appear to us as so many notes of a gamut, as higher or lower degrees in a scale, in a word, as magnitudes. Moreover it would be easy for us to assign each of them its place in the series. For although the extensive cause varies continuously, the changes in the sensation of colour are discontinuous, passing from one shade to another shade. However numerous, then, may be the shades intermediate between the two colours, A and B, it will always be possible to count them in thought, at least roughly, and ascertain whether this number is almost equal to that of the shades which separate Β from another colour C. In the latter case it will be said that Β is equally distant from A and C, that the contrast is the same on one side as on the other. But this will always be merely a convenient interpretation: for although the number of intermediate shades may be equal on both sides, although we may pass from one to the other by sudden leaps, we do not know whether these leaps are magnitudes, still less whether they are equal magnitudes: above all it would be necessary to show that the intermediaries which have helped us throughout our measurement could be found again inside the object which we have measured. If not, it is only by a metaphor that a sensation can be said to be an equal distance from two others.
This is just the case with differences of intensity in sensations of light. Delbœuf's underlying postulate.
Now, if the views which we have before enumerated with regard to luminous intensities are accepted, it will be recognized that the different hues of grey which Delbœuf displays to us are strictly analogous, for our consciousness, to colours, and that if we declare that a grey tint is equidistant from two other grey tints, it is in the same sense in which it might be said that orange, for example, is at an equal distance from green and red. But there is this difference, that in all our past experience the succession of grey tints has been produced in connexion with a progressive increase or decrease in illumination. Hence we do for the differences of brightness what we do not think of doing for the differences of colour: we promote the changes of quality into variations of magnitude. Indeed, there is no difficulty here about the measuring, because the successive shades of grey produced by a continuous decrease of illumination are discontinuous, as being qualities, and because we can count approximately the principal intermediate shades which separate any two kinds of grey. The contrast A Β will thus be declared equal to the contrast Β C when our imagination, aided by our memory, inserts between A and Β the same number of intermediate shades as between Β and C. It is needless to say that this will necessarily be a very rough estimate. We may anticipate that it will vary considerably with different persons. Above all it is to be expected that the person will show more hesitation and that the estimates of different persons will differ more widely in proportion as the difference in brightness between the rings A and Β is increased, for a more and more laborious effort will be required to estimate the number of intermediate hues. This is exactly what happens, as we shall easily perceive by glancing at the two tables drawn up by Delbœuf.23 In proportion as he increases the difference in brightness between the exterior ring and the middle ring, the difference between the numbers on which one and the same observer or different observers successively fix increases almost continuously from 3 degrees to 94, from 5 to 73, from 10 to 25, from 7 to 40. But let us leave these divergences on one side: let us assume that the observers are always consistent and always agree with one another; will it then be established that the contrasts A Β and Β C are equal? It would first be necessary to prove that two successive elementary contrasts are equal quantities, whilst, in fact, we only know that they are successive. It would then be necessary to prove that inside a given tint of grey we perceive the less intense shades which our imagination has run through in order to estimate the objective intensity of the source of light. In a word, Delbœuf's psychophysics assumes a theoretical postulate of the greatest importance, which is disguised under the cloak of an experimental result, and which we should formulate as follows: "When the objective quantity of light is continuously increased, the differences between the hues of grey successively obtained, each of which represents the smallest perceptible increase of physical stimulation, are quantities equal to one another. And besides, any one of the sensations obtained can be equated with the sum of the differences which separate from one another all previous sensations, going from zero upwards." Now, this is just the postulate of Fechner's psychophysics, which we are going to examine.
Fechner's psychophysics. Weber's Law.]
Fechner took as his starting-point a law discovered by Weber, according to which, given a certain stimulus which calls forth a certain sensation, the amount by which the stimulus must be increased for consciousness to become aware of any change bears a fixed relation to the original stimulus. Thus, if we denote by Ε the stimulus which corresponds to the sensation S, and by ΔΕ the amount by which the original stimulus must be increased in order that a sensation of difference may be produced, we shall have ΔΕ/E = const. This formula has been much modified by the disciples of Fechner, and we prefer to take no part in the discussion; it is for experiment to decide between the relation established by Weber and its substitutes. Nor shall we raise any difficulty about granting the probable existence of a law of this nature. It is here really a question not of measuring a sensation but only of determining the exact moment at which an increase of stimulus produces a change in it. Now, if a definite amount of stimulus produces a definite shade of sensation, it is obvious that the minimum amount of stimulus required to produce a change in this shade is also definite; and since it is not constant, it must be a function of the original stimulus. But how are we to pass from a relation between the stimulus and its minimum increase to an equation which connects the "amount of sensation" with the corresponding stimulus? The whole of psychophysics is involved in this transition, which is therefore worthy of our closest consideration.
The underlying assumptions and the process by which Fechner's Law is reached.
We shall distinguish several different artifices in the process of transition from Weber's experiments, or from any other series of similar observations, to a psychophysical law like Fechner's. It is first of all agreed to consider our consciousness of an increase of stimulus as an increase of the sensation S: this is therefore called S. It is then asserted that all the sensations ΔS, which correspond to the smallest perceptible increase of stimulus, are equal to one another. They are therefore treated as quantities, and while, on the one hand, these quantities are supposed to be always equal, and, on the other, experiment has given a certain relation ΔΕ = ⨍(E) between the stimulus Ε and its minimum increase, the constancy of ΔS is expressed by writing ΔS = C ΔE/⨍(E), C being a constant quantity. Finally it is agreed to replace the very small differences ΔS and ΔΕ by the infinitely small differences dS and dE, whence an equation which is, this time, a differential one: dS = C dE/⨍(E). We shall now simply have to integrate on both sides to obtain the desired relation24: S=C ⨍Eo dE/⨍(E). And the transition will thus be made from a proved law, which only concerned the occurrence of a sensation, to an unprovable law which gives its measure.
Without entering upon any thorough discussion of this ingenious operation, let us show in a few words how Fechner has grasped the real difficulty of the problem, how he has tried to overcome it, and where, as it seems to us, the flaw in his reasoning lies.
Can two sensations be equal without being identical?
Fechner realized that measurement could not be introduced into psychology without first defining what is meant by the equality and addition of two simple states, e.g. two sensations. But, unless they are identical, we do not at first see how two sensations can be equal. Undoubtedly in the physical world equality is not synonymous with identity. But the reason is that every phenomenon, every object, is there presented under two aspects, the one qualitative and the other extensive: nothing prevents us from putting the first one aside, and then there remains nothing but terms which can be directly or indirectly superposed on one another and consequently seen to be identical. Now, this qualitative element, which we begin by eliminating from external objects in order to measure them, is the very thing which psychophysics retains and claims to measure. And it is no use trying to measure this quality Q by some physical quantity
