---------- Forwarded message ---------- From: Jan Mycielski <firstname.lastname@example.org> Date: Dec 24, 2005 3:14 PM Subject: Questions about history To: reuben hersh <email@example.com>
I had a similar experience. I found that the area of any spherical cap (yarmulka) on any sphere equals pi r^2 (just like in R^2), where r is the distance from the top of the cap to its perimeter in R^3. Then somebody told me that this is known but I was not able to trace its history. Do you know how to find it? (The theorem follows easily from Archimedes' theorem about the preservation of areas of countries on the sphere when they are projected from the central axis of a circumscribed cylinder into this cylinder, so he could have known it.) Erdos had the idea of God's book in which all the shortest proofs of theorems are written. It would be very useful if a human attempt to write such a book existed. But presumably it would have to be a collective effort. Of course the MAA publications could be a big source. Perhaps the books of G.F.Simmons come close, but he is more utilitarian.