Spinning ancient mathematical yarns

The Infinite in the Finite
March 15, 1996

It is not easy to write an accessible book about mathematics. You can water down the mathematical content to the point at which the book ceases to be challenging, you can focus on the people and slide over the technical content of their ideas, you can link the mathematics to applications, or to philosophy, and focus on those I or you can follow Alistair Macintosh Wilson and approach mathematics through its history.

He tells us that when he was a boy he wanted a mathematics book that read like a story, and this of course is where history has a considerable advantage. History is by and large a collection of stories, and while it is certainly possible to ruin material as promising as Henry VIII by insisting that your readers memorise the dates of his wives, you need to have serious talent as a thoroughgoing bore to carry it off. The youthful Wilson sought the ideal mathematical storybook for years, but in the end he decided that the only solution was to write one himself.

The result is both readable and unusual. Its historical period is mainly that of the ancient Greeks and their intellectual successors, which at first sight puts it into the category of books that make mathematics accessible by concentrating on material that is at least 1,000 years old. I have always felt that this is a bit of a cheat, for fairly obvious reasons, but there is no need to rehearse them here because they do not apply. Although the framework is ancient history, many of the issues are modern.

The Infinite in the Finite does have faults. One is an over-affectionate attention to detail - extensive proofs of geometric theorems and lengthy algebraic calculations. These will not appeal to those who believe every formula halves a book's sales. I am not suggesting we should always follow this market-oriented style of thinking ("Professor Einstein, if you would only omit the c-squared and leave just E=m I am sure your theories would be far more acceptable to the public"), but do we really need things like "we now find that BF.CF/BF.FI= (CH+OF)/OF=(BC+CG)/CG =BG/CG"? To be fair, at this point Wilson is explaining Heron's formula for the area of a triangle in terms of its sides, which is pretty ambitious, but even so . . .

The book also has many virtues. The technique of introducing an interesting piece of mathematics by means of a historical tale works rather well. Thus the idea that numbers have hidden patterns emerges - not surprisingly - from a story about the Pythagoreans; but it is also linked more fancifully to Zeus's alleged problems in creating the world. In consequence Zeus becomes entangled in an explanation of the symmetry groups of regular polyhedra. This did not really work for me, but I suspect that readers who have not encountered groups before will find this passage entirely natural.

Occasionally the approach becomes confusing when fact and fiction blur or history is re-ordered to suit exposition. For example a chapter about proof and why it is necessary begins by showing how the eye can easily be deceived, using the celebrated dissection of an 8x8 square into what looks like a 5x13 rectangle. (8x8=64 but 5x13=65: where did the extra square come from?) Wilson explains that the same trick can be played with any three consecutive Fibonacci numbers, going on to say that "the Pythagoreans knew this wouldn't do at all". Anyone not paying attention - which included me - would get the impression that the Pythagoreans knew of this particular dissection. Then the date for Fibonacci (around 1200) would perhaps sink in I Of course, by "this" Wilson means "the need to treat visual evidence with caution". It would have been helpful to say so.

Sometimes, too, I felt that we were being presented with pop history rather than the real thing - Eric Temple Bell rather than Otto Neugebauer. Thus we are told that Fibonacci is the nickname of Leonardo of Pisa (true though with reservations) and that it means "son of a good man" (an alternative version is that Leonardo's dad was named Bonaccio). All this is moot, however, because there is no evidence that the nickname was ever used in Leonardo's day. It is his nickname, but it used not to be: it seems to have been invented by Guillaume Libri in the 19th century. Like Newton's apple and Alfred's cakes, the Fibonacci factoid is too good a story to let a few facts get in its way. On the other hand, Wilson deserves due credit for noting that Hippasus of Metapontum "must have been the kind of person you have to drown twice" - once, allegedly, for revealing how to make a dodecahedron and once, even more allegedly, for proving that C2 is irrational.

I raise a few concerns about historical reliability not to put you off the book, but to make it clear just what you are getting, which is a fine yarn spun from mathematical threads. The yarn begins with the Egyptians and Babylonians, spends most of its time wrapped up with the ancient Greeks, and ends by getting tangled up in the Indians and the Arabs.

I particularly enjoyed those last two chapters. Everything is accessible with a smattering of high school trigonometry and algebra, and a nodding acquaintance with Euclidean geometry - not so much its content as its notational conventions and discursive style. Despite this, I'm inclined to characterise this book as one that will help people who are attracted to mathematics get to grips with history, rather than one that will help people who are attracted to history get to grips with mathematics - but whichever it is, it is a brave and largely successful attempt to weave both subjects together in an entertaining and informative manner.

Ian Stewart is professor of mathematics, University of Warwick. In 1995 he was awarded the Royal Society's Michael Faraday Medal.

The Infinite in the Finite

Author - Alistair Macintosh Wilson
ISBN - 0 19 853950 9
Publisher - Oxford University Press
Price - £25.00
Pages - 524

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