Ptolemy's geocentric system of the universe
Crystal spheres
Ptolemy is said to have believed that all his deferents and epicycles were carved out of a solid, ultratransparent matter. Is this actually possible? Let us consider each of the classes of problem in turn.
The Sun and the Stars
From a geocentric and stationary Earth, the Sun and Stars appear to revolve round the Earth once a day. More precisely, the Stars revolve around the Earth in 23 hours and 56 minutes while the Sun revolves slightly slower completing its revolutuion in 24 hours.
Moreover, the Sun does not follow the celestial equator in its retrograde wanderings through the stars, it follows a line called the ecliptic which is inclined to the equator by 23½º. In addition, owing to the eccentricity of the Earth's orbit, the Sun does not move along this line at a totally constant speed. On the other hand this effect is very small (the eccentricity of the Earth's orbit is only 1.6%) and it is not clear to me that Ptolemy would have had the means to detect it anyway. After all, to Ptolemy, noon was the time when the shadow of a sundial pointed due south. He would have no way of knowing that at some times of the year, his 'noon' was as much as 8 minutes too early or too late as he had no accurate clocks.
In this, and all subsequent analyses, I shall assume that Ptolemy ignored the eccentricity of the Earth's orbit and that the orbit of the Sun round the Earth is circular and uniform. The system is complicated enough as it is!
Now, how shall we implement the Sun's orbit in crystal?
First, the Earth and its atmosphere are contained within a sphere of solid crystal which has two pins attached above the poles. Lets call this sphere E. This sphere is surrounded by an enormously thick spherical shell (Z) which rotates once every 23 hours 56 min and which carries the 'fixed stars'. Embedded in this shell is a second concentric shell (S) which carries the Sun. This shell also has pins which fix its axis with respect to Z and, of course, the two axes thus defined are inclined at an angle of 23½º. S has a retrograde motion with respect to Z whose period is 1 year.
Naturally, S cuts Z into two independent bits but Ptolemy has to make the assumption that the inner and outer bits of Z move as one, even though they are not actually constrained to do so. In all the following discussions, I shall take the view that Z (and all its bits) is stationary which E rotates inside it. (This point of view was, in fact suggested, by several astromomers after Ptolemy but it was not widely adpted except as a mathematical convenience because it throws out the one physical reality of which absolutely everyone was convinced namely the fact that the Earth was obviously stationary!)
The Moon
As we have seen, in order to account for the motion of the Moon round the Earth, Ptolemy needs an offset deferent (MD) and a small epicycle (ME). This is not straightforward because the semi-major axis of the Moon's orbit precesses round once every 8 years. This means that the offset must also precess at the same rate. We can, however implement the offset as an epicycle (MO) embedded within a concentric shell which has a retrograde motion almost equal to the motion of the deferent.
Within this offset epicycle is embedded the actual epicycle which carries the Moon itself.