HIPPOCRATES’S QUADRATURE OF LUNES 185 
two expressions separated by ‘ or ’ may no doubt refer not to 
one but to two different fallacies. But if ‘ the quadrature by 
means of lunes ’ is different from Hippocrates’s quadratures of 
limes, it must apparently be some quadrature like the second 
quoted by Alexander (not by Eudemus), and the fallacy attri 
buted to Hippocrates must be the quadrature of a certain lune 
plus a circle (which in itself contains no fallacy at all). It seems 
more likely that the two expressions refer to one thing, and that 
this is the argument of Hippocrates’s tract taken as a whole. 
The passage of Alexander which Simplicius reproduces 
before passing to the extract from Eudemus contains two 
simple cases of quadrature, of a lune, and of lunes plus a semi 
circle respectively, with an erroneous inference from these 
eases that a circle is thereby squared. It is evident that this 
account does not represent Hippocrates’s own argument, for he 
would not have been capable of committing so obvious an 
error; Alexander must have drawn his information, not from 
Eudemus, but from some other source. Simplicius recognizes 
this, for, after giving the alternative account extracted from 
Eudemus, he says that we must trust Eudemus’s account rather 
than the other, since Eudemus was ‘nearer the times’ (of 
Hippocrates). 
The two quadratures given by Alexander are as follows. 
1. Suppose that AB is the diameter of a circle, D its centre, 
and AC, GB sides of a square 
inscribed in it. 
On AG as diameter describe 
the semicircle AEG. Join GB. 
Now, since 
= 2 AG 2 , 
and circles (and therefore semi 
circles) are to one another as the squares on their diameters, 
(semicircle AGB) — 2 (semicircle AEG). 
But (semicircle AGB) — 2(quadrant ADC); 
therefore (semicircle A EG) = (quadrant ADC). 
If now we subtract the common part, the segment AFC, 
we have (lune AEGF) = A ADC, 
and the lune is ‘ squared ’.