Lobachevsky threw out Euclid's parallel rule and found a whole new geometry
For two thousand years, mathematicians tried to prove Euclid's parallel postulate from his other axioms. Nikolai Lobachevsky, a professor in provincial Kazan, took the opposite route in 1826: he assumed the rule was false and built a consistent geometry where triangles hold less than 180 degrees. The St. Petersburg Academy turned his paper down.
Euclid's fifth postulate, in John Playfair's version, says that through a point off a given line exactly one line can be drawn that never meets it. Lobachevsky allowed more than one such line. The resulting hyperbolic geometry has an angle of parallelism that changes with the point's distance from the line, and every triangle's angles add up to under 180 degrees. He first presented it to Kazan's physics and mathematics department on 23 February 1826, then printed it as On the Origin of Geometry in 1829 and 1830.
The Hungarian János Bolyai reached the same idea independently, and Gauss had privately worked along similar lines but published nothing. Claims that Gauss steered Lobachevsky are wrong; the two never corresponded beforehand, though Gauss later admired his work. The field is often called Bolyai–Lobachevskian geometry, and it later fed into differential geometry. William Kingdon Clifford dubbed him the Copernicus of Geometry.
Born in 1792 in or near Nizhny Novgorod, Lobachevsky lost his father, a land-survey clerk, at seven and moved with his mother to Kazan. He won a scholarship to Kazan University, then only a few years old, where he studied under Johann Bartels, a former teacher of Gauss. In 1811 a spiteful supervisor accused the student of godlessness. He nonetheless rose to full professor at 30 and rector in 1827. With his wife Varvara he had perhaps eighteen children, of whom about seven reached adulthood.
He also found a way to approximate roots of algebraic equations, known in Russia as the Lobachevsky method, and defined a function as a pairing between two sets of real numbers. Dismissed in 1846, nearly blind and unable to walk by the early 1850s, he died poor in 1856. A lunar crater and an asteroid carry his name, and Tom Lehrer's 1953 comic song borrowed it purely because it scanned well.
Source: Nikolai Lobachevsky
