## Essay d'Analyse sur les Jeux de Hazard. Seconde edition. Revue & augmentée de plusieurs Lettres.

Paris: J. Quilau, 1713.

Extremely rare presentation copy inscribed by the author of the greatly enlarged second edition of the first separately published textbook of probability. This second edition is arguably more important than the first, as it contains the first publication of extensive correspondence between Montmort and Johann I and Nicholas Bernoulli, which includes, among other things, the first ever treatment (pp. 409-412) of a problem in game theory in the modern sense, and even expressed in terms of the ‘minimax method’ used by Von Neumann and Nash two centuries later (Henny, p. 502). Montmort’s solution of this game was included in a letter he wrote to Nicholas Bernoulli on 13 November 1713, but it was given to him by James Waldegrave (1684-1741), later the first Earl Waldegrave, a British diplomat living in Paris who himself published nothing in mathematics. Montmort also discusses the famous ‘St. Petersburg Paradox’, often said to have been first stated by Daniel Bernoulli in 1738. “This correspondence in turn provided an incentive for Nicholas to publish [in 1713] the *Ars conjectandi* of his uncle Jakob I Bernoulli (1655-1705)” (DSB). “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). In this second edition, which is more than twice the length of the first, “the section on combinatorics in the first edition is much expanded and made into the first part (pp. 1-72) of the second edition; then follow the three parts from the first edition with additions and generalizations and, finally, as the fifth part, 132 pages of letters between Montmort and John and Nicholas Bernoulli … The correspondence with Nicholas Bernoulli is fascinating reading. It has the form of a series of challenges of increasing difficulty. It usually begins with a problem set by Montmort, who illustrates the solution by a numerical example without disclosing his proof; Nicholas Bernoulli then provides a proof not only of the original problem but also of a generalization” (Hald, pp. 287 & 292).

*Provenance*: Inscribed on verso of front fly-leaf: “Pour Monsieur Siltri (?) // Par son tres humble et tres obeisant serviteur et malade // Remond de Montmort.”

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’ – how to divide the stakes in a game of two players which is terminated before either player has won; 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” (DSB).

“The second edition of the *Essay* is a greatly improved version of the first, not only because Montmort collected the combinatorial theorems in an introductory part, but also because he added various proofs and generalizations and his correspondence with the Bernoullis. There is no doubt that Montmort was much influence by Nicholas Bernoulli and that the improvements owe much to collaboration with him” (Hald, pp. 291-292).

The second edition begins with an *Avertissement* (pp. xxv-xlii), not present in the first, “Montmort explains that he omitted some proofs in the first edition to stimulate the curiosity of the reader – he certainly succeeded with respect to Nicholas Bernoulli and de Moivre. He adds that he has now included these proofs at the request of some friends.

“In the *Avertissement* Montmort mentions the publication of *De Mensura Sortis* by de Moivre in 1712 [i.e., 1711, in the *Philosophical Transactions*] and quotes from the preface:

‘Huygens was the first that I know who presented rules for the solution of this sort of problems, which a French author has very recently well illustrated with various examples; but these distinguished gentlemen do not seem to have employed that simplicity and generality which the nature of the matter demands; moreover, while they take up many unknown quantities, to represent the various conditions of gamesters, they make their calculation too complex; and while they suppose that the skill of the gamesters is always equal, they confine this doctrine of games within limits too narrow.’

“It is of course a great injustice to characterize Montmort’s work by the remark that he has well illustrated the rules developed by Huygens with various examples. Montmort vigorously refutes the critical remarks by de Moivre, and on pp. 361-370 he gives a ‘review’ of *De Mensura Sortis* and a comparison with his own work. De Moivre regretted his remarks, and in the preface of the *Doctrine of Chances* (1718) he wrote,

‘However, had I allowed my self a little more time to consider it, I had certainly done the Justice to its Author, to have owned that he had not only illustrated Huygens’s Method by a great variety of well chosen Examples, but that he had added to it several curious things of his own Invention … Since the printing of my Specimen [*De Mensura Sortis*], Mr. de Montmort, Author of the *Analyse des jeux de Hazard*, Published a Second Edition of that Book, in which he has particularly given many proofs of his singular Genius, and extraordinary Capacity; which Testimony I give both to Truth, and to the Friendship with which he is pleased to Honour me.’

