In a recent paper, Arun Bala [Ba] argues against the case I made in my Copernicus paper [Bl]. Here I reply to his arguments point by point.

> Blåsjö’s argument that Copernicus as a skilled geometer could easily have discovered the Tusi couple leaves unanswered the question why it was not discovered much earlier in Europe by astronomers working with the Ptolemaic model, or by other astronomers in the Arabic tradition before al-Tusi. (72)

It is not strange that earlier astronomers did not discover the Tusi couple if they had no reason to do so and if it served no purpose to them. Copernicus and al-Tusi were trying to eliminate the equant, and in this context the Tusi couple was valuable to them. But if earlier astronomers were not trying to eliminate the equant they had no reason to make this discovery. So if Bala’s argument has any weight it comes only from the following point:

> In fact al-Tusi himself was motivated to make this discovery because he accepted Ibn al-Haytham’s demand nearly two centuries earlier that astronomers should do away with the equant. But the fact that in the Arabic world it required two centuries to accomplish this discovery makes Blåsjö’s claim that any skilled geometer could easily have discovered the theorem extremely implausible. (72)

This would be a good argument if in fact two centuries worth of skilled astronomers worked hard trying to eliminate the equant. But nothing of the sort has been proved. The fact that Ibn al-Haytham said that someone ought to do so doesn’t mean that everyone spent two hundred years trying. “How influential Ibn al-Haytham’s treatise was is not certain” [Sw, 45]. And in any case the conflict between equant and uniform circular motion had already been obvious to everyone since antiquity.

> Blåsjö seems to presume that Copernicus would have independently discovered the Urdi lemma. Again this is extremely implausible since if he had done so he would have mentioned it as a discovery of a new mathematical theorem. (73)

Would he? Why should we believe this completely unsubstantiated assertion? In reality, the Urdi Lemma is not an interesting “new mathematical theorem.” It is a piece of applied mathematics interesting only in its specific context. In applied mathematics one constantly derives various relations relevant to the specific matter at hand without calling them “new mathematical theorems” in the abstract.

> It is particularly striking that although Copernicus deployed the Tusi couple and the Urdi lemma in his astronomical model, he did not make any mention of the fact that he independently discovered these theorems quite unknown to the Greeks. (73)

There is nothing “striking” about this. Copernicus does state and prove the Tusi couple theorem. The notion that he should have stopped and said “oh, by the way, I discovered this” is absurd. How many mathematical treatises have you read where, in the middle of the mathematical exposition, the author chimes in and says “hey, I discovered this, you know”? If this is a required mark of originality we would have to infer that Archimedes and Gauss, for example, never made any mathematical discoveries either, because such remarks are lacking in their works too.

Furthermore, Copernicus may well have thought that the Greeks did know the theorem. A passage in Proclus suggests as much, and in fact Copernicus himself explicitly cites this passage in this connection (albeit only many years after first using the Tusi couple). [Ve] [diB]

> Even after Ibn al-Haytham pointed to the problem [of variation in distance in Ptolemy’s lunar model], it took centuries of work by a chain of astronomers for a solution to be found. … Despite his undoubted genius al-Haytham was unable to solve the problem which was taken up by a string of mathematical astronomers who followed him, including al-Tusi and Urdi, until it was finally solved by Ibn al-Shatir more than two centuries later. (74)

I believe this is false. I believe Ibn al-Haytham did not in fact point this out at all. Although Ibn al-Haytham criticised Ptolemy on other points, “Curiously, he says nothing about the great variation of distance in the lunar theory, the one case in which the model … is obviously incorrect” [Sw, 45]. Nor am I aware of any long “string” of works devoted to trying to resolve this issue before Ibn al-Shatir.

> Since Ptolemy himself would have been aware of the problem, it makes one wonder why a solution that appears so obvious to Blåsjö would not have been adopted by Ptolemy himself. (74)

I agree that Ptolemy would most likely have been aware of this, but it is less clear to what extent he would have been interested in addressing it. Ptolemy’s model is basically just as good as that of Copernicus and Ibn al-Shatir as long as one cares only about the angular position of the moon, which is the essential thing for virtually all astronomical purposes. Their improvement is only needed if one is concerned not only with meeting the traditional needs of computational astronomy but also with arriving at a physically “true” model. Since the latter concern is completely separable from traditional technical astronomy, Ptolemy could quite possibly have considered it more or less irrelevant.

> Blåsjö’s conclusion is questionable because there are many ways of constructing non-Ptolemaic astronomical models that do away with the equant by deploying the Tusi-couple and the Urdi lemma. The al-Shatir model is merely one possible approach. In the sixteenth century Shams al-Din al-Khafri (died 1550) produced four such different models for Mercury’s motion. None of them were similar to the others in mathematical construction but all of them were able to account for the same set of observations. This raises the question: Why did Copernicus propose a model so similar to the Ibn al-Shatir model and not any of the other possible models that al-Khafri had devised? (75)

In fact “Copernicus considered no less than four models for Mercury” [Sw, 405], not just the one which can be interpreted as equivalent to Ibn al-Shatir. So Copernicus had a bunch of models and Islamic astronomers had a bunch of models–––what does that prove? Nothing.

Incidentally, it is perhaps not surprising that Copernicus’s models would have more in common with Ibn al-Shatir’s than with al-Khafri’s. Copernicus and Ibn al-Shatir both correct Ptolemy’s lunar model in the same way. This involves getting rid of Ptolemy’s “crank” arrangement (i.e., the small circle near the center of the orbit in [Bl, figure 4]) for moving the body back and forth in terms of distance from the earth. In this way they eliminate the distance problem in Ptolemy’s theory. Copernicus’s Mercury model does something very much analogous to Ptolemy’s Mercury model: here too a “crank” construction has been eliminated [Bl, figure 5]. Is it so surprising, then, that Copernicus and Ibn al-Shatir have much in common when it comes to both the Moon and Mercury, when these cases are in some respects analogous? Meanwhile, al-Khafri differs from them in that his lunar model retains the absurd distance implications of Ptolemy’s model [Sa, 24]; and this is perhaps not the only respect in which his work is a step backwards from Ibn al-Shatir.

REFERENCES

[Ba] Arun Bala, The Scientific Revolution and the Transmission Problem, Confluence: Online Journal of World Philosophies, Issue 4, 2016. [link]

[Bl] Viktor Blåsjö, A Critique of the Arguments for Maragha Influence on Copernicus. Journal for the History of Astronomy, 45(2), 2014, pp. 183–195. [link]

[diB] Mario di Bono, Copernicus, Amico, Fracastoro and Tusi’s Device: Observations on the Use and Transmission of a Model, Journal for the History of Astronomy, XXVI, 1995, 133-154. [link]

[Sa] George Saliba, A Sixteenth-Century Arabic Critique of Ptolemaic Astronomy: The Work of Shams al-Din al-Khafri, Journal for the History of Astronomy, XXV, 1994, 15-38. [link]

[Sw] Noel Swerdlow and Otto Neugebauer, Mathematical Astronomy in Copernicus’ De Revolutionibus, Springer, 1984.

[Ve] I. N. Veselovsky, Copernicus and Nasir al-Din al-Tusi, Journal for the History of Astronomy, IV, 1973, 128-130. [link]