regular intensive gradation. The world in which an organism possessed of such an eye would live might then be very light, light, moderately light, hardly light, or not light: the sensations of light would range between a maximal and a zero intensity, precisely as our own sensations of noise range between maximal loudness and silence, — that is, no noise at all. The organ- ism would not distinguish, as we do, between light and dark, but would simply experience varying degrees of light. In the absence of an adequate light »timulus U would not, a.s we doa see dark, but would see nothing. Cf. § 15.
§ 63. Mental Measurement. — The psychological problem of the intensity of sensation is bound up, historically, with a much wider question, the question of the possibility of mental measurement. Every science tries to state its facts and to formulate its laws as precisely as possible, that is, in quantitative terms, in measured amounts. Thus, it is not enough to say that gravitation is a force which the earth exerts upon every particle of matter; it is not even enough to say that the force is proportional to the mass of the material body, but independent of the particular kind of matter of which it is composed: the physicist goes further, and measures the force of gravitation in terms of accelera- tion. Physics and chemistry are, indeed, from end to end, quantitative or measuring sciences. But biology, too, tries to measure: the modern biologist measures the range of variation shown by the members of a species, gives numer- ical expression to the factors that determine heredity, and so on. And psychology has to face the same issue. There are facts of mind and there are laws of mind: can, then, these facts and laws be quantified? can we measure mind? Now the question of the possibility of mental measurement has been chiefly discussed, as was said above, with reference to the intensity of sensation. Here, therefore, is the fitting place to take it up. We shall ask what measurement means; we shall ask in what sense, and to what extent, measurement may be applied in psychology; and we shall draw our illustrations from the study of sense-intensity, as denned in the preceding § 62. — Whenever we measure, in any department of science, we compare a given magnitude with some conventional unit of the same kind, and determine how many times the unit is contained in the magnitude. If we say, for instance, that 208 The Intensity of Sensation a certain line is 5 cm. long, we mean that we have compared the line with the conventional unit of length, 1 cm., and have found that it contains this unit five times over. All measurement thus implies three given terms: the two limit- ing points of the magnitude to be measured (beginning and end, top and bottom, extreme right and extreme left, zero and maximum), and a third point lying at unit-distance from the one or the other limiting point.
The intensities of sensation lie, as we have said, along a straight line which extends from a zero-point to a point of maximal strength. Here is a magnitude with limiting points. In order to measure sense-intensity, — the intensity of sensations of light or tone or noise, of pressure or taste or smell, — we have, first, to establish these two points, defi- nitely and accurately, and secondly to determine the unit of intensive measurement, the standard subdivision of the total line.
It is important to realise that the unit of measurement is always a conventional unit; its choice is simply a matter of practical con- venience. Scientific men are now generally agreed that the unit of physical space shall be the cm., the unit of time the sec, and the unit of mass the gr. There is, however, nothing absolute about these units. The metric system makes calculation easy, relates the three fundamental quantities in a very simple manner; but that is its sole, as it is its sufficient, claim to acceptance.
And just as the unit of measurement is conventional, and the pace or the span, the ounce or the pound, will give us perfectly valid measures of space and mass, so also may our selection of the magnitude to be measured be arbitrary or conventional. The ordinary postal balance weighs up to 16 oz.; the ordinary kitchen scales weigh up to 4 lb. We can measure — we can express by the number of contained units from zero to maximum — any mag- nitude that chance throws in our way.
But this means that our experiments with the cups of sugared m water (§62) was a quantitative experiment, a measurement of mind. We had, as our arbitrarily chosen magnitude, the sense-distance from the moderately sweet a to the syrupy b. Then we bisected that distance, by finding the sweet c that lay midway for sensation between a and b. The half-distance was our arbitrary unit; and we can write, in terms of it, ab = 2 ac = 2 cb, just as with a unit of 1 ft. we can say that the regular carpenter's rule contains 2 ft. The three points by help of which we have measured, the two limiting points a, b upon the intensive line of sweets and the point c which lies at unit-distance from the lower point a, can be estab- lished, for comparative purposes, by a statement of the relative amounts of sugar and water in the three solutions.
