[Ill.u.s.tration: Fig. 53.--Angles affect the apparent length of lines.]

In Fig. 53 the three horizontal lines are of equal length but they appear unequal. This must be due primarily to the size of the angles made by the lines at the ends. Within certain limits, the greater the angle the greater is the apparent elongation of the central horizontal portion. This generalization appears to apply even when the angle is less than a right angle, although there appears to be less strength to the illusions with these smaller angles than with the larger angles. Other factors which contribute to the extent of the illusion are the positions of the figures, the distance between them, and the juxtaposition of certain lines. The illusion still exists if the horizontal lines are removed and also if the figures are cut out of paper after joining the lower ends of the short lines in each case.

[Ill.u.s.tration: Fig. 54.--The horizontal line appears to tilt downward toward the ends.]

[Ill.u.s.tration: 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 a.s.sumption 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 Muller-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.

[Ill.u.s.tration: Fig. 56.--The Muller-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 Muller-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 Muller-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 s.p.a.ce. If the small dot is objected to as providing an additional Muller-Lyer figure of the empty s.p.a.ce, this dot may be omitted. As a subst.i.tute 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 s.p.a.ce.

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.

[Ill.u.s.tration: Fig. 57.--Combined influence of angles and contrasting lengths.]

[Ill.u.s.tration: Fig. 58.--Two equal oblique lines appear unequal because of their different positions.]

At this point a digression appears to be necessary and, therefore, Fig. 59 is introduced. Here the areas of the two figures are equal. The judgment of area is likely to be influenced by juxtaposed lines and therefore, as in this case, the lower appears larger than the upper one. Similarly two trapezoids of equal dimensions and areas may be constructed. If each is constructed so that it rests upon its longer parallel and one figure is above the other and only slightly separated, the mind is tempted to be influenced by comparing the juxtaposed base of the upper with the top of the lower trapezoid. The former dimension being greater than the latter, the lower figure appears smaller than the upper one. Angles must necessarily play a part in these illusions, although it is admitted that other factors may be prominent or even dominant.

[Ill.u.s.tration: Fig. 59.--An illusion of area.]

This appears to be a convenient place to insert an illusion of area based, doubtless, upon form, but angles must play a part in the illusions; at least they are responsible for the form. In Fig. 60 the five figures are constructed so as to be approximately equal in area. However, they appear unequal in this respect. In comparing areas, we cannot escape the influence of the length and directions of lines which bound these areas, and also, the effect of contrasts in lengths and directions. Angles play a part in all these, although very indirectly in some cases.

[Ill.u.s.tration: Fig. 60.--Five equal areas showing the influence of angles and contrasting lengths.]

To some extent the foregoing is a digression from the main intent of this chapter, but it appears worth while to introduce these indirect effects of the presence of angles (real or imaginary) in order to emphasize the complexity of influences and their subtleness. Direction is in the last a.n.a.lysis an effect of angle; that is, the direction of a line is measured by the angle it makes with some reference line, the latter being real or imaginary. In Fig. 61, the effect of diverting or directing attention by some subtle force, such as suggestion, is demonstrated. This "force"

appears to contract or expand an area. The circle on the left appears smaller than the other. Of course there is the effect of empty s.p.a.ce compared with partially filled s.p.a.ce, but this cannot be avoided in this case. However, it can be shown that the suggestions produced by the arrows tend to produce apparent reduction or expansion of areas. Note the use of arrows in advertis.e.m.e.nts.

[Ill.u.s.tration: Fig. 61.--Showing the effect of directing the attention.]

Although theory is subordinated to facts in this book, a glimpse here and there should be interesting and helpful. After having been introduced to various types and influences, perhaps the reader may better grasp the trend of theories. The perspective theory a.s.sumes, and correctly so, that simple diagrams often suggest objects in three dimensions, and that the introduction of an imaginary third dimension effects changes in the appearance of lines and angles. That is, lengths and directions of lines are apparently altered by the influence of lines and angles, which do not actually exist. That this is true may be proved in various cases. In fact the reader has doubtless been convinced of this in connection with some of the illusions already discussed. Vertical lines often represent lines extending away from the observer, who sees them foreshortened and therefore they may seem longer than horizontal lines of equal length, which are not subject to foreshortening. This could explain such illusions as seen in Figs. 4 and 5. However this theory is not as easily applied to many illusions.

According to Thiery's perspective theory a line that appears nearer is seen as smaller and a line that seems to be further away is perceived as longer. If the left portion of _b_, Fig. 56, be reproduced with longer oblique lines at the ends but with the same length of horizontal lines, it will appear closer and the horizontal lines will be judged as shorter. The reader will find it interesting to draw a number of these portions of the Muller-Lyer figure with the horizontal line in each case of the same length but with longer and longer obliques at the ends.

