The work was dictated by Lobachevsky, who was by then blind, in French, the previous year for the commemoration proceedings planned for February , but publication was delayed by the Tsar about a disagreement over the date of the University's founding. Numerous deductions, notably that of the fundamental equation of hyperbolic geometry, are given in other forms: until his last days, despite his precarious health, Lobachevsky sought to perfect the geometry he had constructed" Kagan. Original printed wrappers, unopened spine rubbed with loss at ends. Verlag: Hayez, Bruxelles Beschreibung: Hayez, Bruxelles, Couverture rigide.

Edition originale. Lobachevsky, which sums up in detail, and in the necessary cases develops, the results of his earlier work. It is from this memoir that one can draw the most completely information on the global scientific, worldoutlook and philosophical views of this great mathematician. In this work the fundamental notions of geometry are discussed in detail: adjacency, cuts and the definition of the notions of the notion of point connected with them, lines, surfaces, and also the basic theorems on perpendicular straight lines and planes, relations in triangles, linear and angular measures, measuring of areas, an others.

Starting from more general fundamental premises compared with earlier works , a theory of parallel straight lines is constructed in detail. The fundamental equations of the imaginary geometry are introduced. As a whole, in this work Lobachecsky establishes the precise axiomatic foundations of geometry and defines the principles of its logical development, accompanying them with the corresponding foundational results in each of the fields he considered"], Kline [p.

However, the paper was never printed and was lost. He gave his approach to non-Euclidean geometry in a series of papers, the first two of which were published in Kazan journals. The first was entitled " On the Foundations of Geometry" and appeared in Verlag: Paris: Gauthier-Villars Beschreibung: Paris: Gauthier-Villars, First edition, in the original printed wrappers, of the first translation of Lobachevsky's 'Geometrische Untersuchungen zur Theorie der Parallellinien' Berlin, ; Dibner , the first separately published work on non-Euclidean geometry.

His gift for languages was used to evaluate and frequently to expound or translate important foreign mathematical writings. Of great importance were his successful efforts to overcome the long-standing failure of mathematicians to appreciate the significance of non-Euclidean geometry. The following year he also published Bolyai's appendix on non-Euclidean geometry, which was translated into Italian by Battaglini in I Lobachevsky appeared in In Sotheran's catalogue no. Original printed wrappers, uncut spine mostly work away, covers soilded and foxed, minimal foxing internally , former owner's signature on front wrapper.

Verlag: Reimer, Berlin Beschreibung: Reimer, Berlin, First edition. 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. His fundamental paper was read to his colleagues in Kazan in but he did not publish the results until when a series of five papers appeared [in Russian] in the Kazan University Courier [O nachalakh geometrii, ]" PMM. A work with the same title, Voobrazhaemaya geometriya, was published in Russian in , but according to Sommerville p. 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.

He called such a system "imaginary geometry," proceeding from an analogy with imaginary numbers. 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. 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. 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.

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. 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 ; Novye nachala geometrii s polnoi teoriei parallelnykh ; Voobrazhaemaya geometriya ; Primenenie voobrazhaemoi geometrii k nekotorym integralam With the exception of Voobrazhaemaya geometriya, these early Russian works on non-Euclidean geometry were not translated until the last years of the 19th century.

Lobachevsky published a final summary work in Russian, Pangeometria, to mark the jubilee of the University of Kazan in ; 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. They deal only with Lobach.

Verlag: Universitetskai a tip, Kazan Beschreibung: Universitetskai a tip, Kazan, First edition, incredibly rare offprint, of this important book-length memoir on the foundations of calculus and real analysis by the first inventor of non-euclidean geometry. The Kazan publications of Lobachevsky are exceptionally rare, even in Russian collections. OCLC lists the Harvard copy only; we are not aware of any other copy having appeared in commerce.

It may be observed that Lobachevsky's works in other areas of mathematics were either directly relevant to his geometry as his calculations on definite integrals and probable errors of observation or results of his studies of foundations of mathematics as his works on the theory of finites and the theory of trigonometric series. His work on these problems again for the most part paralleled that of other European mathematicians.

