their direction. Of course, it must be admitted that some of these
illusions might be classified under those of "contrast" and others. In
fact, it has been stated that classification is difficult but it appears
logical to discuss the effect of angles in this chapter apart from the
divisions presented in the preceding chapters. This decision was reached
because the effect of angles could be seen in many of the illusions which
would more logically be grouped under the classification presented in the
preceding chapters.
[Illustration: 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.
[Illustration: Fig. 54.--The horizontal line appears to tilt downward
toward the ends.]
[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 Mueller-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 Mueller-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 Mueller-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 Mueller-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 Mueller-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.
[Illustration: Fig. 57.--Combined influence of angles and contrasting
lengths.]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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
analysis 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 space
compared with partially filled space, 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 advertisements.
[Illustration: 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 assumes, 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
Mueller-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 space 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
Mueller-Lyer figure appears divided into two unequal parts.
Somewhat analogous 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 analysis 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-homogeneous 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
homogeneity 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 assume 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
assumed, and if the assumptions 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 balls in the same
plane, but one appears larger than the other. With binocular vision the
balls 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 balls, 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 circumstances 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 class. 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-balls, and by stressing the eyelids.
However, these are not important and are merely mentioned in passing.
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 compass, 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 compass-points are very convenient for making a demonstration
of this pronounced illusion. The relation of size and distance easily
accounts for the illusion.
Obviously this type of illusion cannot be illustrated 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 classes
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 circumstances
attending visual illusions of depth are not the same for various cases,
and the illusions vary widely in their features.