We may take another illustration of the same thing. Let the horizontal line in Fig. 25 represent the complete scale of sensation intensity in the sphere of noise. And suppose that we have given the two noises m and o, a weaker and a stronger noise due, perhaps, to the falling of two ivory balls from different heights upon ebony plates. By selecting from a number of intermediate noises, we may deter- mine the noise n that lies midway for sensation between in and o. We may then write mo = 2 mn = 2 no. That estab- lished, we can take the distance no as given, and can compare 110 with distances above 0, until we finally reach a point p such that no = op. We may then write nip = 3 mn. Again, we can take 11111 as given, and compare it with distances below m, until we reach the point / at which Im = mn. We may then write Ip — 4 Im. And we can evidently go on to determine q, r,...and k,j,...in the same manner. So that, if we continue the procedure as far as possible towards the limits of the horizontal line of the Fig., we shall finally have measured the entire range of noise-intensities in terms of an arbitrary unit. Between the limits of the faintest and the loudest noise there will be so-and-so many steps, or distances, of the unit-magnitude mn. This is measurement of mind. — In neither of the above instances, however, has the measurement p P 210 The Intensity of Sensation been methodical. We took any sweet-distance ab, and any noise- distance mo. It would have been more methodical to determine, at the outset, the two end-points of the total line of sweets and noises, to determine what are called the liminal and the terminal intensities of sweet-sensation and noise-sensation. Moreover, we took as our units of measurement — and —. Now we do not know, in the first place, whether or not these units fit the scale, whether they will divide it up without remainder. And, in the second place, we have no reason to think that other psychologists will adopt them: they do not recommend themselves, in any way, for general use, as comparable with the c.g.s. units of the physical sciences.
§ 64. Liminal and Terminal Stimuli. — The sense-organ, like any other mechanism, has a certain inertia, offers a certain amount of frictional resistance to stimulus; and has also a definite capacity, transmits so much energy and no more. Hence in all of the sense-departments there are stimuli that are too weak to be sensed, and in all there comes a point beyond which we cannot increase the inten- sity of sensation by any further increase of stimulus, but get the same response, over and over again, until the organ breaks down.
Instances of subliminal stimuli are not far to seek. Some lights are too faint to be seen: there are stars that, even on the darkest night, remain invisible to the naked eye. Some sounds are too weak to be heard: we know that the clock in the tower is ticking, because we see the hands move; but we have to climb the stairs to hear it. Some pressures are too slight to be felt: we have no knowledge, from the skin, of the flake of cigar-ash that has fallen upon our hand: and so on.
Maximal stimuli fall less commonly within the range of our ex- perience. It is, however, easy to assure oneself that there is a point beyond which sugar cannot further sweeten or quinine make more bitter, and that a continued increase of pressure, after it has carried § 64. Liminal and Terminal Stimuli 211 the sensation of pressure to a certain height, is felt not as pressure but as pain. Dazzling lights and deafening noises set a like limit to the functional capacity of eye and ear.
The magnitude of stimulus which evokes the sensation at the lower end of the intensive scale, the first term of an intensive series, is known technically as the liminal stimu- lus. It may be determined as the stimulus which gives a positive result, evokes a sensation, in one-half of a long series of observations, while in the remaining one-half the result is negative or doubtful. This value, so the mathe- maticians tell us, is as nearly as possible identical with the magnitude of stimulus which, if all sources of error were completely eliminated, would call forth a barely perceptible sensation. Since, however, the liminal stimulus is not a constant but a variable, it cannot in strictness be repre- sented by any single value, not even by the most probable value; its formula must always be written x ±y, where x is the most probable value of the stimulus and ±y in- dicates its range of variation.
The magnitude of stimulus which evokes the sensation at the upper end of the intensive scale, the last term of an intensive series, is known technically as the terminal stim- ulus. Theoretically, it may be determined in the same way; in practice it is rarely approached, out of regard for the integrity of the organ.
A variable quantity is a quantity which varies with change of the conditions under which an observation is made. Thus, a measure- ment in physics may vary with temperature, with humidity, with stress, as well as with the delicacy of graduation of the measuring instrument. The experimenter seeks, so far as possible, to keep all the conditions constant while a measurement is in progress; but even so there will be a slight range of variation. And the 212 The Intensity of Sensation result is always stated in a qualified way, with reference to the conditions.1 The liminal stimulus is, in this sense, a variable; and its varia- tion is due partly to the sense-organ and partly to the brain. When we are tired, for instance, our sense-organs are dulled and our general disposition is unfavourable to close work; the liminal stimulus is, accordingly, much larger than it is when we are fresh. But even under the best conditions there is fluctuation. The or- ganic mechanism, made up of sense-organ and brain, is extraor- dinarily complicated, and complex machinery gets out of order more easily than simple. Besides, the organic mechanism is plastic, not rigid; it is influenced by all sorts of things, — directly by nutritive factors, indirectly by the state of all the rest of the organism. The wonder is, indeed, not that the liminal stimulus should be a variable quantity, but that it should be so nearly constant, for normal persons, as in fact it turns out to be.
