He continued:
If this were so then the motions of the deferents will have to be irregular by themselves, sometimes speeding up and at other times slowing down. And that is impossible according to the principles of this science (uṣūl hādhā al-'ilm). ... If one were to admit these kinds of impossibilities in this discipline (ṣinā'a), then it would all be baseless, and it would have been sufficient to say that each planet has one concentric sphere only, and any other eccentric or epicyclic sphere would be an unnecessary addition.[276]
The simple solution that was proffered by Ghars al-Dīn, for the voluntary motions of the spheres, and yet allowing for their predictability, simply stated:
Where would the need be for the particular spheres that you (meaning the Ptolemaic astronomers) have posited, which you have up till now failed to correct, with all the contrivances and circumvention implied by them? Let us then say that each planet has one sphere that moves by its own volition, sometimes speeding up, other times slowing down, becomes stationary, moves forward, and retrogrades, etc. What adds to its being natural is the fact that it follows a specific pattern.[277]
Incidentally, Ghars al-Dīn's solution of the problem of predictability and yet allowing for volition, by allowing the spheres to "follow a specific pattern", is reminiscent of the concept of custom ('āda) that was offered about 500 years earlier by Ghazālī who also had the same predicament of allowing miracles to take place and yet have the continuity of the world and the predictability that continuity entailed.[278] Dare we suggest here solutions derived from religious texts being applied to astronomical texts, as Ghars al-Dīn seems to be doing?
For the astronomer Naṣīr al-Dīn al-Ṭūsī (d. 1274), the necessity of developing a new mathematical theorem in order to resolve the Ptolemaic predicament of the latitudinal motion of the planets had other "unintended" philosophical consequences. When Ptolemy wished to allow the inclined plane (really the equator of the carrying sphere) of the lower planets of Venus and Mercury to oscillate north and south of the ecliptic as the planetary epicycles of those planets moved from the extreme north to the extreme south, he proposed to attach the tips of the diameter of that inclined plane to two small circles that were placed perpendicularly to the plane of the ecliptic. Ptolemy imagined that in this fashion he could have the diameter's tips move along those circles and as a result they would generate the required oscillating motion that will in turn explain the latitudinal motion.
At that point, Ṭūsī exclaimed that the kind of speech that Ptolemy was using was outside the craft of astronomy.[279] Not only because such attachments of the diameter's tips would produce a wobbling motion when it performed the required latitudinal motion, but because that same wobble would first destroy the longitudinal component of the motion that was painstakingly calculated and accounted for with the rest of the predictive mathematical model. Second, it would introduce into the celestial realm oscillating types of motions or motions that were not in complete circles. This last requirement would violate the very essence of the Aristotelian definition of the celestial spheres.
Ṭūsī proposed to resolve the problem by the introduction of his own theorem, now called the Ṭūsī Couple, which allowed for the solution of both of Ptolemy's problems: first it allowed for the oscillating motion as a result of complete circular motions, and second it avoided the necessary wobbling that was required by the Ptolemaic suggestion. With one theorem both problems were solved at once.
As an unintended consequence, this theorem confronted the Aristotelian dogmatic separation of the celestial world from the sublunar one. Aristotle had separated those two worlds on the basis of the nature of motion that pertained to either one of them. Linear motion was natural to the sublunar world, while the celestial world only moved in circular motion. Ṭūsī's theorem now presented the most glaring counter example. For here we have, with Ṭūsī's Couple, a universe in which, under the right conditions, linear motion could necessarily result from two circular motions. This would not only make the Aristotelian division of those two worlds completely artificial, that is unnatural in the Aristotelian sense, but it would also make the Aristotelian characterization of generation and corruption as a by-product of the contrary linear motions, particular to the sublunar world, also artificial and completely arbitrary. Furthermore, since the Ṭūsī Couple could also demonstrate that the oscillatory linear motion, which was produced by the two uniform circular motions, was necessarily continuous and uniform, then the requirement that there be a moment of rest between ascending and descending directions of oscillatory motion was also cast in doubt.[280]
Ṭūsī did not make any of those critiques of the Aristotelian universe at the time when he proposed his new theorem, for then he was more concerned with the damage Ptolemaic latitudinal motion was inflicting on the longitudinal motion. But his commentators, starting with his immediate student and collaborator, Quṭb al-Dīn al-Shīrāzī (d. 1311), noticed all the "unintended" philosophical implications the theorem managed to produce.[281] He articulated his observation concerning the moment of rest between two contrary motions in the following terms:
This could be used as a proof for the absence of rest between two motions, one going up and one going down. This is obvious. And the one who asserts that there must be rest between the two motions cannot deny the possibility of such motions by the celestial bodies simply because he believes there must be rest and rest is not possible for the celestial objects. This is so because we shall use it whenever there is an ascending motion and a descending one as we shall see in the forthcoming discussion. We couldn't be blamed if we also used it to disprove that principle [i.e. the Aristotelian principle of rest between two opposing motions], as can be witnessed from observation. For if we drill a hole in the bottom of a bowl whose edge is circular, but of unequal height above its base, and if we pass a thread through the hole and attach a heavy object to it. Then if we move the other edge of the taut thread along the edge of the bowl, the heavy object will descend and ascend on account of the variation in the height of the bowl's edge, in spite of the fact that it does not come to rest because the mover does not come to rest by assumption.[282]
This example of producing oscillatory motion as a result of continuous circular motion is a variation on another example, dealing with the very same notion of rest between two contrary motions that was already offered by the twelfth-century philosopher Abū al-Barakāt al-Baghdādī (d. 1152). Al-Baghdādī stipulated that one could produce such oscillatory motion by drilling a hole in the middle of a ruler and passing a thread through that hole. If one were to attach at one end of the thread a plumb line, and hold the other end with his hand, then as one moved his hand continuously from one end of the ruler to the other, the plumb line would oscillate up and down without coming to rest in between the contrary motions since the cause of those motions did not come to rest.[283]
277
Ghars al-Dīn, Aḥmad b. Khalīl al-Ḥalabī (d. 1563),
278
Al-Ghazālī,
279
See Saliba, "Role of the Almagest commentaries", reprinted in Saliba,
280
The repercussion of this theorem in terms of categories of motion is treated in greater detail in Saliba, "Aristotelian Cosmology", p. 263f.
281
See G. Saliba and E. S. Kennedy, "The Spherical Case of the Ṭūsī Couple",
283
Abū al-Barakāt al-Baghdādī,