My introduction to real analysis came from G.H. Hardy’s*A Course of Pure Mathematics,* which, in 1908, became the first rigorous English university text on analysis. However, as with many subsequent books on analysis, it conforms well to the description given by David Bressoud, who says:

The traditional course begins with a discussion of the properties of the real numbers, moves on to continuity, then differentiability, integrability, and finally infinite series, culminating in a rigorous proof of Talyor series…This is the right way to view analysis, but it is not the right way to teach it.

Bressoud believes that a better way (the best way?) to teach analysis is rooted in the historical issues that have shaped its development, which is why he includes the word ‘radical’ in the title. The aim is to dispel the myth that mathematical ideas and methods are perfectly formed in the moment of their conception. Accordingly, Bressoud’s ‘Darwinian’ approach reveals how, over long periods of time, the central ideas of analysis have evolved by processes of experiment and scholarly disputation (which, of course, is true of mathematics as a whole).

This philosophy is evident in the very first chapter, which explains the controversial nature of Fourier’s work on trigonometric series. Later in the book, it is shown how those ideas led to the exposure of many of the difficulties and paradoxes arising from infinite summations, and eventually led to a re-examination of the foundations of calculus. Following this theme, there is study of Gauss’s work on hypergeometric series, Cauchy’s analysis, and ideas from Riemann and Weierstrass. But the focal point of the book is the intervention of Dirichlet who, in 1829, resolved the concerns over Fourier’s ideas. Yet, although the emphasis is on the period 1807 to 1829, there are many references to earlier and later times. For example, there is Archimedes and his influence upon Cauchy. There is mention of Newton and Leibniz — and their sternest critic, Bishop Berkeley. Reference is also made to the work of more recent mathematicians, such as Dedekind, Babbage, Riemann, Bertrand Russell — and, of course, G.H. Hardy. Along with much biographical detail, the errors and successes of almost a hundred mathematicians come under varying degrees of scrutiny.

In the first half of the book, the material is presented in accordance with the historical progression of its invention. The daunting formality of typical analysis texts is replaced with a discursive style of narrative that motivates the introduction of the related concepts and methods. In particular, introduction of the dreaded ε–δ techniques is preceded by much discussion and many heuristic activities on limits, so that, when these Greek characters eventually appear, they seem like the final pieces in a jig-saw, rather than harbingers to a nightmare.

Another innovative feature is the nature of the exercises, which often involve the reader in the methods of the mathematicians under discussion. As well as the many carefully graded problem sets, there are many open-ended problems that suggest ramifications of the main analytic ideas (I particularly like the exercises relating to Archimedean treatment of infinite summations and the later exercises on grouping and rearrangements).

Supplementary to the main text is the author’s well organised website, which facilitates graphical and numerical investigations via Mathematica and Maple; and it also provides access to investigative projects and further historical commentary etc. Throughout the book, readers are frequently prompted to refer to relevant features of this resource.

So, is this book to suitable as a *main introductory text* for use with undergraduate students? Any such decision should take account of various factors:

- For purposes of motivation and pre-requisite knowledge, prior insight into analytic ideas via a good course on calculus is strongly recommended.
- The historical insights provided here would be all the more appreciated if students had previously undertaken an introductory course on the history of mathematics.

Finally, while the text fulfills its main aims regarding ‘history informing pedagogy’, readers can gain only partial insights into the mathematics of earlier times. Take, for instance, the matter of notation, which appears throughout this book in very modern form. Nowadays, l’Hospital’s theorem is expressed in function notation, whereas l’Hospital himself used infinitesimals, and had no general function concept with which to work. Looking at the very first version of this theorem, on page 207 of the 1768 edition of his calculus textbook, it bears no apparent relationship to that contained in this book. The same applies to first published version of the fundamental theorem of calculus as given by Isaac Barrow, whose proof was based upon classical geometry and used no algebraic symbolism whatever.

So far, this review may contain nothing new for those familiar with the first (1994) edition, so what changes have been made for this 2006 version? The main ones are:

- Conceptual changes in chapter 2, including the reconciliation of the approaches of Archimedes and Cauchy to infinite series and a rigorous introduction to convergence and a new section on geometric series
- Chapter 4 has been reorganised and some material has been transferred to the website
- Many new exercises added.

As for errors and other quibbles, I found only one misprint (exercise 2.1.2), but the discussion of infinity on page 29 could have been fuller. Simply to say that ∞ is ‘not a number’ is to discount the work of transfinite arithmeticians and takes no account of historical perceptions (e.g. l’Hospital in the very first book on analysis). Moreover, the comments on integration in chapter 6 give the impression that it wasn’t until the 19^{th} century that it was seen in any way other than anti-differentiation, but Barrow established the connection between tangency and quadrature as early as 1670.

In summary, the book is written in a clear, lively style and, unlike many traditional courses, it doesn’t overload students with a welter of concepts and techniques (no Heine-Borel theorem here). But it is by no means an easy option and provides a stern, but meaningful, challenge for its readers. It is highly recommend to any lover of mathematics, for whatever purpose.

Peter Ruane has now escaped the bureaucratic confines of higher education, where he spent a working life training primary and secondary mathematics teachers.