The exact determination of the liminal stimulus, that is, of the amount of mechanical energy required to arouse a sense-organ to response, is a very delicate and difficult matter, and a knowledge of methods and results is of interest only to the special student. It must suffice here to say that, for most of the organs, the measure- ment has been made.2 In ordinary laboratory practice it is enough to take a rough determination in empirical terms. Thus, 1 If there is anything constant in the world, it would seem to be the length of the standard metre, which is the unit of reference for all linear measure- ment in physics. Yet we are told that " from the result of many years of comparison at the Bureau International [in Paris], the conclusion is reached that the length of a standard can be absolutely guaranteed to an exactitude of about 0.2 micron at all usual temperatures" (\V. Hallock and II. T. Wade, Outlines of the Evolution of Weights and Measures and the Metric System, 1906, 256). In this statement, the measurement is qualified, first by reference to temperature, and secondly by a statement of the range of variability; a micron is a millionth of a metre. A measure that is correct to one five- millionth is, to all intents and purposes, a constant; it would be to all intents and purposes a constant if the variation were far greater. In strictness, however, it is a variable.
2 See, for instance, S. P. Langley, Energy and Vision, Philosophical Maga- zine, xxvii., 1S89, 1.
§6$. Just Noticeable Difference as Unit of 'Measurement 213 a couple of hours of methodical work will settle the question from what height a leaden shot of a given weight must fall upon a glass FlG. 26. Acoumeter for determining the Stimulus Limen of Noise. SS, set- sorews (two of the three are shown), supporting a wooden platform. M, mi- crometer screw at the centre of the platform, with scale beside it. FF, spring for- ceps, lying on the head of M, and carrying a small shot for dropping on the glass plate above W. The shot rebounds, and falls noiselessly into a padded trough.
plate if the noise is to be just audible to an observer seated 10 m. away. Results of this kind are useful, as means of comparison, but have no general scientific value.
§ 65. The Just Noticeable Difference as the Unit of Meas- urement.— We denned the liminal stimulus, or the just noticeable stimulus, as that magnitude of stimulus which evokes a sensation in one-half of a long series of obser- vations, while in the remaining one-half the result is nega- tive or doubtful. If, now, we take a second stimulus of the same magnitude, and gradually increase its intensity by very small amounts; and if, at every step of this progression, we compare the sensations set up by the two stimuli: then we shall presently arrive at a stimulus- difference which behaves in the same way as the liminal 214 The Intensity of Sensation stimulus itself. We shall, that is, come to a difference which is perceived as a difference in one-half of a long series of observations, while in the remaining one-half there is no perceptible difference, or the observer is in doubt. And the procedure may be repeated, again and again, until we have traversed the whole or a large part of the intensive scale.
It has been suggested that this difference, which is known as the just noticeable difference of stimuli, or as the differential limen of sensation, may be regarded as the natural unit of the scale of sensible intensity. The zero- point of the scale is given with the liminal stimulus, or (as it is called, with reference to sensation) the stimulus limen. The end-point of the scale is given with the terminal stimulus. The units of the scale will then be given with the series of just noticeable differences as defined above. For, it is said, the just noticeable differences cor- respond to least distances upon the sense-scale, minima of sensible distance. Now least distances, being the smallest possible distances at which sensations can be distinguished, are necessarily equal distances; and equal distances are the very things that we are in search of, to furnish the subdivisions of our mental scale.
Logically, however, this argument is not sound. It is by no means self-evident that least steps, at various parts of the sense-scale, should also be equal steps. A given difference between sensations might be the least per- ceptible difference, and yet, as compared with another least perceptible difference from another part of the scale, might be larger or smaller. The equality of just notice- able differences must, then, be proved; it cannot be as- sumed. The appeal lies to the results of experiment.
We shall see in the next § 66 that the results of experiment are ambiguous. Nevertheless, the preponderance of the experi- mental evidence is, in the writer's judgment, very definitely in favour of the equality of the just noticeable differences; the dis- crepant results can be accounted for in terms of known sources of error. We return to the point later.
