# Nikolai Lobachevskii

## Lobachevskii, Nikolai Ivanovich

Born Nov. 20 (Dec. 1), 1792, in Nizhny Novgorod, now the city of Gorky; died Feb. 12 (24), 1856, in Kazan. Russian mathematician and founder of non-Euclidean geometry; materialist thinker and advocate of university and public education.

The son of a minor government official, Lobachevskii spent almost his entire life in Kazan. He studied there, first in the *Gymnasium* (1802–07) on a government stipend and later at the University of Kazan (1807–11). He displayed outstanding abilities at an early age. He received a master’s degree from the university upon completion of studies in 1811 and remained there to work. In 1814 he became an adjunct professor, in 1816 an extraordinary professor, and in 1822 an ordinary (full) professor. Despite the reactionary atmosphere that prevailed during the curatorship of M. L. Magnitskii, Lobachevskii conducted intensive scientific and pedagogical work (he taught mathematics, physics, and astronomy) and purchased equipment for the physics laboratory and books for the library in St. Petersburg. He was head librarian for ten years (from 1825), was in charge of the observatory, and was elected dean of the physics and mathematics department (1820–22, 1823–25). But the clashes with the curator intensified: Lobachevskii advocated scientific materialist ideas in his teaching.

During these years, Lobachevskii sought a rigorous construction of the foundations of geometry. Student notes of his lectures (of 1817), where he attempted to prove Euclid’s parallel postulate, have been preserved; however, he rejected this attempt in the manuscript of his textbook *Geometry* (1823). In “Surveys of the Teaching of Pure Mathematics,” which covered the years 1822–23 and 1824–25, Lobachevskii indicated the “heretofore unconquerable” difficulty of the problem of parallelism and the necessity of adopting as basic concepts in geometry concepts obtained directly from nature. Finally, overcoming age-old traditions, he created a new geometry known as Lobachevskian geometry.

On Feb. 7, 1826, Lobachevskii submitted for publication to the *Uchenye zapiski* (Proceedings) of the physics and mathematics section the paper “A Concise Outline of the Foundations of Geometry With a Rigorous Proof of the Theorem of Parallels” (in French). On February 11 it was examined and reviewers were appointed. Lobachevskii himself presented the argument at the section’s meeting on February 12. But the paper was not published. The manuscript and the comments have not been preserved, but Lobachevskii incorporated the paper itself in the article “On the Foundations of Geometry” in the journal *Kazanskii vestnik* (1829–30), which constituted the first publication in world literature on non-Euclidean geometry.

Proceeding from the search for unconditional rigor and clarity in the foundations of geometry, Lobachevskii considered Euclid’s parallel postulate as an arbitrary limitation and as too rigid a requirement, one limiting the possibilities of the theory describing the properties of space. He replaced the postulate with a broader and more general requirement: in a plane, more than one line not intersecting another line can be drawn through a point not on the given line in the plane (essentially at least one line if we take into account the limiting case).

The new geometry worked out by Lobachevskii differs substantially from Euclidean geometry, but for large values of some constant *R* (the radius of curvature of space) that enters into the formulas the deviation becomes insignificant.

In accordance with his materialist approach to the study of nature, Lobachevskii believed that only scientific experiment can determine which of the geometries prevails in physical space. Using the latest astronomical data of the day, he concluded that the number *R* is very large and that deviations from Euclidean geometry, if they exist, lie within the limits of measurement errors. Thus, the practical suitability of Euclidean geometry was substantiated. Moreover, Lobachevskii demonstrated the applicability of his geometry in other branches of mathematics, namely, in mathematical analysis for calculating definite integrals.

Lobachevskii’s report coincided with Magnitskii’s dismissal. Lobachevskii was highly esteemed by the new curator, M. N. Musin-Pushkin. Lobachevskii was elected rector of the university in 1827 and occupied that position for 19 years. The university thrived as never before under his guidance. His program was reflected in his remarkable speech ‘On the Most Important Subjects of Education” (1828; published 1832), in which he outlined the ideal of the harmonious development of the personality, emphasized the social significance of upbringing and education, and discussed the role of the sciences and the scientist’s duty to his country and people.

