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Title: The kaleidoscope
Author: David Brewster
Release date: September 14, 2024 [eBook #74417]
Language: English
Original publication: London: John Camden Hotten, 1870
Credits: deaurider and the Online Distributed Proofreading Team at https://www.pgdp.net (This book was produced from images made available by the HathiTrust Digital Library.)
*** START OF THE PROJECT GUTENBERG EBOOK THE KALEIDOSCOPE ***
Transcriber’s Notes:
Underscores “_” before and after a word or phrase indicate _italics_
in the original text.
Equal signs “=” before and after a word or phrase indicate =bold= in
the original text.
Small capitals have been converted to SOLID capitals.
Illustrations have been moved so they do not break up paragraphs.
Old or antiquated spellings have been preserved.
Typographical and punctuation errors have been silently corrected.
The list of works by Sir David Brewster has been moved form the first
page to the end of the text.
[Illustration]
THE KALEIDOSCOPE,
ITS HISTORY, THEORY, AND CONSTRUCTION.
WITH ITS APPLICATION TO THE FINE AND USEFUL ARTS.
BY
SIR DAVID BREWSTER, K.H., M.A., D.C.L.,
F.R.S., V.P.R.S., EDIN., M.R.I.A., F.G.S., F.R.A.S.,
ASSOCIATE OF THE IMPERIAL INSTITUTE OF FRANCE, HONORARY OR
CORRESPONDING MEMBER OF THE ACADEMIES OF PETERSBURGH, VIENNA, BERLIN,
COPENHAGEN, STOCKHOLM, BRUSSELS, GOTTINGEN, MODENA, AND OF THE
NATIONAL INSTITUTE OF WASHINGTON, AND PRINCIPAL OF THE
UNITED COLLEGES OF ST. SALVATOR AND ST. LEONARD, ST. ANDREWS.
_Nihil tangit quod non ornat._
[Illustration]
Third Edition, greatly enlarged.
WITH FIFTY-SIX WOOD ENGRAVINGS AND ONE PLATE.
LONDON:
JOHN CAMDEN HOTTEN, PICCADILLY.
1870.
CONTENTS.
PAGE
INTRODUCTION—HISTORY OF THE KALEIDOSCOPE, 1
CHAP. I.—PRELIMINARY PRINCIPLES RESPECTING THE
EFFECTS OF COMBINING TWO PLAIN MIRRORS, 9
II.—ON THE PRINCIPLES OF THE KALEIDOSCOPE,
AND THE FORMATION OF SYMMETRICAL PICTURES BY THE
COMBINATION OF DIRECT AND INVERTED IMAGES, 16
III.—ON THE EFFECTS PRODUCED BY THE MOTION OF
THE OBJECT AND THE MIRRORS, 26
IV.—ON THE EFFECTS PRODUCED UPON THE SYMMETRY OF
THE PICTURE BY VARYING THE POSITION OF THE EYE, 37
V.—ON THE EFFECTS PRODUCED UPON THE SYMMETRY
OF THE PICTURE BY VARYING THE POSITION OF THE OBJECT, 46
VI.—ON THE INTENSITY OF THE LIGHT IN
DIFFERENT PARTS OF THE FIELD, AND ON THE EFFECTS
PRODUCED BY VARYING THE LENGTH AND BREADTH OF THE
REFLECTORS, 51
VII.—ON THE CONSTRUCTION AND USE OF THE SIMPLE
KALEIDOSCOPE, 59
VIII.—ON THE SELECTION OF OBJECTS FOR THE
KALEIDOSCOPE, AND ON THE MODE OF CONSTRUCTING THE
OBJECT-BOX, 67
IX.—ON THE ILLUMINATION OF TRANSPARENT OBJECTS
IN THE KALEIDOSCOPE, 76
X.—ON THE CONSTRUCTION AND USE OF THE TELESCOPIC
KALEIDOSCOPE, FOR VIEWING OBJECTS AT A DISTANCE, 81
XI.—ON THE CONSTRUCTION AND USE OF POLYANGULAR
KALEIDOSCOPES, IN WHICH THE REFLECTORS CAN BE FIXED
AT ANY ANGLE, 88
1. Bate’s Polyangular Kaleidoscope with Metallic
Reflectors, 89
2. Bate’s Polyangular Kaleidoscope with Glass
Reflectors, 94
XII.—ON THE CONSTRUCTION AND USE OF ANNULAR AND
PARALLEL KALEIDOSCOPES, 98
1. Mr. Dollond’s Universal Kaleidoscope, 100
2. Ruthven’s Universal Kaleidoscope, 102
XIII.—ON THE CONSTRUCTION AND USE OF POLYCENTRAL
KALEIDOSCOPES, 105
1. On combinations of four mirrors forming a square, 107
2. On combinations of four mirrors forming a
rectangle, 109
3. On combinations of three reflectors at angles
of 60°, 109
4. On combinations of three reflectors at angles
of 90°, 45°, and 45°, 111
5. On combinations of three reflectors at angles
of 90°, 60°, and 30°, 112
XIV.—ON KALEIDOSCOPES IN WHICH THE EFFECT
IS PRODUCED BY TOTAL REFLEXION FROM THE INTERIOR
SURFACES OF TRANSPARENT SOLIDS, 114
XV.—ON THE APPLICATION OF THE KALEIDOSCOPE TO
THE MAGIC LANTERN, SOLAR MICROSCOPE, AND CAMERA
OBSCURA, 117
XVI.—ON THE CONSTRUCTION OF KALEIDOSCOPES WHICH
COMBINE THE COLOURS AND FORMS PRODUCED BY
POLARIZED LIGHT, 122
XVII.—ON THE CONSTRUCTION OF STEREOSCOPIC
KALEIDOSCOPES, 126
XVIII.—ON THE CONSTRUCTION OF MICROSCOPIC
KALEIDOSCOPES, 128
XIX.—ON THE CHANGES PRODUCED BY THE KALEIDOSCOPE, 131
XX.—ON THE APPLICATION OF THE KALEIDOSCOPE TO
THE FINE AND USEFUL ARTS, 134
1. Architectural Ornaments, 137
2. Ornamental Painting, 141
3. Designs for Carpets, 144
XXI.—ON THE PHOTOGRAPHIC DELINEATION OF THE
PICTURES CREATED BY THE KALEIDOSCOPE, 148
XXII.—ON THE ADVANTAGES OF THE KALEIDOSCOPE AS
AN INSTRUMENT OF AMUSEMENT, 154
XXIII.—HISTORY OF THE COMBINATIONS OF PLANE
MIRRORS WHICH HAVE BEEN SUPPOSED TO RESEMBLE
THE KALEIDOSCOPE, 162
1. Baptista Porta’s multiplying speculum, 164
2. Kircher’s combination of plane mirrors, 168
3. Bradley’s combination of plane mirrors, 175
APPENDIX, 185
ON THE KALEIDOSCOPE.
INTRODUCTION.
HISTORY OF THE KALEIDOSCOPE.
The name Kaleidoscope, which I have given to a new Optical Instrument,
for creating and exhibiting beautiful forms, is derived from the Greek
words χαλός, _beautiful_; εἶδος, _a form_; and σχοπέω, _to see_.
The first idea of this instrument presented itself to me in the year
1814, in the course of a series of experiments on the polarization of
light by successive reflexions between plates of glass, which were
published in the _Philosophical Transactions_ for 1815, and which the
Royal Society did me the honour to distinguish by the adjudication of
the Copley Medal. In these experiments, the reflecting plates were
necessarily inclined to each other during the operation of placing
their surfaces in parallel planes; and I was therefore led to remark
the circular arrangement of the images of a candle round a centre,
and the multiplication of the sectors formed by the extremities of
the plates of glass. In consequence, however, of the distance of the
candles, &c., from the ends of the reflectors, their arrangement was
so destitute of symmetry, that I was not induced to give any farther
attention to the subject.
On the 7th of February 1815, when I discovered the development of the
complementary colours, by the successive reflexions of polarized light
between two plates of gold and silver, the effects of the Kaleidoscope,
though rudely exhibited, were again forced upon my notice; the
multiplied images were, however, coloured with the most splendid tints;
and the whole effect, though inconceivably inferior to the creations
of the Kaleidoscope, was still far superior to anything that I had
previously witnessed.
In giving an account of these experiments to M. Biot on the 6th of
March 1815, I remarked to him, “that when the angle of incidence (on
the plates of silver) was about 85° or 86°, and the plates almost
in contact, and inclined at a very small angle, the two series of
reflected images appeared at once in the form of two curves; and that
the succession of splendid colours formed a phenomenon which I had no
doubt would be considered, by every person who saw it to advantage, as
one of the most beautiful in optics.” These experiments were afterwards
repeated with more perfectly polished plates of different metals, and
the effects were proportionally more brilliant: but notwithstanding
the beauty arising from the multiplication of the images, and the
additional splendour which was communicated to the picture by the
richness of the polarized tints, it was wholly destitute of symmetry,
as I was then ignorant of those positions for the eye and the objects,
which are absolutely necessary to produce that magical union of parts,
and that mathematical symmetry throughout the whole picture, which,
independently of all colouring, give to the visions of the Kaleidoscope
the peculiar charm which distinguishes them from all artificial
creations.[1]
[1] An account of the experiments above alluded to was given in
the _Analyse des Travaux de la Classe des Sciences Mathématiques
et Physiques de l’Institut Royal de France, pendant l’année 1815_,
par M. le Chev. Delambre, p. 29, &c. The colours produced by
repeated reflexions from plates of silver are those of _Elliptical
Polarization_, and are explained at great length in my paper on that
subject, published in the _Philosophical Transactions_ for 1830.
Although I had thus combined two plain mirrors, so as to produce highly
pleasing effects, from the multiplication and circular arrangement of
the images of objects placed at a distance from their extremities, yet
I had scarcely made a step towards the invention of the Kaleidoscope.
The effects, however, which I had observed, were sufficient to prepare
me for taking advantage of any suggestion which experiment might
afterwards throw in the way.
In repeating, at a subsequent period, the very beautiful experiments
of M. Biot, on the action of homogeneous fluids upon polarized light,
and in extending them to other fluids which he had not tried, I found
it most convenient to place them in a triangular trough, formed by two
plates of glass cemented together by two of their sides, so as to form
an acute angle. The ends being closed up with pieces of plate glass
cemented to the other plates, the trough was fixed horizontally, for
the reception of the fluids. The eye being necessarily placed without
the trough, and at one end, some of the cement, which had been pressed
through between the plates at the object end of the trough, appeared
to be arranged in a manner far more regular and symmetrical than I
had before observed when the objects, in my early experiments, were
situated at a distance from the reflectors. From the approximation to
perfect symmetry which the figure now displayed, compared with the
great deviation from symmetry which I had formerly observed, it was
obvious that the progression from the one effect to the other must
take place during the passage of the object from the one position to
the other; and it became highly probable, that a position would be
found where the symmetry was mathematically perfect. By investigating
this subject optically, I discovered the leading principles of the
Kaleidoscope, in so far as the inclination of the reflectors, the
position of the object, and the position of the eye, were concerned.
I found, that in order to produce perfectly beautiful and symmetrical
forms, three conditions were necessary.
1. That the reflectors should be placed at an angle, which was an
_even_ or an _odd_ aliquot part of a circle, when the object was
regular, and similarly situated with respect to both the mirrors; or
the _even_ aliquot part of a circle when the object was irregular, and
had any position whatever.
2. That out of an infinite number of positions for the object, both
within and without the reflectors, there was _only one_ where perfect
symmetry could be obtained, namely, when the object was placed in
contact with the ends of the reflectors. This was precisely the
position of the cement in the preceding experiment with the triangular
trough.
3. That out of an infinite number of positions for the eye, there was
_only one_ where the symmetry was perfect, namely, as near as possible
to the angular point, so that the circular field could be distinctly
seen; and that this point was the _only one_ out of an infinite number
at which the uniformity of the light of the circular field was a
maximum, and from which the direct and the reflected images had the
same form and the same magnitude, in consequence of being placed at the
same distance from the eye. This, also, was the position in which the
eye was necessarily placed when looking through the fluid with which
the glass trough was partially filled.
Upon these principles I constructed an instrument, in which I fixed
_permanently_, across the ends of the reflectors, pieces of coloured
glass, and other irregular objects; and I showed the instrument in this
state to some members of the Royal Society of Edinburgh, who were much
struck with the beauty of its effects. In this case, however, the forms
were nearly permanent; and slight, though beautiful, variations were
produced by varying the position of the instrument with respect to the
source of light.
The great step, however, towards the completion of the instrument
remained yet to be made; and it was not till some time afterwards that
the idea occurred to me _of giving motion to objects, such as pieces
of coloured glass, &c., which were either fixed or placed loosely in
a cell at the end of the instrument_. When this idea was carried into
execution, and the reflectors placed in a tube, and fitted up on the
preceding principles, the Kaleidoscope, in its _simple form_, was
completed.
In this form, however, the Kaleidoscope could not be considered as
a general philosophical instrument of universal application. The
least deviation of the object from the position of symmetry at the
end of the reflectors, produced a deviation from beauty and symmetry
in the figure, and this deviation increased with the distance of the
object. The use of the instrument was therefore limited to objects in
contact with the ends of the reflectors, or held close to them, and
consequently to objects, or groups of objects, whose magnitudes were
less than its triangular aperture.
The next, and by far the most important step of the invention, was
to remove this limitation, and to extend indefinitely the use and
application of the instrument. This effect was obtained by employing a
draw tube, containing a convex lens, or, what is better, an achromatic
object-glass of such a focal length, that the images of objects, of
all magnitudes and at all distances, might be distinctly formed at the
end of the reflectors, and introduced into the pictures created by the
instrument in the same manner as if they had been reduced in size, and
placed in the true position in which alone perfect symmetry could be
obtained.
When the Kaleidoscope was brought to this degree of perfection, it was
impossible not to perceive that it would prove of the highest service
in all the ornamental arts, and would, at the same time, become a
popular instrument for the purposes of rational amusement. With these
views I thought it advisable to secure the exclusive property of it
by a Patent;[2] but in consequence of one of the patent instruments
having been exhibited to some of the London opticians, the remarkable
properties of the Kaleidoscope became known before any number of them
could be prepared for sale. The sensation excited by this premature
exhibition of its effects is incapable of description, and can be
conceived only by those who witnessed it. “It very quickly became
popular,” says Dr. Roget, in his excellent article on the KALEIDOSCOPE
in the _Encyclopædia Britannica_, “and the sensation it excited in
London throughout all ranks of people was astonishing. It afforded
delight to the poor as well as the rich; to the old as well as the
young. Large cargoes of them were sent abroad, particularly to the East
Indies. They very soon became known throughout Europe, and have been
met with by travellers even in the most obscure and retired villages in
Switzerland.” According to the computation of those who were best able
to form an opinion on the subject, no fewer than two hundred thousand
instruments were sold in London and Paris during three months. Out of
this immense number there were perhaps not one thousand constructed
upon scientific principles, and capable of giving anything like a
correct idea of the power of the Kaleidoscope; and of the millions who
have witnessed its effects, there is perhaps not a hundred individuals
who have any idea of the principles upon which it is constructed, who
are capable of distinguishing the spurious from the real instrument, or
who have sufficient knowledge of its principles to be able to apply it
to the numerous branches of the useful and ornamental arts.
Under these circumstances I have thought it necessary to draw up
the following short treatise, for the purpose of explaining, in as
popular a manner as I could, the principles and construction of the
Kaleidoscope; of describing the different forms in which it is fitted
up; of pointing out the various methods of using it as an instrument
of recreation; and of instructing the artist how to employ it in the
numerous branches of the useful and ornamental arts to which it is
applicable.
[2] As this Patent, in so far as the simple Kaleidoscope is concerned,
was to a great extent infringed, it has been supposed that it was
reduced in a Court of Law. The validity of the Patent was never
questioned by any lawyer, or any philosopher acquainted with its theory
and construction, as will appear from the opinion of four competent
judges, given in the _Appendix_.
In a trial for the infringement of a Patent several years ago, a
distinguished judge (we believe it was Judge Alderson) stated it as
a fact, that the Patent for the Kaleidoscope had been set aside in a
Court of Law. The party whose case was prejudiced by this erroneous
assertion, applied to me for an affidavit, by which he was enabled to
contradict it in Court, and remove any unfavourable impression it might
have made upon the jury.
CHAPTER I.
PRELIMINARY PRINCIPLES RESPECTING THE EFFECTS OF COMBINING TWO PLAIN
MIRRORS.
The principal parts of the Kaleidoscope are two reflecting planes, made
of glass, or metal, or any other reflecting substance ground perfectly
flat and highly polished. These reflectors, which are generally made
of plate glass, either rough ground on their outer side, or covered
with black varnish, may be of any size, but in general they should be
from four or five to ten or twelve inches long; their greatest breadth
being about an inch when the length is six inches, and increasing in
proportion as the length increases. When these two plates are put
together at an angle of 60°, or the sixth part of a circle, as shown
in Fig. 1, and the eye placed at the narrow end =E=, it will observe
the opening =A O B= multiplied six times, and arranged round the centre
=O=, as shown in Fig. 2.
[Illustration: FIG. 1.]
[Illustration: FIG. 2.]
In order to understand how this effect is produced, let us take a
small sector of white paper of the shape =A O B=, Fig. 2, and having
laid it on a black ground, let the extremity =A O= of one of the
reflectors be placed upon the edge =A O= of the sector. It is then
obvious that an image =A O _b_= of this white sector of paper will be
formed behind the mirror =A O=, and will have the same magnitude and
the same situation behind the mirror as the sector =A O B= had before
it. In like manner, if we place the edge =B O= of the other reflector
upon the other side =B O= of the paper sector, a similar image =B O
_a_= will be formed behind it. The origin of three of the sectors seen
round =O= is therefore explained: the first, =A O B=, is the white
paper sector seen by direct vision; the second, =A O _b_=, is an image
of the first formed by _one_ reflexion from the mirror =A O=; and the
third is another image of the first formed by _one_ reflexion from the
other mirror =B O=. But it is well known, that the reflected image of
any object, when placed before another mirror, has an image of itself
formed behind this mirror, in the very same manner as if it were a new
object. Hence it follows, that the image =A O _b_= being, as it were, a
new object placed before the mirror =B O=, will have an image =_a_ O α=
of itself formed behind =B O=; and for the same reason the image =B O
_a_= will have an image =_b_ O β= of itself, formed behind the mirror =A
O=, and both these new images will occupy the same position behind the
mirrors as the other images did before the mirrors.
