This is a reprint of the 1982 Dover edition of a book that was first published by Penguin Books in 1955. It is an account of some of the more ‘surprising and stimulating branches of mathematics’ together with ‘an analysis of the mathematical mind’. The author’s approach is claimed to be ‘geared to the layman whose grasp of things mathematical may be a bit precarious’.

I first read *Prelude to Mathematics* in the early 1960s, as a follow-up to Sawyer’s first book, *Mathematician’s Delight*, which was more like an introduction to high school mathematics. Having previously been taught the subject as a series of methods, Sawyer’s books encouraged me to think more widely about mathematics and, in that respect, his books were innovative publications in the 1950s and early 1960s.

At the time of the first edition, however, topics such as linear algebra, group theory and finite geometries were just making there way into the undergraduate curriculum, and the sparse amount of associated literature was geared to the needs of maths specialists. Consequently, non-specialist readers of this book would have been surprised and stimulated, not only by such relatively new mathematical ideas, but also by Sawyer’s philosophical observations on mathematics.

Regarding the process of generalisation, for example, Sawyer endeavours to show that ‘higher mathematics is simpler than elementary mathematics’, which he illustrates by comparing projective geometry with Euclidean geometry. He also discusses the notion of ‘truth’ and physical reality — as opposed to mathematical consistency. The aesthetic value of mathematics is contrasted with its practical utility, and he encapsulates his approach to isomorphism with Poincaré’s dictum that ‘Mathematics is the art of giving the same name to different things’.

So, after a fifty-year gap between reading the first and second editions of this book, how does it now seem? The main difference between the two editions is that the material on group theory was updated for the second edition but other than that there is little ostensible difference between the two versions. Inevitably, however, the book now appears a little dated in various respects. For example, there is reference to the slide rule as the main computational tool, and mathematical practitioners are principally referred to as being men or schoolboys. Less importantly, the topic of determinants is introduced prior to matrices and there is no reference to topological thinking.

Overall, this book stands the test of time as being one of the most imaginatively composed introductions to non-elementary mathematics. I thoroughly enjoyed it on a second reading, and recommend it for inclusion on reading lists for students at the high school level and beyond.

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