Figure 4.1
The equivalence of the eccentric and epicyclic models for the sun.
If Ptolemy was talking, as he seemed to be doing so clearly, about Aristotelian spheres that moved uniformly in place, then both alternatives that he proposed for the motion of the sun suffered from other Aristotelian considerations as we have seen before: First, the eccentric model of Ptolemy, would propose that there is a center of heaviness in the universe around which the most obvious luminary, the sun, would move, which was different from the Earth that was taken by Aristotle to be the center of heaviness par excellence. That is contrary to all the Aristotelian assumptions about the composition of the universe, and about the need to have an Earth at the center of the universe, not only as a center of heaviness to which all other elements would "naturally" gravitate or recede from, but as the fixed center of the sphere of the universe which is as essential to it as any fixed center is to a rotating mathematical sphere.
The second alternative, the epicyclic model, also assumed the existence of a sphere, out in the celestial realm, which had its own relatively fixed center of heaviness, different from that of the universe, and thus would make the element ether, of which all the celestial bodies are made, a complex element rather than the simple element Aristotle defined it to be.
These are the obvious contradictions that gave rise to the medieval discussions about eccentrics and epicycles, and about their undesirability in general, which we have already mentioned. And as we have already seen, they irked Ibn al-Shāṭir (d. 1375) of Damascus enough so that he would try to resolve them, as we shall repeat below. As for Ptolemy, and despite his insistence on the Aristotelian feature of uniform motion, he remained absolutely silent on these other Aristotelian considerations. In fact, he proceeded as if nothing was wrong and went ahead to assess the merits of each of the two models with respect to the criteria of simplicity. From that perspective he judged the eccentric model to be simpler on account of the fact that it involved one motion instead of two.[248] As far as he was concerned this instrumental reason was enough to allow him this temporary lapse of memory that there were other Aristotelian conditions to be met.
In fact, the situation became even worse as he proceeded. For although he could offer two options for the case of the solar motions, either an eccentric or an epicyclic one, when it came to the other planets he knew quite well that he would no longer have these options. He would have to use both eccentrics as well as epicycles in order to account for their more complex motions. Without any reference to Aristotelian cosmology, or any recollection that it was his guiding cosmology from the beginning, he went on to say: "... for bodies which exhibit a double anomaly both of the above hypotheses [meaning the eccentric and the epicyclic] may be combined, as we shall prove in our discussion of such bodies..."[249]
Although he had his own pedagogical reasons to do so, Ptolemy moved on to discuss the motion of the moon before discussing the motion of the other planets. It may be worth mentioning at this point that Ptolemy grouped the planets together in terms of the predictive mathematical model that he devised for their motions, of course with total disregard for Aristotelian cosmology as we just saw. As a result, he treated the motions of the sun, the moon, and mercury, separately and with separate models for each, and then grouped the other "upper" planets, Saturn, Jupiter, Mars, and Venus together and described their motion with one model. Furthermore, in the final arrangement of his presentation of the models, he also had to abandon the principle of simplicity and presented the models in the following order: sun, moon, upper planets, and mercury.
For the purpose of illustrating the underlying Aristotelian cosmological problems that these models entailed, I shall readopt the principle of simplicity and proceed to expose the problems with the model for the upper planets next, before I pass on to the models of the moon and mercury.
The motions of the upper planets Saturn, Jupiter, Mars, and Venus, described in Book IX of the Almagest and grouped together in IX, 6, can be briefly summarized in the following manner (figure 4.2): Each of the upper planets was supposed to be carried by an epicycle, attached to it in the same fashion a crown is attached to and carried by the ring, to use medieval descriptions.
Figure 4.2
Ptolemy's model for the upper planets. The observer is at point O. The planet P is carried on an epicycle with center C, which is in turn carried by the deferent with center T. Note that the deferent rotates uniformly around the equant E and not around its own center T.
The epicycle was itself carried within the shell of and by an eccentric sphere called the deferent, here represented by a simple circle with center T. Each of those spheres moved uniformly in order to account for the anomalistic and mean motions respectively. But in order to account for the observations properly Ptolemy had to assume that the carrying deferent sphere did not move uniformly around its center T, nor around the Earth which was at the center O of the universe still, but around another point E, called later the equant point, or the center of the equalizer of motion when described by Ptolemy as a sphere. Without any proof of any sort, Ptolemy went on to stipulate that the equant point was located away from the deferent's center, by the same distance as the center of the deferent itself was removed from the center of the universe, and on the opposite side. That is the eccentricity OT was equal to TE.[250] With this arrangement, the deferent's uniform motion around its equant accounted for the mean motion of the planet, and the epicycle's motion around its own center accounted for the anomalistic motion, and thus the phenomena were sufficiently saved.
But from the Aristotelian perspective, this new predictive model for the positions of the planets did not only violate the Aristotelian presuppositions once with the adoption of the eccentric sphere, but twice with the adoption of the epicyclic one as well. It even went beyond that to violate, in a very serious fashion, the very mathematical property of a sphere. With Ptolemy's new assumption that there could be a physical sphere that could move uniformly, in place, around an axis that did not pass through its center, it became very clear that the same assumption would require us to abandon completely the very concept of the mathematical sphere and its defining properties. The only axis around which a physical sphere could move uniformly in place was the one that had to pass through the fixed center of the sphere; otherwise it could not stay in place.
Even if Ptolemy could have satisfied the Aristotelian conditions by avoiding the eccentric sphere and accounted for it by another epicycle, as he did with the case of the sun, and even if he had allowed himself the license to use epicycles, arguably against the Aristotelian conception of the simplicity of the element ether, as was done later by Ibn al-Shāṭir of Damascus (1375), for example, still his requirement that any physical sphere could move uniformly around an axis that did not pass through its center would make the existence of such a sphere physically impossible. And as was clearly stated by Ibn al-Haitham later on, and already quoted before, we do not live in an imaginary world where such spheres may only exist in the mind, but in a very real one whose motions had to be accounted for.
250
See the objections of Jābir Ibn Aflāḥ against this particular point in the Ptolemaic model, and his proposed solution for it, in Swerdlow, "Jābir Ibn Aflāḥ's Interesting Method", supra.