(2) The only inconsistency in the strain-distance complex is with Baldwin in E. He reported that the more distant group appeared rather as an undifferentiated mass whose number was not so well obtained, while in the near the individuals were significant. He seemed to be in the midst of these. The case seems analogous to that of the observer whose introspection was reported under Table I, and who at first accepted what we may call the objective analysis, by which the scattered group gave up more distinct objects than the compact; but later attempting voluntarily to disintegrate the compact, found a bewildering confusion in the task that made this group seem very numerous, and brought about in the end an exact reversal of tendency. (3) Can we now separate in the results between the influences of strain and of distance? So far we have regarded them as one complex. But the introspections speak merely of the space-characters of the objects, Hutchison agreeing with Baldwin that the more remote group is judged as an area rather than as a collection of definite objects. (4) The almost entire absence of correct judgments in these experiments adds new evidence to that of the immediately preceding experiments in proof of the insignificance of the actual numerical relation for the judgment of relative number.

IV. THE INFLUENCE OF FACTORS OUTSIDE OF THE OBJECTS AND IN OTHER SENSE-FIELDS

The One-Group Apparatus was employed, and cards in general corresponding to those where area was not in question,—white-circle groups equal in size and irregular in inner distribution, which was not duplicated on the same card, though the resulting distribution-error was formally eliminated in the usual way. The usual care was taken to fill the group-area homogeneously. The small-difference cards were retained at first; but on later discovering the possibility of duplication a few supplementary experiments were added.

1. The Influence of Touch.

The apparatus employed to give the touch-stimulus consisted in a long lever attached to the armature of a small electro-magnet. In the end of the lever was inserted at right angles a wooden peg, cork-tipped. In view of the other conditions of the experiment a convenient spot for the application of the stimulus was found to be the forehead where it curves backward above the right eye. The apparatus was supported by rods and clamps upon a long upright steel rod set in an iron base and placed behind the chairs of the observers. The same rod carried a head-rest, designed not as a support but merely to show the observer that he had returned to the original position after he had bent forward to record judgment. Where two observers were used at once two sets of this apparatus were employed, with the magnets in a single circuit governed by a floor-button. The touch-stimulus was made to coincide as closely as possible with the appearance of a given group.

In view of the practical remoteness of this factor from the object of judgment the experimentation here took two forms,—one in which the observer was passive toward the touch-stimulus; the other in which the effort was made closely to associate the touch with the visual group by imagining the group to be responsible for the touch. For the passive method the touch was given irregularly now on the first and now on the last, but as many times on one as on the other.

For the active method, it was given always on the last group. This constancy was held to favor the active association of touch and particular group. The constant time-error was guarded against by experiments in which no modifying factor was introduced. A and B of Table VIII present the results of the passive and active methods respectively. C and D repeat A with duplication of groups,—C with the usual (1/25 sec.), D with a longer, exposure. These last sets were taken that the factor of touch might be studied when the objective conditions of the strong distribution influence should have been removed. It might prove that a factor swamped in the former situation might emerge into effectiveness.

The following summary gathers the chief facts of Table VIII: (1) Touch appears practically without effect in A. (2) In B, the results for touch seem again insignificant; but comparison with the control-results, to isolate touch from time-order, while it shows no marked change for Angier, does show for the others that touch was effective in determining the direction of error by difference-values, in the two cases of 10.2 and 14 per cent. The active method seems to be slightly more favorable to the influence of touch. (3) The duplication of the groups in C gives a large increase to the apparent effectiveness of touch, which is considerably diminished but not destroyed by the lengthening of the exposure in D. (4) The introspection for A indicates that touch under these experimental conditions has little subjective importance for the judgment of number. It is sometimes quite unnoticed. Angier made a possible exception in its favor in cases of great hesitancy where it added "importance" to the group with which it occurred. Usually he felt little doubt. With Shaw the touch was at first distracting but later indifferent. Johnston's notes indicate rather more effect. The touch prevented strict attention to the figure impression whereby the space-intervals in that group lost in value. Later it lost its confusing effect. Here seems to be subjective tendency, but not enough to predominate in results.

