220 
SPECIAL PROBLEMS 
handsj; of later and abler geometers, since it gives a method 
of approximating, with any desired degree of accuracy, to the 
area of a circle, and lies at the root of the method of exhaustion 
as established by Eudoxus. As regards Hippocrates’s quadra 
ture of lunes, we must, notwithstanding the criticism of 
Aristotle charging him with a paralogism, decline to believe 
that he was under any illusion as to the limits of what his 
method could accomplish, or thought that he had actually 
squared the circle. 
The squaring of the circle. 
There is presumably no problem which has exercised such 
a fascination throughout the ages as that of rectifying or 
squaring the circle ; and it is a curious fact that its attraction 
has been no less (perhaps even greater) for the non-mathe 
matician than for the mathematician. It was naturally the 
kind of problem which the Greeks, of all people, would take 
up with zest the moment that its difficulty was realized. The 
first name connected with the problem is Anaxagoras, who 
is said to have occupied himself with it when in prison. 1 
The Pythagoreans claimed that it was solved in their school, 
‘ as is clear from the demonstrations of Sextus the Pythagorean, 
who got his method of demonstration from early tradition ’ 2 ; 
but Sextus, or rather Sextius, lived in the reign of Augustus 
or Tiberius, and, for the usual reasons, no value can be 
attached to the statement. 
The first serious attempts to solve the problem belong to 
the second half of the fifth century b.c. A passage of 
Aristophanes’s Birds is quoted as evidence of the popularity 
of the problem at the time (414 b.c.) of its first representation. 
Aristophanes introduces Meton, the astronomer and discoverer 
of the Metonic cycle of 19 years, who brings with him a ruler 
and compasses, and makes a certain construction ‘ in order that 
your circle may become square ’. 3 This is a play upon words, 
because what Meton really does is to divide a circle into four 
quadrants by two diameters at right angles to one another ; 
the idea is of streets radiating from the agora in the centre 
1 Plutarch, De exil. 17, p. 607 F. 
2 Iambi, ap. Simpl. in Categ., p. 192, 16-19 K., 64 b 11 Brandis. 
3 Aristophanes, Birds 1005.