A difficulty now presents itself in accounting for the formation of the
last or sixth sector, =α O β=. Mr. Harris, in the XVIIth Prop. of his
Optics, has evaded this difficulty, and given a false demonstration of
the proposition. He remarks, that the last sector, =α O β= is produced
“by the reflexion of the rays forming either of the two last images”
(namely, =_b_ O β= and =_a_ O α=); but this is clearly absurd, for the
sector =α O β= would thus be formed of two images lying above each
other, which is impossible. In order to understand the true cause of
the formation of the sector =α O β=, we must recollect that the line =O
E= is the line of junction of the mirrors, and that the eye is placed
any where in the plane passing through =O E= and bisecting =A O B=.
Now, if the mirror, =B O=, had extended as far as =O β=, the sector =α
O β= would have been the image of the sector =_b_ O β=, reflected from
=B O=; and in like manner, if the mirror =A O= had extended as far as
=O α=, the sector =α O β= would have been the image of the sector =_a_
O α= reflected from =A O=; but as this overlapping or extension of the
mirrors is impossible, and as they must necessarily join at the line
=O E=, it follows, that an image =α O _e_=, of only half the sector
=_b_ O β=, viz., =_b_ O _r_=, can be seen by reflexion from the mirror
=B O=; and that an image =β O _e_=, of only half the sector =_a_ O α=,
viz., =_a_ O _s_=, can be seen by reflexion from the mirror =A O=.
Hence it is manifest, that the last sector, =α O β=, is not, as Mr.
Harris supposes, _a reflexion from either of the two last images_, =_b
o_ β=, =_a o_ α=, but is composed of the images of two half sectors,
one of which is formed by the mirror =A O=, and the other by the mirror
=B O=.
Mr. Harris repeats the same mistake in a more serious form, in his
second Scholium, § 240, where he shows that the images are arranged
in the circumference of a circle. The two images =D=, _d_, says he,
coincide and make but one image. Mr. Wood has committed the very same
mistake in his second Corollary to Prop. XIV., and his demonstration
of that Corollary is decidedly erroneous. This Corollary is stated in
the following manner:—“When _a_ (the angle of the mirrors) is a measure
of 180° _two images coincide_,” and it is demonstrated, that _since
two images of any object_ =X= (Fig. 2) _must be formed_, viz., one by
each mirror, and since these two images must be formed at 180° from
the object =X=, placed between the mirrors, that is, at the same point
_x_, it follows that the two images must coincide. Now, it will appear
from the simplest considerations, that the assumption, as well as the
conclusion, is erroneous. The image _x_ is seen by the last reflexion
from the mirror =B O E=, and another image _would be seen at x_, if the
mirror =A O E= had extended as far as _x_; but as this is impossible,
without covering the part of the mirror =B O E=, which gives the first
image _x_, there can be only one image seen at _x_. When the object =X=
is equidistant from =A= and =B=, then one-half of the last reflected
image _x_ will be formed by the last reflexion from the mirror =B O=,
and the other half by the last reflexion from the mirror =A O=, and
these two half images will join each other, and form a whole image at
_e_, as perfect as any of the rest. In this last case, when the angle
=A O B= is a little different from an even aliquot part of 360°, the
eye at =E= will perceive at _e_ an appearance of two incoincident
images; but this arises from the pupil of the eye being partly on one
side of =E= and partly on the other; and, therefore, the apparent
duplication of the image is removed by looking through a very small
aperture at =E=. As the preceding remarks are equally true, whatever be
the inclination of the mirrors, provided it is an even aliquot part of
a circle, it follows,—
1. That when =A O B= is ¼, ⅙, ⅛, ⅒, ¹/₁₂, etc., of a circle, the number
of reflected images of any object =X=, is 4 - 1, 6 - 1, 8 - 1, 10 - 1,
12 - 1.
2. That when =X= is nearer one mirror than another, the number of
images seen by reflexion from the mirror to which it is nearest will
be ⁴/₂, ⁶/₂, ⁸/₂, ¹⁰/₂, ¹²/₂, while the number of images formed by the
mirror from which =X= is most distant will be ⁴/₂ - 1, ⁶/₂ - 1, ⁸/₂ -
1, ¹⁰/₂ - 1; that is, an image more always reaches the eye from the
mirror nearest =X=, than from the mirror farthest from it.
3. That when =X= is equidistant from =A O= and =B O=, the number of
images which reaches the eye from each mirror is equal, and is always
4 - 1, 6 - 1, 8 - 1, 10 - 1, 12 - 1
----- ----- ----- ------ ------
2 2 2 2 2
which are fractional values, showing that the last image is composed of
two half images.
When the inclination of the mirrors, or the angle =A O B=, Fig. 3,
is an _odd_ aliquot part of a circle, such as ⅓, ⅕, ⅐, ⅑, etc., the
different sectors which compose the circular image are formed in the
very same manner as has been already described; but as the number of
_reflected sectors_ must in this case always be _even_, the line =O E=,
where the mirrors join, will separate the two last reflected sectors,
=_b_ O _e_=, =_a_ O _e_=. Hence it follows,—
[Illustration: FIG. 3.]
1. That when =A O B= is ⅓, ⅕, ⅐, ⅑, etc., of a circle, the number of
reflected images of any object is 3 - 1, 5 - 1, 7 - 1, etc., and,—
2. That the number of images which reach the eye from each mirror is
3 - 1 5 - 1 7 - 1,
-----, -----, -----,
2 2 2
which are always even numbers.
Hitherto we have supposed the inclination of the mirrors to be
_exactly_ either an even or an odd aliquot part of a circle. We shall
now proceed to consider the effects which will be produced when this is
not the case.
If the angle =A O B=, Fig. 2, is made to increase from being an _even_
aliquot part of a circle, such as ⅙th, till it becomes an _odd_
aliquot part, such as ⅐th, the last reflected image =β O α=, composed
of the two halves =β O _e_=, =α O _e_=, will gradually increase, in
consequence of each of the halves increasing; and when =A O B= becomes
⅐th of the circle, the sector =β O α= will become double of =A O B=,
and =α O _e_=, =β O _e_= will become each complete sectors, or equal to
=A O B=.
If the angle =A O B= is made to vary from ⅙th to ⅕th of a circle, the
last sector =β O α= will gradually diminish, in consequence of each of
its halves, =β O _e_=, =α O _e_=, diminishing; and just when the angle
becomes ⅙th of a circle, the sector =β O α= will have become infinitely
small, and the two sectors, =_b_ O β=, =_a_ O α=, will join each other
exactly at the line =O _e_=, as in Fig. 3.
CHAPTER II.
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=.
CHAPTER III.
ON THE EFFECTS PRODUCED BY THE MOTION OF THE OBJECT AND THE MIRRORS.
Hitherto we have considered both the object and the mirrors as
stationary, and we have contemplated only the effects produced by the
union of the different parts of the picture. The variations, however,
which the picture exhibits, have a very singular character, when either
the objects or the mirrors are put in motion. Let us, first, consider
the effects produced by the motion of the object when the mirrors are
at rest.
[Illustration: FIG. 9.]
If the object moves from =X= to =O=, Fig. 9, in the direction of
the radius, all the images will likewise move towards =O=, and the
patterns will have the appearance of being absorbed or extinguished in
the centre. If the motion of the object is from =O= to =X=, the images
will also move outwards in the direction of the radii, and the pattern
will appear to develop itself from the centre =O=, and to be lost or
absorbed at the circumference of the luminous field. The objects that
move parallel to =X O= will have their centre of development, or their
centre of absorption, at the point in the lines =A O=, =B O=, _a_ =O=,
_b_ =O=, etc. where the direction in which the images move cuts these
lines. When the object passes across the field in a circle concentric
with =A B=, and in the direction =A B=, the images in all the sectors
formed by an even number of reflexions will move in the same direction
=A B=, namely, in the direction β _b_, _a_ α; while those that have
been formed by an odd number of reflexions will move in an opposite
direction, namely, in the directions _a_ =B=, =A= _b_. Hence, if the
object moves from =A= to =B=, the points of absorption will be in the
lines =B O=, α =O=, and _b_ =O=, and the points of development in the
lines =A O=, _a_ =O=, and β =O=, and _vice versa_, when the motion of
the object is from =B= to =A=.
If the object moves in an oblique direction _m n_, the images will move
in the directions _m t_, _o n_, _o p_, _q t_, _q p_, and _m_, _o_, _q_,
will be the centres of development, and _n_, _p_, _t_, the centres of
absorption; whereas, if the object moves from _n_ to _m_, these centres
will be interchanged. These results are susceptible of the simplest
demonstration, by supposing the object in one or two successive points
of its path _m n_, and considering that the image must be formed at
points similarly situated behind the mirrors; the line passing through
these points will be the path of the image, and the order in which the
images succeed each other will give the direction of their motion.
Hence, we may conclude in general,
1. That when the path of the object cuts both the mirrors =A O= and =B
O= like _m n_, the centre of absorption will be in the radius passing
through the section of the mirror to which the object moves, and in
every alternate radius; and that the centre of development will be in
the radius passing through the section of the mirror from which the
object moves, and in all the alternate radii: and,
2. That when the path of the object cuts any one of the mirrors and
the circumference of the circular field, the centre of absorption will
be in all the radii which separate the sectors, and the centre of
development in the circumference of the field, if the motion is towards
the mirror, but _vice versa_ if the motion is towards the circumference.
When the objects are at rest, and the Kaleidoscope in motion, a new
series of appearances is presented. Whatever be the direction in which
the Kaleidoscope moves, the object seen by direct vision must always
be stationary, and it is easy to determine the changes which take
place when the Kaleidoscope has a progressive motion over the object.
A very curious effect, however, is observed when the Kaleidoscope has
a rotatory motion round the angular point, or rather round the common
section of the two mirrors. The picture created by the Instrument
seems to be composed of two pictures, one in motion round the centre
of the circular field, and the other at rest. The sectors formed by an
odd number of reflexions are all in motion in the same direction as
the Kaleidoscope, while the sector seen by direct vision, and all the
sectors formed by an even number of reflexions, are at rest. In order
to understand this, let =M=, Fig. 10, be a plane mirror, and =A= an
object whose image is formed at _a_, so that =_a_ M = A M=. Let the
mirror =M= advance to =N=, and the object =A=, which remains fixed,
will have its image _b_ formed at such a distance behind =N=, that =_b_
N = A N=; then it will be found that the space moved through by the
image is double the space moved through by the mirror; that is, =_a b_
= 2 M N=. Since =M N = A M - A N=, and since =A M = _a_ M=, and
=A N = _b_ N=, we have =M N = _a_ M - _b_ N=; and adding =M N= or
its equal =_b_ M + _b_ N= to both sides of the equation, we obtain
=2M N = _a_ M - _b_ N + _b_ N + _b_ M=; but =-_b_ N + _b_N = 0=,
and =_a_ M + _b_ M = _a b_=; hence =2M N = _a b_=. This result may be
obtained otherwise, by considering, that if the mirror =M= advances
_one inch_ towards =A=, one inch is added to the distance of the image
_a_, and one subtracted from the distance of the object; that is, the
difference of these distances is now two inches, or twice the space
moved through by the mirror; but since the new distance of the object
is equal to the distance of the new image, the difference of these
distances, which is the space moved through by the image, must be two
inches, or twice the space described by the mirror.
[Illustration: FIG. 10.]
Let us now suppose that the object A advances in the same direction as
the mirror, and with twice its velocity, so as to describe a space =A
α = 2 M N = _a b_=, in the same time that the mirror moves through =M
N=, the object being at α when the mirror is at =N=. Then, since =A
α = _a b_= and =_b_ N = A N=, the whole =α N= is equal to the whole
=_a_ N=, that is, =_a_= will still be the place of the image. Hence it
follows, _that if the object advances in the same direction as the
mirror, but with twice its velocity, the image will remain stationary_.
[Illustration: FIG. 11.]
If the object =A= moves in a direction opposite to that of the mirror,
and with double its velocity, as is shown in Fig. 11; then, since _b_
would be the image when =A= was stationary, and when =M= had moved to
=N=, in which case =_a b_ = 2 M N=, and _bʹ_ the image when =A= had
advanced to =α= through a space =A α = 2 M N=, we have =_b_ N = A N=, and
=_bʹ_ N = α N=, and, therefore,= _b bʹ_ = A N - α N = A α = 2 M N=, and
=_a b_ + _b bʹ_= or its equal =_a bʹ_ = 4 M N=. Hence it follows, _that
when the object advances towards the mirror with twice its velocity,
the image will move with four times the velocity of the mirror_.
If the mirror =M= moves round a centre, the very same results will be
obtained from the very same reasoning, only the angular motion of the
mirror and the image will then be more conveniently measured by parts
of a circle or degrees.
[Illustration: FIG. 12.]
Now, in Fig. 12, let =X= be a fixed object, and =A O=, =B O=, two
mirrors placed at an angle of 60° and moveable round =O= as a centre.
When the eye is applied to the end of the mirrors (or at =E=, Fig. 1),
the fixed object =X=, Fig. 12, seen by direct vision will, of course,
be stationary, while the mirrors describe an arch =X= of 10° for
example; but since =A O= has approached =X= by 10°, the image of =X=
formed behind =A O= must have approached =X= by 20°, and consequently
moves with twice the velocity in the same direction as the mirrors.
In like manner, since =B O= has receded 10° from =X=, the image of
=X= formed by =B O= must have receded 20° from =X=, and consequently
must have moved with twice the velocity in the same direction as the
mirrors. Now, the image of =X= in the sector =_b_ O β= is, as it were,
an image of the image in =B O _a_= reflected from =A O=. But the image
in =B O _a_= advances in the same direction as the mirror =A O= and
with twice its velocity, hence the image of it in the sector =_b_ O β=
will be stationary. In like manner it may be shown, that the image in
the sector =_a_ O α= will be stationary. Since =α O _e_= is an image of
=_b_ O _r_= reflected from the mirror =B O=, and since all images in
that sector are stationary, the corresponding images in =α O _e_= will
move in the same direction =α β= as the mirrors; and for the same reason
the images in the other half-sector =β O _e_= will move in the same
direction; hence, the image of any object formed in the last sector =α
O β= will move in the same direction, and with the same velocity as the
images in the sectors =A O _b_=, =B O _a_=.
By a similar process of reasoning, the same results will be obtained,
whatever be the number of the sectors, and whether the angle =A O B= be
the even or the odd aliquot part of a circle. Hence we may conclude,
1. That during the rotatory motion of the mirrors round =O=, the
objects in the sector seen by direct vision, and all the images of
these objects formed by an even number of reflexions are at rest.
2. That all the images of these objects, formed by an odd number of
reflexions, move round =O= in the same direction as the mirrors, and
with an angular velocity double that of the mirrors.
3. That when the angle =A O B= is an _even_ aliquot part of a circle,
the number of moving sectors is equal to the number of stationary
sectors, a moving sector being placed between two stationary sectors,
and _vice versa_.
4. That when the angle =A O B= is an _odd_ aliquot part of a circle,
the two last sectors adjacent to each other are either both in motion
or both stationary, the number of moving sectors being greater by one
when the number of sectors is 3, 7, 11, 15, etc., and the number of
stationary sectors being greater by one when the number of sectors is
5, 9, 13, 17, etc. And,
5. That as the moving sectors correspond with those in which the images
are inverted, and the stationary ones with those in which the images
are direct, the number of each may be found from the table given in
page 24.
When one of the mirrors, =A O=, is stationary, while the other, =B O=,
is moved round, and so as to enlarge the angle =A O B=, the object =X=,
and the image of it seen in the stationary mirror =A O=, remain at
rest, but all the other images are in motion receding from the object
=X=, and its stationary image; and when =B O= moves towards =A O=, so
as to diminish the angle =A O B=, the same effect takes place, only
the motion of the images is towards the object =X=, on one side, and
towards its stationary image on the other. These images will obviously
move in pairs; for, since the fixed object and its stationary image are
at an invariable distance, the existence of a symmetrical arrangement,
which we have formerly proved, requires that similar pairs be arranged
at equal distances round =O=, and each of the images of these pairs
must be stationary with regard to the other. Now, as the fixed object
is placed in the sector =A O B=, and its stationary image in the
sector =A O _b_=, it will be found that in the semicircle =M _b e_=,
containing the fixed mirror, the
1st reflected image and direct object, }
2d 3d reflected image }
4th 5th } are stationary with
6th 7th } respect to each other.
8th 9th }
while in the same semicircle =M _b e_=, the
1st reflected image and 2d reflected image }
3d 4th }
5th 6th } are movable with
7th 8th } respect to each other.
9th 10th }
On the other hand, in the semicircle =M _a e_=, containing the movable
mirror, the phenomena are reversed, the images which were formerly
stationary with respect to each other being now movable, and _vice
versa_.
In considering the velocity with which each pair of images revolves,
it will be readily seen that the pair on each side, and nearest the
fixed pair, will have an angular velocity _double_ that of the mirror
=B O=; the next pair on each side will have a velocity _four_ times as
great as that of the mirror; the next pair will have a velocity _eight_
times as great, and the next pair a velocity _sixteen_ times as great
as that of the mirror, the velocity of any pair being always double the
velocity of the pair which is adjacent to it on the side of the fixed
pair. The reason of this will be manifest, when we recollect what has
already been demonstrated, that the velocity of the image is always
double that of the mirror, when the mirror alone moves towards the
object, and quadruple that of the mirror when both are in motion, and
when the object approaches the mirror with twice the velocity. When =B
O= moves from =A O=, the image in the sector =B O _a_= moves with twice
the velocity of the mirror; but since the image in =_b_ O β= is an
image of the image in =B O _a_= reflected from the fixed mirror =A O=,
it also will move with the same velocity, or twice that of the mirror
=B O=. Again, the image in the sector =_a_ O α=, being a reflexion of
the stationary image in =A O _b_= from the moving mirror, will itself
move with double the velocity of the mirror. But the image in the next
sector =α O β= is a reflexion of the image in =_b_ O β= from the moving
mirror =B O=; and as this latter image has been shown to move in the
direction =_b_ β=, with twice the velocity of the mirror =B O=, while
the mirror =B O= itself moves towards the image, it follows that the
image in =α O β= will move with a velocity four times that of the
mirror. The same reasoning may be extended to any number of sectors,
and it will be found that in the semicircle =M _b e_=, containing the
fixed mirror,
The {2 and 3} 2} times the
images {4 and 5} reflexions, move with 4} velocity of
formed {6 and 7} 8} the mirror;
by {8 and 9} 16}
whereas in the semicircle =M _a e_=, containing the movable mirror,—
The {1 and 2} 2 { times the
images {3 and 4} reflexions, move with 4 { velocity of
formed {5 and 6} 8 { the mirror;
by {7 and 8} 16{
a progression which may be continued to any length.
