PASCAL AND FERMAT. 11 



exhibits a complete statement of all the cases which can occur in 

 a game of six points. 



14. Pascal then proceeds to another topic. He says 



Je n'a pas le temps de vous envoyer la demonstration d'une difficulte 

 qui etonnoit fort M. de Mere : car il a tres-bon esprit, mais il n'est pas 

 geometre ; c'est, comme vous savez, un grand defaut; etmeme ilne com- 

 prend pas qu'une ligne mathematique soit divisible a I'infini, et croit 

 fort bien entendre qu'elle est composee de points en nombre fini, et 

 jamais je n*ai pu Ten tirer ; si vous pouviez le faire, on le rendroit 

 parfait. II me disoit done qu'il avoit trouve faussete dans les nombres 

 par cette raison. 



The difficulty is the following. If we undertake to throw a 

 six with one die the odds are in favour of doing it in four throws, 

 being as 671 to 625 ; if we undertake to throw two sixes with two 

 dice the odds are not in favour of doing it in twenty-four throws. 

 Nevertheless 24 is to 86, which is the number of cases with two 

 dice, as 4 is to 6, which is the number of cases with one die. 

 Pascal proceeds 



"Voilk quel etoit son grand scandale, qui lui faisoit dire hautement 

 que les propositions n'etoient pas constantes, et que I'arithmetique se 

 d^mentoit. Mais vous en verrez bien aisement la raison, par les prin- 

 cipes o^ vous etes. 



15. In Pascal's letter, as it is printed in Fermat's works, the 

 name de Mere is not given in the passage we have quoted in the 

 preceding article ; a blank occurs after the 21. It seems, however, 

 to be generally allowed that the blank has been filled up correctly 

 by the publishers of Pascal's works : Montmort has no doubt on 

 the matter ; see his p. XXXII. See also Gouraud, p. 1 ; Lubbock 

 and Drinkwater, p. 41. But there is certainly some difficulty. For 

 in the extract which we have given in Art. 11, Pascal states that 

 M. de Mere could solve one problem, celle des des, and seems to 

 imply that he failed only in the Problem of Points. Montucla 

 says that the Problem of Points w^as proposed to Pascal by the 

 Chevalier de Mer^, " qui lui en proposa aussi quelques autres sur le 

 jeu de des, comme de detemiiner en combien de coups on pent 

 parier d'amener une rafle, &c. Ce chevalier, plus bel esprit que 