In the meantime: Why, it may be asked, should we not appeal to introspection? Why should we not directly compare two just noticeable differences, from different regions of the intensive scale, and see if they are alike or different? — For the simple reason that they are the results of measurement. If mere ob- servation were enough, we should not need to measure at all, in any field of science. If we could estimate the sixtieths of a circle, it would not be necessary to mark off the clock-face into minutes; if we could estimate spaces of so many feet, it would not be necessary to secure architects' plans before we built a house. The just noticeable difference is not determined by a single introspective observation, and cannot be carried in the head as a standard of magnitude: it is the calculated result of a long series of introspective observations, and stands for a most probable or representative value. The whole object of measure- ment is to carry accuracy into fields in which mere observation, simple estimation, is inaccurate.
§ 66. Weber's Law. — If we determine a series of just noticeable differences, in the middle region of the inten- sive scale, we find a very simple relation between change of sensation and increase of stimulus. At the beginning, where the stimuli are relatively weak, only a small addition is required to effect a noticeable increase in the intensity of sensation; as the series progresses, the additions become larger and larger; and towards the end, where the stimuli are relatively strong, the largest additions are needed. And this progressive increase of the stimulus-increment is uniform: so that, in general, the series of least sense- 216 The Intensity of Sensation distances corresponds to a series of stimulus-increments that are, approximately, equal fractions of the original stimulus. Thus, if we start with the stimulus 10, and find a just noticeable difference with the stimulus n, then when we come to 20 we shall find a difference with 22, when we come to 30 we shall find it with 33, and so on.
If, therefore, we could regard all just noticeable differ- ences as equal, — all least sense-distances as equal sense- distances, — we could sum up the results of our experiment by saying that an arithmetical series of sense-distances corresponds to a geometrical series of stimulus-values. We should have, on the side of sensation, a series of in- tensities o1, 1, 2, 3, 4,..., lying at points equidistant upon the intensive scale; and we should have on the side of stimulus a progression of the order R, R{\ + r), R{\ +?')2, R(i+r)s,..., where R is the first stimulus taken (here the stimulus 10, correlated with the sensible intensity o), and r is a certain fractional part of R (here, one-tenth).
We can put the question to the test of experiment. Let us take, for instance, the case of sensations of light. We know, from many investigations, that a succession of just noticeable differences of light-sensation is paralleled by a geometrical series of physical light-stimuli. Now we have recourse to larger, supraliminal differences of light-sensa- tion. We set up on the colour-mixer (Fig. 4) three com- pound discs of black and white paper. The two outer discs are adjusted to give on rotation a dark and a light 1 This o is not, of course, the zero-point of intensity of sensation at large: it is only the zero-point of our arbitrarily selected scale, and therefore stands for the intensity of sensation with which the experiments begin. — If we are measuring a table with a foot-rule, we begin with o, in just the same way; but we do not mean that space at large begins where our rule begins.
grey respectively; they remain constant throughout the experiment. The proportion of black and white in the middle disc is varied, until a grey is obtained which lies, for sensation, midway between the extremes. Our three rotating discs then show two equal sense-distances, of much more than liminal extent. What of the stimuli? The stimuli, measured by means of the photometer, prove to form a geometrical series; their photometric values differ, not by equal amounts, but by relatively equal amounts.
Here, however, is the answer to our question. Here we have an arithmetical series of sense-distances, two succes- sive distances guaranteed equal by introspection, corre- sponding to a geometrical series of light-stimuli. Since, then, we found a geometrical series of stimuli correspond- ing to our series of just noticeable differences of sensa- tion, it follows that these just noticeable differences must themselves be psychologically equal. The just noticeable difference may be accepted as the unit of the intensive scale.
A strictly methodical procedure is as necessary in this case as it was in the case of the just noticeable difference: we do not dis- cover the equality of the two sense-distances by direct introspec- tion, but we calculate the most probable point of equality from a long series of introspective observations. The difference in the two experiments is this: that in the determination of the just noticeable difference the observer reports the likeness or differ- ence of two sensations, whereas in the present experiment he reports the likeness or difference of two sense-distances. Hence we have, with supraliminal differences, an introspective control that is lacking for the liminal.
