SECT. 9.] Modes of establishing the Groups or Series. 83 



occur once in four times. It cannot be proved indeed in this 

 way that they need ever occur at all. 



9. The formula, then, not being demonstrable a priori, 

 (as might have been concluded,) can it be obtained by ex 

 perience ? To a certain extent it can ; the present experience 

 of mankind in pence and dice seems to show that the smaller 

 successions of throws do really occur in about the proportions 

 assigned by the theory. But how nearly they do so no one 

 can say, for the amount of time and trouble to be expended 

 before we could feel that we have verified the fact, even for 

 small numbers, is very great, whilst for large numbers it 

 would be simply intolerable. The experiment of throwing 

 often enough to obtain heads ten times has been actually 

 performed by two or three persons, and the results are given 

 by De Morgan, and Jevons 1 . This, however, being only 

 sufficient on the average to give heads ten times a single 

 chance, the evidence is very slight ; it would take a con 

 siderable number of such experiments to set the matter 

 nearly at rest. 



Any such rule, then, as that which we have just been 

 discussing, which professes to describe what will take place 

 in a long succession of throws, is only conclusively proved by 

 experience within very narrow limits, that is, for small repe 

 titions of the same face ; within limits less narrow, indeed, 

 we feel assured that the rule cannot be flagrantly in error, 

 otherwise the variation would be almost sure to be detected. 

 From this we feel strongly inclined to infer that the same 

 law will hold throughout. In other words, we are inclined 

 to extend the rule by Induction and Analogy. Still there 

 are so many instances in nature of proposed laws which hold 

 within narrow limits but get egregiously astray when we 



1 Formal Logic, p. 185. Principles of Science, p. 208. 



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