Chapter VII: Section II: showed that the shorter filled distances are (3)

All of the subjects except _R_ were conscious of more or less of a _strain_, which varied during the intervals, and was by some felt to be largely a tension of the chest and other muscles, while others felt it rather indefinitely as a 'strain of attention.' The characteristics of this tension feeling were almost always different in the second interval from those in the first, the tension being usually felt to be more _constant_ in the second interval. In experiments of the third group a higher degree of tension was felt in awaiting a light tap than in awaiting a heavy one.

Evidently, in all these cases, the effect of a _difference_ between two stimulations was to introduce certain changes in sensation _during_ the interval which they limited, owing to the fact that the subject expected the difference to occur. Thus in the third group of experiments there were, very likely, in all cases changes from sensations of high tension to sensations of lower, or vice versa. It is probable that, in the experiments of the second group, there were also changes in muscular sensations, partly those of eye muscles, partly of chest and arm muscles, introduced by the change of attention from one point to another. At any rate, it is certain that there were certain sensation changes produced during the intervals by changes of locality.

If, then, we assume that the introduction of additional sensation change into an interval lengthens it, we are led to the conclusion that psychological time (as distinguished from metaphysical, mathematical, or transcendental time) is perceived simply as the quantum of change in the sensation content. That this is a true conclusion is seemingly supported by the fact that when we wish to make our estimate correspond as closely as possible with external measurements, we exclude from the content, to the best of our ability, the general complex of external sensations, which vary with extreme irregularity; and confine the attention to the more uniformly varying bodily sensations. We perhaps go even further, and inhibit certain bodily sensations, corresponding to activity of the more peripherally located muscles, that the attention may be confined to certain others. But attention to a dermal stimulation is precisely the condition which would tend to some extent to prevent this inhibition. For this reason we might well expect to find the error in estimation more variable, the 'constant error' in general greater, and the specific effects of variations which would affect the peripheral muscles, more marked in 'tactual' time than in either 'auditory' or 'optical' time. Certainly all these factors appear surprisingly large in these experiments.

It is not possible to ascertain to how great an extent subject _Sh_ inhibited the more external sensations, but certainly if he succeeded to an unusual degree in so doing, that fact would explain the absence of effect of stimulation difference in his case.

Explanation has still to be offered for the variable effect of intensity difference upon the _second_ interval. According to all subjects except _Sn_, there is a radical difference in attitude in the two intervals. In the first interval the subject is merely observant, but in the second he is more or less reproductive. That is, he measures off a length which seems equal to the standard, and if the stimulation does not come at that point he is prepared to judge the interval as 'longer,' even before the third stimulation is given. In cases, then, where the judgment with equal intensities would be 'longer,' we might expect that the actual strengthening or weakening of the final tap would make no difference, and that it would make very little difference in other cases. But even here the expectation of the intensity is an important factor in determining tension changes, although naturally much less so than in the first interval. So we should still expect the lengthening of the second interval.

We must remember, however, that, as we noticed in discussing the experiments of Group 2, there is complicated with the lengthening effect of a change the _bare constant error_, which appears even when the three stimulations are similar in all respects except temporal location. Compare _WWW_ with _SSS_, and we find that with all five subjects the constant error is decidedly changed, being even reversed in direction with three of the subjects.

Now, what determines the direction of the constant error, where there is no pause between the intervals? Three subjects reported that at times there seemed to be a slight loss of time after the second stimulation, owing to the readjustment called for by the change of attitude referred to above, so that the second interval was begun, not really at the second stimulation, but a certain period after it. This fact, if we assume it to be such, and also assume that it is present to a certain degree in all observations of this kind, explains the apparent overestimation of the first interval. Opposed to the factor of _loss of time_ there is the factor of _perspective_, by which an interval, or part of an interval, seems less in quantity as it recedes into the past. The joint effect of these two factors determines the constant error in any case where no pause is introduced between _ST_ and _CT_. It is then perfectly obvious that, as the perspective factor is decreased by diminishing the intervals compared, the constant error must receive positive increments, _i.e._, become algebraically greater; which corresponds exactly with the results obtained by Vierordt, Kollert, Estel, and Glass, that under ordinary conditions long standard intervals are comparatively underestimated, and short ones overestimated.

On the other hand, if with a given interval we vary the loss of time, we also vary the constant error. We have seen that a change in the intensity of the stimulations, although the relative intensity of the three remains constant, produces this variation of the constant error; and the individual differences of subjects with regard to sensibility, power of attention and inhibition, and preferences for certain intensities, lead us to the conclusion that for certain subjects certain intensities of stimulation make the transition from the receptive attitude to the reproductive easiest, and, therefore, most rapid.

