Visual Illusions: Their Causes, Characteristics and Applications

VISUAL ILLUSIONS

THEIR CAUSES, CHARACTERISTICS
AND APPLICATIONS

BY
M. LUCKIESH

DIRECTOR OF APPLIED SCIENCE, NELA RESEARCH LABORATORIES,
NATIONAL LAMP WORKS OF GENERAL ELECTRIC CO.

AUTHOR OF “COLOR AND ITS APPLICATIONS,” “LIGHT AND SHADE
AND THEIR APPLICATIONS,” “THE LIGHTING ART,” “THE
LANGUAGE OF COLOR,” “ARTIFICIAL LIGHT—ITS
INFLUENCE UPON CIVILIZATION,”
“LIGHTING THE HOME,” ETC.

100 ILLUSTRATIONS

NEW YORK
D. VAN NOSTRAND COMPANY
EIGHT WARREN STREET
1922

Copyright, 1922, by
D. VAN NOSTRAND COMPANY

PREFACE

Eventually one of the results of application to the analysis and measurement of the phenomena of light, color, lighting, and vision is a firmly entrenched conviction of the inadequacy of physical measurements as a means for representing what is perceived. Physical measurements have supplied much of the foundation of knowledge and it is not a reflection upon their great usefulness to state that often they differ from the results of intellectual appraisal through the visual sense. In other words, there are numberless so-called visual illusions which must be taken into account. All are of interest; many can be utilized; and some must be suppressed.

Scientific literature yields a great many valuable discussions from theoretical and experimental viewpoints but much of the material is controversial. The practical aspects of visual illusions have been quite generally passed by and, inasmuch as there does not appear to be a volume available which treats the subject in a condensed manner but with a broad scope, this small volume is contributed toward filling the gap.

The extreme complexity of the subject is recognized, but an attempt toward simplicity of treatment has been made by confining the discussion chiefly to static visual illusions, by suppressing minor details, and by subordinating theory. In other words, the intent has been to emphasize experimental facts. Even these are so numerous that only the merest glimpses of various aspects can be given in order to limit the text to a small volume. Some theoretical aspects of the subject are still extremely controversial, so they are introduced only occasionally and then chiefly for the purpose of illustrating the complexities and the trends of attempted explanations. Space does not even admit many qualifications which may be necessary in order to escape criticism entirely.

The visual illusions discussed are chiefly of the static type, although a few others have been introduced. Some of the latter border upon motion, others upon hallucinations, and still others produced by external optical media are illusions only by extension of the term. These exceptions are included for the purpose of providing glimpses into the borderlands.

It is hoped that this condensed discussion, which is ambitious only in scope, will be of interest to the general reader, to painters, decorators, and architects, to lighting experts, and to all interested in light, color, and vision. It is an essential supplement to certain previous works.

M. LUCKIESH
November, 1920.

CONTENTS

LIST OF ILLUSTRATIONS

VISUAL ILLUSIONS

I
INTRODUCTION

Seeing is deceiving. Thus a familiar epigram may be challenged in order to indicate the trend of this book which aims to treat certain phases of visual illusions. In general, we do not see things as they are or as they are related to each other; that is, the intellect does not correctly interpret the deliverances of the visual sense, although sometimes the optical mechanism of the eyes is directly responsible for the illusion. In other words, none of our conceptions and perceptions are quite adequate, but fortunately most of them are satisfactory for practical purposes. Only a part of what is perceived comes through the senses from the object; the remainder always comes from within. In fact, it is the visual sense or the intellect which is responsible for illusions of the various types to be discussed in the following chapters. Our past experiences, associations, desires, demands, imaginings, and other more or less obscure influences create illusions.

An illusion does not generally exist physically but it is difficult in some cases to explain the cause. Certainly there are many cases of errors of judgment. A mistaken estimate of the distance of a mountain is due to an error of judgment but the perception of a piece of white paper as pink on a green background is an error of sense. It is realized that the foregoing comparison leads directly to one of the most controversial questions in psychology, but there is no intention on the author’s part to cling dogmatically to the opinions expressed. In fact, discussions of the psychological judgment involved in the presentations of the visual sense are not introduced with the hope of stating the final word but to give the reader an idea of the inner process of perception. The final word will be left to the psychologists but it appears possible that it may never be formulated.

In general, a tree appears of greater length when standing than when lying upon the ground. Lines, areas, and masses are not perceived in their actual physical relations. The appearance of a colored object varies considerably with its environment. The sky is not perceived as infinite space nor as a hemispherical dome, but as a flattened vault. The moon apparently diminishes in size as it rises toward the zenith. A bright object appears larger than a dark object of the same physical dimensions. Flat areas may appear to have a third dimension of depth. And so on.

Illusions are so numerous and varied that they have long challenged the interest of the scientist. They may be so useful or even so disastrous that they have been utilized or counteracted by the skilled artist or artisan. The architect and painter have used or avoided them. The stage-artist employs them to carry the audience in its imagination to other environments or to far countries. The magician has employed them in his entertainments and the camoufleur used them to advantage in the practice of deception during the recent war. They are vastly entertaining, useful, deceiving, or disastrous, depending upon the viewpoint.

Incidentally, a few so-called illusions will be discussed which are not due strictly to errors of the visual sense or of the intellect. Examples of these are the mirage and certain optical effects employed by the magician. In such cases neither the visual sense nor the intellect errs. In the case of the mirage rays of light coming from the object to the eye are bent from their usual straight-line course and the object appears to be where it really is not. However, with these few exceptions, which are introduced for their specific interest and for the emphasis they give to the “true” illusion, it will be understood that illusions in general as hereinafter discussed will mean those due to the visual mechanism or to errors of judgment or intellect. For the sake of brevity we might say that they are those due to errors of visual perception. Furthermore, only those of a “static” type will be considered; that is, the vast complexities due to motion are not of interest from the viewpoint of the aims of this book.

There are two well-known types of misleading perceptions, namely illusions and hallucinations. If, for example, two lines appear of equal length and are not, the error in judgment is responsible for what is termed an “illusion.” If the perceptual consciousness of an object appears although the object is not present, the result is termed an “hallucination.” For example, if something is seen which does not exist, the essential factors are supplied by the imagination. Shadows are often wrought by the imagination into animals and even human beings bent upon evil purpose. Ghosts are created in this manner. Hallucinations depend largely upon the recency, frequency, and vividness of past experience. A consideration of this type of misleading perception does not advance the aims of this book and therefore will be omitted.

The connection between the material and mental in vision is incomprehensible and apparently must ever remain so. Objects emit or reflect light and the optical mechanism known as the eye focuses images of the objects upon the retina. Messages are then carried to the brain where certain molecular vibrations take place. The physiologist records certain physical and chemical effects in the muscles, nerves, and brain and behold! there appears consciousness, sensations, thoughts, desires, and volitions. How? and, Why? are questions which may never be answered.

It is dangerous to use the word never, but the ultimate answers to those questions appear to be so remote that it discourages one from proceeding far over the hazy course which leads toward them. In fact, it does not appreciably further the aims of this book to devote much space to efforts toward explanation. In covering this vast and complex field there are multitudes of facts, many hypotheses, and numerous theories from which to choose. Judgment dictates that of the limited space most of it be given to the presentation of representative facts. This is the reasoning which led to the formulation of the outline of chapters.

Owing to the vast complex beyond the physical phenomena, physical measurements upon objects and space which have done so much toward building a solid foundation for scientific knowledge fail ultimately to provide an exact mathematical picture of that which is perceived. Much of the author’s previous work has been devoted to the physical realities but the ever-present differences between physical and perceptive realities have emphasized the need for considering the latter as well.

Illusions are legion. They greet the careful observer on every hand. They play a prominent part in our appreciation of the physical world. Sometimes they must be avoided, but often they may be put to work in various arts. Their widespread existence and their forcefulness make visual perception the final judge in decoration, in painting, in architecture, in landscaping, in lighting, and in other activities. The ultimate limitation of measurements with physical instruments leaves this responsibility to the intellect. The mental being is impressed with things as perceived, not with things as they are. It is believed that this intellectual or judiciary phase which plays such a part in visual perception will be best brought out by examples of various types of static illusions coupled with certain facts pertaining to the eye and to the visual process as a whole.

In special simple cases it is not difficult to determine when or how nearly a perception is true but in general, agreement among normal persons is necessary owing to the absence of any definite measuring device which will span the gap between the perception and the objective reality. Illusions are sometimes called “errors of sense” and some of them are such, but often they are errors of the intellect. The senses may deliver correctly but error may arise from imagination, inexperience, false assumptions, and incorrect associations, and the recency, frequency, and vividness of past experience. The gifts of sight are augmented by the mind with judgments based upon experience with these gifts.

The direct data delivered by the visual sense are light, intensity, color, direction. These may be considered as simple or elemental sensations because they cannot be further simplified or analyzed. At this point it is hoped that no controversy with the psychologist will be provoked. In the space available it appears unfruitful to introduce the many qualifications necessary to satisfy the, as yet uncertain or at least conflicting, definitions and theories underlying the science of psychology. If it is necessary to add darkness to the foregoing group of elemental visual sensations, this will gladly be agreed to.

The perceptions of outline-form and surface-contents perhaps rank next in simplicity; however, they may be analyzed into directions. The perception of these is so direct and so certain that it may be considered to be immediate. A ring of points is apparently very simple and it might be considered a direct sense-perception, but it consists of a number of elemental directions.

The perception of solid-form is far more complex than outline-form and therefore more liable to error. It is judged partially by binocular vision or perspective and partly by the distribution of light and shade. Colors may help to mold form and even to give depth to flat surfaces. For example, it is well known that some colors are “advancing” and others are “retiring.”

Perhaps of still greater complexity are the judgments of size and of distance. Many comparisons enter such judgments. The unconscious acts of the muscles of the eye and various external conditions such as the clearness of the atmosphere play prominent parts in influencing judgment. Upon these are superposed the numerous psycho-physiological phenomena of color, irradiation, etc.

