ON THE EFFECTS PRODUCED UPON THE SYMMETRY OF THE
PICTURE BY VARYING THE POSITION OF THE EYE.
It has been taken for granted in the preceding chapters, not only that
the object seen by direct vision is in a state of perfect junction
with the images of it formed by reflexion; but that the object and
its images have the same apparent magnitude, and nearly the same
intensity of light. As these conditions are absolutely necessary to the
production of symmetrical and beautiful forms, and may be all effected
by particular methods of construction, we shall proceed to investigate
the principles upon which these methods are founded, in so far as the
position of the eye is concerned.
When any object is made to touch a common looking-glass in one or more
points, the reflected image does not touch the object in these points,
but is always separated from it by a space equal to the thickness
of the glass, in consequence of the reflexion being performed by
the posterior surface of the mirror. The image and the object must
therefore be always disunited; and as the interval of separation must
be interposed between all the reflected images, there cannot possibly
exist that union of forms which constitutes the very essence of
symmetry. In mirror-glass there is a series of images reflected from
the _first_ surface, which unite perfectly with the object, and with
one another. When the angles of incidence are not great, this series
of images is very faint, and does not much interfere with the more
brilliant images formed by the metallic surface. As the angles of
incidence increase, the one series of images destroys the effect of the
other, from their overlapping or imperfect coincidence—an effect which
increases with the thickness of the glass; but when the reflexions are
made at very oblique incidences, the images formed by the metallic
surface become almost invisible, while those formed by the first
surface are as brilliant and nearly as perfect as if the effect of the
posterior metallic surface had been entirely removed. In the following
observations, therefore, it is understood that the images are reflected
either from a polished metallic surface, or from the first surface of
glass.
[Illustration: FIG. 13.]
In order to explain the effects produced upon the symmetry of the
picture by a variation in the position of the eye, we must suppose the
object to be placed at a small distance from the end of the mirror.
This position is represented in Fig. 13, where =A E= is a section of
the mirror in the direction of its length; =M N O P= an object placed
at a distance from the extremity =A= of the mirror, and _m n o p_,
its image seen by an eye to the right hand of =E=, and which, by the
principles of catoptrics, will be similar to the object and similarly
situated with respect to the mirror =A E=. Now, if the eye is placed at
ε, it will see distinctly the whole object =M N O P=, but it will only
see the portion _n r s o_ of the image cut off by drawing the line =ε
A _r_= through the extremity of the mirror, so that there cannot be a
symmetrical form produced by observing at the same time the object =M
N O P= and this portion of its image; and the deviation from symmetry
will be still greater, if =M N O P= is brought nearer the line =B A=,
for the image _m n o p_ will be entirely included between the lines =A
_r_= and =A B=, so that no part whatever of the image will be visible
to an eye at =ε=. As the eye of the observer moves from ε to _e_, the
line =ε A _r_= will move into the position =_e_ A _x_=, and when it has
reached the point =e=, the whole of the image _m n o p_ will be visible.
The symmetry, therefore, arising from the simultaneous contemplation
of the object and its image will be improved; but it will still be
imperfect, as the image will appear to be distant from the plane of the
mirror, only by the space _m x_, while the distance of the object is
=M _x_=. As the eye moves from =_e_= to =E=, the line =_e_ A _x_= will
move into =E A B=, and the object and its image will seem to be placed
at the equal distances =M B=, =_m_ B= from the plane of the mirror,
and will therefore form a symmetrical combination. When the object is
moved, and arrives at =B A= the image will touch the object, and they
will form one perfect and united whole, whatever be the shape of the
line =M P=. Hence we conclude, _that when an object is placed at a
little distance from the extremity of a plain mirror, its image formed
by reflexion from the mirror cannot unite with the object in forming a
conjoined and symmetrical picture, unless the eye is in the plane of
the mirror_.
[Illustration: FIG. 14.]
When two mirrors, therefore, are combined, as in Fig. 14, the eye
must be in the plane of both, in order that the object and its image
may have a symmetrical coincidence, and therefore it must be at the
point =E= where the two planes cut each other. The necessity of this
position, and the effects of any considerable deviation from it, will
be understood from Fig. 14, where =A O B= is the angle formed by the
mirrors, and =M N= the place of the object. Then if the eye is placed
at ε, the aperture =A O B= will be projected into =_a b_ ω= upon a
plane passing through =M N= and at right angles to =E Oʹ=; but the
orthographic projection of =A B O= upon the same plane is =Aʹ Bʹ Oʹ=,
or, what is the same thing, the reflecting surfaces of which =A O=, =B
O= are sections, will, when prolonged, cut the plane passing through
=M N= in the lines =Aʹ Oʹ=, =Bʹ Oʹ=; hence, rays from the objects
situated between =Aʹ Oʹ Bʹ= and =_a_ ω _b_= cannot fall upon the
mirrors =A O E=, =B O E=, or images of these objects cannot be formed
by the mirrors. The images, therefore, in the different sectors formed
by reflexion round =O= as a centre, cannot include any objects without
=Aʹ Oʹ Bʹ=; and since the eye at ε sees all the objects between =Aʹ Oʹ
Bʹ= and =_a_ ω _b_=, there can be no symmetry and uniformity in the
picture formed by the combination of such an object with the images
in the sectors. When the eye descends to _e_, the aperture =A O B=
is projected into =_aʹ oʹ bʹ_=, which approaches nearer to =A O B=;
but for the reasons already assigned, the symmetry of the picture is
still imperfect. As the eye descends, the lines =_aʹ oʹ_, _bʹ oʹ_=
approach to =Aʹ Oʹ=, =Bʹ Oʹ=, and when the eye arrives at =E=, a point
in the plane of both the reflecting surfaces, the projection of the
aperture =A O B= will be =Aʹ Oʹ Bʹ=, and the images in all the sectors
will be exactly similar to the object presented to the aperture.
