The title of this delightful little book is taken from a letter from Isaac Newton to his contemporary Robert Hooke written in 1675: "If I have seen further (than others) it is by standing on the shoulders of giants." In essence, this slim volume is a series of thought-provoking essays demonstrating the apparently fortuitous relevance of seemingly obscure pieces of mathematics for the progress of physics.

In support of this thesis, Malcolm Lines quotes Nobel prizewinning physicist Eugene Wigner, one of the pioneers of the application of group theory to quantum physics, who was awed by the almost supernatural association between the newly discovered physical world and previously unapplied abstract mathematics. Wigner wrote: "We are in a position similar to that of a man who was provided with a bunch of keys and who, having opened several doors in succession, always hit on the right key with the first or second trial. He became sceptical concerning the uniqueness of the coordination between keys and doors."

Lines wisely eschews further pondering on any mystical connection between mathematics and the real world and is content to let his 11 "case studies" speak for themselves. After reading them, I am not sure that all of these stories entirely support the claim of the cover blurb that they "illustrate the manner in which mathematics has come to the aid of physics and even pointed to directions for future research". Some of them are at least as good at demonstrating how research in mathematics was stimulated by the success of physicists in using hitherto dubious mathematical procedures or of finding curious physical phenomena.

The book begins with a discussion of the difference between physicists and mathematicians. The great physicist Einstein supposedly said to the great mathematician Poincare: "I considered taking up mathematics but decided against it because of its lack of connection with the real world and the impossibility of telling what is important." Poincare is said to have retorted: "In my youth I was seriously tempted to become a physicist, but I decided against it because in physics it is impossible to tell what is true." Lines illustrates the difference between a physicist's version of proof and that of a mathematician with a "Feynman story" that I had not heard - Richard Feynman's "proof" of Fermat's last theorem. Feynman conducted some "experimental" tests of Fermat's conjecture and concluded that physicists should be just as happy to accept the validity of his proof as they are to accept any other experiment in physics. For mathematicians, Feynman's "proof" probably raises as many questions as it answers. In fact, as Lines tells us, a rigorous proof of Fermat's last theorem now appears to have been found by Princeton mathematician Andrew Wiles.

The topics covered by the book span an enormous range of mathematics and physics, together with a wide assortment of mathematicians and scientists. The mathematics ranges from the origins of number theory and differential calculus to group theory and topology. The cast of mathematicians includes classic figures such as the tragic Evariste Galois, whose contribution to posterity was written down the night before he died in a duel at the age of 20, as well as modern greats such as Roger Penrose, famous not only for his contributions to singularity theory and general relativity but also for his curious "Penrose tiles". The scientific applications range from the structure of glasses and polymers, through quasicrystals and fractons, to quantum chaos and supersymmetry. Physicists from Newton to Einstein and Schrodinger make their appearance along with more recent contributors such as meteorologist Edward Lorenz and physicist Mitchell Feigenbaum, with their contributions to the modern theory of chaos. Instead of adopting a "gee-whiz" approach, Lines has attempted to explain as simply as possible the basic mathematical features of each problem and, in the main, succeeds admirably (though I do have a few minor academic quibbles). This is not, however, a popular science book of the type that contains only one equation: I suspect some of the material will be quite challenging for some of today's science undergraduates. For those willing to make the effort, Lines succeeds in illuminating many interesting and exciting corners of mathematics and physics.

Tony Hey is head of the department of electronics and computer science, University of Southampton.

## On the Shoulders of Giants

Author - Malcolm E. Lines

ISBN - 0 7503 0104 X and 0103 1

Publisher - Institute of Physics Publishing

Price - £49.50 and £15.00

Pages - 288

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