164 Measurement of Belief. [CHAP. vi. 



36. A word or two of explanation may be added about 

 the expression employed above, the proportion in the long 

 run. The run must be supposed to be very long indeed, in 

 fact never to stop. As we keep on taking more terms of the 

 series we shall find the proportion still fluctuating a little, 

 but its fluctuations will grow less. The proportion, in fact, 

 will gradually approach towards some fixed numerical value, 

 what mathematicians term its limit. This fractional value 

 is the one spoken of above. In the cases in which deductive 

 reasoning is possible, this fraction may be obtained without 

 direct appeal to statistics, from reasoning about the con 

 ditions under which the events occur, as was explained in 

 the fourth chapter. 



Here becomes apparent the full importance of the dis 

 tinction so frequently insisted on, between the actual irregular 

 series before us and the substituted one of calculation, and 

 the meaning of the assertion (Ch. I. 13), that it was in the 

 case of the latter only that strict scientific inferences could 

 be made. For how can we have a limit in the case of 

 those series which ultimately exhibit irregular fluctuations ? 

 When we say, for instance, that it is an even chance that 

 a given person recovers from the cholera, the meaning of 

 this assertion is that in the long run one half of the persons 

 attacked by that disease do recover. But if we examined 

 a sufficiently extensive range of statistics, we might find 

 that the manners and customs of society had produced such 

 a change in the type of the disease or its treatment, that we 

 were no nearer approaching towards a fixed limit than we 

 were at first. The conception of an ultimate limit in the 

 ratio between the numbers of the two classes in the series 

 necessarily involves an absolute fixity of the type. When 

 therefore nature does not present us with this absolute fixity, 

 as she seldom or never does except in games of chance (and 



