JAMES BERXOULLI. 63 



In precisely tlie same manner we may find jS's chance at any 



stage of the game ; and his chance at the beginning of the game 



will be 



h"" (g^ - If) 



It will be observed that the sum of the chances of A and B at 

 the beginning of the game is unitif. The interpretation of this 

 result is that one or other of the players must eventually win 

 all the counters; that is, the play must terminate. This might 

 have been expected, but was not assumed in the investigation. 



The formula which James Bernoulli here gives will next come 

 before us in the correspondence between Nicolas Bernoulli and 

 Montmort; it was however first published by De Moi\Te in his 

 De Mensiira Soiiis, Problem ix., where it is also demonstrated. 



108. We may observe that Bernoulli seems to have found, 

 as most who have studied the subject of chances have also found, 

 that it was extremely easy to fall into mistakes, especially by 

 attempting to reason without strict calculation. Thus, on his 

 page 15, he points out a mistake into which it would have been 

 easy to fall, nisi nos calculus aliud clocuisset He adds, 



Qao ipso proin monemiir, ut cauti siraiis in jiidicando, 'nee ratio- 

 cinia nostra super qiiacunque statim aiialogia in rebus deprehensji fun- 

 dare suescamus; quod ipsum tamen etiam ab iis, qui vel maxinie sapere 

 videntur, nimis frequenter fieri solet. 



Again, on his page 27, 



Quae quideiu eum in finem hie adduce, ut palam fiat, quam parum 

 fideudum sit ejusmodi ratiociniis, qu?e corticem tantuiu attingunt, nee 

 in ipsam rei naturam altius penetrant; tametsi in toto vitse usu etiam. 

 apud sapientissimos quosque nihil sit frequentius. 



Again, on his page 29, he refers to the difficulty which Pascal 

 says had been felt by M. de * * * *, whom James Bernoulli calls 

 Anonymus quidam coetera subacti judicii Yir, sed Geometriae 

 expers. . James Bernoulli adds, 



Hac enim qui imbuti sunt, ejusmodi erai'Tto^avetai minime moran- 

 tur, probe conscii dari innumera, qua3 admoto calculo aliter se habere 

 comperiuntur, quam initio apparebaut; ideoque sedulb cavent, juxta id 

 quod semel iterumque monui, ne quicquam analogiis temere tribuant. 



