## Géométrie imaginaire.

Berlin: Reimer, 1837.

First edition, journal issue, of the first account of any part of Lobachevsky’s revolutionary discovery of non-Euclidean geometry to be published in a Western European language. “The researches that culminated in the discovery of non-Euclidean geometry arose from unsuccessful attempts to ‘prove’ the axiom of parallels in Euclidean geometry. This postulate asserts that through any point there can be drawn one and only one straight line parallel to a given straight line. Although this statement was not regarded as self-evident and its derivation from the other axioms of geometry was repeatedly sought, no one openly challenged it as an accepted truth of the universe until Lobatchewsky published the first non-Euclidean geometry … In Lobatchewsky’s geometry an infinity of parallels can be drawn through a given point that never intersect a given straight line … His fundamental paper was read to his colleagues in Kazan in 1826 but he did not publish the results until 1829-30 when a series of five papers appeared [in Russian] in the Kazan University Courier [*O nachalakh geometrii*, 1829-30]” (PMM). A work with the same title, *Voobrazhaemaya geometriya*, was published (in Russian) in 1835, but according to Sommerville (p. 28) this French version was “Written previous to the Russian paper bearing the same title, 1835”. In it, “he built up the new geometry analytically, proceeding from its inherent trigonometrical formulas and considering the derivation of these formulas from spherical trigonometry to guarantee its internal consistency” (DSB). Lobachevsky shows that all the analytical and geometrical theorems in non-Euclidean geometry follow from these formulas.

“In his early lectures on geometry, Lobachevsky himself attempted to prove the fifth postulate [i.e., the axiom of parallels]; his own geometry is derived from his later insight that a geometry in which all of Euclid’s axioms except the fifth postulate hold true is not in itself contradictory. He called such a system “imaginary geometry,” proceeding from an analogy with imaginary numbers. If imaginary numbers are the most general numbers for which the laws of arithmetic of real numbers prove justifiable, then imaginary geometry is the most general geometrical system. It was Lobachevsky’s merit to refute the uniqueness of Euclid’s geometry, and to consider it as a special case of a more general system.

“In Lobachevskian geometry, given a line *a* and a point *A* not on it, one can draw through *A* more than one coplanar line not intersecting *a.* It follows that one can draw infinitely many such lines which, taken together, constitute an angle of which the vertex is *A*. The two lines, *b* and *c*, bordering that angle are called parallels to *a* and the lines contained between them are called ultraparallels, or diverging lines; all other lines through *A* intersect *a*. If one measures the distance between two parallel lines on a secant equally inclined to each, then, as Lobachevsky proved, that distance decreases indefinitely, tending to zero, as one moves farther out from *A* … A comparison of Euclidean and Lobachevskian geometry yields several immediate and interesting contrasts [notably that] for all triangles in the Lobachevskian plane the sum of the angles is less than two right angles” (DSB). In Euclidean geometry the sum of the angles of a triangle equals two right angles, and in spherical geometry it is always greater.

Of particular interest are the curves called ‘horocycles.’ These are the limiting curves of the circles that share a tangent at a given point, as the radius of the circles tends to infinity. In Euclidean geometry this limiting curve would be a straight line, but in Lobachevskian geometry it is a new kind of curve. By rotating the horocyle around the line perpendicular to the tangent, Lobachevsky obtained a ‘horosphere’ and he proved the remarkable fact that the geometry on a horosphere is Euclidean, so that Euclidean geometry is in a sense contained within non-Euclidean geometry.

“Working from the geometry (and, hence, trigonometry) of the Euclidean plane on horospheres, Lobachevsky derived trigonometric formulas for triangles in the Lobachevskian plane … Comparing these formulas with those of spherical trigonometry on a sphere of radius *r*, … Lobachevsky discovered that the formulas of trigonometry in the space he defined can be derived from formulas of spherical trigonometry if the sides of triangles are regarded as purely imaginary numbers or, put another way, if the radius *r* of the sphere is considered as purely imaginary … In this Lobachevsky saw evidence of the non-contradictory nature of the geometry he had discovered” (*ibid*.).

