VI.31 Nicolai Ivanovich Lobachevskii
b. Nizhni Novgorod (formerly Gorki), Russia, 1792;
d. Kazan, Russia, 1856
Non-Euclidean geometry
Lobachevskii came from a poor background, but his mother was able to have him enrolled at the local Gymnasium (or high school) on a scholarship in 1800. In 1805 the Gymnasium was made the kernel of the new University of Kazan, and in 1807 Lobachevskii began to study there. The university had just appointed Martin Bartels as Professor of Mathematics, and Bartels not only trained Lobachevskii well, but protected him from trouble with the authorities when Lobachevskii was suspected of atheism. Eventually, Lobachevskii graduated not with the ordinary degree but with a Master’s qualification, and his career as a professional mathematician began.
In 1826, after a reform of the university, Lobachevskii gave a public lecture: “On the principles of geometry, with a rigorous demonstration of the theory of parallels.” The manuscript of this talk is now lost, but it probably marked the start of Lobachevskii’s awareness of a non-Euclidean geometry. Lobachevskii was soon elected Rector of the University of Kazan, a post he occupied with distinction for thirty years, helping to protect the university from a cholera epidemic in 1830, to rebuild it after a fire in 1841, and generally to expand its library and other facilities.
In the 1830s he also wrote his major works, on a geometry different in only one respect from Euclidean geometry. He called it imaginary geometry and it is known today as non-Euclidean geometry. In the new geometry, given a line in a plane and a point not on the line there are two lines through the point that are asymptotic to the given line (one in each direction); these two lines separate the lines through the point which meet the given line from those that do not. Lobachevskii called these the parallels to the given line through the given point. Starting from this definition, he gave formulas for the new trigonometry of triangles, and showed that these formulas reduce to the familiar formulas of plane Euclidean trigonometry when the triangles are very small. He extended his results to describe a geometry of three dimensions, thus making it clear that his new geometry could be a geometry of space, and attempted, inconclusively, to measure the parallax of stars in order to determine whether his imaginary geometry gave a more accurate account of space than Euclidean geometry.
He published these conclusions in lengthy papers in Russian in the Journal of Kazan University, but they drew only a relentlessly hostile review from Ostrogradskii, a much better known mathematician in Saint Petersburg. He published in French in a German journal in 1837, in German in a booklet of 1840, and again in French in 1855, but to little avail. GAUSS [VI.26] appreciated the booklet of 1840 and in 1842 had Lobachevskii made a corresponding member of the Göttingen Academy of Sciences, but this was to be the only acclaim Lobachevskii received in his lifetime.
Lobachevskii’s final years were marked by terrible financial and mental decline. Such was the chaos of his household that Lobachevskii’s biographers have been unable to establish the number of children born into it, but it may well have been fifteen or even eighteen.
Further Reading
Gray, J. J. 1989. Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, second edn. Oxford: Oxford University Press.
Lobachetschefskij, N. I. 1899. Zwei geometrische Abhandlungen, translated by F. Engel. Leipzig: Teubner.
Rosenfeld, B. A. 1987. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. New York: Springer.