Now finally, as regards the apparent failure of the change in _SSW_ to lengthen the second interval, for which we are seeking to account; the comparatively great loss of time occurring where the change of attitude would naturally be most difficult (that is, where it is complicated with a change of attention from a strong stimulation to the higher key of a weak stimulation) is sufficient to explain why with most subjects the lengthening effect upon the second interval is more than neutralized. The individual differences mentioned in the preceding paragraph as affecting the relation of the two factors determining the constant error, enter here of course to modify the judgments and cause disagreement among the results for different subjects.

Briefly stated, the most important points upon which this discussion hinges are thus the following: We have shown--

1. That the introduction of either a local difference or a
difference of intensity in the tactual stimulations limiting
an interval has, in general, the effect of causing the
interval to appear longer than it otherwise would appear.

2. That the apparent exceptions to the above rule are, (_a_)
that the _increase_ of the local difference in the first
interval, the stimulated areas remaining unchanged, produces a
slight _decrease_ in the subjective lengthening of the
interval, and (_b_) that in certain cases a difference in
intensity of the stimulations limiting the second interval
apparently causes the interval to seem shorter than it
otherwise would.

3. That the 'constant error' of time judgment is dependent
upon the intensity of the stimulations employed, although the
three stimulations limiting the two intervals remain of equal
intensity.

To harmonize these results we have found it necessary to assume:

1. That the length of a time interval is perceived as the
amount of change in the sensation-complex corresponding to
that interval.

2. That the so-called 'constant error' of time estimation is
determined by two mutually opposing factors, of which the
first is the _loss of time_ occasioned by the change of
attitude at the division between the two intervals, and the
second is the diminishing effect of _perspective_.

It is evident, however, that this last assumption applies only
to the conditions under which the results were obtained,
namely, the comparison of two intervals marked off by three
brief stimulations.

* * * * *

PERCEPTION OF NUMBER THROUGH TOUCH.

BY J. FRANKLIN MESSENGER.

The investigation which I am now reporting began as a study of the fusion of touch sensations when more than two contacts were possible. As the work proceeded new questions came up and the inquiry broadened so much that it seemed more appropriate to call it a study in the perception of number.

The experiments are intended to have reference chiefly to three questions: the space-threshold, fusion of touch sensations, and the perception of number. I shall deny the validity of a threshold, and deny that there is fusion, and then offer a theory which attempts to explain the phenomena connected with the determination of a threshold and the problem of fusion and diffusion of touch sensations.

The first apparatus used for the research was made as follows: Two uprights were fastened to a table. These supported a cross-bar about ten inches from the table. To this bar was fastened a row of steel springs which could be pressed down in the manner of piano keys. To each of these springs was fastened a thread which held a bullet. The bullets, which were wrapped in silk to obviate temperature sensations, were thus suspended just above the fingers, two over each finger. Each thread passed through a small ring which was held just a little above the fingers. These rings could be moved in any direction to accommodate the bullet to the position of the finger. Any number of the bullets could be let down at once. The main object at first was to learn something about the fusion of sensations when more than two contacts were given.

Special attention was given to the relation of the errors made when the fingers were near together to those made when the fingers were spread. For this purpose a series of experiments was made with the fingers close together, and then the series was repeated with the fingers spread as far as possible without the subject's feeling any strain. Each subject was experimented on one hour a week for about three months. The same kind of stimulation was given when the fingers were near together as was given when they were spread. The figures given below represent the average percentage of errors for four subjects.

Of the total number of answers given by all subjects when the fingers were close together, 70 per cent. were wrong. An answer was called wrong whenever the subject failed to judge the number correctly. In making out the figures I did not take into account the nature of the errors. Whether involving too many or too few the answer was called wrong. Counting up the number of wrong answers when the fingers were spread, I found that 28 per cent. of the total number of answers were wrong. This means simply that when the fingers were near together there were more than twice as many errors as there were when they were spread, in spite of the fact that each finger was stimulated in the same way in each case.

A similar experiment was tried using the two middle fingers only. In this case not more than four contacts could be made at once, and hence we should expect a smaller number of errors, but we should expect still to find more of them when the fingers are near together than when they are spread. I found that 49 per cent. of the answers were wrong when the fingers were near together and 20 per cent. were wrong when they were spread. It happens that this ratio is approximately the same as the former one, but I do not regard this fact as very significant. I state only that it is easier to judge in one case than in the other; how much easier may depend on various factors.