TABLE VIII

Results in B for the subjects separately were as follows: Angier 4%, Johnston 8.2%, Shaw 5%, all, so far as they went, in favor of the touch-group. Control experiments to determine the time-error gave the following results: Angier 6.8% in favor of the group last seen, Johnston 2.2%, and Shaw 9% in favor of the first group.

Some further introspective evidence appears in connection with the active method of B. Angier confirms his earlier account exactly. Usually the factors of distribution practically associated with number determine the judgment promptly; but in cases of doubt the touch is felt to add to its group something that appears as number-value. Johnston's subjective situation seems a little complicated. I may summarize thus: (a) The connection between the touch and its group being established, that group seems smaller, as being, together with the touch, somewhere nearly equal to the first. (b) The connection established and touch failing to come, that group seems smaller. (c) The connection not established and attention being concentrated on the visual impression, the touch-group feels much larger. The curious attitude in (a) results in a discounting in advance of the actual number. This done, the touch adds numerical value to its group. In (c) the effort at abstraction appears to emphasize the second (touch) group. Later, he reported similarly that the touch-stimulus seemed to add to the number of circles in its group even when the judgment favored the other group; and that "any outside stimulus connected with the one of two exposures tends to lose its own significance and be translated into number of dots to help the accompanying exposure to equal or exceed the first." The touch-group is felt to have more significance through association with an idea of superior energy or greater motor impulse.

Of the character of the influence exerted by the touch, Shaw reported that there seemed to be a diminution in the size of the first group and something extra in the second. More specifically, this effect appeared at times as an added circle at the right of the second (touch) group. He thought that this effect was overruled by the real bases of number-judgment which he summarized as "size, regularity, density, etc."

These notes show a definite tendency on the part of the touch-stimulus to break in upon the course of the number-judgment ordinarily determined by the practical association of a specific group of factors with number. That this result gets no more marked registration in the percentages is apparently due to the strength of these customary associations.

(5) The extent to which the distribution-error complicates the present study is shown by the prompt increase in effectiveness of the touch-stimulus when the groups were duplicated, as in C and D.

2. The Influence of Hearing.

The scheme of the experimentation upon this factor conformed in general to that of Section IV, 1. But a new sort of differentiation was possible in the auditory field, and one more readily suggestive of numerousness, perhaps, in that by use of an electric bell a rapid succession of sounds could be given with one group while with the other a single sound could be produced. An actual numerical difference in the auditory field might fuse with the factor of relative visual number and determine the judgment to its direction. These results are set down in B of Table IX. The same set of cards was used in the One-Group Apparatus for these experiments as for those of Section IV, 1. In these two sections of Table IX the observers did not know on which group the sound or the particular sound would be given; but any possible disturbing effect of this irregularity was formally eliminated as in Section IV, 1. The experiments of Table IX, C, repeat those of A with duplication of groups; and D repeats those of C with longer exposure.

The sound for A was that of a small organ-pipe (Ut 4) blown by mouth. As in A of the preceding table the observers did not know in a given experiment with which group the sound would be given, but, as before, it was given the same number of times with the first as with the last. For B the multiplied sound was produced by an electric bell with a wooden gong. This was adopted in preference to metal because of the prompt ceasing of the sound after the stroke,—a very necessary condition when this sound accompanied the first group, that it might be clearly connected with its own group. A metal gong was used for the single sound, that the two might not be too unequal in loudness. Its vibrations were deadened by a rubber band, and each bell was controlled by a floor-button. For C and D a higher sound, from the same pipe unstopped, was used in preference to the former, for the reason that in certain experiments performed just previously the lower sound had been used and was presumably very familiar. So in order that the sound might be brought, if possible, afresh to the attention, the change was made.

