LOBACHEVSKY, Nikolai.

Воображаемая геометрія (Voobrožajemaja geometrija) [Imaginary Geometry]. Kazan: at the University Press, 1835. [With:] Примѣненіе воображаемой геометріи къ нѣкоторымъ интегралаъ (Primenenije voobrožajemoj geometrii k nekotorym integralam) [Application of Imaginary Geometry to Certain Integrals]. Kazan: at the University Press, 1836.

Kazan: at the University Press, 1835; 1836.

First edition, extraordinarily rare offprints, of both parts of Lobachevsky’s second publication on non-Euclidean, or imaginary, geometry, following his epoch-making O nachalakh geometrii (1829–30). Hitherto, only journal issues or extracts of these articles have been accessible; Rick Watson described the journal issues as exceptionally rare, but these offprints are much rarer still. We have been unable to locate any other copies (for comparison, only one offprint of O nachalakh geometrii is known to exist), and it is likely that these are the only copies outside Russia. Our offprints are from the library of Friedrich Engel, who together with Paul Stäckel was responsible for translating and editing Lobachevsky’s works for a Western audience. The Voobrozhaemaya geometriya represents the issue of two millennia of criticism of Euclid’s Elements: as the DSB notes, it was Lobachevsky’s achievement to refute the uniqueness of Euclidean geometry and to recast it as a special case of a more general system. The non-Euclidean researches grew out of two thousand years of failed attempts to prove the parallel postulate — the assertion that through any point only one straight line can be drawn parallel to a given line — and, as the PMM editors observe, no one openly challenged that postulate as a received truth of the universe until Lobachevsky published his fundamental paper of 1826, eventually printed as a series of five articles in the Kazan University Courier in 1829–30, in which an infinity of parallels could be drawn through a given point without ever intersecting a given line. In O nachalakh geometrii Lobachevsky proceeded axiomatically, deriving the logical consequences of denying the parallel postulate. But this did not prove that imaginary geometry was free of inconsistencies and so did not, strictly speaking, prove the existence of non-Euclidean geometry. That problem is resolved in the present works. The DSB describes the change of method: where the earlier papers had defined imaginary geometry on an a priori basis, beginning from the negation of Euclid’s fifth postulate and unfolding the principal tenets of the new geometry without defining it as a structure, Voobrazhaemaya geometriya built up the system analytically, starting from its inherent trigonometrical formulas and using their derivation from spherical trigonometry to guarantee internal consistency; the sequel, Primenenie voobrazhaemoi geometrii k nekotorym integralam, then applied geometrical reasoning in Lobachevskian space to the evaluation of known integrals (as a check that the new method gave valid results) and afterwards to new, previously uncalculated integrals. Both works appeared in the Uchenye zapiski (Scientific memoirs) of Kazan University, a journal Lobachevsky himself had founded. OCLC lists one copy of the 1835 offprint (at Johns Hopkins) and none of the 1836. RBH lists one copy of the 1835 paper (Christie’s, 27 November 2019, lot 49, £150,000), described as an offprint although it had only the journal title page, not the separately-printed title page of our copy (the pagination of offprints and journal publications is identical).

Provenance: 1. Friedrich Engel (1861-1941), German mathematician (inscription on flyleaf dated 30 April 1898). Engel studied under Felix Klein at Leipzig, where he worked, first as lecturer and later as professor of mathematics, from 1885 to 1904. In 1904 he succeeded his friend Eduard Study as full professor at Greifswald, and in 1913 he went in the same capacity to Giessen, where, after his retirement in 1931, he continued to work until his death. Engel translated the works of Lobachevsky from Russian into German, thus making these works more accessible. With Paul Stäckel he wrote a history of non-Euclidean geometry, Theorie der Parallellinien von Euklid bis auf Gauss (1895); 2. Egon Ullrich (1902-57), Austrian mathematician (20th-century inscription on flyleaf). Ullrich was associate professor at the University of Giessen from 1936 and full professor there from 1940, where he attended lectures by his colleague Friedrich Engel; 3. Dr. Martin Sändig, Wissensch.-techn. Antiquariat (Wiesbaden book label on front pastedown).

In his early Kazan lectures on geometry Lobachevsky had himself attempted a proof of the fifth postulate; his own geometry grew from his later realisation that the system in which every other Euclidean axiom holds, but the fifth postulate fails, is not in itself contradictory. He named the resulting system imaginary geometry, by analogy with imaginary numbers: as imaginary numbers are the most general numbers under which the laws of real arithmetic remain consistent, imaginary geometry is the most general geometrical system. The achievement, the DSB emphasises, lay in dethroning Euclid’s geometry from uniqueness and exhibiting it as a special case of a more general structure.

