[Illustration: Fig. 55.--The horizontal line appears to sag in the middle.] In Fig. 54 the horizontal straight line appears to consist of two lines tilting slightly upward toward the center. This will be seen to be in agreement with the general proposition that the sides of an angle are deviated in the direction of the angle. In this case it should be noted that one of the obtuse angles to be considered is _ABC_ and that the effect of this is to tilt the line _BD_ downward from the center. In Fig. 55 the horizontal line appears to tilt upward toward its extremities or to sag in the middle. The explanation in order to harmonize with the foregoing must be based upon the assumption that our judgments may be influenced by things not present but imagined. In this case only one side of each obtuse angle is present, the other side being formed by continuing the horizontal line both ways by means of the imagination. That we do this unconsciously is attested to by many experiences. For example, we often find ourselves imagining a horizontal, a vertical, or a center upon which to base a pending judgment. A discussion of the influence of angles must include a reference to the well-known Müller-Lyer illusion presented in Fig. 56. It is obvious in _a_ that the horizontal part on the left appears considerably longer than that part in the right half of the diagram. The influence of angles in this illusion can be easily tested by varying the direction of the lines at the ends of the two portions. [Illustration: Fig. 56.--The Müller-Lyer illusion.] 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.