Courses in the history of mathematics are a welcome addition to the undergraduate maths curriculum. Part of their popularity lies in the fact that students ``may find that a little history will serve them as light relief from the rigours of algebra''. This attitude is misconceived. The history of any theoretical subject is the story of the emergence and development of ideas, and an understanding of these ideas is crucial if the student is to appreciate their history.
In the first half of Luke Hodgkin's book, the focus is on mathematical cultures. There are two chapters on Greek maths and one each relating to Babylonian, Chinese and Islamic cultures. In the second half, attention shifts to shorter historical periods or specific concepts: the scientific revolution, calculus and non-Euclidean geometries. Each chapter starts with an immensely useful discussion of the primary and secondary literature.
The introduction is a lengthy apology for adding another book to a crowded market. Although Hodgkin asserts that he ``does not set out to argue a case'', he rightly has no time for those who adhere to an outdated Eurocentrism. But I do not find the evidence he presents convincing. One of his guides for the non-European roots of mathematics is George Joseph's The Crest of the Peacock, reviews of which have not inspired my confidence in its scholarly quality. But Hodgkin dismisses such criticism as ``the fashionable nonsense school of reviewing''. This allusion to a book by Alan Sokal and Jean Bricmont suggests that Hodgkin thinks their well-argued critique of postmodern and feminist abuse of science is initimately connected with ``current anti-Islamic trends in the West''.
Hodgkin makes a convincing case for the importance of Islamic mathematics. But the emphasis ought to be on tracing the mathematical thinking, not on questions of cultural superiority.
Later chapters focus even more on priority disputes instead of mathematical content. Concerning calculus, students would benefit from learning how mathematical rigour and, say, the notion of limit evolved. By contrast, the question, ``Did the Indians have a version of Calculus in the Middle Ages?'' is - in the words of Gian-Carlo Rota - one of palaeontology, not history.