The discus and domino effects

This is a splendid compilation of eye-catching illustrations of mathematical modelling. Those in the title are interspersed with others from various branches of mechanics (almost none of which is to be found in traditional textbooks), economics, biology and lots of different sports, and each has a useful bibliography. Robert Banks's theme of mathematical universality is exemplified by his discussion of athletics, which ranges from how to run a race optimally to how to assess performances in the Olympics most fairly.

The book's style will make it accessible, almost uniformly and instantaneously, to inquisitive sixth-formers or undergraduates who do not have much background except elementary calculus. Indeed, it will not teach them much more than elementary calculus. Rather, it will attract the attention of bright students who cannot imagine how mathematics could ever be useful. Whereas the level is comparable to that of many fireside books of mathematical puzzles, here the puzzles concern practical situations that are transparently crying out for quantitative analysis to provide the means to gain insight and useful predictions. For students, this will be much the most important attribute of the book.

The author's hectic pace enables him to cover applications broad enough to do justice to the modern spirit of applied mathematics. But this is achieved at the loss of quality control over the models he proposes. Newton's equations of mechanics are banged down with the same gay abandon as those of Malthusian growth or traffic flow. Worse, there is no space to show how successive refinement of a model often goes hand in hand with the development of exciting new theories, to the mutual benefit of science and mathematics. This interplay is at the heart of applied mathematics, and there are some golden opportunities that could have been used to illustrate it in this book. Stirling's formula for factorials, the Kutta-Joukowski condition for lift, delay equations and models for population dynamics all come out of the blue with no mention that they are the starting point for beautiful mathematical theories that far transcend the model from which they were spawned. And nowhere is there mention of the key role that computation plays in any modern "adventure in applied mathematics".

So this book must come with a warning label; the student reader may be in danger of never seeing the true power of mathematics as a tool of understanding so much of what goes on in this world. Let us hope that Banks is working on a follow-up containing as much mathematical imagination as this does applications imagination. The awareness created by books written in his style and with his vision may be a key ingredient in restoring the intellectual status of mathematics in the US and Europe to its previous level.

John Ockendon is lecturer, Centre for Industrial and Applied Mathematics, University of Oxford.