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XIXth century mathematics. Mathematical logic. Algebra. Number theory. Probability theory. Ed. by A. N. Kolmogorov and A. P. Yushkevich. (Математика {XIX} века. Математическая логика. Алгебра. Теория чисел. Теория вероятностей.) (Russian) Zbl 0523.01001

Moskva: “Nauka”. 255 p. R. 2.10 (1978).
This volume is the first of a series under the general editorship of A. N. Kolmogorov and A. P. Yushkevich to cover the history of 19th century mathematics. The first chapter, “Mathematical logic” by Z. A. Kuzicheva (pp. 11–38), makes a bad start, as she writes almost entirely on the tradition of Boolean algebra as it led to algebraic logic. “Algebra and algebraic number theory” (pp. 39–122) then follows, principally from the hands of I. G. Bashmakova and A. N. Rudakov. The usual ground is covered: Gauss’s Disquisitiones, Abel and Galois, linear algebra with Cayley and Sylvester, and so on: the essential roots in analysis for many of these developments are once again passed over by historians. “Problems in the theory of numbers” (pp. 123–183) by E. P. Ozhigova with A. P. Yushkevich concentrates on quadratic and related forms, analytic number theory, and transcendental numbers; overlap with the previous chapter is somewhat uneasy. The final chapter, “Theory of probability”, by B. V. Gnedenko and O. B. Sheĭnin attends to the special aspects and applications of the subject, but somewhat overlooks the specifically legal components.
The volume contains illustrations of some principal figures, and a name index; but the lack of a subject index is unfortunate. If this first volume is to be typical of the series, then we will have here a source which lies nicely between general histories of mathematics and specialist monographs. While the text sits intellectually closer to the former than the latter source (the citation of specialist secondary Western literature is patchy), it is a welcome addition to the scanty literature on the 19th century.

MSC:

01A05 General histories, source books
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A55 History of mathematics in the 19th century
03-03 History of mathematical logic and foundations
11-03 History of number theory
15-03 History of linear algebra
60-03 History of probability theory