[441] Cf. Sully: Mind, x, 494-5; Bradley: ibid. xi, 83; Bosanquet: ibid. xi, 405.
[442] The judgment becomes easier if the two couples of terms have one member in common, if a—b and b—c, for example, are compared. This, as Stumpf says (Tonpsychologie, i, 131), is probably because the introduction of the fourth term brings involuntary cross-comparisons with it, a and b with d, b with c, etc., which confuses us by withdrawing our attention from the relations we ought alone to be estimating.
[443] J. Delbœuf: Éléments de Psychophysique (Paris, 1883), p. 64. Plateau in Stumpf, Tonpsych., i, 125. I have noticed a curious enlargement of certain 'distances' of difference under the influence of chloroform. The jingling of the bells on the horses of a horse-car passing the door, for example, and the rumbling of the vehicle itself, which to our ordinary hearing merge together very readily into a quasi-continuous body of sound, have seemed so far apart as to require a sort of mental facing in opposite directions to get from one to the other, as if they belonged in different worlds. I am inclined to suspect, from certain data, that the ultimate philosophy of difference and likeness will have to be built upon experiences of intoxication, especially by nitrous oxide gas, which lets us into intuitions the subtlety whereof is denied to the waking state. Cf. B. P. Blood: The Anæsthetic Revelation, and the Gist of Philosophy (Amsterdam, N. Y., 1874). Cf. also Mind, vii, 200.
[444] Op. cit. p. 126 ff.
[445] Stumpf, pp. 111-121.
[446] Stumpf, pp. 116-7. I have omitted, so as not to make my text too intricate, an extremely acute and conclusive paragraph, which I reproduce here: "We may generalize: Wherever a number of sensible impressions are apprehended as a series, there in the last instance must perceptions of simple likeness be found. Proof: Assume that all the terms of a series, e.g. the qualities of tone, c d e f g, have something in common,—no matter what it is, call it X; then I say that the differing parts of each of these terms must not only be differently constituted in each, but must themselves form a series, whose existence is the ground for our apprehending the original terms in serial form. We thus get instead of the original series a b c d e f ... the equivalent series Xα, Xβ, Xγ,... etc. What is gained? The question immediately arises: How is α β γ known as a series? According to the theory, these elements must themselves be made up of a part common to all, and of parts differing in each, which latter parts form a new series, and so on ad infinitum, which is absurd."
[447] The most important ameliorations of Fechner's formula are Delbœuf's in his Recherches sur la Mesure des Sensations (1873), p. 35, and Elsas's in his pamphlet Über die Psychophysik (1886) p. 16.
[448] Reversing the order is for the sake of letting the opposite accidental errors due to 'contrast' neutralize each other.
[449] Theoretically it would seem that it ought to be equal to the sum of all the additions which we judge to be increases divided by the total number of judgments made.
[450] J. Delbœuf, Éléments de Psychophysique (1883), p. 9.