De Montmort: Essay d'Analyse

Florence Nightingale David published Games, Gods & Gambling in 1962. In this classic work she described the life and work of Pierre-Rémond de Montmort. Below we present a version of an extract from this description.

The life of Pierre-Rémond de Montmort, after a stormy start, was a simple, happy one. He was born in Paris on October 27th, 1678, the second of three sons of François and Marguerite Rallu Rémond who were of the nobility. François Rémond intended his son to study law, there being a vacant magistracy waiting for him, but, his biographer says, the son was contemptuous of any restraint, and left home, travelling first to England and then to the Low Countries and Germany. At the house of his cousin in Germany he found a copy of Malebranche's La Recherche de la Vérité, and on reading it appears to have suffered a minor conversion. He was inspired to return home and make his peace with his father. This was in 1699. His father died shortly afterwards leaving him, at the age of 22, a large fortune. He does not, however, appear to have plunged into the dissipated life which was thought natural for a young nobleman of this time to lead. His conversion stood him in good stead and he continued to occupy his time with "pious exercises" and with studying philosophy and mathematics with Father Nicolaus de Malebranche who was then in the House of the Oratory of Saint-Honoré de Paris. After learning some mathematics Montmort went to England again (in 1700), on purpose to make the acquaintance of the English mathematicians and in particular of Newton. This visit gave further impetus to his study of mathematics, and he came back to Paris to pursue his studies in algebra, geometry and the new calculus, which he found "thorny". [One may speculate here about the new calculus. Montmort may have heard talk of it in London, but he uses Leibniz's

d

notation which would imply that he learnt it from a non-English source.]

Montmort succeeded his elder brother as a canon of Notre-Dame about this time, but he did not occupy his stall for long. He bought the estate of Montmort in 1704 and two years later (1706) he resigned his canonry in order to marry the great-niece of the Duchess of Angoulême. His mathematical interests had suffered to some extent from his ecclesiastical duties, but during the time of his canonry he had printed at his own expense several scientific works, including Newton on the quadrature of curves. After his marriage he settled down on his country estate and set himself to work on the theory of probability. Quite why he chose this topic is not known, for he was no gambler. It was known in France and known to Montmort, if only from the Éloge of Jacob Bernoulli, that Jacob had, at the time of his death, left the manuscript of a book on the subject. Possibly Montmort had a contact with a pupil of Jacob - it is thought he did not meet Nicolaus Bernoulli until 1709 - and this contact inspired him to pursue the new calculus with its fascinating sidelines of the summation of infinite series and the manipulation of binomial coefficients. It seems unlikely that he would have taken it up without some impetus of this kind. He himself says:-

Several of my friends have urged me for a long time to see if I could not determine by algebra what is the advantage of the Banker in the game of Pharaoh. I would not have dared undertake this research ... if the success of M Bernoulli had not incited me for more than two years to try to calculate the different chances in this game ... . This gave me the idea of getting to the bottom of this matter and the desire to make up to the Public in some fashion for the loss that it has sustained in being deprived of the excellent work of M Bernoulli ... .

[Since he had published a number of the scientific works of others at his own expense it is a little curious that he did not try to publish l'excellent ouvrage de M Bernoulli.]

The results of his researches were published in the Essay d'Analyse sur les Jeux de Hasard printed in Paris in 1708. This, the first edition, often passed over because of the greater length of the second edition and the incorporation of the Montmort-Bernoulli correspondence, is worthy of comment. As I have noted, one does not know quite what started Montmort off, but it is known that he wrote it in the comparative isolation of his country estate and it represents therefore the attempts of a competent mathematician, given the outline of the problems discussed by Jacob Bernoulli, to solve them using the latest mathematical techniques, and to extend the field of application of these techniques to card games. This first edition is Montmort himself speaking; the second edition, while much more comprehensive, is probably a mixture of Montmort and Nicolaus Bernoulli. The fact that he wrote at all is probably a fortunate one for the probability calculus. That a nobleman of France and an ex-canon of Notre-Dame should find such problems worthy of speculation and not an impious study would give the subject a certain cachet and air of respectability which left lesser mortals free to work undisturbed. Even so, Montmort found it necessary to write an apologia for having spent his time working on such problems and he wrote this with a self-conscious rectitude:-

