It is a relief to learn at the outset that, despite the title, the editor of this book does not believe an equation can be intrinsically beautiful. Rather, it is like a poem whose power, in the finest examples, lies in its perfect expression of layer upon layer of meanings, some unrecognised even by the author. A humbler analogy would be my boyhood volume of Aesop in which I would read, at the end of a fable, something such as "Moral: Look before you leap". So we might picture Einstein, after a feat of mental gymnastics, writing "Moral: E = mc 2 ". It is a pithy guide to future action and a reminder of where it comes from and why it is worth knowing; and, as with a great poem, or Aesop's fables for that matter, every generation discovers something new and significant.
Some discoverers of great equations did not simply overlook their implications, they positively resisted them. Planck may have introduced the quantum but he was never happy with Einstein's exposition of the need for light, and ultimately all vibrations, to exhibit the characteristics of both waves and particles. In due course Schrödinger extended Einstein's concept in his wave mechanics and was dismayed to find he had released the genie of indeterminacy - and Einstein shared his dismay. By contrast, Dirac trusted his almost miraculously beautiful electron theory to the point of turning its most unwelcome weakness - particles of negative energy - into its great success, prediction of the positron's existence. This was perhaps the start of his growing devotion to mathematical beauty as a guide to truth, so that he could not accept renormalisation theory as anything but makeshift, despite its astonishing success in accounting for tiny discrepancies between his equation and experimental measurements. Such failures of appreciation do nothing to derogate great achievements but serve to reveal how discoveries can close some doors only to open others.
This sort of physics, and other chapters dealing with field theory and relativity, make up more than half the book. I regret that less rarefied branches of physics do not appear. I would have preferred, for instance, that the ideas expressed in such a simple, elegant and fertile form as Bragg's equation, the foundation of X-ray crystallography, should replace the only equation in the book that I think unworthy of inclusion. Drake's equation, which supposedly provides a basis for estimating the number of alien intelligences accessible by radio signals, is neither beautiful nor useful. It contains symbols for quantities that cannot be guessed even approximately for lack of information and, worse still, seems to take for granted that the evolution of life elsewhere (if anywhere) has followed the same path as that on earth, with mankind's special senses and interests the crowning glory. Alfred Adler, reviewing a conference on the subject, called it "a travesty of all that is taken seriously by men and women who love and value science and intellect". Oliver Morton, who wrote this chapter, at least deserves credit for including this quotation.
Naturally, physics is the science most likely to generate important equations, but the book casts its net wider. There are chapters by John Maynard Smith on the use of game theory to contrast different patterns of evolution, Robert May on the pervasive occurrence and still rather immature study of determinate chaos, Aisling Irwin on the chemistry of ozone destruction in the upper atmosphere and Igor Aleksander on Shannon's information theory. Though little in the book is easy reading, the majority of the authors have succeeded in making most of their accounts intelligible to the careful reader. Roger Penrose has sound advice on how to cope with the difficult bits - read, note and pass on boldly - which he puts into practice in expounding the really hard case of general relativity and non-Euclidean geometry without losing contact with the observational evidence. I do not understand it all, but I have never read a clearer exposition of the principles.
Even if one cannot hope to achieve mastery of a topic, especially when it is an imaginative triumph, by reading a single chapter, the authors manage to show that the subtle concepts are not wholly beyond the grasp of ordinary folk. Occasionally, one is confounded by the omission of essential detail; thus Arthur Miller quotes Schrödinger's equation in the compact form Ĥ Ψ = E Ψ without explaining that it is shorthand for a complicated, frequently repellent family of differential equations that yield only rarely to analytical solution. The beauty lies in the thought that produced the equation, not in its calligraphy. The hardest of all the chapters is Christine Sutton's on the Yang-Mills equation. It introduces sophisticated mathematics to express concepts altogether foreign to everyday experience; she is to be congratulated on a heroic attempt to make the crooked straight.
Most of the authors are scientists or historians and philosophers of science, who have taken pains to write clearly and economically; two are journalists who are in command of their material. They all, as well as the editor Graham Farmelo, deserve our thanks for a compilation so well organised and presented that a reader, dizzily perched on intellectual heights, must agree with Steven Weinberg's final burst of enthusiasm for the great equations, "which may outlast even the beautiful cathedrals of earlier years".
Sir Brian Pippard is emeritus professor of physics, University of Cambridge.
It Must Be Beautiful: Great Equations of Modern Science
Editor - Graham Farmelo
ISBN - 1 86207 479 8
Publisher - Granta Books
Price - £20.00
Pages - 283