Gilbert and Sullivan's Major-General may have known many, but the only cheerful fact about the square on the hypotenuse that is familiar to most of us is that it is equal to the sum of the squares on the other two sides. Jacob Bronowski described this discovery as "the most significant single theorem in the whole of mathematics". For more than two millennia the theorem has been attached to the name of Pythagoras, who may (or, more likely, may not) have been the first to provide rigorous proof, but the result was certainly known in Mesopotamia long before his time.

This imaginative account of the "lives and times" of the theorem over 4,000 years begins with Aldous Huxley's six-year-old Archimedes sketching a visual proof of the theorem, and moves fluently between the Old Babylonians, 3rd-century China, 7th-century India, the Soviet Union and the present day. We learn that Elisha Scott Loomis collected 367 distinct proofs of the theorem. US president James Garfield found an original proof (four years before his election and subsequent assassination), as did Emma A. Coolidge, a blind Bostonian girl, information about whom the authors have gathered by inspired detective work. When a Milwaukee schoolteacher in 1924 challenged his class to find a new proof, Alvin Knoer succeeded, and he has provided the only known account of the creative process involved. More recently, the eminent computer scientist Edsger Dijkstra formulated a result that he modestly regarded as four times as rich as the original.

We are led to approach the theorem from many angles. There are geometric proofs, algebraic ones and some of great ingenuity: one argument, presented as "Pythagoras made difficult", uses differential equations. My favourite proof involves turning a plane diagram into a torus: appropriately, the authors relate this to a type of window, whose French name *le vasistas* apparently comes from the words spoken by German soldiers on first seeing it. Geometric diagrams abound: remarkably, one of them is transformed into a saucy wartime cover for the magazine *La Vie Parisienne*. Throughout we have a succession of fascinating and wide-ranging titbits, both mathematical and literary. The magpie reader will find many gems in this volume.

There is a lot of actual mathematics here and it is generally presented clearly: readers will want to have pencil and paper to hand. We are given enough mathematical detail to satisfy most readers, but the level of mathematics required is not unduly demanding. Some of the technical material is consigned to optional end-of-chapter appendices. The authors' writing style is florid, and while I admired their elegance, I felt that sometimes a less mannered presentation would have helped me to focus better on the mathematics. Ellen Kaplan's charming illustrations break up the text and add to the appeal.

The title of the book is well chosen - this is not a history of the Theorem of Pythagoras. The discussion of its use by the Old Babylonians, for example, is highly speculative, of interest for its insights into the mathematics rather than as history. The book's afterword, which quotes the puzzled exclamation of Jane Austen's Catherine Morland "that history should be so dull, for a great deal of it must be invention", is important in indicating the spirit in which it should be read.

This book should be regarded as an extended mathematical *jeu d'esprit*, a witty ramble through entertaining territory offering many provocative insights, and anyone approaching it in that spirit will find a wealth of interesting material. Many mathematicians will enjoy it, but I suspect it will hold less appeal to historians.

## Hidden Harmonies: The Lives and Times of the Pythagorean Theorem

By Robert Kaplan and Ellen Kaplan. Bloomsbury, 304pp, £20.00. ISBN 9781596915220. Published 7 March 2011

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