APPENDIX E. 239 



arithmetical mean is more likely to be the true measurement than 

 any other quantity that can be named. 



This assumption cannot be justified in vital phenomena. For ex- 

 ample, suppose we endeavour to match a tint ; Weber's law, in its 

 approximative and simplest form, of Sensation varying as the 

 logarithm of the Stimulus, tells us that a series of tints, in which 

 the quantities of white scattered on a black ground are as 1, 2, 4, 

 8, 16, 32, &c., will appear to the eye to be separated by equal in- 

 tervals of tint. Therefore, in matching a grey that contains 8 por- 

 tions of white, we are just as likely to err by selecting one that has 

 16 portions as one that has 4 portions. In the first case there 

 would be an error in excess, of 8 units of absolute tint ; in the 

 second there would be an error in deficiency, of 4. Therefore, an 

 error of the same magnitude in excess or in deficiency is not equally 

 probable in the judgment of tints by the eye. Conversely, if two 

 persons, who are equally good judges, describe their impressions of 

 a certain tint, and one says that it contains 4 portions of white and 

 the other that it contains 16 portions, the most reasonable conclu- 

 sion is that it really contains 8 portions. The arithmetic mean of 

 the two estimates is 10, which is not the most probable value ; it 

 is the geometric mean 8, (4 : 8 : : 8 : 16), which is the most probable. 



Precisely the same condition characterises every determination by 

 each of the senses; for example, in judging of the weight of bodies or 

 of their temperatures, of the loudiiess and of the pitches of tones, 

 and of estimates of lengths and distances as wholes. Thus, three 

 rods of the lengths a, b, c, when taken successively in the hand, 

 appear to differ by equal intervals when a : b : : b : c, and not when 

 a b b - c. In all physiological phenomena, where there is on the 

 one hand a stimulus and on the other a response to that stimulus 

 Weber's or some other geometric law may be assumed to prevail 

 in other words, the true mean is geometric rather than arithmetic. 



The geometric mean appears to be equally applicable to the ma- 

 jority of the influences, which, combined with those of purely vital 

 phenomena, give rise to the events with which sociology deals. It is 

 difficult to find terms sufficiently general to apply to the varied topics 

 of sociology, but there are two categories which are of common oc- 

 currence in which the geometric mean is certainly appropriate. The 

 one is increase, as exemplified by the growth of population, where an 



