parallel postulate

The parallel postulate is the fifth and most controversial of Euclid's postulates set forth in the Greek geometer's great work, Elements. To later mathematicians, the parallel postulate always seemed less obvious than the other four and many attempts were made to derive it from them, but without success. In 1823, Janos Bolyai and Nikolai Lobachevsky independently realized that entirely self-consistent types of non-Euclidean geometry could be created in which the parallel postulate doesn't hold. Karl Gauss had made the same discovery earlier but kept the fact secret.