Ibn al-Haytham was born, possibly in 965, in Basra, then on the Persia to Iraq border, now in Iraq, and died, possibly in 1040, in Cairo, Egypt. MacTutor hedges both dates explicitly with "possibly." He is known by two epithets marking his geographic biography, al-Baṣrī (from Basra) and al-Miṣrī (from Egypt), and to the Latin scholarly tradition as Alhazen, a Latinised form of his given name al-Ḥasan.
Principal treatises
- Kitāb al-Manāẓir ("Book of Optics"), MacTutor identifies this seven volume work as his most important. Book IV formulates the laws of reflection and describes an instrument for measuring reflection from plane, spherical, cylindrical, and conical mirrors; Book V contains what is now called Alhazen's problem, solved by six geometric lemmas; Book VII treats refraction. The work was translated into Latin as Opticae thesaurus Alhazeni in 1270 and shaped European optics for centuries afterward.
- Opuscula, the collection that contains the congruence system problem discussed below.
- Analysis and Synthesis, examines mathematical problem solving method itself, and (per MacTutor) attempts to characterise the converse of Euclid's even perfect number theorem.
The congruence problem
The problem, as stated by MacTutor:
To find a number such that if we divide by two, one remains; if we divide by three, one remains; if we divide by four, one remains; if we divide by five, one remains; if we divide by six, one remains; if we divide by seven, there is no remainder.
This is structurally a modern olympiad Chinese Remainder Theorem problem. The interesting part is how he approaches it. MacTutor records that "in number theory al-Haytham solved problems involving congruences using what is now called Wilson's theorem", the statement that if \(p\) is prime then \(1 + (p-1)!\) is divisible by \(p\). John Wilson formulated the theorem in the West in 1770; Ibn al-Haytham was using its reasoning more than seven centuries earlier.
Alhazen's problem
In Book V of the Optics, Ibn al-Haytham asks: given a spherical mirror, a light source, and an observer, where on the mirror does the reflection occur? Reduced to its geometric essentials, this is a construction problem that resists elementary solution. His approach via six geometric lemmas is the historical origin of what modern geometry texts call Alhazen's problem.
On the perfect number claim
MacTutor reports that "Rashed claims ibn al-Haytham was the first to state this converse [of Euclid's even perfect number theorem], although the statement does not appear explicitly in ibn al-Haytham's work." The parenthetical is crucial. It identifies the claim as a scholarly inference by a named modern historian. We preserve the full qualification whenever this priority claim appears.
Related curriculum links in the curriculum
Ibn al-Haytham's mathematics reaches MOTA's number theory strand through the congruence system problem, and its geometry strand through the construction now bearing his Latin name. His full presence in the shorter Heritage narrative is on the About page.
Sources
- MacTutor History of Mathematics, University of St Andrews, Ibn al-Haytham biography (dates hedged "possibly 965 to possibly 1040," Basra and Cairo, the two epithets and Alhazen derivation, exact statement of the congruence problem, Wilson's theorem language, Rashed's hedged claim on the perfect number converse, Alhazen's problem via six lemmas, Latin translation as Opticae thesaurus Alhazeni 1270). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Haytham/
- Encyclopædia Britannica, Kitāb al-Manāẓir entry (secondary verification of the treatise title, its seven volume structure, and its historical impact via the 1270 Latin translation). https://www.britannica.com/topic/Kitab-al-manazir