Actually, there is a very relevant 1980 paper in Mathematics Magazine, by Scott Brodie, about Archimedes' axioms. Here is the Math Reviews review of it, by H.S.M. Coxeter: << Archimedes' treatise on the sphere and cylinder uses the axioms of Euclid along with five new ones. These include a statement to the effect that if two curves in the plane both intersect a line at the same two points so that each curve, together with the line, bounds a convex set, and if one of these convex sets is a proper subset of the other, then the curve bounding the larger set is the longer of the two. The author uses calculus to prove this statement in modern terms.
As Rich suggested, this is indeed extremely close to Cauchy's theorem; in fact this axiom looks logically equivalent to it. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele