THE CONCEPTUAL WORLD 



17 



cannot be further decomposed are necessarily equal 

 to each other ; if, for instance, the atoms represent 

 the ultimate units into which we break up the matter 

 called oxygen, then these atoms are all equal to each 

 other. Therefore the increments of loudness are 

 equal to each other. 



If we plot these equal increments of loudness as 

 the dependent variable S in a graph, and the amplitude 

 of the vibrations of the atmosphere as the independent 

 variable E, we can obtain the following curve If we 

 investigate this we 

 shall find that a cer- 

 tain relation exists be- 

 tween the " values ' 

 of the sensation and 

 the values of the stim- 

 uli that correspond to 

 them ; a regular in- 

 crease in the loudness 

 of the sensation cor- 

 responds to a regular 

 increase in the log- 

 arithms of the strength of the stimuli. Let S = the 

 sensation, E the stimulus, and C and Q constants ; 

 then 



5 = C log ; 



FIG. 3. 



so that we seem to establish a mathematical relation 

 between the intensity of our sensations and the 

 intensity of the stimuli that give rise to those sensations, 

 but this relation depends on the assumption that what 

 we call " minimal perceptible differences " of sensation 

 are numerical differences that are equal to each other, 

 and this is, of course, an assumption that cannot 

 possibly be proved. 