Before concluding this chapter, it may be proper to mention a very
remarkable effect produced by moving the two plain mirrors along one of
two lines placed at right angles to each other. When the aperture of
the mirrors is crossed by each of the two lines, the figure created by
reflexion consists of two polygons with salient and re-entering angles.
By moving the mirrors along one of the lines, so that it may always
cross the aperture at the same angle, and at the same distance from the
angular point, the polygon formed by this line will remain stationary,
and of the same form and magnitude; but the polygon formed by the
other line, at first emerging from the centre, will gradually increase
till its salient angles touch the re-entering angles of the stationary
polygon; the salient angles becoming more acute, will enclose the
apices of the re-entering angles of the stationary polygon, and at last
the polygon will be destroyed by truncations from its salient angles.
When the lines cross each other at a right angle, the salient angles
of the opening polygon can never touch the salient angles of the
stationary polygon, but always its re-entering angles. If the lines,
however, form a less angle than the complement of the angle formed
by the mirrors, then the salient angles of the opening polygon may
touch the salient angles of the stationary polygon, by placing the
mirrors so as to form re-entering angles in the polygon. When the
lines form an angle between 90° and the complement of the angle formed
by the mirrors, the salient angle of the opening polygon may be made
to touch the salient angles of the stationary one, but in this case
the stationary polygon can have no re-entering angles. The preceding
effects are finely exemplified by the use of a Kaleidoscope with a
draw-tube and lens described in Chapter X., and by employing the
vertical and horizontal bars of a window, which may be set at different
angles, by viewing them in perspective.
CHAPTER IV.
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.
CHAPTER V.
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.
CHAPTER VI.
ON THE INTENSITY OF THE LIGHT IN DIFFERENT PARTS OF
THE FIELD, AND ON THE EFFECTS PRODUCED BY VARYING
THE LENGTH AND BREADTH OF THE REFLECTORS.
When we look through a Kaleidoscope in which the mirrors are placed
at an angle of 18° or 22½°, the eye will perceive a very obvious
difference in the intensity of the light in different parts of the
field. If the inclination of the mirrors be about 30°, and the eye
properly placed near the angular point, the intensity of the light is
tolerably uniform; and a person who is unaccustomed to the comparison
of different lights, will find it extremely difficult to distinguish
the direct sector from the reflected ones. This difficulty will be
still greater if the mirrors are made of finely polished steel, or
of the best speculum metal, and the observer will not hesitate in
believing that he is looking through a tube whose diameter is equal to
that of the circular field. This approximation to uniformity in the
intensity of the light in all the sectors, which arises wholly from the
determination of the proper position of the eye, is one of the most
curious and unexpected properties of the Kaleidoscope, and is one which
could not have been anticipated from any theoretical views, or from
any experimental results obtained from the ancient mode of combining
plain mirrors. It is that property, too, which gives it all its value;
for, if the eye observed the direct sector with its included objects
distinguished from all the rest by superior brilliancy, not only would
the illusion vanish, but the picture itself would cease to afford
pleasure, from the want of symmetry in the light of the field.
[Illustration: FIG. 18.]
Before we proceed to investigate the effects produced by a variation in
the length of the reflecting planes, it will be necessary to consider
the variation of the intensity of the light in different parts of the
reflected sectors. In the direct sector =A O B=, Fig. 2, the intensity
of the light is uniform in every part of its surface; but this is far
from being the case in the images formed by reflexion. In Fig. 17, take
any two points _m_, _o_, and draw the lines _m n_, _o p_, perpendicular
to =A O=, and meeting =β O= in _n_ and _p_. Let =O E=, Fig. 18, be a
section of the reflector =A O= seen edgewise, and let =O _p_=, =O _n_=,
be taken equal to the lines _m n_, _o p_, or the height of the points
_n_, _p_, above the plane of the reflector =A O=. Make =O R= to =R E=
as =O _p_= is to =E _e_= the constant height of the eye above the
reflecting plane, and =O _r_= to =_r_ E= as =O _n_= to =E _e_=, and the
points =R, _r_=, will be the points of incidence of the rays issuing
from _p_ and _n_; for in this case =O R _p_ = E R _e_=, and =O _r n_ =
E _r e_=. Hence it is obvious, that =E R _e_= is less than =E _r e_=,
and that the rays issuing from _p_, by falling more obliquely upon
the reflecting surface, will be more copiously reflected. It follows,
therefore, that the intensity of the light in the reflected sector =A
O β= is not uniform, the lines of equal brightness, or the _isophotal_
lines, as they may be called, being parallel to the reflecting surface
=A O=, and in every sector parallel to the radius, between the given
sector and the reflecting surface by which the sector is formed.
As it is easy from the preceding construction to determine the angles
at which the light from any points _m_, _n_, is reflected, when the
length =O E= of the reflectors, and the position of the eye at =E= is
given, we may calculate the intensity of the light in any point of
the circular field by means of the following table, which shows the
number of rays reflected at various angles of incidence, the number
of incident rays being supposed to be 1000. Part of this table was
computed by Bouguer for plate glass not quicksilvered, by means of a
formula deduced from his experiments. By the aid of the same formula I
have extended the table considerably.
_Table showing the quantity of light reflected at various angles of
incidence from plate glass._
+---------------+---------------+
| Complement of | Rays Reflected|
| the Angles | out |
| of Incidence. | of 1000. |
+---------------+---------------+
| 2½° | 584 |
| 5 | 543 |
| 7 | 474 |
| 10 | 412 |
| 12½ | 356 |
| 15 | 299 |
| 20 | 222 |
| 21 | 210 |
| 25 | 157 |
| 26 | 149 |
| 30 | 112 |
| 31 | 105 |
| 34 | 85 |
| 35 | 79 |
| 36 | 74 |
| 37 | 69 |
| 38 | 65 |
| 39 | 61 |
| 40 | 57 |
| 46 | 40 |
| 50 | 34 |
| 55 | 29 |
| 60 | 27 |
| 70 | 25 |
| 80 | 25 |
| 90 | 25 |
+---------------+---------------+
In order to explain the method of using the table, let us suppose that
the angle of incidence, or =O R _p_=, Fig. 19, is 85°: then the number
of rays in the corresponding point π of the reflected sector =A O _b_=
(Fig. 17) will be 543. By letting fall perpendiculars from the points
μ, π, upon the mirror =B O=, and taking =O _p_=, =O _n_=, Fig. 19,
equal to these perpendiculars, we may ascertain the angles at which the
light from the points μ, π, suffer a second reflexion from the mirror
=B O=. Let the angle for the point π be 10°, then the number of rays
out of 1000 reflected at this angle, according to the table, is 412;
but as the number of rays emanating from π, and incident upon =B O=, is
not 1000, but only 543, we must say as 1000: 412 = 543: 224, the number
of rays reflected from =B O=, or the intensity of the light in a point
in the line =O _bʹ_= corresponding to π.
[Illustration: FIG. 19.]
The preceding method of calculation is applicable only with strictness
to the two sectors =A O _b_=, =B O _a_=, formed by one reflexion, for
the intensity of the light in the other sectors which are formed by
more than one reflexion, must be affected by the polarization which
the light experiences after successive reflexions; for light which has
acquired this property is reflected according to laws different from
those which regulate the reflexion of direct light.
When the mirrors are metallic, the quantity of reflected light is also
affected by its polarization, but it is regulated by more complicated
laws.
In Kaleidoscopes made of plates of glass, the last reflected image
=β O ω=, Fig. 2, is more polarized than any of the rest, and is
polarized in a plane perpendicular to =X E=, or in the same manner as
if it had been reflected at the polarizing angle from a vertical plane
parallel to =X E=.
[Illustration: FIG. 20.]
Let us now consider what will take place by a variation in the length
of the reflecting planes, the angular extent of the field of view
remaining always the same. If =A O E=, =A O Eʹ=, Fig. 20, be two
reflecting plates of the same breadth =A O=, but of different lengths,
it is manifest that the light which forms the direct sector must be
incident nearer the perpendicular, or reflected at less obliquities in
the short plate than in the long one, and, therefore, that a similarly
situated point in the circular field of the shorter instrument, will
have less intensity of light than a similarly situated point in a
larger instrument. But in this case, the field of view in the short
instrument is proportionally enlarged, so that the comparison between
the two is incorrect. When the long and the short instrument have equal
apparent apertures, which will be the case when the plates are =A O E=,
=Aʹ O Eʹ=, then similarly situated points of the two fields will have
exactly the same intensity of light.
This will be better understood from Fig. 19, where =O E= may represent
the long reflector and =Oʹ E= the short one. Then, if these two have
exactly the same aperture, or a circular field of the same angular
magnitude, the rays of light which flow from two given points, _p_,
_n_, of the long instrument, will be reflected at a certain angle
from the points =R=, _r_; but as the points _pʹ_, _nʹ_, are the
corresponding points in the field of the shorter instrument, the rays
which issue from them will be reflected at the same angles from the
points =R=, _r_, the eye being in both cases placed at the same point
_e_. Hence it is obvious, that the quantity of reflected light will
in both cases be the same, and, therefore, that there is no peculiar
advantage to be derived, in so far as the light of the field is
concerned, by increasing the length of the reflectors, unless we raise
the eye above _e_, till every part of the pupil receives the reflected
rays.
There is, however, one advantage, and a very important one, to be
derived from an increase of length in the mirrors, namely a diminution
of the deviation from symmetry which arises from the small height of
the eye above the plane of the mirrors, and of the small distance
of the objects from the extremity of the mirrors. As the height of
the eye must always be a certain quantity, =E _e_=, Fig. 17, above
the angular point =E=, whatever be the length of the reflectors, it
is obvious, that when the length of the reflectors is =_e_ O=, the
deviation from symmetry will be only =P _oʹ_=, whereas when the length
of the reflectors is reduced to =_eʹ_ O=, the height of the eye =_eʹ_
Eʹ= being still equal to =_e_ E=, the aberration will be increased
to =P _o_=. This advantage is certainly of considerable consequence;
but in practice the difficulty of constructing a perfect instrument,
increases with the length of the reflectors. When the plates are long,
it is more difficult to get the surface perfectly flat; the risk of a
bending in the plates is also increased, which creates the additional
difficulty of forming a good junction, on which the excellence of the
instrument so much depends. By augmenting the length of the reflectors,
the quantity of dust which collects between them is also increased,
and it is then very difficult to remove this dust, without taking the
instrument to pieces. From these causes it is advisable to limit the
greatest length of the reflectors to seven or eight inches.
CHAPTER VII.
ON THE CONSTRUCTION AND USE OF THE SIMPLE KALEIDOSCOPE.
In order to construct the Kaleidoscope in its most simple form, we
must procure two reflectors, about five, six, seven, or eight inches
long. These reflectors may be either rectangular plates, or plates
shaped like those represented in Fig. 1, having their broadest ends
=A O=, =B O=, from one to two inches wide, and their narrowest ends
=_a_ E=, =_b_ E=, half an inch wide. If the reflectors are of glass,
the newest plate glass should be used, as a great deal of light is
lost by employing old plate glass, with scratches or imperfections
upon its surface. The plate glass may be either quicksilvered or not,
or its posterior surface may be ground, or covered with black wax, or
black varnish, or anything else that removes its reflective power.
This, however, is by no means absolutely necessary, for if the eye
is properly placed, the reflexions from the posterior surface will
scarcely affect the distinctness of the picture, unless in very intense
lights. If it should be thought necessary to extinguish, as completely
as possible, all extraneous light that may be thrown into the tube
from the posterior surface of the glass plates, that surface should be
coated with a varnish of the same refractive and dispersive power as
the glass.
If the plates of glass have been skilfully cut with the diamond, so as
to have their edges perfectly straight, and free from chips, two of
the edges may be placed together, as in Fig. 17 (p. 49), or one edge
of one plate may be placed against the surface of the other plate, as
shown in the section of Mr. Bates’s Kaleidoscope. But if the edges are
rough and uneven, one of them may be made quite straight, and freed
from all imperfections, by grinding it upon a flat surface, with very
fine emery, or with the powder scraped from a hone. When the two plates
are laid together, so as to form a perfect junction, they are then to
be placed in a brass or any other tube, so as to form an angle of 45°,
36°, 30°, or any even aliquot part of a circle. In order to do this
with perfect accuracy, direct the tube containing the reflectors to
any line, such as _m n_, Fig. 2, placed very obliquely to one of the
reflectors =A O=, and open or shut the plates till the figure of a star
is formed, composed of 8, 10, or 12 sectors, or with 4, 5, or 6 points,
corresponding to angles of 45°, 36°, and 30°. When all the points of
the star are equally perfect, and none of the lines which form the
salient and re-entering angles disunited, the reflectors must be fixed
in that position by small arches of brass or wood =A B=, =_a b_=, Fig.
21, filed down till they exactly fit the space between the open ends of
the plates. The plates must then be kept in this position by pieces or
wedges of cork or wood, or any other substance pushed between them and
the tube. The greatest care, however, must be taken that these wedges
press lightly upon the reflectors, for a very slight force is capable
of bending and altering the figure even of very thick plates of glass.
When the reflectors are thus placed in the tube, as in Fig. 21, their
extremities =_a_ E=, =_b_ E=, next the eye, must reach to the very end
of the tube, as it is of the greatest importance that the eye get as
near as possible to the reflectors. The other ends of the reflectors
=A O=, =B O=, must also extend to the other extremity of the tube, in
order that they may be brought into contact with the objects which are
to be applied to the instrument. In using transparent objects the cell
or box which contains them may be screwed into the end of the tube, so
as to reach the ends of the reflectors, if they happen to terminate
within the tube; but an instrument thus constructed is incapable of
being applied to opaque objects, or to transparent objects seen by
reflected light.
[Illustration: FIG. 21.]
If the plates are narrower at the eye-end, as in Fig. 21, the angular
point =E= should be a little on one side of the axis of the tube, in
order that the aperture in the centre of the brass cap next the eye
may be brought as near as possible to =E=. When the plates have the
same breadth at both ends, the angular point =E= will be near the
lower circumference of the tube, as it is at =O=; and in this case it
is necessary to place the eye-hole out of the centre, so as to be a
little above the angular point =E=. This construction is less elegant
than the preceding; but it has the advantage of giving more room for
the introduction of a feather, or a piece of thin wood covered with
leather, for the purpose of removing the dust which is constantly
accumulating between the reflectors. In some instances the plates have
been put together in such a manner that they may be taken out of the
tube, for the purpose of being cleaned; but though this construction
has its advantages, yet it requires some ingenuity to replace the
reflectors with facility, and to fix them at the exact inclination
which is required. One of the most convenient methods is to support the
reflector in a groove cut out of a solid cylinder of dry wood of nearly
the same diameter as the tube; and after a slip of wood, or any other
substance, is placed along the open edges of the plates, to keep them
at the proper angle given by the groove, the whole is slipped into the
tube, where it remains firm and secure from all accident.
If the length of the reflectors is less than the shortest distance at
which the eye is capable of seeing objects with perfect distinctness,
it will be necessary to place at the eye-end =E= a convex lens, whose
focal length is equal to, or an inch or two greater than, the length
of the reflectors. By this means the observer will see with perfect
distinctness the objects placed at the object end of the Kaleidoscope.
This lens, however, must be removed when the instrument is to be used
by persons who are short-sighted.
The proper application of the objects at the end of the reflectors
is now the only step which is required to complete the simple
Kaleidoscope. The method of forming, selecting, and mixing the objects,
will be described in the next chapter. At present, we shall confine our
attention to the various methods which may be employed in applying them
to the end of the reflectors.
[Illustration: FIG. 22.]
The first and most simple method consists in bringing the tube about
half an inch beyond the ends of the reflectors. A circular piece of
thin plane glass of the same diameter as the tube, is then pushed into
the tube, so as to touch the reflectors. The pieces of coloured glass
being laid upon this piece of glass when the tube is held in a vertical
position, another similar disc of plane glass, having its outer surface
ground with fine emery, is next placed above the glass fragments, being
prevented from pressing upon them, or approaching too near the first
plane glass by a ring of copper or brass; and is kept in its place
by burnishing down the end of the tube. The eye being placed at the
other end of the instrument, the observer turns the whole round in
his hand, and perceives an infinite variety of beautiful figures and
patterns, in consequence of the succession of new fragments, which are
brought opposite the aperture by their own gravity, and by the rotatory
motion of the tube. In this rude state, however, the instrument is by
no means susceptible of affording very pleasing exhibitions. A very
disagreeable effect is produced by bringing the darkest sectors, or
those formed by the greatest number of reflexions, to the upper part
of the circular field, and though the variety of patterns will be very
great, yet the instrument is limited to the same series of fragments,
and cannot be applied to the numerous objects which are perpetually
presenting themselves to our notice. These evils can be removed only
by adopting the construction shown in Fig. 22, in which the reflectors
reach the very end of the tube. Upon the end of the tube _a b_, _c
d_, Fig. 22, is placed a ring of brass, _m n_, which moves easily upon
the tube _a b c d_, and is kept in its place by a shoulder of brass
on each side of it. A brass cell, =M N=, is then made to slip tightly
upon the moveable ring _m n_, so that when the cell is turned round by
means of the milled end at =M N=, the ring _m n_ may move freely upon
the tube. The fragments of coloured glass, etc., are now placed in a
small object-box, as it may be called, consisting of two glasses, the
innermost of which, _m n_, is transparent, and the other ground on
the outside =P=, and kept at the distance of ⅛th or ⅒th of an inch by
a brass rim: this brass rim generally consists of two pieces, which
screw into one another, so that the object-plate can be opened by
unscrewing it, and the fragments changed at pleasure. This object-box
is placed at the bottom of the cell =M N=, as shown at =O P=, and the
depth of the cell is such as to allow the side =O= to touch the end
of the reflectors, when the cell is slipped upon the ring _m n_. When
this is done, the instrument is held in one hand with the angular point
=E=, Fig. 21, downwards, which is known by a mark on the upper side
of the tube between _a_ and _b_, and the cell is turned round with
the other hand, so as to present different fragments of the included
glass before the aperture =A O B=. The tube may be directed to the
brightest part of the sky in the day-time, or in the evening to a
candle, or an Argand Lamp, so as to transmit the light directly through
the coloured fragments; but it will always be found to give richer and
more brilliant effects if the tube is directed to the window-shutter,
a little to one side of the light, or is held to one side of the
candle—or, what is still better, between two candles or lamps placed
as near each other as possible. In this way the picture created by the
instrument is not composed of the harsh tints formed by transmitted
light; but of the various reflected and softened colours which are
thrown into the tube from the sides and angles of the glass fragments.
