The per cent recorded in the no-tendency class is an average of all per cents below 10, whether in favor of the one or the other of the two remaining classes. This is true for all the following tables.

The following facts are presented by the several parts of Table I: (1) The large per cents of difference show that area is to a large extent a determinant of the judgment of relative number. (2) Different subjects show opposite tendencies. (3) A comparison of the results of individual subjects through the four series shows that this opposition in tendency occurs in the same subject at different times. The introspective notes of one of these subjects show the internal process of change from one tendency to the other. It consists in a gradual increase of analytic activity toward the compact group. At first glance the composition of the scattered group was more evident; but when attention was fairly turned toward the compact, the inability to isolate objects made them seem very numerous. The importance of a coöperating subjective factor is here evident. (4) Out of a possible 57 cases there are but 10 showing no tendency.

2. The Influence of the Internal Arrangement. As before, the Two-Group Apparatus was employed; and the factor was studied in three aspects.

A. The material consisted of two groups of gray circles (Normal Gray Darker, Prang) covering equal areas. In one group this area was filled homogeneously, in the other the circles were gathered into nuclei. In order that there might be exactly the same relation of parts when the cards were reversed, each group was so arranged on a diagonal axis of symmetry from upper left to lower right corner that each half repeated the other in reverse order. Otherwise the arrangement of circles was irregular.

Six cards only were used,—four (20 to 20), one (17 to 23), and one (23 to 17). Slight differences among them occurred in the arrangement of the equality-cards, which might help to counteract any incipient reasoning from sameness of appearance to sameness of number. The large increase in the difference-values is accounted for in part by the fact that the cards (19 to 20) and (20 to 19) were omitted, and for this reason: that when an observer tended largely to favor a particular group, the introduction of a card in which that group was objectively greater would mean an increase in the number of correct judgments; whereas the introduction of two objectively equal groups for the others would increase considerably the number of erroneous judgments.

B. The numerical character of the cards here used shows a return to the usual. The material was like that of A, except for a new internal arrangement. Here the area of one group was filled homogeneously, while that of the other contained a pattern of this sort: An ellipse just contained within the boundaries of the normal area; a circle in each of the four corners of that area; and in the centre a diamond formed of four circles. Numerical changes were always confined to the ellipse, as less open to counting than the rest of the figure.

C. Material and method repeat B but with another internal arrangement. One group, as before, showed an area homogeneously filled, and irregularly, as usual. The other group carries to an extreme the distinction of open and filled space made prominent in the other groups of this table by massing the circles in an outline completely enclosing the area and in a diagonal from upper left to lower right corner. The outline did not show even spacing; more circles were crowded in one part than in others, that counting might be more difficult.

That there might be no attempt to remember cards, in all cases where there were twenty objects in the homogeneous group the same kind of irregular arrangement was repeated. This is a different method from that employed in A. Since the length of exposure was so short and the arrangement in the group irregular, either one is probably as good as the other.

D. The material used for B and C had a kind of regularity, since definite patterns were used. The introspection of the observers showed, however, that the patterns as such were not in question in the judgment, but rather the vacancies and the crowding. With the Two-Group Apparatus an arrangement in parallel lines could rather easily be counted, but with the One-Group Apparatus and its means for instantaneous exposure this difficulty was to some extent overcome. The arrangement of the objects in parallel lines was therefore adopted and matched against the irregularity of an accompanying group. The same size of group-area was kept, but the small difference-cards were omitted. There was no other change beyond those made necessary by the apparatus and already indicated on an earlier page. The length of exposure was 125 sec. The bearing of this time-factor on the results will be considered in a later section.

