66 JAMES BERNOULLI. 



taken c at a time. He also draws various simple inferences from 

 the rule. He digresses from the subject of this part of his book to 

 resume the discussion of the Problem of Points ; see his page 107. 

 He gives two methods of treating the problem by the aid of 

 the theory of combinations. The first method shews how the 

 table which he had exhibited in the first part of the A7'S Con- 

 jectandi might be continued and the law of its terms expressed; 

 the table is a statement of the chances of A and B for winning 

 the game when each of them wants an assigned number of points. 

 Pascal had himself given such a table for a game of six points ; 

 an extension of the table is given on page 16 of the Ars Con- 

 jectandi, and now James Bernoulli investigates general expressions 

 for the component numbers of the table. From his investigation 

 he derives the result which Pascal gave for the case in which one 

 player wants one point more than the other player. James Ber- 

 noulli concludes this investigation thus ; Ipsa solutio Pascaliana, 

 quae Auctori suo tantopere arrisit. 



James Bernoulli's other solution of the Problem of Points is 

 much more simple and direct, for here he does make the application 

 to which we alluded in Art. 101^ Suppose that A wants m points 

 and B wants 7i points ; then the game will certainly be decided in 

 m + n — 1 trials. As in each trial A and B have equal chances 

 of success the whole number of possible cases is 2"'"^""\ And 

 A wins the game if B gains no point, or if B gains just one point, 

 or just two points,... or any number up to w — 1 inclusive. Thus 

 the number of cases favourable to A is 



! + ;. + _-_ + ^ + ... + ^^^ ^ 



where //< = m -f w — 1 . 



Pascal had in effect advanced as far as this; see Art. 23: but 

 the formula is more convenient than the Arithmetical Triangle. 



114. In his fifth Chapter James Bernoulli considers another 

 question of combinations, namely that which in modern treatises is 

 enunciated thus : to find the number of homogeneous products of 

 the r^^ degree which can be formed of n symbols. In his sixth 

 Chapter he continues this subject, and makes a slight reference to 



