QUANTITATIVE ESTIMATES OF SENSORY EVENTS 309 



will endeavour to confine attention here to the essential principles of 

 Fechner's theory and examine these in the light of the general principles 

 of measurement. 



Let us assume a stimulus to be increased from zero intensity by steps 

 each of which is a j.n.d. Let us denote the j.n.d. at intensity / by A/. 

 The total stimulus intensity at any stage is, of course, the sum of all the 

 A/s which have been added up to that stage. From our experiments we 

 can express A/ as some function of /, say A/ =/(/). 



When the j.n.d. A/^ is added to the stimulus /i there is an increase in 

 intensity of sensation which we denote by AS^. Similarly when the j.n.d. 

 A/2 is added to the stimulus I^ there is an increment of sensation AS,, 

 and in general, the stimulus increment A/ when added to the stimulus / 

 produces an increment AS in the sensation intensity. Fechner's first 

 principle is that the total sensation 5 corresponding to the stimulus / may 

 be regarded as the sum of all the ASs corresponding to all the A/s which 

 have been added to produce the stimulus. If we know the relative magni- 

 tudes of the various A5s we then, by this principle, have a scale of magni- 

 tude for S. Of course we have no a priori knowledge of the relative 

 magnitudes of the sensation increments corresponding to j.n.ds. at diflterent 

 intensity levels, and Fechner's second principle is that we are free to 

 postulate an arbitrary quantitative relation between AS, the liminal 

 increment of sensation, and S, the total sensation, say AS = ffiS), where 

 9(5) is some arbitrary function. 



We then have, by postulate, AS = 9(5), and from j.n.d. experiments 

 A/ =/(/). 



^^^"'^^ ws) = m 



treating the small quantities A5 and A/ as differentials we get the 

 differential equation 



dS^ ^ dl 



9(5) /(/) 



and by solving this equation we obtain the relation between 5 and /, that 

 is to say, the relation between sensation intensity and stimulus intensity. 



I have put Fechner's second principle in its most general form. Fechner 

 himself propounded it in the special form in which 9(5) = k : that is to 

 say, he postulated that all the A5s are equal. Plateau and others have 

 suggested the form 9(5) = kS, equivalent to the postulate that AS/S 

 is constant at all parts of the scale. Much discussion has centred round 

 which of these forms of 9(5) is most in accordance with facts. It does 

 not appear to have been noticed that the very possibility of a factual criterion 

 being applied to discriminate between the two functions is inconsistent 

 with either of them forming a true basis of measurement, for, as we have 

 seen, a scale of measurement is independent of any facts other than those 

 created by the necessary and sufficient conventions postulated in defining 

 the required association between number and magnitude for the scale in 

 question. If Fechner's second principle is to be accepted it is immateria) 

 what form is given to the arbitrary function 9(5), and Fechner was quite 

 justified in adopting the siniplest one. 



Now we note that Fechner. aimed at measuring sensation intensity as 

 an A magnitude in terms of units of its own kind : his two principles imply 

 both a criterion of addition and a criterion of equality. In Fechner's own 

 form of the second principle the criterion of equality is stated explicitly, 



