420 THE PRINCIPLES OF SCIENCE. 



Arithmetic mean 50 per cent. 



Geometric 41 



Harmonic 33 



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 I j2~ or about 1*07, that is, the increase 

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

 when the quantities differ but little, the arithmetic and 

 geometric means are approximately the same. Thus the 

 arithmetic mean of I'ooo and rooi is rooo5, and the 

 geometric mean is about 1-0004998, the difference being 

 of an order inappreciable in almost all scientific or prac- 

 tical processes. Even in the comparison of standard weights 

 by Gauss' method of transposition 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, we 

 must still distinguish between its two uses where it 

 defines 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 specific gravity of a homo- 

 geneous piece of gold there is a certain definite ratio 

 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 on the other hand the mean density of the 



