Hyperbolic geometry—an alternative geometry to Euclidean geometry, in which Euclid's first four axioms are true but the fifth axiom, the parallel postulate, is not. Hyperbolic geometry was discovered almost accidentally in the 19th century. Its discoverers were not looking for an alternative geometry, but instead were trying to prove that such a geometry could not exist. Ever since Euclid first set out his five basic assumptions (axioms) of geometry around 300 BC, a debate had raged over whether the fifth axiom is truly a basic assumption, or whether it can in fact be proven using the other four axioms; this would make it not an assumption, but a theorem. Many mathematicians tried to prove the fifth axiom from the other four, but sooner or later each such "proof" was found to have some hole in it.

In the middle of the 19th century the mathematicians...