In vision judgments are quickly made and the process apparently is largely outside of consciousness. Higher and more complex visual judgments pass into still higher and more complex intellectual judgments. All these may appear to be primary, immediate, innate, or instinctive and therefore, certain, but the fruits of studies of the psychology of vision have shown that these visual judgments may be analyzed into simpler elements. Therefore, they are liable to error.

That the ancients sensed the existence or possibility of illusions is evidenced by the fact that they tried to draw and to paint although their inability to observe carefully is indicated by the absence of true shading. The architecture of ancient Greece reveals a knowledge of certain illusions in the efforts to overcome them. However, the study of illusions did not engage the attention of scientists until a comparatively recent period. Notwithstanding this belated attention there is a vast scientific literature pertaining to the multitudinous phases of the subject; however, most of it is fragmentary and much of it is controversial. Some of it deals with theory for a particular and often a very simple case. In life complex illusions are met but at present it would be futile to attempt to explain them in detail. Furthermore, there have been few attempts to generalize and to group examples of typical phenomena in such a manner as to enable a general reader to see the complex fabric as a whole. Finally, the occurrence and application of illusions in various arts and the prominence of illusions on every hand have not been especially treated. It is the hope that this will be realized in the following chapters in so far as brevity of treatment makes this possible.

Doubtless thoughtful observers ages ago noticed visual illusions, especially those found in nature and in architecture. When it is considered that geometrical figures are very commonly of an illusory character it appears improbable that optical illusions could have escaped the keenness of Euclid. The apparent enlargement of the moon near the horizon and the apparent flattened vault of the sky were noticed at least a thousand years ago and literature yields several hundred memoirs on these subjects. One of the oldest dissertations upon the apparent form of the sky was published by Alhazen, an Arab astronomer of the tenth century. Kepler in 1618 wrote upon the subject.

Philosophers of the past centuries prepared the way toward an understanding of many complexities of today. They molded thought into correct form and established fundamental concepts and principles. Their chief tool was philosophy, the experimental attack being left to the scientists of the modern age. However, they established philosophically such principles as “space and time are not realities of the phenomenal world but the modes under which we see things apart.” As science became organized during the present experimental era, measurements were applied and there began to appear analytical discussions of various subjects including optical illusions. One of the earliest investigations of the modern type was made by Oppel, an account of which appeared in 1854. Since that time scientific literature has received thousands of worthy contributions dealing with visual illusions.

There are many facts affecting vision regarding which no theory is necessary. They speak for themselves. There are many equally obvious facts which are not satisfactorily explained but the lack of explanation does not prevent their recognition. In fact, only the scientist needs to worry over systematic explanations and theoretical generalizations. He needs these in order to invade and to explore the other unknowns where he will add to his storehouse of knowledge. A long step toward understanding is made by becoming acquainted with certain physical, physiological, and psychological facts of light, color, and lighting. Furthermore, acquaintance with the visual process and with the structure of the eye aids materially. For this reason the next two chapters have been added even at the risk of discouraging some readers.

In a broad sense, any visual perception which does not harmonize with physical measurements may be termed an “illusion.” Therefore, the term could include those physical illusions obtained by means of prisms, lenses, and mirrors and such illusions as the mirage. It could also include the physiological illusions of light and color such as after-images, irradiation, and contrast, and the psycho-physiological illusions of space and the character of objects. In fact, the scope of the following chapters is arbitrarily extended to include all these aspects, but confines consideration only to “static” illusions.

In a more common sense attention is usually restricted to the last group; that is, to the psycho-physiological illusions attending the perception of space and the character of objects although motion is often included. It should be obvious that no simple or even single theory can cover the vast range of illusions considered in the broad sense because there are so many different kinds of factors involved. For this reason explanations will be presented wherever feasible in connection with specific illusions. However, in closing this chapter it appears of interest to touch upon the more generally exploited theories of illusions of the type considered in the foregoing restricted sense. Hypotheses pertaining to illusions are generally lacking in agreement, but for the special case of what might be more safely termed “geometrical-optical illusions” two different theories, by Lipps and by Wundt respectively, are conspicuous. In fact, most theories are variants of these two systematic “explanations” of illusions (in the restricted sense).

Lipps proposed the principle of mechanical-esthetic unity, according to which we unconsciously give to every space-form a living personality and unconsciously consider certain mechanical forces acting. Our judgments are therefore modified by this anthropomorphic attitude. For example, we regard the circle as being the result of the action of tangential and radial forces in which the latter appear to triumph. According to Lipps’ theory the circle has a centripetal character and these radial forces toward the center, which apparently have overcome the tangential forces during the process of creating the circle, lead to underestimation of its size as compared with a square of the same height and breadth. By drawing a circle and square side by side, with the diameter of the former equal to the length of a side of the latter, this illusion is readily demonstrated. Of 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.

II
THE EYE

Helmholtz, who contributed so much toward our knowledge of the visual process, in referring to the eye, once stated that he could make a much better optical instrument but not a better eye. In other words, the eye is far from an ideal optical instrument but as an eye it is wonderful. Its range of sensitiveness and its adaptability to the extreme variety of demands upon it are truly marvelous when compared with instruments devised by mankind. Obviously, the eye is the connecting link between objective reality and visual perception and, therefore, it plays an important part in illusions. In fact, sometimes it is solely responsible for the illusion. The process of vision may be divided into several steps such as (1) the lighting, color, character, and disposition of objects; (2) the mechanism by which the image is formed upon the retina; (3) various optical defects of this mechanism; (4) the sensitiveness of the parts of the retina to light and color; (5) the structure of the retina; (6) the parts played by monocular and binocular vision; and (7) the various events which follow the formation of the image upon the retina.

The mechanism of the eye makes it possible to see not only light but objects. Elementary eyes of the lowest animals perceive light but cannot see objects. These eyes are merely specialized nerves. In the human eye the optic nerve spreads to form the retina and the latter is a specialized nerve. Nature has accompanied this evolution by developing an instrument the—eye—for intensifying and defining and the whole is the visual sense-organ. The latter contains the most highly specialized nerve and the most refined physiological mechanism, the result being the highest sense-organ.

Fig. 1.—Principal parts of the eye.

A, Conjunctiva; B, Retina; C, Choroid; D, Sclera;
E, Fovea; F, Blind Spot; G, Optic Nerve;
H, Ciliary Muscle; I, Iris; J, Cornea; K, Ligament.

The eye is approximately a spherical shell transparent at the front portion and opaque (or nearly so) over the remaining eighty per cent of its surface. The optical path consists of a series of transparent liquids and solids. The chief details of the structure of the eye are represented in [Fig. 1]. Beginning with the exterior and proceeding toward the retina we find in succession the cornea, the anterior chamber containing the aqueous humor, the iris, the lens, the large chamber containing the vitreous humor, and finally the retina. Certain muscles alter the position of the eye and consequently the optical axis, and focusing (accommodation) is accomplished by altering the thickness and shape, and consequently the focal length, of the lens.

The iris is a shutter which automatically controls to some degree the amount of light reaching the retina, thereby tending to protect the latter from too much light. It also has some influence upon the definition of the image; that is, upon what is termed “visual acuity” or the ability to distinguish fine detail. It is interesting to compare the eye with the camera. In the case of the camera and the photographic process, we have (1) an inverted light-image, a facsimile of the object usually diminished in size; (2) an invisible image in the photographic emulsion consisting of molecular changes due to light; and (3) a visible image developed on the plate. In the case of the eye and the visual process we have (1) an inverted light-image, a facsimile of the object diminished in size; (2) the invisible image in the retinal substances probably consisting of molecular changes due to light; and (3) an external visible image. It will be noted that in the case of vision the final image is projected outward—it is external. The more we think of this outward projection the more interesting and marvelous vision becomes. For example, it appears certain that if a photographic plate could see or feel, it would see or feel the silver image upon itself but not out in space. However, this point is discussed further in the next chapter.

In the camera and photographic process we trace mechanism, physics, and chemistry throughout. In the eye and visual process we are able to trace these factors only to a certain point, where we encounter the super-physical and super-chemical. Here molecular change is replaced by sensation, perception, thought, and emotion. Our exploration takes us from the physical world into another, wholly different, where there reigns another order of phenomena. We have passed from the material into the mental world.

The eye as an optical mechanism is reducible to a single lens and therefore the image focused upon the retina is inverted. However, there is no way for the observer to be conscious of this and therefore the inverted image causes no difficulty in seeing. The images of objects in the right half of the field of view are focused upon the left half of the retina. Similarly, the left half of the field of view corresponds to the right half of the retina; the upper half of the former to the lower half of the latter; and so on. When a ray of light from an object strikes the retina the impression is referred back along the ray-line into the original place in space. This is interestingly demonstrated in a simple manner. Punch a pin-hole in a card and hold it about four inches from the eye and at the same time hold a pin-head as close to the cornea as possible. The background for the pin-hole should be the sky or other bright surface. After a brief trial an inverted image of the pin-head is seen in the hole. Punch several holes in the card and in each will be seen an inverted image of the pin-head.

The explanation of the foregoing is not difficult. The pin-head is so close to the eye that the image cannot be focused upon the retina; however, it is in a very favorable position to cast a shadow upon the retina, the light-source being the pin-hole with a bright background. Light streaming through the pin-hole into the eye casts an erect shadow of the pin-head upon the retina, and this erect image is projected into space and inverted in the process by the effect of the lens. The latter is not operative during the casting of the shadow because the pin-head is too close to the lens, as already stated. It is further proved to be outward projection of the retinal image (the shadow) because by multiplying the number of pin-holes (the light-sources) there are also a corresponding number of shadows.

