Jump to content

Nikolai Lobachevsky

From Wikipedia, the free encyclopedia
(Redirected from Lobachevsky)
Nikolai Lobachevsky
Николай Лобачевский
Portrait by Lev Kryukov [ru], c. 1839
Born1 December 1792[a][b]
Makaryev,[c] Makaryevsky Uyezd, Nizhny Novgorod Governorate,[d][7][e] Russian Empire (now Makaryevo, Nizhny Novgorod Oblast, Russia)
Died24 February 1856 (1856-02-25) (aged 63)
Kazan, Kazan Governorate, Russian Empire (now Tatarstan, Russia)
EducationKazan University (MSc, 1811)
Known forLobachevskian geometry
Scientific career
Academic advisorsJ. C. M. Bartels[1][2]
Notable studentsNikolai Brashman[1]

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, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula.

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



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 Russian and Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya.[11][12][h] He was one of three children. When he was seven, his father, a clerk in a land-surveying office, died, and Nikolai moved with his mother to Kazan. Nikolai Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807, and then received a scholarship to Kazan University,[11][12] which had been 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 the German mathematician Carl Friedrich Gauss (1777-1855).[11] Lobachevsky received a Master of Science in physics and mathematics in 1811. In 1814 he became a lecturer at Kazan University, in 1816 he was promoted to associate professor. In 1822, at the age of 30, he became a full professor,[11][12] teaching mathematics, physics, and astronomy.[12] He served in many administrative positions and became the rector of Kazan University[11] in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children (eighteen according to his son's memoirs, though 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 and was buried in Arskoe Cemetery, Kazan.

In 1811, in his student days, Lobachevsky was accused by a vengeful supervisor of atheism (Russian: признаки безбожия, lit.'signs of godlessness').[14][15][16][17]



Lobachevsky's main achievement is the development (independently from János Bolyai) of a non-Euclidean geometry,[12] 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 only one line through the point not intersecting the given 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 periodical 'Kazan University Course Notes' as On the Origin of Geometry (О началах геометрии) between 1829 and 1830. In 1829 Lobachevsky wrote a paper about his ideas 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 wrongly claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss, which is untrue. Gauss himself appreciated Lobachevsky's published works 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.[18]

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)[19] and Pangeometry (1855).[20][i]

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:[22]

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.

Russian 1 rouble coin commemorating the 200th anniversary of Lobachevsky's birth, 1992.


Stamp of 1956 marking the centenary of Lobachevsky's death

A fictional mathematician named "Lobachevsky" is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his 1953 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".[23]

In Poul Anderson's 1969 fantasy novella "Operation Changeling" – which was later expanded into the novel Operation Chaos (1971) – a group of sorcerers navigate a non-Euclidean universe with the assistance of the ghosts of Lobachevsky and Bolyai.

Roger Zelazny's science fiction novel Doorways in the Sand contains a poem dedicated to Lobachevsky.

In the sitcom 3rd Rock from the Sun, "Dick and the Single Girl" (season 2 episode 24) original aired in the US on May 11 1997, Sonja Umdahl (Christine Baranski)[24] a forgotten colleague transferring to teach at another university gives as the reason behind her departure that Columbia is the only holder of Nikolai Lobachevsky's manuscripts.[25]


  • Kagan V. F. (ed.): N. I. Lobachevsky – Complete Collected Works, Vol. I–IV (Russian), Moscow–Leningrad (GITTL), (1946–51).
    • Vol. I: Geometrical Researches on the Theory of Parallels (1840); On the Origin of Geometry (1829–30).
    • Vol. II: New Principles of Geometry with 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
Also in: Seth Braver Lobachevski illuminated, MAA 2011.

See also



  1. ^ This is the date given by Kagan (1957)[3] and Andronov (1956)[4] (the latter gives 1 December [O.S. 20 November] 1792).
  2. ^ Older sources in Russian—e.g., Popov (1857)[5]—give 1793 rather than 1792, while the Dictionary of Scientific Biography (1970) gives December 2, 1792. Further information on Lobachevsky's birthdate can be found in Papadopoulos (2010)[6] However, note that page 207 incorrectly converts 1792-11-20 to 1792-12-02 instead of 1792-12-01
  3. ^ spelled 'Makaryev' not 'Makaryevo' at that time; nominative form in Russian: Макарьев
  4. ^ Not to be confused with Makaryev in Makaryevsky uezd, Kostroma Governorate
  5. ^ Other sources in Russian—e.g., A. A. Andronov (1956)—give the city of Nizhny Novgorod rather than the Governorate as his birthplace[8][9]
  6. ^ Bell (1986) attributes this quote to another mathematician, William Kingdon Clifford.[10]
  7. ^ 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
  8. ^ Ivan Maksimovich Lobachevsky (Jan Łobaczewski in Polish) came from a Polish noble family of the Jastrzębiec and Łada coats-of-arms, and was classified as a Pole in Russian official documents[13]
  9. ^ Lobachevsky dictated two versions of that work, a first one in Russian, and a second one in French.[21]


