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April 26

law of cosines in France and al-Kashi

Apparently the law of cosines is sometimes known as the formule d'Al-Kashi in France (after Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (c. 1380 – 1429), the Persian astronomer and computer of trigonometric tables). Does anyone know who invented/popularized this attribution / this name for the identity, and when? –jacobolus (t) 01:07, 26 April 2023 (UTC)[reply]

The use of the digraph ⟨sh⟩ in French texts to represent /ʃ/ is relatively recent. If the law was known earlier by the Persian eponym, it was more likely known as théorème d'al-Kachi. But I only found recent uses by any spelling, like here, from 2000, la formule dite d'Al Kashi. --Lambiam 09:25, 26 April 2023 (UTC)[reply]

The French Wikipedia article says this "appeared in the 1990s in school textbooks", but doesn't give a useful source. I can only find 2 independent sources in English using the name, both by French authors (the other sources I could find in English just copied from Wikipedia), so I am removing the name from the lead section of the law of cosines article, while also expanding the relevant part of the history section to directly quote al-Kāshī's book so readers can judge for themselves. –jacobolus (t) 20:55, 26 April 2023 (UTC)[reply]

First mention I could find of a formule d'Al-Kashi in Google Books was in this 1988 book by Yvonne and René Sortais. I think based on the appearance that it might be a school book of some sort, although feel free to correct me if I am wrong. There may be some earlier mentions, but without clearer context from the book previews, I was unable to tell. GalacticShoe (talk) 12:08, 27 April 2023 (UTC)[reply]

The 1988 book is a course book, part of a series for continuing education or the instruction of teachers. --Lambiam 14:24, 27 April 2023 (UTC)[reply]

A late 19th century source mentioning al-Kāshī calls him Ghiath-Eddin Djemchid Al-Kâchâni, but says nothing about solving triangles or the law of cosines. ––jacobolus (t) 21:22, 26 April 2023 (UTC)[reply]

Hm, that's got me thinking; would you happen to know by any chance what the name of the actual work in which al-Kāshī proved the law of cosines is? If so, the year of its popularization in French may be related to some translation into French, or mention within some French media, of said document. GalacticShoe (talk) 12:12, 27 April 2023 (UTC)[reply]

According to the French Wikipedia article Al-Kashi, "Al-Kashi is credited with stating this theorem in his book Miftah al-hisab (« Clé de l'arithmétique »)." --Lambiam 14:38, 27 April 2023 (UTC)[reply]

I recently added a direct quotation and some context to Law of cosines § History (a section which is still somewhat incomplete). –jacobolus (t) 14:52, 27 April 2023 (UTC)[reply]