Many of their arguments have the form:
Something is the same in Copernicus and Ibn al‐Shatir/Tusi. This is surprising if they are independent.
To which I reply: No, it’s not. They are independent only in a very limited sense, for they are both thoroughly dependent on the exact same Ptolemaic tradition. Therefore many agreements are to be expected. The possibility of this explanation for their agreements is neglected in the following passages:
> Roberts’ 1957 article established that Copernicus’ moon model had the same structure as Ibn al‐Shatir’s. A single correspondence like this might have been dismissed as surprising but happenstance. (20)
That is not my argument. Rather, I argue that it is not surprising at all.
> [In Tusi and Copernicus,] the orientation of the outer circles and their radii are the same, which is curious if the two authors were drawing figures independently. (23)
It is not surprising because they are simply following Ptolemaic tradition. For instance, the epicycle is in the top left quadrant, which also seems to be the quadrant favoured by Ptolemy for showing epicycles in general position (as when he introduces them in Figs. 3.5, 3.6 of Toomer’s translation, and most times after that).
> And the senses of rotation of the major circles are the same. (23) … the large circle rotates counterclockwise … But the same results would follow by reversing the directions of rotations, leading to an alternative diagram for Figure 4, in which the smaller circle would appear to the right of the reference diameter of the larger circle. So Tusi’s proof embodies a choice about which direction everything should move. … This choice is perhaps influenced by the convention of reading Arabic script from right to left. (28)
The large circle rotates counterclockwise in Ptolemy’s basic epicycle model as well (ibid.). This seems to me a more plausible explanation than the direction of Arabic script. Thus this is not an “unexplained coincidence” (29), but simply the default expectation on my hypothesis.
We come now to the lettering argument. In my paper I discredited the argument that agreements between the digram letterings used by Tusi and Copernicus prove dependence. The simple fact of the matter is: In Copernicus’s proof the lettering is exactly the alphabetical order following the order in which the points occur in the proof, just as in every proposition of Euclid. So his lettering is the obvious and natural one, and there are no grounds whatsoever for trying to argue that there is some kind of remarkable or unexplained coincidence involved here.
Barker & Heidarzadeh spend much time trying to resurrect the lettering argument, but their efforts are fundamentally misguided because they have not taken into account the obvious fact that the lettering of mathematical diagrams generally follow a natural numerical/alphabetical ordering corresponding to the order in which the points occur in the proof. Let us call this The Rule. All of Barker & Heidarzadeh’s arguments are based on ignoring The Rule. Thus:
> Blåsjö, Goddu and Di Bono have denied any correspondence here, on the grounds that the choices of lettering are to be expected given conventions in the Islamic and Latin mathematical communities. As an initial way to evaluate their claim, let us compare Copernicus’ diagram with the next three versions to appear in Europe after the publication of De revolutionibus, which appeared in 1568, 1589 and 1596 (Fig. 10). Suppose we represent the positions of the letters in the order used by Copernicus as 12345678. The 1568 version is then 12435867. Magini’s from 1589 will be 41352687 and Maestlin’s version from 1596 will be 12435687. From these examples, we are unable to identify any convention plausibly shared by all of these authors, beyond the use of letters from the beginning of the alphabet. (37-38)
This is obvious nonsense in light of The Rule. First of all it is absurd to look only at the diagrams in isolation, as Barker & Heidarzadeh do, since The Rule pertains to the proofs. But let’s say for the sake of argument that some examples like these can be found which violate The Rule. Would this prove anything? Of course not. No one has claimed that The Rule is absolutely universal. Of course sometimes mathematicians deviate from The Rule, for instance because of later revisions or alterations of an original draft, or for the sake of agreement with other figures in the same work. Nevertheless it is an undeniable fact that The Rule is extremely well entrenched in the mathematical literature.
> there is still no explanation for the fact that the sequence and placement of Copernicus’s letters follow those of Tusi, and that this is very unlikely to be chance or coincidence (41) ... The convention for lettering figures using the Roman alphabet would dictate a completely different set of letters from the Arabic convention. However, the sequence followed in placing the letters is even more important. After a letter has been inserted to designate a point, the author faces several possible choices of which point to designate by the next letter. Even if the choice of letters could be explained there would be no reason for the authors to follow the same sequence of applying letters to individual points. These issues combine to make it highly unlikely that the diagrams are independent. (41)
There is an explanation, namely The Rule (which Copernicus follows exactly and Tusi at least roughly).