TABLE II

The several parts of Table II give us the following facts: (1) The judgment of relative number is very markedly a function of the internal arrangement. (2) The marked tendency among the observers to favor the homogeneous in A and B meets a check in C. Recalling the direction of difference between C and the other sets, that in C the gradually increasing contrast between the inner vacancies and the filling reaches a maximum, we may suspect that these vacancies begin to seem no longer a part of the group-situation, while the compactness of the filling, where it does occur, is thrust prominently forward. The notes of the observers confirm this suspicion. (3) The results of the different subjects show that the shifting of tendencies occurs as before. (4) As to the way in which regularity functions in the judgment, the notes of one observer are very clear. The blank spaces in the irregular are noticeable, he says, which is not true of the regular, where, on the contrary, one has a feeling of compactness of figure. I am able to confirm this character of the spaces by my experience outside this experiment. A simple pattern is very easily apprehended and irrelevancies of the background dismissed. Increase its complexity to a maximum, as in the case of an irregular group, and I am almost at a halt to isolate objects from their fellows and maintain them apart, yet together. The background is hardly to be shut out. This is probably due to the absence of a centrally excited image of the group. The object and the not-object run together. (5) The position of the single observer in the no-tendency class of D was marked subjectively by great difficulty in forming a judgment. The groups seemed incomparable, the vividness of form excluding the perception of number.

3. The Influence of Complexity in Group-Composition.

Complexity of group-content was attained by introducing objects of different colors; so there was not a clean isolation of factors. By comparing these results with those recorded in Table IV, A, we shall be able somewhat roughly to make allowance for the factor of mere color.

Sets of 132 experiments each from sixteen observers were obtained for each of these factors. Unfortunately the distribution-error was not eliminated. Later experiments showed the importance of this factor, and, in consequence, the impossibility of interpreting the results under consideration. So new experiments were performed under the proper conditions, but at a time when only a few of the first observers could be used. Their results from the earlier series are given in Table III, A. The exclusion of the small-difference cards from the later series (Table III, B) and the consequent increase of the number of experiments on objective equality prevent comparison.

The material in A consisted of two groups of circles of the usual size, one Normal Gray (Prang), the other of three colors—Red, Yellow Orange Shade 2 (Bradley), Light Blue Blue Green (Prang). The intent was to equalize the two groups in brightness. When the observers were questioned about the relative brightness, supporters were found for all three possible opinions. So it seems probable that the groups did not differ widely in this respect. As nearly as the number-condition would allow, the three colors were represented equally in the group; and they were distributed so as to make the whole as homogeneous as possible.

In B the changes were the correction for distribution as described in the introduction to this section, and the substitution of equality-cards for those of slight numerical difference. In addition, three other colors replaced those of A, in the interest of regulated brightness and more pleasing æsthetic effect. These were, in the Bradley system of broken spectrum scales, A-Red, medium; A-Yellow Orange, dark; A-Blue Green, dark. With these exceptions B was like A. The observers were all inclined to consider the gray brighter than the mixed. The Two-Group Apparatus was used.

TABLE III

The following facts may be gathered from Table III: (1) The tendency to overestimate the gray is due in part at least to the additional factor of complexity in the other group, as is shown by the markedly changed tendencies in Table IV, A, where a solid color takes the place of the mixed colors. The actual colors involved in the two cases are different, to be sure, and necessarily so, and this difference may of course be invoked as the cause, as well as a possible difference in brightness between the gray and mixed in III, B. The introspective notes help us here. One observer felt that he favored the gray primarily because there was a tendency to consider but one color in the mixed. Another was drawn toward the gray because it seemed definite and consistent. For both of these observers æsthetic elements were involved in favor of the gray. The latter found also that the greater brightness of the gray gave it a larger area. With a third subject the fact of variety was felt as decidedly important; but his notes show a conflict between this factor and that of distribution which was the conscious basis for his normal judgment.

4. The Influence of Differences in the Kind of Objects.

A. The Factor of Color. The material in A 1 consisted of two groups of circles of the usual size, one Normal Gray (Prang), the other Red (Bradley). The attempt was made by this choice to equalize the brightness. The size and shape of the group-area were those of the smaller area of Table I, C and D. In A 2 the only changes were the correction for distribution, as described in the introduction to this section, and the substitution of equality-cards for those of slight numerical difference. The Two-Group Apparatus was used.

