GUSTAV THEODOR FECHNER

(1801-1887)

ELEMENTS OF PSYCHOPHYSICS

Translated from the German* by
HERBERT SIDNEY LANGFELD

VII. THE MEASUREMENT OF SENSATION

WEBER'S law, that equal relative increments of stimuli are proportional to equal increments of sensation, is, in consideration of its generality and the wide limits within which it is absolutely or approximately valid, to be considered fundamental for psychic measurement. There are, however, limits to its validity as well as complications, which we shall have carefully to examine later. Yet even where this law ceases to be valid or absolute, the principle of psychic measurement continues to hold, inasmuch as any other relation between constant increments of sensation and variable increments of stimulus, even though it is arrived at empirically and expressed by an empirical formula, may serve equally well as the fundamental basis for psychic measurement, and indeed must serve as such in those parts of the stimulus scale where Weber's law loses its validity. In fact such a law, as well as Weber's law, will furnish a differential formula from which may be derived an integral formula containing an expression for the measurement of sensation.

This is a fundamental point of view, in which Weber's law, with its limitations, appears, not as limiting the application of psychic measurement, but as restricted in its own application toward that end and beyond which application the general

From G. F. Fechner's Elemente der Psychophysik, Leipzig, i860; unverand. Aufl. 1889.

principle of psychic measurement nevertheless continues to hold. It is not that the principle depends for its validity upon Weber's law, but merely that the application of the law is involved in the principle.

Accordingly investigation in the interest of the greatest possible generalization of psychic measurement has not essentially to commence with the greatest possible generalization of Weber's law, which might easily produce the questionable inclination to generalize the law beyond its natural limitation, or which might call forth the objection that the law was generalized beyond these limits solely in the interest of psychic measurement; but rather it may quite freely be asked how far Weber's law is applicable, and how far not; for the three methods which are used in psychic measurement are applicable even when Weber's law is not, and where these methods are applicable psychic measurement is possible.

In short, Weber's law forms merely the basis for the most. numerous and important applications of psychic measurement, but not the universal and essential one. The most general and more fundamental basis for psychic measurement is rather those methods by which the relation between stimulus increments and sensation increment in general is determined, within, as well as without, the limits of Weber's law; and the development of these methods towards ever greater precision and perfection is the most important consideration in regard to psychic measurement.

And yet a great advantage would be lost, if so simple a law as Weber's law could not be used as an exact or at least sufficiently approximate basis for psychic measurement; just such an advantage as would be lost if we could not use the Kepler law in astronomy, or the laws of simple refraction in the theory of the dioptric instruments. Now there is just the same difficulty with these laws as with Weber's law. In the case of Kepler's law we abstract from deviations. In the case of simple lens refraction we abstract from optical aberration. In fact they may become invalid as soon as the simple hypotheses for which they are true no longer exist. Yet they will always re

main decisive for the principle relation with which astronomy and dioptrics are concerned. Weber's law may in like manner, entirely lose its validity, as soon as the average or normal conditions under which the stimulus produces the sensation are unrealized. It will always, however, be decisive for these particular conditions.

Further, just as in physics and astronomy, so can we also in psychic measurement, neglect at first the irregularities and small departures from the law in order to discover and examine the principle relations with which the science has to do. The existence of these exceptions must not, however, be forgotten, inasmuch as the finer development and further progress of the science depends upon the determination and calculation of them, as soon as the possibility of doing so is given.

The determination of psychic measurement is a matter for outer psychophysics and its first applications lie within its boundary; its further applications and consequences, however, extend necessarily into the domain of inner psychophysics and its deeper meaning lies there. It must be remembered that the stimulus does not cause sensation directly, but rather through the assistance of bodily processes with which it stands in more direct connection. The dependence, quantitatively considered of sensation on stimulus, must finally be translated into one of sensation on the bodily processes which directly underlie the sensation in short the psycho-physical processes; and the sensation, instead of being measured by the amount of the stimulus, will be measured by the intensity of these processes. In order to do this, the relation of the inner process to the stimulus must be known. Inasmuch as this is not a matter of direct experience it must be deduced by some exact method. Indeed it is possible for this entire investigation to proceed along exact lines, and it cannot fail at some time or other to obtain the success of a critical study, if one has not already reached that goal.

Although Weber's law, as applied to the relation of stimulus to sensation, shows only a limited validity in the domain of outer psychophysics, it has, as applied to the relation of sensa

tion to kinetic energy, or as referred to some other function of the psycho-physical process, in all probability an unlimited validity in the domain of inner psychophysics, in that all exceptions to the law which we find in the arousal of sensation by external stimulus, are probably due to the fact that the stimulus only under normal or average conditions engenders a kinetic energy in those inner processes proportional to its own amount. From this it may be foreseen, that this law, after it has been restated as a relation between sensation and the psychophysical processes, will be as important, general, and fundamental for the relations of mind to body, as is the law of gravity for the field of planetary motion. And it also has that simplicity which we are accustomed to find in fundamental laws of nature.

Although, then, psychic measurement depends upon Weber's law only within certain limitations in the domain of outer psycho-physics, it may well get its unconditional support from this law in the field of inner psychophysics. These are nevertheless for the present merely opinions and expectations, the verification of which lies in the future.

XIV. THE FUNDAMENTAL FORMULA AND THE MEASUREMENT FORMULA

Although not as yet having a measurement for sensation, still one can combine in an exact formula the relation expressed in Weber's law, --that the sensation difference remains constant when the relative stimulus difference remains constant, with the law, established by the mathematical auxiliary principle, that small sensation increments are proportional to stimulus increments. Let us suppose, as has generally been done in the attempts to preserve Weber's law, that the difference between two stimuli, or, what is the same, the increase in one stimulus, is very small in proportion to the stimulus itself. Let the stimulus which is increased be called B, the small increase dB, where the letter d is to be considered not as a special magnitude, but simply as a sign that dẞ is the small increment of

B. This already suggests the differential sign. The relative

dß
B'

stimulus increase therefore is On the other hand, let the

sensation which is dependent upon the stimulus ẞ be called y, and let the small increment of the sensation which results from the increase of the stimulus by dß be called dy, where d again simply expresses the small increment. The terms dß and dy are each to be considered as referring to an arbitrary unit of their own nature.

According to the empirical Weber's law, dy remains constant dB when remains constant, no matter what absolute values В dB and B take; and according to the a priori mathematical auxiliary principle the changes dy and dß remain proportional to one another so long as they remain very small. The two relations may be expressed together in the following equation:

where x is a constant (dependent upon the units selected for y and B). In fact, if one multiplies Bd and ẞ by any number, so long as it is the same number for both, the proportion remains constant, and with it also the sensation difference dy. This is Weber's law. If one doubles or triples the value of the variation dB without changing the initial value ß, then the value of the change dy is also doubled or tripled. This is the mathematical кав В

principle. The equation dy= therefore entirely satisfies both Weber's law and this principle; and no other equation satisfies both together. This is to be called the fundamental formula, in that the deduction of all consequent formulas will be based upon it.

The fundamental formula does not presuppose the measurement of sensation, nor does it establish any; it simply expresses the relation holding between small relative stimulus increments and sensation increments. In short, it is nothing more than Weber's law and the mathematical auxiliary principle united and expressed in mathematical symbols.