When the pattern remains fixed in any position of the instrument, a
variety of beautiful changes may be effected by making the end of the
tube revolve round a candle or a bright gas flame, placed near the
object-plate. The general pattern remains the same, but its colours
vary both in their position and intensity, as the light falls upon
different sides of the fragments of glass.
In the preceding method of applying the objects to the reflectors,
the fragments of coloured glass are introduced before the aperture,
and pass across it in concentric circles; and as the fragments always
descend by their own gravity, the changes in the picture, though
infinite in number, constantly take place in a similar manner. This
defect may be remedied, and a great degree of variety exhibited in
the motion of the fragments, by making the object-plates rectangular
instead of circular, and moving them through a groove cut in the cell
at =M N=, in the same manner as is done with the pictures or sliders
for the magic lantern and solar microscope. By this means the different
fragments that present themselves to the aperture may be made to pass
across it in every possible direction, and very interesting effects may
be produced by a combination of the rotatory and rectilineal motions of
the object-plate. When the object or objects are fixed, and the tube
with the reflectors moved round a centre, as described in Chapter
II.,[3] we have the same succession of symmetrical pictures; but in
this case every alternate sector is stationary, and the same number in
motion, the moving figures always changing their form, and assuming
that of the figures in the stationary sectors, which of course change,
while the ends of the mirror pass over the fixed objects.
[3] See Chapter X.
When the simple Kaleidoscope is applied to opaque objects, such as a
seal, a watch-chain, the seconds hands of a watch, coins, pictures,
gems, shells, flowers, leaves, and petals of plants, impressions
from seals, etc., the object, instead of being held between the eye
and the light, must be viewed in the same manner as we view objects
through a microscope, being always placed as near the instrument as
possible, and so as to allow the light to fall freely upon the object.
The object-plates, and all transparent objects, may be viewed in this
manner; but the most splendid exhibition of this kind is to view minute
fragments of coloured glass, and objects with opaque colours, etc.,
placed in a flat box, the bottom of which is made of mirror-glass. The
light reflected from the mirror-glass, and transmitted through the
transparent fragments, is combined with the light reflected both from
the transparent and opaque fragments, and forms an effect of the finest
kind.
As dust is apt to collect in the angle formed by the reflectors, it may
be removed when the reflectors are fixed, either by the end of a strong
feather, or blown away with a pair of bellows. When the dust is lodged
upon the face of the reflectors, it should be removed by a piece of
soft leather.
CHAPTER VIII.
ON THE SELECTION OF OBJECTS FOR THE KALEIDOSCOPE, AND
ON THE MODE OF CONSTRUCTING THE OBJECT-BOX.
Although the Kaleidoscope is capable of creating beautiful forms
from the most ugly and shapeless objects, yet the combinations which
it presents, when obtained from certain shapes and colours, are so
superior to those which it produces from others, that no idea can be
formed of the power and effects of the instrument, unless the objects
are judiciously selected.
When the inclination of the reflectors is great, the objects, or the
fragments of coloured glass, should be larger than when the inclination
is small; for when small fragments are presented before a large
aperture, the pattern which is created has a spotted appearance, and
derives no beauty from the inversion of the images, in consequence of
the outline of each separate fragment not joining with the inverted
image of it.
The objects which give the finest outlines by inversion, are those
which have a curvilineal form, such as circles, ellipses, looped
curves like the figure 8, curves like the figure 3 and the letter S;
spirals, and other forms, such as squares, rectangles, and triangles,
may be applied with advantage. Glass, both spun and twisted, and
of all colours, and shades of colours, should be formed into the
preceding shapes; and when these are mixed with pieces of flat coloured
glass, blue vitriol, native sulphur, yellow orpiment, differently
coloured fluids, enclosed and moving in small vessels of glass, etc.,
they will make the finest transparent objects for the Kaleidoscope.
When the objects are to be laid upon a mirror plate, fragments of
opaquely-coloured glass should be added to the transparent fragments,
along with pieces of brass wire, of coloured foils, and grains of
spelter. In selecting transparent objects, the greatest care must be
taken to reject fragments of opaque glass, and dark colours that do
not transmit much light; and all the pieces of spun glass, or coloured
plates, should be as thin as possible.
When the objects are thus prepared, the next step is to place them
in the object-box. The distance between the interior surfaces of the
two plane glasses, of which the object-plates are generally composed,
should be as small as possible, not exceeding ⅛th of an inch. The
outermost of these glasses has its external surface rough ground, or
is what is called a _grey glass_, the principal use of which is to
prevent the lines of external objects, such as the bars of the window,
or the outlines of the illuminating flame from being introduced into
the picture. When a strong light is used, a circular disk of fine thin
paper placed outside of the object-box may be advantageously employed
in place of the ground glass. The thickness of the transparent glass
plate next the reflectors should be just sufficient to keep the glass
from breaking; and the interior diameter of the brass rings into which
the transparent and the grey or ground plates of glass are burnished,
should be so great that no part of the brass rim may be opposite the
angular part of the reflectors during the rotatory motion of the
cell. If this precaution is not attended to, the central part of the
pattern, where the development of new forms is generally the most
beautiful, will be entirely obliterated by the interposition of the
brass rim. Instead of using transparent or grey glasses on the sides
of every object-box, some of the boxes should be made with disks of
flint glass, the interior surface of which have been stamped while in
a state of fusion with a sort of pattern, or with curved lines of a
pleasing form. In others, the outer surface alone of the plate next the
reflectors might be thus formed. An object-box might also be formed
of disks of glass, one side of which is colourless, and the other
coloured, some of the coloured portions being ground away irregularly,
as in certain Bohemian articles of glass; and the colour in one disk
may be complementary to that in the other. In object-boxes of this
kind, pieces of coloured glass may also be placed. When the two parts
of the object-boxes thus constructed are screwed or fixed together,
the box should be nearly two-thirds filled with the mixture of regular
and irregular objects, already mentioned. If they fall with difficulty
during the rotation of the cell, two or three turns of the screw
backward, when there is a screw, will relieve them; and if they fall
too easily, and accumulate, by slipping behind one another, the space
between the glasses may be diminished by placing within the box another
glass in contact with the grey glass.
When the object-box, now described, is placed in the cell, and examined
by the Kaleidoscope, the pictures which it forms are in a state
of perpetual change, and can never be fixed, and shown to another
person. To obviate this disadvantage, an object-box with fixed objects
generally accompanies the instrument; the pieces of spun and coloured
glass are fixed by a transparent cement to the inner side of the
glass of the object-plate, next the eye, so that the patterns are all
permanent, and may be exhibited to others. After the cell has performed
a complete rotation, the same patterns again recur, and may therefore
be at any time recalled at the pleasure of the observer. The same
patterns, it is true, will have a different appearance, if the light
falls in a different manner upon the objects, but its general character
and outline will, in most cases, remain the same.
The object-boxes which have now been described, are made to fit
the cell, but at the same time to slip easily into it, so that
they themselves have no motion separate from that of the cell. An
object-plate, however, of a less diameter, called the vibrating
object-plate, and containing loose objects, is an interesting addition
to the instrument. When the Kaleidoscope is held horizontally, this
small object-plate vibrates on its lower edge, either by a gentle
motion of the tube, or by striking it slightly with the finger; and the
effect of this vibration is singularly fine, particularly when it is
combined with the motion of the coloured fragments.
Another of the object-boxes, in several of the instruments, contains
either fragments of colourless glass, or an irregular surface of
transparent varnish or indurated Canada balsam. This object-box gives
very fine colourless figures when used alone; but its principal use
is to be placed in the cell between an object-box with bright colours
and the end of the instrument. When this is done, the outline of the
pieces of coloured glass are softened down by the refraction of the
transparent fragments, and the pattern displays the finest effects of
soft and brilliant colouring. The colourless object-box supplies the
outline of the pattern, and the mass of colour behind fills it up with
the softest tints.
Some of the object-boxes are filled with iron or brass wires, twisted
into various forms, and rendered broader and flatter in some places by
hammering. These wires, when intermixed with a few small fragments of
coloured glass, produce a very fine effect. Other object-plates have
been made with pitch, balsam of tolu, gum lac, and thick transparent
paints; and when these substances are laid on with judgment, they
form excellent objects for the Kaleidoscope. Lace has been introduced
with considerable effect, and also festoons of beads strung upon wire
or thread; but pieces of glass, with cut and polished faces, are ill
fitted for objects. When the object-box is wide, certain insects may
be introduced temporarily, without killing or injuring them, and
the crystallization of certain salts from their solution, and other
chemical changes, may be curiously exhibited.
Hitherto we have supposed all colours to be indiscriminately adopted in
the selection of objects; but it will be found from experience, that
though the eye is pleased with the combination of various objects,
yet it derives this pleasure from the beauty and symmetry of the
outline, and not from the union of many different tints. Those who are
accustomed to this kind of observation, and who are acquainted with the
principles of the harmony of colours, will soon perceive the harshness
of the effect which is produced by the predominance of one colour, by
the juxtaposition of others, and by the accidental union of a number;
and even those who are ignorant of these principles, will acknowledge
the superior effect which is obtained by the exclusion of all other
colours except those which harmonize with each other.
In order to enable any person to find what colours harmonize with each
other, I have drawn up the following table, which contains the harmonic
colours.
Deepest Red, Blue and Green equally mixed.
Red, Blue and Green, with most Blue, mixed.
Orange Red, Blue mixed with much Indigo.
Orange, Blue and Indigo, the Indigo predominating.
Orange-Yellow, Indigo unmixed.
Yellow, Violet and Indigo nearly in equal portions.
Greenish Yellow, Pale Violet.
Green, Violet.
Greenish Blue, Violet and Red in equal portions.
Blue, Orange Red, Red.
Indigo, Orange-Yellow.
Violet, Green.
It appears from the preceding table that _Bluish Green_ harmonizes with
_Red_, or, in other words, _Red_ is said to be the _accidental colour_
of _Bluish Green_, and _vice versa_. These colours are also called
_complementary colours_, because the one is the complement of the
other, or what the other wants of white light; that is, when the two
colours are mixed, they will always form white by their combination.[4]
[4] See the article _Accidental Colours_, in the EDINBURGH
ENCYCLOPÆDIA, vol. i. p. 88.
The following general method of finding the harmonic colours will
enable the reader to determine them for tints not contained in the
preceding table. Let =A B=, Fig. 23, be the prismatic spectrum,
containing all the seven colours, namely, _Red_, _Orange_, _Yellow_,
_Green_, _Blue_, _Indigo_, and _Violet_, in the proportion assigned to
them by Sir Isaac Newton, and marked by the dotted lines. Bisect the =A
B= at _m_, so that =A _m_= is equal to =B _m_=, and let it be required
to ascertain the colour which harmonizes with the colour in the Indigo
space at the point _p_. Take =A _m_=, and set it from _p_ to _o_,
and the colour opposite _o_, or an orange-yellow, will be that which
harmonizes with the indigo at _p_. If _p_ is between _m_ and =A=, then
the distance =A _m_= must be set off from _m_ towards _n_.
[Illustration: FIG. 23]
In order to show the method of constructing object-boxes on the
preceding principles, we shall suppose that the harmonic colours of
orange-yellow and indigo are to be employed. Four or five regular
figures, such as those already described, must be made out of
indigo-coloured glass, some of them being plain, and others twisted.
The same number of figures must also be made out of an orange-yellow
glass; and some of these may be drawn of less diameter than others,
in order that tints of various intensities, but of the same colour,
may be obtained. Some of these pieces of spun glass, of an indigo
colour, may be intertwisted with fibres of the orange-yellow glass. A
few pieces of white flint-glass, or crystal spun in a similar manner,
and intertwisted, some with fibres of orange-yellow, and others with
fibres of indigo glass, should be added; and when all these are joined
to some flat fragments of orange-yellow glass, and indigo-coloured
glass, and placed in the object-plate, they will exhibit, when applied
to the Kaleidoscope, the most chaste combinations of forms and colours,
which will not only delight the eye by the beauty of their outline,
but also by the perfect harmony of their tints. By using the thin and
highly-coloured films or flakes of decomposed glass, very brilliant and
beautiful patterns are produced. These films may be placed either upon
a mirror plate or upon black wax, and they may be placed among other
objects, or fixed in movable cells. By applying the Kaleidoscope to
crystals in the act of formation, shooting out in different directions,
symmetrical patterns are instantaneously created.
The effect produced by objects of only one colour is perhaps even
superior to the combination of two harmonic colours. In constructing
object-plates of this kind, various shades of the same colour may be
adopted; and when such objects are mixed with pieces of colourless
glass, twisted and spun, the most chaste and delicate patterns are
produced; and those eyes which suffer pain from the contemplation of
various colours, are able to look without uneasiness upon a pattern in
which there is only one.
In order to show the power of the instrument, and the extent to which
these combinations may be carried, I have sometimes constructed a
long object-plate, like the slider of the magic lantern, in which
combinations of all the principal harmonic colours followed one another
in succession, and presented to the eye a series of brilliant visions
no less gratifying to some persons, and to some others even more
gratifying, than those successions of musical sounds from which the ear
derives such intense delight.
We cannot conclude this chapter without noticing the fine effects which
are produced by the introduction of carved gems, and figures of all
kinds, whether they are drawn or engraved on opaque, or transparent
grounds. The particular mode of combining these figures will be pointed
out in a subsequent chapter.
CHAPTER IX.
ON THE ILLUMINATION OF TRANSPARENT OBJECTS IN THE KALEIDOSCOPE.
When the Kaleidoscope is directed to the sky, or to a luminous object,
such as a gas flame, or the flame of a candle, a uniform tint is seen
through the pieces of glass, or other transparent fragments that
have flat surfaces, and there is a certain degree of hardness in
their outlines. When the instrument is not opposite the flame, but
directed to one side of it, the light enters the transparent fragments
obliquely, and a much finer effect is produced. The pattern, indeed,
changes very considerably by making the Kaleidoscope move round the
flame. An excellent effect is obtained, as we have already stated,
by directing the tube between two bright lights; and the richness of
the symmetrical pattern increases with the number of lights which
illuminate the objects. As it would be inconvenient to adopt such a
mode of illumination, it becomes of importance to have some contrivance
attached to the instrument, by which we can illuminate the objects by
light falling upon them in different directions.
[Illustration: FIG. 24.]
The simplest method of thus illuminating the objects, is to fix on the
end of the Kaleidoscope the portion of a metallic or silvered-glass
cone. The light of a bright flame, placed in front of the cone, will
be reflected from its interior surface, and fall obliquely on the
fragments of coloured glass. In many cases the effect will be increased
by placing in the base or mouth of the cone a circular stop, or opaque
disk, in order to prevent any light from falling directly upon the
objects, their oblique illumination being produced solely by the
rays reflected from the interior surface of the cone. This will be
understood from the annexed figure, where =M A N C= is a portion of
the cone, fixed to the end =E F= of the Kaleidoscope, and _m n o p_
the object-box. If the angle formed by the sides of the cone is such
that a ray of the sun’s light falling upon =M=, the upper margin of the
reflecting surface of the cone, is reflected to _o_, the lower side
of the object-box, then all the rays of a beam of the sun’s light
incident upon the upper half of the conical surface, will be reflected
upon the object-box; and, for the same reason, if a ray falling on
=N= is reflected to _m_, all the other rays falling on the lower half
of the conical surface will be reflected upon the object-box, and
illuminate obliquely the objects which it contains.
In the Kaleidoscopes of more recent construction, the object-box is
made transparent throughout,—the plates of glass _m n_, _o p_ being
fixed in a cylindrical case of glass _m n o p_, so that rays =R R=,
either parallel or diverging, may be reflected from the cone =A B C D=,
and after passing through the transparent cylindrical rim _m n o p_,
illuminate the objects. Another cone =A M N C=, with its angle less
than a right angle, may be joined to =A B D C=, so as to throw the rays
obliquely, into _m n_ and _o p_, and also, if desired, upon the front
glass _m o_ of the object-box, the direct rays being excluded by an
opaque disk =S S=. In this construction, the outer face of the glass
_m o_ should not be ground, as it would prevent the admission of the
light to the objects, the exclusion of external objects, the purpose
for which the grey glass is required, being effected by the stop =S
S=. The illuminating cone may be of tin, or, what is much better, of
plated copper, which reflects more light than any other metal, and it
must be so attached to the tube containing the reflectors, as to have a
rotatory motion.
The same kind of lateral illumination may be obtained from polyhedral
cones or hemispheres of solid glass. If =A B C D=, Fig. 25, is the
section of a polyhedral cone of flint or plate glass, a portion _m
n o p_ is cut out of its base =A B=, to form an object-box for the
reception of the pieces of coloured glass, or other objects. The sides
_m n_, _m o_, _o p_, being highly polished, rays of light, either
parallel, as emanating from the sun, or diverging from artificial
sources of light, will be refracted and fall obliquely upon the faces
of the object-box, and illuminate its contents with the irregular
prismatic spectra which are formed by refraction. The apex =D C E= may,
in some cases, be cut off, and the polygonal section =D E= blackened in
order to prevent the introduction of direct light, and act as the stop
=S S= in the preceding figure.
[Illustration: FIG. 25.]
