## Essay d'Analyse sur les Jeux de Hazard.

Paris: J. Quilau, 1708.

A fine copy, of the first separately published textbook of probability. “In 1708 [Montmort] published his work on Chances, where with the courage of Columbus he revealed a new world to mathematicians” (Todhunter, p. 78). “The *Essay* (1708) is the first published comprehensive text on probability theory, and it represents a considerable advance compared with the treatises of Huygens (1657) and Pascal (1665). Montmort continues in a masterly way the work of Pascal on combinatorics and its application to the solution of problems on games of chance. He also makes effective use of the methods of recursion and analysis to solve much more difficult problems than those discussed by Huygens. Finally, he uses the method of infinite series, as indicated by Bernoulli (1690)” (Hald, p. 290). “Montmort’s book on probability, *Essay d’analyse sur les jeux de hazard*, which came out in 1708, made his reputation among scientists” (DSB). Based on the problems set forth by Huygens in his *De Ratiociniis in Ludo Aleae* (1657) (published as an appendix to Frans van Schooten’s *Exercitationum mathematicarum*), the *Essay* spawned Abraham de Moivre’s two important works *De Mensura Sortis* (1711) and *Doctrine of Chances* (1718), as well as Jacob I Bernoulli’s celebrated *Ars Conjectandi* (1713). ABPC/RBH list just two copies of this first edition (Christie’s 1981 and Hartung 1987).

The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the ‘Problem of points’; this was published in Fermat’s *Varia Opera* (1679). Pascal included his solution as the third section of the second part of his 36-page *Traité du triangle arithmétique* (1665), which was essentially a treatise on pure mathematics. “Huygens heard about Pascal’s and Fermat’s ideas [on games of chance] but had to work out the details for himself. His treatise *De ratiociniis in ludo aleae* … essentially followed Pascal’s method of expectation. … At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are Bernoulli’s *Ars conjectandi* (1713), Montmort’s *Essay d'analyse sur les jeux de hazard* (editions in 1708 and 1711 [i.e., 1713]) and De Moivre’s *Doctrine of Chances* (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermat’s combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascal’s method of expectations.” (Glenn Schafer in *Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences* (1994), Grattan-Guiness (ed.), p. 1296).

“It is not clear why Montmort undertook a systematic exposition of the theory of games of chance. Gaming was a common pastime among the lesser nobility whom he frequented, but it had not been treated mathematically since Christiaan Huygens’ monograph of 1657. Although there had been isolated publications about individual games, and occasional attempts to come to grips with annuities, Jakob I Bernoulli’s major work on probability, the *Ars conjectandi*, had not yet been published. Bernoulli’s work was nearly complete at his death in 1705; two obituary notices give brief accounts of it. Montmort set out to follow what he took to be Bernoulli’s plan …

“[Montmort] continued along the lines laid down by Huygens and made analyses of fashionable games of chance in order to solve problems in combinations and the summation of series. For example, he drew upon the game that he calls “treize,” in which the thirteen cards of one suit are shuffled and then drawn one after the other. The player who is drawing wins the round if and only if a card is drawn in its own place, that is, if the *n*th card to be drawn is itself the card *n*. In the generalized game, the pack consists of *m* cards marked in serial order …

“The greatest value of Montmort’s book lay perhaps not in its solutions but in its systematic setting out of problems about games, which are shown to have important mathematical properties worthy of further work. The book aroused Nikolaus I Bernoulli’s interest in particular and the 1713 edition includes the mathematical correspondence of the two men. This correspondence in turn provided an incentive for Nikolaus to publish the *Ars conjectandi* of his uncle Jakob I Bernoulli …

“The work of De Moivre is, to say the least, a continuation of the inquiries of Montmort. Montmort put the case more strongly—he accused De Moivre of stealing his ideas without acknowledgment. De Moivre’s *De mensura sortis* appeared in 1711 and Montmort attacked it scathingly in the 1713 edition of his own *Essay*. Montmort’s friends tried to soothe him, and largely succeeded. He tried to correspond with De Moivre, but the latter seldom replied. In 1717 Montmort told Brook Taylor that two years earlier he had sent ten theorems to De Moivre; he implied that De Moivre could be expected to publish them” (DSB).

“The value of Montmort’s work resides partly in his scholarship. He was well-versed in the work of chance of his predecessors (Pascal, Fermat, Huygens), met Newton on one of a number of visits to England, corresponded with Leibniz, but remained on good terms with both sides during the strife between their followers. The summation of finite series is an element of Montmort’s mathematical interests which enters into his probability work and distinguishes it from the earlier purely combinatorial problems arising out of enumeration of equiprobable sample points. Although the *Essay* to a large extent deals with the analysis of popular gambling games, it focuses on their mathematical properties and is thus written for mathematicians rather than gamblers … Montmort’s best-known contribution to elementary probability is a result connected with the card games *Rencontre, Treize*, and *Snap*, in which *n* distinct objects are assigned a specific order, while *n* matching objects are assigned a random order … Montmort also worked with Nicolaus [I Bernoulli] on the problem of duration of play in the gambler’s ruin problem, possibly prior to de Moivre, and at the time the most difficult problem solved in this subject area” (Heyde & Seneta, *Statisticians of the Centuries*, p. 53).

“Pierre Rémond de Montmort (1678-1719) was born into a wealthy family of the French nobility. As a young man he traveled in England, the Netherlands, and Germany. Shortly after his return to Paris in 1699 his father dies and left him a large fortune. He studied Cartesian philosophy under Malebranche and studied the calculus on his own. … Montmort corresponded with Leibniz whom he greatly admired. He was also on good terms with Newton whom he visited in London. In 1709 he printed 100 copies of Newton’s *De Quadratura* at his own expense … through John Bernoulli, he also offered to print *Ars Conjectandi*. He was on friendly terms with Nicholas Bernoulli and Brook Taylor” (Hald, pp. 286-7). The Royal Society elected Montmort a Fellow in 1715 and the *Academic Royale des Sciences* made him an associate member (as he was not a resident of Paris) the following year.

For detailed accounts of the work see David, *Games, Gods and Gambling* (1962), Chap. 14; Hald, *A History of Probability and Statistics and their Applications before 1750*, Chap. 18; Todhunter, *History of the Theory of Probability* (1867), Chap. 7. Sotheran's 3059 (‘rare’).

4to (261 x 197 mm), pp [i-iii] iv-xxiv, 189, [3], with three folding plates.Contemporary vellum, red morocco title label to spine, engraved book plate of Sir Francis Hopkins to front paste-down. Uniform very light browning throughout, paper flaw in G3. Rare in such good condition.

Item #3894

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Price:
$12,000.00
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