HIPPOCRATES’S QUADRATURE OF LUNES 199 
this third hypothesis the fallacy would lie in the fact that 
the sum of the circle and the lune is squared, and not the 
circle alone.’ 
If, however, the reference of Aristotle was really to Hip 
pocrates’s last quadrature alone, Hippocrates was obviously 
misjudged; there is no fallacy in it, nor is Hippocrates likely 
to have deceived himself as to what his proof actually 
amounted to. 
In the above reproduction of the extract from Eudemus 
I have marked by italics the passages where the writer follows 
the ancient fashion of describing points, lines, angles, &c., with 
reference to the letters in the figure: the ancient practice was 
to write to arj/ieiov k<p’ <p (or e<p’ ov) K, the (point) on which (is) 
the letter K, instead of the shorter form to K arjpGov, the 
point K, used by Euclid and later geometers; rj 1(f) rj AB 
(evOeia), the straight line on which (are the letters AB, for 
■}] AB {eiiOeia}, the straight line AB; to rpiycovov to k<f)’ ov 
EZH, the triangle on which (are the letters) EFG, instead of 
to EZH Tpiyoivov, the triangle EFG; and so on. Some have 
assumed that, where the longer archaic form, instead of the 
shorter Euclidean, is used, Eudemus must be quoting Hippocrates 
verbatim; but this is not a safe criterion, because, e.g., Aristotle 
himself uses both forms of expression, and there are, on the 
other hand, some relics of the archaic form even in Archimedes. 
Trigonometry enables us readily to find all the types of 
Hippocratean lunes that can 
be squared by means of the 
straight line and circle. Let 
ACB be the external circum 
ference, ABB the internal cir 
cumference of such a lune, 
r, r' the radii, and 0, 0' the 
centres of the two arcs, 6, 6' 
the halves of the angles sub 
tended by the arcs at the centres 
respectively. 
Now (area of lune) 
= (difference of segments ACB, ABB) 
= (sector OAGB-AAOB)-(sector 0'ABB —A AO'B) 
— r 2 Q — r' 2 6' + \ (r' 2 sin 2 6' — r 2 sin 2 6).