A similar effect will be obtained from a polyhedral solid, of a
hemispherical form, as shown in Fig. 26, where =A C=, =C D=, =D E=, =B
F=, =F G=, =G E=, are polished facets, by which parallel or diverging
rays immediately before it, or incident in any lateral direction, may
be refracted so as to illuminate by the prismatic rays the objects in
the box _m n o p_. The front =D E C= may have an apex, as in Fig. 25,
or may be made spherical to act as a condensing lens, the surface of
which may be blackened when necessary, for the purpose of excluding
the direct light. These illuminators may be attached in various ways
to the tube containing the reflectors, so to have a rotatory motion in
front of them.
[Illustration: FIG. 26.]
When the light is strong, a circular disk of fine grained white
paper may be advantageously placed upon the outer face _m o_ of the
object-box.
CHAPTER X.
ON THE CONSTRUCTION AND USE OF THE TELESCOPIC
KALEIDOSCOPE, FOR VIEWING OBJECTS AT A DISTANCE.
We have already seen, in explaining the principles of the Kaleidoscope,
that a symmetrical picture cannot be formed from objects placed at
any distance from the instrument. If we take the simple Kaleidoscope,
and holding an object-box in contact with the reflectors, gradually
withdraw it to a distance, the picture, which is at first perfect in
every part, will, at the distance of one-tenth of an inch, begin to be
distorted at the centre, from the disunion of the reflected images;
the distortion will gradually extend itself to the circumference, and
at the distance of eighteen inches, or less, from the reflectors, all
the symmetry and beauty of the pattern will disappear. An inexperienced
eye may still admire the circular arrangement of the imperfect and
dissimilar images; but no person acquainted with the instrument could
endure the defects of the picture, even when the slightest distortion
only is visible at the centre.
As the power of the Kaleidoscope, therefore, in its simple form, is
limited to transparent objects, or to the outline of opaque objects
held close to the aperture of the reflectors, it becomes a matter of
consequence to extend its power by enabling it to produce perfectly
symmetrical patterns from opaque objects, from movable or immovable
objects at a distance, or from objects of such a magnitude that they
cannot be introduced before the opening of the reflectors. Without such
an extension of its power, the Kaleidoscope might only be regarded as
an instrument of amusement; but when it is made to embrace objects of
all magnitudes, and at all distances, it takes its place as a general
philosophical instrument, and becomes of the greatest use in the fine,
as well as the useful arts.
[Illustration: FIG. 27.]
In considering how this change might be effected, it occurred to
me, that if =M N=, Fig. 27, were a distant object, either opaque or
transparent, it might be introduced into the picture by placing a lens
=L L=, single or achromatic, at such a distance before the aperture =A
O B=, that the image of the object may be distinctly formed in the air,
or upon a plate of glass, the inner side of which was finely ground,
and in contact with the ends =A O=, =B O= of the reflectors, the plane
passing through =A O B=. By submitting this idea to experiment, I found
it to answer my most sanguine expectations. The image formed by the
lens at =A O B= became a new object, as it were, and was multiplied
and arranged by successive reflexions in the very same manner as if
the object =M N= had been reduced in the ratio of =M L= to =L A=, and
placed close to the aperture.
[Illustration: FIG. 28.]
The Compound or Telescopic Kaleidoscope is therefore fitted up as
shown in Fig. 28, with two tubes, =A B=, =C D=. The inner tube, =A B=,
contains the reflectors as in Fig. 27, and at the extremity =C=, of the
outer tube =C D=, is placed a lens which, along with the tube, may be
taken off or put on at pleasure. The focal length of this lens should
always be much less than the length of the outer tube =C D=, and should
in general be such that it is capable of forming an image at the end
of the reflectors, when =A B= is pulled out as much as possible, and
when the object is within three or four inches of the lens. When it is
required to introduce into the picture very large objects placed near
the lens, another lens of a less focal length should be used; and when
the objects are distant, and not very large, a lens, whose principal
focal length is nearly equal to the greatest distance of the lens from
the reflectors, should be employed.
When this compound Kaleidoscope is used as a simple instrument for
viewing objects held close to the aperture, the tube =A B= is _pushed
in as far as it will go_, the cell with the object-plate is slipped
upon the end =C= of the outer tube, and the instrument is used in the
same way as the simple Kaleidoscope.
In applying the compound Kaleidoscope to distant objects, the cell is
removed and the lens substituted in its place. The instrument is then
directed to the objects, and the tube =A B= drawn out till the inverted
images of the objects are seen perfectly distinct, or in focus, and
the pattern consequently perfectly symmetrical. When this is done, the
pattern is varied, both by turning the instrument round its axis, and
by moving it in any direction over the object to which it is pointed.
When the object is about four inches from the lens, the tube requires
to be pulled out as far as possible, and for greater distances it must
be pushed in. The points suited to different distances can easily
be determined by experiment, and marked on the inner tube, if it
should be found convenient. In most of the instruments there is, near
the middle of the tube =A B=, a mark which is nearly suited to all
distances beyond _three feet_. The object-plates held in the hand, or
the mirror-box placed upon a table, at a distance greater than five
or six inches, may be also used when the lens =L= is in the tube. The
furniture of a room, books and papers lying on a table, pictures on
the wall, a blazing fire, the moving branches and foliage of trees and
shrubs, bunches of flowers, horses and cattle in a park, carriages in
motion, the currents of a river, waterfalls, moving insects, the sun
shining through clouds or trees, and, in short, every object in nature
may be introduced by the aid of the lens into the figures created by
the instrument.
The patterns which are thus presented to the eye are essentially
different from those exhibited by the simple Kaleidoscope. Here the
objects are independent of the observer, and all their movements are
represented with the most singular effect in the symmetrical picture,
which is as much superior to what is given by the simple instrument,
as the sight of living or moving objects is superior to an imperfect
portrait of them. When the flame of a blazing fire is the object, the
Kaleidoscope creates from it the most magical fireworks, in which the
currents of flame which compose the picture can be turned into every
possible direction.
In order to mark with accuracy the points on the tube =A B=, suited to
different distances, the instrument should be directed to a straight
line, inclined like _m n_, Fig. 3, to the line bisecting the angular
aperture =A O B=, and brought nearer to the centre =O= of the field.
The perfect junction of the reflected images of the line at the points
_mʹ nʹ_, &c., so as to form a star, or a polygon with salient and
re-entering angles, will indicate with great nicety, that the tube has
been pulled out the proper length for the given distance. In this way,
a scale for different distances, and scales for different lenses, may
be marked on the tube.
In the construction of the Tele-Kaleidoscope, as it may be called, the
greatest care must be taken to have the lens of sufficient magnitude.
If it is too small, the field of view will not coincide with the
circular pattern, that is, the centre of the circular pattern will
not coincide with the centre of the field; and this eccentricity will
increase as the distance of the lens from the reflectors is increased,
or as the object introduced into the picture approaches to the
instrument. The boundary of the luminous field is also an irregular
outline, consisting of disunited curves. These irregularities are
easily explained. When the lens is too small, the luminous field is
bounded by the brass rim in which the glass is fixed; and as this brass
rim is at a distance from the reflectors, the portion of it presented
to the angular aperture cannot be formed by successive reflexions into
a continuous curve; and for the same reason, the upper sectors of the
luminous field are larger than the lower ones, and consequently the
centre of the pattern cannot coincide with that of the field. In order
to avoid these defects, therefore, the diameter of the lens should be
such, that when it is at its greatest distance from the reflectors, the
field of view may be bounded by the arch =A B=, Fig. 13, and not by the
brass rim which holds the lens. This may be readily known by removing
the eye-glass, and applying the eye at =E= when the lens is at its
greatest distance. If the eye cannot see the brass rim, then the lens
is sufficiently large; but if the brass rim is visible, the lens is too
small, and must be enlarged till it ceases to become visible. Sometimes
the lens has been made so small that the brass rim is seen not only at
=A B=, but appears also above the angular point =O=, and produces a
dark spot in the centre of the picture.
Instead of using two tubes, a lens is sometimes fitted into a tube
about an inch longer than the focal length of the glass, and this tube
is slipped upon the object end =A B O=, Fig. 21. This mode of applying
the lens is, however, inferior to the first method, as there is little
room for adjusting it to different distances; whereas with the long
tube all objects at a greater distance than four inches from the lens
may be introduced into the picture—a property which possesses very
peculiar advantages.
The extension of the instrument to distant objects is not the only
advantage which is derived from the use of the lens. As the position
for giving perfect symmetry is rather within the extremities of the
reflectors than without them; and as it is impossible to place movable
objects within the reflectors, we are compelled to admit a small
error, arising principally from the thickness of the objects, and from
the thickness of the plate of glass which is necessarily interposed
between the objects and the reflectors. The compound Kaleidoscope,
however, is entirely free from this defect. The image of a distant,
or even of a near object, can be formed within the reflectors, and in
the mathematical position of symmetry; while, at the same time, the
substitution of the image for the object itself, enables us to produce
all the changes in the picture which the motion of the object could
have effected, merely by turning the instrument round its axis, or by
moving it horizontally, or in any other direction across the object.
This instrument may be advantageously placed upon a stand like a
telescope, and may either have a partial motion of rotation by means
of a ball and socket, as shown in the figure, or what is better, a
complete motion of rotation round the axis of the tube =C D=, within a
brass ring, occupying the place of the ball and socket.[5]
[5] An instrument called _The Improved Kaleidoscope_ has been recently
brought out in Paris. It is merely the Telescopic Kaleidoscope
_deteriorated_. It consists of a lens _fixed_ at the distance of
about two inches in front of the reflectors, and can therefore give
symmetrical pictures of objects only at _one distance_, while it cannot
be used as an ordinary Kaleidoscope. The instrument described in the
preceding page, with a lens that can be slipped off, is a much better
Kaleidoscope.
CHAPTER XI.
ON THE CONSTRUCTION AND USE OF POLYANGULAR
KALEIDOSCOPES, IN WHICH THE REFLECTORS CAN BE FIXED
AT ANY ANGLE.
In all the preceding instruments, the reflecting planes are fixed at
an invariable angle, which is some even aliquot part of 360°; and
therefore, though the forms or patterns which they create are literally
infinite in number, yet they have all the same character, in so far
as they are composed of as many pairs of direct and inverted images
as half the number of times that the inclination of the reflectors is
contained in 360°.
It is therefore of the greatest importance, in the application of the
Kaleidoscope to the arts, to have it constructed in such a manner, that
patterns composed of any number of pairs of direct and inverted images
may be created and drawn. With this view, the instrument may be fitted
up in various ways, with paper, cloth, and metallic joints, by means
of which the angle can be varied at pleasure; but the most convenient
methods are shown in the Figures from Fig. 29 to 35, inclusive, which
represent two different kinds of Polyangular Kaleidoscopes, as made by
the late Mr. R. B. Bate, Optician, London, who had devoted much time
and attention to the perfection of this species of Kaleidoscope.
_Bate’s Polyangular Kaleidoscope with Metallic Reflectors._
[Illustration: FIG. 29.]
[Illustration: FIG. 30.]
The three Figures, viz., 29, 30, and 31, represent the Polyangular
Kaleidoscope with metallic reflectors, as made by Mr. Bate. Fig. 29
shows the complete instrument, when mounted upon a stand; Fig. 30
is a section of it in the direction of its length; Fig. 31 is a
transverse section of it through the line =S T=, Fig. 30, and Fig. 32
shows the lens of the eye-hole =E=. The tube of this instrument is
composed of two cones, =M M=, =N N=, Fig. 30, connected together by a
middle piece or ring, =R R=, into which they are both screwed. These
two cones enclose two highly polished metallic reflectors, =A O=, =B
O=, Fig. 31, only one of them, viz., =B O E=, being seen in Fig. 30.
One of these reflectors, =B O E=, is fixed to the ring =R R=, by the
intermediate piece =K G L=. The reflector is screwed to this piece by
the adjustable screws =K=, =L=; and the piece =K G L= is again fixed
to the ring =R R=, by two screws seen above and below =G=, in Fig. 31.
Hence the tube, consisting of the cones =M M=, =N N=, and the ring =R
R=, are immovably connected with the mirror =B O E=. The surface of the
reflector =B O E= is adjusted by the screws at =K= and =L=, till it
passes accurately through the axis of the cones and ring as seen in
Fig. 31. The other reflector =A O=, is fixed to an outer ring _r r_,
by means of an intermediate piece, similar to =K G L=, the arm =F= of
which, corresponding to =G=, passes through an annular space or open
arch, of more than 90°, cut out of the circumference of the inner ring
=R R=. The arm =F= is fixed to the outer ring _r r_ by two screws,
seen above and below =F=; and the reflector =A O= is fixed to the bar
corresponding to =K L=, Fig. 30, by similar screws, for the purpose of
adjusting it.
[Illustration: FIG. 31.]
[Illustration: FIG. 32.]
The lower edge =O E= of the reflector =B O E= extends about the 15th of
an inch below the axis of the cones, as represented by the dotted line
in Fig. 30; but the lower edge =O E= of the other reflector =A O E=,
which is finely ground to an acute angle, forming a perfectly straight
and smooth line, is placed exactly in the axis of the cones, so as just
to touch a line in the reflector =A O E=, which coincides with the axis
of the cone, and to form a junction with that line in every part of
the two meeting planes. The very nice adjustments which are necessary
to produce so exact a motion are effected by the screws corresponding
to =K= and =L=.
If we now fix the outer ring _r r_ into the ring of a stand =S T=,
so as to be held fast, and turn the cones with the hand, we shall
give motion to the reflector =B O=, so as to place it at any angle
we please, from 0° to 90°; and during its motion through this arch,
the junction of the two reflectors must remain perfect, if the
touching lines are adjusted, as we have described them, to the axis
of motion, which must also be the axis of the cones and rings. If, on
the contrary, we take away the stand, and, holding the instrument in
the hand by either of the cones =M=, =N=, turn the ring =R R= with
the other, we shall give motion to its reflector =A O=, and produce a
variation in the angle in the same manner as before. The same effect
may be produced by an endless screw working in teeth, cut upon the
circumference of the outer ring _r r_.
In order to enable the observer to set the reflectors at once to any
even aliquot part of a circle, or so as to give any number of pairs of
direct and inverted images, the most convenient of the even aliquot
parts of the circle are engraven upon the ring _r r_; so that we have
only to set the index to any of these parts, to the number 12, for
example, and the reflectors will then be placed at an angle of 30° (12
× 30 = 360°), and will form a circular field with _twelve_ luminous
sectors, or a star with _six_ points, and consequently a pattern
composed of _six_ pairs of direct and inverted images.
As the length of the plates is only about five inches, it is
necessary, excepting for persons very short-sighted, to have a convex
lens placed in front of the eye-hole =E=, as shown in Figs. 30 and
32. A brass ring containing a plane glass screws into the outer ring
=C D=, when the instrument is not in use; and there is an object-box
containing fragments of differently coloured glass. This object-box
consists of two plates of glass, one ground and the other transparent,
set in brass rims. The transparent one goes nearest the reflector, and
the brass rim which contains it screws into the other, so as to enclose
between them the coloured fragments, and regular figures of coloured
and twisted glass. A loose ring surrounds this object-box; and when
this ring is screwed into the circular rim =C D=, the object-box can be
turned round so as to produce a variety of patterns, without any risk
of its being detached from the outer cone.
In applying this instrument to opaque objects, such as engravings,
coins, gems, or fragments of coloured glass laid upon a mirror, the
aperture of the mirrors is laid directly over them, the large cone
=M M= having been previously unscrewed, for the purpose of allowing
the light to fall freely upon the objects. This property of the
Kaleidoscope is of great importance, as in every other form of the
instrument opaque objects must be held obliquely, and therefore at
such a distance from the reflectors as must affect the symmetry of the
pattern.
As the perfection of the figures depends on the reflectors being kept
completely free of dust, particularly at their junction, where it
naturally accumulates, the greatest facility is given by the preceding
construction in keeping them clean. For this purpose, the large cone
must be unscrewed; the reflectors having been previously closed, by
turning the index to 60 on the ring. They are next to be opened to
the utmost, and the dust may in general be removed by means of a fine
point wrapped in clean and dry wash-leather. If any dust, however,
still adheres, the small screw in the side of the ring opposite
to the index should be removed, and the smaller cone, =N N=, also
unscrewed. By easing the supporting screws of either of the reflectors,
their touching sides will separate, so as to allow a piece of dry
wash-leather to be drawn between them. When every particle of dust has
been thus removed, the metals should be re-adjusted and closed before
the cones are replaced; both of which should be screwed firmly into the
ring =R R=.
As the axis of motion in the preceding construction is necessarily the
axis of the cones and rings, the diameter of these cones and rings must
everywhere be double the breadth of the reflectors. From this cause,
the tube, and consequently the object-box, are wide, and the instrument
is, to a certain degree, not very portable. This defect is completely
avoided in another Polyangular Kaleidoscope constructed by Mr. Bate,
upon entirely different principles, which we shall now proceed to
describe.
_Bate’s Polyangular Kaleidoscope with Glass Reflectors._
[Illustration: FIG. 33.]
[Illustration: FIG. 34.]
[Illustration: FIG. 35.]
[Illustration: FIG. 36.]
A section of the whole of this instrument, in the direction of its
length, is shown in Fig. 33. A section through =M N= or =O P=, near
the eye-end, is shown in Fig. 35, Fig. 34 representing the mode of
supporting the fixed reflector, and Fig. 36 the mode of supporting
the movable reflector. The tube of the Kaleidoscope, in Fig. 33, is
represented by _b c d e f g h_, and consists of two parts, _b c g h_,
and _c d e f g_. The first of these parts unscrews from the second,
and the second contains all the apparatus for holding and moving the
reflectors. At the parts =M N O P=, of the tube, are inserted a short
tube, a section of which is shown in Fig. 34. The object of these tubes
is to support the fixed mirror =A O=, which rests with its lower end
=O= upon the piece of brass _t_. It is kept from falling forwards by
the tongue _r_, connected with the upper part _s s_, and from falling
backward by the piece of cork =Q=, which may be removed at any time,
for the purpose of taking out and cleaning the reflectors. This
little tube is fixed to the outer tube by the screws _s_, _s_. The
contrivance for supporting and moving the second reflector =B O=, is
shown in Fig. 36, in section; and a longitudinal view of it is given
in Fig. 33. The mirror =B O= lies in an opening, cut into two pieces
of brass, =_v_ B _p_=, one of which is placed at =M N=, and the other
at =O P=. These two pieces of brass are connected by a rod _m n_,
Fig. 33; and in the middle of this rod there is inserted a screw _k_,
which passes through the main tube _c d e f g_, into a broad milled
ring _w w_, which revolves upon the tube. As the screw _k_, therefore,
fastens the ring _w w_ to the rod _m n_, the reflector =B O= will be
supported in the tube by the ring _w w_. The lower part of the mirror
=B O=, or rather of the brass piece =_v_ B _p_=, rests at _y_, upon
the piece of watch-spring _x y z_, fastened to the main tube at _z_.
