huvgp:ns. 2.3 



of which that work is composed. Two English translations of the 

 treatise have been published ; one which has been attributed to 

 Motte, but which was probably by Arbuthnot, and the other by 

 W. Browne. 



31. The treatise contains fourteen propositions. The first pro- 

 position asserts that if a player has equal chances of gaining a sum 

 represented by a or a sum represented by b, his expectation is 

 ^ (a + b). The second proposition asserts that if a player has equal 

 chances of gaining a or 6 or c, his expectation is J (a + 6 + c). The 

 third proposition asserts that if a player has 2^ chances of gaining a 



and q chances of gaining b, his expectation is — . 



i^ + 2' 



It has been stated with reference to the last proposition : 



*' Elementary as this truth may now appear, it was not received 



altogether without opposition." Lubbock and Drinhwater, p. 42. 



It is not obvious to what these words refer; for there does not 



appear to have been any opposition to the elementary principle, 



except at a much later period by D'Alembert. 



82. The fourth, fifth, sixth, and seventh propositions discuss 

 simple cases of the Problem of Points, when there are two players ; 

 the method is similar to Pascal's, see Art. 12. The eiirhth and 

 ninth propositions discuss simple cases of the Problem of Points 

 when there are ^/i?'e^ players ; the method is similar to that for two 

 players. 



83. Huygens now proceeds to some questions relating to dice. 

 In his tenth proposition he investigates in how many throws a 

 player may undertake to throw a six with a single die. In his 

 eleventh proposition he investigates in how many throws a player 

 may undertake to throw twelve with a pair of dice. In his 

 twelfth proposition he investigates how many dice a player must 

 have in order to undertake that in one throw two sixes at least 

 may appear. The thirteenth proposition consists of the following 

 problem. A and B play with two dice ; if a seven is thrown, 

 ^1 wins; if a ten is thrown, B Avins; if any other number is 

 thrown, the stakes are divided : compare the chances of A and B. 

 They are shewn to be as 13 is to 11. 



