in ‘Uchenya zapiski’. Kazan: at the University, 1835.

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. ‘The researches that culminated in the discovery of non-Euclidean geometry arose from unsuccessful attempts to ‘prove’ the axiom of parallels in Euclidean geometry. … In Lobatchewsky’s geometry an infinity of parallels can be drawn through a given point that never intersect a given straight line … His fundamental paper was read to his colleagues in Kazan in 1826 but he did not publish the results until 1829-30 when a series of five papers appeared [in Russian] in the Kazan University Courier’ (PMM). Lobachevsky ‘built up the new geometry analytically, proceeding from its inherent trigonometrical formulas and considering the derivation of these formulas from spherical trigonometry to guarantee its internal consistency. […] If imaginary numbers are the most general numbers for which the laws of arithmetic of real numbers prove justifiable, then imaginary geometry is the most general geometrical system. It was Lobachevsky’s merit to refute the uniqueness of Euclid’s geometry, and to consider it as a special case of a more general system’ (DSB). Lobachevsky produced a continuation to this article, which was published in the following issue of the Kazan University Proceedings: Primenenie voobrazhaemoi geometrii k nekotorym integralam [Application of Imaginary Geometry to Certain Integrals], Kazan, University Press, 1836. See PMM 293, see Norman I, 1379. Octavo (222 x 131mm). With a folding engraved plate (faint spotting on the plate). Early 20th-century half leather, flat spine gilt, green strait grain cloth sides (corners rubbed). Provenance : some pencil underlining.