d'alembert. 259 



need of the second throw. Thus he makes only three cases, 



2 

 namely, H, TH, TT\ therefore the chance is ^. 



o 



Similarly in the case of three throws he makes only four cases, 

 namely, H, TH, TTE, TTT: therefore the chance is |. The 

 common theory would make eight equally likely cases, and obtain 

 - for the chance. 



465. In the same article D'Alembert notices the Petersburg 

 Problem. He refers to the attempts at a solution in the Com- 



mentarii Acad Petrop. Vol. v, which w^e have noticed in 



Arts. 889 — 393 ; he adds : mais nous ne savons si on en sera satis- 

 fait ; et il y a ici quelque scandale qui merite bien d'occuper les 

 Algebristes. D'Alembert says we have only to see if the expecta- 

 tion of one player and the corresponding risk of the other really 

 is infinite, that is to say greater than any assignable finite number. 

 He says that a little reflexion will shew that it is, for the risk 

 augments wdth the number of throws, and this number may by the 

 conditions of the game proceed to any extent. He concludes that 

 the fact that the game may continue for ever is one of the reasons 

 wdiich produce an infinite expectation. 



D'Alembert proceeds to make some further remarks w^hich are 

 repeated in the second volume of his Ojniscides, and which will 

 come under our notice hereafter. We shall also see that in the 

 fourth volume of his Opuscules D'Alembert in fact contradicts the 

 conclusion which w^e have just noticed. 



466. We have next to notice the article Gageure, of the 

 Encyclopedie; the volume is dated 1757. D'Alembert says he wall 

 take this occasion to insert some ver}^ good objections to what he 

 had given in the article Croix ou Pile. He says, Elles sont de 

 M. Necker le fils, citoyen de Geneve, professeur de Mathematiques 

 en cette ville, ... nous les avons extraits d'une de ses lettres. The 

 objections are three in number. First Necker denies that D'Alem- 

 bert's three cases are equally likely, and justifies this denial. 

 Secondly Necker gives a good statement of the solution on the 



17—2 



