have given measurements by method 1 which cover the most extensive series, and moreover apply to six different colors of light. These experiments (performed in Helmholtz's laboratory, apparently,) ran from an intensity called 1 to one which was 100,000 times as great. From intensity 2000 to 20,000 Weber's law held good; below and above this range discriminative sensibility declined. The increment discriminated here was the same for all colors of light, and lay (according to the tables) between 1 and 2 per cent of the stimulus.* Delboeuf had verified Weber's law for a certain range of luminous intensities by method 4; that is, he had found that the objective intensity of a light which appeared midway between two others was really the geometrical mean of the latter's intensities. But A. Lehmann and afterwards Neiglick, in Wundt's laboratory, found that effects of contrast played so large a part in experiments performed in this way that Delboeuf's results could not be held conclusive. Merkel, repeating the experiments still later, found that the objective intensity of the light which we judge to stand midway between two others neither stands midway nor is a geometric mean. The discrepancy from both figures is enormous, but is least large from the midway figure or arithmetical mean of the two extreme intensities. Finally, the stars have from time immemorial been arranged in 'magnitudes' supposed to differ by equalseeming intervals. Lately their intensities have been gauged photometrically, and the comparison of the subjective with the objective series has been made. Prof. J. Jastrow is the latest worker in this field. He finds, taking Pickering's Harvard photometric tables as a basis, that the ratio of the average intensity of each magnitude' to that below it decreases as we pass from lower to higher magnitudes, showing a uniform departure from Weber's law, if the method of equal-appearing intervals be held to have any direct relevance to the latter.t

* Berlin Acad. Sitzungsberichte, 1888, p. 917. Other observers (Dobro. wolsky, Lamansky) found great differences in different colors.

+ See Merkel's tables, loc. cit. p. 568.

American Journal of Psychology, I. 125. The rate of decrease is small but steady, and I cannot well understand what Professor J. means by saying that his figures verify Weber's law.

Sounds are less delicately discriminated in intensity than lights. A certain difficulty has come from disputes as to the measurement of the objective intensity of the stimulus. Earlier inquiries made the perceptible increase of the stimulus to be about of the latter. Merkel's latest results of the method of just perceptible differences make it about for that part of the scale of intensities during which Weber's law holds good, which is from 20 to 5000 of M.'s arbitrary unit. Below this the fractional increment must be larger. Above it no measurements were made.

For pressure and muscular sense we have rather divergent results. Weber found by the method of just-perceptible differences that persons could distinguish an increase of weight of when the two weights were successively lifted by the same hand. It took a much larger fraction to be discerned when the weights were laid on a hand which rested on the table. He seems to have verified his results for only two pairs of differing weights,† and on this founded his 'law.' Experiments in Hering's laboratory on lifting 11 weights, running from 250 to 2750 grams showed that the least perceptible increment varied from for 250 grams to for 2500. For 2750 it rose to again. Merkel's recent and very careful experiments, in which the finger pressed down the beam of a balance counterweighted by from 25 to 8020 grams, showed that between 200 and 2000 grams a constant fractional increase of about was felt when there was no movement of the finger, and of about when there was movement. Above and below these limits the discriminative power grew less. It was greater when the pressure was upon one square millimeter of surface than when it was upon seven.‡

Warmth and taste have been made the subject of similar investigations with the result of verifying something like Weber's law. The determination of the unit of stimulus is, however, so hard here that I will give no figures. The results may be found in Wundt's Physiologische Psychologie, 3d Ed. I. 370-2.

* Philosophische Studien, v. 514-5.

+ Cf. G. E. Müller: Zur Grandlegung der Psychophysik, §§ 68–70. Philosophische Studien, v. 287 ff.

The discrimination of lengths by the eye has been found also to obey to a certain extent Weber's law. The figures will all be found in G. E. Müller, op. cit., part II, chap. x, to which the reader is referred. Professor Jastrow has published some experiments, made by what may be called a modification of the method of equal-appearing differences, on our estimation of the length of sticks, by which it would seem that the estimated intervals and the real ones are directly and not logarithmically proportionate to each other. This resembles Merkel's results by that method for weights, lights, and sounds, and differs from Jastrow's own finding about star-magnitudes.*

If we look back over these facts as a whole, we see that it is not any fixed amount added to an impression that makes us notice an increase in the latter, but that the amount depends on how large the impression already is. The amount is expressible as a certain fraction of the entire impression to which it is added; and it is found that the fraction is a well-nigh constant figure throughout an entire region of the scale of intensities of the impression in question. Above and below this region the fraction increases in value. This is Weber's law, which in so far forth expresses an empirical generalization of practical importance, without involving any theory whatever or seeking any absolute measure of the sensations themselves. It is in the

Theoretic Interpretation of Weber's Law

=

that Fechner's originality exclusively consists, in his assumptions, namely, 1) that the just-perceptible increment is the sensation-unit, and is in all parts of the scale the same (mathematically expressed, 4s const.); 2) that all our sensations consist of sums of these units; and finally, 3) that the reason why it takes a constant fractional increase of the stimulus to awaken this unit lies in an ultimate law of the connection of mind with matter, whereby the quantities of our feelings are related logarithmically to the quantities of their objects. Fechner seems to find something inscrutably sublime in the existence of an ultimate 'psychophysic' law of this form.

