WEBER'S LAW is the principal generalization of that branch of scientific investigation which has come to be known as " psycho-physics." According to Fechner, who has done most to prosecute these inquiries and to consolidate them under a separate name, " psycho-physics is an exact doctrine of the relation of function or dependence between body and soul." In other words, it is throughout an attempt to submit to definite measurement the relation of physical stimuli to the resulting psychical or mental facts, and forms an important department of experimental psychology. It deals with the quantitative aspects of mental facts—their intensity or quantity proper and their duration. Physical science enables us, at least in the case of some of the senses, to measure with accuracy the objective amount of the stimulus, and introspection enables us to state the nature of the subjective result. Thus we are able to say whether a stimulus produces any psychical, result, and can fix in that way the minimum sensibile or " threshold of consciousness " for each of the senses. In like manner (though with less accuracy, owing to the dis-turbing nature of the conditions) we can fix the sensational maximum, or upper limit of sensibility, in the different senses, that is to say, the point beyond which no increase of stimulus produces any appreciable increase of sensation. We thus determine, as Wundt puts it, the limit-values between which changes of intensity in the stimulus are accompanied by changes in sensation. But the central inquiry of psycho-physics remains behind. Between the quantitative minimum and the quantitative maximum thus fixed can we discover any definite relation between changes in the objective intensity of the stimuli and changes in the intensity of the sensations as estimated by consciousness 1 The answer of psycho-physics to this in-quiry is given in the generalization variously known as "Weber's law," " Fechner's law," or the "psycho-physical law," which professes to formulate with exactitude the rela-tions which exist between change of stimulus and change of sensation.

As we have no means of subjectively measuring the absolute intensity of our sensations, it is necessary to de-pend upon the mental estimate or comparison of two or more sensations. Comparison enables us to say whether they are equal in intensity, or if unequal which is the greater and which is the less. But as they approach equality in this respect it becomes more and more difficult to detect the difference. By a series of experiments, therefore, it will be possible, in the case of any particular individual, to determine the least observable difference in intensity between two sensations of any particular sense. This least observable difference is called by Fechner the " Unterschiedsschwelle," or " difference-threshold," that is to say, the limit of the discriminative sensibility of the sense in question. That such a " threshold," or least observable difference, exists is plain from very simple examples. Very small increases may be made in the objective amount of light, sound, or pressure—that is, in the physical stimuli applied to these senses—without the subject on whom the experiment is made detecting any change. It is further evident that, by means of this " Unterschiedsschwelle," it is possible to compare the discriminative sensibility of different individuals, or of different senses, or (as in the case of the skin) of different parts of the same sense organ : the smaller the difference observable the finer the discriminative sensibility. Thus the discrimination of the muscular sense is much more delicate than that of the sense of touch or pressure, and the discriminative sensibility of the skin and the retina varies very much according to the parts of the surface affected. Various methods have been adopted with a view to determine these minima of discriminative sensibility with an approach to scientific precision. The first is that employed by Weber himself, and has been named the method of just observable differ-ences. It consists either in gradually adding to a given stimulus small amounts which at first cause no perceptible difference in sensation but at a certain point do cause a difference to emerge in consciousness, or, vice versa, in gradually decreasing the amount of additional stimulus, till the difference originally perceived becomes imper-ceptible. By taking the average of a number of such results, the minimum may be determined with tolerable accuracy. The second method is called by Fechner the method of correct and incorrect instances. When two stimuli are very nearly equal the subject will often fail to recognize which is the greater, saying sometimes that A is greater, sometimes that B is greater. When in a large number of trials the right and wrong guesses exactly balance one another we may conclude that the difference between the two stimuli is not appreciable by the sense. On the other hand, as soon as the number of correct guesses definitely exceeds half of the total number of cases, it may be in-ferred that there is a certain subjective appreciation of difference. This method was first employed by Vierordt. The third method, that of average errors, is very similar to the one just explained. Here a certain weight (to take a concrete example) is laid upon the hand of the person experimented upon, and he is asked, by the aid of subjec-tive impression alone, to fix upon a second weight exactly equal to the first. It is found that the second weight sometimes slightly exceeds the first, sometimes slightly falls below it. Whether above or below is of no con-sequence to the method, which depends solely on the amount of the error. After a number of experiments, the different errors are added together, and the result being divided by the number of experiments gives us the average error which the subject may be calculated upon to make. This marks the amount of stimulus which is just below the difference-threshold for him. This method was first employed by Fechner and Volkmann. The different methods were first named, and the theory of their applica-tion developed by Fechner, in his Elemente der Psycho-physik (1860).

