July 19, 1900] 



NATURE 



279 



jries of just observable differences in sensation we must always 

 lid the same fraction— one-third of the weight — at each constant 

 ;ep of the series. 

 Now Fechner assumed that these just perceivable increments 

 f sensation are all of the same value, or are constant ; in which 

 ise they form an arithmetical series — that is to say, one that 

 - produced by successive additions of the same amount. But 

 :.e corresponding series of stimuli are not in arithmetical pro- 

 tession, since the successive increments are not of the same 

 mount. The increase is, however, always by the same pro- 

 •rtional amount. Each successive stimulus has to be multi- 

 lied by a constant factor, \. The series, therefore, forms an 

 fderly sequence in geometrical progression. 

 .We thus reach what is known as the Weber-Fechner formula, 

 ' which the relation of stimulus to sensation is expressed in 

 antitative terms. It may be thus stated :— To obtain an 

 jlthmetical series of sensations a geometrical series of stimuli 

 -required. To give the former, equal increments of sensation 

 added ; to obtain the latter we must multiply the successive 

 nuli by a constant factor. 



[t must be admitted, however, that the results of a great 

 iber of carefully-conducted observations are by no means in 

 itisfactory accordance with this formula. Hering and his 

 pils have shown that for very small stimuli, lying near the 

 eshold of sensation, both stimulus and sensation increase very 

 irly pari passu in arithmetical progression. The Weber- 

 :hner formula cannot, therefore, at present be regarded as 

 jre than an approximation to the truth. 



Jn extracting the Weber-Fechner formula from the data 

 "")rded by observations on the method of least observable 

 iference, it is necessary to piece together the results observed 

 jly and in succession. But from the nature of the field of 

 sion it is possible to obtain a series of increments of stimulus 

 lich shall afford a scale of sensation visible as a whole and at 

 glance. In the current number of the Psychological Review 

 io\. vii. No. 3, p. 217; I have published in detail the results 

 "an investigation "On the relation of stimulus to sensation 

 visual impressions," by which I have been led to suggest a 

 Kiification of the Weber-Fechner formula. Stripped as far 

 possible of technicalities, the method and results may be 

 re briefly described. 

 It is well known that if a disc with white and black sectors 

 rapidly rotated, the effect on the eye is a uniform grey. If 

 ^ He white sectors are proportionally small, occupying, for 

 Example, only 5 per cent, of the disc, the effect is that of a very 

 dark grey ; if they are relatively large, occupying, say, 90 per 

 cent, of the disc, the effect is that of a very light grey. With 

 such sectors the same proportional amount of white is introduced 

 in all parts of the disc, so as to give in each case the same 

 shade of grey throughout its whole extent. But it is possible to 

 introduce varying proportions of white from centre to circum- 

 ference, and when this is done the rotating disc no longer 

 presents all over its surface the same uniform shade of grey, 

 but shows varying shades. Let us now endeavour to reduce 

 these varying shades to order. Let us arrange the proportions 

 of white stimulus which we introduce, in such a way as to leave 

 a ring of full black (with no white) at the circumference, and to 

 give a ring of full white (with no black) near the centre, and 

 between these extremes to obtain a perfectly smooth and even 

 gradation of shades of grey from one so dark as to be scarcely 

 distinguishable from black, to one so light as to be scarcely 

 distinguishable from white. We may then, when the disc is 

 rapidly rotating, run our eye from white near the centre, 

 through deepening and deepening grey, to black at the circum- 

 ference, with nowhere any observable jump in sensation — 

 nowhere, so to speak, a steeper slope of change than elsewhere ; 

 as if, in fact, we were passing along a perfectly even inclined 

 plane of sensation from the lowest depth of black to the extreme 

 height of white. If we succeed in this — and it is by no means easy 

 of attainment — we shall have secured an arithmetical series of 

 sensation. From one end to the other we have at successively 

 equal distances constant increments of white sensation, just as in 

 ascending a uniform incline we gain equal increments of 

 height for every yard we progress towards our goal. This even 

 slope of sensation is produced by the juxtaposition of all the 

 least observable differences whose sum gives the full scale. 

 Having obtained this result we are able to ascertain, by careful 

 angular measurements of the proportional areas of white at 

 different parts of our disc, the exact amounts of stimulus which 

 are affecting the eye from these several parts. We may, for 



NO. 1603, VOL. 62] 



example, subdivide the area of the disc lying between the inner 

 white circle and the outer ring of black, by drawing nine con- 

 centric circles equidistant from each other, and at these nine 

 distances make angular measurements of the proportional 

 amounts of white to black ; and then, by plotting, sweep a 

 curve of stimulus through points representing these measured 

 amounts. 



When these amounts are tabulated and dealt with by appro- 

 priate mathematical methods, it is found that they are not in 

 accordance with the Weber-Fechner formula. Nor does a disc 

 prepared in accordance with this formula give the smooth and 

 evenly-graded incline of an arithmetical series in sensation. 

 For details the reader may be referred to the paper in which the 

 observations and calculations are set forth. The accompanying 

 figure gives the results plotted in a curve on the graphic method. 

 The dotted steps indicate the nine measured increments. The 

 vertical distance of any point on the curve, measured from 

 below, upwards, gives the percentage of sensation. The hori- 

 zontal distance, measured from left to right, gives the corre- 

 sponding percentage of white stimulus. The law which results 

 from a discussion of these observations, and of others where red, 



StirrLuliLS 



orange and blue stimuli were used instead of white (each of 

 which gives a different curve on the same principle), may be 

 thus formulated : — For constant increments of sensation the cott' 

 comitant increments of stimulus are in geometrical progression. 

 This differs from the Weber-Fechner formula in assigning the 

 geometrical progression to the successive increments of 

 stimulus. 

 The subjoined table gives the increments and sums of 



White on Black. 



