The Development of Arabic Mathematics: Between Arithmetic and Algebra

Revisiting Al-Samaw’al’s table of binomial coefficients: Greek inspiration, diagrammatic reasoning and mathematical induction

“…Al-Samaw'al's diagram for Proposition 5 (which, in modern terms, asserts that (ab) 4 = a 4 b 4 ) has a similar (but not identical) structure 41 to the one we saw for the related Proposition 2 (which says that (cd) 3 alphabetically in the order in which they arise in al-Samaw'al's exposition. Thus alSamaw'al begins by naming the numbers (a and b) and their product c = ab.…”

“…After much scholarly discussion, this type of reasoning is generally realized as a spectrum of techniques which include at one end the method of generalizing examples and incomplete induction 51 through to formal mathematical induction at the other. Rashed [41], building on the work of Freudenthal [22], highlights three precursors of mathematical induction which he labels R 1 , R 2 and R 3 . Two of these come from Freudenthal (see [41, p. 73]).…”

An Historical - Didactic Introduction to Algebra

“…His most important work, "al-kitab al-mukhtasar fi hisab al-jabr wa'l-muqabalah", provided to modern languages a very popular used term, the term "algebra". The word "algebra" comes from "al-jabr", one of the used operations for solving quadratic equations, as described in his book [19,20]. The term "algorithm" is derived from the Latin transcription of the name of this Persian mathematician, considered one of the first authors who made reference to this concept [21].…”

The Decagonal Tomb Tower at Maragha and Its Architectural Context: Lines of Mathematical Thought

“…Mathematics in the eleventh and twelfth centuries in Iran comprised precisely these topics, as well as dissections and algebraic and geometric approaches to the solution of quadratic and cubic equations [Rashed 1994]. The evolutionary development of mathematical thinking that expanded upon the corpus of works of Euclid, Apollonius and others, translated from Greek into Arabic, likely found expression in the contemporary phenomena of experimentation with architectural structures and decoration, discussed above, but this remains to be demonstrated.…”

“…Other authors [Cromwell 2009;Lee 1987;Fleurent 1992;Kaplan 2005] ascribe similar procedures to the practical construction of such patterns using different terminology, including "polygons in contact" [Kaplan 2005] and "star and polygon compositions" [Necipoºlu 1995: 92]. The actual processes that craftsmen used in the twelfth century have not been determined definitively, but mathematicians at that time endeavored to understand relationships between geometry and algebra [Rashed 1994] and actively sought demonstrations for proofs based on Euclidean models [Kheirandish 2008]. Dissections were part of the mathematical curriculum, as were compass and straightedge constructions [Berggren 2007;Rashed 2005].…”