It is particularly in games of chance that the weakness of the human mind appears and its leaning towards superstition ... . There are those who will play only with packs of cards with which they have won, with the thought that good luck is attached to them. Others on the contrary prefer packs with which they have lost, with the idea that having lost a few times with them it is less likely that they will go on losing, as if the past can decide something for the future ... . Others refuse to shuffle the cards and believe they must infallibly lose if they deviate from their rules. Finally there are those who look for advantage where there is none, or at least so small as to be negligible. Nearly the same thing can be said of the conduct of men in all situations of life where chance plays a part. It is the same superstitions which govern them, the same imagination which rules their method of procedure and which blinds their fears and hopes. Often they abandon a small, certain, wealth in order to run fearfully after a greater: and often they wilfully give up well-founded expectations in order to conserve a thing the value of which is nothing like those which they neglect. The general principle of these superstitions and errors is that most men attribute the distribution of good and evil and generally all the happenings in this world to a fatal power which works without order or rule. They believe that it is necessary to appease this blind divinity that one calls Fortune, in order to force her to be favourable to them in following the rules which they have imagined. I think therefore it would be useful, not only to gamesters but to all men in general, to know that chance has rules which can be known, and that through not knowing these rules they make faults every day, the results of which with more reason may be imputed to themselves than to the destiny which they accuse ... . It is certain that men do not work honestly as hard to obtain what they want as they do in the pursuit of Fortune or Destiny ... . The conduct of men usually makes their good fortune or their bad fortune, and wise men leave as little to chance as possible.

There is much more along these lines.

In this first edition of the Essay d'Analyse 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.

It may happen that Pierre, having won several times and beginning again at one has not enough cards in his hand to complete the thirteen, etc., etc.

He begins by assuming Pierre has two cards and one opponent, Paul. Then Pierre has three cards, four, and finally any number. Next he argues generally by building up his chances. Thus:-

If we call S the chance we want, the number of cards Pierre has being denoted by p, and let g be Pierre's chance when the number of cards he holds is p - 1, and let d be the chance when his number of cards is p - 2, we have

$S = \large\frac{g ((p-1)+d)}{p}\normalsize$ .

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, and such that the numerator is the series of numbers which are found in the Table (i.e. the Arithmetic Triangle) in the perpendicular column which corresponds to p, beginning with p, and the denominator the series of products $p \times p - 1 \times p - 2 \times p - 3 \times p - 4 \times p - 5$ ; so that, cancelling out the common terms, we have for Pierre's chance the very simple

$\large\frac{1}{1}\normalsize - \large\frac{1}{1.2}\normalsize + \large\frac{1}{1.2.3}\normalsize - \large\frac{1}{1.2.3.4}\normalsize + \large\frac{1}{1.2.3.4.5}\normalsize - \large\frac{1}{1.2.3.4.5.6}\normalsize +$ etc.

Let us suppose a logarithm of which the subtangent is unity. We will take on this curve a constant ordinate = 1 and another ordinate smaller = 1 - y. We will call x the abscissa contained between these two ordinates and we will have $dx = dy/(l -y)$ and

           $x = y + \large\frac{1}{2}\normalsize y^{2} + \large\frac{1}{3}\normalsize y^{3} + \large\frac{1}{4}\normalsize y^{4}$ + etc.,

whence by the method of inversion of series

           $y = x - \large\frac{1}{1.2}\normalsize x^{2} + \large\frac{1}{1.2.3}\normalsize x^{3} - \large\frac{1}{1.2.3.4}\normalsize x^{4} + \large\frac{1}{1.2.3.4.5}\normalsize x^{5} -$ etc.

which, putting x = 1, becomes

           $\large\frac{1}{1}\normalsize - \large\frac{1}{1.2}\normalsize + \large\frac{1}{1.2.3}\normalsize - \large\frac{1}{1.2.3.4}\normalsize + \large\frac{1}{1.2.3.4.5}\normalsize -$ etc.

He gives a more general way of getting this series which he says he has obtained from a paper of Leibniz (Leipzig, 1693) in which is the problem Un logarithme étant donnée, trouver le nombre qui lui correspond. This is possibly the first exponential limit in the calculus of probability, but having achieved it Montmort can't make much use of it. He contents himself by remarking:-

One could make several interesting remarks about these series but that would take us outside the present subject and would lead us too far away.