The following results appear in Table IV, A 1 and A 2: (1) While complexity seemed on the whole to diminish apparent number, red noticeably increases it. Some of the observers report that group as more vivid and interesting. One observer compensated by emphasizing the gray in attention, as his results showed. (2) If one ask how the color red functioned in the judgment, the reply must apparently be, by its brightness and vividness. The mixed group functioned in a double way, as vivid and so more numerous, as fragmentary and so fewer.

TABLE IV

B. The Factor of Size. The Two-Group Apparatus was used, the material consisting of India ink circles (13 to 12 mm. line) on a background of granite paper. This paper was chosen here and for the experiments of Table I, D, to get a suitable mean between too sharp contrast and sufficient distinctness. The circles in the one group were ten mm. in diameter; in the other seven mm. The two areas were approximately equal, and of the same size as that of the more compact group in Table I. This is in fact the standard size throughout these studies in Relative Number, wherever area is not in question. The small-difference cards were included.

Two sources of possible complication must be considered. It is unavoidable that the factor of differences in compactness should enter and that clean results on the basis of object-size be denied. Our interpretation must not fail to consider this fact. Because of this, it seems unlikely that a distribution-error should arise; so the usual precaution to eliminate it was omitted both here and in the study of area (Table I). Distribution affects the appearance of the vacant spaces. When differences in the amount are by the conditions inevitably prominent, differences in the conformation may be safely regarded as of minimal vividness.

The following results appear in Table IV, B: (1) The illusion of numerical inequality is marked for many subjects. (2) The judgment is quite possibly a function of the two factors—object-size and group-vacancies. If we recall the fact that the small-object group is more scattered than the other, we shall note that the leading class here is like the leading class in Table I, and we may fairly reckon this factor as of importance in the issue. Of the incomplete introspective notes on this question, those of only one observer speak clearly for the size. He says: "There is an overpowering feeling of predominance in case of the large and I must judge for them. The large space covered seems an important factor. The longer I reflect upon the relative numbers the more numerous seem the larger, that is, they appear to increase over the small after the exposure. It is hard to give judgments of equal in most cases."

C. The Factor of Form. The material consisted of a group of circles, each of the same size as in former material; and a group of squares, each approximately equal to a circle of the other group. These were made of Prang's gray paper (Normal Gray Darker) and pasted upon a black background. The areas of the two groups were approximately equal. The squares were set irregularly except for those in the corners, where the edges were placed parallel to the edges of the card. The Two-Group Apparatus was used and the small-difference cards included.

In this material, again, the formal elimination of the distribution-error was not attempted. The striking difference in the conformation of the vacancies through the form-differences of the objects probably makes the repetition of the exact positions insignificant. Still the fact must be noted.

The results appear in Table IV, C. (1) The illusion of numerical inequality is here again marked for many observers. (2) The introspective notes are not on the whole very illuminating as to the basis of judgment. One observer, who favored circles, found that the appearance of more orderly arrangement in squares made them seem few. Another, who favored squares, found, on the contrary, the more regular the more numerous, and thought that the squares may have seemed more regular. A third, who favored circles, found the squares better individualized, with whom a fourth agreed in both respects, who also was influenced by the apparently greater bulkiness of the squares. Fewer could go into a given area. A fifth, on the other hand, who found the circles better individualized, still favored them. So we have these observers apparently doing the same thing under opposite conditions, and the opposite thing under the same conditions. Here indeed is a situation for any theory. So far as we can learn from the foregoing, the form may influence the judgment merely through its space-characteristics, but possibly also through the vividness of intrinsic interest.

D. The Factor of Complexity. The One-Group Apparatus was used in this work and results were obtained for two different lengths of exposure, 125 sec. and 14 sec. The material differed, in that to the centres of the circles of one group were added small Red (Bradley) circles (6 mm.). With the apparatus used, the color was not very effective, the brightness contrast between dark centres and white periphery being chiefly prominent. The total group-brightness was of course diminished by those centres. The small-difference cards were omitted.

