It is not difficult to understand why knowing the history of mathematics is both interesting and important to the lives of many research mathematicians. It is also not difficult to see why it is hard to write about this history. Very few historians have a good enough understanding of mathematics to write histories which answer the questions that mathematicians want to know the answer to, but on the flip side very few mathematicians know enough about historical scholarship to truly be able to understand the mathematical history. This task is somewhat easier when studying the mathematicians of the 18th or 19th century, whose culture we understand fairly well, but as Eleanor Robson writes in one of the essays in the collection under review: "we are already more or less familiar with Galois's cultural background, or Newton's ... this allows us to contextualize the mathematical content of their work ... but when we start to study mathematics from cultures whose languages, social practices, and common knowledge we do not share, we have to work considerably harder at positioning it within a historically and mathematically plausible framework."
Sherlock Holmes in Babylon, published by the Mathematical Association of America, collects forty-four articles on the history of mathematics which have been published in the various MAA journals such as the Monthly and Mathematics Magazine over the last century. The book breaks these articles down into four sections, each of which has a well-written introduction and afterword to put the articles in the section better into context. The first section covers Ancient Mathematics, and begins with R. Creighton Buck's 1980 article which lends its title to the entire collection. It is a nice introduction, as the author describes "the intellectual challenge of reconstructing pieces of a culture from random fragments of the past" — in his case looking at Babylonian tablets to try to reconstruct the number system used in that culture.
The next few articles cover similar ground: there are articles about Diophantus and Hypatia. There are articles about Liu Hui and Chinese mathematicians, including the description of an inductive construction of π dating back to 263 AD. The section concludes with several articles about the mathematics of the indigenous people of North America, including the detailed descriptions of the number system of the Mayas and the Incas.
The second section of the book is entitled "Medieval and Renaissance Mathematics", and it starts by reminding us that although not much progress was made in Europe on mathematics during the Middle Ages, the same cannot be said of the rest of the world. In particular, there was quite a bit happening in India, and the first several articles of this section detail some of that progress, including Nilakantha's discovery in the late fifteenth century of a power series for arctangent, and in turn a series formula for π. Victor Katz has an excellent article detailing how some of the ideas that Newton and Liebniz would later discover for their calculus actually turn up hundreds of years earlier in Islamic and Indian mathematics. In particular, he writes, "the area formula had been developed in Egypt around the year AD 1000" although they did not appear to be interested in polynomials of degree five or more or more general functions.
The remaining articles in this section are mostly devoted to several classic works of mathematical writing, including Juan Diez's Sumario Compendioso, which is considered to be the first mathematics book published in the Western Hemisphere, Leonardo of Pisa (aka Fibonacci)'s classic Book of Squares, and Rafael Bombelli's Algebra. The article dealing with the latter, by Abraham Arcavi and Maxim Bruckheimer, leads the reader through a sixteenth century algorithm for computing square roots, pointing out the differences in "language, notation, and also the spirit" in mathematical writing from that era.
The third section of the book is devoted to the mathematics of the seventeenth century. It is highlighted by two articles written by Judith Grabiner, one on Descartes' Geometry, and another on the history of the derivative. It also includes E.A. Whitman's excellent article on the cycloid, the curve first 'discovered' by Galileo and later considered in various contexts by a wide range of mathematicians including Fermat and Pascal. Other articles are dedicated to the mathematics of Mercator, Cotes, Roberval, Gregory, Liebniz, and of course Isaac Newton, who is at the center of five of these articles.
The fourth and final section of the book deals with the eighteenth century, and in particular with the mathematics that was developed in the wake of the introduction of calculus as well as the mathematics of Leonhard Euler. Articles on the former topic include another by Grabiner, this one on Maclaurin's Treatise on Fluxions, which extended Newton's calculus and had great influence on Lagrange and Euler, among others. The section on Euler opens with a piece by J.J. Burckhardt which attempts to summarize all of Euler's mathematics — a task that I do not envy — and continues with an article on the history of the number e, both how it was first discovered and how it was later computed. There are also articles dealing with Euler's attempts to prove the Fundamental Theorem of Algebra, and his visions of generalizing calculus to functions of several variables and how it was eventually realized by Schwartz two centuries later.
Sherlock Holmes in Babylon is a potpourri and as such the articles vary quite a bit in style. Some, such as William Dunham's article dealing with Jakob Bernoulli's proof that the Harmonic Series diverges, involve lots of concrete mathematics. Others are all history with nary an equation to be found — a feature that makes this book quite accessible to a broad audience, something I saw firsthand when my mother-in-law, who is about as far from a mathematician as you can get, picked up my copy of the book and had trouble putting it down. Some of the articles are nicely illustrated, with pictures of the primary source material, while others are just text. One thing that does not vary significantly is the quality and readability of the articles — the editors of the book (Marlow Anderson, Victor Katz, and Robin Wilson) deserve quite a bit of credit for choosing very interesting articles.
In fact, my main complaint about the book is not what is in the book but rather what is not. The articles are by their nature quite short, and much of the time the article would end before my curiosity on the subject did. Luckily, in addition to the bibliographies of the individual articles, the editors include afterwords to each section which recommend even more articles and other books that the reader can go to in order to satisfy such curiosity. I also felt that the book ended too soon, historically speaking. I would be interested in reading similar accounts of the mathematics of the nineteenth and twentieth centuries, and I hope that a second volume with articles covering more recent history will appear before too long. Finally, I was happy to see the inclusion of significant coverage of non-European mathematics in the first two sections, but disappointed that there were no such articles in the second half of the book, or even a mention of what was happening mathematically in Asia, Africa, or the Americas after the mid-1600's.
One interesting feature of the book is that the articles were written throughout the twentieth century, and many date back to the beginning of that time, giving them an extra layer of interest. W.C. Eells' 1913 article on "Number Systems of the North American Indians", in particular, stands out not only for its actual content (which is quite interesting), but as a primary source material on the history of the history of mathematics. He discusses his subject in a way that would verge on the politically incorrect by 2004 standards, but presumably was seen as appropriate at the time.
Finally, I should note that many of the articles in the book would be of interest not only to mathematicians but also to our students. In particular, I am already planning that the next time I teach a course in number theory I will show the students the final article in the book: Harold Edwards' account of Euler's statement of the quadratic reciprocity theorem, and the calculations that brought it about. I imagine that anyone reading this book will find similar uses inside the classroom. But even if you are not likely to use this book to influence your teaching, I cannot imagine anyone who will not find at least a few of these articles to contain new and interesting information, and as such it is a book worth adding to your collection.
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 firstname.lastname@example.org