Once he had generalized that theorem, he knew he had found a fecund theorem that could be used whenever one needed to produce linear motion as a result of combined circular motions. He then went on to produce, in the same Tadhkira, a formal proof for the theorem, as in figure 4.7, and later applied it to construct two of his alternative models: the lunar model and the model for the upper planets. In this fashion he then managed to solve the equant problem in the two respective Ptolemaic models for those planets.
The success of this theorem had widespread repercussions. It ended up being used by almost every serious astronomer that followed Ṭūsī, including the Renaissance astronomers such as Copernicus and his contemporaries, as we have already hinted and shall see again in more detail later on. In contrast to Copernicus however, the only place where Ṭūsī failed to apply his Couple, was in the case of the planet Mercury, whose behavior was quite challenging for Ṭūsī as we have already seen. When discussing that planet's motions in particular, Ṭūsī unambiguously declared that although he succeeded in solving the equant problem of the models of the moon and the upper planets, he was hoping to complete his task later on by solving the equant problem of Mercury, to which he had no new things to add at the time.
Figure 4.8
Shīrāzī's lunar model. By taking a new deferent whose center was halfway between the center of the world O and the center of the Ptolemaic deferent J, Shīrāzī compensated for that by introducing a new small epicycle, with center H, whose radius was equal to the same distance between the two centers of the respective deferents. By making the small epicycle move at the same speed as the new deferent, and in the same direction he managed to satisfy the conditions for 'Urḍī's Lemma, which could now be applied to lines HE and OF, thus making line EO always parallel to line HF, and making the epicycle center C appear to be moving around the center of the universe O.
Ṭūsī's student and colleague Quṭb al-Dīn al-Shīrāzī (d. 1311) made use of 'Urḍī's Lemma twice, once in developing his lunar model, and the other time when he adopted the same model for the upper planets as that of 'Urḍī In the lunar model (figure 4.8), he avoided the use of the Ptolemaic equant by bisecting the eccentricity of Ptolemy's deferent for the moon, and adjusting for it by positing a small circle at the circumference that satisfied the same conditions 'Urḍī's small circle satisfied in the case of the upper planets. That is, he allowed the small circle to move at the same speed as the deferent and in the same direction thus satisfying the condition of having two interior angles equal, and thus produced the parallel lines. The new arrangement, as suggested by Shīrāzī, made sure that the small circle, which moved uniformly around the center of its own deferent, also had the tip of its radius seem as if it was moving uniformly around the center of the world, which in turn was the observational condition Ptolemy's model satisfied. Thus by positing the Ptolemaic lunar epicycle at the tip of that radius the moon would then move around its own epicycle, but at the same time was seen to satisfy the same observational conditions it satisfied in Ptolemy's case.
In the case of the planet Mercury, Shīrāzī gave some nine models that could describe the motion of this planet. Those models are detailed in two of his most famous works, the Nihāyat al-idrāk fīdirāyat al-aflāk (the ultimate understanding regarding the comprehension of the spheres), and the Tuḥfa Shāhīya (the Royal Gift). In a still later work (Fa'altu fa-lā talum, meaning "I had to do it thus don't blame"), he signaled that seven of those models were faulty. Furthermore, one of the last two was also wrong. But the determination of which one was left unsaid so that Shīrāzī could test the intelligence of his readers, as he boldly claimed. The chosen model, which he finally claimed was the correct one, involved the use of two sets of the Ṭūsī Couple, arranged in such a way that it could successfully avoid the use of the Ptolemaic equant but preserved its effect and the conditions it entailed. That is, the final center of Mercury's epicycle seemed as if it moved around the point designated as the equant by Ptolemy, without ever having to have that motion come about as the product of uniform circular motions of any sphere around an axis that did not pass through its center.
Shīrāzī did not offer any new theorems, but obviously he benefited from his two contemporary astronomers, and deployed their theorems to the best of his abilities. One wonders why, for example, he opted to use 'Urḍī's model for the upper planets instead of the equally good model of Ṭūsī. But one has to also acknowledge that even if we cannot answer the question in the present circumstances, we can certainly affirm that Shīrāzī had at least two options and that he chose the one that deployed 'Urḍī's Lemma for his solution of the lunar model as well as the model of the upper planets, and reserved the double use of the Ṭūsī Couple for his Mercury model. One has to also acknowledge that Shīrāzī's double use of the Ṭūsī Couple for the planet Mercury, was in itself a significant step, not only because he succeeded where Ṭūsī had already declared failure, but because he seems to have put into wider circulation a novel idea, such as the double use of the Ṭūsī Couple, which was itself a remarkable departure from the accepted Ptolemaic astronomy when it used once. This remarkable achievement of Shīrāzī does not only put him at the forefront of the astronomers of his time, but allows us to see how novel ideas began to take hold in the scientific culture. They apparently succeeded when they began to be accepted and deployed by their contemporaries.
Shīrāzī insisted that one could begin to think of solving the observational behaviors of the planets by applying different mathematical techniques and producing more than one mathematical model. At the same time though, Shīrāzī was still under the impression that some mathematical solutions are more "true" than others. This attitude will become considerably important later on when we consider yet another conceptual shift as the one taken in the works of Shams al-Dīn al-Khafrī (d. 1550). Properly speaking, Shīrāzī's work lies at the beginning of a tradition that began to seek alternative mathematical solutions to the same physical problem. But at that early stage, the tradition still sought true mathematical solutions that could properly describe the motions of the planets. This tradition would not mature until the time of Khafrī. But just by seeking alternative mathematical solutions, and thus new ways of thinking about the problems, allows us to group Shīrāzī with 'Urḍī and Ṭūsī, who also created new shifts in the articulation of responses to Ptolemaic astronomy.
But because Shīrāzī also tried to group together a series of solutions which were offered by his predecessors, a series that he called uṣūl (principles/ hypotheses)[261], and which included such concepts as the eccentric versus the epicyclic models as two such principles, he can also be considered as the forerunner of the work of someone like Ibn al-Shāṭir (d. 1375), who came about half a century later, and who also used the solutions of his predecessors, and also globally called them uṣūl, as in his taṣḥīḥ al-uṣūl (correction of principles). In the case of Ibn al-Shāṭir, he too went beyond his predecessors and managed to succeed where they failed, again as in the case of the Mercury model that was correctly solved by Ibn al-Shāṭir when Ṭūsī had failed to accomplish that. But Ibn al-Shāṭir did more still, and deliberately carved new directions for his astronomy that also proved to be very productive for the Renaissance scientists.
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Now see Robert Morrison for the latest edition and translation of Shīrāzī's chapter dealing with the