Nikolai Lobachevsky

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Nikolai Lobachevsky : biography

December 1, 1792 – February 24, 1856

Career

Lobachevsky’s main achievement is the development (independently from János Bolyai) of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid’s fifth postulate from other axioms. Euclid’s fifth is a rule in Euclidean geometry which states (in John Playfair’s reformulation) that for any given line and point not on the line, there is one parallel line through the point not intersecting the line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true. This idea was first reported on February 23 (Feb. 11, O.S.), 1826 to the session of the department of physics and mathematics, and this research was printed in the UMA (Вестник Казанского университета) in 1829–1830. Lobachevsky wrote a paper about it called A concise outline of the foundations of geometry that was published by the Kazan Messenger but was rejected when it was submitted to the St. Petersburg Academy of Sciences for publication.

The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry. Lobachevsky replaced Playfair’s axiom with the statement that for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part. He developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a hyperbolic triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of differential geometry which has many applications. Hyperbolic geometry is frequently referred to as "Lobachevskian geometry" or "Bolyai-Lobachevskian geometry".

Some mathematicians and historians have wrongfully claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss, which is untrue – Gauss himself appreciated Lobachevsky’s published works very highly, but they never had personal correspondence between them prior to the publication. In fact out of the three people that can be credited with discovery of hyperbolic geometry – Gauss, Lobachevsky and Bolyai, Lobachevsky rightfully deserves having his name attached to it, since Gauss never published his ideas and out of the latter two Lobachevsky was the first who duly presented his views to the world mathematical community.

Lobachevsky’s magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry (1835–1838). He also wrote Geometrical Investigations on the Theory of Parallels (1840) and Pangeometry (1855).

Another of Lobachevsky’s achievements was developing a method for the approximation of the roots of algebraic equations. This method is now known as the Dandelin–Gräffe method, named after two other mathematicians who discovered it independently. In Russia, it is called the Lobachevsky method. Lobachevsky gave the definition of a function as a correspondence between two sets of real numbers (Peter Gustav Lejeune Dirichlet gave the same definition independently soon after Lobachevsky).

Works

  • Kagan V.F.(ed.): N.I.Lobachevsky – Complete Collected Works, Vols I-IV (Russian), Moscow-Leningrad (GITTL) 1946-51
    • Vol.I Geometrical investigations on the theory of parallel lines; On the foundations of geometry (1829–30).
    • Vol.II New foundations of geometry with a complete theory of parallels. (1835–38)
    • Vol.III Imaginary geometry (1835); Application of imaginary geometry to certain integrals (1836); Pangeometry (1856).
    • Vol.IV Works on other subjects.
English translations
  • Geometrical investigations on the theory of parallel lines. Halstead G.N.(tr) 1891. Reprinted in Bonola: NonEuclidean Geometry 1912, Dover reprint 1955.
  • Pangeometry. D.E. Smith: Source Book of Mathematics. McGraw Hill. Dover reprint
  • Nikolai I. Lobachevsky, Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society, 2010.

Impact

E.T.Bell in his book Men of Mathematics wrote about Lobachevsky’s influence on the following development of mathematics: The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other ‘axioms’ or accepted ‘truths’, for example the ‘law’ of causality which, for centuries, have seemed as necessary to straight thinking as Euclid’s postulate appeared till Lobatchewsky discarded it. The full impact of the Lobatchewskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobatchewsky the Copernicus of Geometry, for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.