Kamal al-Din al-Farisi

Kamal al-Din Abu'l Hasan Muhammad ibn al-Hasan al-Farisi worked in optics and number theory. MacTutor explicitly identifies him as an Islamic mathematician, with birth about1260 and death about1320 in Tabriz. These remain approximations, without an invented Hijri equivalent. Individual Muslim identity is documented; the separately declared Sunni museum frame is not a new personal denominational certificate ³.

He studied with Qutb al-Din al-Shirazi. Russell places his arrival in Tabriz possibly before1290. Questions about refraction prompted consultation; the teacher obtained a book by Ibn al-Haytham from a distant country. This is not a complete itinerary, nor evidence that al-Farisi visited every place from which a book arrived ⁴.

The Tanqih al-Manazir went beyond elementary commentary. It distinguished the predecessor's statements from the reviser's and could dispute a conclusion. Russell reports competing completion dates: before1290, or after1302 and before1311. These limits are retained rather than replaced by an exact date from an unread summary ⁵.

His water-sphere rainbow model represents a drop with a transparent water-filled vessel. Two refractions surround one or two internal reflections. The approximation involving glass remains explicit, as does the failure of his physical colour explanation. Russell also discusses shadows, eclipses, burning spheres, eye anatomy and reported experiments on a ram's eye. These are biographical interests, not instructions to reproduce those procedures ^{6 7}.

Sunni — declared frame: the documented Islamic identity is separate from priority attribution. Academic account: MacTutor reports Boyer's preference for the teacher's discovery and pupil's elaboration, and Rashed's objection. No final verdict is imposed. Russell finds no evidence of a Latin translation, while discussing the comparison with Theodoric ^{8 9}.

MacTutor describes a new proof of Thabit ibn Qurra's amicable-number theorem in the Tadhkira al-ahbab fi bayan al-tahabb. It reports the pairs220/284 and17296/18416. Hogendijk argues that Thabit knew the latter pair earlier, so this is not an unqualified first-discovery claim for al-Farisi, nor an exclusive attribution to Euler because that name later became attached to the pair ¹⁰.

The account includes establishing the primality of1151 with lemmas including an application of Eratosthenes' sieve, and an approach involving factorisation and combinatorial ideas. Whether he proved, or attempted to prove, unique factorisation is a scholarly question reported through Rashed and another study, not something settled here by inspecting his original argument ¹¹.

MacTutor also describes connections between polygonal numbers and binomial coefficients, with early induction-style arguments linking repeated triangular-number sums to combinations. It records his assertion that a sum of two positive integer fourth powers cannot be a fourth power, without a proof. That does not become a proof of Fermat's general theorem, and every biographical observation is not automatically a separate invention ¹².

The actual readings here are MacTutor's biography and Russell's browser article body, not the Arabic originals, manuscripts or every study in their bibliographies. Russell warns of errors in the Hyderabad Tanqih edition. A copy date is not a birth or discovery date, and no surviving physical apparatus has been independently inspected ¹³.

MacTutor describes the Memorandum for friends and its factorization/combinatorial proof approach to Thabit ibn Qurra's amicable-number rule, with220/284 and17296/18416. It attributes the unique-factorization interpretation to Rashed and notes disagreement about proof completeness. Neither absolute proof priority nor first discovery of the larger pair is awarded ¹².

Bozhuyuk's TDV entry differs from the earlier name and approximate dates: Hasan ibn Ali (may God be pleased with him) ibn Hasan,1267 and1319, from a reported student's colophon. The alternative is visible without erasing the earlier fields' attribution to MacTutor/Russell or claiming original collation. TDV identifies the treatise and a1982edition not read here ¹⁴.