The focus will then be on those astronomers who felt that they needed to invent new concepts, or more concretely new mathematical theorems, in order to solve the problems of Ptolemaic astronomy, and not that much on those who followed them by incorporating the latest theorems to build upon them in order to create new planetary models of their own. Of those who introduced such new theorems, the names of Mu'ayyad al-Dīn al-'Urḍī (d. 1266) and Naṣīr al-Dīn al-Ṭūsī (d. 1274) stand out on account of the fact that each of them supplied his own mathematical theorem while undertaking to overhaul the fundamental features of Ptolemaic astronomy.
'Urḍī thought that Ptolemy's models for the sun were adequate enough, and that Ptolemy's choice of the simple eccentric model was innocuous enough, that it did not deserve any special transformation. Neither 'Urḍī nor Ptolemy ventured to state explicitly what he really thought of the epicyclic model, posited by Ptolemy at least as an alternative to the "offensive" eccentric model. One suspects that the ubiquitous use of epicycles in all other planetary models, and the impossibility of their replacement, may have made their use a necessity that could not be avoided. But no one was willing to defend the use of those epicycles explicitly. Their final theoretical solution would not come until a century later with the works of Ibn al-Shāṭir (d. 1375) as we shall soon see.
As for the motions of the moon and Mercury, and the notorious equants in both models, in addition to the prosneusis point in the case of the moon, 'Urḍī felt that he could not let things be. Instead he decided to take advantage of the similarities between the two models, and tried to reconfigure them by adopting three new steps. First he decided to shift the directions of the motions of the various spheres. Then he adjusted the magnitudes of those motions. And finally he tried, in a global way, to avoid the Ptolemaic handicaps that plagued both models by making all spheres move uniformly on axis that passed through their centers. At this point he still restricted himself to the mathematics that was available to Ptolemy from Euclid's Elements, for example, without having to offer any new mathematical propositions of his own. There were times though when he would venture to say that he faulted Ptolemy for his inability to theorize (ḥads) properly, but would still express his full admiration for Ptolemy's observational and mathematical control of the data. Such shifts in theorizing, as long as they did not involve the introduction of new material like the trigonometric functions, were quietly introduced nevertheless without much fuss.
But when it came to the model of the upper planets, 'Urḍī felt that the Ptolemaic model was no longer redeemable, and thus had to be reconfigured in a fundamental way. It was there that he proposed to introduce what has now become known in the literature as 'Urḍī's Lemma in order to resolve the very thorny issue of the equant problem. At this point, 'Urḍī's concern was no longer focused on the cosmological choices of eccentrics versus epicyclic models, but was focused on the more fundamental equant stipulation which forced the very sphere that was supposed to carry out the motion of the epicycle to loose its sphericity. This physical impossibility could not be tolerated, and still pretend to carry out astronomical theorizing, as these astronomers saw their functions to be. Instead, 'Urḍī approached the problem of the model of the upper planets with the mathematically rigorous manner it deserved.
After demonstrating the physical failings of Ptolemy's model, he went on a tangent and said that in order to theorize better about the motions of those planets, he needed to introduce a new theorem, the statement of which could be rephrased thus: Given any two equal lines that form equal angles with a base line, either internally or externally, the line joining the extremities of those two lines would be parallel to the base line.[258]
Taken on its own, 'Urḍī's Lemma looked like a generalization of Apollonius's theorem, in that the equal angles needed for the proof of the parallelism of the end line with the base line are no longer restricted to the exterior angles used in the construction of the epicyclic model. Instead 'Urḍī could show that the internal equal angles would produce the same effect of parallelism and thus could be used to rectify the instance of the equant without losing its observational value that had forced Ptolemy to adopt it in the first place.
Figure 4.6
'Urḍī's model for the upper planets. By defining a new deferent with a center at K, halfway between the center of the Ptolemaic deferent T and the equant D, 'Urḍī allowed that deferent to carry a small epicycle whose radius was equal to TK = KD. He made the small epicycle move at the same speed as the new deferent, and in the same direction. By applying his own lemma, 'Urḍī could demonstrate that line ZD, which joined the tip of the radius of the small epicycle to the equant, would always be parallel to line KN, which joined the center of the new deferent to the center of the small epicycle. He could also show that point Z, the tip of the radius of the small epicycle, came so close to the point O, which was the center of the Ptolemaic epicycle, that the two points could not be distinguished. Then it was easy to see that the uniform motion of O that Ptolemy thought took place around point D was indeed a uniform motion around point N which in turn moved uniformly around K, thus making Z appear to be moving uniformly around D and satisfying the Ptolemaic observations.
Instead of assuming (figure 4.6) that the epicycle is carried by a deferent that moved uniformly around an axis that did not pass through its center, as was done by Ptolemy, 'Urḍī shifted the center of his new deferent to a point K, which was located halfway between the center of the old Ptolemaic deferent T and the equant point D. He then allowed this new deferent to carry a small epicycle whose radius was equal to half the Ptolemaic eccentricity, or equal to the same magnitude by which the center of the deferent was shifted in the first place. The small epicycle moved at the same speed as the old Ptolemaic deferent, and in the same direction, and in turn carried the Ptolemaic epicycle. The combination of the equal motions allowed the lines joining the extremities of the small epicycle's radius to points K and D respectively to be always parallel. This made the center of the Ptolemaic epicycle, now carried at the extremity of 'Urḍī's small epicycle, look like it was moving uniformly around the Ptolemaic equant. In fact it moved around the center of its own small epicycle, and the center of that epicycle, in turn, moved around the center of the new deferent. With all spheres now moving uniformly, in place, around axis that passed through their centers, 'Urḍī managed to avoid the absurdity of the Ptolemaic equant altogether, and, at the same time, still retain its observational value, as was required by the Ptolemaic observations.
'Urḍī's Lemma, introduced through the mechanism of the small circle in the model for the upper planets, proved to be a very useful tool for other astronomers and at other occasions as well. A whole host of astronomers ended up using it in order to construct their own alternative models to those of Ptolemy. Astronomers such as Quṭb al-Dīn al-Shīrāzī (d. 1311) used it in his lunar model. While Ibn al-Shāṭir of Damascus (d. 1375) ended up using it in more than one of his own models. 'Alā al-Dīn al-Qushjī (d. 1474) used it in his model for the planet Mercury, and Shams al-Dīn al-Khafrī (d. 1550) made a double use of it in his own model for the upper planets. Finally, Copernicus (d. 1543) used it for the same model of the upper planets. As it turned out, this mathematical tool became very fecund in the construction of all sorts of responses to Greek astronomy.
258
For a full original statement of this theorem see G. Saliba, "The Original Source", reprinted in Saliba