In uniting by the convergency of the optic axes two dissimilar pictures, as shewn in Fig. 18, the solid cone mn ought to appear at mn much nearer the observer than the pictures which compose it. I found, however, that it never took its right position in absolute space, the base mn of the solid seeming to rest on the same plane with its constituent pictures ab, ab, whether it was seen by converging the axes as in Fig. 18 or in Fig. 22. Upon inquiring into the reason of this I found that the disturbing cause was simply the simultaneous perception of other objects in the same field of view whose distance was known to the observer.

In order to avoid all such influences I made experiments on large surfaces covered with similar plane figures, such as flowers or geometrical patterns upon paper-hangings and carpets. These figures being always at equal distances from each other, and almost perfectly equal and similar, the coalescence of any pair of them, effected by directing the optic axes to a point between the paper-hanging and the eye, is accompanied by the instantaneous coalescence of them all. If we, therefore, look at a papered wall without pictures, or doors, or windows, or even at a considerable portion of a wall, at the distance of three feet, and unite two of the figures,—two flowers, for example, at the distance of twelve inches from each other horizontally, the whole wall or visible portion of it will appear covered with flowers as before, but as each flower is now composed of two flowers united at the point of convergence of the optic axes, the whole papered wall with all its flowers will be seen suspended in the air at the distance of six inches from the observer! At first the observer does not decide upon the distance of the suspended wall from himself. It generally advances slowly to its new position, and when it has taken its place it has a very singular character. The surface of it seems slightly curved. It has a silvery transparent aspect. It is more beautiful than the real paper, which is no longer seen, and it moves with the slightest motion of the head. If the observer, who is now three feet from the wall, retires from it, the suspended wall of flowers will follow him, moving farther and farther from the real wall, and also, but very slightly, farther and farther from the observer. When he stands still, he may stretch out his hand and place it on the other side of the suspended wall, and even hold a candle on the other side of it to satisfy himself that the ghost of the wall stands between the candle and himself.

In looking attentively at this strange picture some of the flowers have the aspect of real flowers. In some the stalk retires from the plane of the picture. In others it rises from it. One leaf will come farther out than another. One coloured portion, red, for example, will be more prominent than the blue, and the flower will thus appear thicker and more solid, like a real flower compressed, and deviating considerably from the plane representation of it as seen by one eye. All this arises from slight and accidental differences of distance in similar or corresponding parts of the united figures. If the distance, for example, between two corresponding leaves is greater than the distance between other two corresponding leaves, then the two first when united will appear nearer the eye than the other two, and hence the appearance of a flower in low relief, is given to the combination.

In continuing our survey of the suspended image another curious phenomenon often presents itself. A part of one, or even two pieces of paper, and generally the whole length of them from the roof to the floor, will retire behind the general plane of the image, as if there were a recess in the wall, or rise above it as if there were a projection, thus displaying on a large scale the imperfection in the workmanship which otherwise it would have been difficult to discover. This phenomenon, or defect in the work, arises from the paper-hanger having cut off too much of the margin of one or more of the adjacent stripes or pieces, or leaving too much of it, so that, in the first case, when the two halves of a flower are joined together, part of the middle of the flower is left out, and hence, when this defective flower is united binocularly with the one on the right hand of it, and the one on the left hand united with the defective one, the united or corresponding portion being at a less distance, will appear farther from the eye than those parts of the suspended image which are composed of complete flowers. The opposite effect will be produced when the two portions of the flowers are not brought together, but separated by a small space. All these phenomena may be seen, though not so conveniently, with a carpet from which the furniture has been removed. We have, therefore, an accurate method of discovering defects in the workmanship of paper-hangers, carpet-makers, painters, and all artists whose profession it is to combine a series of similar patterns or figures to form an uniformly ornamented surface. The smallest defect in the similarity or equality of the figures or lines which compose a pattern, and any difference in the distance of single figures is instantly detected, and what is very remarkable a small inequality of distance in a line perpendicular to the axis of vision, or in one dimension of space, is exhibited in a magnified form at a distance coincident with the axis of vision, and in an opposite dimension of space.