The dynamic theory of Lipps gives an important role to the inner activity of the observer, which is not necessarily separated from the objects viewed, but may be felt as being in the objects. That is, in viewing a figure the observer unconsciously separates it from surrounding s.p.a.ce and therefore creates something definite in the latter, as a limiting activity. These two things, one real (the object) and one imaginary, are balanced against each other. A vertical line may suggest a necessary resistance against gravitational force, with the result that the line appears longer than a horizontal one resting in peace. The difficulty with this theory is that it allows too much opportunity for purely philosophical explanations, which are likely to run to the fanciful. It has the doubtful advantage of being able to explain illusions equally well if they are actually reversed from what they are. For example, gravity could either contract or elongate the vertical line, depending upon the choice of viewpoint.

The confusion theory depends upon attention and begins with the difficulty of isolating from illusory figures the portions to be judged. Amid the complexity of the figure the attention cannot easily be fixed on the portions to be judged. This results in confusion. For example, if areas of different shapes such as those in Fig. 60 are to be compared, it is difficult to become oblivious of form or of compactness. In trying to see the two chief parallel lines in Fig. 38, in their true parallelism the attention is being subjected to diversion, by the short oblique parallels with a compromising result. Surely this theory explains some illusions successfully, but it is not so successful with some of the illusions of contrast. The fact that practice in making judgments in such cases as Figs. 45 and 56 reduces the illusion even to ultimate disappearance, argues in favor of the confusion theory. Perhaps the observer devotes himself more or less consciously to isolating the particular feature to be judged and finally attains the ability to do so. According to Auerbach's indirect-vision theory the eyes in judging the two halves of the horizontal line in _a_, Fig. 56, involuntarily draw imaginary lines parallel to this line but above or below it. Obviously the two parts of such lines are unequal in the same manner as the horizontal line in the Muller-Lyer figure appears divided into two unequal parts.

Somewhat a.n.a.logous to this in some cases is Brunot's mean-distance theory.

According to this we establish "centers of gravity" in figures and these influence our judgments.

These are glimpses of certain trends of theories. None is a complete success or failure. Each explains some illusions satisfactorily, but not necessarily exclusively. For the present, we will be content with these glimpses of the purely theoretical aspects of visual illusions.

VII

ILLUSIONS OF DEPTH AND OF DISTANCE

Besides the so-called geometrical illusions discussed in the preceding chapters, there is an interesting group in which the perception of the third dimension is in error. When any of the ordinary criteria of relief or of distance are apparently modified, illusions of this kind are possible. There are many illusions of this sort, such as the looming of objects in a fog; the apparent enlargement of the sun and moon near the horizon; the flattening of the "vault" of the sky; the intaglio seen as relief; the alteration of relief with lighting; and various changes in the landscape when regarded with the head inverted.

Although some of the criteria for the perception of depth or of distance have already been pointed out, especially in Chapter III, these will be mentioned again. Distance or depth is indicated by the distribution of light and shade, and an unusual object like an intaglio is likely to be mistaken for relief which is more common. An a.n.a.lysis of the lighting will usually reveal the real form of the object. (See Figs. 70, 71, 72, 73, 76 and 77.) In this connection it is interesting to compare photographic negatives with their corresponding positive prints.

Distance is often estimated by the definition and color of objects seen through great depths of air (aerial perspective). These distant objects are "blurred" by the irregular refraction of the light-rays through non-h.o.m.ogeneous atmosphere. They are obscured to some degree by the veil of brightness due to the illuminated dust, smoke, etc., in the atmosphere.

They are also tinted (apparently) by the superposition of a tinted atmosphere. Thus we have "dim distance," "blue peaks," "azure depths of sky," etc., represented in photographs, paintings, and writings.

Incidentally, the sky above is blue for the same general reasons that the atmosphere, intervening between the observer and a distant horizon, is bluish. The ludicrous errors made in estimating distances in such regions as the Rockies is usually accounted for by the rare clearness and h.o.m.ogeneity of the atmosphere. However, is the latter a full explanation?

To some extent we judge unknown size by estimated distance, and unknown distance by estimated size. When a person is viewing a great mountain peak for the first time, is he not likely to a.s.sume it to be comparable in size to the hills with which he has been familiar? Even by allowing considerable, is he not likely to greatly underestimate the size of the mountain and, as a direct consequence, to underestimate the distance proportionately? This incorrect judgment would naturally be facilitated by the absence of "dimness" and "blueness" due to the atmospheric haze.

Angular perspective, which apparently varies the forms of angles and produces the divergence of lines, contributes much information in regard to relative and absolute distances from the eye of the various objects or the parts of an object. For example, a rectangle may appear as a rhomboid.

It is obvious that certain data pertaining to the objects viewed must be a.s.sumed, and if the a.s.sumptions are incorrect, illusions will result.

These judgments also involve, as most judgments do, other data external to the objects viewed. Perhaps these incorrect judgments are delusions rather than illusions, because visual perception has been deluded by misinformation supplied by the intellect.

Size or linear perspective is a factor in the perception of depth or of distance. As has been stated, if we know the size experience determines the distance; and conversely, if we know the distance we may estimate the size. Obviously estimates are involved and these when incorrect lead to false perception or interpretation.