These methods have been chiefly applied to determine the relation of the difference-threshold to the absolute magni-tude of the stimuli employed. For a very little reflexion tells us that the smallest perceivable difference is not an amount whose absolute intensity is constant even within the same sense. It varies with the intensity of the stimuli employed. We are unable, for example, to recognize slight differences in weight when the weights compared are heavy, though we should be perfectly able to make the distinction if the weights compared were both light. Ordinary observation would lead us, therefore, to the conclu-sion that the greater the intensity of the original stimulus at work the greater must be the increase of stimulus in order that there may be a perceptible difference in the resulting sensation. E. H. Weber (1795-1878) was the first (after a prolonged series of experiments on the sensa-tions of sight, hearing, and touch) to clothe this generality with scientific precision by formulating the law which has since gone by his name. The purport of the law is that, in order that the sensational difference may remain un-changed, the increase of stimulus must maintain the same proportion to the intensity of the preceding stimulus. The smallest perceptible difference is therefore not absol-utely the same, but it remains relatively the same, that is, it remains the same fraction of the preceding stimulus. For example, if we can distinguish 16 oz. and 17 oz., we shall be able to distinguish 32 oz. and 34 oz. but not 32 oz. and 33 oz., the addition being in each case ^ of the preceding stimulus. This fraction (supposing it to be the difference-threshold of the muscular sense) remains a con-stant, however light or however heavy the weights com-pared. The law may be formulated thus :—The difference between any two stimuli is experienced as of equal magni-tude, in case the mathematical relation of these stimuli remains unaltered. Or, otherwise expressed, In order that the intensity of a sensation may increase in arithmetical progression, the stimulus must increase in geometrical pro-gression. It is also expressed by Fechner in the form— The sensation increases as the logarithm of the stimulus.

The law has been variously interpreted. Fechner him-self designated it the psycho-physical law, and treated it as the fundamental formula of the relation between body and mind, thus assigning to it an ontological dignity and significance. But in this " psycho-physical" interpreta-tion of his results he has not had a numerous following. Wundt interprets the law in a purely "psychological" sense, making it a special instance of the general law of relativity which governs our mental states. Introspection can give us no information as to the absolute intensity of the stimulus; for a stimulus is known in consciousness only through its sensational resultant. Hence, he argues, we can only compare one psychical state with another, and our standard of measurement is therefore necessarily a relative one ; it depends directly upon the preceding state with which we compare the present. Others have attempted to give the law a purely physical or " physiological" explanation. Instead of holding with Fechner that the law expresses a recondite relation between the material and the spiritual world, they prefer to regard the quanti-tative relation between the last physical antecedent in the brain and the resultant mental change as prima facie one of simple proportion, and to treat Weber's law as holding between the initial physical stimulus and the final action of the nerve-centres. According to this interpretation, the law would be altogether due to the nature of nervous action. As a nerve, says Sully, after a temporary degree of stimulation temporarily loses its sensibility, so the greater the previous stimulation of a nerve the greater is the additional stimulus required to produce an appreciable amount of sensation.

Weber's law, it must be added, holds only within certain limits. In the " chemical " senses of taste and smell experiments are almost impossible. It is not practicable to limit the amount of the stimulus with the necessary exactitude, and the results are further vitiated by the long continuance of the physiological effects. The same considerations apply with still more force to the organic sensations, and the results in the case of temperature sensations are completely uncertain. The law is approxi-mately true in the case of sight, hearing, pressure, and the muscular sense,—most exactly in the case of sound. As this is the sense which affords the greatest facilities for measuring the precise amount of the stimulus, it may per-haps be inferred that, if we could attain the same exacti-tude in the other senses, with the elimination of the numerous disturbing extraneous influences at work, the law would vindicate itself with the same exactitude and certainty. It is further to be noted, however, that even in those senses in which it has been approximately verified, the law holds with stringency only within certain limits. The results are most exact in the middle regions of the sensory scale ; on the contrary, when we approach the upper or lower limit of sensibility, they become quite un-certain.

Literature.—Weber's investigations were given to the public in his articles in Wagner's Handworterbuch derPhysiologie. Fechuer's Elemente der Psycho-physik, referred to above, was published in 1860, and contains an elaborate exposition of the whole subject. He replied to his critics in two later works, In Sachen der Psycho-physik (1877) and Revision der Hauptpunkte der Psycho-physik (1882). Delboeuf s Étude psychophysique (1873), and G. E. Midler's Zur Grundlegung der Psycho-physik, are also important documents ;
and the subject is fully treated in Wundt's Physiologische Psychologie and in Ladd's Physiological Psychology, which is based upon Wundt. Two English monographs may be referred to as containing important criticisms of the law, —the paper of Messrs Dewar and M'Kendrick in the Transactions of the Royal Society of Edinburgh for 1878, and Mr James Ward's "Attempt to Interpret Fechner's Law," in Mind, i. 452 sq. (A. SE.)