A little practice will enable the observer to realize and to maintain the singular binocular vision which replaces the real picture.[35] The occasional retention of the picture after one eye is closed, and even after both have been closed and quickly reopened, shews the influence of time over the evanescence as well as over the creation of this class of phenomena. On some occasions, a singular effect is produced. When the flowers or figures on the paper are distant six inches, we may either unite two six inches distant, or two twelve inches distant, and so on. In the latter case, when the eyes have been accustomed to survey the suspended picture, I have found that, after shutting or opening them, I neither saw the picture formed by the two flowers twelve inches distant, nor the papered wall itself, but a picture formed by uniting all the flowers six inches distant! The binocular centre (the point to which the optic axes converged, and consequently the locality of the picture) had shifted its place, and instead of advancing to the real wall and seeing it, it advanced exactly as much as to unite the nearest flowers, just as in a ratchet wheel, when the detent stops one tooth at a time; or, to speak more correctly, the binocular centre advanced in order to relieve the eyes from their strain, and when the eyes were opened, it had just reached that point which corresponded with the union of the flowers six inches distant.

We have already seen, as shewn in Fig. 22, that when we fix the binocular centre, that is, converge the optic axes on a point beyond the dissimilar pictures, so as to unite them, they rise into relief as perfectly as when the binocular centre, as shewn in Fig. 18, is fixed between the pictures used and the eye. In like manner we may unite similar pictures, but, owing to the opacity of the wall and the floor, we cannot accomplish this with paper-hangings and carpets. The experiment, however, may be made with great effect by looking through transparent patterns cut out of paper or metal, such as those in zinc which are used for larders and other purposes. Particular kinds of trellis-work, and windows with small squares or rhombs of glass, may also be used, and, what is still better, a screen might be prepared, by cutting out the small figures from one or more pieces of paper-hangings. The readiest means, however, of making the experiment, is to use the cane bottom of a chair, which often exhibits a succession of octagons with small luminous spaces between them. To do this, place the back of the chair upon a table, the height of the eye either when sitting or standing, so that the cane bottom with its luminous pattern may have a vertical position, as shewn in Fig. 25, where mn is the real bottom of the chair with its openings, which generally vary from half an inch to three-fourths. Supposing the distance to be half an inch, and the eyes, l, r, of the observer 12 inches distant from mn, let lad, lbe be lines drawn through the centres of two of the open spaces a, b, and rbd, rce lines drawn through the centres of b and c, and meeting lad, lbe at d and e, d being the binocular centre to which the optic axes converge when we look at it through a and b, and c the binocular centre when we look at it through b and c. Now, the right eye, r, sees the opening b at d, and the left eye sees the opening a at d, so that the image at d of the opening consists of the similar images of a and b united, and so on with all the rest; so that the observer at l, r no longer sees the real pattern mn, but an image of it suspended at mn, three inches behind mn. If the observer now approaches mn, the image mn will approach to him, and if he recedes, mn will recede also, being 1½ inches behind mn when the observer is six inches before it, and twelve inches behind mn when the observer is forty-eight inches before it, the image mn moving from mn with a velocity one-fourth of that with which the observer recedes.

