**Nikolai Lobachevsky** published his work on non-Euclidean geometry, the first account of the subject to appear in print.

- There the boys attended Kazan Gymnasium, financed by government scholarships, with Nikolai Ivanovich entering the school in 1802.
- In 1807 Lobachevsky graduated from the Gymnasium and entered Kazan University as a free student.
- Lobachevsky was highly successful in all courses he took ...
- A skilled teacher, Bartels soon interested Lobachevsky in mathematics.
- Since Euclid's Elements and his theory of parallel lines are discussed in detail in Montucla's book, it seems likely that Lobachevsky's interest in the Fifth Postulate was stimulated by these lectures.
- Lobachevsky received a Master's Degree in physics and mathematics in 1811.
- Lobachevsky had experienced difficulties during this period at the University of Kazan.
- Lobachevsky bought equipment for the physics laboratory, and he purchased books for the library in St Petersburg.
- The atmosphere now changed markedly and Musin-Pushkin found in Lobachevsky someone who could work with him in bringing important changes to the university.
- In 1827 Lobachevsky became rector of Kazan University, a post he was to hold for the next 19 years.
- The University of Kazan flourished while Lobachevsky was rector, and this was largely due to his influence.
- There was a marked increase in the number of students and Lobachevsky invested much effort in raising not only the standards of education in the university, but also in the local schools.
- Owing to resolute and reasonable measures taken by Lobachevsky the damage to the University was reduced to a minimum.
- For his activity during the cholera epidemic Lobachevsky received a message of thanks from the Emperor.
- Despite this heavy administrative load, Lobachevsky continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics.
- However Lobachevsky was not lucky in his marriage.
- After Lobachevsky retired in 1846 (essentially dismissed by the University of Kazan), his health rapidly deteriorated.
- Lobachevsky did not try to prove this postulate as a theorem.
- Lobachevsky categorised euclidean as a special case of this more general geometry.
- On 11 February 1826, in the session of the Department of Physico-Mathematical Sciences at Kazan University, Lobachevsky requested that his work about a new geometry was heard and his paper A concise outline of the foundations of geometry was sent to referees.
- The text of this paper has not survived but the ideas were incorporated, perhaps in a modified form, in Lobachevsky's first publication on hyperbolic geometry.
- In 1834 Lobachevsky found a method for the approximation of the roots of algebraic equations.
- This method is today called the Dandelin-Gräffe method since Dandelin also independently investigated it, but only in Russia does the method appear to be named after Lobachevsky who is the third independent discoverer.
- In 1837 Lobachevsky published his article Géométrie imaginaire Ⓣ(Imaginary geometry) and a summary of his new geometry Geometrische Untersuchungen zur Theorie der Parellellinien Ⓣ(Geometric Investigations on the theory of parellelism) was published in Berlin in 1840.
- This coincidence has prompted speculation that both Lobachevsky and Bolyai were led to their discoveries by Gauss.
- There are other claims made about Lobachevsky and the discovery of non-euclidean geometry which have been recently refuted.
- The story of how Lobachevsky's hyperbolic geometry came to be accepted is a complex one and this biography is not the place in which to go into details, but we shall note the main events.
- In 1866, ten years after Lobachevsky's death, Hoüel published a French translation of Lobachevsky's Geometrische Untersuchungen Ⓣ(Geometric Investigations on the theory of parellelism) together with some of Gauss's correspondence on non-euclidean geometry.
- Beltrami, in 1868, gave a concrete realisation of Lobachevsky's geometry.
- Weierstrass led a seminar on Lobachevsky's geometry in 1870 which was attended by Klein and, two years later, after Klein and Lie had discussed these new generalisations of geometry in Paris, Klein produced his general view of geometry as the properties invariant under the action of some group of transformations in the Erlanger Programm.
- There were two further major contributions to Lobachevsky's geometry by Poincaré in 1882 and 1887.
- Perhaps these finally mark the acceptance of Lobachevsky's ideas which would eventually be seen as vital steps in freeing the thinking of mathematicians so that relativity theory had a natural mathematical foundation.

Born 1 December 1792, Nizhny Novgorod (was Gorky from 1932-1990), Russia. Died 24 February 1856, Kazan, Russia.

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Algebra, Analysis, Astronomy, Geometry, Group Theory, Origin Russia

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive