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

November 20, 2014 Leave a comment Go to comments

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

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).

Website: http://www.cs.mcgill.ca/~dirk/workshop

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