In Lobachevskian geometry, given a line a and a point A not on it, more than one coplanar line through A can be drawn that does not intersect a. In fact infinitely many can: taken together they form an angle with vertex at A, the two boundary lines b and c being called parallels to a and the lines between them ultraparallels or diverging lines, while all other lines through A intersect a. The distance between two parallel lines, measured on a secant equally inclined to each, decreases indefinitely toward zero as one moves farther from A. The most striking contrast with Euclid is that the angle-sum of every triangle in the Lobachevskian plane falls short of two right angles. Indeed, Lobachevsky proved in O nachalakh geometrii that, if the angles of the triangle are α, β, and γ, its area is

π – α – β – γ,

so that α + β + γ is always less than π. In Euclidean geometry the sum of the angles of a triangle equals two right angles, and in spherical geometry it is always greater.

Lobachevsky derived trigonometric formulas for triangles in his plane and compared them with the corresponding formulas of spherical trigonometry on a sphere of radius r. He found that his formulas can be obtained from the spherical ones simply by treating the sides of the triangle as purely imaginary numbers, or equivalently by treating the radius r as purely imaginary. He saw in this transformation a powerful indication that the new geometry was free of contradiction.

In the introduction to Voobrazhaemaya geometriya, Lobachevsky 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. Voobrazhaemaya geometriya deals only with Lobachevskian trigonometry, omitting the underlying synthetic geometry which had been treated in detail in O nachalakh geometrii. By treating the subject in this way, Lobachevsky hoped to put to rest any doubt that the assumptions on which his geometry rests could ever lead to a contradiction.

Jeremy Gray has analysed Lobachevsky’s confidence in the consistency of his system as resting on two convictions. The first was a starting point in the analysis of the motion of rigid bodies; the second was a finishing point in the language of trigonometry and the calculus. Lobachevsky sought formulae because he believed that geometry was at bottom a science of measurement, that measurements are numbers, and that numbers are related to one another by formulae whose validity was an algebraic question quite apart from any geometrical interpretation. He therefore inferred — and Gray notes that this is logically a gap in the argument — that his new geometry was self-consistent because he had a workable system of trigonometric formulae, that the same formulae described Euclidean geometry when their parameter was let grow indefinitely large, and that they reduced to spherical trigonometry when the parameter was made purely imaginary.

In the first part of Voobrazhaemaya geometriya (equations 1-15), Lobachevsky derives the formulas of non-Euclidean trigonometry relating the lengths of the sides and angles of a triangle. For any number a, he defines a number a’ by

sin a’ = 1/cosh a.

Then, in a triangle with angles A, B, C and opposite sides a, b, c,

sin A tan a’ = sin B tan b

1− cos b’ cos c’ cos A = sin b’ sin c’ / sin a’

cos A + cos B cos C = sin B sin C / sin a’

cot A sin C sin b’ + cos C = cos b’ / cos a’.

He notes that the derivations of these formulas work in exactly the same way whether he adopts the hypothesis that the sum of the angles of a triangle is greater than π, as in spherical geometry, or less than π, as in imaginary geometry. Since his method produces the known formulas of spherical trigonometry, he takes this as evidence that his formulas for imaginary trigonometry must be correct.

In the second part Lobachevsky turns to the differential geometry of the two-dimensional imaginary plane and the three-dimensional imaginary space, specifically the calculations of lengths, areas and volumes in his imaginary geometry. He notes that this had been done in O nachalakh geometrii using methods of pure geometry, but he now wants to proceed using infinitesimal calculus methods. He was able to do this because he had shown in the first part of Voobrazhaemaya geometriya that the equations of imaginary trigonometry reduce to the equations of Euclidean trigonometry for triangles with infinitesimally small sides. He used this latter circumstance to establish the expressions for arc elements, surface elements, and volume elements valid in the new geometry, and then computed the area of a quadrilateral, the circumference of a circle, the surface area and volume of a sphere, and the volume of a cone with a flat base.

In Voobrazhaemaya geometriya Lobachevsky also gives applications of his imaginary geometry to the computation of definite integrals. Lobachevsky’s integrals represent areas of surfaces and volumes of bodies in twoand three-dimensional non-Euclidean space. Computing the area or the volume of the same object in different ways turns out to be an efficient way of finding attractive formulae for some definite integrals. This is analogous to the way one would apply the elementary geometry of the Euclidean circle of radius 1 to the evaluation of the integral

∫ √(1 – x²) dx

(taken between 0 and 1), noting that this is equal to the area of a quadrant of the circle (i.e., π/4). In each formula that Lobachevsky obtains using non-Euclidean geometrical methods, and verifiable by pure analysis, he sees a new confirmation of the logical consistency of his geometrical theory.

