Al-Nayrīzī's own proof of Euclid's parallel postulate (Q2737635)

This article is devoted to the study of one treatise by the ninth-century mathematician-astronomer by the name of Abū al-'Abbās al-Nayrīzī (d.c. 922 AD). NEWLINENEWLINENEWLINEIt contains an edition of the Arabic text of the said treatise, an English translation, and a very cursory commentary. NEWLINENEWLINENEWLINEThe treatise in question deals with the notorious Euclidean postulate regarding parallel lines, and is part of a long tradition that had started in late antiquity and continued into Islamic civilization and into Europe well into the nineteenth century when it was finally shown that giving this postulate up would lead to non-Euclidean geometry. As it becomes clear to anyone studying the history of non-Euclidean geometry, there is a direct connection between this long series of attempts to prove the fifth Euclidean postulate before the nineteenth century and the final crowning of that activity by the development of non-Euclidean geometry. NEWLINENEWLINENEWLINEIn the present article, al-Nayrīzī's attempt at the proof of the fifth postulate is clearly reproduced in an edition that is based on two extant manuscripts, together with an introduction that puts it in its larger historical context by tracing its roots and detecting its influences. The importance of this investigation is that it demonstrates very clearly the extent to which mathematicians and astronomers of the early Islamic times were not only appropriating Greek scientific thought but that they were debating it and, in this instance, attempting to perfect it just as that thought was being translated into Arabic or very soon thereafter. The works of Greek authors who worked on the same problem which are no longer extant in Greek are preserved here in a round about way through the debate they generated in Islamic sources such as this treatise. NEWLINENEWLINENEWLINEThe only blemish in the final product that is published in this article is the admission of the article's author that he allowed himself to adapt ``the orthography [of the Arabic MSS] to modern usage but \dots{} have not corrected grammatical errors''. One is left wondering as to why this attitude was adopted when the production of a critical edition could have been easily achieved. Alas, the opportunity was missed.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00039].