286 10. EUCLIDEAN GEOMETRY
Laertius quotes a letter allegedly from Archytas to Dionysus urging that Plato be
released. Plato returned to the Academy in 360 and remained there for the last 13
years of his life. He died in 347.
Archytas. Archytas, although a contemporary of Plato, is counted paradoxically
among the "pre-Socratics" in philosophy; but that is because he worked outside
Athens and continued the earlier Pythagorean tradition. Archytas' solution of the
problem of two mean proportionals using two half-cylinders intersecting at right
angles was mentioned above. In his Symposium Discourses, Plutarch claimed that
for that reason Plato also lamented that the disciples of Eudoxus,
Archytas, and Menaechmus attacked the duplication of a solid by
building tools and machinery hoping to get two ratios through the
irrational, by which it might be possible to succeed, [saying that
by doing so they] immediately ruined and destroyed the good of
geometry by turning it back toward the physical and not directing
it upward or striving for the eternal and incorporeal images, in
which the god is ever a god.Although the sentiment Plutarch ascribes to Plato is consistent with the ideals
expressed in the Republic, Eutocius reports one such mechanical construction as
being due to Plato himself. From his upbringing as a member of the Athenian elite
and from the influence of Socrates, Plato had a strong practical streak, concerned
with life as it is actually lived.^12 Platonic idealism in the purely philosophical
sense does not involve idealism in the sense of unrealistic striving for perfection.
Archytas and Philolaus provided the connection between Pythagoras and Plato,
whose interest in mathematics began some time after the death of Socrates and
continued for the rest of his life. Mathematics played an important role in the
curriculum of his Academy and in the research conducted there, and Plato himself
played a leading role in directing that research. Lasserre (1964, P- 1?) believes that
the most important mathematical work at the Academy began with the arrival of
Theaetetus in Athens around 375 and ended with Eudoxus' departure for Cnidus
around 350.
The principle that knowledge can involve only eternal, unchanging entities led
Plato to some statements that sound paradoxical. For example, in Book 7 of the
Republic he writes:
Thus we must make use of techniques such as geometry when we
take up astronomy and let go of the things in the heavens if we
really intend to create something intrinsically useful and practical
in the soul by mastering astronomy.If Plato's mathematical concerns seem to be largely geometrical, that is prob-
ably because he encountered Pythagoreanism at the time when the challenges dis-
cussed above were still current topics. (Recall the quotation from the Republic in
(^12) In the famous allegory of the cave in Book 7 of the Republic, Plato depicts the nonphilosophical
person as living in a cave with feet in chains, seeing only flickering shadows on the wall of the
cave, while the philosopher is the person who has stepped out of the cave into the bright sunshine
and wishes to communicate that reality to the people back in the cave. While he encouraged his
followers to "think outside the cave," his trips to Syracuse show that he understood the need to
make philosophy work inside the cave, where everyday life was going on.