212 JOHN BERNOULLI. 



It will be more convenient to defer any account of the section 

 in Simpson until we examine Lagrange's memoir, and then we will 

 state what Simpson gave in 17 o7. 



372. The fourth volume of the collected edition of John Ber- 

 noulli's works, which was published in 1742 has a section entitled 

 De Alea, sive Arte Conjectandi, Prohlemata qucedam; this section 

 occupies pages 28 — 33 : it contains seven problems. 



373. The first and second problems are simple and well- 

 known ; they are solved completely. The third problem relates to 

 the game of Bowls ; John Bernoulli gives, without demonstration, 

 the result which had already been published ; see Montmort, 

 page 248, and the Doctrine of Chances, page 117. 



374. The fourth problem contains an error. John Bernoulli 

 sa3"s that if 2n common dice are thrown, the number of ways in 

 which the sum of the marks is 7n is 



(7n-l) (7^-2)(7n-3)...(5yz + l) . 

 1.2.3.4 ... (2?z-l) * 



this amounts to asserting that the expression here given is the co- 

 efficient of x"" in the expansion of 



. (ic + a?' -I- a;' + a;' + x\+ x^ : 



in fact however the coefficient is a series of which the above ex- 

 pression is only the first term. 



375. The fifth and sixth problems involve nothing new in 

 principle ; John Bernoulli gives merely the numerical results which 

 would require long calculation to verify. The seventh problem 

 does not seem intelligible. 



