1 LIBRARY OF THE University of California. Class PSYCH. LIBRARY A EXPERIMENTAL STUDIES IN PSYCHOLOGY AND PEDAGOGY Editor: LIGHTNER WITAIER UNIVERSITY OF PENNSYLVANIA III. THE APPLICATION OF STATISTICAL METHODS TO THE PROBLEMS OF PSYCHOPHYSICS THE APPIJCATION OF STATISTICAL METHODS TO THE PROBLEMS OF PSYCHOPHYSICS BY F. M. URBAN, Ph.D. HARRISON FELLOW FOR RESEARCH University of Pennsylvania FROM THE LABORATORY OF PSYCHOLOGY PHILADELPHIA, PA. THE PSYCHOLOGICAL CLINIC PRESS 1908. -'< f», ^ TOWN PRINTING CO. PHILADELPHIA Br£3 7 PSYCH. LIBRARY PREFACE The following study deals with the methods which serve for the determination of the threshold of difference. These methods can be easily adapted to the determination of the maximum and minimum stimulation, so that it seemed justified to refer in the title of the book to the problems of psychophysics in general. These problems are treated by such propositions of, the calculus of probabilities as apply to the results of observations of statistical numbers of relative frequency. The term "statistical method" has two slightly different meanings. The first refers to the method of collecting observations on a group of individuals or on one indi- vidual at different times. The data obtained in this way must be subjected to some kind of treatment and the term "statistical method" refers in its second meaning to every algorithm for the numerical evaluation of these data or to every definite method l^y which conclusions can be drawn from them. One may say in gen- eral that a problem is treated statistically, if the data are collected with a view to determining certain numbers of frequency which serve as a basis for further deductions. Some of the ideas expounded in this book date back a consid- erable time. The considerations of the last chapter originated in the course of a study of some physical phenomena and they were fol- lowed up later on in connection with astudy of thelawof Gomperz- Makeham. Most -of the formulae of the third chapter have been known to me for several years, and the idea of a purely formal treatment of the results of psychological experiments came to me in the course of the experiments of Mr. Kobilecki, which were reported lately in the "Psychologische Studien". These small ideas, however, would have remained undeveloped for a long time, had not Prof. Witmer brought to my knowledge an experimental arrangement which could be adapted to the purposes of this in- vestigation. It is so extremely easy to have brilliant ideas in psychology that theoretical discussions justly meet with a certain v 180105 VI PREFACE distrust, if they are not supported by adequate experimental ma- terial. The impossibility of deciding on an experimental pro- cedure which would yield suitable material for the test of the theo- retical deductions was the chief reason for not publishing this an- alysis of the psychophysical methods before. This delay of more than five years proved to be a very fortunate circumstance, not only because the growth of such ideas is very slow, but also be- cause the problems of psychophysics have been the subject of several important publications in the last years. Wundt, Mliller, Lipps and Titchener, have treated of the psychophysical methods in the last few years; especially by the work of Titchener is the literature on this problem opened up as it was never before. A similar great help for the analysis of the psychophysical methods was found in Czuber's work on the calculus of probabilities and in those parts of the "Encyklopadie der Mathematischen Wissen- schaften" which deal with the calculus of probabilities and its application to statistics. Comparatively little reference to litera- ture is made in this book. Constant reference to work previ- ously done was out of the question, because it would have made the book too voluminous, and occasional quotations are of little or no use. Add to this that the present investigation has not the same starting point as the treatises of the previous authors, and that, therefore, it would have been necessary to argue against views which in themselves are interesting and of merit. This would have given an entirel}^ erroneous impression of my opinion of the value of these views, and hence it seemed best to describe in separate papers the different phases of the development of the psychophysical methods. In so far as the English literature is concerned this presentation will be given in one of the reports on "Die Psychologie in Amerika", which appear from time to time in the Archiv far die gesammte Psi/chologie. In working out the numerical results it was made a rule to re- peat the computation independently whenever it was not possible to apply a thoroughgoing check. The course of the calculations is given in great detail. This was done in order to illustrate the theoretical deductions by numerical examples, and to show that the actual application of the methods described here is simple and that it is shorter than the methods used at present. This PREFACE VII remark refers especially to the method of just perceptible differ- ences, which requires four times as many experiments as the new method to give the result with the same degree o.' accuracy. Great care was taken to present all the deductions in as simple a form as possible. Mr. Titchener's work has set a standard for such treatment, and it may be hoped that the following consid- erations will be intelligible to everyone who has gone through the "Manual". This does not mean t4iat all the theorems used are spoken of in this work, but that they are such that they might be understood by ever}' one who can read Mr. Titchener's book. This rule of avoiding complicated deductions made necessary the cur- tailing of a demonstration in the third chapter at the place where reference is made to Bruns's theorem of the conservation of the 0(^)-type. The complete solution of the problem would have required a long demonstration of a very technical character, in which some of the more complicated functions are used. It therefore seemed best to give it at some other place. TABLE OF CONTENTS. PAGE. Ch.^pter I. Description of the Experiments i Chapter II. The Statistical Numbers of Relative Frequency 19 Chapter III. On The Method of Just Perceptible Dif- ferences 40 Chapter IV. The Equality Cases 99 Chapter V. The Psychometric Functions 106 Chapter VI. A General Inquiry Concerning the Psy- chometric Functions 139 PROBLEMS OF PSYGHOPEYSrCS CHAPTER I. DESCRIPTION OF THE EXPERIMENTS. The results of a series of experiments on lifted weights are the basis of our discussion. The weights were hollow brass cylin- ders 7.5 cm. in diameter and 3 cm. in height, closed at one end. The weight of the metal of the cylinders as they came from the shop was slightly less than the smallest weight to be used: by pouring different quantities of melted paraffine into the cylin- ders sufficient to make each cylinder a little heavier than de- sired and then scraping out small quantities of the hardened paraffine, the weight could be adjusted very exactly. The ex- actitude of this adjustment finds a limit only in the exactitude of the balance used. The variations which occur in weights prepared in this way are inconsiderable; they were observed regularly and Table 1 (Appendix p. 173) gives the variations which occurred in winter and spring 1907. The first column of this table gives the number by which each weight could be identified. Since the size and the external appearance of all the weights was the same, it was necessary to distinguish them in some way; this was accomplished by stamping small numbers in the centre of the upper side of every cylinder. These num- bers could be seen only on close inspection. The second column of Table 1 gives the weight in grams to which the c^dinders were adjusted before the experiments were started; this adjustment as well as the other observations was made with a balance which showed differences of 1 mgr. The differences between the observed weight and the adjusted weight are given in milligrams for every day of control, the dates of which are given at the heads of the columns. The plus sign indicates an 1 2 miOBLEMS OF PSYCHOPHVSICS increase of weight, the minus sign a loss of weight. Weights in which a positive or negative difference of more than 10 mgr. from the standard was observed, were readjusted; this, however, happened only three times in the period of three months for which the results are reported. The last column of the table gives the sum of all the variations, understanding by variations the positive or negative difference which the weight has under- gone between two observations. The cylinder No. 3, for instance, was adjusted for the weight of 100 gr. and the observed differ- ences from this standard are given in Table 1 as -3, 3, -5 mgr. for the first three observations. This means that on these days the weight of this cylinder was found to be 99.997, 100.003 and 99.995 gr. We have, therefore, an increase of 6 mgr. between the first and the second observation, and a decrease of 8 mgr. between the second and the third. The sum of all the varia- tions does not exceed 31 mgr. for anyone of the weights and on the average it is less than 17 mgr. A glance at the table shows that the deviations from the adjusted weight are negative in the majority of cases, but it is worth while noticing that the variations between the single observations were negative in 46 eases and positive in 45 cases, whereas in 23 cases no variations could be detected. The original adjustment of the weights was made in March, 1906. It seems that the cylinders suffer a loss of weight in the beginning, but that they undergo little variation afterwards. The smallness of these variations is chiefly due to the use of anhygroscopic materials, as may be seen by a comparison with the variations of weights which are made of materials more susceptible to hygroscopic influences. For the sake of this comparison Table 2 (Appendix p. 173) is given which contains similar observations for four weights of two patterns. The weights which are called in this table I and II are Cattell weights and belong to the set of weights which were used by Fullerton and Cattell in the series of experi- ments which are discussed in the publication of these authors.* Weights III and IV are solid wooden blocks filled with lead for adjustment. The table shows the variations of these four *FuLLERTON and Cattell, The Perception of Smill Difjcrences, 1S9-', pp. 116-129. DESCRIPTION OF THE KXPEHIMEXTS 3 weiiihts in the period from April 26 till May 10; the upper part of the table gives the results of the observations in grams and the lower part of the table gives the variations between two observations. It may be seen from the table that the varia- tions of the wooden blocks are by far the largest; they are two or three times as large as those of the Cattell weights. These weights are not fit for experimentation which aims at any ac- curacy not only because the variations are ver}' large, but also because it is difficult to correct a negative variation. The vari- ations of the two Cattell weights are considerably smaller, but the variations of one weight in three weeks are almost twice as great as the variations of all our IS weights in the course of a little less than three months. It need not be remarked that all the weights were kept under the same conditions, the disad- vantage being on the side of our weights which were exposed to rough handling in the course of the experimentation. These cylinders were not only exactly alike to sight but also, what is perhaps more important, to touch.''' The metallic sur- face of the weights was polished so that it gave the same impres- sion of smoothness no matter where it was touched. Since all the weights, furthermore, were of the same metal and care was taken ta keep them under the same conditions of temperature (by keeping all the weights in the same room and by not touch- ing with the warm hand one weight more frequently than the others) an influence of differences of temperature was avoided. The only quality by which one cylinder differed from another was its weight. These weights were arranged along the circumference of a round turning table at regular intervals, the position of each weight *The sensations of touch are an important component of the sensation of weight, and for this reason it is necessary to keep them constant, if the pur- pose of the experiments is an analysis of the sensations of weight. The im- portance of the sensation of touch for the judgment of the weight of a body was recognized by Fechner, Elemente der Psychophysik, \'ol. I, p. 19V1, and this observation was confirmed lately by A. Lehm.\nn, Beiirdge zur Psychodynamik der Gewichtsempfindungen, Arch. f. d. ges. Psycliologie, Vol. 6, 1906, pp. 446-448, who found that variations of the sensations of touch may cause differences in the estimation of weights, which may be equal to those caused by differences of { of the intensity of the stimulus. 4 PROBLEMS OF PSYCHOPHYSICS being determined by one of the numbers 1-14; a great number of variations in the order of the stimuli may be obtained by interchanging the position of these 14 weights. A general feat- ure of all the arrangements used in our experiments was that weights of 100 gr. were placed at the odd numljers. The weights placed at the even numbers had to be compared with these standard weights. The weights Avere always lifted with the right hand. The right forearm of the subject rested on a firm table in such a position that the hand from the wrist extended over the edge. The subject was sitting at this table which was placed so that the hand of the subject was vertically over one of the weights of the turning table. The turning table with the weights on it was shut off from the view of the subject by means of a screen. The height of the turning talkie could be regulated so that the weights were within easy reach of the sub- ject. The lifting was done entirely from the wrist and no part of the arm above this joint was moved. The height of lifting was not regulated, but the subject was instructed to lift the weights in such a way as seemed best in order to obtain an ac- curate judgment. This was not an equivocal instruction, since every subject was given a large amount of practice before the actual experiments were begun, so that the subject could find the way of lifting which suited him best. The speed of the move- ments of the hand did not seem to vary much after the subject had settled down to a certain way of lifting. Most subjects favored an excursion of the hand of approximately 5-7 cm.; one subject preferred to lift his hand considerably higher, and an- other adopted a very small excursion of 2-3 cm. This is in so far of importance as the time of the movements was kept con- stant and the excursion of the hand gives an indication of the velocity with which the weights were lifted. The height of lifting depends to some extent on the flexibility of the hand at the wrist. The movements of the hand were regulated by a metronome of Malzel, which beat 92 times a minute and every fourth beat of which was marked by the stroke of the bell. On the stroke of the bell the hand of the subject went down and grasped the weight; on the second beat the hand was lifted; on the third DESCRIPTION OF THE EXPERIMENTS 5 beat the weight was put back on the table, and on the fourth beat the hand returned to its original position. After the weight was put l)ack the operator had sufficient time to turn the table and to bring the next weight directly below the hand of the sub- ject. The experiments thus could go on without interruption and were continued for five complete turns of the table, so that in such a series five judgments were given on every pair of com- parison weights. The position of the hand of the subject was the same in lifting all the weights. The judgments referred to the second weight of the pair. The possible judgments are that the second weight is heavier or lighter than the standard, or that Ijoth weights are equal. The sub- ject was allowed to express the degree of his confidence by an- necting one of the numerals 1, 2, or 3 to his judgment. The unit indicated an ordinary degree of confidence and was omitted^ 2 and 3 indicated higher and highest degrees of confidence, but it was found that the subjects rarely went beyond 2. For the so-called doubtful and equality cases the following arrange- ment was made. It is a fact that those cases where the judg- ment is absolutely doubtful diminish in number when the practice of the subject is increased.* These cases alone ought to be called doubtful cases, because in the other class of cases the judgment is "equal" and the judgment is not doubtful at all. Cases where the subject was unable to form a judgment occurred in the preliminary series, but they were rare in the actual inves- tigation and the experiments were repeated whenever this hap- pened. When no difference could be detected the subject was instructed to make a guess whether the second weight was heavier or lighter, but to mark these cases by the words "heavier guess" or "lighter guess." The "guess "-judgments are theoretically^ i. e. by their definition, identical with the equality cases, but practically we met with some difficulties. Most subjects had in the beginning of their training some difficulty in distinguish- ing the "guess "-judgments from judgments which expressed *Cfr. WuNDT, Physiol. Psychologic, 5 ed., 1903, Vol. 1, p. 482; G. E. MvELLEK and Fr. Schumann, Vber die psychologischen Grundlagen der Ver gkichung gchobener Gewichie, Arch..}, d. ges. Physiologic. Vol. 45, 1889, p. 40 6 PKOBLEM.S OF PSYCHOPHYSICS the recognition of an existing difference with a very low degree of certainty. Later on all the subjects but one (subject III) found no difficulty in separating these two classes of judgments. This subject almost never seemed to have the impression of equality of the stimuli and he consistently gave "guess "-judg- ments when he did not feel quite sure; in these cases he mostly gave the judgment "heavier guess". Since it was not possible to eradicate this habit by a very considerable amount of prac- tice given to this subject, it was decided not to interfere with his way of judging. All the judgments given were recorded. For this purpose special record sheets were prepared which contained seven rows of five places each. Every comparison weight was given one line, so that the 35 experiments of running the table around five times could be recorded on one blank. If the comparison weights are written down in the same order in which they are placed on the table, every column of the record gives the exper- iments of one turn of the table and the rows contain the judg- ments given on one comparison weight in the five turns of the table. The following example will make clear this way of 104 92 h h h, h h 5 h hg 1 1 hg 3 108 h^ h h h h 5 88 1 hg 1 1 1 1 96 h h h hg h h 5 100 hg h h h, 5 84 hg 1 1 1 1 1 6 6 4 4 5 keeping the records. The letters h and 1 stand for the judg- ments "heavier" and "lighter." The "guess "-judgments were DESCRIPTION OF THE EXPERIMENTS 7 recorded by hg and Ig, according to whether the judgment "heavier guess" or "lighter guess" was given. The first cohimn of the record gives the comparison weight; it is not necessary to put down the standard weight because it is the same for all the pairs. In the 35 spaces of such a table of five columns all the judgments given during the five turns of the taljle may be recorded by entering everv judgment as it is given in its place running from the top of the columns downward. The numbers in the last column refer to the number of "heavier "-judgments given on this weight, no discrimination being made between "hg" and simple "h "-judgments. The numbers at the bot- tom of the columns do the same for the columns i. e. they give the number of "heavier "-judgments given in one turn of the table without distinguishing between guesses and ordinary judg- ments. This experimental arrancement made it necessary to have three persons present: The subject, the operator of the table and the recorder. The recorder of course became acquainted with the arrangement of the comparison stimuli, and it was found very soon that this knowledge was of disturbing influence, if the recorder had to serve as subject in the series where the same arrangement was used. For this reason nol:)ody was used as recorder in a series before he had finished it as a sui^ject; as a rule the recording was done by the conductor of the experiments, who could not avoid having some knowledge of the order of the comparison weights anyhow, or by an assistant who did not act as a subject. In this way it was tried to keep from the subject all knowledge about the actual relation of the stimuli, and the ex- perimental arrangement fairly excludes the possibility of the subject's unwittingly learning what this relation is. Even the knowledge of the order in which the pairs are presented does not interfere with the proper performance of the experiments, if attention is devoted to the experiments and not to the order of the succession of the judgments. There were some cases — very few in number — where the subject reported that the idea was impressed upon him he knew the order of the pairs. Cases of this type were investigated on the spot and it was found that this impression of the subject was only seldom justified. These 8 PROBLEMS OF PSYCHOPHYSICS cases, as a matter of fact, are perhaps more interesting as ex- amples of memory illusions and of the origin of faulty impres- sions based on incomplete observations. After the series re- produced above, for example, the subject reported having been conscious that there was in the series a succession of weights described by the judgments ho, 1, hg, and the subject offered to pick out these weights. A number of trials was immediately given, but the subject could not make up his mind and finally gave up this task as impossible. Then the record of this series was surveyed, and it was found that a succession of judgments as described by the subject did not occur a single time and that the hj-judgments were given on different weights. This exam- ple points out two facts: 1, that memory is very unreliable for events on which attention is not fixed and 2, that ^he degree of confidence with which a judgment is given does not depend solely on the amount of objective difference, because the degree of confidence with which the judgments on a certain pair of comparison weights are given varies considerably. The latter fact is well illustrated by a few cases where hj and lo-judg- ments were given on the same pair of comparison weights inside the same series. These cases were Subject St., April 2, 1906, I A, No. 8 (h 2 and 1 j on 96) ; Subject W., June 8, 1906, IV A, No. 4 (h ^ and 1 2 on 100) ; Subject B., Oct. 23, 1906, I, No. 2 (h 2 and 1 2 on 108). Great confidence or lack of confidence are no criteria of the reliability of the judgment, and it is not likely that the con- fidence in the judgment is in a simple relation to the difference between the stimuli. There are cases on record where the subject complained of very little or no confidence at all, and where a survey of the results shows that the judgments in this series have fewer mistakes than any other series of the same day. In some cases, on the contrary, this lack of confidence DESCRIPTION OF THE EXPERIMENTS 9 was in so far justified as the results showed a considerable number of erroneous judgments.* There are two points in our experimental procedure which need some explanation; the first is the regulation of the movement of the hand by the metronome, the second is the taking of the experiments in series of 35. The regular sound stimuli may have some influence on the judgment, and the length of one series may tend to produce states of fatigue or variations of attention which make results of different parts of one series not directly com- parable. It seems that only the second possibility requires special attention, l)ecause the use of the metronome for regulating the movements of the hand has been found advantageous and unobjectionable in several previous investigations. It will be shown in the second chapter that there exists no evidence of an ... X appreciable influence of making the experiments in series of 35. There are several ways in which the beats of the metronome might possibly influence the judgments: (l.) Regular acoustical stimuli produce a feeling processf which may be the immediate cause for judgments in a certain di- rection. It is very easy to observe this waving up and down of feelings when listening to the beats of a metronome, but these feelings are by far less noticeable if attention is primarily con- centrated on the performance of the experiments. (2.) The beats of the metronome may influence attention, which is the psychical function affected by the rythmic motion of lift- ing the weights. *rhe first part of this observation contradicts the statement of Fuller- ton and Cattell that with an increase of the difference of the intensities of the stimuli our judgment changes from complete uncertainty to confidence. Judgments on the same amount of objective difi'erence are given with a very variable degree of confidence. Of course it is possible to say that on the aver- age our confidence increases with the difference of the intensities, but taking such an ayerage is not a well defined process. The latter part of the obser- vation agrees with a remark of these investigators {On Ike Perception of Small Difjerences, 1892, p. 126 sqs.) in so far as it shows that accuracy is not necessarily proportional to subjective certainty, an observation to which these authors give the piquant turn that those subjects who are most confi- dent are the least accurate of all. jW. WuNDT, Phys. Psych. 5 ed., 1903, Vol. 3, p. 23. 10 PUOHLEMS OF PSYC'HOPH Y.SICS (3.) The Jicousticul stimuli may interfere with the mechanical contraction of the muscles in such a way as to reinforce or in- hibit the action of the muscles. The observations of Bowditch and Warren/'^ Cleghorn,t HofbauerJ, Yerkes^f and others show that the effect of two stimuli which are applied in not too great an interval is never equal to the sum of the effects of the stimuli, but that they interfere in such a way as to inhibit or reinforce each other. There is little doubt that there is the possibility of such an influence of the regular acoustical stimuli on the contraction of the muscles, although one can not say at present what effect on the judgment of weight such an interfer- ence of stimuli may have. We have more positive information in regard to the second point, which was the object of an experimental investigation of Smith. § For our present purpose, however, it is less im- portant to have an analysis of the factors which decide the judg- ment on the weight of a body, than to arrange the experiments in such a way that the conditions for all the judgments are the same, no experiment being favored by an influence which is not at work in the others too. Similar considerations hold good for the possibility of an in- fluence of the length of a single series. Besides fatigue and shift- ing of attention there is the possibility that the stimulation pro- duced by the first lifting does not die out immediately, but tliat it interferes with the next stimulation, so as to inhibit it or rein- force it. This possibility was considered recently by Lehmann. || This author supposes that such an influence exists and he gives *H. P. Bowditch and J. W. Warren, The Knee-jerk and its Physiologi- cal Modifications, Journ. of Physiology, Vol. 9, p. 60, 1890. fA. CuEGHORN, The Reinforcement of Voluntary Muscular Contractions, Amcr. Journ. of Physiology, Vol. 1, p. 336, 1898. XL- HoFBAUER, Interferenz zwischen verschiedenen Impulsen im Cen- tralnervensystem, Arch. f. d. ges. Physiologie, 1897, Vol. 68, p. 546. "jR. M. YerkeS, Bahnung und Hemmung auf tactile Reize durch akustische Reize beim Frosche, Arch. f. d. ges. Physiologie, Vol. 107, 1905. §Margaret Keiver Smith, Rythmus und Arbeit, Phil. Stud. Vol. 16, 1900, pp. 71-133, 197-305. II A. Lehm.\nn, Beitrdge zur Psychodynamik der Gcwichtsempjindungen, Arch. f. d. ges. Psyclwlogie Vol. 6, 1906, pp. 425-499 DESCRIPTION OF THE EXPERIMENTS 1 1 a mathematical expression for it. An influence of the type Lehmann describes \vill not impair the experiments Init it will go in as a constant factor. Indeed since the interval between two successive stimulations is constant, no matter whether the preceding stimulus inhibits or reinforces the following, this influence will tend to produce a state where the amount of in- hil)ition or reinforcement is the same for all the weights. This influence, therefore, will be distributed equally over all parts of the series of experiments. The first weight in every pair was 100 gr.; it was compared with weights of 84, 88, 92, 96, 100, 104, 108 gr. in those series the results of which are discussed here. It will be noticed that the series of comparison weights is not equally extended above and below the standard. The motive for the choice of this series of comparison weights was that it was observed in the preliminary experiments that all the subjects had the tendency to give a ma- jority of "heavier"-judgments on the comparison of 100 with 100. It does not seem profitable to use an equal number of posi- tive and negative differences, because one would get a very great percentage of "heavier"-judgments in the upper part of the series and a small percentage of "lighter "-jud,2;ments in the lower part, or one would suffer from the opposite inconvenience as the case may be one of overestimation or underestimation of the second stimulus. It came out as a result of our experiments that for some piu'poses, e. g. for the determination of the thres- hold by the method just perceptible differences, the use of at least one weight beyond 108 gr.is desirable, and that with some subjects the use of a weight as far down as 84 ma}' be dispensed with. Presenting these seven pairs of stimuli five times to the sub- ject in the way described takes approximately three minutes (3'S"). After the series was finished the subject was given a rest of some five minutes before another series of experiments was begun. This arrangement enabled us to make six or seven series of experiments in the course of one hour. It was, there- fore, possible to obtain in one hour of experimentation 30 to 35 judgments on each one of the pairs. The order in which the pairs were presented was not changed before 100 judgments on every difference were obtained. Such a group of 100 experi- 12 PROBLEMS OF PSYCHOPHYSICS ments was marked with a Roman numeral. It furthermore was found convenient to divide the results in groups of 50 experiments, which were marked by the corresponding Roman numerals and by the Arabic numerals 1 and 2; III2, thus, refers to the group of experiments which comprises those from the 51st to the 100th experiment inclusive in the third series. The data of five different series were used for the present in- vestigation. Two of these series were made in spring 1906, the others in the year 1906-07. The order of the pairs was this: la 96, 104, 108, 84, 92, . 100, 88 IVa 104, 92, 108, 88, 96, 100, 84 I 96, 104, 108, 84, 92, 100, 88 III 84, 104, 96, 100, 92, 108, 88 IV 84, 88, 92, 96, 100, 104, 108. The series made in spring 1906 are marked by the letter a.* Nineteen subjects were experimented on, but only the results of those are used for the puspose of the present study who have gone through series I, III and IV. The number of these sub- jects was seven, three of whom also made series la, and IVa. The subjects will be spoken of as subjects I, II, III, IV, V, VI, VII. With the exception of subject V all were males. Sub- ject III was 42 years; the age of the others was between 21 and 30. Subjects I, II and III have gone through all the series and we have for each one of them 450 judgments on every pair of comparison weights, since only the second part of series la was available. t The other subjects gave 300 judgments on every pair. The total number of experiments for each one of the first three subjects is 3,150, and for each one of the others 2,100, so that our discussions are based on the results of 17,850 experi- ments. This number is great, but by no means excessive; it will be seen that it is sufficient for most purposes. *The plan of these series of experiments was the outcome of several pre- liminary investigations undertaken by the Psychological Laboratory of the University of Pennsylvania. They were conducted by Professor Witmer, from whom the records were obtained for'use in this investigation. fThe results of lal show the influence of practice and were, therefore, no t used for the purpose of this study. Series Ila, Ilia and II contain experi- ments which are not comparable to those of the other series, because the experimental conditions (time of lifting, interval between the pairs, order of standard and comparison weight) were different. DESCRIPTION OF THE EXPERIMENTS 13 It is easy to see that the arraa,ii;einent of our experiments is a modification of the one described by Martin* and more recently used also by Smith,t the modifications having the purpose of eliminating the space error. The principles for the choice of this experimental arrangement were laid down by Miil- ler and Schumann, whom it seemed safe to follow in this respect. The difference of the arrangement used by Martin and by Smith from the one used in our experiments consists chiefly in that that these investigators used a stationary table and only one standard. Having the weights placed on a stationary table implies that the so-called Fechnerian space error is taken in, whereas it seemed desirable to avoid this error in our experiments. The use of only one standard weight causes a rise in the temperature of the metal owing to the fact that this weight is touched very frequently with the warm hand. This is to be avoided for two reasons: First, because differences of temperature might influence in some way our judgments of weight, and second because the standard weight can be recognized by being warmer,^ whicl\ introduces an almost incontrolable in- fluence. The use of only one standard weight may be less ob- jectionable if the original Cattell weights or other weights are used which are made of materials which are bad conductors of heat. In experiments in which the turning table is used, it is almost indispensable to use several standard weights. 'The purpose of our experiments was in so far different from that of Martin and Miiller, as we wanted to have as few disturbing influences as possible and Martin and MiiUer were chiefly interested in the analysis of the time and space error. It seems that the time error may be kept constant by *Martin' und Mueller, 7.iir Analyse dcr Unterschiedsempfindliclikeit, 1899, p. 3 sq. tM.\RGARET KeiveR Smith, Rythmus und Arbeit, Phil. Stud. Vol. IG, 1900, pp. 96-98. l.\ cisuil observation of the increase of the temperature of the stand- ard weight by frequent contact with the skin was made by MuELLER and ScHU.M.\N.N' Uber die psychologischen Grundlagen der Vergleichung gehohener Grufichte, .\rch. f. d. ges. Pky.dologie, Vol. 4.5, 18S.), p. 112) who, however, did not change their experimental procedure. 14 PROBLEMS OF PSYCHOPHYSICS means of regulating the movement, whereas a similar supposi- tion can not be made for the space error unless very definite instructions are given, which it is hard to follow strictly in an extended series of experiments. For this reason it seemed best to take in the time error as a whole, but to avoid the space error. The question whether the time error may be eliminated by an appropriate arrangement of the computation is of second- ary nature, and one may doul)t whether there is any great use in doing so. In this respect it seems that our way of experi- menting is a slight improvement over Miiller's method, in so far as it makes it possible to avoid the space error without further inconvenience and to perform the experiments in com- paratively short time. It is not possible to pass an equally favorable judgment on our modification of the series of judg- ments of which the subject has the choice. It has been mentioned that one subject could not be trained to the proper use of the "guess "-judgments and it seems to come out as a result of the experiments that the guesses are more likely to be correct than incorrect. This indicates that the "g "-judgments were also for the other subjects not equality judgments in the proper sense of the term, meaning that the subject had nothing in his sensa- tion to go upon in the formation of his judgment, but that they were judgments of a low degree of certainty. This is a fact which, perhaps, is not void of interest but which has no direct bearing On the problem of measuring the accuracy of sensations and, after having verified this fact, it will be best not to use this distinction between "heavier guess" and "lighter guess". The only natural judgment in these cases is that of equality between the two stimuli. The possibility of giving a "guess "-judgment aids the subject in his natural tendency not to commit himself. It seems that a similar remark must he made in regard to letting the subject express the degree of confidence with which the judgments are given. The meaning of the terms "an ordi- nary degree of certainty," "a high degree of certainty" and "a very high degree of certainty" seems to be clear at first and not requiring any special definition. In the actual experimen- tation one finds very soon that it is difficult to give these terms a definite meaning, an observation which was also made by DESCRIPTION OF THE EXPERIMENTS 15 one of Miiller's best trained observers.* It seems that there is some danger of mixing up two different problems, the prob- lem of the degree of subjective confidence and that of the accur- acy of sensation. It certainly would be not only an interesting but also a very important problem to find their relation, but it seems that the factors which determine the degree of subjective certainty are not yet as well understood as they might be, and that other experiments might be used more profitably for set- tling this problem. The problem of finding a measurement of the accuracy of sensations is difficult enough and it need not be made more complicated by attacking it from the most difficult side. It seems advisable for this reason not to use other judg- ments than "heavier," "lighter" and "equal," unless one in- tends to connect the study of the exactitude of sensations with some other problem. The degree of subjective confitlence with which the judgments were given was disregarded entirely in the working out of the results for the present study. The Tables 3-9 (Appendix pp. 174-177) contain the results of the experiments. Each subject is given a separate table which contains the -judgments for all the seven pairs of compar- ison weights for every series separately. The numbers in the column T are the numbers of "guess"-judgment3 which are the sum of the "hg"- and the "lg"-judgments. The numbers under the heading "h" and "1" give the number of "heavier" and "lighter "-judgments. The sum of the numbers T, h, and 1 in one line of a column is equal to 50. The different series are marked in the way explained above. The numbers at the bot- tom of the columns in the row marked ^ give the sum of the numbers in a column. The numbers of these tables, as they stand here, illustrate plainly the one fact that the results of psychological experiments show a great varial^ility. The greatest amount of care was taken to keep all the conditions of the experiments constant and there arise, nevertheless, variations of very considerable size. This indicates on how terribly uncertain a basis most of the results rest which are gained from psychological experiments. ♦Martin und Mueller, 1. c p. 9. 16 PROBLEMS OF PSYCHOPHYSICS Pul:>lications are not scarce which cUiira to demonstrate the in- fluence of a certain factor on a psychical phenomenon by observed differences in rehitive frequencies amounting to 4 or 5%. It is very obvious to raise the objection to these investigations that a similar or perhaps even a greater variation of the results might have been obtained, if the experiments had been repeated without varying the condition the influence of which was to be demonstrated. The next observation is that there does not exist apparently a difference of the intensities which is always jvidged in the same way. A negative difference of 16 grams is by far beyond the difference which is called the threshold, but there occur some cases in all the subjects where this difference could not l^e de- tected or was judged the wrong way. A difference which is such that greater differences are always judged in the same way is a fiction, the fact is that the judgments of one Subject on the same difference vary, and it is impossible to foresee in which judgment a certain experiment will result. This is not a new observation as e. g. Miiller and Cattell* in their discus- sions of the notion of the threshold have' laid stress on the fact that a difference which is always judged in one way does not exist. Lately the view was taken by Kobileckif that "there must be something wrong with the experiments" if this occurs. This view, if once taken, is of course irrefutable, because it lays down as a criterion of the correctness of the experiments, that they must conform with the requirement of a difference of intensities which is always judged in the same way, e. g. that it is alwaj's possible to find a difference which is judged equal. The only objection is that it is not possible to fiad another fault with the experiments in question than that their results do not agree with one's definition of a correct experiment, and that it does *There is a slight diiTerence in the positions of Catteil and Miiller. Mill ler takes it as a given fact that there does not exist a difference which is always judged in the same way, whereas Catteil assumes the suppositions of the theory or errors and argues that the assumption of such a difference is con- tradictory. fSTAXiSLAUS KoBYLECKi, Uber die Wahrnehnibarkcit plotzlicher Druck- anderungen, Psychologische Studien, Vol. I, 1906, p. 293, where the reference to Mailer's writings may he found. DESCKIPTIOX OF THE EXPERIMENTS 17 not seem to be quite justified to discard experiments which were performed with just the same care as others, merely because their outcome was not such as it ought to be according to a defi- nition laid down beforehand. These facts seem to suggest the following view. If the sub- ject is required to give judgments on the comparison of two stimuli his judgments will vary, so that one can not possibly foresee what the judgment will be in any particular experiment. The only thing we know beforehand is that, if the experiment succeeds, the judgment vnLl belong to one of those classes of judgments which were admitted. This is the formal character of random events and we introduce the notion that there exists a definite probability for every class of judgment, that the com- parison of certain two stimuli by a given subject under well defined conditions ^\'ill result in a judgment of this class. If we admit the judgments "heavier", "lighter" and "equal" we will have the probabilities p, q, r that the comparison of a given difference will result in the corresponding judgment. The numerical values of these probabilities may vary from subject to subject, and it is. a problem of investigation whether they remain constant for the same subject and the same difference, or whether they are subjected to temporal variations.* For *Wreschner defines "trustworthiness" of a sensation (Zuverlassigkeit) as the expectations which a single sensation or a judgment on it has to be confirmed in the case of repetition under conditions which are as nearly alike as possible. {Methodologische Beitrage zu psychophysischen Messungen, Schrijten der Geseilschaft jiir psychologische Forschung, 1895, p. 25 "Hier- unter (unter Zuverlassigkeit) wollen wir die Aussichten verstehen, welche eine einmalige Empfindung oder deren Beurtheilung hat, in einem Wieder- holungsfalle, der unter moglichst gleichen Versuchsbedingungen stattfindet- bestattigt zu werden....). The correct analysis of this notion, of which Wreschner makes but Uttle use, might have led to the introduction of the notion of a probabiHty of judgments of certafti type. W'reschner's definition refers merely to the subjective side of the notion of a probability, neglecting the objective definition which alone is suitable for the mathematical treat- ment. For the explanation of a mathematical probability as a measure- ment of the degree of certainty see v. Kries, Die Principien der Wahrschein- lichkeiisrechnung, 18.S6, and STfMPF, Vber den Begrifj der mathematischen W ahrscheinlichkeit, Berichte der bayrischcn Akademie, Phil. Kl. 1892. 18 PROBLEMS OF PSYCHOPHYSICS positive differences which are considerable the value of the prob- ability of a ''greater "-judgment comes very near to the unit and the probabilities of other judgments correspondingly are very small. For negative differences which are considerable the probability of a "smaller "-judgment is only little different from the unit and the probabilities of the judgments "greater" and "equal" are very close to zero. Data like those embodied in the Tabl-es 3-9 are observations on the repeated realization of chance events and they may be used as empirical determin- ations of the underlying probabilities. The question is, how far do the results comply with this notion of a probability of a judgment of certain type, and how much can* be done with it for the solution of the problems of psychophysics. CHAPTER II. ON THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY. A statistical investigation ma}- be confined to a description of the conditions and a statement of the facts which were ob- served. Such a statement is in itself valualile and not void of interest because it contains the description of a fact, but as long as this fact is not connected with other facts its statement is not so much knowledge as the material for the future acquisition of knowledge. On this ground one even can not conclude that under similar conditions results will 1)e obtained which resemble those of the first series of observations. It is, indeed, out of question to reproduce exactly the same conditions and, since one does not know anything about the conditions which necessitate the re- sult, one can not positively say that only the observed conditions are of importance and one must resign the hope to foretell future results. But the main interest of all investigations is to know, whether the same, or at least similar results will be obtained in a future repetition of the observation. Before such a statement can 1)6 made it is necessary to form one's views about the causes which were at work to produce the first result. This can be done by demonstrating the causal relation between the results of the observation and a certain group of conditions l)y means of the method of experimental variation of the conditions. The application of this method is comparatively simple in those eases where the conditions under which we want to okserve a phenomenon are well under our control. This method becomes more laborious if our control of the conditions is not complete enough to enable us to establish causal relations; in these cases it becomes necessary to introduce new notions. One forms the hypothesis that the result is due to a complex of conditions which remain constant and another group of con- ditions which are variable. The variations of the latter group 20 • PROBLEMS OF PSYCHOPHYSICS are supposed to have random character, i. e. they do not show a recognizable law or regularity. The group of constant condi- tions is represented liy those conditions which one can vary or keep constant at will, or which l:)y their nature do not vary, or are supposed not to vary. The result of these constant and varia])le conditions, then, is determined in so far as it must have a certain general character, but it is subjected to quantitative or qualitative variations. The constant conditions determine not only the class, to which an event which depends on them belongs, but they determine also the amount of variation which may occur in individual cases. Degrees of variation may define classes which are sub-classes of the class. Individuals of these sul)-classes occur with different relative frequency, which is a fraction, the denominator of which gives the number 'of all individ- uals of the class, and the numerator the numl^er of the individ- uals of the sub-class. This number, which is smaller than one, has the character of a mathematical prol:)ability. Every group of conditions which gives certain numerical probabilities to every sub-class is called a system of causes, or abbreviated a cause of an event of the class. Different systems of causes produce different results in so far as they may 1, produce events which ])elong to entirely different classes, or 2, that they give numer- ically different probabilities to the sub-classes of the same type of events. This terminology differs from the common usage of the word cause which designates a group of conditions which necessarily produces a certain event; here we mean by this word a group of conditions which gives a certain probability to the event in question. This group of conditions may or may not be susceptible of further analysis, but as long as further infor- mation is not supplied, it is characterized only by the numerical value of the probability of the event. Two groups of conditions which give the same probability to the same event are not dis- tinguishable on this basis.