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

Archive for History of Exact Sciences 69 (6):537-576 (2015)
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Abstract

In a famous passage from his al-Bāhir, al-Samaw’al proves the identity which we would now write as (ab)n=anbn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}(ab)n=anbn(ab)^n=a^n b^n\end{document} for the cases n=3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}n=3,4n=3,4\end{document}. He also calculates the equivalent of the expansion of the binomial (a+b)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}(a+b)n(a+b)^n\end{document} for the same values of n and describes the construction of what we now call the Pascal Triangle, showing the table up to its 12th row. We give a literal translation of the whole passage, along with paraphrases in more modern or symbolic form. We discuss the influence of the Euclidean tradition on al-Samaw’al’s presentation, and the role that diagrams might have played in helping al-Samaw’al’s readers follow his arguments, including his supposed use of an early form of mathematical induction.

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