In the second half of the first part 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. He illustrates this principle by considering a pack of 40 playing cards, "mélées à discrétion," the court cards being excluded. If the pack is dealt, the chance that the first four cards will be the four aces is established as

\large\frac{1.2.3.4}{40.39.38.37}\normalsize

The generalisation of this idea causes some difficulty and certainly calls for some sort of notation. Thus (Proposition XIV):-

Let there be any number of cards whatever composed -of an equal number of aces, of twos, of threes, of fours, etc. Pierre wagers that in drawing a given number of cards from this pack at random he will have so many singletons, so many doubles, so many triples, so many quadruples, so many quintuples, etc.

[We omit David's description of the mathematics which Montmort uses to analyse this, mainly since Montmort uses a notation which is essentially impossible to reproduce!]

In the second part of his treatise Montmort discusses the game of Quinquenove and the game of Hazard, remarking about the latter that the game is known only in England. This, from the references in medieval French literature, is unlikely, but since the rules have been a matter for discussion we set down here Montmort's version of them:-

This game is played with two dice like Quinquenove. Let us call Pierre the die-thrower and suppose Paul represents all the other players. Pierre throws the dice until he has obtained either 5, 6, 7, 8 or 9. Any of these numbers, whichever turns up first, is Paul's chance. Then Pierre throws the dice again to obtain his own chance. He may have the numbers 4, 5, 6, 7, 8, 9 or 10, so that he has two more possibilities than Paul, namely 4 and 10.

(i) If Pierre, after having given Paul a chance which is 6 or 8, throws with his second the same number, or twelve, he wins. But if he throws two aces, or a two and an ace, or eleven, he loses.

(ii) If he has thrown for Paul the number 5 or 9, and in the following throw he gets the same number he wins: but if he throws two aces or a two and an ace, or eleven or twelve, he loses.

(iii) If he has given Paul the chance 7, and he throws the same number at the next throw, or eleven, he wins. But if he throws two aces, a two and an ace, or twelve, he loses.

(iv) Pierre having obtained a permissible number different from that of Paul, he will win if, when he throws again, he throws his chance before throwing that of Paul, and he will lose if he throws Paul's number before throwing his own.

(He is definite here about the number of dice being two, and this appears to be the number used, except in Italy. The references in Dante and Galileo's problem suggest that Hazard may have been played in Italy with three dice. [The game of Three Dice as described by Montmort bears a distinct resemblance to his game of Hazard.] Montmort gives the chances of Peter and Paul according to the rules he has laid down 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. He doesn't bother with the rules (they must have been entirely established by this time) and he calculates several simple chances but remarks that in the majority of situations the solution cannot be found. Remembering the intricacies of the game, one is inclined to agree with him. Apart from further complicated calculations on games involving 2, 3, 4, 5, 6, 7, . . . dice, in which the principles of calculating a conditional probability have been already laid down, using his cumbersome combinatorial notation, he does not appear to achieve anything else which is new. From an historical point of view, however, there is interest in his game of Nuts.

I have remarked earlier that 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. 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.

His exposition of the chances involved is quite clear (he just refers back to the Arithmetic Triangle), with the advantage to the nut-thrower of

\large\frac{3}{256}\normalsize

. After some moral reflection he goes on:-

I think I should add that this problem was posed by me to a Lady, who gave me almost immediately the correct solution using the Arithmetic Triangle. But this table is useful only by chance, for if the nuts, instead of having two faces, had more than that, say four, this table would not be useful, and the problem would be less easy than the preceding.

[Two reflections occur to me. It would be interesting to know who the noblewoman was who had a facility in applying the Arithmetic Triangle to games of chance. The principles of the calculation of probabilities must have been generally known among educated persons.]

Having solved the problem for four-sided nuts he concludes his book with Huygens' five problems, and some reflections on the games of Her, Ferme and Tas. It is fairly clear that Montmort modelled his book on what he thought was the plan followed by Jacob Bernoulli. For he writes:-

If I was going to follow M Bernoulli's project I should have added a fourth part where I applied the methods contained in the first three parts to political, economic, and moral problems. What has prevented me is that I do not know where to find the theories based on factual information which would allow me to pursue my researches ... . To speak exactly nothing depends on chance; when one studies nature one is soon convinced that its CREATOR moves in a smooth, uniform way which bears the stamp of infinite wisdom and prescience. Thus to attach to the word "chance" a meaning which conforms with true philosophy one must think all things are regulated according to certain laws, those which we think dependent on chance being those for which the natural cause is hidden from us. Only after such a definition can one say that the life of man is a game where chance reigns.

