The Principles of Psychology, Vols. 1-2

" The sound of the clock, the light of the stars, the pressure of the pound, these are all stimuli to our senses, and stimuli whoso 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.

u 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 pre-exist ing stimulation of the organ is small, but must be so much the larger; the greater the pre-existing stimulation is. From this in a general way we can perceive the connection between the stimulus and the feeling it excites. At least thus much appears, that the law of dependence is not as simple a one as might have been expected beforehand. The simplest relation would obviously be that the sensation should increase in identically the same ratio as the stimulus, thus that if a stimulus of strength one occasioned a sensation one, a stimulus of two should occa sion sensation two, stimulus three, sensation three, etc. But if this simplest of all relations prevailed, a stimulus added to a pre-existing strong stimulus ought to provoke as great an increase of feeling as if it were added to a pre-existing weak stimulus ; 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. This we know not to be the case : the stars are invisible by day, the addition they make to our sensation then is unnoticable, whereas the same addition to our feeling of the twi light is very considerable indeed. 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 ex periences expressed themselves in one kind of fact, that the same differ ence 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 observa tions if we start with an arbitrary strength of stimulus, notice what sensation it gives us, and then s\,e how much we can increase the stim ulus 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 per ceived 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 per ceptible alteration of sensation, we shall have a series of figures in which is immediately expressed the law according to which the sensa tion alters when the stimulation is increased. ..."

easy to make in the spheres of light-, sound-, and pressuresensation. . . . Beginning with the latter case, "We find a surprisingly simple result. The barely sensible ad dition 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 originally 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 his seventh lecture (from which our extracts have been made) 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 sen sation 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 sen sation to the stimulus that calls it forth was first discovered by the physiologist Ernst Hcinrich Weber to obtain in special cases. Gustav Theodor Fechner first proved it to be a law for all departments of sen sation. Psychology owes to him the first comprehensive investigation of sensations from a physical point of view7, the first basis of an exact Theory of Sensibility."

So much for a general account of what Fechner calls Weber's law. The ' exactness ' of the theory of sensibility to which it leads consists in the supposed fact that it gives the means of representing sensations by numbers. The unit of any kind of sensation will be that increment which, 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 physi cal stimulus of which it is a function. For if we know how much of the stimulus it will take to give a barely percep tible sensation, and then what percentage of addition to the stimulus will constantly give a barely perceptible incre ment 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 sensa tional 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 corre spond 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 log arithmic 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 sen sation 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 wre call the stimulus R, and the constant factor C, we get the formula

which is what For-hnor calls the psychophysischer Maasforniel. This, in brief, is Fechner's reasoning, as 1 under stand it. The Maasformd admits of mathematical development in various directions, and has given rise to arduous discus sions 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. Mean while 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 disceruibility, 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 Feehner's formula are Delbceuf s In his Recherches sur la Mesure des Sensations (1873), p. 85, and Elsus's in his pamphlet Uber die Psychophysik (1886) p. 10.

increment of the sensation is without taking the average ol 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. Feclmer 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 -j- d until you distinctly feel the effect of the subtraction ; * call the difference here the ratio of this quantity to the original 8 (or rather to J3 + d — d') is what Fechner calls the difference-threshold. This difference-threshold should be a constant fraction (no matter what is the size of 8) 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 theo retically to be.f

Of course the sensibility is small when the least dis cernible difference is large, and vice versa ; in other words, it and the difference-threshold are inversely related to each oilier. (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. f 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 sen sations 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 rep resents 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 6, 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. Feclmer made use of this method to ascertain his own power of discriminating differences of weight, record ing 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 Feclmer to be the 'reciprocal' of the discriminative sensi bility 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. Delbceuf and Wundt have experimented with larger differences oy means of what Wundt calls the Methode tier mittleren Abstufungen, and what we may call (4) The Method of Equalappear ing Intervals. This con sists in so arranging three stimuli in a series that the inter vals 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 incre ments 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, Yolkmann, Aubert, Masson, Helmholtz, and Krapelin find figures varying from J or J to y^-y 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 dis criminative 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 con stant, 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) betAveen 96 and 4096, beyond which in tensity no experiments were made.* Konig and Brodhun

