290 10. EUCLIDEAN GEOMETRYThe method of exhaustion. Eudoxus' second contribution is of equal importance
with the first; it is the proof technique known as the method of exhaustion. This
method is used by both Euclid and Archimedes to establish theorems about areas
and solids bounded by curved lines and surfaces. As in the case of the definition of
proportion for incommensurable magnitudes, the evidence that Eudoxus deserves
the credit for this technique is not conclusive. In his commentary on Aristotle's
Physics, Simplicius credits the Sophist Antiphon (480-411) with inscribing a poly-
gon in a circle, then repeatedly doubling the number of sides in order to square the
circle. However, the perfected method seems to belong to Eudoxus. Archimedes
says in the cover letter accompanying his treatise on the sphere and cylinder that
it was Eudoxus who proved that a pyramid is one-third of a prism on the same
base with the same altitude and that a cone is one-third of the cylinder on the
same base with the same altitude. What Archimedes meant by proof we know: He
meant proof that meets Euclidean standards, and that can be achieved for the cone
only by the method of exhaustion. Like the definition of proportion, the basis of
the method of exhaustion is a simple observation: When the number of sides in an
inscribed polygon is doubled, the excess of the circle over the polygon is reduced
by more than half, as one can easily see from Fig. 13. This observation works to-
gether with the theorem that if two magnitudes have a ratio and more than half
of the larger is removed, then more than half of what remains is removed, and this
process continues, then at some point what remains will be less than the smaller
of the original two magnitudes (Elements, Book 10, Proposition 1). This principle
is usually called Archimedes' principle because of the frequent use he made of it.
The phrase if two magnitudes have a ratio is critical, because Euclid's proof of the
principle depends on converting the problem to a problem about integers. Since
some multiple (n) of the smaller magnitude exceeds the larger, it is only a matter
of showing that a finite sequence oj, 02,... in which each term is less than half of
the preceding will eventually reach a point where the ratio á& : áú is less than 1 jn.
The definition of ratio and proportion allowed Eudoxus/Euclid to establish all
the standard facts about the theory of proportion, including the important fact that
similar polygons are proportional to the squares on their sides (Elements, Book 6,
Propositions 19 and 20). Once that result is achieved, the method of exhaustion
makes it possible to establish rigorously what the Pythagoreans had long believed:
that similar curvilinear regions are proportional to the squares on similarly situated
chords. In particular, it made it possible to prove the fundamental fact that was
being used by Hippocrates much earlier: Circles are proportional to the squares
on their diameters. This fact is now stated as Proposition 2 of Book 12 of the
Elements, and the proof given by Euclid is illustrated in Fig. 14.