- THE EARLIEST GREEK GEOMETRY 275
FIGURE 3. Hippocrates' quadrature of a lune, acording to Simplicius.doing so, first invoking familiar surfaces such as cones and cylinders, which could
be generated by moving lines on circles, and intersecting them with planes so as
to get the conic sections that we know as the ellipse, parabola, and hyperbola.
These curves made it possible to solve two of the three problems (trisecting the
angle and doubling the cube). Later, a number of more sophisticated curves were
invented, among them spirals, the cissoid, and the quadratrix. This last curve got
its name from its use in squaring the circle. Although it is not certain that the
Pythagoreans had a program like the one described here, it is known that all three
of these problems were worked on in antiquity.
Squaring the circle. Proclus mentions Hippocrates of Chios as having discovered
the quadratures of lunes. In fact this mathematician (ca. 470-ca. 410 BCE), who
lived in Athens at the time of the Peloponnesian War (430-404), is said to have
worked on all three of the classical problems. A lune is a figure resembling a
crescent moon: the region inside one of two intersecting circles and outside the
other. In the ninth volume of his commentary on Aristotle's books on physics,
the sixth-century commentator Simplicius discusses several lunes that Hippocrates
squared, including the one we are about to discuss. After detailing the criticism
by Eudemus of earlier attempts by Antiphon (480-411) to square the circle by the
kind of polygonal approximation we discussed in Chapter 9, Simplicius reports one
of Hippocrates' quadratures (Fig. 3), based on Book 12 of Euclid's Elements. The
result needed is that semicircles are proportional to the squares on their diameters.
Simplicius' reference to Book 12 of Euclid is anachronistic, since Hippocrates
lived before Euclid; but it was probably well known that similar segments are
proportional to the squares on their bases. Even that theorem is not needed here,
except in the case of semicircles, and that special case is easy to derive from the the-
orem for whole circles. The method of Hippocrates does not achieve the quadrature
of a whole circle; we can see that his procedure works because the "irrationalities"
of the two circles cancel each other when the segment of the larger circle is removed
from the smaller semicircle.
In his essay On Exile, Plutarch reports that the philosopher Anaxagoras worked
on the quadrature of the circle while imprisoned in Athens. (He was brought there
by Pericles, who was eventually compelled to send him away.) Other attempts are
reported, one by Dinostratus (ca. 390-ca. 320 BCE), who is said to have used the
curve called (later, no doubt) the quadratrix (squarer), invented by Hippias of Elis
(ca. 460-ca. 410 BCE) for the purpose of trisecting the angle. It is discussed below
in that connection.
Doubling the cube. Although the problem of doubling the cube fits very naturally
into what we have imagined as the Pythagorean program, some ancient authors