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



attempt to push them to great lengths, that we must give at 

 best but a qualified assent to the truth of the formula. 



10. The object of the above reasoning is simply to 

 show that we cannot be certain that the rule is true. Let 

 us now turn for a minute to consider the causes by which 

 the succession of heads and tails is produced, and we may 

 perhaps see reasons to make us still more doubtful. 



It has been already pointed out that in calculating proba 

 bilities a priori, as it is called, we are only able to do so by 

 introducing restrictions and suppositions which are in reality 

 equivalent to assuming the expected results. We use words 

 which in strictness mean, Let a given process be performed ; 

 but an analysis of our language, and an examination oJ 

 various tacit suppositions which make themselves felt the 

 moment they are not complied with, soon show that our real 

 meaning is, Let a series of a given kind be obtained ; it is to 

 this series only, and not to the conditions of its production, 

 that all our subsequent calculations properly apply. The 

 physical process being performed, we want to know whether 

 anything resembling the contemplated series really will be 

 obtained. 



Now if the penny were invariably set the same side up 

 permost, and thrown with the same velocity of rotation anc 

 to the same height, &c. in a word, subjected to the same 

 conditions, it would always come down with the same side 

 uppermost. Practically, we know that nothing of this kind 

 occurs, for the individual variations in the results of the 

 throws are endless. Still there will be an average of these 

 conditions, about which the throws will be found, as it were, 

 to cluster much more thickly than elsewhere. We should be 

 inclined therefore to infer that if the same side were always 

 set uppermost there would really be a departure from the 

 sort of series which we ordinarily expect. In a very large 