To carry the point still further I took only two bullets, one over the second phalanx of each middle finger. When the fingers were spread the two were never felt as one. When the fingers were together they were often felt as one.

The next step was to investigate the effect of bringing together the fingers of opposite hands. I asked the subject to clasp his hands in such a way that the second phalanges would be about even. I could not use the same apparatus conveniently with the hands in this position, but in order to have the contacts as similar as possible to those I had been using, I took four of the same kind of bullets and fastened them to the ends of two æsthesiometers. This enabled me to give four contacts at once. However, only two were necessary to show that contacts on fingers of opposite hands could be made to 'fuse' by putting the fingers together. If two contacts are given on contiguous fingers, they are quite as likely to be perceived as one when the fingers are fingers of opposite hands, as when they are contiguous fingers of the same hand.

These results seem to show that one of the important elements of fusion is the actual space relations of the points stimulated. The reports of the subjects also showed that generally and perhaps always they located the points in space and then remembered what finger occupied that place. It was not uncommon for a subject to report a contact on each of two adjacent fingers and one in between where he had no finger. A moment's reflection would usually tell him it must be an illusion, but the sensation of this illusory finger was as definite as that of any of his real fingers. In such cases the subject seemed to perceive the relation of the points to each other, but failed to connect them with the right fingers. For instance, if contacts were made on the first, second and third fingers, the first might be located on the first finger, the third on the second finger, and then the second would be located in between.

So far my attention had been given almost entirely to fusion, but the tendency on the part of all subjects to report more contacts than were actually given was so noticeable that I concluded that diffusion was nearly as common as fusion and about as easy to produce. It also seemed that the element of weight might play some part, but just what effect it had I was uncertain. I felt, too, that knowledge of the apparatus gained through sight was giving the subjects too much help. The subjects saw the apparatus every day and knew partly what to expect, even though the eyes were closed when the contacts were made. A more efficient apparatus seemed necessary, and, therefore, before taking up the work again in 1900, I made a new apparatus.

Not wishing the subjects to know anything about the nature of the machine or what could be done with it, I enclosed it in a box with an opening in one end large enough to allow the subject's hand to pass through, and a door in the other end through which I could operate. On the inside were movable wooden levers, adjustable to hands of different width. These were fastened by pivotal connection at the proximal end. At the outer end of each of these was an upright strip with a slot, through which was passed another strip which extended back over the hand. This latter strip could be raised or lowered by means of adjusting screws in the upright strip. On the horizontal strip were pieces of wood made so as to slide back and forth. Through holes in these pieces plungers were passed. At the bottom of each plunger was a small square piece of wood held and adjusted by screws. From this piece was suspended a small thimble filled with shot and paraffine. The thimbles were all equally weighted. Through a hole in the plunger ran a thread holding a piece of lead of exactly the weight of the thimble. By touching a pin at the top this weight could be dropped into the thimble, thus doubling its weight. A screw at the top of the piece through which the plunger passed regulated the stop of the plunger. This apparatus had three important advantages. It was entirely out of sight, it admitted of rapid and accurate adjustment, and it allowed the weights to be doubled quickly and without conspicuous effort.

For the purpose of studying the influence of weight on the judgments of number I began a series of experiments to train the subjects to judge one, two, three, or four contacts at once. For this the bare metal thimbles were used, because it was found that when they were covered with chamois skin the touch was so soft that the subjects could not perceive more than one or two with any degree of accuracy, and I thought it would take entirely too long to train them to perceive four. The metal thimbles, of course, gave some temperature sensation, but the subject needed the help and it seemed best to use the more distinct metal contacts.

In this work I had seven subjects, all of whom had had some experience in a laboratory, most of them several years. Each one took part one hour a week. The work was intended merely for training, but a few records were taken each day to see how the subjects progressed. The object was to train them to perceive one, two, three, and four correctly, and not only to distinguish four from three but to distinguish four from more than four. Hence five, six, seven, and eight at a time were often given. When the subject had learned to do this fairly well the plan was to give him one, two, three, and four in order, then to double the weight of the four and give them again to see if he would interpret the additional weight as increase in number. This was done and the results were entirely negative. The subjects either noticed no difference at all or else merely noticed that the second four were a little more distinct than the first.

The next step was to give a number of light contacts to be compared with the same number of heavy ones--the subject, not trying to tell the exact number but only which group contained the greater number. A difference was sometimes noticed, and the subject, thinking that the only variations possible were variations of number and position, often interpreted the difference as difference in number; but the light weights were as often called more as were the heavy ones.

