Career
Lobachevsky's main achievement is the development (independently from János Bolyai) of a non-Euclidean geometry, also referred to as Lobachevskian geometry. In contrast to Bolyai's work, Lobachevsky's work contained only hyperbolic 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 Euclid's parallel postulate 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; a famous consequence is that the sum of angles in a triangle must be less than 180 degrees. Non-Euclidean geometry is now in common use in many areas of mathematics and physics, such as general relativity; and hyperbolic geometry is now often 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 (Dirichlet gave the same definition independently soon after Lobachevsky).
Read more about this topic: Nikolai Lobachevsky
Famous quotes containing the word career:
“What exacerbates the strain in the working class is the absence of money to pay for services they need, economic insecurity, poor daycare, and lack of dignity and boredom in each partners job. What exacerbates it in upper-middle class is the instability of paid help and the enormous demands of the career system in which both partners become willing believers. But the tug between traditional and egalitarian models of marriage runs from top to bottom of the class ladder.”
—Arlie Hochschild (20th century)
“I seemed intent on making it as difficult for myself as possible to pursue my male career goal. I not only procrastinated endlessly, submitting my medical school application at the very last minute, but continued to crave a conventional female role even as I moved ahead with my male pursuits.”
—Margaret S. Mahler (18971985)
“A black boxers career is the perfect metaphor for the career of a black male. Every day is like being in the gym, sparring with impersonal opponents as one faces the rudeness and hostility that a black male must confront in the United States, where he is the object of both fear and fascination.”
—Ishmael Reed (b. 1938)