The observer resuming the position in the figure where his eyes, l, r, are twelve inches distant from mn, let us consider the important results of this experiment. If he now grasps the cane bottom at mn, his thumbs pressing upon mn, and his fingers trying to grasp mn, he will then feel what he does not see, and see what he does not feel! The real pattern is absolutely invisible at mn, where he feels it, and it stands fixed at mn. The fingers may be passed through and through between the real and the false image, and beyond it,—now seen on this side of it, now in the middle of it, and now on the other side of it. If we next place the palms of each hand upon mn, the real bottom of the chair, feeling it all over, the result will be the same. No knowledge derived from touch—no measurement of real distance—no actual demonstration from previous or subsequent vision, that there is a real solid body at mn, and nothing at all at mn, will remove or shake the infallible conviction of the sense of sight that the cane bottom is at mn, and that dl or dr is its real distance from the observer. If the binocular centre be now drawn back to mn, the image seen at mn will disappear, and the real object be seen and felt at mn. If the binocular centre be brought further back to f, that is, if the optic axes are converged to a point nearer the observer than the object, as illustrated by Fig. 18, the cane bottom mn will again disappear, and will be seen at uv, as previously explained.

This method of uniting small similar figures is more easily attained than that of doing it by converging the axes to a point between the eye and the object. It puts a very little strain upon the eyes, as we cannot thus unite figures the distance of whose centre is equal to or exceeds 2½ inches, as appears from Fig. 22.

In making these experiments, the observer cannot fail to be struck with the remarkable fact, that though the openings mn, mn, uv, have all the same apparent or angular magnitude, that is, subtend the same angle at the eye, viz., dlc, dre, yet those at mn appear larger, and those at uv smaller, than those at mn. If we cause the image mn to recede and approach to us, the figures in mn will invariably increase as they recede, and those in uv diminish as they approach the eye, and their visual magnitudes, as we may call them, will depend on the respective distances at which the observer, whether right or wrong in his estimate, conceives them to be placed,—a result which is finely illustrated by the different size of the moon when seen in the horizon and in the meridian. The fact now stated is a general one, which the preceding experiments demonstrate; and though our estimate of magnitude thus formed is erroneous, yet it is one which neither reason nor experience is able to correct.

It is a curious circumstance, that, previous to the publication of these experiments, no examples have been recorded of false estimates of the distance of near objects in consequence of the accidental binocular union of similar images. In a room where the paper-hangings have a small pattern, a short-sighted person might very readily turn his eyes on the wall when their axes converged to some point between him and the wall, which would unite one pair of the similar images, and in this case he would see the wall nearer him than the real wall, and moving with the motion of his head. In like manner a long-sighted person, with his optical axes converged to a point beyond the wall, might see an image of the wall more distant, and moving with the motion of his head; or a person who has taken too much wine, which often fixes the optical axes in opposition to the will, might, according to the nature of his sight, witness either of the illusions above mentioned.

Illusions of both these kinds, however, have recently occurred. A friend to whom I had occasion to shew the experiments, and who is short-sighted, mentioned to me that he had on two occasions been greatly perplexed by the vision of these suspended images. Having taken too much wine, he saw the wall of a papered room suspended near him in the air; and on another occasion, when kneeling, and resting his arms on a cane-bottomed chair, he had fixed his eyes on the carpet, which had accidentally united the two images of the open octagons, and thrown the image of the chair bottom beyond the plane on which he rested his arms.

After hearing my paper on this subject read at the Royal Society of Edinburgh, Professor Christison communicated to me the following interesting case, in which one of the phenomena above described was seen by himself:—“Some years ago,” he observes, “when I resided in a house where several rooms are papered with rather formally recurring patterns, and one in particular with stars only, I used occasionally to be much plagued with the wall suddenly standing out upon me, and waving, as you describe, with the movements of the head. I was sensible that the cause was an error as to the point of union of the visual axes of the two eyes; but I remember it sometimes cost me a considerable effort to rectify the error; and I found that the best way was to increase still more the deviation in the first instance. As this accident occurred most frequently while I was recovering from a severe attack of fever, I thought my near-sighted eyes were threatened with some new mischief; and this opinion was justified in finding that, after removal to my present house, where, however, the papers have no very formal pattern, no such occurrence has ever taken place. The reason is now easily understood from your researches.”[36]

Other cases of an analogous kind have been communicated to me; and very recently M. Soret of Geneva, in looking through a trellis-work in metal stretched upon a frame, saw the phenomenon represented in Fig. 25, and has given the same explanation of it which I had published long before.[37]

Before quitting the subject of the binocular union of similar pictures, I must give some account of a series of curious phenomena which I observed by uniting the images of lines meeting at an angular point when the eye is placed at different heights above the plane of the paper, and at different distances from the angular point.