So far as the primary object of this part of the experiment is concerned the results are negative, but incidentally the process of training brought out some facts of a more positive nature. It was early noticed that some groups of four were much more readily recognized than others, and that some of them were either judged correctly or underestimated while others were either judged correctly or overestimated. For convenience the fingers were indicated by the letters _A B C D_, _A_ being the index finger. The thumb was not used. Two weights were over each finger. The one near the base was called 1, the one toward the end 2. Thus _A12 B1 C2_ means two contacts on the index finger, one near the base of the second finger, and one near the end of the third finger. The possible arrangements of four may be divided into three types: (1) Two weights on each of two fingers, as _A12 B12, C12 D12_, etc., (2) four in a line across the fingers, _A1 B1 C1 D1_ or _A2 B2 C2 D2_, (3) unsymmetrical arrangements, as _A1 B2 C1 D2_, etc. Arrangements of the first type were practically never overestimated. _B12 C12_ was overestimated once and _B12 D12_ was overestimated once, but these two isolated cases need hardly be taken into account. Arrangements of the second type were but rarely overestimated--_A2 B2 C2 D2_ practically never, _A1 B1 C1 D1_ a few times. Once the latter was called eight. Apparently the subject perceived the line across the hand and thought there were two weights on each finger instead of one. Arrangements of the third type were practically never underestimated, but were overestimated in 68 per cent. of the cases.

These facts in themselves are suggestive, but equally so was the behavior of the subject while making the answers. It would have hardly done to ask the person if certain combinations were hard to judge, for the question would serve as a suggestion to him; but it was easy to tell when a combination was difficult without asking questions. When a symmetrical arrangement was given, the subject was usually composed and answered without much hesitation. When an unsymmetrical arrangement was given he often hesitated and knit his brows or perhaps used an exclamation of perplexity before answering, and after giving his answer he often fidgeted in his chair, drew a long breath, or in some way indicated that he had put forth more effort than usual. It might be expected that the same attitude would be taken when six or eight contacts were made at once, but in these cases the subject was likely either to fail to recognize that a large number was given or, if he did, he seemed to feel that it was too large for him to perceive at all and would guess at it as well as he could. But when only four were given, in a zigzag arrangement, he seemed to feel that he ought to be able to judge the number but to find it hard to do so, and knowing from experience that the larger the number the harder it is to judge he seemed to reason conversely that the more effort it takes to judge the more points there are, and hence he would overestimate the number.

The comments of the subjects are of especial value. One subject (Mr. Dunlap) reports that he easily loses the sense of location of his fingers, and the spaces in between them seem to belong to him as much as do his fingers themselves. When given one touch at a time and told to raise the finger touched he can do so readily, but he says he does not know which finger it is until he moves it. He feels as if he willed to move the place touched without reference to the finger occupying it. He sometimes hesitates in telling which finger it is, and sometimes he finds out when he moves a finger that it is not the one he thought it was.

Another subject (Dr. MacDougall) says that his fingers seem to him like a continuous surface, the same as the back of his hand. He usually named the outside points first. When asked about the order in which he named them, he said he named the most distinct ones first. Once he reported that he felt six things, but that two of them were in the same places as two others, and hence he concluded there were but four. This feeling in a less careful observer might lead to overestimation of number and be called diffusion, but all cases of overestimation cannot be explained that way, for it does not explain why certain combinations are so much more likely to lead to it than others.

In one subject (Mr. Swift) there was a marked tendency to locate points on the same fingers. He made many mistakes about fingers _B_ and _C_ even when he reported the number correctly. When _B_ and _D_ were touched at the same time he would often call it _C_ and _D_, and when _C_ and _D_ were given immediately afterward he seemed to notice no difference. With various combinations he would report _C_ when _B_ was given, although _C_ had not been touched at the same time. If _B_ and _C_ were touched at the same time he could perceive them well enough.

The next part of the research was an attempt to discover whether a person can perceive any difference between one point and two points which feel like one. A simple little experiment was tried with the æsthesiometer. The subjects did not know what was being used, and were asked to compare the relative size of two objects placed on the back of the hand in succession. One of these objects was one knob of the æsthesiometer and the other was two knobs near enough together to lie within the threshold. The distance of the points was varied from 10 to 15 mm. Part of the time the one was given first and part of the time both were given together. The one, whether given first or second, was always given about midway between the points touched by the two. If the subject is not told to look for some specific difference he will not notice any difference between the two knobs and the one, and he will say they are alike; but if he is told to give particular attention to the size there seems to be a slight tendency to perceive a difference. The subjects seem to feel very uncertain about their answers, and it looks very much like guess-work, but something caused the guesses to go more in one direction than in the other.

