Heilbron is a great writer -- eloquent and clear, without excess. Just look at the title of his book: it gets across its aims and contents in 6 words. If, then, the book is long, that is because he has a lot to tell us and his story is different from many others: "Most of the so-called geometry books now used in schools are not agents of culture, wisdom, or even education, however, but opportunities to exercise students in the use of pocket calculators" (Preface). But readers are not let off lightly, just listening to this great story. There is something for everyone and lots for anyone who wants to join in, and he pushes for participation: there are scores of illuminating exercises, many of them based on historical texts or procedures, or taken from rich sources like the Ladies' Diary, and most of these examples are followed by comments and help addressed to the reader.
He also offers us many things he calls APS, an acronym I think that he would like to be as well-known as QED; these are Ad pleniorem scientiam, "'for fuller understanding', as the scholastic philosophers used to say, when multiplying examples of the concepts so far introduced" (p. 59). Perhaps I should give an example, which is not entirely straightforward to do since most things in the book come with a figure or illustration. But the figure for the very first and simplest APS is just some lines radiating from a point, with some of the angles numbered. Then
APS 2.1.1. Figure 2.1.21 contains several pairs of supplementary, and vertical angles. Can you name them all? (p. 59).
('Vertical angle' is not a description that is familiar to me, so I had to check his meaning: "Vertical angles are the two angles at the opposite sides of the vertex of two straight lines" -- I understand that this is US usage.)
Next, an exercise. Here the choice of possible meaningful examples is even more restricted, since the associated figures are generally more difficult to describe, but the following one illustrates a lot of typical features:
Exercise 5.5.6. (p. 253) When travelling in Northern India, the Arab philosopher Al-Biruni invoked a new way to measure the circumference of the earth. 'It does not involve walking in deserts', he said, in recommending it. Instead, you must climb a high mountain. [Berggren, Episodes (1986), 142-3.] Can you guess his method?
and this is followed by 17 lines of discussion:
A (Fig 5.5.5.) is the mountain top, from which ... [and here follows a discussion of the method and al-Biruni's data]. Columbus knew some of the Arabs' results, which he preferred to Christian ones, since by assuming incorrectly that they referred to Italian miles, he derived from them the additional evidence that the world was smaller than anyone else imagined. Cf. Jordan, Zeits. Verm., 18 (1889), 100-9, and Hamner, ibid., 38 (1909), 721-3.].
This last example brings out several features of the book. First, its strong historical point of view; in fact, the book starts with a substantial historical chapter, and many of the straight mathematical passages are enhanced by a historical comment. As part of this history, the book is full of references to sources, some of them recondite. Here the Berggren reference is to his Episodes in the Mathematics of Medieval Islam, an excellent book that I recommend readers who might be interested in the topic to consult, and the date gives some idea of what it might be like and how easily available it will be, but Zeits. Verm. (1889) is a different kettle of fish! The articles are in German and the journal is Zeitschrift zur Vermessungswesen; perhaps the curious reader might like to check if it is in their library. Heilbron's Bibliography is 8 pages of fine type, well more than 200 items, and many things in it might take a long time to find, even for someone with access to the best of libraries; I, a book and bibliography lover, know of perhaps 60 of them, and have looked at about 40 of these. However let me assure the reader that this uncharacteristically excessive aspect of the book can be completely ignored; here, for example, almost everybody will carry away only the information that al-Burini and Columbus are people associated with the story, and that is sufficient.
Next, the problem itself: This one is not at all difficult, but his collection range from the very straightforward to fiendishly difficult, so something for everyone, but the reader is not left struggling; Heilbron will describe as much of the solution as he thinks is needed, though this means that a strong-minded reader should not immediately read this part. Another illustration of this is in the last 3 pages of the book, devoted to what he calls the Tantalus problem, an innocuous-looking question about the angles in a triangle with three lines drawn inside. This apparently (first?) appeared in the Washington Post in 1995: "I contacted about 40 geniuses around the nation and they all gave me insights about the problem without being able to solve it" wrote its author. We are led through a simpler version, then the full thing, and left with the conclusion which can refer both to the problem and the book itself: "When Alison read this proof, she observed that it demonstrated one thing very well: it does not take 40 geniuses, or even one, to do problems in plane geometry. All that is required is patience, resourcefulness, and the methods taught in this book" (pp. 292-5). Strong-minded readers who get there by themselves will be able to feel a glow of real satisfaction!
This last quotation also refers the the most important content of the book: a rattling good introduction to plane geometry based on Euclidean methods, interspersed with brief summaries of what has been covered and how this centres on material in the first six books of the Elements. There are also other lists of important words and other information. None of this is oppressive, but a convenient and well-arranged way of bringing together what has just been done.
So do I like everything? Well I am a historian of Greek mathematics, among other things, and while I like and enjoy the history that he uses profusely throughout the book, I do not really like his style of history. The opening historical chapter, 49 pages of which perhaps 15 are illustrations, gives a summary of geometry across the ages by summarising or quoting from commentators of all kinds on the subject. This material is put together into a well-constructed story, with references to the sources, some of them recondite like the example above, but with very little indication of how representative, reliable, or influential these sources might be. In other words, it belongs to the style of history as an assembly of material from the past. Moreover his past geometry looks very much like our present, and that is indeed why he includes it in his book. I would have preferred less quotes and references to generally inaccesible material of unknown reliability, and more discussion. But that is my choice, and I know that some find it heavy going. Certainly Heilbron's procedure is much better adapted to his aims -- for which, once again, see his title.
David Fowler is a Reader in Mathematics at the University of Warwick, UK. His principal interest is the history of mathematics, especially Greek mathematics, on which he has published a book The Mathematics of Plato's Academy: A New Reconstruction, of which a new augmented edition has just appeared (Clarendon Press, Oxford, 1999); it contains a 21-page Bibliography containing more than 450 items!