ON THE EFFECTS PRODUCED UPON THE SYMMETRY OF THE
PICTURE BY VARYING THE POSITION OF THE OBJECT.
Having ascertained the proper position of the eye, we shall now proceed
to determine the position of the object.
If the object is placed within the reflectors at any point =D=, Fig.
16, between their object end =O=, and their eye-end =E=, a perfectly
symmetrical picture will obviously be formed from it; but the centre of
this picture will not be at =O=, the centre of the luminous sectors,
but at the point =D=, where the object is placed, or its projection
_d_, so that we shall have a circular luminous field enclosing an
eccentric circular pattern. Such a position of the object is therefore
entirely unfit for the production of a symmetrical picture, unless the
object should be such as wholly to exclude the view of the circular
field, formed by the reflected images of the aperture =A O B=.
As the point =D= approaches to =O=, the centre _d_ of the symmetrical
picture will approach to =O=, and when =D= coincides with =O=, the
centre of the picture will be at =O=, and all the images of the
object placed in the plane =A O B= will be similarly disposed in all
the sectors which compose the circular field of view. Hence we may
conclude, that a perfectly symmetrical pattern cannot be exhibited
in the circular field of view, when the object is placed between =O=
and =E=, or anywhere within the reflectors. If the eye could be placed
exactly at the angular point =E=, so that every point of the line =E O=
should be projected upon =O=, then the images would be symmetrically
arranged round =O=; but this is obviously impossible, for the object
would, in such circumstances, cease to become visible when this
coincidence took place. But independent of the eccentricity of the
pattern, the position of the object within the mirrors prevents that
motion of the objects, without which a variation of the pattern cannot
be produced. An object between the reflectors must always be exposed
to view, and we cannot restrict our view to one-half, one-third, or
one-fourth of it, as when we have it in our power to move the objects
across the aperture, or the aperture over the objects.
[Illustration: FIG. 16.]
Another evil arising from the placing of the objects within the
mirrors, is, that we are prevented from giving them the proper degree
of illumination which is so essential to the distinctness of the last
reflexions. The portions of the mirrors, too, beyond the objects, or
those between =D= and =O=, are wholly unnecessary, as they are not
concerned in the formation of the picture. Hence it follows, that the
effects of the Kaleidoscope cannot be produced by any combination of
mirrors, in which the objects are placed within them.
Let us now consider what will happen, by removing the object beyond
the plane passing through =A O B=. In this case the pattern will lose
its symmetry from two causes. In the first place, it is manifest, as
already explained, that as the eye is necessarily raised a little above
the point =E=, and also above the planes =A O E=, =B O E=, it must see
through the aperture =A O B= a portion of the object situated below
both of these planes. This part of the object will therefore appear
to project beyond the point, or below the plane where the direct and
reflected images meet. If we suppose, therefore, that all the reflected
images were symmetrical, the whole picture would lose its symmetry in
consequence of the irregularity of the sector =A O B= seen by direct
vision. But this supposition is not correct; for since the image _m
n_, Fig. 3, seen by direct vision does not coincide with the first
reflected images _mnʹ_, _nmʹ_, it is clear that all the other images
will likewise be incoincident, and, therefore, that the figure formed
by their combination must lose its symmetry, and, consequently, its
beauty.
As the eye must necessarily be placed above a line perpendicular to the
plane =A B O= at the point =O=, it will see a portion of the object
situated below that perpendicular continued to the object. Thus, in
Fig. 16, if the eye is placed at _e_ above =E=, and if =M N= is the
object placed at the distance =P O=, then the eye at _e_ will observe
the portion =P Oʹ= of the object situated below the axis =P O E=, and
this portion, which may be called the aberration, will vary with the
height =E _e_= of the eye, and with the distance =O P= of the object.
[Illustration: FIG. 17.]
Let us now suppose =E _e_= and =O P= to be constant, and that a
polygonal figure is formed by some line placed at the point =Q= of the
object =M N=. Then if =P Q= is very great compared with =P Oʹ=, the
polygonal figure will be tolerably regular, though all its angles will
exhibit an imperfect junction, and its lower half will be actually,
though not very perceptibly, less than its upper half. But if =Q=
approaches to =P=, =P Oʹ= remaining the same, so that =P Oʹ= bears a
considerable ratio to =P Q=, then the polygonal figure will lose all
symmetry, the upper sectors being decidedly the largest, and the lowest
sectors the smallest. When =Q= arrives near =P=, the aberration becomes
enormous, and the figure is so distorted, that it can no longer be
recognised as a polygon.
The deviation from symmetry, therefore, arising from the removal
of the object from the extremity of the reflectors, increases as
the object approaches to the centre of the luminous sectors or the
circular field, and this deviation becomes so perceptible, that an eye
accustomed to observe and admire the symmetry of the combined objects,
will instantly perceive it, even when the distance of the object or =P
O= is less than the twentieth part of an inch. When the object is very
distant, the defect of symmetry is so enormous, that though the object
is seen by direct vision, and in some of the sectors, it is entirely
invisible in the rest.
The principle which we have now explained is of primary importance
in the construction of the Kaleidoscope, and it is only by a careful
attention to it that the instrument can be constructed so as to give
to an experienced and fastidious eye that high delight which it never
fails to derive from the exhibition of forms perfectly symmetrical.
From these observations it follows, that a picture possessed of
mathematical symmetry, cannot be produced unless the object is placed
exactly at the extremity of the reflectors, and that even when this
condition is complied with, the object itself must consist of lines
all lying in the same plane, and in contact with the reflectors. Hence
it is obvious, that objects whose thickness is perceptible, cannot
give mathematically symmetrical patterns, for one side of them must
always be at a certain distance from =O=. The deviation in this case
is, however, so small, that it can scarcely be perceived in objects of
moderate thickness.
In the simple form of the Kaleidoscope, the production of symmetrical
patterns is limited to objects which can be placed close to the
aperture =A O B=; but it will be seen in the sequel of this treatise,
that this limitation may be removed by an optical contrivance, which
extends indefinitely the use and application of the instrument.