* If we know that in a series of obser- vations on a certain event the result of a single observation *This may be illustrated by the following example. It was found in an investigation on the estimation of long time intervals by a great number of u ntrained subjects, that zero was a favored numeral occurring at the last place much more frequently than any other numeral. An other investiga- THE STATISTICAL NUMUERS OF RELATIVE FREQUENCY 21 depends on a system of causes which is characterized by the numerical value of the probability of the event, we are not only able to state the most probable outcome of the series, but we also can assign a definite probability with which we may expect a given deviation from the most probable result. We base our expectation that a repetition of the series of experiments will give similar results on the identity of the conditions which we know and on the supposition of the random character of the influences which we do not know. Repetitions of the same series of observations, however, do not give exactly the same result and the question arises necessar- ily, whether the differences between these results are due to chance variations or whether they are due to a change in the underly- ing system of causes. This question, as a matter of fact, is the all important problem of every statistical investigation which applies the method of experimental variation of the conditions. A certain phenomenon, e. g. a mental state, is observed under certain conditions and from these observations statistical num- bers of relative frequency are derived. One or several of the conditions which we have so far under control as to vary them at will may be susceptible of qualitative or quantitative varia- tion. Keeping all the other elements constant we vary these conditions and we want to know, what the influence of this change is. This influence is characterized only by the differences of tion on the estimation of short time intervals by a small number of highly trained subjects showed a similar great frequency of zero. If nothing else had been known than the percentages expressing the great frequency of zero, one had been obliged to conclude that similar conditions were at work in both cases, the conclusion of identical conditions beii.g made impossible by difi"er- ences in the numerical values of the percentages. Collateral evidence, how- ever, showed that the conditions involved were absolutely difi'erent. (See the author's article on "Systematic Errors in Time-Estimation." Amer. Journ. Psychology, 1907, Vol. 18.) The collateral evidence in this case had the form of diflerences in the percentages of the other numerals, espec- ially of 5 and this circumstance indicated the possibility and necessity of further psychological analysis. It is not necessary that the collateral evi- dence must have the character of statistical numbers of relative frequency; introspective data or our general knowledge of the processes involved may make one hypothesis more likely than the others. 22 PROBLEMS OF PSYCHOPH Y8ICS the observed numbers of relative frequency in the two series. The influence of the variation of the concUtion is demonstrated, if we can convince ourselves that the difference is due to a differ- ent numerical value of the prolDability of the event in the second series and not to chance variations. It is the object of further consideration to form a view on the nature of this influence, and it will depend on the particular kind of problem one has to deal with which views will seem acceptable. The real difficulty of the treatment of the statistical data consists in the demonstra- tion that the difference of results is due to a different system of causes. The outcome of the comparison of the two series con- sists in the difference of certain numerical values, and similar differences would have been found if the conditions had not been changed. Are these differences then to be attributed to chance variations, or are they due to differences in the objective condi- tions? The answer usually given is that the differences must be ac- counted for by a change of the objective conditions, if they are unequivocal and considerable. By the term "unequivocal" one means that the differences must always have the same sign i. e. there must be uniform decrease or uniform increase, but not increase alternating with decfease, if the group of conditions which we investigate varies in one way. One finds it rather hard to comply with this condition in actual investigations and one must not be too strict in the application of this requirement, because even with phenomena which are as constant as the size- weight illusion one finds exceptions. As a rule one will be satis- fied if the percentage of those cases which do not comply with the requirement of an unequivocal variation is small. A simi- lar vagueness lies in the requirement of considerable differences. It is a matter of course that one can not speak of absolute cUffer- ences but only of relative or percentual differences. The abso- lute differences between the results of two statistical series must increase if the numl)er of observations becomes greater, even if we have to deal with no other but chance variations, as it follows from the theorem of Bernoulli. A difference which may be considerable in one case will be inconsiderable in another case and, furthermore, there exists no fixed limit where a nu- THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY 23 merical difference begins to be considerable. Tlie notion of a considerable difference is too indefinite to be serviceable for dis- tinguishing between two results. The particular character of the notions used does not permit a strict answer, but only the statement of the view which is more likely, because it has the greater probability. On the basis of the results one may form a number of different hypotheses which may explain both results but which have a different degree of probability. It is obvious at once that no difference between the results, no matter how great it may be, can be a strict dem- onstration of a difference in the underlying systems of causes. Since the result of a single experiment is not strictly determined as far as our knowledge goes, it follows that one has only a cer- tain probability with which one may expect the result to fall within given limits, and it is always possible that the opposite of an event which is only probable arrives, no matter how near to the unit the probability may be. Deviations of every si.^e are possible, only the probabilities of their occurrence are differ- ent. The first requirement for the prediction of future results is that the observed numbers of relative frequency have not only the formal but also the material character of mathematical prob- abilities. Such a number has the formal character of a proba- bility, if it is a fraction the denominator of which gives the num- ber of all the cases and the numerator the number of all those cases which are favorable to the event. The ratio of the number of the judgments "heavier" to the total number of judgments given on a pair of weights has the formal char- acter of a mathematical probability. The percentage of these judgments, and the ratio of the number of "heavier"-judgments to the number of "lighter "-judgments have the character of func- tions of mathematical probabilities (lOOp and — -). Exam- ples of numbers of relative frequency which have aot the formal character of mathematical probabilities, nor of functions of such, are the ratio of the number of births which take place in a country during a year to the number of inhabitants of this coun- try, or the ratio of those men of a country wlio choose a certain 24 PROBLEMS OF PSYCHOPHYSICS profession to the number of inhabitants.* Numbers of this kind must not be treated as mathematical probabilities and certain important conclusions cannot be drawn from them. Neither the theorem of Bernoulli for constant probabil- ities, nor the theorem of Poisson for variable probabilities can avail for demonstrating that a statistical number of relative frequency has the material character of a probability. The theorems of the calculus of probabilities, indeed, do not contain any statement about actual events, but they give only the for- mal character of those events to which the attribute of random- ness belongs. The only possible w^ay of demonstrating that a statistical number of relative frequency has the material char- acter of a mathematical probability is to show that this number approaches in a certain way a definite limit, if the number of observations is increased indefinitely. Let us suppose we have n series of observations on an event E, each one comprising s observations of w^hich m^, m^, mn have given the result E. Then the first requirement says that the fractions s s ' s must be grouped around a number p in a certain way. It fol- lows from the fact that p is the limit of the numbers — - that s they will be clustered around p more closely than around any ♦For use of the birth rate see Laplace, Th eorie analytique de la probability art. 30, 31. Statistical numbers of the second type were used by Mr. Cat- TELL in his study of American men of science {A Statistical Study of Amer- ican Men of Science, III; Science, December 7, 1906, N. S. Vol. XXIV, No. 623.) These numbers have not the formal - character of a mathematical probabiUty because among the men living in a country there are also those who have chosen their career, so that the number of inhabitants does not represent a totality of cases which may go one way or the other. The choice of profession of all the men born in a country during a certain period of time represents more nearly a set of cases which depend on chance. It is not likely that such a number would have the material character of the probability of becoming e. g. a scientist for a child born in this country, because the denom- inator of such a ratio ought to be the number of all those circumstances which decide the choice of a man's life work, a totality of cases to which no definite number can be assigned. THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY 25 other value. As to the law of the distribution of these numbers there are two cases possible which are fundamentally different for the interpretation of the event E: either the distribution follows the (I>(^)-law or it does not. The interpretation will be different according to the following cases. (I.) The distribution follows the (;')-law. 1. The distribution is such that the number of deviations, the absolute value of which is smaller or equal to y, y being any value, •«L-p\ ir) ='^J e dx , (1) o where n is the number of series of observations and where h is a constant called the coefficient of precision which is given by h=J—±— . (2) \2P{1-P) This is the case of a distribution according to Bernoulli's theorem. This will be e. g. the distribution of observations, where the single observation consists in drawing a ball from an urn which contains black and white balls only, if the ball is put back after its color was registered. If p is not known one must in ■ take the arithmetical mean of the numbers — * . s n )i s "^ ' (3) I - 1 and one obtains for the corresponding coefficient of precision h' = J— ^ . (4) V 2a(i-a) 26 PROBLEMS OF PSYCHOPHYSICS Statistical events of this type are similar to the drawing of balls from an urn which contains the same number of black balls and white balls throughout the series. The results of such a series may be regarded as the products of a constant system of under- lying causes represented by those conditions which determine the probability of the event, and of another system which is sub- jected to chance variations. A statistical series of this kind is called by W. Lexis* a typical series of normal dispersion. We are allowed to suppose that a repetition of the observation will give a result which complies with the expectations which we base on the calculus of probabilities. (2.) The distribution follows the O ( ;-)-law but the precis- ion required by formula 1 is not equal to that given by formula 4. (a.) We have In this case the values \ 2a {i-a) — ' -J) are less frequent in the neighbor- s hood of p than they were in (1). The series, however, is typical, because the value p is fit to represent the entire group of the I Til ■ ' numbers j — --'P\- Lexis explains this case by supposing I ^ I that we have not to deal with a constant probability, but with one which undergoes chance variations. Events of this type may be compared to drawings from a set of urns which were filled with white and black balls in the following way. A coin is tossed up n times and every time when head appears a white ball is put into the first urn and when tail appears a black ball is put into it. This process is continued until every urn contains n balls, and then from every urn the same number of drawings is made, the ball being replaced after its color is registered. f The *W. Lexis, Zur Theorie der Massenersckeinungen in der menschlichen Ges- ellschaft, 1877, p. 22. Cfr. E. Czuber, Die Entwicklung der Wahrschein- lichkeiistheorie , Jahreshericht der Deutschen Mathematikervereinigung 1898, Vol. 7, p. 85, 233, and the same author, Wahrscheinlichkeitsrechnung, 1G03, p. 131. fFor another example see E. Czuber, Wahrscheinlichkeitsrechnung, 1903, p. 131. THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY 27 result of the drawings from one urn depends on the probability of the appearance of a white ball due to the number of black and white balls which it contains, and the total outcome depends on the combination of two chance events. The result of this com- bination is a distribution according to the (^)-law.* Events of this type are chance events in the proper sense of the word and the result of one observation does not depend on that of any of the previous observations. (b.) The differences s are distributed as required by formula (1.) but the constant h is greater than h' computed by formula (4.) In this case the differences are clustered more closely around p than Bernoulli's theorem requires. Series of this type are called typical series with less than normal dis- persion. The single results are not independent from each other but there exists a relation between them. According to Lexis such a distribution is due to a law or a norm, which tends to pro- duce a certain numerical value of the relative frequency. Events of this kind are not independent from each other. (II.) The distribution of the terms s does not follow the (;')-law. We speak in this case of an irregular distri- bution! and we call such a series of observations a symptomatic series. J These series indicate a change in the system of under- lying causes. If the change occurs always in the same direction we may speak of an evolutoric series ; if phases of increase alternate with phases of decrease we may speak of an undulatory variation and, finally, if the same changes occur after constant time intervals we may speak of periodic changes or variations. It follows from this discussion that the interpretation of a sta- *This is a special case of what Bruns calls the conservation of the of the Tables 18-24 (Appendix pp. 181-184). Dividing h' by h" gives the value of the coefficient of di- vergence which is given in the tables in the columns Q. A glance at these numbers shows that they vary not inconsiderably, and that the range and the amount of variation is very different for the different subjects. The averages of the coefficients of divergence are given here for the different subjects: Subject II 1.01 VII 1.13 I 1.22 VI 1.39 IV 1.50 V 1.62 III 1.69 These numbers seem to indicate that we have to deal in the cases of subjects IV, V and III with a slightly overnormal dispersion. It may be doubtful whether subjects I and VI, are cases of normal or slightly overnormal dispersion, but for the subjects II and VII it is almost certain that their result show a normal dispersion. For subject II, for instance, 7 of the Q's are above and 7 are below the unit and if one disregards the result for 10 4h, which is clearly out of bounds, the results vary between 0.76 and 1.43. With this one may compare the numbers for the coefficients of diver- gence for the rate of male and female births, which were found by Lexis* in his first series of observations. They are based on the statistics of the birth rates in 34 Prussian counties during 24 months. This gives 34 series of 24 observations each; one series consists of from 639 to 4,766 observations. The values of the Q's vary between 0.72 and 1.47 and 19 of them are beyond and 15 below the unit. The range of variation is therefore a little larger than in our experiments, if one omits the result for 104h, and our results compare very favorably even if one does not *W. Lexis, Das Geschlechtsvcrhaltiiiss der Geborenen und die Wahrschein- liohkeitsrechnung, pp. 216-245; Zur Theorie der Massenerscheinungen, pp 64.78; the results are reproduced in E. Czuber, Wahrscheinlichkeitsrechnung und ihre Anivcndung, 1903, p 324. THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY 33 omit it. The arithmetical mean of the Q's in Lexis's observa- tions is 1.09, approaching more closely the less satisfactory result for subject VII than that for subject II. The slight devi- ations from the unit may be well accounted for by the inaccuracy in the determination of h". Considering the fact that Lexis's results are based on 24 series of observations each one contain- ing at least 13 times as many experiments than ours (the aver- age of the number of "experiments" in one series of Lexis is approximately 2,300 or 46 times as many than in our experiments) one will be justified in believing that the results for the subjects II, VII, I and perhaps also those for subject VI indicate a normal dispersion. The dispersion in the results, for the other subjects may be considered to be overnormal. We formulate this result in this way: The statistical numbers of relative frequency which one obtains in psychological experiments show for some subjects a normal dispersion. According to our pre- vious discussion we will conclude that the group of conditions on which the formation of a judgment depends may remain fairly constant for certain subjects. The answer of the question whether or not we have to deal with a typical probability cannot be undertaken on the basis of our present material; it must be postponed until further results are at hand. The demonstration of the normal dispersion of the results is interesting in several respects. At first, the number of examples in which a normal dispersion could be demonstrated is not great. Besides this pretium raritatis there comes in the practical conclu- sion that experimental psychology is certainly capable of be- coming as much of a science as any other branch of knowledge which takes its material from statistical results. The fact that psychology has to deal with probabilities of normal dispersion gives this science a distinguishing feature of which not many statistical sciences can boast at present. The normal dispersion of the numbers of relative frequency also proves that the instabil- ity of mental states is not so very great as some writers were in- clined to believe. For some time the view was very popular among certain philosophers, that a mental state may be connected with very widely different physical conditions, and that the same groups of physical conditions may be connected with different 34 PROBLEMS OF PSYCHOPHYSICS mental states. Our results show that at least the latter part of this statement is not quite correct, for in some subjects the re- sults are such as they would be, if they were chance phenomena under the influence of a constant group of conditions. Training in psychological experiments seems to have the tendency of produc- ing in a sul^ject the ability of reacting under certain influences, and we may define a subject with training in psychological experiments as a subject, which is capable of reacting in such a way that the reactions are due to a constant group of conditions w^hich are so far under the control of the subject that they may be varied at the will of the experimenter. Subject II who has the smallest coefficient of divergence, had several years experience in psycho- logical experiments. Subjects VII and I had considerable abil- ity for this work, whereas subject III was the man who could not be trained to the proper use of the "guess "-judgments, and sub- ject V was a young lady with little practice in experimental work. The observed values of the coefficients of divergence prove that there is no appreciable change in the results which are taken at different times for at least some subjects. The next question is whether there do not occur any changes within a series. Be- sides the variability of the conditions of a single lifting (e. g. variations in the velocity or height of lifting) there is one circum- stance which might be of influence on the results: Namely the taking of the experiments in groups of 35. It takes about three minutes to perform such a series and one may ask whether this does not produce an influence (fatigue, strain of attention,) which puts the judgments in the latter part of one series under conditions which are entirely or partially different from those in the preceding part of the series. One may try to settle this question by means of the introspective evidence that there was or that there was not an appreciable influence of fatigue or lack of attention. It is very likely that this question refers to con- ditions which are not favorable to introspection, and we take the attitude that an objection raised against a series of experi- ments on the ground of introspective evidence makes the series suspect, but that the absence of any such objection based on introspection does not put the result beyond doubt. The introspective evidence, as a matter of fact, forms only one part THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY 35 of the outcome of the experiments. The objective records are the other part, and it would be one-sided to neglect one part for the sake of the other. This attitude is similar to that which one takes towards sets of measurements which are the results of observations of a physical quantity, where an opinion about the value of the observations formed in collecting the data must be verified by a minute examination of the results in order to establish the objective value of the .results. There were in our experiments no spontaneous complaints from the part of the subjects as to the length of the experiments and cautious ques- tioning never revealed any discomfort caused by the protracted experimentation. Introspective evidence against our experiments is therefore not at hand. A change in the conditions, which is not noticed by introspec- tion, must be expressed by a variation of the results i.e. in our case by a different frequency of the "heavier", "lighter" and "guess"- judgments in the different parts of one series of 35 experiments. We shall make this investigation on the numbers for the combined h and hg judgments. It happens that the records of the single series are naturally divided in five groups, each one containing the judgments given during one turn of the table, i. e. one judg- ment for every pair of comparison weights. The number of "heavier "-judgments given during one turn of the table can be found by counting these judgments in one column of the records. Dividing the number of "heavier "-judgments in the 1,2, ....5 col- umn by the total number of judgments of a column one finds the relative frequency of these judgments during the first, second. ....fifth part of a series. These five numbers of relative fre- quency are, for one individual, only slightly different, as may be seen from Table 25 (Appendix p. 185). The numerals at the head of the columns (1-5) refer to the five turns of the table in one series and the numbers in one column give the relative frequency of the "heavier "-judgments during the first, second,.. ..fifth turn of the table for the different subjects. The variations of the numbers referring to one person are small and they are not systematic, increase and decrease interchanging without regularity. The total number of judgments given during one particular turn of the table was for the subjects I, II and III, 840 and for 36 PROBLEMS OF PSYCHOPHYSICS the other subjects 560. The numbers in one line of Table 25 give the ratios — {i= 1, 2, 3, 4, 5), where ni; is the number of s "heavier "-judgments in the columu i. Let us call the arith- metical mean of these numbers (referring to one subject) p a:id designate by ^( — '-/>) the sum of the squares of the d3vi- f in ■ ations of the numbers — ' from their mean. We define further- <. more, by q the probability of a judgment which is not a "heavier" judgment so that q= 1-p, and we designate by u the mean error of the arithmetical mean of the numbers — * -ViPe^ These numbers may be found from the data of the preceding table and are given under the corresponding headings of Table 27. They serve for the determination of the coefficient of divergence Q, which in problems of this kind is found by the formula 6.^/.+ -^'-^™'-^). (8) We determine, furthermore, a quantity M which is defined by M -Vfi-^e-^)- «) This quantity is called the component of physical variation* because it indicates the amount of variations of the underlying ♦This term (Physische Schwankungskomponente) is due to W. Lexis, Ubcr die Theorie der Stabilitat statistischer Reihen, Jahrbtich ^ .National Okon- omie u. Statistik, Vol. 32, 1879. L. v. BorTkewiTsch, Das Gcstez der kleinen Zahlen, 1898, p. 30, calls the quantity M "absoluter Fehlerexcedent. " Cfr. E. CzuBER, Wahrscheinlichkeitsrechnung, 1903, pp. 317-321. THE STATISTICAL NUMHERS OF RELATIVE FREQUENCY 37 complex of causes which determines the probability of the event in question in the different parts of the series. M and Q are in a simple relation shown by the formula where the factor u' represents the quantity These relations may be used for checking the computation by the first formula for M. We will illustrate the course of the computation by working out the results for the first subject. The total number of judgments in each one of the five columns was 840, and the numbers of "heavier" and "heavier guess" judgments are given in the following little table under the heading m^. By dividing these numbers by 840 one obtains the relative frequencies of these judgments in the five columns; the arithmetical mean of these numbers is found to be p = 0.5374. The column under the head- ing — *- — h gives the deviations from the arithmetical mean. ^ 840 ^ ^ The sum of the positive deviations is 0.0449 and the sum of the negative deviations is 0.0450; this difference is due to the abbreviation of the result. The last column of the table gives the squares of the deviations, the sum of which is found to be 0.00202773. 472 461 444 459 424 840 0.5619 0.5488 0.5250 0.5464 0.5048 840 -^b-P \ 840 P)' + 0.0245 0.00060025 + 0.0114 0.00012996 -0.0124 0.00015376 + 0.0090 0.00008100 -0.0326 0.00106276 m- 840 2.6869 ^/j^_ \. ^ 0.00202773 840 p= 0.5374 I \l{'^^-py=^- 5 ^8ao ^/ 00040555 840 38 PROBLEMS OF PSYCHOPHYSICS Dividing the sum of the squares of the deviations by 5.4=20 and taking the square root one finds the number w = /y/ 0.000 1039 = 0.0101, the mean error of the arithmetical mean. Now one proceeds to find M by means of formula 9. log 839=2.92376 log 840=2.92428 Difference =-0.00052 log 0.00040555= 0.60804-4 Sum (regarding • the sign) = 0.60752-4 Dividing by 2 = 0.30376-2 = log M M= 0.020126 The computation of the quantity Q by formula (8.) and the check of the computation of M may be effected in this way. log 0.00040555=0.60804-4 log 0.5374 = 0.73030-1 log 839 =2.92376 log 0.4626 = 0.66521-1 Sum =0.53180-1 Sum =0.39551-1 = log pq =0.39551-1 (Q^- -1) = log pq Difference = 0. 13629= log Q=V 2-1369 = 1.462 log (Q^-l) = 0.13629 log \/Q'-l= 0-06814 log pq = 0.39551-1 log 840 =2.92428 Difference =0.47123-4 Dividing by 2 =0.23562-2 logv'Q'-l =0.06814 Sum = 0.30376-2= log M, as above. THE STATISTICAL NUMBERS OF RELATIVE FREQUENCY 39 In Table 26 (Appendix p. 185) data are given whicli facilitate the control of the computation; the denotations used in this table are the same as those used in the text. The final results for the coefficients of divergence and for the components of phys- ical variation are given in Table 27, (Appendix p. 185) under the corresponding headings. It must be kept in mind that in problems like this, where the coefficient of divergence is computed by formula (8), the result must necessarily be a value greater than the unit, and one can require only that it does not exceed this value too much. The value Q=\/2= 1-414 is the limit which ought not to be exceeded, because in this case the physical variation is equal to the accidental variation. Only in two of our subjects do the Q's exceed this limit (I and VI) the margin being small (0.048 and 0.164.) The average of all the Q's is 1.345 which comes very near the case of equality of physical and accidental variation the difference being on the safe side. It will be noticed that for the subjects II and VII, for whom we have found above a normal dispersion, the values of Q come nearest to 1.414, the differences being 0.006 and 0.018. From this it follows that there is no difference between the re- sults of experiments made at different parts of a series of 35. CHAPTER III. ON THE METHOD OF JUST PERCEPTIBLE DIFFERENCES.* The method of just perceptible differences is the oldest of all psychophysical methods. After attention was called to the fact that our judgments on the equality of two stimuli are not exact in any strict sense of the word, it was obviously possible to investigate the limits within which two stimuli may vary with- out a difference being perceived. This was indeed the problem of Lambert t and BouguerJ, the first investigators who tried to determine this range of objective variability with subjective equality for optical sensations. When Lambert lighted a screen by a candle placed in front of the centre and tried to find the range inside of which there was no appreciable difference in the intensity of the light, he had in his result a quantity essentially identical with the threshold of difference as determined by the method of just perceptible differences. This method came more into prominence by the experiments of Weber and, later on, of Fechner, who used the notion of a just perceptible difference for his theory of the measurement of sensations. The notion of the threshold became identical with that of the just perceptible difference, which was defined as the smallest difference which could be perceived. The fact that smaller differences may be per- ceived and that larger differences may not be detected was ex- plained by accidental errors. The notion of the just perceptible difference thus became the fundament of quantitative psychology and this method is used directly or indirectly in most of the investigations which deal with Weber's Law or with the problem *A short abstract of this chapter was given in the Psychological Review, Vol. 14, July, 1907, pp. 244-253. ILambert, Photomeiria, sive de mcnsura et gradibus luminis, colorum et umbrae, 1760, in OsTWALD's Klassiker d. ex. Naiurwiss, 31-33. iBouGUER, Traits d'optique sur la gradation de la lumikre, \7C<0. 40 METHOD OF JUST PERCEPTIBLE DIFFERENCES 41 of the accuracy of sense perception, in spite of the fact that there is at present a tendency to substitute one of the error methods for the method of just perceptible differences in accurate work. The form in which the method of just perceptible differences was used by the first investigators was open to many criticisms and it lasted some time until this method obtained the form in which it is used to-day. The merit of this perfection is chiefly due to Fechner, who also gave the name to this method. In determining the threshold by the method of just perceptible differences* one starts from two stimuli which are equal and, keeping one stimulus constant, one increases the other until a difference is perceived. This difference is put down as the result of an observation on the just perceptible positive difference. The second step consists in finding the just imperceptible posi- tive difference. This is done by allowing a supraliminal differ- ence to decrease until it ceases to be perceptible. The largest difference which is not noticed is put down as an observation of the just imperceptible positive difference. The just percep- tible and the just imperceptible positive differences are, gener- ally speaking, not identical. In a similar way one finds the just perceptible and the just imperceptible negative differences. In an investigation which aims at some accuracy it is indispens- able to make a considerable number of experiments in order to eliminate the errors of observation which express themselves in differences of the numerical results. It seems to l)e' justified to treat the results by the algorithm of the method of least squares,. since one considers the just perceptible difference a physical quantity the observations of which are subjected to chance errors. The probable error of the final result may serve as an indication of the greater or smaller variability of the psycho- physical conditions. The arithmetical mean of the just percep- *For the description of this method see: Fechxer, Uber die psychischen Massprincipien, Phil. Stud. Vol. IV, 1888, p. 161 sqs. besides the "Eletnente" Vol. I and II and the "Revision" ; Wundt, Das ]]'ehersche Gesetz uttd die Methode der Minimaldnderungen, Phil. Stud. Vol. I, pp. 556 sqs. Physiolo- gische Psychologic, 5 ed., \'ol. I, pp. 470 sq., 475-479; TitchenER, Experi- mental Psychology, 1905, Vol. II, Part I, pp. 55-69, and Part 2, pp. 99-143 with historical notes. 42 PROBLEMS OF PSYCHOPHYSICS tible positive and of the just imperceptible positive difference defines the threshold in the direction of increase, and the arithmetical mean of the jusc perceptible and of the just im- perceptible negative difference gives the threshold in the direc- tion of decrease.* The practical application of this method meets with two very peculiar difficulties which impair its serviceability and necessi- tate a change in the experimental procedure. It seems that it is essential for the result, by which steps the threshold is ap- proached. After the first rough determination is made, one tries to get a more accurate one by using smaller intermediate steps. In experimenting with this new series one notices very soon that the subject is more apt to perceive smaller differences in the determination of the just perceptible dif^'erenc^, and not to perceive larger differences in the determination of the just im- perceptible difference than he was before. Usually one explains this fact by the influence of expectation, because the subject knows that an imperceptible difference is to be increased and a perceptible difference is to be diminished. In order to avoid this inconvenience one has employed two means. The first was to use only trained subjects who are free from the influence of expectation, and the second consisted in applying the stimuli in irregular order. The first way cannot always be used, because one frequently is obliged to experiment on untrained ot^alf- trained subjects, and this requirement of training on tlte part of the subject, furthermore, disposes of the possibility of using this method for practical purposes. The determination whether the sensitivity of a patient is below or above the normal must necessarily be made on untrained subjects, and until now no other method but the method of just perceptible differences could be used for this purpose. The method of presenting the comparison stimuli in irregular order has the inconvenience that the results cannot be worked out by the algorithm of the method *The threshold m the direction of increase and that in the direction of decrease are frequently combined in order to obtain a generalized threshold or to eliminate constant errors. This procedure, which was first suggested by VoLKMANN, gi\es a result which seemed to be not entirely clear in its signification; see Titchner, /. c. Part 2, p. 112 sqs. METHOD OF JUST PERCEPTIBLE DIFFERENCES 43 of just perceptible differences. This modification of the method of just perceptible differences is called the method of irregular variation. The results obtained in this way are usually worked out by an algorithm similar to that of the error methods.* The second difficulty is due to the fact that frequently a differ- ence is not noticed after a smaller one has been perceived in a series for the determination of the just perceptible difference, or that a difference is noticed after a larger difference has been imperceptible in a determination of the just imperceptible differ- ence.! The question arises, what must be done with such results? Some investigators take the view that these series must be ruled out and that only those series must be kept for the final computation, where no such inversion occurs and where the two classes of judgments are strictly sep- arated. Another possibility consists in not going beyond the first difference which is perceived or which fails to be perceived, thus avoiding these dubious cases. It is obvious of course that one may try to escape this difficulty by taking very large inter- mediate steps, but usually one is afraid of doing this, because one seems to renounce the hope of obtaining a determination of the threshold the accuracy of which may compare favorably with that of determinations by one of the error methods. To these practical difficulties comes the theoretical problem of finding the relation between the results of the method of just perceptible differences and those of the error methods, especially of the method of right and wrong cases. This question is a very urgent one since the individual experiments are the same for *\VuNDT, Physiologische Psychologic, 5 ed., 1902, Vol. I, p. 478. Some criticism of Wundt's view on the relation of the method of just perceptible differences to the method of irregular variation may be found in E. B. Holt, Classification of Psychophysical Methods, Psych. Review, Vol. XI, Nov. 1904, p. 348, who contends that the method of irregular variation does not ap- proach the error methods, as Wundt says, but that it is identical with them. He furthermore remarks that the method of just perceptible differences is not a method, if this word is used in its proper sense, but the statement of the intention to find a significant value for the threshold. tSeveral examples of such series were published lately by Wilhelm Specht, Da^ Verhalten von U titer schiedsschwelle und Reizschuelle im Gebiete des Gehorsinnes, Archiv. j. d. ges. Psychologic, Vol. 9, 1907, p. 207. 44 PROBLEMS OF PSYCHOPHYSICS both methods. The difference merely consists in using several pairs of comparison stimuli in a particular order in the method of just perceptible differences, or in any order in the method of irregular variation, and only one pair in the method of right and wrong cases. To this comes that the results of the error methods, as well as those of the method of just perceptible differences seem to confirm Weber's Law, in spite the fact that they are not comparable among each other. Some investigators believed that the results of the method of right and wrong cases were in no direct relation to those of the method of just perceptible differences, but others tried to establish such a relation by math- ematical formulae; this relation however proved to be of very complicated nature. Of course one might take the view that one of these methods is not legitimate, but this view, which is somewhat narrow, would be justifiable only if the interpretation of the results of either one of these methods were absolutely clear. The difficulties of the method of just perceptible differ- ences, which were mentioned above, are considered strong argu- ments against the use of this method and at present many, if not most psychologists would favor the error methods against the method of just perceptible differences, if they had to choose between them, in spite the fact that the foundations of the method of right and wrong cases are little known and not very well under- stood. For the development of the psychophysical methods one circumstance proved to be of fundamental importance: The introduction of the theory of errors by Mbbius and Fechner, which necessitates the division of the judgments into the two classes of correct and wrong cases. The theory of errors of observation gave some insight into the nature of the error meth- ods and one could hope to find the relation of these methods to the method of just perceptible differences, because the Gaussian coefficient of precision (the "mensura j.recisionis") seemed to give a measure of the accuracy of sensations similar to that afforded by the smallest perceptible difference. It escaped attention for a long time that the application of the theory of errors of observation, though helpful for certain purposes, en- tirely excludes the notion of a just perceptible difference. It METHOD OF JUST PERCEPTIBLE DIFFERENCES 45 is the merit of Jastrow and Cattell to liave called attention to this fact. The theory, indeed, starts from the supposition that the probability of every error is a function of the si^e of this er- ror; the theory makes certain assumptions as to the nature of this dependence, which imply that no error, no matter how large, is impossible although its probability may be very small. One also finds that the greatest error which is likely to be committed in a certain series of observations depends on the number of observations, so that the more extended the series is the greater the largest error becomes which is likely to be committed. Cat- tell, supposing that the errors in judgments on differences of intensity of two stimuli follow the same law as the errors of ob- servation,* concludes that there does not exist a just percep- tible difference in any absolute sense of the term, because the smallest difference beyond which all the judgments of a series are correct depends on the number of observations in this series. The supposition that there exists a difference which is always perceived is, therefore, in contradiction with the fundamental supposition of the method of right and wrong cases. Cattell stands in his demonstration entirely on the ground of the theory of errors of observation and he does not go beyond it. His arguments prevail against the criticisms of his view, some of which miss entirely the point of his argument, e. g. the experi- mental demonstration that one may make the difference be- tween two light intensities so small that it cannot be perceived. It seems that the source of these difficulties is the introduction of the distinction between correct and wrong judgments. This is a logical category. What is immediately given are not right and wrong judgments but the judgments "greater", "equal" and 'smaller"; the correctness or incorrectness of the judgments is a secondary feature. By introducing this distinction between correct and incorrect judgments it becomes necessary to dispose in some way of the equality cases, which as a matter of fact were so troublesome a feature in the method of right and wrong cases, that some investigators have tried to get rid of them by not allowing the subject to pass the judgment "equal." The imme- *G. S. FuLLERTOX and J. McKeen C.\TTELL, On the Perception of Small Differences, 1892, p. 12 sqs. 46 PROBLEMS OF PSYCHOPHYSICS diate data of experiments for the determination of the sensitivity of a subject are the observed frequencies of the cases in which the judgments "greater", "equal" and "smaller" were passed. We will try to base our judgment on the sensitivity of a subject on these percentages for various differences, without obliterating one feature of the results by eliminating a class of judgments, and without introducing the logical category of right and wrong judgments. We shall use for this purpose the notion of the probability of a judgment of certain type, which was introduced in the preceding chapters. Let us suppose a subject compares n pairs of stimuli which have one stimulus, the standard, in common. Let the order in which the comparison stimuli are presented be the same for all the pairs, e. g. let the standard be the first stimulus of every pair Let us call the different stimuli with which the standard is compared r^, r,, .... r^, and let us suppose that the "pairs are presented in such an order that ril2 The stimulus rj^ was the first to be judged "greater" N^ times, where The algorithm of the method of just perceptible differences pre- scribes to take the arithmetical mean of all these determinations 48 PROBLEMS OF PSYCHOPHYSICS for the final determination of the just perceptible positive differ- ence. This average is given by In a considerable number of determinations every stimulus r^ will tend to occur in a number of times proportional to its prob- ability P]^. The most probable result for a great number of determinations of the just perceptible positive difference is there- fore T=r,P,+ r,P, + ....