It occurs to me that Archimedes, with his love of statics, might have phrased this in terms of the center of mass of a hemispherical shell: it is halfway between the center of the shell and the place where the shell meets its axis of symmetry. Jim Propp On Thu, Jul 25, 2019 at 10:25 AM James Propp <jamespropp@gmail.com> wrote:
One can prove that the expected distance from a random point on the surface of a sphere to the equatorial plane is half the radius. Assuming we could rephrase this claim in a form that Archimedes would recognize, how would he have proved it?
As an example of the kind of proof I would like to see, consider the proposition that the expected distance from a random point in a disk to the boundary of the disk is 1/3 of the radius. One can prove this using the formula for the volume of a cone. (I came up with this myself but I’m sure others have too.)
Further examples of the kind of proof I have in mind are Archimedes’ determination of the surface area and volume of the sphere.
Jim Propp