116 MONTMORT. 



changed the enunciation into another quite different, and yet had 

 really solved the problem according to Huygens's enunciation. By 

 the corrections which he made in his second edition, Montmort 

 shewed that he admitted the justice of the objections urged against 

 his solutions of the second and fifth problems; in the case of 

 the third problem he retained his original opinion; see his 

 pages 292, 805. 



John Bernoulli next notices the solution of the Problem of 

 Points, and gives a general formula, to which we have referred in 

 Art. 173. Then he adverts to a problem which Montmort had 

 not fully considered; see Art. 185. 



200. John Bernoulli gives high praise to Montmort's work, 

 but urges him to extend and enrich it. He refers to the four 

 problems which Montmort had proposed for investigation ; the 

 first he considers too long to be finished in human life, and the 

 fourth he cannot understand : the other two he thinks might be 

 solved by great labour. This opinion seems singularly incorrect. 

 The first problem is the easiest of all, and has been solved without 

 difficulty; see Article 161 : perhaps however John Bernoulli took 

 it in some more general sense; see Montmort's page 308. The 

 fourth problem is quite intelligible, and a particular case of it is 

 simple ; see Art. 193. The third and fourth problems seem to be 

 far more intractable. 



201. A letter to Montmort from Nicolas Bernoulli occupies 

 pages 299 — 303. This letter contains corrections of two mistakes 

 which occurred in Montmort's first edition. It gives without de- 

 monstration a formula for the advantage of the Banker at Pharaon, 

 and also a formula for the advantage of the Banker at Bassette ; 

 Montmort quoted the former in the text of his second edition ; 

 see Art. 157. Nicolas Bernoulli gives a good investigation of the 

 formulae which occur in analysing the game of Treize ; see Art. 161. 

 He also discusses briefly a game of chance which we will now 

 explain. 



202. Suppose that a set of players A, B, C, D, ... undertake 

 to play a set of I games with cards. A is at first the dealer, there 

 are m chances out of on + n that he retains the deal at the next 

 game, and n chances out of m + 7i that he loses it ; if he loses the 



