Bolyai, János (1802-1860)

Hungarian mathematician, one of the founders of non-Euclidean geometry. He was the son of Wolfgang Bolyai. He was one of the first to see Euclidean geometry as only one case, and that others were possible. In a paper written in 1823 he described a geometry in which several lines can pass through the point P without intersecting the line L.

Bolyai was born in Koloszvár, Hungary (now Cluj-Napoca, Romania), and was taught mathematics by his father. In 1818 he entered the Royal College of Engineers in Vienna, and on graduation joined the army, but retired in 1833.

By about 1820, János Bolyai had become convinced that a proof of Euclid's postulate about parallel lines was impossible; he began instead to construct a geometry which did not depend upon Euclid's axiom. He developed his formula relating the angle of parallelism of two lines with a term characterizing the line. In his new theory Euclidean space was simply a limiting case of the new space, and János introduced his formula to express what later became known as the space constant.

János described his new geometry in a paper of 1823 called ‘The absolute true science of space’. Wolfgang sent it to German mathematician Karl Gauss, who replied that he had been thinking along the same lines for more than 25 years. János's paper was printed as an appendix to his father's Tentamen juventutem (1832-33). Its publication received little attention and he subsequently discovered that Nikolai Lobachevsky had published an account of a very similar geometry (also ignored) in 1829.