## ‘De Mensura Sortis, seu, de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus,’ pp. 213-264 in Philosophical Transactions, Vol. 27, No. 329. For the months of January, February, and March 1711.

London: printed for H. Clements … and W. Innys … and D. Brown, [1712].

First edition of de Moivre’s first published work on probability, and the first original work on the subject published in Britain, a precursor to his *Doctrine of Chances* which appeared seven years later. De Moivre’s paper occupies a complete issue of the *Philosophical Transactions*, as offered here. “[De Moivre’s] work on the theory of probability surpasses anything done by any other mathematician except Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Daniel Bernoulli’s theorem by the aid of Stirling’s theorem” (Cajori, p. 245). “The only systematic treatises on probability printed before 1711 were Huygens’ *De ratiociniis in ludo aleae* (1657) and Pierre Rémond de Montmort’s *Essay d’analyse sur les jeux de hazard* (1708). Problems which had been posed in these two books prompted de Moivre’s earliest work and, incidentally, caused a feud between Montmort and de Moivre on the subject of originality and priority” (DSB). “Nearly all of *De Mensura Sortis* was later incorporated into de Moivre’s book *The Doctrine of Chances* (1718, 1738, 1756), which was the most important textbook on probability theory until the publication of Laplace’s *Théorie Analytique des Probabilités* (1812). In the preface de Moivre states that he began his work on probability theory at the exhortation of Francis Robartes, who asked him to solve the division problem for two gamesters playing bowls and also to find the probability of getting certain given faces as the outcome of a given number of throws with a die. He also states that he had previously read the books by Huygens and Montmort ‘but these distinguished gentlemen do not seem to have employed that simplicity and generality which the nature of the matter demands.’ Furthermore he writes that ‘while they suppose that the skill of the gamesters is always equal, they confine this doctrine of games within limits too narrow.’ Finally his remarks about Montmort may be read as if Montmort had used only the method of Huygens on some new examples. These rash remarks naturally provoked a dispute with Montmort” (Hald 1984, pp. 230-1). “The most remarkable of de Moivre’s contributions [in *De mensura sortis*] are his derivation of the ruin probability in Huygens’ fifth problem; his use of the Poisson approximation to solve the binomial equation B(*c, n, p*) = ½ with respect to *n*; his solution of the occupancy problem by means of the method of inclusion and exclusion, and the algorithm for the continuation probability in the duration of play for the ruin problem. Furthermore, he gives without proof the probability of getting a given number of points by throwing any given number of dice and the probability of ruin when one of the players has infinitely many counters. The only contemporary evaluation of these impressive results is the critical review given by Montmort in a letter of 5 September 1712 to Nicholas Bernoulli, about a month after Montmort had received a copy of the paper from de Moivre … [Montmort] recognizes de Moivre’s priority to the Poisson approximation, to Robartes' problem, and to the algorithm for finding the continuation probability in the problem of the duration of play” (Hald 2003, pp. 403-4). No copies in auction records.

De Moivre’s interest in probability was awakened by Francis Robartes (1649-1718), Member of Parliament and scion of an aristocratic family. In 1692, Robartes wrote a manuscript on two probability problems that he presented to the Royal Society but never published, and in the following year he succeeded in publishing another paper on probability. In 1710 Robartes helped John Harris with his article entitled “Play” in Harris’s scientific dictionary *Lexicon Technicum*. Robartes devised an algorithm that Harris used to extract the appropriate terms in a binomial expansion in order to solve the problem of the division of stakes. At some point over the years 1708 to 1710, Robartes received a copy of Montmort’s *Essay*, which he showed to de Moivre. He also gave de Moivre three challenge problems of his own devising to work on. “Once de Moivre had solved the first problem, within a day of Robartes posing it, Robartes gave de Moivre the other two problems to work on, while at the same time encouraging him to write on probability. The encouragement proved fruitful. De Moivre finished his manuscript on probability during a holiday that he spent at a country house, possibly Robartes’s. On June 11, 1711, de Moivre submitted his manuscript to the Royal Society. The Society’s Journal Book quietly marked the beginning of a new era for probability in England with the note, ‘Mr. De Moivre presented a Treatise Intituled, de Probabilitate Eventum in Ludo Alea, This Treatise was Ordered to be printed in the Transactions.’ The treatise, with the title *De Mensura Sortis *or ‘Of the measurement of lots,’ comprises an entire issue (Number 329) of *Philosophical Transactions*. At 52 journal pages, it is more than three times longer than anything else de Moivre had written to that date” (Bellhouse, p. 70).

