2 8o THE SCIENCE OF LOGIC 



Although we cannot be sure that the result of any particular 

 throw will be a head, or a six, we can be sure that if we kept 

 on throwing indefinitely, the heads obtained would be , or the 

 sixes ^, of the whole series of trials. The reason for such certitude 

 about the results of a hypothetical indefinite series of experiences 

 is a negative one : because, namely, any other result would be un 

 accountable, would be an effect without a cause. But in any definite 

 series of experiences we are prepared to find that the actual results 

 may deviate somewhat from what the apriori probability points to 

 from i or \ in the cases just mentioned. We are justified, how 

 ever, in expecting further, that (i) the greater the total number 

 of experiences the more closely will the actual results approximate 

 to the a priori probability; though (2) in a larger number of ex 

 periences we are prepared to find that the deviation from the a 

 priori probability is in itself greater than in a smaller number of 

 experiences. In other words, as the series of trials extends, the 

 absolute magnitude of the possible deviations will also increase ; 

 but its relative magnitude, its proportion to the total number of 

 trials, will decrease : so that the actual results tend progressively 

 towards the realization of the antecedent probability. 



Within what limits, in any given series of experiences, should 

 the actual results fluctuate around the antecedent probability ? What 

 is the largest deviation that we should expect, in a given set of 

 trials, from a known or assumed antecedent probability? If we 

 could solve this problem, we could in some degree eliminate 

 chance, and conclude that the excessive deviation must be due to the 

 operation of some cause (264). &quot; If we are able to say,&quot; writes Fr. 

 Joyce, 1 &quot; that in a definite number of trials, certain limits will not 

 be overstepped, and if it is discovered that the result is totally at 

 variance with our mathematical estimate, then it becomes clear that 

 we were mistaken in our view as to the nature of the case. . . . 

 A loaded die gives, it is true, very irregular results. Were it not 

 so, it would be detected at once, and so defeat its own purpose. 

 But there comes a point at which, after a certain number of trials, 

 the mathematician is able to say that the appearances of the six 

 exceed the limits of mere chance, and that the very act of throw 

 ing must tend to turn that face uppermost.&quot; 



A noted mathematician, James Bernouilli (1650-1705), devoted 

 many years to the study of this whole question of deviations from 



1 op. dt. P . 378- 



