The astronomical developments that took place after the time of Ibn al-Haitham have special significance for another reason. Not only do they illustrate the continuity of the earlier critical tradition, but also demonstrate the kind of new questions that began to emerge, and the similarity between those questions and the ones that were raised later on during the European Renaissance.
Naṣīr al-Dīn al-Ṭūsī (d. 1274), who was mentioned before in connection with the various critiques of the Ptolemaic text of the Almagest, had his own doubts about the cosmological issues that have been raised so far. In his Taḥrīr al-majisṭī (completed in 1247), he only criticized Ptolemy sporadically. But in his later work, the Tadhkira (completed in 1260), he devoted much longer sections to the cosmological questions and proceeded to formulate his own mathematical models to replace those of Ptolemy. We shall have occasion to return to Ṭūsī's reformed models later. For now, and in the context of the encounter with the Greek tradition, the remarks he made in his Taḥrīr should give us an idea of his thoughts on the subject around the middle of the thirteenth century.
By comparing Ṭūsī's various works it becomes apparent that he began to ponder the importance of the cosmological issues for the first time when he was composing the Taḥrīr, a book that was devoted to the production of a useful, updated version of the Almagest, thus naturally offering an ideal occasion to voice his own reservations about the book he was re-editing.
In the Taḥrīr, and while discussing the lunar model of Ptolemy, Almagest [V, 2], Ṭūsī concluded that section with the following remark: "As for the possibility of a simple motion on a circumference of a circle, which is uniform around a point other than the center, it is a subtle point that should be verified."[213] Doubtless, this is the same irregularity that was mentioned before in the context of the equant problem, i.e. the absurdity arising from the situation when a sphere is forced to move uniformly, in place, around an axis that did not pass through its center.
Furthermore, in the case of the prosneusis point of the lunar model, Ṭūsī simply said: "This motion is similar to the motion of the five [planets] in the inclination and the slanting, as will be shown later on, except that that is a motion in latitude while this one is in longitude. One must look into the possibility of the existence of complete circular motions that would produce such observable motions [i.e. similar to the oscillating prosneusis motion of the epicyclic diameter.] Let that be verified."[214] One could easily see how this perplexity could have been at the origin of Ṭūsī's thinking, and which later led him to invent his famous mathematical theorem, now called the Ṭūsī Couple in the literature. The theorem itself achieved just that: an oscillatory motion produced by a combination of two circular motions.
In fact the latitudinal motion of the planets shares many features with the motions of the lunar spheres, and in particular with the slanting of the prosneusis point. That is, the oscillation of the axis, which marks the beginning of the epicyclic motion of the moon, is similar to the oscillation of the inclined planes of the lower planets. And it was this particular Ptolemaic theory of planetary latitudes that exhausted Ṭūsī's patience. He saved his sharpest criticism just for that notion. In a nutshell, Ptolemy accounted for the motion of the planetary inclined planes in latitude by suggesting that one could affix the tips of the diameters of those planes to a pair of small circles along which the tips of the diameters of the said planes would move. And as soon as he suggested those small circles he knew that he was not abiding by the accepted principles, as we hinted before, and thus felt that he had to justify his solution in the following manner: "Let no one, considering the complicated nature of our devices, judge such hypotheses to be over-elaborated. For it is not appropriate to compare human [constructions] with divine, nor to form one's beliefs about such great things on the basis of very dissimilar analogies.''[215] To this, Ṭūsī could only say:
This statement is, at this point, extraneous to the art [of astronomy] (khārij 'an al-ṣinā'a). For it is the duty of those who work in this art to posit circles and parts that move uniformly in such a way that all the varied observed motions would result as a combination of these regular motions. Moreover, since the diameters of the epicycles had to be carried by small circles so that they could be moved northward and southward, which also entailed that they would be moved as well from the plane of the eccentric [i.e. the deferent] so that they would no longer point to the direction of the ecliptic center, nor would they be parallel to the specific diameters in the plane of the ecliptic, but they would rather be swayed back and forth in longitude by an amount equal to their latitude, that, is contrary to reality. One could not even say that this variation is only felt in the case of the latitude, and not in the longitude, because they are equal in magnitude and equally distant from the center of the ecliptic.[216]
In the context of this very criticism of Ptolemy, Ṭūsī did not only redefine the function of the astronomer with respect to the observations and the mathematical methods with which these observations should be explained, but he went on to propose a new theorem that could resolve this specific predicament of Ptolemy. The new theorem, which was expressed in the
Taḥrīr in a preliminary fashion only to be developed further into the just-mentioned Ṭūsī Couple later on in the Tadhkira, will be revisited in the sequel when we return to the context of the non-Ptolemaic models that were constructed for the specific purpose of formulating alternatives to Ptolemaic astronomy.
Returning to the shukūk tradition, we note that about three centuries later, by the end of the fifteenth century, when the classical narrative had already preached the death of Islamic science, the problems (shukūk/ishkālāt) of Ptolemaic astronomy continued to attract the attention of the working astronomer. In fact those very problems became so famous, and so widespread by then, that they were taken up on their own and made into subjects of individual works, in a manner reminiscent of the specialized shukūk of Rāzī and Ibn al-Haitham almost half a millennium before.
One such fifteenth-century work (of about forty folios in one manuscript) was composed by Muḥyī al-Dīn Muḥammad b. Qāsim, known as al-Akhawayn (d. ca. 1500). The title of the work is simply al-ishkālāt fī 'ilm al-hay'a (Problems in the Science of Astronomy), and seems to have been taken from the first sentence of the book which followed the usual introduction. The sentence began immediately with the enumeration of the famous problems of astronomy. By al-Akhawayn's count, those problems were reducible to seven, and they were all to be found in the received Ptolemaic astronomy.
Al-Akhawayn's treatise began thus:
Know that the famous problems relating to the science of astronomy (al-ishkālāt fī 'ilm al-hay'a) in regard to the configurations of the spheres are seven. The first is (the problem) of speeding up, slowing down, and mean motion... The second (concerns the appearance) of planetary bodies being sometimes small, and sometimes large. The third (concerns) the stations, retrograde, and direct motion... The fourth (concerns) uniform motion around a point different from the center of the mover. That is, when a mover moves another body in circular motion and the second body covers equal angles in equal times around a point other than the center of its mover. The fifth (concerns) a motion that is uniform around a specific point as it draws near to that point and moves away from it. The sixth (concerns) the slanting of the direction of the diameter of one sphere that is moved by another sphere from the center of that sphere (meaning the moving sphere)... The seventh (concerns) the lack of complete revolutions among the celestial motions as will be explained in detail.[217]