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*Numbers and Figures* contains six chapters illuminating topics in number theory by providing a connection to geometry. Part A of each chapter provides a very gentle introduction to the topic that should be accessible to any mathematically interested undergraduate. In Part B the topic is followed up in considerable depth. Any of the chapters could form the basis for a rewarding project in a number theory or combinatorics class; reading the whole book would make a beguiling independent study or seminar class.

Chapter Two, for example, opens with several instances of Simpson paradox. There is a lovely two-dimensional geometric illustration for the case of the paradox using diagonals of parallelograms. Simple cases of the paradox lead directly to consideration of Farey addition. Part B opens by asking about rational approximations to irrational numbers. We get a quick proof of Dirichlet’s theorem: any irrational can be approximated by a rational to within for infinitely many different values of Conversely, algebraic irrationals cannot be too closely approximated by rationals (Liouville’s theorem), which gives a simple proof of the transcendence of Liouville’s number. After an interlude on Farey sequences and Ford circles, we get a proof of Hurwitz’s, best possible, improvement to Dirichlet’s theorem which puts a factor of . in the denominator. Farey sequences are used to generate rational approximations, connections are drawn to the Basel problem and the Fibonacci numbers.

Every chapter is similarly rich with deep and beautiful ideas not usually explored in the standard undergraduate curriculum. The book is self-contained, everything is proven, and each chapter concludes with a list of suggested readings and a small number of challenging exercises.

*The History of Mathematics: A Source-Based Approach* describes the history of mathematics with an incredible richness of detail. But, and this is the first of two defining and exceptional features of the book, the authors explicitly and constantly keep in the foreground historical questions: How do we know this? And, How confident can we be that we are correct? Of course, the answer to the first question is that our predecessors left documents and we can read them. And that brings us to the second exceptional feature: the book includes excerpts of primary sources with the expectation that we will read them as a historian would. The authors are making the point that to understand history one must struggle with source material and try to understand what the author of it is saying in his or her own terms.

This, the second volume of a two-volume set, takes the reader from the invention of calculus to the beginning of the twentieth century. The initial discoverers of calculus are thoroughly investigated, and special attention is paid to Newton’s *Principia*. The eighteenth century is presented primarily in terms of the development of calculus and its applications. Mathematics blossomed in the nineteenth century and the book explores progress in geometry, analysis, foundations, algebra, and applied mathematics, especially celestial mechanics. The reader learns not only the history of mathematical ideas, but also how to think like a historian. These volumes were designed as textbooks for the authors’ Open University course, but they can also be profitably read by mathematicians.