To take an example: If stimulus A just falls short of producing a sensation, and if r be the percentage of itself which must be added to it to get a sensation which is barely perceptible—call this sensation 1—then we should have the series of sensation-numbers corresponding to their several stimuli as follows:
Sensation 0 = stimulus A;
Sensation 1 = stimulus A (1 + r);
Sensation 2 = stimulus A (1 + r)2;
Sensation 3 = stimulus A (1 + r)3;
.....
Sensation n = stimulus A (1 + r)n.
The sensations here form an arithmetical series, and the stimuli a geometrical series, and the two series correspond term for term. Now, of two series corresponding in this way, the terms of the arithmetical one are called the logarithms of the terms corresponding in rank to them in the geometrical series. A conventional arithmetical series beginning with zero has been formed in the ordinary logarithmic tables, so that we may truly say (assuming our facts to be correct so far) that the sensations vary in the same proportion as the logarithms of their respective stimuli. And we can thereupon proceed to compute the number of units in any given sensation (considering the unit of sensation to be equal to the just perceptible increment above zero, and the unit of stimulus to be equal to the increment of stimulus r, which brings this about) by multiplying the logarithm of the stimulus by a constant factor which must vary with the particular kind of sensation in question. If we call the stimulus R, and the constant factor C, we get the formula
S = C log R,
which is what Fechner calls the psychophysischer Maasformel. This, in brief, is Fechner's reasoning, as I understand it.
The Maasformel admits of mathematical development in various directions, and has given rise to arduous discussions into which I am glad to be exempted from entering here, since their interest is mathematical and metaphysical and not primarily psychological at all.[447] I must say a word about them metaphysically a few pages later on. Meanwhile it should be understood that no human being, in any investigation into which sensations entered, has ever used the numbers computed in this or any other way in order to test a theory or to reach a new result. The whole notion of measuring sensations numerically, remains in short a mere mathematical speculation about possibilities, which has never been applied to practice. Incidentally to the discussion of it, however, a great many particular facts have been discovered about discrimination which merit a place in this chapter.
In the first place it is found, when the difference of two sensations approaches the limit of discernibility, that at one moment we discern it and at the next we do not. There are accidental fluctuations in our inner sensibility which make it impossible to tell just what the least discernible increment of the sensation is without taking the average of a large number of appreciations. These accidental errors are as likely to increase as to diminish our sensibility, and are eliminated in such an average, for those above and those below the line then neutralize each other in the sum, and the normal sensibility, if there be one (that is, the sensibility due to constant causes as distinguished from these accidental ones), stands revealed. The best way of getting at the average sensibility has been very minutely worked over. Fechner discussed three methods, as follows:
(1) The Method of just-discernible Differences. Take a standard sensation S, and add to it until you distinctly feel the addition d; then subtract from S + d until you distinctly feel the effect of the subtraction;[448] call the difference here d'. The least discernible difference sought is d + d'/2; and the ratio of this quantity to the original S (or rather to S + d - d') is what Fechner calls the difference-threshold. This difference-threshold should be a constant fraction (no matter what is the size of S) if Weber's law holds universally true. The difficulty in applying this method is that we are so often in doubt whether anything has been added to S or not. Furthermore, if we simply take the smallest d about which we are never in doubt or in error, we certainly get our least discernible difference larger than it ought theoretically to be.[449]
Of course the sensibility is small when the least discernible difference is large, and vice versâ; in other words, it and the difference-threshold are inversely related to each other.
(2) The Method of True and False Cases. A sensation which is barely greater than another will, on account of accidental errors in a long series of experiments, sometimes be judged equal, and sometimes smaller; i.e., we shall make a certain number of false and a certain number of true judgments about the difference between the two sensations which we are comparing.