*Elements of Mathematics: From Euclid to **G**ödel**,* by John Stillwell. 440 pp. illustrations, bibliography, and index. $39.95 hardcover, ISBN: 978-0-691-17168-5. Princeton, NJ: Princeton University Press, 2016.

Before I even opened this book, the title intrigued me. *Elements of Mathematic*s: ‘What *Elements*? What *Mathematics*?,’ I wondered. ‘Is this a foundations book: sets, functions, logic, ...?’ Further, the historical associations of the subtitle, *From Euclid to Gödel* – from an apparently complete, accepted mathematics to one open to question and doubt – seemed to foretell a “plot.” My suspicions were pleasantly assuaged. The mathematics in question is basic: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic; all, initially, at an introductory level. Still there is a plot here: author John Stillwell uses his broad and impressive command of mathematics to transport the reader through each topic and to a higher level of understanding and questioning. He enforces a rethinking of the mathematics that we think we know and understand. Stillwell was inspired in this project by Felix Klein’s *Elementary Mathematics from an Advanced Standpoint* (1908), written in a time of philosophical and mathematical transition and focused on the question, “What is mathematics all about?” Stillwell, too, leaves his reader with this worthy question.

Stillwell’s writing is clear and direct, embellished by illustrative anecdotes and examples. I found the reading enjoyable. While the accompanying advertisement solicits readers “from high school students to professional mathematicians,” a level of sophistication and experience with mathematics is required that I doubt would be possessed by young students. On the other hand, I strongly recommend a reading of this book to professional mathematicians and, especially, instructors of the subject. The transcendence from the mathematically mundane to substantive advances comes quickly: arithmetic and computation are clarified by the use of Turing machines; geometry supersedes Euclid’s theories with a consideration of tangent and vector spaces and the *arithmetization* of the discipline. In the survey of combinatorics, the reader moves from Euler’s study of polyhedra in 1752 to the rise of topology and graph theory, and is soon “climbing trees” to the heights of König’s Infinity Lemma and Brouwer’s Fixed Point Theorem. Similar transitions take place in the discussions of each of the elementary subjects covered. The concept of “reverse mathematics,” proceeding from a theorem back to the axioms supporting it, is introduced and pursued. The importance of “infinity” as a basic principle of mathematical understanding is highlighted.

In summary, I consider *Elements of Mathematics *to be a thought-provoking, informative, and worthwhile read and a valuable reference. However, the trait for which I would recommend the book most highly is the author’s embedding of concepts and accomplishments into their historical setting. The historical referencing is done simply but effectively, with no laborious relating of history. Names and dates are supplied as a part of a movement or evolution. For example, the identification of Gauss, Riemann, and Beltrami accompany the conceptual lineage of non-Euclidean geometry. The “Four Color Problem” is traced from its origins in 1852 to Kempe in 1879, to Heawood in 1890, to Appel and Haken in 1976, and similarly for each movement of mathematical discovery and refinement. The reader is made to realize that many people, over time, were involved in shaping mathematics into its present form. Stillwell, who has previously written on mathematics history, in *Mathematics and its History* (2010), provides a template for all instructors who wish to humanize their mathematic teaching by employing some history. If you want to teach mathematics with its history, this is a way to do it!

*Editor's note:* For another discussion of the same book, see the review by Mark Hunacek in *MAA Reviews.*