Lobachevsky’s first lecture on non-Euclidean geometry, “Exposition succincte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles,” was delivered (in French) to the Kazan department of physics and mathematics at a meeting held on 23 February 1826. French was the language of scientific discourse in Russia but Lobachevsky strongly advocated the use of the Russian language and published his first four works in his native tongue: *O nachalakh geometrii* (1829-30); *Novye nachala geometrii s polnoi teoriei parallelnykh* (1835-38); *Voobrazhaemaya geometriya* (1835); *Primenenie voobrazhaemoi geometrii k nekotorym integralam* (1836). With the exception of *Voobrazhaemaya geometriya*, these early Russian works on non-Euclidean geometry were not translated until the last years of the 19^{th} century. Lobachevsky published a final summary work in Russian, *Pangeometria*, to mark the jubilee of the University of Kazan in 1855; this was translated into French in the following year.

As we have already noted, according to Sommerville the offered work was written before *Voobrazhaemaya geometriya*. A comparison of this work with ‘Géométrie imaginaire’ shows that the former is not a straightforward translation of the latter, although the content of the two works is very similar. They deal only with Lobachevskian trigonometry, omitting the underlying synthetic geometry which had been treated in detail in *O nachalakh geometrii*, but giving applications to the computation of definite integrals. Lobachevsky’s integrals represent areas of surfaces and volumes of bodies in two- and three-dimensional non-Euclidean space. Computing the area or the volume of the same object in different manners turns out to be an efficient way of finding attractive formulae for some definite integrals. But besides the intrinsic value of the results obtained, there are several reasons why Lobachevsky worked out these computations. First of all, using non-Euclidean geometry for the computation of integrals was a way of showing the usefulness of non-Euclidean geometry in another branch of mathematics, namely analysis. At another level, drawing consequences of the new axiom system, like finding values of known integrals using these new methods was a way of checking that the new geometric system was not self-contradictory. This was a major concern for Lobachevsky, which he addresses in the introduction to ‘Géométrie imaginaire’. He writes that he feels that he was unable to deal with the subject in the necessary detail in *O nachalakh geometrii*. Many results were stated there without proof, and this may have led some readers to doubt the truth of his work. In this work he therefore retraces the path taken in the earlier work, but this time starting with the fundamental equations of Lobachevskian trigonometry and deriving their consequences in more detail than before. By doing so he hopes to put to rest any doubt that the assumptions on which his geometry rests could ever lead to a contradiction. “Lobachevsky’s work was little heralded during his lifetime. M. V. Ostrogradsky, the most famous mathematician of the St. Petersburg Academy, for one, did not understand Lobachevsky’s achievement, and published an uncomplimentary review of *O nachalakh geometrii* (“On the Principles of Geometry”); the magazine *Syn otechestva* soon followed his lead, and in 1834 issued a pamphlet ridiculing Lobachevsky’s paper” (*ibid*.).

“Lobachevsky was the son of Ivan Maksimovich Lobachevsky, a clerk in a land-surveying office, and Praskovia Aleksandrovna Lobachevskaya. In about 1800 the mother moved with her three sons to Kazan, where Lobachevsky and his brothers were soon enrolled in the Gymnasium on public scholarships. In 1807 Lobachevsky entered Kazan University, where he studied under the supervision of Martin Bartels, a friend of Gauss, and, in 1812, received the master’s degree in physics and mathematics. In 1814 he became an adjunct in physical and mathematical sciences and began to lecture on various aspects of mathematics and mechanics. He was appointed extraordinary professor in 1814 and professor ordinarius in 1822, the same year in which he began an administrative career as a member of the committee formed to supervise the construction of the new university buildings. He was chairman of that committee in 1825, twice dean of the department of physics and mathematics (in 1820-1 and 1823-5), librarian of the university (1825-35), rector (1827-46), and assistant trustee for the whole of the Kazan educational district (1846-55).

“In recognition of his work Lobachevsky was in 1837 raised to the hereditary nobility; he designed his own familial device (which is reproduced on his tombstone), depicting Solomon’s seal, a bee, an arrow, and a horseshoe, to symbolize wisdom, diligence, alacrity, and happiness, respectively. He had in 1832 made a wealthy marriage, to Lady Varvara Aleksivna Moisieva, but his family of seven children and the cost of technological improvements for his estate left him with little money upon his retirement from the university, although he received a modest pension. A worsening sclerotic condition progressively affected his eyesight, and he was blind in his last years” (*ibid*.).

See PMM 293; Norman I, 1379 [*O nachalakh geometrii*]; Sommerville p. 28.

Pp. 295-320 and Plate II in Journal für die reine und angewandte Mathematik, Bd. 17. 4to (250 x 218 mm), pp. iv, 394 with three folding plates. Contemporary mottled boards with leather lettering-piece on spine (extremities a little rubbed).

Item #4453

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Price:
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