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’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, 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 one stating that there is more than one line that can be extended through any given point 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.

Read about his life and work: http://teslaradioconspiracy.blogspot.com