Alexander Masters
IGUM AND IGIBUM
Unknown Quantity: A Real and Imaginary History of Algebra
By John Derbyshire (Atlantic Books 382pp £22)

John Derbyshire's Unknown Quantity is everything a popular mathematics book should be: gentle, chatty, anecdotal and full of mind-aching equations. It is a history of algebra - the study of number systems, things such as quadratic equations, and of everything that is the bane of schoolchildren's lives.

Babylonian tax inspectors liked quadratic equations, which are useful for finding areas of things. The more you could determine about the land a man owned - not just its total area but all its little shapes - the more efficiently you could dun him for Sodom-and-Gomorrah era VAT. Derbyshire includes a blissful problem in quadratics (written in cuneiform, but to be chanted in hoodoo) found on a clay tablet from 1800 BC, the time of Hammurabi:

The igibum exceeded the igum by 7
What are the igum and the igibum?
12 is the igibum, 5 the igum.

Algebra is filled with Lewis Carroll-ishness and poetry.

What's the volume of that minaret? How can we make another even fatter/taller/more thrusting one, without using more stones? For this you need a cubic equation. The Persian poet Omar Khayyam, author of the Rubaiyat, began the first serious investigation of examples of these, but what mathematicians wanted was a general solution to all cubic questions. Then, instead of having to agonise through every particular case, which might take days of effort, they could simply slot the basic information into a standard formula and out would pop the answer. Thousands of years later, a swinish Italian called Cardano published the solution. A gambler and diviner (one of his insights was that 'a woman with a wart upon her left cheek, a little to the left of the dimple, will eventually be poisoned by her husband'), he also worked out the general formula for the quartic - useful for those odd people who want to investigate volumes in the fourth dimension.

There's a sense of remorselessness about the process after this: next will come equations called the quintic, sextic, heptic, etc etc. Why should it ever stop? Why should anyone but the very peculiar - or those who live in the fifth, sixth and seventh dimensions - bother? Because equations with these sorts of terms crop up all the time in physics, and in finance and computing and making car engines and aeroplanes and, no doubt, the assembly lines needed to produce this copy of Literary Review and the knife and fork with which you ate your breakfast. Mathematicians pursue algebraic results for the fun of the chase, in the same way that commuters pursue sudoku puzzles; the rest of us depend on them for the survival of our comfortable way of life.

The quintic is the Snark of mathematics. It was hunted across Europe until it was finally killed off by a 26-year-old Norwegian called Niels Abel, who starved to death shortly after. But the quintic was a Boojum, you see. Unlike the equations that had gone before, Abel proved that it has no general solution. The reason why this is the case, as the French student Everiste Galois showed, is infinitely more important than the failure of the result. A day after he wrote down the explanation for this boojumish fact, he was shot dead, in a duel, aged twenty-one.

Historians of mathematics are always complaining that mathematicians are a dry and uninteresting lot; but it's not so. Algebra has been powered by numerous astonishing characters and absurd situations. The beautiful virgin Hypatia, the first known woman mathematician (there are only three, in this book), was pulled from her chariot by an enraged mob and had her flesh scraped from her bones with oyster shells. (Women and algebra have not always been kind to each other. George Boole, who developed an algebraic system for logic, died because his wife threw buckets of icy water over him when he was in bed with a chill.) Alexandre Grothendieck is the most recent curious fellow: in his prime he knocked down policemen and won the top mathematics prize, the Fields Medal. Now he lives in total retirement in the Pyrenees, pondering how to survive on dandelion soup.

The best parts of Unknown Quantity are not the anecdotes, but the sums. In asides throughout the text, and in special chapters he titles 'Math Primers', John Derbyshire cleverly chooses one or two simple mathematical examples to illustrate horridly difficult ideas and, using metaphor and fine writing, investigates them closely. Vector spaces, algebraic geometry, imaginary numbers, group theory, field theory, matrices ... These sections are worth reading twice just for the pleasure of being able to say covenish phrases like 'the ideal of a polynomial ring' without feeling you're turning into a dotty about to bother people on buses.

Now that I've finally finished this distracting review, I'm going to re-puff the pillows on my bed, and study the formulae in the book properly.