VI.34 János Bolyai
b. Klausenburg, Transylvania, Hungary (now Cluj, Romania), 1802;
d. Marosvásárhely, Hungary (now Tirgu-Mures, Romania), 1860
Non-Euclidean geometry
János Bolyai’s father Farkas Bolyai taught him mathematics at home, using the first six books of EUCLID’S [VI.2] Elements and EULER’S [VI.19] Algebra. Between 1818 and 1823 János studied at the Royal Engineering Academy in Vienna, and then served as an engineer in the Austrian Army for ten years, before retiring on a pension as a semi-invalid. Probably inspired by his father’s attempts to prove the parallel postulate, a key assumption in Euclidean geometry, but very much against the advice of his father, János also attempted to prove it. But in 1820 he switched direction and attempted to show that there could be a geometry independent of the parallel postulate. By 1823 he believed he had succeeded, and after much subsequent discussion father and son agreed to publish the son’s ideas as a twenty-eight-page appendix to his father’s two-volume work on geometry in 1832.
In this appendix, Bolyai started from a new definition of parallels, according to which, given a line in a plane and a point not on the line, there are many lines through the point that do not meet the given line. Of these lines, there are two that are asymptotic to the given line (one in each direction), and Bolyai called these the parallels to the given line through the given point. He went on to derive many results that follow from this assumption in the geometry of two and three dimensions, and gave formulas for the new trigonometry of triangles. He showed that these formulas reduce to the familiar formulas of plane Euclidean geometry when the triangles are very small. He also found a surface in his three-dimensional geometry in which geometry is Euclidean. He concluded that there were logically two geometries and that it remained undecided which one corresponded to reality. He also showed that in his new geometry it was possible to construct a square equal in area to a given circle, thus accomplishing a feat that was widely (and, as was later shown, correctly) believed to be impossible in Euclidean geometry.
A copy of the book was sent to GAUSS [VI.26], who eventually replied on March 6, 1832, that he could not praise the work, for “to praise it, would be to praise myself,” going on to claim that the methods and results in the appendix agreed with his own work over the previous thirty-five years, although he was “very glad that it was just the son of my old friend, who takes the precedence of me in such a remarkable manner.” This endorsement of the validity of János’s ideas pleased the father but infuriated the son, and soured relations between father and son for several years. They did eventually resume an uncomfortable relationship, which persisted until Farkas’s death in 1856.
János Bolyai published virtually nothing else, and his discovery was not appreciated in his lifetime. Indeed, it is unclear that anyone but Gauss ever read it, but specific comments about it that Gauss left behind led mathematicians back to it, and it was translated into French by Hoüel in 1867 and into English in 1896 (reprinted in 1912 and 2004).
Further Reading
Gray, J. J. 2004. János Bolyai, Non-Euclidean Geometry and the Nature of Space. Cambridge, MA: Burndy Library, MIT Press.