86 Modes of establishing the Groups or Series. [CHAP. iv. 



themselves, owing to the infinite complications produced by 

 the casual variations in throwing; but the long ones may 

 suffer, unless their interest be consciously or unconsciously 

 regarded at the outset. 



12. The advice, Only try long enough, and you will 

 sooner or later get any result that is possible, is plausible, 

 but it rests only on Induction and Analogy; mathematics do 

 not prove it. As has been repeatedly stated, there are two 

 distinct views of the subject. Either we may, on the one 

 hand, take a series of symbols, call them heads and tails; 

 H, T, &c. ; and make the assumption that each of these, and 

 each pair of them, and so 011, will occur in the long run with 

 a regulated degree of frequency. We may then calculate 

 their various combinations, and the consequences that may 

 be drawn from the data assumed. This is a purely algebraical 

 process; it is infallible; and there is no limit whatever to the 

 extent to which it may be carried. This way of looking at 

 the matter may be, and undoubtedly should be, nothing 

 more than the counterpart of what I have called the substi 

 tuted or idealized series which generally has to be introduced 

 as the basis of our calculation. The danger to be guarded 

 against is that of regarding it too purely as an algebraical 

 conception, and thence of sinking into the very natural 

 errors both of too readily evolving it out of our own con 

 sciousness, and too freely pushing it to unwarranted lengths. 

 Or on the other hand, we may consider that we are treat 

 ing of the behaviour of things; balls, dice, births, deaths, 

 &c.; and drawing inferences about them. But, then, what 

 were in the former instance allowable assumptions, become 

 here propositions to be tested by experience. Now the whole 

 theory of Probability as a practical science, in fact as any 

 thing more than an algebraical truth, depends of course upon 

 there being a close correspondence between these two views 