+r^P^ (H.) This relation shows that the result of the method of just percep- tible differences depends on the probabilities of the judgments of different types and that, therefore, its basis is identical with that of the error methods. The second remark which we have to make regards equation I. The value of P does not change if the order of the terms q^, qj, qk.i p^ is changed, because a product is independent of the order of its factors. It does not matter either if a stimulus r^^^ is given before r^, because P]j + s does not enter into relation I. This means that the prob- ability of a stimulus being the smallest on which a " greater "- judgment is given does not depend on the order in which the pairs are presented. The method of just perceptible differences, therefore, is not tied down to the rule of presenting the stimuli in the order of their magnitude. Two interpretations may be given to relation II. The first is based on the notion of the mathematical expectation; its psychol- ogical bearing is less obvious but it is moie serviceable for cer- tain practical purposes. Multiplying each stimulus with the probability that it will be obtained as a result of the method of just perceptible differences gives what is called the mathematical expectation for this stimulus being the threshold, and the sum of these products for all the pairs of the series gives the mathe- matical expectation for the entire series. The mathematical expectation for N repetitions of the same event is N times that for a single event and taking the results of many observations for its determination gives the result a greater exactitude. This interpretation of the algorithm of the method of just percep- METHOD OF JUST PERCEPTIBLE DIFFERENCES 49 tible differences is independent from any particular hypothesis as to the law of distribution. One may try to make its meaning clear in this way. Let us suppose that A pays to B a sum of money proportional to the intensity of the comparison stimulus which is obtained as a determination of the just perceptible positive difference, how much must B pay to A in order to in- duce him to make this agreement and to make it a fair wager? Relation II gives the answer that B's payment must be propor- tional to T, because in this case the expectation of A is equal to that of B. The second interpretation is based on the signification of the arithmetical mean for symmetrical distributions. Taking the average of a series of observations means that one tries to determine the most probable value, if the distribution is sym- metrical. The most probable result of one determination by the series t^, rj, r^^ is the one for which the product of formula I is a maximum. Pj^ is a maximum if or introducing the corresponding expressions i = A- 2 i = k-i i = k P^..^ n q,P^ + , n q,....{lU.) i=l i=i i=i By splitting up this relation into two and eliminating the com- mon factors we obtain the conditions Pk.i<1k.iPk and Pi,>qi,Pi,^^ which are identical with -^^^-h.. (IV.) It is, furthermore, a fact that the probability of a "heavier "- judgment becomes the greater the smaller negativ3 differences, and the greater positive differences of the stimuli become, so that relation IV must be simultaneous with Pk-i < Pk< Pk + i (IVa.) This relation shows that the position of the maximum of P;^ depends on the probability of a "h3avier "-judgment for this 50 PROBLEMS OF PSYCHOPHYSICS pair and on those for the pairs immediately preceding and im- mediately following. The formulae IV and IVa give the condi- tions, that the value of a certain Pj^ is greater than those in its neighborhood. If these conditions are fulfilled it does not fol- low that Pk is the absolute maximum (i. e. greater than any other value of the series), but if Pj^ is the absolute maximum the conditions IV and IVa must be fulfilled. Let us suppose that the first k stimuH i\, i\, r^ are chosen in such a way that P, Pk to the caseT(r;^)(^-e) which determines the relation ^■(^k + x)> i^ (V-) ^'^^^ 1 + 2^ This fraction is smaller than 1 and it may represent, therefore, a mathematical probability. It remains to show that it may 52 PROBLEMS OF PSYCHOPHYSICS represent the probability of a "greater "-judgment on one of the stimuli which may be chosen as the k+1 pair of our series. These stimuli have to satisfy the relation We find for the greatest value of the difference ^(rj^)-^(rjj^J l-2e {l-2ey \-e- l+2e 2(l + 2g) which is always negative. From this it follows that it is always possible to find a stimulus rjj + j>rj^ which satisfies the relation V. as long as p(^)-law is only a special case of symmetrical distributions. Under what conditions will the values i\ Nj^. be distributed symmetrically around their mean? The nature of this distribution obviously depends on the values of the P's. The P's are constituted of the p's and of the q's of the pairs and since it depends on our choice which pairs we will use, no a priori statement is possible in regard to any particular series unless one knows the pairs and the respective probabilities of "h" judgments. Generally the distribution will not be symmetrical, but it may very well be that a particular series has a symmetrical distribution or one which approaches this type. A priori one even cannot make the supposition that the distribution is regular i. e. that it shows an uninterrupted increase at first and, after having attained a maximum, an uninterrupted decrease. Indeed, if the compar ison stimuli are picked out entirely at random it may very well be, that the P's increase in value at first, then after having reached a secondary maximum decrease and increase again later on. This will be the case if there are in the series two or more comparison stimuli with only slightly different probabilities, which are smaller than ^, for the appearance of a "'greater- "judg- ment. By taking the stimuli in equal intervals one is to some extent guarded against this eventuality, but if one was unfortunate in the choice of the comparison stimuli one cannot eliminate this influence by any amount of care in the perform- ance of the experiments, or by the combination of any number of observations. It is not possible either to eliminate the influ- ence of a skew distribution by taking the arithmetical mean of a great number of experiments. On account of the fact that one cannot make any statement about the distribution, it is impos- sible to say whether the final result of an individual series of 54 PROBLEMS OF PSYCHOPHYSICS experiments by the method of just perceptible differences has the character of being the most probable value of the threshold or not. The aspect of the problem is entirely different if one has to deal not with the results of one series of comparison stimuli, but with the results of several series with different arrangements of the pairs of comparison stimuli. Owing to the beautiful theorem which Bruns calls the conservation of the ^ {■jr)-type this mixture of independent distributions tends to produce the (p ( ^) -distribution. We have therefore to deal in this case with a symmetrical distribution and the arithmetical mean is the most probable value. In the apparently trivial caution "not to approach the threshold always by the same steps" lies to a large extent the justification of the algorithm of the method of just perceptible differences. Most investigators, as a matter of fact, used different series of comparison stimuli for their deter- minations of the threshold, and this mixture of distributions produced a symmetrical distribution which gave the results a signification independent of the comparison stimuli which had been used. One might have surmised that the results of the method of just perceptible differences must have a definite signifi- cation l^ecause the results of different investigators are in gen- eral agreement in spite of large individual differences. It is hardly imaginable that a regularity like Weber's Law could have been found and verified by independent observations, if every single investigator had determined another quantity, as would have been the case if the results depended essentially on the choice of the comparison stimuli. A very important prac- tical caution for the application of the method of just perceptible differences follows from these considerations: Do not use always the same series of comparison stimuli, but use as many different arrangements as can be done conveniently. In an extended series of observations one will go about systematically in choosing the arrangements. In agreement with the procedure suggested above one will use a series of stimuli of the same difference, but instead of beginning with i\ one will use a stimulus slightly different, say r', so that the second arrangement will be METHOD OF JUST PERCEPTIBLE DIFFERENCES 55 r'3 = r\ + 2d r', = r\+(n-l)d If one intends to use k different arrangements one will choose the stimuli so that no point of the interval is favored which can be done by making r',-r^ = ^. If we have succeeded in choosing d in such a way that none of the pairs complies with relation V (which can be done by choosing d not too small) the P's will increase at first up to a certain maxi- mum, which is reached for the stimulus rj^, and the value of r for which p = ^ must be in one of the intervals r^_^ i\ or r;. i\ + ^. If we have chosen our comparison stimuli in such a way as not to favor any value, there will be an even probability that the stimulus for which P is a maximum is greater or smaller than the stimulus for which p = i. From this it follows that in a large number of determinations of the just perceptible positive difference made with different series of comparison stimuli, the most probable result is given by that amount of difference for which there exists the probability h that the judgment "greater" will be given. Relation IV explains the surprising fact that the just percep- tible difference seems to become smaller, if new comparison stim- uli are interpolated. The result of such an interpolation may be as follows: As a rule this interpolation will be made system- atically and one will not only not avoid interpolating new pairs in the neighborhood of the point where one expects to find the threshold, but one will be careful to do so since this is the region in which one is most interested. This interpolation of new pairs may produce a shifting of the most probal^'.e value, which will no: be the same in the new series as it was in the first. In regard to this change the following rule holds: If the most prob- able result varies it must become smaller necessarily. Indeed, let there be three stimuli rj..,, r;^ and rj^.^, for wliich the prob- abilities of "greater "-judgments are Pi^.,, p;^ and Pk + i- The probabilities that one of these stimuli will be obtained as a deter- 56 PROBLEMS OF PSYCHOPHYSICS miiiation of the just perceptible difference are P^.,, P^ and P^^., respectively. If there exists the relation ■Pk-i<^k>-Pk + i we also must have ^k-i Pk + i- Now let us interpolate a hew stimulus r'j^ between r^., and t^ and another stimulus r'i^_^i between rj^ and r;^.^! so that ^k-i < ''k< 'k<»'k + i< ^k + i where p'^ and p'k + i stand for the probabilities of^ "greater"- judgments on the new stimuli. It follows from p'k + iPk + i- l-Z'k This means that the position of the maximum is not changed by interpolating new stimuli between the stimulus for which P is a maximum and the subsequent stimulus. This fact is not sur- prising since no stimulus higher than rj^ enters into the formula for Pj.. No general statement is possible in regard to the relations _%— >/>!, and p\ p' as well as l-Z'k-i Fk-i <^^^^. It must be decided by formula V which of the 1-^k-i values Pi-, Pk' and P^ is the maximum. It follows from these considerations that the most probable result for the determi- nation of the just perceptible difference may be changed by the interpolation of new stimuli, and if it is changed it must become smaller. This indicates that there is a certain danger in taking intermediate steps which are very small and, generally a little vaguely speaking, one ought not to take intervals which are METHOD OF JUST PERCEPTIBLE DIFFERENCES 57 too small. It will be see later that the size of the interval de- pends on the number of experiments one is willing to make and that there exists a principle by which one's choice of the intervals may be guided. Now let us suppose that the interpolation of new stimuli was carried through in such a way as not to change the position of the maximum of the P's, but that the number of stimuli preced- ing this maximum has been increased. This will have the effect of adding some more factors to the product in formula I, and since every factor is smaller than the unit the values of the P's in the new series must be smaller than those in the first, series. This means that it is less likely that an individual result will give the most probable result, which stands out less distinctly in the new series than it did in the first one and the probability of smaller results has increased correspondingly. ■ These considerations show that the effect of the interpolation of new pairs of comparison stimuli must necessarily be a more or less considerable diminution of the just perceptible difference. By this it is not intended to say that all the observed diminutions of the just perceptible difference are entirely due to this circum- stance, because there is some evidence that the attitude of the subject is not quite the same in judging differences in a series with small intervals than it is in series with large intervals. The study of the influence of such a variation of the conditions on attention is a separate problem, but our considerations show that the interpolation of new pairs of comparison stimuli may have the effect of diminishing the just perceptible difference, even if the subject is not influenced at all by the new condi- tions. The lowering of the threshold is not only not an argu- ment for the ruling out of a particular series, but one would have to be suspicious of a change in the attitude of the subject, if the interpolation of new pairs would constantly fail to have the effect described. Until now w'e have confined our discussion to the just percep- tible positive difference. The complete method of just percep- tible differences requires the determination of three other quanti- ties: The just imperceptible positive difference, the just percep- tible negative difference and the just imperceptible negative 58 PROBLEMS OF PSYCHOPHYSICS difference. All these quantities are expressed by formulae analogous to formula II, so that the theoretical discussions based on this relation are valid for all the quantities in question. Presenting our series of pairs of stimuli in the order r^^, r^^.i, r,, where T^>r^_i> >r2>ri we will have a determination of the just imperceptible positive difference in the first stimulus on which the judgment "greater" is not given, all the preced- ing pairs being judged "greater". Using the same denotation as before we find for the probability that the stimulus r^ will be obtained as a determination of the just imperceptible positive difference because p^ Pn.i Pk + i is the probability that on all the stim- uli greater than i\ a "greater "-judgment will be given and q^ is the probability that such a judgment will not be given on r^. These probabilities F^' are different for the different stimuli and in a great number of experiments the most probable result is that every stimulus will be obtained as a determination of the just imperceptible positive difference in a number of times pro- portional to this probability. The most probable result of these determinations of the just imperceptible positive difference is therefore ' r = }\P\ + r,P', + + >\,P'^ (VII.) Formulae VI and VII are analogous to I and II. If the stimu- lus rj, has the greatest probability for being obtained as a result of a determination of the just imperceptible positive difference we must have or introducing the expressions for these probabilities PnPn-l Pk%.l Pn Pn-1 Pk + 2 (Jk + 1 which are identical with Pk%-i<9k and /^k + i9k>% + i (VIII.) The relations VIII and formula VI may be used for showing that the result of a particular series depends essentially on our METHOD OF JUST PERCEPTIBLE DIFFERENCES 59 choice of the comparison stimuli, but that it is independent from the order in which the pairs are presented. This demonstra- tion requires only a repetition of our above considerations in- troducing the p's in the place of the q's and vice versa. We will consider only the effect of an interpolation of new pairs of com- parison stimuli in that part of a series where the maximum of F'y. is situated. Let us suppose we have three stimuli Tj^.i, Tjj and ri^^i for which the probabilities of "greater "-judgments are Pi^.i, pj^ arill p^ + i, and let their probabilities of being ob- served as results of the just imperceptible positive difference be P'k-i> P'k a^d P'k + i respectively. If F'^ is a maximum we must have the relations g,,< -^^ and q^> J^±i-. 1-^k l-9k + i as it follows from the relations VIII. Now let us interpolate a new stimulus r'l^.i between t-^_j^ and r,^ and another stimulus r'k + i between r^^ and rj^ + j, so that we have ^k-i<^''k.i<''k<^'k + i<'k + i and, therefore, also /'k-i ^'k-i > '/k > 9'k + 1 > 'Zk + 1 We may conclude from this relation that o' < ^'^ ^I k-i^- l-9k but we cannot conclude that 9k > '^'^^^ 1-9'k + i From this it follows that the interpolation of new pairs of com- parison stimuli has no effect on the most probable value of the 60 PROBLEMS OF PSYCHOPHYSICS determination of the just imperceptible positive difference, if stimuli are interpolated which are smaller than the stimulus which has the greatest probability. The interpolation of inten- sities which are larger than the most probable result in the first series may or may not have an effect on the situation of the maximum of probabilities; if the position of the maximum of probabilities has been changed in the new series it must have shifted towards higher values. These considerations bear out the fact, which was observed before, that the result of a determin- ation of the threshold by the method of just perceptible differ- ences depends somewhat on the size of the intervals which are used. It is, therefore, not necessarily a sign of incomplete train- ing of the subject or of his inability to direct his attention on the comparison of the stimuli, if series with small differences fail to give the same result as series with large differences. We supposed in our demonstration that the new series with smaller intervals contained all the stimuli of the old series and some new ones in addition, whereas one frequently will use series of differ- ent stimuli. This case was not considered, because it is not possible to make a general statement about the comparative results of two series, unless some other data are at hand. Our method has, furthermore, the advantage of showing the imme- diate effect of the interpolation of new stimuli. It depends on the distribution of the V\ whether the algorithm of the method of just perceptible differences leads to the deter- mination of the most probable value for the just imperceptible positive difference or not. Also in this case it is indispensable to have the results of a great number of different series in order to warrant the hypothesis of a symmetrical distribution. The arith- metical mean has the character of the most probable value if this condition is complied with. The formula for P'j^ is exactly analogous to the formula for F^, the p's and q's being inter- changed. By repeating our considerations one can show that P'ij is a maximum independent of our choice of the following stimuli if qk = i or since Pk + '7k = lj i^ Pk = i- '^^^ stimulus for which Pk = i is the most probable result for the determina- tion of the just perceptible difference, and we see that the most probable results for the determination of the just perceptible METHOD OF JUST PERCEPTIBLE DIFFERENCES 61 and of the just imperceptible positive difference are identical. Tiie determination of the just perceptible positive difference is based on the probabilities of the stimuli smaller than r^ and the determination of the just imperceptible positive difference is based on the stimuli beyond this value. One, therefore, niust expect to find differences between the results for these values for every particular series of experiments. Taking the average of the just perceptible and of the just imperceptible positive difference for the final determination of the threshold in the direction of increase, has the signification of basing one's deter- mination of the difference for which there exists the probability i that a "greater "-judgment will be given on all the results of the series, and not only on the results for the stimuli in the lower part of the series, as the determination of the just percep- tible difference does, or on those of the upper part, as the deter- mination of the just imperceptible difference does. For the determination of the just perceptible and the just imperceptible negative difference we have to consider the prob- abilities of "smaller "-judgments. Designating by Wi>«2> >u^ the probabilities that in the comparison of the standard with the stimuli r^, r^, r,^ a "smaller "-judgment will be given we have 1 — u^ = z\ 1 — U.^ = V2 for the probabilities that a "smaller "-judgment will not be given. The stimulus r^ is a determination of the just perceptible neg- ative difference, if all the stimuli greater than rj^ were judged "greater" or "equal" and i\ is judged "smaller". The prob- ability of this compound event is f^k = i''nVl ^"k + lWk (X-) 62 PROBLEMS OF PSYCHOPHYSICS The most probable result of a great number of determinations of the just perceptible negative difference is S = r,U,^r,lJ, + + rJJ^ (XL) The just imperceptible negative difference is defined as the small- est intensity on which a judgment is given which is not a "smaller "-judgment, all the stimuli of smaller intensities being judged "smaller." The probability that the stimulus rj^ will be obtained as a determination of the just imperceptible nega- tive difference is equal to the compound probability that all the stimuli of smaller intensity are judged "smaller" and that on this stimulus a judgment is given which is not a "smaller "-judg- ment. This gives U\ = u^u^....U],_^i\ '. (XII.) The most probable result of a great number of observations on the just imperceptible negative difference is S' = r,U\ + r,U\ + + ;,t/', (XIII.) Formula X and XII are analogous to those which we obtained for the probabilities of a stimulus being observed as a result of the determination of the just perceptible and of the just imper- ceptible positive difference. One may find from these formulae the relations which must be satisfied if a stimulus v^ has a max- imum of probability. The investigation of these relations leads to the conclusion, that the theoretical values of the just percep- tible and of the just imperceptible negative difference are the same. The just perceptible negative difference is that intensity of the comparison stimulus for which there exists the probabil- ity \ that a "smaller "-judgment will be given. The just imper- ceptible negative difference is that intensity for which there exists the probability ^ that a "smaller "-judgment will not be given. The average of these quantities is a more accurate de- termination of the intensity for which there exists the probabil- ity \ for a "smaller "-judgment. The result of the method of just perceptible differences has, therefore, a meaning perfectly well definable in terms of the probabilities of judgments of certain type. We admit three METHOD OF JUST PERCEPTIBLE DIFFERENCES 63 types of judgments. There is a certain realm for the intensities of the comparison stimuli inside of which neither the judgments "greater" nor the judgments "smaller" have a probability greater than the sum of the two other types of judgments. Out- side of this realm, however, one of these judgments has a prob- ability exceeding ^. We have explained the reasons which suggest the adoption of the method of irregular variation. The results obtained by this method can be worked out by the method of just perceptible differences, if one takes care to record all the judgments given and if all the comparison stimuli are presented to the subject in one round i. e. that a pair is not presented a second time before all the other pairs were presented. From the records one picks out the stimulus which is the smallest on which a "greater "- judgment was given and the largest on which such a judgment was not given. The first serves for the determination of the just perceptible positive difference, the second for the determin- ation of the just imperceptible positive difference. The next step is to find the greatest stimulus on which a "smaller "-judg- ment is given and the smallest stimulus on which such a judg- ment is not given; these stimuli are determinations of the just perceptible and of the just imperceptible negative difference. The complete records, therefore, may be worked out by the method of just perceptible differences and, since they give the percentages of correct and wrong judgments, also by any one of the error methods. The necessity of recording all the judg- ments given is avoided in that form of the method of just percep- tible differences, which is recommended by most writers, where the outcome of an entire series is reduced to one number. This little saving of clerical work is quite inconsiderable when com- pared with the work actually spent in experimenting, and it is by far outweighed by the possibility of impairing the results through the influence of expectation on the part of the subject. In the complete records one also has data which may serve other purposes than the determination of the threshold. The possi- bility of presenting the pairs in different order in every round, furthermore, gives a means of eliminating influences due to the arrangement of the series. 64 PROBLEMS OF PSYCHOPHYSICS Under the supposition that the judgment on one pair is not influenced by the judgments on any of the preceding pairs, i. e. that the order in which the pairs are presented is irrelevant, the result of the method of just perceptible differences is identical with that of the method of irregular variation. Both methods determine the intensities for which there exists the probability ^ that on the comparison with the standard stimulus the judg- ments "greater" or "smaller" will be given. These methods are entirely independent of any hypothesis as to the law of distri- bution of correct and wrong cases, and this is a great superior- ity over the method of right and wrong cases. The essential feature of the error methods is that one attempts to find certain constants, which enable one to find the relative frequency of correct cases for any difference from the oljserved relative fre- quency of these judgments on one difference or on several dif- ferences, as it was suggested by Miiller and Titchener. It makes but little difference whether one uses for this purpose the simple law of Gauss, or Fechner's double sided law of Gauss, or whether one may use one of Pearson's formulae or any other rela- tion, which by its nature is fit to represent a law of distribution. Among all these different distributions the law of Gauss must necessarily retain a very prominent place for all practical pur- poses, not only because it depends only on one parameter, but also because it is until now the best understood of all the laws of distribution and finally, because it does not suffer from little shortcomings as e. g. the discontinuity of the second derivative in Fechner's generalisation of this law. The method of right and wrong cases may be characterized as a method of interpo- lation under the supposition of a particular law of distribution. The scope of this method is by far more ambitious than that of the method of just perceptible differences or of the method of irregular variation, but it can be reached only .by means of a definite hypothesis which may be dispensed with for either one of the two methods. It still remains to consider what the limits are inside of which we may expect the result of the method of just perceptible differ- ences with a given probability. It seems very obvious to formu- late the problem in this way: METHOD OF JUST PERCEPTIBLE DIFFERENCES 65 The result r, may be expected with the probability P^; a ti „ a it (I ti n i( T) '■2 ^2' The result r„ may be expected with the probability P^^. One makes N trials each one of which must result in ri, or in r,, or .... or in r^^. Which is the probability that the sum of all the results will be S ? One may try to find the most probable result and the limits within which one may expect a deviation with a given probability, just in the same way as one derives these quantities for Bernoulli's theorem. This formulation is nothing but a generalisation of the so-called problem of Moivre. The algebraical difficulties of this problem soon become very great and it does not seem that there exists at present a general solution. In any case it is not likely that the solution will be simple. The second way of formulating the problem starts from the consideration that the most probable outcome of a series of N experiments will be that r^ occurred NP^ times, rj occurred NPj times and so on, and that the theorem of Bernoulli gives the probabilities for the deviations from these most probable results. One therefore may compute the probable deviations and find the probable error of the final result from them. This way leads to a result, but it seems to be the most elegant and most profitable way to start from our first interpretation of formula II and to apply the following theorem of Tchebit- cheff.* Let x, y, z, .... be any magnitudes which may assume different values, each one with a certain probability, and let the values of x be Xj, x,, — x^ to which correspond the probabilities Pv Pi, •••• Pk so that k I P^ = 1; *TcHEBiTCHEFF, Joumal de Liouville, (2) Vol. 12, 1867, p. 177. 66 PROBLEMS OF PSYCHOPHYSICS in the same way let the values of y be Uv Vz, •••• ih with the probabilities Qi, q2, •••• 9i so that and so on. We then have ^- k 1 I p^x. = a, I q^y^=b, 1 1 for the mathematical expectations or averages of x, y, k 1 i'9k = l, and for the mean values of the squares. Under these conditions the theorem holds, that the probability P that the arithmetical mean ^k + Vk+---- of the observed values of x, y, z, ... is contained between the lim- its a + b+ ... . _ 1 /a, + 6,+ . . .. _a' + b'+ .... n t y n n and a + 6+.... is larger than 1- + IV^ 1+&1+ a^ + 6? + This theorem may serve for the determination of the ac- curacy of the results of the method of just perceptible differ- METHOD OF JUST PERCEPTIBLE DIFFERENCES 67 ences. Let us suppose that there were h different series of comparison stimuli, the first arrangement consisting of the comparison stimuli which have the probabilities for being obtained as determina- tions of the just perceptible difference F^, r^, .... "ki in the second arrangement the stimuli r\, r'„ .... r\ were used which had the probabilities pt pr pi r .^f r 2, ■■■■ -r ] respectiveh", for being obtained as determinations of the just perceptible difference; and so on. We then have k 1 1 I for the mathematical expectations in the single series, and k lP,r,^ = a„ Ir\P\ = h„.... for the mean values of the square. With the first arrangement we make n experiments, with the second n' experiments, with the third n" experiments, ...., so that n + n' + n"+ ....=N. It will liave the same effect for the calculation, if we take the mean value and the mean square for each series as often as this series was given. We then have a probability P that the arith- metical mean nr^ + n'r'^. + .... N 68 PROBLEMS OF PSYCHOPHYSICS of the observed values will be contained between the limits fv- na + n'b+ ... 1 Jna^^n'b,+ . . . . na' + n'b'+. N t \ N N and na + n'b+ .. .. _ 1 La, + n'b^+ . . . . _na' +n'b'' + . . . . N t V ^ ^ t^ which is greater than 1 - — . This formula shows the ad- N vantage of using great numbers of experiments, because the dif- t^ ference 1- — approaches the unit if N increases and t remains constant. This means that it is possible to choose the number of experiments so large that one may expect with- a probability as little different from the unit as one pleases, that the actual result will be within given limits from the calculated result. In regard to the limits of accuracy it may be allowed to make the following comparison of the method of just perceptible dif- ferences with the error methods. The limits of accuracy of an empirical determination of a probability depend on the coeffi- cient of precision in Bernoulli's theorem, which is a function of the probability of the event. The coefficient of precision has a minimum for the probability \, so that events which may be compared with the tossing up of a coin have the smallest pre- cision. The coefficient of precision increases as the value of the probability of the event approaches zero or the unit. The accu- racy of the method of just perceptible differences depends on empirical determinations of the P's, which are constituted of the p's which are used in the error methods. Formula II shows that the P's are always smaller than the corresponding p's for the same stimulus except the trivial case k=l where they are eqiud. This shows that in the observation of events based on the P's a greater precision may be expected than for a series of observations based on the p's. The method of just perceptible differences and the method of irregular variations make use of the P's, but the error methods use the p's, so that a number of observations made by the method of right and wrong cases will METHOD OF JUST PERCEPTIBLE DIFFERENCES 69 give a smaller precision than the same number of observations by the method of just perceptible differences or by the method of irregular variation. These methods do not undertake as much as the method of right and wrong cases does, but what they do they do with a great amount of precision. On the basis of these considerations we may give the follow- ing description of the method of just perceptible differences. Prepare a series of comparison stimuli which cover an interval in the beginning of which the subject gives a very high percent- age of "smaller "-judgments and at the end of which there exists a similar high probability for "greater "-judgments. Be careful to apply the standard stimulus and the comparison stimulus always in the same order so as to keep the time error con- stant. Present the pairs in any order and record all the judg- ments given. If the investigation is somewhat extended it is necessary to exhaust the possible orders systematically, so as not to favor any single one. Use different arrangements of com- parison stimuli. All the stimuli of the same arrangement must be gone through, before the same stimulus is presented to the subject another time. Find from the records the smallest com- parison stimulus in every series on which the judgment "great- er" was given, and the largest stimulus on which another but a "greater "-judgment was given. A comiDarison stimulus is the smallest in a series to be judged "greater", if all the compari- son stimuli of smaller intensity were judged "smaller" or "equal" and if this stimulus was judged "greater." A compari- son stimulus is the largest on which a "greater "-judgment was not given, if on this stimulus either one of the judgments "smaller" or "equal" is passed and if all the larger comparison stimuli were judged "greater". These are the data for the determination of the just perceptible and of the just impercep- tible positive difference. Comljine the results of the different series by taking the arithmetical mean of all the ol)servations for the determination of the just perceptible positive difference and for that of the just imperceptible positive difference. The average of the just perceptible and of the just imperceptible positive difference gives that amount of difference, for which there exists the probability h that the judgment "greater" will be 70 PROBLEMS OF PSYCHOPHYSICS given. This value is called the threshold in the direction of in- crease. Find from the same records the largest stimulus on which the judgment "smaller" was given, and the smallest stimulus on which a "smaller "-judgment was not given. A stimulus is the largest of a series on which the judgment "smaller" was given, if this stimulus was judged smaller and if all the larger differences were, judged "equal" or "greater". A stimulus is the smallest on which the judgment "smaller" was not given, if this stimulus was judged "equal" or "greater" and if all the smaller stimuli were judged "smaller". Find the just perceptible and the just imperceptible negative difference by taking the averages of all the observations and combine the results. The final result is that amount of difference for which there exists the probability ^ that the judgment "stnaller" will be given. This quantity is called the threshold in the direction of decrease. The result of the so-called method of just perceptible differ- ences has, therefore, a signification which does not depend on the notion of a difference which is always perceived. The result of this method is expressible in the fundamental terms of the error methods and it was seen that this method has several decisive advantages over the other methods Avhich use the same type of experiments for the measurement of sensitivity. The method of just perceptible differences has the feature of an eminenth^ practical method, in so far as the experiments required are very simple, as its algorithm is extremely easy and its degree of accuracy is great, so that only a relatively small number of observations is required. For all the ordinary purposes of a de- termination of the thi;eshold one may use the method in the form described by the previous investigators and one will obtain serviceable results. For all purposes, however, which require some accuracy one will not fail to keep a complete record of all the judgments given. It will be seen later that such a set of re- sults can be treated rnore exhaustively by another method, which combines the advantages of the error methods with those of the method of just perceptible differences and which yields the result of the latter method almost without any work. This description of the method of just perceptible differences METHOD OF JUST PERCEPTIBLE DIFFERENCES 71 may be illustrated by the example of our experiments on lifted weights. The judgments "greater" and "smaller" correspond TO our judgments ''heavier" and "lighter", and the "guess"-judg- ments of our experiments are equivalent to the equality cas.s. From the observed frequencies of these three types of judgments we may find their probabilities for the differences which were used. The theorem of Bernoulli gives the limits of accuracy of these determinations. These probabilities are the p's in our formulae and we may find from them the "value of the P's for every difference used in the experiments by formula II. The first business is to arrange the results in the way shown in Table 2S (Appendix p. 186). Each one of our seven subjects is given a double column, the first column of which bears the heading "h" and the second the heading "not-h." The column "h" gives the observed relative frequencies of "heavier "-judgments for every difference, and the second column gives the difference of this number from the unit, i. e. the relative frequencies of judgment? which are not "heavier "-judgments. From these numbers one finds the values of the P's, beginning at the top of the table, successively; these values are given in Table 29 (Appendix p. 186). This computation does not take much time if one considers that the product in formula II contains the q's for all the preceding stimuli. One divides conveniently the com- putation in two steps, the first of which consists in finding the sum of the logarithms of the q's and the second in adding the logarithm of the corresponding pj.. A glance at the results in Table 29 shows that the numbers increase at first and decrease after having attained a maximum. The rise and fall of the numbers is uniform not being interrupted by secondary maxima except for subject V where there is a slight notch at 88. The absence of interruptions is doubtlessly due to the fact that the intervals between the comparison stimuli used in the experi- ments are rather large. It will be noticed furthermore, that the maximum of the P's coincides with one of the stimuli at the beginning or at the end of the interval in which the value p = ^ is found. It so happens that the maximum is at the end of the interval for the subjects I to VI, and that the maximum i> found at the beginning of the interval only for the subject 72 PROBLEMS OF PSYCHOPHYSICS VII. The sum of all the P's is given at the bottom of the table. These sums are slightly different from the unit, which indicate that there is a probability which is not negligible that the entire series may be presented to the subject without giving a result for the determination of the threshold. This circum- stance affords a very convenient way of checking the result of the computation. A series of experiments will not give a result for the determination of the just perceptible positive difference, if no "heavier "-judgment is given on any one of the stimuli. The probability of this event is identical with the pro- bability that on all the stimuli judgments are given which are not "heavier "-judgments, and this probability is equal to the product of all the q's. For the computation of the last P one has to form the product of all the q's with the exception of the q for the last stimulus. The product of all the q's, there- fore, may be found by one single multiplication and the result must add up to one with the sum of all the P's. The products of the q's are given in the line marked R. The sum of ^ and R is equal to the unit except in those cases where it was necessary to correct the individual values of the P's in such a way that their sum involved an error of a unit at the last decimal place. We may make the remark that it is necessary to start out with four correct decimals in order to get the final results correct within two decimals. The next step is to find the values of r^P;.and of rj^'Pj^, which also may be done conveniently by successive multiplication. Table 30 (Appendix p. 187) gives the value of rj^Pj^. The num- bers of this table show that the significant values are clustered around a certain point, which is different for the different sub- jects, and that the values of these products decrease very rap- idly on both sides of this point. This indicates that the influ- ence of adding new stimuli at the ends of the series is inconsid- erable. It seems, however, that our series is not extended enough for a determination of the just perceptible positive difference, be- cause the values at the bottom of the table are considerably greater than those at the upper end of the table. We conclude that it would be advisable in a series of experiments which has the main ' purpose of ascertaining the threshold of difference to add at METHOD OV JUST PERCEPTIHLE DIFFERENCES 73 least one more stimulus at the upper end of our series of stimuli. This is also suggested by the fact that the values of R in Table 29 are by no means inconsiderable, so that the chances are that one will not only have a relatively great number of resultless ex- periments, but one also will neglect some terms which are of influ- ence for the determination of the threshold of difference. The value of rj^Pj^ for r=108 gives an indication of the possible influ- ence of the next term and the following comparison shows that the values of R and influence of the last term go parallel to some extent. Subject Value of R r P for 108 gr VII 0.0189 9.2772 VI 0.0128 10.4976 II 0.0050 5.0868 I 0.0025 4.3092 III 0.0017 4.1148 IV 0.0017 2.4732 V 0.0009 1.6524 It is advisable to have in every series at least two stimuli for which the probability of a "heavier"-judgment is very close to the unit because the product of the q's becomes very small in this case. A further observation which is impressed by Table 30 is the marked asymmetry in the distribution of the values. With the exception of subject V the values of i\ F^ increase slowly and show a very steep descent, whereas in the subject V the course is opposite. The skewness is quite unmistakable also in this case, but the slope in the ascent and descent of the values is not as steep for this subject as the descent of the values for the other subjects. The sum of all the r^ Pj,'s is given at the bottom of Table 30. This value is the just perceptible positive difference for this series. If we were dealing with symmetrical distributions this value would give the amount of difference for which there exists the probability ^ that it will l^e judged "greater", but independently from a supposition about tlio form of the distrifjution this number has the signification of the mathematical expectation. It is noteworthy that all these valu3S are smaller than 100 gr, which 74 ' PROBLEMS OF PSYCHOPHYSICS indicates that in the comparison of a 100 gr standard with a 100 gr comparison weight a number of "heavier "-judgments will be given which exceeds 50%. The numbers in our table vary- between 96.9139 and 99.8164, and it is, perhaps, not a chance incident that the smallest value is that of the female observer. Her value is by more than 1 gr smaller than the lowest value of the other subjects. From the values of v^ P^ given in Table 30 we find the values of Tj.