Weber's Law.—I said that the intensity of the sensation increases by slower steps than those by which its exciting cause increases. If there were no threshold, and if every equal increment in the outer stimulus produced an equal increment in the sensation's intensity, a simple straight line would represent graphically the 'curve' of the relation between the two things. Let the horizontal line stand for the scale of intensities of the objective stimulus, so that at 0 it has no intensity, at 1 intensity 1, and so forth. Let the verticals dropped from the slanting line stand for the sensations aroused. At 0 there will be no sensation; at 1 there will be a sensation represented by the length of the vertical S¹—1, at 2 the sensation will be represented by S²—2, and so on. The line of S's will rise evenly because by the hypothesis the verticals (or sensations) increase at the same rate as the horizontals (or stimuli) to which they severally correspond. But in Nature, as aforesaid, they increase at a slower rate. If each step forward in the horizontal direction be equal to the last, then each step upward in the vertical direction will have to be somewhat shorter than the last; the line of sensations will be convex on top instead of straight.
[Fig. 2] represents this actual state of things, 0 being the zero-point of the stimulus, and conscious sensation, represented by the curved line, not beginning until the 'threshold' is reached, at which the stimulus has the value 3. From here onwards the sensation increases, but it increases less at each step, until at last, the 'acme' being reached, the sensation-line grows flat. The exact law of retardation is called Weber's law, from the fact that he first observed it in the case of weights. I will quote Wundt's account of the law and of the facts on which it is based.
"Every one knows that in the stilly night we hear things unnoticed in the noise of day. The gentle ticking of the clock, the air circulating through the chimney, the cracking of the chairs in the room, and a thousand other slight noises, impress themselves upon our ear. It is equally well known that in the confused hubbub of the streets, or the clamor of a railway, we may lose not only what our neighbor says to us, but even not hear the sound of our own voice. The stars which are brightest at night are invisible by day; and although we see the moon then, she is far paler than at night. Every one who has had to deal with weights knows that if to a pound in the hand a second pound be added, the difference is immediately felt; whilst if it be added to a hundredweight, we are not aware of the difference at all....
"The sound of the clock, the light of the stars, the pressure of the pound, these are all stimuli to our senses, and stimuli whose outward amount remains the same. What then do these experiences teach? Evidently nothing but this, that one and the same stimulus, according to the circumstances under which it operates, will be felt either more or less intensely, or not felt at all. Of what sort now is the alteration in the circumstances upon which this alteration in the feeling may depend? On considering the matter closely we see that it is everywhere of one and the same kind. The tick of the clock is a feeble stimulus for our auditory nerve, which we hear plainly when it is alone, but not when it is added to the strong stimulus of the carriage-wheels and other noises of the day. The light of the stars is a stimulus to the eye. But if the stimulation which this light exerts be added to the strong stimulus of daylight, we feel nothing of it, although we feel it distinctly when it unites itself with the feebler stimulation of the twilight. The poundweight is a stimulus to our skin, which we feel when it joins itself to a preceding stimulus of equal strength, but which vanishes when it is combined with a stimulus a thousand times greater in amount.
"We may therefore lay it down as a general rule that a stimulus, in order to be felt, may be so much the smaller if the already preëxisting stimulation of the organ is small, but must be so much the larger, the greater the preëxisting stimulation is.... The simplest relation would obviously be that the sensation should increase in identically the same ratio as the stimulus.... But if this simplest of all relations prevailed, ... the light of the stars, e.g., ought to make as great an addition to the daylight as it does to the darkness of the nocturnal sky, and this we know to be not the case.... So it is clear that the strength of the sensations does not increase in proportion to the amount of the stimuli, but more slowly. And now comes the question, in what proportion does the increase of the sensation grow less as the increase of the stimulus grows greater? To answer this question, every-day experiences do not suffice. We need exact measurements, both of the amounts of the various stimuli, and of the intensity of the sensations themselves.
"How to execute these measurements, however, is something which daily experience suggests. To measure the strength of sensations is, as we saw, impossible; we can only measure the difference of sensations. Experience showed us what very unequal differences of sensation might come from equal differences of outward stimulus. But all these experiences expressed themselves in one kind of fact, that the same difference of stimulus could in one case be felt, and in another case not felt at all—a pound felt if added to another pound, but not if added to a hundredweight.... We can quickest reach a result with our observations if we start with an arbitrary strength of stimulus, notice what sensation it gives us, and then see how much we can increase the stimulus without making the sensation seem to change. If we carry out such observations with stimuli of varying absolute amounts, we shall be forced to choose in an equally varying way the amounts of addition to the stimulus which are capable of giving us a just barely perceptible feeling of more. A light to be just perceptible in the twilight need not be near as bright as the starlight; it must be far brighter to be just perceived during the day. If now we institute such observations for all possible strengths of the various stimuli, and note for each strength the amount of addition of the latter required to produce a barely perceptible alteration of sensation, we shall have a series of figures in which is immediately expressed the law according to which the sensation alters when the stimulation is increased...."
Observations according to this method are particularly easy to make in the spheres of light, sound, and pressure. Beginning with the latter case,