De Moivre begins *De Mensura Sortis *with two basic definitions from which many of his results are derived. The first comes directly from Huygens’s *De ratiociniis*. For two players, A and B, contending for a stake of value *a*, A has *p *chances to win and B has *q*. The expected value for each player follows what Huygens obtained: *ap*/( *p *+ *q*) for A and *aq*/( *p *+ *q*) for B. The second definition may have come from Edmond Halley or Francis Robartes. If an event can happen in *p *ways and fail to happen in *q*, and a second event can happen in *r *ways and fail to happen in *s*, then all the chances for events happening or failing are in the product (*p *+ *q*)(*r *+ *s*) or *pr *+ *qr *+ *ps *+ *qs*. For example, *pr *is the number of ways both events can happen and *ps *is the number of ways that the first event happens and the second fails. This is the approach that Halley used in evaluating joint life annuities in his 1693 paper on mortality data from the city of Breslau ...

“De Moivre finishes the introduction by saying that if the first event is repeated *n *times, then the total number of chances in the game is given in the binomial expression (*p *+ *q*)* ^{n}*. When this expression is expanded, it may be written as a sum containing terms of the form

*p*multiplied by an appropriate coefficient, where

^{i}q^{n-i}*i*represents the number of times the event happens and

*n*–

*i*represents the number of times it fails [the sum of the first

*c*terms of this expansion is denoted

*B*(

*c, n, p*)] … The binomial expansion becomes the motif for the paper … At the beginning (nine of the first ten problems) there are some simple variations on the use of the expansion of (

*p*+

*q*)

*and then at the end (the last seven problems) the expansion is used to solve a very complex problem, the problem of the duration of play. In the middle, there are several solutions to a number of challenge problems taken from various sources, including the three from Robartes” (Bellhouse, p. 73).*

^{n}Problems 1, 3 and 4 are relatively straightforward applications of the binomial distribution. Problem 1 is to find the chance of throwing an ace two or more times in 8 throws with a single die. Problem 3 is to determine the chances of A and B winning a single game, supposing that A can give B two games out of three. Problem 4 is similar.

In problems 5 to 7 de Moivre considers the problem of finding the number of trials that gives an even chance for getting at least one success but fewer than some given number of successes, *c* say. This means he has to solve the equation *B*(*c – *1*, n, p*) = ½ for *n* when *c* and *p* are given. De Moivre considers the two extreme cases *p* = ½ and *p* tends to 0, of which the first is easy by symmetry. De Moivre shows that, as *p *tends to 0, *B*(*c – *1*, n, p*) tends to *e ^{-m}*multiplied by the sum of the first

*c*terms in the series expansion of

*e*in powers of

^{m}*m*, where

*m = np*/(1 –

*p*) [de Moivre was not able to express the result this way because our notation for the exponential function had not yet been invented.] This result, the ‘Poisson approximation’ to the binomial distribution, played a very important role in later developments. “There has been some discussion of whether it is reasonable to contend that de Moivre found the Poisson distribution” (Hald 1984, p. 231), more than a century before Simeon-Denis Poisson.

Problems 2, 3, 4, and 10 are on the division of stakes, or ‘problem of points’. “Consider a series of games with two players, A and B, where in each game A has probability *p* and B probability *q* = 1 – *p *of winning a point. If play stops when A lacks *a* points and B lacks *b *points in winning, how should the stake be divided between them? De Moivre proves that A’s probability of winning equals the sum of the last *b* terms of the expansion of (*p + q*)^{a+b}^{-1}, and B’s probability of winning equals the remaining *a* terms. This result had already been derived by Johann Bernoulli in 1710 in a letter to Montmort, but it was not published until 1713 … In Problem 8 de Moivre generalizes to *k* players, say, and gives the solution as the sum of the appropriate terms of the multinomial (*p*_{l} + *p*_{2} + ... + *p*_{k})^{n+l-k}, *n* being the total number of points lacking. He points out that certain terms have to be divided among the players depending on the permutation of the *p*’s.

