88 Modes of establishing the Groups or Series. [CHAP. IV. 



having only arithmetical calculations to connect them to 

 gether. I wish, for instance, to find the chance of throwing 

 heads three times running with a penny. All I have to do 

 is first to ascertain the possible number of throws. Permu 

 tations tell me that with two things thus in question (viz. 

 head and tail) and three times to perform the process, there 

 are eight possible forms of the result. Of these eight one 

 only being favourable, the chance in question is pronounced 

 to be one- eighth. 



Now though it is quite true that the actual calculation of 

 every chance problem must be of the above character, viz. an 

 algebraical or arithmetical process, yet there is, it seems to 

 me, a broad and important distinction between a material 

 science which employs mathematics, and a formal one which 

 consists of nothing but mathematics. When we cut our 

 selves off from the necessity of any appeal to experience, we 

 are retaining only the intermediate or calculating part of the 

 investigation; we may talk of dice, or pence, or cards, but 

 these are really only names we choose to give to our symbols. 

 The H s and T s with which we deal have no bearing on ob 

 jective occurrences, but are just like the a? s and y s with 

 which the rest of algebra deals. Probability in fact, when so 

 treated, seems to be absolutely nothing else than a system of 

 applied Permutations and Combinations. 



It will now readily be seen how narrow is the range of 

 cases to which any purely deductive method of treatment can 

 apply. It is almost entirely confined to such employments 

 as games of chance, and, as already pointed out, can only be 

 regarded as really trustworthy even there, by the help of 

 various tacit restrictions. This alone would be conclusive 

 against the theory of the subject being rested upon such a 

 basis. The experimental method, on the other hand, is, in the 

 same theoretical sense, of universal application. It would 