The foregoing not only illustrates the inversion of the image but again emphasizes the fact that we do not see retinal images. Even the “stars” which we see on pressing the eye-lid or on receiving a blow on the eye are projected into space. The “motes” which we see in the visual field while gazing at the sky are defects in the eye-media, and these images are projected into space. We do not see anything in the eye. The retinal image impresses the retina in some definite manner and the impression is carried to the brain by the optic nerve. The intellect then refers or projects this impression outward into space as an external image. The latter would be a facsimile of the physical object if there were no illusions but the fact that there are illusions indicates that errors are introduced somewhere along the path from and to the object.

It is interesting to speculate whether the first visual impression of a new-born babe is “projected outward” or is perceived as in the eye. It is equally futile to conjecture in this manner because there is no indication that the time will come when the baby can answer us immediately upon experiencing its first visual impression. The period of infancy increases with progress up the scale of animal life and this lengthening is doubtless responsible and perhaps necessary for the development of highly specialized sense-organs. Incidentally, suppose a blind person to be absolutely uneducated by transferred experience and that he suddenly became a normal adult and able to see. What would he say about his first visual impression? Apparently such a subject is unobtainable. The nearest that such a case had been approached is the case of a person born blind, whose sight has been restored. This person has acquired much experience with the external world through other senses. It has been recorded that such a person, after sight was restored, appeared to think that external objects “touched” the eyes. Only through visual experience is this error in judgment rectified.

Man studies his kind too much apart from other animals and perhaps either underestimates or overestimates the amount of inherited, innate, instinctive qualities. A new-born chick in a few minutes will walk straight to an object and seize it. Apparently this implies perception of distance and direction and a coördination of muscles for walking and moving the eyes. It appears reasonable to conclude that a certain amount of the wealth of capacities possessed by the individual is partly inherited, and in man the acquired predominates. But all capacities are acquired, for even the inherited was acquired in ancestral experience. Even instinct (whatever that may be) must involve inherited experience. These glimpses of the depths to which one must dig if he is to unearth the complete explanations of visual perception—and consequently of illusions—indicate the futility of treating the theories in the available space without encroaching unduly upon the aims of this volume.

Certain defects of the optical system of the eye must contribute toward causing illusions. Any perfect lens of homogeneous material has at least two defects, known as spherical and chromatic aberration. The former manifests itself by the bending of straight lines and is usually demonstrated by forming an image of an object such as a wire mesh or checkerboard; the outer lines of the image are found to be very much bent. This defect in the eye-lens is somewhat counteracted by a variable optical density, increasing from the outer to the central portion. This results in an increase in refractive-index as the center of the lens is approached and tends to diminish its spherical aberration. The eye commonly possesses abnormalities such as astigmatism and eccentricity of the optical elements. All these contribute toward the creation of illusions.

White light consists of rays of light of various colors and these are separated by means of a prism because the refractive-index of the prism differs for lights of different color or wave-length. This causes the blue rays, for example, to be bent more than the red rays when traversing a prism. It is in this manner that the spectrum of light may be obtained. A lens may be considered to be a prism of revolution and it thus becomes evident that the blue rays will be brought to a focus at a lesser distance than the red rays; that is, the former are bent more from their original path than the latter. This defect of lenses is known as chromatic aberration and is quite obvious in the eye. It may be demonstrated by any simple lens, for the image of the sun, for example, will appear to have a colored fringe. A purple filter which transmits only the violet and red rays is useful for this demonstration. By looking at a lamp-filament or candle-flame some distance away the object will appear to have a violet halo, but the color of the fringe will vary with accommodation. On looking through a pin-hole at the edge of an object silhouetted against the bright sky the edge will appear red if the light from the pin-hole enters the pupil near its periphery. This optical defect of the eye makes objects appear more sharply defined when viewed in monochromatic light. In fact, this is quite obvious when using yellow glasses. The defect is also demonstrated by viewing a line-spectrum focused on a ground glass. The blue and red lines cannot be seen distinctly at the same distance. The blue lines can be focused at a much less distance than the red lines. Chromatic aberration can account for such an illusion as the familiar “advancing” and “retiring” colors and doubtless it plays a part in many illusions.

The structure of the retina plays a very important part in vision and accounts for various illusions and many interesting visual phenomena. The optic nerve spreads out to form the retina which constitutes the inner portion of the spherical shell of the eye with the exception of the front part. Referring again to [Fig. 1], the outer coating of the shell is called the sclerotic. This consists of dense fibrous tissue known as the “white of the eye.” Inside this coating is a layer of black pigment cells termed the choroid. Next is the bacillary layer which lines about five-sixths of the interior surface of the eye. This is formed by closely packed “rods” and “cones,” which play a dominant role in the visual process. A light-sensitive liquid (visual purple) and cellular and fibrous layers complete the retinal structure.

The place where the optic nerve enters the eye-ball and begins to spread out is blind. Objects whose images fall on this spot are invisible. This blind-spot is not particularly of interest here, but it may be of interest to note its effect. This is easily done by closing one eye and looking directly at one of two small black circles about two inches apart on white paper at a distance of about a foot from the eye. By moving the objects about until the image of the circle not directly looked at falls upon the blind-spot, this circle will disappear. A three-foot circle at a distance of 36 feet will completely disappear if its image falls directly upon the blind-spot. At a distance of 42 inches the invisible area is about 12 inches from the point of sight and about 3 to 4 inches in diameter. At 300 feet the area is about 8 feet in diameter. The actual size of the retinal blind-spot is about 0.05 inch in diameter or nearly 5 degrees. Binocular vision overcomes any annoyance due to the blind-spots because they do not overlap in the visual field. A one-eyed person is really totally blind for this portion of the retina or of the visual field.

The bacillary layer consists of so-called rods and cones. Only the rods function under very low intensities of illumination of the order of moonlight. The cones are sensitive to color and function only at intensities greater than what may be termed twilight intensities. These elements are very small but the fact that they appear to be connecting links between the retinal image and visual perception, acuity or discrimination of fine detail is limited inasmuch as the elements are of finite dimensions. The smallest image which will produce a visual impression is the size of the end of a cone. The smallest distance between two points which is visible at five inches is about 0.001 inch. Two cones must be stimulated in such a case. Fine lines may appear crooked because of the irregular disposition of these elemental light-sensitive points. This apparent crookedness of lines is an illusion which is directly due to the limitations of retinal elements of finite size.

The distribution of rods and cones over the retina is very important. In the fovea centralis—the point of the retina on the optical axis of the eye—is a slight depression much thinner than the remainder of the retina and this is inhabited chiefly by cones. It is this spot which provides visual acuteness. It is easily demonstrated that fine detail cannot be seen well defined outside this central portion of the visual field. When we desire to see an object distinctly we habitually turn the head so that the image of the object falls upon the fovea of each eye. Helmholtz has compared the foveal and lateral images with a finished drawing and a rough sketch respectively.

The fovea also contains a yellow pigmentation which makes this area of the retina selective as to color-vision. On viewing certain colors a difference in color of this central portion of the field is often very evident. In the outlying regions of the retina, rods predominate and in the intermediate zone both rods and cones are found. Inasmuch as rods are not sensitive to color and cones do not function at low intensities of illumination it is obvious that visual impressions should vary, depending upon the area of the retina stimulated. In fact, many interesting illusions are accounted for in this manner, some of which are discussed later.

It is well known that a faint star is seen best by averted vision. It may be quite invisible when the eye is directed toward it, that is, when its image falls upon the rod-free fovea. However, by averting the line of sight slightly, the image is caused to fall on a retinal area containing rods (sensitive to feeble light) and the star may be readily recognized. The fovea is the point of distinct focus. It is necessary for fixed thoughtful attention. It exists in the retina of man and of higher monkeys but it quickly disappears as we pass down the scale of animal life. It may be necessary for the safety of the lower animals that they see equally well over a large field; however, it appears advantageous that man give fixed and undivided attention to the object looked at. Man does not need to trust solely to his senses to protect himself from dangers. He uses his intellect to invent and to construct artificial defenses. Without the highly specialized fovea we might see equally well over the whole retina but could not look attentively at anything, and therefore could not observe thoughtfully.

When an image of a bright object exists upon the retina for a time there results a partial exhaustion or fatigue of the retinal processes with a result that an after-image is seen. This after-image may be bright for a time owing to the fact that it takes time for the retinal process to die out. Then there comes a reaction which is apparent when the eye is directed toward illuminated surfaces. The part of the retina which has been fatigued does not respond as fully as the fresher areas, with the result that the fatigued area contributes a darker area in the visual field. This is known as an after-image and there are many interesting variations.

The after-image usually undergoes a series of changes in color as well as in brightness as the retinal process readjusts itself. An after-image of a colored object may often appear of a color complementary to the color of the object. This is generally accounted for by fatigue of the retinal process. There are many conflicting theories of color-vision but they are not as conflicting in respect to the aspect of fatigue as in some other aspects. If the eye is directed toward a green surface for a time and then turned toward a white surface, the fatigue to green light diminishes the extent of response to the green rays in the light reflected by the white surface. The result is the perception of a certain area of the white surface (corresponding to the portion of the field fatigued by green light) as of a color equal to white minus some green—the result of which is pink or purple. This is easily understood by referring to the principles of color-mixture. Red, green, and blue (or violet) mixed in proper proportions will produce any color or tint and even white. Thus these may be considered to be the components of white light. Hence if the retina through fatigue is unable to respond fully to the green component, the result may be expressed mathematically as red plus blue plus reduced green, or synthetically a purplish white or pink. When fatigued to red light the after-image on a white surface is blue-green. When fatigued to blue light it is yellowish.

Further mixtures may be obtained by directing the after-image upon colored surfaces. In this manner many of the interesting visual phenomena and illusions associated with the viewing of colors are accounted for. The influence of a colored environment upon a colored object is really very great. This is known as simultaneous contrast. The influence of the immediately previous history of the retina upon the perception of colored surfaces is also very striking. This is called successive contrast. It is interesting to note that an after-image produced by looking at a bright light-source, for example, is projected into space even with the eyes closed. It is instructive to study after-images and this may be done at any moment. On gazing at the sun for an instant and then looking away, an after-image is seen which passes in color from green, blue, purple, etc., and finally fades. For a time it is brighter than the background which may conveniently be the sky. On closing the eyes and placing the hands over them the background now is dark and the appearance of the after-image changes markedly. There are many kinds, effects, and variations of after-images, some of which are discussed in other chapters.