TABLE IX

The results of these experiments may be summarized as follows: (1) The figures give evidence of but two cases out of eleven where sound was influential. (2) Duplication of groups is not effective in developing evidence of the influence of sound. (3) Increased length of exposure works, as in former cases, to lessen the influence of the modifying factor. (4) The introspections are to the effect that the sound seems to be entirely without influence upon the judgment, beyond the distraction it brings in the earlier stages of work. Sometimes it dropped wholly out of consciousness. Sometimes the distraction seemed to last longer. One observer reported, when D was taken, that he felt as if the sound sometimes increased and sometimes decreased the apparent numerousness. In some other experiments not directly upon this point, but later to be reported, a sound was used; and one observer reported that it seemed to become functionally connected with certain gaps in the groups, as though the puff had blown a hole in the group. Here its effect was of course to emphasize negative factors. It appears thus that the sound might function in opposite directions at different times, somewhat in accord with the particular character of the visual presentation. We should expect, then, to have percentages that look insignificant. (5) We shall not have failed to notice the difference between touch and auditory stimuli in the feeling of influence upon the number-judgment. If we seek a cause for the superior influence of touch, we may perhaps find it in the fact that practical experience has trained us to disregard in any case of judgment such simultaneous presentations as were employed for auditory stimuli; while a definite tap upon the brow is a rather unusual experience likely to attract notice to itself in spite of attempts at abstraction. As one observer said, who took part in both kinds of experiments, the touch seemed more "intimate."

3. The Influence of Kinæsthetic Impression.

The method consisted in the employment of active effort upon a fist dynamometer or a wooden handle during the appearance of one of the groups. The handle was preferable because noiseless. The effort was made with the left hand because the right was used in recording. The amount of it was left to the observer's regulation, with the one instruction that its presence be made decidedly evident but without too great fatigue. The cards of Section IV 1 and 2 were used in the One-Group Apparatus. Similarly again the experiments were repeated with the duplicate-group cards. I present no table here because the figures show practically no influence of the effort. On one subject 176 experiments were made; on a second 132; on a third 88.

It is interesting to note here certain results obtained from one observer when he was in what he described as an active attitude toward the groups, in which he seemed to rouse himself to an unusual pitch of concentration upon the visual situation. This was evidently a condition of increased effort to abstract. Without abstraction he gave 26 to 6 in favor of the effort while with abstraction this tendency had fallen off to 30 to 17. The strength of the tendency is thus strongly indicated. Another observer felt a kind of motor difference between the groups; he expected the effort-group to look larger and felt additionally excited, a scattered activity, while he was passive toward the other group. Perhaps this account puts a little meaning into his small per cent. That his power of abstraction was effective here is hinted by his remark that he felt a difference in the groups even when he judged them equal. The third observer found no subjective evidence that effort modified his judgment.

V. THE "ERRORS" OF EXPERIMENTATION

Throughout the foregoing experiments has been involved the possibility of some one of the three "errors" of experimentation, those of time, space, and distribution, and sometimes all three. Their effect on the results, if it existed, was, to be sure, eliminated in the well-known way, but their existence, if actual, would raise an interesting problem. It was possible, in the case of every group of experiments, to rearrange the tables in such a way as to bring out the evidence for any tendency to overestimate, for instance, the first group as against the second, the right as against the left, or one kind of irregular distribution as against another.

The distribution-error must have a word of explanation. It refers to a tendency to give more wrong judgments in favor of one kind of irregular distribution than of the other kind with which, in a given card, it is mated. In the construction of a set of cards several forms of irregular internal arrangement were used, in order that the judgment might not be one merely of form, and of course on any given card the forms were not the same. Elimination of the effect of these form-differences from the results involved the appearance of any given one as many times in connection with one of the two contrasting factors studied in a given experiment as with the other. Thus two sets of forms were carried through an experimental series—a source of error indeed, but avoidable only by such means as were used to escape the effects of the space-error. Analysis would show which, if either, of the two sets received more judgments in its favor, resulting in further evidence as to the extent to which the judgment of relative number is a function of distribution, and as to the fineness of discrimination for such differences.

Now the tables, when thus rearranged, show that these errors exist to a surprisingly large extent. In many cases their causes, whatever they are, seem to be the controlling factors in the judgment of relative number.

Barring the experiments of Section III, in which the space-error has largely been accounted for, I now propose to gather in one survey all the results of those analyses that have given us the information of the existence of these errors, and all the material of later tables that bears on this point, and to test them by further experimentation. I will begin with the space-error.

