Nikolai Lobachevsky

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Nikolai Lobachevsky
Portrait by Lev Kryukov (c. 1843)
Born December 1, 1792[1][2]
Makaryev, Makaryevsky uezd, Nizhny Novgorod Governorate,[3][4] Russian Empire
(now Makaryevo, Nizhny Novgorod Oblast, Russia)
Died February 24, 1856 (aged 63)
Kazan, Kazan Governorate, Russian Empire
(now Tatarstan, Russia)
Nationality Russian
Fields Geometry
Alma mater Kazan University
Academic advisors J. C. M. Bartels[5][6]
Notable students Nikolai Brashman[5]
Known for Lobachevskian geometry

Nikolai Ivanovich Lobachevsky (Russian: Никола́й Ива́нович Лобаче́вский; IPA: [nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj]; 1 December [O.S. 20 November] 1792 – 24 February [O.S. 12 February] 1856) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry.

William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.[7][8]


Nikolai Lobachevsky was born either in or near the city of Nizhny Novgorod in the Russian Empire (now in Nizhny Novgorod Oblast, Russia) in 1792 to parents of Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya.[9][10][11] He was one of three children. His father, a clerk in a land surveying office, died when he was seven, and his mother moved to Kazan. Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807 and then received a scholarship to Kazan University,[9][10] which was founded just three years earlier in 1804.

At Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels, a former teacher and friend of German mathematician Carl Friedrich Gauss.[9] Lobachevsky received a master's degree in physics and mathematics in 1811. In 1814, he became a lecturer at Kazan University, in 1816 he was promoted to associate professor, and in 1822, at the age of 30, he became a full professor,[9][10] teaching mathematics, physics, and astronomy.[10] He served in many administrative positions and became the rector of Kazan University[9] in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children (eighteen according to his son's memoirs, while only seven apparently survived into adulthood). He was dismissed from the university in 1846, ostensibly due to his deteriorating health: by the early 1850s, he was nearly blind and unable to walk. He died in poverty in 1856.

He was an atheist.[12][13]


Lobachevsky's main achievement is the development (independently from János Bolyai) of a non-Euclidean geometry,[10] 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. Although three people—Gauss, Lobachevsky and Bolyai—can be credited with discovery of hyperbolic geometry, Gauss never published his ideas, and Lobachevsky was the first to present his views to the world mathematical community.[14]

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)[15] and Pangeometry (1855).[16][17]

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).


E.T. Bell wrote about Lobachevsky's influence on the following development of mathematics in his 1937 book Men 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 until Lobachevsky discarded it. The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky 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.[18]


In popular culture[edit]

Annual celebration of Lobachevsky's birthday by participants of Volga's student Mathematical Olympiad
  • Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his Songs by Tom Lehrer album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as "Plagiarize!", as long as one is always careful to "call it, please, research". According to Lehrer, the song is "not intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons".[19]
  • In Poul Anderson's 1969 fantasy novella "Operation Changeling" – which was later expanded into the fix-up novel Operation Chaos (1971) – a group of sorcerers navigate a non-Euclidean universe with the assistance of the ghosts of Lobachevsky and Bolyai. The story also contains the line, "Nikolai Ivanovich Lobachevsky is his name," possibly a nod to the Tom Lehrer song.
  • Roger Zelazny's science fiction novel Doorways in the Sand contains a poem dedicated to Lobachevsky.


  • 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. G. B. Halsted (tr.). 1891. Reprinted in Bonola: NonEuclidean Geometry 1912. Dover reprint 1955.
Also in: Seth Braver Lobachevski illuminated , MAA 2011.
  • Pangeometry. D.E. Smith: Source Book of Mathematics. McGraw Hill. Dover reprint.
  • New foundations of geometry with a complete theory of parallels. G. B. Halsted (tr.) [20]
  • Nikolai I. Lobachevsky, Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society. 2010.