At the same time, the change from comparison of sensations to comparison of sense-distances may have a decided influence upon the observer's judgment. We said above (§ 65) that the result* 218 The Intensity of Sensation of experiment were ambiguous. As a matter of fact, several recent investigations have led to the result that equal supraliminal dis- tances, determined in the way just described, do not contain — as on our view they should — equal numbers of just noticeable differences, but that, on the contrary, the higher of the two con- tains fewer just noticeable differences than the lower.1 And this result has been interpreted to mean that the just noticeable differ- ence is a magnitude that increases with increase of stimulus, so that it cannot serve as the unit of measurement. However, another interpretation is possible. The upper distance, which contains the fewer just noticeable differences, may in reality be shorter than the lower; the observer's judgment that the two dis- tances are equal may be erroneous. For one very dangerous source of error, in experiments upon the comparison of supra- liminal sense-distances, is that the observer tends to judge, not in terms of sensation, but in terms of stimulus. He thinks, not of the light-sensations, but of the grey papers; not of the sounds heard, but of the heights from which the balls must have fallen to give those sounds (§ 62). If this error, which is known techni- cally as the stimulus-error, creeps into the observations, then the stimuli which delimit the two sense-distances are likely to form, not a geometrical, but an arithmetical series. The consequence is plain. The upper distance must now contain fewer just notice- able differences than the lower; it is not psychologically, but only physically, equal to the lower. The observer, who was called upon to space out intensities of sensation, has really spaced out, in the light of his everyday experience, characters or properties of material things; and his spacing has, naturally, led to an approximate physical equality. — This, in general, is the writer's explanation of the discrepancies in the experimental results.
The law that equal sense-distances correspond to rela- tively equal differences of stimulus is known as Weber's Law. It has been found to hold, at least approximately 1 See, e.g., W. Ament, Uebtr das Verhaltnis der ebenmerklichen zu den ubermerklichen Unterschieden bei Licht- und Sckallintensilaten, in Wundt's Philosophische Studien, xvi., 1900, 135.
and within a certain middle region of the intensive scale, for intensities of noise and tone, of light, of pressure, of various kinaesthetic com- plexes (lifted weights, move- ments of the arm, movements of the eyes), and of smell. Its validity in the fields of taste and of temperature is doubtful. It may possibly hold for affection (§ 73), as well as for sensation; but no experimental test in the sphere of feeling has as yet been made.
Fig. 27. Generalised representation of the relation between 6" and R formu- lated in Weber's Law. Equal sense- steps are marked off as abscissas, and the corresponding ^-values are entered as ordinates.
In 1834 the German physioloperformed some experiments with weights and visual distances which seemed to establish a constancy of the relative differ- ential limen. He accord- ingly concluded that " what we perceive, when we are discriminating between ob- jects, is not their absolute difference, but rather the proportion which the differ- ence bears to their magni- tude." G.T. Fechner(i8oi- 18S7) gave the law a precise phrasing, and put it to elabo- rate experimental test. Al- though Fechner's modesty led him to name it after FlG. 28. Pair of black-and-white discs for the demonstration of Weber's Law. The brightness of the left-hand disc increases, from the centre towards the periphery, in geometrical progression; that of the right- hand disc increases in arithmetical pro- gression.— A. Kirschmann, American Journal of Psychology, vii., 1896, 386 ff.; E. C. Sanford, A Course in Experimental Psychology, 1898, 335 f.
220 The Intensity of Sensation Weber, we might more correctly term it Fechner's Law or the Weber-Fechner Law.
Fechner formulated the law in the equation S=c log R, where £ stands for intensity of sensation, R for stimulus, and c for a constant factor. Fechner's understanding of the formula was wrong; he fell into the very common error which we discussed in § 62. The formula itself, however, may be retained. Here is its derivation in terms of supraliminal sense-distances.
We know, from our experimental results, that the magnitude of a sense-distance is dependent upon the quotient of the two R that limit it. Let the dependence be expressed by the mathematical sign of function, /. Then we have, for two successive sense- distances, the equations: 5W(f Adding these equations, we get R But we know, again from our experiments, that: and, of course, Rx We have, then, finally Now the only continuous function that can satisfy an equation of this form is, as we learn from the mathematical text-books, a logarithmic function. Hence we may write (inserting a constant factor, c, to indicate our choice of some particular logarithmic system) ■ § 6y. Theory of Weber s Law 221 S2S3 = c log Or, in general, if 6"0 and R0 denote the S and R with which we start, and 5 and R themselves denote any other sensation and its corresponding stimulus-value: R Ro And, lastly, if we denote the intensive ^-distances reckoned from an initial S0 by S, and the ^-intensities calculated in terms of the corresponding R0 by R, we have simply: The formula may also be derived, though not without help from the calculus, in terms of liminal sense-distances or just noticeable differences.