The results are recorded in Table IV, D 1 and D 2. (1) The illusion is apparently strong. (2) The amounts of the difference-values show that the shorter exposure is more favorable to the illusion. (3) The introspective notes indicate that both brightness and complexity functioned in the judgment. Two observers, both of whom show large tendencies, were not conscious of any influence of complexity. One of these did find differences in brightness important; and in favoring the darker group his results exactly coincide with those of Table IV, E, where this factor is under direct consideration. A third found the complex group interesting. With a fourth the complex group developed in number amazingly during the few moments after exposure and had an appearance of great intricacy, often seeming to be in active movement. A fifth observer too felt that its numerical character depended on its complexity.

E. The Factor of Brightness. Hitherto the absolute arrangement of the objects in any two groups compared, where this factor has not been the object of enquiry, has been in the two cases different, though with respect to irregularity alike. This course was governed by a desire to avoid the substitution of a form-judgment for one on number, through recognition of the fact that both groups had identical forms. The resulting distribution-error I tried to eliminate in the usual way. Tests toward the end of these studies showed that there was no danger from this source. Errors seemed about as frequent as before. No observer made any comment on the fact, except one who through his official connection with the laboratory work knew that the test would be made sometime, but not exactly when. During many of the experiments he did not perceive the likeness of form; and when he did the numerical judgment arose without connection with that factor, as was shown by the feeling that the two groups were unequal in number. He called the relative fewness of the first group a case of "perspective effect." This must have significance for any account of the time-error; but by no means carries with it its own interpretation.

One welcome result of these tests was their assurance that I might without fear further simplify the experimental conditions by avoiding the possibility of a distribution-error. The material for these experiments on brightness therefore profited by this possibility. Each card had a different specific irregularity, but always in duplicate. In choosing the degree of brightness-difference Prang's brightest shade of normal gray was found as dark as could be conveniently perceived with the artificial light of the One-Group Apparatus. The contrast between this shade and white was quite evident enough for the purpose. The small-difference cards were omitted.

Table IV, E, shows the decisive character of the results. The observers fall all into one class in favoring the darker group, and by a large difference-value. The following introspection of one observer shows the extent to which the factors of brightness and number fuse: "I frequently lose sight of time-order. It is a question of number and not one of light-intensity, and if called upon to state which group came first I might not be able to answer. In equality-judgments the difference of light comes out distinctly."

5. The Influence of Complexity of Environment.

The material prepared for these experiments certainly lays stress upon relative, not absolute, complexity; for the conditions were satisfied by placing 5 mm. strips of white paper, equal in length to the width of a group, a few millimetres off at the top and the bottom of the groups that were on one side of the cards. The One-Group Apparatus was used and the small-difference cards omitted.

TABLE V

44 experiments with two subjects.
88 experiments with two subjects.

Exposure = 125 sec.

The results are recorded in Table V. (1) The drift of tendency is toward the group with the more complex environment. No one markedly favors the other group. (2) The notes of the observers indicate that the added strips functioned through their effect upon the apparent area of their group. The observers all found the dimensions increased; but with some, apparently by contrast, the added height brought out sharply the narrowness. One observer found this true in general; another, when the barred group came first. The latter says: "The unbarred group, coming first, appears to reflect its compact character on the barred one, when it comes, so that it does not look so attenuated and strange." Here the image brought over to the second took the width of the second somewhat out of relation to its illusory height, whereas in the reverse order the contrast relation was fully maintained.

III. THE INFLUENCE OF FACTORS PRESENTED IN OTHER SENSE-FIELDS BY THE OBJECTS WHOSE NUMBER IS IN QUESTION

A very simple apparatus was employed. The objects whose number was in question were bright steel balls (3/8 inch) thrown loosely into square black frames, 13 cm. inside, placed side by side on a black-topped table. The experiments were performed in series of 30. The groups were kept equal numerically, with this exception, that into each series were introduced four experiments where the groups were so unequal that the observer could have no question as to the correctness of his judgment and the existence of objective differences. This numerical superiority was given to each group alternately, and judgments on it were, of course, excluded from the results. The actual numbers employed in a series varied between 35 and 60 in accordance with the following scheme:

