Bolyai (bō´lyoi), family of Hungarian mathematicians. The father, Farkas, or Wolfgang, Bolyai, 1775–1856, b. Bolya, Transylvania, was educated in Nagyszeben from 1781 to 1796 and studied in Germany during the next three years at Jena and Göttingen, where he began a lifelong friendship with Carl F. Gauss. From 1804 to 1853 he was professor of mathematics at Maros Vásárhely. His primary interest was in the Euclidean parallel postulate. His principal work, the Tentamen (1832–33), inspired by his mathematically gifted son János, is an attempt at a rigorous and systematic foundation of geometry (Vol. I) and of arithmetic, algebra, and analysis (Vol. II).
János, or Johann, Bolyai, 1802–60, b. Koloszvár, Transylvania, was educated by his father in Maros Vásárhely and from 1818 to 1822 in Vienna, where he received military training at the imperial engineering academy. In 1820 he began to work in a direction that ultimately led him to a non-Euclidean geometry. In 1823, after vain attempts to prove the Euclidean parallel postulate, he developed his system by assuming that a geometry could be constructed without the parallel postulate. His theory of absolute space was published as an appendix to his father's Tentamen and constituted the sole work published in his lifetime.