In Primenenie voobrazhaemoi geometrii k nekotorym integralam Lobachevsky takes the application of imaginary geometry to the computation of definite integrals much further. He begins with a study of the ‘Lobachevsky function’ (pp. 7-10),

L(x) = – ∫ log(cos t) dt,

the integral being taken from 0 to x. Many of the later volume computations are expressed in terms of this function. Conversely, the fact that a given volume can be expressed by different integrals is used to derive identities between the values of the function L whose arguments are linked by trigonometric relations.

Lobachevsky then gives a discussion of polar coordinates in the non-Euclidean plane, and his first application to the evaluation of a trigonometric integral (pp. 22-25). He then turns to integration in three dimensions, and the application to the evaluation of a ‘transcendental’ integral (which involves both trigonometric and exponential functions).

With these preparations in place, Lobachevsky devotes the next part of the work to the computation of the volumes of cones in three-dimensional non-Euclidean geometry (pp, 34-61). These cones need not have a circular base, and oblique cones are also considered. Of particular importance for what comes later is his calculation of the volumes of ‘asymptotic’ cones, whose generating lines are parallel to a straight line that is perpendicular to the base of the cone (so that the vertex of the cone is at ‘infinity’).

The analysis of cones is applied in the main part of the book (pp. 61-137), which is devoted to the computation of the volumes of non-Euclidean tetrahedra, the three-dimensional analogues of triangles in the plane – a tetrahedron is a cone whose base is a triangle. Various cases are considered, according to whether the faces of the tetrahedra are right-angled triangles or not, and whether or not any of the vertices are at infinity (this means that the sides of the tetrahedron ‘passing through’ this vertex are parallel lines). The volumes of finite tetrahedra are deduced, by a process of addition and subtraction, from those of asymptotic tetrahedra, which have one vertex at infinity.

Lobachevsky’s main result is a formula for the volume of an ‘orthotetrahedron’ – a tetrahedron three of whose dihedral angles (the angles between intersecting faces) are right-angles (these are the three-dimensional analogues of right-angled triangles in the plane). If α, β, and γ are the other dihedral angles of the orthotetrahedron, Lobachevsky proves that its volume is:

¼{L(α + θ – π/2) – L(α – θ – π/2) + L(β – θ) – L(β + θ)

+ L(γ + θ – π/2) – L(γ – θ – π/2) – 2L(θ)},

where θ is the angle between 0 and π/2 such that

tan θ = (cos²β – sin²α sin²γ)^½ / cosα cosγ.

Since any tetrahedron can be ‘cut up’ into orthotetrahedra (just as any triangle can be cut up into two right-angled triangles), one can now compute the volume of any tetrahedron; the volume of any polyhedron can be computed by cutting it up into tetrahedra (just as any polygon can be cut up into triangles).

Lobachevsky’s results on tetrahedra are of particular interest in view of much more recent developments in geometry and topology. In his 1895 paper ‘Analysis situs’ Henri Poincaré showed that three-dimensional manifolds (geometric objects that look locally like pieces of three-dimensional space) could be constructed by ‘gluing together’ the faces of certain polyhedra. Using polyhedra in non-Euclidean space results in a manifold that is ‘locally like’ non-Euclidean space – such a manifold is now known as a ‘hyperbolic manifold’. Modern developments in the theory of hyperbolic manifolds began with the work of William Thurston, for which he was awarded the Fields medal in 1983. He showed that a wide class of three-dimensional manifolds are hyperbolic and emphasized the importance of the volume of a hyperbolic manifold.

The work concludes (pp. pp. 138-166) with a list of the 50 types of integral that Lobachevsky evaluates by using non-Euclidean methods. Some idea of their complexity can be gained from the very first integral in the list:

∫ ∫ dωdx(e^x– e^-x)F’[a(e^x+ e^-x) + bcosω(e^x– e^-x)] = - πF[2√(a² – b²)] / √(a² – b²),

where a² > b², the first integral is taken from 0 to π, the second from 0 to infinity, and F is any reasonable function (and F’(x) = dF/dx).

