Eudoxos completed the generalization of
proportion theory, one of the principal
intellectual efforts of the previous 50 years,
and developed fundamental techniques for
comparing figures by approximating figures
leading to a reductio. They are among the
most enduring achievements of ancient
Greek mathematics. Our knowledge of his
work on mathematics comes principally
from four sources: P’ claim that
Eudoxos expanded the number of general
theorems; scholia to E’s Elements
claiming Eudoxos as the author of Book 5
on proportion theory and to Elements 12.2
(circles are as the square on their diagonals)
and 12.10 (a cone is^1 / 3 a cylinder with the
same height and base); A’
comment in the introductions to On the
Sphere and Cylinder and Method that Eudoxos
proved that the pyramid is^1 / 3 a prism with
the same height and base (= Elements 13.3–
7) and the cone/cylinder theorem; and finally E’ claim that Eudoxos pro-
duced a solution to the double mean proportion problem: given a, b, to find x, y so that a : x
= x : y = y : b, using curved lines, which Eratosthene ̄s found impractical and E (our
source for Eratosthene ̄s) found too garbled to reproduce.
From these sources, a general understanding of 4th c. BCE mathematics, and traces
especially in A, Archime ̄de ̄s, and T, we can reconstruct some of
Eudoxos’ mathematical ideas. The method by which theorems from Euclid, Elements 12 are
proved, inappropriately called “the method of exhaustion,” approximates the compared
figures by inscribed figures whose relations are known. Then it proves by contradiction that
the approximated figures must be in the same relation. The reductio builds on two implicit
principles: (1) given two comparable magnitudes A, B, A > B or A = B or A < B (connectiv-
ity), (2) given comparable magnitudes, A and B, and a magnitude C, there is an X such that
A : B = C : X (existence of 4th proportional), and one fundamental theorem: (3) given A, B,
if A > B and more than half is taken away from A, and so continuously from the remainder,
there will eventually be left a magnitude X, such that X < B (a bisection principle proved in
Elements 10.1). He also uses a theorem based on (2): (4) if X > B and A : B is a ratio, then
there is a Y, such that Y < A, X : A = B : Y (cf. Elements 5.14). Principles (2) and (4) are not
used in the cone/cylinder theorem.
As an example of the structure of the method, for which there are several other forms,
suppose that one needs to prove that A : B = C : D. The proof involves two theorems. In the
first, one proves for an approximating class of figures, a, b of A, B, that a : b = C : D. In the
second, one assumes that A : B C : D, in which case, by (2), there is an X, A : X = C : D,
where X < B or X > B, by (1). For the first part of the proof, suppose X < B. One now finds,
by construction, ai, bi, such that ai < A and bi < B, where ai : bi = C : D, by the first part of
the proof, and B-bi < B – X by (3), so that X < bi < B. But since ai : bi = C : D = A : X and ai
< A, it follows that bi < X. This is a contradiction, so that A : B C : D. We can take
Eudoxos of Knidos © Budapest Museum
EUDOXOS OF KNIDOS