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Formulating figurate numbers

The multiplication formula for figurate numbers (or binomial coefficients) we use today appeared in western Europe in verbal form in the late 1500s and in symbolic form in the early 1600s. In this paper, we first recount the early history of figurate numbers and especially of multiplicative means for computing them. We then focus on the development of multiplication formulas for figurate numbers in the late sixteenth and early seventeenth centuries by Cardano, Faulhaber, Briggs, and Harriot. Throughout the paper, we invite the reader to consider what it means to ‘have a formula for’ a mathematical relationship. Indeed, the story of figurate number formulas is interesting not only in its own right but because it provides rich fodder for a broader discussion of mathematical formulas. A preliminary version of this paper was presented during the July 2008 quadrennial meeting of the International Study Group on Relations Between History and Pedagogy of Mathematics, and appeared in the proceedings of that conference.