## Osnovania matematicheskoy teorii veroyatnostey [Foundations of the mathematical theory of probability].

St. Petersburg: Imperial Academy of Sciences, 1846.

First edition, very rare, of the first Russian work on probability. “The prime impetus for the initial development in the 1820s of probability theory in the Russian Empire (putting aside the eighteenth-century contributions of Leonhard Euler and Daniel Bernoulli) was the need for a proper basis for actuarial and demographic work, and for the statistical treatment of observations generally. Pierre Simon Laplace’s classic work on probability (*Théorie analytique des probabilités*, 1812), which initiated the Paris school of probabilistic investigations, not only laid foundations for the subject, but also contained applications to real-world situations. Its ideology was brought to the Russian Empire, partly in response to the statistical needs mentioned above, by Viktor Yakovlevich Bunyakovsky (1804–89) … Bunyakovsky’s prime achievement was the first treatise on probability in the Russian language [the offered work]. Its aim was the simplification and classification of existing theory; its lasting achievement was the creation of a Russian probabilistic terminology” (Seneta, Russian probability and statistics before Kolmogorov, Section 10.6 in: *Companion Encyclopedia of the History and philosophy of the Mathematical Sciences*, pp. 1325–26). Bunyakovsky sought to adapt Laplace’s *Théorie analytique des probabilités* (1812) for Russian mathematicians and statisticians. He applied Laplace’s theory to applied mathematics and statistics, and in particular to the statistical control of quality. His work also discussed the analysis of election results and legal decisions, demographics, population increase, compiling mortality tables, and much else. Bunyakovsky studied in Paris where he attended lectures by Laplace. In 1825 he received his doctorate under Cauchy’s supervision. Upon his return to St. Petersburg he devoted the rest of his life to research and teaching. He was elected vice-president of the St. Petersburg Academy of Sciences and held the post for 25 years. Among his numerous outstanding pupils was Pafnuty Lvovich Chebychev (1821-1894), one of Russia’s greatest mathematicians. The present copy of Bunyakovsky’s book is bound up with Chebyshev’s fundamental work on approximation theory and orthogonal polynomials, *Sur l’interpolation dans le cas d’un grand nombre de données fournies par les observations* (St Petersburg: Imperial Academy of Sciences, 1859), in which he introduced the famous ‘Chebyshev polynomials’. No other copy of either work located in auction records.

“Buniakovsky’s book [the offered work] is his main contribution to the theory of probability. Here (p. ii) he stated that, while following Laplace, he had sought to simplify its exposition. Buniakovsky also expressed a justified hope that he succeeded in making easier the study of the *Théorie analytique*, a classic which ‘is intelligible [only] to very few readers’ …

“In the Introduction (p. 3), Buniakovsky indicated that some events were more likely than others and called probability the measure of likelihood … Following Laplace (p. 176), he maintained that ‘the analysis of probabilities considers and quantitatively estimates even such phenomena … which, due to our ignorance, are not subject to any suppositions’ …

“If factor *x* in the expression for the expectation of a continuous random variable *X* is replaced by log *x*, the new quantity will be the ‘moral expectation’ of *X*. Daniel Bernoulli made use of moral expectation, if not the term itself, in order to study the Petersburg paradox, an imaginary game of chance whose investigation by means of mathematical expectation patently contradicted common sense. He also noted that an equal distribution of a given cargo on two ships increases the moral expectation of the freight owner’s capital as compared with the transportation of the cargo on a single ship. Buniakovsky (pp. 103-122) described Bernoulli’s reasoning and proved the validity of his remark …

“Following Buffon and Laplace, Buniakovsky (pp. 137-143) considered two versions of the celebrated problem concerning the ‘Buffon needle’: a needle falls from above on a number of equally spaced parallel lines; it is required to determine the probability that the needle intersects a line … Buniakovsky solved one more problem of this kind. This time he considered the fall of the needle on a system of congruent equilateral triangles and determined the probability that the needle should intersect at least one side of the system …

“Buniakovsky (pp. 132-137) solved a problem unusual for his time by calculating the probability that the equation

*x*^{2} + *px + q* = 0

with coefficients *p* and *q* having random integral values ±1, ±2, …, ±*m*, has real roots … Buniakovsky noted that, as *m* tends to infinity, the probability sought tends to 1 …

“’Given the position of two squares on … a chess-board, it is required to determine the probability that a castle standing on one of these squares reaches the other one in *x *moves’ (pp. 143-147) … The castle is to move over the board at random, but in accordance with the rules of the game … Thus, Buniakovsky considered a problem concerning generalized random walks. In spite of its elementary nature, this fact deserves to be placed on record. Indeed, it is possible to identify a number of games of chance with a one-dimensional random walk of a particle. However, random walks in their proper sense were hardly considered before Buniakovsky …

“Buniakovsky appended to his treatise a study of military losses (pp. 455-469). In 1850, he published it as a separate memoir. Let *n* soldiers be selected at random from all the men in a detachment *N*, and suppose that by a certain moment of an engagement *i* of these *n* men are put out of action. What will be the *probable *number of casualties, Buniakovsky asked (p. 456) … The *probable* number of casualties will be *iN/n*, and the main question thus reduces itself to determining the probability that this number will belong to a certain interval …

“Buniakovsky (p. 36) maintained that, according to [the law of large numbers], the difference between the theoretical and statistical probabilities tends to zero as the number of trials increases … Again, referring to Laplace and obtaining, practically speaking, the same result, Buniakovsky derived the De Moivre-Laplace integral limit theorem (with a correction term), calling it the Bernoulli theorem … The Poisson form of the law of large numbers did not earn recognition all at once; Buniakovsky (p. 35), however, was one of the first to mention it.

