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In a famous passage from his al-Bāhir, al-Samaw’al proves the identity which we would now write as  for the cases . He also calculates the equivalent of the expansion of the binomial  for the same values of  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.

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