In all these figures the influence of angles is obvious. This does not mean that they are always solely or even primarily responsible for the illusion. In fact, the illusion of Poggendorff ([Fig. 46]) may be due to the incorrect estimation of certain linear distances, but the angles make this erroneous judgment possible, or at least contribute toward it. Many discussions of the theories or explanations of these figures are available in scientific literature of which one by Judd[2] may be taken as representative. He holds that the false estimation of angles in the Poggendorff figure is merely a secondary effect, not always present, and in no case the source of the illusion; furthermore, that the illusion may be explained as due to the incorrect linear distances, and may be reduced to the type of illusion found in the Müller-Lyer figure. Certainly there are grave dangers in explaining an illusion on the basis of an apparently simple operation.
In [Fig. 56], b is made up of the two parts of the Müller-Lyer illusion. A small dot may be placed equally distant from the inside extremities of the horizontal lines. It is interesting to note that overestimation of distance within the figure is accompanied with underestimation outside the figure and, conversely, overestimation within the figure is accompanied by underestimation in the neighboring space. If the small dot is objected to as providing an additional Müller-Lyer figure of the empty space, this dot may be omitted. As a substitute an observer may try to locate a point midway between the inside extremities of the horizontal lines. The error in locating this point will show that the illusion is present in this empty space.
In this connection it is interesting to note some other illusions. In [Fig. 57] the influence of several factors are evident. Two obviously important ones are (1) the angles made by the short lines at the extremities of the exterior lines parallel to the sides of the large triangle, and (2) the influence of contrast of the pairs of adjacent parallel lines. The effect shown in [Fig. 53] is seen to be augmented by the addition of contrast of adjacent lines of unequal length.
An interesting variation of the effect of the presence of angles is seen in [Fig. 58]. The two lines forming angles with the horizontal are of equal length but due to their relative positions, they do not appear so. It would be quite misleading to say that this illusion is merely due to angles. Obviously, it is due to the presence of the two oblique lines. It is of interest to turn to [Figs. 25], [26] and various illusions of perspective.
Fig. 57.—Combined influence of angles and contrasting lengths.
Fig. 58.—Two equal oblique lines appear unequal because of their different positions.