It is, sadly, not just possible but common to do an entire university degree in mathematics and not be exposed to anything of its history. And when that is covered it is almost always tacked on as a separate course rather than being integrated into the subject. Mathematics and Its History is not a text for such an add-on course, a traditional history of mathematics, but rather a presentation of core aspects of the major strands of mathematics within their historical context. Its goal is to get the reader to understand the development of different areas of mathematics and their methods.

The progression is roughly chronological, though not strictly so; the discussion of infinity in Greek mathematics, for example, looks forward to Dedekind cuts. The coverage makes no pretence at being comprehensive, but spans most of mathematics, with chapters on geometry, number theory, infinity, projective geometry, calculus, complex numbers, differential and non-Euclidean geometry, algebraic number theory, topology, the classification of finite simple groups, logic and computability, and combinatorics.

These chapters make no attempt to be systematic either historically or conceptually, but present a few important or central results or theorems. So the chapter "Analytic Geometry" looks at Descartes' conception of algebraic curves, Newton's classification of cubics, Bézout's Theorem ("not properly proved until long after the theory itself had been abandoned"), and the arithmetization of geometry. The chapter "Simple Groups" touches on finite simple groups, the Mathieu groups, continuous groups, the simplicity of SO(3), Lie groups and algebras, and the Monster. Exercises accompany each section. These typically work through key results or simple cases or examples of them; more complex results are broken down into steps, with guidance, to make them accessible.

Stillwell includes a few reproductions from original texts — Newton's classification of cubics, for example — but uses modern notation and terminology throughout, making no attempt to survey sources let alone explain the specialist skills needed to read source documents. Taking up maybe a fifth of the book, there are short biographies of key mathematicians at the end of each chapter, covering some forty mathematicians in total, from Pythagoras to Erdös. These are potentially readable by someone without any mathematical training, but there are plenty of books available for anyone just after potted "great mathematicians" biographies.

There is quite a substantial amount of mathematics in Mathematics and Its History, as the material Stillwell selects for attention is often the more interesting, "advanced" results. It won't be too demanding for those already familiar with most of it — a nice recapitulation, perhaps filling in a few gaps as well as adding historical insight — but it's not aimed at a student approaching higher mathematics for the first time; it would hardly make sense as an introduction to (say) group theory. When it comes to teaching mathematics in its historical context, Mathematics and Its History is an original, engaging and effective book, which I think would be enjoyed by students, lay readers with the right background, or indeed mathematicians themselves.

March 2019

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