BOLYAI, JNOS

born Dec. 15, 1802, Kolozsvr, Hung. [now Cluj, Rom.] died Jan. 27, 1860, Marosvsrhely, Hung. [now Trgu Mures, Rom.] Hungarian mathematician and one of the founders of non-Euclidean geometrygeometry that does not include Euclid's axiom that only one line can be drawn parallel to a given line through a point not on the given line. Although Bolyai knew nothing of mathematics at the age of 10, by the age of 13 he had mastered calculus and analytic mechanics under the tutelage of his father, the distinguished mathematician Farkas Bolyai. He also became an accomplished violinist at an early age and later was renowned as a superb swordsman. He studied at the Royal Engineering College in Vienna (181822) and served in the army engineering corps (182233). The elder Bolyai's fanatic preoccupation with proving Euclid's parallel axiom infected his son, and, despite his father's warnings, Jnos persisted in his own search for a solution. In 1820 he concluded that a proof was probably impossible and began developing a geometry that did not depend on Euclid's axiom. In 1823 he sent his father a draft of Appendix Scientiam Spatii Absolute Veram Exhibens (Appendix Explaining the Absolutely True Science of Space), a complete and consistent system of non-Euclidean geometry. Before his work was published, Bolyai found that he had largely been anticipated by Carl Gauss of Germany. This was a profound blow to Bolyai, even though Gauss had no claim to priority since he had never felt enough confidence in his findings to publish them. Bolyai allowed the Appendix to be published with his father's Tentamen Juventutem Studiosam in Elementa Matheseos Purae Introducendi (1832; An Attempt to Introduce Studious Youth to the Elements of Pure Mathematics), but the essay went unnoticed by other mathematicians. In 1848 he discovered that N.I. Lobachevsky had published an account of virtually the same geometry in 1829. Although Bolyai continued his mathematical studies, the importance of his work was never recognized in his lifetime. In addition to work on his non-Euclidean geometry, he developed a geometric concept of complex numbers as ordered pairs of real numbers.