(Lobachevsky believed that another form of geometry existe...)
Lobachevsky believed that another form of geometry existed, a non-Euclidean geometry, and this 1840 treatise is his argument on its behalf. Line by line in this classic work he carefully presents a new and revolutionary theory of parallels, one that allows for all of Euclid’s axioms, except for the last. This 1891 translation includes a bibliography and translator George B. Halsted’s essay on elliptic geometry.
The Foundations of Geometry: Works on Non-Euclidean Geometry
(Neither general relativity (which revealed that gravity i...)
Neither general relativity (which revealed that gravity is merely a manifestation of the non-Euclidean geometry of spacetime) nor modern cosmology would have been possible without the almost simultaneous and independent discovery of non-Euclidean geometry in the 19th century by three great mathematicians - Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss (whose ideas were later further developed by Georg Friedrich Bernhard Riemann).
Nikolai Ivanovich Lobachevsky was a Russian mathematician. He was a founder of non-Euclidean geometry, which he developed independently of János Bolyai and Carl Gauss.
Background
Nikolai Ivanovich Lobachevsky was born on December 1, 1792, in Nizhny Novgorod, Russian Empire. His father Ivan Maksimovich Lobachevsky, worked as a clerk in an office which was involved in land surveying. His mother was Praskovia Alexandrovna Lobachevskaya.
Lobachevsky grew up in a family of moderate means. Family relations, however, were evidently strained. After 1791 his mother was effectively divorced from her husband, Ivan Lobachevsky. Modern analysis of previously unknown archival materials shows that Lobachevsky’s father was most likely Sergey Shebarshin, a graduate of Moscow State University who worked as a surveyor and rose to the rank of titular councilor. After Shebarshin’s death, the economic circumstances of the family worsened.
Lobachevsky was one of three sons in this poor family. When Lobachevsky was seven years of age his father died and, in 1800, his mother moved with her three sons to the city of Kazan in western Russia on the edge of Siberia.
Education
In Kazan Lobachevsky and his brothers attended Kazan Gymnasium, financed by government scholarships. He entered the school in 1802. In 1806 Lobachevsky graduated from the Gymnasium and entered Kazan University as a free student in 1807. Kazan State University had been founded in 1804, the result of one of the many reforms of the emperor Alexander I, and it opened in the following year, only two years before Lobachevsky began his undergraduate career. His original intention was to study medicine, but he changed to study a broad scientific course involving mathematics and physics.
One of the excellent professors who had been invited from Germany was Martin Bartels who had been appointed as Professor of Mathematics. Bartels was a school teacher and friend of Gauss, and the two corresponded. A skilled teacher, Bartels soon interested Lobachevsky in mathematics. We do know that Bartels lectured on the history of mathematics and that he gave a course based on the text by Montucla. Since Euclid's Elements and his theory of parallel lines are discussed in detail in Montucla's book, it seems likely that Lobachevsky's interest in the Fifth Postulate was stimulated by these lectures. Lobachevsky received a Master's Degree in physics and mathematics in 1811.
In 1814 Nikolai Lobachevsky received the degree of the adjunct of pure mathematics and permission to teach independently. He gave courses on number theory, arithmetic, algebra, trigonometry, integral and differential calculus, plane and spherical geometry, mechanics, physics, and astronomy. From 1816 he was professor extraordinarius. In 1819 the Kazan regional board of education instituted a xenophobic policy, and the German faculty left. The resulting shortage of professors led to rapid advancement in Lobachevsky’s career and in 1822 he became a professor ordinarius. Lobachevsky bought equipment for the physics laboratory, and he purchased books for the library in St Petersburg. He was appointed to important positions within the university such as the dean of the Mathematics and Physics Department between 1820 and 1825 and head librarian from 1825 to 1835. He also served as Head of the Observatory and was clearly strongly influencing policy within the University.
In 1826 Tsar Nicholas I became ruler and introduced a more tolerant regime. In that year Magnitski was dismissed as curator of Kazan University and a new curator M. N. Musin-Pushkin was appointed. The atmosphere now changed markedly and Musin-Pushkin found in Lobachevsky someone who could work with him in bringing important changes to the university. In 1827 Lobachevsky became rector of Kazan University. Lobachevsky was elected to this nonremunerated position six successive times, occupying it for 19 years. The following year he made a speech (which was published in 1832) On the most important subjects of education and this gives clearly what were the ideas in his educational philosophy. He encouraged in every possible way the dissemination of education in the extensive Kazan district. He was instrumental in stopping the spread of a virulent cholera epidemic (1830-1831) among the teachers and students of the university by means of a rigid quarantine, and by bold personal action, he saved the university from a devastating fire that swept Kazan in 1842.
The University of Kazan flourished while Lobachevsky was rector, and this was largely due to his influence. There was a vigorous program of new building, with a library, an astronomical observatory, new medical facilities and physics, chemistry, and anatomy laboratories being constructed. He pressed strongly for higher levels of scientific research, and he equally encouraged research in the arts, particularly developing a leading center for Oriental Studies. There was a marked increase in the number of students and Lobachevsky invested much effort in raising not only the standards of education in the university, but also in the local schools.
In 1846 Lobachevsky resigned the post of a rector and was named an assistant trustee of the regional board of education. In later years he went blind, fell seriously ill, and lost his beloved son (1852), yet he continued his scholarly work, dictating his last work, “Pangéométrie,” in French in 1855.
