78 INDUCTION. 



twice as likely to have existed in this case, and to 

 have been the cause which produced M. 



For, since A exists in nature twice as often as B ; 

 in any 300 cases in which one or other existed, A has 

 existed 200 times and B 100. But either A or B 

 must have existed wherever M is produced : therefore 

 in 300 times that M is produced, A was the pro- 

 ducing cause 200 times, B only 100, that is, in the 

 ratio of 2 to 1. Thus, then, if the causes are alike in 

 their capacity of producing the eifect, the probability 

 as to which actually produced it, is in the ratio of their 

 antecedent probabilities. 



CASE II. Reversing the last hypothesis, let us 

 suppose that the causes are equally frequent, equally 

 likely to have existed, but not equally likely, if they 

 did exist, to produce M : that in three times that 

 A occurs, it produces that effect twice, while B, in 

 three times, produces it only once. Since the two 

 causes are equally frequent in their occurrence ; in 

 every six times that either one or the other exists, A 

 exists three times and B three times. A, of its three 

 times, produces M in two ; B, of its three times, 

 produces M in one. Thus, in the whole six times,, M 

 is only produced thrice ; but of that thrice it is pro- 

 duced twice by A, once only by B. Consequently, 

 when the antecedent probabilities of the causes are 

 equal, the chances that the effect was produced by 

 them are in the ratio of the probabilities that if they 

 did exist they would produce the effect. 



CASE III. The third case, that in which the causes 

 are unlike in both respects, is solved by what has pre- 

 ceded. For, when a quantity depends upon two other 

 quantities, in such a manner that while either of them 

 remains constant it is proportional to the other, it 

 must necessarily be proportional to the product of 



