The fundamental proof is in AlmagestXII, 1, and a full exposition is in NeugebauerO., A history of ancient mathematical astronomy (Berlin-Heidelberg-New York, 1975), 190–1, 267–70.
2.
See the comparisons with the Earth in Neugebauer, op. cit.191. The condition can also be applied to show that as viewed from the Sun, the Moon will never appear to move retrograde since the ratio of the radii of the orbits of the Moon and Earth, about 1/400, is far less than the ratio of their velocities of about 1/13.
3.
Retrograde motion would not occur in a system in which r1/r2 ≤ P1/P2 = ω2/ω1,. but by PrincipiaI, iv, cor. v and vii, this implies a centripetal force as 1/rn with n ≤ 1, rather far from the inverse-square.
