Years ago, several of my colleagues in the university’s mathematics department told me that many high school students no longer study classic plane geometry, the kind of instruction that emphasizes careful logic applied to axioms, postulates, and theorems. I asked, “When is this done now?“ I was told: “Later, often in the third year of university studies. It is their first serious ‘proof’ course.” I then realized why so many students have such a tough time surviving the required introductory discrete mathematics course in the computer science curriculum: they lack the preparation of understanding and doing proofs.

Stillwell’s book on the history of proof in mathematics is a superb review of the history of proof across the mathematical subdisciplines. There are a total of 16 chapters and a 13-page section of references at the end, all within slightly more than 400 pages. The chapter topics are ordered chronologically, but each chapter’s presentation extends the discussion into later time periods to show how a proof technique is applicable to investigations beyond those in which it emerged. For example, some of the post-Euclid developments in geometry matured in later centuries in projective geometry as Renaissance artists employed perspective in their work.

The book begins with three chapters on geometry spanning the time period before Euclid through Euclid’s geometry, including the historical beginnings of proof by induction and mathematical series, and post-Euclid plane geometry. The traditional mathematics of computation continues with three chapters on algebra, algebraic geometry, and calculus. Among the topics introduced in these chapters are quadratic and cubic equations; linear algebra; groups, fields, and rings; infinite series and sums, and infinitesimals.

The next ten chapters cover more specialized topics--number theory; the fundamental theorem of algebra; non-Euclidean geometry; topology; arithmetization; set theory; axioms for numbers, geometry, and sets; axiom of choice; logic; and incompleteness. The history of graph theory is well treated in the topology chapter as an illustration of revitalizing the visual elements in proof. Computation is part of the logic chapter that includes both propositional and predicate logic and the Turing machine. Arithmetization includes the completeness of R; the concepts of point, line, and space; and continuity and differentiability. The discussion of proof in set theory is marked by the concept of the infinite. Complex numbers are included in non-Euclidean geometry. These are only a handful of the topics covered in the last ten chapters.

The history of proof is a challenging subject to cover in a little over 400 pages. The book is driven by examples that are supplemented with useful figures. It would be an excellent resource in the teaching of mathematics, for example, as an introduction to the content and processes in areas of mathematics unfamiliar to the reader. Even when the reader is familiar with a particular subject, the placement of the methods of proof in historical context is valuable. John Stillwell has written or edited many books on mathematics and the history of mathematics. This is an excellent addition to his body of work.