252 OF THE THEORY OF PROBABILITIES. [CHAP. XVI. 



possible cases is diminished, while the'number of favourable cases 

 remains unaltered, in exactly the same proportion will the pro- 

 babilities of any events to which these cases have reference be 

 increased. And as the new hypothesis, viz., the diminution of 

 the number of possible cases without affecting the number of 

 them which are favourable to the events in question, increases 

 the probabilities of those events in a constant ratio, the relative 

 measures of those probabilities remain unaltered. If the principle 

 we are considering be then, as it appears to be, inseparably in- 

 volved in the very definition of probability, it can scarcely, of 

 itself, conduct us further than the attentive study of the defini- 

 tion would alone do, in the solution of problems. From these 

 considerations it appears to be doubtful whether, without some 

 aid of a different kind from any that has yet offered itself to our 

 notice, any considerable advance, either in the theory of proba- 

 bilities as a branch of speculative knowledge, or in the practical 

 solution of its problems can be hoped for. And the establish- 

 ment, solely upon the basis of any such collection of principles as 

 the above, of a method universally applicable to the solution of 

 problems, without regard either to the number or to the nature 

 of the propositions involved in the expression of their data, 

 seems to be impossible. For the attainment of such an object 

 other elements are needed, the consideration of which will occupy 

 the next chapter. 



