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.