If this were the only work on mathematical history to become required reading as part of mathematics degree programmes, then it would already be a giant step forward. This is because the majority of students will graduate in mathematics from the myriad of British and American universities knowing almost nothing about its history.

This book by Steven Krantz consists of twenty-two chapters, arranged in the chronological order of their historical themes. The first chapter discusses the work of Pythagoras, Euclid and Archimedes, and the last one begins with a short biography of Alan Turing. This chapter concludes with a brief introduction to Turing machines and cryptography. Many intermediate chapters concentrate upon the work of a particular mathematician (e.g. Newton, Gauss, Riemann), while others focus upon historical aspects of a particular area of mathematics, such as complex numbers, Zeno’s paradox or prime numbers etc. As such, the structure of the book is compatible with its title.

The make-up of the book is also consistent with the author’s declaration that it is ‘unabashedly mathematical’. In practice, this means that most of the chapters consist of an historical introduction followed by a substantial number of examples and exercises expressed in modern mathematical notation. Some of the exercises require the reader to replicate the method of a particular historical figure, such as Cardano’s technique for solving cubic equations, or Archimedes’ method for calculating the area of an ellipse.

This philosophy, combined with an approach to mathematical history that is ‘episodic’, can lead to certain historical discontinuities (although not inevitably so). For example, in the chapter devoted to Descartes, there is little indication of his innovatory development of algebraic notation, which, as much as anything, helped him to formulate his revolutionary geometric ideas. Similar comments apply to the chapter on Hypatia’s work on conic sections. In this case, discussion of Dandelin spheres would have bridged the mathematical gap between the Greek idea of conics as sections of a cone, and the subsequent synthetic definition involving focus and directrix (due to Pappus, I believe).

Another notable omission is mention of Euler’s reformulation of the calculus based upon his clearer notion of functions, and his powerful algebraic notation for handling them. This effectively freed much of mathematics from the long-standing over-reliance upon geometrical methods. Consequently, it has to be one of the major turning points in mathematical history. For this, and the above reasons, Steven Krantz’s book might have been more aptly titled ‘Episodes in the History of Mathematics’,

In terms of the overall balance of contents, it’s surprising that a book commencing with a dedication to the eminent geometer, Marvin J Greenberg, should contain so little reference to geometrical ideas. Only 15% of the book is directly concerned with geometry. In particular, projective geometry and the birth of non-Euclidean geometry receive no mention.

Nonetheless, this book is a source of fascinating reading, and it could well succeed where other approaches have failed. That is to say, because it doesn’t immerse the incipient historian too suddenly into the depths of historical detail (such as exposure to archaic notations found in original sources), it could form a less daunting challenge for those who wish to include aspects of history into undergraduate mathematics courses. Copies of this book should be included in all university and college libraries, and all teachers of mathematics should find it of interest. I certainly did.

Peter Ruane first adopted an historical approach to the teaching of mathematics with primary school children in the 1960s and 1970s. For this purpose, he used a series of school texts called *Let’s Explore Mathematics*, by L. G. Marsh (long out of print, but used copies are available via Amazon UK).