Figure 6.1
The lunar model of Ibn al-Shāṭir and Copernicus.
But this finding opened the door to further investigation. It was also Neugebauer who widened the circle of research and began to look for other similarities of ideas between the works of Renaissance scientists and the scientists of the Islamic world. And it was in that context that he revisited a chapter from the Tadhkira fī 'ilm al-hay'a, of Naṣīr al-Dīn al-Ṭūsī (d. 1274), which had already been translated into French by Bernard Carra de Vaux in 1893 and published under the title "Les spheres célestes selon Nasîr-Eddîn Attûsî."[311] In this chapter, which was originally written by Ṭūsī in 1260-61, Ṭūsī formalized as well as generalized and now supplied a rigorous mathematical proof to the famous theorem that has come to be known in the literature as the Ṭūsī Couple, which we have seen before.
We have also seen that the first articulation of this theorem, had already been proposed in 1247, in yet another work by Ṭūsī, the Taḥrīr al-majisṭī (Redaction of the Almagest), which is yet to be edited and published. And, as I have noted, this earlier first articulation was specifically proposed in order to respond to the failings of the Ptolemaic latitude theory of the planets. This background of the theorem was neither mentioned by Ṭūsī nor was it signaled by Carra de Vaux, on his own, nor was it apparently known to Neugebauer who did not work on the Arabic manuscripts directly.
Ṭūsī's Couple, as we have seen, offered a general solution to the problem of generating linear motion from a combination of circular motions. It was expressed in terms of the motion of two spheres, usually called in the Arabic astronomical literature that followed al-kabīra wa-l-ṣaghīra (the large and the small). As has been noted, one of those spheres was taken to be twice the size of the other, and in the initial setting the spheres were taken to be internally tangent at one point. With the motion of the larger sphere at any speed, and the motion of the smaller sphere at twice that speed, in the opposite direction, the point of tangency was then found to oscillate along the diameter of the larger sphere, thus producing the required linear motion. In 1260-61, after supplying the formal mathematical proof to this theorem, Ṭūsī went on to use it in the lunar model and then in the model for the upper planets, as we have already seen.
Carra de Vaux's translation of this chapter gave all the contents of the original Arabic in a rather faithful French, but was then concluded with an assessment by Carra de Vaux himself. In it, and on the basis of his encounter with this particular work of Ṭūsī, de Vaux summed up the general character of Arabic astronomy. In de Vaux's time, and because very little else was known then from the astronomical production of the Islamic civilization from these later periods, de Vaux was emboldened to say, that while Arabic astronomy did not hold Ptolemy's work with much regard [an understatement indeed about a chapter that was devoted specifically to critiquing the problems in Ptolemaic astronomy], it did not, on its own, have enough "génie" to transform astronomy altogether, and instead suffered from a general "faiblesse" and "mesquinerie" that did not allow it to develop further. From such a statement, one has to draw the conclusion that de Vaux could not fully appreciate the importance of the chapter that he was translating at the time. We shall have occasion to return to this issue when we speak about the so-called age of decline of Islamic science.
With a completely different attitude, and being immersed in the mathematical astronomy of Copernicus at the time, Neugebauer could immediately see the essence of Ṭūsī's problem, because he could also see that it was the same problem that was faced by Copernicus later on, in De Revolutionibus III, 4.[312] Both astronomers needed to utilize a mechanism that allowed them to generate linear motion from circular motion or combinations thereof, as I have said several times already. And both used the same Couple, except for one difference: Ṭūsī knew that he was introducing a new theorem in 1247[313] and again in 1260-61, which was nowhere to be found in any earlier Greek source, and said so, while Copernicus silently went ahead and described the same theorem and produced a very similar proof as we shall see, without mentioning that he had invented the theorem or the proof himself, nor that he had seen it in any other source. He only mentions the vague reference to a statement by Proclus,[314] referring to the latter's commentary on Euclid, in which Proclus says that linear motion could be gotten from circular motion. But for those who read Proclus closely will immediately realize that Proclus was talking about curved lines and straight lines being produced from one another and not oscillating motion resulting from complete circular motion as was required by Ṭūsī and Copernicus after him.
By 1973, Willy Hartner discovered a remarkable feature in Copernicus's proof of the same theorem.[315] By comparing Ṭūsī's proof, which was completed in 1260-61, with that of Copernicus, which was published in 1543, Hartner discovered that the two proofs (figure 6.2) carried the same alphabetic designators for the essential geometric points. That is, where Ṭūsī's proof designated a specific point with the Arabic letter "alif", Copernicus's proof signaled that same point with the equivalent phonetic Latin letter "A", where Ṭūsī, had "bā'", Copernicus had "B", etc., except in one case where Ṭūsī had "zain" and Copernicus has "F". On the basis of the letter correspondences, letter to letter, Hartner ventured to say that Copernicus must have known about Ṭūsī's work while in Italy. The implication that was also spelled out by Hartner was that Copernicus must have had access to Ṭūsī's work in some indirect form, since as far as we know neither Copernicus could read Arabic, nor was Ṭūsī's text, in which the theorem appeared, was ever translated into Latin. To Hartner, it meant that Copernicus must have recruited someone who could explain to him the diagram, while he took notes and used those notes later when he came to write the De Revolutionibus.
Figure 6.2
Proofs of the Ṭūsī Couple from the works of Ṭūsī (left) and Copernicus (right), showing the identity of the lettering of the diagrams. Wherever Ṭūsī had alif Copernicus had A, and wherever Ṭūsī had bā' Copernicus had B, and so on, except that where Ṭūsī had zain for the center of the smaller sphere Copernicus had F. See figure 6.3.
In a more recent reassessment of Hartner's results, I added the Arabic manuscripts evidence to account for the variation between the "zain" and the "F" in the two proofs.[316] By comparing Arabic manuscripts from the medieval period, and noting that the two Arabic letters "zain" and "fā"' that were usually used to designate geometric points, the similarities between those two letters were in fact so close that it would be quite easy for someone, who was not experienced enough with Arabic manuscript traditions, to misread the "zain" for a "fā'" (figure 6.3). I ventured to say that either Copernicus himself or someone sitting next to him, looking at an Arabic text of the proof of Ṭūsī, simply misread the "zain" in the original Arabic manuscript for a "fā'", thus leading Copernicus to introduce the sole variation in the lettering of the two proofs.
312
Nicolaus Copernicus,
313
In response to Ragep's assessment in Ragep,
314
For a treatment of this problem and the vague reference by Copernicus to "some people" before him who had used the theorem, and the connection to the statement by Proclus, see Ragep's longer discussion, together with several references, of this particular point (
315
Willy Hartner, "Copernicus, the Man, the Work, and Its History",
316
See G. Saliba, "Re-visiting the Astronomical Contacts between the World of Islam and Renaissance Europe: The Byzantine Connection" (forthcoming).