The Least Perceptible Différence ^With the dif

ferential weight which in Exp. 1 gave the nearest to 75 per cent correct judgments, make 100 consécutive trials. Eecord 1 for right or x for wrong in each trial.

3. The Least Perceptible Différence în the Double Weight. — Put the glass used into the other one. Double the differential weight. With this doubled weight and doubled incrément make 100 trials in the same manner as in Exp. 2. Compare the per cent of right cases in Exp. 2 and Exp. 3. If Weber's law applies, the per cent of error should be approximately equal in the two cases.

There are mathematical formùlae by which we may calculate from the above results what incrément is needed in order to yield any particular degree of certainty. It is customary to use 75 per cent of right judgments as a norm or standard, as it lies half-way between no knowledge and absolute certainty in this method. Thus, if the glass in Exp. 2 weighs 200 grams

that is, as he felt it would run by chance, and suppose that he had given the order "Second heavier" three times in succession, would not the observer require more than an ordinary degree of assurance before he would pronounce a fourth one also of the same order ? Yet according to chance, and in the flipping of the coin, the number of times head has been up has not the slightest influence on the next throw. On account of our community of ideas there is always a more or less subconscious tendency présent to anticipate the order of a so-called chance séries which is determined by one human will for another.

Another important variable is the bodily attitude taken by the observer at the table. The attitude of attention is a bodily attitude of muscular tension. If the observer rests against the back of the chair he will make a much poorer record than if he sits up with body erect and firm, bending his head gently toward the lifting hand.

96 WEBERS LAW

and the differential weight is 12 grams, and this ratio (6:100) gives 72 per cent of right judgments; then, by referring to a simple table,* we find that the incrément must be 14 grams in order to yield a certainty of 75 per cent right judgments. This computation is not needed in the présent case because we may hère compare the percentages directly.

Working with compact hard-rubber blocks, which are more favorable than the glasses for fine discrimination, the average constant for a good observer is about 1 :24. The ratio obtained in this test is probably considerably larger, mainly for two reasons — the bulkiness of the glassjt and the lack of practice of both expérimenter and observer.

Similar ratios may be worked ont for différent conditions in this sensé and for différent sensés. Each individual has his own ratio, which is an expression of his Personal équation.

The principle of its application is illustrated in Fig. 13, which is drawn for the constant ratio 1 :24. 16, 25, 50, 64, etc., represent selected standards and .6, 1.0, 2.0, 2.6, etc., the respective incréments that would be just perceptible.

This illustration and the above experiment hâve référence to sensation-différence. To be more spécifie,

* Sanford, "Expérimental Psyehology," p. 354 ; or Titchener, "Expérimental Psyehology," Quantitative, Inatructors' Manual, p.

fThe more compact the weight is the finer the discrimination will be. The discrimination will be finer if a thick glass is used rather than a thin one, because for the thin glass the bulk is proportionally greater than the weight. Both glasses seem to be lighter than they really are.

the law stated in the first paragraphs mîght be restated thus : "To increase the sensation-différence in arithmetieal ratio, it is necessary to increase the stimulus-différence in geometrical ratio." There is a parallel aspect of Weber's law which has référence in a more direct way to sensation-magnitudes, greater than the least perceptible: "To increase the sensation in arithmetical ratio, it is necessary to increase the stimulus in geometrical ratio." This latter form of the law has great practicaJ significance in our classification of magnitudes by sense-perception.

co

• FiG. 13.

Suppose we hâve a thousand pebbles which range in weight from 1 gram up to 100 grams and we are required to sort them into five piles, according to weight, so divided that we shall be equally sure for each pile that the sorting is right. If we could resort to the use of accurate scales, the piles would range in arithmetical ratio ; each pile would cover a range of 20 grams. But not so when we judge the weight by lifting. Weber's law enables us to predict what the grouping should be.

An excellent illustration of this second aspect of the law is found in the astronomer's classification of the stars. Stars are classified into a comparatively small number of groups, usually fifteen, according to their perceived magnitudes. Astronomers assume that this classification foUows Weber's law ; namely, that the physical magnitudes (the amounts of light emitted) do not stand in a direct arithmetical ratio, but in a geometrical ratio, in the grouping. Each group is supposed to stand in a certain ratio in the foUowinggroup, say 1:10.

A psychologist attempted to verify this hypothesis. He produced, in the psychological laboratory, artificial stars whose physical magnitudes could be measured, and then he grouped thèse empirically into five groups on the basis of perceived magnitude and ascertained their average candle-power by physical measurements. The results ran as f ollows :

Magnitude j or group I II III IV V

Average candle-power . . . .269 146 83 45 21

When one considers the crudeness of a measurement of this kind, one feels that thèse resTilts show a f air approximation to the law.*

Another type of démonstration of Weber's law may

* There is at least a very close analogy to Weber's law in any practical grouping of the stars. Thus from Kapteyn's récent work we leam that within a sphère the radius of which is 560 light-years there will be found:

