Lobachevsky, Nikolai Ivanovich (1792-1856)

Russian mathematician who founded non-Euclidean geometry, concurrently with, but independently of, Karl Gauss in Germany and János Bolyai in Hungary. Lobachevsky published the first account of the subject in 1829, but his work went unrecognized until Georg Riemann's system was published. In Euclid's system, two parallel lines will remain equidistant from each other, whereas in Lobachevskian geometry, the two lines will approach zero in one direction and infinity in the other. In Euclidean geometry the sum of the angles of a triangle is always equal to the sum of two right angles; in Lobachevskian geometry, the sum of the angles is always less than the sum of two right angles. In Lobachevskian space, also, two geometric figures cannot have the same shape but different sizes. Lobachevsky was born at Nizhni-Novgorod and studied at the University of Kazan, Tatarstan. He taught there from 1814, becoming professor 1822 and rector of the university 1827-47. He also took on administrative work for the government. Lobachevsky developed non-Euclidean geometry between 1826 and 1856. He came to see that it was not contradictory to speak of a geometry in which all Euclid's postulates except the fifth held true. By including imaginary numbers, he made geometry more general, and Euclid's geometry took on the appearance of a special case of a wider system. The clearest statement of Lobachevsky's geometry was made in the book Geometrische Untersuchungen zur Theorie der Parallellinien, published in Berlin 1840. His last work was Pangéométrie 1855.