Two were called less than one .... 16% of the times given.
" " " equal to .... 48% " "
" " " greater than .... 36% " "

Approximately half of the time two were called equal to one, and if there had been no difference in the sensations half of the remaining judgments should have been that two was smaller than one, but two were called larger than one more than twice as many times as one was called larger than two. There was such uniformity in the reports of the different subjects that no one varied much from this average ratio.

This experiment seems to indicate a very slight power of discrimination of stimulations within the threshold. In striking contrast to this is the power to perceive variations of distance between two points outside the threshold. To test this the æsthesiometer was spread enough to bring the points outside the threshold. The back of the hand was then stimulated with the two points and then the distance varied slightly, the hand touched and the subject asked to tell which time the points were farther apart. A difference of 2 mm. was usually noticed, and one of from 3 to 5 mm. was noticed always very clearly.

I wondered then what would be the result if small cards set parallel to each other were used in place of the knobs of the æsthesiometer. I made an æsthesiometer with cards 4 mm. long in place of knobs. These cards could be set at any angle to each other. I set them at first 10 mm. apart and parallel to each other and asked the subjects to compare the contact made by them with a contact by one card of the same size. The point touched by the one card was always between the points touched by the two cards, and the one card was put down so that its edge would run in the same direction as the edges of the other cards. The result of this was that:

Two were called less, 14 per cent.
" " " equal, 36 " "
" " " greater, 50 " "

I then increased the distance of the two cards to 15 mm., the other conditions remaining the same, and found that:

Two were called less, 11 per cent.
" " " equal, 50 " "
" " " greater, 39 " "

It will be noticed that the ratio in this last series is not materially different from the ratio found when the two knobs of the æsthesiometer were compared with one knob. The ratio found when the distance was 10 mm., however, is somewhat different. At that distance two were called greater half of the time, while at 15 mm. two were called equal to one half of the time. The explanation of the difference, I think, is found in the comments of one of my subjects. I did not ask them to tell in what way one object was larger than the other--whether longer or larger all around or what--but simply to answer 'equal,' 'greater,' or 'less.' One subject, however, frequently added more to his answers. He would often say 'larger crosswise' or 'larger lengthwise' of his hand. And a good deal of the time he reported two larger than one, not in the direction in which it really was larger, but the other way. It seems to me that when the two cards were only 10 mm. apart the effect was somewhat as it would be if a solid object 4 mm. wide and 10 mm. long had been placed on the hand. Such an object would be recognized as having greater mass than a line 4 mm. long. But when the distance is 15 mm. the impression is less like that of a solid body but still not ordinarily like two objects.

In connection with the subject of diffusion the _Vexirfehler_ is of interest. An attempt was made to develop the _Vexirfehler_ with the æsthesiometer. Various methods were tried, but the following was most successful. I would tell the subject that I was going to use the æsthesiometer and ask him to close his eyes and answer simply 'one' or 'two.' He would naturally expect that he would be given part of the time one, and part of the time two. I carefully avoided any suggestion other than that which could be given by the æsthesiometer itself. I would begin on the back of the hand near the wrist with the points as near the threshold as they could be and still be felt as two. At each successive putting down of the instrument I would bring the points a little nearer together and a little lower down on the hand. By the time a dozen or more stimulations had been given I would be working down near the knuckles, and the points would be right together. From that on I would use only one point. It might be necessary to repeat this a few times before the illusion would persist. A great deal seems to depend on the skill of the operator. It would be noticed that the first impression was of two points, and that each stimulation was so nearly like the one immediately preceding that no difference could be noticed. The subject has been led to call a thing two which ordinarily he would call one, and apparently he loses the distinction between the sensation of one and the sensation of two. After going through the procedure just mentioned I put one knob of the æsthesiometer down one hundred times in succession, and one subject (Mr. Meakin) called it two seventy-seven times and called it one twenty-three times. Four of the times that he called it one he expressed doubt about his answer and said it might be two, but as he was not certain he called it one. Another subject (Mr. George) called it two sixty-two times and one thirty-eight times. A third subject (Dr. Hylan) called it two seventy-seven times and one twenty-three times. At the end of the series he was told what had been done and he said that most of his sensations of two were perfectly distinct and he believed that he was more likely to call what seemed somewhat like two one, than to call what seemed somewhat like one two. With the fourth subject (Mr. Dunlap) I was unable to do what I had done with the others. I could get him to call one two for four or five times, but the idea of two would not persist through a series of any length. He would call it two when two points very close together were used. I could bring the knobs within two or three millimeters of each other and he would report two, but when only one point was used he would find out after a very few stimulations were given that it was only one. After I had given up the attempt I told him what I had been trying to do and he gave what seems to me a very satisfactory explanation of his own case. He says the early sensations keep coming up in his mind, and when he feels like calling a sensation two he remembers how the first sensation felt and sees that this one is not like that, and hence he calls it one. I pass now to a brief discussion of what these experiments suggest.

