## Stephen Menn – Eudoxus’ theory of proportion and his method of exhaustion

The Montreal Inter-University Workshop on the History and Philosophy

of Mathematics presents:

Stephen Menn (McGill):

Eudoxus’ theory of proportion and his method of exhaustion

Friday, November 21, 2014

McGill University, Leacock Buiding, Room 927. 3:30-5:30pm

Abstract:

Euclid in Elements V gives an astonishingly rigorous and logically

complex formulation of the theory of proportion, proving such

propositions as “alternation” (if A:B::C:D then A:C::B:D) for all

magnitudes, before applying them to lines and areas in Elements VI. It

is very hard to see what could have motivated, or led to the discovery

of, such a complex set of proofs of propositions that might easily be

taken for granted (notably “if A>B then A:C>B:C”). (T.L. Heath’s

story, that this theory was provoked by a foundational crisis caused

by the discovery of incommensurables, was refuted 80 years ago by

Oskar Becker.) A possible clue comes from an anonymous scholiast who

says that much ofElements V goes back to Eudoxus (a collaborator in

Plato’s Academy, 50–100 years before Euclid). We know, on better

grounds, that Eudoxus invented the “method of exhaustion” used by

Euclid in Elements XII to prove e.g. that circles are to each other as

the squares on their diameters, and that a cone is one-third the

volume of a cylinder with the same base and height. It is easier to

explain the origin of the method of exhaustion than of the Euclidean

theory of proportion, and if, as is often thought, the two theories

were somehow linked for Eudoxus, this might help us understand the

proportion theory, but it is remarkably difficult to explain how the

two theories were connected. Building on work of Wilbur Knorr, which

distinguishes an earlier Eudoxian theory of proportion (surviving in

Archimedes Equilibrium of Planes I) from Euclid’s theory in Elements

V, I offer a reconstruction, first of how Eudoxus could have been led

to discover his theory of proportion in connection with the method of

exhaustion, and then of how Euclid could have been led to develop his

theory of proportion out of Eudoxus’.

For more information, please contact:Gregory Lavers (Concordia),

Mathieu Marion (UQÀM), Jean-Pierre Marquis (UdM), Dirk Schlimm (McGill).