Non-Euclidean Geometry

For nearly two thousand years mathematicians tried to prove Euclid's fifth postulate, concerning parallel lines, from his other four, and repeatedly produced proofs that turned out to smuggle in an assumption equivalent to the postulate itself. Girolamo Saccheri in 1697 tried assuming the postulate false and searching for a contradiction, unknowingly deriving theorems of non-Euclidean geometry along the way but wrongly concluding he had found one. Carl Friedrich Gauss became convinced between 1792 and 1817 that the postulate was genuinely independent of the others and developed a geometry admitting more than one parallel line through a point, but never published his conclusions, fearing the controversy they would provoke. Janos Bolyai discovered the same territory independently and published a twenty four page account in 1825, writing that he had created a strange new world out of nothing; Nikolai Lobachevsky published the first full account in 1829, and Eugenio Beltrami built the first consistent model of it in 1868, finally proving the parallel postulate genuinely independent of Euclid's other axioms. Bernhard Riemann's 1854 lecture generalized the whole concept of geometry still further, treating it as a space with enough structure to measure length, and shaped a wealth of geometries that followed.