Having reached this point he decides that while the rules of probability can be applied to the game of life, the chances of this game are too difficult to compute, much as it is too difficult to compute the value of a throw in backgammon:-

The reasons and the different motives that men are able to have in order to move one way rather than another make it difficult to find out how they will act. Often they do not know where their own interest lies ... . Caprice serves them rather than wisdom.

M de Montmort obviously has very little idea of what Jacob Bernoulli was going to put in the fifth part of the Ars Conjectandi.

The first edition of the Essai d'Analyse was published in 1708. In 1709 Nicolaus Bernoulli presented his thesis De Arte Conjectandi in Jure for the degree of doctor of law at Basel University and then set off on his travels. When he went to Paris he met and became friendly with Montmort, staying with him on his country estate for some three months and afterwards keeping up a long correspondence with him. It was through Nicolaus perhaps that Montmort also corresponded with Johann Bernoulli, Jacob' brother and Nicolaus' uncle, who had succeeded Jacob in the chair of mathematics at Basel. This correspondence was also friendly in character, and on the whole Montmort seems to have been a likeable person. He had a wide circle of correspondents among mathematicians of all countries, including Newton and Leibniz, exchanging with them news about mathematical problems and discussing solutions to the problems of the day (and fray). It is true that there is an echo of a stubborn temper in his running away from home at the age of 18, but this did not reappear in his later years. The greater part of his mathematical work was done at his house in the country. He appears to have been capable of great concentration since Malebranche recounted how he sat working with people playing the clavichord while his sons ran about the room and teased him. He was said to be quick-tempered and given to short, sharp bursts of anger, but it seems that because of his sweet nature he was very soon afterwards sorry and a little shamefaced. It is therefore more than a little strange that he reacted so sharply to de Moivre.

Francis Robartes, a fellow of the Royal Society of London and afterwards Earl of Radnor, was obviously interested in the calculation of probabilities since he wrote in 1693 a note "An Arithmetical Paradox, concerning the Chances of Lotteries" (Phil. Trans., 17). In 1711 Abraham de Moivre wrote "De Mensura Sortis, seu de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus" which was also published in the Philosophical Transactions (vol. 27) and was afterwards expanded a little to form the first edition of the Doctrine of Chances. It was to this De Mensura Sortis that Montmort reacted so sharply. De Moivre wrote about this first memoir:-

The occasion of my undertaking this subject was chiefly owing to the Desire and Encouragement of the Honourable Francis Robartes, ... who, upon occasion of a French tract, called l'Analyse des Jeux de Hasard, which had lately been published, was pleased to propose to me some Problems of much greater difficulty than any he had found in that Book; which having solved to his satisfaction, he engaged me to methodise these Problems, and to lay down the Rules which had led me to their Solution .... Huygens, first, as I know, set down rules for the solution of the same kind of problem as those which the new French author illustrates freely with diverse examples. But these famous men do not seem to have been accustomed to that simplicity and generality which the nature of the thing demands. For while they talk of many unknown quantities so that they may represent various conditions of play, they set out their intricate calculations too meagrely, and while they always use equal skill in games, they contain this theory of games between much too narrow limits ... .

The words from "Huygens" onwards are translated by the present writer from the Latin of De Mensura Sortis and they are not fair comment. De Moivre's first attempts were largely derivative from this first edition of Montmort, and they do not, on the whole, display the masterly treatment of the subject which is evinced in the Miscellanea Analytica of 1730 or the third edition of the Doctrine of Chances (1756). We note these words of de Moivre because he behaved in an unfair way to Montmort and gave him provocation to retaliate. What is so surprising is that Montmort reacted so strongly.

