V« 
K 
182 THE ELEMENTS DOWN TO PLATO’S TIME 
Hippias of Elis, the famous sophist already mentioned (pp. 2, 
23-4), was nearly contemporary with Socrates and Prodicus, 
and was probably born about 460 b.c. Chronologically, there 
fore, his place would be here, but the only particular discovery 
attributed to him is that of the curve afterwards known as 
the quadratrix, and the quadratrix does not come within the 
scope of the Elements. It was used first for trisecting any 
rectilineal angle or, more generally, for dividing it in any 
ratio whatever, and secondly for squaring the circle, or rather 
for finding the length of any arc of a circle; and these prob 
lems are not what the Greeks called ‘ plane ’ problems, i. e. 
they cannot be solved by means of the ruler and compasses. 
It is true that some have denied that the Hippias who 
invented the quadratrix can have been Hippias of Elis; 
Blass 1 and Apelt 2 were of this opinion, Apelt arguing that at 
the time of Hippias geometry had not got far beyond the 
theorem of Pythagoras. To show how wide of the mark this 
last statement is we have only to think of the achievements 
of Democritus. We know, too, that Hippias the sophist 
specialized in mathematics, and I agree with Cantor and 
Tannery that there is no reason to doubt that it was he who 
discovered the quadratrix. This curve will be best described 
when we come to deal with the problem of squaring the circle 
(Chapter YII); here we need only remark that it implies the 
propositi Oil .-that the lengths of arcs in a circle are proportional 
to the angles subtended by them at the centre (Euch VI. 33). 
The most important name from the point of view of this 
chapter is Hippocrates of Chios. He is indeed the first 
person of whom it is recorded that he compiled a book of 
Elements. This is lost, but Simplicius has preserved in his 
commentary on the Physics of Aristotle a fragment from 
Eudemus’s History of Geometry giving an account of Hippo 
crates’s quadratures of certain ‘ lunules ’ or luues. 3 This is one 
of the most precious sources for the history of Greek geometry 
before Euclid; and, as the methods, with one slight apparent 
exception, are those of the straight line and circle, we can 
form a good idea of the progress which had been made in the 
Elements up to Hippocrates’s time. 
1 Fleckeisen's Jahrhuch, cv, p. 28. 
2 Beiträge zur Gesch. cl. gr. Philosophie, p. 379. 
3 Simpl, in Phgs., pp. 60. 22-68. 32, Diels.