Hence we conclude in general, _that when an object is placed at any
distance before two mirrors inclined at an angle, which is an even
aliquot part of 360°, the symmetry of the picture is perfect, when the
eye, considered as a mathematical point, is placed at_ =E=, and that
_the deviation from symmetry increases as the eye recedes from_ =E=
_towards_ =ε=.
If the object were a mathematical surface, all the parts of which were
in contact with the extremities =A O=, =B O= of the mirrors, then it is
easy to see that the symmetry of the picture will not be affected by
the deviation of the eye from the point =E=, and, in consequence of the
enlargement of the sector, seen by direct vision. The symmetry of the
picture, is, however, affected in another way, by the deviation of the
eye from the point =E=.
We have already seen, that, in order to possess perfect symmetry, an
object must consist of two parts in complete contact, one of which is
an inverted image of the other. But in order that an object possessing
perfect symmetry may appear perfectly symmetrical, four conditions
are required. The two halves of the object must be so placed with
respect to the eye of the observer, that no part of the one half shall
conceal any part of the other; that whatever parts of the one half
are seen, the corresponding parts of the other must also be seen;
and that the corresponding parts of both halves, and both halves
themselves, must subtend the same angle at the eye. When we stand
before a looking-glass, and hold out one hand so as to touch it, the
hand will be found to conceal various parts of its image; and, in
some positions of the eye, the whole image will be concealed, so that
a symmetrical picture cannot possibly be formed by the union of the
two. If the eye is placed so obliquely to the looking-glass, that the
hand no longer interferes with its image, it will still be seen, that
parts of the hand which are not directly visible, are visible in its
reflected image, and therefore that a symmetrical picture cannot be
created by the union of two parts apparently dissimilar. If the eye of
the observer is placed near his hand, so that he can see distinctly
both the hand and its image, the angular magnitude of his hand is much
greater than that of its image; and therefore, when the two are united,
they cannot form a symmetrical object. This will be better understood
from Fig. 15. When the eye is placed at ε, the object =M N O P= is
obviously nearer than its image _m n o p_, and must therefore appear
larger; and this difference in their apparent magnitudes will increase
as the eye rises above the plane of the mirror =A E=. As the eye
approaches to =E=, the distances of the object and its image approach
to an equality; and when the eye is at =E=, the object =M N O P=, and
its image _m n o p_, are situated at exactly the same distance from the
eye, and therefore have the same angular magnitude. Hence it follows,
that when they are united, they will form a perfectly symmetrical
combination.
[Illustration: FIG. 15.]
When the eye is placed in the plane of both the mirrors, the field of
view arising from the multiplication of the sector =A O B=, Fig. 14,
will be perfectly circular; but as the eye rises above the plane of
both the mirrors, this circle will become a sort of ellipse, becoming
more and more eccentric as the eye comes in front of the mirrors, or
rises in the direction =E ε=. If the observer were infinitely distant,
these figures would be correct ellipses; but as the eye, particularly
when the mirrors are broad, must be nearly twice as far from the last
reflected sector as from the sector seen by direct vision, the field
of view, and consequently every pattern which it contains, must be
distorted and destitute of beauty, from this cause alone.
Hitherto we have alluded only to symmetry of form, but it is manifest
that before the union of two similar forms can give pleasure to the
eye, there must be also a symmetry of light. If the object =M N O P=,
Fig. 15, is white, and its image _m n o p_ black, they cannot possibly
form, by their combination, an agreeable picture. As any considerable
difference in the intensity of the light will destroy the beauty of the
patterns, it becomes a matter of indispensable importance to determine
the position of the eye, which will give the greatest possible
uniformity to the different images of which the picture is composed.
It has been ascertained by the accurate experiments of Bouguer, that
when light is reflected perpendicularly from good plate glass, only 25
rays are reflected out of 1000; that is, the intensity of the light of
any object seen perpendicularly in plate glass, is to the intensity
of the light of its image as 1000 to 25, or as 40 is to 1. When the
angle of incidence is 60°, the number of reflected rays is 112, and the
intensities are nearly as 9 to 1.—When the angle of incidence is 87½°,
the number of reflected rays is 584, and the intensities are nearly
as 17 to 10, so that the luminousness of the object and its image
approach rapidly to an equality. It is therefore clear, that, in order
to have the greatest uniformity of light in the different images which
compose the figure, the eye must be placed as nearly as possible in the
plane of both the mirrors, that is, as near as possible to the angular
point.—But as it is impracticable to have the eye exactly in the plane
of both mirrors, the images formed by one reflexion must always be
less bright than the direct object, even at the part nearly in contact
with the object. The second and third reflexions, etc., where the rays
fall with less obliquity, will be still darker than the first; though
this difference will not be very perceptible when the inclination of
the mirrors is 30° or upwards, and the eye placed in the position
already described.
It is a curious circumstance, that the positions of the eye which are
necessary to effect a complete union of the images—to represent similar
parts of the object and its images—to observe the object and its image
under the same angular magnitude, and to give a maximum intensity of
light to the reflected images—should all unite in the same point. Had
this not been the case, the construction of the Kaleidoscope would have
been impracticable, and hence it will be seen how vain is the attempt
to produce beautiful and symmetrical forms from any combination of
plain mirrors in which this position of the eye is not a radical and
essential principle.