Let ac, bc, Fig. 26, be two lines meeting at c, the plane passing through them being the plane of the paper, and let them be viewed by the eyes successively placed at e‴, e″, e′, and e, at different heights in a plane, gmn, perpendicular to the plane of the paper. Let r be the right eye, and l the left eye, and when at e‴, let them be strained so as to unite the points a, b. The united image of these points will be seen at the binocular centre d‴, and the united lines ac, bc, will have the position d‴c. In like manner, when the eye descends to e″, e′, e, the united image d‴c will rise and diminish, taking the positions d″c, d′c, dc, till it disappears on the line cm, when the eyes reach m. If the eye deviates from the vertical plane gmn, the united image will also deviate from it, and is always in a plane passing through the common axis of the two eyes and the line gm.

If at any altitude em, the eye advances towards acb in the line eg, the binocular centre d will also advance towards acb in the line eg, and the image dc will rise, and become shorter as its extremity d moves along dg, and, after passing the perpendicular to ge, it will increase in length. If the eye, on the other hand, recedes from acb in the line ge, the binocular centre d will also recede, and the image dc will descend to the plane cm, and increase in length.

The preceding diagram is, for the purpose of illustration, drawn in a sort of perspective, and therefore does not give the true positions and lengths of the united images. This defect, however, is remedied in Fig. 27, where e, e′, e″, e‴ is the middle point between the two eyes, the plane gmn being, as before, perpendicular to the plane passing through acb. Now, as the distance of the eye from g is supposed to be the same, and as ab is invariable as well as the distance between the eyes, the distance of the binocular centres oO, d, d′, d″, d‴, p from g, will also be invariable, and lie in a circle odp, whose centre is g, and whose radius is go, the point o being determined by the formula

Hence, in order to find the binocular centres d, d′, d″, d‴, &c., at any altitude, e, e′, &c., we have only to join eg, e′g, &c., and the points of intersection d, d′, &c., will be the binocular centres, and the lines dc, d′c, &c., drawn to c, will be the real lengths and inclinations of the united images of the lines ac, bc.

When go is greater than gc there is obviously some angle a, or e″gm, at which d″c is perpendicular to gc.

This takes place when

When o coincides with c, the images cd, cd′, &c., will have the same positions and magnitudes as the chords of the altitudes a of the eyes above the plane gc. In this case the raised or united images will just reach the perpendicular when the eye is in the plane gcm, for since

GC = GO,   Cos. A = 1 and   A = 0.

When the eye at any position, e″ for example, sees the points a and b united at d″, it sees also the whole lines ac, bc forming the image d″c. The binocular centre must, therefore, run rapidly along the line d″c; that is, the inclination of the optic axes must gradually diminish till the binocular centre reaches c, when all strain is removed. The vision of the image d″c, however, is carried on so rapidly that the binocular centre returns to d″ without the eye being sensible of the removal and resumption of the strain which is required in maintaining a view of the united image d″c. If we now suppose ab to diminish, the binocular centre will advance towards g, and the length and inclination of the united images dc, d′c, &c., will diminish also, and vice versa. If the distance rl (Fig. 26) between the eyes diminishes, the binocular centre will retire towards e, and the length and inclination of the images will increase. Hence persons with eyes more or less distant will see the united images in different places and of different sizes, though the quantities a and AB be invariable.

