From that perspective, this relatively simple lemma proved to be an epoch maker, just like the Ṭūsī Couple, which will be mentioned later. And like the Ṭūsī Couple too, once it was discovered, it allowed several generations of astronomers to think differently about Ptolemaic astronomy, and about the possibilities with which this astronomy could be reformed.
As far as 'Urḍī was concerned, it turned out that with one new theorem, and with small adjustments to the directions and magnitudes of motions, he could reconfigure the whole body of Ptolemaic astronomy, and still produce his own configuration that was free of the absurdities of the equant and the like. In that regard he ended up playing a pivotal role in the development of Arabic astronomy, a role comparable only to that of Naṣīr al-Dīn al-Ṭūsī (d. 1274) as we shall soon see. His work on the planetary latitudes, however, like the work of all the other medieval astronomers, up to and including Copernicus, it could not resolve the major issues with Ptolemaic astronomy as elegantly as it could solve the longitudinal component.
'Urḍī's former boss at the Marāgha observatory came to his own solution of the equant problem in a slightly different fashion. To him, the problem was not fundamentally a problem of an epicycle that moved uniformly around an equant point, thus creating the physical absurdity, but was more a problem of a uniform motion that was observed from varying distances thus appearing to be non-uniform. One way of thinking about it was to allow the center of the epicycle in the model for the upper planets to move uniformly while at the same time still allow it to draw close to the point of Ptolemy's equant when close to the apogee, and move away while at perigee. This motion would in effect duplicate the phenomenon that Ptolemy said was exhibited by the observations. Therefore one could solve the problem if he/she could devise a way in which a body moving in uniform circular motion could still be allowed to come close to a specific point and draw away from it while at the same time retain the uniform circular motion undisturbed. The net effect would be that the body would be perceived to move at varying speed in an oscillating motion with respect to that point when, in fact, it would in itself continue to move in uniform circular motion. The problem was to achieve an oscillating motion in the realm of spheres that were all supposed to move uniformly around their own centers.
The idea of an oscillating motion resulting from circular motion seems to have occurred to Ṭūsī when he was tackling the problem of the Ptolemaic latitude theory. This was apparently the same circumstance under which Copernicus reached the same connection between the two phenomena.[259] Later on in the Commentariolus, and while describing the motions of the planet Mercury, Copernicus goes further by clearly making reference to the second connection between the motions of Mercury and the motions in latitude. At that point he does in fact describe the same Ṭūsī Couple that he used for his own Mercury Model as being related to the motions that he had already described in the latitude theory.[260] This very connection between the genesis of the Ṭūsī Couple and the motion in the latitude theory first came about when Ṭūsī had already noted, some three centuries before Copernicus, in his Taḥrīr that the oscillating motions described by Ptolemy in the latitude theory could be accounted for by a combination of two circular motions. Once construed as such, the net effect of the motion of the Ṭūsī Couple could in addition account for the Ptolemaic statement regarding the inclined planes of the lower planets Mercury and Venus, which were supposed to seesaw in order to produce the latitudinal motion of these planets. The elegance and superiority of this solution of the oscillating motion, through a deployment of a Ṭūsī Couple, becomes very clear when we remember Ptolemy's alternative suggestion of having the tips of the diameter of the inclined plane be attached to two "small circles" so that he could achieve that seesawing motion — a motion that would, at the same time, destroy the longitudinal motion on account of the resulting wobbling necessitated by the "small circles." It was in that context that Ṭūsī felt that Ptolemy's speech was outside the craft of astronomy.
Instead Ṭūsī suggested that one could achieve a better seesawing effect, without having to accept the necessary result of wobbling. And in order to do that, Ṭūsī then produced a rudimentary construction of two small circles of his own, which were fitted in such a way that one of them rode on the circumference of the other, and had the tip of the diameter of the inclined plane attached to the circumference of the second circle as well. When the motions of the two circles were supposed to be in such a way that the one riding on the circumference moved at twice the speed as the other one and in the opposite direction, then the point at the very tip of the circumference of the riding circle, i.e. the tip of the diameter of the inclined plane, would end up oscillating along the joint diameter of the two circles as a result of their uniform circular motions. This produced at once an oscillating motion from two combined uniform circular motions, and allowed the tip of the circumference of the riding circle to oscillate along a straight line, thus keeping it from wobbling. The combined effect of Ṭūsī's two circles successfully produced a straight motion by combining two circular motions, a result that was to have tremendous effects on later astronomers.
About 13 years after he wrote the Taḥrīr (that is, around 1260 or 1261), Ṭūsī developed the idea further in his al-Tadhkira fī al-Hay'a (Memoir on Astronomy), and produced it in the form of a theorem, that is now called the Ṭūsī Couple (figure 4.7). He did reach the same conclusion a few years earlier when he included the same theorem in his Persian text Dhayl-i Mu'inīya, whose date of composition is still uncertain but has to be sometime between the publication of the Taḥrīr in 1247 and the Tadhkira in 1260/61, where the theorem is fully stated and proved.
Figure 4.7
The Ṭūsī Couple. If two spheres such as AGB and GHD touched internally at point A, and if AGB's diameter was twice as large as that of GHD, and if the larger sphere moved uniformly in the direction indicated, and the smaller sphere moved in the opposite direction, at twice the speed, then point A would oscillate up and down the diameter of the larger sphere AB.
The theorem itself spoke of two spheres, instead of circles, one of them twice the size of the other, and placed in such a way that the smaller sphere was internally tangent to the bigger sphere as in figure 4.7 (1). Then Ṭūsī went on to prove that when the larger sphere moved uniformly at any speed, while the smaller one also moved uniformly, but in the opposite direction, at twice the speed, then the common point of tangency would end up oscillating along the diameter of the larger sphere.
259
In his discussion of the latitudinal motion of the planets, Copernicus makes the following remark: "If indeed this motion of libration takes place in a straight line, it is still possible that such a motion be composed from two spheres." Swerdlow,
260
Describing the connection between the oscillating epicyclic center of Mercury and the oscillating motion of the latitude plane of the planets we described above, Copernicus says: "For by this composite motion, the center of the larger epicycle is carried on a straight line, just as we have explained concerning latitudes that are librated", Swerdlow,