10 PASCAL AND FERMAT. 



the second player none. If they proceed to play for a point the 

 condition is that if the first player gains it the players will be in 

 the situation first examined, in which the first player is entitled to 

 5Q pistoles ; if the first player loses the point each player has then 

 a point, and each is entitled to 32 pistoles. Thus if they do not 

 wish to play, the first player would say to the second " Give me 

 the 82 pistoles of which I am certain and divide the remainder of 

 the 56 pistoles equally, that is, divide 24 pistoles equally." Thus 

 the first player will have the sum of 32 and 12 pistoles, that is 

 44 pistoles, and consequently the second will have 20 pistoles. 



13. Pascal then proceeds to enunciate two general results 

 without demonstrations. We will give them in modern notation. 



(1) Suppose each player to have staked a sum of money 

 denoted by A ; let the number of points in the game be n+ 1, and 

 suppose the first player to have gained n points and the second 

 player none. If the players agree to separate without playing 



A 



any more the first player is entitled to 2 A — ~ . 



(2) Suppose the stakes and the number of points in the game 

 as before, and suppose that the first player has gained one point 

 and the second player none. If the players agree to separate 

 without playing any more, the first player is entitled to 



, 1 . 3 . 5 . . . (2n - 1) 



■^2.4.6... 2/1 • 



Pascal intimates that the second theorem is difficult to prove. 

 He says it depends on two propositions, the first of which is purely 

 arithmetical and the second of which relates to chances. The 

 first amounts in fact to the proposition in modern works on 

 Algebra which gives the sum of the co-efficients of the terms in 

 the Binomial Theorem. The second consists of a statement of 

 the value of the first player's chance by means of combinations, 

 from which by the aid of the arithmetical proposition the value 

 above given is deduced. The demonstrations of these two results 

 may be obtained from a general theorem which will be given later 

 in the present Chapter ; see Art. 23. Pascal adds a table which 



