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 ’.