Sposob uvi︠e︡rsitʹsi︠a︡ v izchezanīi bezkonechnykh strok i priblizhatʹsi︠a︡ k znachenīi︠u︡ funkt︠s︡īĭ ot vesʹma bolʹshikh chisel [Cyrillic] [A method for ascertaining the convergence of infinite series and for obtaining approximate values of functions of a large number of variables].

Kazan: Universitetskai︠a︡ tip. 1835.

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. “As early as 1835, Lobachevsky showed in a memoir [the offered work] the necessity of distinguishing between continuity and differentiability” (Cajori, History of Mathematics, p. 421). This paper was published in the Scientific Memoirs of Kazan University, 1835, No. 2, pp. 211-342. 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. It is, for example, worth noting that in his algebra Lobachevsky suggested a method of separating roots of equations by their repeated squaring, a method coincident with that suggested by Dandelin in 1826 and by Gräffe in 1837. His paper on the convergence of trigonometric series, too, suggested a general definition of function like that proposed by Dirichlet in 1837. (Lobachevsky also gave [in the offered paper] a rigorous definition of continuity and differentiability, and pointed out the difference between these notions)” (DSB).

“The mathematicians of the eighteenth century did not touch the question of the relation between continuity and differentiability, presuming silently that every continuous function is eo ipso a function having a derivative. Ampère tried to prove this position, but his proof lacked cogency. The question about the relation between continuity and differentiability awoke general attention between 1870 and 1880, 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 1835: ‘A method for ascertaining the convergence, etc.’ [the offered paper]. 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. 242). Lobachevsky’s interest in the foundations of calculus seems to have been awakened when he read Bernard Bolzano’s important 1817 work Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werten, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege, in which he gave the first rigorous proof of the intermediate-value theorem (see Grattan-Guinness, p. 52, note).

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, … such that u(p1) = 2-1, u(p2)= 2-2, u(p3) = 2-3… Then, the infinite series u(1) + u(2) + u(3) + … converges if and only if series p12-1 + p22-2 + p32-3 + … converges.

Lobachevsky also treats the problem of expressing functions by infinite products. He asserts that, if a function F(x) vanishes only when x = f (n), for n = 1, 2, 3, ... , then,

F(x) = A(1-x/f(1))(1-x/f(2))(1-x/f(3)) …

where A is a constant. 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. One year later, Lobachevsky devoted a whole memoir to this subject, ‘Primenenie voobrazhaemoi geometrii k nekotorym integralam’ (‘Application of Imaginary Geometry to Certain Integrals’).

“Lobachevsky’s non-Euclidean geometry was the product of some two millennia of criticism of the Elements. 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. 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” (DSB).

Engel (ed.), Zwei geometrische Abhandlungen, Leipzig, B. G. Teubner, 1898-9, pp. 417-9; I. Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann, 1970; G. B. Halsted, ‘Biology and Mathematics,’ Twelfth Annual Report of the Ohio State Academy of Science, 1903, pp. 239-47; A. Vasiliev, Nicolai Ivanovich LobachevskyAddress pronounced at the commemorative meeting of the Imperial University of Kazan, October 22, 1893, G. B. Halsted (tr.), 1894.

8vo, pp. 134. Original printed wrappers with printed title enclosed in decorative border (wrappers damaged with partial loss of border, though not affecting text, first few leaves chipped at corners).

Item #3652

Price: $8,000.00