262 d'alembeet. 



both parties A and B, and any other rule would be unfah^ to one 

 or the other. 



472. D'Alembert concludes from his remarks that when the 

 probability of an event is very small it ought to be regarded and 

 treated as zero. For example he says, suppose Peter plays with 

 James on this condition ; a coin is to be tossed one hundred times, 

 and if head appear at the last trial and not before, James shall give 

 2^^^ crowns to Peter. By the ordinary theory Peter ought to give 

 to James one crown at the beginning of the game. 



D'Alembert says that Peter ought not to give this crown 

 because he will certainly lose, for head will appear before the 

 hundredth trial, certainly though not necessarily. 



D'Alembert's doctrine about a small probability being equi- 

 valent to zero was also maintained by Buffon. 



473. D'Alembert says that we must distinguish between what 

 is metaphysically possible, and what is physically possible. In the 

 first class are included all those things of which the existence is not 

 absurd ; in the second class are included only those things of which 

 the existence is not too extraordinary to occur in the ordinary 

 course of events. It is metaphysically possible to throw two sixes 

 with two dice a hundred times running ; but it is physically impos- 

 sible, because it never has happened and never will happen. 



This is of course only saying in another way that a very small 

 chance is to be regarded and treated as zero. DAlembert shews 

 however, that when we come to ask at what stage in the diminu- 

 tion of chance we shall consider the chance as zero, we are in- 

 volved in difficulty ; and he uses this as an additional argument 

 against the common theory. 



See also Mill's Logic, 1862, Vol. ii. page 170. 



474. D'Alembert says he will propose an idea which has 

 occurred to him, by which the ratio of probabilities may be 

 estimated. The idea is simply to make experiments. He ex- 

 emplifies it by supposing a coin to be tossed a large number of 

 times, and the results to be observed. We shall find that this 

 has been done at the instance of Buffon and others. It is need- 

 less to say that the advocates of the common Theory of Proba- 



