Nikolay Ivanovich Lobachevsky

Russian mathematician

• Born: December 1, 1792
• Birthplace: Nizhny Novgorod, Russia
• Died: February 24, 1856
• Place of death: Kazan, Russia

Lobachevsky was the boldest and most consistent founder of a post-Euclidean theory of real space. His persistence in holding open his revolutionary line of inquiry into the reality of geometry helped to set the stage for the radical discoveries of twentieth century theoretical physics.

Early Life

Nikolay Ivanovich Lobachevsky (luh-buh-CHAYF-skuh-ih), whose parents were a minor government clerk and an energetic woman of apparently no education, was a member of what most nearly corresponded to a middle class in preemancipation Russia. As a government-supported student, he was recorded in school as a raznochinets (person of miscellaneous rank—not a noble, a peasant, or a merchant). Despite the early marriage of his mother, Praskovya, to the collegiate registrar Ivan Maksimovich Lobachevsky, the evidence strongly suggests that Nikolay and his two brothers were the illegitimate sons of an army officer and land surveyor, S. S. Shebarshin. Ivan Maksimovich Lobachevsky died when Nikolay, the middle son, was only five years old. The widowed Praskovya left Nizhny Novgorod and moved eastward along the Volga to the provincial center of Kazan. She enrolled all three boys in the local gymnasium (preparatory school). Nikolay attended the school between 1802 and 1807.

Lobachevsky’s student years at Kazan University (1807 to 1811) were a time when Russia was eager to learn from the West and to give more than it had received. Lobachevsky was awarded Kazan’s first master’s degree for his thesis on elliptic movement of the heavenly bodies. He worked closely with Johann Martin Bartels, who had earlier discovered and taught Carl Friedrich Gauss, a great mathematician of the day.

Lobachevsky taught at Kazan University from 1811 until his mandated but most unwilling retirement in 1846. The tenure of Mikhail Magnitskii as curator from 1819 to 1826 was the school’s most difficult period. A religious fanatic who attempted to give this particularly science-oriented university the atmosphere of a medieval monastery, Magnitskii was imprisoned in 1826 for his gross incompetence. He was particularly suspicious of the philosophy of Immanuel Kant. All the distinguished German professors left; for a time, the young Lobachevsky carried the burden of providing all the advanced lectures in mathematics, physics, and astronomy alone. His own development and integrity were only strengthened during this phase. It did Lobachevsky no harm that he too was anti-Kantian; he completely disagreed with Kant’s view that Euclidean geometry was proof of the human mind’s inborn sense of lines, planes, and space.

Life’s Work

As a young professor in 1817, Lobachevsky was intrigued by the problem of Euclid’s fifth postulate, which implies the possibility of infinitely parallel lines. More technically, one may draw a single line through a given point on a given plane that will never intersect another given line on the same plane. On one hand, this is not a simple axiom that has no need of proof. On the other, it cannot be proved. Two thousand years of general satisfaction with Euclidean geometry had seen many vain attempts to prove the fifth postulate and thereby give this geometry its final perfection. Such attempts became particularly frenzied in the eighteenth century. A rare few thinkers began to entertain the idea that the postulate was wrong, but they denied it even to themselves.

From 1817 to 1822, Lobachevsky made repeated attempts to prove the fifth postulate, already resorting to non-Euclidean concepts such as an axiom of directionality. Once he perceived the hidden tautology of even his best attempts, he concluded that the postulate must be wrong and that geometry must be put on a new foundation.

In addition to the resistance of intellectual tradition, Lobachevsky could expect little support in a country whose ruling house saw itself as the very embodiment of stability and conservatism. The unsettling implications of losing true parallelism and rocking the foundations of classical geometry were as unwelcome in czarist Russia as they could possibly have been anywhere in the world. This resistance makes Lobachevsky’s boldness all the more impressive. Simultaneously and independently, two leading mathematicians of the day—Gauss in Germany and János Bolyai in Hungary—were facing the same conclusion as Lobachevsky. Despite their secure reputations, both refrained from pursuing the implications of negating the fifth postulate, correctly assessing that the world was not ready for it.

To exacerbate the radicalism of his approach in a highly religious country, Lobachevsky, though not an atheist, was a materialist in a most fundamental and original sense of the word. In his mathematical syllabus for 1822, he made the extraordinary statement:

We apprehend in Nature only bodies alone; consequently, concepts of lines and planes are derived and not directly acquired concepts, and therefore should not be taken as the basis of mathematical science.

In 1823, Lobachevsky’s full-length geometry textbook Geometriya was submitted to school district curator Magnitskii, who sent it to the St. Petersburg Academy of Sciences for review. The text was emphatically rejected, and Lobachevsky’s difficulties with the academy began. A subsequent manuscript, “O nachalakh geometrii” (1829-1830; on the elements of geometry), was also submitted to the academy. Not only did the academy reject the manuscript but also an academician’s flawed critique was fed to the popular press, which turned it into a lampoon of Lobachevsky.

The date February 7, 1826, marks the official debut of Lobachevskian geometry as an independent theory. On that day, Lobachevsky submitted to his department his paper entitled “Exposition succincte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles.” It was rejected for publication, as his colleagues ventured no opinion on it. Other major works continued to be largely ignored.