While the eyes at e″ are running along the lines ac, bc, let us suppose them to rest upon the points ab equidistant from c. Join ab, and from the point g, where ab intersects gc, draw the line ge″, and find the point d″ from the formula

Hence the two points a, b will be united at d″, and when the angle e″gc is such that the line joining d and c is perpendicular to gc, the line joining d″c will also be perpendicular to gc, the loci of the points d″d″, &c., will be in that perpendicular, and the image dc, seen by successive movements of the binocular centre from d″ to c, will be a straight line.

In the preceding observations we have supposed that the binocular centre d″, &c., is between the eye and the lines ac, bc; but the points a, c, and all the other points of these lines, may be united by fixing the binocular centre beyond ab. Let the eyes, for example, be at e″; then if we unite a, b when the eyes converge to a point, Δ″, (not seen in the Figure) beyond g, we shall have

and if we join the point Δ″ thus found and c, the line Δ′c will be the united image of ac and bc, the binocular centre ranging from Δ″ to c, in order to see it as one line. In like manner, we may find the position and length of the image Δ‴c, Δ′c, and Δc, corresponding to the position of the eyes at e‴e and e. Hence all the united images of ac, bc, viz., cΔ‴, cΔ″, &c., will lie below the plane of abc, and extend beyond a vertical line ng continued; and they will grow larger and larger, and approximate in direction to cg as the eyes descend from e‴ to m. When the eyes are near to m, and a little above the plane of abc, the line, when not carefully observed, will have the appearance of coinciding with cg, but stretching a great way beyond g. This extreme case represents the celebrated experiment with the compasses, described by Dr. Smith, and referred to by Professor Wheatstone. He took a pair of compasses, which may be represented by acb, ab being their points, ac, bc their legs, and c their joint; and having placed his eyes about e, above their plane, he made the following experiment:—“Having opened the points of a pair of compasses somewhat wider than the interval of your eyes, with your arm extended, hold the head or joint in the ball of your hand, with the points outwards, and equidistant from your eyes, and somewhat higher than the joint. Then fixing your eyes upon any remote object lying in the plane that bisects the interval of the points, you will first perceive two pair of compasses, (each by being doubled with their inner legs crossing each other, not unlike the old shape of the letter W). But by compressing the legs with your hand the two inner points will come nearer to each other; and when they unite (having stopped the compression) the two inner legs will also entirely coincide and bisect the angle under the outward ones, and will appear more vivid, thicker, and larger, than they do, so as to reach from your hand to the remotest object in view even in the horizon itself, if the points be exactly coincident.”[38] Owing to his imperfect apprehension of the nature of this phenomenon, Dr. Smith has omitted to notice that the united legs of the compasses lie below the plane of abc, and that they never can extend further than the binocular centre at which their points a and b are united.

There is another variation of these experiments which possesses some interest, in consequence of its extreme case having been made the basis of a new theory of visible direction, by the late Dr. Wells.[39] Let us suppose the eyes of the observer to advance from e to n, and to descend along the opposite quadrant on the left hand of ng, but not drawn in Fig. 27, then the united image of ac, bc will gradually descend towards cg, and become larger and larger. When the eyes are a very little above the plane of abc, and so far to the left hand of ab that ca points nearly to the left eye and cb to the right eye, then we have the circumstances under which Dr. Wells made the following experiment:—“If we hold two thin rules in such a manner that their sharp edges (ac, bc in Fig. 27) shall be in the optic axes, one in each, or rather a little below them, the two edges will be seen united in the common axis, (gc in Fig. 27;) and this apparent edge will seem of the same length with that of either of the real edges, when seen alone by the eye in the axis of which it is placed.” This experiment, it will be seen, is the same with that of Dr. Smith, with this difference only, that the points of the compasses are directed towards the eyes. Like Dr. Smith, Dr. Wells has omitted to notice that the united image rises above gh, and he commits the opposite error of Dr. Smith, in making the length of the united image too short.

If in this form of the experiment we fix the binocular centre beyond c, then the united images of ac, and bc descend below gc, and vary in their length, and in their inclination to gc, according to the height of the eye above the plane of abc, and its distance from ab.