I think I have an answer . . .
What is the average distance between two points
on the surface of a sphere with a radius ?
With no loss of generalization, let one point be at the North Pole of the sphere.
. . Let be the South Pole.
Let the other point be
Consider the great circle through and
It could pass through Greenwich Mean Time
. . (discovered by the Armenian explorer, Prime Meridian).
* o *
* * P
* * - - - - * E
* O r *
* o *
* * *
Point can be anywhere on the arc
Since half the points are above the Equator and half are below,
. . the average distance would be: .
Therefore, average distance .
. . just as Haytham predicted . . .