This spring presses the face of the reflector =B O= against the ground
and straight edge of the other reflector =A O=, so as always to effect
a perfect junction in every part of their length:—The apparatus for
both reflectors is shown in Fig. 35. An arch of about 45° is cut out
of the main tube, so as to permit the screw _k_ to move along it; and
hence, by turning the broad ring _w w_, the reflector =B O= may be
brought nearly to touch the reflector =A O=, and to be separated from
it by an arch of 45°, so as to form every possible angle from nearly
0° to 45°, which is a sufficient range for the Kaleidoscope. The main
tube terminates in a small tube at =E=, upon which may be screwed, when
it is required, a brass cap _e f_, containing a convex lens. A short
tube, or cell, _a a a a_, for containing the object-boxes, slips upon
the end of the tube, and should always be moved round from right to
left, in order that the motion may not unscrew the portion of the tube
_b c g h_, upon which it moves. When the instrument is used for opaque
objects, the end piece, _b c g h_ of the tube, screws off, so as to
admit the light freely upon the objects.
The advantages which the Polyangular Kaleidoscopes possess over all
others are—
1st, That patterns of any number of sectors, from the
simplest to the most complicated, can be easily
obtained.
2d, That the reflectors can be set, with the most
perfect accuracy, to an even aliquot part of a
circle.
3d, That the reflectors can be at any time completely
cleaned and freed from all the dust that
accumulates between them, and the instrument
rendered as perfect as when it came from the hands
of the maker.
In order to apply this Kaleidoscope to distant objects, or make it
telescopic,[6] a piece of tube with a lens at the end of it is put upon
the end piece, _b c g h_, and may be suited to different distances
within a certain range.
[6] See Chapter X.
CHAPTER XII.
ON THE CONSTRUCTION AND USE OF ANNULAR AND PARALLEL KALEIDOSCOPES.
In the instruments already described, the pictures which they create,
though they may be made of various outlines, have all a centre to which
the reflected images are symmetrically related. The same instruments
give an annular pattern, or a pattern returning into itself, and
included between two concentric circles, by keeping the objects from
the central part of the aperture; but as such a pattern can never have
its greatest radius more than the breadth of the mirror, and as annular
patterns of a very great radius, where the eye can see only a portion
of them at a time, are often required, it becomes of importance to
adapt the Kaleidoscope for this species of ornament.
[Illustration: FIG. 37.]
Let =A C B D=, Fig. 37, be two plane mirrors, and let their
inclination be measured by the angle =A O B=; then, if the eye is
placed between =C= and =D=, it will observe the reflected images of the
objects which are placed before the aperture =A C B D=, arranged, in
the annular segment =M A B N=, round =O=, as a centre. The effect is
exactly the same as if the reflectors had been continued to =O=, with
this difference only, that the annular segment can never be complete.
This defect in the segment arises from two causes: When the centre =O=
is near =C D=, the defect is occasioned by the want of a reflecting
surface to complete the ring, and not from any want of light in the
reflected images; but when the centre =O= is remote from =C D=, the
defect arises from the want of light in the last reflexions, as well as
from the want of a reflecting surface.
The theory of the Annular Kaleidoscope is exactly the same as that of
the common instrument; and therefore all the contrivances for producing
symmetrical pictures, from near and distant objects, are applicable to
this instrument. As the picture, however, never can return into itself,
it is of no importance that the angle =A O B= be the aliquot part of a
circle, the picture being equally complete at all angles. In order to
have the most perfect symmetry with this Kaleidoscope, the eye should
be placed at =E=, between the nearest ends of the reflectors, as it
will there be nearer the plane of both reflectors than in any other
position. If the two mirrors are brought nearer each other, so that
their surfaces always pass through the point O, the deviation from
perfect symmetry will diminish as the eye becomes more and more in the
plane of both; and for the same reason the light of the field will be
more brilliant.
When the point =O= is infinitely distant, the two reflectors become
parallel to each other, as in Fig. 38, and the series of reflected
images extends in a straight line, forming beautiful rectilineal
patterns for borders, &c. In this position of the reflectors the eye
should be placed in the centre at =E=, and the symmetry of the picture
and the light of the field will increase as the distance of the
reflectors diminishes, or as their length is increased.
[Illustration: FIG. 38.]
Two different kinds of instruments have been constructed on the
preceding principles, the one by Mr. Dollond, and the other by Mr. John
Ruthven, both of which possess very valuable properties.
_Mr. Dollond’s Universal Kaleidoscope._
[Illustration: FIG. 39.]
[Illustration: FIG. 40.]
The instrument constructed by Mr. Dollond is represented in Figs. 39,
40, and 41, in section, and is intended to unite the properties of a
common Kaleidoscope, in which the reflectors are inclined at an angle
of 30°, and also those of an Annular and a Parallel Kaleidoscope.
Fig. 39 represents the reflectors, etc., when they act as a common
Kaleidoscope; and Fig. 40 shows them when they form a parallel
Kaleidoscope, an annular Kaleidoscope being formed when they have an
intermediate position. The tube of the instrument is shown, in section,
by =T T=; and to this tube is fixed, by the screws _s s_, a frame of
metal, _a b_, to which the reflectors are fastened. The reflectors,
which are made of the finest speculum metal, are shown at =A O=, =B
O=, and are attached to plates of brass, _c d_, _c d_, whose breadth
exceeds that of the reflectors so as to allow their extremities to
descend below the point =O=, Fig. 39. A double spring, _y x x y_, is
placed in the tube, so as to press upon the back of the reflectors, and
keep them in contact, as shown in Fig. 39, and is sufficiently elastic
as to allow them to open, as in Fig. 40. The milled head =M N=, which
passes through the lower part of the tube, carries, at its lower end, a
very eccentric button or wheel, the least diameter of which is seen at
_m_, Fig. 39, and the greatest at _m_, Fig. 40. In the first position
it has allowed the reflectors to come into contact at =O=. In the other
position it has forced them open into the position of parallelism. By
turning the milled head, the lower ends (=O O=) of the reflectors may
be brought to any distance less than =O O=, so as to form an annular
Kaleidoscope. The eye-end of the instrument is shown in Fig. 41. The
lens =E= is placed in a slider, =C D=, which is to be moved according
to the position of the reflectors, being a little above =O=, in Fig.
39, opposite the centre of the tube in Fig. 40, and at an intermediate
position in the intermediate position of the reflectors.
[Illustration: FIG. 41.]
This instrument is attached to a stand with a draw-tube, which screws
into the bottom of a mahogany box. The object-plates, and the lens for
introducing distant objects, are placed at the end of the instrument,
in the same manner as those of the usual construction. Particular kinds
of objects are selected for giving rectilineal borders.
_Ruthven’s Universal Kaleidoscope._
[Illustration: FIG. 42.]
[Illustration: FIG. 43.]
The instrument constructed by Mr. Ruthven is also a Universal
Kaleidoscope, which unites the properties of a Polyangular one with
those of Annular and Parallel Kaleidoscopes. Its construction will be
understood from Figs. 42, 43, and 44, where =A B E F G H= represents a
frame of iron or brass, which slips into the tube. The two sides =A B=,
=F H=, of this frame, are kept together by _four_ cross pieces, _a b_,
_c d_, etc., the other two corresponding to these being invisible in
the figure. The two reflectors, the ends of which are seen at =Aʹ O=,
=Bʹ O=, are each fixed to a plate of metal _p p_, a section of which is
seen in Fig. 44. Each plate of metal has four cylindrical pins, _p_,
_p_, etc., both on its upper and under edge. The two pins nearest the
ends of the reflectors pass through openings in the cross pieces _a b_,
_c d_. On the top of the frame is placed a plate of brass =M N Q P=, in
which are cut grooves _e f_, _g h_, _k l_, _m n_; _e f_ and _k l_, and
also _g h_ and _m n_, being parallel to each other. This plate can be
moved forwards and backwards between the cross pieces _a b_, _c d_,
by means of a small screw =S S=, working in a female screw fixed upon
the edge =N O= of the plate, and as the middle pins _p p_, attached to
the plates which carry the reflectors, pass through the grooves, any
change in the position of the plate =M Q=, produces a change in the
distance, _p p_, of the pins, and consequently in the distance of the
upper edges of the reflectors. By turning the screw =S S=, therefore,
the upper edges of the reflectors may be either brought into contact,
or separated to a distance regulated by the inclination of the grooves
_e f g h_. A similar plate with a similar screw is placed upon the
lower edges of the reflectors, so that we are furnished with the means
of giving the plates any inclination to each other, or placing them at
any distance within certain limits. For example, if the lower edges of
the plates are in contact, we can vary the angle of their inclination
by separating or closing their upper edges by means of the upper screw.
By separating the lower edges, we give them the position for annular
patterns, and by making the distance of the lower and upper edges the
same, we obtain from them rectilineal patterns; and the figures of
these annular and rectilineal patterns may be either contracted or
expanded, by altering the distance of the plates when in this parallel
position.
[Illustration: FIG. 44.]
CHAPTER XIII.
ON THE CONSTRUCTION AND USE OF POLYCENTRAL KALEIDOSCOPES.
Hitherto we have considered the effects of combining two reflectors,
by means of which the reflected images are arranged around one
centre, either visible or invisible; but it must be obvious, from
the principles already explained, that very singular effects will
be obtained from the combination of three or more reflectors. As
in instruments of this kind the reflected images are arranged
round several centres, we have distinguished them by the name of
_Polycentral_.
As 90° is the greatest angle which is an even aliquot part of 360°,
and as all regular polygons, with a greater number of sides than four,
must have their interior angles greater than 90°, it follows, that
symmetrical pictures cannot be created by any number of reflectors
greater than four, arranged like the sides of a regular polygon. If the
polygon is irregular, and consists of four sides, or more, then one of
its angles must exceed 90°, and consequently it cannot give symmetrical
patterns. In constructing Polycentral Kaleidoscopes, we are limited to
combinations of four or three reflectors.
The only modes in which we can combine four reflectors, are so as
to form a hollow square, or a hollow rectangle; but though these
combinations afford regular patterns, from their angles being even
aliquot parts of 360°, yet these figures are composed merely of a great
number of squares, or rectangles, the point where every four squares
or rectangles meet being the centre of a pattern. Those, however, who
may wish to construct such instruments, must make the plates as narrow
as possible at the eye-end, so as to bring the eye, as much as can be
done, into the plane of all the four reflectors.
In combining three reflectors, the limitation is nearly as great; but
the effect of the combination is highly pleasing. Since the angles at
which the reflectors must be placed are even aliquot parts of 360°,
such as 90°, 60°, 45°, 36°, 25-¹⁰/₁₄° 22½°, 20°, 18°, etc., which are
the quotients of 360°, divided by the even numbers, 4, 6, 8, 10, 12,
14, 16, 18, 20, etc.; and since the reflectors are combined in the
form of a prism, the section of which is everywhere a triangle, the
sum of whose angles is 180°, we must select any three of the above
even aliquot parts which amount to 180°; and when the reflectors are
combined at these angles, they will afford forms perfectly symmetrical.
Now, it is obvious, that these conditions will be complied with when
the angles are—
90° + 45° + 45° = 180°
90° + 60° + 30° = 180°
60° + 60° + 60° = 180°
The Polycentral Kaleidoscopes are therefore limited to _five_ different
combinations, namely,—
1. Four reflectors of equal breadth, forming a square.
2. Four reflectors, two of which are broader than the other
two, and form, a rectangle.
3. Three reflectors at angles of 90°, 45°, and 45°.
4. Three reflectors at angles of 90°, 60°, and 30°.
5. Three reflectors at angles of 60°, 60°, and 60°.
1. _On combinations of four mirrors forming a square._
[Illustration: FIG. 45.]
The first of these Kaleidoscopes is represented in Fig. 45, where =A
B=, =B C=, =C D=, =D A=, are the four equal and similar reflectors
placed accurately at right angles to each other. If we consider the
effect only of the two reflectors =A B=, =B C=, and regard =A D=, =D C=
as only the limits of the aperture, it is obvious, from the principles
explained in Chap. II., that we shall have a regular figure =D _m k
h_ D=, composed of four squares, one of which =D B= is seen by direct
vision; other two =A _l_=, =C _i_= formed by one reflexion from each
mirror; and the fourth =B _k_= composed of two half squares, each half
being formed by a second reflexion from each mirror. In like manner,
if we suppose =A D=, =D C= to act alone, they will form a square
pattern =B _b d f_=, composed like the last; and the same result will
be obtained by supposing =B A=, =A D=, and =B C=, =C D= to act alone.
The combination of these effects will produce a square _a d g k_,
composed of nine squares, four of which, formed by second reflexions,
are placed at the angles; other four formed by first reflexions in
the middle; and one, seen by direct vision, in the centre. Hence,
it follows, that the light of the different squares is symmetrical
as well as the patterns, a property which does not belong to all
polycentral instruments. The pattern, however, does not terminate with
the square _a d g k_, but extends indefinitely on all sides till the
squares become invisible, from the extinction of the light by repeated
reflexions. In order to discover the law according to which the squares
succeed each other, we shall examine in what manner a still larger
square =E F G H= is completed round the central square, seen by direct
vision. By considering every square in the large square _a d g k_ as
an object placed before the four reflectors, and recollecting that the
reflected images must be similarly situated behind the reflectors, we
shall find that the larger square =E F G H= is completed by images that
have suffered two, three, and four reflexions, as marked in the figure,
and that all these are symmetrically arranged with regard to the
central square. The squares which are crossed with a dotted diagonal
line, are those composed of two halves, each half being formed by a
different reflector. When a Kaleidoscope is formed out of the preceding
combination, the aperture, or the breadth of the plates next the eye,
should not exceed one-sixth of an inch. The effect is very pleasing
when the reflectors are accurately joined and nicely adjusted, and when
distant objects are introduced by means of a lens.
2. _On combinations of four mirrors forming a rectangle._
When the reflectors are of different breadths, so as to form a
rectangle, the very same effects are produced as in the preceding
combination, with this difference only, that the images are all
rectangular, in place of being square.
3. _On combinations of three reflectors at angles of 60°._
[Illustration: FIG. 46.]
When three reflectors are combined at angles of 60°, as shown at =A O
B=, Fig. 46, they form an equilateral triangle, and therefore all the
images will also be equilateral triangles. The figure =C D E F G H=,
which is a truncated equilateral triangle, is obviously composed of
three hexagonal patterns, of which the sectors, or rather triangles,
are arranged round the three centres =A=, =O=, =B=; the triangle =A O
B= being common to all the three. The three triangles, adjacent to the
sides of =A O B=, are formed by one reflexion from each mirror. The
three which spring from the vertices =A=, =O=, =B=, of the triangle,
consist of two halves, each of which is formed by three reflexions,
the last reflexion of the one half being made from one of the nearest
mirrors, and that of the other half from the other nearest mirror. If
we consider the formation of a more extended figure, =I L M P N K=,
which is also a truncated equilateral triangle, with its truncations
corresponding to the sides of the former figure, we shall find that it
has been completed by an addition, to each side of the former, of three
equilateral triangles, two of which are formed by three reflexions, and
the third, consisting of two halves, formed by four reflexions. This
figure consists of three entirely separate hexagons, =I C A O H K=, =L
D A B E M=, and =B O G N P F=, all of which are formed of reflected
images;—of one triangle =A O B= seen by direct vision,—and of three
triangles =A C D=, =B E F=, =O G H=, consisting of half sectors.
In constructing this Kaleidoscope, which, like the two former, has the
equally luminous images symmetrically arranged round the aperture =A O
B=, it is unnecessary to shape all the reflectors with accuracy. When
two of them, both of which have a greater breadth than is wanted, are
placed together, with the edge of the one resting upon the face of the
other, the third reflector, which must be ground with great accuracy to
the desired shape of the tapering equilateral prism, may then be placed
so that each of its edges rests upon the faces of the other two. When
this instrument is nicely executed with metallic plates, and when all
the junctions are perfect, the effects which it produces are uncommonly
splendid.
4. _On combinations of three reflectors at angles of 90°, 45°, and 45°._
[Illustration: FIG. 47.]
The effect produced by the combination of three reflectors at angles
of 90°, 45°, and 45°, is shown in Fig. 47. The two reflectors =A O=,
=A B= produce a pattern =C D B I=, composed of eight triangles; the
reflectors =B O=, =B A=, likewise give a pattern =A F G H=, composed of
eight triangles; and the reflectors =A O=, =O B=, give a pattern =A B H
I=, composed of four triangles. The triangle =I H K= is an image formed
by three reflexions, one half of it being a reflexion of half of =A I
_a_=, from the mirror =B O=, and the other half a reflexion of half of
=B H _b_=, from the mirror =A O=; and the triangle =D E F= consists of
two half images, which are reflexions of the two half images in =I O
H=. The remaining triangle =D L F= is a reflexion of =I K H=, from the
mirror =A B=, and is therefore formed by four reflexions.
As the three mirrors are not symmetrically placed, with regard to each
other, the equally luminous images are not arranged symmetrically
round the open triangle =A O B=, as in the preceding combinations.
The effect is, however, very pleasing, and all the reflected images
included in the figure =C L G K= are sufficiently bright.
5. _On combinations of three reflectors at angles of 90°, 60°, and 30°._
[Illustration: FIG. 48.]
The most complicated combinations of three reflectors is represented
in Fig. 48. In the first combination, all the angles were equal; in
the second, two of the angles only were equal; but in the present
combination, none of them are equal. The field of view, represented in
the figure by =D E H L M P=, is a truncated rhomb, consisting of no
fewer than _thirty-one_ images of the aperture =A O B=. The figure is
composed of two hexagons =D E F B R C=, =R B K L M N=, every division
of the hexagon consisting of two reflected images, and of two rhombs =C
R N P=, =F B K H=, each of which is composed of four reflected images.
