DE MOIVRE. 163 



however, some details which the reader might be expected to fill 

 up for himself, and also the method of Brook Taylor. On the 

 other hand, the last nine pages of the discussion in the third 

 edition were not in the first edition ; these consist of explanations 

 and investigations with the view of enabling a reader to determine 

 numerical results for any number of players, supposing that at 

 any stage it is required to stop the play and divide the money 

 deposited equitably. This part of the problem is peculiar to 

 De Moivre. 



The discussions which De Moivre gives of the particular 

 cases of three players and four players are very easy and satis- 

 factory ; but as a general solution his method seems inferior to 

 that of Nicolas Bernoulli. We may remark that the investigation 

 for three players given by De Moivre will enable the student to 

 discover how Montmort obtained the results which he gives with- 

 out demonstration for three players ; see Art. 209. De Moivre 

 determines a pla^^er's expectation by finding first the advantage 

 resulting from his chance of winning the whole sum deposited, and 

 then his disadvantage arising from the contributions which he 

 may have had to make himself to the whole sum deposited ; the 

 expectation is obtained by subtracting the second result from the 

 first. Montmort determined the expectation by finding, first the 

 advantage of the player arising from his chance of winning the 

 deposits of the other two players, and then the disadvantage 

 arising from the chance which the other two players have of 

 winning his deposits ; the expectation is obtained by subtracting 

 the second result from the first. 



The problem will come before us again as solved by Laplace. 



296. Problem XLVI. is on the game of Hajzard; there is no 

 description of the game here, but there is one given by Montmort 

 on his page 177 ; and from this description, De Moivre's solution 

 can be understood : his results agree with Montmort's. Pro- 

 blem XLVII. is also on Hazard ; it relates to a point in the game 

 which is not noticed by Montmort, and it is only from De Moivre's 

 investigation itself that we can discover wliat the problem is, 

 which he is considering. With respect to this problem, De Moivre 



says, page 165, 



11—2 



