SECT. 7.] Modes of establishing the Groups or Series. 81 



therefore, if we scrutinize our language, we shall find that 

 any supposed a priori mode of stating a problem is little else 

 than a compendious way of saying, Let means be taken for 

 obtaining a given result. Since it is upon this result that 

 our inferences ultimately rest, it seems simpler and more 

 philosophical to appeal to it at once as the groundwork of 

 our science. 



7. Let us again take the instance of the tossing of a 

 penny, and examine it somewhat more minutely, to see what 

 can be actually proved about the results we shall obtain. 

 We are willing to give the pence fair treatment by assuming 

 that they are perfect, that is, that in the long run they show 

 no preference for either head or tail ; the question then 

 remains, Will the repetitions of the same face obtain the 

 proportional shares to which they are entitled by the 

 usual interpretations of the theory ? Putting then, as before, 

 for the sake of brevity, H for head, and HH for heads twice 

 running, we are brought to this issue ; Given that the 

 chance of H is J, does it follow necessarily that the chance of 

 HH (with two pence) is J ? To say nothing of H ten times 

 occurring once in 1024 times (with ten pence), need it occur 

 at all ? The mathematicians, for the most part, seem to think 

 that this conclusion follows necessarily from first principles ; 

 to me it seems to rest upon no more certain evidence than a 

 reasonable extension by Induction. 



Taking then the possible results which can be obtained 

 from a pair of pence, what do we find ? Four different 

 results may follow, namely, (1) HT, (2) HH, (3) TH, (4) TT. 

 If it can be proved that these four are equally probable, that 

 \ is, occur equally often, the commonly accepted conclusions 

 will follow, for a precisely similar argument would apply to 

 all the larger numbers. 



8. The proof usually advanced makes use of what is 

 v. 6 



