Rich wrote: << . . .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?

...

Actually, it seems easy to verify this in-betweenness for the circle directly. a) For a point outside the circle, its two tangent segments have lengths that total more than the circle's arclength between them. This amounts to showing that tan(theta) > theta for 0 < theta < pi/2. I believe Archimedes was capable of verifying this inequality even without calculus and modern terminology for trig functions. b) For a side of the inscribed polygon, that must be shorter than the circle's arc between the side's endpoints. a) and b) together prove the in-betweenness of the circle's length w.r.t. any in- & circumscribed polygons. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele