HIPPOCRATES’S QUADRATURE OF LUNES 197 
circumference the arc of a semicircle, but also (2) the lime 
in which the outer circumference is greater, and (3) the lune in 
which it is less, than a semicircle. 
‘ But he also squared the sum of a lune and a circle in the 
following manner. 
‘ Let there he two circles about K as centre, such that the 
square on the diameter of the outer is 6 times the square on 
that of the inner. 
‘Let a {regular) hexagon ABCDEF he inscribed in the 
inner circle, and let KA, KB, KG he joined from the centre 
and produced as far as the circumference of the outer circle. 
Let GH, HI, GI he joined.’ 
[Then clearly GH, HI are sides of a hexagon inscribed in 
the outer circle.] 
‘About GI [i.e. on GI] let a segment he circumscribed 
similar to the segment cut off by GH. 
‘ Then GH = 3 GH 2 , 
for GI 2 + (side of outer hexagon) 2 = (diam. of outer circle) 2 
= 4 GH 2 . 
[The original states this in words without the help of the 
letters of the figure,] 
‘Also GH 2 = GAB 2 . 
had squared one particular lune of each of three types, namely those 
which have for their outer circumferences respectively (1) a semicircle, 
(2) an arc greater than a semicircle, (8) an arc less than a semicircle, he 
had squared all possible lunes, and therefore also the lune included in his 
last quadrature, the squaring of which (had it been possible) would 
actually have enabled him to square the circle. The question is, did 
Hippocrates so delude himself? Heiberg thinks that, in the then 
state of logic, he may have done so. But it seems impossible to believe 
this of so good a mathematician ; moreover, if Hippocrates had really 
thought that he had squared the circle, it is inconceivable that he 
would not have said so in express terms at the end of his fourth 
quadrature. 
Another recent view is that of Bjdrnbo (in Pauly-Wissowa, Tieal-Ency- 
dopiidie, xvi, pp. 1787-99), who holds that Hippocrates realized perfectly 
the limits of what he had been able to do and knew that he had not 
squared the circle, but that he deliberately used language which, without 
being actually untrue, was calculated to mislead any one who read him 
into the belief that he had really solved the problem. This, too, seems 
incredible ; for surely Hippocrates must have known that the first expert 
who read his tract would detect the fallacy at once, and that he was 
risking his reputation as a mathematician for no purpose. I prefer to 
think that he was merely trying to put what he had discovered in the 
most favourable light; but it must be admitted that the effect of his 
language was only to bring upon himself a charge which he might easily 
have avoided.