During Lobachevskii’s rectorship a complex of auxiliary buildings, including a library, an astronomical observatory, physics and chemistry laboratories, an anatomical theater, and a clinic, was constructed between 1832 and 1840. Lobachevskii founded *Uchenye zapiski Kazanskogo universiteta* (1834) and greatly encouraged scientific and scholarly publishing. The level of the scientific, scholarly, and academic work was raised and the number of students increased. The university became an important center of Oriental studies. Lobachevskii also invested considerable effort in improving the organization of teaching in the district’s *Gymnasiums* and schools. His concern for the university was displayed with particular clarity during times of natural disasters (the cholera epidemic in 1830, the Kazan fire in 1842). But the rectorship did not prevent Lobachevskii from teaching, and over the years he lectured on analytic mechanics, hydromechanics, the integral calculus, differential equations, mathematical physics, and the calculus of variations. From 1838 to 1840 he conducted popular scientific lectures on physics for the general public. Students highly valued his lectures.

However, Lobachevskii’s scientific ideas were not understood by his contemporaries. His work *On the Foundations of Geometry*, which was presented to the Academy of Sciences by the university council in 1832, was given a negative evaluation by M. V. Ostrogradskii, and a short anonymous article mocking Lobachevskii appeared in the reactionary journal *Syn otechestva* in 1834. Despite this, Lobachevskii did not cease to develop his geometry. Various works by him appeared between 1835 and 1838, and in 1840 his book *Geometric Researches* (in German) was published in Germany. This perseverance for scientific truth distinguishes Lobachevskii from two of his contemporaries, who also discovered non-Euclidean geometry. The Hungarian mathematician J. Bolyai published his work in 1832, after Lobachevskii. But, gaining no support from his contemporaries, he did not continue his investigations. The German mathematician K. F. Gauss was also familiar with the foundations of non-Euclidean geometry. But he did not develop them further and did not publish them because of the fear of being misunderstood. Without doing so in print, however, he highly praised Lobachevskii’s works, and on his proposal Lobachevskii was elected a corresponding member of the Göttingen Academic Society in 1842.

Lobachevskii also obtained a number of valuable results in other branches of mathematics. For example, in algebra he worked out a new method of the approximate solution of equations, and in mathematical analysis he obtained a number of subtle theorems on trigonometric series and refined the concept of continuous function.

In 1846, Lobachevskii was virtually dismissed from the university. He was appointed an unpaid assistant to the new curator and deprived of the rectorship. His health declined. However, domestic tragedy—the death of his son—financial difficulties, and incipient blindness could not break Lobachevskii’s spirit. He wrote his last work, *Pangeometry*, one year before his death, dictating the text.

Lobachevskii died unrecognized. The studies of E. Beltrami (1868), F. Klein (1871), H. Poincaré (1883), and other mathematicians played a major role in gaining acceptance for Lobachevskii’s works. The University of Kazan and the Kazan Physics and Mathematics Society did much to clarify the import of Lobachevskii’s ideas and to publish his works on geometry. Broad recognition came only by the 100th anniversary of Lobachevskii’s birth—an international prize was established and a monument was unveiled in Kazan (1896).

### WORKS

*Poln. sobr. soch*., vols. 1–5. Moscow-Leningrad, 1946–51.

*Izbr. trudy po geometrii*. Moscow-Leningrad, 1956.

### REFERENCES

Vasil’ev, A. V.*Lobachevskii*. St. Petersburg, 1914.

Kagan, V. F.

*Lobachevskii*, 2nd ed. Moscow-Leningrad, 1948. (With bibliography.)

Laptev, B. L. “Velikii russkii matematik.”

*Vestnik vysshei shkoly*, 1967, no. 12.

*Istoriko-matematicheskie issledovaniia*, issues 3, 4, 6, and 11. Moscow-Leningrad, 1950–58. (A number of articles.)

Modzalevskii, L. B.

*Materialy dlia biografii N. I. Lobachevskii*. Moscow-Leningrad, 1948.

B. L. LAPTEV