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This paper is concerned with (a) a brief outline of Chebyshev's life; (b) certain background material in connection with Chebyshev's work on approximation and integration in finite terms; (c) the question of whether he already was in Paris in 1842, 10 years prior to his established and presumed first trip to Western Europe; d) his contacts with Catalan, Liouville, Hermite, Lucas, d'Ocagne, Laussedat, and Dwelshauvers-Dery; (e) his contacts with Dirichlet, Borchardt, Kronecker, and Weierstrass; and (f) Chebyshev's last two trips to the West. It is argued that the great Russian scientist did not work in isolation at St. Petersburg. Instead, he was in personal contact, at least until 1884, with many of the greatest European scientists of the time.

In this paper we trace the early development of a method for finding the approximate solutions —called the WKB solutions—for a class of ordinary differential equations of second order. We also analyze the attempts made by the various contributors to this method to substantiate their results. These approximating solutions were subsequently shown to be asymptotic in the sense of Poincare. Also presented and examined are the several methods used to deal with the “connection problem” which arises in the use of the WKB solutions.

One of the significant contributions of Chinese mathematicians is the method of solving numerical equations of higher degree. A number of scholarly works have established similarities between ancient and medieval Chinese root-extraction procedures and Horner's method of solving a numerical equation of any degree. The conceptual development of the Chinese methods, which began with the procedures of extracting square and cube roots during the Han dynasty, culminated in the solution of higher numerical equations in the 13th and the beginning of the 14th centuries. This paper attempts to show the intimate role played by the triangular array of numbers (known in the West as the Pascal triangle) in the process of the development of the Chinese methods, especially when the original geometrical concept was being replaced by the more general algebraic method.