Joshua wrote: << On Thu, Aug 14, 2008 at 12:40 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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.
I think that's the hard part -- how do you show that? How did Archimedes do it?
Consider the picture of the circle with one external point and one of its two tangent segments to the circle. My guess is that Archimedes considered a very small arc of the circle, and its radial projection onto the tangent segment. The very small segment is almost straight, whereas its projection onto the tangent segment is exactly straight but with endpoints clearly farther apart (from the radial projection). I can easily imagine Archimedes thinking in terms of the limit, where the small slightly rounded arcs became infinitesimal but straight. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele