♦ HIPPOCRATES OF CHIOS 
183 
It would appear that Hippocrates was in Athens during 
a considerable portion of the second half of the fifth century, 
perhaps from 450 to 430 в.C. We have quoted the story that 
what brought him there was a suit to recover a large sum 
which he had lost, in the course of his trading operations, 
through falling in with pirates; he is said to have remained 
in Athens on this account a long time, during which he con 
sorted with the philosophers and reached such a degree of 
proficiency in geometry that he tried to discover a method of 
squaring the circle. 1 This is of course an allusion to the 
quadratures of lunes. 
Another important discovery is attributed to Hippocrates. 
He was the first to observe that the problem of doubling the 
cube is reducible to that of finding two mean proportionals in 
continued proportion between two straight lines. 2 The effect 
of this was, as Proclus says, that thenceforward people 
addressed themselves (exclusively) to the equivalent problem 
of finding two mean proportionals between two straight lines. 3 
(a) Hippocrates s quadrature of tunes. 
I will now give the details of the extract from Eudemus on 
the subject of Hippocrates’s quadrature of lunes, which (as 
I have indicated) I place in this chapter because it belongs 
to elementary ‘plane’ geometry. Simplicius says he will 
quote Eudemus ‘word for word’ (ката A飿r) except for a few 
additions taken from Euclid’s Elements, which he will insert 
for clearness’ sake, and which are indeed necessitated by the 
summary (memorandum-like) style of Eudemus, whose form 
of statement is condensed, ‘ in accordance with ancient prac 
tice’. We have therefore in the first place to distinguish 
between what is textually quoted from Eudemus and what 
Simplicius has added. To Bretschneider 4 belongs the credit of 
having called attention to the importance of the passage of 
Simplicius to the historian of mathematics ; Allman 5 was the 
first to attempt the task of distinguishing between the actual 
1 Philop. in Phys., p. 31. 3, Vitelli. 
2 Pseudo-Eratosthenes to King Ptolemy in Eutoc. on Archimedes (vol. 
iii, p. 88, Heib.). 
3 Proclus on Eucl. I, p. 218. 5. 
4 Bretschneider, Die Geometric and die Geometer vor Eukhdes, 1870, 
pp. 100-21. 
5 Hermathena, iv, pp. 180-228; Greek Geometry from Thales to Euclid, 
pp. 64-75.