ON THE PRINCIPLES OF THE KALEIDOSCOPE, AND THE
FORMATION OF SYMMETRICAL PICTURES BY THE
COMBINATION OF DIRECT AND INVERTED IMAGES.
The principles which we have laid down in the preceding chapter must
not be considered as in any respect the principles of the Kaleidoscope.
They are merely a series of preliminary deductions, by means of which
the principles of the Instrument may be illustrated, and they go no
farther than to explain the formation of an apparent circular aperture
by means of successive reflexions.
All the various forms which nature and art present to us, may be
divided into two classes, namely, _simple_ or _irregular_ forms, and
_compound_ or _regular_ forms. To the first class belong all those
forms which are called picturesque, and which cannot be reduced to two
forms similar, and similarly situated with regard to a given point; and
to the second class belong the forms of animals, the forms of regular
architectural buildings, the forms of most articles of furniture and
ornament, the forms of many natural productions, and all forms, in
short, which are composed of two forms, similar and similarly situated
with regard to a given line or plane.
Now, it is obvious that all compound forms of this kind are composed
of a direct and an inverted image of a simple or an irregular form;
and, therefore, every simple form can be converted into a compound
or beautiful form, by skilfully combining it with an inverted image
of itself, formed by reflexion. The image, however, must be formed
by reflexion from the first surface of the mirror, in order that the
direct and the reflected image may join, and constitute one united
whole; for if the image is reflected from the posterior surface, as
in the case of a looking-glass, the direct and the inverted image can
never coalesce into one form, but must always be separated by a space
equal to the thickness of the mirror-glass.
If we arrange simple forms in the most perfect manner round a centre,
it is impossible by any art to combine them into a symmetrical and
beautiful picture. The regularity of their arrangement may give some
satisfaction to the eye, but the adjacent forms can never join, and
must therefore form a picture composed of disunited parts.
The case, however, is quite different with compound forms. If we
arrange a succession of similar forms of this class round a centre, it
necessarily follows that they will all combine into one perfect whole,
in which all the parts either are or may be united, and which will
delight the eye by its symmetry and beauty.
In order to illustrate the preceding observations, we have represented
in Figs. 4 and 5 the effects produced by the multiplication of single
and compound forms. The line _a b c d_, for example, Fig. 4, is a
simple form, and is arranged round a centre in the same way as it would
be done by a perfect multiplying glass, if such a thing could be made.
The consecutive forms are all disunited, and do not compose a whole.
Fig. 5 represents the very same simple form, _a b c d_, converted into
a compound form, and then, as it were, multiplied and arranged round a
centre. In this case every part of the figure is united, and forms a
whole, in which there is nothing redundant and nothing deficient; and
this is the precise effect which is produced by the application of the
Kaleidoscope to the simple form _a b c_.
[Illustration: FIG. 4.]
[Illustration: FIG. 5.]
The fundamental principle, therefore, of the Kaleidoscope is, that
it produces symmetrical and beautiful pictures, by converting simple
into compound or beautiful forms, and arranging them, by successive
reflexions, into one perfect whole.
This principle, it will be readily seen, cannot be discovered by any
examination of the luminous sectors which compose the circular field of
the Kaleidoscope, and is not even alluded to in any of the propositions
given by Mr. Harris and Mr. Wood. In looking at the circular field
composed of an even and an odd number of reflexions, the arrangement
of the sectors is perfect in both cases; but when the number is odd,
and the form of the object simple, and when the object is not similarly
placed with regard to the two mirrors, a symmetrical and united picture
cannot possibly be produced. Hence it is manifest, that neither the
principles nor the effects of the Kaleidoscope could possibly be
deduced from any practical knowledge respecting the luminous sectors.
