Pangeometrie, oder die auf einer allgemeinen und strengen Theorie der Parallelen gegründeten Hauptsätze der Geometrie.

Berlin: Georg Reimer, 1858.

First German edition of Lobachevskii’s last publication on non-Euclidean geometry (first published in Kazan 1856).

The work was dictated by Lobachevskii, who was by then blind, in French, the previous year for the commemoration proceedings planned for February 1855, but publication was delayed by the Tsar over a disagreement over the date of the University’s founding. In the meantime, the Russian version was prepared and published, more or less simultaneously with the French. This German edition is little known: it is not mentioned in Friedrich Engel’s ‘Zwei geometrische Abhandlungen’ (1898) and Heinrich Liebmann claims in his own German translation (1902) that it is the first.

With the title Pangeometrie Lobachevskii emphasizes the universality of his ‘imaginary geometry’ and gives his most concise formulation of a geometry free of the parallel postulate. In this work Lobachevskii applies differential and integral calculus to non-Euclidean geometry and develops important refinements of his earlier works. “Numerous deductions, notably that of the fundamental equation of hyperbolic geometry, are given in other forms: until his last days, despite his precarious health, Lobachevskii sought to perfect the geometry he had constructed” (Kagan). The work ends with Lobachevskii discussing the geometry of nature and the necessity of experimentally determining if space is in fact non-Euclidean.

“In his early lectures on geometry, Lobachevskii himself attempted to prove the fifth postulate; his own geometry is derived from his later insight that a geometry in which all of Euclid’s axioms except the fifth postulate hold true is not in itself contradictory. He called such a system imaginary geometry, proceeding from an analogy with imaginary numbers. 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 Lobachevskii’s merit to refute the uniqueness of Euclid’s geometry and to consider it a special case of a more general system” (B.A. Rosenfeld in DSB).

8vo (228 x 139 mm), in: 'Archiv für wissenschaftliche Kunde von Russland', Volume 17, no. 3, pp 397-456, the complete issue (pp [2:title] 341-498) offered here in the original green printed wrappers, uncut and unopened a very fine copy. Rare as the single issue in wrappers.

Item #2889

Price: $2,800.00