“The remaining part of the *Avertissement* is taken up by a history of probability theory before 1713” (Hald, p. 291).

The first section of the 1713 edition of Montmort’s book is devoted to combinatorial analysis. This material had been treated in less detail in the first edition, and the results scattered among the treatment of games of chance. “Montmort writes that Pascal (1665) has given the best and most comprehensive exposition of combinatorics. Pascal's work consists of separate sections on the binomial coefficients, the figurate numbers, the binomial expansion, combinations, and the sums of powers of integers, respectively, which entail many repetitions. Montmort gives an integrated theory of these topics, much like Bernoulli's [in *Ars Conjectandi*], but for some results with simpler proofs following Pascal … Montmort’s combinatorial analysis is deeper and more comprehensive than Bernoulli's and requires more perseverance of the reader because of its greater difficulty. Furthermore, because Montmort does not write in the elaborate pedagogical style of Bernoulli, his text did not become as popular as Bernoulli’s” (Hald, pp. 292-3 & 297).

Parts II-IV of the 1713 edition follow parts I-III of the first, but including additional proofs and generalizations. In part II, “Montmort begins by finding the chances involved in various games of cards. He discusses such simple games as Pharaoh, Bassette, Lansquenet and Treize, and then, not so fully or successfully, Ombre and Picquet. The work is easy to read in that he prefaces each section with the rules of the game discussed, so that what he is trying to do can be explicitly understood. Possibly he found it necessary to do this because different versions of the games were in vogue, but this does not always occur to other writers. Having set down the rules, he solves simple cases in a method somewhat reminiscent of Huygens, and then takes a plunge into a general solution which appears to be correct but is not always demonstrably so. The *Problèmes divers sur le jeu du treize* are interesting indeed in that he gives the matching distribution and its exponential limit. Treize has survived today as the children's game of Snap.

‘*The players draw first of all as to who shall be the Bank. Let us suppose that this is Pierre, and the number of players whatever one likes. Pierre having a complete pack of *52* shuffled cards, turns them up one after the other. Naming and pronouncing one when he turns the first card, two when he turns the second, three when he turns the third, and so on until the thirteenth which is the King. Now if in all this proceeding there is no card of rank agreeing with the number called, he pays each one of the Players taking part and yields the Bank to the player on his right. But if it has happened in the turning of the thirteen cards that there has been an agreement, for example turning up an ace at the time he has called one, or a two at the time he has called two, or three when he has called three, he takes all the stakes and begins again as before calling one, then two, etc.*’

“He begins by assuming Pierre has two cards and one opponent, Paul. Then Pierre has three cards, four, and finally any number [say, *p*] … He gives *p* successively values 1, 2, ... , 13 and calculates Pierre's chance at each stage. It is, however, the remarks on this which are interesting. After his calculations he says:

‘*The preceding solution furnishes a singular use of the figurate numbers *(*of which I shall speak later*)*, for I find in examining the formula, that Pierre's chance is expressible by an infinite series of terms which have alternate + and – signs … we have for Pierre's chance the very simple*

1*/*1* – *1*/*1*.*2* + *1*/*1*.*2*.*3* – *1*/*1*.*2*.*3*.*4* + *1*/*1*.*2*.*3*.*4*.*5* – *1*/*1*.*2*.*3*.*4*.*5*.*6* + etc.*’

“This is possibly the first exponential limit in the calculus of probability … [the sum of the series is 1/*e*, where *e* is the base of natural logarithms].

