A History of Mathematics: An Introduction

Victor Katz’s A History of Mathematics: An Introduction is already well known as a comprehensive textbook in the history of mathematics, courageously covering material from ‘ancient civilizations’ to ‘computers and their applications’ in just under 900 pages. In content the third edition (2009) remains essentially the same as the second edition (1998) but it has also been revised and updated. What is new?

Most significantly, since working on the previous edition Katz has edited The Mathematics of Egypt, Mesopotamia, China, India, and Islam, and this has led to substantial changes to A History of Mathematics, with new sections on each of those five regions. Thus new scholarship on the ancient and medieval world has been rapidly introduced into a widely used textbook. In the better known field of Greek mathematics, Euclid’s Elements, so essential to an understanding of later European mathematics, now benefits from a longer treatment than before, and there is also new discussion of the Archimedes palimpsest and the discoveries that have arisen from it.

For later periods of history, where the focus moves to western Europe, there is are helpful separations of material so that, for example, Viète and Stevin, who were so very different in motivation and output, now each have their own subsections; similarly, the mature calculus of Newton and Leibniz is treated separately from the earlier seventeenth-century ‘beginnings of calculus’.

Katz’s revised discussion of eighteenth-century calculus is also differently arranged, with a new section on translating the methods of Newton’s Principia into differential calculus. For the eighteenth century, probability, algebra, and geometry now have a whole chapter each, and there is also a new chapter on probability in the nineteenth century, so that it is now possible to follow the distinct threads of algebra, analysis, probability, and geometry through the second half of the book. The volume ends with a new discussion of twentieth-century solutions to some old problems: Fermat’s last theorem, the four-colour problem, and the Poincaré conjecture.

Despite the welcome updating, some of the problems of earlier editions remain. One is the translation of historical mathematics into modern notation. Of course this is a useful, sometimes necessary, thing to do to aid understanding, but to do it without ever returning to the original texts obscures historical reality. Katz’s account of Newton’s discovery of the general binomial theorem, for example (pp. 547–548), claims that Newton did so by means of an elegant modern formula. This bears little relation to the manuscript evidence of Newton pursuing a lengthy process of trial and error and empirical observation, with a formula of sorts emerging only at the end.

A second problem, which compounds the first, is the lack of references, making it very difficult for readers to return to original sources for themselves. Thus, in the context just discussed of Newton’s binomial theorem, Katz tells us (p. 550) that the infinite series for arcsin appeared for the first time in Europe in De Analysi. A library search for a book of this name, however, will reveal nothing. Newton’s ‘De analysi’, written in 1669 was unpublished for many years (and which, therefore, in keeping with a widely used convention, I write in quotes rather than italic). When it was finally printed, in 1711, it was under the title of Analysis per Quantitatum, Series, Fluxiones, ac Differentias. This, like many other historical mathematical texts, is now available online, a development of inestimable value for historians of mathematics, but in order to identify and find such texts the student needs accurate dates and titles, which Katz all too rarely gives.

Lists of references are provided at the end of each chapter, but some are now a little dated. The reference list for algebra in the eighteenth century, for instance (pp. 684–685), lists as ‘recent’ three publications from 1973, 1984, and 1985, and cites only the second edition (why not the first?) of Maclaurin’s A Treatise of Algebra (1748).

Updating a book of this length is, of course, a major undertaking, but it is a pity that some of these shortcomings of earlier editions could not have been addressed in the latest round of revisions. Nevertheless, Katz’s A History of Mathematics remains, as it has been for some years, the most comprehensive textbook available in the history of mathematics, and for this reason alone is a valuable resource for students and teacher alike.

Jacqueline Stedall is lecturer in the history of mathematics and a Fellow of The Queen’s College, Oxford.