In this combination, as in the last, the equally luminous sectors
are not symmetrically arranged round the centre =O= of the figure.
In the rhomb =C R N P=, for example, the four images are formed by
three, four, and five reflexions; whereas in the corresponding rhomb
=F H K B=, they are formed by two, three, and four reflexions. The
effects produced by a Kaleidoscope constructed in this manner are very
beautiful, particularly when the reflectors are metallic. In the four
figures which represent the different combinations of the reflectors,
the small figures indicate the number of reflexions by which each image
is produced.
CHAPTER XIV.
ON KALEIDOSCOPES IN WHICH THE EFFECT IS PRODUCED BY
TOTAL REFLEXION FROM THE INTERIOR SURFACES OF
TRANSPARENT SOLIDS.
When light is incident upon the most perfectly polished metals, a very
considerable quantity of it is absorbed, and even when the reflexion is
made at the greatest obliquities, there is a very manifest difference
in the intensity of the direct and the reflected pencil. In the total
reflexion of light from the second surfaces of transparent bodies, the
loss of light is very inconsiderable, and the reflexion is made with a
degree of brilliancy far surpassing that of the most resplendent metals.
[Illustration: FIG. 49.]
In constructing a Kaleidoscope upon this principle, we must procure
a piece of glass entirely free from veins, and cut it into the form
shown in Fig. 49.[7] The two surfaces =B O E=, =A O E=, must be
inclined at an angle which is the even aliquot part of a circle. They
must be ground perfectly flat and highly polished, and the junction
=O E= must be made as fine as possible. The upper surface =A B E=
should be rough-ground, and the side =A B O=, and the side at =E=,
should be parallel and well polished. If the glass is colourless and
good, the eye, when placed at =E=, will see the very same appearance
as in the simple Kaleidoscope; and objects placed at =A B O= will be
arranged into the same beautiful figures. The only defects attending
this form of the Kaleidoscope, are the loss of light occasioned by its
passing through a mass of solid glass, not perfectly transparent, and
the difficulty of obtaining a perfect junction of the two reflecting
planes. The first of these evils is, however, counterbalanced by
the great intensity of the light which suffers total reflexion; and
the second does not exist when the Kaleidoscope is intended to give
rectilineal or annular patterns.
[7] When this chapter was written (1818), it was very difficult to
procure glass sufficiently homogeneous for this purpose: but it can now
be procured from the Glass Works of Messrs. Chance & Co., at Smethwick,
near Birmingham.
In the construction of instruments of this kind, it is necessary to
make the prism of glass longer than the distance at which the eye
can see objects with perfect distinctness; that is, if the eye is
capable of seeing objects distinctly at the distance of five inches,
it will not perceive the same objects distinctly when they are placed
at the end of a prism of glass five inches long. This singular effect
arises from a property of plain lenses or pieces of plain glass, in
consequence of which, they cause divergent rays to diverge from a point
nearer the lens or plate, than that from which they radiated. It will
therefore be more convenient, for many reasons, to make the glass prism
only two or three inches long, and obtain distinct vision by means
of a lens placed at the eye-end of it; but, for the reason already
mentioned, the focal length of the lens must be less than the length of
the glass prism. The lens may even be joined to the prism, by grinding
the eye-end into a spherical form, but the degree of convexity must be
calculated upon the principles already stated.
The solid form of the Kaleidoscope is peculiarly fitted for polycentral
instruments, as we have only to polish the side, which would otherwise
have been left rough, the prism being supposed to be cut to the angles
which are necessary to give symmetrical forms, according to the
principles stated in Chapter XIII.
CHAPTER XV.
ON THE APPLICATION OF THE KALEIDOSCOPE TO THE MAGIC
LANTERN, SOLAR MICROSCOPE, AND CAMERA OBSCURA.
[Illustration: FIG. 50.]
In the various forms of the Kaleidoscope which have been described
in the preceding chapters, the pictures which it creates are visible
only to one person at a time; but it is by no means difficult to fit
it up in such a manner as to exhibit them upon a wall to any number of
spectators. The necessary limitation of the aperture at the eye-end
of the instrument, however, is hostile to this species of exhibition,
as it requires a very intense light for the purpose of illuminating
the objects. The general principle of the apparatus requisite for this
purpose is shown in Fig. 50, where =C D G F= is the tube containing
the reflectors =A O E=, etc. The objects from which the pictures are
to be created are placed in the cell =C D=, which may be made either
to have a rotatory movement round the axis of the tube, or to slide
through a groove, like the sliders of a magic lantern. These objects
are powerfully illuminated by a lens =B=, which concentrates upon them
the direct light of the lamp or candle =H=, and also the part of the
light which is reflected from the mirror =M N=. At the eye-end =E=
of the Kaleidoscope, is placed a lens =L L=, close to the end of the
reflectors, and having its centre coincident with the centre of the
aperture at =E=. In order that this lens may form behind it an image =P
P= of the objects placed in the object-plate =C D=, its focal length
must be less than the length =A E= of the plates. If the focal length
of =L L= is so small as one-half of =A E=, then it follows, from the
principles of optics, that the distance =L P= at which the image is
formed behind the lens, will be precisely equal to the distance =A E=
of the object; but this is obviously too small a distance, for the
diameter of =P P= would be equal only to the apparent diameter of the
circular aperture of the Kaleidoscope, or to twice =A O=. Hence it is
necessary, that the focal length of the lens =L L= be less than =A E=,
and greater than half of =A E=. Two-thirds, or three-fourths of =A E=
will be found to be a suitable focal length; for if it is larger than
this, the image will be formed upon the wall or screen at too great a
distance from the instrument.
When the instrument is thus fitted up, an enlarged image of the pattern
will be thrown upon the wall, which must be covered with white paper,
or some white ground, in order to exhibit the colours to advantage. By
turning the object-plate round its centre, or, if it is a rectilineal
one, by pushing it through the groove, and at the same time giving it
a rotatory motion, the pattern on the wall will undergo every possible
transformation, and exhibit to the spectators, in a magnified form,
all those variations which have been observed by applying the eye to
the Kaleidoscope.
When the preceding apparatus is used in daylight, so that the
objects are illuminated by the rays of the sun, the mirror =M N= is
unnecessary. The Kaleidoscope, etc., must, however, be attached to the
part of the frame of a Solar Microscope, which goes into the aperture
in the window-shutter.
As the most brilliant light is obtained from the burning of oxygen,
either by itself, or along with coal gas, as in the Bude light,[8] a
lamp of this kind is peculiarly fitted for displaying the pictures
of the Kaleidoscope to a number of spectators. One of Mr. Bate’s
Polycentral Kaleidoscopes was, many years ago, fitted up with a lamp
of this kind, for exhibition, at the lectures on natural philosophy,
delivered at Guy’s Hospital, by that eminent chemist, the late William
Allen, Esq., F.R.S.
The patterns which are created by the Telescopic Kaleidoscope from
natural objects, or from objects independent of the instrument, may, in
like manner, be exhibited to several spectators at once. If the objects
are in a room, such as bunches of flowers, statues, human figures, or
large pictures, they must be placed in one apartment, and strongly
illuminated. The lens must then be placed upon the end =A O= of the
Kaleidoscope, the object-plate =C D= having been removed, and must be
so adjusted that the images of the objects may fall exactly upon the
end =A O= of the reflectors. The objects may be placed at any distance
from the lens, from six inches to twelve feet, according to their
magnitude, and the pictures will be exhibited with great distinctness
and effect upon the wall of the other apartment. If a blazing fire is
employed, the most brilliant display of fireworks may be exhibited.
When the objects are out of doors, such as trees, shrubs, etc., the
Kaleidoscope, with its two lenses, must be fixed in the circular
opening of a window-shutter, and the picture received upon white paper,
or any other suitable ground, as in the Camera Obscura.
[8] This light was first proposed by myself. See _Edinburgh Review_,
April 1833, vol. lvii. p. 192.
Similar effects may be produced in a portable Camera Obscura, by
placing the apparatus =C F L L G D= in the moveable drawer of that
instrument. If the lens =L L= is of such a focal length as to admit
the formation of the image within the instrument, the picture will
be finely displayed upon the ground glass, and may be copied with
considerable exactness.
In the preceding applications of the Kaleidoscope, the great difficulty
to be overcome arises from the smallness of the aperture which can
be obtained at the eye-end of the reflectors. If we take a larger
aperture, for the purpose of gaining more light, the light of the
reflected images is diminished by this very circumstance, and the
picture loses its symmetry at the centre. The only method by which we
can remove this evil is to lengthen the reflectors, and consequently
increase their breadth in the same proportion. Let us suppose, for
example, that when the reflectors are five inches long, we can safely
employ an aperture _one-fourth_ of an inch in diameter; then if the
plates are made ten inches long, we may use an aperture _two-fourths_
in diameter, the symmetry continuing as complete, and the light of
the reflected images as intense, as when their length was only five
inches. By increasing the length of the reflectors, therefore, we
increase also the quantity of light; but, unfortunately, this increase
of length is unfavourable to the other properties of the instrument;
for we must now use a lens =L L= of a great focal length, which will
render it necessary to receive the image at a great distance from the
instrument.
In order to render the effect as brilliant as possible, the inclination
of the reflectors should, in the present case, never exceed 30°, and
might be even 36° or 45°. The objects should be selected as thin as
possible, and none with dark tints ought to be admitted into the
object-boxes.
CHAPTER XVI.
ON THE CONSTRUCTION OF KALEIDOSCOPES WHICH COMBINE THE
COLOURS AND FORMS PRODUCED BY POLARIZED LIGHT.
In the preceding chapters we have supposed that the objects are
illuminated by common light, and that the forms which compose the
symmetrical figure are those of material bodies. If we employ polarized
light we may introduce into the Kaleidoscopic figures, the splendid
colours produced by crystallized bodies, and also the forms which
these colours assume, round the optical axes of crystals, or through
different thicknesses of the doubly-refracting substance. The part of
the polarizing apparatus which polarizes the light, may be a large
Nicol’s prism, or a bundle of thin glass plates, or a single plate of
black glass, fixed at the object end of the Kaleidoscope. The analysing
part of the apparatus may be a Nicol’s prism, or plates of the sulphate
of iodo-quinine, discovered by Dr. William Herapath, of Bristol.
Owing to the thickness of a Nicol’s prism, it is not well fitted for
the analyser, as it prevents the eye of the observer from getting
sufficiently near the small eye-hole of the Kaleidoscope. The plates
of the sulphate of the iodo-quinine, are therefore peculiarly adapted
for analysers; and when they can be obtained of the same size as the
angular aperture of the Kaleidoscope, with fixed reflectors, or of the
whole circular aperture when the reflectors are movable, they will also
form the best polarizers.
The crystals which are to give the colours and forms produced by
polarized light and its subsequent analysis, may be either _uniaxal_
crystals, such as calcareous spar, or quartz, or beryl, or _biaxal_
crystals, such as selenite, topaz, mica, arragonite, nitre, etc.[9]
These crystals must be placed at the end of the reflectors, and when
they transmit polarized light, their brilliant colours and forms will
vary by turning the cell which contains them, or by giving a motion
of rotation to the analyser. Thin films, or laminæ of selenite of
different thicknesses, and generally of such a thickness as gives the
bright rings of the _second_ order of colours in Newton’s scale, may
be placed in a narrow cell or object-box, and may have their outlines
of various curvatures, so as to combine both form and colour in the
Kaleidoscopic figure. Different forms may also be obtained by using
pieces of colourless glass of different shapes, or pieces of thin wire
bent into a variety of curves. If the outlines of the pattern are to be
obtained from pieces of glass or wire, the films of selenite might be
cemented to one of the glass plates of the object-box, so as to have
their axes lying in different directions.
[9] See my _Treatise on Optics_, Edit. 1853, Chaps. XXVIII, XXIX.
The coloured figures produced by glass quickly cooled, might also
be advantageously employed, and, likewise, the remarkable forms of
circular crystals and crystalline groups, when they are sufficiently
large to be seen by the naked eye.
Very beautiful objects may be made by cementing a plate of sulphate of
lime (_selenite_), 0.01818 of an inch thick, to a plate of glass, and
cutting out, upon a turning-lathe, grooves or bands of such different
depths, as to give different colours by polarized light. Beautiful
patterns may be thus executed in bands or lines of different colours;
but for the purposes of the Kaleidoscope, it will be sufficient to have
curves or portions of curves of such different forms and curvatures, as
will produce agreeable figures by their combination. These curves and
irregular forms of any kind, may be scratched or excavated in the plate
of selenite by the point of a sharp knife, and afterwards polished, or
even deepened, by the action of water, which has the property of slowly
dissolving the selenite.
In order to observe the effects produced by polarized light, it is
not necessary to have the polarizing and analysing apparatus attached
to the Kaleidoscope. The polarizer, in the form of a plate of black
glass, or a bundle of plates of common window-glass, may be laid on
the table so as to reflect the light into the Kaleidoscope at an angle
of 56°, and the analyser may be held in one hand, and the Kaleidoscope
in the other. In this case, however, it would be better to have the
Kaleidoscope fixed, in order that the observer may have the use of his
left hand, to turn the object-box, which contains the doubly-refracting
crystal.
Without the use of polarized light, very fine forms, and these
splendidly coloured, may be obtained by means of the coloured rings
exhibited in the _Iriscope_.[10] When two such systems of rings are
formed by breathing through a tube upon black glass, over which soap
has been rubbed and subsequently wiped off, by a piece of chamois
leather, they form a double system, similar to the biaxial system in
mica and topaz, with curves of various shapes, which exhibit beautiful
combinations in the Kaleidoscope.
[10] _Treatise on Optics_, p. 120.
CHAPTER XVII.
ON THE CONSTRUCTION OF STEREOSCOPIC KALEIDOSCOPES.
If we apply the Kaleidoscope to any statue or architectural ornament,
or any other solid object represented, photographically, on a
transparent binocular slide, the figures will be combined into a
flat symmetrical pattern, as shown in a future chapter. But if, in
the lenticular stereoscope, we place a Kaleidoscope between each of
the two semi-lenses and the statue, or other object in the binocular
slide, we shall then see the statue or other object in full relief
in the symmetrical figure. This, in a rude form, is the Stereoscopic
Kaleidoscope.
In order to construct the instrument independently of the stereoscope,
we have only to combine two equal Kaleidoscopes, with their reflecting
mirrors equally inclined to each other, and place at the eye-end of
them two semi-lenses or quarter lenses, at the distance of two and a
half inches, and having their focal lengths equal to the length of the
stereoscope. If the two right and left eye photographs, to which we
apply them, are opaque, upon paper or silver plate, an opening must
be left above the object end of the reflectors, of sufficient size to
allow light to be thrown upon the photographs. When the figures are
transparent, this aperture must be closed.
If the objects in the binocular slides represent either bas-reliefs
or alto-relievos, or statues, or vases, or portions of any solid
bodies, such as animals, plants, or flowers, which can be combined into
symmetrical patterns, they will appear in their true relief in the
Stereoscopic Kaleidoscope. In this way the artist may avail himself
of the instrument in designing circular Gothic windows, the circular
decorations of ceilings, and rectilineal or curvilineal belts, that
are to be cut out of metals, marble, freestone, or wood, or formed of
plaster of Paris, or metals susceptible of fusion.[11]
[11] See my _Treatise on the Stereoscope_, Chap. xi. p. 86.
If we combine two Telescopic Kaleidoscopes, in the manner already
described, so that the centres of their object-lenses are distant two
and a half inches, and receive the right and left eye pictures of real
objects upon disks of ground glass, with the ground side touching
the ends of the reflectors, these objects, though themselves in true
relief, will be reduced to plane pictures on the ground glass, and
again brought into stereoscopic relief, and combined into symmetrical
patterns by the instrument. The effect thus produced is different from
what we should see were the ground glass removed, and the image of the
object formed in the air at the end of the reflectors, for the same
reason that the picture of an object in the stereoscope is different
from what it appears if viewed directly by the eyes.
CHAPTER XVIII.
ON THE CONSTRUCTION OF MICROSCOPIC KALEIDOSCOPES.
The name of _Microscopic Kaleidoscope_ may be given to the instrument,
under two forms, namely, when it is made to produce symmetrical
patterns from microscopic objects, or when it is made so short that a
lens of a high power is necessary at one end of the reflectors, to see
distinctly, and magnify the objects at the other end. In both these
forms I have often constructed them so small as one inch and one inch
and a half in length. As the Kaleidoscope, in this minute state, has
been applied both in this country and abroad, as a female ornament, we
shall proceed to point out the best method of constructing it.
Since the aberration from symmetry increases, as the length of the
reflectors is diminished, and since the light of the field diminishes
from the same cause, it becomes extremely difficult to obtain correct
figures, and uniformity of light in small instruments. In order to
overcome these difficulties, as far as possible, the reflectors should
be metallic, and may be either made of polished steel or polished
speculum metal. The inclination at which they are fixed should not be
less than 36° or 45°; and the eye-hole, which should not exceed ¹/₁₅th
of an inch in diameter, must be placed as near as possible to the
angular point. Since the aberration from symmetry increases with the
distance of the object from the reflectors, and is much augmented in
small instruments, the greatest care must be taken to have the objects
at the least possible distance from the reflectors. To accomplish this,
the objects themselves should be as thin and slender as they can be
made; the colours should be brilliant and not gloomy; and they should
be separated from the reflectors by a thin film of the most transparent
mica, which is superior to glass of equal thickness, even if it could
be got, from its extreme toughness and elasticity. The mica, indeed,
is easily scratched, but if this should take place to any extent, it
can easily be replaced by a new film. It would even be of consequence
to bend the mica into a slight concavity, so as to permit the objects
to lie rather within than without the extremity of the reflectors. In
order to see the pattern with perfect distinctness, a lens must be
placed at the end of the instrument; the focal length of this lens,
however, must not be exactly equal to the distance of the objects from
the eye, but as much greater as possible, so that the eye, by a little
exertion, may be able to obtain distinct vision. The reason of this
will be understood, by considering that the images of the objects,
seen by reflexions, are thrown to a greater distance, as it were, from
the eye, and could not therefore be seen distinctly by using a lens
adjusted exactly to the nearest part of the picture. Consequently,
the focal length of the lens must be a mean between the distances of
different parts of the picture, that is, a little greater than that
which is suited to the sector seen by direct vision.
