302 THE PRINCIPLES OF SCIENCE. [CHAP. 



In all calculations concerning the average rate of 

 progress of a community, or any of its operations, the 

 geometric mean should be employed. For if a quantity 

 increases 100 per cent, in 100 years, it would not on the 

 average increase 10 per cent, in each ten years, as the 

 10 per cent, would at the end of each decade be calculated 

 upon larger and larger quantities, and give at the end of 

 100 years much more than 100 per cent., in fact as much 

 as 159 per cent. The true mean rate in each decade 

 would be J^~ or about ro/, that is, the increase would 

 be about 7 per cent, in each ten years. But when the 

 quantities differ very little, the arithmetic and geometric 

 means are approximately the same. Thus the arithmetic 

 mean of rooo and rooi is rooc>5, and the geometric mean 

 is about i '0004998, the difference being of an order in- 

 appreciable in almost all scientific and practical matters. 

 Even in the comparison of standard weights by Gauss' 

 method of reversal, the arithmetic mean may usually be 

 substituted for the geometric mean which is the true result. 



Regarding the mean in the absence of express qualifica- 

 tion to the contrary as the common arithmetic mean, \ve 

 must still distinguish between its two uses where it 

 gives with more or less accuracy and probability a 

 really existing quantity, and where it acts as a mere 

 representative of other quantities. If I make many 

 experiments to determine the atomic weight of an element, 

 there is a certain number which I wish to approximate to, 

 and the mean of my separate results will, in the absence 

 of any reasons to the contrary, be the most probable 

 approximate result. When we determine the mean 

 density of the earth, it is not because any part of the earth 

 is of that exact density ; there iray be no part exactly 

 corresponding to the mean density, and .as the crust of the 

 earth has only about half the mean density, the internal 

 matter of the globe must of course be above the mean. 

 Even the density of a homogeneous substance like carbon 

 or gold must be regarded as a mean between the real 

 density of its atoms, and the zero density of the interven- 

 ing vacuous space. 



The very different signification of the word " mean " in 

 these two uses was fully explained by Quetelet, 1 and the 



1 Letters on the Theory of Probabilities, transl. by Dowries, Part ii. 



