But misreadings and variations are on their own very useful for detecting textual transmissions. For as a result of these reading "mistakes" I became quite confident about the conclusion just drawn: that Copernicus was either himself working from an Arabic manuscript where he mistook the "zain" for a "fā'" which is unlikely since we do not know that he knew any Arabic at all, or that he was reading, at least the diagram, with someone else's help who made the same mistake. Furthermore, the complete conformity of the other geometric points between the two proofs now makes the issue of coincidence and independent discovery a most unlikely scenario.
Figure 6.3
A medieval Arabic manuscript exhibiting the similarities between the letters zain = Z and fā' = F.
Therefore, not only did Copernicus apparently seek to solve the same problem of the Greek astronomical tradition by adopting the same approach (that is, adding a mathematical theorem that allowed for the generation of linear motion from circular motion); he also used a theorem that had been invented by Ṭūsī about 300 years earlier, and supplied a proof that was very similar to the one supplied by Ṭūsī, with a slight modification in protocol, but still adhered to the same geometric points that were used by Ṭūsī in the original proof. All of this cannot be mere coincidence, as some people still like to think. And future research both in the world and works of Copernicus, as well as the world and works of the Arabic-writing astronomers who preceded him will, I am sure, eventually uncover the exact route by which Copernicus learned of the earlier astronomical findings of the Muslim astronomers.
Mu'ayyad al-Dīn al-'Urḍī (d. 1266) was a colleague of Ṭūsī, and a distinguished astronomer and engineer in his own right. His distinguished fame must have been the deciding factor for Ṭūsī when he hired him to build the observational instruments for the famous Marāgha Observatory.[317] The observatory itself was established in 1259 in the city of Marāgha in northwest modern-day Iran, under the patronage of the Ilkhānid monarchs.[318]Because of the concentration of the astronomers who worked at the observatory, and because of the recently-found connection between their works and the work of Copernicus, this observatory has now become very famous in the secondary literature. 'Urḍī's fame, however, was obviously based on his most important work which was simply called Kitāb al-hay'a (Book on Astronomy).[319] In it he attempted to revamp the whole of Greek astronomy, having been obviously motivated by the same considerations, which had been discussed for generations before him within the intellectual circles of the Islamic Civilization. The most important problem for the time was still encapsulated in the discussion of the inadmissibility of the equant sphere, on account of the well-known absurdity this concept produced.
Trying his own hand at the resolution of this problem, 'Urḍī proposed a new simple theorem which allowed him to reconstruct the Ptolemaic model[320] for the upper planets by adding new spheres and epicycles, but still accounted for the same observations that were reported by Ptolemy, without having any of the absurdities that were adopted by Ptolemy. In 'Urḍī's model, all the spheres moved uniformly in place around axis that passed through their centers. One could therefore say that in this model 'Urḍī managed to avoid the use of the Ptolemaic equant, but did not avoid accounting for its essential observational effects.
The theorem itself (figure 6.4), now known as 'Urḍī's Lemma, is extremely simple. It stipulates that for any two lines (such as AG and BD) that are equal in length and that form equal angles with a base line AB, either internally or externally, the line DG, joining the other extremities of these two lines, is parallel to the base line AB.
Figure 6.4
A general representation of the four cases of 'Urḍī's Lemma as it appeared in the original manuscripts. This illustrates the four cases of possible equations between internal and external angles.
He first used it in his lunar model, and then he incorporated it in the same model for the upper planets that he had borrowed from 'Urḍī, theorem and all. In both of Shīrāzī's works, which were written few years apart in the 1280s, 'Urḍī's model for the upper planets remained to be Shīrāzī's favorite model, despite the fact that both of the said works of Shīrāzī were themselves written as commentaries on the work of Ṭūsī, and not that of 'Urḍī. By preferring "'Urḍī's Lemma over the solutions that were offered by Shīrāzī's very own teacher Ṭūsī, which made use of the Ṭūsī Couple, Shīrāzī's choice can only be taken as a testament to the popularity of 'Urḍī's Lemma.
Ibn al-Shāṭir (d. 1375), who lived a full century later, followed suit. After using the equivalent of the Apollonius theorem to shift the eccentricities to epicyclic attachments, as we have already seen, in order to return to strict geocentric cosmology, he added to what can be called the Apollonius epicycle another 'Urḍī epicycle in order to account for the motion around the equant as was done by 'Urḍī. In essence, Ibn al-Shāṭir's model for the upper planets is the same as that of 'Urḍī, except for the transposition of the eccentricity that was used by 'Urḍī, and which was one and a half times as large as that of Ptolemy, to an epicycle with the same radius. The rest of the model preserved the same properties. That is, it deployed the same dimensions for the 'Urḍī epicycle, and the same motion conditions, exactly as was done by 'Urḍī (figure 6.5).
As a new tool, 'Urḍī's Lemma, was also used by other astronomers and in new areas of application as well, as we have also seen before, most notably by ''Alā al-Dīn al-Qushjī (d. 1474) and by Shams al-Dīn al-Khafrī (d. 1550) in their respective constructions of their models for the motion of the planet Mercury. Both of these astronomers could assume that they had this new tool in their repertoire, to use it whenever they pleased. The fact that it was used by the previous astronomers for some 200 years must have been taken as a proof that first, it withstood the test of time, and second, that it was a more general form of the Apollonius theorem. It clearly allowed for the transposition of eccentricities to deferent circumferences. But most importantly, it also allowed for the transposition of reference points for uniform motions, such as the motion of the equant, or any other center of motion that was required by the observations.
And since Copernicus had used the same model for the upper planets that was used by Ibn al-Shāṭir (figure 6.6), with the additional transposition of the center of the universe to the sun of course, in that sense Copernicus too ended up using 'Urḍī's Lemma, as Ibn al-Shāṭir had done before him.
318
On the motivation of the Ilkhānids to construct such an institution, see G. Saliba, "Horoscopes and Planetary Theory: Ilkhanid Patronage of Astronomers", lecture delivered at a colloquium organized by the Los Angeles County Museum, June 2003, and to appear in the proceedings of the colloquium.