born , Dec. 1 [Nov. 20, Old Style], 1792, Nizhny Novgorod, Russia

died Feb. 24 [Feb. 12, Old Style], 1856, Kazan

Russian mathematician and founder of non-Euclidean geometry, which he developed independently of János Bolyai and Carl Gauss. (Lobachevsky's first publication on this subject was in 1829, Bolyai's in 1832; Gauss never published his ideas on non-Euclidean geometry.)

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 1829–30 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 (1561–1626), 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 (1834–36). His research interests also included the theory of probability, integral calculus, mechanics, astronomy, and meteorology.

The real significance of 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 U.S.S.R. Academy of Sciences (now the Russian Academy of Sciences), but in 1992 the awarding of the medal reverted to Kazan State University.

"Nikolay Ivanovich Lobachevsky." Encyclopædia Britannica. 2004. Encyclopædia Britannica Premium Service.

18 July 2004 <http://www.britannica.com/eb/article?eu=49824>.

Eugenia Ellis, Design Arts, Drexel University

At the time of his death, the American architect Claude Bragdon (1866-1946) was working on a book entitled The Veil of M_y_, which he described as a retelling of a tale told in The Stars and the Earth; or, Thoughts Upon Space, Time, and Eternity—a book given to him as a child by his father that had been written by Felix Eberty (1812-1884) and published anonymously in 1846. In brief, in the first part of the book through using the phenomenon of light the author shows how the past may be present to God simultaneously with its physical manifestation in the present. This same logic undergirded Einstein’s “discovery” of the Theory of Relativity, to which he attributed his imagining when he was sixteen-years-old that he was riding a beam of light. In the second part, the author demonstrates the unity of the Creator by showing the unity of creation as the embodiment of a single thought that occupies neither Space nor Time, which are modes of human perception. This paper will investigate how Claude Bragdon’s architectural theosophy and four-dimensional geometric demonstrations illustrate a simultaneity of thought originating in ancient Eastern religious philosophy that was reiterated in early 20th century imaginings of space/time relationships and scientific relativity.

'**Dimensions of the Pure Imagination**': Pavel Florenskii's The Analysis of Space and Time in the Fine Arts

Elizabeth English, School of Architecture, Tulane University

Pavel Florenskii was a major thinker in scientific, literary and artistic circles in Russia in the early 20th century—a mathematician, philosopher, physicist, mystic, art historian, and Orthodox priest. His dream was to create a system of metalogic having a similar relation to ordinary logic as non-Euclidean has to Euclidean geometry, derived by negating certain axioms of Aristotelian logic according to the method developed by mathematician Nikolai Lobachevskii. Florenskii developed theories of an “imaginary space” not only related to Lobachevskii’s “imaginary geometry” but also a locus for the essential being of imaginary numbers, spiritual understanding and the creative activities of the human imagination. In his book Analiz prostranstvennosti i vremeni v khudozhestvenno-izobrazitel'nikh proizvedeniiakh (The analysis of space and time in the fine arts), Florenskii outlined the importance of the concept of the fourth dimension in the synthesis of art, mathematics and religion in imaginary space, and argued the inherent necessity of the fourth dimension as a condition of being in the world.

Art and the Fourth Dimension

Lobachevsky, born in 1792, was a pretty bright kid. He started school young and at age 14 he entered the new university of Kazan. He excelled at math, earning academic honors, but before graduating he almost got kicked out of school for playing too many practical jokes. But by 1811, at age 18, he received his master’s degree and then stayed at Kazan for 40 years, ultimately becoming the rector (the head guy) at the school that almost expelled him.

Besides teaching math, and physics, and astronomy, Lobachevsky filled other odd jobs at Kazan, such as organizing the library and the museum. Nevertheless he found time to pursue some original mathematical thinking, and by 1826 he’d produced, in the words of E. T. Bell, “one of the great masterpieces of all mathematics and a landmark in human thought.”

Lobachevsky had worked out the basics of a new geometry, based on the proposition that you could draw at least two parallel lines through a point not on the first line. He first delivered his ideas in an 1826 lecture; they were published (obscurely) a few years later. He called his creation “imaginary geometry” and claimed that it was superior to Euclid’s.

“In [Euclidean] geometry I find certain imperfections which I hold to be the reason why this science . . . can as yet make no advance from that state in which it came to us from Euclid,” Lobachevsky wrote later in a book explaining his system.

He was not the only mathematician to explore the non-Euclidean realm at that time. Equal credit for discovering the new geometry sometimes goes to the Hungarian Janos Bolyai, son of Gauss’s friend Farkas. Bolyai’s approach was similar to Lobachevsky’s, showing that Euclid’s geometry was not the only possible description of the world. Or if it was, maybe there was another world. “Out of nothing,” the younger Bolyai wrote, “I have created a strange new universe.”

He published his version in 1832, as an appendix to his father’s math textbook. It was not a good place to attract a lot of attention, so Bolyai’s work went unnoticed for years—as did, for the most part, Lobachevsky’s.

From: Strange Matters: Undiscovered Ideas at the Frontiers of Space and Time (2002)

Joseph Henry Press (JHP)

Copyright H. David Marshak, All Rights Reserved