Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian
mathematician and
geometer, known primarily for his work on
hyperbolic geometry, otherwise known as
Lobachevskian geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
, and also for his fundamental study on
Dirichlet integrals
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
: \int_0^\inf ...
, known as the
Lobachevsky integral formula
In mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions.
One of those is the improper integral of the sinc function over the positive real line,
: \int_0^\inf ...
.
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in hi ...
called Lobachevsky the "
Copernicus
Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulate ...
of Geometry" due to the revolutionary character of his work.
Biography
Nikolai Lobachevsky was born either in or near the city of
Nizhny Novgorod in the
Russian Empire (now in
Nizhny Novgorod Oblast
Nizhny Novgorod Oblast (russian: link=no, Нижегородская область, ''Nizhegorodskaya oblast''), is a federal subject of Russia (an oblast). Its administrative center is the city of Nizhny Novgorod. It has a population of 3,310 ...
,
Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eight ...
) in 1792 to parents of Russian and
Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya.
[Victor J. Katz. ''A history of mathematics: Introduction''. Addison-Wesley. 2009. p. 842.][ Stephen Hawking. ''God Created the Integers: The Mathematical Breakthroughs that Changed History''. Running Press. 2007. pp. 697–703.] 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
Kazan (Volga region) Federal University (russian: Казанский (Приволжский) федеральный университет, tt-Cyrl, Казан (Идел буе) федераль университеты) is a public research uni ...
,
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
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(1777-1855).
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
Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who pr ...
,
teaching mathematics, physics, and astronomy.
He served in many administrative positions and became the
rector
Rector (Latin for the member of a vessel's crew who steers) may refer to:
Style or title
*Rector (ecclesiastical), a cleric who functions as an administrative leader in some Christian denominations
*Rector (academia), a senior official in an edu ...
of Kazan University
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
Atheism, in the broadest sense, is an absence of belief in the existence of Deity, deities. Less broadly, atheism is a rejection of the belief that any deities exist. In an even narrower sense, atheism is specifically the position that ther ...
( ru , признаки безбожия , translation = signs of godlessness
).
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
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
's
fifth postulate from other
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s. Euclid's fifth is a rule in Euclidean geometry which states (in
John Playfair
John Playfair FRSE, FRS (10 March 1748 – 20 July 1819) was a Church of Scotland minister, remembered as a scientist and mathematician, and a professor of natural philosophy at the University of Edinburgh. He is best known for his book ''Illu ...
'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 as ''On the Origin of Geometry'' (''О началах геометрии'') in 1829–1830 (Kazan University Course Notes). 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
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle and ...
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.
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
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
of the
roots of
algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. 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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
as a correspondence between two sets of real numbers (
Peter Gustav Lejeune Dirichlet gave the same definition independently soon after Lobachevsky).
Impact
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.
Honors
*
1858 Lobachevskij, an asteroid discovered in 1972, was named in his honour.
* The lunar crater
Lobachevsky was named in his honor.
*
Lobachevsky Prize
The Lobachevsky Prize, awarded by the Russian Academy of Sciences, and the Lobachevsky Medal, awarded by the Kazan State University, are mathematical awards in honor of Nikolai Ivanovich Lobachevsky.
History
The Lobachevsky Prize was established ...
, a
mathematics award by the
Kazan State University
Kazan (Volga region) Federal University (russian: Казанский (Приволжский) федеральный университет, tt-Cyrl, Казан (Идел буе) федераль университеты) is a public research uni ...
.
* The
Lobachevsky University
The National Research State University of Nizhny Novgorod named after N.I. Lobachevsky, UNN (Russian: Нижегородский национа́льный иссле́довательский госуда́рственный университ ...
was named in his honor.
In popular culture
* 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
obachevsky'scharacter" and the name was chosen "solely for
prosodic reasons".
[Liner notes, "The Tom Lehrer Collection", ''Shout!'' Factory, 2010]
* In
Poul Anderson's 1969
fantasy novella "Operation Changeling" – which was later expanded into the novel ''
Operation Chaos
Operation CHAOS or Operation MHCHAOS was a Central Intelligence Agency domestic espionage project targeting the American people from 1967 to 1974, established by President Lyndon B. Johnson and expanded under President Richard Nixon, whose mission ...
'' (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.
*
Morris Panych
Morris Stephen Panych (born 30 June 1952) is a Canadian playwright, director and actor.
Early life
Panych was born in Calgary, Alberta and grew up in Edmonton, Alberta. He studied at Northern Alberta Institute of Technology, and the Universit ...
's
Governor General's Award
The Governor General's Awards are a collection of annual awards presented by the Governor General of Canada, recognizing distinction in numerous academic, artistic, and social fields.
The first award was conceived and inaugurated in 1937 by th ...
-winning play, ''
Girl in the Goldfish Bowl'', references Lobachevsky repeatedly as the focus of Owen's geometry obsession.
*
Vladimir Nabokov mentions Lobachevsky in his
Lectures on Russian Literature.
Works
* 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:
*
Geometrical Researches on the Theory of Parallels'.
G. B. Halsted (tr.). 1891. Reprinted in Roberto Bonola: ''Non-Euclidean Geometry: A Critical and Historical Study of its Development''. 1912. Dover reprint 1955.
: Also in: Seth Braver ''Lobachevski illuminated'',
MAA 2011.
*
Pangeometry'. translated by Henry P. Manning: in
D. E. Smith ''A Source Book in Mathematics''. McGraw Hill 1929. Dover reprint, pp. 360–374.
*
New Principles of Geometry with Complete Theory of Parallels'. G. B. Halsted (tr.). 1897.
* Nikolai I. Lobachevsky, ''Pangeometry'', translator and editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society. 2010.
See also
*
1858 Lobachevskij
*
Gauss–Bolyai–Lobachevsky space
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
*
Hyperbolic geometry
*
Hyperboloid structure
*
Non-Euclidean geometry
*
Lobachevsky (crater)
*
Lobachevsky function
*
Lobachevsky Medal
*
Lobachevsky University
The National Research State University of Nizhny Novgorod named after N.I. Lobachevsky, UNN (Russian: Нижегородский национа́льный иссле́довательский госуда́рственный университ ...
*
Upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
References
External links
*
*
Website dedicated to Lobachevsky
Nikolaj Ivanovič Lobačevskij - Œuvres complètes, tome 2– Gallica-Math
Lobachevsky State University of Nizhny Novgorod
{{DEFAULTSORT:Lobachevsky, Nikolai Ivanovich
1792 births
1856 deaths
19th-century mathematicians from the Russian Empire
Mathematicians from Kazan
Geometers
Russian atheists
Russian people of Polish descent
Burials at Arskoe Cemetery
Kazan Federal University faculty