The first two numbers in the first line signify that when an open distance of 4 cm. was taken, an adjacent open distance of 4.7 cm. was judged equal; but when the adjacent space was filled, 5.3 cm. was judged equal. Each number in the column of filled distances represents an average of five judgments. All of the contacts in Table I. were made simultaneously; in Table II. they were made successively.
In the next series of experiments the illusion was approached from an entirely different point of view. The two points representing the open space were given on one arm, and the filled space on a symmetrical part of the other arm. I was now able to use a much wider range of distances, and made many variations in the weights of the points and the number that were taken for the filled distance.
However, before I began this second series, in which one of the chief variations was to be in the weights of the different points, I made a brief preliminary series of experiments to determine in a general way the influence of pressure on judgments of point distances. Only three distances were employed, four, six and twelve centimeters, and three weights, twelve, twenty and forty grams. Table III. shows that, for three men who were to serve as subjects in the main experiments that are to follow, an increase in the weight of the points was almost always accompanied by an increase in the apparent distance.
TABLE III.
| Distances. | 4 cm. | 6 cm. | 12 cm. | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Weights (Grams). | 12 | 20 | 40 | 12 | 20 | 40 | 12 | 20 | 40 |
| R. | 3.9 | 3.2 | 3.0 | 6.2 | 5.6 | 5.3 | 11.4 | 10.4 | 9.3 |
| F. | 4.3 | 4.0 | 3.6 | 6.1 | 5.3 | 5.5 | 12.3 | 11.6 | 10.8 |
| B. | 4.1 | 3.6 | 3.1 | 6.0 | 5.7 | 5.8 | 12.0 | 10.2 | 9.4 |
| P. | 4.3 | 4.1 | 3.7 | 5.9 | 5.6 | 5.6 | 13.1 | 11.9 | 10.7 |
In the standard distances the points were each weighted to 6 grams. The first three figures signify that a two-point distance of 4 cm., each point weighing 6 grams, was judged equal to 3.9 cm. when each point weighed 12 grams. 3.2 cm. when each point weighed 20 grams, etc. Each figure is the average of five judgments.
Now the application of this principle in my criticism of Parrish's experiments, and as anticipating the direction which the following experiments will take, is this: if we take a block such as Parrish used, with only two points in it, and weight it with forty grams in applying it to the skin, it is plain that each point will receive one half of the whole pressure, or twenty grams. But if we put a pressure of forty grams upon a block of eight points, each point will receive only one eighth of the forty, or five grams. Thus, in the case of the filled space, the end points, which play the most important part in the judgment of the distance, have each only five grams' pressure, while the points in the open space have each twenty grams. We should, therefore, naturally expect that the open space would be overestimated, because of the decided increase of pressure at these significant points. Parrish should have subjected the blocks, not to the same pressure, but to a pressure proportional to the number of points in each block. With my apparatus, I was easily able to prove the correctness of my position here. It will be seen in Tables IV. to VIII. that, when the sum of the weights of the two end points in the open space was only just equal to the sum of the weights of all the points in the filled space, the filled space was underestimated just as Parrish has reported. But when the points were all of the same weight, both in the filled and the open space, the filled space was judged longer in all but the very short distances. For this latter exception I shall offer an explanation presently.
Having now given an account of the results of this digression into experiments to determine the influence of pressure upon point distances, I shall pass to the second series of experiments on the illusion in question. In this series, as has been already stated, the filled space was taken on one arm and the open on the other, and then the process was reversed in order to eliminate any error arising from a lack of symmetry between the two regions. Without, for the present, going into a detailed explanation of the statistics of this second series of experiments, which are recorded in Tables IV., V., VI., VII. and VIII., I may summarize the salient results into these general conclusions: First, the short filled distance is underestimated; second, this underestimation of the filled space gradually decreases until in the case of the filled distance of 18 cm. the judgments pass over into pronounced overestimations; third, an increase in the number of points of contact in the shorter distances increases the underestimation, while an increase in the number of points in the longer distance increases the overestimation; fourth, an increase of pressure causes an invariable increase in the apparent length of space. If a general average were made of the results given in Tables IV., V., VI., VII. and VIII., there would be a preponderance of evidence for the conclusion that the filled spaces are overestimated. But we cannot ignore the marked tendencies in the opposite direction for the long and the short distances. These anomalous results, which, it will be remembered, were also found in our first series, call for explanation. Several hypotheses were framed to explain these fluctuations in the illusion, and then some shorter series of experiments were made in different directions with as large a number of variations in the conditions as possible, in the hope of discovering the disturbing factors.
TABLE IV.¹
4 Centimeters.