At this time the English, commanded by Marlborough and under the shadow of the Grand Alliance, were marching about Europe defeating the French armies, and in 1708 and 1709 they were knocking at the gates of France. It has been said of this period that the sciences were never at war, but the mystic attachment of every Frenchman for the soil of France may have caused in Montmort a reaction against the émigré Frenchman in London, especially when he was given just cause. It may also be said that one generally reacts most violently to one's own faults displayed in others, and it is possible that Montmort was not entirely easy in his conscience about his own publication; for it may have been the thought that the Ars Conjectandi was unpublished at the time of issue of the Essai d'Analyse, and that he himself had learnt much of Jacob' work from Nicolaus, which caused him to resent de Moivre, The Ars Conjectandi, still unfinished, was at last published in 1713, eight years after the death of Jacob, and it was followed in 1714 by the second edition of the Essai d'Analyse. The preface of the first edition was repeated, followed by an Avertissement in which Montmort wrote:-

The author did me the honour of sending me a copy. ... M Moivre was right to think I would need his book to reply to the criticism he made of mine in his introduction. His praiseworthy intention of boosting and increasing the value of his work has led him to disparage mine and to deny my methods the merit of novelty. As he imagined he could attack me without giving me reason for complaint against him, I think I can reply to him without giving him cause to complain against me. ...

And reply he does over many pages, setting out the history from Pascal and Fermat onwards and always in what might be described as polemical fashion. This sort of behaviour is foreign to Montmort and it is, as far as I know, the only time in which he so indulged himself. Usually he carefully took the middle road in argument, as may be seen in his careful neutrality at the time of the calculus controversy. On one side was the army of English mathematicians and on the other Johann Bernoulli, with Leibniz in the background. Each side tried to get Montmort to join them, but his philosophical and ecclesiastical training stood him in good stead and he managed to avoid getting embroiled. On a visit to England in 1715 he became reconciled to de Moivre and was made a Fellow of the Royal Society. In Paris in 1716 he was made a member of the Académie des Sciences. He frequently visited Paris for business reasons and on the last of these trips he caught smallpox and died of it on October 7th, 1719.

The second edition of the Essai d'Analyse is more than twice as long as the first and reflects to a certain extent the maturity of thought on the subject which came to the author, possibly as a result of his conversations and exchange of letters with Nicolaus Bernoulli. The history in the Avertissement is interesting if only because it indicates how little was known of the story of the development of the calculation of chances. According to Montmort, Pascal originated most of combinatory theory and Pascal and Fermat the calculation of chances. He gives a clear review of Huygens' and Jacob Bernoulli's work, with many eulogistic references also to Leibniz. (If Montmort is a fair example, and in this instance he probably is, Leibniz seems to have been greatly venerated by the continental mathematicians.) Possibly because he has been somewhat scathing about de Moivre and English mathematics he gives some space to a memoir by Craig, which is interesting only because of later French work on the credibility of witnesses. Montmort does not, I think, know how to take this memoir. He writes to begin with:-

I have found in the Phil. Trans. a memoir in which it is proposed to estimate the probability of men speaking the truth, whether in speech or in writing: but can one find this? If an emphatic yes gives a semblance of truth $\large\frac{a}{b}\normalsize$ , an emphatic yes of an emphatic yes will give a semblance $\large\frac{a}{b}\normalsize .\large\frac{c}{d}\normalsize$ of truth if the witness of the second is not of the same strength, and $\large\frac{a^2}{b^2}$ if they hold the same authority, ... which is obvious. But what can one conclude from this, and how can one apply these theories? I think that this is impossible. It has however been undertaken ... by an English mathematician. ... The book of which I speak has for title Philosophiae Christianae Principia Mathematica. M Craig is the author. ... The Author tries in the main to prove, against the Jews, the story of Jesus Christ and to demonstrate to libertines that the choice they have made in preferring the pleasures of this world ... to the expectation of those ... who follow the law of the Evangelist, is not reasonable and does not accord with their true interests. ...

In spite of his not believing that Craig's calculations are possible he feels that they are of considerable theoretical interest:-

For myself I find the design of M Craig pious and worthy of praise, and the execution of it as good as it can be, but I believe this work much more suitable as an exercise for mathematicians than as a means to convert the Jews or the incredulous. One can certainly conclude after reading this treatise that the Author is very ingenious, that he is a great mathematician and highly intelligent. The clarity of mathematics and the saintly obscurity of the faith are two entirely opposing things: I do not think that anyone will ever succeed in combining them.

