Books VI-VIII of the Almagest did not bother Ibn al-Haitham very much. Instead he moved very quickly to Book IX where the issue of the equant is discussed. In Almagest IX, 2, Ptolemy made the explicit statement that the upper planets moved in a uniform circular motion, just like the other planets he had discussed before. But by Almagest IX, 5, Ptolemy had laid the foundation for the equant problem when he insisted that "we find, too, that the epicycle centre is carried on an eccentre which, though equal in size to the eccentre which produces the anomaly, is not described about the same centre as the latter."[193]
The point that Ptolemy was trying to make at that occasion was that the two spheres, whose combined motion was responsible for the motion of the planet, were distinct spheres: one, the deferent, simply carried the epicycle of the planet, and the second, taken to be equal to the deferent in size, was responsible for the uniform motion of the planet's epicycle, but explicitly stating that the motion of the last sphere did not take place around the same center as the deferent. It was the center of the latter fictitious sphere, the sphere of uniform motion, that was later called the equant. In chapter IX, 6 of the Almagest, Ptolemy went on to describe much more clearly the equant center. There he defined it as a point along the line of apsides such that its distance above the center of the deferent was equal to the distance of the deferent's center from the center of the world.
Moreover, the line connecting this equant point to the center of the epicycle, when extended, constituted the line from which the mean motion of the epicycle was measured. In effect, this said that the deferent sphere, which carried the epicycle, was forced to move uniformly around a center, now called the equant, other than its own center, which was physically impossible.
By then, Ibn al-Haitham seems to have obviously realized the seriousness of the problem, as his following statement indicates: "What we have reported is the truth of what Ptolemy had established for the motion of the upper planets; and that is a notion that necessitates a contradiction."[194] This was in fact the contradiction between the physical reality of the celestial spheres and the mathematical model that was supposed to represent them. For as Ptolemy had accepted the uniform motion of the upper planets, the epicyclic centers of those planets were carried by deferents, which were supposed to move in this uniform motion. But with the equant proposition, one was told that the epicyclic center described equal arcs in equal times, i.e. moved uniformly, around a center that was not the center of the deferent that carried it.
But by Ptolemy's own proof in Almagest III, if a body moved uniformly around one point it could not move uniformly around any other point. Therefore the epicyclic center, as stipulated by Ptolemy, must move non- uniformly around the center of its own carrier, the deferent. And since the equant sphere was a fictitious sphere, and thus could not produce any perceptible motion of its own, as was often repeated by Ibn al-Haitham, the only sphere that could produce a real motion was that of the deferent, and that was now proved to be moving non-uniformly around its own center. This contradicts the assumption of uniform motion that was accepted by Ptolemy in the first place, hence the contradiction that was realized by Ibn al-Haitham. The other alternative was to assume that the same physical sphere, the deferent, could move uniformly around an axis that did not pass through its own center which was physically impossible, for it was exactly the physical absurdity mentioned before.
All the other models of the Almagest, except the model of the sun, which had problems of its own, shared this absurd feature of the equant. In the case of the moon, its epicycle too was also supposed to be carried on a deferent that moved in such a way that the epicyclic center of the moon did not describe equal arcs around its own deferent center, in equal times, but rather around the center of the world. That was in essence requiring a sphere to move uniformly around an axis that did not pass through its own center as well, which was exactly the point of the equant problem.
Mercury's model, which was considerably more complicated than the other planetary models, shared this feature as well. There too, the deferent that carried the epicycle of Mercury moved in such a way that its motion was not uniform around the deferent's center but around a point that was along the line of apsides half way between the center of the world and the center of another director sphere that carried the deferent sphere of Mercury.
Furthermore, both in the case of Mercury as well as the case of the upper planets, Ptolemy did not even attempt to demonstrate how he arrived at the location of the equant. It was simply stated to occupy such and such a position without any further discussion as to why, or any proof, as would have been expected in a mathematical science such as astronomy. It was this issue in particular that gave rise to the question, which was raised by another Andalusian astronomer by the name of Jābir b. Aflaḥ (middle of the twelfth century) and was singled out in his own research.[195]
From all those Ptolemaic configurations, Ibn al-Haitham could draw only one conclusion: that they were all extraneous to the field of astronomy. This much was even admitted by Ptolemy himself in Almagest IX, 2, where he had stated, in no ambiguous terms, that he was using a configuration that was contrary to accepted principle (khārija 'an al-qiyās as in the Arabic translation of the text, or "not from some readily accepted principle" as in Toomer's translation of the Almagest). From that admission, Ibn al-Haitham could then only conclude with a rebellious voice against the whole of Ptolemaic astronomy, articulated in the following terms:
[Since Ptolemy] had already admitted that his assumption of motions along imaginary circles was contrary to [the accepted] principles, then it would be more so for imaginary lines to move around assumed points. And if the motion of the epicyclic diameter around the distant center [i.e. the equant] was also contrary to [the accepted] principles, and if the assumption of a body that moved this diameter around this center was also contrary to [the accepted] principles, for it contradicted the premises, then the arrangement, which Ptolemy had composed for the motions of the five planets, was also contrary to [the accepted] principle. And it is impossible for the motion of the planets, which was perpetual, uniform, and unchanging to be contrary to [the accepted] principles. Nor should it be permissible to attribute a uniform, perpetual, and unchanging motion to anything other than correct principles, which are necessarily due to accepted assumptions that allowed no doubt. Then it became clear, from all that was demonstrated so far, that the configuration, which Ptolemy had established for the motion of the five planets, was a false configuration (hay'a bāṭila), and that the motions of these planets must have a correct configuration, which included bodies moving in a uniform, perpetual, and continuous motion, without having to suffer any contradiction, or be blemished by any doubt. That configuration must be other than the one established by Ptolemy.[196]
195
See Swerdlow, "Jābir Ibn Aflaḥ's Interesting Method for Finding the Eccentricities and Direction of the Apsidal Line of a Superior Planet", in