Models of elegance, not fashion

Can there ever be a real breakthrough in the public understanding of mathematics? Conventional wisdom suggests not. After all, the standard advice to any aspiring author of a popular maths book is pretty bleak: do not mention maths in the title, do have lots of pictures of fractals, do go for the human-interest angle (for example, Evariste Galois getting shot), and for goodness sake do not include any equations if you can possibly avoid it.

So when a book called The Equations arrived on my desk, I was intrigued from the start. The author, a distinguished theoretical physicist at Amsterdam University, claims that banishing equations when popularising science is like asking someone to explain art without showing pictures. We are soon told, then, that in this book the equations themselves will be the focus of attention.

And they are not just any old equations, either. The book presents, briefly but uncompromisingly, many of the deepest equations of nature, encapsulating classical dynamics, electromagnetism, thermodynamics and kinetic theory, fluid dynamics, relativity, quantum mechanics, elementary particle physics and the theory of superstrings. And each of the fundamental equations has a page to itself; you just cannot miss it.

Let me say at once that I learnt a great deal from the author's broad-brush approach. I had not realised, for example, that the typical size of superstrings is none other than the Planck distance, a length scale concocted more than a hundred years ago from three of the fundamental constants of nature. And I detected few inaccuracies (a somewhat absurdly missing minus sign; a confused account of incompressible flow). The physics is very readable, with connections between different parts of the subject made well. And because the book is so short, the reader has a real chance to see "the big picture".

But what of the equations themselves? Most of the deepest equations of nature are differential equations, so any author seeking to present them must either assume some familiarity with elementary calculus or at least sketch some of the necessary mathematical background. Sander Bais opts for the second route by including a short mathematical "toolkit" early on, but this strikes me as rather less successful. For example, we read that when handling equations "what you do to the left-hand side, you've got to do to the right-hand side". True enough, but I seriously doubt whether anyone who benefits from this will make very much of Euler's number e when it appears, completely undefined and without explanation, just a few pages later, to say nothing of the full symbolism of vector calculus in, say, Maxwell's equations.

True, the book does not claim to be for the elusive "general reader", and I suspect it is best suited to someone who already has a reasonable background in calculus, though not necessarily physics. For the non-scientific reader, the accounts of the physics may well be readable enough, but I suspect that the equations will be so iconic as to be largely incomprehensible.

In the end, however, there is no doubt that The Equations is a bold and original book. And, in my view, imagination of this kind will be needed in the future if maths and science education is to avoid sliding gradually into lightweight stuff about what Newton ate for breakfast.

David Acheson is fellow in mathematics, Jesus College, Oxford.