A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Date: 2015 Mar 22, 18:57 -0700
"As an ex-teacher, I would take points off for not showing your work."
Es an ex-student, however, I would take points off the evaluation sheet of a teacher who presents his material in an overly complicated form.
The largest circle on a sphere is a great circle. All of them are the same size. The diameter as defined by David Fleming of any of them is just half of a great circle flipped 90 deg. So that ratio is clearly 1 : 2.
And nobody who uses conformal projections of the sphere onto a flat sheet of paper for his daily coastal work doubts that the sphere can be locally approximated by a plane without ever dreaming of L'Hopital. So the rules and formulas of plane geometry hold on the sphere for reasonably small neighbourhoods. Which makes the sought ratio 1 : Pi.
I would actually give David Fleming extra points for understanding the problem in the first place. I had problems understanding what was meant by "diameter", as well as with the term "prime rational number". In fact, I understood Karl's problem only after David Fleming had solved it.