- THE EARLIEST GREEK GEOMETRY 283
FIGURE 11. A mechanical device for drawing the conchoid of Nicomedes.the late nineteenth and early twentieth centuries, urged that they be studied as a
regular part of the curriculum (Beman and Smith, 1930).
1.5. Challenges to Pythagoreanism: the paradoxes of Zeno of Elea. Al-
though we have some idea of the geometric results proved by the Pythagoreans, our
knowledge of their interpretation of these results is murkier. How did they conceive
of geometric entities such as points, lines, planes, and solids? Were these objects
physically real or merely ideas? What properties did they have? Some light is shed
on this question by the philosophical critics of Pythagoreanism, one of whom has
become famous for the paradoxes he was able to spin out of Pythagorean principles.
In the Pythagorean philosophy the universe was generated by numbers and
motion. That these concepts needed to be sharpened up became clear from critics
of the Pythagorean school. It turned out that the Pythagorean view of geometry
and number contained paradoxes within itself, which were starkly pointed out by
the philosopher Zeno of Elea. Zeno died around 430 BCE, and we do not have
any of his works to rely on, only expositions of them by other writers. Aristotle,
in particular, says that Zeno gave four puzzles about motion, which he called the
Dichotomy (division), the Achilles, the Arrow, and the Stadium. Here is a summary
of these arguments in modern language, based on Book 6 of Aristotle's Physics.
The Dichotomy. Motion is impossible because before an object can arrive at its
destination it must first arrive at the middle of its route. But before it can arrive
at the middle, it must travel one-fourth of the way, and so forth. Thus we see that
the object must do infinitely many things in a finite time in order to move.
The Achilles. (This paradox is apparently so named because in Homer's Iliad the
legendary warrior Achilles chased the Trojan hero Hector around the walls of Troy,
overtook him, and killed him.) If given a head start, the slower runner will never be
overtaken by the faster runner. Before the two runners can be at the same point at
the same instant, the faster runner must first reach the point from which the slower
runner started. But at that instant the slower runner will have reached another