1.   50 each
2.   45   "
3.   50   "
4.   55   "
5.   60 to 40
6.   50 each
7.   45   "
8.   40   "
9.   45   "
10. 50   "
11. 55   "
12. 60   "
13. 40 to 60
14. 50 each
15. 45   "
16. 40   "
17. 35   "
18. 40   "
19. 45   "
20. 50   "
21. 55   "
22. 60   "
23. 60 to 45
24. 50 each
25. 45   "
26. 50   "
27. 60   "
28. 55   "
29. 50   "
30. 45 to 60

The time of a single exposure—in this case two groups at once—was 3 sec. measured by a stop-watch. As to the arrangement of the balls, care was taken that they should not be massed in one place, but scattered somewhat homogeneously over the space within the frames. The illumination was daylight, so managed that shadows cast by the balls were reduced to a minimum. The observer sat close to the table with the groups directly in front of him. He either kept his eyes closed between experiments or held a small screen before them. Sometimes he merely turned away. The operator worked from the opposite side of the table, taking care to make the necessary noises as little suggestive as possible. The observers agreed that they were not consciously influenced by the manipulation.

The progress of these experiments disclosed an astonishing space-error. So far as was conveniently possible the usual technique of elimination was employed.

1. The Influence of Active Pressure.

In this study the groups were differentiated in this way: With one hand the observer rolled the balls of one group under his fingers, while the other group was presented to vision only. The method of observation consisted in rapidly and lightly rolling the balls under the fingers a few times and then surveying both groups visually for the remainder of the exposure, judgment being given on the visual number.

Evidently there is much that is rough about this procedure. Pressure and kinæsthetic factors are lumped off together; the length of the touch-stimulus was not exactly determined; and there is the possibility that the visual stimulation from the group touched is weakened. To be sure the method prevents any great difference in the latter respect; and if we are guarded in our interpretation, something of interest may be learned.

There appeared to be no convenient way to eliminate the space-error. The right hand was used with the right group and the left with the left. So here again interpretation must be circumspect.

TABLE VI

Turning to the results in Table VI, A, we find the following: (1) The influence of the pressure-kinæsthetic complex practically does not appear; while the space-error shows a marked tendency that, for the two observers, is in opposite directions. (2) On the other hand, the notes of one observer show that in his case at least the face value of the table is erroneous. To this effect he says in substance that he can make a more accurate estimate of the number in the group touched. He tries to ignore these sensations of touch, but with ill success in the case of the left hand, where clumsiness not only makes it difficult to touch the balls gently but also to keep them under the fingers, which often feel the ground-space. For this cause the group seems small in number. Clearly enough, then, it is the space-error that tells the story of the effect of the added stimuli on this observer, only it must not be interpreted as space-error. The pressure functioned in the judgment through its numerical aspect. But the positive effect with the right hand was turned to a negative with the left through its emphasis of vacancies. The high difference-value in the space-column becomes thus a striking evidence of the effect of pressure, and the results are accounted for without reference to the kinæsthetic factor. The other observer felt that the active pressure was relatively indifferent. (3) The entire absence of correct judgments on the objectively equal groups shows to what a surprising extent other factors have modified the numerical.

2. The Influence of Unevenness in Active Pressure-Feeling.

In the experiments of this section the groups differed in this way, that one rested on the smooth table-top while the other had for its bottom a coarse wire mesh covered with black cloth. The balls of the one rolled smoothly beneath the fingers while the other balls moved lumpily over their mesh. Both hands were used—each for the group on its side; and the method of observation and length of exposure agreed with those conditions in the preceding section, except that the balls were rolled a little more vigorously that the factor studied might come clearly into consciousness. The groups were interchanged for half the number of series. This could not of course completely eliminate the space-error, since kinæsthetic differences in the limbs remained uncompensated. In general the criticism in the preceding section is again applicable.