Lobachevsky’s first lecture on non-Euclidean geometry, his Exposition succincte des principes de la géométrie, was delivered in French to the Kazan department of physics and mathematics on 23 February 1826. French was the established language of scientific discourse in Russia, but Lobachevsky strongly advocated for the use of Russian and published his first four works in his native tongue: O nachalakh geometrii (1829–30), Voobrazhaemaya geometriya (1835), Primenenie voobrazhaemoi geometrii k nekotorym integralam (1836), and Novye nachala geometrii s polnoi teoriei parallelnykh (1835–38). With the exception of Voobrazhaemaya geometriya, which appeared in slightly modified French dress in 1837 as Géométrie imaginaire in the Journal für die reine und angewandte Mathematik (Bd. 17, 295–320), these early Russian works were not translated until the late nineteenth and early twentieth century. The Primenenie voobrazhaemoi geometrii was rendered into German by Heinrich Liebmann in 1904, and that translation appears still to be the only complete one. Liebmann himself observed (p. vi) that of all Lobachevsky’s writings the Primenenie most fully bears out Gauss’s verdict that his treatises resemble a tangled jungle. In 1840 Lobachevsky published a summary work in German, the Geometrische Untersuchungen über die Theorie der Parallellinien, which is not a translation of any of the Russian works.

Lobachevsky’s work was little heralded during his lifetime. M. V. Ostrogradsky, the most famous mathematician of the St Petersburg Academy, did not grasp the achievement and published an uncomplimentary review of O nachalakh geometrii; the magazine Syn otechestva followed his lead and in 1834 issued a pamphlet ridiculing Lobachevsky’s paper. Gauss, who had received a copy of the Geometrische Untersuchungen from the author, spoke to him flatteringly of the work, learned Russian in order to read the originals, and supported his election to the Göttingen Gesellschaft der Wissenschaften, but he never publicly commented on the discovery.

The next decisive step came from Riemann, whose habilitation lecture of 1854, Über die Hypothesen, welche der Geometrie zu Grunde liegen, was published posthumously in 1866. Developing Gauss’s idea of the intrinsic geometry of a surface, Riemann introduced the notion of a multidimensional curved space — now Riemannian space — in which it subsequently became clear that Lobachevskian space is the Riemannian space of constant negative curvature.

The presence of specifically Lobachevskian geometry is felt in modern physics through the isomorphism between the group of motions of Lobachevskian space and the Lorentz group. That isomorphism makes Lobachevskian geometry directly applicable to a range of problems in relativistic quantum physics. Within the framework of general relativity, the question of the geometry of the real world, to which Lobachevsky had devoted so much thought, was finally answered: the geometry of the universe is that of variable curvature, which on average lies much closer to Lobachevsky’s than to Euclid’s.

Lobachevsky was the son of Ivan Maksimovich Lobachevsky, a clerk in a land-surveying office, and Praskovia Aleksandrovna Lobachevskaya. About 1800 his mother moved with her three sons to Kazan, where Lobachevsky and his brothers were soon enrolled in the Gymnasium on public scholarships. In 1807 he entered Kazan University, studying under Martin Bartels, a friend of Gauss, and in 1812 took the master’s degree in physics and mathematics. He became an adjunct in 1814 and began lecturing on various aspects of mathematics and mechanics, was appointed extraordinary professor that same year, and rose to professor ordinarius in 1822 — the 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 chaired that committee in 1825, served twice as dean of the department of physics and mathematics (1820–1 and 1823–5), as university librarian (1825–35), as rector (1827–46), and as assistant trustee for the whole Kazan educational district (1846–55).

In recognition of his work Lobachevsky was raised in 1837 to the hereditary nobility, designing for himself a familial device, reproduced on his tombstone, that combined Solomon’s seal, a bee, an arrow, and a horseshoe to symbolise wisdom, diligence, alacrity, and happiness. In 1832 he made a wealthy marriage to Lady Varvara Aleksivna Moisieva, but his seven children and the cost of technological improvements to his estate left him with little money on his retirement from the university despite a modest pension. A worsening sclerotic condition progressively affected his eyesight, and he was blind in his last years.

References: Gray, Worlds Out of Nothing (2007) — Kagan, Lobatchevski Travaux de Géométrie 2 & 3 — Liebmann, Imaginäre Geometrie und Anwendung der imaginären Geometrie auf einige Integrale (1904) — Sommerville, Bibliography of Non-Euclidean Geometry, p. 26 — W. P. Watson, Cat. 4 no. 57 & Cat. 10 no. 91 (the original Kazan material noted as exceptionally rare even in Russian collections) — See PMM 293 & Norman 1379 for O nachalakh geometrii.

8vo (220 x 135 mm), pp. [ii], 88, with one folding plate; [ii], 166, with one folding plate (titles transliterated in pencil, intermittent spotting predominately in the first and last leaves of each work). 19th-century quarter black cloth with marbled paper over paper boards, original printed wrappers bound in (extremities rubbed, light scratches on covers, first work's wrappers with small tears and one repaired, second work’s wrappers with stains).

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Item #6525

Price: $195,000.00