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

Abstract

In a famous passage from his al-Bahir, al-Samaw'al proves the identity which we would now write as (ab)^n = a^n b^n for the cases n = 3; 4. He also calculates the equivalent of the expansion of the binomial (a + b)^n 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.