QUANTITATIVE ESTIMATES OF SENSORY EVENTS 311 



equal by this general criterion of equality. We have no right to make this 

 assumption for there is no operation in the determination of a j.n.d. series 

 which corresponds to the operation of judging equality of sensations in the 

 ordinary way. In fact it is really meaningless to enquire if the A^s are 

 equal on any scale applicable to sensation intensities, for the criterion of 

 equality applicable to sensation intensities is not applicable to liminal incre- 

 ments at different intensities. Failure to realise this from the first is doubt- 

 less due to failure to realise that a practical operation for establishing 

 equality is part of the definition of any magnitude. Equality of sensation 

 intensities is established when we compare stimuli neither of which appears 

 greater than the other. But A5s at different parts of the j.n.d. series cannot, 

 from their nature, be experienced under the same degree of stimulation. 

 The operation of establishing equality of sensation intensities is inapplicable, 

 not merely in practice but in principle, to members of the A5 series. 

 From the point of view of measurement it is therefore impossible to regard 

 the A5s as samples of the magnitude S. Thus Fechner's first principle, 

 that we may regard iS as the sum of a series of A.Ss, is invalid, and also, of 

 course, his identification of the ratio AS/AI with the differential coefficient 

 of S with respect to / ; for, of course, the small quantities constituting the 

 numerator and denominator of a differential coefficient must differ in size 

 only, and not in nature, from any other samples of the two variables. They 

 must be measurable on the same scales as other and larger samples, which 

 means that the phenomenal relation defining equality and the phenomenal 

 operation defined as addition must be the same for the differentials as for 

 all other samples of the variables. As this is not true of A5, it cannot be 

 regarded as a small quantity of the magnitude 5 and A S/AI has no relation 

 to a differential coefficient. 



This conclusion, derivable wholly from the principles of measurement, 

 confirms those psychologists who have argued, from quite different premises, 

 that we cannot regard a sensation S as analysable into a series of small 

 quantities added together like the millimetres in a metre stick. A milli- 

 metre and a metre are samples of the same magnitude and differ only in 

 size, but the AS of j.n.d. experiment and S are not : they differ in nature, 

 as they cannot be defined by the same kind of relations. 



We could go on much longer examining Fechner's principles in detail, 

 but any one of the objections we have already discussed is sufficient to show 

 that these principles cannot lead to the measurement of any magnitude at 

 all, either of the A or B class. 



Another type of psycho-physical experiment we must consider is that in 

 which a series of three or more stimuli are graded by the method of ' niean 

 gradations.' The principle may be illustrated by an example from vision. 

 The observer is presented with a series of patches of light, all of the same 

 colour, whose intensities are under his control, and is asked to adjust their 

 brightnesses until they form a series so that the ' seeming disparity ' between 

 each one and the next in the series is the same. Brown {The Essentials of 

 Mental Measurement, 191 1. 2) calls this 'seeming disparity' a 'sense- 

 distance.' When this experiment is performed a relation can be established 

 between the grading so effected and the grading in terms of stimulus intensity 

 (photometric units). The results of experiments of this kind for vision 

 and other senses are usually interpreted as establishing a relation between a 

 psychological magnitude — sense-distance — and stimulus intensities. The 

 grading is supposed to consist of equal sense-distances, and the relation 

 found between these equal sense-distances and the corresponding stimulus 

 intervals is regarded by most psycho-physicists as providing a basis on 

 which a quantitative relation between sensation intensity and stimulus 



