16 PASCAL AND FERMAT. 



20. There is another letter £i'oni Fermat to Pascal which is 

 not dated. It relates to a simple question which Pascal had pro- 

 posed to Fermat. A person undertakes to throw a six with a die 

 in eight throws ; supposing him to have made three throws with- 

 out success, what portion of the stake should he be allowed to take 

 on condition of giving up his fourth throw ? The chance of success 

 is J, so that he should be allowed to take J of the stake on con- 

 dition of giving up his throw. But suppose that we wish to esti- 

 mate the value of the fourth throw before any throw is made. The 

 first throw is worth J of the stake ; the second is worth J of what 

 remains, that is -^ of the stake ; the third throw is worth i of w^hat 

 now remains, that is -ff^ of the stake ; the fourth throw is worth 

 J of what now remains, that is -^-ff-Q of the stake. 



It seems possible from Format's letter that Pascal had not dis- 

 tinguished between the two cases ; but Pascal's letter, to which 

 Format's is a reply, has not been preserved, so that we cannot 

 be certain on the point. 



21. We see then that the Problem of Points was the prin- 

 cipal question discussed by Pascal and Fermat, and it was certainly 

 not exhausted by them. For they confined themselves to the case 

 in which the players are supposed to possess equal skill; and their 

 methods would have been extremely laborious if applied to any 

 examples except those of the most simple kind. Pascal's method 

 seems the more refined ; the student will perceive that it depends 

 on the same principles as the modern solution of the problem 

 by the aid of the Calculus of Finite Differences ; see Laplace, 

 Theorie...cles Proh. page 210. 



Gouraud awards to Format's treatment of the problem an 

 amount of praise which seems excessive, whether we consider that 

 treatment absolutely or relatively in comparison with Pascal's ; see 

 his page 9, 



22. We have next to consider Pascal's Traite du triangle 

 arithmetique. This treatise was printed about 1G5-4, but not 

 pubhshed until 1665 ; see Montucla, p. 387. The treatise will be 

 found in the fifth volume of the edition of Pascal's works to which 

 we have already referred. 