As the intensity of illumination of a landscape, for example, decreases toward twilight, the retina diminishes in sensibility to the rays of longer wave-lengths such as yellow, orange, and red. Therefore, it becomes relatively more sensitive to the rays of shorter wave-length such as green, blue, and violet. The effects of this Purkinje phenomenon (named after the discoverer) may be added to the class of illusions treated in this book. It is interesting to note in this connection that moonlight is represented on some paintings and especially on the stage as greenish blue in color, notwithstanding that physical measurements show it to be approximately the color of sunlight. In fact, it is sunlight reflected by dead, frigid, and practically colorless matter.

Some illusions may be directly traced to the structure of the eye under unusual lighting conditions. For example, in a dark room hold a lamp obliquely outward but near one eye (the other being closed and shielded) and forward sufficiently for the retina to be strongly illuminated. Move the lamp gently while gazing at a plain dark surface such as the wall. Finally the visual field appears dark, due to the intense illumination of the retina and there will appear, apparently projected upon the wall, an image resembling a branching leafless tree. These are really shadows of the blood vessels in the retina. The experiment is more successful if an image of a bright light-source is focused on the sclerotic near the cornea. If this image of the light-source is moved, the tree-like image seen in the visual field will also move.

The rate of growth and decay of various color-sensations varies considerably. By taking advantage of this fact many illusions can be produced. In fact, the careful observer will encounter many illusions which may be readily accounted for in this manner.

It may be said that in general the eyes are never at rest. Involuntary eye-movements are taking place all the time, at least during consciousness. Some have given this restlessness a major part in the process of vision but aside from the correctness of theories involving eye-movements, it is a fact that they are responsible for certain illusions. On a star-lit night if one lies down and looks up at a star the latter will be seen to appear to be swimming about more or less jerkily. On viewing a rapidly revolving wheel of an automobile as it proceeds down the street, occasionally it will be seen to cease revolving momentarily. These apparently are accounted for by involuntary eye-movements which take place regardless of the effort made to fixate vision.

If the eyelids are almost closed, streamers appear to radiate in various directions from a light-source. Movements of the eyelids when nearly closed sometimes cause objects to appear to move. These may be accounted for perhaps by the distortion of the moist film which covers the cornea.

The foregoing are only a few of the many visual phenomena due largely to the structure of the eye. The effects of these and many others enter into visual illusions, as will be seen here and there throughout the chapters which follow.

III
VISION

A description of the eye by no means suffices to clarify the visual process. Even the descriptions of various phenomena in the preceding chapter accomplish little more than to acquaint the reader with the operation of a mechanism, although they suggest the trend of the explanations of many illusions. At best only monocular vision has been treated, and it does not exist normally for human beings. A person capable only of monocular vision would be like Cyclops Polyphemus. We might have two eyes, or even, like Argus, possess a hundred eyes and still not experience the wonderful advantages of binocular vision, for each eye might see independently. The phenomena of binocular vision are far less physical than those of monocular vision. They are much more obscure, illusory, and perplexing because they are more complexly interwoven with or allied to psychological phenomena.

The sense of sight differs considerably from the other senses. The sense of touch requires solid contact (usually); taste involves liquid contact; smell, gaseous contact; and hearing depends upon a relay of vibrations from an object through another medium (usually air), resulting finally in contact. However, we perceive things at a distance through vibration (electromagnetic waves called light) conveyed by a subtle, intangible, universal medium which is unrecognizable excepting as a hypothetically necessary bearer of light-waves or, more generally, radiant energy.

It also is interesting to compare the subjectiveness and objectiveness of sensations. The sensation of taste is subjective; it is in us, not in the body tasted. In smell we perceive the sensation in the nose and by experience refer it to an object at a distance. The sensation of hearing is objective; that is, we refer the cause to an object so completely that there is practically no consciousness of sensation in the ear. In sight the impression is so completely projected outward into space and there is so little consciousness of any occurrence in the eye that it is extremely difficult to convince ourselves that it is essentially a subjective sensation. The foregoing order represents the sense-organs in increasing specialization and refinement. In the two higher senses—sight and hearing—there is no direct contact with the object and an intricate mechanism is placed in front of the specialized nerve to define and to intensify the impression. In the case of vision this highly developed instrument makes it possible to see not only light but objects.

As we go up the scale of vertebrate animals we find that there is a gradual change of the position of the eyes from the sides to the front of the head and a change of the inclination of the optical axes of the two eyes from 180 degrees to parallel. There is also evident a gradual increase in the fineness of the bacillary layer of the retina from the margins toward the center, and, therefore, an increasing accuracy in the perception of form. This finally results in a highly organized central spot or fovea which is possessed only by man and the higher monkeys. Proceeding up the scale we also find an increasing ability to converge the optic axes on a near point so that the images of the point may coincide with the central spots of both retinas. These changes and others are closely associated with each other and especially with the development of the higher faculties of the mind.

Binocular vision in man and in the higher animals is the last result of the gradual improvement of the most refined sense-organ, adapting it to meet the requirements of highly complex organisms. It cannot exist in some animals, such as birds and fishes, because they cannot converge their two optical axes upon a near point. When a chicken wishes to look intently at an object it turns its head and looks with one eye. Such an animal sees with two eyes independently and possibly moves them independently. The normal position of the axes of human eyes is convergent or parallel but it is possible to diverge the axes. In fact, with practice it is possible to diverge the axes sufficiently to look at a point near the back of the head, although, of course, we do not see the point.

The movement of the eyes is rather complex. When they move together to one side or the other or up and down in a vertical plane there is no rotation of the optical axes; that is, no torsion. When the visual plane is elevated and the eyes move to the right they rotate to the right; when they move to the left they rotate to the left. When the visual plane is depressed and the eyes move to the right they rotate to the left; when they move to the left they rotate to the right. Through experience we unconsciously evaluate the muscular stresses, efforts, and movements accompanying the motion of the eyes and thereby interpret much through visual perception in regard to such aspects of the external world as size, shape, and distance of objects. Even this brief glimpse of the principal movements of the eyes indicates a complexity which suggests the intricacy of the explanations of certain visual phenomena.

At this point it appears advantageous to set down the principal modes by which we perceive the third dimension of space and of objects and other aspects of the external world. They are as follows: (1) extent; (2) clearness of brightness and color as affected by distance; (3) interference of near objects with those more distant; (4) elevation of objects; (5) variation of light and shade on objects; (6) cast shadows; (7) perspective; (8) variation of the visor angle in proportion to distance; (9) muscular effort attending accommodation of the eye; (10) stereoscopic vision; (11) muscular effort attending convergence of the axes of the eyes. It will be recognized that only the last two are necessarily concerned with binocular vision. These varieties of experiences may be combined in almost an infinite variety of proportions.

Wundt in his attempt to explain visual perception considered chiefly three factors: (1) the retinal image of the eye at rest; (2) the influence of the movements of one eye; and, (3) the additional data furnished by the two eyes functioning together. There are three fields of vision corresponding to the foregoing. These are the retinal field of vision, the monocular field, and the binocular field. The retinal field of vision is that of an eye at rest as compared with the monocular field, which is all that can be seen with one eye in its entire range of movement and therefore of experience. The retinal field has no clearly defined boundaries because it finally fades at its indefinite periphery into a region where sensation ceases.

It might be tiresome to follow detailed analyses of the many modes by which visual perception is attained, so only a few generalizations will be presented. For every voluntary act of sight there are two adjustments of the eyes, namely, focal and axial. In the former case the ciliary muscle adjusts the lens in order to produce a defined image upon the retina. In axial adjustments the two eyes are turned by certain muscles so that their axes meet on the object looked at and the images of the object fall on the central-spots of the retina. These take place together without distinct volition for each but by the single voluntary act of looking. Through experience the intellect has acquired a wonderful capacity to interpret such factors as size, form, and distance in terms of the muscular movements in general without the observer being conscious of such interpretations.

Binocular vision is easily recognized by holding a finger before the eyes and looking at a point beyond it. The result is two apparently transparent fingers. An object is seen single when the two retinal images fall on corresponding points. Direction is a primary datum of sense. The property of corresponding points of the two retinas (binocular vision) and consequently of identical spatial points in the two visual fields is not so simple. It is still a question whether corresponding points (that is, the existence of a corresponding point in one retina for each point in the other retina) are innate, instinctive, and are antecedent of experience or are “paired” as the result of experience. The one view results in the nativistic, the other in the empiristic theory. Inasmuch as some scientists are arrayed on one side and some on the other, it appears futile to dwell further upon this aspect. It must suffice to state that binocular vision, which consists of two retinas and consequently two fields of view absolutely coördinated in some manner in the brain, yields extensive information concerning space and its contents.

After noting after-images, motes floating in the field of view (caused by defects in the eye-media) and various other things, it is evident that what we call the field of view is the external projection into space of retinal states. All the variations of the latter, such as images and shadows which are produced in the external field of one eye, are faithfully reproduced in the external field of the other eye. This sense of an external visual field is ineradicable. Even when the eyes are closed the external field is still there; the imagination or intellect projects it outward. Objects at different distances cannot be seen distinctly at the same time but by interpreting the eye-movements as the point of sight is run backward and forward (varying convergence of the axes) the intellect practically automatically appraises the size, form, and distance of each object. Obviously, experience is a prominent factor. The perception of the third dimension, depth or relative distance, whether in a single object or a group of objects, is the result of the successive combination of the different parts of two dissimilar images of the object or group.

