§ 2. Interpretation.—The explanation of the facts described has been much discussed. One hypothesis is that increase in the intensity of the stimulus fails to produce an increase in the intensity of the sensation until the increment is a certain fraction of the original stimulus. On this hypothesis the sensation ought to vary by leaps and bounds at certain fixed points. The reason why no unlikeness in the sensation is discernible before these points are reached is that no unlikeness in the sensation exists. This view may be definitely rejected. There are no such fixed points of transition. Whatever the intensity of the original sensation may be, the same relative increment is required to make unlikeness discernible. In gradually increasing the intensity of the stimulus, it is not found that there are certain points at which change in sensation becomes perceptible in such a way that any pair of stimuli gives rise to distinguishable sensations, if they lie at opposite sides of the point of transition, however closely they may approach it. As a matter of fact, a sensation A may be indistinguishable from B, and B from C, and yet A may be distinguishable from C. If discernible unlikeness in sensation were coextensive with actual unlikeness, this would be impossible. Another objection is that the power of discriminating very small degrees of unlikeness is greatly improved by practice, and varies greatly with the concentration of attention. It seems improbable that these conditions should have so great an effect on the actual intensity of sensation produced by the stimulus.

Another explanation is that adopted by Fechner. He rightly holds that the sensation varies with the stimulus even when the variation is not perceptible. It becomes perceptible when the degree of variation has passed a certain limit. So far, we may follow him. But he also holds that the increase in intensity of sensation required to constitute a discernible unlikeness is not relative but absolute, so that the variations of stimulus form a geometrical series, while the corresponding variations of the sensation form an arithmetical series. In estimating weight by means of pressure, if we begin with an ounce, we must add a third of an ounce before any unlikeness is discernible; if we begin with a pound, we must add a third of a pound before any unlikeness is discernible. In both cases, according to Fechner the increase in the intensity of the pressure-sensations is not relatively the same but absolutely the same. There are very serious objections to this view. If we compare the weight of an ounce with no weight at all, according to Fechner, the degree of unlikeness between the two experiences ought to be strictly proportional to the difference between the intensity of sensation produced by one ounce, and the complete absence of pressure sensation. In other words, it ought to be proportional to the absolute intensity of pressure produced by one ounce. But as a matter of fact, the unlikeness between the zero value of a sensation and any finite value is infinite. Hence, for this limiting case, Fechner's interpretation breaks down. There is a difficulty in testing it in other cases, because of the peculiar nature of intensive magnitude. Intensive magnitude is indivisible. We cannot subtract a fainter sound from a louder so as to be able to point to a certain degree of loudness as the mathematical remainder. Hence we cannot in such cases immediately test Fechner's contention that the degree of unlikeness between two sensations is simply proportional to their mathematical difference,—to the remainder which would be left if one could be subtracted from the other. But there are other cases of the application of Weber's law in which this difficulty does not present itself. Weber's law holds good of extensive as well as intensive magnitude, and it also holds good of number. If we compare a line two inches long with a line three inches long, and then compare a line six inches long with a line seven inches long, according to Fechner the degree of unlikeness between the two inch line and the three inch line ought to be identical with the degree of unlikeness between the six inch line and the seven inch line. In both cases the mathematical difference is the same—one inch. This is true from the psychological as well as from the physical point of view. For if we suppose the lines to be presented to the eye under similar conditions, the mode in which an inch affects the retina in the one case may be virtually identical with the mode in which it affects the retina in the other case. The inches are not only equal as measured by a rule ; they also appear equal as they are presented to consciousness. We are therefore dealing with psychical, and not merely with physical, magnitudes. But in spite of the fact that 3 — 2 = 1, and that 7 — 6 also = 1, there is a greater degree of unlikeness between the line of two inches taken as a whole, and that of three inches taken as a whole, than there is between the line of six and that of seven inches. The same holds for least perceptible degrees of unlikeness. If we have to increase the length of a line of six inches by a certain amount in order that the unlikeness may be just discernible, we must increase the length of a line of two inches, not by the same amount, but in the same proportion, in order that the unlikeness may be just discernible. Number as well as extension affords illustration. If we lay a group of three counters on the table beside a group of two, and if we then lay a group of eight beside a group of seven, it is clear that there is a greater resemblance between the group of eight and the group of seven than there is between the group of three and the group of two. Yet in both cases the mathematical difference is the same— one counter ; and it may appear to be the same as presented to consciousness. The principle holds also for magnitudes which are not directly perceived, but thought of. Everybody recognises that a billion and one is more like a billion than eleven is like ten. So in the ordinary dealings of life, if we have to pay or receive sums amounting to hundreds of pounds, we feel that it does not matter about odd pence; but a penny more or less is by no means negligible if the sum to be paid or received is under a shilling.

We may then conclude that degree of unlikeness between the visible quantities is neither identical with their mathematical difference nor proportioned to it.

In the case of intensive magnitudes, such as the loudness of a sound, or the brightness of a light, there is, properly speaking, no mathematical difference, because we cannot divide such magnitudes into parts, so as to find a numerical equivalent for each, and subtract the one from the other. None the less, there may be in intensive magnitude something analogous to the mathematical difference. The velocity of a moving body is an intensive magnitude; but it is a magnitude which can be represented by a number which is a function of the space traversed and the time which it takes to traverse it. It may thus be treated as if it were an extensive magnitude capable of addition and subtraction. There is no reason why the intensity of sensation should not be conceived in the same way. At any rate, the mere fact that we are dealing with intensive magnitude does not in itself constitute an insuperable objection to the abstract possibility of such a mode of treatment. Hence there is in principle no objection to Fechner's attempt to correlate increased intensity of sensation with increased intensity of stimulus. But he was overhasty in supposing that equal degrees of unlikeness involved equal absolute differences of quantity in the sensation. On the contrary, the analogy of extensive magnitude seems to show that degree of unlikeness is correlated with relative, not absolute, differences in intensity of sensation. Fechner's problem is yet to be solved. We do not yet know the law which connects increase in the strength of the stimulus with corresponding increments of sensation. We cannot yet assign a number which shall represent degrees of loudness or brightness, as the number obtained by dividing the sum of units of time into the sum of units of space represents velocity.