very rare first edition of lobachevsky's seminal work on non-euclidean geometry.
Kazan, in modern Tatarstan, was home to an important university, founded in 1804, whose students were to include Tolstoy and Lenin. Lobachevsky (1792-1856) was one of its earliest students who went on to be appointed professor, rector and librarian. He was responsible for building the old library which is now named after him.
Lobachevsky was a polymath but his reputation (outside Kazan) rests on his invention of non-Euclidean geometry. "Lobatchewsky challenged the assumption that Euclid's parallel postulate or, what is equivalent, the hypothesis of the right angle, is necessary to a consistent geometry, and he backed his challenge by producing a system of geometry based on the hypothesis of the acute angle in which there is not one parallel through a fixed point given a straight line but two. Neither of Lobatchewsky's parallels meets the line to which both are parallel, nor does any straight line drawn through the fixed point and lying within the angle formed by the two parallels. This apparently bizarre situation is 'realized' by the geodesics on a pseudo-sphere... Lobatchewsky abolished the necessary 'truth' of Euclidean geometry" (E.T. Bell, Men of Mathematics, New York 1937, pp.305-306).
He first announced this in 1826 at a meeting of the Physical-Mathematical Society of Kazan, which was then published in five articles in various issues of Kazanskii Vestnik for 1829 and 1830; it was his first published work. His research was not widely known until the appearance of an article in German in 1840, "Geometrische Untersuchungen zur Theorie der Parallellinien" in the Journal für die reine und angewandte Mathematik.
"The presence of specifically Lobachevskian geometry is felt in modern physics in the isomorphism of the group of motions of Lobachevskian space and the Lorentz group. This isomorphism opens the possibility of applying Lobachevskian geometry to the solution of a number of problems of relativist quantum physics. Within the framework of the general theory of relativity, the problem of the geometry of the real world, to which Lobachevsky had devoted so much attention, was solved; the geometry of the real world is that of variable curvature, which is on the average much closer to Lobachevsky's than to Euclid's" (DSB VIII, p.434).
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