The Russian mathematician Nikolai Ivanovich Lobachevskii (1792-1856) was one of the first to found an internally consistent system of non-Euclidean geometry. His revolutionary ideas had profound implications for theoretical physics, especially the theory of relativity.

Nikolai Lobachevskii was born on Dec. 2 (N.S.; Nov. 21, O.S.), 1792, in Nizhni Novgorod (now Gorkii) into a poor family of a government official. In 1807 Lobachevskii entered Kazan University to study medicine. However, the following year Johann Martin Bartels, a teacher of pure mathematics, arrived at Kazan University from Germany. He was soon followed by the astronomer J. J. Littrow. Under their instruction, Lobachevskii made a permanent commitment to mathematics and science. He completed his studies at the university in 1811, earning the degree of master of physics and mathematics.

In 1812 Lobachevskii finished his first paper, "The Theory of Elliptical Motion of Heavenly Bodies." Two years later he was appointed assistant professor at Kazan University, and in 1816 he was promoted to extraordinary professor. In 1820 Bartels left for the University of Dorpat (now Tartu in Estonia), resulting in Lobachevskii's becoming the leading mathematician of the university. He became full professor of pure mathematics in 1822, occupying the chair vacated by Bartels.

## Euclid's Parallel Postulate

Lobachevskii's great contribution to the development of modern mathematics begins with the fifth postulate (sometimes referred to as axiom XI) in Euclid's *Elements.* A modern version of this postulate reads: Through a point lying outside a given line only one line can be drawn parallel to the given line.

Since the appearance of the *Elements* over 2, 000 years ago, many mathematicians have attempted to deduce the parallel postulate as a theorem from previously established axioms and postulates. The Greek Neoplatonist Proclus records in his *Commentary on the First Book of Euclid* the geometers who were dissatisfied with Euclid's formulation of the parallel postulate and designation of the parallel statement as a legitimate postulate. The Arabs, who became heirs to Greek science and mathematics, were divided on the question of the legitimacy of the fifth postulate. Most Renaissance geometers repeated the criticisms and "proofs" of Proclus and the Arabs respecting Euclid's fifth postulate.

The first to attempt a proof of the parallel postulate by a *reductio ad absurdum* was Girolamo Saccheri. His approach was continued and developed in a more profound way by Johann Heinrich Lambert, who produced in 1766 a
theory of parallel lines that came close to a non-Euclidean geometry. However, most geometers who concentrated on seeking new proofs of the parallel postulate discovered that ultimately their "proofs" consisted of assertions which themselves required proof or were merely substitutions for the original postulate.

## Toward a Non-Euclidean Geometry

Karl Friedrich Gauss, who was determined to obtain the proof of the fifth postulate since 1792, finally abandoned the attempt by 1813, following instead Saccheri's approach of adopting a parallel proposition that contradicted Euclid's. Eventually, Gauss came to the realization that geometries other than Euclidean were possible. His incursions into non-Euclidean geometry were shared only with a handful of similar-minded correspondents.

Of all the founders of non-Euclidean geometry, Lobachevskii alone had the tenacity and persistence to develop and publish his new system of geometry despite adverse criticisms from the academic world. From a manuscript written in 1823, it is known that Lobachevskii was not only concerned with the theory of parallels, but he realized then that the proofs suggested for the fifth postulate "were merely explanations and were not mathematical proofs in the true sense."

Lobachevskii's deductions produced a geometry, which he called "imaginary, " that was internally consistent and harmonious yet different from the traditional one of Euclid. In 1826, he presented the paper "Brief Exposition of the Principles of Geometry with Vigorous Proofs of the Theorem of Parallels." He refined his imaginary geometry in subsequent works, dating from 1835 to 1855, the last being *Pangeometry.* Gauss read Lobachevskii's *Geometrical Investigations on the Theory of Parallels,* published in German in 1840, praised it in letters to friends, and recommended the Russian geometer to membership in the Göttingen Scientific Society. Aside from Gauss, Lobachevskii's geometry received virtually no support from the mathematical world during his lifetime.

In his system of geometry Lobachevskii assumed that through a given point lying outside the given line at least two straight lines can be drawn that do not intersect the given line. In comparing Euclid's geometry with Lobachevskii's, the differences become negligible as smaller domains are approached. In the hope of establishing a physical basis for his geometry, Lobachevskii resorted to astronomical observations and measurements. But the distances and complexities involved prevented him from achieving success. Nonetheless, in 1868 Eugenio Beltrami demonstrated that there exists a surface, the pseudosphere, whose properties correspond to Lobachevskii's geometry. No longer was Lobachevskii's geometry a purely logical, abstract, and imaginary construct; it described surfaces with a negative curvature. In time, Lobachevskii's geometry found application in the theory of complex numbers, the theory of vectors, and the theory of relativity.

## Philosophy and Outlook

The failure of his colleagues to respond favorably to his imaginary geometry in no way deterred them from respecting and admiring Lobachevskii as an outstanding administrator and a devoted member of the educational community. Before he took over his duties as rector, faculty morale was at a low point. Lobachevskii restored Kazan University to a place of respectability among Russian institutions of higher learning. He cited repeatedly the need for educating the Russian people, the need for a balanced education, and the need to free education from bureaucratic interference.

Tragedy dogged Lobachevskii's life. His contemporaries described him as hardworking and suffering, rarely relaxing or displaying humor. In 1832 he married Varvara Alekseevna Moiseeva, a young woman from a wealthy family who was educated, quick-tempered, and unattractive. Most of their many children were frail, and his favorite son died of tuberculosis. There were several financial transactions that brought poverty to the family. Toward the end of his life he lost his sight. He died at Kazan on Feb. 24, 1856.

Recognition of Lobachevskii's great contribution to the development of non-Euclidean geometry came a dozen years after his death. Perhaps the finest tribute he ever received came from the British mathematician and philosopher William Kingdon Clifford, who wrote in his *Lectures and Essays,* "What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobachevsky to Euclid."

## Further Reading on Nikolai Ivanovich Lobachevskii

There is no definitive biography of Lobachevskii in English. Useful works include E.T. Bell, *Men of Mathematics* (1937); Veniamin F. Kagan, *N. Lobachevsky and His Contributions to Science* (trans. 1957); and Alexander S. Vucinich, *Science in Russian Culture,* vol. 1: *A History to 1860* (1963). Valuable for treating Lobachevskii's geometry in historical perspective are Roberto Bonola, *Non-Euclidean Geometry: A Critical and Historical Study of Its Developments* (trans. 1955); A. D. Aleksandrov, "Non-Euclidean Geometry, " in *Mathematics: Its Content, Methods, and Meaning,* vol. 3, edited by A.D. Aleksandrov, A. N. Kolmogorov, and M.A. Lavrentev (trans. 1964); and Carl B. Boyer, *A History of Mathematics* (1968).