His paper on the convergence of trigonometric series, too, suggested a general definition of function like that proposed by Dirichlet in Lobachevsky also gave [in the offered paper] a rigorous definition of continuity and differentiability, and pointed out the difference between these notions " DSB. The question about the relation between continuity and differentiability awoke general attention between and , when Weierstrass gave an example of a function continuous within a certain interval and at the same time having no definite derivative within this interval non-differentiable.

Meanwhile, Lobachevski already in the thirties showed the necessity of distinguishing the 'changing gradually' in our terminology: continuity of a function and its 'unbrokeness' now: differentiability. With especial precision did he formulate this difference in his Russian Memoir of 'A method for ascertaining the convergence, etc.

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A function changes gradually when its increment diminishes to zero together with the increment of the independent variable. A function is unbroken if the ratio of these two increments, as they diminish, goes over insensibly into a new function, which consequently will be a differential-coefficient. Integrals must always be so divided into intervals that the elements under each integral sign always change gradually and remain unbroken" Halsted, p.

This work includes an extensive discussion of infinite series. Much of this parallels the contributions of western European mathematicians, but it includes a new convergence criterion, now known as 'Lobachevsky's test': Let u x be a function defined for all positive values of x, which decreases as x increases and which approaches zero as x increases without limit.

There are numbers p1, p2, p3,. Lobachevsky also treats the problem of expressing functions by infinite products. But, as he later realised, this assertion is not correct without further assumptions. Much space is also devoted in this memoir to definite integrals, motivated by the computation of areas and volumes in Lobachevskian geometry.

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One year later, Lobachevsky devoted a whole memoir to this subject, 'Primenenie voobrazhaemoi geometrii k nekotorym integralam' 'Application of Imaginary Geometry to Certain Integrals'. Geometers had historically been concerned primarily with Euclid's fifth postulate. This postulate is equivalent to the statement that given a line and a point not on it, one can draw through the point one and only one coplanar line not intersecting the given line.

Throughout the centuries, mathematicians tried to prove the fifth postulate as a theorem. In his early lectures on geometry, Lobachevsky himself attempted to prove the fifth postulate; 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.

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" DSB. Engel ed. Teubner, , pp.

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Halsted, 'Biology and. Verlag: Georg Reimer, Berlin Beschreibung: Georg Reimer, Berlin, First German edition of Lobachevskii's last publication on non-Euclidean geometry first published in Kazan The work was dictated by Lobachevskii, who was by then blind, in French, the previous year for the commemoration proceedings planned for February , but publication was delayed by the Tsar over a disagreement over the date of the University's founding. This German edition is little known: it is not mentioned in Friedrich Engel's 'Zwei geometrische Abhandlungen' and Heinrich Liebmann claims in his own German translation that it is the first.

With the title Pangeometrie Lobachevskii emphasizes the universality of his 'imaginary geometry' and gives his most concise formulation of a geometry free of the parallel postulate.

In this work Lobachevskii applies differential and integral calculus to non-Euclidean geometry and develops important refinements of his earlier works. The work ends with Lobachevskii discussing the geometry of nature and the necessity of experimentally determining if space is in fact non-Euclidean. It was Lobachevskii's merit to refute the uniqueness of Euclid's geometry and to consider it a special case of a more general system" B. Rare as the single issue in wrappers. Contemporary blank, blue wrappers original?

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A closed tear and a bit of staining to back wrapper and some tears and scratches to spine. Internally very nice and clean. Presumably not an off-print, as there are stitching-holes to the margins, indicating that it has been removed from a volume, although the wrappers could look original, certainly contemporary. Scarce first printing of Lobachavsky's main contribution to his second most important field after non-Euclidean geometry , namely infinite series, more specifically trigonometric series.

## Mahirwan Mamtani

This constitutes one of Lobachevsky's earliest papers and the one in which he presents his new results in the theory of trigonometric series. It is here that he gives his definition of a function as a correspondence between two sets of real numbers, the same definition that Dirichlet some three years later discovers independently of Lobachevsky and is given the general credit for. This important paper was published in the Scientific Memoirs of the Kazan University.

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