^ Pj^ immediately. These numbers and their sums are given in Table 31 (Appendix p., 187). The sum of these numbers gives the mean value of the squares, a number which is needed for the application of the theorem of Tchebitcheff . Subtracting from the sums in this table the square of the numbers at the bottom of the preceding table and taking the square root from the result gives the coefficient which we need for the com- putation of the probability of a deviation by the theorem of Tcheb- itcheff. We obtain in this way the numbers 15.11, 8.64, 5.97, 6.23, 5.20, 12.30 and 14.41 for our seven subjects I. II,.... VII. The most probable result for the determination of the just per- ceptible positive difference and the probabilities for deviations from it may be represented in this way: There exists a proba- t^ bility greater than 1- — that the observed value will fall with- n in the limits 15.11 15.11 , , 99.76 and 99.76 -H for the subject I; t t 8.64 ^ 8.64 ^^ ^, 98.31 " 98.31+- " " " II; ^ 97 5 97 99.82-- " 99.82+- " " " III; t t 97.98 __^:^ - 97.98+ --''- - " " IV; t t METHOD OF JUST PERCEPTIBLE DIFFERENCES 75 5 20 5 ''O 96.91 ^ and 96.91+ '" for the subject V; t t ^ 99.21 - 12.30 99.21 + 12.30 VI; 98.76 - 14.41 98.76 + 14.41 VII. These numbers may be used in two ways; either one may ask how many experiments are needed in order to find the result in- side of a certain interval with a given probability, or one may ask what is the probability that a given deviation will be observed in a certain number of experiments. This affords the possibility of comparing the theoretical results with the actual observa- tions. We may ask, for instance, what is the interval inside of which we may expect the result with the probability ^, if for each of the subjects I, II and III 400 and for every other subject 300 observations on the just perceptible positive difference were made. For the first three subjects we find t = \/200 = 14.14 and for the other subjects t = \/l75 = 13.23. The observed diff- erences from the theoretical value may be found from Table 39 (Appendix p. 191), and they may be compared with the results of the computation. The deviation from the most probable result inside of which the observed result may be expected with a probability greater than ^ is for the I subject 1.07, the observed difference is 0.60 " " II " 0.61 " " 0.40 " " III " 0.42 " " 0.32 " u jy « 0.47 " " 0.54 " " • V " 0.55 " " 0-78 « „ Y « 0.55 " " 0.78 " " VI " 0.91 " " 1.20 " " VII " 1.09 " " 2.11 76 PROBLEMS OF PSYCHOPHYSICS Four of the observed differences are greater and three are smaller than the theoretical result. We would have to expect such a result for an event the probability of which is approximately \. It is noteworthy that the observed and the theoretical deviations are of the same order of magnitude. In order to obtain the empirical determination of the just per- ceptible positive difference we have to go back to the records of the experiments, and we have to find in every column that weight which was the smallest on which a "heavier "-judgment was given. In the example of a record sheet which we have given on page 6 we find in the first column 92, in the second column 96, in the third column 96, in the fourth column 100 and in the fifth column 96 as the smallest weight on which the judgment "heavier" is given. In this way one may find for each comparison weight, how many times it happened that it was judged "heavier" when all the smaller weights were judged "lighter" or "equal". These results are given in the Tables 32-38 (Appendix pp. 188-191). Each subject is given a separate table, which contains the results of every series of 100 experiments in a double column. We have, therefore, 400 determinations of the just perceptible positive dif- ference for the subjects I, II, III and 300 determinations for each of the other subjects. The numbers in the columns under the head- ing Njj give for every weight the number of times it was obtained as a determination of the just perceptible positive difference, and the numbers opposite in the next column give the product of the weight multiplied by the number of times it was observed as the just perceptible difference (rj^ Nj^). This is the number which must be taken for every weight according to the al- gorithm of the method of just perceptible differences. The final determination is obtained by dividing the sum of the rj^ Njj by the total number of observations, i. e. by the sum of the Nj^. The sums are given at the bottom of the tables in the line marked ^ and the results of the final determination of the just percep- tible positive difference are given in the line marked "Average". It will be remarked that the sum of the Nj. is not always equal to 100. This is a consequence of the fact that the sum of the Pj^ is not exactly equal to the unit. The number of those cases which did not give a result for the determination of the just per- METHOD OV JUST PERCEPTIBLK DIFFERENCES I I ceptible positive difference is small, as it may be expected since the smallness of the differences of the sums of the P's from the vmit indicates, that it is a rather rare event that all the pairs are presented to the subject without a single weight being judged "heavier". This event happened most frequently with the sub- jects II, VI and VII, for whom the difference of the sum of the P's from the unit is largest. The comparison of the observed results for the determination of the just perceptible difference with the calculated results given in Table 30 shows that the coincidence of the observed results with the theoretical results is very satisfac- tory even with a relatively small number of observations. An analysis of this coincidence is given in Table 39 (Appendix p. 191). where it is shown how much the result of the observations differs from the theoretical result in every series of 100 experiments. The deviations are very small and they comply with the test of Tchebitcheff's theorem in a very high degree. In a small ma- jority of cases the observed result is found inside the interval for which there exists a probability greater than h. The last column of this table gives the combined result of all the observa- tions. The numbers are found from the data of Table 40 (Ap- pendix p. 192), which shows the results of the determination of the just perceptible difference if the results of all the series are combined. The signification of Nj^ and rj^Ni, is the same as in the previous tables. Every subject is given a double column and the numbers Nj^ in this table are equal to the sum of the cor- responding values given above in the table for the same subject. A test of the correctness of the computation consists in compar- ing the sum of rj^Nj, which is equal to the sum of the correspond- ing values in the former table for the same subject. In close agreement with the results which were found above we find in the majority of cases the value of the just perceptible positive difference below 100, and in those cases where it exceeds 100 the difference is very small. The term "just perceptible positive difference ' ' is, therefore, somewhat misleading, because the actual difference between the two stimuli will be negative in most cases. This is due to the fact that the time error was not eliminated in our experiments. One must keep in mind that we use the term "just perceptible positive difference" merely as an abbreviation 78 PROBLEMS OF PSYCHOPHYSICS for the comparipon stimulus of the pair, which is obtained as a result of a determination of the just perceptible positive difference by the method of just perceptible differences. We know that there exists for this stimulus the probability h that the com- parison with the standard will result in the judgment "greater". It is not to be feared that a serious misunderstanding may arise from this terminology, and for this reason it does not seem ad- visable to change a terminology which is more or less commonly in use. The negative sign of this difference indicates that one must not start from objective equality of the stimuli when applying the method of just perceptible differences but from subjective equal- ity, a fact which was recognised alsa by previous investigators of this method. The judgment which is passed on the compari- son of the stimuli depends on a number of conditions among which the actual difference of the stimuli is only one. We have to distinguish between the actual difference and the effective difference, to use a term of G. E. Miiller, understanding by this term the difference of a stimulus from, subjective equality. The next step to be taken in working out the data by the method of just perceptible differences is the computation of the just im- perceptible positive difference. The necessary data for this com- putation are given in Table 28. Beginning at the bottom of the table we have to form successively the products of the p's and multiply them by the corresponding values of c^. In this way one finds the numbers given in Table 41 (Appendix p. 192), which are the P's for the computation of the just imperceptible posi- tive difference. The P's for 108 are, of course, identical with the probability that on this weight a judgment will be given which is not a "heavier"-judgment. The course of the numbers in this table is similar to that of the numbers in Table 29. It is perfectly regular, not a single break in the ascent or descent oc- curring throughout the entire table. In some cases the ascent is very rapid the maximum being reached almost with a jump (subject I, III and IV), in other cases the ascent and descent take place with approximately equal rapidity (e. g. subject V), so that without further investigation one almost might speak of a symmetrical distribution. The values of the P's become very small for 88 and 84, and since the values for 108 are rather large, METHOD OF JUST PERCEPTIBLE DIFFERENCES 79 it makes the impression as if the table of the distribution of the P's were cut off at the upper end. A consequence of the small values of P for the last differences of the table is that the values of R are insignificant for all the seven subjects. In no case has R a counting figure on the fourth decimal place. The sum of the P's may differ by the unit of the last decimal place from one owing to necessary corrections of the single P's. Multiplying the numbers of Table 41 with the corresponding r's one finds the values r,^ F^, which are given in Table 42 (Ap- pendix p. 193). It becomes evident from this table that the series of comparison stimuli is extended enough in the direction of decrease, but that it is too short in the upper part. The lower part of this table shows very well how rapidly the influence of a comparison stimulus becomes insignificant, if it is somewhat dis- tant from the most probable result. The sum of all the terms ^k ^k gives the just imperceptible positive difference, which is given at the bottom of Table 42 in the line marked^. It will not be void of interest to compare the theoretical values of the just perceptible and of the just imperceptible positive differences. 1 1^ 1 p p t Just perceptible Just imperceptible LUJCL' I, positive difference. positive difference I 99.7642 99.1388 II 98.3050 99.3640 III 99.8164 98.7522 IV 97.9791 98.1866 \^ 96.9139 97.3702 \T 99.2082 100.5176 VTI 98.7590 100.9592 . These numbers show that the just imperceptible positive difference is smaller than the just perceptil^le positive difference in the case of two subjects (I and III) and that it is larger in all the other cases. Several investigators have made the obser- vation that, with their subjects, the just imperceptible positive differences had the tendency to be smaller than the just percep- tible positive differences. This fact was attributed to the influ- ence of attention and it seemed to be an objection against the method of just perceptible differences. Our results bear out, what 80 ' PROBLEMS OF PSYCHOPHYSICS one easily can verify by theoretical considerations, that in some cases the just perceptible positive difference will be fj;reater than the just imperceptible difference and that in other cases the oppo- site will take place. It depends on the chance influence of the choice of the comparison stimuli which difference will be greater. Multiplying the values r^Pj^ by r^ gives the values v^'P^^ which are given in Table 43 (Appendix p. 193). The mean value of the squares is needed for the determination of the limits inside of which the result may be expected to fall with a probability t^ exceeding 1- — . These intervals are 3.86 3.86 99.14 and 99.14 + ^ for the subject I; t t 4.58 4.58 99.36 " 99.36+ " " " II; •? 71 3 71 98.75 -_lii- " 98.75 + " " " III: 98:19- ^'^^ " 98.19 + -^^^^ " " " IV; t 4.64 t 5.10 t 4.24 5 10 97.37-——- " 97.51 + ^^^^ " " " V; 4.24 100.52 "100.52 + " " " VI; 4 5'^ 4 52 100.96 '-^- "100.96+-^ " " " VII. t t The coefficients which determine the size of the interval are smaller for the just imperceptible positive difference than for the just perceptible positive difference, and we may expect that METHOD OF JUST PERCEPTIBLE DIFFERENCES 81 the coincidence of the observed results with the theoretical results is even closer than it was in the first case. For the empirical determination of the just imperceptible posi- tive difference we have to find from the records how many times every weight was the largest on which a judgment was given which was not a "heavier "-judgment. In the example of a record sheet on p. 6 we have 100 in the first column, 92 in the second column, 92 in the third column, 96 in the fourth column and 92 in the last column as empirical deterininations of the just imper- ceptible positive difference. Counting the records over in this way one obtains the data for the construction of the Tables 44-59 (Appendix pp. 194-197), which are similar in their construction to the Tables 32-38. Each subject is given a separate table, which contains in every double column the results of a series of 100 experiments. The numbers in the column Nj^ refer to the number of times a weight was obtained as a determination of the just imperceptible positive difference. The numbers in the col- umns rj^Nj. give the product of the weight with the number of times it was obtained as a determination of the just imperceptible positive difference. The numbers at the bottom of the tables give the sums of all the numbers in the corresponding columns. The averages of the numbers r,^ Nj^ are the empirical determina- tions of the just imperceptible positive difference in these partic- ular series. The results of the observations agree very well not only with the theoretical results but also with the results of the empirical determinations of the just perceptible positive differ- ence. We have seen that in the case of a symmetrical distril)u- tion the just perceptible positive difference is that amount of difference for which there exists the probability \ that a "heav- ier "-judgment will be given, and that under the same condition of a symmetrical distribution the just imperceptible positive differ- ence is that amount of difference for which there exists the probability \ that a "heavier "-judgment will not be given. This is of course the same quantity in both cases and empir- ical determinations of the just perceptible positive difference and of the just imperceptiljle positive difference must give results which are sensibly equal. The comparison of these numbers is given in Table 51 (Appendix p. 197), which contains in every 82 PROBLEMS OF PSYCHOPHYSICS double column the results for the just perceptible and for the just imperceptible positive difference of one series of 100 experi- ments. The columns under the headinf^ A give the results of the determination of the just perceptible positive difference, and those under the heading B give the just imperceptible positive differ- ence. The largest difference between the corresponding numbers in the columns A and B is 2.92 gr (subject III, series I), and the smallest is 0.04 gr for the same subject in the series IVa and in the series IV. The differences between these values are not large, and small differences, furthermore, interchange with large differences; it is, however, remarkable that the just perceptible difference is greater than the just imperceptil)le difference in a large majority of the cases. The results of all the series of 100 experiments each may be combined and a determination of the just imperceptible difference may be oljtfiined from all the data. The combined results are given and worked out in Talile 52 (Appendix p. 198), which is constructed in the same way as Table 40 giving the results for each one of the 7 subjects in one double column. It is of interest to notice that the sum of the N's differs from the total number of experiments for two subjects (IV and VII). This indicates that it is rather rare that the series is gone through without one weight bei;ig judged "not-h." This agrees with Tal)le 41 al- though the observed values are somewhat larger than we may expect on the basis of the numbers of this table. The results of the final determination of the just imperceptible positive difference compare in this way with those for the just percep- tible positive difference. „ , . Just perceptible Just imperceptiljle .^ oUDjeci. . . Tff •,• !•«■ JJirierence. positive difference, positive difference. I 100.36 98.84 . +1.52 II 98.71 98.71 0.00 HI 99.50 99.67 -0.17 IV 98.52 97.97 -fO.55 V 97.69 97.00 +0.69 VI 100.41 100.25 +0.16 VII 100.87 98.37 +2.50 . METHOD OF JUST I'EUCKPTIH LK DIFFEKKXCES 83 A Striking feature of this comparison is tlic sign of the diff- erences. The just perceptible positive difference is almost invari- ably larger than the just imperceptible positive difference. This observation agrees with the results of previous investigators of the method of just perceptible differences, who as a rule attributed this fact to the particular order in which the comparison stimuli were presented. It was supposed that a small stimulus was more readily perceived, if it was preceded b}^ several other small stimuli of decreasing intensity, because attention was better adapted by the preceding stimulation. This explanation can not hold for our experiments because the stimuli were presented in very different order and an adaptation of attention, there- fore, cannot take place. We now turn to the description of our ol)servati()ns on the just perceptible and on the just imperceptible negative difference. The empirical determinations of the probabilities of a " lighter "- judgment are the data which we need for this computation. These numbers are given in Table 53 (Appendix p. 198) for each subject in one double column. The numbers in the columns marked "1" give the probabilities of "lighter " -judgments for the various differences. The numbers in the columns mark- ed "not-1" give the probabilities of judgments which are not ''lighter "-judgments. These probabilities are of course equal to the differences of the numbers "1" from the unit. The num- bers "1" and "not-l" are the values u and v in our theoretical deductions and we find from them the U's by successive multipli- action beginning at the bottom of the table. The values of U and u are identical for the comparison weight 108 gr, as may be seen from Table 54 (Appendix p. I99j, where the values of the U's are given. The increase and decrease of the U's is uniform throughout the whole table. The sums of the U's are very close to the unit as may be seen from the values of R, which are given at the bottom of table. It is instructive to compare the values of R in this table with those of Table 29. 84 PROBLEMS OF PSYCHOPHYSICS Subject. Value of R in Table 29. Value of R in Table 54 I 0.0025 0.0022 . II 0.0050 0.0010 III 0.0017 0.0000 IV 0.0017 0.0003 V 0.0009 0.0005 VI 0.0128 0.0004 VII 0.0189 0.0002 . The values of R for the just perceptible positive difference are throughout greater than those for the just perceptible nega- tive difference, and only for the first subject does the latter value come anywhere near the first. This indicates that the series of comparison weights is better adapted for the determination of the just perceptible negative difference. The largest value of R in Table 54 is 0.0022 or approximately g^- Since R gives the probability that the series of stimuli will be presented to the subject without giving a result for the method of just percepti- ble differences — a fact which one will try to avoid for obvious reasons — one may be satisfied with this series of stimuli in a de- termination of the threshold which does not aim at a higher ac- curacy than it is attainable in some 500 experiments. For the subjects III and IV one even could leave out the comparison stimlus 84 without seriously impairing the efficiency of the series. This observation indicates that the series of comparison stimuli must be adapted to the individual whose sensitivity is to be tested, and that superfluous experimenting may be avoided by an appropriate choice of the stimuli. The values of the u's serve for the computation of the products i\ V^ and r^.-Uk^ which are given in Tables 55 and 56 (Appendix p. 199 sq). which are needed for the calculation of the just perceptible negative difference and for the determination of the reliability of this determination. The data of these two tables allow to represent the determination of the just perceptible negative difference in the following way. There t^ exists a probability greater than 1 that the result of an observation of this quantity will fall within the limits METHOD OF JUST Pf:HCEPTIBLE DIFFERENCES 85 93.08 - -^— and 93.08 + — — for the subject I; t t 7.89 7.89 95.28 •' 95.28 + " " " II; t t 97.40--^^ " 97.40 + -5:^ " " " III; t t 5.74 5.74 95.23--'' "95.23 + " " " IV; t t 5.21 5.21 94.20 " 94.20+ — " " " V; t t 95.25-^^ " 95.52 + ^^^ " " " VI; 96.04-^^^ "'94.04 + —- " " " VII. t t These values are found in the same way as the corresponding values previously given, namely by the algorithm 1 I^Pzk-rV^'kPv-i^'kP^y. k The signification of these numbers is similar to that of the numbers given in Tables 30 and 31, so that it does not need be discussed any more. We proceed immediately to give the re- sults of the observations on the just perceptible negative difference A stimulus will be recorded as an observation of the just percep- tible negative difference, if it is the largest on which the judg- ment "lighter" was given, all the larger stimuli being judged "heavier" or "equal." In the sample of a record sheet given 86 PROBLEMS OF PSYCHOPHYSICS on p. G we have in the first column 88, in the second 84, in the third column 92, in the fourth column 92 and in the fifth col- ums 88 as determinations of the just perceptible negative dif- ference. These results of our experiments are given in the Tables 57-63 (Appendix pp. 200-203), which are constructed in the same wa}' as the corresponding tables for the just perceptible positive dffference. The results for each subject are given in a separate table, which contains the results of every series of 100 experiments in one double column. The headings N,^ and ri.Ni. have the same signification as before. The sums of Ni^ and those of the products Ti^ N^ are given at the bottom of every col- umn and the final result of the determination of the just percep- tible negative difference is given in the line marked "Average". The following table (Table 64, Appendix p. 204) gives the deter- mination of the just perceptible negative difference if all the results are combined. Our theoretical considerations have shown that we may expect the results to fall within rather narrow limits. The results verify this expectation in so far as our observations show not only a close coincidence with the theoretical results, but the results of the different series of 100 experiments of the same subject also show very little variation from each other. This coincidence is illustrated by the data of Table 65 (Appendix p. 205), which gives under the heading IVa, I, III, IV the results for the different series, and in the column "Total" the result of the determination of the just perceptible negative difference for the combined results. If the coincidence of the observed results with the theoretical results is great in the series of 100 experi- ments one cannot help calling this coincidence for the more extended series surprising. The maximum of the deviations is 0.18 gr, the minimum is 0.00 and the average is 0.076 gr. The number of the positive and that of the negative deviations are approximately equal. This shows that the method of just per- ceptible differences is capable of great exactitude under favorable circumstances. We now turn to the study of the just imperceptible negative difference, the data for the computation of which we find in Table 53 (Appendix p. 198). Applying the formula based on the de- finition of the just imperceptible negative difference one finds MKTHOD OF .IIST PERCEPTIBLE DIFFEREXCES 87 by successive multiplication, beginning at the top of Table 53, the values of the U's, which are given in Table 66 (Appendix p. 205). The values of U for 84 are identical with the probabilities of a "not-T'-judgment for this weight. The sums of the U's are only slightly different from the unit, indicating that the proba- bility is very small that a series will end resultless. In one case (subject I) it happened that the sum of the U's exceeds one by the unit of the last decimal place, -which is due to the necessity of correcting the single values of the U's. The course of the in- crease and decrease of the U's is regular throughout the entire table. From these values of the U's one finds the products rj^Ujj and Tj^^Uj^, which are given in the following two' tables(Tables 67 and 68, Appendix p. 206). The sum of the r^JJ^^ is the de- termination of the just imperceptible negative difference. These values compare thus with the just perceptible negative difference: Subject. Just percept i negative differ ble ence. Just imperceptible negative difference. I 93.0849 93.5089 II 95.2780 94.4648 III 97.4041 98.2963 IV 95.2281 95.5572 V 94.1966 94.7417 VI 95.5218 95.0922 VII 96.0379 95.5472 . The just perceptible negative difference is smaller than the just imperceptible difference in four cases and larger in three cases. There is in our results no indication of a constant rela- tion Ijetween the just perceptible and the just imperceptible negative difference. The data of the last two tables may be used for representing the results in this way. There exists a probabil- itv greater than 1 that the result of the observation will fall n within the limits PROBLEMS OV PSYCHOPHYSICS 4.46 , 4.46 , , , . 93.51 and 98.51 + for the subject I; t t 4.71 4.71 94.46 " 94.46+ - " " " ,11; t t 4 99 4 99 98.30--' - " 88.30 + —^- " " " HI; Xi t 5 67 5 67 94.74 _J:^_ " 94.74+ ' " " " IV; t t 4 19 4 19 94.74- ■ " 94.74+ '- " " " V; t t 4 93 95.09 ' " 95.09 + ^^^ " " " VI; t 4 78 4.78 95.55 " 95.55+ " " " VII. The coefficients which determine the limits within which we may expect the result are small, so that a close coincidence of the theoretical results with the observed results may be expected. The just imperceptible negative difference is the smallest stim- ulus of a series on which a judgment is given which is not a "light- er"-judgment. In the example of a record on p. 6, we have in the first column 84, in the second column 88, in the third column 96, in the fourth column 96 and in the fifth column 92 as determi- nations of the just imperceptible negative difference. The re- sults of our experiments are given in Tables 69-75 (Appendix pp. 207-209,211), in the same way as in the previous tables and the sig- nification of the letters is also the same. The next table (Table 76, Appendix p. 210) gives the combined results for the determi- nation of the just imperceptible negative difference. In order METHOD OF JUST PRECEPTIBLE DIFFERENCES 89 to make the comparison between the theoretical and the observ- ed results easier Table 77 (Appendix p. 211) was constructed which shows the coincidence of these quantities. The differences between the observed and the theoretical results in the different series of 100 experiments are small, and the differences between the theoretical results and the results of the combined series (see Ta])le 77 under the heading "Totals") are so inconsiderable that they ma}^ be neglected for all practical purposes; the only- exception is subject VII, for whom this difference is somewhat considerable. This is due to the fact that for this subject all the deviations from the most probable result are positive. This large difference between the observed and the theoretical values for subject VII may be looked at as just as much of a chance as the very close coincidence for the subjects III and VI. It is of importance to compare the results of the observations on the just imperceptible negative difference with those of the obser- vations of the just perceptible negative difference. This serves also the purpose of getting material for answering the question whether it is an advantage to take the arithmetical mean of the just perceptible and of the just imperceptible difference. iubiect . Just percept] ible Just impercepti ible 1 llTT^T'£M negati ve differ ence. negative difference. xyiiicrci I 93.16 93.82 0.66 II 95.24 94.71 0.53 III 97.45 98.32 0.87 IV . 95.05 96.07 1.02 V 94.22 94.91 0.69 VI 95.36 95.04 0.32 VII 96.04 97.44 1.40 The difference between these quantities are small and in this case it will be a decided advantage to take the arithmetical mean as the representative value, as it is prescribed by the method of just perceptible differences. We shall make the following con- siderations on the method of determining the threshold as the arithmetical mean of the just perceptible and of the just imper- ceptible differences. Ther3 are only two cases possible: either 90 PROBLEMS OF PSYCHOPHYSICS the stimulus for which there exists the probability h that a "greater" or that a "smaller "-judgment will be given lies in the interval between the empirical determinations of the just perceptible and of the just imperceptil.)le difference, or the empir- ical determinations lie both on the same side of this value. The arithmetical mean comes closer to the true value than at least one of the determinations in both cases. Add to this that it is very desirable to have the final results in as simple a form as pos- sible, and there will remain little doubt that the combination of the just perceptible and of the just imperceptible difference is a very serviceable means for the determination of the threshold in the direction of increase and in the direction of decrease. These facts are independent of the theoretical consideration that the just perceptible and the just imperceptible differences are empir- ical determination of the same quantity, so that their average has the character of being the most probable result. The two fol- lowing tables serve the purpose of showing the superior accuracy of the mean of the just perceptible and of the just imperceptible differences. Table 78 (Appendix p. 212 sq.) is a composite table of the individual results of all the series. The results for every subject are given in one group. The results of the observa- tions on the just perceptible positive difference, on the just imper- ceptible positive difference, on the just perceptible negative dif- ference and on the just imperceptible negative difference are given on one line for every series of 100 experiments. The theoretical values of all these quantities are given for every subject in the top line of the group. The column marked "Observed values" contains the results of the observations and the numbers opposite to these give the differences between the observed and the theoret- ical values. These differences are given the positive sign if the observed value is greater than the theoretical value, and the negative sign if it is smaller. The averages at the bottom of the columns refer to the differences taken regardless of sign. The data of Table 79 (Appendix p. 214) are arranged in a similar way, the only difference being that the numbers do not refer to just perceptible and just imperceptible differences, but to the averages of the just perceptible and just imperceptible difference, i. e. to the thresholds in the direction of increase and to those in the di- METHOD OF JUST PERCEPTIBLE DIFFERENCES 91 rection of decrease. The comparison of the averages of the differ- ences between the theoretical and the observed values, the differ- ences being taken regardless of sign, with those of the preceding table shows, indeed, that at least one of the just perceptil^le and of the just imperceptil)le tlifferences has a larger sum of the devia- tions of the observed from the calculated value than the corres- ponding threshold. The threshold in the direction of decrease, for instance, is the average of the |ust perceptible and of the just imperceptible negative difference. For subject III the devia- tions of the observed values from the theoretical value of the threshold in the direction of decrease are on the average 0.495 (Table 79), which is smaller than 0.5S0, the average of these devia- tions for the just imperceptible negative difference, but which is greater than 0.4SS (Table 78), the average of the deviations for the just perceptiljle negative difference for the same subject. It happens in four cases (the positive thresholds for the subjects I, II, III and \ll) that the sum of the deviations for the thres- hold is smaller than any one of those for the corresponding just perceptible and just imperceptible differences. It is easy to see that the probability of this event is equal to the prol^ability that the theoretical value lies in the interval between the just percep- tible and the just imperceptible difference, and that it is nearer to the arithmetical mean of the observations than to either one of them. The prol:)aliility of either one of these events is ^ and the probability of the compound events is, therefore, {. Our results show that this event happened in 4 out of 14 cases, which is exactly what we have to expect. This coincidence is one more argument that the conditions of randomness of events are very closely realized in our experiments. The final evaluation of the results of the method of just percep- tible differences is given in the Tables SO and 81 (Appendix p. 215). The just perceptible and the just imperceptible differences are combined to find the threshold in the direction of increase and decrease. The results are given up to the second decimal place. Table 80 gives the theoretical results and the following table gives the observed results. The coincidence of the the- oretical with the observed results is very close for all the sub- jects except for subject VII, for whom it is less satisfactory. The 92 PROBLEMS OF PSYCHOPHYSICS two thresholds define an interval outside of which there exists a probability exceeding ^either for a "heavier" or for a "lighter"- judgment. We may express this fact shortly by saying that for differences beyond the threshold there exists a probability exceeding ^ that they will be recognized. Inside of this interval one can not expect either a "heavier" or a "lighter"-judgment with an even or more than even probability, and for this reason it is, perhaps, appropriate to call this interval the interval of uncertainty. We give here a comparison of the theoretical and of the observed length of this interval. Theoretical value Observed value Subject, I II III IV V VI VII of interval of of interval of Difference uncertainty. uncertainty. 6.15 6.11 -0.04 3.96 3.74 -0.22 1.43 1.70 + 0.27 2.69 2.6S -0.01 2.67 2.78 + 0.09 4.56 5.12 + 0.56 4.07 2.88 -1.19. The observed values exceed the theoretical values in three cases, and they fall short of them in four cases. The average of all the deviations (taken regardless of sign) is 0.34. It is important to have an idea of the accuracy of the determination of this quan- tity, because the interval of uncertainty is the basis of the psychophysical measurements afforded by the method of just perceptible differences, as it is used to-day. Indeed, let us call the intensity of the stimulus which corresponds to subjective equality r, the just perceptible positive difference r'^ and the just imperceptible positive difference r''^. We then have ^o=i (^o' + O and A^o = ''o-''- METHOD OF JUST PERCEPTIBLE DIFFERENCES 93 Callius: the just perceptible negative difference r'^ and the just imperceptible difference r''^ we find in the same way and The quantities A^u^nd A^o are combined as a rule into an aver- age, which is called the mean threshold of difference and which is defined by A'- = i(A'u+A''o). Introducing the values r, r^^, r^ we obtain A '' = H'' - ' u + ' o- »') = H ''o- ^u) and we see from this formula that the mean threshold of difference is nothing else but half the length of the interval of uncertainty. The method of just perceptible differences prescribes to take this quantity as a measurement of the accuracy of sensation, putting this accuracy inversely proportional to the threshold of difference. It is of interest to notice that the value r has dropped out from the formula for the final result of the method of just perceptible differences. It is, therefore, not necessary to determine it and, as a matter of fact, special care must be devoted to the determi- nation of the point of subjective equality in those investigations where only the threshold in the direction of decrease or the one in the direction of increase is determined, or where the amount of overestimation or underestimation is of interest. Observa- tions on the point of subjective equality are only an incident in the method of just perceptible differences. There is nothing in the data of this method which may serve as a definition of the objective difference of stimuli which corresponds to subjective equality, and it is quite impossible that this method gives any indications about it, because its algorithm entirely disregards the "equality"-judgments and a judgment on the difference which corresponds to subjective equality can hardly be based on any- thing else but on the "equality"-judgments. 94 PROBLEMS OF PSYCHOPHYSICS Our discussions show that the result of the experimental pro- cedure which is called the method of just perceptible differences is expressible in terms of probalnlities of judgments belonging to certain types, and that the method even gains in lucidity by doing so. These judgments are, of course, either correct or in- correct and there does not exist a fundamental difference be- tween this method and the error methods in so far as both use the same data. There may be, however, a very striking difference in so far as the attitude of the subject is concerned. The results of every psychological experiment may be worked out in two ways. In the first case all one is after are statistical numbers of rela- tive frequency. The introspections of the subject are of little interest and they are of importance only in so far as they control the correct performance of the experiments. The- only state- ment ^yhich the subject has to make in regard to his mental states is: "I complied with the conditions of the experiment." The sub- ject is regarded as an instrument which automatically shows when its function is not such as it ought to be. The experimental re- sults which one obtains serve for the determination of those factors which are of influence on the function of the subject and the analysis of this function is the only purpose of the inves- tigation. The second w^ay of performing psychological experi- ments is to lay stress primarily on the introspections of the sub- ject. The numbers of relative frequency which are gained in such experiments are of interest merely in so far as they give an indication as to the Constance or variability of the conditions under • which a certain psychical state arises. It may seem advis- able to characterize the first way of experimenting as the psy- chophysical attitude, and the second as the psychological atti- tude in the restricted sense of the w^ord. Both are of equal importance and neither one must be neglected at the expense of the other, but confusion is inevitable if they are not strictly sep- arated. If one uses the method of just perceptible differences as a purely psychological method the only purpose of the ex- perimental arrangement is to help the subject in finding a stim- ulus on which he may give the judgment that it is the smallest difference which he can perceive. The observations which are given alongside with this judgment or in the course of reaching .METHOD OF JUST PERCEPTIBLE DIFI'ERENCES 95 this judgment are by far more important than the numerical value of the observed difference. The variations of the single observations are, perhaps, more interesting than the final result, because they give an indication of the variability of the men- tal state under observation. The purpose of the experiments is absolutely different if one uses the method of just perceptible differences as a purely psychophysical method. In this case one wants to form one's view about the process of judging differ- ences by means of the observed numbers of relative frequency. These numbers are the onl}^ outcome of the experimentation and they must be put to the best possible use, since it must be re- quired that all is got out of the results what is in them. The error-methods are used primarily as psychophysical methods, whereas the method of just perceptible differences may be easily adapted for both puiposes. The notion of a just perceptible difference was the object of frequent attacks which were met by an equally stubborn defence, because this notion seemed to be the indispensable basis of quan- titative psychology. Fechner, who utilized this notion for this purpose, had given it the signification of an increment of sensa- tion thus inaugurating the view that a sensation is the sum of very small sensations. This view in connection with Weber's observation of the Constance of the relative threshold leads to the establishment of Fechner's fundamental formula for psychical measurement. This theory combines in a peculiar way the feat- ures of the psychophysical and of the psychological attitude. The notion of a difference which the subject just perceives, indeed, is a purely psychological one, and it cannot possibly be found in the statistical numbers of relative frequency, unless it is deprived of its psychological character and transformed into the notion of a difference for which there exists the probal^lity h that it will l)e recognized. There is no objection against the view that a sensation is constituted of elementary sensations from the psycho- physical point of view, because one understands in this case b}^ sensation nothing but a certain process, which may very well be the sum of other processes. The question merely is whether this hypothesis is serviceable, but the objection that a sensation is given as a whole and not as the sum of smaller sensations is 96 PROBLEMS OF PSYCHOPHYSICS very obvious and very decisive for the psychological point of view. This objection was raised very soon against Fechner's formula, but there were on the other hand certain physiological facts concerning the theory of sensations which spoke in favor of this view. The possibility of composing sensations of their elements seemed to give the possibility of exhausting the mani- foldness of sensations by a limited number of simple sensations. For these simple sensations one might postulate different phys- iological processes thus reducing the study of sensations to that of a limited number of elementary processes. The principle of specific sense energ}^, thus, became more or less consciously a factor in favor of Fechner's view. At the same time the notion that a sensation is the sum of elementary sensations opened up the way for the application of the theory of errors. The sensation depends only on the combination of elementary sen- sations and it depends on chance which combination is aroused by a certain stimulus. The limits inside of which the result may be expected with a given probability determine the accu- racy or precision of a sensation.* On this basis one may try find the probability of correct judgments for given differences of the intensities of two stimuli. The way in which the solution of this problem was attempted may be illustrated by the following example. There are two rifles, A and B, in a fixed position in such a way that A is aimed at a point which is somewhat higher than the point at which B is aimed. This difference is known. These rifles are fired a great number of times and one observes in each case whether the bullet of the rifle A struck a higher point than that of the rifle B. It is required to form a judgment on the precision of the rifles on the basis of these data. The solution of *The expression, "precision of sensation" is taken from Robert MuellER Uehcr die Gmndlagen der Riclitigkeit der Sinnesaussagc, Journal j. Psychologie u. Neurologie, Vol. Ill, 1904, pp. 112-126, but essentially this notion was used already by Fullerto.nt and C.\TTELL /. c p. 12, "Owing to the complex phys- ical, physiological and psychological antecedents of the perception, the same stimulus is not always accompanied by the same sensation. There is a nor- mal error of observation ...." These authors introduce in the further por- gress of their discussion the hypothesis that this normal error of observation is the sum of elementary errors, a hypothesis which leads of course to the prob- ability integral. MKTHOD OF JUST PEKCKPTIULE DIFFEREN'CES 97 this problem requires the (;')-funotion and one ol)tains formulae wliich are analogous to those used in the method of right and wrong cases. The agreement with the theory of specific sense energy and the possibility of obtaining a safe starting point for the quantita- tive methods in psychology were strong arguments in favor of the view that a sensation is built up from sensational elements. Our considerations show that it is fiot necessary to conceive of the result of the method of just perceptible differences as deter- mining a sensational element, and they leave to this difference merely the dignity of possessing a distinguished numerical value of the prol^ability with which a certain class of judgments is passed. Our analysis shows that it is not necessary to base the theory of psychophysical measurement on this physiological theory, a blow which at the present time is more serious for this theory than for the deductions of quantitative psychology. We see that very diverse interests came into play in the prob- lem of Fechner's fundamental formula. In this formula psy- chophysical and psychological views are intermingled in a very peculiar way, and it seems that this combination of view, points which are essentially different, is the source of the difficulties which arise in the interpretation of this formula. It is indispen- sable to separate the psychological and the psychophysical attitude strictly and to work out both consistently. The statistical num- bers of relative frequency of the different judgments are immedi- ately given and accessible to everybody. The sensations of the subject are not known to anybody but to himself, and we have to disregard them entirely for the purpose of an objective treatment of the results in the same way as we disregard the inner states of a body with which we deal in physics. These considerations show clearly that we do not measure sensations in psychological ex- periments, neither do we determine the relations of sensations to certain differences of stimuli. What we determine are the probabilities of certain events and the nature of their dependence on certain physical conditions, just in the same way as we do not measure heat but certain linear magnitudes. This does not in- terfere with our being able to solve all the problems of psychophys- ics just in the same way as if we could measure sensations. All 1 98 PROBLEMS OF PSYCHOPH YSICS the prol)lems which are treated in quantitative psychology are accessible either Ijy one of the error methods or by the method of just perceptible differences, or by both. We have seen that the result of the method of just perceptible differences is ex- pressible in terms of probabilities, and so are the results of the error methods. It follows from this that every problem which can be treated in psychology at the present time is expressible in terms of probabilities of the different judgments. i CHAPTER IV. THE EQUALITY CASES; In our experiments the cases in which a "guess"-) udgment was given are equivalent to equality cases. The instructions given to the subject call for a "guess"-judgment when no dif- ference between the two stimuli could be detected; there ought to be nothing in the sensation to go upon in the formation of the judgment that either one of the weights is heavier. It is of some interest to study the number of times the second weight was guessed to be lighter or heavier when the subject believed to be unable to detect a difference. We will study the dis- tribution of the guesses between the "hg" and "l2:"-judgmentsby means of the ratio of thenumberof "hg"-judgments tothe number of "Ig "-judgments, because only the relative frequency is of inter- est, and because this ratio allows us to express the results by one number, thus reducing the size of the 'table by one-half. The necessary data are given in Tables 3-9 (Appedix pp. 174-177). These ratios are given in Table 82 (Appendix p. 215) for all the sul^jects except subject III for each pair of comparison weights separately. Numbers of one column refer to the same subject and numbers in one line refer to the same pair of comparison weights. The results for subjects III are not given, because this subject did not succeed in conforming himself to the condi- tions of the experiments in so far as the use of the guesses was concerned. This subject gave very few "guess"-judgments, and those which he gave were almost exclusively "hg"-judgments. An inspection of Table 82 shows that the numbers in the upper third of the table are, generally speaking, smaller than 1 and that they are in the lower part of the table greater than 1 with- out a single exception. This means that for negative differences the "lighter-guesses" are in the majority and that for positive differences considerably more "heavier-guesses" are given. 