Some years ago I participated in a summer seminar at Yale on "The
Mathematical Sciences in Antiquity," with Prof. Asger Aaboe. If my memory
serves me correctly, he spoke about Ptolemy's late writing moving in the
direction of treating his geocentric picture as a representation of reality,
complete with specific values for the various circle radii and including the
requirement that no two planetary domains would overlap,
thereby excluding the possiibiility of collisions, etc.
Sorry, but I have no references handy at the moment.
YES, this is accurate. The relevant reference here is probably an article
by Bernie Goldstein, "The Arabic Version of Ptolemy's Planetary Syntaxis,"
Trans. Amer. Phil. Soc. 57 (4), 1967, pp. 1-12. Ptolemy assumed
(apparently) the reality of spheres corresponding to the mathematical
devices he employed to explain the details of planetary appearances. He
assumed, eg., that the minimum distance of Jupiter equalled the maximum
distance of Mars, from the earth of course, as required by his
epicycle/deferent theory. No interplanetary voids were allowed. He ends up
getting about 20,000 earth radii to the stellar sphere. In modern numbers
(the exact value for the radius of the earth in the ancient world was never
certain) this would be 80M miles, giving the size of the universe ("world")
as 160M miles in diameter. Not large by modern standards, but enormous by
human standards, as he realized. One appreciate even more his statement
(elsewhere) that the earth is an insigificant speck, which fact (among
others) gives the lie to the modern myth that Copernicus "degraded" us by
moving us out of the world's center, a claim I cannot support from
historical sources.
Ted Davis