As an object approaches, the axes of the eyes converge more and more and the eye-lens must be thickened more and more to keep the object in focus.

As stated in Chapter III, we have learned to interpret these accompanying sensations of muscular adjustment. This may be demonstrated by holding an object at an arm's length and then bringing it rapidly toward the eyes, keeping it in focus all the time. The sensations of convergence and accommodation are quite intense.

The two eyes look at a scene from two different points of view respectively and their images do not perfectly agree, as has been shown in Figs. 2 and 3. This binocular disparity is responsible to some degree for the perception of depth, as the stereoscope has demonstrated. If two spheres of the same size are suspended on invisible strings, one at six feet, the other at seven feet away, one eye sees the two b.a.l.l.s in the same plane, but one appears larger than the other. With binocular vision the b.a.l.l.s appear at different distances, but judgment appraises them as of approximately equal size. At that distance the focal adjustment is not much different for both b.a.l.l.s, so that the muscular movement, due to focusing the eye, plays a small part in the estimation of the relative distance. Binocular disparity and convergence are the primary factors.

Some have held that the perception of depth, that is, of a relative distance, arises from the process of unconsciously running the point of sight back and forth. However, this view, unmodified, appears untenable when it is considered that a scene illuminated by a lightning flash (of the order of magnitude of a thousandth of a second) is seen even in this brief moment to have depth. Objects are seen in relief, in actual relation as to distance and in normal perspective, even under the extremely brief illumination of an electric spark (of the order of magnitude of one twenty-thousandth of a second). This can also be demonstrated by viewing stereoscopic pictures with a stereoscope, the illumination being furnished by an electric spark. Under these circ.u.mstances relief and perspective are quite satisfactory. Surely in these brief intervals the point of sight cannot do much surveying of a scene.

Parallax aids in the perception of depth or distance. If the head be moved laterally the view or scene changes slightly. Objects or portions of objects previously hidden by others may now become visible. Objects at various distances appear to move nearer or further apart. We have come to interpret these apparent movements of objects in a scene in terms of relative distances; that is, the relative amount of parallactic displacement is a measure of the relative distances of the objects.

The relative distances or depth locations of different parts of an object can be perceived as fluctuating or even reversing. This is due to fluctuations in attention, and illusions of reversible perspective are of this cla.s.s. It is quite impossible for one to fix his attention in perfect continuity upon any object. There are many involuntary eye-movements which cannot be overcome and under normal conditions certain details are likely to occupy the focus of attention alternately or successively. This applies equally well to the auditory sense and perhaps to the other senses.

Emotional coloring has much to do with the fixation of attention; that which we admire, desire, love, hate, etc., is likely to dwell more in the focus of attention than that which stirs our emotions less.

A slight suggestion of forward and backward movements can be produced by successively intercepting the vision of one eye by an opaque card or other convenient object. It has been suggested that the illusion is due to the consequent variations in the tension of convergence. Third dimensional movements may be produced for binocular or monocular vision during eye-closure. They are also produced by opening the eyes as widely as possible, by pressure on the eye-b.a.l.l.s, and by stressing the eyelids.

However, these are not important and are merely mentioned in pa.s.sing.

An increase in the brightness of an object is accompanied by an apparent movement toward the observer, and conversely a decrease in brightness produces an apparent movement in the opposite direction. These effects may be witnessed upon viewing the glowing end of a cigar which is being smoked by some one a few yards away in the darkness. Rapidly moving thin clouds may produce such an effect by varying the brightness of the moon. Some peculiar impressions of this nature may be felt while watching the flashing light of some light-houses or of other signaling stations. It has been suggested that we naturally appraise brighter objects as nearer than objects less bright. However, is it not interesting to attribute the apparent movement to irradiation? (See Chapter VIII.) A bright object appears larger than a dark object of the same size and at the same distance. When the same object varies in brightness it remains in consciousness the same object and therefore of constant size; however, the apparent increase in size as it becomes brighter must be accounted for in some manner and there is only one way open. It must be attributed a lesser distance than formerly and therefore the sudden increase in brightness mediates a consciousness of a movement forward, that is, toward the observer.

If two similar objects, such as the points of a compa.s.s, are viewed binocularly and their lateral distance apart is altered, the observer is conscious of a third dimensional movement. Inasmuch as the accommodation is unaltered but convergence must be varied as the lateral distance between the two, the explanation of the illusion must consider the latter.

The pair of compa.s.s-points are very convenient for making a demonstration of this p.r.o.nounced illusion. The relation of size and distance easily accounts for the illusion.

Obviously this type of illusion cannot be ill.u.s.trated effectively by means of diagrams, so the reader must be content to watch for them himself. Some persons are able voluntarily to produce illusory movements in the third dimension, but such persons are rare. Many persons have experienced involuntary illusions of depth. Carr found, in a series of cla.s.ses comprising 350 students, 58 persons who had experienced involuntary depth illusions at some time in their lives. Five of these also possessed complete voluntary control over the phenomenon. The circ.u.mstances attending visual illusions of depth are not the same for various cases, and the illusions vary widely in their features.