99 100 PROBLEMS OF PSYCHOPHYSICS There exists, therefore, more thap. an even probabihty that a "guess "-judgment will be correct. This seems to indicate that the subject in giving a judgment on the equality of two stimuli is liable to overlook certain data of the sensation which would l)e suf- ficient to decide his judgment. It remains undecided whether this is generally true of all "ecjuality "-judgments, or whether it is due to the terminology used in our experiments. This termi- nology, indeed, favors the mistaken use of a judgment of the equality of the stimuli given with all positiveness for a judgment of a very faintly perceived difference. With all the subjects, however, there is strong introspective evidence that the "guesses" were indeed judgments on perceived equality, so that one may favor the view that similar conditions will be found also if an- other terminology is used. It is important in the choice of the terminology by which one allows the suliject to express his judg- ments to consider that the subject has the natural tendency not to commit himself. This tendency must not be encouraged by allowing him the choice of a type of judgment to which he may resort as a subterfuge for judgments given with a low degree of confidence. It was seen in the discussion of the method of just perceptible differences that the threshold of increase and the threshold of decrease enclose an interval inside of which neither "heavier"- judgments, nor "lighter''-judgments have a probability equal to or surpassing ^. This method gives no indication about that difference which corresponds to subjective equality. It is evi- dent that the determination of the point of subjective equality must be based on the equality cases. The peculiar difficulties of this problem become apparent by the data of Table 83 (Appendix p. 215). This table gives the numljers of relative frequency of the " g"-judgments for every pair of comparison weights. For the subject I, for instance, these numbers set in with the frequency 0.0622 for 84, increase constantly to 0.4644 for the comparison of 100 gr with 100 gr and after having reached this maximum they drop down rapidly to 0.0911 for 104 and 0.0533 for 108. Which difference shall we take as the representative value for the case of subjective equality? The problem thus arising is frequently met with in quantitative psychology and in other EQUALITY CASES 101 biometrical sciences. The values most frequently cliosen as representative for tlie entire set of results are the median, the mode and the arithmetical mean. It is obvious that the median can not be used in our case, because the range of the differences used is determined not by objective conditions but by the chance element of subjective choice. It would be, perhaps, different if this range -were so wide that the proximal and the distal terms of the table of distribution had a frequency equal to or near to zero. The determination of the mode meets with considerable algebraical difficulties and its position does not throw much light on this problem as will be seen in the following chapter. There remains, therefore, only the arithmetical mean to be considered here. Let us make, for the piu'pose of this discussion, the following considerations. Suppose we make a determination of the standard weight with a very crude balance. The set of weights with which we have to compare the standard weight of unknown intensity may consist of the weights 84, 88, 92, 96, 100, 104, and 108 gr. We make with this instrument as many comparisons of every weight with the standard as we make experiments with a subject. Our balance will show in some cases that the standard weight is heavier than the comparison weight, in other cases it will be found to be lighter and, since our instrument is very in- accurate, we will not be able to detect any difference at all in a certain ntmiber of cases. Those cases where no difference can be detected are noted as determinations of the weight of the stand- ard, and the discrepancies between the single observations will have to be eliminated by the method of least squares. Making these determinations with different instruments and known intensities of weight we shall obtain results which not only allow us to make a statement about the comparative accuracy of the balances, but which also enable us to tell whether our instruments are affected by a constant error. Disregarding entirely the "heavier" and "lighter "-judgments we suppose that our instrument has shown the standard to be equal to every one of the weights as many times as otir subject could not detect a difference between the weights. The total numlier of experiments is the same in both cases. 102 PROBLEr-S OF PSYCHOPHYSICS A set of results arranged for this purpose is given in Table 84 (Appendix p. 216). The columns Nj^ give the number of times each stimulus seemed to be equal to the standard, i.e. was ob- tained as a determination of its weight. These results must be regarded as determinations of different pondus* of the same weight and the column under the heading Nj^rj^ gives the number with which each observation comes down for the determination of the arithmetical mean. The sums of these numbers are given at the bottom of the columns in the line marked -T. Dividing the sum of the Nj^rj^ by the sum of the Nj^ one obtains the most probable value for the determination of the weight of the stand- ard; these numbers are given for each subject in the line marked "Average". One then proceeds to find in the usual way the deviations of the observations from the mean and the sum of their weighted squares obtaining the final result in the form: I 96.10 ± 1.44 II 96.43 + 1.68 III 97.78 ± 1.36 IV 97.15 ± 1.52 V 96.98 + 1.38 VI 97.65 + 1.42 vu 98.03 A- 1.58 We would not hesitate to say that instruments which give such results are not only very inaccurate, but that they are also af- fected by a considerable constant error. The difference of the general mean from 100 gr would have to be taken as the constant error which in all our cases is negative, so that we have clearly to deal only with subjects who overestimate the second weight. The comparison of the accuracy can not be based directly on the numbers given above for the probable error of the mean, because *The use of the English term "weight" would necessitate constant circum- locutions and involve the possibility of a mistake, because this term would be used in two different meanings. It seemed appropriate for this reason to use the Latin term whenever the word "weight" had to be used in the meaning which it has in the theory of least squares. EQUALITY CASES 103 the different series are not equally extended, so that the deter- minations for the different subjects have not the same pondus. The relative pondera may be found by the following considera- tion. It was necessary to make 3150 experiments with subjeet III in order to obtain 16S determinations of the weight of the standard, whereas with subject VI only 2100 experiments were needed to obtain 401 determinations. The pondus which we give to the different determination; depends on the amount of work required for obtaining the result^, and since all the experi- ments were made ^^•ith equal care, we put the pondus propor- tional to the number of experiments performed. If the pondus of the determinations for the first three subjects is 1, that for the subjects I^'-^TI will be only §. It is indifferent whether we base our comparison of the accuracy of the determinations of the weight of the standard by the different subjects on the coefficient of precision or on any other quantity which is derived from it. We choose the probable error of a single determination, this being directly comprobable for all the subjects. The values of the probable errors of a single determination for the different subjects are given here along with the length of the interval of uncertainty determined above by the method of just percep- tible differences. Subject. Probable Error. Interv 'al of Uncertainty I 38.37 6.15 II 35.79 3.96 III 17.69 1.70 IV 26.69 2.69 V 24.07 2.67 VI 34.82 4.56 VII 38.05 4.07. These results show that in general a large probable error and a large interval of uncertaint}' are found in the same subject. This correlation will become clearer if we arrange our seven subjects first by the size of their probable errors and then by the length of their intervals of uncertainty. Beginning with the subject who has the largest probable error and with the one who has the 104 PROBLEMS OF PSYCHOPHYSIGS greatest interval of uncertainty we obtain the following two arrangements. Order of the subjects Order of the subjects by probable error. by interval of uncertainty I I VII VI II VII VI II IV IV V V III ^ III. This little table shows that these two orders go parallel to a cer- tain extent. The subject who appears first in one order has also the first place in the second order, and. the subject who is last in one order has the same place in the other order. There are, furthermore, two subjects (IV and V) who appear at the same places in both arrangements. The other three subjects appear at different places in the two arrangements, but subjects II and VII appear in the same succession, so that in so far as the suc- cession is concerned only subject VI is out of his place. Sub- ject VII is on second place and subject II is on third place in the order by the size of the probable error, and these subjects have third and fourth place in the order by the length of the inter- val of uncertainty. This means that only one out of seven sub- jects appears out of the order of succession in the two arrange- ments. We may express this result by saying that the size of the probable error in general goes parallel to the length of the interval of uncertainty. The calculation of the probable error is based on the sum of the weighted squares of the deviations from the arithmetical mean. G. Fr. Lipps based a calculation of the threshold on the same value, and he found that the square of the quantity which he calls the threshold must be smaller than three times the sum of the squares of the deviations. Our results bear out in a general way that this quantity may be used as a measure- ment of sensations, because the order of the subjects when EQUALITY CASES 105- they are arranged by the size of this quantity will be similar to the one if the subjects are arranged by the length of the inter- val of uncertainty. The same holds good for any other measure- ment of sensations which is based on the sum of the squares of the deviations multiplied by a constant factor. CHAPTER V. The psychometric functions. Until now we treated the probabilities of the judgments for various amounts of difference in the intensities of the stimuli as unrelated quantities. It is a very natural supposition that these probabilities depend on the amount of difference between the standard and the comparison stimulus. We immediately give to this supposition the form in which it is useful for further treatment by assuming that these probabilities are analytic functions of the difference between the stimuli.* There exist, therefore, three different functions one of which expresses the probabilities of "lighter "-judgments, the second the probabili- ties of "heavier "-judgments and the third the probabilities of "equality "-judgments as depending on the amount of difference between the stimuli. This is, indeed, nothing but a formulation of the fact that the probabilities must depend in some way on the amount of difference between the stimuli. We have to ex- pect for large negative differences large percentages of " lighter "- judgments and small percentages of "equality "-judgments and of "heavier "-judgments. In a similar way we expect with a high probability "heavier "-judgments for considerable positive differences, for which differences the probabilities of ."equality "- judgments and of "heavier "-judgments are small. We make no definite hypothesis about the nature of this dependence but we merely try to see how much one can say about the results with the minimum addition of theoretical suppositions. The func- tions which give the dependence of the probabilities of the judg- ments on the differences of the stimuli may be called the functions *This supposition has not the character of a specific hypothesis, but that of a general supposition necessary for mathematical treatment. A dis- cussion of the importance of this far reaching assumption is given in the last chapter. 106 THE PSYCHOMETRIC FUNCTIONS 107 of the distribution of these judgments, or in view of the import- ance which they have for the problems of quantitative psychology one may call them the psychometric functions. This term is an imitation of the term "biometric function" which is in common use for functions which represent the mortality for the different ages. There are as many psychometric functions as one admits types of judgments. In our experiments we have to distinguish the psychometric function for the " heavier", for the "lighter" and for the "guess "-judgments. The number of psychometric func- tions would be increased considerably if the degree of subjec- tive confidence were taken in consideration. Our problem, then, is formulated in this way. The values of a function were observed for a series of differences and the results must serve for the determination of the function; what is the course of this function and what are its values for other differences? The answer is relatively simple if the nature of the function is known. Such a function, indeed, must depend on a certain number of parameters, which can be directly de- termined if the number of observations is equal to their number, and the most probable values of which can be determined by the method of least squares if the number of observations ex- ceeds the number of the constants of the function. If the num- ber of observations is smaller than the number of parameters, the problem is undetermined. If the function is not known, one finds the values for other differences by interpolation. The interpolation may be effected either by means of Newton's method of differences or by Lagrange's formula of interpola- tion. Both methods are based on the assumption that a set of n observations can be represented by an algebraic function of degree (n-1), but even if this supposition is not justified (the function being of higher than the (n-1)'^^ degree), one may take the values given by these formulae as values of the simplest function by which one can account for the n different values.* *The term "the simplest function" is due to Gauss, Theoria intcrpola- tionis mcthodo nuva iractata, Werke, Vol. 3, p. 275 "Quae praecedunt sup- positione innituntur, functionem X ultra potestatem x"^*^ non egredi, sive differentiam m ^^™ una cum superioribus evanescere, in quo casu methodus interpolationis rigorose vera est. Si vero ilia suppositio locum non habet, 108 PROBLEMS OF PSYCHOPHYSICS The results of our experiments when brought into the form in which they are given in Table 85 (Appendix p. 217) represent sets of seven observations on the three psychometric functions for each one of our seven subjects. The observations are made for the standard weight of 100 gr and with the seven comparison weights of 84, 88, 92, 96, 100, 104, and 108 gr. We call the results of these observations a^, a^, a^, a^, a^, a^, and a^. The table gives these values for all the subjects under the headings G, H and L, so that we have the data for the determination of all the psy- chometric functions. The general expression for Lagrange's formula is (X-X,) (X-X3) -.(x-xj (x-x,) (X-X3) ■■■■(x-xj (Xi-X2)(Xi-X3)....(Xi-xJ^l (X2-X,) (X2-X3) ....(jX-xj"' •■*• ^ (X-Xj) (x-x,) ....(x-x^.i) ^ ' (X^-Xj) (x^-x^) ....(x„-xi^.j) where x^ x,, x^^ are the values for which the observations were made and a^, a.^, aj^ are the results of these observations. Introducing the data of our experiments inthisformula we obtain ^ , (x-88) (x-92) (x-96) (x-100) (x-104) (x-108) ^ \^) : — z — .- — 7^ — -z^ — :r: a,- 4. 8. 12. 16. 20. 24 (x-84) (x-92) (x-96) (x-100) (x-104) (x-108) 4. 4. 8. 12. 16. 20 a,+ (x-84) (x-88) (x-96) (x-100) (x-104) (x-108) "~ 8. 4. 4. 8. 12. 16 (x-84) (x-88) (x-92) (x-100) (x-104) (x-108) ■«4 + 1^. O. '±. '±. O. 1^ (x-84) (x-88) (x-92) (x-96) (x-104) (x-108) "^ 16. 12. 8. 4. 4. 8. ■ ^'" rx-84) (x-88) rx-92^ rx-96) Tx-lOO^ Tx-lOS) 20. 16. 12. 8. 4. 4 _^ (x-84) (x-88) (x-92) (x-96) (x-100) (x-104) ^ 24720. 16. 12. 8. 4 ^" interpolatio eo tantummodo tendit, ut loco functionis X functio simplicissim eruatur, per quam valoribus A, B, C, D,.,. satifiat." THE PSYCHOMETRIC FUNCTIONS 109 This equation of sixth degree assumes the value a, for x = 84, the value a, for x = S8, ...and the value a^ for x=108. Intro- ducing for X any particular value one obtains the corresponding value of the psychometric function. It may be allowed to make the following remarks in regard to the theoretical and prac- tical application of this formula. It is at once obvious that this equation can not possibly represent the psychometric function in its entire course. This function," indeed, is by its nature of representing a mathematical probability limited to values be- tween zero and one, the limits included. Values greater than the unit and smaller than zero can not be admitted. An al- gebraic function has the limit co for x= x , i. e. it assumes very great values for considerable values of its variable and one, therefore, must not undertake to extrapolate by this formula. The occurrence of values greater than the unit and of negative | values inside of the realm of the observations has only sympto- matic significance. The practical application of Lagrange's formula is extremely simple and easy although a little laborious. It is a matter of course that one must arrange the computation systematically whenever one has to treat an experimental material which is some- what extended. In this respect it is of importance that the a's are the same throughout the whole computation for a psychometric function of one subject. The interpolation for different values of x will be effected with the same values of the a's but with different values of the coefficients. The values of these coefficients, on the other hand, do not depend on the a's and remain the same for dif- ferent subjects. It is, therefore, possible to arrange the computa- tion in two ways, if one has to treat the results for different sub- jects with whom the same differences were used. The first way is to attack the results for all the subjects at once, making the interpolation for one value of x for all the subjects at the same time. The other way is to carry out all the interpolations for one subject before beginning those for another. Every inter- polated value of our tables is the result of seven multiplications, two additions and one substraction. This work is reduced con- siderably by the logarithmic computation which consists almost of nothing else but copying logarithms, which were looked up once no PROBLEMS OF PSYCHOPHYSICS for all, and performing simple additions, if the computation is ar- ranged appropriately. The only inconvenience of this work is that there are no thoroughgoing checks, so that the calculation must simply be done over again. The results of this computation are shown in the Tables 86-92 (Appendix pp. 218-221). The values of the psychometric functions for the "equality" and " heavier "- judgments were obtained by direct interpolation and the values for the "lighter"-judgments were found by subtraction from ' one. The tables show the values of the psychometric functions for the "heavier", for the "lighter" and for the "equality"-] udg- ments for the values between 84 and 108 gr with intervals of 1 gr, so that three values are interpolated between two observa- tions. The column under the heading G gives the values of the psychometric function for the "equality"-judgments,-and the let- ters H and L at the head of the third and second column stand in abbreviation of the psychometric function for the "heavier" and of that for the "lighter"-judgments. The general trend of these numbers conforms to our expectation about the course of the psy- chometric functions. The numbers in the column G set in with low values and gradually rise up to a certain maximum, and after this is attained they decrease again. The numbers H set in with very low values and increase at first slowly then more rapidly assuming ver}^ high values for positive differences. The mere inspection of the tables shows two striking features. The first is the occurrence of negative values and of values greater than the unit in the results of all the subjects except for the sub- jects IV and VI. These breaks occur at the extreme ends of the tables. The second remarkable feature is the smallness of the values in the column G. These numbers never come any- where near the unit and only for subject I do they approach the value ^. For none of the other subjects do they exceed the value 0.4 and for tw^o subjects they do not reach the value 0.2 (subjects III and V). This confirms our previous observation that one can not speak in any absolute sense of the word of a point of subjective equality, because for this value one would have to require an at least even probability for the "equality"- judgments. There remain still two possibilities of defining the point of subjective equality. In the first case the probability THE PSYCHOMETRIC FUNCTIONS 111 of an equality judgment is greater than the probability of either a "greater" or a "smaller"-] udgment although it remains smaller than h and, therefore, smaller than the sum of the probabilities of the "greater" and "smaller"-judgments. The interval of sub- jective equality is represented in this case by those differences in the intensities of the stimuli for which the outcome of a com- parison is more likely to be an "equality '-'judgment than either a "greater" or a "smaller"-judgment, but for which the equality judgments are less frequent than the two classes together. There still remains the difficulty of defining the point of subjec- tive equality, because one has the choice between the centre of the interval of subjective equality and the value for which the probability of an '^equality"-judgment is a maximum. An in- terval of subjective equality exists only if there is an interval inside of which the psychometric function of the "equality "-judg- ments rises above the psychometric functions of the "greater" and of the "smaller"-judgments. If there is no such interval one must define the point of subjective equality as the value for which the psychometric function of the "equality"- judgments attains its maximum. This definition does not imply that "equal- ity"-judgments are more probable than "greater" or "smaller"- judgments, but it merely says that for this point the judgments of equality are more frequent than for any other difference of the stimuli. It is not necessary that there exists an interval of subjective equality in which the psychometric function of the "equality"-judgments rises above the psychometric functions of the "greater" or "smaller "-judgments. This definition of the point of subjective equality is the most general, because it does not make any assumption about the course of the psychometric func tions. The further analysis of the results of our experiments is con- siderably facilitated by the graphical representation of the psy- chometric functions as given in the Charts I-VII. On the lines XX' of the charts the intensities of the comparison stimuli are represented, the unit of the linear measurement corresponding to 1 gr. The ordinates represent the probabilities of the different judgments, the unit of the Y axis being chosen ten times as large as that of the abscissa. The lines NN' are drawn in the unit of 112 PROBLEMS OF PSYCHOPHYSICS Chart I. 84 88 A gQ B/oo Chart II. /04 m THE PSYCHOMETRIC P^UNCTIONS 113 M" 92 96 A idoB Chart III. 92 A 96 Chart IV. 114 PROBLEMS OF PSYCHOPHYSICS ~9l ^4 B'96 "JSo Chart V. -N' THE PSYCHOMETRIC FUNCTIONS 115 Chart VII. distance from XX'. The psychometric function for the "lighter"- judgments are represented in each chart by the curve passing through the point A'; the psychometric function for the "heavier"- judgments is represented by the curve passing through B', and the curve passing through the points E and E' represents the course of the psychometric function for the "equality"-] udg- ments. The point A marks the values of x for which the psy- chometric function for the "lighter"-judgments assumes the value \; A' is the corresponding point on the curve. The point B gives the intensity for which the psychometric function for the "heavier" judgments assumes the value h; B' is the corre- sponding point on the curve. The lines AA' and BB' have the length h. The general course of the curves is the same for all sub- jects. The tracings show at once whether the functions assume negative values or values greater than the unit, because in these cases the curves cross the lines XX' or NN'. We see, furthermore, that the course of the curves is in most cases rather irregular, in- crease alternating with decrease; only for the subjects II, IV and VI does it come near to regularity. There is, however, for all the subjects an interval inside of which all the three psychome- tric functions behave regularly and this interval lies approxi- mately in the middle of the charts. We conclude from this fact that in our experiments the conditions are not quite the same for large and for small differences. This conclusion is in per- 116 PROBLEMS OF PSYCHO PHYSICS feet agreement with introspective evidence and with the obser- vation of previous investigators that the influence of attention in series of comparison stimuli which contain only small differences is different from that in series which contain large differences intermingled with small ones. The points for which the psycho- metric functions for the "heavier" and those for the "lighter"- judgments assume the value J are always inside the interval in which these functions are regular. A close inspection of the curves in these intervals shows that the psychometric functions for the "heavier" and for the "lighter "-judgments differ only very slightly from straight lines. Perry's "test of the stretched thread" discloses scarcely any difference of the curves from straight lines in these intervals. This circumstance suggests the following very handy deter- mination of the value for which the psychometric functions for the "heavier" or for the "lighter "-judgments assume the value +, values of which we know that they have the signification of being the most probable result of the threshold in the direction of increase or of decrease as determined by the method of just per- ceptible differences. The complete solution of this problem would require to solve the equation F(x)=J. The work of solving this equation by one of the methods of approximation is insignifi- cant when compared with the amount of work spent in setting up the equation by the formula of Lagrange, because the actual setting up of the equation requires a great number of long mul- tiplications in which it is difficult to avoid mistakes. It is a very fortunate circumstance that the conditions for finding the ap- proximate value for which p = ^ are such, that we can avoid this long computation. One sees at once from the original data given in Table 85, in which interval the value in question must be sought. One includes the value for which p = Mn narrower and narrower limits by successive interpolations, until the interval is small enough to interpolate on a straight line. The charts show that it is not necessary to go in our results beyond inter- vals of 1 gr. In order to show the handiness of this method we will illustrate the course of this computation by finding the just perceptible positive difference for subject V from the data of Table 85 without having recurrence to those of Table 90, which THE PSYCHOMETRIC FUNCTIONS 117 contains the values of the psychometric functions for this sub- ject. The data of Table 85 show that the difference for which there exists the probability ^ for a "heavier"-judgment must lie somewhere between 92 and 96. The numbers of relative frequency of the "heavier "-judgments (0.1600 for 92 and 0.5133 for 96) suggest that this value will be found in the upper part of the interval nearer to 96 than to 94. We interpolate by Lagrange's formula the value for 95. Introducing x = 95 into the formula given above on p. IDS we find the following values of the coefficients: 7. 3.-1.-5.-9.-13 273 is the coefficient of ai = 0.0267; 4. ^ i. 12. 16. 20 .24 11. 3.-1.-5 .-9. -13 4.. 4. 8. 12. 16. 20 11. 7.-1.-5 .-9. -13 8. 4. 4. 8. ] [2. 16 11. 7. 3.-5.- -9.- -13 12. 8. 4. 4. 8. 12 11, 7. 3.-1. -9.- -13 16. 12.8.4. 4. 8 11. 7.3.-1.- -5.- -13 20. 16. 12.; 8.4 :. 4 11. 7. 3.-1 .-5. -9 65536 1287 32768 15015 65536 15015 16384 " a2 = 0.0233; " a3 = 0.1600; " a,=0.5133; 9009 ^, 65536 " " -^ = 0-6800; 1001 32768 " ae = 0.8700; 231 =_ " " " a=0 94'^'^ 24. 20. 16. 12. 8. 4 65536 ' ^-^^66. It is, perhaps, worth while noticing that all the denominators are powers of 2. This is a consequence of our stimuli being chosen equidistant in intervals of 4 gr. This apparently trivial circum- stance facilitates the computation not inconsiderably, and it is generally recommendable to use intervals which are powers of 2. In some cases it may be more convenient to make use of the method of differences for effecting the interpolation and if 118 PROBLEMS OF PSYCHOPHYSICS ^ one has to deal with equidistant values distributed over an inter- val which is a power of 2 one can apply the very handy formula for the "interpolation in the middle." We begin conveniently by forming the sum of the negative terms, which are in this case the second, fifth and seventh terra. The logarithmic computation may be arranged thus: log 0.0233 =0.36736-2 log 1287 =3.10958 Sum =1.47694 log 32768 =4.51545 Difference =0.96149-4= log 0.00092 log 0.6800 =0.83251-1 log 9009 =3.95468 Sum =3.78719 log 65536 =4.81648 Difference =0.97071 -2 = log 0.09348 log 0.9433 =0.97465-1 log 231 =2.36361 Sum =2.33826 log 65536 =4.81648 Difference =0.52178 -3 = log 0.00332. These are the three negative terms; their sum is 0.00092 0.09348 0.00332 Si = 0.09772 THE PSYCHOMETRIC FUXCTIONS 119 By a similar computation one may find the positive terms and form their sum. Ft)r this purpose one has to form the products for the first, third, fourth and sixth term. log 0.0267 =0.42651-2 log 273 =2.43616 Sum =0.86267 log 65536 =4.81648 Difference =0.04619-4 = log 0.00011 log 0.1600 =0.20412-1 log 15015 =4.17653 Sum =3.38065 log 65536 =4.81648 Difference =0.56417-2 = log 0.03666 log 0.5133 =0.71037- 1' log 15015 =4.17653 Sum =3.88690 log 16384 =4.21442 Difference =0.67248- 1= log 0.47041 log 0.8700 =0.93952-1 log 1001 =3.00043 Sum =2.93995 log 32768 =4.515^5 Difference =0.42450-2 = log 0.02658. 120 PROBLEMS OF PSYCHOPHYSICS This gives for the sum of the positive terms 0.00011 0.03666 0.47041 0.02658 S,= 0.53376 The final result of the interpolation is found by subtracting the sum of the negative terms from the sum of the positive terms, which gives §2=0.53376 Si= 0.09772 0.43604. We find, therefore, that the value of the psj^chometric function of the "heavier "-judgments for 95 is for this subject 0.43604 or rather 0.4360 since the last figure is not exact. This is the value given in Table 90 (Appendix p. 220); all the values given in the tables were found by a similar computation. We see that the value of the psychometric function for 95 is smaller than ^ and for 96 it is greater than h The value of x for which the psy- chometric function assumes the value h lies therefore in this in- terval. The difference between the values of the function for 95 and 96 is 0.0773 and the difference of 0.4360 from 0.5000 is 0.0640, and since 0.0640:0.0773 = 0.83 we have for the value in ques- tion the approximate result 95.83. The same computation may be used for finding the just perceptible negative difference. These results are given for all the subjects in the Tables 93 and 94 (Ap- pendix p. 221). The first two columns give the observed re- sults of the threshold, and the values calculated by the algorithm of the method of just perceptible differences. The last column is marked "Value found by interpolation" and contains the re- sults calculated in the way just described. The results of the ob- servations and the values calculated by the different methods are sensibly equal in all the cases. The differences between the observed results and the values found by interpolation are, gen- THE PSYCHOMETRIC FUNCTIONS 121 erally spciikin*:, iireater than those between the observed results and the values found by the algorithm of the method of just per- ceptil)le differences. The value found by interpolation coincides only in the case of subject IV so closely with the observed result that the difference is consideral)ly smaller than that of the re- sult calculated by the algorithm of the method of just percep- tible differences, although the difference between the latter re- sult and the observed result is only 0.16. This fact that the re- sult of the method of just perceptible differences coincides better with the o])servations is not surprising^ because in our experi- ments only one set of differences was used so that the chance influence of the choice of the set of comparison stimuli is very great. To this fact it is due that the method of just perceptible differences gives not the exact values for which p = ^ or u = ^. The results of the interpolation allow us to determine the interval of uncertainty in a new way; we give these quantities alongside with those found by the algorithm of the method of just percep- tible differences. Interval of uncertainty Interval of uncertaintv ibject. by method of just perceptible differences. by interpolation. I 6.15 7.69 II 3.96 4.33 III 1.43 1.57 IV 2.69 3.02 V 2.67 2.08 VI 4.56 5.22 vu 4.07 5.44. The interpolated values deserve more credit than the results of the method of just perceptible differences, if the latter is not based on the results of experiments with different sets of com- parison stimuli. The values found by interpolation are greater than the results of the method of just perceptible differences for all the subjects except for subject V. It is not void of interest to order the subjects by the size of the interval of uncertainty and to compare this order with the one which is obtained if the sub- 122 PROBLEMS OF PSYCHOPH YHICS jects are ordered by the size of their probable errors which were calcuhited in the hist chapter. Subjects arranged by Subjects arranged by probable error. interval of uncertainty found by interpolation. I I VII \ll II VI VI II IV IV V V III III. The order of all the subjects is the same in both -series except for the subjects II and VI who change places, The probable errors for those two subjects are only slightly different and one may account for this break in the regularity of the two series by chance disturbances of the empirical determination. If this explanation is accepted one comes to the conclusion that the probable error derived from the "equality "-judgments by the method described in the preceding chapter gives a measure- ment of the sensations which goes parallel to that afforded by the threshold. We turn now to the study of the psychometric function for the "equality"-judgments. The curves representing these func- tions comply with our expectation in so far as their general trend is to rise at first and then to decrease after having attained a maxi- mum. They have in common with the curves representing the two other psychometric functions that their course is irregular in the extreme parts of the charts. The fact that the curves do not rise to any great height is only an expression of the ob- servation which was made previously that the relative frequency of the equality cases is very low even in that interval where it is greatest. It is interesting to make the following remark. The points E and E' in the charts mark the points of intersection with the curves for the psychometric functions of the "lighter" and of the "heavier"-judgments, the letter E being given to the point of intersection on the left. The curve for the "equality'-' THE PSYCHOMETRIC FUNCTIONS 123 cases rises above the curves for the ' heavier" an J for tlie "lighler"- jiulgments in those cases wliere the pohit of intersection with the curve for the "lighter"-] udgnients Hes on the left; this state- ment is, of course, restricted to points of intersection which lie withhi the interval of regularity. The point of intersection of the psychometric function for the "equality"-] udgments with the curve for the "lighter"-judgments lies on the left of the point of intersection with the curve for thq ' heavier"-] udgments only in the cases of the subjects I, VI, and VII. There exists for these subjects an interval of subjective equality, which in the charts is marked by the letters E and E', within which there ex- ists a greater probability for an "equality"-judgment than for any other kind of judgment. One must make the distinction be- tAveen an interval of uncertainty and an interval of subjective equality. All subjects have an interval of uncertainty, but only those subjects have an interval of subjective equality for whom there exists an interval inside of which the probability of an ''equality"-judgment exceeds the probabilities for "heavier" or "lighter"-j udgments. The curves for the "equality"-judgments approach more or less the type of a bell shaped curve inside an interval which is rather extended. The curve for the "equality"-j udgments of subject IV resembles as much a probability curve as one may ex- pect from an empirical determination; there occurs only one break at the extreme upper end of the curve. The curve for subject II comes almost as near to the type of a probability curve. With the exception of subject I all the curves approach the type of a probability curve within a considerable interval and even for this subject the course of this curve comes very near to that of a probability curve within the interval from 98 to 104. Outside the interval of regularity one obtains as a rule values which are too high, a fact which may be explained by the remark made above, that according to the observations of previous investiga- tors the psychophysical conditions for the judgments on the com- parison of two weights are not the same in series which contain only small differences as they are in series which contain large and small differences. Variations in the adaptation of attention are possibly the cause of this fact. 124 PROBLEMS OF PSYCHOPH YSICS It is of some importance and not void of interest to find the maximum value of the psychometric function of the "equality"- judgments. The situation of this maximum throws some light on the question of what we have to understand by the point of subjective equality, and the maximum value of the psychometric function of the "equality"-judgments can be put to a very in- teresting use. The exact determinations of the value of x for which the psychometric function attains its maximum requires that the function be actually set up by Lagrange's equation. The maximum is found by differentiating this equation and solv- ing the resulting equation of fifth degree. This process is very laborious and it seems to be advisable' to use the following meth- od. One finds in the tables of the psychometric function of the equality judgments the three values of x for which the function is greatest. Let us call the value whicli precedes the maximum A, the maximum value B and the third value C, and let the corresponding values of x be x,, x^ and Xg. These values are found approximately in the middle of the tables and they are always in the interval of regularity, so that all the values preceding A are smaller than A and all the values following B are smaller than B. We lay through these three points a parabola /(x)=ax' + fox + c in such a way that x^is the origin of the coordinates. This gives /(-1)=A /(0)=B /(I =C which is s system of three linear equations for the determination of the three coefficients a, b and c. The solution gives A + C a = h = 2 C-A -B c = B. The maximum of the function'f (x) is found by /'(x,„)=2ax^ + 6 = 0"' which gives for x,„ b ^"'^~ 2^ THE PSYCHOMETRIC FUNCTION'S 125 or if we express the values of a and 1) In- A, B and C A-C "' 2(A+C-2B) The maximum of the psychometric function is then determined by Xj + Xj^. This value lies on the side of x, or X3 according to whether x„^ has the negative or the positive sign. (1.) The sign of x^ depends on the sign of the difference A-C if B is the greatest value, and the maximum .value obviously is situated nearer to the value for which the function assumes the greater value. (2.) If A is the greatest value, the difference A-C is posi- tive and the sign of x^ depends on the difference A + C-2 B. This difTerence is necessarily negative for a concave function decreas- ing, throughout the interval, i. e. for a function which decreases more rapidly than a straight line, as it is the case of the psychome- tric function of the "equality "-judgments in the interval of regu- larity for values of x which are greater than the value for which the maximum is attained, x^ is negative and the maximum of the psychometric function lies on the side of Xj. (3.) If C is the greatest of our three values the difference A-C has the minus sign and A + C-2 B must be negative, because the psychometric function of the "equality"-judgments is an increasing concave function inside the interval of regularity for values of x which are smaller than the value for which the maximum is attained. Xj^ is positive in this case and the maximum of the psychometric function lies on the side of Xg. The course of this simple calculation may be illustrated by working out the results for sul)ject I. We find in Table 86 that the psychometric function of the "equality"-judgments for this subject has the greatest values for 97, 98 and 99. We therefore have to put A =0.4664 which is the value of the function for x = 97, 5 = 0.4827 " " " " " " x = 98, C = 0.4S46 " " " " " " x = 99. These quantities serve for the determination of x^^. We find from these data A-C = 00182 2(A+C -2 B) =0.0288 x,„=+0.63. The maximum of the psychometric function is, therefore, situated at 98.63. 120 PROBLEMS OV PSYCHOPHYSICS It is of some importance to get an idea about the degree of ap- proximation of this calculation. It is, of course, possible to de- termine by purely theoretical considerations the error which is due to the approximate representation of the course of a func- tion of sixth degree by a parabola, but it is perhaps best to show by an example what this error may bo. This example will show that a considerable amount of work is required in order to ob- tain a determination not quite as good as ours, if the maximum is found by differentiating the equation set up by Langrange's formula. This is due to the fact that the coefficients of the equa- tion in our example were not determined accurately enough, be- cause if they had been, the determination of the maximum would be absolutely exact. This example serves also the ulterior pur- pose of showing that the application of Lagrange's formula is a little cumbersome for some purposes although, or rather because, this formula represents the smallest possible theoretical addition. The idea suggests itself that a more definite assumption might lead to simpler results. It seems convenient to postpone the dis- cussion of this question for a future publication, although we might be in the possession of the solution of this problem. We will determine the position of the maximum of the psy- chometric function of the "equality "-judgments for the subject I by setting up this equation by the formula of Lagrange. The expression of Lagrange's formula has the form (x-88) (x-92) (x-96) (x-100) (x-104) ( x-lOS) F(x) .0.0622- 4. 8. 12. 16. 20. 24 (x-84) (x-92) (x-96 ) (x-100) (x-104) (x-108) " 471l'8. 12. 16. 20 .0.1244 + _ (x-84) (x-88) (x-96) (x-100) (x-104) (x-108) 8. 4. 4. 8. 12. 16 (x-84) (x-88) (x-92) (x-100) (x-104) (x-108) ^ ,,,^ . .0.4422 -h 12. 8. 4. 4. 8. 12 THE PSYCHOMETRIC FUNCTIONS (X-S4 (X-S8 (x-92) (x-96) (x-104) (x-108) + -^^ .0.4644- 16. 12. 8. 4. 4. 8 (x -84) (x-8 8) (x-92) ( x-96) (x-100) (x-108) ^ ^^^ 20. 16. 12^ 8. 4. 4. (x-84) (x-88) (x-92) (x-96) (x-100) (x-104) + -^ ^ ~ -^ .0.0533. 24. 20. 16. 12. 8. 4 It is of advantage for the actual setting ii ^ of the equation to write the formula in this way <^(x) . 0.0622 0(x) 0.1244 0(x) 0.3311 x-8F'2'^3^5"x-88 T'^.3.5 "^ x-92 ~2'\3 ~ (f>)x) 0.422 (/)(x) 0.4644 c6(x) 0.911 + x-96 2'^3' x-100 2'^3 x-104 4^5,3.5 ^(x) 0.0533 x-108 2'^3^5 where .0(x) = (x-84) (x-S8) (x-92) (x-96) (x-100) (x-104) (x-108) Carrying out all the multiplications indicated gives ^^^ ^and contain the coefficients of the powers mk(x-Xk) of X which stand at the head of each column. The terms THE PSYCHOMETRIC FUN'CTIONS 120 ^tJS^'- are positive and the term '2k9v- _ are neg- X - 1132.778365833333333333 + 18411.8220 83973.644081250000000000 1342295.9550 109864.878525000000000000 1732084.7544 787.446871250000000000 12271.2590 195758.747843333333333333 3105063.7904 165770.332198750000000000 2632256.2980 - 29988.415644583333333333 + 472807.4924 - 29988.415644583333333333 X - 13089.584145000000000000 + 210899.0520 144341 .302600000000000000 2290673.2850 8339.445453750000000000 + 130683.8610 165770.332198750000000000 +2632256.2980 98.571 is a possible maximum of the function. This result is by 0.06 smaller than that of our approximate determination. These two determinations of the maximum may be compared with the actual course of the function in the interval from 98.55 to 98.67. By effecting the interpolation in the usual way we find the fol- lowing results a,^(x) : (X - 84)m, .000000021091037326 a,{x) : (X- 92 jm, .000001684061686196 a,(x) : (x-lOO)TOs .000002362060546875 a,^(x) : (x-108)m, .000000018073187934 x' X* .0000 1 240 1 52994791 6 + .0030354220980 1 3889 .000976755777994789 .23568780 1 106770834 .00 I 35 10986328 1 2500 .32 1 504785 1 56250000 .0000 1 1 9327799479 1 .002392890082465277 .393836835937500000 + 29.010490347222222222 30.283901985677083333 2185.368642187500000000 40.738552734375000000 2899.140356250000000000 .299274641927083333 21.031865381944444444 - 1132.778365833333333333 + 18411.8220 83973.644081250000000000 1342295.9550 109864.878525000000000000 1732084.7544 787.446871250000000000 12271.2590 1, .000004085286458331 2, .000003437296549477 .002350449218749996 .001980424804687499 .562620898443500000 .474690136718759000 71.717586197916666666 60.586975781249999999 5 1 34 .55 1 354 1 66666666666 4342.969052864583333333 195758.747843333333333333 165770.332198750000000000 3105063.7904 2632256.2980 Equation: Derivative: .000000647989908854 .000003887939453124 .0003700244 1 4062497 + .08793076 1 724750000 .001850122070312485 +.351723046899000000 11.130610416666666666 + 791.582301302083333333 33 .39 1 S3 1 250000000000 1 583 . 1 64602604 1 66666667 299SS.415644583333333333 2998S .4 1 5644583333333333 + 472807.4924 a,^:x) : ( x- 88.)