R�solution des �quations num�riques et alg�bre: ?araf-al-din al-T?s?, vi�te

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

“…53-64]. For example, al-Samaw'al himself later used the propositions and binomial table, that we study in this paper, to develop methods for extracting nth roots [40]. The arithmetical methods mentioned here are being contrasted with geometrical methods, although as we shall see these approaches are mingled together in the demonstrations.…”

“…Thus four arbitrary numbers, a, b, c and d, are listed in order in the top row, while the relevant products of pairs of these numbers, e = ab, f = ac, g = bd and h = cd, are listed in the bottom row. 40 It is tempting to imagine a teacher drawing and labelling each line as each new number is encountered during the demonstration.…”

“…For Al-Bāhir we refer to the Aya Sofya 2718 manuscript since this is the only manuscript with completed diagrams. 40 Just as in Arabic writing, these diagrams are usually read from right to left. By contrast, al-Samaw'al's diagrams for Propositions 3 and 4 (the binomial expansions for n = 3 and n = 4) consist of a single line segment AB representing the number that has been divided into two parts AC and CB ( Figure 6).…”

Horner versus Holdred: An Episode in the History of Root Computation

“…Florian Cajori [15] pointed out that the method was anticipated by Paolo Ruffini [68; 69], in Italy. Long before then related techniques were known to Chinese and Arabic writers; see for example, Martin Nordgaard [60,[54][55], Paul Luckey [48; 49], Adolf Youschkevitch [45, 41-48, 242-248; 81, 76-80, 169], Youschkevitch and Boris Rosenfeld [82], Carl Boyer [10,[268][269][270], Ulrich Libbrecht [46,, Roshdi Rashed [65], and Lennart Berggren [7,[48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63]. The main aim of the present paper is to show that even in England Horner was anticipated, in fact by Theophilus Holdred [36].…”

The Mathematics of Egypt, Mesopotamia, China, India, and Islam. A sourcebook

“…In my view, however, this document is interesting in two respects. On the one hand, according to the reconstruction carried out in [Rashed, 1974[Rashed, , 1986], Tusi's tabular layout for solving equations is very similar to al-Kashi's layout for the computation, when the latter presents the extraction of higher degree roots. This remark illustrates why a classification of source material according to modern categories may obscure how close some topics were to the original practitioners.…”