have given measurements by method 1 which cover the most extensive series, and moreover apply to six different colors of light. These experiments (performed in Helmholtz's laboratory, apparently,) ran from an intensity called 1 to one which was 100,000 times as great. From intensity 2000 to 20,000 Weber's law held good ; below and above this range discriminative sensibility declined. The incre ment discriminated here was the same for all colors of light, and lay (according to the tables) between 1 and 2 per cent of the stimulus.* Delbceuf had verified Weber's law for a certain range of luminous intensities by method 4 ; that is, he had found that the objective intensity of a light which appeared midway between two others was really the geometrical mean of the latter's intensities. But A. Lehmann and afterwards Neiglick, in Wuudt's laboratory, found that effects of contrast played so large a part in experiments performed in this way that Delboeuf's results could not be held conclusive. Merkel, repeating the experiments still later, found that the objective intensity of the light which we judge to stand midway between two others neither stands midway nor is a geometric mean. The discrepancy from both figures is enormous, but is least large from the midway figure or arithmetical mean of the two extreme in tensities, t Finally, the stars have from time immemorial been arranged in ' magnitudes ' supposed to differ by equalseeming intervals. Lately their intensities have been gauged photometrically, and the comparison of the subjec tive with the objective series has been made. Prof. J. Jastrow is the latest worker in this field.

He finds, taking Pickering's Harvard photometric tables as a basis, that the ratio of the average intensity of each ' magnitude ' to that below it decreases as we pass from lower to higher magni tudes, showing a uniform departure from Weber's law, if the method of equal-appearing intervals be held to have any direct relevance to the latter.:}: ~ * Berlin AcfidTSitz"iuigsberichte, 1888, p. 917. Other observers (Dobro. wolsky, Lamausky) found great differences in different colors.

f American Journal of Psychology, i. 125. The rate of decrease is small but steady, and I cannot well understand what Professor J. means by saying that his figures verify Weber's law. Sounds are less delicately discriminated in intensity than lights. A certain difficulty has come from disputes as to the measurement of the objective intensity of the stimulus. Earlier inquiries made the perceptible increase of the stim ulus to be about ^ of the latter. Merkel's latest results of the method of just perceptible differences make it about •£$ for that part of the scale of intensities during which Weber's law holds good, which is from 20 to 5000 of M.'s arbitrary unit.* Below this the fractional increment must be larger. Above it no measurements were made.

For pressure and muscular sense we have rather divergent results. Weber found by the method of just-perceptible differences that persons could distinguish an increase of weight of ^j- when the two weights were successively lifted by the same hand. It took a much larger fraction to be discerned when the weights were laid on a hand which rested on the table. He seems to have verified his results for only two pairs of differing weights, t and on this founded his ' law.' Experiments in Hering's laboratory on lifting 11 weights, running from 250 to 2750 grams showed that the least perceptible increment varied from g*T for 250 grams to ^ for 2500. For 2750 it rose to ^ again. Merkel's recent and very careful experiments, in which the finger pressed down the beam of a balance counterweighted by from 25 to 8020 grams, showed that between 200 and 2000 grams a constant fractional increase of about T^ was felt when there was no movement of the finger, and of about fa when there was movement. Above and below these limits the discriminative power grew less. It was greater when the pressure was upon one square millimeter of sur face than when it was upon seven.J

Wo.rmih and taste have been made the subject of similar investigations with the result of verifying something like Weber's law. The determination of the unit of stimu lus is, however, so hard here that I will give no figures. The results may be found in Wundt's Physiologische Psychologie, 3d Ed. I. 370-2. The discrimination of lengths by the eye has been found also to obey to a certain extent Weber's law. The figures will all be found in G. E. Miiller, op. cit., part n, chap, x, to which the reader is referred. Professor Jastrow has published some experiments, made by what may be called a modification of the method of equal-appearing differ ences, on our estimation of the length of sticks, by which it would seem that the estimated intervals and the real ones are directly and not logarithmically proportionate to each other. This resembles Merkel's results by that method for weights, lights, and sounds, and differs from Jastrow' s own finding about star-magnitudes.*