It has long been known that two points near together on the skin are often perceived as one. It has been held that in order to be felt as two they must be far enough apart to have a spatial character, and hence the distance necessary for two points to be perceived has been called the 'space-threshold.' This threshold is usually determined either by the method of minimal changes or by the method of right and wrong cases.

If, in determining a threshold by the method of minimal changes--on the back of the hand, for example, we assume that we can begin the ascending series and find that two are perceived as one always until the distance of twenty millimeters is reached, and that in the descending series two are perceived as two until the distance of ten millimeters is reached, we might then say that the threshold is somewhere between ten and twenty millimeters. But if the results were always the same and always as simple as this, still we could not say that there is any probability in regard to the answer which would be received if two contacts 12, 15, or 18 millimeters apart were given by themselves. All we should know is that if they form part of an ascending series the answer will be 'one,' if part of a descending series 'two.'

The method of right and wrong cases is also subject to serious objections. There is no lower limit, for no matter how close together two points are they are often called two. If there is any upper limit at all, it is so great that it is entirely useless. It might be argued that by this method a distance could be found at which a given percentage of answers would be correct. This is quite true, but of what value is it? It enables one to obtain what one arbitrarily calls a threshold, but it can go no further than that. When the experiment changes the conditions change. The space may remain the same, but it is only one of the elements which assist in forming the judgment, and its importance is very much overestimated when it is made the basis for determining the threshold.

Different observers have found that subjects sometimes describe a sensation as 'more than one, but less than two.' I had a subject who habitually described this feeling as 'one and a half.' This does not mean that he has one and a half sensations. That is obviously impossible. It must mean that the sensation seems just as much like two as it does like one, and he therefore describes it as half way between. If we could discover any law governing this feeling of half-way-between-ness, that might well indicate the threshold. But such feelings are not common. Sensations which seem between one and two usually call forth the answer 'doubtful,' and have a negative rather than a positive character. This negative character cannot be due to the stimulus; it must be due to the fluctuating attitudes of the subject. However, if the doubtful cases could be classed with the 'more than one but less than two' cases and a law be found governing them, we might have a threshold mark. But such a law has not been formulated, and if it had been an analysis of the 'doubtful' cases would invalidate it. For, since we cannot have half of a sensation or half of a place as we might have half of an area, the subject regards each stimulation as produced by one or by two points as the case may be. Occasionally he is stimulated in such a way that he can regard the object as two or as one with equal ease. In order to describe this feeling he is likely to use one or the other of the methods just mentioned.

We might say that when the sum of conditions is such that the subject perceives two points, the points are above the threshold, and when the subject perceives one point when two are given they are below the threshold. This might answer the purpose very well if it were not for the _Vexirfehler_. According to this definition, when the _Vexirfehler_ appears we should have to say that one point is above the threshold for twoness, which is a queer contradiction, to say the least. It follows that all of the elaborate and painstaking experiments to determine a threshold are useless. That is, the threshold determinations do not lead us beyond the determinations themselves.

In order to explain the fact that a person sometimes fails to distinguish between one point and two points near together, it has been suggested that the sensations fuse. This, I suppose, means either that the peripheral processes coalesce and go to the center as a single neural process, or that the process produced by each stimulus goes separately to the brain and there the two set up a single activity. Somewhat definite 'sensory circles,' even, were once believed in.

If the only fact we had to explain was that two points are often thought to be one when they are near together, 'fusion' might be a good hypothesis, but we have other facts to consider. If this one is explained by fusion, then the mistaking of one point for two must be due to diffusion of sensations. Even that might be admissible if the _Vexirfehler_ were the only phenomenon of this class which we met. But it is also true that several contacts are often judged to be more than they actually are, and that hypothesis will not explain why certain arrangements of the stimulating objects are more likely to bring about that result than others. Still more conclusive evidence against fusion, it seems to me, is found in the fact that two points, one on each hand, may be perceived as one when the hands are brought together. Another argument against fusion is the fact that two points pressed lightly may be perceived as one, and when the pressure is increased they are perceived as two. Strong pressures should fuse better than weak ones, and therefore fusion would imply the opposite results. Brückner[1] has found that two sensations, each too weak to be perceived by itself, may be perceived when the two are given simultaneously and sufficiently near together. This reënforcement of sensations he attributes to fusion. But we have a similar phenomenon in vision when a group of small dots is perceived, though each dot by itself is imperceptible. No one, I think, would say this is due to fusion. It does not seem to me that we need to regard reënforcement as an indication of fusion.

