Lobačevskian Geometry

Abstract

Centuries of attempts to prove the parallel postulate led to the discovery of non-Euclidean geometry made at the beginning of the 19th century. This discovery was first published by the great Russian mathematician and professor at Kazan University Nikolaĭ Ivanovič Lobačevskiĭ in the paper On the principles of geometry (O načalah geometrii. Kazan, 1829) [333. vol. 1, pp. 185–261]. The first public announcement about this discovery was made during a meeting of the division of the physicomathematical sciences of Kazan University and took the form of a lecture entitled A brief exposition of the principles of geometry including a rigorous proof of the theorem on parallels (Exposition succincte des principles de la Géométrie avec une démonstration rigoureuse du théoremè des parallèles). Lobačevskiĭ notes that he drew on this lecture for the first part of the memoir “On the principles of geometry.” In the beginning of this part he writes: Who would not agree that a Mathematical discipline must not start out with concepts as vague as those with which we, in imitation of Euclid, begin Geometry, and that nowhere in Mathematics should one tolerate the kind of insufficiency of rigor that one was forced to allow in the theory of parallel lines. … The initial concepts with which any discipline begins must be clear and reduced to the smallest possible number. It is only then that they can provide a firm and adequate foundation for the discipline. Such concepts must be learned by the senses — the inborn ones must not be trusted [333, vol. 1, pp. 185–186]. KeywordsEuclidean GeometryCross RatioFractional Linear TransformationSpherical TriangleLimit CircleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.