d'alembert. 261 



468. In various places in his Ojyuscules Mathematiques D'Alem- 

 bert gives remarks on the Theory of ProbabiUties. These remarks 

 are mainly directed against the first principles of the subject which 

 D'Alembert professes to regard as unsound. We will now examine 

 all the places in which these remarks occur. 



469. In the second volume of the Opuscules the first memoir 

 is entitled Reflexions silt le calcul des Prohabilites ; it occupies 

 pages 1 — 25. The date of the volume is 1761. D'Alembert 

 begins by quoting the common rule for expectation in the Theory 

 of Probability, namely that it is found by taking the product of the 

 loss or gain which an event will produce, by the probability that 

 this event will happen. D'Alembert says that this rule had been 

 adopted by all analysts, but that cases exist in which the rule 

 seems to fail. 



470. The first case wdiich D'Alembert brings forward is that 

 of the Petersburg Problem; see Art. 389. By the ordinary theory 

 A ought to give B an infinite sum for the privilege of playing 

 with him. D'Alembert says. 



Or, independamment de ce qu'une somme infinie est line cliimere, 

 11 n'y a personne qui vouliit douner pour jouer a ce jeu, je ne dis pas 

 une somme infinie, mais meme une somme assez modique. 



471. D'Alembert notices a solution of the Petersburg Problem 

 which had been communicated to him by un Geometre celebre 

 de I'Academie des Sciences, plein de savoir et de sagacite. He 

 means Fontaine I presume, as the solution is that which Fontaine 

 is known to have given ; see Montucla, page 403 : in this solution 

 the fact is considered that B cannot pay more than a certain sum, 

 and this limits what A ought to give to induce B to play. D'Alem- 

 bert says that this is unsatisfactory ; for suppose it is agreed that 

 the game shall only extend to a finite number of trials, say 100 ; 

 then the theory indicates that A should give 50 crowns. D'Alem- 

 bert asserts that this is too much. 



The answer to D'Alembert is simple ; and it is very well put in 

 fact by Condorcet, as we shall see hereafter. The ordinary rule is 

 entitled to be adopted, because in the long run it is equally fair to 