?n, .000000253092447916 a, - 33.39183125000000x^ + 15S3.164602604166667X - 29988.4156445833333 Putting this equation of fifth degree equal to zero shows that 98.571 is a possible maximum of the function. This result is by 0.06 smaller than that of our approximate determination. These tw^o determinations of the maximum may be compared with the actual course of the function in the interval from 98.55 to 98.67. By effecting the interpolation in the usual way we find the fol- lowing results 130 PROBLEMS OF PSYCHOPHYSICS Value of the psychometric Value of X. fun ctionof the"equality"- jud gments for subject I. 98.55 0.48600 98.57 0.48602 98.59 0.48605 98.61 0.48605 98.63 0.48604 98.65 0.48603 98.67 0.48602. This little table shows that neither 98.57 nor 98.63 is the maximum of the function. The maximum lies in the interval between 98.59 and 98.61 nearer to the second value than to the first; 98.63 is,- therefore, a more exact determination of the maximum than 98.57. Such a difference, however, is insignificant for our purpose, be- cause the maximum value of the psychometric function of the "equality"-] udgments is of greater interest than the value of x for which the maximum is attained. The result of the approximate determination of the maximum certainly falls in the neighbor- hood of the real maximum, and the determination of the greatest value which the psychometric function attains is not considerably affected by a small error, because a function varies only little in the neighborhood of its maximum. The results of the approximate determination of the position of the maximum of the psychometric function of the equality cases for our seven subjects are given here alongside with the arithmeti- cal means of the "equality"-judgments which were found pre- viously. ean Difference 4-2.51 + 1.14 + 2.73 + 0.19 -1.09 + 1.58 -1.23. )ject Maximum Arithmetics I 98.61 96.10 II 97.57 96.43 III 100.51 97.78 IV 97.34 97.15 V 95.89 96.98 IV 99.23 97.65 Vll 96.80 98.03 THE PSYCHOMETRIC FUNCTIONS 131 The maximum comes nearest to the arithmetical mean for the results of subject IV; this is only a consequence of the fact that the curve for this subject is very regular and approaches the sym- metrical type. It is interesting to notice that the interval inside of which the maximum values vary for the seven subjects is by far greater than the interval within which the arithmetical means vary. The smallest value of the maximum is 95.89 and the largest is 100.51, which gives 4.62 for the length of the interval. The arithmetical means vary between 96.10 and 98.01, or in an interval of 1.91. The values of x for which the psychometric function for the "equality "-judgment attains its maximum are those values which have the highest probability of being judged equal to the standard stimulus. This value has to be taken as a determination of the point of subjective equality. It is of interest to notice that the point of subjective equality does not coincide with the arithmetical mean. The value for which the function reaches the maximum is in five cases greater and in two cases smaller than the arithmetical mean; the difference between these two values is 1.50 on the average. The intensities of the compar- ison stimulus for which the maximum values are reached are gen- erally smaller than the standard weights of 100 gr; only subject III is an exception since for this subject the maximum is attained for 100.51. This indicates that the second weight is overesti- mated as a rule. The amount of overestimation, however, is smaller if we take the value for which the "equality "-judgments have the highest probability as representative of the point of subjective equality, than it is if we take the arithmetical mean. The next step is to find the maximum value of the function. This value is found by introducing the value of x for which the function attains its maximum into Lagrange's formula. The values of x for which the psychometric function of the "equality" judgments reaches its maximum are different for the different subjects. For this reason the computation must be arranged in a way slightly different from the one described above and it is not possible to attack the results for all the subjects at once. The computation is considerably facilitated by the circumstance that the observations are made for all the subjects on the same comparison stimuli. It is a consequence of this fact that the con- 182 PROBLEMS OF PSYCHOPHYSICS stant denominators of the terms in Lagrange's formula are the same for all the subjects. Let us call these denomiators mj,m2, m^, so that M; is the constant denominator of the coefficient of a;. Inworkingout the results for subject I we begin by find- ing the logarithms of these quantities and obtain log m, = 6.46969 log /«, = 5.69154 log m3 = 5.29360 log m, = 5.16866 log m, = 5.29360 log mg = 5.69154 log 7/17 = 6.46969. We have found by our abbreviated method that the psychometric function of the equality judgments for the subject I reaches its maximum for x' = 98.63. We have to form the products (x'-88) (x'-92) (x'-96) (x'-lOO) (x'-104) (x'-108) (x'-84) (x'-92) (x'-96) (x'-lOO) ■x'-104) (x'-108) (x'-84) (x'-88) (x'-96) (x'-lOO) (x'-104) (x'-108) (x'-84) (x'-88) (x'-92) (x'-lOO) (x'-104) (x'-108) (x'-84) (x'-88) (x'-92) (x'- 96) (x'-104) (x'-108) (x'-84) (x'-88) (x'-92) (x'- 96) (x'-lOO) (x'-108) (x'-84) (x'-88) (x'-92) (x'- 96) (x'-lOO) (x'- 104) The simplest way to find these products is to form the product P= (x' - 84) (x' - 88) (x' - 92) (x' - 96) (x' - 100) (x'- 104) (x' - 108) and to subtract the single factors. Introducing the value 98.63 for x' we obtain: log 14.63 = 1.16524 log 10 63 = 1 .02653 log 6.63 = 0.82151 log 2.63 = 0.41996 log 1.37 = 0.13672 log 5.37 = 0.72997 log 9.63 = 0.97174 logP =5.27167 TUK PSYCHOMETRIC FUNCTIONS 13:^ The logarithmic computation takes the foUowinii form: log ai =0.79379-2 log m , =6.46969 logP =5.27167 I, =4.06546 2", =7.63498 log 14.63 = 1.16524 i", = 7.63493 Ji-J, =0.43053-4 =log 0.00027 log 32 =0.09482-1 logP =5.27167 I, =4.36649 I^ =6.71807 0.64842- 3 =log 0.00445 ).51996- logP =5.27167 log as =0.51996-1 I, =4.79163 J, =6.11511 = 0.67652-2 = log 0.04748 loga^ =0 64562-1 logP =5.27167 I, =4.91729 I^ =5.58862 1,-1, =0.32867-1 =log 0.21314 log as =0.66689-1 logP =5.27167 I, =4.93856 J, =5.43032 logm, = 5.69154 log 10.63 = 1 02653 i", = 6.71807 logmg =5.29360 log 6.63 =0.82151 I, =6.11511 logm, =5.16866 log 2.63 =0.41996 I. =5.58862 logm^ =5.29360 log 1.37 =0.13672 I, =5.43032 X-J, =0.50824-1= log 0.32229 134 PROBLEMS OF PSYCHOPHYSICS logae =0.95952-2 losP =5.27167 log me =-5.69154 log 5.37 =0.72997 Ji =4.23119 2, =6.42151 1,-1. =0.80968-3= log 0.00645" log a; =0.72673-2 loffP =5.27167 2i =3.99840 I. =7.44143 I, =6.42151 logm^ =6.46969 log 9.68 =0.97174 J, = 7.44143 Ir^2 =0.55697-4 = log 0.00036 The terms resulting from a,, a3 and ag have to be taken with the negative sign; the sum of these terms is 0.00027 0.04748 0.00645 So = 0.05420. The sum of the terms a,, a^, aj, and a^, which have the positive sign, is 0.00445 0.21314 0.32229 0.00036 S, =0.54024 S, = 0.05420 S,- 8, =0.48604. The difference 81-83 = 0.4860 is the value which the psychome- tric function of the "equality"-judgments for subject I attains at the point x = 98.63. THE PSYCHOMETKIC FUXCTIOIVS 18o We sive here tlie results for all our subjects indieatin.ti the value for which the maximum is reached and the value which the psy- cliometric function attains at this point. Subject I II III I\' V YI VII Position of Maximum value of the the maximum psy chometric function 98.63 0.4860 97.57 0.2667 100.51 0.1508 97.34 0.2111 95.89 0.1969 99.23 0.3771 96.80 0.3797. This table shows clearly that the values of the psychometric func- tion of the "equality"-) udgments do not exceed A and, except for subject I, they even do not come anywhere near this value. Let us arran.se our seven subjects by the maximum value which their psychometric function for the "eciuality"-judgments attains, and let us compare this order with those which we obtain when we arrange the subjects (1) by the length of the interval of uncer- tainty determined by the method of interpolation, (2) by the length of the interval of uncertainty determined by the method of just perceptible chfferences, and (3) by the proliable error as determined in the preceding chapter. Order of subjects by maximum of the psychometric function of the "G"-judgments. I VII M II IV V III Order of subjects by interval of uncertainty de- termined by interpolation. I VII VI II IV V III Order of subjects by interval of un- certainty deter- mined by method of just percepti- ble differences. I VI VII II IV V III Order of sub- jects by prob- able error. I VII II VI IV V III. 136 PROBLEMS OF PSYCHOPHYSICS This table shows that the order of the subject is the same in the first and second column; every subject appears at the same place in these two columns. We conclude from this fact that the max- imum value of the psychometric function may seA^e as a measure of the sensitivity of a subject just as well as the interval of un- certainty as determined by interpolation. The order of the suli- jects by the length of the interval of uncertainty as determined by the method of just perceptible differences and the order by the pro])able error are slightly different from those preceding. All the four orders, however, become absolutely identical if sub- ject VI is omitted. This subject is the only one who is out of his place. It is a matter of course that the determinations of the in- terval of uncertainty by the method of just perceptible differences and by interpolation must give similar results, because both meth- ods determine the same quantity and their results must coincide within the limits of the accuracy of an empirical determination. It is more surprising that these results coincide with those obtained by the determinations of the probable error and of the maximum value of the psychometric function of the "equality"-judgments. These two quantities depend entirely on the "equality "-judg- ments, whereas the threshold in the cUrection of increase depends on the probability of "greater"-judgments and the threshold in the direction of decrease depends on the probability of "smaller"- judgments. The threshold, therefore, does not depend directly on the frequency of the "equality"-judgments. The prol)able error and the maximum of the psychometric function of the"equal- ity"- judgments, furthermore, are found by algorithms which have nothing in common. We conclude from the fact, that the results of these four different methods coincide, that there must exist some kind of a relation between the values of the different psychometric functions, and there must also exist a relation be- tween the probable error and the maximum value of the psy- chometric function of the "equality"-judgments. From these facts one can draw certain conclusions as to the nature of the three psychometric functions. The fact that the length of the inter- val of uncertainty, the probable error and the maximum of the psychometric function of the "equality"-judgments are quantities each one of which may serve as a measure of the sensitivity of a THE PSYCHOMETRIC FUNCTIONS 137 subject is an important point for the special study of the psycho- metric functions. One would be free to make an appropriate hypothesis for the explanation of this fact, if one chooses to do so. It is, however, doubtful whether there is any great use in introducing a hypothesis at this point. One reason for introduc- ing a hypothesis consists in simplif\dng the practical application of a method. The determination of the limits of the interval of uncertainty by interpolation and the* computation of the probable error are so handy, that it is not likely that any hypothesis could simplify them very much or define a measure of sensation which is more readily to be found. Another reason for introducing a hypothesis is the interest which we take in connecting different empirical facts, because we gain a new insight by establishing relations between facts. There exists, however, no such need at present and it seems, therefore, to be best to leave this fact for further analysis without mixing it up with any hypothesis. Such a hypothesis would refer very likely to the nature of the dependence of the probabilities of the different classes of judgments on the intensity of the comparison stimulus. A definite assumption on the nature of the psychometric function of the "equality"-cases is indispensable for those sciences which use the results of repeated observations on the same quantity. Psychology has at present no practical interest in making such an assumption. There re- mains the theoretical interest in the problem, which can be satis- fied only by the results of further analysis and not by a hypothesis. The special study of the psychometric functions will be the ob- ject of a future publication. We may sum up the peculiarly ad- vantageous position of psychology by saying that the dependence of the probabilities of the different classes of judgments on the intensity of the comparison stimulus is an object of investigation for psychology, and that it need not be made the object of a hy- pothesis. This problem can be treated only accidentally in other sciences, becau.se all that is needed there is a fixed standard by which the accuracy of different determinations can be compared, and a definite algorithm for the combination of o]:>servations which do not give identical values for the same quantity. We have seen that the probable error and the maximum prob- ability for an "equality "-judgment may be used for a measure- 138 PROBLEMS OF PS YCHOPHYSICS ment of the accuracy of sensation in the same way as the inter- val of uncertainty. These quantities are entirely based on the probabihties of the "equality "-cases. This fact is, perhaps, a lit- tle surprising if one remembers that this class of judgments was more or less of a difficulty for the psychophysical methods, a difficulty which seemed to be so serious that some investigators recommended suppressing these judgments entirely. CHAPTER VI. A GENERA f. INQUIRY CONCERNING THE PSYCHOMETRIC FUNCTIONS. We have avoided introducing into the alwve discussion any one of those notions which in most of the current text-books of psvcholog}" are proclaimed as indispensable presuppositions of this science. There is, for this reason, no necessity for philosophi- cal considerations in our argumentation. The notions of which we have made use are by no means different from those used e. g. in the theory of life insurance or in the formal theory of population. There is, however, one point in our considerations which seems to throw considerable light on the epistemological nature of .some scientific methods and which for this reason de- serves some attention. The point in question is the introduction of the assumption that the probabilities of the different judgments are analytic functions of the amount of difference of the intensities of the stimuli. This assumption leads to the construction of the notion of a psychometric function which was found useful for many pur- poses. We confined our discussions to the considerations of the case that an algebraic function of sixth degree expresses the re- sults of our observations on seven values, because this assump- tion is the smallest possible theoretical addition. The degree of this equation depends merely on the number of the values for which observations were made and the mathematical expression may be adapted to any number of observations whatsoever. Inter- polating by Lagrange's formula has not the character of a definite hypothesis on the nature of the psychometric function, but it is rather a means of completing a set of observations. The method of expressing a set of n observations by an algebraic function of degree n-1 lends itself more readily to different results than any other hypothesis; it is of course indifferent whether the values of these functions are determined by Lagrange's formula of inter- 139 140 PROBLEMS OF PSYCHOPHYSICS polation or whet lier Newton's method of differences is used. We have seen, on the other hand, that an algebraic function can be only a preliminary result, because it cannot possibly give the dependence of the probabilities of the judgments of different types for all the values of the comparison stimulus, since an al- gebraic function must assume values greater than the unit or smaller than zero. The question as to the nature of this depend- ence of the probabilities on the intensity of the comparison stim- ulus can be settled definitively only by experimental evidence; as long as no facts are at hand one is free to introduce anv hy- pothesis which may seem appropriate. The following consider- ations will show that some caution is recommended, because the assumption of a definite form of the psychometric function is a far reaching hypothesis. The immediate result of the observations is a set of results for a greater or smaller number of observed differences. The num- ber of observations must be of course sufficient to determine the constants of the psychometric function the form of which is sup- posed to be known; if the number of observations is greater than the number of parameters in the psychometric functions one may combine the results and determine the most prol^able value of the constants, but the problem remains indeterminate if the num- ber of observations is smaller than the number of constants of the function. After the constants of the function are found one can give the probabilities for all the intensities of the comparison stimulus. This means that a finite number of observations — in many cases a very small number — is sufficient to serve as a basis for an infinity of statements. The probabilities of the dif- ferent classes of judgments depend on the psychophysical condi- tion of the subject under observation, and in the absence of more complete information we can characteiize the psychophysical condition of a subject only hy the probabilities with which the different classes of judgments may be expected for various differ- ences. It would seem that an infinite number of observations were needed for this purpose, because there is no reason a priori why the group of causes which gives a certain probability to the judgment "greater" on the difference x should be in any connec- tion with the group of causes which determine this probability THE PSYCHOMETRIC FUNCTIONS 141 for the difference y. The psychophysical conditions of a subject which determine the probabilities of the judgments of different classes on a given difference come obviously under the heading of what is usually called an attribute or a quality. The existence of a psychometric 'function which can be determined by a finite number of observations implies that there exist certain relations between the different qualities, so that all the qualities can be expressed by a certain number of them. It is obvious that no experience whatsoever can be the basis of such a statement, and it follows from this that the notion of a psychometric function is not a result of experience, but the expression of the methodolog- ical assumption that there exist relations of some .kind between the dift'erent qualities of the subject. There is no doubt that the conclusions drawn from this assumption are in agreement with experience and one therefore can only ask: Why is it that one can conclude from one quality of an object to another quality? Why must an object have certain qualities because it has certain other qualities? The formal character of the logical process is that of a subsumption. The general expression of the psychometric function is the major which is specified by introducing special values. The difficulty is not of logical but of epistemological nature aiid refers to the way in \vhich the psychometric function is established. It is quite obvious that we never can reach a proposition of such generality by induction, because the psycho- metric function makes a statement about every element of a class which is of the order of the continuum, and, as a matter of fact, the number of observations on which this statement is based is rather small. It is, therefore, not a generality of the empirical type. It is, furthermore, clear that the relations which exist be- tween the different qualities of a subject are not of the causal type, if this word is used in its common meaning which refers to a succession of phenomena. Qualities of an object are simultane- ous, not successive, and the relation of qualities, therefore, can- not possibly be of the causal tj^pe. It does not avail, either, to take the view that the qualities which determine the probabilities of a judgment of a certain type for different intensities of the comparison stimulus must follow- a certain law, because they are qualities of the same sul)ject. The only rigorous condition that 142 PROBLEMS OF PSYCHOPHYSICS must be satisfied by qualities of the same object is that they are not contradictory. There is nothing in the quahties which deter- mine the prol^iliihty of a "heavier "-judgment for the difference x which impHes, that there must be a certain other probabihty for another difference y. Such a conchision can only be made after the dependence between these qualities is established l)y means of the psychometric function, and one will tr}^ in vain to formu- late the problem in such a way, that the probability of a "heav- ier "-judgment for the difference y follows from that for the differ- ence X In- logical processes, if one does not use the notion of a psychometric function. This problem, which is decidedly of philosophic nature, is little treated in the literature and it does not seem that anybody has given a solution besides that of Kant. This solution may be formulated in this way. The empirical datum are certain ob- served values which, in themselves, are not a law nor are they in relation to each other. Such a relation can be established merely by representing these data in a possible intuition.* The empiri- cal representative of this intuition, in our case, is supplied by representing the results by points in a plain and by a curve drawn through them. The representation in a pure intuition makes it possible to find a law of the generality of the psychometric functions. Without trying to refute or to confirm Kant's solu- tion we will try to show by the following considerations that the question of the relation between the qualities of an object implies a problem of verj^ great generality, which contains the epistemolog- ical problem of the principle of causality as a special case. The totality of those facts which constitute experience are the material which science works on. There is no primary dis- tinction between these facts, all of which stand on the same level of reality, but a great number of secondary distinctions comes in, which are introduced for the sake of the . description of the phenomena, because the task of giving an exhaustive de- *The word intuition has become customary as a translation of the German term " Anschauung, " but this translation is not quite appropriate in so far as this word carries the side meaning of "as if by inspiration". For this reason it would he advisable to invent a new word for the translation of this purely technical terra. THE PSYCHOMETRIC FUNCTIONS HH scription is made easier by classifyiiifz; the events into (groups. The term "science" is frequently used in two slightly different significations. In the first signification this term designates a group of propositions which describe a certain part of the realm of experience, but in its second meaning this term is applied to every effort or activity which has the purpose of establishing such a system. We shall speak of science only in the first signi- fication as applied to a set of propositions which describe a given part of experience. We may speak of a system of science, if its single propositions are connected in such a way that all the prop- ositions may be deduced from a set of propositions. We may speak in this case of a systematic science in contra-distinction to descriptive sciences which may be able to give an accurate de- scription of all the phenomena in theii* realm of experience, but which are not able to deduce all their propositions from a funda- mental set of propositions. A system of science is economical if the logical processes involved ai'e reduced to the smallest pos- sil)le number. The quality of being economical involves that the set of fundamental propositions is reduced to the smallest possible number; i. e. to those propositions which are necessary and sufficient to deduce all the propositions of the system. Prop- ositions of this type are independent from each other; this quality can be proved by showing that not all the propositions of the system can be deduced, if one of the fundamental propositions, or a group of them, is omitted. Every proposition of a set of independent propositions may be substituted by its logical oppo- site without the conclusions drawn from this new set l)y purely logical processes being contradictory among themselves. The log- ical truth of a system of science is the consistence of any one of its propositions, or of any group of propositions, with any other proposition of the same system, or of any proposition which may be derived by purely logical processes from the same set of fiui- damental propositions. A set of fundamental propositions is log- ically neither true nor false''' if its elements are independent of each other. *For this reason they are sometimes called propositional functions, see B. Russell, Principles of Mathematics, 13, 353; A. N. Whitehead, Intro- duction logiquc a la geometric, Revue de Metaphysique et de Morale, \'ol. 15, 1907, p. 35. 144 PROBLEMS OP' PSYCHOPHYSICS The greatest and most perfect example of such a system of science is geometry, which has in so far a distinguished position among the other sciences as it can state at the start the proposi- tions which will exhaust its field of work. It is due to the investi- gations into the foundations of geometry that one has acquired an insight into the relations of the single propositions to the original suppositions of a science, since it was seen that systems of consist- ent propositions ma}' be derived from fundamental sets of proposi- tions in which one proposition is superseded by its opposite. Physical sciences have at present not yet reached this high state of development, but there is little doul)t that similar sets of neces- sary and sufficient propositions may lie constructed for the differ- ent branches of mathematical ph}-sics, as it was required l:)y Hil- bert.* It is more doubtful whether it will be possible to construct similar sets for the biological sciences, because one has not yet found principles of high generality, and it is e. g. by no means an easy task to state all the principles which are common to phys- ical and chemical physiology. The fact remains that one tries to construct systems which cover a certain range of the field of work. These systems have nothing to do with the existence or non- existence of the objects they speak of. A system of science be- comes applicable to a certain part of experience only by means of the statement, that the objects contained therein are such as those which are spoken of in this science. This application to experience is the test of the empirical truth of a system of science. A system of science is empirically true if an empirical fact corres- ponds to every one of its propositions, and the system is complete if there corresponds to every empirical fact of the realm of expe- rience in question a certain proposition of the system. As a rule one concludes from the fact that certain empirical events comply with the requirements of the system that all its conclusions must be. true,' but one also can argue from the agreement of the conse- **D. HiLBERT, Gdttinger Nadir. Math.-phys. Kl. 1900, p. 272 (also Arch, f. Math. u. Phys. Vol. i, (3 ser.) 1901, p. 62); O. HoeLDER {.\nschauung und Denkcn in der Geometrie, 1900) suspects that the deductions of mechanics might possibly h^ not as pure as those of geometry. THE PSYCHOMETRIC FUNCTIONS 145 quenccs with experience to the truth of the system.* This logical process is under the ,i:;eneral principle that all the conclu- sions drawn by merely logical processes from a proposition which is empirically true are empirically true. This may be illustrated by the example of geometry. Those ol)jects which we call spatial manifoldnesses l)elong to three different types. The sets of fun- damental propositions required for the description of these differ- ent manifoldnesses are identical in, all but one proposition and, since this proposition is independent from the others, one may derive different systems of geometry, in which every proposition is consistent with any other proposition, or group of propositions, of the same system. All these systems, therefore, are logically true, but only one of them has empirical truth too. These three manifoldnesses differ in regard to a certain quantity, the so-called Gaussian measure of curvature, which is capable of empirical determination and the question which of the three systems of geometry is empirically true is reduced to the qiiaestio facti: Which is the value of the measure of curvature of empirical space? Tho.se measurements which show that physical space has a curva- ture which can differ from zero only b}^ a quantity which is smaller than the smallest which can be detected by our instruments of measurement, is the empirical warrant that all the conclusions of Euclidean geometry are empirically true. After a system of science is established and its empirical basis is put into proper light, it is possible to state it entirely as a series of consequences of a simple empirical fact, but our subjective confidence in such systems depends materially on the agreement in which its prop- ositions are found to be with experience. The explanation of a *L. BoLTZMAXN, Uber die Entwicklung der Mcthoden der tlieoretischen Physik in neiiercr Zeit, Jahresbericht d. deutschen Mathematiker-Vereinigung, Vol. 8, 1899, p. 89, says that our confidence in the fundamental equations of electrodj'namics is not so much based on Ampere's experiments, which allow us to state them as on the general agreement of experiment with all the conclusions which may be drawn from them. Tlie case of geometry as science of space is very similar. Nobody doubts geometry but not because one depends so much on those facts which show that actual space, if it can be regarded as a geometrical manifoldness, is a Euclidean manifoldness, but because all the propositions of geometry have been found to correspond to geometrical facts relating to einpirical space 146 PROBLEMS OF PSYCHOPHYSICS phenomenon which belongs to such a system of science consists in the reduction of the proposition which is the description of this phenomenon to the system of fundamental propositions i. e. in showing how this proposition may be obtained as a logical conclusion from them.* The calculus of probabilities, similarly, is a system of proposi- tion derived by, purely logical processes from the notion of a mathematical probability, which has its logical foundation in the hypothetical judgment. There exists no connection between this system and actual events and one tries in vain to deduce any such relation by means of the theorem of Bernoulli or of the theo- rem of Poisson. These propositions become applicable to a cer- tain realm of experience only by means of the statement: The events of this realm of experience have the character of ran- domness in the mathematical sense of the word. It follows from this statement that all the propositions of the calculus of pro- babilities must be true for this group of events, but the truth of every single proposition is based and derived from this one empirical fact that the events have random character. The more frequent practice of concluding from the agreement of all the inferences of the calculus of probabilities with experience to the randomness of the events stands of course on the same epistemological level. The set of fundamental proposition may be subjected to criti- cism or revision, but as long as one stands on the ground of this science one has no possibility, and no need, to doubt the truth of the set of fundamental propositions. All one can do is to state the fact that the conclusions drawn from it coincide or do not coincide with experience. It is not always possible to state the fundamental principles in such a way that one can be sure of their empirical truth; this is the case mostly when the facts science has to work on are yet imperfectly known. In these cases one accepts the formulation which fits best to the facts known; propositions *C. F. Gauss, Magnetismus und M j-^ueto meter, Werkc, Vol. 5, p. 315. " Unter Erklaren versteht aber der Naturforscher nichts anderes, als das Zuriickfiihren auf moglichst wenige und moglichst einfache Grundgesetze, iiber die er nicht weiter hinaus kann, sondern sie schlechthin fordern muss, und aus ihnen die Erscheinungen vollstandig ableitet." THE P.SYt'HOMKTUIC FINC'TIOXS 147 of I his kintl are hypotheses of generalizing cliaracter. Systems whicli are l)ase(l on generalizations from incomplete data are likely to undergo changes due to better information. The cor- rections which our views have to undergo under the influence of the knowledge of new facts must render the system again the one that fits the facts best. This requirement is based on the methodological principle that a proposition based on the obsei'va- tion of empirical facts holds goocU for experience in general as long as no contrary instances are known. The system of science which fits best the facts known has, therefore, the smallest probability of being obliged to undergo a correction under the influence of new information. This rule may l)e formulated, that one chooses among systems of hypothetical character the one which has the greatest stability. It was mentioned above that the difficulties which one encoun- ters in the construction of these systems of science are very differ- ent in the difTerent realms of experience. The totality of all the data which constitute experience may be divided into two classes. The first class comprises all those events which are interpreted as indicating a mental life similar to our own; the second class is constituted by all the events which are not interpreted in this way. Let us designate events of the fii'st class by the capital letters A, B, C, , events of the second class by the small letters a, b, c, and let us use greek letters if we speak of an event without laying stress on its belonging to one of these two classes. The description of the phenomena a, b, c, has made steady progress, but that of the phenomena of the class A, B, C, was found to be singularly difficult. The elements of the class a, b, c, show a certain constancy in their arrangement which makes classification comparatively easy, and which is very important for the study of the succession of these phenomena, understanding by succession the temporal order of the piienomena. This succession is under the general rule of the law of causality which states that there are certain definite rules which regulate the order in which a group of events //, v, o, .... is followed by other events .... /«', v', o' , If the rules are such that a certain group .... p, a, r is always followed by the same group//, a' ,-' , ...., we may say that the succession shows uniformity, but if the 148 PROBLEMS OF PSYCHOPHYSICS rules are such that a group of events .... p, a, z, .... may be fol- lowed at different times by different gioups, although the group which follows it, is well defined, we may say that the succession is regular. The law of causality in this general formulation does not discriminate between events of the classes A, B, C .... and a, 1), c, The difficulty of establishing rules of succession depends to a large extent on the possibility of identif3dng groups of events and on their stability, which consists chiefl}' in their not being broken up or interrupted by other events. The num- ber of rules for the succession of the events of the class a, b, c, .... is so great and that for the class A, B, C, .... is so small, that one has tried to utilize this fact for laying down a general distinction between these two classes of events. Leaving the metaphysical side of this view out of question (that the class A, B, C, .... is made up of those events for which.it is character- istic that they never can be brought under the law of causality) this definition is certainly not serviceable for empirical purposes, because the apparent lack of rules of the succession is also found in other events (events of random character) which we do not classify among A, B, C, Drawing the line of demarcation at those points where the limits of our actual knowledge are, has the inconvenience that'the class A, B, C, .... is cut down by every progress of science. The logical outcome of such a procedure is the denial of the existence of this class, the elements of which are found only outside of the realm of science, a denial, which in turn is confronted with the fact that these events, whatever their general importance may be, are most intimately known and of greatest immediate interest for us. The difference between these two classes does not exist for primitive thinking, which is anthropomorphic to such an extent that all the events seem to belong to the class A, B, C, The above mentioned elimination of this class belongs only to a com- paratively late stage of development. This development is brought about by the consistent and continuous use of certain notions the origin of which we do not discuss here and the metaphysical bear- ing of which we leave aside. In the first rank among these no- tions stand the ideas of substance and causality. Substance is defined as that, the idea of which is the absolute subject of our THE PSYCHOMETRIC FUNCTIONS 140 judgments and which is, therefore, not a determination of any- thing. Substance is not the predicate of any object, but it is the subject to which all attributes refer. All phenomena are regarded as attributes of something, and this is substance which remains unaltered by the change of its determinations. The difference be- tween the two classes of events being established, the phenomena were referred to two entirely different substances, and the problem was to define them and to answer the questions which arise from the application of these notions to experience. The bearer of all the predicates of the class a, b, c, .... can be defined as what is move- able in space, and the term used for this notion is matter. The substance underlying the phenomena A, B, C, .... is characterized by thinking. These are the characteristic qualities of the materia extensa and of the materia cogitans, from which their other deter- minations follow. These notions of mind and matter met with very different suc- cess. On the notion of matter one can build up systemsof almost inexhaustible fertility, but all the efforts to deduce anything from the notion of a thinking substance have been in vain. In the realm of experience this notion is of no use, and beyond this it leads only to endless controversies. The success of the one notion, however, produced a favorable prejudice in favor of the other and numberless attempts were made to define it in such a way that the difficulties which it implies may be avoided. We may call every theory which uses the notion of substance an ontological hypothesis, because it expresses a view on the nature of things. Any ontological theory must take one of the following views. 1.) There exists only one substance, because the other can be reduced to it; 2.) there exist two substances, one of which produces the phenomena A, B, C, .... and the other the phenomena a, b, c, The first h3pothesis is the monistic view, which may be formulated as materialism or as spiritualism of which materialism is the more important partly for reason of the consistency of the system and partly for the number of its followers. Materialism (spiritualism) is the ontological view that extended matter (thinking substance) is the only existing thing. Both forms of the monistic hypothesis meet with one common difficulty, but against materialism there is an argument 150 PROBI.EMS OF PSYCHOPHYSICS available to wliich spiritualism is not open. This argument may be called the epistemological argument against materialism. It is based on the fact, that what is immediately given to us is con- sciousness and that matter is known to us merely by our ideas. To reduce mental life to matter is therefore an explanation per ignotius.-''- Every monistic system is called upon to explain how a substance may produce phenomena so fundamentally different in their qual- ities. Materialism can explain everything but thought; spiritual- ism can explain everything but matter. The materia cogitans can not be reduced to the materia e.rtensa nor vice versa. Materialism and spiritualism are equally open to this argument which may l)e called the ontological argument against monism. This argument is greath^ used in the different refutations of materialism, f find Leibnitz's example of the millj is very likely the first exhibition of it. No new argument against materialism was found in spite of great effort and, as F. A. Lange remarks, T[ it is always the same hit, the impossilnlity to reduce the psychic to the physic, which deals a knock-out blow to materialism. *The trend of this argument goes back to K.vntt K. d. r. V., Werke, ed. Hartenstein, Vol. 3, pp. 606, 607 (omitted in the second edition.) The full statement of this subtle argument is due to Schopenhauer, IT. a. 11'. u. V. Werke, ed. Griesebach, Vol. 1, p. 62 sqs. " Der Materialismus ist also der Versuch das unmittelbar Gegebene aus dem mittlebar Gegebenem zu erklaren." This argument was adopted also by Riehl, Schuppe, Bergmann, Adickes and others; see L. BussE, Geist und K or per, Seek und Leib, 1903, pp. 15-17, who states the argument in full and gives references to further literature. tMaterialism has two forms: 1.) The psychic is a particular kind of matter; 2.) the psychic is a type of motion. The latter view may be formu- lated more cautiously in this way: The effects of the psychic are equivalent to motion. It seems that this formulation avoids the ontological argument but in this formulation, materialism has lost entirely its ontological charac- ter taking the jjlienomenalistic point of view that a force is known to us merely by its efl'ects. JOne finds this argun;:it very frequently.- The last statement is that of C. L. Herrick, The Nature of the Soul, Psych. Rev. Vol. XIV, 1907, i). 208, where the argument is attributed to Rabier. "IF. A. L.\NGE, History of .Materialism (Engl. Trans!.,) 1881, Vol. Ill, iJ. 329. For the critique of other arguments against materialism L. BussE, Geist und Korper, Sccic und Leib, 1903, pp. .50-61. THE PSVCHOMKTUIC FIXCTIONS 151 Tlie diialistic Iwpotheses are in turn coufroutecl with the neces- sity of explaining how tw'O substances which are so fundamentally different can influence one another. The relation of these two substances is the problem to be explained by dualism; the intrin- sic difficulties of this view are so great that the authors who pro- fess these ideas introduce them, very frequently, as the one hy- pothesis against which the smallest number of arguments tells.* 8uch a view, even if full credit is given to all the arguments of the authors, has of course not the character of the most probable, but only of the least objectionable hypothesis, and it amounts almost to a renunciation of every theoretical intelligibility to introduce the incomprehensible right at the beginning of one's explanation.! The dualistic systems occur in the following three forms: 1.) Interactionalism, 2.) occasionalism and 3.) parallelism, the first and third form having two different types. *Prof. [AMES, for instance, introduces in his "Principles of Psychology" the notion of a soul in this way at the end of a long discussion of other views; the trend of Busse's argumentation is similar. fThe same objection holds against the view that one has to consider it as a fact that the psychic influences the physical, and vice versa, and that the incomprehensibility of such an influence, which makes this influence equiva- lent to a miracle, is no objection, " because a miracle that happens every day ceases to be a miracle." This view was taken by Stumpf and Jerusalem. This argument proves too much. It is for instance also available in defence of occasionalism, which is thoroughly acceptable if one is satisfied with resolv- ing the events in an uninterrupted chain of miracles. The argument, however, misses the following point: The difficulty of the problem lies in the correct definition of notions which must be serviceable for the descri])tion of certain facts, about which is little doubt in so far the quaestio facti goes. The notions used until now do not serve this purpose of a correct and contradicticjnless description. In other problems one would dispose of notions which lead to contradictions and try new ones. If the notions which are available should not meet with better success one would suspect either that the problem is insolvable, or that our present means are not suificient for the solution. In the first case one is satisfied with the demonstration that the problem cannot be solved and one leaves the problem aside, as it was done with the i^roblems of squaring the circle and of constructing the perpetuum mol)ile. An example of the second possibility is the problem of n bodies, o( which it was shown lately that our present means are not sufllcient for a general solution. The pecuUarity of the mind-body problem lies in its connection with other highly important questions, which make it desirable to solve the prol^lem in one way or the otiier. 