“In the second half of [part II] on Piquet, Ombre, etc. he interpolates a section on problems in combinations. This is all quite sound mathematics, although he takes a very long time to establish the Arithmetic Triangle. The principle of conditional probability, often attributed to de Moivre but probably dating back to the controversy between Huygens and Hudde, is used with facility and understanding …

“In [part III] Montmort discusses the game of Quinquenove and the game of Hazard, remarking about the latter that the game is known only in England … Montmort gives the chances of [the two players in Hazard] and then describes another game, which he says has no name and so he dubs it the game of Hope, and gives some calculations on this also. Backgammon however rather defeats him …

“From an historical point of view, however, there is interest in his game of Nuts … divination among primitive tribes is (and was) carried out by casting pebbles, grain, or nuts, etc. It is also still a puzzle that the same ritual of divination was used in games to while away the idle hour. That this duality of purpose was probably universal, not just European, appears likely from Montmort's discussion on *Problème sur le Jeu des Sauvages, appellee Jeu des Noyaux*. He writes:

‘*Baron Hontan mentions this game in the second book on his travels in Canada, p. *113 [*Nouveaux voyages dans l’Amérique septentrionale, 1703*]*. This is how he explains it. It is played with eight nuts black on one side and white on the other. The nuts are thrown in the air. If the number of black is odd, he who has thrown the nuts wins the other gambler's stake. If they are all black or all white he wins double stakes, and outside these two cases he loses his stake*’” (David, pp. 144-150).

Part III also contains a discussion of a ‘random walk’ problem (this is not in the first edition). “On pp. 248-257 and p. 366 of the second edition, there follows an analysis of the problem of points in a game of bowls. According to de Moivre, this problem was posed to him by Francis Robartes, and de Moivre gave the solution of a special case in *De Mensura Sortis*. Robartes’ problem was generalized by Montmort as follows: Players A and B play a game of bowls, A with *m* bowls and B with *n*. The skill of A is to the skill of B as *r* to *s*. In each game the winner gets a number of points equal to the number of his bowls that are nearer to the jack than any of the loser’s. If the play is interrupted when A needs *a *points and B *b* points in winning, how should the stake be divided equitably between them? … In modern terminology, Robartes’ problem may be described as a random walk in two dimensions, the horizontal steps being of length 1 or 2, …, or *m*, and the vertical steps being of length 1or 2, …, or *n*. A wins if the random walk crosses the vertical line through (*a*,0) before crossing the horizontal line through (0,*b*)” (Hald, pp. 308-9).

Part IV of the 1713 edition treats the five exercises for the reader stated by Huygens in his *De ratiociniis in ludo aleae*. The most interesting of these problems is the last, which is perhaps the first statement of the Gambler’s Ruin problem: ‘A and B start with 12 balls and continue to throw three dice on the condition that if 11 is thrown A gives a ball to B and if 14 is thrown B gives one to A*. *The game is to continue until one or other has all the balls.’

The most important addition to the second edition is the Montmort/Bernoulli correspondence, which constitutes part V. Montmort sent a copy of his *Essay *to John Bernoulli who responded with a letter of 17 March 1710. Bernoulli writes that the *Essay* contains many beautiful, interesting, and useful results and goes on with many detailed remarks on the games Pharaon, Lansquenet, and Treize, on Huygens’ problems, on the multinomial coefficient, the problem of points, and the duration of play … John Bernoulli lent his copy of the *Essay *to his nephew Nicholas, who wrote some important remarks which were sent to Montmort together with John’s letter. A correspondence ensued of which the *Essay *contains seven letters from Nicholas and six from Montmort. In one of these letters (p. 322), Montmort asks the Bernoullis’ permission to include their letters in the new edition of the *Essay* which he is planning … In his last letter, Bernoulli writes that in return for the many problems posed by Montmort he shall now pose five problems for Montmort to solve … The fourth problem is as follows: Player B throws successively with an ordinary die and gets *x* crowns from A if a six occurs for the first time at the *x*th throw; find B’s expectation … The fifth problem, known today as the St. Petersburg Problem, is obtained from the fourth by letting the prize increase in geometric progression, for example as 2^{x - }^{1}. This leads to an infinite expectation … The last of the published letters is from Montmort to Bernoulli and dated 15 November 1713. Montmort writes that his friend, Mr. Waldegrave, is taking care of the printing of the new edition of the *Essay *and that it is nearly finished.

