An Experimental Study of Angular Subtension

Abstract

The confusion arising over judgements of the sizes of angular forms is described. Three experiments are reported. In the first, the apparent-sizes of acute and obtuse angles are estimated. In the second, judgements of the sizes assumed by upwardand downward-and left-and right-facing angles are compared. The third experiment is concerned with tactile reproduction of angles of specified sizes. In discussion an attempt is made to clarify the logical and empirical problems involved.

Judgment of angle size: An experimental appraisal

“…This latter overall relationship remains constant, but the absolute differences are reduced at either end of the range, in comparison with the band of 47 to 135 deg. This corresponds to the position in Fisher's (1969) Fig. 4, with the differences in his case narrowed around 90 deg by the nonperceptual consequences of using 5-deg intervals and a free-estimation response mode (see also Chapanis, 1951).…”

“…A randomizing procedure determined whether the vertex was up/down, right/left, or a combination of these for diagonally presented angles. Fisher (1969) had reported that the direction of the vertex had no effect on judgments obtained in his situation.…”

“…Interest in the problem of describing and explaining the characteristic over-and underestimation of the size of angles has been renewed with papers by Beery (1968) and Fisher (1969). Until these studies, the conventional view, stemming from Helmholtz, Titchener, Wundt, Ladd, and Woodworth, that acute angles are overestimated and obtuse angles underestimated was generally accepted.…”

Constant errors occur in matched reproduction of angles even when likely biases are eliminated

“…The logical flaw in this procedure has been pointed out by Fisher (1969) (italics in original): "Suppose, for example, that when shown a 40°angle a subject judges it to be one of 45°. When attempting to reproduce it he will construct two lines which he believes to represent an angle of 45°.…”

Zöllner illusion as perceptual enlargement of acute angle

“…The law essentially expresses the amount of illusion as a function of the size relationships of the stimulus figure and predicts relative values for the qualitative characteristics of illusion functions. One instance of the law concerns the perceptual enlargement or overestimation of an acute angle (Wundt, 1896;Berliner & Berliner, 1948;Fisher, 1969). The overestimation of an acute angle (0) is given by…”