The early history of the `Pascal' triangle
University of Canterbury, New Zealand.
The famous triangular array of binomial coefficients, nowadays commonly referred to as the Pascal triangle, was in fact known to mathematicians predating Pascal, some by many centuries. For instance, around 1150 the Islamic mathematician Al-Samaw'al included such a triangle in his mathematical treatise Bahir Fi al-Jabr (Splendid Book of Algebra). In the famous passage which concerns this triangle, Al-Samaw'al demonstrates the identity which we would write as (ab)^n=a^n b^n for the cases n=3,4. After this he computes the expansion of the binomial (a+b)^n for the same values of n and outlines the construction of a triangle of binomial coefficients up to its 12th row. In addition, several statements Al-Samaw'al makes in this passage point to the recognition of an early form of mathematical induction. We will examine the features of this early triangle of binomial coefficients and revisit the question of how this text is to be understood in the historical development of mathematical induction.