Rich wrote: << Quoting Dan Asimov <dasimov@earthlink.net>:
on a theorem of Cauchy:
Theorem: Given two convex curves A,B in the plane with B lying inside A, then length(B) < length(A).
Is it fair to give this one to Cauchy? Archimedes must have assumed something like it to estimate the circumference of a circle as being between the in- & circumscribed polygons. Was Cauchy just bringing the rigor up to modern standards, or is some non-obvious insight required?
I don't know the history of this theorem -- and hadn't realized Archimedes knew it. (How did he show this in the rigor of his day?) (Cauchy's proof goes something like this: First, show that if C is a convex curve in the plane, and you average the lengths of its projections onto lines of all directions, then that average is just half the length of the curve. (This is proved by first proving it for an interval, then approximating a convex curve by a convex polygon.) Now the above Theorem follows readily. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele