16 On certain kinds of Groups or Series. [CHAP. I. 



are not, strictly speaking, exceptions. A type, that is, which 

 shall be in the fullest sense of the words, persistent and 

 invariable is scarcely to be found in nature. The full import 

 of this conclusion will be seen in future chapters. Attention 

 is only directed here to the important inference that, although 

 statistics are notoriously of no value unless they are in suffi 

 cient numbers, yet it does not follow but that in certain cases 

 we may have too many of them. If they are made too ex 

 tensive, they may again fall short, at least for any particular 

 time or place, of their greatest attainable accuracy. 



12. These natural uniformities then are found at 

 length to be subject to fluctuation. Now contrast with them 

 any of the uniformities afforded by games of chance ; these 

 latter seem to show no trace of secular fluctuation, however 

 long we may continue our examination of them. Criticisms 

 will be offered, in the course of the following chapters, upon 

 some of the common attempts to prove a priori that there 

 must be this fixity in the uniformity in question, but of its 

 existence there can scarcely be much doubt. Pence give 

 heads and tails about equally often now, as they did when 

 they were first tossed, and as we believe they will continue 

 to do, so long as the present order of things continues. The 

 fixity of these uniformities may not be as absolute as is 

 commonly supposed, but no amount of experience which we 

 need take into account is likely in any appreciable degree to 

 interfere with them. Hence the obvious contrast, that, 

 whereas natural uniformities at length fluctuate, those af 

 forded by games of chance seem fixed for ever. 



13. Here then are series apparently of two different 

 kinds. They are alike in their initial irregularity, alike in 

 their subsequent regularity ; it is in what we may term 

 their ultimate form that they begin to diverge from each 

 other. The one tends without any irregular variation 



