This question is a follow-up to How much did the ancient Greeks know of non-Euclidean geometry?, where it was claimed in a comment that "the earliest inklings [of hyperbolic geometry] came from medieval Islamic mathematicians." Is there any evidence for such a claim? I am hoping for something more serious than attributing the earliest inklings of group theory to the Maians because they made pots possessing reflectional symmetry. As I suspect, some such claims may have been made by Rashed who also attributed infinitesimal calculus to scholars far earlier than is generally considered reliable, but I am looking forward to being surprised by some solid evidence.

For example, Yushkevich died in 1993. In 1996, Rashed included an article by Yushkevich in a volume he published. If Yushkevich held such sentiments, they would presumably appeared in articles he published while alive. Note that he was born in 1906 and had plenty of oportunity to do so (he had hundreds of publications listed in MathSciNet alone).

Rozenfeld and Yushkevich have a book appropriately entitled "theory of parallels in medieval East" (no mention of hyperbolic geometry). Here is the relevant part of a MathSciNet review:

"The scientists of the Arabic caliphate already at the time of caliph al-Ma'moun (813–833) realized a shortcoming in the Elements of Euclid. The first in the caliphate to pay attention to the peculiar role of the fifth postulate of Euclid was the great scholar of Central Asia, al-'Abbas al-Jauhari , member of the Academy of al-Ma'moun. Later on the same attempts were made but were unsuccessful. Perhaps the greatest level of the theory of parallel lines in the East was reached in the works of 'Omar Khayyam and Nasir ad-Din at-Tusi . The information from these works was adopted by European scientists of the XVI–XVIII centuries. More or less all of this is fully expounded in the book under review."

So we are apparently talking about attempts to prove the parallel postulate (rather than either hyperbolic geometry or non-Euclidean geometry).

If there is any evidence of "inklings of hyperbolic geometry" I am still interested in seeing it.

Proclus dealt with the theory of parallels centuries earlier, so if anything "the earliest inklings" are misattributed.

I have a 1912 translation of Bonola's book, translated by Carlslaw. Bonola discusses Proclus starting on page 2. On page 3, Bonola mentions that Proclus refers to a work by Geminus from 1st century BCE. By page 7, Bonola gets to the Arabian Commentary of Al-Nirizi (9th Century) ... and is attributed to Aganis. In footnote 3 on page 7, Bonola mentions a scholarly disagrement about whether Aganis is Geminus or not. On pages 9-10 Bonola emphasizes the derivative nature of the work of Al-Nirizi. On page 10, Bonola deals with Nasir-Eddin's proof of the parallel postulate. All this material is in a chapter I entitled "attempts to prove the parallel postulate." Chapter II, entitled "Forerunners of Non-Euclidean geometry", starts with Saccheri.


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From Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447–494 [470], Routledge, London and New York:

Three scientists, Ibn al-Haytham, [Omar] Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts.

(emphasis mine, extracted from Wikipedia transcription of the reference in hyperbolic geometry)

These Islamic scholars were not trying to come up with hyperbolic geometry in the sense that this was not invented yet. Khayyam was trying to prove the parallel postulate and in doing so he came up with things like the Saccheri quadrilateral, foreshadowing results that would be found later with the development of non-Euclidean geometry.

I think that your concern that the editor misinterpreted Yushkevich is just overcaring about the terminology. The paragraph above is not saying that Khayyam (or any other for that matter) went against his time and created hyperbolic geometry. This was not the case. It is just a statement that when his findings are reworked in modern terms, they could indeed be thought of as "emboidying" the "first few theorems of the hyperbolic and the elliptic geometries".

In the end it all depends on how close has a development to be in order to be considered an "inkling". As you say for that matter, Proclus did a commentary on the parallel postulate before Medieval Scholars, should he be the considered to be first inkling? The threshold of what counts or not as an "inkling" is in the eye of the beholder.


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