course, the square has a greater area than the circle and it is difficult to determine the effect of this disparity in area. Figure 60 where the areas of the circle and square are equal and consequently the height of the former is considerably greater than the latter, is of interest in this connection. By experimenting with a series of pairs consisting of a circle and a square, varying in dimensions from equal heights to equal areas, an idea of the “shrinking” character of the circle becomes quite apparent.

Wundt does not attribute the illusion to a deception or error of judgment but to direct perception. According to his explanation, the laws of retinal image (fixation) and eye-movement are responsible. For example, vertical distances appear greater than horizontal ones because the effort or expenditure of energy is greater in raising the eyes than in turning them through an equal angle in a horizontal plane. Unconscious or involuntary eye-movements also appear to play a part in many linear or more accurately, angular illusions, but certainly Wundt’s explanation does not suffice for all illusions although it may explain many geometrical illusions. It may be said to be of the “perceptive” class and Lipps’ theory to be of the “judgment” or “higher-process” class. As already stated, most of the other proposed explanations of geometrical illusions may be regarded as being related to one of these two theories. There is the “indistinct vision” theory of Einthoven; the “perspective” theory of Hering, Guye, Thiéry, and others; the “contrast” theory of Helmholtz, Loeb, and Heyman; and the “contrast-confluxion” theory of Müller-Lyer. In order not to discourage the reader at the outset, theories as such will be passed by with this brief glimpse. However, more or less qualified explanations are presented occasionally in some of the chapters which follow in order to indicate or to suggest a train of thought should the reader desire to attempt to understand some of the numerous interesting illusions.