> The similarities between Copernicus’ diagram for the Tusi couple in De revolutionibus and Tusi’s original figure in the Tadhkira are so great that the occurrence of an accidental similarity would require a gross violation of probabilities. There is the improbability that Copernicus would select the same (non-sequential) pattern of letters for the points in his figure that correspond to Tusi’s own. There is the separate improbability that these letters would be attached to the same points in the same order. And there is the additional peculiarity that he would also have to accidentally have arrived at a pattern that allowed the transcribed Arabic letters to be filled out in just such a way that continuous lettering could be restored in the Latin alphabet, while simultaneously picking the same orientation and senses of rotation for the circles in his figure. (54)
Not so. In all of this Copernicus is simply following standard conventions, as shown above. As is Tusi, to a large extent, so no wonder there are many agreements between them.
> Why has there been such persistent refusal to acknowledge Copernicus’ intellectual debt to Islam? (42)
There hasn’t. “Why has there been such persistent enthusiasm to accept such a debt?” seems to me the more apt question. I’m not the one with a Canada Research Chair or a professorship at Columbia, Chicago or Caltech.
> In both theory and practice the Latin astronomical tradition before Copernicus is permeated with Islamic influences. This hunger for Islamic materials – rightly viewed as superior to Latin counterparts – is the most important factor in the intellectual context in which the reform of astronomy in the West, and the education and career of Copernicus, take place. It is this background that makes transmission, rather than independent discovery, a priori more historically credible. To make independent discovery plausible, its proponents need to deny, discount or ignore this context. (55)
I do not deny this general picture, but the question is whether the Tusi material was part of this tradition as transmitted. It seems to me that according to this “a priori” prediction one might reasonably expect explicit references to and Latin translations of Tusi’s work, of which there are none at this time.
> To insist, in the face of the steadily accumulating evidence for East-West exchange that the Tusi device, and the other mathematical techniques found in Copernicus, were invented independently in Europe, valorizes European exceptionalism and the lone genius model of scientific history. It is our hope that the history of science can dispense with both of these outmoded viewpoints. (56)
This doesn’t make any sense. My whole argument is based on stressing the similarities between Copernicus and the Maragha astronomers. They belonged to the same tradition, worked on the same problems, and came up with the same conclusions. How could anyone conclude from this that I am “valorising European exceptionalism”? Clearly I am arguing for the exact opposite: that neither tradition was exceptional, but rather they they were both just a natural continuation of classical astronomy. If anyone is arguing for exceptionalism it is Barker & Heidarzadeh. They are the ones arguing that the Maragha tradition did something Copernicus could not. That is the only exceptionalism involved here.
Nor does my argument imply or assume that Copernicus was a “lone genius.” On the contrary I am arguing that the models Copernicus and the Maragha school have in common are quite natural and basic. It certainly doesn't take a “genius” to think of them. And of course everyone agrees that Copernicus was not “lone” but rather “a leading member of the second generation of European astronomers to actively undertake the reform of astronomy” (56).
The tradition Copernicus is “borrowing” from “took 200 years to develop” (57), Barker & Heidarzadeh claim, suggesting that it is unrealistic for a “lone genius” to come up with all of this by himself. This is a dubious argument first of all in that obviously those 200 years contain much more besides the particular details that are found also in Copernicus, so it in no way follows that Copernicus copied 200 years’ worth of material. But let’s say for the sake of argument that the Maragha models used by Copernicus are too advanced for a single person in the European 16th century to come up with. Would it not then follow a fortiori that he could not have come up with the idea of heliocentrism either, since this is an even more advanced step? If the “outmoded” idea of the “lone genius” is as ridiculous and unthinkable as Barker & Heidarzadeh claim, then how could this or any comparable innovation ever have entered scientific thought?
In conclusion, Barker & Heidarzadeh provide no grounds for altering my published assessment in any way.