1 Btar giving from 100,000 to 10,000 timee the light of our sun

26 stars " " 10 000

be made in terms of discrimination-time ; that is, the time that it takes to perceive a différence. It rests on the postulate that if différences are equally difficult to perceive, the discrimination-time for ail such cases should be the same. Take, for example, an incrément of one tenth on différent weights. There is an apparatus in the psychological laboratory which consists of a System of weights and balances which may be attached to an instrument that measures time in hundredths of a second. To verify the law, we might measure the discrimination-time for an incrément of 1 gram on 10, 5 grams on 50, 9 grams on 90, etc. ; and, if the law holds, the discrimination-times should be equal. We may take a good illustration of the same type from sight — the perception of the différence in the length of lines. Suppose we hâve three sets of Unes, Fig. 14, and we consider the middle one in each group

FiG. 14.

the standard for that group. We see in the figure that the longer and the shorter lines differ from the standard by one fifth in ail cases. Therefore, the différence should be equally clear in ail the cases, according to Weber's law. To measure the discrimination-time, we arrange the apparatus so that a compared line is exhibited immediately after the standard line, and then measure for each group the time it takes the observer to perceive whether the compared line is longer or shorter than the standard. If the law holds, other things being

equal, this discrimination-time should be equal for ail the standards within a normal range so long as the ratio is constant.

Thèse three démonstrations of Weber's law — the application to the least perceptible différence, the application to the classification of larger différences in sensemagnitudes, and the application to discrimination-time — might be repeated in varions respects for other sensés.

We know this world in terms of four aspects of expérience, which are the four attributes of sensation, namely, quality, intensity, duration, and extensity. Weber's law is primarily a law of the intensity of sensations, but it applies in some respects to each of the other attributes.

It is generally conceded that the law applies approximately to sensation-intensities in ail the sensés. We hâve drawn our expérimental illustrations from the kinsesthetic sensations in the perception of weight. The classification of the stars is an illustration from the intensity of light.*

A clear case of the opération of Weber's law is found

in the intensity of sound. It is perhaps equally rigid

for the intenstity of pressure sensations. It has been

demonstrated within a narrow range for taste and smell.

Although it has not been worked out fully for the

intensity of pain and température sensations, it prob-

ably applies, to some extent, there also.

* Fechner's cloud experiment is the classical inustration in terms of the intensity of light, or brightness. It is to the effect that if we look at a partly clouded sky and sélect a spot where we can just detect a différence in the shading of the two clouds by the naked eye and then look at this through darkening glasses of différent densities, the différence will remain equally clear.

WEBEK'S LAW 101

The opération of the law in the other sense-attributes is not so clear. It bas been demonstrated for visual qualities (color), for arm-space, eye-movements, the duration ôf sounds, etc., but even where it applies to sense-attributes other than intensity it bas référence to sense-magnitude.

The most gênerai évidence of the law is in terms of the least perceptible différence, as in the above experiments on weight-discrimination. Its application to larger sense-magnitudes is always more doubtf ul.

The law can apply only within a middle or normal range of sensation-intensities in any sensé. It does not hold fôr very faint sensations, nor for excessively strong sensations.

Much dépends upon the method of the test. Thus, the application of the law to lines, Fig. 14, is much more likely to give a positive démonstration by the discrimination-time test than by the classification test ; indeed, we should hardly expect any évidence of it by the classification method, because that method introduces factors which may interfère with its opération. We must always bear in mind that every case of discrimination is a complex opération (Jepending upon scores of variables, and the law could at best apply absolutely only to one set of conditions.

The greatest gain in an experiment of this sort should be in the révélation of the complexity and the interrelations of mental laws even in such a simple opération as the^ lifting of a weight. Such révélations help us to analyze and understand our daily activities and should resuit in greater efficiency. We foUow Weber's law

unconsciously in many of our well-regulated actions in which we are guided by our sensés.

Aside from the démonstration of Weber's law, thèse experiments illustrate a common form of measurement in ail the sensés ; namely, the détermination of the least perceptible différence (l, p. rf.).* The two most elementary forms of measurement in ail the sensés are (1) the threshold of sensitivity and (2) the L p. d, or threshold of discrimination. In measuring sensitivity we détermine the smallest quantity of any stimulus — a color, a pressure, an odor, a sound, etc. — ^that can be sensed. In measuring discrimination we détermine the smallest increase (or decrease) that can be perceived on any given strength of stimulus. The former is chiefly a measure of the delicacy of the organism, while the latter is chiefly a measure of the intellectual capacity for using sense-differences. Thus, if one observer obtained the discrimination ratio 1 :12 and the other 1 :18 in the above test, we hâve in thèse ratios a measure of the relative usefulness of the sensé of weight for the two persons.

The L p. d. may be used as a sort of foot-rule in the quantitative study of a great variety of processes. Thus, with it, we may measure features of fatigue, rhythm, attention, memory, mental development, race différences, the effect of a stimulant, the power of émulation, etc., provided always that we bear in mind that in such cases we measure only one f eature, the discrimi-

*The term "just notioeable différence" (;. «. d.) is often used in place of L p. d.; so also is the term "discrimination limen'' {d. l,), many authors using the term "limen" for threshold.

nation, and that our conclusions must not go beyond this. To illustrate the réservation, suppose that the expérimenter sets himself the task of determining what the mental effect of a cup of coffee is. Among other measurements he may include the discrimination-tests in the various sensés and for différent attributes in each sensé, and af ter a long séries of measurements he should be able to say: My discrimination in such and such activities is heightened to such and si^ch a degree •for such and such a period after drinking coffee, and then there is a reaction which shows at such and such a time in such and such a degree for such and such of the activities.