“Buniakovsky devoted more than 60 pages of his treatise to the mathematical treatment of observations. At first he studied the distribution of the arithmetical mean and, in general, of a linear function of errors of observation. Following Laplace and supposing that the errors were distributed over a finite interval either uniformly or according to an arbitrary even law, Buniakovsky proved the relevant limit theorems … In addition, he briefly described the method of least squares …

“Buniakovsky allotted another 60 pages of his treatise to the treatment of the results of elections, to the study of testimonies and legends and of decisions passed by tribunals. Suppose that out of *s *witnesses whose testimonies have the same probability of truth *p* exceeding ½, *r* maintain that a certain fact did occur whereas the rest declare the opposite. The probability that the first group of witnesses tells the truth … coincides with the probability of a unanimous statement made by 2*r – s* people … Thus, Buniakovsky continued, the case of *x* = 212 and *r* = 112 is equivalent to having *r = s* = 12 … Buniakovsky borrowed his numerical example from Laplace who used it to illustrate decisions arrived at by a jury consisting of 12 (or 212) members, provided that for each juror the probability of making a mistake was variable [but less than ½] … In addition, Buniakovsky took into account the [prior] probability of the fact being testified to, as though considering the testimony of a new witness. Allowing for the possibilities that the witnesses are mistaken and deceived or are mistaken and tell the truth, etc. (four cases in all) and, following Laplace, he determined the probability that the fact had actually happened … Buniakovsky paid special attention to the case of an unlikely event. Suppose (p. 314) that two eye-witnesses maintain that letters selected from an alphabet of 36 letters made up the word *Moskva*. Assuming that the witnesses were equally trustworthy, that the letters were drawn at random, and finally that the total number of reasonable six-letter Russian words was 50,000, Buniakovsky determined the probability that the witnesses’ account was true …

“Buniakovsky included in his treatise a good essay on the history of probability … Buniakovsky was one of the first to publish a study of this kind … Laplace himself described the same subject in a section of his *Essai philosophique sur les probabilités*, but his exposition was hardly successful: he rarely referred to definite sources and, furthermore, the complete lack of formulas impeded reading …

“Buniakovsky (pp. 173-213) discussed the main problems of population statistics. He described various methods of compiling mortality tables; he studied the increase of population resulting, in particular, from the weakening of, or deliverance from, a certain cause of mortality; he calculated the expected and probable durations of marriages (and associations) … In addition, Buniakovsky solved two special problems:

- Suppose that, in a given nation,
*p*boys and*q*girls were born during a certain period. [Determine] the probability that a newly-born baby will be a boy. - Denote the population of a small part of a country by
*m*, the number of yearly births in this part by*n*, and by*N*the total number of yearly births in the country. It is required to estimate the entire population of the country, roughly equal to*mN/n*…

“The chief difficulty in both these problems was to estimate the appropriate integrals and, especially, to represent the values of the incomplete beta-functions by the integral of the exponential function of a negative square. Like Laplace, Buniakovsky was content with fairly low accuracy in his calculations. Later mathematicians, who strove for much greater precision, had to overcome considerable obstacles …

“Buniakovsky’s contemporaries did not follow up on his concrete achievements but his contributions for a few decades exerted exceptional influence on the teaching of the theory of probability in Russia. ‘This thorough and clearly written source [the offered work], one of the best in European mathematical literature on the theory of probability, considerably helped to disseminate interest in this discipline among Russian mathematicians and to raise the importance of its teaching in Russian universities to a higher level as compared to the academic institutions of other nations’ [anonymous review of the present work in *Finsk. Vestnik* 16 (1847), pp. 39-44] … Markov [The bicentennial of the law of large number (1914), p. 162] considered Buniakovsky’s writing a ‘beautiful work’, and Steklov [Markov, *Izv. Russ. Akad. Nauk* 16 (1922), p. 177] believed that for his time Buniakovsky had compiled ‘a complete and outstanding treatise’” (Sheynin).

Sheynin, ‘On V. Y. Buniakovsky’s work in the theory of probability,’ *Archive for the History of Exact Science* 43 (1991), pp. 199-223.

4to (273 x 211mm), pp. [iv], xvii [-xx], 478, [2], with lithographed plate (some foxing, ownership stamp on margins of several leaves). Contemporary quarter sheep (rubbed and worn, small piece at head of spine missing).

Item #5961

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Price:
$7,500.00
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