Pascal's Arithmetical Triangle: The Story of a Mathematical Idea

The subtitle of this book, "The story of a mathematical idea", describes its flavor very well. It was first published in 1987 by Charles Griffin & Company Limited, London; the Johns Hopkins edition is unchanged from the original aside from correction of misprints and the addition of an epilogue with additional references. The author begins his story with a rather thorough discussion of the figurate numbers, a theme he carries throughout the text. He starts with defining f_2,n, the n^th figurate number in 2 dimensions (we have altered the notation a bit to make it easier to represent on a web page), along with the fundamental identity

f_2,1=1,       f_2,n = f_2,n-1 + n.

He then extends this notion by considering f_3,n, the n^th figurate number in three dimensions. He notes that f_3,n = 1, while

f_3,n = f_3,n-1+ f_2,n.

This latter identity looks suspiciously like Pascal's identity used for the binomial coefficients. The reader sees the first hint of a connection.

Edwards then presents a very nice history of the arithmetical triangle before Pascal. Combinatorial rules are traced back to Pappus (ca. 320) and Cardano (1501-1576). An entire chapter is devoted to contributions from India. Bhaskara (ca. 1150) gave a table for finding the number of arrangements of r things of one kind and n - r things of another. Bhaskara's examples are taken from language (six long or short syllables) or medicine (six tastes). These can still be interesting examples for the 21^st Century classroom.

Edwards then turns to the West to discuss the contributions of Tartaglia (1500-1557), Cardano (1501-1576) and Mersenne (1588-1648). We see that these mathematicians were looking at the arithmetical triangle from a combinatorial point of view.

One interesting feature of this book is the inclusion of numerous facsimiles of pages from original works. For example, page 39 has a page from Tartaglia's General trattato of 1556 while on page 44 we find the combinatorial triangle from Cardano's 1570 Opus novum. This makes the text even richer in appreciation of the history of the topic.

This book also includes, as you would expect, a fairly intensive discussion of Pascal's work. We see that Pascal's 1654 tract entitled Traité du triangle arithmétique actually had two parts. Part I presented the fundamental ideas of the triangle. This is the part often seen in source books such as Struik's A Source Book in Mathematics: 1200-1800. Part II discussed uses of the arithmetical triangle. Edwards devotes an entire chapter to this latter part. This is one of the most thorough presentations of Pascal's work on the arithmetical triangle that this reviewer has seen.

The final chapters discuss applications in analysis as well as binomial and multinomial distributions. Chapter Ten presents a discussion of Bernoulli's Ars Conjectandi, a very pleasing addition to this volume.

Each Chapter includes extensive Notes. These Notes and the corresponding list of references are particularly noteworthy.

Who should read this book? My first reaction is to say "Everybody!", but that may be a bit extreme. Anyone teaching a discrete mathematics course or a combinatorics must read this book. Not only will it give them substantially more background than the average textbook, but they will also gain a much better perspective on the Arithmetical Triangle, its history and application. This volume would also be suitable for a text within a course in the history of mathematics. In particular, if someone were teaching such a course in a seminar format (perhaps only 1 or 2 credit hours) this would be an excellent choice for a concentration. It might also be combined with Graph Theory 1736-1936 by Biggs, Lloyd and Wilson for a seminar in the history of discrete mathematics

Herb Kasube (hkasube@hilltop.bradley.edu) is Associate Professor of Mathematics at Bradley University in Peoria, Illinois. A member of both CUPM (the MAA Committee on the Undergraduate Program in Mathematics) and CRAFTY (the CUPM Subcommittee on Calculus Reform and the First Two Years). His mathematical interests lie in the history of mathematics.