THE PROPERTIES OF MIRRORS.
ALMOST every one in his younger days has possessed and broken that
pretty instrument known as the kaleidoscope. His researches into its
construction no doubt taught him that it consisted of a cylindrical tube
in tin or cardboard, with a moveable cap at one end and a small hole at
the other. In the interior of the tube were found three long glasses,
blackened on the back, placed at an angle, and kept in position by
pieces of cork. The moveable cap was provided with two circular pieces
of glass, one ground and the other transparent, between which were
placed a number of pieces of coloured glass. On holding the instrument
up to the light and looking through the eye-hole, a beautifully coloured
star was seen whose form and hue changed by simply shaking the tube.
The kaleidoscope was invented by Sir David Brewster, and is exceedingly
simple in principle. We all know that if a luminous object, such as a
taper, is placed before a mirror, it gives forth rays of light in all
directions. Amongst these luminous rays, those that fall on the surface
of the mirror are, of course, reflected in such a manner that the angle
of reflection is equal to the angle of incidence. If another mirror be
placed at right angles to the first, and an object be put in the angle,
the image of it will be multiplied four times. If the angle be
diminished to 60°, six reflections will be seen, and so on. A
symmetrical figure is constantly obtained, forming in one case a cross
composed of four similar portions; in the other a triple star, the
halves of each ray being similar. It is the symmetry of the figure that
gives the pleasing effect. In the ordinary kaleidoscope the angle made
by the reflecting surfaces is thirty degrees, and a star of six rays is
formed, the halves of each ray being alike. The figures formed in the
kaleidoscope are simply endless; and if the space between the glasses in
the moveable cap be filled with bits of opaque as well as transparent
substances, the varieties of light and shade may be added to those of
colour. It was at one time the fashion to copy the images formed in the
kaleidoscope as patterns for room papers, muslins, curtains, shawls, and
other similar fabrics, but thanks to the spread of artistic taste in
this country the decorative designer now relies more on his own talent
than any aid he may receive from optical instruments.
Plane mirrors, as we have seen, reflect objects upright and symmetrical,
reversing only the sides. Concave mirrors reverse them, and if they are
not placed exactly in the proper focus, distort them by making one
portion appear smaller than the other; while convex mirrors reflect them
in an upright position, but also similarly slightly distorted. But when
the mirror is not a portion of a sphere, like those whose properties we
have been considering, the distortion is increased to so great an extent
as to deform the object so that it is difficult to recognise its nature
from its reflection. We all know the distortion that our face undergoes
when reflected from the shining surface of a teapot or spoon, and the
cylindrical mirrors that hang in the shop windows of many opticians are
the source of much amusement to the passers by, whose physiognomies are
shown to them either lengthened to many times their natural size, or
widened to an extent that is ludicrously hideous, according to the
position in which the mirror is hung. Such distortions are known to
opticians as _anamorphoses_, from two Greek words signifying the
destruction of form; and distorted drawings used to be sold at one time
which when reflected from the surface of the cylindrical mirror, became
perfectly symmetrical. Anamorphic drawings may be also made, which when
looked at in the ordinary manner appear distorted, but when viewed from
a particular point have their symmetry restored to them. With a little
knowledge of drawing, it is not difficult to produce these in great
variety.
Suppose the portrait in fig. 62 to be divided horizontally and
vertically by equidistant lines comprehended within the square A B C D.
[Illustration:
Fig. 62.
]
Upon a second piece of paper draw the figure shown in fig. 63 in the
following manner. Draw the horizontal line _a b_ equal to A B (fig. 62),
and divide it into the same number of parts. Through the centre draw a
perpendicular line to V, and cross it by a line _e d_ parallel to _a b_.
Lastly, draw V S horizontal to _e d_. The length of the two lines _e_ V
and S V is quite arbitrary, but the longer you make the former in
proportion to the latter the greater will be the distortion of the
drawing. Now draw the lines V 1, V 2, V 3, and V 4, and join S to _a_.
Wherever S _a_ crosses the divisions 1, 2, _e_, 3, 4, and _b_, draw a
horizontal line, parallel of course with _a b_. You will thus have a
trapezium _a b c d_ divided into as many spaces as the square A B C D in
fig. 62, and it now remains to fill them in with similar portions of the
figure. Thus, for instance, the nose is in the fourth vertical division,
starting from the left, and in the third and fourth counting from the
top; in order, therefore, to make it occupy so lengthened a space it
must be considerably distorted by the pencil. It will be readily seen
also that the more numerous the spaces into which the square is divided,
the easier it will be to draw the distorted picture. It is by this means
that the _anamorphosis_ shown in fig. 63 has been drawn.
The next thing to do is to find the point of view from which we can see
the figure in its natural proportions. This will be found to be at a
distance above the point V equal to the line V S. In order to complete
the experiment it is simply necessary to place the distorted picture in
a horizontal position, and fix a piece of cardboard vertically at the
point V. If a hole be punched in it at a distance from V equal to S, and
the drawing be looked at through it, the whole of the parts will fall
into symmetry immediately.
The experiment may be tried first with fig. 63, the hole being made
rather large, and the eye placed at a distance of from 3 to 4 inches.
It would be difficult, without having recourse to geometrical formulæ,
to explain how it happens that by placing the eye at a particular point
the distorted lines of the drawing become symmetrical; but perhaps a
mechanical demonstration will help to make this difficult subject a
little plainer.
[Illustration:
FIG. 63.—Anamorphosis.
]
Draw in outline any figure upon a piece of cardboard, and make a series
of pin-holes along the most prominent lines of the drawing, taking care
that they are pretty close together. Place the perforated card in a
vertical position on a sheet of paper, so that the rays from a candle or
lamp may fall, on the flat surface beneath. On looking at the luminous
figure formed from the drawing, you will find that it is as much
distorted as the lady’s head in fig. 63, and that the lower you place
the candle the greater will be the deformity. You may if you please,
trace the luminous figure on the paper, and the result will appear
distorted when looked at in the ordinary manner, but symmetrical when
viewed from the point at which the flame of the candle was placed.
In the foregoing experiments we have spoken of the anamorphic drawings
as being placed in a horizontal position, but they may be looked at just
as well vertically, the card with the hole being in this instance
horizontal. It is also not necessary that the point of sight (V, fig.
63) should be in the centre of the picture; it may be placed at one side
or the other, care being taken to draw all the divisional lines so that
they meet at this particular spot. A few experiments with a candle and a
perforated figure will soon show the student how to accomplish this.
Anamorphoses by reflection may be prepared, if this principle is carried
out, which appear a mass of confused lines until they are reflected in a
cylindrical mirror. Formerly opticians were accustomed to construct
anamorphoses which became symmetrical pictures when viewed in a conical
mirror; but the fashion for such toys appears to have gone out. Such
drawings were extremely difficult to make, and the mirrors, having to be
ground and polished with great care, were very expensive.
Some experimentalists have carried the subject so far that, by looking
at the drawing of an object in particular positions, it changed into
quite a different subject. In the cloister of an abbey that once existed
in Paris, there were two anamorphoses of this kind. They were the work
of a certain Father Niceron, who has left behind him a treatise in Latin
on optical wonders, entitled _Thaumaturgus Opticus_, which contains a
long essay on anamorphoses. One of these pictures represented St. John
the Evangelist writing his Gospel; the other Mary Magdalene. When looked
at in the ordinary manner, they appeared to be landscapes; but when the
observer placed himself in a particular position, they changed into the
figures we have mentioned.
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