We have all heard the complaints from our students, and the characterization from our non-mathematician friends, that 'higher mathematics doesn't involve any numbers'. Of course, we all know that this is far from the truth and that numbers turn up in even the most abstract and theoretical branches of mathematics. Steven R. Finch's new book, Mathematical Constants, contains six-hundred-plus pages of information about some of the more complicated numbers that arise in a wide range of ways throughout mathematics.
This book is published in Cambridge University Press's "Encyclopedia of Mathematics and Its Applications" series, and this is exactly where it belongs. The book is an encyclopedia collecting short (generally 3-4 pages long) essays on a wide range of constants. Each essay attempts to put the constant into the proper historical and mathematical context. Some of the essays also describe how (and indeed if) it can be computed, although this is often of secondary importance.
Chapter One is entitled "Well-Known Constants", and includes a variety of facts about e, π, the square-root of 2, and the other numbers that probably came into your head when you read the title of this book. While I might quibble with how well known some of these constants (such as Madelung's constant, -1.615542..., which describes the electrostatic potential at the origin due to placing unit charges of alternating sign at all non-zero integral points of the plane) are, these essays set the very accessible tone that is found throughout the book.
Chapter Two has essays on constants that arise in various parts of number theory, such as the Hardy-Littlewood constant (actually a family of constants obtained by taking products of various formulae evaluated at all prime numbers), Brun's constant (the sum of the reciprocals of all twin primes, 1.90216...), and thirty-one others. The next two chapters go in an entirely different direction: Chapter Three discusses constants associated with analytic inequalities, and Chapter Four deals with constants associated with the approximation of functions.
The book then shifts gears again, and has a chapter entitled "Constants Associated with Enumerating Discrete Structures" which consists of 25 essays on the behavior of constants such as the number of non-isomorphic abelian groups of order n and the Pythagorean Triple constants. A number of these essays also deal with data structures and random walks, topics that would be particularly interesting to computer scientists.
Chapter Six discusses constants that arise when one looks at iterations of functions and Chapter Seven is about constants in complex analysis. The final chapter is about constants that arise in geometry, including the so-called 'coverage constants', such as the area of the smallest square which would cover any path of length one, and the Moving Sofa constant which calculates the largest area of a sofa which can be moved around a corner. These three chapters, as well as the two on constants from analysis, are shorter than the ones on number theory and combinatorics, both in terms of the number and the length of the essays they contain. One gets the feeling that Finch ran out of steam at the end, or had less interest in these topics, and included them more out of a desire for the encyclopedia to be complete than out of a deep interest.
The book concludes with several superb indices, and several constants the author added at the last minute because "the following results are too beautiful to be overlooked". Each of the essays has its own bibliography, several of which run longer than the essays themselves, so that when your curiosity about some of these constants are piqued, Finch gives you many places you can look for further information. This is the real strength of the book: it gives just enough information about each of the constants to make the reader curious, and then helps them figure out where to look to learn even more.
As I said before, this book is an encyclopedia. And just as you wouldn't go pick up your copy of the G-volume of the Encyclopedia Britannica to read on a lazy Sunday afternoon, this book was not designed to be read cover-to-cover in one sitting. Most of the essays stand alone, and the author succeeds in making at least the first paragraphs of each one accessible to even a strong undergraduate student while still containing many deep results. This allows you to jump in at any point in the book and read those essays hat catch your eye while skipping those that don't.
The author's clear and engaging style makes the book a pleasure to read, but I must admit that I am not sure for whom this book would be useful. In particular, I imagine that the reader who knows they are looking for information on, say, Hayman-Korenblum constants will know most of what is included in this volume as well as where else to look for more information. Finch makes an attempt to overcome this problem by including an index of all of the constants organized by their actual value rather than by discipline, presumably so that if you ran across the number 1.8823126... in your research you could note if there was a connection with the geometric probability constants. However, in the end I think that Finch has created a book that might not be particularly useful, but one that is quite fun to flip through. This is not a book that I would keep in a library to be used for research, but one to keep in a departmental lounge, so that people could flip through it over tea and learn a few fascinating facts.
Darren Glass is a VIGRE Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at email@example.com.