In order to explain the formation of the symmetrical picture shown in
Fig. 5, we must consider that the simple form _m n_, Fig. 2, is seen
by direct vision through the open sector =A O B=, and that the image
_n o_, of the object _m n_, formed by one reflexion in the sector =B
O _a_=, is necessarily an inverted image. But since the image _o p_,
in the sector =_a_ O α=, is a reflected and consequently an inverted
image of the _inverted image_, _m t_, in the sector =A O _b_=, it
follows, that the whole _n o p_ is an inverted image of the whole _n
m t_. Hence the image _n o_ will unite with the image _o p_, in the
same manner as _m n_ unites with _m t_. But as these two last unite
into a regular form, the two first will also unite into a regular or
compound form. Now, since the half =β O _e_= of the last sector =β O α=
was formerly shown to be an image of the half sector =_a_ O _s_=, the
line _q v_ will also be an image of the line _o z_, and for the same
reason the line _v p_ will be an image of _t y_. But the image _v p_
forms the same angle with =B O= or _n q_ that _t y_ does, and is equal
and similar to _t y_; and _q v_ forms the same angle with =A O= that
_o z_ does, and is equal and similar to _o z_. Hence, =O _o_ =
_o q_=, and =O _y_ O _v_=, and therefore _q v_ and _v p_ will form
one straight line, equal and similar to _t q_, and similarly situated
with respect to =B O=. The figure _m n o p q t_, therefore, composed of
one direct object, and several reflected images of that object, will
be symmetrical. As the same reasoning is applicable to every object
extending across the aperture =A O B=, whether simple or compound, and
to every angle =A O B=, which is an even aliquot part of a circle, it
follows,—
1. That when the inclination of the mirror is an _even_ aliquot part of
a circle, the object seen by direct vision across the aperture, whether
it is simple or compound, is so united with the images of it formed by
repeated reflexions, as to form a symmetrical picture.
2. That the symmetrical picture is composed of a series of parts, the
number of which is equal to the number of times that the angle =A O B=
is contained in 360°. And—
3. That these parts are alternately direct and inverted pictures of the
object; a direct picture of it being always placed between two inverted
ones, and, _vice versa_, so that the number of direct pictures is equal
to the number of inverted ones.
When the inclination of the mirrors is an _odd_ aliquot part of 360°,
such as ⅕th, as shown in Fig. 3, the picture formed by the combination
of the direct object and its reflected images is symmetrical only under
particular circumstances.
If the object, whether simple or compound, is similarly situated with
respect to each of the mirrors, as the straight line 1, 2 of Fig. 6,
the compound line 3, 4, the inclined lines 5, 6, the circular object 7,
the curved line 8, 9, and the radial line 10, =O=, then the images of
all these objects will also be similarly situated with respect to the
radial lines that separate the sectors, and will therefore form a whole
perfectly symmetrical, whether the number of sectors is odd or even.
[Illustration: FIG. 6.]
But when the objects are not similarly situated with respect to each of
the mirrors, as the compound line 1, 2, Fig. 8, the curved line 3, 4,
and the straight line 5, 6, and, in general, as all irregular objects
that are presented by accident to the instrument, then the image formed
in the last sector =_a_ O _e_=, Fig. 7, by the mirror =B O=, will not
join with the image formed in the last sector =_b_ O _e_=, by the
mirror =A O=. In order to explain this with sufficient perspicuity, let
us take the case where the angle is 72°, or ⅕th part of the circle,
as shown in Fig. 7. Let =A O=, =B O=, be the reflecting planes, and
_m n_ a line, _inclined to the radius which bisects the angle_ =A O
B=, so that =_o m_ > _o n_=; then _m nʹ_, _n mʹ_, will be the images
formed by the first reflexion from =A O= and =B O=, and _nʹ mʺ_, _mʹ
nʺ_, the images formed by the second reflexion; but by the principles
of catoptrics, =o _m_ = o _mʹ_ = O _mʺ_=, and =O _n_ = O _nʹ_
= O _nʺ_=, consequently since =O _m_= is by hypothesis greater than
=O _n_=, we shall have =O _mʺ_= greater than =O _nʺ_=; that is, the
images _mʹ nʺ_, _nʹ mʺ_, will not coincide. As =O _n_= approaches to
an equality with =O _m_=, =O _nʺ_= approaches to an equality with =O
_mʺ_=, and when =O _m_ = O _n_=, we have =O _nʺ_ = O _mʺ_=, and at
this limit the images are symmetrically arranged, which is the case
of the straight line 1, 2 in Fig. 6. By tracing the images of the
other lines, as is done in Fig. 8, it will be seen, that in every case
the picture is destitute of symmetry when the object has not the same
position with respect to the two mirrors.
