Montmort's Essay d'analyse sur les jeux de hazard
For a long time Mathematicians themselves have bragged that they are able by their methods to discover in the natural Sciences, all the truths which are within the reach of the human mind; and it is certain that by the marvellous mixture that they have made during fifty years of Mathematics with Physics, they have forced men to comprehend that this which they claim to the advantage of Mathematics is not without basis. What glory would there be for Science if it could yet serve to regulate judgments and to lead men into the practice of the things of life!
The eldest of the Messers Bernoulli [Jacob Bernoulli], both so well-known in the scholarly world, has not thought that it was impossible to bring Mathematics to this point. He had undertaken to present Rules in order to judge the probability of future events, and of which the knowledge is hidden to us, either in Games, or in the other things of life where chance alone has a part. The title of this Work must be De arte conjectandi, l'art de diviner. A premature death has not permitted him to play the last hand.
Mr Fontenelle and Mr Saurin have each given a short Analysis of this Book; the first in the Histoire de l'Academie (1705); the other in the Journal des Sçavans of France (1706). Here is, according to these two Authors, what was the plan of this Work. Mr Bernoulli divided it into four Parts; in the first three he gave the solution to various Problems relating to Games of chance: one must find there many new things on infinite series, on combinations and permutations, with the solution of the five Problems proposed a long time ago to Mathematicians by Mr Huygens. In the fourth Part, he applied the methods that he had given in the first three, to resolve many moral, political and civil questions.
No one has informed us what are the Games of which this Author determined the parts, nor what subjects of politics and morals he had undertaken to clarify; but as surprising as this project should appear, there is reason to believe that this scholarly Author had perfectly executed it. Mr Bernoulli was too superior to the others in order to wish to impose on them, he was of that small number of rare men who are capable of innovation, and I am persuaded that he had achieved all those things that the title of his Book promised.
Nothing delays more the advancement of the Sciences, and puts a greater obstacle to the discovery of hidden truths, than the mistrust in which we have of our strengths. The greater part of those things which may appear impossible are only so for lack of allowing the human intellect all the extent that it can achieve.
Already a long time ago many of my Friends had prompted me to test if Algebra could not be used to determine what is the advantage of the Banker in the Game of Pharaon. I had never dared to undertake this research, because I knew that the number of all the many possible arrangements of fifty-two cards, surpasses more than one hundred thousand millions of times the number of the grains of sand that the globe of the earth could contain; and it did not appear possible to me to untangle, in a number so vast, those arrangements which are advantageous to the Banker, from those which are contrary or indifferent to him. I would still hold this prejudice if the success of the late Mr Bernoulli had not invited me there some years ago to seek the different chances of this Game. I was happier than I had dared hoped, because beyond the general solution of this Problem, I perceived the paths that it was necessary to follow in order to discover an infinity of similar, or even much more difficult Problems. I knew that one could go quite far in this country where no person had yet set foot; I flattered myself that one could make an ample harvest of truths equally curious and novel; this gave to me the thought to work at the foundation of this matter, and to desire to compensate in some sort the Public from the loss that it would get if it were deprived of the excellent Work of Mr Bernoulli. Diverse reflections have confirmed me in this decision.
JOC/EFR January 2015
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