What were the most important equations of 20th-century science? Here are 11 essays, aimed at non-specialists, each centred on a different equation. The authors are distinguished in their various fields: six of them are scientists, with the others being science writers or historians of science.

Six of the equations, all of which appear chronologically, are from physics, more specifically “fundamental” physics. The first essay is by the book’s editor, Graham Farmelo of the Science Museum in London. Entitled “A revolution with no revolutionaries”, it tackles the Planck-Einstein equation E = hf, which connects frequency with energy. It also serves as an account of the early days of quantum theory, although there was perhaps a missed opportunity here to mention modern examples of the Planck distribution, such as the cosmic-microwave-background radiation.

The second essay – by science historian Peter Galison – discusses Einstein’s equation E = mc². Unfortunately, it does not contain as much detail about the special theory of relativity as one might expect. There is more information about nuclear fission, which is, after all, only mildly dependent on relativity.

The third chapter, by Roger Penrose, is on the general theory of relativity. One of the more detailed essays in the book, it contains several equations that the author tells us we may skip at a first reading.

Do not be put off by the odd title of the next chapter: “Erotica, aesthetics and Schrödinger’s wave equation”. Here historian Arthur I Miller continues the story of the development of quantum theory.

Physicist Frank Wilczek’s essay “A piece of magic” is about the Dirac equation, with the last part of the chapter taking us on to quantum electrodynamics and quantum chromodynamics (QCD) – the theories of the electromagnetic and the strong forces.

Christine Sutton concludes the physics element of the book with “Hidden symmetry”, which tackles the Yang-Mills equation, electroweak unification and (again) QCD. There is, however, nothing on condensed-matter physics: presumably it is too difficult to think of great equations in that field.

The seventh essay, by Igor Aleksander, is on Claude Shannon’s equations that answer the questions of what is information and what inhibits its transmission.

In the last four contributions, the character of the equations changes. There is an essay by journalist Oliver Morton on the Drake equation, which gives the expected number of sources of communication from extraterrestrial intelligences as a product of conditional probabilities. This is certainly not profound in the sense that the first seven equations are. As Farmelo comments, Drake’s formula “brought some coherence into a field potentially rife with woolliness”. I wonder, however, if a different equation would have represented astronomy better, such as the Hubble law of the recession of the galaxies.

Next there are two biological essays. One, by John Maynard Smith, concerns the application of game theory to animal behaviour. The other, by Robert May, president of the Royal Society, is about the “quadratic map” as a model of biological populations, and about the discovery of chaotic behaviour in this equation. A feature of each of these contributions is that the authors are largely describing their own work.

The final essay, by journalist Aisling Irwin, is about the discovery – and the chemistry – of the hole in the ozone layer. There is an “equation” here, but it is only used in the chemical sense to describe a reaction like O₃ –> O₂ + O.

I found all the essays good to read. This is a volume to dip into and to pick and choose from, rather than necessarily to read through consecutively. The various authors have approached their tasks in different ways: some readers will prefer one style, others another.

The book prompts intriguing questions. In particular, have any deserving equations been left out? Should, for example, Heisenberg’s commutation relation between position and momentum have been included? Perhaps Farmelo will one day give us Great Equations of 19th-century Science, or perhaps Great Principles in Science?

Another question is whether great equations are really so important in science? Some affirmative answers can be found in the book. Frank Wilczek, for example, includes the following quote by Heinrich Hertz on Maxwell’s equations: “One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.”

Steven Weinberg, meanwhile, has the following to say in a wise afterword: “When an equation is as successful as Dirac’s, it is never simply a mistake. It may not be valid for the reason supposed by its author, it may break down in new contexts, and it may not even mean what its author thought it meant. We must continually be open to reinterpretations of these equations. But the great equations of modern physics are a permanent part of scientific knowledge, which may outlast even the beautiful cathedrals of earlier ages.”

I am tempted to end my review with this inspiring passage, but it is only fair to point out Weinberg’s reference to physics, so that his words may not apply to all of the equations in this book.

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