[Illustration: FIG. 7.]
[Illustration: FIG. 8.]
This result may be deduced in a more simple manner, by considering
that the symmetrical picture formed by the Kaleidoscope contains half
as many pairs of forms as the number of times that the inclination
of the mirrors is contained in 360°; and that each pair consists of
a direct and an inverted form, so joined as to form a compound form.
Now the compound form made up by each pair obviously constitutes a
symmetrical picture when multiplied any number of times, whether even
or odd; but if we combine so many pair and half a pair, two direct
images will come together, the half pair cannot possibly join both with
the direct and the inverted image on each side of it, and therefore
a symmetrical whole cannot be obtained from such a combination. From
these observations we may conclude,—
1. That when the inclination of the mirrors is an _odd_ aliquot part
of a circle, the object seen by direct vision through the aperture
unites with the images of it formed by repeated reflexions, and forms
a complete and symmetrical picture, only in the case when the object
is similarly situated with respect to both the mirrors; the two last
sectors forming, in every other position of the object, an imperfect
junction, in consequence of these being either both direct or both
inverted pictures of the object.
2. That the series of parts which compose the symmetrical as well as
the unsymmetrical picture, consists of direct and inverted pictures of
the object, the number of direct pictures being always equal to half
the number of sectors increased by one, when the number of sectors is
5, 9, 13, 17, 21, etc., and the number of inverted pictures being equal
to half the number of sectors diminished by one, when the number of
sectors is 3, 7, 11, 15, 19, etc., and _vice versa_. Hence, the number
of direct pictures of the object must always be odd, and the number of
inverted pictures even, as appears from the following table:—
Inclination Number Number of Number of
of the of Inverted Direct
Mirrors. Sectors. Pictures. Pictures.
120° 3 2 1
72 5 2 3
51³/₇ 7 4 3
40 9 4 5
32⁸/₁₁ 11 6 5
27⁹/₁₃ 13 6 7
24 15 8 7
21³/₁₇ 17 8 9
18¹⁸/₁₉ 19 10 9
17⅐ 21 10 11
3. That when the number of sectors is 3, 7, 11, 15, 19, etc., the two
last sectors are inverted; and when the number is 5, 9, 13, 17, 21,
etc., the two last sectors are direct.
When the inclination of the mirrors is not an aliquot part of 360°,
the images formed by the last reflexions do not join like every other
pair of images, and therefore the picture which is created must be
imperfect. It has already been shown at the end of Chap. I. that when
the angle of the mirrors becomes greater than an even or less than an
odd aliquot part of a circle, each of the two incomplete sectors which
form the last sector becomes greater or less than half a sector. The
image of the object comprehended in each of the incomplete sectors
must therefore be greater or less than the images in half a sector;
that is, when the last sector =β O α=, Fig. 2, is greater than =A O
B=, the part _q v_ in one half must be the image of more than _o z_,
and _v p_ the image of more than _t y_, and _vice versa_, when =β O α=
is less than =A O B=. Hence it follows that the symmetry is imperfect
from the image in the last sector being greater or less than the other
images. But besides this cause of imperfection in the symmetry, there
is another, namely, the disunion of the two images _q v_ and _v p_.
The angles =O _q v_= and =O _o p_= are obviously equal, and also the
angles =O _p v_=, =O _p o_=; but since the angle =β O α=, or= _q_ O
_p_=, is by hypothesis greater or less than =_p_ O _o_=, it follows
that the angles of the triangle =_q_ O _p_= are either greater or less
than two right angles, because they are greater or less than the three
angles of the triangle =_p_ O _o_=. But as this is absurd, the lines _q
v_, _v p_, cannot join so as to form one straight line, and therefore
the completion of a perfect figure by means of two mirrors, whose
inclination is not an aliquot part of a circle, is impossible. When the
angle =β O α= is greater than =_p_ O _o_=, or =A O B=, the lines _q v_,
_v p_, will form a re-entering angle towards =O=, and when it is less
than =A O B=, the same lines will form a salient angle towards =O=.