One wonders a little why Jacob Bernoulli's projected fifth part of the "Ars Conjectandi" should have been considered practicable whereas Mr Craig's design was not. The remainder of the Avertissement is principally concerned with the story of the calculation of chances during the years immediately preceding his own writing, and although more or less accurate it is possibly unduly flattering to Leibniz.

The main body of the second edition contains a great deal of new material and has incorporated at the end letters which passed between the Bernoullis, Nicolaus and Johann, and Montmort. It is clear that the Bernoullis helped considerably with this second edition, clarifying Montmort's ideas for him and contributing much in the way of summation of series. Jacob was very good at summing series, so that this type of mathematical exercise was easy for Nicolaus and Johann. Todhunter gives a clear, full account of the majority of the mathematics, and there is little to add but to make the story complete we repeat a little of what he has already described in detail. The theorems on combinations are, in the second edition, brought forward to the beginning of the treatise. This, in a way, is symbolic of the change of treatment throughout the book. Montmort has come to maturity and discusses, wherever he is able, the general solution to the problem he sets himself, rather than beginning with special cases and then plunging into a general statement which is often unsupported by mathematical argument. He still retains his combinatorial notation and introduces the symbol

[q]

for the

q

th figurate number. The expansion involved in the solution of his Proposition XVI:-

Throwing randomly any number whatever, d, of dice, of which the number of faces, f, may be also whatever one wishes, to find what is the number of ways of throwing a given number p,

illustrates both his general method of attack and introduces his new notation. In the first edition he gives, presumably by exhaustive enumeration, the chances involved in throwing with six-sided dice the numbers 1, 2(1), ..., 8. Now he starts off [finding a formula that] will express the looked-for number.

This formula, the differences of zero series, had been reached by de Moivre in De Mensura Sortis in 1711. It is a generalisation of Proposition XXXI of the first edition, but Montmort had no idea then (1708) as to how it might be solved generally. De Moivre probably took the problem from the first edition, generalised it, and gave the solution with no indication of method of proof. It is just possible that he was following Montmort's own method and guessed the general result from the particular cases, but in the light of his undoubted analytic powers this is unlikely. Montmort reached the solution by himself also, since a letter to Johann Bernoulli in 1710 shows that he had already obtained it.

The generalisations of the various topics discussed in the first edition are interesting, without adding anything particularly new to the probability calculus, although the various methods for the summation of series show the skill of the Bernoullis in that part of algebra. The matching distribution is presented with a proof of the general case; this proof was not given in the first edition, implying that Montmort either guessed the original solution or was dissatisfied with his first method of proof. The proof which he now gives is due to Nicolaus Bernoulli, but he repeats his own exponential limiting form. The Problem of Points solved in full generality with two players of unequal skill is presented again with the help of Niholaus. I shall consider this in the discussion of the Doctrine of Chances since the problem did not cease to be of interest for many years. There are the first fumblings towards the questions of annuities, and the analysis of the chances in one or two further games of chance such as Her, not previously given. The most interesting addition is however the printing in full of the letters between himself and the Bernoullis; these are at the end of the book and occupy over one hundred pages. They begin after Nicolaus had left Paris and returned to Basel. Johann writes in a lofty way and obviously enjoys pointing out that Montmort had missed writing down the sum of a geometric progression on at least two occasions. He does not show any great capacity as a probabilist. Both Johann and Nicolaus take as their focus of comment the first edition of the Essai d'Analyse, but it is Nicolaus who comes forward with helpful and sometimes new solutions. Montmort obviously published the long series of letters because he wanted Nicolaus to have the credit of the results he had worked out. Much of what Nicolaus did was a generalisation of problems which had been proposed by Montmort or his uncle Jacob. He does, however, also set out the problem which later was to become famous as the St Petersburg problem, and he also sets out his uncle's "golden theorem" as if it were his own, adding "I recall that my uncle has demonstrated a similar thing in his treatise Ars Conjectandi now being printed at Basel". Nicolaus was an excellent mathematician, with the algebraic diversity and versatility which could be expected from anyone of ability who had been trained in part by Jacob. One may be pardoned perhaps for wondering whether many of his ideas were not derived from the same source.