“In Problems 16 and 17 he gives the solution of Robartes’s problem: the division problem for two gamesters playing bowls. In each game B, say, gets a number of points equal to the number of his bowls which is nearer to the jack than any of A's bowls. By combinatorial methods de Moivre finds the probability of getting *i *points in a single game assuming that the players have the same number of bowls and are of the same skill. The division problem is then solved by recursion” (Hald 1984, pp. 231-2).

Problems 11, 12, and 13 are related to the first two of the five problems Huygens posed at the end of his *De ratiociniis*. “In these problems the players take turns in a specified order until one of them wins. De Moivre gives the solution as the sum of an infinite series” (*ibid*., p. 232).

Problem 15 is ‘Waldegrave’s problem’ (James Waldegrave (1684-1741), later the first Earl Waldegrave, was a British diplomat living in Paris who himself published nothing in mathematics). “Let there be *n* + 1 players, A_{1}, …, A_{n}_{+1 }of equal skill. Players A_{1} and A_{2} play a game, and the loser pays a crown to a common stock and does not enter the play again until all the other players have played; the winner plays against A_{3} and the loser pays a crown to the stock, and so on. If the winner of the first game beats all the rest, the play is finished; if not, the play goes on, each player coming in again in turn until one player has beaten in succession all the other players, and he then receives all the money in the stock.The problem is to determine

- the probability of each player winning the stock;
- the expectation of each player; and
- the probability of a given duration of the play.

“In a letter to Bernoulli of I0 April 1711, Montmort writes that the problem has been proposed to him and also solved by Waldegrave for three players. Independently, de Moivre formulated and solved the problem for three p1ayers in *De Mensura Sortis* (1712)” (Hald 2003, p. 378). “De Moivre solves this problem by means of conditional expectations. First he supposes that A beats B in the first game. On this assumption the play may end with A as winner in the second, fifth, eighth, ... game. The probabilities of A for reaching these games and winning are ½, (½)^{4}, (½)^{7}, … Hence the expected stake plus fines may be found and subtracting A’s expected fine his (conditional) expectation results. Under the same assumption B’s expectation is obtained. The unconditional expectation is then found as the average of the two conditional expectations” (Hald 1984, p. 232).

Problems 18 and 19 are ‘occupancy problems’, the third type of problem posed to de Moivre by Robartes: Find the probability *p _{n}* that

*f*specified faces occur at least once in

*n*throws with a die having

*k*faces. De Moivre calculates

*p*by means of the method of ‘inclusion and exclusion’. In Problem 19 he solves the equation

_{n}*p*= ½ under the assumption that

_{n}*f*is small compared to

*k*.

Problem 9 is a generalization of the ‘gambler’s ruin problem,’ the fifth of Huygens’s problems in *De ratiociniis*. “Consider two players A and B having *a* and *b* counters, respectively. In each game A has probability *p* and B has probability *q* = 1 – *p* of winning and the winner gets a counter from the loser. The play continues until one of the players is ruined. What is the probability of A being ruined? Huygens’s fifth problem is obtained for *a = b* = 12” (*ibid*.). Problems 20-26 are a continuation of the discussion of the ruin problem: what is the probability that the play ends at the *n*th game or before? Problem 25 is the case when A has infinitely many counters; de Moivre states the result without proof.

“Although the publication date is given as 1711, *De Mensura Sortis *was not in print until 1712. Shortly after its publication, de Moivre sent copies of the issue to several people in England, including Edmond Halley, Isaac Newton, and de Moivre’s fellow chess player at Slaughter’s Coffeehouse, the Earl of Sunderland. De Moivre’s friend, Pierre des Maizeaux handled several copies that were bound for the Continent. Using his connections in the Republic of Letters, des Maizeaux sent copies of *De Mensura Sortis *to Abbé Jean-Paul Bignon, at that time the French minister of state with responsibility for the Académie Royale des Sciences. Bignon wrote to des Maizeaux on September 24, 1712, saying that the copies he received had been distributed. He also enclosed a letter from Montmort to de Moivre thanking him for his treatise; the letter has not survived. Whatever he thought personally about de Moivre’s treatise, Montmort was adhering to the code of civility in the Republic of Letters by sending the letter of thanks. Other people on the Continent receiving copies were Nicolaus Bernoulli, Johann Bernoulli, and Pierre Varignon. Johann Bernoulli received his copy via William Burnet, a younger son of Gilbert Burnet, Bishop of Salisbury; Bernoulli had asked Burnet to obtain a copy for him” (Bellhouse, p. 71).

