Formulating figurate numbers

Abstract

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.

Acknowledgements

I am grateful to Janine Stilt, of the University of Redlands, for her assistance in preparing the figures for this article, and to Sandra Richey, of the University of Redlands Library, for obtaining books and journals for me from libraries near and far. I thank the British Library, University of Delaware Library, and Huntington Library for the use of manuscripts, copies of manuscripts, and rare books. Finally, I thank Snezana Lawrence for her encouragement and assistance.

Notes

¹ Viète used the terms ‘triangulo-triangular numbers’ for 1, 5, 15, 35, … and ‘triangulo-pyramidal numbers’ for 1, 6, 21, 56, … in his Ad angularium sectionum analyticen theoremata (Universal theorems on the analysis of angular sections), published posthumously in 1615 (Witmer 1983, 433, 435), and seems to have been the first mathematician to do so (Edwards 1987, 9). Fermat used the first of these two terms in letters to Mersenne and Roberval in 1636 (Mahoney 1994, 231). Harriot, however, called the numbers 1, 5, 15, 35, … ‘triangle piramidall’ at British Library Add MS 6782, f. 38.

² The former is my translation of Maurolico's Latin at his Book I, page a; the latter, Edwards' more modernized translation of Maurolico's Proposition 7 at Book 1, page 5 (Edwards 1987, 15). The statement also appears as Proposition 22 at Book I, page 9.

³ Mysterium arithmeticum is not paginated. ‘Numerum Bestiae … 666’ appears on the title page, which I have numbered 1. With this numbering, the table of generalized triangular numbers is at page 9.

⁴ The charts appear at pages 10 and 11 (my numbering and translation).

⁵ Pascal's arithmetic triangle was printed on a fold-out page just before the title page of his Traité du triangle arithmetique. It is reproduced in Edwards 1987, ii, and, with labels translated from French to English, in Katz 2009, 492, Pascal 1991, 447, and Struik 1969, 22.