Measure of the Intensity of Sound. 175 



branch of the subject.) It is easy to prove, from facts of ordi- 

 nary knowledge, that this hypothesis is impossible. For in- 

 stance, a single diapason stop of soft quality, in the organ, 

 may be taken to represent a definite mechanical intensity. If 

 this be added to a mechanical intensity of less, or not very 

 much greater amount, the addition produces a finite difference 

 of sensation. If, however, the full organ is being used, so that 

 the existing mechanical intensity is very much greater than 

 that added by the single soft stop, the addition of the latter 

 does not affect the sensation at all. The first hypothesis 

 therefore falls to the ground. 



Similar conditions may be applied to any definite loud and 

 soft sounds ; but the above is sufficient to establish the point. v> 



The second hypothesis in point of simplicity would appear 

 to be', that the difference of the numbers which express the 

 measure of the sensation is proportional to the ratio of the 

 mechanical intensities. This is an hypothesis often practically 

 employed in a vague sort of way. I shall endeavour to trace 

 it to its consequences, and show that it is necessary that the 

 measure of sensation thus defined should be connected with 

 another measure of sensation. The second measure introduced 

 is that in which small differences of mechanical intensity cor- 

 respond to small differences of sensation ; and this latter mea- 

 sure is related to the mechanical intensity, according to Fech- 

 ner's law. 



Let \ be the measure of sensation, differences of which are 

 proportional to the ratio of the mechanical intensities, so that 



Let 



\ 1 -\ =k^ (i) 



J-o 



\i=\+A\ ; l!=I + AI, 

 Aq = A-j J-o = ■*-• 



Then (i) becomes 



A , 7 I + AI 



A\=& — - — 



AI 



=*( 1+ t) C"0 



If we diminish AI indefinitely, A\ tends to the value h, 

 which cannot vanish consistently with the fundamental assump- 

 tion. X cannot, therefore, be such a measure of sensation as 

 we ordinarily recognize, since differences of sensation must 

 ultimately diminish indefinitely with differences of the mecha- 

 nical intensity. 



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