[1] Brückner, A.: 'Die Raumschwelle bei Simultanreizung,'
_Zeitschrift für Psychologie_, 1901, Bd. 26, S. 33.

My contention is that the effects sometimes attributed to fusion and diffusion of sensations are not two different kinds of phenomena, but are identical in character and are to be explained in the same way.

Turning now to the explanation of the special experiments, we may begin with the _Vexirfehler_.[2] It seems to me that the _Vexirfehler_ is a very simple phenomenon. When a person is stimulated with two objects near together he attends first to one and then to the other and calls it two; then when he is stimulated with one object he attends to it, and expecting another one near by he hunts for it and hits upon the same one he felt before but fails to remember that it is the same one, and hence thinks it is another and says he has felt two objects. Observers agree that the expectation of two tends to bring out the _Vexirfehler_. This is quite natural. A person who expects two and receives one immediately looks about for the other without waiting to fixate the first, and therefore when he finds it again he is less likely to recognize it and more likely to think it another point and to call it two. Some observers[3] have found that the apparent distance of the two points when the _Vexirfehler_ appears never much exceeds the threshold distance. Furthermore, there being no distinct line of demarcation between one and two, there must be many sensations which are just about as much like one as they are like two, and hence they must be lumped off with one or the other group. To the mathematician one and two are far apart in the series because he has fractions in between, but we perceive only in terms of whole numbers; hence all sensations which might more accurately be represented by fractions must be classed with the nearest whole number. A sensation is due to a combination of factors. In case of the _Vexirfehler_ one of these factors, viz., the stimulating object, is such as to suggest one, but some of the other conditions--expectation, preceding sensation, perhaps blood pressure, etc.--suggest two, so that the sensation as a whole suggests _one-plus_, if we may describe it that way, and hence the inference that the sensation was produced by two objects.

[2] Tawney, Guy A.: 'Ueber die Wahrnehmung zweier Punkte
mittelst des Tastsinnes mit Rücksicht auf die Frage der Uebung
und die Entstehung der Vexirfehler,' _Philos. Stud._, 1897, Bd.
XIII., S. 163.

[3] See Nichols: 'Number and Space,' p. 161. Henri, V., and
Tawney, G.: _Philos. Stud._, Bd. XI., S. 400.

This, it seems to me, may account for the appearance of the _Vexirfehler_, but why should not the subject discover his error by studying the sensation more carefully? He cannot attend to two things at once, nor can he attend to one thing continuously, even for a few seconds. What we may call continuous attention is only a succession of attentive impulses. If he could attend to the one object continuously and at the same time hunt for the other, I see no reason why he should not discover that there is only one. But if he can have only one sensation at a time, then all he can do is to associate that particular sensation with some idea. In the case before us he associates it with the idea of the number two. He cannot conceive of two objects unless he conceives them as located in two different places. Sometimes a person does find that the two objects of his perception are both in the same place, and when he does so he concludes at once that there is but one object. At other times he cannot locate them so accurately, and he has no way of finding out the difference, and since he has associated the sensation with the idea of two he still continues to call it two. If he is asked to locate the points on paper he fills out the figure just as he fills out the blind-spot, and he can draw them in just the same way that he can draw lines which he thinks he _sees_ with the blind-spot, but which really he only _infers_.

Any sensation, whether produced by one or by many objects, is one, but there may be a difference in the quality of a sensation produced by one object and that of a sensation produced by more than one object. If this difference is clear and distinct, the person assigns to each sensation the number he has associated with it. He gives it the name two when it has the quality he has associated with that idea. But the qualities of a sensation from which the number of objects producing it is inferred are not always clear and distinct. The quality of the sensation must not be confused with any quality of the object. If we had to depend entirely on the sense of touch and always remained passive and received sensations only when we were touched by something, there is no reason why we should not associate the idea of one with the sensation produced by two objects and the idea of two with that produced by one object--assuming that we could have any idea of number under such circumstances. The quality of a sensation from which number is inferred depends on several factors. The number itself is determined by the attitude of the subject, but the attitude is determined largely by association. A number of facts show this. When a person is being experimented on, it is very easy to confuse him and make him forget how two feel and how one feels. I have often had a subject tell me that he had forgotten and ask me to give him two distinctly that he might see how it felt. In other words, he had forgotten how to associate his ideas and sensations. In developing the _Vexirfehler_ I found it much better, after sufficient training had been given, not to give two at all, for it only helped the subject to perceive the difference between two and one by contrast. But when one was given continually he had no such means of contrast, and having associated the idea of two with a sensation he continued to do so. The one subject with whom I did not succeed in developing the _Vexirfehler_ to any great extent perceived the difference by comparing the sensation with one he had had some time before. I could get him, for a few times, to answer two when only one was given, but he would soon discover the difference, and he said he did it by comparing it with a sensation which he had had some time before and which he knew was two. By this means he was able to make correct associations when otherwise he would not have done so. It has been discovered that when a subject is being touched part of the time with two and part of the time with one, and the time it takes him to make his judgments is being recorded, he will recognize two more quickly than he will one if there is a larger number of twos in the series than there is of ones. I do not see how this could be if the sensation of two is any more complex than that of one. But if both sensations are units and all the subject needs to do is to associate the sensation with an idea, then we should expect that the association he had made most frequently would be made the most quickly.

