The misfortunes endured by contemporary scholars are as nothing compared to those of their mathematical forbears. Just consider: Archimedes was murdered by Roman soldiers, Galois was killed in a duel, Isaac Barrow survived an attack by pirates in the Mediterranean and Pavel Urysohn and James Lighthill were separately drowned in the English channel. But surely the most horrendous fate seems to have been that suffered by Jan de Witt, whose body was torn to shreds by an angry mob. The results of this barbarous act are depicted in a painting by Jan de Baen, and readers of MAA reviews who are prone to nightmares are advised not to seek it out online — nor by any other means.

But not since the time of Pythagoras has anyone been persecuted because of their *mathematical* ideas, and de Witt was no exception. Being a Dutch statesman and mathematician, Jan de Witt was murdered in 1672 during political strife arising from the Franco-Dutch war. But he also wrote what is said to be the first textbook on analytic geometry. It was published in 1659, which was twenty-two years after the appearance of Descartes’ *La Geometrie *— and thirteen years before he met his gruesome end.

Rather like l’Hospital and Maria Agnesi, Jan de Witt’s main contribution to mathematics emerged from his role as an expositor of current mathematical thought (algebraic geometry in his case). He did this with some support from van Schooten during a period when he was heavily involved in matters of government. But he was also a pioneer with respect to actuarial mathematics — a fact that seems consistent with his role as a national administrator.

De Witt’s *Elementa Curvarum Linearum* consisted of two volumes, the first of which set the scene for the second. The geometrical content of the first volume was presented in the traditional manner of Euclid, and it emphasised the relationship between conics and solid figures. The second volume was written for the purpose of advancing Descartes’ ideas, and its use of algebra is very much in the same style. It provides a classification of conics with respect to relevant equations and frees them from their dependency on a three-dimensional geometric setting.

Descartes’ book begins by setting out the principles whereby line segments can be represented algebraically, and there is no explicit use of coordinate axes. On the other hand, de Witt plunges straight into the classifcation of conics by examining the equations that define them. He also conceived a method of generating conics by means of pencils of lines, which is rather like that of used in synthetic projective geometry. Another innovation was the explicit use of one coordinate axis and consistent use the variables x and y in his defining equations. The introduction of the term *directrix *into the vocabulary of coordinate geometry is also ascribed to de Witt.

The book under review is basically a translation of de Witt’s second volume, which, in turn, provides additional insights on Descartes’ ideas. Albert Grootendorst, the original editor, prepared the English translation of the first volume, which was published by Springer in 2000. But owing to his death in 2004 he was unable to complete the translation of this second volume. Subsequently, Jan Arts completed the translation of the Latin text into English and Reinie Erné translated additional material from Dutch to English. The fourth editor, Miente Bakker undertook the remaining editorial work, including the provision of an index.

A longish introduction by Albert Grootendorst sets the scene for de Witt’s text. It summarises the relevant geometrical work of Greek scholars up to Pappus, and it outlines the development of algebra due to Viète and Descartes. The early history of analytic geometry is portrayed with reference to the achievements of Fermat, Descartes, Wallis and various others. Consequently, this introduction by Grootendorst forms a sound historical basis for the subsequent examination of de Witt’s presentation of analytic geometry.

Two other very useful aspects of the book are the second and fourth chapters. Chapter two summarises the contents of de Witt’s second volume. It restates de Witt’s theorems in modern notation. In the space of twenty-six pages the reader acquires a very good idea of the flavour and structure of *Liber Secundus*. Chapter 4 provides deeper commentary on de Witt’s mathematical thinking and the process of translating it into English.

Facsimiles of each of the original pages of de Witt’s Latin text are faced by the English version, and the notation is accurately replicated. One major difference is that de Witt didn’t use the equal sign and nor did he employ index notation. Nonetheless, the reader can clearly see the connection between the modern notation used by the editors and the early algebraic mode of expression used by de Witt (which was typical of Descartes).

My overall opinion is that this book surpasses D. E. Smith’s translation of Descartes’ *La Géométrie *— if only because of the large amount of additional material that it contains. But this helps the reader to gain real insight into the mathematics of the 17^{th} century, and the editors must therefore be congratulated on the production of a text that should be included in every university library (to say the least).

Peter Ruane has retired from the very pleasant task of preparing students to meet the challenge of teaching mathematics in primary and secondary schools in the UK.