Figure 6.5
Ibn al-Shāṭir's model for the upper planets, clearly showing the incorporation of 'Urḍī's Lemma.
Copernicus, however, did not apparently realize the full significance of the two components of Ibn al-Shāṭir's model (the Apollonius and the 'Urḍī components), and simply used the model as a whole, by transposing it to heliocentrism as we just said. As a result he did not feel that he had to produce a formal proof for the 'Urḍī component as he had done with the Ṭūsī Couple. It was Kepler (1630) who wrote to his teacher Maestlin (1631) to ask him specifically about this omission on the part of Copernican astronomy, as was already demonstrated by Anthony Grafton.[321] And it was Maestlin who supplied the proof of that specific case of the 'Urḍī Lemma which applied to the model of the upper planets, without supplying the general proof as 'Urḍī had done.
For our purposes, the almost unconscious use of 'Urḍī's Lemma by Copernicus, in a construction that was identical to that of Ibn al-Shāṭir, minus heliocentrism of course, must raise doubts about Copernicus's awareness of the roots of all the mathematical techniques that were put at his disposal. Would he have proposed this very new theorem? And would he have offered a formal proof of it as was done by 'Urḍī, and as he did for the complementary Ṭūsī Couple that he also had to use, had this new theorem not been at his disposal from the Islamic sources? I doubt that very much.
Figure 6.6
A schematic representation of the model for the upper planets as conceived by Ptolemy, 'Urḍī, Ibn al-Shāṭir, Copernicus, and Khafrī. If one thinks of the radii of the spheres as vectors, all the models predict the same position for the planet P.
But this example of the use of Ibn al-Shāṭir's model by Copernicus does not even begin to illustrate the extent of the technical interdependence between the two astronomers. For in addition to the identical construction of the lunar model, which we already discussed before, and now the identity of the model for the upper planets, Ibn al-Shāṭir and Copernicus also used identical techniques for resolving the last model of the classical planetary theory (the Mercury model).
If one were to compare Copernicus's model for the planet Mercury to that of Ibn al-Shāṭir, and if one were to allow for the simple mathematical transposition from geocentrism to heliocentrism and vice versa, one will be struck by the similarities between the works of the two astronomers. In this instance, both Ibn al-Shāṭir and Copernicus used a construction of a mathematical model that deployed in its last connection the use of a Ṭūsī Couple, in order to allow for the planet's epicycle to be brought close to the Earth, at the two perigees which were observed by Ptolemy, and to recede away at the apogee. The complete agreement on the technique of achieving this oscillatory motion, while moving the epicycle nearer and farther, raises the question of the possible influence of one astronomer over the other, especially when we already know of the other similarities that we have already witnessed in the other contexts. But the case of the Mercury model in particular brings some remarkable evidence for the case of the interdependence between the two astronomers; this evidence elevates the discussion of the similarities to a whole new level.
When Swerdlow studied the first version of Mercury's model in Copernicus's Commentariolus, which was itself written before 1514, he immediately realized that Copernicus was not aware of the full significance of the model he was describing. For example, Copernicus thought that the planet would have its largest orbit (i.e. the size of its epicycle would look the largest) at quadrature (i.e. when the center of the epicycle—or the Earth in Copernicus's language — was at 90° away from the apogee) while the model itself would predict two such largest appearances when the center of the epicycle, or the Earth, was at 120° on either side of the apogee, exactly as Ibn al-Shāṭir's and the Ptolemaic models would have predicted, and not at 90° as Copernicus now claimed. Having realized that, Swerdlow said:
Copernicus's model for Mercury which, like his other planetary models, is identical to Ibn al-Shāṭir's model except for the heliocentric representation of the second anomaly, is based on exactly this separation of the equation of center from the motion of the center of the eccentric in Ptolemy's model.[322]
While discussing the point, Swerdlow went on to explain why Copernicus did not seem to realize where his model would produce Mercury's closest position to the Earth (figure 6.7):
There is something very curious about Copernicus's description... Copernicus apparently does not realize that the model was designed, not to give Mercury a larger orbit (read epicycle) when the Earth (read center of the epicycle) is 90° from the apsidal line, but to produce the greatest elongations when the Earth (read center of the epicycle) is ±120° from the aphelion (apogee).[323]
Figure 6.7
A model depicting the motion of the planet Mercury as described by Ibn al-Shāṭir. Copernicus adopted the same model without fully realizing the manner in which it functioned. Copernicus seems not to have realized that the apparent size of an object depended on the size of the object and on the object's distance from the observer. It seems that Copernicus confused the size of the planet's orbit, as marked by the dashed circles, with its appearance for an observer at point O. Although the planet's orbit indeed reaches its greatest size when the epicyclic center is 90° from the apogee, for an observer at point O the dashed epicycle does not appear the largest at that point, as Copernicus contends. Rather, it appears largest when the epicyclic center reaches ±120° from the apogee, as would be predicted by the observations of Ptolemy which were followed by Ibn al-Shāṭir, and as can be seen from the comparison between the maximum elongation angles at 90° (solid lines) and at 120° (dashed lines).
With all these problems laid bare, Swerdlow concluded:
This misunderstanding must mean that Copernicus did not know the relation of the model to Mercury's apparent motion. Thus it could hardly be his own invention for, if it were, he would certainly have described its fundamental purpose rather than write the absurd statement that Mercury "appears" to move in a larger orbit when the Earth is 90° from the apsidal line. The only alternative, therefore, is that he copied it without fully understanding what it was really about. Since it is Ibn al-Shāṭir's model, this is further evidence, and perhaps the best evidence, that Copernicus was in fact copying without full understanding from some other source, and this source would be an as yet unknown transmission to the west of Ibn al-Shāṭir's planetary theory.[324]
Later, while assessing Copernican astronomy in the context of Renaissance astronomy, Swerdlow returned to this very point of the connection between Copernicus and his predecessors, and particularly to the problems in the Mercury modeclass="underline"
The transmission of their [meaning the Marāgha astronomers] inventions from Arabic in the East to Latin in the West is obscure. Yet Copernicus's lunar and planetary theory in longitude in the Commcntariolus, right down to the additional complications for Mercury, is that of Ibn al-Shāṭir in nearly every detail, except for the heliocentric arrangement and the extraction of parameters from the Alfonsine Tables, and it is hard to believe in light of so many and such complex identities that Copernicus was entirely without knowledge of his predecessors.[325]
321
See Anthony Grafton, "Michael Maestlin's Account of Copernican Planetary Theory",
325
Noel Swerdlow, "Astronomy in the Renaissance", in