If the feeling of twoness or of oneness is anything but an inference, why is it that a person can perceive two objects on two fingers which are some distance apart, but perceives the same two objects as one when the fingers are brought near together and touched in the same way? It is difficult to see how bringing the fingers together could make a sensation any less complex, but it would naturally lead a person to infer one object, because of his previous associations. He has learned to call that _one_ which seems to occupy one place. If two contacts are made in succession he will perceive them as two because they are separated for him by the time interval and he can perceive that they occupy different places.

When two exactly similar contacts are given and are perceived as one, we cannot be sure whether the subject feels only one of the contacts and does not feel the other at all, or feels both contacts and thinks they are in the same place, which is only another way of saying he feels both as one. It is true that when asked to locate the point he often locates it between the two points actually touched, but even this he might do if he felt but one of the points. To test the matter of errors of localization I have made a few experiments in the Columbia University laboratory. In order to be sure that the subject felt both contacts I took two brass rods about four inches long, sharpened one end and rounded off the other. The subject sat with the palm of his right hand on the back of his left and his fingers interlaced. I stimulated the back of his fingers on the second phalanges with the sharp end of one rod and the blunt end of the other and asked him to tell whether the sharp point was to the right or to the left of the other. I will not give the results in detail here, but only wish to mention a few things for the purpose of illustrating the point in question. Many of the answers were wrong. Frequently the subject would say both were on the same finger, when really they were on fingers of opposite hands, which, however, in this position were adjacent fingers. Sometimes when this happened I would ask him which finger they were on, and after he had answered I would leave the point on the finger on which he said both points were and move the other point over to the same finger, then move it back to its original position, then again over to the finger on which the other point was resting, and so on, several times. The subject would tell me that I was raising one point and putting it down again in the same place all of the time. Often a subject would tell me he felt both points on the same finger, but that he could not tell to which hand the finger belonged. When two or more fingers intervened between the fingers touched no subject ever had any difficulty in telling which was the sharp and which the blunt point, but when adjacent fingers were touched it was very common for the subject to say he could not tell which was which. This cannot be because there is more difference in the quality of the contacts in one case than in the other. If they were on the same finger it might be said that they were stimulating the same general area, but since one is on one hand and one on the other this is impossible. The subject does not think the two points are in the same place, because he feels two qualities and hence he infers two things, and he knows two things cannot be in the same place at the same time. If the two contacts were of the same quality probably they would be perceived as one on account of the absence of difference, for the absence of difference is precisely the quality of oneness.

These facts, together with those mentioned before, seem to me to indicate that errors of localization are largely responsible for judgments which seem to be due to fusion or diffusion of sensations. But they are responsible only in this way, they prevent the correction of the first impression. I do not mean that a person never changes his judgment after having once made it, but a change of judgment is not necessarily a correction. Often it is just the contrary. But where a wrong judgment is made and cannot be corrected inability to localize is a prominent factor. This, however, is only a secondary factor in the perception of number. The cardinal point seems to me the following:

Any touch sensation, no matter by how many objects it is produced, is one, and number is an inference based on a temporal series of sensations. It may be that we can learn by association to infer number immediately from the quality of a sensation, but that means only that we recognize the sensation as one we have had before and have found it convenient to separate into parts and regard one part after the other, and we remember into how many parts we separated it. This separating into parts is a time process. What we shall regard as _one_ is a mere matter of convenience. Continuity sometimes affords a convenient basis for unity and sometimes it does not. There is no standard of oneness in the objective world. We separate things as far as convenience or time permits and then stop and call that _one_ which our own attitude has determined shall be one.