Wonderbras, inflation and π

That π is a suitable thread on which to hang a popular mathematical exposition is obvious: its history acts automatically as an organising principle for such a book. Its evolution begins with the ancient geometers (to whom π was an elusive and approximate fraction just more than 3), runs through the age of analysis and calculus (to whose practitioners it was the limit of an infinite series) and on to the modern computer age (in which it gives insight into normal numbers and is used for checking software).

The evolution of e has already been treated in just such a way (e: The Story of a Number, by Eli Maor) But π has advantages over e. First, its history runs parallel with most of mathematics. Second, one encounters it first in geometry rather than in analysis. Most of us can relate to "circumference = 2πr", and as such π is accessible to most general readers.

The downside to this accessibility is that π attracts to itself all manner of Sharawaggi and trivia, much as a sock in the country attracts burs, as well as those cranks to whom a little learning proves a dangerous thing and who consider that because they can grasp the tail they own the donkey.

Π may not be as prolific in its mathematical implications as e, which at once branches into areas as superficially varied as banking, radioactivity, differential equations, probability and self-similarity; and yet the way that π is handled at a given time in history acts also as a rough index of mathematical sophistication. If people round you tell you that pi is 3, your time machine has delivered you well into the past or - what could be just as bad - into the hands of over-literal interpreters of 1 Kings 7:23.

In general, The Joy of pi follows such a natural exposition, but it somehow fails to capitalise on its strengths. It does its bookish duty, though not as enthusiastically as its title would suggest. Much of what you would expect and is available in other texts is trotted out here: π's early history among the Egyptians, Greeks and Arabs; the birth of analysis and calculus; the present-day race to compute π to a degree of precision absurd from a non-mathematical point of view. There are chapters on π the symbol, on circle-squaring and mnemonics for π, as well as the customary ritualistic nod in the direction of Ramanujan.

What is lacking among this Frankenstein collection of π-facts is a sense of proportion: a sense of what is trivial and what is deep. Take for example Viete, who generalised the Archimedean method of finding upper and lower bounds on the value of π by pinning a circle between escribed and inscribed polygons, and so expressed π as an infinite product. His equation stands at the junction of the old concept of number as constant and the modern one of number as variable. Yet there is little in the tone of Blatner's writing to indicate what a milestone this was.

Most of this material will already be familiar to π-freaks. The chapter on the Chudnovsky brothers is, however, interesting and new. David and Gregory are number theorists originally from Ukraine who built a supercomputer from scratch in a Manhattan apartment to crunch their way through the digits of π. They have now set up the Institute for Mathematics and Advanced Supercomputing in Brooklyn Polytechnic, New York.

One senses the author is more at ease with this subject (he is an expert on computing) than with some of the other mathematical areas and the writing here is more natural. Even so it would have been good to get a deeper insight into why π - and the circle-squarers - are irrational.

The text is cluttered by a proliferation of circular and rectangular sidebars fitted unaesthetically on the page and of a size bearing little relation to the amount of text inside them. The wide range of font sizes and line spacings this necessitates and the varying shades of dull green tints fatigue and irritate. What a difference the inclusion of a graph of the precision to which π was known at different periods of history would have made. And instead of cluttering successive pages with the successive digits of π, might it not have been better to number the pages clearly?

I suspect the book will prove attractive to confirmed π-freaks, but will not encourage many newcomers to the club.

The Universe and the Teacup has no pseud evolutions of π cluttering its pages. In fact, apart from the chapter on innumeracy - for without a few numbers how can one demonstrate our natural an-arithmeticity? - hardly a number appears in the book (except, of course, for the page numbering), though K. C. Cole has such an easy command of her subject that the reader knows where she is without a plan, just by the point that has been reached in the discourse.

The chapters are just the right length, mostly of the order of ten pages long (right for a story well told, and fitting amply into one's attention span, leaving plenty of time for questions afterwards), and they cover topics as immediate as voting, probability, fairness and symmetry.

The book does what Cole sets out to do, namely to show that mathematics is "not about numbers so much as a way of thinking". Hear! Hear! For how can the intelligent layperson get even a tentative handle on maths when she is under a misapprehension as to what the subject is about? Many still have the impression it is to do with calculation.

Cole has the insight to see that our difficulties with numeracy, probability, risk assessment and so on are generally due not to innate stupidity, but rather to a hard-wired difference between the way our brains think and the way that reality works.

As well as being in line with modern thought on the brain's intrinsic irrationality, admitting this "World Blindness" at once frees us to be interested. Too many books on mathematics, by stressing achievements and success, alienate rather than involve.

Why is this book so effortlessly engaging and charming? Cole is interested not just in her subject but also in the people she wants to share it with. She does not patronise, she shares. She allows nothing to break the rapport established with the reader, who is interested in how to divide cakes into envy-free portions, in overpopulation, in what is meant (if anything) by "beyond reasonable doubt" and in whether a surge in Wonderbra sales in 1994 caused inflation. These concrete discussions glide effortlessly into the abstractions of symmetry and quantum mechanics in the company of Einstein, Noether and Weyl. The experience is like learning to swim with someone you trust.

Cole's book is to mathematical literacy what Monty Roberts is to horse-training. May her book prove a passport for many into that land from which bad teaching and fear have alienated them but which is rightfully theirs.

Chris Maslanka presents Puzzle Panel on BBC Radio 4.