JAMES BERNOULLI. 67 



the doctrine of the number of divisors of a given number; for 

 more information he refers to the works of Schooten and WaUis, 

 which we have already examined ; see Arts. 42, 47. 



115. In his seventh Chapter James Bernoulli gives the for- 

 mula for what we now call the number of permutations of n things 

 taken c at a time. In the remainder of this part of his book he 

 discusses some other questions relating to permutations and com- 

 binations, and illustrates his theory by examples. 



116. The third part of the Ars Conjectandi occupies pages 

 138 — 209; it consists of twenty-four problems which are to illus- 

 trate the theory that has gone before in the book. James Ber- 

 noulli gives only a few lines of introduction, and then proceeds to 

 the problems, which he says, 



...nullo fere habito selectu, prout in adversariis reperi, proponam, prre- 

 niissis etiam vel intersj)ersis nonnuUis facilioribus, et in quibua nidlus 

 combiiiationum usus apparet. 



117. The fourteenth problem deserves some notice. There 

 are two cases in it, but it will be sufficient to consider one of 

 them. A is to throw a die, and then to repeat his throw as many 

 times as the number thrown the first time. A is to have the 

 whole stake if the sum of the numbers given by the latter set of 

 throws exceeds 12; he is to have half the stake if the sum is 

 equal to 12; and he is to have nothing if the sum is less than 

 12. Required the value of his expectation. It is found to be 



^Y^^rr , Avliich is rather less than ^ . After giving the connect 



solution James Bernoulli gives another which is plausible but 

 false, in order, as he says, to impress on his readers the necessity 

 of caution in these discussions. The following is the false solution. 



A has a chance equal to -x of throwing an ace at his first trial; 



in this case he has only one throw for the stake, and that throw 

 may give him with equal probabihty any number between 1 and 6 



inclusive, so that we may take ^ (1 + 2 + 34-44-5+6), that is 



31, for his mean throw. We may observe that 3^ is the Arith- 



5—2 



