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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.