If we look back over these facts as a whole, we see that it is not any fixed amount added to an impression that makes us notice an increase in the latter, but that the amount depends on how large the impression already is. The amount is expressible as a certain fraction of the entire impression to which it is added ; and it is found that the fraction is a well-nigh constant figure throughout an entire region of the scale of intensities of the impression in ques tion. Above and below this region the fraction increases in value. This is Weber's law, which in so far forth expresses an empirical generalization of practical importance, without involving any theory whatever or seeking any absolute measure of the sensations themselves. It is in the

Theoretic Interpretation of Weber s Law that Fechner's originality exclusively consists, in his as sumptions, namely, 1) that the just-perceptible increment is the sensation-unit, and is in all parts of the scale the same (mathematically expressed, As — const.) ; 2) that all our sensations consist of sums of these units ; and finally, 3) that the reason why it takes a constant fractional increase of the stimulus to awaken this unit lies in an ultimate law of the connection of mind with matter, whereby the quantities of our feelings are related logarithmically to the quantities of their objects. Fechner seems to find something in scrutably sublime in the existence of an ultimate 'psychophysic ' law of this form.

These assumptions are all peculiarly fragile. To begin with, the mental fact which in the experiments corresponds to the increase of the stimulus is not an enlarged sensation, but a judgment that the sensation is enlarged. What Fechner calls the ' sensation ' is what appears to the mind as the objective phenomenon of light, warmth, weight, sound, impressed part of body, etc. Fechner tacitly if not openly assumes that such a judgment of increase consists in the simple fact that an increased number of sensation-units are present to the mind; and that the judgment is thus itself a quantitatively bigger mental thing when it judges large differences, or differences between large terms, than when it judges small ones. But these ideas are really absurd. The hardest sort of judgment, the judgment which strains the attention most (if that be any criterion of the judgment's * size '), is that about the smallest things and differences. But really it has no meaning to talk about one judgment being bigger than another. And even if we leave out judgments and talk of sensations only, we have already found ourselves (in Chapter YI) quite unable to read any clear meaning into the notion that they are masses of units combined. To introspection, our feeling of pink is surely not a portion of our feeling of scarlet ; nor does the light of an electric arc seem to con tain that of a tallow-candle in itself. Compound things contain parts ; and one such thing may have twice or three times as many parts as another. But when we take a sim ple sensible quality like light or sound, and say that there is now twice or thrice as much of it present as there wras a moment ago, although we seem to mean the same thing as if we were talking of compound objects, we really mean something different.

We mean that if we were to arrange the various possible degrees of the quality in a scale of serial increase, the distance, interval, or difference between the stronger and the weaker specimen before us would seem about as great as that between the weaker one and the beginning of the scale. It is these KELATIONS, these DIS TANCED, ivhich ice are measuring and not the composition of the qualities themselves, as Feclmer thinks. Whilst if we turn to objects which are divisible, surely a big object may be known in a little thought. Introspection shows moreover

that in most sensations a new kind of feeling invariably ac companies our judgment of an increased impression ; and this is a fact which Fechner's formula disregards.* But apart from these a priori difficulties, and even sup posing that sensations did consist of added units, Feclmer's assumption that all equally perceptible additions are equally great additions is entirely arbitrary. Why might not a small addition to a small sensation be as perceptible as a large addition to a large one ? In this case Weber's law would apply not to the additions themselves, but only to their perceptibility. Our noticing of a difference of units in two sensations would depend on the latter being in a fixed ratio. But the difference itself would depend directly on that between their respective stimuli. So many units added to the stimulus, so many added to the sensation, and if the stimulus giew in a certain ratio, in exactly the same ratio would the sensation also grow, though its perceptibility grew according to the logarithmic law.t

If J stand for the smallest difference which we perceive, then we should have, instead of the formula As = const., which interprets all the facts of Weber's law, in an entirely different theoretic way from that adopted by Fechner.J The entire superstructure which Feclmer rears upon the * Cf. Stumpf , Tonpsychologie, pp. 397-9. " One sensation cannot be a multiple of another. If it could, we ought tc be able to subtract the one from the other, and to feel the remainder by itself. Every sensation pre sents itself as an indivisible unit." Professor von Kries, in the Vierteliahrschrift fur wiss. Philosophic, vi. 257 ff., shows very clearly the a surdity of supposing that our stronger sensations contain our weaker ones as parts They differ as qualitative units. Compare also J. farmery in - - - - • ««oo% - 1-j.i 4jt. j. Ward in Mind,