As already stated, the perception of distance, size, and form is based partly upon monocular and partly upon binocular vision, and the simple elements upon which judgments of these are based are light, shade, color, intensity, and direction. Although the interpretation of muscular adjustments plays a prominent part in the formation of judgments, the influences of mathematical perspective, light, shade, color, and intensity are more direct. Judgments based upon focal adjustment (monocular) are fairly accurate at distances from five inches to several yards. Those founded upon axial adjustment (convergence of the two axes in binocular vision) are less in error than the preceding ones. They are reliable to a distance of about 1000 feet. Judgments involving mathematical perspective are of relatively great accuracy without limits. Those arrived at by interpreting aerial perspective (haziness of atmosphere, reduction in color due to atmospheric absorption, etc.) are merely estimates liable to large errors, the accuracy depending largely upon experience with local conditions.

The measuring power of the eye is more liable to error when the distances or the objects compared lie in different directions. A special case is the comparison of a vertical distance with a horizontal one. It is not uncommon to estimate a vertical distance as much as 25 per cent greater than an actually equal horizontal distance. In general, estimates of direction and distance are comparatively inaccurate when only one eye is used although a one-eyed person acquires unusual ability through a keener experience whetted by necessity. A vertical line drawn perpendicular to a horizontal one is likely to appear bent when viewed with one eye. Its apparent inclination is variable but has been found to vary from one to three degrees. Monocular vision is likely to cause straight lines to appear crooked, although the “crookedness” may seem to be more or less unstable.

The error in the estimate of size is in reality an error in the estimation of distance except in those cases where the estimate is based directly upon a comparison with an object of supposedly known size. An amusing incident is told of an old negro who was hunting for squirrels. He shot several times at what he supposed to be a squirrel upon a tree-trunk and his failure to make a kill was beginning to weaken his rather ample opinion of his skill as a marksman. A complete shattering of his faith in his skill was only escaped by the discovery that the “squirrel” was a louse upon his eyebrow. Similarly, a gnat in the air might appear to be an airplane under certain favorable circumstances. It is interesting to note that the estimated size of the disk of the sun or moon varies from the size of a saucer to that of the end of a barrel, although a pine tree at the horizon-line may be estimated as 25 feet across despite the fact that it may be entirely included in the disk of the sun setting behind it.

Double images play an important part in the comparison of distances of objects. The “doubling” of objects is only equal to the interocular distance. Suppose two horizontal wires or clotheslines about fifty feet away and one a few feet beyond the other. On looking at these no double images are visible and it is difficult or even impossible to see which is the nearer when the points of attachment of the ends are screened from view. However, if the head is turned to one side and downward (90 degrees) so that the interocular line is now at right angles (vertical) to the horizontal lines, the relative distances of the latter are brought out distinctly. Double images become visible in the latter case.

According to Brücke’s theory the eyes are continuously in motion and the observer by alternately increasing or decreasing the convergence of the axes of the eyes, combines successively the different parts of the two scenes as seen by the two eyes and by running the point of sight back and forth by trial obtains a distinct perception of binocular perspective or relief or depth of space. It may be assumed that experience has made the observer proficient in this appraisal which he arrives at almost unconsciously, although it may be just as easy to accept Wheatstone’s explanation. In fact, some experiences with the stereoscope appear to support the latter theory.

Wheatstone discovered that the dissimilar pictures of an object or scene, when united by means of optical systems, produce a visual effect similar to that produced by the actual solid object or scene provided the dissimilarity is the same as that between two retinal images of the solid object or scene. This is the principle upon which the familiar stereoscope is founded. Wheatstone formulated a theory which may be briefly stated as follows: In viewing a solid object or a scene two slightly dissimilar retinal images are formed in the two eyes respectively, but the mind completely fuses them into one “mental” image. When this mental fusion of the two really dissimilar retinal images is complete in this way, it is obvious that there cannot exist a mathematical coincidence. The result is a perception of depth of space, of solidity, of relief. In fact the third dimension is perceived. A stereoscope accomplishes this in essentially the same manner, for two pictures, taken from two different positions respectively corresponding to the positions of the eyes, are combined by means of optical systems into one image.

Lack of correct size and position of the individual elements of stereoscopic pictures are easily detected on combining them. That is, their dissimilarity must exactly correspond to that between two views of an object or scene from the positions of the two eyes respectively ([Fig. 2]). This fact has been made use of in detecting counterfeit notes. If two notes made from the same plate are viewed in a stereoscope and the identical figures are combined, the combination is perfect and the plane of the combined images is perfectly flat. If the notes are not made from the same plate but one of them is counterfeit, slight variations in the latter are unavoidable. Such variations will show themselves in a wavy surface.

The unwillingness of the visual sense to combine the two retinal images, if they are dissimilar to the extent of belonging to two different objects, is emphasized by means of colors. For example, if a green glass is placed over one eye and a red glass over the other, the colors are not mixed by the visual sense. The addition of these two colors results normally in yellow, with little or no suggestion of the components—red and green. But in the foregoing case the visual field does not appear of a uniform yellow. It appears alternately red and green, as though the colors were rivaling each other for complete mastery. In fact, this phenomenon has been termed “retinal rivalry.”

The lenses of the stereoscope supplement eye-lenses and project on the retina two perfect images of a near object, although the eyes are looking at a distant object and are therefore not accommodated for the near one (the photographs). The lenses enlarge the images similar to the action of a perspective glass. This completes the illusion of an object or of a scene. There is a remarkable distinctness of the perception of depth of space and therefore a wonderful resemblance to the actual object or scene. It is interesting to note the effect of taking the two original photographs from distances separated by several feet. The effect is apparently to magnify depth. It is noteworthy that two pictures taken from an airplane at points fifty feet or so apart, when combined in the stereoscope, so magnify the depth that certain enemy-works can be more advantageously detected than from ordinary photographs.

Stereoscopic images such as represented in [Fig. 2] may be combined without the aid of the stereoscope if the optical axes of the eye can be sufficiently converged or diverged. Such images or pictures are usually upon a card and are intended to be combined beyond the plane of the card, for it is in this position that the object or scene can be perceived in natural perspective, of natural size, of natural form, and at natural distance. But in combining them the eyes are looking at a distant object and the axes are parallel or nearly so. Therefore, the eyes are focally adjusted for a distant object but the light comes from a very near object—the pictures on the card. Myopic eyes do not experience this difficulty and it appears that normal vision may be trained to overcome it. Normal eyes are aided by using slightly convex lenses. Such glasses supplement the lenses of the eye, making possible a clear vision of a near object while the eyes are really looking far away or, in other words, making possible a clear image of a near object upon the retina of the unadjusted eye. Stereoscopic pictures are usually so mounted that “identical points” on the two pictures are farther apart than the interocular distance and therefore the two images cannot be combined when the optical axes of the eyes are parallel or nearly so, which is the condition when looking at a distant object. In such a case the two pictures must be brought closer together.

Fig. 2.—Stereoscopic pictures for combining by converging or diverging the optical axes.

Fig. 3.—Stereoscopic pictures.

In [Figs. 2] and [3] are found “dissimilar” drawings of the correct dissimilarity of stereoscopic pictures. It is interesting and instructive to practice combining these with the unaided eyes. If [Fig. 2] is held at an arm’s length and the eyes are focused upon a point several inches distant, the axes will be sufficiently converged so that the two images are superposed. It may help to focus the eyes upon the tip of a finger until the stereoscopic images are combined. In this case of converging axes the final combined result will be the appearance of a hollow tube or of a shell of a truncated cone, apparently possessing the third dimension and being perceived as apparently smaller than the actual pictures in the background at arm’s length. If the two stereoscopic pictures are combined by looking at a point far beyond the actual position of [Fig. 2], the combined effect is a solid truncated cone but perceived as of about the same size and at about the same distance from the eye as the actual diagrams. In the latter case the smaller end of the apparent solid appears to be nearer than the larger end, but in the former case the reverse is true, that is, the smaller end appears to be at a greater distance. The same experiments may be performed for [Fig. 3] with similar results excepting that this appears to be a shell under the same circumstances that [Fig. 2] appears to be a solid and vice versa. A few patient trials should be rewarded by success, and if so the reader can gain much more understanding from the actual experiences than from description.

The foregoing discussion of vision should indicate the complexity of the visual and mental activities involved in the discrimination, association, and interpretation of the data obtained through the eye. The psychology of visual perception is still a much controverted domain but it is believed that the glimpses of the process of vision which have been afforded are sufficient to enable the reader to understand many illusions and at least to appreciate more fully those whose explanations remain in doubt. Certainly these glimpses and a knowledge of the information which visual perception actually supplies to us at any moment should convince us that the visual sense has acquired an incomparable facility for interpreting the objective world for us. Clearness of vision is confined to a small area about the point of sight, and it rapidly diminishes away from this point, images becoming dim and double. We sweep this point of sight backward and forward and over an extensive field of view, gathering all the distinct impressions into one mental image. In doing this the unconscious interpretation of the muscular activity attending accommodation and convergence of the eyes aids in giving to this mental picture the appearance of depth by establishing relative distances of various objects. Certainly the acquired facility is remarkable.

IV
SOME TYPES OF GEOMETRICAL ILLUSIONS

No simple classification of illusions is ample or satisfactory, for there are many factors interwoven. For this reason no claims are made for the various divisions of the subject represented by and in these chapters excepting that of convenience. Obviously, some divisions are necessary in order that the variegated subject may be presentable. The classification used appears to be logical but very evidently it cannot be perfectly so when the “logic” is not wholly available, owing to the disagreement found among the explanations offered by psychologists. It may be argued that the “geometrical” type of illusion should include many illusions which are discussed in other chapters. Indeed, this is perhaps true. However, it appears to suit the present purpose to introduce this phase of this book by a group of illusions which involve plane geometrical figures. If some of the latter appear in other chapters, it is because they seem to border upon or to include other factors beyond those apparently involved in the simple geometrical type. The presentation which follows begins (for the sake of clearness) with a few representative geometrical illusions of various types.