152 PROBLEMS OF PSYCHOPHYSICS Iiiteractionalism is found as the theory of an influxus physicus or as the theory of an influxus psychicus or as a combination of these two theories. The first type of the third form of the dual- istic hypothesis is the pre-estabhshed harmony, and the second is psychophysical parallelism in the proper sense, the principle of which is that mental and physical phenomena are independent from each other, so that one of them cannot be reduced to the other, and that they go parallel without being in causal relation. Psychophysical parallelism was stated in many different ways,, but the characteristic feature of parallelistic systems is the alleged impossibility to reduce psychical to physical phenomena or rice versa and the lack of causal relations between them. A certain form of parallelism (universal parallelism in Busse's terminology) approaches closely the view of the pre-established harmony. Parallelism is called upon to explain the intercourse of two "spir- its,' ' as e. g. in conversation, but this argument avails also against the other dualistic theories except interactionalism. The dualistic systems meet with a peculiar difficulty which is caused by the application of the category of substance. This difficulty consists in the impossibility of demonstrating strictly the existence of other thinking beings, the term "thinking being" referring a subject which is the bearer of conscious states. There is no necessity for the assumption of conscious beings besides the one thinking individual, because what we perceive are phe- nomena belonging to the class of those which are attributed to the materia extcnsa, and only our own conscious states are imme- diately given to us. This view is known under the name of ab- solute idealism or transcendental egotism. The scholastic for- mulation of this view maintains that we perceive only the effects of conscious states and that the conclusion from the effect to the cause is not certain. This argument seems to be inevitable un- der the assumption of a substance which underlies the thinking process and the force and peculiarity of this argument is, per- haps, best characterized b}^ Schopenhauer's remark that trans- cendental egotism remains an unconquerable position which, how- ever, as a serious point of view can be found only in the mad- house. This much disputed argument seems to give a pecufiar position to the mental states and it is not void of interest that it THE PSYCHOMETRIC FUNCTIONS 153 may be applietl to objects of every description. This Ijecomes obvious at once when a question is raised like this: How do we know that there is magnetic substance in a magnet? A magnet is defined by the qualities of bodies in general and in addition to these by all those reactions which are characteristic for the mag- netic state. There is nothing about a substance in the data and all we know about a magnetic body is exhausted with their de- scription. In the same way we da not know of intelligence in a thinking individual except by its reactions which, however, are not as clearly defined as those characteristic for magnetism, and which are, furthermore, less stable. This lack of stability is the reason why the notion of substance is of so little use in the treat- ment of psychical phenomena and, no matter whether we regard with Mach the notion of a substance as a hypothesis introduced for the sake of explanation, or whether we consider it with Kant as one of those notions which we are bound to apply to experi- ence — two views which are by no means so very widely different and which may be varied in detail considerably — the fact remains that the notion of substance, though very useful in physics, is of no practical avail in psychology. In physics the notion of substance leads to that of matter, as that which is extended, moveable and impermeable, but when appUed to mental states it can be used only as the notion of an indefinite "something"^ which is of no use. It is a very obvious idea to try how much one can do without this notion. In order to avoid all these diflficulties one may try to formulate the problem in such a way that it loses its ontological character, retaining only the general problem of finding relations between- phenomena. Events are the only immediate datum of experi- ence and one may try to find relations between them. The complete description of an event requires that all the events which are connected with it according to a definite rule are de- scribed. A mental .state is connected not only with events of the class A, B, C, .... but also with events of the class a, b, c, .... which must not be omitted in a complete description. Psycho- physical parallelism, thus, gets a very simple expression on tliis ground. The general principle of psychophysical parallelism is that the events A, B, C, .... are in a relation with the events 154 PROBLEMS aF PSYCHOPHYSICS a, b, c, .... and that a mental phenomenon A is not explained before those events of the class a, b, c, .... are described with which A is in a definite relation. This principle does not make any definite assumption about the nature of this relation, but it is rather the exjjression of the fact that the phenomena of the classes A, B, C, .... and a, b, c, .... constitute together the realm of experi- ence, in which we ma}' look out everywhere for connections be- tween the events. A practical consecjuence of this principle is that the investigation of the regularities of the connection between events, especially of events of the class A, B, C, ....,must not stop at any boundary inside the realm of experience. The existence of definite rules for the connection of events is a supposition of the possibility of constructing scientific systems, because it would not be possible otherwise to deduce an exhaustive system from a finite number of principles The supposition that there are no dis- connected events, i. e. wonders, in the entire realm of experience is the condition of a complete description of the phenomena. In the description of an element of the class A, B, C, .... , e. g. a color sen- sation, the problem of psychophysical parallelism is not exhausted with the answer to the cjuestion: Which are the phenomena of the class a, b, c, . .. which are connected with this event accord- ing to a definite rule? The class of phenomena the description of which is required by this question contains among other pro- cesses those which take place in the optic nerves and in the occip- ital lobes under the influence of retinal stimulation. This answer shows in itself that this description is only a part of the answer to the more general problem which does not restrict the events to those of the second class, and which requires the description of all the events with which the first event is in relation. Events ■of the class A, B, C, .... have the same claim as those of the class a, b, c, . .. and the view that psychology,!, e. the science which deals primarily with the events A, B, C, ...., should be resolved into a special science of the events a, b, c, ....,e. g. into brain phys- iology, is not essentially superior to the opposite view that the events a, b, c, .... are indifferent for psychology, although it may lead at present to a greater number of propositions, because the relations of the second class are better known. The statement that mental phenomena depend on physical phenomena is fre- THE PSVCH(3.METRIC FUNCTIONS 155 quently nothinji else but an expression of the principle of psycho- physical parallelism. Sometimes, however, this statement is ijiven the more definite meaning that spatially well defined groups of events of the class a, b, c, .. . are in constant relation with certain events of the class A, B, C, .... so that the event A does not occur if ....m, n, o, .... did not occur, and that .... m, n, o, .... occur when A is observed. Such a statement contains a special law and is necessaril}^ the product of observation. The best known example of a relation of this type is the so-called principle of cortical localization. The term "principle" is well chosen if this proposition is given the meaning that there exist such groups of events .... m, n, o, .... for every event of the class A, B, C, This proposition is of higher generality than a result of observa- tions can l3e and it deserves the name of a principle, because it lays down a rule for an important part of physiology. The term "prin- ciple of cortical localization," however, is not well chosen, if it is applied to that group of propositions which are the outcome of all the investigations along this line. This group of propositions is the expression of empirical facts and the term "empirical law" or "law" is more appropriate. This relation between events A, B, C, .... and events of the class a, b, c, .... seems to be very mys- terious, if this relation is conceived as a relation between two fundamentally different substances, but this mystery disappears if the purely phenomenological point of view is taken. Both events stand on the same level and their relation is not more mysterious than that between any two others. The existence of definite relations between spatially well defined groups of physical phenomena and certain mental phenomena shows that it is l)y no means an impossible task, involving a contradiction, to establish relations between phenomena of the class A, B, C, .... and those of the class a, b, c, * The contradiction only comes in w'hen the phenomena are referred to substances which are funda- mentally different. Treating the data of experience in a purely *.\n analysis of a very complicated group of sensation into its elements was given by E. M.^ch, Analyse dcr Etnpfinduiigcn, 1902, pp. 32, sq. This author describes in this book as well as in his " Thermodynamics" and in his " .Mechanics" the motives which may induce us to give to the objects a reality independent of our sensations. 156 PROBLEMS OF PSYCHOPHYSICS phenomenological way without making use of the idea of sub- stance shows that the barrier which separates the class A, B, C, .... from the class a, b, c, .... it not insurmountable, or rather that the problem of establishing relations between the events of these two classes is not essentially different from the problem of finding relations between other groups of phenomena. The inadequacy of the substitution of relations l)etween sub- stances for the purely phenomenological relations between events is most apparent in the problem of the relation of the psychical to the phj'sical. The view that all events are modifications of substances may be called the theory of mechanical or substantial causality; this view implies that substances are the ultimate reality of everything. The shortest expression of this view that substance is the factor undergoing changes is given in the sentence: All causing is effecting (Alles Wirken ist ein Bewirken).'^= This view is eminently anthropomorphic, because it takes the will act as the type of a cause. The difficulty of conceiving of the rela- tion between thinking substance and extended substance is by no means the only shortcoming of the notion of substantial causality. This view, in fact, leads to many other difficulties and contradic- tions as e. g. to the statement that every judgment must express a relation between substances, or between a sulDstance and its accidentibus. The only type of judgments admissible under this supposition are those judgments in which the relation of a sul)- ject to its predicates are determined. This classification, how- ever, does not provide for the so-called impersonal judgments to which this definition does not apply. This fact shows that the notion of substantial causality does not exhaust even those phenomena which have no direct bearing on the mind-body prob- lem. It is, therefore, advisable, to see whether the notion of substantial causality cannot be superseded ,by another. We shall use the notion of relations between phenomena; causality, then, is conceived as a type of order between events. *The exponents of this theory are fairly numberless, and we cjuote only as an example C. A. Strong, 117;;' the Mind has a Body, 1903, p. 75, "A voHtion seems the very type of a cause." The most recent exposition of this view was given by Professor Frank Thilly in his article " Caiisality," Philosoph- ical Review, Vol. 16, 1907. THE PSYCHOMETRIC FUNCTIONS 157 The term relation in its broadest signification has the meaning of a one-to-one relation between two elements; this notion if it is confined to ciuantities coincides with the notion of a mathemat- ical function in the sense of Cauchy and Dirichlet. Such a func- tion is a rule by which a value of the dependent variable is ad- joined to every admissible value of the independent variable. The most general form of a relation is that of a non-uniform, discontinuous function. This definition is too general to be treated by our present means with advantage, and one restricts the investigations first to the uniform, continuous functions, and later on it becomes necessary to restrict it still further to the analytic functions. A continuous function may be given by a numerable, infinite set of conditions, but this is not the case of the general discontinuous functions. These functions are char- acterized by an innumerable, infinite set of conditions, what is the same as to say that they cannot be defined. In consideration of the fact that Fourier's analysis shows, how certain discontin- uous functions may be represented by a sum of continuous functions one might have taken the view that these general functions may be represented, or approximated, by analytic functions. This view, however, would hardly have been a rea- sonable, expectation, and it was made impossible lately by the discovery of Baire, that every function which can be represented as the limit of continuous functions must be a function with not more than punctual discontinuities. If it should happen that a function of more than punctual discontinuities occurs in na- ture, it would be not only impossible to represent it in the usual way but it even would be impossible to approximate it. An event of this type could not be described in the same terms as ordinary events. A classification of events of this type cannot lie exhaustive and it is not possible to deduce systems of science from a limited number of propositions, because there is no war- rant that one may not find at any moment an event which does not reseml)le in anything any element of previous experience. *See BoREL, L^qons' sur la theorie de^ fonctions; Lbbesgue, Leqons sur r integration; B.^IRE, Les fonciions discontinues; Philip E. B. Joi-rdain The Development of the Theory of Transfinite Numbers, Archiv der Mathematik und Pkysik, Ser. III. Vol. 10, 1905, pp. 2.54-281. 158 PROBLEMS OF PSYCHOPHYSICS It may be a question under which conditions systems of science are deducible from a limited set of propositions, using only a limited number of fundamental ideas or notions, but there is no doubt that such systems cannot possibly be exhaustive of a cer- tain realm of experience if non-uniform, discontinuous functions are admitted. Winter* has tried to argue against the possil^ility of deriving a system of science from a limited number of funda- mental notions by means of the fact that mathematics in its nat- ural progress constantly introduces new notions, the definition of which is apparently more or less arbitrary. Such is e. g. the origin of the elliptic functions which are obtained by increasing the order of a function in a certain integral which leads to known functions by the unit. Against this argumentation one may say that the way pointed out by Winter is the way of the discov- ery of new^ functions, but that every function can be derived by starting from the series which gives the definition of the function. This series is constituted by no other l)ut known operations, so that the new function is expressible by an algorithm which con- tains no other but elementar}'- operations of which there is only a limited number. The introduction of new functions, therefore, is not an argument against the possibility of a >ystem being built up on a limited number of notions, as long as the new functions can l^e reduced to the forms of relations already known. This possibility is not given for functions which belong to the general type of a function in the sense of Cauchy and of Dirichlet. We have introtluced above the cUstinction between regularity and unifoi'mity of events. Uniformity is the more special case of regularity, because it may be that the rules of the succession of phenomena vary in such a way that different rules of succession are adjoined to the same phenomenon at different times. The type of order which is applied to the study of natural phenomena is the one of exclusive uniformity, requiring that the same group of events under given conditions is alwaj's followed by a certain other group. These relations are most commonly found in mechan- ics, and for this reason one frequently calls this type of relation *M. Winter, Suy I' introduction logique a la theoric dcs fonctioiis, Revue de Meta physique et de Morale, Vol. 15, 1907, pp. 205-209. THE PSYCHOMETRIC FUNCTION'S lo!) mechanical causality. The simplest example of events of this kind is the push and pull of one mass on another mass, and the task of constructing; a system of science in which the events are causally determined was frequently formulated in this way, that every event must be analyzed into motions of masses which are moved by the push and pull of other masses and which have no power of their own to influence their motion {vis a tergo).'^ The beauty of this conception is that the influence of one mass on the motion of another mass seems to be immediately intelligible by the analogy of the influence of our body on its surroundings. The consideration of mechanical phenomena, however, leads only to one class of relations, and it was seen at a very late time that the problem thus restricted excluded all relations which do not belong to a comparatively small class. The class of relations which are admitted for the description of natural phenomena are the so-called analytic functions. The view that all phenomena are under the principle of mechanical causality requires that natural phenomena are described merely in terms of analytic functions. These functions have, besides some other minor properties, the following distinguishing peculiarities: They are single valued, continuous, differentiate, they can be developed into a power series, they admit of an analytic continuation, and they are solutions of differential equations. These peculiarities make the analytic functions so precious for the study of natural phenomena. The conditions in infinitesimally small intervals of time and space are so simple, that we can express them with comparative ease by differential equations, the integration of which leads necessarily to analytic functions. Having established such a relation in no matter how small an interval of time and space, we can determine and foresee every phase of the later de- *The value of this view is the question at issue in the discussions between the adherents of the energetic view and those of the atomistic hypothesis. The denial of the indispensability and of the adequacy of the atomistic hy- pothesis is the contention of the energists (E. Mach, Die Mcchanik m I'mcr Entuicklung, p. 486 " Dass alle Vorgange mechanisch zu erklaren sind halten wir fiir ein Vorurtheil,") whereas the adherents of the atomistic hypothesis point out that the energetic view does not show a success in the explanation of natural phenomena nearly equal to that of the atomistic hypothesis. 160 PROBLEMS OF PSYCHOPHYSICS velopment of the process, because the function which describes this process has an analytic continuation. The course of events, which is the course of nature, is uniquely determined so far as it is characterized by analytic functions. Events which are char- acterized by analytic functions are completely under our control and none of the details of such events can escape detection. Physics deals exclusively with processes which may be described by analytic functions and the success of this science sug.ejested the view that all events, if properly analyzed, must lead to this type of relations.* If it is true that there are in nature no other but analytic functions, then there exists indeed the possibility of exhausting the description of the world by the future progress of science. The observation of an event in no matter how small an interval of time and of space permits to predict the future and *We quote only B. Riemann, Uber die Darstellbarkeit einer Function durch eine trigonometrische Reilie, M'erke, ed. Weber, 1892, p. 237: "Durch die Arbeit Dirichlet's ward einer grossen Menge wichtiger analytischer Unter- suchungen eine feste Grundlage gegeben In der That fiir alle Falle der Natur, um welche es sich allein handelt, war sie vollkommen erledigt, denn so gross auch unsere Unvvissenheit dariiber ist, wie sich die Krafte und Zu- stande der Materie im UnendUchkleinen andern, so konnen wir doch sicher annehmen, dass die Functionen, auf die sich Dirichlet's Untersuchung nicht erstreckt,' in der Natur nicht vorkommen." The functions which Diriclilet's investigation does not cover are the functions with an infinite number of maxima and minima in a finite interval, and the supposition that there are in nature no other but analytic function comes out clearly in Riemann's words. The impossibility of the occurrence of functions with an infinite number of maxima and minima in the description of the motion of a point can be seen in this way. A body which moves along a curve can approach a point in the neighborhood of which there is an infinity of maxima and minima only with always decreasing velocity, and a movement from this point is impossible (See A. KoEPKE, Difjerentirbarkeit und AnschaulichkeH von Functionen. Math. Annalen, Vol. 29, 1887, pp. 137-140.) The reason for the requirement of a difi'erentiable function is that a motion which is characterized by a func- tion without a derivative would have to take place without a definite velocity or acceleration. Some of the modern writers have introduced diilerent labil- ity as a specific requirement, e. g. Heumholtz, V orlcsiingen iiber die Dyna- mik diskreter Massenpunkte {Vorlesungen nber theorctische Physik, Vol. 1, 1898,) p. 7, sq.; L. BoLTZMANN, Vorlesungen iiber die Prinzipe der Mechanik, 1897, pp. 10-13, who says that differentiability ought to be introduced as one of the requirements of mechanics. I k THE PSVCHOMETKIC FUNCTIONS IGl to State every event of the past. The events at any particuhir point of time and space depend in a very complicated but perfectly definite way on all the other events, and it is merely a question of ability how much of the world's course in general one may be able to deduce from one single fact no matter how trivial it may be. The perfect knowledge of all these relation would be the realisa- tion of Laplace's mechanical ideal, which is only a consequence of the supposition of the exclusive occurrence of analytic functions in nature i. e. of the absolute intelligibility of all natural phenomena. ►Special mechanical principles, as e. g. the theorem of the conser- vation of energy, depend on the principle of general causality formulated as the exclusive occurrence of analytic functions in nature, and they do not hold generally if other functions are admitted. '■■'■ This view that all events are characterized by ana- lytic functions and that ever^'thing can be treated by the same scientific method was called materialism by Alexejef¥,t but since this term is used for so many widely different notions and *As long as no sets of necessary and sufficient axioms of mechanics are CLinstructed it is very hard to tell whether a proposition has the character of a principle or of a law. Sigw.\rt {Logik 2, ed. 1893 \'ol. 2, p. 644) says he cannot convince himself that Galilei's inertia and Newton's universal at- traction are necessary consequences of the principles, so that he would be inclined to regard them as natural laws. This may be true for the special form of the Newtonian potential but it is not equally obvious for the law of inertia which was frequently formulated as a negative expression of the prin- ciple of causality. For literature on the problem of inertia see L. Lang Das Inertialsystem vor clem Forum der Nafuriiissenscliajt, Phil. StiuL Vol. 20 1902, pp. 1-71 . The proposition of the conservation of the energy w-as regard- ed by Hemholtz as a consequence of the causalistic conception of the w'orld, whereas other investigators try not very successfully to look at it as a datum of experience, e. g. Erich Becher, Das Gesefz von der Erhaltung, der Energie efc' Ztschr. f. Psychologie, Yo\. 46, 1907, p. 98. On the principle of the con- servation of energy see W. Wundt, Physiologische Psychologic, 5, ed. Vol. '^^, 1903, p. 634. A discussion of the views on this question as advocated by different authors is given by L. BussE, Geist und K or per, Secle und Leib, 1903, p. 119, 124-126. fW. G. AlEXEJEFF, Die arithmologische und ivahrscheinlichkeitstheore- tische Kausalitdt etc., Ztschr. /. Philosophie u. Pddagogik, Vol. 14, Nov. 1905. pp. 50-55 speaks of "Materialismus oder die rein physikalische Betrachtungs- weise des Geistigen." We use only the second term of the two suggested by Alexejeff. 102 PROBLEMS OF PSYCHOPHYSICS since it is primarily a term for an ontological view, it may seem best to find another name for it. The term "mechanistic view" would be appropriate, if it were not frequently used for the desig- nation of the atomistic hypothesis, and it is, perhaps, best to call this view after physics, the science which most successfully uses it. By the term "physical point of view" we intend to designate the view that there are no other but analytic functions to be found in nature.'^' The problem of the description of mental life calls for the es- tablishment of the laws of the succession of these phenomena and the broadest solution of this problem would be to find a one- to-one relation between every moment of time and a certain element of the class A, B, C, .... , but leaving the question open whether this relation has the character of an analytic function or not. The way of proceeding may be illustrated by the fol- lowing example taken from physical science. The temperature of a liquid can be raised by bringing it into contact with a body of higher temperature. The temperatures of the body and of the liquid vary until the flow of heat comes to rest at a certain time, but in every moment the liquid has a certain temperature as well as the body, so that there is a one-to-one relation l^etAveen the moments of time and the temperatures of the liquid. The ac- curate description of this process is supplied by the mathematical relation which shows how the temperature of the liquid depends on time. A similar example of psychology would be found in the description of the succession of the mental states under the *It happens not unfrequently that the supposition that an event must have an analytic expression is superseded by the other supposition that it must be a simple expression. An example may be found in B. A. GotiLD, Use of the Sine-formula for the Diurnal Variations of Temperature. Amer. Jl of Science, Vol. 119, 1880, who tries to show (p. 217) the importance of finding general expressions, because an expression which can represent n observa- tions but depends on a number of parameters smaller than n is a real natural law. A discussion of tliis view may be found in H. Burkh.\rdt, Entwick- lungen nach oscillierenden Reihcn, Jahreshericht d. deuischen Mathematiker Vereinigung, 1902, pp. 249, 250. Burkhardt's opinion that this view is char- acteristic of the American for Anglo-American conception of natural science is perhaps not entirely justified. The conception that a natural law must be a simple law leads of course to a very flat rationalism. THE PSYCHOMETRIC FUNCTIONS 163 infiuonce of an incoming stimulus; the complete solution of tins proljlem would give a one-to-one relation between every moment of the time under consideration and a certain mental content. There is no doubt about it that mental phenomena are functions of time and the difficulty merely consists in defining these func- tions. We dismiss from the start the supposition that the nature of this relation is known before hand. It seems to be very obvious to draw the following erroneous conclusion. There exists a one-to-one relation between the elements A, B, C, .... and the moments of time — no specific hypothesis being made about the nature of this relation — but there exists also a definite relation between the moments of time and the elements a, b, c, One supposes that the latter relation must belong to the causal type i. e. that it must be expressible in analytic functions. Instead of merely concluding that there must be a relation of some kind between the elements A, B, C, .... and the elements a, b, c, ...., one may try to specialize these relations to the type of analytic functions. This conclusion is of course unwar- ranted, because the combination of an analytic function with a non-analytic function does not give an analytic function. We may illustrate the far reaching importance of this supposition by Mr. Montague's theory of the specious present.* This author starts from the observation that physical and psychical events are both functions of time, and he immediately proceeds to form the derivative of the function which gives the subjective phenomena as depending on the objective phenomena; to this (quantity he *\\'. P. MoNTAGiE, .4 Tkcury oj Time-l'erccption, Amer. Jo of Psychology. Vol. XV, 1904, pp. 1-13. Mr. Montague came very near to considerations like those expounded in this chapter, but instead of inquiring into the principles of the problem he unfortunately undertook to solve a very special and very difficult question. Some of the weaknesses of Mr. Montague's argumenta- tion are shown in the author's note on " The Application of Calculus to Menial Phenomena," The Jo of Phil. Psych, and Scient. Methods, \o\. II, 1905, pp. 16-18, but only a general allusion is made to the fact that the weaknesses of Mr. Montague's theory are the necessary outc(mie of a certain definite hy- pothesis on the relation of the pliysic and the psycliic. 164 proble:ms of psychophysics gives a certain psychological interpretation/'' Thus the suppo- sition crept in that the dependence of mental on physical events is given by differentiable functions. This supposition excludes at once a very wide range of possibilities, and it contains the defi- nite statement that the type of order of the events of the class A, B, C, .... is the same as that of the events a, b, c, The reason why this rather obvious fact could be overlooked lies in the peculiar formulation which the author gives to his problem. He introduces both types of events as functions of time and con- cludes that the events A, B, C, .... must be functions of a, b, c, There is nothing to say against this conclusion, if the hpyothesis is not made that there exists a derivative of this function. If one makes the supposition that the events of the class A, B, C, .... and those of the class a, b, c, .... are analytic functions of time, one must necessarily accept the conclusion that the events A, B, C, .... are analytic functions of the events a, b, c, This suppo- sition certainly had not been made, if it had been seen what the. bearing of the hypothesis is that mental events are analytic func- tions of physical phenomena. Physics leads necessarily to analytic functions for the descrip- tion of natural phenomena, because they are the solutions of differential equations and it is obvious that all the problems which deal with events which cannot be described by them must escape notice. This caused the question whether the description of phe- nomena as supplied by physics is exhaustive, and on what ground the supposition rests that there are no other but analytic functions in nature. This question was raised by F. Klein, f who gave a very ingenious answer which is based on the distinction between functions which are given empirically and their ideal mathematical expression, a distinction which this author had introduced previously. J The empirical representation of a func- *Mr. Montague's interpretation of the derivative did riot remain uncontra- dicted; a discussion of this topic may be found in E. B. Holt, Jo of Phil. Psych, and Scient. Methods, Vol. 1, 1904, pp. 320-323 and Mr. Montague's answer in the same journal, Vol. 1, 1904, pp. 378-382. tF. Klein, in his lectures of 1901, Vorlesungcn ilber Differential und In- iegralrechnung, eine Revision der Principien, ed. 1907, pp. 129-139. JF. Klein, Uber den allgemeinen Functionsbegriff und desscn Darsiellung durch eine willkurliche Curve, Math. Ann. Vol. 22, 1883, pp. 249-259. THE PSYCHOMETRIC FUNCTIONS 165 tion is never an ideal curve but an area of finite extension ami what is given empirically are neither analytic nor non-analytic functions, but only approximations. The distinction between analytic and non-analytic functions refers merely to their idealized mathematical representation. This solution cannot be discussed here and also another solution given by J. Boussinesq can be but briefly mentioned.* Boussinesq starts from the view that the exact description of a phenomenonns given by its differential equation with which its integral is equivalent, because purely logical processes are needed for finding the integral of a differen- tial equation. These solutions are given by analytic functions which determine the process everywhere except at certain points, the branch points, where the choice between different courses comes in. The course of events is not defined at these points and it is there, at the branch points, that Boussinesq looks out for the manifestations of mental life, more especially of free will, which manifests itself by the undetermined choice between equal possibilities represented by the branches of the curve. This view has not found many followers and it stands out more as a curious example of the ingenuity of its promoter. The third attempt at a solution of our problem has a direct bearing on the question treated here, and this view is formidable by the number of distinguished followers it has found. The starting point for these considerations is found in the fact that certain phenomena which are the product of human will-decision show a remarkable stability in their averages, if they are taken in large groups, no matter how much chance may have influenced the single decisions. These results of demography and statistics of *J. BoussixESQ, Co7iciliation du veritable d Hcrmi ni ime mccanique etc. Recucil de la Societe des Sciences dc Lilies, Vol. VI, 187S and C. R. Vol. XCIV, p. 208. Boussinesq treats the same problem in his book "Application des poteniicls a I' etude de I' equilibre et du monvemcnt," 1885, pp. 699-704, where he attempts to show that functions which have no derivative can be treated in the same way as other functions by considering functions which differ from them only in a very slight degree. In this attempt he approaches Klein's conception of a " Functiotissireifen" by making it a matter of preference which one of a certain group of functions, which difTer from a given function by less t'lan a certain quantity, one will treat. 166 PROBLEMS OF PSYCHOPHYSICS morals have always aroused great interest and were frequently used as arguments in favor of universal mechanical causality against free will. Instead of drawing these conclusions one must stop considering the conditions under which events, which are taken in large groups, can possibly show regularity in the mean values. In regard to this question there exists a famous propo- sition due to Tchebitcheff, one of the promoters of the view in question, that the chief condition for the applicability of mean values is that the single events must be independent from each other. The results admit of a correct interpretation if the events are independent from each other, but if they are not independent i. e. if they are causally connected, one cannot apply the theorems of the calculus of probabilities. From the fact that we find regu- larity in large groups of human actions we must conclude that they are independent i. e. that they are not in a causal relation, and since all events of the causal chain are inter-connected we must conclude that will decisions are not causally necessitated. The law of causality is, therefore, not the only one and not all events of nature are characterized by analytic functions. In this way the old argument against free will is turned into an argument for it. The argumentation as expounded is due to the so-called Moscow school of idealism as represented by P. L. Tchebitcheff, Nekrassow, W. G. Alexejeff, and the head of the school N. W. Bugajeff. These men have also materially promoted the study of the "half-analytic" functions which originate by the combin- ation of analytic with non-analytic functions, and in which these authors see a more adequate expression of the relation of the physic and the psychic, and important works on the theory of probabilities are due to them. Alexejeff has the merit of mak- ing these investigation generally accessible by several CJerman papers.* .*\V. G. Alexejeff, Ubcr die Entwicklung des Begnffes der hoherenarith- mologischcn Geselzmdssigk-eiten in Natur-und Geistesivissenscliaft, Vierteljahrs- schrift f. wiss. Phil. u. Soz., Vol. 28, 1904, pp. 72-93; "N. W. Bugajew und die idealistischen Probleme der Moskaiier mathematischen Scliule," the same jour- nal, Vol. 29, 1905, pp. 335-367; and "Die arithmologische und wakrschein- Uchkeitstheoretische Causalitdt etc.," Ztsclir. f. Philosotyhie u. Padagogik, Vol. 14, (2.) 1906, pp. 50-55. THE PSYCHOMETRIC FUNCTION'S 107 This argument proves too much, because it applies to all kind of random events among which there are many classes of events which doubtlessly are causally necessitated. There are events which we can submit successfully to the treatment by the calculus of probabilities, although the single events are causally necessitated or even logically determined. There cannot be any doubt in these cases that an independence of the single events does not exist in any absolute sense of the word, but large groups of events comply with the requirements of the calculus of probabilities in a very high degree. Let us take the following examples. Whether high water arrives at London Bridge in the first, second, third or fourth quarter of an hour of the morning is the very type of an event which is causally necessitated. The single events depend entirely on physical conditions, but the results of a long series of obser- vations conform, nevertheless, with the requirements of the cal- culus in a verv high degree.* The following example of events which comply almost perfectly with the requirements of the cal- culus of probabilities, although they are logically determined, is due to Bruns. The distribution of zeros at the last place of the logarithms in the columns (of 60 numbers each) of Vega's Thesaurus Logarithmorum shows that the numbers of those col- umns in which zero occurs 1, 2, .... 60 times is very nearly such as it ought to be, if the occurrence of the numbers at the last place were entirely a matter of chance. f It is quite obvious that the *F. Y. Edgevvorth, The Law of Error, Part II, Transactions of the Cam- bridge Philosophical Society, Vol. 20, 190-5, p. 128, 129. fH. Bruns, Wahrscheinlichkeiisrechnuug iind Kollektivmasslclire, 1905, pp. 8, 279 sqs. Bruns had chosen this example on purpose, because in the case of the occurrence of the dilTerent numerals at certain places of the h^g- arithms one cannot possibly speak of chance in the common sense of the word. The observation that tables of logarithms are material for the study of chance events was made by Gauss, Einige Bemerkungen zu Vegas Thesaurus Log- arithmorum, ]]'erke, Vol. Ill, p. 260. Gauss remarks that the last decimal of the logarithms of the tangent cannot be equal to the last decimal of the difference of the logarithms of the sine and cosine, if the deviations of the loga- rithms of the sine and of the cosine from the correct values have not the same sign, and if their sum is greater than \. The probability of this event is \. Gauss found by counting over the tables of K(')hler that this event happened in 2.50 out of 900 cases. 168 PROBLEMS OF PSYCHOPHYSICS occurrence of a certain numeral at the tenth phice of the logarithms of consecutive numbers is not an event which has random char- acter in an absolute sense of the word, since we can determine the event in every case by the rules of arithmetic. The fact is that randomness of the events, which can be submitted to the treatment by the calculus of probabilities, consists in the absence of a knowable law. This, as a matter of fact, is the conclusion from the test whether a certain group of events has random character or whether it has not, that the agreement with the rules of the calculus of probabilities indicates the absence of a recog- nizable law, and the lack of agreement with the results of the calculus indicates the presence of a constant influence which may be investigated. This type of probability, w'hich may be called randomness by excessive complication, has been recognized al- ready by Kepler, as it was pointed out recently by Bruns.* It is exactly this type of probability which we use for the working out of the results of our experiments on lifted weights. Every single event is causally necessitated, but its conditions are so com- plicated that we must be satisfied with this very general type of relations which form the propositions of the calculus of probabil- ities. A mathematical probability is defined as a fraction the numerator of which gives the number of cases which are favorable to the event, and the denominator of which gives the total num- ber or possible cases. The outcome of every single case must be well defined. We have seen above that the numbers of relative frequency for the judgments of different types on the comparison of two weights have the formal and material character of prob- *H. Bruns, /. c. p. 7. The place in question is J. Kepler, Dc stclla nova in pede Serpentarii, Opera, ed., Frisch, Vol. II, p. 714 "Improvidi sunt, qui hos (tesserarum jactus) plane fortuitos, hoc est avairiovr^ esse putant: sin autem suum casum omni causa privant, nondum ejus exemplum dixerunt in lessens. Quare hoc jactu Venus cecidit, illo canis? Nimirum lusor hac vice tessellam alio latere aripuit, aliter manu condidit, aliter intus agitavit, alio impetu animi manusve projecit, aliter interflavit aura, alio loco alvei impegit. Nihil hie est, quod sua causa caruerit, siquis ista subtilia possitcon- sectari. " The example of the acetarium on the previous pages of Kepler serves as illustration against certain ancient systems of philosophy that all possible combinations of events must be exhausted in an infinite interval of time. THE PSYCHOMETRIC FUNCTIONS 169 abilities. The iiuinber of cases which are favorable to the event, e. g. to the formation of the judgment "heavier", is represented by a finite or infinite number of groups of conditions each one of which leads necessarily to the formation of the judgment that the second weight is heavier. The number of possible cases is rep- resented by the totality of conditions in which a judgment is given according to the rules of the experiments. The ratio of these two numbers gives the probability in question. Only under the supposition that there exists a definite (finite or infinite) num- ber of possibilities each one of which leads necessarily to a certain result can we form the notion of the probability of a "heavier"- judgment. We conclude from the fact that the notion of a math- ematical probability can be successfully applied to physical events, that they have very nearly the character of those events which we treat in the system of idealized propositions which we call the calculus of probabilities. The proposition that a certain well defined group of physical events has the character of random events in the mathematical sense of the word warrants the application of the rules of the calculus to these events. This statement plays the same role for this group of events as the state- ment that physical space is a three-dimensional Euclidean mani- foldness plays for the application of ordinary geometry to physi- cal space. The conformity of events with the rules of the calcu- lus of probabilities is so far from being an argument for the lack of causation of these events, that it is an argument for uniform causality. No statement about the future outcome of obser- vations were possible, if every single event were not causally necessitated.* *This conception is a consequence of basing the idea of probability on the hypothetical judgment, as it was done by Sigwart, Logik, Vol. II, p. 30.5- 320. For the logical details the reader must be referred to this book and to the treatises of K. Stumpf, Uber den Begriff der mathematischen Wahrschein- lichkeit, Bcr. d. bayr. Ak. {Phil. Kl.) 1892; J. v. Kries, Die Principien der Wahrscheinlicbkeitsrechnung, 18S6; H. Bruns, Wahrscheinlichkcitsrechnung und Kollektivmasslehre, 1906, and to the repeatedly quoted works of E. Czit- ber. a different view was expressed lately by Mr. Gomperz, Uber dir W ahrscheinlichkeit der W illensentscheidungen, Sitzungsberichie d. Kaiserlichen Akademie der Wissensckafien zii Wicn, Phil. Hist. Kl. Vol. 149, III. Abh. who tried to find a new conception of free will and began by showing that the sta- 170 PROBLEMS OF PSYCHOPHYSICS The process of finding the numerical values of these probabili- ties is perfectly well defined and stands on the same level with other empirical ol)servations. The theorem of Bernoulli gives the limits of exactitude of such observations, and it depends on our will to make our results as exact as we choose. The empirical datum of a determination of a probability is a realm inside of which we may expect the result with a given probability. The limits of the exactitude of the observation can be made smaller than any given quantity by increasing the number of observa- tions. These observations may be made for different intensities of the comparison stimulus. The res,ult of these experiments is that a higher probability of a "greater "-judgments corresponds to higher intensities of the comparison stimulus, that greater probabilities of "smaller "-judgments correspond to smaller in- tensities of the comparison stimulus, and that the probabilities of the "equality "-judgments increase at first and decrease after having attained a certain maximum. These empirical data do not differ essentially from the results of any other experimental investigation, which show how certain quantities depend on the variation of another quantity. These data are subjected to a process of idealization, the first step of which is to assume that an absolutely exact observation would lead to a definite numerical value and an hyjjothesis is made which these values are. In tistical regularities do not prove anything in favor of causal necessitation of the single events. " If one were to cast a die 60,000 times every year and if there were every 3'ear among the results approximately 10,000 aces, nobody will conclude that in every single case the ace had to come out necessarily." This view, which Mr. Gomperz does not support by arguments, is refuted by the examples mentioned above, where events about the causation of which there cannot be the slightest doubt comply with the requirements of the cal- culus of probabilities. Mr. Gomperz's papers is marred by some slips (" an al- most infinite probability" 1. c. p. 13) which indicate that the author is not in- timately acquainted with the theory of probabiHties, a supposition which is sup- ported by the fact that a " mathematical friend" had to give the formula p.l 6 of his paper, which is easily found by calculating the point of intersection of X 2 y 2 x-a 2 V ^ '-(^)=land(— -) + (^-- s, ^ as' ■ s X -= y 2 x-a ^ y ^ s , the two ecHpses ( — ) + ( ) =1 and ( )-[-(_—) =1 and putting — =q Compare the review of Mr. Gomperz's paper in Ztchr. f. Psychologie, Nov 1900 Vol. 43, p. 318. THE PSYCHOMETllIC FUNCTIONS 171 the case of observations on probabilities the result is determined by the theorem of Bernoulli, in empirical sciences this result is given as a rule by the most probable value as determined by the method of least squares. The next step of idealization refers to the rule of dependence between the variations of the quantities observed. The observations on a " greater"-] udgment show that these probabilities increase with the intensity of the comparison stimuli. This observation holds only for the finite set of obser- vations which were really made. One interpolates between the elements of this finite set of observations the elements of an in- finite set which is of the order of the continuum. The elements of the infinite set are supposed to follow the same law as the ob- served elements. This supposition is based on the methodolog- ical principle mentioned above that unknown phenomena are sup- posed to follow the same law as observed phenomena of the same class, as long as no contrary instances are known. There is no difference between our treatment of psychological observations anil the methods by which physical observations are treated, and we may say in general that the mathematical representation of empirical observations is nothing else than an idealization of ex- perience. APPENDIX 173 TABLE 1. Variations of the Weights used i>f the experiments (in milligrams) ^Corrected. u 1 2 Weight (in grams) February 23, 07. March 18, 07. Is" April 8,07. < 2 '1 ^ . a o > 1 100 - 5 - 5 - 5 - 9 - 6 - 4 - 1 - 8 19 2 100 + 2 + 2 + 2 + 2 + 3 + 4 + 5 + 5 3 3 100 -3 + 3 + 5 + 2 - 7 - 4 - 6 - 2 31 4 100 - 1 - 1 4 5 100 - 4 - 4 - 3 - 4 - 3 - 1 - 3 11 10 100 - 4 - 3 - 6 — 7 - 8 _ 2 - 4 - 6 16 11 100 - 4 - 4 - 7 - 3 1^ - 3 - 4 - 7 IS 12 100 - 4 + 4 - 5 - 5 - o - 6 - 4 - 9 25 6 104 ^ - 4 - 6 - 6 - S — 7 - 6 - 8 8 7 108 - 4 - 6 - 6 - 9 - 9 - 8 -10 8 20 92 - 3 - 3 - 3 - 7 + 1 - 7 - 1 26 21 96 - 3 - 3 - 2 - 5 - 3 - 9 - 5 16 51 84 + 2 + 3 + 3 + 2 + 5 - 3 + 5 21 .52 88 - 4 - 4 - 4 - 5 - 1 -10 - 5 19 23 90 + 13* + 1 + 1 - 2 - 6 - 3 11 24 94 - 4 + 14* + 6 + 6 + 4 - 4 + 2 40 25 98 + 14* - 2 + 2 - 1 - 2 + 2 16 26 102 + 3 + 4 + 5 + 5 + 4 + 5 12 TABLE 2. Date of observation I II III IV April 26. 07 100.308 104.364 103.003 114-227 29, 100.423 104.488 103.434 114.781 May 2. 100.299 104.360 103.159 114.709 6. 100.357 104.439 103.357 114.747 ■ " 10, 100.654 104.704 104.280 115.342 Observed variations Sum of variations 1.259 174 PROBLEMS OF PSYCHOPHYSICS M in m - -' O o o o c ■^ -^ ° CO ^ Tf ^ o CO _M o '-' O ■-^ o w o '-' o ^ "5 •^ CO (N IM (N -^ o o 2 H O lO CO CO (N ■«< '^ -^ o s „ I OrHO-^Or-i.-lC^JC (M 1-- I^ CO ■* O-HiofNtNOOO I .