Books on the history of mathematics come in so many different formats nowadays. Most familiar are the introductory (general) histories and the numerous mathematical biographies. There are also thematic histories devoted to very specific topics (such as the history of π) or to the mathematics of a specific historical period or geographical region. Adding to the mix, we have publications that focus on the history of certain branches of mathematics —calculus, trigonometry, series, vector algebra, and so on. Yet another group consists of the so-called episodic histories that cover a range of seemingly disparate topics. But, as will be later explained, this particular book, by Enrique González-Velasco, overlaps several of the above categories

The book basically consists of the history of six well-known branches of mathematics: trigonometry, logarithms, complex numbers, calculus, infinite series and a particular aspect of real analysis (i.e. convergence). The author says the six topics were chosen because there were prospective high-school teachers taking the course from which this book has evolved. Another reason for this selection is said to be the presence of interlinking themes that provide insights into the history of more general mathematical ideas.

The themes themselves may be mathematical or they may relate to the work of particular mathematicians, such as Francois Viéte. For example, the discussion of his algebra begins in the very first chapter, in the context of his work on trigonometry. Further aspects of Viéte’s algebra are revealed in the subsequent history of complex numbers and in the chapter on calculus. Similar comments apply to the work of James Gregory, Isaac Newton, John Wallis and several others. Another example of a major mathematical theme would be infinite series, which come under discussion in the majority of chapters in the book. Consequently, although this book is described as consisting of ‘creative episodes’, it appears as an integrated whole, rather than a set of disconnected chapters.

On the rear cover of the book, there are several eye-catching claims regarding its uniqueness and originality. Such novel features are said to include:

- An original translation of Rafael Bombelli’s work on complex numbers.
- A new analysis of Leibniz’s writing on calculus.
- The first English translation of James Gregory’s proof of the fundamental theorem of calculus.
- An accessible discussion of Gregory’s work on Taylor series forty years before Taylor.
- The first English translation of some of the innovative work by José Anastácio da Cunha on convergence and calculus that was subsequently attributed to Cauchy.

A more general statement by González-Velasco is that his overall approach is also unique in the extent to which it includes extracts from, and analysis of, original writings. This means that he has sought to minimise use of secondary sources, thereby reducing the distorting effects of what he calls ‘history by hearsay’.

Throughout the book there is cultural, political and biographical background as a context for the historical enquiry. There are many illustrations and excerpts from primary sources, and the author provides a feel for the way in which mathematics was created, why it was created, and how it was expressed. This is particularly true of the chapter on calculus, which is shown it to have evolved through the work of several individuals, and not just Fermat, Newton and Leibniz. It conveys the nature of the geometric approach employed by Isaac Barrow, and clearly distinguishes between Newton’s fluxions and the differentials of Leibniz. Consisting of about 130 pages, this chapter is certainly the longest in the book, and it constitutes a most succinctly insightful history of calculus.

The analysis of Leibniz’s work on calculus is pleasantly different to that given in several specialist histories of calculus, and it exposes the reader more fully to his mode of mathematical expression. And the same thing can be said with respect to discussion of James Gregory’s proof of the fundamental theorem of calculus and his work on Taylor series, which, to my knowledge, is more extensive than that in any other standard text (including a recently published history of infinite series).

Other chapters offering fresh historical perspectives include the very short one on infinite series and the more substantial one on the history of logarithms. The latter begins with a description of Napier’s computational work with sines, and it goes on to explain how his collaboration with Briggs led to the use of y = 10^{x} as a superior computational alternative. There is also an account of Saint-Vincent’s notion of hyperbolic logarithms, Newton’s use of binomial series and the chapter concludes with discussion of Euler’s use of infinite series.

On the downside, there are no suggested activities or exercises, and hence little indication as to how students might embark upon historical enquiry. Perhaps those who subsequently employ this text as the basis of a course could suggest that students explore some of the interesting statements within it. For example, it is said on page 230 that ‘Ideas on integral calculus were perfected in the 17^{th} century’ and, with respect to a comment on page 84, students could be asked to explore the degree to which decimal fractions (and the appropriate notation) have their origin in China.

Another reservation concerns the possible use of this book as an introductory historical text. For example, in the very first chapter, on the history of trigonometry, incipient historians are plunged straight into the task of interpreting archaic modes of mathematical expression without, perhaps, being equipped with a more general overview of the development of mathematics through the ages. But for those who have had a more basic introduction to the history of mathematics, this book is a highly recommended addition to the existing literature.

Peter Ruane’s career was centred upon primary and secondary mathematics education.