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Unanswerable: Euclid's pioneering theorems puzzled mathematicians for centuries

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.

Petra Tonder
July 6th, 2011
8:07 AM
Very interesting article on Euclid. But who painted the beautiful portrait? The tendancy to not say seems to be spreading; I have received Christmas cards depicting uncredited nativities. Frustrating!