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First edition, extremely rare offprint, of the foundation work of approximation theory - the problem of how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. "Starting out from the problem of determining the parameters of the driving mechanism of steam engines - also called Watt's parallelogram - in such a manner that the conversion of straight into circular movement becomes as exact as possible everywhere, he was led to the general problem of the uniform approximation of a real analytic function by polynomials of a given degree. The first goal he achieved was the determination of the polynomial of nth degree with given first coefficient which deviates as little as possible from zero over the interval [-1, 1]. Today this polynomial is called a Chebyshev polynomial of the first kind. Further results were presented by Chebyshev in [the offered] work, where he stated a very general problem: that of determining parameters p, q, . . . of a real-valued function F(x, p, q, . . . ) so that over a given interval [a, b] the maximum [value of F] is minimized . . . he was able to prove a generally necessary condition for the solution of the problem. Using this condition he showed that in special cases (polynomial, weighted polynomial and rational approximation) it led to the necessary condition that F has a fixed number of points where it assumes the maximum value. These points are now known as deviation points . . . The aim he sought to achieve with this contribution is to find the polynomial uniformly deviating as little as possible from zero for any number of given coefficients. Later his pupils would work on several problems arising from this general challenge. This remained the determining element of all contributions of the early St. Petersburg Mathematical School on the subject of approximation theory. [The present work] was the only work by Chebyshev devoted to a general problem of uniform approximation theory. But it was followed by a series of more than 40 publications in which he dealt with the solution of special uniform approximation problems, mainly from the theory of mechanisms" (Steffens, The History of Approximation theory (2006), pp. viii-ix). Chebyshev (1821-94) is regarded as the creator of the largest pre-revolutionary school of mathematics in Russia. He is the author of 80 or so publications; they span a wide area of mathematics, namely approximation theory, probability theory, number theory, theory of mechanisms, as well as many problems of analysis and practical mathematics. OCLC lists 4 copies in US (Columbia, Library of Congress, New York Public Library, Huntington). Large 4to, pp. [ii], [1], 2-91, [1, blank]. Pink paper spine strip. Seller Inventory # ABE-1665320812431
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