10 ON THE FOUNDATIONS OF 



When the trials have taken place, we shall have approxi 

 mately, 



+ ^+ ...... + 



m in 



of the souht events. Of these 



* 





had d priori a probability greater than J. Summing these series 

 and dividing the second by the first, we get ^^ , for the ratio 



which the latter class of events bears to the total number. The 

 limit of this, when m is infinite, or when we take an infinite 

 number of sets of trials, is f , which is the received result. 



17. Thus, it appears this result is based upon some thing 

 equivalent to the following assumption : There are an infinity 

 of events whose simple probability a priori is x, and another 

 infinite number for which it is x'. These two infinities bear to 

 one another the definite ratio of equality (x and x' may repre- 

 sent any quantity from to 1). Now in reality, as we have seen, 

 these numbers are not only infinite, but in rerum naturd inde- 

 terminate, and therefore the assumption that they bear to one 

 another a definite ratio is illusory. 



And this assumption runs through all the applications of the 

 theory to events whose causes are unknown. 



This position could be directly proved only by an analysis of 

 the various ways in which this part of the subject has been con- 

 sidered, which would require a good deal of detail. Those who 

 take an interest in the question, may without much difficulty 

 satisfy themselves, whether the view I have taken (which at 

 least avoids the manifest contradictions of the received results) is 

 correct. 



18. I will add only one remark. If in (16) instead of 

 taking one event from each of the trials there specified, we had 

 taken p in succession, and kept account only of those sequences 

 of p events each, which contained none but events of the kind 

 sought ; we should have had of such sequences 



