Lie-in for eureka moments

Why are so many people scared stiff of mathematics? It is well known that Stephen Hawking was advised that each equation in A Brief History of Time would halve the book's sales. And while mathematicians can now be shown getting excited about equations on film and TV, this is only on the strict understanding that no one attempts to explain what the equations mean.

For some mathematicians, including myself, there is no future in any of this. Sooner or later, someone is going to make a major breakthrough, by taking away much of the fear of mathematics and bringing some of the main ideas and pleasures of the subject to a wide public. The past ten years or so have seen several original attempts at this, and Robert and Ellen Kaplan's book The Art of the Infinite is one of the more recent.

The authors have a lot of experience in this area. Nine years ago, they set up the Math Circle at Harvard University - a series of classes for the promotion of pure mathematics, particularly for the young. And I have seen them give a joint lecture in the UK to a much older audience, gradually luring everyone into the mathematics from a combination of good humour, patience and old-fashioned charm.

Some of this comes over in the book, which could be read by anyone who knows a little school algebra and geometry. It starts by reviewing elementary ideas, proceeds through a variety of topics, including prime numbers, infinite series, geometry and complex numbers, and ends with a chapter on Georg Cantor's ideas of the infinite. Each topic is handled quite well, and often in some detail. In the chapter on triangle geometry, for instance, we are led step by step through a number of classic proofs about line intersections.

And there are some nice touches. Apparently, Carl Gauss, aged 18, once made a mathematical discovery before he had even got out of bed, and the authors view this as "proof, if one is needed, that adolescents should be allowed to get up late during vacations".

In a roundabout way, the authors also made me aware of one extraordinary result involving the infinite that I had not come across before. Imagine taking a number x, slightly greater than 1 (the authors take x to be the square root of 2), and then consider x to the power x to the power x to the power x, and so on, for ever. At first sight, perhaps, this cannot possibly lead to a finite result, but it all depends on where you "put the brackets". Sadly, the authors get the brackets wrong, which makes their own account of this particular topic very confused, but I have learnt subsequently that Euler showed that the outcome is indeed finite, provided x is less than 1.44467 and provided the brackets are inserted in the right way.

The Art of the Infinite has a welcoming, chatty style and hand-drawn illustrations. On top of this, the authors sprinkle the mathematics with literary quotes and allusions, and other loose links with the world of the arts. While some will dismiss much of that as purple prose, others will, I imagine, welcome it as a refreshingly different approach to a demanding subject.

David Acheson is fellow of Jesus College, Oxford.