The most important game discussed in the correspondence is ‘Her’ (or ‘Le Her’). “Le Her is a game of strategy and chance played with a standard deck of fifty-two playing cards. The simplest situation is when two players play the game, and the solution is not simply determined even in that situation. Montmort calls the two players Pierre and Paul. Pierre deals a card from the deck to Paul and then one to himself. Paul has the option of switching his card for Pierre’s card. Pierre can only refuse the switch if he holds a king (the highest valued card). After Paul makes his decision to hold or switch, Pierre now has the option to hold whatever card he now has or to switch it with a card drawn from the deck. However, if he draws a king, he must retain his original card. The player with the highest card wins the pot, with ties going to the dealer Pierre … it is obvious that one would want to switch low cards and keep high ones. The key is to find what to do with the middle cards, such as seven and eight” (Bellhouse & Fillion, p. 27).

Bernoulli thought that both players should change cards in the doubtful cases. “Montmort replied that Bernoulli’s solution agrees with his own. However, before asking for Bernoulli’s opinion, he had for some time discussed the problem with two of his friends” (Hald, p. 318). “These were an English gentleman named Waldegrave and an abbot whose abbey was only a league and a half from Château de Montmort (p. 338). Montmort identified Waldegrave only as the brother of the Lord Waldegrave who married the natural daughter of King James II of England” (Bellhouse & Fillion, p. 27). This was James, later 1^{st} Earl Waldegrave (1684-1741). “In his last letter on Her, Montmort writes that he is now convinced that the original considerations by himself and Bernoulli do not represent a solution because they presuppose that [Peter] knows the strategy of [Paul] … [Montmort] concludes that it is impossible to solve the problem. However, before he has finished his letter he receives one from Waldegrave with the solution, which he includes in his letter to Bernoulli [pp. 409-412]” (Hald, p. 319).

Waldegrave introduced the idea of ‘mixed strategies’, a basic feature of modern game theory. If a player has several strategies, instead of selecting one of them the player elects to use strategy 1 in a certain fraction of repeated plays of the game, strategy 2 in another fraction, etc. – like probabilities, these fractions must add up to 1. “Waldegrave considered the problem of choosing a strategy that maximizes a player’s probability of winning, whatever strategy may be chosen by his opponent … The resultant matrix of probabilities for a win by Peter, given the mixed strategy chosen, persuaded Waldegrave that a player could select a strategy assuring him of a certain outcome, while the other player could prevent him from doing better. Waldegrave concluded that Peter should hold cards of 8 and over (and change lower cards) with probability 5/8, and should change cards of 8 and under with probability 3/8. Paul should hold 7 and over with probability 3/8, and change 7 and under with probability 5/8 … Waldegrave’s solution of Le Her was a minimax solution, but he made no extension of his result to other games, and expressed concerns that a mixed strategy ‘does not seem to be in the usual rules of play’ of games of chance. He abandoned mathematics for diplomacy after leaving France for England in 1721” (Dimand & Dimand, p. 122).

Other problems discussed in the Montmort/Bernoulli correspondence include the problem of coincidences, the gambler’s ruin problem, and the problem of points for the game of tennis. Tennis was discussed in a Letter appended to James Bernoulli’s *Ars conjectandi*, but Nicholas had access to it before it was published in 1713.

“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 died 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.

Bellhouse & Fillion, https://arxiv.org/pdf/1504.01950.pdf. David, *Games, Gods and Gambling*, 1962. Dimand & Dimand, *The History Of Game Theory, Volume 1*: *From the Beginnings to 1945*, 1996. Hald, *A History of Probability and Statistics and their Applications before 1750*, 2003. Henny, ‘Niklaus und Johann Bernoullis forschungen auf dem gebiet der wahrscheinlichrechnung in ihrem briefwechel mit Pierre Remond de Montmort.’ In: *Die Werke von Jakob Bernoulli*, Bd. 3 (1975), pp. 457–507. Todhunter, *History of the Theory of Probability*, 1865.

4to (256 x 185 mm), pp. xlii, 414, [2], with five folding engraved plates. Contemporary calf, spine richly gilt with red lettering-piece (discreet repairs to joints and spine ends).

Item #5063

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Price:
$19,500.00
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