Abraham Moivre stemmed from a Protestant family. His father was a surgeon from Vitry-le-François in the Champagne. From the age of five to eleven he was educated by the Catholic Péres de la doctrine Chrètienne. Then he moved to the Protestant Academy at Sedan were he mainly studied Greek. After the latter was forced to close in 1681 for its profession of faith, Moivre continued his studies at Saumur between 1682 and 1684 before joining his parents who had meanwhile moved to Paris. At that time he had studied some books on elementary mathematics and the first six books of Euclid’s elements. He had even tried his hand at Huygens’ 1657 tract without mastering it completely. In Paris he was taught mathematics by Jacques Ozanam who had made a reputation from a series of books on practical mathematics and mathematical recreations. Ozanam made his living as a private teacher of mathematics. He had extended the usual teachings of the European reckoning masters and mathematical practitioners by what was considered fashionable mathematics in Paris. Ozanam enjoyed a moderate financial success due to the many students he attracted. It seems plausible that young Moivre took him as a model he wanted to follow when he had to support himself. After the revocation of the Edict of Nantes in 1685 the Protestant faith was no longer tolerated in France, and hundreds of thousands of Huguenots who had refused to convert to Catholicism emigrated to Protestant countries. Amongst them was Moivre who arrived in England in 1687. There he began his occupation as a tutor in mathematics. He also added a ‘De’ to his name, probably because he wanted to take advantage of the prestige of a (pretended) noble birth in France in dealing with his clients, many of whom were noblemen. An anecdote from this time which goes back to (de) Moivre himself tells that he cut out the pages of Newton’s *Principia *of 1687 and read them while waiting for his students or walking from one to the other – the main function of this anecdote was to demonstrate that de Moivre was amongst the first true and loyal Newtonians and that as such he deserved help and protection in order to gain a better position than that of a humble tutor of mathematics. In 1692 de Moivre met with Edmond Halley and shortly afterwards with Newton. Halley ensured the publication of de Moivre’s first paper on Newton’s doctrine of fluxions in the *Philosophical Transactions* for 1695 and saw to his election to the Royal Society in 1697. Newton’s influence concerning university positions in mathematics and natural philosophy persuaded de Moivre to engage in the solution of problems posed by the new infinitesimal calculus. In 1697 and 1698 he published the polynomial theorem, a generalization of Newton’s binomial theorem, together with application in the theory of series. In 1704 de Moivre began a correspondence with Johann Bernoulli, but Bernoulli’s letters showed de Moivre that he lacked the time and perhaps the mathematical power to compete with a mathematician of this calibre in the new field of analysis. De Moivre ceased his correspondence with Bernoulli after he was made a member of the Royal Society commission to adjudicate in the priority dispute between Newton and Leibniz over the invention of calculus – continuing the correspondence may have made him appear disloyal to the Newtonian cause. When the Lucasian chair in mathematics at Cambridge was given in 1711 on Newton’s recommendation to Nicholas Saunderson, de Moivre realized that this only option was to continue his occupation as a tutor and consultant in mathematical affairs in the world of the coffee houses where he met his clients; additional income he could draw from the publication of books and from translations. He therefore turned to the calculus of games of chance and probability theory which was of great interest for many of his students and where he had few competitors in England.

Hald, ‘A. de Moivre: *De Mensura Sortis* or *On the Measurement of Chance*,’ *International Statistical Review* 52 (1984), pp. 229-262. Hald, *History of Probability and Statistics and their Applications before 1750*, 2003. Bellhouse, *Abraham de Moivre*, 2011. Cajori, *A History of Mathematics*, 1894.

Small 4to (197 x 157 mm), pp. 213-264 (some browning, mostly light but heavier on the title page). Old, possibly contemporary, marbled wrappers (a bit soiled, spine worn).

Item #5154

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Price:
$5,000.00
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