle should 
be taken to be equal to any polygon that can be drawn intermediate 
between the two ultimate polygons, and this gets over Proclus's difficulty.