TABLE X

1. The Space-Error.

Table X presents to us a summary of the values of the space-error tendency. (1) Taken as a whole they fall into all the three classes that are possible; (a) favoring right; (b) favoring left; (c) no marked tendency. (2) There is no observer that does not at some time show a fairly marked tendency. (3) All the observers fall into (c) and all but five into both (a) and (b). (4) More favor the left than the right,—50 to 35. (5) This survey makes it clear that the observers agree neither with themselves nor with each other in the direction of influence exerted by the causes underlying the space-error.

a. Special Experiments to establish the Facts. It might be suspected that irregularities would be more apparent where other factors such as we have been studying enter to complicate the situation from the point of view of pure relative position of the two groups. Table XI presents the answer to this query. The cards used contained groups of gray circles (Gray Darker, Prang) arranged in equal areas of the usual size and shape. The distribution-error was eliminated, though not by duplication, and the small-difference cards were retained. The Two-Group Apparatus was used, with an exposure of 65 sec.

TABLE XI

The results give us again our inevitable three classes, and in many cases a difference-value surprisingly large when we reflect on the simplicity of the conditions. That the omission of complicating features was of importance is shown by the fact that more of the observers (11 out of 14) show a marked error than in any other case. Clearly enough the various factors introduced tend to eliminate the space-error, but when in any case it does enter, it is even then capable of rising to as high a degree on the whole as in the uncomplicated series, as is shown by the fact that in but four cases does the new value surpass the best of the old, and in three of these by a trifling amount.

It is interesting to note that the three cases of minimum space-error show a well-defined tendency to be determined by distribution.

b. Possible Bases of this Error. The outcome of these special experiments is that the factors found in the groups are at least not directly responsible for the situation that we are considering. The divergence among the observers shows this. In hunting after the cause for this apparent influence of side, we look first for changes in the peripheral, and then in the central, processes that precede the judgment. The material used for the experiments of Table XI seems approximately to have equalized all the objective factors in the two groups. How could there be anything further in the peripheral process whereby group could be differentiated from group? The most evident thing is that the visual stimulus is received in a different way from the two groups. There is a definite peripheral mechanism whose factors seem essentially to be two, however variously they may be combined: (a) The relative amount of time given to each group; (b) the order in which the groups are viewed.

The observers were instructed and continually reminded to equalize the amount of attention devoted to the group; but as this is not wholly a voluntary matter, the possibility of failure to conform has to be reckoned with. Experiment must therefore be employed to test the influence of these factors before one can fall back upon a central process as the cause for this tendency to favor a side.

c. Its Relation to Differences between the Groups in Length of Look. The material was the same used for the experiments of Table XI. The method was the same with the following necessary exceptions: The longer exposure was double the shorter (4/5 sec. to 2/5 sec.), and 2/5 sec. elapsed between the two. Further, the experiments were so arranged as to equalize the influence of the order of exposure with respect to both side and relative length. The means for effecting successive exposure took the earlier form described in the introduction to Section II.

TABLE XII

These facts are yielded by Table XII: (1) There are but two classes of observers, as no tendency exists to favor the group of longer exposure. (2) The time-error shows a considerably more marked tendency than the length of look, which is indeed somewhat surpassed by the space-error. (3) The persistence of the space-error, even among those that reveal a tendency in length of exposure, shows that the factor of relative difference in length of look cannot account for it. The persistence of it, too, when the order of exposure is controlled, even though the conditions are not wholly adapted to the study of this latter factor, suggest at least that the space-error is independent of even that order; but into this we shall make special enquiry. (4) The judgment of number is independent of the amount of eye-movement devoted to the fixation of the objects in a group. This conclusion, so far as the actual movement is concerned, is established by the fact that so many observers favor the shorter look; and by all the experiments with the One-Group Apparatus where an exposure of 1/25 sec. was used, since that time was too short to admit of movement. That ideated movement is likewise insignificant appears from the fact of marked error arising in the material where the groups were duplicates. Here no motives to different movements could lie in the material.

d. Its Relation to the Order in which the Groups are viewed. Table XIII gives us the results of the enquiry. The experimental conditions were not changed except as to the length of exposure. Each group was given 3/5 sec.; and half the experiments were performed in the order right-left and half in the reverse order.

TABLE XIII

The results may be thus summarized: (1) The order of exposure is notably influential upon the judgment of relative number, giving the usual three classes, with the tendency to overestimate the last group well in the lead. (2) The persistence of the space-error under these relatively simple conditions shows conclusively that it is not a function of the order of exposure. The two are independent variables.

