More than two thousand years ago, Euclid of Alexandria wrote the most successful textbook of all time. Starting with a few simple assumptions (often called axioms), he proved one result after another — for example that the angles of a triangle add up to 180?. Scholars wondered whether the last of his five axioms — which referred to parallel lines, and sounded more like a theorem than an assumption — wasn't simply a necessary consequence of the other four. Many tried to prove this, and some false proofs were published. I shall give a very convincing one before outlining the history of geometry up to the nineteenth century. That's when three people independently discovered a perfectly consistent geometry in which the Euclid's fifth axiom is not true, and where the angles of a triangle no longer add up to 180?. This new work inspired others and led eventually to the sort of geometry Einstein needed for his theory of gravity.