The Effect of the Location in the Visual Field.—One of the most common illusions is found in the letter “S” or figure “8.” Ordinarily we are not strongly conscious of a difference in the size of the upper and lower parts of these characters; however, if we invert them

the difference is seen to be large. The question arises, Is the difference due fundamentally to the locations of the two parts in the visual field? It scarcely seems credible that visual perception innately appraises the upper part larger than the lower, or the lower smaller than the upper part when these small characters are seen in their accustomed position. It appears to be possible that here we have examples of the effect of learning or experience and that our adaptive visual sense has become accustomed to overlook the actual difference. That is, for some reason through being confronted with this difference so many times, the intellect has become adapted to it and, therefore, has grown to ignore it. Regardless of the explanation, the illusion exists and this is the point of chief interest. For the same reason the curvature of the retina does not appear to account for illusion through distortion of the image, because the training due to experience has caused greater difficulties than this to disappear. We must not overlook the tremendous “corrective” influence of experience upon which visual perception for the adult is founded. If we have learned to “correct” in some cases, why not in all cases which we have encountered quite generally?

Fig. 4.—The vertical line appears longer than the equal horizontal line in each case.

This type of illusion persists in geometrical figures and may be found on every hand. A perfect square when viewed vertically appears too high, although the illusion does not appear to exist in the circle. In [Fig. 4] the vertical line appears longer than the horizontal line of the same length. This may be readily demonstrated by the reader by means of a variety of figures. A striking case is found in [Fig. 5], where the height and the width of the diagram of a silk hat are equal. Despite the actual equality the height appears to be much greater than the width. A pole or a tree is generally appraised as of greater length when it is standing than when it lies on the ground. This illusion may be demonstrated by placing a black dot an inch or so above another on a white paper. Now, at right angles to the original dot place another at a horizontal distance which appears equal to the vertical distance of the first dot above the original. On turning the paper through ninety degrees or by actual measurement, the extent of the illusion will become apparent. By doing this several times, using various distances, this type of illusion becomes convincing.

Fig. 5.—The vertical dimension is equal to the horizontal one, but the former appears greater.

The explanation accepted by some is that more effort is required to raise the eyes, or point of sight, through a certain vertical distance than through an equal horizontal distance. Perhaps we unconsciously appraise effort of this sort in terms of distance, but is it not logical to inquire why we have not through experience learned to sense the difference between the relation of effort to horizontal distance and that of effort to vertical distance through which the point of sight is moved? We are doing this continuously, so why do we not learn to distinguish; furthermore, we have overcome other great obstacles in developing our visual sense. In this complex field of physiological psychology questions are not only annoying, but often disruptive.

As has been pointed out in Chapter II, images of objects lying near the periphery of the visual field are more or less distorted, owing to the structure and to certain defects of parts of the eye. For example, a checkerboard viewed at a proper distance with respect to its size appears quite distorted in its outer regions. Cheap cameras are likely to cause similar errors in the images fixed upon the photographic plate. Photographs are interesting in connection with visual illusions, because of certain distortions and of the magnification of such aspects as perspective. Incidentally in looking for illusions, difficulty is sometimes experienced in seeing them when the actual physical truths are known; that is, in distinguishing between what is actually seen and what actually exists. The ability to make this separation grows with practice but where the difficulty is obstinate, it is well for the reader to try observers who do not suspect the truth.

Illusions of Interrupted Extent.—Distance and area appear to vary in extent, depending upon whether they are filled or empty or are only partially filled. For example, a series of dots will generally appear longer overall than an equal distance between two points. This may be easily demonstrated by arranging three dots in a straight line on paper, the two intervening spaces being of equal extent, say about one or two inches long. If in one of the spaces a series of a dozen dots is placed, this space will appear longer than the empty space. However, if only one dot is placed in the middle of one of the empty spaces, this space now is likely to appear of less extent than the empty space. (See [Fig. 7].) A specific example of this type of illusion is shown in [Fig. 6]. The filled or divided space generally appears greater than the empty or undivided space, but certain qualifications of this statement are necessary. In a the divided space unquestionably appears greater than the empty space. Apparently the filled or empty space is more important than the amount of light which is received from the clear spaces, for a black line on white paper appears longer than a white space between two points separated a distance equal to the length of the black line. Furthermore, apparently the spacing which is the most obtrusive is most influential in causing the divided space to appear greater for a than for b. The illusion still persists in c.

Fig. 6.—The divided or filled space on the left appears longer than the equal space on the right.

An idea of the magnitude may be gained from certain experiments by Aubert. He used a figure similar to a [Fig. 6] containing a total of five short lines. Four of them were equally spaced over a distance of 100 mm. corresponding to the left half of a, [Fig. 6]. The remaining line was placed at the extreme right and defined the limit of an empty space also 100 mm. long. In all cases, the length of the empty space appeared about ten per cent less than that of the space occupied by the four lines equally spaced. Various experimenters obtain different results, and it seems reasonable that the differences may be accounted for, partially at least, by different degrees of unconscious correction of the illusion. This emphasizes the desirability of using subjects for such experiments who have no knowledge pertaining to the illusion.

Fig. 7.—The three lines are of equal length.

Fig. 8.—The distance between the two circles on the left is equal to the distance between the outside edges of the two circles on the right.

As already stated there are apparent exceptions to any simple rule, for, as in the case of dots cited in a preceding paragraph, the illusion depends upon the manner in which the division is made. For example, in [Fig. 7], a and c are as likely to appear shorter than b as equal to it. It has been concluded by certain investigators that when subdivision of a line causes it to appear longer, the parts into which it is divided or some of them are likely to appear shorter than isolated lines of the same length. The reverse of this statement also appears to hold. For example in [Fig. 7], a appears shorter than b and the central part appears lengthened, although the total line appears shortened. This illusion is intensified by leaving the central section blank. A figure of this sort can be readily drawn by the reader by using short straight lines in place of the circles in [Fig. 8]. In this figure the space between the inside edges of the two circles on the left appears larger than the overall distance between the outside edges of the two circles on the right, despite the fact that these distances are equal. It appears that mere intensity of retinal stimulation does not account for these illusions, but rather the figures which we see.

Fig. 9.—Three squares of equal dimensions which appear different in area and dimension.

In [Fig. 9] the three squares are equal in dimensions but the different characters of the divisions cause them to appear not only unequal, but no longer squares. In [Fig. 10] the distance between the outside edges of the three circles arranged horizontally appears greater than the empty space between the upper circle and the left-hand circle of the group.

Fig. 10.—The vertical distance between the upper circle and the left-hand
one of the group is equal to the overall length of the group of three circles.

Illusions of Contour.—The illusions of this type, or exhibiting this influence, are quite numerous. In [Fig. 11] there are two semicircles, one closed by a diameter, the other unclosed. The latter appears somewhat flatter and of slightly greater radius than the closed one. Similarly in [Fig. 12] the shorter portion of the interrupted circumference of a circle appears flatter and of greater radius of curvature than the greater portions. In [Fig. 13] the length of the middle space and of the open-sided squares are equal. In fact there are two uncompleted squares and an empty “square” between, the three of which are of equal dimensions. However the middle space appears slightly too high and narrow; the other two appear slightly too low and broad. These figures are related to the well-known Müller-Lyer illusion illustrated in [Fig. 56]. Some of the illusions presented later will be seen to involve the influence of contour. Examples of these are [Figs. 55] and [60]. In the former, the horizontal base line appears to sag; in the latter, the areas appear unequal, but they are equal.

Fig. 13.—Three incomplete but equal squares.

Illusions of Contrast.—Those illusions due to brightness contrast are not included in this group, for “contrast” here refers to lines, angles and areas of different sizes. In general, parts adjacent to large extents appear smaller and those adjacent to small extents appear larger. A simple case is shown in [Fig. 14], where the middle sections of the two lines are equal, but that of the shorter line appears longer than that of the longer line. In [Fig. 15] the two parts of the connecting line are equal, but they do not appear so. This illusion is not as positive as the preceding one and, in fact, the position of the short vertical dividing line may appear to fluctuate considerably.

Fig. 14.—Middle sections of the two lines are equal.

Fig. 15.—An effect of contrasting areas (Baldwin’s figure).

[Fig. 16] might be considered to be an illusion of contour, but the length of the top horizontal line of the lower figure being apparently less than that of the top line of the upper figure is due largely to contrasting the two figures. Incidentally, it is difficult to believe that the maximum horizontal width of the lower figure is as great as the maximum height of the figure. At this point it is of interest to refer to other contrast illusions such as [Figs. 20], [57], and [59].

Fig. 16.—An illusion of contrast.

A striking illusion of contrast is shown in [Fig. 17], where the central circles of the two figures are equal, although the one surrounded by the large circles appears much smaller than the other. Similarly, in [Fig. 18] the inner circles of b and c are equal but that of b appears the larger. The inner circle of a appears larger than the outer circle of b, despite their actual equality.

Fig. 17.—Equal circles which appear unequal due to contrast (Ebbinghaus’ figure).

Fig. 18.—Equal circles appearing unequal owing to contrasting concentric circles.

In [Fig. 19] the circle nearer the apex of the angle appears larger than the other. This has been presented as one reason why the sun and moon appear larger at the horizon than when at higher altitudes. This explanation must be based upon the assumption that we interpret the “vault” of the sky to meet at the horizon in a manner somewhat similar to the angle but it is difficult to imagine such an angle made by the vault of the sky and the earth’s horizon. If there were one in reality, it would not be seen in profile.

Fig. 19.—Circles influenced by position within an angle.

Fig. 20.—Contrasting angles.

If two angles of equal size are bounded by small and large angles respectively, the apex in each case being common to the inner and two bounding angles, the effect of contrast is very apparent, as seen in [Fig. 20]. In [Fig. 57] are found examples of effects of lines contrasted as to length.

Fig. 21.—Owing to perspective the right angles appear oblique and vice versa.

The reader may readily construct an extensive variety of illusions of contrast; in fact, contrast plays a part in most geometrical-optical illusions. The contrasts may be between existing lines, areas, etc., or the imagination may supply some of them.

Fig. 22.—Two equal diagonals which appear unequal.

Illusions of Perspective.—As the complexity of figures is increased the number of possible illusions is multiplied. In perspective we have the influences of various factors such as lines, angles, and sometimes contour and contrast. In [Fig. 21] the suggestion due to the perspective of the cube causes right angles to appear oblique and oblique angles to appear to be right angles. This figure is particularly illusive. It is interesting to note that even an after-image of a right-angle cross when projected upon a wall drawn in perspective in a painting will appear oblique.

Fig. 23.—Apparent variations in the distance between two parallel lines.

A striking illusion involving perspective, or at least the influence of angles, is shown in [Fig. 22]. Here the diagonals of the two parallelograms are of equal length but the one on the right appears much smaller. That AX is equal in length to AY is readily demonstrated by describing a circle from the center A and with a radius equal to AX. It will be found to pass through the point Y. Obviously, geometry abounds in geometrical-optical illusions.

Fig. 24.—A striking illusion of perspective.

The effect of contrast is seen in a in [Fig. 23]; that is, the short parallel lines appear further apart than the pair of long ones. By adding the oblique lines at the ends of the lower pair in b, these parallel lines now appear further apart than the horizontal parallel lines of the small rectangle.

The influence of perspective is particularly apparent in [Fig. 24], where natural perspective lines are drawn to suggest a scene. The square columns are of the same size but the further one, for example, being apparently the most distant and of the same physical dimensions, actually appears much larger. Here is a case where experience, allowing for a diminution of size with increasing distance, actually causes the column on the right to appear larger than it really is. The artist will find this illusion even more striking if he draws three human figures of the same size but similarly disposed in respect to perspective lines. Apparently converging lines influence these equal figures in proportion as they suggest perspective.

Fig. 25.—Distortion of a square due to superposed lines.

Although they are not necessarily illusions of perspective, [Figs. 25] and [26] are presented here because they involve similar influences. In [Fig. 25] the hollow square is superposed upon groups of oblique lines so arranged as to apparently distort the square. In [Fig. 26] distortions of the circumference of a circle are obtained in a similar manner.

Fig. 26.—Distortion of a circle due to superposed lines.

It is interesting to note that we are not particularly conscious of perspective, but it is seen that it has been a factor in the development of our visual perception. In proof of this we might recall the first time as children we were asked to draw a railroad track trailing off in the distance. Doubtless, most of us drew two parallel lines instead of converging ones. A person approaching us is not sensibly perceived to grow. He is more likely to be perceived all the time as of normal size. The finger held at some distance may more than cover the object such as a distant person, but the finger is not ordinarily perceived as larger than the person. Of course, when we think of it we are conscious of perspective and of the increase in size of an approaching object. When a locomotive or automobile approaches very rapidly, this “growth” is likely to be so striking as to be generally noticeable. The reader may find it of interest at this point to turn to illustrations in other chapters.

The foregoing are a few geometrical illusions of representative types. These are not all the types of illusions by any means and they are only a few of an almost numberless host. These have been presented in a brief classification in order that the reader might not be overwhelmed by the apparent chaos. Various special and miscellaneous geometrical illusions are presented in later chapters.

V
EQUIVOCAL FIGURES

Many figures apparently change in appearance owing to fluctuations in attention and in associations. A human profile in intaglio ([Figs. 72] and [73]) may appear as a bas-relief. Crease a card in the middle to form an angle and hold it at an arm’s length. When viewed with one eye it can be made to appear open in one way or the other; that is, the angle may be made to appear pointing toward the observer or away from him. The more distant part of an object may be made to appear nearer than the remaining part. Plane diagrams may seem to be solids. Deception of this character is quite easy if the light-source and other extraneous factors are concealed from the observer. It is very interesting to study these fluctuating figures and to note the various extraneous data which lead us to judge correctly. Furthermore, it becomes obvious that often we see what we expect to see. For example, we more commonly encounter relief than intaglio; therefore, we are likely to think that we are looking at the former.

Proper consideration of the position of the dominant light-source and of the shadows will usually provide the data for a correct conclusion. However, habit and probability are factors whose influence is difficult to overcome. Our perception is strongly associated with accustomed ways of seeing objects and when the object is once suggested it grasps our mind completely in its stereotyped form. Stairs, glasses, rings, cubes, and intaglios are among the objects commonly used to illustrate this type of illusion. In connection with this type, it is well to realize how tenaciously we cling to our perception of the real shapes of objects. For example, a cube thrown into the air in such a manner that it presents many aspects toward us is throughout its course a cube.

Fig. 27.—Illustrating fluctuation of attention.

The figures which exhibit these illusions are obviously those which are capable of two or more spatial relations. The double interpretation is more readily accomplished by monocular than by binocular vision. [Fig. 27] consists of identical patterns in black and white. By gazing upon this steadily it will appear to fluctuate in appearance from a white pattern upon a black background to a black pattern upon a white background. Sometimes fluctuation of attention apparently accounts for the change and, in fact, this can be tested by willfully altering the attention from a white pattern to a black one. Incidentally one investigator found that the maximum rate of fluctuation was approximately equal to the pulse rate, although no connection between the two was claimed. It has also been found that inversion is accompanied by a change in refraction of the eye.

Fig. 28.—The grouping of the circles fluctuates.

Another example is shown in [Fig. 28]. This may appear to be white circles upon a black background or a black mesh upon a white background. However, the more striking phenomenon is the change in the grouping of the circles as attention fluctuates. We may be conscious of hollow diamonds of circles, one inside the other, and then suddenly the pattern may change to groups of diamonds consisting of four circles each. Perhaps we may be momentarily conscious of individual circles; then the pattern may change to a hexagonal one, each “hexagon” consisting of seven circles—six surrounding a central one. The pattern also changes into parallel strings of circles, triangles, etc.

Fig. 29.—Crossed lines which may be interpreted in two ways.

The crossed lines in [Fig. 29] can be seen as right angles in perspective with two different spatial arrangements of one or both lines. In fact there is quite a tendency to see such crossed lines as right angles in perspective. The two groups on the right represent a simplified Zöllner’s illusion ([Fig. 37]). The reader may find it interesting to spend some time viewing these figures and in exercising his ability to fluctuate his attention. In fact, he must call upon his imagination in these cases. Sometimes the changes are rapid and easy to bring about. At other moments he will encounter an aggravating stubbornness. Occasionally there may appear a conflict of two appearances simultaneously in the same figure. The latter may be observed occasionally in [Fig. 30]. Eye-movements are brought forward by some to aid in explaining the changes.

Fig. 30.—Reversible cubes.

In [Fig. 30] a reversal of the aspect of the individual cubes or of their perspective is very apparent. At rare moments the effect of perspective may be completely vanquished and the figure be made to appear as a plane crossed by strings of white diamonds and zigzag black strips.

The illusion of the bent card or partially open book is seen in [Fig. 31]. The tetrahedron in [Fig. 32] may appear either as erect on its base or as leaning backward with its base seen from underneath.

The series of rings in [Fig. 33] may be imagined to form a tube such as a sheet-metal pipe with its axis lying in either of two directions. Sometimes by closing one eye the two changes in this type of illusion are more readily brought about. It is also interesting to close and open each eye alternately, at the same time trying to note just where the attention is fixed.

The familiar staircase is represented in [Fig. 34]. It is likely to appear in its usual position and then suddenly to invert. It may aid in bringing about the reversal to insist that one end of a step is first nearer than the other and then farther away. By focusing the attention in this manner the fluctuation becomes an easy matter to obtain.

33.—Reversible perspective of a group of rings or of a tube.

Fig. 34.—Schröder’s reversible staircase.

In [Fig. 35] is a similar example. First one part will appear solid and the other an empty corner, then suddenly both are reversed. However, it is striking to note one half changes while the other remains unchanged, thus producing momentarily a rather peculiar figure consisting of two solids, for example, attached by necessarily warped surfaces.

Fig. 35.—Thiéry’s figure.

Perhaps the reader has often witnessed the striking illusion of some portraits which were made of subjects looking directly at the camera or painter. Regardless of the position of the observer the eyes of the portrait appear to be directed toward him. In fact, as the observer moves, the eyes in the picture follow him so relentlessly as to provoke even a feeling of uncanniness. This fact is accounted for by the absence of a third dimension, for a sculptured model of a head does not exhibit this feature. Perspective plays a part in some manner, but no attempt toward explanation will be made.

In [Fig. 36] are two sketches of a face. One appears to be looking at the observer, but the other does not. If the reader will cover the lower parts of the two figures, leaving only the two pairs of eyes showing, both pairs will eventually appear to be looking at the observer. Perhaps the reader will be conscious of mental effort and the lapse of a few moments before the eyes on the left are made to appear to be looking directly at him. Although it is not claimed that this illusion is caused by the same conditions as those immediately preceding, it involves attention. At least, it is fluctuating in appearance and therefore is equivocal. It is interesting to note the influence of the other features (below the eyes). The perspective of these is a powerful influence in “directing” the eyes of the sketch.

In the foregoing only definite illusions have been presented which are universally witnessed by normal persons. There are no hallucinatory phases in the conditions or causes. It is difficult to divide these with definiteness from certain illusions of depth as discussed in Chapter VII. The latter undoubtedly are sometimes entwined to some extent with hallucinatory phases; in fact, it is doubtful if they are not always hallucinations to some degree. Hallucinations are not of interest from the viewpoint of this book, but illusions of depth are treated because they are of interest. They are either hallucinations or are on the border-line between hallucinations and those illusions which are almost universally experienced by normal persons under similar conditions. The latter statement does not hold for illusions of depth in which objects may be seen alternately near and far, large and small, etc., although they are not necessarily pure hallucinations as distinguished from the types of illusions regarding which there is general perceptual agreement.

