U^ALEMBERT. 275 



500, We will now take the first of the two portions, which 

 occupies pages 73 — 105, 



D'Alembert begins with a section Siir le calcul des Prohahilites. 

 This section is chiefly devoted to the Petersburg Problem. The 



chance that head wdll not appear before the n^^' throw is ^ 



on the ordinary theory. D'Alembert proposes quite arbitrarily to 

 change this expression into some other which will bring out a 



finite result for ^'s expectation. He suggests , ^-^ where 



/8 is a constant. In this case the summation wdiich the problem re- 



1 



quires can only be effected approximately. He also suggests ^^^^^ 



id 



^^^d „^^.a(»-i) where a is a constant. 



He gives of course no reason for these suggestions, except 

 that they lead to a finite result instead of the infinite result of 

 the ordinary theory. But his most curious suggestion is that of 



1 ^ 1 , where B and K are constants 



and 17 an odd integer. He says, 



Nous mettons le nombre pair 2 au denominateur de Texposant, afiu 

 que quand on est arrive au nombre n qui donne la probabilite ^gale 

 a zero, on ne trouve pas la probabilite negative, en faisant n plus 

 grand que ee nombre, ce qui seroit clioquant ; car la probabilite ne 

 sauroit jamais ^tre au-dessous de zero, II est vrai qu'en faisant n 

 plus grand que le nombre dont il s'agit, elle devient imaginaire; mais 

 cet inconvenient me paroit moindre que celui de devenir negative;... 



501. D'Alembert's next section is entitled Bur Tanalyse des 

 Jeux. 



D'Alembert first proposes une consideration tr^s-simple et 

 tr^s-naturelle a faire dans le calcul des jeux, et dont M. de Buffon 

 m'a donne la premiere idee, . . . This consideration we will explain 

 when noticing a w^ork by Buffon. D'Alembert gives it in the 

 form which Buffon ought to have given it in order to do justice 

 to his own argument. But soon after in a numerical example 



18—2 