The results appear in Table VI, B. (1) One observer shows a tendency to favor the smoothly rolling group, while the other again shows no tendency. Both have large space-errors of the same character as in A of this table. (2) The introspection of the observer showing no tendency is to the effect that touch plays little or no conscious part in the situation. The other's notes give no hint that the factor studied here was influential; but to the effect on the judgment of touch in general, especially with the right hand, they give clear witness. The touch-sensations, he says, were difficult to ignore. Those from the right hand were more vivid than those from the left; and the right hand seemed more sensitive. Judgment was based on a general feeling of "moreishness" which came promptly. There is nothing to contradict the evidence of the earlier experiments that touch is again influential through its numerical character. (3) Both observers regard factors of distribution as of fundamental importance, though one was inclined at first to insist that there was nothing but number in his judgment. The significance of this unanalyzed feeling will appear in a later section. (4) These results agree with the preceding in the approximate exclusion of correct judgments.

3. The Influence of Active Weight.

The variation here in question consisted in lifting one of the groups during judgment of the relative number in the two groups. The apparatus was made by transforming into trays the frames containing the balls, by putting into these frames wire-mesh bottoms covered with black cloth. They were set each upon four small wooden pillars so that the hand could be easily thrust under the tray. At the signal a given tray was several times raised a little way and lowered, and the judgment formed on the same factor as before. Here again the space-error was not entirely eliminated. Each hand was used with the group on its side, but kinæsthetic differences peculiar to each of the limbs remained. There was always some motion among the balls in the lifted tray, though the gentleness of the lifting prevented the existence of much. This is a radical defect, but one not easily avoided with maintenance of other desirable conditions. Even more serious, as the issue proved, was the failure to control the lifting impulse; yet, as it happens, we are not prevented from getting an experimental answer to our question.

The results are recorded in Table VI, C. (1) They show no apparent effect of the weight, and with one observer the further unusual fact of no space-error. This error is marked enough in the case of the other, and, conforming in direction to that of the preceding sections of this table, allows us in so far to adopt the same interpretation of his results. (2) The introspection of one observer was to the effect that he felt a tendency toward a modification of the number-judgment by weight. It was especially strong when the group was lighter or heavier than was anticipated, the light group seeming less numerous, and the heavy group more. Occasionally he caught himself weighing the second group mentally; and sometimes he had to recover himself from a tendency to make judgments on a wrong basis, presumably that of mere weight. With such a conflict of tendencies the character of the results is not surprising. Particularly important are the opposing tendencies lying in the factor of weight itself. The other observer reported that a very heavy weight exerted an influence that it was hard but not impossible to ignore, while a lighter weight did not effectively enter the situation at all. His earlier inclination to say that there was nothing but number in his judgment inclines one to believe that fusion of factors may have passed beyond the stage of ready analysis. (3) Our analysis has given us reason to believe that active weight has a definite tendency to modify the judgment of relative number.

4. The Influence of Muscular Strain in Observation.

The study was made from the point of view of more than one set of experimental conditions, viz.:

(1) Equal strain (minimum).
(a) Right—left.
(b) Up—down.
(2) Equal strain (maximum) eyes turned.
(3) Strain vs. ease.
(a) Head and eyes turned.
(b) Eyes turned.

The conditions of (1) (a) were exactly those of the earlier experiments with the exception that the groups were undistinguished save by position. In (1) (b) one group was so placed between the other and the observer that there might be as little increased effort as possible in viewing the farther. In the up-down movement more muscles are involved in the lift than in the fall of the eye. So really we have here a case of (3) but not so marked. In (2) the groups were put to the right and left at such distances that, when sitting between, the observer could just take each one in without turning his head. This brought a decided strain upon the eye-muscles. In (3) (a) the groups were separated by the length of the table—a distance of 90 cm.; and the observer placed in alternate series before each; as he was in (3) (b) where the farther group was carried to the limit of vision to be reached without turning the head. Here the strain was like that in (2), but for one group only.

An incompleteness in experimental analysis lies in the impossibility of separating the factors of distance and strain.

TABLE VII

Here are the facts of chief interest: (1) The following tabulation gives us a ready view of the character of the results in Table VII; and shows the extent to which they are consistent: