It was not until the seventeenth century that doing mathematics became an identifiable profession, such that historians today can legitimately talk of ‘mathematicians’ as a community of experts producing mathematical works for one another in various institutional settings. This is the starting premise for Ivor Grattan-Guinness's mammoth edited volume Landmark Writings in Western Mathematics 1640–1940 – a book that takes the reader from René Descartes's La Géométrie (1649) to David Hilbert's and Paul Bernays's Grundlagen der Mathematik (1934–9). Written by a Who's Who? of authors in the history of mathematics, including Grattan-Guinness, Niccolò Guicciardini, June Barrow-Green, Jeremy Gray and Tony Crilly, to mention only a few, the book is composed of seventy-seven articles dealing with eighty-nine pieces of ‘writing’ spanning a wide range of mathematical topics.

Importantly, the book does not contain the original texts under discussion. It is not a compendium of primary sources. It is, rather, a compendium of succinct survey articles that provide the social and educational background of the mathematicians or writers in question, descriptive accounts of the most salient aspects of the piece of writing being analysed, and a brief discussion of the ‘impact’ or effect the piece had on mathematical thought. Thus we get articles on aspects of geometry, calculus, functions and differential equations, algebras, number theory, real and complex analysis, set theory (and its foundations), mechanics, astronomy, probability and statistics, dynamics, mathematical physics and topology, as well as articles in the history of mathematics, including Jean-Etienne Montucla's Histoire des mathématiques (1799–1802), and articles in the ‘social and life sciences’, including pieces on W. S. Jevons's Theory of Political Economy (1871) and Vito Volterra's book on mathematical biology (1931).

Each article starts with a list of all traceable versions of the writing in question, including various editions, translations and reprints. Of the three centuries dealt with here, just over fifty per cent of the articles fall in the hundred years from 1840 to 1940, which is, presumably, a reflection of the increasing professionalization and institutionalization of mathematics during that century relative to the previous two (only ten per cent of the articles fall between 1640 and 1740). The unifying theme for all of these writings is their unique character as ‘landmark’. Admittedly, the manner in which this characteristic is measured is ambiguous, despite Grattan-Guinness's attempts to clarify the selection process in his introduction. Briefly, landmark writings had to have had a wide impact in the mathematical community, broadly defined, and preferably an international impact.

The laudable concision with which some difficult pieces are dealt with in this volume should also be noted. One of the best examples is Albert C. Lewis's article on Sir William Rowan Hamilton's Lectures on Quaternions. Hamilton's lectures are not pleasurable reading by any stretch. Yet Lewis manages fluidly and effortlessly to summarize them with enough technical detail to satisfy historian and mathematician, expert and novice, alike. He contextualizes Hamilton's development of quaternions (represented by the expression a+bi+cj+dk) in the light of pre-existing concerns over the generalization of complex numbers (z=a+bi) in a space of three dimensions. Likewise, Jeremy Gray's account of Bernhard Riemann's posthumous thesis, ‘On the hypotheses which lie at the foundation of geometry’, deconstructs the idea that Riemann's thesis constitutes just another moment in a linear history of non-Euclidean geometers. Gray's account is a much-needed corrective to the standard way in which some historians think of the historical development of non-Euclidean geometry.

Yet, for all this volume's usefulness, there is one important aspect that renders it less than fantastic. The table of contents alone indicates that it is another Great Man history of mathematics. Although Grattan-Guinness is careful to point out that this compendium does not aim to be a ‘general history’ of mathematics over the three centuries involved, the book nonetheless makes one very large general historical claim – that no women were involved in the creation of landmark mathematical ideas. In his introduction, Grattan-Guinness writes that not all important pieces of writing could be included; facetiously, he says, ‘If Your Favourite Writing is missing, dear reader, then we mortals have offended’ (p. xii). Much to my chagrin, however, it is not my favourite piece of writing that is missing, it is my entire gender. Not one female mathematician's work appears throughout the whole of this compendium of landmarkedly brilliant mathematics. The book's editorial board might protest that this is a simple fact of history – no women were mathematicians, or producing anything like important mathematical works, between 1640 and 1940. But surely we recognize today that there were some women involved in mathematics. An obvious place to start would have been with Mary Somerville, the first of two women appointed to the Royal Astronomical Society in 1835 (the other was Caroline Herschel). Somerville's The Connection of the Physical Sciences, published in 1834, discussed the existence of a hypothetical planet perturbing Uranus. The book deeply influenced John Couch Adams in his own researches, playing a significant role in his later discovery of Neptune. We know then that women were doing science and presumably mathematics too. Their voices were not given space to be heard in the centuries in which they worked; it is our duty in this century to rectify that matter rather than reproduce the system of exclusion that has so long defined women's role in formal mathematics. The least I would have expected in a book of this magnitude is a brief account in the introduction of the obvious material inequalities – lack of education, lack of financial and sexual independence, lack of access to resources, and so on – that women faced between 1640 and 1940, which caused them to be excluded from the world of professional research and therefore absent from three hundred years of ‘landmark’ mathematical writing.