240 NATURAL INHERITANCE. 



already large nation tends to receive larger accessions than a small 

 one under similar circumstances, or when a capital employed in a 

 business increases in proportion to its size. The other category is 

 the influences of circumstances or of " milieux " as they are often 

 called, such as a period of plenty in which a larger field or a larger 

 business yields a greater excess over its mean yield than a smaller 

 one. Most of the causes of those differences with which sociology are 

 concerned, and which are not purely vital phenomena, such as those 

 previously discussed, may be classified under one or other of these 

 two categories, or under such as are in principle almost the same. 

 In short, sociological phenomena, like vital phenomena are, as a 

 general rule, subject to the condition of the geometric mean. 



The ordinary law of Frequency of Error, based on the arithmetic 

 mean, corresponds, no doubt, sufficiently well with the observed facts 

 of vital and social phenomena, to be very serviceable to statisticians, 

 but it is far from satisfying their wants, and it may lead to absurdity 

 when applied to wide deviations. It asserts that deviations in excess 

 must be balanced by deviations of equal magnitude in deficiency ; 

 therefore, if the former be greater than the mean itself, the latter 

 must be less than zero, that is, must be negative. This is an impossi- 

 bility in many cases, to which the law is nevertheless applied by sta- 

 tisticians with no small success, so long as they are content to confine 

 its application within a narrow range of deviation. Thus, in respect 

 of Stature, the law is very correct in respect to ordinary measure- 

 ments, although it asserts that the existence of giants, whose height 

 is more than double the mean height of their race, implies the possi- 

 bility of the existence of dwarfs, whose stature is less than nothing 

 at all. 



It is therefore an object not only of theoretical interest but of 

 practical use, to thoroughly investigate a Law of Error, based on the 

 geometric mean, even though some of the expected results may 

 perhaps be apparent at first sight. With this view I placed the fore- 

 going remarks in Mr. Donald Macalister's hands, who contributed 

 a memoir that will be found in the Proc. Royal Soc., No. 198, 1879, 

 following my own. It should be referred to by such mathematicians 

 as may read this book. 