* American J. of Psychology, ш. 44–7.

These assumptions are all peculiarly fragile. To begin with, the mental fact which in the experiments corresponds to the increase of the stimulus is not an enlarged sensation, but a judgment that the sensation is enlarged. What Fechner calls the 'sensation' is what appears to the mind as the objective phenomenon of light, warmth, weight, sound, impressed part of body, etc. Fechner tacitly if not openly assumes that such a judgment of increase consists in the simple fact that an increased number of sensation-units are present to the mind; and that the judgment is thus itself a quantitatively bigger mental thing when it judges large differences, or differences between large terms, than when it judges small ones. But these ideas are really absurd. The hardest sort of judgment, the judgment which strains the attention most (if that be any criterion of the judgment's 'size'), is that about the smallest things and differences. But really it has no meaning to talk about one judgment being bigger than another. And even if we leave out judgments and talk of sensations only, we have already found ourselves (in Chapter VI) quite unable to read any clear meaning into the notion that they are masses of units combined. To introspection, our feeling of pink is surely not a portion of our feeling of scarlet; nor does the light of an electric arc seem to contain that of a tallow-candle in itself. Compound things contain parts; and one such thing may have twice or three times as many parts as another. But when we take a simple sensible quality like light or sound, and say that there is now twice or thrice as much of it present as there was a moment ago, although we seem to mean the same thing as if we were talking of compound objects, we really mean something different. We mean that if we were to arrange the various possible degrees of the quality in a scale of serial increase, the distance, interval, or difference between the stronger and the weaker specimen before us would seem about as great as that between the weaker one and the beginning of the scale. It is these RELATIONS, these DISTANCES, which we are measuring and not the composition of the qualities themselves, as Fechner thinks. Whilst if we turn to objects which are divisible, surely a big object may be known in a little thought. Introspection shows moreover

that in most sensations a new kind of feeling invariably accompanies our judgment of an increased impression; and this is a fact which Fechner's formula disregards.*

But apart from these a priori difficulties, and even supposing that sensations did consist of added units, Fechner's assumption that all equally perceptible additions are equally great additions is entirely arbitrary. Why might not a small addition to a small sensation be as perceptible as a large addition to a large one? In this case Weber's law would apply not to the additions themselves, but only to their perceptibility. Our noticing of a difference of units in two sensations would depend on the latter being in a fixed ratio. But the difference itself would depend directly on that between their respective stimuli. So many units added to the stimulus, so many added to the sensation, and if the stimulus grew in a certain ratio, in exactly the same ratio would the sensation also grow, though its perceptibility grew according to the logarithmic law.†

If 4 stand for the smallest difference which we perceive, then we should have, instead of the formula 48 const.,

which is Fechner's, the formula

As 8

const., a formula

which interprets all the facts of Weber's law, in an entirely different theoretic way from that adopted by Fechner.‡

The entire superstructure which Fechner rears upon the

* Cf. Stumpf, Tonpsychologie, pp. 397-9. "One sensation cannot be a multiple of another. If it could, we ought to be able to subtract the one from the other, and to feel the remainder by itself. Every sensation presents itself as an indivisible unit." Professor von Kries, in the Vierteljahrschrift für wiss. Philosophie, vi. 257 ff., shows very clearly the absurdity of supposing that our stronger sensations contain our weaker ones as parts. They differ as qualitative units. Compare also J. Tannery in Delbœuf's Eléments de Psychophysique (1883), p. 134 ff.; J. Ward in Mind, 1. 464; Lotze, Metaphysik, § 258.

+ F. Brentano, Psychologie, 1. 9, 88 ff.-Merkel thinks that his results with the method of equal-appearing intervals show that we compare considerable intervals with each other by a different law from that by which we notice barely perceptible intervals. The stimuli form an arithmetical series (a pretty wild one according to his figures) in the former case, a geometrical one in the latter at least so I understand this valiant experimenter but somewhat obscure if acute writer.

This is the formula which Merkel thinks he has verified (if I under. stand him aright) by his experiments by method 4.