ON THE UNION OF SIMILAR PICTURES IN BINOCULAR VISION.
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.
[35] A sheet of Queen’s heads may be advantageously used to accustom
the eyes to the union of similar figures.
[Illustration: FIG. 25.]
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 L_ad_, L_be_ be lines drawn through the centres of two of the open
spaces _a_, _b_, and R_bd_, R_ce_ lines drawn through the centres of
_b_ and _c_, and meeting L_ad_, L_be_ 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 _d_L or _d_R 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., _d_L_c_, _d_R_e_, 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]
[36] See _Edin. Transactions_, 1846, vol. xv. p. 663, and _Phil. Mag._,
May 1847, vol. xxx. p. 305.
[37] _Bibl. Universelle_, October 1855, p. 136.
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.
[Illustration: FIG. 26.]
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.
[Illustration: FIG. 27.]
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
GM × AB
GO = GD = —————————
AB + RL.
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
GC
Cos. A = ————.
GO
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 _g_E″, and
find the point _d″_ from the formula
_g_E″ × _ab_
_gd″_ = ———————————-.
_ab_ + RL
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
GE × AB
GΔ″ = ———————;
RL - AB
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.
[38] Smith’s _Opticks_, vol. ii. p. 388, § 977.
[39] _Essay on Single Vision, &c._, p. 44.
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.