S-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, O 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

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 carriagewheels 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 pound weight 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 preexisting stimulation of the organ is small, but must be so much the larger, the greater the preexisting 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 sensa tions 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,

"We find a surprisingly simple result. The barely sensible addition to the original weight must stand exactly in the same proportion to it, be the same fraction of it, no matter what the absolute value may be of the weights on which the experiment is made. . . . As the average of a number of experiments, this fraction is found to be about ; that is, no matter what pressure there may already be made upon the skin, an increase or a diminution of the pressure will be felt, as soon as the added or subtracted weight amounts to one third of the weight origi nally there."

Wundt then describes how differences may be observed in the muscular feelings, in the feelings of heat, in those of light, and in those of sound; and he concludes thus:

"So we have found that all the senses whose stimuli we are enabled to measure accurately, obey a uniform law. However various may be their several delicacies of discrimination, this holds true of all, that the increase of the stimulus necessary to produce an increase

of the sensation bears a constant ratio to the total stimulus. The figures which express this ratio in the several senses may be shown thus in tabular form:

"These figures are far from giving as accurate a measure as might be desired. But at least they are fit to convey a general notion of the relative discriminative susceptibility of the different senses. . . . The important law which gives in so simple a form the relation of the sensation to the stimulus that calls it forth was first discovered by the physiologist Ernst Heinrich Weber to obtain in special cases.'

Fechner's Law. Another way of expressing Weber's law is to say that to get equal positive additions to the sensation, one must make equal relative additions to the stimulus. Professor Fechner of Leipzig founded upon Weber's law a theory of the numerical measurement of sensations, over which much metaphysical discussion has raged. Each just perceptible addition to the sensation, as we gradually let the stimulus increase, was supposed by him to be a unit of sensation, and all these units were treated by him as equal, in spite of the fact that equally perceptible increments need by no means appear equally big when they once are perceived. The many pounds which form the just perceptible addition to a hundredweight feel bigger when added than the few ounces which form the just perceptible addition to a pound. Fechner ignored this fact. He considered that if n distinct perceptible steps of increase might be passed through in gradually increasing a stimulus from the threshold-value till the intensity s was felt, then the sensation of s was composed of n units, which were of the same value all along the line. Sensations once

* Vorlesungen über Menschen u. Thierseele, Lecture VII.

In other words, S standing for the sensation in general, and d for its noticeable increment, we have the equation dS= const. The in

represented by numbers, psychology may become, according to Fechner, an 'exact' science, susceptible of mathematical treatment. His general formula for getting at the number of units in any sensation is S = C log R, where S stands for the sensation, R for the stimulus numerically estimated, and C for a constant that must be separately determined by experiment in each particular order of sensibility. The sensation is proportional to the logarithm of the stimulus; and the absolute values, in units, of any series of sensations might be got from the ordinates of the curve in Fig. 2, if it were a correctly drawn logarithmic curve, with the thresholds rightly plotted out from experiments.

Fechner's psycho-physic formula, as he called it, has been attacked on every hand; and as absolutely nothing practical has come of it, it need receive no farther notice here. The main outcome of his book has been to stir up experimental investigation into the validity of Weber's law (which concerns itself merely with the just perceptible increase, and says nothing about the measurement of the sensation as a whole) and to promote discussion of statistical methods. Weber's law, as will appear when we take the senses, seriatim, is only approximately verified. The discussion of statistical methods is necessitated by the extraordinary fluctuations of our sensibility from one moment to the next. It is found, namely, 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. Our incessant accidental inner alterations 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

crement of stimulus which produces dS (call it dR) meanwhile varies. Fechner calls it the 'differential threshold '; and as its relative value dR R

to R is always the same, we have the equation

= const.