On September 5th, 1712, Montmort wrote to Nicolaus a tirade about De Mensura Sortis which he had just finished reading. He is seething about it because there is nothing new in it if one takes into account the letters between himself and Nicolaus, which were not yet published, as well as the first edition of the Essai d'Analyse:-

... You will find that the problems he discusses, which are not solved, are solved in our letters. Moreover I do not think there is in this work, elsewhere very good, anything new to you, and nothing which will give you pleasure by its originality. ...

The letter, of extreme length, is that of a very angry man. Nicolaus, contrary to the usual Bernoulli practice of joining in battles, here attempts to soothe his friend. He wrote from London on October 11th, 1712, before he had received Montmort's outburst, that:-

I have had the pleasure of often seeing here M de Moivre who has given me a copy of his book De Mensura Sortis. He tells me that he has also sent you a copy and that he is awaiting with impatience your views on his work. You will be surprised to find there many of the problems which we have solved. ...

Nicolaus wrote again from Brussels at the-end of 1712 (December 30th) explaining that Montmort's letter had had to follow him to London and then to the Low Countries and had consequently been delayed. He dealt with Montmort's letter in detail prefacing his remarks with:-

I am content that you have received M de Moivre's book De Mensura Sortis. It is true that nearly all the problems proposed there have been solved either in your book or in our letters. As I knew that M de Moivre awaits with impatience your judgement on his book, I have taken the liberty of sending him the substance of your remarks. ...

[Apart from de Moivre's correspondence with Johann Bernoulli, which has nothing to do with probability, there are no letters written by de Moivre known to the present writer and a search has so far failed to find any. Yet it is not beyond the bounds of possibility that de Moivre wrote to Montmort after receiving a letter from Nicolaus. Montmort would probably have been too angry to answer. Todhunter states that there was a correspondence between Montmort and de Moivre, but I have not been able to trace it.]

Nicolaus is soothing to Montmort but fair to de Moivre, pointing out in several places in his commentary that de Moivre had shown to him his general solutions to various problems when he was in London, and trying to explain that de Moivre had not intended to slight Montmort by his introduction. [ I find this a little difficult to believe]:-

I do not know if M de Moivre has had the intention in his preface of insulting you as you think: for myself I hold that the methods you have given in your book are good enough to solve all the general problems of M Moivre, which for the most part differ from yours only in the generality of the algebraic expression. As I am persuaded that M Moivre himself would do you the justice of recognising that you have taken this subject much farther than M Huygens and M Pascal, who have given only the first elements of the science of chance, and that after them you have been the first who has published general methods for these calculations. ...

(One wonders a little about Jacob Bernoulli.) That Montmort was not appeased is apparent from the Avertissement on which I have commented. Apart from a discussion by Nicolaus on the problem of the parity of the proportion of male and female births, which was being discussed in London at the time of his visit, there is little more of interest in the correspondence, which appears to have finished in 1713 just before the publication of the second edition of the Essai d'Analyse.

With the publication of this second edition Montmort seems to have given up researches on the probability calculus. It may have been that the short history which he wrote about the theory of probability (or possibly the calculus controversy) piqued his curiosity, but he wrote to Nicolaus (August 20th, 1713):-

I would very much like to know what you and your uncle think of the book entitled Commercium Epistolicum, etc., that the Royal Society has had printed to assure to M Newton the glory of having been the first and only one to have invented the new methods. I promise you secrecy if you tell me. Everyone here waits M Leibniz's answer.

It is to be desired that someone would take the trouble of instructing us how and in what order the discoveries in mathematics have succeeded one another and to whom we are obliged for them. We have a history of painting, of music, of medicine, etc. A history of mathematics, and in particular of geometry, would be a very useful work. What pleasure would one not have to see the link-up, the connection between the methods, the relation between the different theories, beginning from the first stirrings up to the present?

Whatever started him off, he undertook the longed-for work himself and at the time of his death was engaged in compiling a history of geometry. None of these manuscripts seem to have survived his death.

Montmort's importance from the probability point of view is possibly not in the new ideas which he introduced but in the algebraic methods of attack. These were perhaps much the same as those of Jacob Bernoulli, but the two mathematicians, coupled with Nicolaus, reinforce one another. They must have given inspiration to many other pure mathematicians, among them de Moivre, who would not have been interested solely in the laborious enumeration of the fundamental probability set.

Last Updated August 2007