SECT. 3.] On certain kinds of Groups or Series. 5 



is not the import of the statement. It rather contemplates 

 our examination of a large number, of a long succession 

 of instances, and states that in such a succession we shall 

 find a numerical proportion, not indeed fixed and accurate at 

 first, but which tends in the long run to become so. In 

 every kind of example with which we shall be concerned we 

 shall find this reference to a large number or succession 

 of objects, or, as we shall term it, series of them. 



A few additional examples may serve to make this plain. 

 Let us suppose that we toss up a penny a great many 

 times ; the results of the successive throws may be conceived 

 to form a series. The separate throws of this series seem to 

 occur in utter disorder; it is this disorder which causes our 

 uncertainty about them. Sometimes head comes, sometimes 

 tail comes ; sometimes there is a repetition of the same face, 

 I sometimes not. So long as we confine our observation to a 

 few throws at a time, the series seems to be simply chaotic. 

 But when we consider the result of a long succession we find 

 a marked distinction; a kind of order begins gradually to 

 emerge, and at last assumes a distinct and striking aspect. 

 We find in this case that the heads and tails occur in about 

 equal numbers, that similar repetitions of different faces do 

 so also, and so on. In a word, notwithstanding the individual 

 disorder, an aggregate order begins to prevail. So again if 

 we are examining the length of human life, the different lives 

 which fall under our notice compose a series presenting the 

 same features. The length of a single life is familiarly un 

 certain, but the average duration of a batch of lives is be 

 coming in an almost equal degree familiarly certain. The 

 larger the number we take out of any mixed crowd, the 

 clearer become the symptoms of order, the more nearly will 

 the average length of each selected class be the same. 

 These few cases will serve as simple examples of a property 



