when the stimulus is increased, we can just barely perceive to be added. The total number of units which any given sensation contains will consist of the total number of such increments which may be perceived in passing from no sensation of the kind to a sensation of the present amount. We cannot get at this number directly, but we can, now that we know Weber's law, get at it by means of the physical stimulus of which it is a function. For if we know how much of the stimulus it will take to give a barely perceptible sensation, and then what percentage of addition to the stimulus will constantly give a barely perceptible increment to the sensation, it is at bottom only a question of compound interest to compute, out of the total amount of stimulus which we may be employing at any moment, the number of such increments, or, in other words, of sensational units to which it may give rise. This number bears the same relation to the total stimulus which the time elapsed bears to the capital plus the compound interest accrued.

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:

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.* 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

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.

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; * call the difference here d+ d' d'. The least discernible difference sought is

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.t

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

Reversing the order is for the sake of letting the opposite accidental errors due to contrast' neutralize each other.

+ 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 num ber of judgments made.

true judgments about the difference between the two sensations which we are comparing.

"But the larger this difference is, the more the number of the true judgments will increase at the expense of the false ones; or, otherwise expressed, the nearer to unity will be the fraction whose denominator represents the whole number of judgments, and whose numerator represents those which are true. If m is a ratio of this nature, obtained by comparison of two stimuli, A and B, we may seek another couple of stimuli, a and b, which when compared will give the same ratio of true to false cases."*

If this were done, and the ratio of a to b then proved to be equal to that of A to B, that would prove that pairs of small stimuli and pairs of large stimuli may affect our discriminative sensibility similarly so long as the ratio of the components to each other within each pair is the same. In other words, it would in so far forth prove the Weberian law. Fechner made use of this method to ascertain his own power of discriminating differences of weight, recording no less than 24,576 separate judgments, and computing as a result that his discrimination for the same relative increase of weight was less good in the neighborhood of 500 than of 300 grams, but that after 500 grams it improved up to 3000, which was the highest weight he experimented with.

(3) The Method of Average Errors consists in taking a standard stimulus and then trying to make another one of the same sort exactly equal to it. There will in general be an error whose amount is large when the discriminative sensibility called in play is small, and vice versa. The sum of the errors, no matter whether they be positive or negative, divided by their number, gives the average error. This, when certain corrections are made, is assumed by Fechner to be the 'reciprocal' of the discriminative sensibility in question. It should bear a constant proportion to the stimulus, no matter what the absolute size of the latter may be, if Weber's law hold true.

These methods deal with just perceptible differences. Delboeuf and Wundt have experimented with larger differ

J. Delbœuf, Eléments de Psychophysique (1883), p. 9.

ences oy means of what Wundt calls the Methode der mittleren Abstufungen, and what we may call

(4) The Method of Equal-appearing Intervals. This consists in so arranging three stimuli in a series that the intervals between the first and the second shall appear equal to that between the second and the third. At first sight there seems to be no direct logical connection between this method and the preceding ones. By them we compare equally per ceptible increments of stimulus in different regions of the latter's scale; but by the fourth method we compare increments which strike us as equally big. But what we can but just notice as an increment need not appear always of the same bigness after it is noticed. On the contrary, it will appear much bigger when we are dealing with stimuli that are already large.

(5) The method of doubling the stimulus has been employed by Wundt's collaborator, Merkel, who tried to make one stimulus seem just double the other, and then measured the objective relation of the two. The remarks just made apply also to this case.

So much for the methods. The results differ in the hands of different observers. I will add a few of them, and will take first the discriminative sensibility to light.

By the first method, Volkmann, Aubert, Masson, Helmholtz, and Kräpelin find figures varying from or too of the original stimulus. The smaller fractional increments are discriminated when the light is already fairly strong, the larger ones when it is weak or intense. That is, the discriminative sensibility is low when weak or overstrong lights are compared, and at its best with a certain medium illumination. It is thus a function of the light's intensity; but throughout a certain range of the latter it keeps constant, and in so far forth Weber's law is verified for light. Absolute figures cannot be given, but Merkel, by method 1, found that Weber's law held good for stimuli (measured by his arbitrary unit) between 96 and 4096, beyond which intensity no experiments were made.* König and Brodhun

* Philos. Studien, rv. 588.