It is easy to fall into the trap of assuming that a mathematician’s fame and influence is directly proportional to the size of his/her written output. Of course, in the case of Euler, this would be no misconception at all. Yet Descartes is one of those who bequeathed us comparatively little written mathematics and yet altered he the course of its history. In fact, *La Géométrie* was his only ‘book’, and even that appeared as an appendix to a larger work. Yes, there are his various lesser known achievements, such as Descartes’ rule of signs, the Descartes circle theorem and the Descartes-Euler theorem but, apart from those, his main mathematical work was his ninety eight page appendix that led to the modernisation of geometry and subsequently facilitated the development of the calculus. In the process, of course, he streamlined the existing modes of algebraic notation, thereby expediting the use of algebra in other areas of mathematics (index notation is due to him). Who else has achieved so much by writing so little?

On the other hand, the amount of Descartes’ mathematical writing is also inconsiderable when compared to his general output on philosophical matters, and he is described on the jacket of this book as being ‘*one of the founders of the* *modern world’*. More specifically, Bertrand Russell considered him as the founder of modern philosophy, and others rank him as one of the giants upon whose shoulders Newton stood for the enhancement of his scientific perspicacity. Whatever the truth of all this, we now have Grayling’s book that provides wider perspectives Descartes’ life and work.

So, based on his own research, Grayling has written this general-interest biography revealing new aspects of Descartes’ life in the context of the turbulent events of his era. Its chronological structure depicts the important stages of his life, beginning with his birth in rural France to his death in Sweden, where he served as tutor to Queen Christina. But Grayling also provides plausible explanations regarding the puzzling gaps in existing biographical knowledge. For example, by study of surviving correspondence, he suggests that Descartes had a hidden life as a Jesuit spy during the Thirty Years War; he also provides evidence pointing to Descartes’ involvement with a secret society (the Rosicrucians). Glimpses of his life as a family man, although fragmentary, contribute to the overall picture of him as a flesh-and-blood character with a pugnacious ability to promote and defend his views; and Grayling explains with great clarity why such qualities were necessary in an era still dominated by medieval beliefs and the confining doctrines of the church.

This book is a most enjoyable read, and it could well serve as an informal introduction to Cartesian philosophy. In terms of mathematics, however, the content is slender indeed; but the relatively sparse amount of commentary that Grayling does provide forms an accurate summary of Descartes’ work. Moreover, it is greatly enlivened by colourful descriptions of his friendship with Mersenne, Desargues and Pascal — and his inharmonious interactions with Fermat (whose work Descartes described as ‘dung’!). Those seeking more detailed knowledge of Descartes’ mathematical output can easily find many accounts in the standard mathematical histories.

Peter Ruane (ruane.pandh@blueyonder.co.uk) is retired from university teaching, where his interests lay predominantly within the field of mathematics education.