CHAP. XVI.] OF THE THEORY OF PROBABILITIES. 251 



ability to discuss such a system as the above, that we are able to 

 resolve problems whose data are the probabilities of any system 

 of conditional propositions ; far less that we can resolve problems 

 whose data are the probabilities of any system of propositions 

 whatever. And, viewing the subject in its material rather 

 than its formal aspect, it is evident, that the hypothesis of exclu- 

 sive causation is one which is not often realized in the actual 

 world, the phenomena of which seem to be, usually, the products 

 of complex causes, the amount and character of whose co-opera- 

 tion is unknown. Such is, without doubt, the case in nearly all 

 departments of natural or social inquiry in which the doctrine of 

 probabilities holds out any new promise of useful applications. 



9. To the above principles we may add another, which has 

 been stated in the following terms by the Savilian Professor of 

 Astronomy in the University of Oxford.* 



"Principle 8. If there be any number of mutually exclusive 

 hypotheses, h l9 A 25 ^ 3 , . . of which the probabilities relative to a 

 particular state of information are p^p^p A , . . and if new infor- 

 mation be given which changes the probabilities of some of them, 

 suppose of k m+l and all that follow, without having otherwise 

 any reference to the rest ; then the probabilities of these latter 

 have the same ratios to one another, after the new information, 

 that they had before, that is, 



P\ : p'z : p'z - - : p'm = PI P* PS ' Pm, 



where the accented letters denote the values after the new infor- 

 mation has been acquired." 



This principle is apparently of a more fundamental character 

 than the most of those before enumerated, and perhaps it might, as 

 has been suggested by Professor Donkin, be regarded as axio- 

 matic. It seems indeed to be founded in the very definition of 

 the measure of probability, as "the ratio of the number of cases 

 favourable to an event to the total number of cases favourable or 

 contrary, and all equally possible." For, adopting this definition, 

 it is evident that in whatever proportion the number of equally 



* On certain Questions relating to the Theory of Probabilities ; by W. F. 

 Donkin, M. A., F. R. S., &c. Philosophical Magazine, May, 1851. 



