The History of Mathematics: A Brief Course

288 10. EUCLIDEAN GEOMETRY

A

G II Β C D Ê

FIGURE 12. A fundamental theorem in the theory of proportion.

Proposition 1 of Book 6 of the Elements.

The solution to the difficulty was provided by Eudoxus of Cnidus (ca. 407-354
BCE), whom Diogenes Laertius describes as "astronomer, geometer, physician, and
lawgiver." He learned geometry from Archytas. Diogenes Laertius cites another
commentator, named Sotion, who said that Eudoxus spent two months in Athens
and attended lectures by Plato. Because of his poverty, he could not afford to live
in Athens proper. He lived at the waterfront, known as the Piraeus, supported
by a physician named Theomedus, and walked 11 km from there into Athens.
Then, with a subsidy from friends, he went to Egypt and other places, and finally
returned, "crammed full of knowledge," to Athens, "some say, just to annoy Plato
for snubbing him earlier." Plato was not in Eudoxus' league as a mathematician;
and if Eudoxus felt that Plato had patronized him in his earlier visit, perhaps
because Plato and his other students were wealthy and Eudoxus was poor, his
desire to return and get his own back from Plato is quite understandable. He must
have made an impression on Plato on his second visit. In his essay On Socrates'
Daemon, Plutarch reports that when the Delians consulted Plato about doubling
the cube, in addition to advising them to study geometry, he told them that the
problem had already been solved by Eudoxus of Cnidus and Helicon of Cyzicus.
If true, this story suggests that the Delians appealed to Plato after Eudoxus had
left for Cnidus, around 350. By that time Plato was a very old man, and perhaps
mellower than he had been a quarter-century earlier during Eudoxus' first stay
in Athens. In Cnidus, Eudoxus made many astronomical observations that were
cited by the astronomer Hipparchus, and one set of his astronomical observations
has been preserved. Although the evidence is not conclusive, it seems that while
he was in Athens, he contributed two vital pieces to the mosaic that is Euclid's
Elements.

The Eudoxan definition oj proportion. The first piece of the Elements contributed

by Eudoxus was the solution of the problem of incommensurables. This solution

is attributed to him on the basis of two facts: (1) Proclus' comment that Euclid