[Larger Image]

Fig. 36.—Illustrating certain influences upon the apparent direction of vision.
By covering all but the eyes the latter appear to be drawn alike in both sketches.

In explanation of the illusory phenomena pertaining to such geometrical figures as are discussed in the foregoing paragraphs, chiefly two different kinds of hypotheses have been offered. They are respectively psychological and physiological, although there is more or less of a mixture of the two in most attempts toward explanation. The psychological hypotheses introduce such factors as attention, imagination, judgment, and will. Hering and also Helmholtz claim that the kind of inversion which occurs is largely a matter of chance or of volition. The latter holds that the perception of perspective figures is influenced by imagination or the images of memory. That is, if one form of the figure is vividly imagined the perception of it is imminent. Helmholtz has stated that, “Glancing at a figure we observe spontaneously one or the other form of perspective and usually the one that is associated in our memory with the greatest number of images.”

The physiological hypotheses depend largely upon such factors as accommodation and eye-movement. Necker held to the former as the chief cause. He has stated that the part of the figure whose image lies near the fovea is estimated as nearer than those portions in the peripheral regions of the visual field. This hypothesis is open to serious objections. Wundt contends that the inversion is caused by changes in the points and lines of fixation. He says, “The image of the retina ought to have a determined position if a perspective illusion is to appear; but the form of this illusion is entirely dependent on motion and direction.” Some hypotheses interweave the known facts of the nervous system with psychological facts but some of these are examples of a common anomaly in theorization, for facts plus facts do not necessarily result in a correct theory. That is, two sets of facts interwoven do not necessarily yield an explanation which is correct.

VI
THE INFLUENCE OF ANGLES

As previously stated, no satisfactory classification of visual illusions exists, but in order to cover the subject, divisions are necessary. For this reason the reader is introduced in this chapter to the effects attending the presence of angles. By no means does it follow that this group represents another type, for it really includes many illusions of several types. The reason for this grouping is that angles play an important part, directly or indirectly, in the production of illusions. For a long time many geometrical illusions were accounted for by “overestimation” or “underestimation” of angles, but this view has often been found to be inadequate. However, it cannot be denied that many illusions are due at least to the presence of angles.

Apparently Zöllner was the first to describe an illusion which is illustrated in simple form in [Fig. 29] and more elaborately in [Figs. 37 to 40]. The two figures at the right of [Fig. 29] were drawn for another purpose and are not designed favorably for the effect, although it may be detected when the figure is held at a distance. Zöllner accidentally noticed the illusion on a pattern designed for a print for dress-goods. The illusion is but slightly noticeable in [Fig. 29], but by multiplying the number of lines (and angles) the long parallel lines appear to diverge in the direction that the crossing lines converge. Zöllner studied the case thoroughly and established various facts. He found that the illusion is greatest when the long parallel lines are inclined about 45 degrees to the horizontal. This may be accomplished for [Fig. 37], by turning the page (held in a vertical plane) through an angle of 45 degrees from normal. The illusion vanishes when held too far from the eye to distinguish the short crossing lines, and its strength varies with the inclination of the oblique lines to the main parallels. The most effective angle between the short crossing lines and the main parallels appears to be approximately 30 degrees. In [Fig. 37] there are two illusions of direction. The parallel vertical strips appear unparallel and the right and left portions of the oblique cross-lines appear to be shifted vertically. It is interesting to note that steady fixation diminishes and even destroys the illusion.

Fig. 37.—Zöllner’s illusion of direction.

The maximum effectiveness of the illusion, when the figure is held so that the main parallel lines are at an inclination of about 45 degrees to the horizontal was accounted for by Zöllner as the result of less visual experience in oblique directions. He insisted that it takes less time and is easier to infer divergence or convergence than parallelism. This explanation appears to be disproved by a figure in which slightly divergent lines are used instead of parallel ones. Owing to the effect of the oblique crossing lines, the diverging lines may be made to appear parallel. Furthermore it is difficult to attach much importance to Zöllner’s explanation because the illusion is visible under the extremely brief illumination provided by one electric spark. Of course, the duration of the physiological reaction is doubtless greater than that of the spark, but at best the time is very short. Hering explained the Zöllner illusion as due to the curvature of the retina, and the resulting difference in the retinal images, and held that acute angles appear relatively too large and obtuse ones too small. The latter has been found to have limitations in the explanation of certain illusions.

This Zöllner illusion is very striking and may be constructed in a variety of forms. In [Fig. 37] the effect is quite apparent and it is interesting to view the figure at various angles. For example, hold the figure so that the broad parallel lines are vertical. The illusion is very pronounced in this position; however, on tilting the page backward the illusion finally disappears. In [Fig. 38] the short oblique lines do not cross the long parallel lines and to make the illusion more striking, the obliquity of the short lines is reversed at the middle of the long parallel lines. Variations of this figure are presented in [Figs. 39] and [40]. In this case by steady fixation the perspective effect is increased but there is a tendency for the parallel lines to appear more nearly truly parallel than when the point of sight is permitted to roam over the figures.

Fig. 38.—Parallel lines which do not appear so.

Fig. 39.—Wundt’s illusion of direction.

Fig. 40.—Hering’s illusion of direction.

Many investigations of the Zöllner illusion are recorded in the literature. From these it is obvious that the result is due to the additive effects of many simple illusions of angle. In order to give an idea of the manner in which such an illusion may be built up the reasoning of Jastrow[1] will be presented in condensed form. When two straight lines such as A and B in [Fig. 41] are separated by a space it is usually possible to connect the two mentally and to determine whether or not, if connected, they would lie on a straight line. However, if another line is connected to one, thus forming an angle as C does with A, the lines which appeared to be continuous (as A and B originally) no longer appear so. The converse is also true, for lines which are not in the same straight line may be made to appear to be by the addition of another line forming a proper angle. All these variations cannot be shown in a single figure, but the reader will find it interesting to draw them. Furthermore, the letters used on the diagram in order to make the description clearer may be confusing and these can be eliminated by redrawing the figure. In [Fig. 41] the obtuse angle AC tends to tilt A downward, so apparently if A were prolonged it would fall below B. Similarly, C appears to fall to the right of D.

Fig. 41.—Simple effect of angles.

This illusion apparently is due to the presence of the angle and the effect is produced by the presence of right and acute angles to a less degree. The illusion decreases or increases in general as the angle decreases or increases respectively.

Although it is not safe to present simple statements in a field so complex as that of visual illusion where explanations are still controversial, it is perhaps possible to generalize as Jastrow did in the foregoing case as follows: If the direction of an angle is that of the line bisecting it and pointing toward the apex, the direction of the sides of an angle will apparently be deviated toward the direction of the angle. The deviation apparently is greater with obtuse than with acute angles, and when obtuse and acute angles are so placed in a figure as to give rise to opposite deviations, the greater angle will be the dominant influence.

Although the illusion in such simple cases as [Fig. 41] is slight, it is quite noticeable. The effect of the addition of many of these slight individual influences is obvious in accompanying figures of greater complexity. These individual effects can be so multiplied and combined that many illusory figures may be devised.

In [Fig. 42] the oblique lines are added to both horizontal lines in such a manner that A is tilted downward at the angle and B is tilted upward at the angle (the letters corresponding to similar lines in [Fig. 41]). In this manner they appear to be deviated considerably out of their true straight line. If the reader will draw a straight line nearly parallel to D in [Fig. 41] and to the right, he will find it helpful. This line should be drawn to appear to be a continuation of C when the page is held so D is approximately horizontal. This is a simple and effective means of testing the magnitude of the illusion, for it is measured by the degree of apparent deviation between D and the line drawn adjacent to it, which the eye will tolerate. Another method of obtaining such a measurement is to begin with only the angle and to draw the apparent continuation of one of its lines with a space intervening. This deviation from the true continuation may then be readily determined.

Fig. 42. The effect of two angles in tilting the horizontal lines.

Fig. 43. The effect of crossed lines upon their respective apparent directions.

A more complex case is found in [Fig. 43] where the effect of an obtuse angle ACD is to make the continuation of AB apparently fall below FG and the effect of the acute angle is the reverse. However, the net result is that due to the preponderance of the effect of the larger angle over that of the smaller. The line EC adds nothing, for it merely introduces two angles which reinforce those above AB. The line BC may be omitted or covered without appreciably affecting the illusion.

Fig. 44.—Another step toward the Zöllner illusion.

In [Fig. 44] two obtuse angles are arranged so that their effects are additive, with the result that the horizontal lines apparently deviate maximally for such a simple case. Thus it is seen that the tendency of the sides of an angle to be apparently deviated toward the direction of the angle may result in an apparent divergence from parallelism as well as in making continuous lines appear discontinuous. The illusion in [Fig. 44] may be strengthened by adding more lines parallel to the oblique lines. This is demonstrated in [Fig. 38] and in other illustrations. In this manner striking illusions are built up.

If oblique lines are extended across vertical ones, as in [Figs. 45] and [46], the illusion is seen to be very striking. In [Fig. 45] the oblique line on the right if extended would meet the upper end of the oblique line on the left; however, the apparent point of intersection is somewhat lower than it is in reality. In [Fig. 46] the oblique line on the left is in the same straight line with the lower oblique line on the right. The line drawn parallel to the latter furnishes an idea of the extent of the illusion. This is the well-known Poggendorff illusion. The upper oblique line on the right actually appears to be approximately the continuation of the upper oblique line on the right. The explanation of this illusion on the simple basis of underestimation or overestimation of angles is open to criticism. If [Fig. 46] is held so that the intercepted line is horizontal or vertical, the illusion disappears or at least is greatly reduced. It is difficult to reconcile this disappearance of the illusion for certain positions of the figure with the theory that the illusion is due to an incorrect appraisal of the angles.