It's always a pleasure to sit down with a new (to me) textbook on the history of mathematics, and this text by David Burton has much to recommend it. One of the features which sets this book apart from its competitors is the attention that Burton pays to the cultural and historical millieu in which various mathematicians lived and worked through the ages. This is illustrated even in the typical timeline one expects to see inside the front cover; Burton compares mathematical advancements with important historical events such as the engraving of the Rosetta Stone in 195 B.C. (which eventually led to the deciphering of Egyptian hieroglyphics and the understanding of Egyptian mathematics) and the production of paper in Bologna in 1293 (allowing for the cheap production of textbooks using carved wooden blocks and, much later, movable type).

Throughout the book Burton pauses to give lengthy discourses on how the events of the time affected the development of mathematics. We learn about the transition of Christianity from persecuted sect to the official religion of Rome and how that led to a neglect (in the west) of the pagan Greek writings on mathematics. We are given a fascinating description of the Carolingian renaissance of the ninth century and how Charlemagne's system of monastery schools led to both the collection and preservation of libraries and to the design of an easy-to-read Roman script that is the precursor of our modern alphabet. We also are treated to a wonderful story of how the modern American mathematical community came into existence, deeply influenced by the "new" German university system of the nineteenth century (which was itself motivated by the Napoleonic defeat of the Prussian army in 1806). These historical discourses help the student to understand the environment of the time and also to relate the material to topics from their classes in European or world history.

A favorite subject of mine is the history of women in mathematics, and I'm always interested to see how textbooks cover this important topic. Burton includes descriptions of a number of lesser-known female mathematicians, such as Mary Somerville (who wrote many expository versions of 19th century scientific texts, among them Laplace's *Mécanique Céleste*), Winifred Edgerton (the first female recipient of an American Ph.D. and one of the founders of Barnard College) and Charlotte Scott (who was denied a degree by Cambridge, yet later directed several doctorates as chair of the mathematics department at Bryn Mawr). Many pages are devoted to the life and times of Emmy Noether and to her interactions with fellow mathematicians. I would have like to have seen a similar in-depth coverage of Sophie Germain. Unfortunately, her life and work are described in only a few paragraphs, and while brief mention is made of her interactions with Gauss, Burton left out Gauss' wonderfully chivalrous description of her after his discovery that his mysterious correspondent was a woman.

This sixth edition contains a new section on mathematical developments in China and Arabia. It troubles me to report that (in at least one area) this addition does not live up to the scholarship that one might expect.

Burton notes in this section that the Chinese mathematician Chu Shih-Chieh (a.k.a. Chu Shi-Kie) gave the first eight rows of Pascal's triangle in 1303, yet he fails to mention that this was done earlier by Yang Hui (around 1261), who himself gives credit to Jia Xian (mid 11th century). Two hundred pages later, he does mention Jia Xian (written Chia Hsia), but not Yang Hui, nor does he refer to his earlier discussion of this topic. When discussing the development of Pascal's triangle among Arabic mathematicians, Burton attributes al-Karajî (early 11th century) with the first description of what we now call the binomial coefficients. Unfortunately, Burton completely mangles the formula for generating these numbers, writing it as

C_{n}^{k} = C_{n-1}^{n-k} C_{n-1}^{k}

instead of the correct (not to mention easier-to-read) form of

C_{n,k} = C_{n-1,k-1} + C_{n-1,k}.

Then, when discussing Pascal's triangle later in the text, Burton ignores al-Karajî entirely and instead credits two other Arab mathematicians.

Would I recommend Burton's text for classroom use? It certainly has many nice features, and each section concludes with an appealing (albeit modest) collection of historically appropriate problems. While the section on Arab and Chinese mathematicians would benefit from some careful editing, I do not see this as being an unsurmountable problem. This book contains so much interesting material, presented in such a lively and entertaining manner, that I feel it would be an excellent reference, and it could also serve well as a primary text. I'll certainly be referring to my copy quite often when teaching my upcoming class on the history of mathematics.

Gregory P. Dresden is Associate Professor of Mathematics at Washington & Lee University in Lexington, VA.