-I I~(NTq(NTj(cOU5IMCO OOC^lCOt^tOlOiiilMCO I —I o o - 1 CO M X r^ r^ CO »-< lO CD 1 lO >o -I CO to X _| -H rt r-^ OS CD 1 (M ^ o ■* O t^ rt CO •fl^ CO 00 oi X (N CD 1^ O ■* --c (M rt rt rt CO CD H CO C^l CO !M C^l CO X CO X 01 CO C-) — 1 rt C5 c CO 05 " o t^ o M — ■ CO CD CO LO ■-H IM 1 c ■^ X (35 w 05 CO CO •* ■* lO M O X X O O -M O --I — 1 c c M ^ H CD CO co"-* (N 0-1 lO O t- 3! O O O CO CO 05 CO -H o O O rt r-l (N (M }< N CO •* "O W 60 J3 (N O CO CO O ^ rt (N O CI H t^ lO t^ t^ IM •* lO t^ N CD 1< 00 - CO t^ CD CO X >o X r^ X 0) o o o o o O C C -H ^ CO ■* M CT e ^ (%] ^ N -1 N ^ 2 - ° !M o o (N CO c o o 1 '^ .« r- ■* C-l CD lO CO Til T)( ■* 1 — 1 5< _bfl IM -' CO -' CO O) IM 1 s BO -^ lO ■* (M CO O IM N ^ 1 s H CO CO lO CO CO -^ lO ■* CO 1 5§ ^ I c-l t^ CO 0> CD !-• O O (M "bo~ — I c-i o ci ^^ rt rt CO -a< CO lO C^ IM t> lO 1 rH ,^ O) CO IM o 05 o ,_) 1 •* '~ \ '^ '^ "^ ^ c-l '"' o 1 (M IM OS Tti lO ,_! w ■* -* 1 <» X! 1 1 (M C4 (M IM CI CI CO CO CO 1 CI BO 1 -^ CD -* t~ lO CO t~ CO ■* 1 % M 1 CO rt lO O X CO 1^ IP ^ 1 '^ J= 1 , 1 •-H 1 CD 1- o ,-^ CO -rs -^ ^ o 1 °^ ^ 1 '^ '"' ""^ 1 2 •-icoior-t^iocO'-it^ TtiCOOCOCO'OXXiO I -^ CDOCO-^t^wSt-COCO I O CO CI o H ,-1 r-l rt -H •^ ic CO CO O I "Oi CO CO CO CO CO —I ^ \ Tj< CO CI 2 °^ CO ■* CO c 1 c i? 1 CO CO CI Cl lO in ■* t- CI 1 a 1 c» 1 \ CD in lo in CO t^ CD t-x 1 c C5Xr-t^xcio^o ^ -H - I X I OCO'^lO'-lCO^^Cl'^ I i-lOOO'-l--iC0'-iC] I --< d'^cocoinoco'^ci ci CO CO O CO CO CO ■* CO CO Cl X CO CO t^ ■^ I '^ I CDt^O5C0O3XC0lOt>- -H d O O --< --I "-I 1-1 CO ^ -^oocO'-''-icicici I cT ClCOClO'-i— 1 '-' O O -H o o o o o 1 ^ (M O (M O •* O O -H o 1 2 C0"3WO'*OO'-iO I -H M Ol Tj( w O — I — I (>) M CO ■*OOt^OC0OOC CO "O O 00 5D Ol C rj uo CO M C5 O — ' 'M C5 O I O TT '«< •* CO ■* —I X i^ r~ t^ -^ o -H t^ o o o o s --o o s s o o to 00 "J" •"J" 60 = o o o o o o o o ° ■1 01 CO .-o 0) CO X _ao 00 CO CO -^ « o 1 CO Ol IM •* CO !N CD OO 1 Ol ^ 1 O t^ C3 CO 00 •o t- t- ■* •* 1 0) 00 00 01 00 C3 1 Tjl TH Tf Tl< ■*! Tt< lO _co_ Ol ■^ O) O O CO Ol 1-1 1 X i? ■-i c H o ^ - ^ (N e c Z ,H iM -■ IN '^ e« -' « ^ « > > 176 PROBLEMS OF PSYCHOPHYSICS o o - o -^ ^ CI - -^ 1 to Si 5< 00 X CI 03 CO 00 CI _5f '^ o o c o o '^ 43 00 ^ ^ o o o 1 s H a> -^ '^ o o o 1 s " ■* IN (N '^ '^ o 1 2 1 « CI ,« § ■<1< to 00 _ho o o '-' o o o -- t- '" CI ^ o t^ IM cq '^ 1 o o ^ t^ ^ ::! o CI CO '-* CO >f5 ^ to o CO 05 C) M ■* « ■* •* § _M '^ ■* o o '^ '^ ^ ^ to in o '-' ■* o ^ ^ t- 03 2 o 0\ C] CO o ^ fe to 05 - o ^ X to w CO CI CO CO 2 W) o •* '^ o CI CO 2 ^ cc "^ to o to CO 05 LO ^ CO •o r- o 00 to (M 'OS 00 00 o CO I- CO ■* 05 CI 1 ^ 1 c3 •« 00 to o 00 •* CO s CO CO '^ o OS CO 2 CO to CO '-' 00 CI g5 ^^ to ~to~ "oo' o ^ C) •^ o '-" = CI CO '-' '^ ^ ^ o o o N '-' ^ ^ CO I^ o o o '^ ^ ^ •* t^ o o C) CI o 00 H 00 ■<1< o •o 05 to to 00 GO CI •« 1 CI o o ^ '-' ■* 00 ^ 1 o o ^ o CI o CO s o o o o -^ o '-' ^ 1 o o -^ = CO o ■* c C o " (M CI ;::; > CI > ^ 2 „ CO ^ O -H -. o 1 '^ A 05 Gi »0 I^ O u'^ CO tJ< ^ ^ ^ ^ 1 '-'3 to 1 C) M CJ O rH rt -* rH 05 ^ to O ■* i-H lO ■<)< s H 00 O "5 CI OS lO 1 s o -a ■* O CI CI U5 t» 1 §5 t^ lO C) 05 CO I~ CO T)< T)< CO CO C5 1 " CI CO O -H O O CO 1 2 to O "O t^ t^ CO 5 o: lO to CS CI o s „ CI 05 t^ 05 CO CI 1 "^l 1 to A U3 to to t^ lO t^ CJ CI CI --H -^ -H 1 to 1 ^ W) U3 to OO O -H l~ u 60 CO 05 05 ■«" -H T}( CO to H CO o r~ ^ CI -H -H rt -1 CI CI C) CI o 03 - to CI to ■* O 05 C) CI CI O CO 'H ^ CO 00 CO o to ^ _M lO O CO CO OS 00 CO CO ^ to >0 00 00 lO CI 5 H rt O -H -1 •* o -H — • CI CI rt CI s CI - CI O CO OS CO 3C CO CO •* CO CO CO CO ^ t^ to CO 'H CI o 2 ^ ■* t^ O t^ CO CO M Si t^ t^ ■* CO CJ OS CI H rH •* ■* O 12 2 to to CI 05 CI lO CO X ■* CO •*■*■* ■* 05 CI -« o to O O »-i o •^ M lO CI to ■ > =M APPENDIX 177 i ^ N O ■'1' ■* -H - 1 £3 J3 CO -"K o O ir: 1 ^ 1 s; J3 T!< « '^ -H rt o 1 =° M r- Irt LO CO (N 1 ^ H 2 * ® ® ■^ N 1 5J o ^ CI x. •* C) CI o 1 2 J5 J3 lO o CO M ■* X o CO T)< CO CI 1 w U5 -H w lO rt '^ 1 2 00 -^ CO LO t^ CO 1 s H n C4 LO O X r~ 1 s - CO u-; ■* CT t^ C) ^ 1 g ^ ci c: (N -1 o C-l LO Tji -< CI CO IS « CO 02 CO X CO •o 1 J:; ^ ^ CJ !M o o CO C) O CO CO » Oi X CI rt CI 1 o 1 X 1 CO •= X — CO o c CO 1 s « O CO CO t^ —1 o 1 IS ^ (M t^ ;::^ X 'H CO 1 s Ol CO CI M ■* UO CI — CI CO 1 C] 05 - CO CO ft- o CI CI CI CI •« t- - CO -H C) CO o ! cj « 1 Mi O ^^ co~ ■* "■*" CO CO o 'iO~ 1 "5 1 CO \ CI cs CO X m X o 1 s 00 X ■* CO •* CO 1^ ■* •* 1 § •« 1 rt o o « o o 1 '^ ip 1 (N •^ CO CO M CO g ^ 1 CO s CI CO -^ o =: H 1 LO •* >.-: o CO CO 05 CI S9 1 c: X ^ ^ K 05 CI •^ 1 -H O o -H O c ^ 1 o o - o o '^ ^ ^ 1 T3S o o o o o o ° o e-1 -^ o o '^ ^ ^ CI s CJ -H W " > > tN - _ ^ _ _ _ r) ^ -1 o CO w :_; '^ i^ d c o o o " o o o o c o d _ _ - _ _ _ ^ _ _ _, -r -r S3 ~1 •o X -* ~: O"- o: 05 9 -9" ooooooooo O CI O CI O CI CI ■<1< o ddsoddooo xo:05ocoqoqoqo505 ddddddddd W i-O -^ dddddddd ooocooooo Cl ■* C ■* CI CO CI CI o cO'-^tCl-»'-^'coco>.o ooooooooo CDX-VXOCIXXO — COOCOCl— iCICI— I ddddddddd OOOOOOOOO CICOOO-T-rOOCO OOOOOOOOO X'CO 0505050505 05 C:0505 ddddddddd ooooooooo rt rt a i> .Si c^ — 1 CI -< c — CI — ■ CI r: 1 •"• > > HH »-. - ^ ;^ > 1< 178 PROBLEMS OF PSYCHOPHYSICS 00 o - 0.000 0.040 0.000 0.000 0.040 0.000 0.000 0.000 0.000 CO O d ^ ocoocoooo S3XOC3XC3r5c:o O o d ccccccooc . o o c - oocoooooo ^ ■* "M X ri -M o o •* c-i -H rt^ ^ rt c o o q d d d d d d d d d CD O O O d J3 ococococo 'MCOOX'S'OOtO q i~ X i^ r^ G3 C5 X 30 ddddddddd CO lO X d CO Cl d - ooooooooo cc X c> oa }. X o o c ^ ooooooooo ^Tt00 lO CO — ' Cl Cl O 1-1 Cl Cl ddddddddd d Cl 3; - •*cixocc£co^3 ICCOOOOOOOO ddddddddd X CO X d " 0.240 O.ISO 0.020 0.040 0.020 0.040 0.040 0.040 0.020 t^ X X - ooooooooo o^oxcooxo OJXOXCOCqO dd^d'^^'^d'^ CO lO LO o d ^ OOOOOOOOO CDOO-fOOOClO c: ^ =: S C c:^ c=0 < .\PPE\DIX 179 2 0.040 0.060 0.020 0.000 0.000 0.000 i o d J3 o o o o o o •<1< O CJ O X X o! X o: o c; m c c c d d d § o - c o o c c: c X Tj> CI O CI X q q q q q q o 5 d X c = c c c: c C O Cl Tf CJ CI c: c; 05 X c: X d d d d d d CO CO X X d 2 " CI X ■* ^ 7 d ci CI CI -• q r.~ q CJ d j:: o o c c c o X X X -T to CI t^ L': u- t^ t^ ■* d d d d d d c c d C3 " O C C C C C: CI C X O X ^ CO > 11 2 > < [ 180 PROBLEMS OF PSYCHOPHYSICS 03 O - o = c c o c O "M C M C-l O c c o o o c d d d d c d o o (M o d ■^ o o o o o o CC X O Tt< c o t- o O O X O: 6 d d d c 6 CO CO 00 00 d o - o o o o o o OC O ■* 'J' o -^ q c o o -H ^ d d d d d d CD CD O d ^ o o o o o o •* o •* 00 T»f t>- C3 oo t- ^ lo d d d d d d CO CO d o o " o o o o o o ■* 00 ^ X -XI T)< (M ^ T-l — 1 C-l M d d d d d d CO o d XI o o o o o o O C-l (M ^ o •* lO lo ic CO CO CO d d d d d d d g " o o o o o o C^l ^ C^l 00 O X lO ■* lO •* ID CO d d d d d d o o d J3 o o o o o o CD to ;d O 0) (M M CO O --I '^ 'M d d d d d d CD 00 d - o o o o o o ■* O O X ® CO CO CO X 1^ CO t>- d d d d d d CD d ^ o o c o o o •* C-l CD C-l ■* O -H -H o O O O d d d d d d CO CO CO o d 00 00 " o o c= o o o 'Ji X tT O CO CO 00 r^ X 05 X 02 d d d d d d CO CO CD 00 d J2 o o o o o o O (M C O C-1 O d d d d d d CO CO l» o d 00 " o o o o c o X CO c o c o 05 00 o o o c d d -^ -^ -^ r-^ CO CO I- d ^ o o o o o o o ■* o o o o o q o q q q d d d d d d o d .Si ^ Cq -H (N rt N V > X o " 0.040 0.000 o.oso o.oso 0.020 0.020 o c-l 00 = 1 ^ c o o o o o Tf Tt O O O •* CO X X 00 35 o d d d d d d 1 o ^ o o o o o o •* CD X ■* •* C O -H o o c o d d d d d d o c CD d J5 o o o o o o O O C^l CO O CO t^ q 00 t^ 00 00 d d d d d d CO d i o o o o c o O O X ■* •* X c-l CO IM ■* r-H — 1 d d d d d d CO CO Ol d J3 o c o o o o ■* X O O X C) T? CO ^* CO ■* CO d d d d d d CD CO CD - o o o o o o O Cl CO X CO C-l •^ lO -^ O CO Tj^ d d d d d d CD O d X o o o o o o CO M CD IM O CO -H O C-l -H IM W d d d d d d o o 1^ d - o o o o o c 00 C-l X X C X CD t^ I- X X >o d d d d d d c o •C (N rH O O O O O O O t^ t- t- O t- t-- o to CD CO C CO iC O O O CD t~ CO O OJ O O O "-I •^. f^ *? o o o o o o o ^, ,- CO -I sc O -H 05 05 05 00 o oi T-; d, d d> o d o o C-l CO CC C-l "* CO C C; '-' '1'. '^ <*. d o o d o c3 CO i> O r- C Q t;- 00 CO C « O O CO S it ^ 00 ci CO in 05 C5 00 ■* CO '-^ P d c d d d o o f, CO O CO C O CO CO CO O CO o p CO M CI CO "-I cc t^ 3 c d' -< lO p 00 p d d d d d d o t, CO CO CO o t-- o «5 ■" '^ =2 S 2 S 05 St. cc p <^ ^ P d a d d d a o CO t- t; g5 g 2 55 5 -35 CO § q ^ CO CD CO p d d d di d o o O CD C5 00 05 I-; S O S 03 I- « CO M O t- N t~ ^ 2 S o C5 05 t^ in --1 P rH d d 6 o c o o (M 1^ N t^ S 12 O p p IM •* CO p d d 6 d d d o 00 CO 2 I- :i 2! S t^ m 00 CD -^ 3^ S L t-- 00 o t^ 00 g; 05 S 00 t- ri; --; p d d d d o o o ^5 5 - ?g ^ s I i § ;: s S ss § d d d d d o o S g :: e:: So o § 05 05 05 t- •" "^ P d d d d o o o ri C C5 IM CO CO O %\ C * '. '^. p d d d d d d d CO o 05 >o >2 •;; S CO CO o o s 5S S d d d d d o o o ■*■*■* o 2 o d o ci CO CO d d d d d d d r-i ^1 O "^ '^ '-'^ "^ d o ^ ^' CO CI p d d d d d d o rji Ci X ^ 1^ O — < CI CO p d d d ■^ d APPENDIX 187 S Q ■y) Tf to CO Cl Cl CI o M M CO o c: I-- o: > to ':o •O CO to £K t^ o lO lO O O CO 00 CI t- o o CO "5 CO CO ffl 00 .-1 CO CO t/l rn 00 CO ■* •* CO o CI X t-< -H •* > CO Tf •* 00 •* CO 05 o lO o CO CJ o ^ •* CI o IM lO to rH CO o 05 1-1 CO CO 00 CO CI •* t^ 00 ■ •* 02 lO CI lO ■o IN 05 O CO ■* CI CO o CI '~' CO 02 CO »— ' •-1 CO N 1-1 ■^ CO- fN o 00 O U5 ■* CI r^ 00 N CJ T-l Tl< CO a > lO on r~ t- 35 Cl CI t^ 05 CO •>»« Ci '^ CI O 00 CI 05 ^ CI CO ^ CI 1^ o ^ T)< 00 ■* o o on ■* 1^ •— 4 t^ O O CO oc CO t~ O ^~* a C3 (X t^ ^ CI CO •>J< 05 OS •* on d CO o ■* V) ^ 00 on C3 o o o o rH -H '"' Ph CI CI CI to O 00 CO to lO -< -H >0 O CI t^ CO i^ lO CO CI CI lo CO OS *~] CI CI OS C) CO 00 o q > 1--^ -^ cb CO CD CO — " ^ • r>^ oi OJ CO 'I' o CO CO ■* I- -H CD C ■* CO OS rH lO lO --I ■* •-' OS T-l CO CO --1 OS CI 00 '■*•<* C CI CI CJ O CZ) Tf X o •-< o w o 00 i~ CO i^ c; iO > CO I^ CO CD -^ CI •* 'S' 00 lo t~ t^ ci d 00 d 00 t^ X t~ •* 1^ I^ CI i-H .-1 CI I- CD '-' "-1 ■* l-H CO CI -H OS X O CO O C CO CO to ■* ■* 1- CI C: t- O o -^i m OS uc lO o »-H > ■* CO CD --I . --' X ■-; t^ ■* .-^ X d ic ci t-^ 00 CO O CO CO OS —1 CO CO -H CI OS I~ CI O CI to CI CO CI OS O CI CO ■* O C -"l" to O --1 >H CI C X X O t^ C O C CI OS o (_H o OS ■* -1 CO I- CO OS ►— ' d X I- CD d ^ ■*' 00 O X X lO CI o ■* OS .-1 lO X CO CO •* OS 1-1 CO CO OS CI CI O CI O N Ttl N CO t^ •«*' OS o CO •* rJH O X »-< O •* t^ (^ (— t CO CO ■-; p i-| 00 CO 'I' f— 1 d •* 1^ ci d OS cs 00 O X OS OS CO X •* CO .-1 T-1 X CI .-> ■<1' LO t^ CI CO CI OS CI O CO O O CO CO o CO O -< X O i-O CO CI CI X CI CD O O OS OS in X in t^ •-; "-^ CO CI ►^ lo •*' d •* 'I' 00 i-o CI ^ in .-1 CI CD CO CO X ^ OS X X c:s -"i" ^ CI CO o ■>!f< X CI to O ■«1" 00 ^ X X OS OS c o o »-t »-l t-t 188 PROBLEMS OF PS YCHOPHYSICS TABLE 32. Observations on the Just Perceptible Positive Difference. SUBJECT L IVa I III IV Tk Nk TkNk Nk TkNk Nk : TkNk Nk l-kNk 84 88 — 3 264 1 88 3 264 92 9 828 8 736 7 644 6 552 96 16 1536 24 2304 20 1920 16 1536 100 15 1500 15 1500 36 3600 40 4000 104 58 6032 43 4472 35 3640 33 3432 108 2 216 7 756 1 108 2 216 I 100 10112 100 10032 100 10000 100 10000 Average 101.12 100.32 100.00 100.00 TABLE 33. Observations on the Just Perceptible Positive Difference. SUBJECT II. Nk [Va l-k Nk I III IV fk Nk rkNk Nk rtNk Nk TkNk 84 4 336 2 168 1 84 1 84 88 — 3 264 4 352 3 . 264 92 8 736 18 1656 7 644 12 1104 96 25 2400 26 2496 27 2592 23 2208 100 30 3000 23 2300 28 2800 37 3700 104 24 2496 19 1976 31 3224 21 2184 108 8 864 6 648 2 216 2 216 I 99 9832 97 9508 100 9912 99 9760 Average 99.31 98.02 99.12 98.59 APPENDIX 189 TABLE 34. Observations on thb Just Perceptible Positive Difference. SUBJECT III. IVa I III IV Tfc Nk TfeNfe Nk TkNk Nk TkNk Nk TkNk 84 1 84 88 4 352 3 264 — 1 88 92 10 920 19 1748 4 368 3 276 96 21 2016 36 3456 8 768 20 1920 100 36 3600 34 3400 32 3200 41 4100 104 28 2912 5 520 56 5824 30 3120 108 1 108 2 216 — 5 . 540 I 100 9908 100 9688 100 10160 100 10044 Average 99.08 96.88 101.60 100.44 TABLE 35. Observations on the Just Perceptible Positive Difference. SUBJECT IV. I III IV kk Nk TkNk Nk TkNk Nk TkNk 84 2 168 4 336 88 2 176 3 264 1 88 92 8 736 20 1840 4 368 96 35 3360 28 2688 27 2592 100 32 3200 24 2400 34 3400 104 22 2288 20 2080 25 2600 108 1 108 3 324 5 540 I 100 9868 100 9764 100 9924 Average 98.68 97.64 99.24 190 PROBLEMS OF PSYCHOPH YSICS TABLE 36. Observations on the Just Perceptible Positive Difference. SUBJECT V. I III IV Tk Nk TkNi Nk TkNk Nk rkNk 84 2 168 1 84 88 1 88 — 3 264 92 9 828 16 - 1472 13 1196 96 29 2784 36 3456 63 6048 100 30 3000 35 3500 17 1700 104 19 1976 12 1248 4 416 108 10 1080 — — I 100 9924 100 9760 100 9624 Average 99.24 97.60 96.24 TABLE 37. Observations on the Just Perceptible Positive Difference. SUBJECT VI. I III IV fk Nk rkNfe Nk TkNk Nk rkNk 84 2 168 — 88 6 528 — — 1 88 92 13 1196 5 460 2 184 96 25 2400 8 768 18 1728 100 • 31 3100 44 4400 20 2000 104 21 2184 37 3848 32 3328 108 1 108 6 648 23 2484 I 99 9684 100 10124 96 9812 Average 97.82 101.24 102.21 APPENDIX 101 TABLE 38. Observations on the Just Perceptible Positive Difference. SUBJECT VII. I III IV Tk Nk TfcNk Nk TfcNk Nk TkNk 84 1 84 1 84 _ _ 88 1 88 1 88 — — 92 7 644 4 368 8 736 96 6 576 18 1728 19 1824 100 35 3500 27 2700 39 3900 104 31 3224 38 3952 30 3120 108 19 2052 7 756 4 432 I 100 10168 96 9676 100 10012 Average 101.68 100.79 100.12 TABLE 39. Differences between Observed and Calculated Values op the Just Perceptible Positive Difference. Subject Differences between observed and calculated values. 1 , Calculated value IVa I III IV ToUl I 99.76 + 1.36 + 0.56 + 0.24 + 0.24 + 0.60 II 98.31 + 1.01 -0.29 + 0.81 + 0.28 + 0.40 III 99.82 -0.74 -0.94 + 1.78 + 0.62 -0.32 IV 97.98 + 0.70 j -0.34 + 1.26 + 0.54 V 96.01 + 2.33 + 0.69 -0.67 + 0.78 VI 99.21 -1.39 + 2.03 + 3.00 + 1.20 VII 98.76 + 2.92 + 2.03 + 1.36 + 2.17 192 PROBLEMS OF PSYCHOPHYSICS > z QO (O « 00 O O O to t^ ■* C^ O Ol •* ,1 ^ t^ ^ — I (M CI .1 ■* O O CC to >o CO 1 > A! M 00 to o to o o o to -^ ti en o CD -"i" .-( o CO 00 lO CO c-i n Tji o a> CO o to CS o 1 M Z C) t- O .1 lO o o C) 10 O) OJ CO cq > (N (N to 00 O O O lO lO 05 GO O ^ 00 Cq CO •* CI C5 to O CO O 00 CO 'H 00 o CO C5 CI to 05 q 05 z CO ■* » CX3 C^ "5 O CO IN X CO -" o o CO > z •«J< 00 -^ O O 00 N O IM -^ ■>* O to t^ iC i-O Oi CD O OS 0> C^ 00 05 CO to >o 05 •o 00 Z CD te CI o o f~ 05 CO 05 03 to o o CO S M Z li ■^ •* CI o o to •<1< 00 O ^^ CO o t^ to t- CO -^ CO CO 00 CO 00 '!< C) o o 00 OS CO 05 Z ,-1 X to lO CO C5 X CO 00 ■>* 'H o o a z li O O O CD O O •* t- 00 •* C5 o oo 2; to CO — ' O CO 00 05 t3< 05 -"I 05 ^ CJ s 05 CO 00 05 z 00 O >0 ^ 00 lO 00 .-1 ■* O -^ 05 rH lO 05 CO « z li to O CO O CD to 1 ^ to C) O t- O) CD 1^ CI CO lO CJ C) I^ O t- "^ o o to CO d o z t^ O CD to C5 CI 1 CO t- O to f-H § T)< 00 CI to O ■* 00 00 00 05 Ci o o o ^ 4) > < > O lO lO C5 O — 1 o O OS O 1" CO CO o 00 OS ■* CJ Tf o o rH -1 CO CI q q q d> o d d o d d o o o o o o o o -i d > t- 1^ 00 CO CI CI -« to to O ■* CO CO o ^ C) X CI ■* o o .I c) CO c> q q q d d d d d d d o o o o o o o o -i d > f, to to to to 00 o to CI d ^ o -^ --< lO d to t^ •* ■* o o --I -CI c) CI q q d d d d d d d OS o OS o OS o OS O d d > o LO 1* ■* 05 CO in O QO O CI t^ O O t^ O CI C» lO CI o o -< CO CO ■-; q q d d d d d d d o o o o o o o o ^ d 5 C5 N lO lO CJ Cl Cl CI CJ CI CO CO to o •<}< o i-O --I X o o o ^ ■* CO o q q d d d d d d d o o o o o o o o ^ d « 0.0956 0.1668 0.3475 0.2757 0.1017 0.0124 0.0003 o o o o o o o o ^ d - O -^ 05 CO •* o o O X CO O O f- o CO 05 05 t- r^ o o o o •* CI o q q d d d d d d d § § o o o q « d X -"J" O CO CI X 1" O O O 05 OS X X i APPENDIX 193 > 19.4400 20.7480 34.9520 21.5904 3.9560 0.2728 0.0000 c» "5 d o i > to 00 O 00 ■* to ■^ CO to O CI •* -1 00 o r- 30 CO CO » o to o q lO 'i; CI q ci co' 00 -H -^r d d •^ M PO c^ to lO d o i > to •>* O CO Cl 1- o CO C — ' CO iO C) ■* c) lo to r~ CO 1" 00 ^ t^ ci q -H q q to ci to to ci CO d ^ i 7.5600 11.2840 32.0370 30.9504 14.5268 1.7864 0.0420 to to 00 00 05 1 1 1 4.5576 10.6288 45.2530 30.0960 7.6544 0.5456 0.0168 CJ CJ "O 00 OS 03 CI O Cl 'J' Cl CJ •* t^ CI 1^ to '^ lO cj Ti< in to i-o c: CI CO CO t- ■* CO q q d t^ ■«)< td d '-' d ^ r^ n c-i o to CO d 05 6.4800 10.2024 49.3860 25.9776 6.4768 0.6160 0.0000 00 00 CO d OJ i 00 •<*i O to CJ 00 '1< O O O C5 » 00 00 ^} O O O •f o ■* o X O C) O X CI to o X CI en o r^ lo 3 o ^ in r- CI to o> o o •-• ^ OS t- m CI CO •«»' o CO > o> m 05 r- to c» O -H ■* O CO Cl Cl CO Cl "^ 00 (M O 00 00 00 to o X r^ o oo ■<)< o o X X o ■* to X o -- o; o -■ o) t- t- t- > -H -1 X t- t^ Tf O ,-1 to m o CO o c) CI CO ■<>< X O tH -H Cl CO CI 00 to o to '»< Cl o CD X -H o m X ^ to ■* ^ o to CO CO in > CO c -■ C 't o o 05 -H CO to CO o to r- o CD Cl CI C CO 'fi o CD CO o in o CO o rt CI CI CI 05 O O O ■* CO CI o c< o to o X in CO X X CO O CO 'to O CI lO > ■^ I.- 1- CI ■* CI in t— « O CO CO -< CO t^ CO CI — 1 t- o w CO m to X -• CI O) CO ^ o --^ CO CI ^ 05 X CI O O X X CI X o in o to ^ CI -H o CI o> o — O -< -H H-t CI CO CO CI Cl o ■* t^ ^ CI in m o ■* X i-H in 0-. O Cl X o ■«»< to ■>!»< -H O X t- rt ■q' CI O) ■«1< X O CJ X to 00 to X X o -^ X in CD 05 r^ o o i-n X CI -< to 1— ( O -H CI X 1- o — t— ( in •^ m c o CO CI •* •^ O t- Tf O 35 05 -H X t O X -H ^ CO CI 05 o to o to to o o 00 O C5 O (35 lO 00 O M ■* Tj< o ■* to o o ti 1 X O to X X CI o •* 05 — • X CO in •* o CO OS to CO 05 C5 in If to o 05 •* in » ^ ■* CI 05 ^ X ■^ O to Cl OO Tl< ^ u O O O 05 05 00 X >-H ,-1 rf 1 194 PROBLEMS OF PSYCHOPH YSICS TABLE 44. Observations on the Just Imperceptible Positive Difference. SUBJECT I. IVa I III IV fk Nk TfeNk Nk TkNfc Nk Nkfk Nk rkNk 84 88 5 440 2 176 1 88 92 9 828 12 1104 9 828 6 552 96 15 1440 13 1248 31 2976 40 3840 100 58 5800 56 5600 44 4400 42 4200 104 5 520 11 1144 11 1144 8 832 108 8 864 6 648 5 540 3 324 I 100 9892 100 9920 100 9888 100 9836 A-Verage 98.92 99.20 98.88 98.39 TABLE 45. Observations on the Just Imperceptible Positive Difference. SUBJECT II. IVa I III [V Tk Nk i-kNk Nk TkNfc Nk l-kNK Nk TkNk 84 1 84 88 1 88 4 352 2 176 3 264 92 13 1196 12 1104 11 1012 14 1288 96 27 2592 20 1920 33 3168 38 3648 100 28 2800 35 3500 45 4500 25 2500 104 17 1768 19 1976 5 520 12 1248 108 14 1512 10 1080 4 432 7 756 I 100 9956 100 9932 100 9808 100 9788 Average 99.56 99.32 98.08 97.88 APPENDIX 195 TABLE 46. Observations on the Just Imperceptible Positive Difference. SUBJECT III. IVa I III IV fk ■Nk TfcNk Nk TfcNk' Nk TkNk Nk TkNk 84 88 92 96 100 104 108 2 10 33 45 10 176 920 3168 4500 1040 2 29 48 14 7 184 2784 4800 1456 756 4 17 43 36 368 1632 4300 3744 1 5 IS 40 31 5 88 460 1728 4000 3224 540 I 100 9804 100 9980 100 10044 100 10040 Average 98.04 99.80 100.44 100.40 TABLE 47. Observations on the Just Imperceptible Positive Difference. SUBJECT IV. I [II IV Tk Nk TfcNk Nk TkNk Nk Nkk 84 1 84 88 2 176 5 440 2 176 92 18 1656 19 1748 18 1656 96 27 2592 25 240 28 2688 100 38 3800 29 2900 36 3600 104 S 832 12 1248 13 1352 108 7 756 9 972 2 216 I 100 9812 100 9792 99 9688 Average 98.12 97.92 97.86 196 PROBLEMS OF PSYCHOPHYSICS TABLE 48. Observations on the Just Imperceptible Positive Difference. SUBJECT V. [ III IV fk Nk TkNk Nk TkNk Nk I-fcNk 84 88 4 352 8 704 92 8 736 19 1748 59 5428 96 24 2304 39 3744 20 1920 100 30 3000 28 2800 10 1000 104 28 2912 6 624 2 208 108 10 1080 4 432 1 , 108 I 100 10032 100 9700 100 9368 Average 100.32 97.00 93.68 TABLE 49. Observations on the Just Imperceptible Positive Difference. Subject VI. I III IV l-k Nk TkNk Nk TkNk Nk TkNk 84 . 88 2 176 92 12 1104 2 184 4 368 96 25 2400 32 3072 12 1152 100 36 3600 44 4400 33 3300 104 14 1456 16 1664 36 3744 108 11 1188 6 648 15 1620 I 100 9924 100 9968 100 10184 Average 99.24 99.68 101.84 APPENDIX 197 TABLE 50. Observations on the Just Imperceptible Positive Difference. SUBJECT VII. I III IV fk Nk I-kNk Nk TkNk Nk J-kNk 84 1 84 1 84 88 4 352 4 352 1 88 92 14 1288 9 828 16 1472 96 21 2016 37 3552 24 2304 100 33 3300 28 2800 42 4200 104 22 2288 17 1768 16 1664 108 5 540 3 324 1 108 y 100 9868 99 9708 100 9836 Average ; 98.68 ; 98.06 1 98.36 TABLE 51. Observed Values of Just Perceptible and Just Imperceptiblb Positive Diffbrencb. Subject. •IVa I III IV A B A B A B A B I II III IV V IV VI, 101.12 99.47 99.08 98.92 99.56 98.04 100.32 98.02 96.88 98.68 99.24 97.82 101.68 99.20 99.32 99.80 98.12 100.32 99.24 98.68 100.00 99.12 101.60 97.64 97.60 101.24 100.79 98.88 100.00 98.08 98.59 100.44 i 100.44 97.92 1 99.24 97.00 i 96.24 99.68 102.21 98.06 100.12 98.36 97.88 100.40 97.86 93.68 101.84 98.36 198 PROBLEMS OF PSYCHOPHYSICS 00 W » C^ O O IM CI ■X to o» 00 r- o CI 1- CD 2; rH t^ lO M CO t^ OJ ■* CO CO I- O lO X '"' > M Z (M 05 0» M CO in 05 05 CO 00 O lO c» m CO •* o •* 5D CO CO 2 I- 1.0 CI O CO ".O rt CO CO CO M 'l- o I 1 .-H CO ^ CO CO CO o o H.4 ■ > i! (M » a; CO CO IN ^ r- CO -^ CO CO 2 CO CO CI 00 o ■^ o o o Z 1^ — 1 CD O 't CI C OS Oi OC t^ CO 1—1 o 1 rH t~ t^ CD CO -H o 05 > M CI CD CO OC CD Lt o 2 CO 2 ■* CI O O O IM •* CI t~ 00 O: CD 00 O CO ■* t^ O CD CO ■^ O CI o > .M ^ 0> >0 O CO CO 00 05 2 lO « O CO rt ^i ,J>! ■* CI CI O ■* CO cc o 2 O CO -• O CD Oi CD CI O CO CD ■* C) .14 r^ C2 t^ O ■-< OS Vi »—< 1 M CO -"i t- CO -< CI o 2 Tt< .14 ■* O O CC O CI o ■* o 2 00 00 O CI o --^ cc .1^ ^ rt CO "C CO 05 X 03 u ■-H rH CO 1— t 1 ■^ -H O O OC CO CO >o o — LO -• CO i^' CO 2 'J' j< Tt< CI ■* O O CO CO o 2 o ^ c o ■* r- CO >o - 1- CO - 3 CO C. 05 CO X o> X to Oi O lO CI o .£<: CO O O CO C4 •* 2 < > o c ! O CO O CO CO o o O CO O CO CO o o CI O O ■* CO ■* CO O ^ CI O r-; CJ 05 d di d> d o d> o - O t- O I- I- o o O CO o o o o c X C5 ^ lO CO CD ■* 05 X t^ ■* CI o o d d d d d d d > o t^ t~ CO O CO CO o CD CO CO O CO CO O CI CO X -1 C5 CO X O ^ CI lO 1-; 05 O! d d d d d d d CO CO t- o r^ r- O CO CO O O CO CO o t~ CO — ' 05 O CO CI C5 X t- r)< CI o o d d d d d d d > o c O CO O O CO t^ o O CO O O CO CO o ■* 1^ O -H CI CO X O O CO 1--; 00 05 05 d d d d d d d -■ o r- o o t- CO o O CD O O CO CO O CD CI O C5 I^ CO CJ o; q t- c^ — q q d d d d d d d > _, ot^coocoo ,■ O CO CO o o o c ■S i d d d t^ o 0.0667 0.1378 0.3000 0.5511 0.7689 0.9044 0.9844 - CO Cl O 05 -^ «0 CD CO CI O X ■- >« "3 CO CO O "^ (^ 05 •-1 S X S ^ CI q q d d d d d <^ o c Tj< T)< o •* X t- M •* •* O ■* t~ CD CO CD -^ CI CO 1^ X 05 o --< ■* 'C 00 q ® d d d d d d d - CD CO O CO Cl CO t~ lO »o O *0 Cl CO CD CO ira X CO CI -:< o 05 00 lO CO ■-; q q d d d d d d d 1 Tj- X CI O O 3" 00 X X O: C-. C O O APPENDIX 19 9 TABLE 54. Values op the U's for the Determinatiom of the Just Perceptible Nbgativb Difference. I II III IV- V VI VII 108 0.0067 0.0156 0.0267 0.0200 0.0200 0.0200 0.0400 104 0.0132 0.0941 0.0584 0.0490 0.0327 0.0653 0.0576 100 0.1198 0.2058 0.3477 0.1955 0.1674 0.1890 0.2406 96 0.2887 0.3073 0.3971 0.3236 0.2262 0.3556 0.3022 92 0.3315 0.2641 0.1479 0.3158 0.3876 0.2652 0.2661 88 0.2054 0.0976 0.0213 0.0887 0.1539 0.0905 0.0838 84 0.0324 0.0145 0.0009 0.0071 0.0117 0.0139 0.0095 I 0.9977 0.9990 1.0000 0.9997 0.9995 0.9995 0.9998 R 0.0022 0.0010 0.0000 0.0003 0.0005 0.0004 0.0002 T^ lBLE 55. V ALUES OP T^ Uij POR the Determination of the Just Perceptible Negative Di fference. 1 I II III 1 >v V VI VII 84 2.7216 1.3104 0.0756 0.5964 0.9828 1.1676 0.7980 88 18.0752 8.5888 1.8744 7.8056 13.5432 7.9640 7.3744 92 30.4980 24.2972 13.6068 28.9436 35.6592 24.3984 24.4812 96 27.7152 29.5008 38.1216 31.0756 21.7152 34.1376 29.0112 100 11.9785 20.5764 34.7685 ' 19.5509 16.7354 18.9030 24.0627 104 1.3728 9.7864 6.0736 5.0960 3.4008 6.7912 5.9904 108 , 0.7236 1.2180 2.8836 2.1600 2.1600 2.1600 4.3200 I 93.0849 95.2780 97.4041 1 95.2281 1 94.1966 95.5218 96.0379 200 PROBLEMS OF PSYCHOPHYSICS O ^» •* IN C IC O ■* 2 O -^ lO X O !M o © M 1^ •«< «D ■* (N N (M ■* O OO 5D g 2 O CO (M M O 2 5^ O CO > CO . tt IN N 52 t" ^ o (N 03 '^ (M i£> -^ iM O N O CO iO ""1 CO Ci O CO o lO O ■* lO ■* OD X CO > lO X CD CO "^ 1^ ^1 '"* W — 1 o ■«> CO CO CO o X 35 X X t^ lO CO — 1 ra o to CO iM — 1 CO c-i ^ !0 00 M 5D O O O CI (^ IM rt t^ O rj< O CO 03 05 ^ lO » X X > O 00 X M O 05 IN •* o CO IN CO m 05 CO 1— ( us 00 CO X "O C^l CO o CD CO cs o> lO ^^ (N IN --1 05 ■^ IN CO CD O •* X o O t^ lO CO O •<3< X o lO -^ CI t^ lO i-O CI CO l_l CO 05 X CO 00 CD "* t^ s CO ■* --I Ci CO --^ -^ CI CO lo in t- CO " o >n -< CO CO OS CO •* •* X O CO o X CO ■># CI CO O lO "9" CD t^ -^ ■* 1^ Tf X ■* t~ t— 1 O X CO O CO I^ lO o *"* o »n in CI t» r- --I o ■-1 in CO CO m —1 CO ■* •H t~ CI X o o -^ Cl C) CI -H 05 •'1' CO O CI O CI X CI ■* t^ CD 05 O — < X t^ -< rH -« K3 in t^ Tf r^ h-» CD CD X CO 00 r- -H ■* 00 o in o f^ CJ X Tt< IN 05 C CO 05 ■>J< t~ o C4 in X CD -" -^ r^ X •fli X N CO O •^i X 1 M X 05 05 C O O •H »-K 1-H > it Cl X CD CO O C) O •o CO 05 r^ o -^ c IN 05 Tji OS ^ CO CI CO c) .-H c CI CO 05 05 X 2 CO -H X —1 —1 CO o rt CO CO -H 05 05 s ■^ Tti CO X O 00 X X ■* CO CO o o o -H c -:< CO o -H ^ CO c<3^ -H X CO O! 05 -H M CO CO CD CI r< T^ CO CO rt 05 05 - z CI ■^ O X o •* in ■^ ■* X o o C) CO X CO o ^ CO -H CI rH X CI C) 05 X o ci 05 2 CO X O X O -H CO CI CJ -H § > 2 O CD ■* O O CI t^ CD CI O T}( CO X 05 CD C) CO -H 05 05 2 in r^ CI o CD CI Tl' N i •«< X CI CO O iJ" X X X 05 05 o o o tN <: APPENDIX 201 TABLE 58. Observations on the Just Perceptible Negative Difference. SUBJECT II. IVa I III IV ffc N. fkNk Nk Tk'Nk Nk TfeNk Nk l-kNk 84 2 168 1 84 2 168 4 336 88 6 528 12 1056 13 1144 17 1496 92 30 2760 23 2116 28 2576 33 3036 96 29 2784 29 2784 24 2304 30 2880 100 21 2100 19 1900 28 2800 14 1400 104 10 1040 13 1352 1 104 2 208 108 2 216 3 324 4 432 I 100 9596 100 9616 100 9528 100 9356 Average 95.96 96.16 95.28 93.56 TABLE 59. Observations on the Just Perceptible Negative Difference. SUBJECT III. IVa I III IV fk Nk TkNk Nk l-kNk Nk l-kNk Nk i-kNk 84 2 168 88 4 352 2 176 1 88 92 17 1564 13 1196 9 828 13 1196 96 38 3648 45 4320 42 4032 40 3S40 100 34 3400 27 2700 45 4500 34 3400 104 5 520 8 832 o 208 10 1040 108 5 540 2 216 2 216 I 100 9652 100 9764 100 9784 100 9780 Average 96.52 ' 1 97.64 97.84 97.80 202 PROBLEMS OF PSYCHOPHYSICS TABLE 60. Observations on the Just Perceptible Negative Difference. SUBJECT IV. I [II IV l-k Nk 1-kNk Nk TkNic Nk TkNk* 84 1 84 1 84 2 168 88 15 1320 6 528 5 440 92 35 3220 36 3312 37 3404 96 28 2688 34 3264 22 2112 100 17 1700 17 1700 24 2400 104 3 312 5 520 6 624 108 1 108 4 432 I 100 9432 99 9408 100 9580 Average 94.32 95.03 95.80 TABLE 61. Observations on the Just Perceptible Negative Difference. SUBJECT V. I III IV Tk Nk i-kNk Nk TkNk Nk rfcNk 84 1 84 3 252 88 10 880 8 704 27 2376 «2 33 3036 30 2760 53 4876 -96 31 2976 28 2688 10 960 100 18 1800 28 2800 4 400 104 6 624 3 312 1 104 108 1 108 3 324 1 108 I 100 9508 100 9588 99 * 9076 Average 95.08 j 1 95.88 91.68 APPENDIX 203 TABLE 62. Observations on the Just Perceptible Negative Difference. SUBJECT VI. I III IV Tk Nk TkNk Nk' i-kNk Nk TkNk S4 1 84 1 84 88 16 1408 7 616 10 880 92 25 2300 32 2944 24 2208 96 32 3072 42 4032 32 3072 100 19 1900 13 1300 23 2300 104 4 416 4 416 11 1144 108 3 324 1 108 I 100 9504 100 9500 100 9604 Average 95.04 95.00 96.04 TABLE 63. Observations on the Just Perceptibi^e Negative Difference. SUBJECT VII. I III IV Tk N], l-k^'k Nk rkNk Nk l-kNk 84 2 168 88 9 792 1 88 16 1408 92 28 2576 24 2208 27 2484 96 28 2688 31 2976 35 3360 100 23 2300 31 3100 16 1600 104 10 1040 5 520 2 208 108 2 216 8 864 2 216 y; 100 9612 100 9756 100 9444 Average 96.12 97.56 94.44 204 PROBLEMS OF PSYCHOPHYSICS 1— 1 > it U 00 00 00 Tti o cc o CO 00 « 'M O CD 05 .^ CI C-1 O O 1^ CI C-l t- » t- ■-' -^ 0) X X IN T* o CD 05 a -Z (M to » ■* O t^ IM C-I t- 05 t- T^ •-< § CO > 00 •* N CO O CO (M CD O lO t^ O t^ M .-H O) •* -H lO 05 ■<)< (M t^ O lO ^ i X IN CO CO in 05 IM CO -H CO lO 05 ■* CO M O lO rt CO > J4 CO O (N ■* O O O CO CO t^ IN O ■* -"f CO 05 CO CO o o o (N t^ X IN ■* Lt. CO 05 c o o ■* ,-< CO >o .-1 o> 2 i COOOCO'^OCDO Q COOOCOCOO'O'* ?1 CONOsOOO-*iO ■* (N 05 00 lO "-I 00 c o O) 2 Tj< CO CO ■* 00 Til 10 \ t^ M lo o lo Oi lO CD ■* C) o o K a. CO t X IN O ■* •* rH rt 00 (N i - 00 (N CD p-1 O ■* •* O CO CO U7 O M (M O 00 IN r- CO CO CO .-1 t~ (N O •* CO o CO CO CO 05 J4 2 IN 0! CO C-l CO CD CO •^ 00 CO 1^ il> X C5 CO AS ■* X IN CO O ■* 00 X X O! 05 O O O c^, ID §> > APPENDIX 205 TABLE 65. Differences between- Observed and Calculated Values of the Just Perceptible Negative Difference. Differences between observed and calculated values. Subject Calculated value IVa I III IV Totals I 93.08 -1.28 -0.80 + 1.35 +1.06 +0.08 II 95.28 + 0.68 + 0.88 0.00 -1.72 -0.04 III 97.40 -0.88 + 0.23 + 0.44 + 0.40 + 0.05 IV 95.23 -0.91 -0.20 + 0.57 -0.18 V 94.20 + 0.88 + 1.68 1 -2.52 + 0.02 VT 95.52 -0.48 -0.52 +0.52 + 0.16 VII 96.04 + 0.08 + 1.52 -1.60 0.00 TABLE 66. Values of the U's for the Determination of the Just Imperceptible Negative Difference. I II III IV V VI VII 84 0.0644 0.0667 0.0044 0.0400 0.0400 0.0267 0.0200 88 0.1351 0.1286 0.0442 0.0736 0.0704 0.1331 0.1012 92 0.3362 0.2414 0.1247 0.2068 0.2669 0.2380 0.2285 96 0.3085 0.3104 0.2480 0.3806 0.4421 0.3071 0.3533 100 0.1368 0.1944 0.3588 0.2362 0.1487 0.2341 0.2178 104 0.0188 0.0528 0.2067 0.0596 0.0309 0.0569 0.0745 108 0.0003 0.0056 0.0128 0.0031 0.0010 0.0040 0.0046 I 1.0001 0.9999 0.9996 0.9999 1.0000 0.9999 0.9999 R 0.0000 0.0001 0.0004 0.0001 0.0000 0.0001 0.0002 206 PROBLEMS OF PSYCHOPHYSICS > C CO O X C O 00 O lO C] O X X « X O CI ^ 1^ Tf C3 :d o o 05 i^ r~ •* T-^ X ^ CO ^ t^ d C^ CO CI CI iq id C2 > 2.2428 11.7128 21.8960 29.4816 23.4094 5.9176 0.4320 CI CI rs o id 05 > o ci X to in to o O LO Tfi -H X CO X CO m lo •* to -H o CO >-; o •* X CI '-; CO CO 'i^ ci Tr CO d Cl ^ rt C3 > O =0 to to O Tl" X O to LO t^ T)< X ■* S r- CI CO c; 03 CO CO •* p lO CD .-; CO CO to oi CO CO CO d ^ CO C) CI t-- UO id o a to CO -# O lO X ■* 05 05 CI X t^ CO CI CO X t~ o r~ o X CO X ■* X X ^ CO d CO -^ CO "-d ^ rH rt CI CO CI CO to 05 CJ X CI ^~* X X X •* O CI X CI CO X X CI -^ Tit O -^ O O •* 35 O CO CO C) t-; •* •* CD i-d '^ ci d d id d ^ Cl Cl -> X Tf to ■* ••ji C5 HH CD X ■* O lO CI ■* 03 X O CO CO "O CI O X CO -"I t^ "-O CO ^ X CI CO CD c> q id -^ d d CO --t o ^ CO ci rt X o lO CO C5 ^ U T)< X IM CO C *^ X X X o o o o o ^ 3 X O X C C TJ< o C CI •* CI O C) Tl< o CI o CI — o O lO 05 ■ ■ rt CD p q X r-; CO p 1— 1 rH CO ■* CO t- id CO ci > ■* X CO LO 1^ O lO lO ■-1 W O CI -H X -H CO CI OS CI ■* o CD O ■* O CD lO CO CI CO O C CO CO ci CI CO CO ■<»' CO LO ,_, CO t- ■* CI 05 ■* CD 00 > x d -^ d d id CO cd X CO ^ CO •* -1 •* CO 1-1 O O X CO CD o rt CI CI CI o O CO CO CO O •* o CI O t- -H CO o ■^ ■* ■^ r^ -^ O i^ '-' CO X > CI -H p CO X CI CO lo CI id d •*'td Ti< ,-< cd X -^t LO t^ X CO " CI lO CI O "* CO a> O) CI ■* .-1 X O ■* CI to O CD ■* C5 O X LO 05 O CO X IC 'i' LO LO O O CO LO LO > CI p CO CO ■^. CD ^ CO ci d d r-^ CI tiJ cd cd X CD LO O CD Tt" CO lO CI lO t^ LO CO CD .-< CO N Cs ■«< X X O O CI CI Ttl CO ■<)< o X o r^ o CO •<9< X CO CO lO CD Ol t^ HH p CI T)< in r-- p CI o H ^ ci id LO t~^ LO d 1^ CO •* lO X X CO •* X CO C CI lO CI rt to T-1 CI CO CI 03 CI ■* CO •* O X •* X LO X O CO O ■* X CI CO t~ O ■* O X -H t^ t—1 p 00 CI p CI p CO 05 ^~* d LO cd d •* — < id d t^ o ^ to •'1' r- CO in Tj( O O X 05 LO cs CI CI •-< X ■* ■* X O O X CI CO CO ■* CD CO O O Oi CO O -^ C5 CO lO •* o: •* •* CI LO -^ CO CO •* 00 *~* ■*' cd Ld cd t-^ cd cd cd LO 'I' •* ■* CD O CO ■* O X X CO CI r^ r^ CI CI .-1 X ^ ■* X CI CO c •* X ^ I. X X 05 C3 o o o APPENDIX 207 TABLE 69. Observations on the Just Imperceptible Negative Difference. SUBJECT 1. IVa I III IV Tk N. • TkNfe Nk ' r^N^ Nk i l-kNk Nk J-kNk 84 7 588 6 504 * 7 ! 588 5 420 88 12 1056 13 1 1144 9 1 792 8 704 92 33 3036 50 4600 33 3036 18 1656 96 39 1 3744 18 1728 36 3456 40 3840 100 9 900 12 1200 13 1300 23 2300 104 1 104 2 208 6 624 V 100 9324' 100 9280 100 9380 100 9544 Average 1 93.24 92.80 j 93.80 95.44 TABLE 70. Observations on the Just Imperceptible Negative Difference. SUBJECT II. IVa - I III IV Tk Nk I-k Nk Nk TkNk Nk ' i-^Nk Nk 1 r^Nj, 84 5 420 8 672 3 252 9 756 88 12 1056 14 1232 13 1144 12 1056 92 19 1748 23 1 2116 21 1932 28 2576 96 35 3360 26 j 2496 38 3648 31 2976 100 19 1900 22 2200 15 1500 15 1.500 104 9 936 5 520 10 1040 4 416 108 1 108 2 216 1 108 I 100 9528 100 9452 100 [ 9516 100 9388 Average 95.28 1 ^^••^^' j 95.16 1 93.88 208 PROBLEMS OF PSYCHOPHYSICS TABLE 71. Observations on the Just Imperceptible Negative Difference. SUBJECT III. IVa I [II IV Tk Nk TkNk Nk TkNk Nk TkNk Nk J-kNk 84 2 168 _ 88 8 704 5 440 92 14 1288 6 552 3 276 11 1012 96 24 2304 21 2016 31 2976 41 3936 100 33 3300 47 4700 61 6100 36 3600 104 18 1872 20 2080 2 208 10 1040 108 1 108 1 108 3 324 2 216 I 100 9744 100 9896 100 9884 100 9804 Average 97.44 98.96 98.84 98.04 TABLE 72. Observations on the Just Imperceptible Negative Difference. SUBJECT IV. I III £W I-k Nk TkNk Nk TkNk Nk TkNk 84 1 84 4 336 5 420 88 9 792 7 616 1 88 92 15 1380 31 2852 11 1012 96 46 4416 36 3456 39 3744 100 21 2100 17 1700 28 2800 104 8 832 5 520 14 1456 108 2 216 I 100 9604 100 9480 100 9736 Average 96.04 94.80 97.36 APPENDIX 209 TABLE 73. Observations on the Just Imperceptible Negative Diffekence. SUBJECT V. I III IV fk Nk TkNk Nk TfcNk Nk Tk Nk 84 2 168 2 168 7 588 88 12 1056 8 264 8 704 92 22 2024 21 1932 31 2852 96 38 3648 44 4224 5.0 4800 100 22 2200 23 2300 4 400 104 4 416 7 728 108 y 100 9512 100 9616 100 9344 Average 95.12 96.16 93.44 TABLE 74. Observations on the Just Imperceptible Negative Difference. SUBJECT VI. - I III IV Tk Nk rkNfc Nk r^Nk Nk TkNk 84 7 588 88 16 1408 13 1144 9 792 92 36 3312 16 1472 25 2300 96 19 1S24 39 3744 35 3360 100 18 1800 29 2900 23 2300 104 4 416 3 312 6 624 108 2 216 I 100 93.48 100 9572 100 9592 Average 93.48 95.72 1 95.92 210 PROBLEMS OF PSYCHOPHYSICS > A! ■* Tf OC O M •* O 'M Tji o ffl to t- CO 00 -^ CO 00 ■* O O CI C) CO Cl Oi Cl 1 OS a, 2 GO W CO -^ CO OC rji ^ O (M 1 o ?3 > iA 2 00 •^ ■* 00 O (M <£> 00 Tf 00 C-1 O lO ^ lO CO O © O CO !M CO I- =0 1-- ^ CI o 2 t^ CO t^ CO O CO IM CO t^ o t- »-i 8 CO > •*•.*< 00 CI o ■* M CJ O t~ O ■* O O 00 CO O --I IM CO M tP 1-1 CI 00 CI 2 ,-1 CO -^ M 02 ^ rH C-1 I- CO 'I' .-( o o CO > O CO •* CO O 00 CO Tf 05 rJH rt O O -^ 00 ■* (M CO CD 00 CJ .-H U5 -H CD (M 00 00 c» q CD o t^ r^ -< 'XI t^ iM j^ rH rH O CI CO M 2 o o CO ►-H « 00-*OOiMOOCO <« ^ ^ ^ C-l (^ Cvj t^ Ji --I CO — 1 t^ lO 1 00 Cl CO CO Cl CO 00 o 2 W CO ■* t^ t^ O t^ 'H CO -H r- in •* •^ O CC IM O O < APPKN'DIX 211 TABLE 75. Observations on the Just Imperceptible Negative Difference. TABLE VII. I III IV Tk Nk rfcNk Nk 1 TkNk Nk i-kNk 84 88 2 176 6 528 92 12 1104 16 1472 19 1748 96 37 3552 32 3072 44 4224 100 32 3200 42 4200 27 2700 104 13 1352 7 728 3 312 108 4 432 3 324 1 108 I 100 9816 100 ! 9796 100 9620 Average 98.16 97.96 96.20 Table 76 will be fou.n'd on the preceding page TABLE 77. Differences between- Observed and Calculated Values op the Just Imperceptibls Negative Difference. Difference between observed and calculated values. Subject Calculated Value IVa I III IV Totals I 1 93.51 j -0.27 -0.71 +0.29 +1.93 + 0.31 II 94.46 + 0.82 +0.06 ' +0.70 -0.58 + 0.25 III 98.30 -0.86 +0.66 +0.54 -0.26 + 0.02 IV 95.56 + 0.48 ! -0.76 +1.80 + 0.51 V 94.74 + 0.38 1 +1.42 -1.30 + 0.17 IV 95.09 -1.61 +0.63 +0.83 +0.05 VII 95.55 j + 2.61 + 2.41 + 0.65 + 1.89 212 PROBLEMS OF PSYCHOPHYSICS E Ji C eti XI c 1 S « 05 O '— ' o CD O O CO (U CI I- CI o (D o o c -^ !C 1 1 + + P lO lO lO lO c 00 O ■* lO £ IM 00 CO O -H O ^ ^ it! 1 1 + + p o o o o Tl" O O ■<*' 05 05 Oi OS o o o o O 00 C5 ■* Oi 05 05 OS -H «D "O r^ CI O i (-) o •o CO LJ o > ,_, ^ O (^ O o O o !^ •"• ^^ »— < ^ O OJ t-H H- < > u > ^ I jDaCqns i-O lO lO lO lO ra C/) X CD •o o o O o + 1 + 1 o o o o oc CI CO 00 CI lO -H 00 CI CI CI 00 00 00 O -H CD rH o I- o d d -H + + + I o o o o CD CD CC « oi ^ CI in lO d fD 03 Ci a a Oi CO ■* ■* ■* 05 ■* 00 00 rt o C) ■* o o T-H r-H + 1 1 1 o o o O CD CI fTi 00 lO 03 o 00 O 03 00 !>. o o 05 8 O 289 815 281 rt o o o 1 + 1 + + 1 § a > CO -< O CD CI C) 00 O ^ lO "cS § 00 05 00 05 Ol 05 > '-'!=;> II losfqns CD ■rf TjH CO CO CO CO LO CI o c C O 1 + 4- 1 o c o c CO ■* Tf ■ > h— . *— ' III issfqi^s APPENDIX 21.3 CO t- « 1 00 o c ■«< t^ 00 o o ^^ i + 1 + o o 1 ■* o 1 o 00 CO <£> •* r^ 1 O m » I + P Q o o o c^ CO o CO o 00 •* in >o 05 o m o S2 a i5 r^ t^ 00 (D O C-l O c-1 CO d 1 c o 1 1 o O 05 OS 00 - 00 r^ r^ Oi Ol 05 re o 50 r- CO IN o o ^H 1 + 1 + o n O 1 00 (D CD ■» 00 t^ o 1 o o OS 1 > 1 00 CO 2 ■* o CO o ^^ »M + + 1 CO CO 1^ 00 00 .-I 00 to >c d .-< ' A I loafqns A l^sfqns M X X •M C^ CO X — 1 o d 1 + + O n X rl M •* l'- a> CO lO o o 05 03 M C^l ^" n C o ^H ■N CO 1 ' + + 03 O o X M M t^ ,— ( M 1 1 "* O O ^ 1 „ > M CI + + IN (M X 03 cq ffl O UJ •fl O -H ^H + + 1 o o O IN CO ■"J- ^ in Tt" CO t^ Tjt 03 o; O o -> o X O CD CO O CO CO IN CO CD 03 o CO IN IN ^H + + + O IN C 03 CO t^ .-^ (-; Q o o > lA psfqng II A 533.-q"s 214 PROBLEMS OF PSYCH(^PH YSICS TABLE 79. Deviations of Observed Values from the Theoretical Results for the Thresholds. Threshold in direction of increase. Threshold in direction of decrease. Theoretical value 99.452 Theoretical value 93.297 ,_, Series Observed value Difference Observed value Difference o 3 IVa I III IV 100.020 99.760 99.440 99.180 + 0.568 + 0.308 -0.012 -0.272 92.520 92.540 94.115 94.770 -0.777 -0.757 + 0.818 + 1.473 Average 0.290 Average 0.956 Theoretical value 98.835 Theoretical value 94.871 f— < U 3 IVa I III IV 99.437 98.670 98.600 98.233 + 0.602 -0.165 -0.235 -0.652 95.620 95.340 95.220 93.720 + 0.749 + 0.469 + 0.349 -2.151 Average 0.414 Average 0.929 Theoretical value 99.284 Theoretical value 97.850 o 3 IVa I III IV 98.560 98.340 101.020 100.420 -0.724 -0.944 + 1.736 + 1.144 96.980 1 98.300 1 98.340 1 97.920 -0.870 + 0.550 + 0.490 + 0.070 Average 1.137 Average 0.495 Theoretial value 98.083 I III IV 98.400 97.780 98.550 + 0.317 -0.303 + 0.467 Average 0.362 Theoretical value 95.393 95.180 94.915 96.580 -0.213 -0.478 + 1.187 Average 0.626 Theoretical value 97 .142 Theoretical value 94.469 > o '2 3 I III IV ' 99.780 97.300 94.960 + 2.638 + 0.158 -2.182 95.100 96.020 92.560 + o.63i + 1.551 -1.909 Average 1.659 Average 1.364 Theoretical value 99.863 Theoretical value 95.307 > I III IV 98.530 100.460 j 102.025 I -1.333 + 0.597 + 2.162 94.260 95.360 1 95.980 -1.147 + 0.053 + 0.673 Average 1.364 Average 0.624 Theoretical value 99.859 Theoretical value 95.793 I III IV 100.180 99.426 99.240 Average + 0.321 -0.433 -0.619 0.458 97.140 97.760 95.320 Average + 1.347 + 1.967 -0.473 1.262 API'KNDIX 215 TABLE SO, THEOKICTK'AL VaI.I'I'.S of IHB Thuhsiiui.us. Subject Threshold in direc- Threshold in ilirec- tion of increase , lion of decrease I 99.45 93.30 II 98. S3 94.87 in 99.28 97.85 IV 98.08 95.39 V 97.14 - 94.47 IV 99.86 i 95.31 vn 99.86 ' 95.79 Interval of uncertainty 6.15 3.96 1.43 2.69 2.67 4.56 407 TAI3LE SI. Observed V.\i.l'es of the Thresholds. Subiect Threshold in direc- } Threshold in direc- Interval of tion of increase tion of decrease uncertainty I 99.60 93.49 (ill II 98.71 94.98 3.74 III 99.58 97.88 1.70 IV 98.24 95.56 , 2.68 y 97.34 94.57 2.78 IV 100.33 95.20 5.12 VII 99.62 96.74 2.88 TABLE 82. R.\TIOS OF THE XU.MBERS OF " HR.\ VI liK-GU ESS " AND " LIGHTER GUESS "-JUDGMENTS. 84 88 92 96 100 104 108 I II TV V VI VII 0.3333 0.6667 0.2500 0.3333 1.0000 0.0000 0.2727 0.8214 1..5000 2.7.500 0.5455 0.4500 0.6021 1.5758 0.5909 1.2105 1.7500 0.6571 1.0101 1.2745 0.8485 4.9000 0.S302 0.9649 3.8605 1.3478 1.0769 5.1429 1.3830 1.4054 2.7273 1.5000 3.0000 28.0000 3.0714 2.6667 3.8000 1.2500 4.0000 10.0000 2.2222 4.2500 TABLE 83. Relative FREguENCiEs of "G" Judgments. Comparison I n III Weight 84 0.0622 0.0444 0.0044 88 0.1244 0.1133 0.0200 92 0.3311 0.1889 0.0600 96 0.4422 0.2578 0,0778 100 0.4644 0.2400 0.1 489 104 0.0911 0.0889 0.0467 108 0.0533 0.0800 0.0156 IV V VI VII 0.0167 0.0133 0,0200 0.0133 0.0.500 0.0500 0,1133 0.0967 0,1167 0.1400 0.2200 0.1933 0,2033 1967 0.3233 0.3733 0,1 SOO 0.1433 0,3733 0.2967 0,0667 0.0967 0.1900 0.1833 0,0500 0.0367 0.0967 0.1400 216 PROBLEMS OF PSYCHOPHYSICS > > U «0 Cl to M O O CO « >0 « O O IM CO c<3 >n « t- ra r^ 10 (M IC O 00 lO 'J' CO 00 CO CO o 00 05 •* Oi X (M m lO M iM ira — ' CO lO •* 0-. 00 CO ■* t- r-i o CI CC O 05 O M CI 0> — ■ CI ;D 05 '— ' lO CO o 05 CO >o CO 05 CO ■* CO t- CI t- a CO CO c; -H ic CI "S > it 2 CO O •* ■* O CO 00 CO CI CD CO C -I x CO CO * CO re C; — — ' CO L-: •* CO ^ 00 00 CO OS 00 o> CD OS •* lO CI O! CO CT: -1 i-H •^ IC ■* CI — 1 CO > l-i J! O O O CO o o o CI CI CI uO O CC CI ■* CO CI 00 ■* O CO -H CO -o .o CI -. 2 05 ^ 2 lO lO >0 — 1 ■* 1.0 o -H CO CO LO CI -H CI 00 CI CD i t-H 2 on M 2 ca 05 t^ lo r- -H t^ (N CO CO M 00 CD M AS 2 O 00 O CO O O tn 00 CO C) CO O CO CO CO ■* 00 -^ 00 r^ Of) -H ■ijl t^ -H O •* CO CI 05 CO CO ■<»■ CO 05 O -H lO CO 00 O CO CI O 00 -^ O ■* CO CO CI 00 00 -"J" O ■* CI lO CI O O O CO Ol CO OS 1^ -^ 03 CI lO CI Tf CO 05 O Ttl W 00 CO CO o CO o> 00 CO C5 OS 05 -H ■* N lO ■* OS O T(i c^ ^ rt M 1- ■* SS « = ° o 30 so 05 M vO « „ N t < .\PPElSrDIX 217 J O CO « o o o o ;0 C^l CO Tf ^^ lO CI 05 05 t- -^ M q c d d d d d d d > X CO r~ t^ r^ O CO o CO CO !D CO O CO O M C-l -- lO r-^ CO CO O O --I CO CO 00 o> d d d d d d d o t^ O t^ CO o t^ o CO O CO CO O CO o Si ira !i< o M CO in o q r^ N -^ o o o o o o o o o J CO CO OS O O O r^ lO lO 00 O O O CO O! lO CD O 00 CO C"l OS o 00 r-; CO q q d d d d d d d X 0.0000 0.0244 0.0711 0.2222 0.4711 0.8933 0.9578 ■* o o 00 O) r^ CO ■^ O O t- 00 CO "O O C-1 CD 1- ^ ■* -^ o w o q ^ c q d d d d d d d .1 1 ►J CO C-) O 35 —1 CO CO CO M O » — ' »0 lO CO CO O T< CO OS -^ 05 X t» •» M q q d d d d d d ci X o M Tf -H CO 05 CD -^ CI 'J' -H CO « "O • c •* CO X t- O X o Tl< -^ » lO "1" X X C rH ^ CI CI C O d d d o d o d Hil J CO CO O CO c^ CO r- lO in O *0 CI CO CO CO lO 00 CO d '^ o 0> 00 lO CO ^ o o d d d d d d d X o 1 CI O 05 Cl CO CO o CI O X CI CO iC c O C-I X CI -- 0-. ■* O O O CI ■•I' X 05 d d d d d d d 0.0622 0.1244 0.3311 0.4422 0.4644 0.0911 0.053;; i 1 ■ 05 o o o hJ Q t^ o t^ t^ o o O CO O CO CO o o X 05 •* iC CO CD ■<>< 05 X t^ •0' C\| o o d d d d d d d i_i X r~ i~ r^ o h- t^ s CO CO CO O CO CO o o o CO w CO o CI O O O -H ■* I- X > o o o o o o o o CO r- CO CO t- CO o CO CD CO CO CO CO o -1 05 O) I- O) X ■v C O ^ CO CI " rt d d d d d d d J CO CO t^ o t- t^ o CO CO CO O CO CO o I, CD — > o: O CO CI OS X I^ TT CI o o o o o o o o o > X i^ CO CO t^ c CO CO CO CO CO CO c; CO CO O CI CD X CI -n- X O O O •-< ■* 1^ 00 d d d d d ^ d o O CO O CO CO o t^ O CO O CO CO O CO CI « CI Cl t' 05 05 o « CI CO CO -H q d d d d d d d J o r^ o o t^ CO o O CO O O CD CO C CO N O 05 I- CO CI 05 05 I- CI -■ q q d d d d d d d ^ 1 > X t- M c CO c; o w CO CO O CO C O CO d CI CO -< X 1- -TJ" o o -^ "O q 00 05 d d d d d d d 1 o CO C C I- CO t^ h; CO O O CD CO CD CO -- in Tj> 05 -a' C5 CO o o -H ^ -. q q d d d d d d o T)i X M CO O ■* X X X 05 05 o o c 218 PROBLEMS OF PSYCHOPHYSICS TABLE 86. Values of the Psychometric Fu.vctioms. SUBJECT L G H L 84 1 0.0622 0.0022 0.9356 85 -0.0167 0.0433 0.9734 86 -0.0039 0.0434 0.9605 87 0.0532 0.0302 0.9166 88 0.1244 0.0200 0.8556 89 0.1926 0.0205 0.7869 90 0.2501 0.0337 0.7162 91 0.2955 0.0577 0.6468 92 0.3311 0.0889 0.5800 93 0.3608 0.1232 0.5160 94 0.3881 0.1576 0.4343 95 0.4152 0.1905 0.3943 96 0.4422 0.2222 0.3356 97 0.4664 0.2554 0.2782 98 0.4827 0.3020 0.2153 99 0.4846 0.3453 0.1701 100 0.4644 04133 0.1223 101 0.4157 0.5033 0.0810 102 0.3347 0.6168 0.0485 103 0.2232 0.7513 0.0255 104 0.0911 0.8956 0.0133 105 -0.0398 1.0294 0.0104 106 -0.1328 1.1186 0.0142 107 -0.1284 1.1125 0.0159 108 0533 0.9400 000(i7 T.\[ BLE 87. Valu ES OF THE PSVCHOME TRic Functions. S J EJECT IL "^ Tk G H L 84 0.0444 0.0222 0.9334 85 0.0614 0.0285 0.9101 86 0.0786 0.0258 0.8956 87 0.0959 0.0227 0.8814 88 0.1133 0.0244 0.8623 89 0.1310 0.0337 0.8353 90 0.1497 0.0516 0.7987 91 0.1690 0.0777 0.7533 92 0.1889 o.iiu 0.7000 93 0.2088 0.1505 0.6407 94 0.2279 0.1946 0.5775 95 0.2447 0.2424 • 0.5129 96 0.257S 0.2933 0.4489 97 0.2654 0.3471 0.3875 98 0.2650 0.4040 0.3301 99 0.257S 0.4643 0.2779 100 0.2400 0.5289 0.2311 101 0.2125 0.5975 0.1900 11)2 0.1761 0.6698 0.1541 U)3 0.1334 0.7436 0.1230 104 0.0889 0.8156 0.0955 105 0.0495 0.8795 0.0710 106 0.0251 0.9255 0.0494 107 0.0294 0.9403 0.0303 108 0.0,800 0.9044 0.0156 APPENDIX 219 TABLE 88. Values of the PsvcHo.\rETRic Function's. rk 84 K.i 86 sr ss 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 10.3 106 107 108 SUBJECT III. 0.0044 0.0000 0.9956 - 0.0398 0.0359 1.0038 -0.0347 0.0403 0.9944 - 0.0080 0.0325 0.9764 0.0201) 0.0244 0.9556 0.043.1 , 0.0226 0.9339 0.0.').")i> 0.0299 0.9142 0.0612 O.0464 0.8924 0.0600 0.0711 0.8689 0.0577 0.1022 ' 0.8401 0.0590 0.1381 0.8029 0.0647 0.1781 ' 0.7573 0.0778 0.2222 0.7000 0.0960 0.2714 0.6326 0.1170 0.3278 0.5552 0.1364 0.3936 0.4700 0.1480 0.4711 0.3800 0.1490 0.5619 0.2891 0.1325 066.55 0.2020 0.0975 0.7786 0.1239 0.0667 0.8933 0.0400 -0.0108 0.99.55 0.0153 -0.0571 1.0636 - 0.0065 -0.0629 1.0657 - 0.0028 0.01.56 0.9578 0.0266 TABLE 89. Values of the Psycho.metric Function'? fk G H 84 0.0167 0.0233 85 0.0386 0.0.532 86 0.04.53 0.0.502 87 0.0469 0.0370 88 0.0.500 0.0267 89 0.0.581 0.0276 90 0.0725 0.0430 91 0.0926 0.0736 92 0.1167 0.1167- 93 0.1425 0.1697 94 0.1672 0.2-294 95 0.1883 0.2925 96 0.20.33 0.3.567 97 0.2106 0.4205 98 0.2092 0.48.39 99 0.1986 0.5464 100 0.1800 0.6100 101 0.1548 0.67.56 102 0.1253 0.7439 103 0.0948 0.8142 104 0.0667 0.8,S33 105 0()44,S 0.9446 106 0.032S 0.9863 107 0.03.37 0.9904 108 0.0500 0.9300 SUBJECT IV. L 0.9600 0.9082 0.9045 0.9161 0.9233 0.9143 0.8845 0.8338 0.7666 0.6878 0.6034 0.5192 0.4400 0.3689 0.3069 0.2.5.50 ().21(MI 0. 1696 0.1308 0.0910 0.0500 0.0106 -0.0191 -0.0241 0.0200 220 PROBLEMS OF PSYf'HOPHYSICS TABLE 90. Values of the Psychometric Functions. SUBJECT V. G H L 84 0.0133 0.0267 0.9600 85 0.0290 0.1116 0.8594 88 0.034S 0.1031 0.8621 87 0.0401 0.0610 0.8989 88 0.0500 0.0233 0.9267 89 0.0663 0.0108 0.9229 90 0.0883 0.0313 0.8804 91 0.1138 0.0833 0.8029 92 0.1400 0.1600 0.7000 93 0.1637 0.2514 0.5849 94 0.1822 0.3465 0.4713 9.5 0.1935 0.4360 3705 96 0.1967 0..-)133 0.2900 97 0.1917 0.5744 0.2339 98 0.1797 0.6200 0.2003 199 0.1626 0.6530 0.1'844 TOO 0.1433 0.6800 0.1767 101 0.124S 0.7088 0.1664 102 0.1 198 0.7473 0.1329 103 0.1003 0.S009 0.0988 104 0.0967 0.8700 0.0333 105 0.0967 0.9469 - 0.0436 108 0.0946 1.0121 -0.1067 107 0.0800 1.0299 -0.1099 . ■ 108 0.0367 0.9433 0.0200 TABLE 91. Values of the Psychometric Functions. SUBJECT VL Tk G H L 84 0.0200 0.0067 0.9733 85 0.0235 0.0171 0.9594 86 0.0476 0.0207 0.9317 87 0.0800 0.0218 0.8982 88 0.1133 0.0233 0.8634 89 0.1440 0.0272 0.8288 90 0.1814 0.0347 0.7839 91 0.1962 0.0465 0.7573 92 0.2200 0.0633 0.7167 93 0.2442 0.0853 0.6705 94 0.2697 0.1129 0.6174 f5 0.2965 0.1458 0.5577 96 0.3233 0.1867 0.4900 97 0.3479 0.2338 0.4183 98 0.3670 0.2883 0.3447 99 0,3768 0.3504 0.2728 100 0.3733 0.4200 0.2067 101 0.3532 0.4963 0.1505 102 0.3148 0.5779 0.1073 103 0.2591 0.6616 0.0793 104 0.1900 0.7433 0.0667 105 0.1181 0.8166 0.0653 106 0.0599 0.8728 0.0673 107 0.0408 0.9001 0.0591 108 0.0967 0.8833 0.0200 APPENDIX 221 TABLE 92. V.M.UES OF THE PSYCHOMETRIC FUNCTIONS. SUBJECT VII. I"k G H L 84 0.0133 0.0067 0.9800 85 O.llVH) - 0.0242 0.9052 86 0.1366 -0.0252 0.8886 87 O.llSl -0.0119 0.8938 88 0.0967 0.0067 0.8966 89 0.0903 0.0248 0.S849 90 0.1059 " 0.0406 0.8535 91 0.1422 0.0538 0.8040 92 0.1933 0.0667 0.7400 93 0.2504 0.0819 0.6677 94 0.3045 0.1024 0.5931 95 0.3474 0.1310 0.5216 96 0.3733 0.1700 0.4567 97 0.3793 0.2205 0.4002 98 0.3657 0.2828 0.3515 99 0.3362 0.3557 0.3081 100 0.2967 0.4376 0.2666 101 0.2550 0.5223 0.2227 102 0.2187 0.6078 0.1735 103 0.1941 0.6878 0.1181 104 0.1833 0.7567 0.0600 105 0.1816 0.8086 0.0098 106 0.1743 0.8386 -0.0129 107 0.1337 0.8430 0.0233 108 0.1400 0.8200 0.0400 TABLE 93. Calculated axo Observed Values of the Threshold in the Direction of Increase Result of Method of Just Perceptible Difference Value found by Interpolation Observed Calculated I 99.60 99.45 100.95 II 98.71 98.83 99.55 III 99.58 99.28 100.32 IV 98.24 98.08 98.26 V 97.35 97.14 95.83 VI 100.33 99.86 101.04 VII 99.63 99.86 100.74 TABLE 94. Calculated \sa OBSEiiVEo Values of the Th.^eshold in the Direction of Deckease Subject Result of Method of Just Perceptible Difference Value found by Intcrpolalion Observed Calculated I 93.49 93.30 93.26 11 94.98 94.87 95.20 III 97.88 97.85 98.65 IV 95.56 95.39 95.24 V 94.57 94.47 93.75 VI 95.20 95.31 95.82 VII 96.74 95.79 95.33 14 DAY USE RETURN TO DESK FROM WHICH BORROWED This book is due on the mj^g^g^ ^ , on the date toMMflRinewed Renewed books are subjea to immediate recall elow, or 3ISSI01S DURlNfi ^ZyC^^Z^ "OeT'BTVj /7f.i.t^J^L^Lyr.^ J — nm 1 i ^ IQ COT 2 9 OCT 2 1 1962 -tOAI4 StP" t^-W\- — mrm~i96r" oct 1 1 RK'D -b m pSYCHOtO^ ^^^^?&««- LD 21-50w-6,'59 (A2845sl0)476 General Library , University of California Berkeley U C BERKELEY LIBRARIES CD2TM2fiT17 \ *:• ;--* ^ L