Al-Samaw'al al-Maghribī

Al-Samaw'al was born around 1130 in Baghdad and died around 1180 in Marāgha, in modern Iran. His career touched Baghdad, Syria (including Aleppo), Kohistan, and Azerbaijan. MacTutor's transliteration of his death place is Maragha; the more common contemporary transliteration is Marāgha, adopted here, and confirmed by multiple independent sources including the McGill Islamic Studies biographical dictionary and Roshdi Rashed's 2021 critical edition.

Principal treatise

Al-Bāhir fi'l-jabr ("The Brilliant in Algebra") is his principal mathematical work. It is organised into four books, and the contents of each, as MacTutor summarises them, are:

1. Premises, multiplication, division, and extraction of roots.
2. Extraction of unknown quantities; quadratic equations; indeterminate equations of the type "find \(x^n\) so that \(ax^n\) is a square" and "find \(x^n\) so that \(ax^n + bx^{n-1}\) is a square"; a description of the binomial theorem with coefficients given by Pascal's triangle; an early form of induction; and the sum of squares identity.
3. Arithmetic of irrational magnitudes, following Book X of Euclid's Elements. Includes a worked rationalisation of \(\sqrt{30} / (\sqrt{2} + \sqrt{5} + \sqrt{6})\).
4. Classification of problems (into necessary, possible, and impossible) and the combinatorics problem discussed below.

The sum of squares identity

In Book 2, Al-Samaw'al gives the closed form

\(1^2 + 2^2 + 3^2 + \cdots + n^2 \;=\; \tfrac{1}{6} n(n+1)(2n+1)\).

Every olympiad student meets this identity at some point, proves it by induction, and then reaches for it repeatedly in counting problems. It is one of the standard tools of the trade.

The ten unknowns combinatorics problem

Book 4 poses a genuinely suited to an olympiad combinatorics problem: find ten unknown quantities, given the 210 equations that specify their sums taken six at a time. Al-Samaw'al works out the 504 consistency conditions the system must satisfy, and classifies problems of this general kind as necessary, possible, or impossible depending on their internal consistency. The problem, together with the classification scheme applied to it, is a direct ancestor of the modern olympiad genre in which one recovers a set of quantities from symmetric combinatorial data.

The inductive method, credit where it is due

Book 2 contains an inductive argument in the same style as Al-Karajī's , verify the small cases, assert the pattern extends. MacTutor is explicit about the attribution here, and we quote it directly:

He was not the first to use this form of recursive reasoning, since al-Karaji had used similar methods.

This is the most important point on this page. The Al-Bāhir preserves Al-Karajī's earlier method, and Al-Samaw'al credits the technique to him. The tradition develops through citation and continuity. This matters in a training academy because the scholars of this period took the crediting of predecessors seriously, as part of what they considered rigorous work.

On the arithmetisation of algebra

MacTutor reports that "Al-Samawal was the first to give this development a precise description," referring to his characterisation of algebra as "operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known." This is presented as MacTutor's paraphrase of Al-Samaw'al's own words. It identifies a specific formulation and keeps broader priority claims open.

Related curriculum links in the curriculum

Al-Samaw'al reaches MOTA's algebra strand (through the binomial theorem, Pascal's triangle coefficients, and the arithmetisation move) and its combinatorics strand (through the ten unknowns problem). The shorter Heritage narrative on the About page gives the summary version.

Sources

1. MacTutor History of Mathematics, University of St Andrews, Al-Samaw'al biography (c. 1130 Baghdad to c. 1180 Marāgha, the four book structure of Al-Bāhir fi'l-jabr, the sum of squares identity, the ten unknowns and 210 equations combinatorics problem with 504 consistency conditions, the explicit statement that Al-Samaw'al was "not the first to use this form of recursive reasoning, since al-Karaji had used similar methods"). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Samawal/
2. Roshdi Rashed's 2021 De Gruyter critical edition of Al-Samaw'al's Al-Bāhir (reviewed in the Bulletin critique des Annales islamologiques), secondary verification of the treatise title, of Al-Samaw'al's biographical dates, and of his death location as Marāgha. https://journals.openedition.org/bcai/7543
3. McGill University Islamic Sciences RASI biographical dictionary entry on Al-Samaw'al al-Maghribī (secondary verification of the death location Marāgha). https://islamsci.mcgill.ca/RASI/BEA/Samaw'al_BEA.pdf
4. Roshdi Rashed, "L'induction mathématique: al-Karajī, as-Samaw'al," Archive for History of Exact Sciences, vol. 9 (1972), pp. 1 to 21, the primary modern study of the induction method as it appears in Al-Karajī and Al-Samaw'al. https://www.semanticscholar.org/paper/L%27induction-math%C3%A9matique%3A-al-Karaj%C4%AB%2C-as-Samaw%27al-Rashed/fc3c72d557fd6fd904f768fef11170b0ee470cad