The Genius of Geometry

David Brooks is political flavour of the month this side of the Atlantic, so in thrall is Westminster to Barack Obama

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In an episode of his TV panel show, QI, Stephen Fry asked: “Who wrote the greatest textbook of all time?” After the obligatory witty responses, someone mentioned the Greeks, and Fry said, yes indeed, it was Euclid, whose Elements, written around 300 BC, has served as a foundation for learning geometry ever since.

But who was Euclid? No one knows, though we usually think of him working in that great city Alexandria, whose great library became the world’s foremost centre of scholarship.

Before Euclid, results in geometry had been a bit hit and miss. Some, such as Pythagoras’s theorem, were known with certainty, but others were more hazy, and some, like an Egyptian formula for the area of a four-sided figure, were downright wrong. Euclid started from scratch. He wrote the world’s first mathematics text that laid out its assumptions before proving anything. From a few basic definitions and assumptions he then built a cathedral of geometric reasoning that still stands.

Euclid proved theorems-for example that the angles of a triangle add up to 180˚ — all from five basic assumptions about geometry in the plane. The first was that between any two points there’s a straight line, and the next three were also uncontroversial. But the fifth caused concern because it sounded more like a theorem than an assumption. It said that if two lines were not both perpendicular to a third line, then they’d meet if extended far enough — in other words they weren’t parallel. Surely this could be proved from the other four assumptions?

As centuries ticked by, Romans took over from Greeks, Alexandria’s library was destroyed, and scholarship moved elsewhere. In Baghdad in the early 9th century, Euclid was translated into Arabic, and then in 1120 an Englishman named Adelard of Bath went to Spain, got hold of an Arabic translation and translated that into Latin.

For many centuries, scholars writing in Arabic, and later in Latin, tried to show that Euclid’s fifth axiom was a necessary consequence of the other four, and therefore superfluous. To give just one late example, a brilliant Italian mathematician named Giovanni Girolamo Saccheri published a book proving the fifth axiom unnecessary. It appeared in 1733, the same year he died in his mid sixties. He was wrong, and Euclid was right all along, though it wasn’t until a century later that anyone could prove this.

Attempts continued until, quite suddenly, three people independently discovered a “non-Euclidean” plane where Euclid’s first four axioms held true but the fifth one failed. In this new geometry the angles of a triangle added up to less than 180˚— the larger the triangle, the smaller the sum. Spectacular stuff, but the scholarly community wasn’t ready for it, and of the three people who made the discovery, one kept quiet. This was Carl Friedrich Gauss (1777-1855), one of the top three mathematicians of all time, a man on a par with Archimedes and Newton. Gauss decided to leave his discovery to be published posthumously, but the other two were young men, a Hungarian, Janos Bolyai (1802-1860), and a Russian, Nikolai Lobachevsky (1792-1856), who pursued publication.

Neither achieved the fame they deserved but their work led to new ideas in geometry, eventually creating the background for Einstein’s general theory of relativity, where the geometry of the universe is influenced by gravitation. 

Given Euclid’s extraordinary mathematical legacy, his useful axioms and complete proofs for a whole string of wonderful theorems, it is frustrating that no one knows who he was.

The date and place of his birth are unknown, as are the date and circumstances of his death. The internet hosts pictures of a bearded, older man, but they’re all made centuries, if not millennia, after the fact. There is no contemporary portrait, sculpture, correspondence or commentary. Nothing. And that’s not simply because he lived so long ago, because there are plenty of earlier Greek writers about whom we know far more: Aeschylus, Sophocles and Plato, to name a few. All of them lived to be 70 or 80-Sophocles to 90 — and continued to produce brilliant work well beyond middle age. Could it be that Euclid did his work while a young man, then disappeared? That is certainly true of another extraordinary mathematician-this time from the 20th century — Alexander Grothendieck.

Grothendieck was born in Berlin in 1928, spent his career in France and, as far as I know, now lives in solitude in the Pyrenees. He wrote several foundational treatises on a branch of mathematics called algebraic geometry. It was seminal work, undertaken when he was a professor at the French Institut des Hautes Etudes Scientifiques, but his radical political and pacifist views made him uncomfortable working in what he regarded as une cage dorée, and in 1970, still in his early forties, he quit.

In early 2010 he wrote to a former student saying that any work published in his absence was done without permission, it should not be copied, and libraries possessing it should destroy it. Is this a cry for purity in a messy world, influenced by his early life as an adoptee, and subsequently a stateless refugee?

Euclid was certainly known in the ancient world as the author of the Elements, but did he perhaps abandon the gilded cage of that great library at Alexandria, leaving no trace except for his mathematics? We know that Apollonius of Perga (c.262-190 BC) refers to Euclid’s work, and that Pappus of Alexandria (active in the 4th century AD) claimed Apollonius had studied under Euclid’s pupils in Alexandria. But that’s about all.

It may seem odd that anyone who achieves greatness can simply retire to solitude, but mathematicians inhabit an abstruse world, and can leave almost as abruptly as they enter. I’ll leave the last words to the great Isaac Newton, who discovered the universal force of gravity, invented calculus, and explained Kepler’s observations about planetary orbits thus:

“I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

What Euclid thought of his own work we have no idea, but at a distance of more than 2,000 years the jury is unanimous: Elements is the greatest textbook of all time.