The real significance of Nikolai Lobachevsky’s geometry was not fully understood and appreciated until the work of the great German mathematician Bernhard Riemann on the foundations of geometry (1868) and the proof of the consistency of non-Euclidean geometry by his compatriot Felix Klein in 1871. In the late 19th century Kazan State University established a prize and a medal in Lobachevsky’s name. Beginning in 1927 the Lobachevsky Prize was awarded by the Soviet Union Academy of Sciences (now the Russian Academy of Sciences), but in 1992 the awarding of the medal reverted to Kazan State University.
(Lobachevsky believed that another form of geometry existe...)
1840
Religion
Nikolai Ivanovich Lobachevsky was an Atheist and considered that most of the scientists shared this view.
Politics
Lobachevsky wasn't involved in politics though he used protectorate of such Russian political figures as Senator Mikhail Musin-Pushkin.
Views
In February 1826 Lobachevsky presented to the physico-mathematical college the manuscript of an essay devoted to "the rigorous analysis of the theorem on parallels," in which he may have proposed either a proof of Euclid’s fifth postulate (axiom) on parallel lines or an early version of his non-Euclidean geometry. This manuscript, however, was not published or even publicly discussed by the college, and its content remains unknown. Lobachevsky gave the first public exposition of the ideas of non-Euclidean geometry in his paper "On the principles of geometry," which contained fragments of the 1826 manuscript and was published in 1829-1830 in a minor Kazan periodical. In his geometry, Lobachevsky abandoned the parallel postulate of Euclid, which states that in the plane formed by a line and a point not on the line it is possible to draw exactly one line through the point that is parallel to the original line. Instead, he based his geometry on the following assumption: In the plane formed by a line and a point not on the line it is possible to draw infinitely many lines through the point that are parallel to the original line. It was later proved that his geometry was self-consistent and, as a result, that the parallel postulate is independent of Euclid’s other axioms - hence, not derivable as a theorem from them. This finally resolved an issue that had occupied the minds of mathematicians for over 2,000 years; there can be no parallel theorem, only a parallel postulate. Lobachevsky called his work "imaginary geometry," but as a sympathizer with the empirical spirit of Francis Bacon, he attempted to determine the "true" geometry of space by analyzing astronomical data obtained in the measurement of the parallax of stars. A physical interpretation of Lobachevsky’s geometry on a surface of negative curvature (see the figure of a pseudosphere) was discovered by the Italian mathematician Eugenio Beltrami, but not until 1868.
From 1835 to 1838 Lobachevsky published "Imaginary geometry," "New foundations of geometry with the complete theory of parallels," and "Application of geometry to certain integrals." In 1842 his work was noticed and highly praised by Gauss, at whose instigation Lobachevsky was elected that year as a corresponding member of the Royal Society of Göttingen. Although Lobachevsky was also elected an honorary member of the faculty of Moscow State University, his innovative geometrical ideas provoked misunderstanding and even scorn. The famous Russian mathematician of the time, Mikhail Ostrogradskii, a member of the St. Petersburg Academy, as well as the academician Nicolaus Fuss, spoke pejoratively of Lobachevsky’s ideas. Even a literary journal managed to accuse Lobachevsky of "abstruseness." Nevertheless, Lobachevsky continued stubbornly to develop his ideas, albeit in isolation, as he did not maintain close ties with his fellow mathematicians.
In addition to his geometry, Lobachevsky obtained interesting results in algebra and analysis, such as the Lobachevsky–Gräffe method for computing the roots of a polynomial (1834) and the Lobachevsky criterion for convergence of an infinite series (1834-1836). His research interests also included the theory of probability, integral calculus, mechanics, astronomy, and meteorology.
Quotations:
"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world."
Membership
Nikolai Lobachevsky was a member of the Göttingen Academy of Sciences.
Göttingen Academy of Sciences
,
Germany
Personality
Lobachevsky was an impractical manager who jeopardized his financial position by purchasing the estate while living on a pension; that he had no time to look after the estate and took little interest in it; that he was left in poverty and ignored by the local officials. Health and financial difficulties added to the heavy burdens he had to bear over his last years. His great mathematical achievements were not recognized in his lifetime, and he died without having any notion of the fame and importance that his work would achieve.
Physical Characteristics:
After Lobachevsky retired in 1846 (essentially dismissed by the University of Kazan), his health rapidly deteriorated. Soon after he retired, however, his favorite eldest son died and Lobachevsky was hit hard by this tragedy. The illness that he suffered from became progressively worse and led to blindness.
Quotes from others about the person
"He [Lobachevsky] challenged an axiom." - Albert Einstein
Interests
Philosophers & Thinkers
Karl Friedrich Gauss
Connections
In 1832 Lobachevsky married Varvara Alexeyevna Moiseyeva who came from a wealthy family. At the time of his marriage, his wife was a young girl while Lobachevsky was forty years old. The couple lived in a big three-story house and received a lot of guests with lavish hospitality. However, Lobachevsky was not lucky in his marriage. The marriage gave them seven children (eighteen according to his son's memoirs, while only seven apparently survived into adulthood). The children and the cost of technological improvements for his estate left him with little money upon his retirement.