2. The Time-Error.

In pursuit of our enquiry we must survey the facts as they are given in the various experiments already reported and later to be reported. These facts are gathered into Table XIV, which furnishes the following items of significance: (1) All the observers, with the exception of Rouse, show at some time a definite tendency. One case only is given for him in this table, but other experiments not included in the tables from which the present is drawn confirm this fact by the ratio 29 to 30. (2) There is a rather striking consistency in the several observers. (3) The predominance of the last group is marked.

TABLE XIV

a. Relation of the Error to the Absolute Length of the Total Exposure. Table XV is set to answer this question. It is unsatisfactory in that but two observers took part in both XIII and XV. The material used for judgment consisted of the same cards used in the earlier experiment, but presented now in the One-Group Apparatus. The time of exposure was changed from 3/5 sec. to 1/25 sec. for each group. As the space-error was eliminated, the tendency to a time-error, if present at all, would presumably have freer play.

But the difference-values of the new table are for the most part very small. We have thus the further fact about the time-error that, under the conditions studied, it appears to be independent of the absolute length of exposure, when the groups are equal in this respect. To this we may add another fact drawn from Table XIV, that with the One-Group Apparatus the time-error is greater on the whole where the groups are differentiated by other factors. Thirdly, the values for Table XIII show that with all complicating factors withdrawn, except the differences in position, the error is at a maximum. This may be significant of the effect of space-differences upon that error, or, more probably, be due to the general difference between work by daylight and work in a dark room by artificial light. We shall be better able to consider this later.

TABLE XV

•   88 experiments with each of two subjects.
• 176 experiments with one subject.
• 154 experiments with one subject.
•   66 experiments with one subject.
•   Exposure = 125 sec.

3. The Distribution-Error.

The last of our three "errors" of experimentation is now before us. We may recall once more the meaning the term has had for us in these studies. It points to a tendency discovered by the use of those cards where all objective factors were in the course of a series equalized,—a tendency to mass one's judgments in favor of a particular arrangement of the circles; though each group had been constructed with a view to filling the given area as homogeneously as an irregular arrangement would allow.

As in the two "errors" preceding, so here we must get possession of the facts that gave rise to the present enquiry. Table XVI presents them to us, gathered out of all the tables wherein such a tendency has been technically reckoned with. But first a few words of explanation are needed to make the new table intelligible. Two sets of results are found in its two parts. In each set the particular group-arrangements employed and the frequency of their appearance are exactly the same. The two sets differ, as their headings suggest, in that the material for the second set was formed out of the first by replacing the small-difference cards by those having equal groups. Such a change as this might affect the proportion of judgments given in favor of the two sets of arrangements in a particular series, and these new results are, therefore, no longer fully comparable with the earlier ones. In presenting the directions of tendency in the results, it is impossible here, as in all the similar cases throughout the tables, to name a factor as a standard in whose favor all the judgments in the plus column should be understood as given,—impossible for this reason that, because the very method by which the circles were distributed in the groups, the experimenter was unable to satisfy himself as to the significant differences in the arrangements. All the results, however, when analyzed on this basis, were recorded consistently, so that consistencies and agreements among the observers might be readily apparent. We can now understand in part what Table XVI has to say to us.

TABLE XVI

Here as elsewhere the per cents recorded indicate the average per cent of difference in favor of a given class.

Here are the facts, first of A: (1) The only lapses from consistency are confined to two observers; and in both these cases there is but a single break in a uniform trend. (2) With three exceptions all agree in the trend of their difference-values. Of these three—Holt, Hylan, and Moore—the last furnishes but one significant value, and so must be left out of the reckoning on this point. (3) Of the 64 cases, 50 rise above 10%, some far beyond, showing the importance for the judgment of relative number of this factor of distribution. (4) Of the 50 cases, 45 agree in tendency. (5) That with this surprising agreement we have still a few exceptions, adds another item to the growing array of evidence on behalf of the importance of some subjective factor for the number-judgment. As to the nature of this factor we are yet in the dark. (6) To these facts B of this same table adds the further information that the observers inconsistent in the old are not consistent in the new, while the consistent still maintain their record.

a. Analysis of the Experimental Conditions of Distribution. At once we are interested to enquire for the factors underlying these results. To put ourselves upon the right track we must first consider what factors are involved in any such arrangement of objects as we have used in the material for these studies, and then, more precisely, we may ask in what way such arrangements could differ significantly. Finally, by an experimental trial-and-error process, we may solve our problem.

