440 



which evidence is of a cumulative character, as where different posi- 

 tive assertions are made of the absolute truth of a fact, — but to cases 

 in which different modes of considering a subject lead us to assign 

 different probabilities to a fact or hypothesis, and it is our object to 

 take between these probabilities an average. 



For example if A and B, whose veracity, that is the probability 

 of their speaking truth, has a given value, both affirm that the 

 event E has occurred, the formula does not apply. The proper, 



at any rate the received, formula for such a problem is ^ ^ 



p q + l-p . l — q 



But if the question be the probability that Sir Philip Francis was 

 the author of Junius' Letters, and p and q be the probabilities 

 derived from external and internal evidence, then the formula 



applies. 



aJ p Q -^ ij l-p . l-q 



In that portion of the memoir, which is introductory to the de- 

 monstration of the above results, the author explains the grounds of 

 his method for the solution of questions in the theory of probabilities 

 whose elements are logical. They are briefly the following : — 



\stf He defines the mathematical probability of an event, as the 

 ratio which the number of cases or hypotheses favourable to that 

 event bears to the whole number of cases conceivable, supposing 

 that to none of those cases the mind is entitled to attach any pre- 

 ference over any other. 



2dly, He remarks, that when the probabilities of simple events con- 

 stitute our only data, no knowledge whatever being given of their 

 connection, we can thence, by virtue of the definition, determine the 

 probability of any logical combination of them, either absolutely or 

 conditionally. 



^dly, He postulates that when the data are not the probabilities of 

 simple events, we can only grasp them and apply them to the calcu- 

 lation of probability by regarding them, not as primary, but as de- 

 rived from some hypothesis in which the data are the probabilities of 

 simple events, and to which, accordingly, we may apply the princi- 

 ples already referred to, as flowing from the very definition of pro- 

 bability. The probabilities of the simple events in the hypothesis 

 must, of course, be determined in accordance with the original data. 

 At this stage the question arises, How shall such an hypothesis 

 be lawfully framed? To this question the following answer is given. — 



