Nikolai Lobachevsky

Labeling it "imaginary geometry," Nikolai Lobachevsky published the first complete text on non-Euclidean geometry in 1829. Many have since heralded him as the Copernicus of geometry. Just as Copernicus had challenged the long-established theory that the earth was the center of the universe, Lobachevsky's alternative view of geometry was in direct contradiction to the long-revered work of Euclid. Referring to Lobachevsky, Einstein said, "He dared to challenged an axiom."

Lobachevsky spent most of his life at the University of Kazan, near Siberia. Founded by Czar Alexander I in 1804, Kazan was Europe's easternmost center of higher education. Among its first student, Lobachevsky received his master's degree in mathematics and physics in 1811 at the age of eighteen. He remained at Kazan teaching courses for civil servant until 1816, when he was promoted to full professorship.



Geometry received Nikolai's special attention. Being inquisitive, Lobachevsky made several attempts to prove the Parallel Postulate; he failed at each. He then  proceed to examine the consequences of substituting an alternative to Euclid's postulate of a unique parallel; Lobachevsky assumed that more than one parallel could be drawn through a point. Living in distant isolation from the learning capitals of Europe, Lobachevsky was unaware of  the similar approaches taken by Gauss and Bolyai.

Poincare disk - as a visualization

Lobachevsky's idea of geometry based on an axiom in opposition to Euclid began to take from 1823, when he drew up an outline for a geometry course he was teaching. In 1826, he gave a lecture and presented a paper incorporating his belief in the feasibility of a geometry based on axiom different from Euclid's. Although this paper has been lost (as have so many in the history of mathematics), it was the first recorded attempt to breach Euclid's bastion. In 1829, the monthly academic journal of the University of Kazan printed a series of his work titled On Foundation of Geometry; this publication is considered by many to be the official birth of the radically new, non-Euclidean Geometry.

As with anything new and at odds with status quo, Lobachevsky's work was not embraced with open arms. The St. Petersburg Academy rejected it for publication in its scholarly journal and printed an uncomplimentary review. In contrast to Gauss, who did not have the courage to print, and Bolyai, who did not have the fortitude to face his opponents, Lobachevsky remained undaunted. He proceeded in his work, expanding Foundation into New Elements of Geometry, with a Complete Theory of Parallels, which was also published in Kazan's Academic journal of 1835. Shortly thereafter, his work began to be recognized outside of Kazan; he was published in Moscow, Paris, and Berlin.

In 1846, when Gauss received a copy of Lobachevsky's latest book, Geomatrical Investigations on the Theory of Parallels (which contained 61 pages), he wrote to a collegue, "I have had occasion to look through again that little volume by Lobachevsky. You know that for fifty-four years now I have held the same conviction. I have found in Lobachevsky's work nothing that is new to me, but the development is made in a way different from that which I have followed, and certainly by Lobachevsky in a skillful way and truly geometrical spirit." However Gauss did not give public approval to Lobachevsky's work.

As is the case with those who pioneer ideas and art forms that are incomprehensible to the world at large, the radical of geometry, Lobachevsky and Bolyai, never received full recognition of the value of their work during their lifetimes. However, they opened the door for a host of new ideas in geometry adn in the axiomatic system in general.

--------------------------------------------------------------------------
wiki : http://en.wikipedia.org/wiki/Nikolai_Lobachevsky