See also[edit]


  1. ^ This is the date given by V. F. Kagan's 1957 book N. Lobachevsky and His Contribution to Science (first published in Russian in 1943), p. 26, and A. A. Andronov's 1956 article "Где и когда родился Н.И.Лобачевский" ("Where and when was Lobachevsky born?") (the latter gives 1 December [O.S. 20 November] 1792).
  2. ^ Older sources in Russian—e.g., A. F. Popov, "Воспоминания о службе и трудах проф. Казанского университета Н. И. Лобачевского" ("Memoirs of the Service and Work of N. I. Lobachevsky"), 1857—give 1793 rather than 1972, while the Dictionary of Scientific Biography (1970) gives December 2, 1792. Further information on Lobachevsky's birthdate can be found in: Athanase Papadopoulos (ed.), Nikolai I. Lobachevsky. Pangeometry, European Mathematical Society. 2010, pp. 206–7.
  3. ^ See "К 150-летию со дня смерти Н.И.Лобачевского" ("On the 150th anniversary of the death of N. Lobachevsky") by G. M. Polotovsky, PDF page 3: "Н.И.Лобачевский родился в Макарьевском уезде Нижегородской губернии в 1793 году" (quoting A. F. Popov (1857)); page 4: "[В.Ф.Каган (1943)] местом рождения называет Макарьев".
  4. ^ Other sources in Russian—e.g., A. A. Andronov (1956)—give the city of Nizhny Novgorod rather than the Governorate as his birthplace; see also Lobachevsky's biography at the website of the Lobachevsky Nizhny Novgorod State University Museum and Andrey Kalinin's article "Чье имя носит университет" ("After whose name the University has been named").
  5. ^ a b Nikolai Lobachevsky at the Mathematics Genealogy Project
  6. ^ Athanase Papadopoulos (ed.), Nikolai I. Lobachevsky. Pangeometry, European Mathematical Society. 2010, p. 208.
  7. ^ Bell, E. T. (1986). Men of Mathematics. Touchstone Books. p. 294. ISBN 978-0-671-62818-5.  Author attributes this quote to another mathematician, William Kingdon Clifford.
  8. ^ This is a quote from G. B. Halsted's translator's preface to his 1914 translation of The Theory of Parallels: "What Vesalius was to Galen, what Copernicus was to Ptolemy that was Lobachevsky to Euclid." — W. K. Clifford
  9. ^ a b c d e Victor J. Katz. A history of mathematics: Introduction. Addison-Wesley. 2009. p. 842.
  10. ^ a b c d e Stephen Hawking. God Created the Integers: The Mathematical Breakthroughs that Changed History. Running Press. 2007. pp. 697–703.
  11. ^ Ivan Maksimovich Lobachevsky (Jan Łobaczewski in Polish) came from a Polish noble family of Jastrzębiec and Łada coats of arms, and was classified as a Pole in Russian official documents; Jan Ciechanowicz. Mikołaj Łobaczewski - twórca pangeometrii. Rocznik Wschodni. Issue 7–9. 2002. p. 163.
  12. ^ Bardi, Jason (2008). The Fifth Postulate: How Unraveling a Two Thousand Year Old Mystery Unraveled the Universe. John Wiley & Sons. p. 186. ISBN 978-0-470-46736-7. His stubbornness, reported atheism, and genius supported his rise as a champion of the proletariat. To the Soviets, Lobachevsky represented not just the greatness of the common man, emerging from a humble background as he did, he also was a revolutionary of sorts. 
  13. ^ "The History of Science". Soviet Science. Taylor & Francis. p. 329. Though Lobachevsky appears to have invented non-Euclidean geometry without the help of the Almighty, he built a church on the instructions of the University council. It is said that he was an atheist. 
  14. ^ O'Connor, John J.; Robertson, Edmund F., "Nikolai Lobachevsky", MacTutor History of Mathematics archive, University of St Andrews .
  15. ^ The 1914 English translation by George Bruce Halsted is available at "". The University of Michigan Historical Mathematics Collection. Retrieved 2012-12-17. 
  16. ^ The 1902 German translation by Heinrich Liebmann is available at "". The University of Michigan Historical Mathematics Collection. Retrieved 2012-12-17. 
  17. ^ Lobachevsky dictated two versions of that work, a first one in Russian, and a second one in French (Papadopoulos 2010, p. v).
  18. ^ Bell, E. T. (1986). Men of Mathematics. Touchstone Books. p. 336. ISBN 978-0-671-62818-5. 
  19. ^ Liner notes, "The Tom Lehrer Collection", Shout! Factory, 2010
  20. ^ "New foundations of geometry with a complete theory of parallels". Yale university library digital collections. Retrieved 23 August 2015. 

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