Nevertheless, the new school district curator who replaced Magnitskii, Count Mikhail Musin-Pushkin, was sufficiently impressed with Lobachevsky to make him rector of Kazan University in 1827. Thus began Lobachevsky’s dual life as a brilliantly successful local administrator and a frustrated intellectual pioneer kept outside the pale of the St. Petersburg establishment. During his tenure as rector, Lobachevsky built Kazan University into an outstanding institution of high standards. He founded the scientific journal Uchenye zapiski , in which he published many of his works and that has flourished to the present day.

In 1846, Lobachevsky’s life in the sphere of action fell apart, as he received a succession of blows: Musin-Pushkin was transferred to the St. Petersburg school district; the request to forestall Lobachevsky’s mandatory retirement was denied; his eldest son, Aleksei, died of tuberculosis at the age of nineteen; his wife became seriously ill; his wife’s half brother, dispatched to handle the sale of two distant estates, gambled away both the Lobachevskys’ money and all of his own; and Lobachevsky’s eyesight began to deteriorate. In the last year of his life, he was virtually blind, yet he dictated his best and strongest work, Pangéométrie (1855-1856). His views had evolved from rejection of Euclidean parallelism into a vision of reality that anticipated theories of the curvature of space and was validated by Albert Einstein’s general theory of relativity.

Significance

When Nikolay Ivanovich Lobachevsky’s ideas first caught the imagination of a wide audience during the late nineteenth century, he was dubbed “the Copernicus of geometry,” partly because of his Slavic origin (Nicolaus Copernicus was Polish), but far more because of the profound reorientation of thought that he set in motion. Lobachevsky forced the scale of earthly dimension as the measure of the universe off its pedestal, as Copernicus had earlier shattered the illusory status of Earth as the center of the solar system. This upheaval, which initially met with great resistance, forced the mind to focus on awesome phenomena that were not so much abstract as invisible to the human eye. Lobachevsky promoted bold and fruitful speculation about the nature of reality and space.

Pangéométrie, Lobachevsky’s crowning work, opens: “Instead of beginning geometry with the line and plane, as is usually done, I have preferred to begin with the sphere and the circle.” For this geometry, there are no straight lines or flat planes, and all lines and planes must curve, however infinitesimally. Yet, while pointing to modern concepts of the curvature of space, Lobachevsky does not abolish Euclidean geometry. In some areas, his geometry and Euclid’s coincide. However, the latter is a limited case, whose relative certainties hold true on a merely earthly scale. In the conclusion to Pangéométrie, Lobachevsky correctly predicted that interstellar space would be the proving ground for his theory, which he saw not as an abstruse logical exercise but as the real geometry of the universe.

Bibliography

Bell, E. T. Men of Mathematics. New York: Simon & Schuster, 1937. Reprint. 1986. Chapter 16, “The Copernicus of Geometry,” focuses on Lobachevsky’s life and mathematical discoveries.

Bonola, Roberto. Non-Euclidean Geometry: A Critical and Historical Study of Its Development. Translated by H. S. Carslaw. Mineola, N.Y.: Dover, 1955. Up-to-date for its time, and still relevant to the general reader. Focuses on a basic exposition of Lobachevsky’s theories without the highly sophisticated applications thereof. Contains several relevant appendixes.

Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries: Development and History. 3d ed. New York: W. H. Freeman, 1993. Includes information about Lobachevsky’s discovery of the hyperbolic geometric.

Kagan, Veniamin Fedorovich. N. Lobachevsky and His Contributions to Science. Moscow: Foreign Language Press, 1957. A solid, basic account by one of the chief Russian experts on Lobachevsky. Omits most of the human-interest material to be found in Kagan’s 1944 biography. Includes a bibliography, necessarily of primarily Russian materials.

Kulczycki, Stefan. Non-Euclidean Geometry. Translated by Stanisław Knapowski. Elmsford, N.Y.: Pergamon Press, 1961. Another introduction for the general reader, which updates but does not supplant Bonola.

Mlodinow, Leonard. Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace. New York: Free Press, 2001. Lobachevsky’s contributions to geometry are briefly discussed in this book focusing on the work of five other mathematicians.

Shirokov, Pëtr Alekseevich. A Sketch of the Fundamentals of Lobachevskian Geometry. Edited by I. N. Bronshtein. Translated by Leo F. Boron and Ward D. Bouwsma. Groningen, Netherlands: P. Noordhoff, 1964. Written in Russian during the 1940’s, it appears to have been aimed at the secondary-school mathematics student.

Smogorzhevsky, A. S. Lobachevskian Geometry. Translated by V. Kisin. Moscow: Mir, 1982. Partly accessible to the general reader but of particular interest to the serious student of mathematics. Emphasizes specific mathematical applications of Lobachevsky’s theories.

Vucinich, Alexander. “Nikolay Ivanovich Lobachevsky: The Man Behind the First Non-Euclidean Geometry.” Isis 53 (December, 1962). A substantial, well-written article, abundantly annotated to point the reader in the direction of all the basic sources, which are primarily in Russian. Highlights some avenues not mentioned elsewhere, such as Lobachevsky’s role in the mathematization of science. Includes a balanced account of Lobachevsky’s life.