The groups of objects in our material were arranged in an area marked out in each corner by a circle. Within this area the circles were set irregularly, with the result that the group, as a mass of objects distinguished from a homogeneous background, had a more or less irregular outline whose irregularity varied with different internal arrangements. Within its outlines this area presented a mixed pattern of bright and dark. While the total enclosure marked off by the corner circles was always the same and theoretically the relative amounts of brightness and darkness in equal groups was likewise the same, yet practically differences, more or less slight, might enter through the changing character of the rude outlines whose ideal completeness could scarcely be brought out of a black background by the uninitiated. The amount of this difference is sometimes surprising to one whose chief thought of the group has been as vignetted in process of construction. As the objects are pushed toward the edges the central spaces open out; as they are withdrawn toward the interior gaps appear in the margin.

It is not a very easy task to fill an area with objects in irregular arrangement in such a way that no sections of vacancy or filling stand out by contrast against the remainder of the same element. To succeed in this is to fill the area homogeneously. But the chances are good that some vacant patch will get slightly the better of its neighbors or some section of circles will gather a little more closely than the surrounding circles; or perhaps a gap in the outline will be unexpectedly intrusive. Now in a given area the circles of one part cannot become more thickly massed without a corresponding enlargement of the vacancies of the other parts, and of course the converse is as true; but this theoretical situation may be quite out of ken at the moment when the group is seen. Either member of this pair of complements may stand out vividly in the field and its fellow quite escape perception. The very nicety with which in practical affairs we have to make a reliable comparison of this sort shows what suspicion of accuracy the off-hand judgment has bred. And further, the widening of a gap or thickening of the filling in one small part of a group may give a complementary loss to the rest of the group small enough to be unperceived when distributed throughout the larger section.

Two factors must therefore be considered as possibly significant in moving the judgment,—vacancies and filling; and with the former must be reckoned indrawing of the outline. Psychologically, increase in the prominence of either of these factors would be all one with their objective increase. With respect to the direction of their influence upon the judgment of number the increase of vacancies must signify the waning, and the increase of filling the waxing, of the objective number in the group.

It is in advance altogether probable that the results gathered into Table XVI were brought about by these two factors, at least in large part. And we have also in these factors the possibility of two types; for as we saw above, increased vacancies in one part involves increase of filling in another, and conversely. So the interesting question turns upon the altogether disproportional representation of types. Which is the type of the majority?

b. Experimental Test of Hypotheses. The question was put to the test of experiment. This was done by using groups in which now vacancies and now filling were objectively emphasized in contrast with the usual homogeneous group. First the vacancies. A set of cards was prepared after the method previously used to eliminate the distribution-error without duplication of groups on any one card. (See Section II.) In the present case, however, the two sets of arrangements were definitely differentiated as already indicated. One set had a homogeneously filled area, the other a prominent vacancy within or gap in the edge. The size of these variations was kept pretty close to the limit of noticeableness, that the increase in compactness of the other portions might be as slight as possible. It was experimentally necessary to free the material as far as might be from ambiguity, and practically important to avoid rousing the suspicions of the observers and the resulting reflections. It seemed very likely that the strength of the tendency shown by the distribution-error was due to its appearance in situations where the observers knew that other factors were being tested.

The general method already described was used in preparing the groups that gave objective prominence to compacted parts of the filling. To fulfil the conditions outlined above was here even more difficult than in the first set; and the cause will appear in the sequel. The small-difference cards were omitted and the One-Group Apparatus used.

A further attempt was made to head off reflection by a subterfuge. It had been found that, among the factors whose influence on the judgment had been studied, hearing had been as little effective as any. So the small stopped pipe used for those experiments was again brought into service and the error resulting eliminated in the usual way. Incidentally our new tables will thus give us further information about the effect of this factor, though of course under conditions that are theoretically highly unfavorable, since we are forcing upon the attention of the observers other factors that experience has shown them only too ready to seize upon. So if a tendency traceable to the factor of hearing should appear, we ought perhaps to give it somewhat more than its face value.

TABLE XVII