- THE EARLIEST GREEK GEOMETRY 285
to their bases, or the theorem (Book 12, Proposition 2) that the areas of two circles
are proportional to the squares on their diameters. Even the simplest constructions,
such as the construction of a square equal in area to a given rectangle or the
application problems mentioned above, may require the concept of proportionality
of lines. Because of the importance of the theory of proportion for geometry, the
discovery of incommensurables made it imperative to give a definition of proportion
without relying on a common measure to define a ratio.
Fowler (1998) argues for the existence of a Pythagorean theory of proportion
based on anthyphairesis, the mutual subtraction procedure we have now described
many times.^9 He makes a very telling point (p. 18) in citing a passage from
Aristotle's Topics where the assertion is made that having the same antanairesis is
tantamount to having the same ratio. Fowler takes antanairesis to be a synonym of
anthyphairesis. Like Gray (quoted above), Knorr (1975) argues that the discovery
of irrationals was not a major "scandal," and that it was not responsible for the
"geometric algebra" in Book 2 of Euclid. While arguing that incommensurability
forced some modifications in the way the Pythagoreans thought about physical
magnitudes, he says (p. 41):
It is thus thoroughly obvious that, far from being in a state of
paralysis, fifth- and fourth-century geometers proceeded with their
studies of similar figures as if they were still unaware of the foun-
dational consequences of the existence of incommensurable lines.1.7. The influence of Plato. Plato is still held in high esteem by philosophers,
and it is well recognized that his philosophy contains a strong mathematical ele-
ment. But since Plato was a follower of Socrates, who was almost entirely concerned
with questions of ethics and the right conduct of life,^10 his interest in mathematical
questions needs to be explained. Born in 427 BCE, Plato served in the Athenian
army during the Peloponnesian War. He was also a devoted follower of Socrates.
Socrates enjoyed disputation so much and was so adept at showing up the weakness
in other people's arguments that he made himself very unpopular. When Athens
was defeated in 404 BCE, Plato sided with the party of oligarchs who ruled the
city temporarily. When the democratic rule was restored, the citizens took revenge
on their enemies, among whom they counted Socrates. Plato was devastated by
the trial and execution of Socrates in 399 BCE. He left Athens and traveled to
Italy, where he became acquainted with the Pythagorean philosophy. He seems to
have met the Pythagorean Philolaus in Sicily in 390. He also met the Pythagorean
Archytas at Tarentum (where some Pythagoreans had fled to escape danger at Cro-
ton). Plato returned to Athens and founded the Academy in 387 BCE. There he
hoped to train the young men^11 for public service and establish good government.
At the behest of Archytas and a Syracusan politician named Dion, brother-in-law
of the ruler Dionysus I, Plato made several trips to Syracuse, in Sicily, between 367
and 361 BCE, to act as advisor to Dionysus II. However, there was virtual civil war
between Dion and Dionysus, and Plato was arrested and nearly executed. Diogenes(^9) Fowler avoids as far as possible using the phrase Euclidean algorithm.
(^10) The Socrates depicted by Plato is partly a literary device through which Plato articulated his
own thoughts on many subjects that the historical Socrates probably took little notice of.
(^11) In his writing, especially The Republic, Plato argues for equal participation by women in
government. There is no record of any female student at his Academy, however. His principles
were far in advance of what the Athenians would tolerate in practice.