  1. ^ a b Nikolai Lobachevsky at the Mathematics Genealogy Project
  2. ^ Athanase Papadopoulos (ed.), Nikolai I. Lobachevsky. Pangeometry, European Mathematical Society. 2010, p. 208.
  3. ^ V. F. Kagan, N. Lobachevsky and His Contribution to Science, 1957 (first published in Russian in 1943), p. 26
  4. ^ A. A. Andronov, "Где и когда родился Н.И.Лобачевский" ("Where and when was Lobachevsky born?"), 1956
  5. ^ 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
  6. ^ Athanase Papadopoulos (ed.), Nikolai I. Lobachevsky. Pangeometry, European Mathematical Society. 2010, pp. 206–7.
  7. ^ "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".
  8. ^ Lobachevsky's biography at the website of the Lobachevsky Nizhny Novgorod State University Museum
  9. ^ Andrey Kalinin's article "Чье имя носит университет" ("After whose name the University has been named")
  10. ^ Bell, E. T. (1986). Men of Mathematics. Touchstone Books. p. 294. ISBN 978-0-671-62818-5.
  11. ^ a b c d e Victor J. Katz. A history of mathematics: Introduction. Addison-Wesley. 2009. p. 842.
  12. ^ a b c d e Stephen Hawking. God Created the Integers: The Mathematical Breakthroughs that Changed History. Running Press. 2007. pp. 697–703.
  13. ^ Jan Ciechanowicz. Mikołaj Łobaczewski - twórca pangeometrii. Rocznik Wschodni. Issue 7–9. 2002. p. 163.
  14. ^ Петров, Юрий Петрович (2012) [2005]. История и философия науки. Математика, вычислительная техника, информатика. Учебноее пособие. Saint Petersburg: БХВ-Петербург. p. 62. ISBN 9785941576890. Retrieved 18 August 2022. В 'шнуровой книге' университета сохранилась запись, что Лобачевский 'в значительной мере явил признаки безбожия'.
  15. ^ 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.
  16. ^ "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.
  17. ^ Kramer, Edna E. (1982) [1970]. "Mathematical Reasoning from Eudoxus to Lobachevsky". The Nature and Growth of Modern Mathematics (corrected reprint ed.). Princeton: Princeton University Press. pp. 56–57. ISBN 9780691023724. Retrieved 18 August 2022. It was [Kondyrev's] responsibility to supervise the students and to report their conduct to the principal. Kondyrev avenged himself by submitting very bad reports on Lobachevsky, even to the extent of accusing him of atheism, a charge which was not at all justified but which might have had tragic consequences for Lobachevsky.
  18. ^ O'Connor, John J.; Robertson, Edmund F., "Nikolai Lobachevsky", MacTutor History of Mathematics Archive, University of St Andrews
  19. ^ The 1914 English translation by George Bruce Halsted is available at "Quod.lib.umich.edu". The University of Michigan Historical Mathematics Collection. Retrieved 2012-12-17.
  20. ^ The 1902 German translation by Heinrich Liebmann is available at "Quod.lib.umich.edu". The University of Michigan Historical Mathematics Collection. Retrieved 2012-12-17.
  21. ^ Papadopoulos 2010, p. v
  22. ^ Bell, E. T. (1986). Men of Mathematics. Touchstone Books. p. 336. ISBN 978-0-671-62818-5.
  23. ^ Liner notes, "The Tom Lehrer Collection", Shout! Factory, 2010
  24. ^ Lennon, Madison (June 16, 2019). "Screen Rant, 10 Stars You Forgot Appeared On 3rd Rock From The Sun". Screen Rant. Retrieved February 27, 2024.
  25. ^ "3rd Rock from the Sun, "Dick and the Single Girl" (season 2 episode 24)". Imdb.com. Retrieved February 26, 2024.