When small Kaleidoscopes are made with four or with three mirrors, the
preceding directions are equally applicable, the greatest care being
taken that the reflectors taper nearly to a point at the eye-end, so
as not to leave an aperture greater than ⅟₁₅th of an inch in diameter.
When they are made of solid glass, the focal length of the lens must be
determined from the principles contained in the last chapter.
The preceding instruments may be fitted up with a draw-tube and lens,
and when it is required to introduce drawings of pictures and statues,
small microscopic photographs may be employed.
CHAPTER XIX.
ON THE CHANGES PRODUCED BY THE KALEIDOSCOPE.
The property of the Kaleidoscope, which has excited more wonder,
and therefore more controversy than any other, is the number of
combinations or changes which it is capable of producing from a small
number of objects. Many persons, entirely ignorant of the nature of the
instrument, have calculated the number of forms which may be created
from a certain number of pieces of glass, upon the ordinary principles
of combination. In this way it follows, that twenty-four pieces of
glass may be combined 1,391,724,288,887,252,999,425,128,493,402,200
times—an operation, the performance of which would require hundreds of
thousands of millions of years, even upon the supposition that twenty
of them were performed every minute. This calculation, surprising as
it appears, is quite false, not from being exaggerated, but from being
far inferior to the reality. It proceeds upon the supposition that
_one_ piece of glass can exhibit only _one_ figure, and that _two_
pieces can exhibit only _two_ figures, whereas it is obvious that the
two pieces, though they can only be combined in two ways, _in the same
straight line_, yet the one can be put _above_ and _below_ the other,
as well as upon its right side and its left side, and may be joined,
so that the line connecting their centres may have an infinite number
of positions with respect to a horizontal line. It follows, indeed,
from the principles of the Kaleidoscope, that _if only one object is
used, and if that object is a mathematical line without breadth, the
instrument will form an infinite number of figures from this single
line_. The line may be placed at an infinite number of distances from
the centre of the aperture, and equally inclined to the extremities
of the reflectors. It may be inclined at an infinite variety of
angles to the radii of the circular field, and it may be placed in an
infinite variety of positions parallel to any radius. In all these
cases, the Kaleidoscope will form a figure differing in character and
in magnitude. In the first case, all the figures are polygons of the
same character, but of different sizes. In the second case, they are
stars, differing from each other in the magnitude of their salient and
re-entering angles; and in the third case, they form imperfect figures,
in which the lines unite at one extremity and are open at the other.
If, instead of supposing a mathematical line to be the object, we
take a _single piece_ of coloured glass, with an irregular outline,
we shall have no difficulty in perceiving, from experiment, that an
infinite variety of figures may be created from it alone. This system
of endless changes is one of the most extraordinary properties of
the Kaleidoscope. With a number of loose objects, it is impossible
to reproduce any figure which we have admired. When it is once lost,
centuries may elapse before the same combination returns. If the
objects, however, are placed in the cell, so as to have very little
motion, the same figure, or one very near it, may, without difficulty,
be recalled; and if they are absolutely fixed, the same pattern will
recur in every revolution of the object-plate.
CHAPTER XX.
ON THE APPLICATION OF THE KALEIDOSCOPE TO THE FINE AND
USEFUL ARTS.
If we examine the various objects of art which have exercised the
skill and ingenuity of man, we shall find that they derive all their
beauty from the symmetry of their form, and that one work of art excels
another in proportion as it exhibits a more perfect development of this
principle of beauty. Even the forms of animal, vegetable, and mineral
bodies, derive their beauty from the same source. The human figure
consists of two halves, one of which is the reflected image of the
other; and the same symmetry of form presents itself in the shapes of
almost all the various tribes of animated beings. In the structure of
vegetables, the principle of symmetry is less perfectly developed. From
the extreme delicacy and elasticity of its parts, a plant, regularly
constructed, would have lost all its symmetry from the influence of
gravitation, or from the slightest breath of wind; and therefore a
symmetrical combination of parts has been effected only in its leaves
and flowers. When the laws of crystallization are allowed to perform
their functions uncontrolled, the beautiful geometrical forms which
they create are marked with the most perfect regularity. Even their
physical properties are symmetrically related to some axis or fixed
line; and though all their functions are performed in utter silence and
repose, yet their physiology, if we may apply that name to the actions
of apparently dead matter, is not less wonderful than that which
embraces the busy agencies of animal and vegetable bodies.
The irregular forms which are the foundation of picturesque beauty
constitute a single exception to the general law, and therefore the
landscape painter is the only artist who is not professionally led to
the study of that species of beauty which arises from the inversion and
multiplication of simple forms.
When we consider the immense variety of professions connected both with
the fine and the useful arts, in which the creation of symmetrical
ornaments forms a necessary part, we cannot fail to attach a high
degree of utility to any instrument by which the operations of the
artist may be facilitated and improved. We are disposed to imagine
that no machine is really useful, unless it is directly employed in
providing for our more urgent wants. This, however, is a vulgar error.
An engine which forms the head of a pin, has, in reality, as much
importance as an engine for raising water, or for manufacturing cloth;
for in these cases the three machines have the same object, which
is merely that of abridging manual labour. The water would still be
raised, and the cloth and pins manufactured, if the machines did not
exist; but the machinery insures us a more regular supply of these
articles, and enables us to receive them at a cheaper rate.
The operations of machinery have, however, a still higher character in
comparison with those of individual exertion, when they enable us to
obtain any article, either of necessity or of luxury, in a more perfect
state. In this case, the machine effects what is beyond the reach of
manual labour; and instead of being the mere representative of animal
force, it exhibits a concentration of talent and skill which could not
have been obtained by uniting the separate exertions of living agents.
When we consider, that in this busy island thousands of individuals are
wholly occupied with the composition of symmetrical designs, and that
there is scarcely any profession into which these designs do not enter
as a necessary part, so as to employ a portion of the time of every
artist, we shall not hesitate in admitting, that an instrument must
have no small degree of utility which abridges the labour of so many
individuals. If we reflect further on the nature of the designs which
are thus composed, and on the methods which must be employed in their
composition, the Kaleidoscope will assume the character of the highest
class of machinery, which improves at the same time that it abridges
the exertions of individuals. There are few machines, indeed, which
rise higher above the operations of human skill. It will create, in a
single hour, what a thousand artists could not invent in the course of
a year; and while it works with such unexampled rapidity, it works also
with a corresponding beauty and precision.
The artist who forms a symmetrical design, is entirely ignorant of
the effect till it is completed—and if the design is to be embodied
in coloured materials, or in stone, or any other solid substance, he
has no means of predicting the final effect which it is to produce.
Every result, in short, is a matter of uncertainty: and when the work
is completed, it must remain as it is. The art of forming designs,
therefore, is in a state of extreme imperfection; and a more striking
proof of this could not be obtained than from the servility with which
we copy, at the present moment, the mouldings and ornaments of Greek
and Gothic architecture, and the decorations which embellish the
furniture, the dresses, and the utensils of the Romans.
If the Kaleidoscope had been an instrument which merely enabled us to
project upon a plane surface a variety of designs of the same character
as those which the artist forms with his pencil, it would still have
been an instrument of great utility. But it does much more than this.
When properly constructed, and rightly applied, it exhibits the _final_
effect of the design, when executed in the best manner; and it does
this, not only by embodying the very materials out of which the reality
is to be produced; but by exhibiting, instead of lights and shades, the
very eminences and depressions which necessarily exist in every design
the parts of which lie in various planes.
In proceeding to point out the practical methods of obtaining these
effects from the Kaleidoscope, we take it for granted that the artist
has one or other of the correct instruments described in the preceding
chapters, and that they are mounted upon a stand, and furnished with
Dr. Wollaston’s Camera Lucida, for enabling him to copy the designs
which he wishes to perpetuate.
1. _Architectural Ornaments._
Almost every public and private edifice, with the exception of
picturesque cottages, and buildings erected for the purposes of
defence, has a regular form, consisting of two halves, one of which is
the inverted image of the other. The inferior parts of the building,
such as the doors and windows, have the same regular character; and
hence it necessarily follows, that all the decorations, whether in
the form of rectilineal borders, circular patterns, or groups of
figures, should not only have the same symmetry, but should also
be symmetrically related to the bisecting line which separates the
building into two halves. If, for example, a rectilineal border,
surrounding a building like a belt, consists of a pattern, or of
lines inclined in one direction, such a border is not symmetrically
related to the vertical and horizontal lines of which the building
consists. Hence it will follow, that sculptures, representing an
action of any kind, or statues representing living objects, when
they are sufficiently large to be seen at the same time with the
whole building, can never connect themselves with its regular
outline. If these sculptures, or statues, are inverted so as to form
a Kaleidoscope pattern, like the beautiful sculpture in the door of
the temple of Jun-wassa,[12] they may then be employed without the
risk of destroying the general symmetry of the edifice. These remarks
are equally applicable to every object which derives its beauty from
symmetry; and it is curious to observe the numerous deviations from
this principle, and the bad effects they produce on some of the finest
vases and ornaments of the Romans.[13] If Mr. Cockerell’s ingenious
theory[14] of the original composition of the statues of Niobe and her
children be correct, the mode of grouping the figures will show how
much the artist was disposed to sacrifice every other kind of effect,
to obtain something like a symmetrical group within the pediment. The
gradual increase in the height of the statues towards the middle of the
tympanum, and their inclination on both sides towards the same point,
form strong proofs of Mr. Cockerell’s hypothesis, and afford a singular
example, the only one with which we are acquainted, of an attempt to
reconcile the apparently incongruous effect of a real picture and a
symmetrical group. Had the statues been confined to one-half of the
tympanum, while the other half was a reflected image of the first,
we are persuaded, that though the effect, as a picture, would have
been diminished, yet the effect, as a part of the temple, would have
been greatly increased. A very remarkable example of this species of
symmetry is shown in the fine painting of the Four Sibyls, by Raphael,
which is now in the church of Santa Maria della Pace, at Rome.
[12] See _Edinburgh Encyclopædia_, Art. Civil Architecture, plate CLI.
[13] See Hope’s _Costume of the Ancients_, vol. ii. plates 230, 264,
278, figs 1 and 298, fig. 2.
[14] See _Journal of the Royal Institution_, vol. v. p. 99.
After the architect has fixed upon the nature and character of his
ornaments, he must cut, upon the surface of a large stone, or place in
relief upon it, the elements of a variety of patterns. These elements
need not be exact representations of any object, or any portion of
it, though, in some cases, an approximation to this may be desirable.
When this stone is set in a vertical position, and so that the light
may fall obliquely upon its surface, for the purpose of giving light
and shade to the pattern, the Kaleidoscope should be placed exactly
opposite the stone, and at a distance from it corresponding to the
magnitude of the pattern which is wanted. The tube containing the
lens or lenses, being put on, or the inner tube being drawn out, if
the instrument consists of two complete tubes, it must be adjusted
to the distance of the stone, or till an image of the stone is formed
at the end of the reflectors. When this adjustment is perfect, the
Kaleidoscope must be directed to the carved part of the stone, out of
which it is proposed to form the pattern; and by slight changes in its
position, by turning it round its axis, and by varying the inclination
of the reflectors, an immense variety of the most beautiful designs
will be exhibited, in the finest relief, and as perfect as if they had
been carved out of the stone by the most skilful workman. The architect
has therefore only to select from the profusion of designs which are
thus presented to him; and when he has made his choice, he may either
copy it photographically, or with his eye, or by means of the Camera
Lucida; or he may trace upon the stone the projection of the angular
aperture of the instrument, in order that, in the execution of his
work, he may have constantly before his eyes the real element out of
which the picture is created. If, in the course of this selection, the
picture should become capable of improvement, either by giving it depth
in particular parts, or by altering the outline, this alteration can
be easily made, and its effect throughout the whole ornament will be
instantly seen.
If the architect is desirous to introduce into his ornament a natural
object, such as a leaf, he may first try the effect which it will
produce when applied in its natural state to the instrument, and he
may then carve either the whole or the half of it in stone, and then
examine what will be its final effect. If one half of the leaf is an
inverted image of the other half, it is necessary only to carve one
half of it, and place the reflectors at an angle contained as many
times in a circle as twice the number of times that he wishes the
whole leaf to be multiplied: for example, if the whole leaf is to be
multiplied six times, the angle of the mirrors must be ¹/₁₂th of 360°,
or 30°. The very same effect would be obtained by applying the whole
leaf to the instrument when the inclination of the reflectors is 60°,
or ⅙th of 360°. But when the whole leaf is not symmetrical in itself,
or consists of two dissimilar halves, it must be applied in its entire
state to the instrument.
In the formation of circular Gothic windows, the architect will find
the Kaleidoscope a most important auxiliary. By applying it to a
mullion drawn upon paper, with a portion of the curves which he wishes
to introduce, or by placing it upon various ornamental parts of the
drawing of a Gothic cathedral, he will obtain combinations to which he
has never observed the slightest approximation.
In designing the decorations for ceilings, which are generally made of
plaster of Paris, the same method should be followed as that which we
have described for architectural ornaments.
2. _Ornamental Painting._
In ornaments carved out of stone or marble, or formed from plaster of
Paris, the idea of colour does not enter into the consideration of
the architect. The forms, however, which are necessary in ornamental
painting, are always associated with colour; and therefore, in the
invention and selection of these forms, the Kaleidoscope performs a
double task. While it creates the outline, it at the same time fills it
up with colour; and by representing the effect of the two in a state
of combination, it enables the artist to judge of the harmony of his
tints, as well as of the proportion of his forms.
In the decoration of public halls and galleries, there is no species of
ornament more appropriate than those which consist in the combination
of single figures, or of groups of heads, which are either directly
or metaphorically associated with the history or object of the
institution. Regular historical paintings on the ceiling of a room are
quite incompatible with the symmetrical character of a public gallery.
If they are well executed, they can never be seen to advantage, and
therefore their individual effect is lost, while, from their very
nature, they cannot possibly produce that general effect as an ornament
which good taste imperiously requires. In employing regularly combined
groups of figures, there is sufficient scope given to the powers of
the artist, while the systematic arrangement of his work prevents it
from interfering with the general character of the place which it is to
embellish.
The effects which the Kaleidoscope develops, when applied to the
representations of living objects, will, we have no doubt, give very
great surprise to those who have not previously examined them. In
order to enable the reader to form some notion of them, we have given,
in the annexed Plate, a series of reduced figures, taken principally
from the antique. In order to separate an individual figure from the
rest, we have only to cut an opening of nearly the same size in a
piece of paper, and lay it upon the surface of the plate, so as to
conceal all the adjoining figures, and permit the required figure to
be seen through the aperture. By applying a Kaleidoscope, in which the
inclination of the mirrors does not exceed 30°, the figure will be
combined into a fine pattern, exhibiting, perhaps, the head and a part
of the body in every sector; while the hands, or the lower extremities,
are thrown into the stiff part of the design. The singular ease and
grace with which the figure necessarily rises out of the formal part
of the pattern, and with which it connects itself with the general
picture, produces a new effect, which, so far as we can learn, no
artist had ever attempted to produce.
In order to group these single figures with perfect accuracy, the
Kaleidoscopes constructed by Mr. Bate should be employed, as both his
instruments have a contrivance which allows the light to fall freely
upon the surface of the picture.[15]
[15] See p. 93.
[Illustration: FIG. 51.]
When the figures which we wish to introduce are larger than the
aperture of the Kaleidoscope, we must use the lens, and place them at
such a distance as to reduce them to the proper magnitude.
The effect which is produced by these simple outlines will convey some
idea of the beauty which must characterize the designs when the figures
are finely shaded, or chastely coloured.
By the application of the lens, paintings or statues of any size may be
reduced and admitted into the figure.
In order to convey to the reader some idea of the effect produced by
the Kaleidoscope in grouping figures, we have given in Fig. 51 a design
created by the instrument. The inclination of the mirrors by which the
figures were arranged is 36°, or ⅒th of a circle; and therefore the
object is multiplied _ten_ times, so as to give _five_ pair of direct
and inverted images.
3. _Designs for Carpets._
There is none of the useful arts to which the creations of the
Kaleidoscope are more directly applicable than the manufacture of
carpets. In this case, the manufacturer requires not merely the outline
of a design, but a design filled up with the most brilliant colours;
and upon the nature of the figure which he selects, and the tints with
which he enriches it, will depend the beauty of the effect which is
produced. A carpet, indeed, is in general covered with a number of
Kaleidoscope designs, arranged in lines parallel to the sides of the
apartment; and while this instrument creates an individual pattern, it
may also be employed, by the assistance of the lens, in exhibiting the
effect or arranging or grouping these individual patterns, according
to the form of the apartment, and other circumstances which should
invariably be attended to.
When a plasterer ornaments the ceiling of a room, the figure which
he chooses is always related to the shape of the ceiling, and varies
according as it is circular, elliptical, square, or rectangular. In
like manner, a carpet should always have a relation to the form of
the apartment, not only in the shape and character of the individual
designs, but in the mode in which they are combined into a whole.
Although the designs given by the Kaleidoscope are in general circular,
yet, when they are once drawn, their outline may be made either
triangular, square, rectangular, elliptical, or of any other form that
we please, without destroying their beauty. The outline of the pattern
may be varied in the instrument, by varying the shape of the part of
the tube or aperture which bounds the field of view at the widest
end of the angular aperture; but it is only at certain inclinations
of the reflectors that any of the regular figures can be produced in
this way. If the bounding line is circular, the field of view will be
a circle; if the bounding line is rectilineal, and equally inclined
to the reflectors, the field of view will be a regular polygon, of as
many sides as the number of times that the angle of the reflectors is
contained in 360°; if the bounding line is rectilineal, but placed
at right angles to one of the reflectors, the figure will still be
a regular polygon, but its number of sides will be equal to _half_
the number of times that the angle of the reflectors is contained in
360°. Hence it follows, that a square field may be obtained in two
ways, either by placing the mirrors at an angle of 45°, and making the
bounding line perpendicular to one of the reflectors; or by inclining
the mirrors 90°, and making the bounding line equally inclined to
both reflectors;—and that a triangular field may be obtained, either
by inclining the mirrors 60°, and setting the bounding line at right
angles to one of the reflectors, or by making the inclination 120°, and
placing the bounding line at an angle of 60° and 30° to the reflectors.
An elliptical field may be obtained, by giving the bounding line the
shape of one quarter of an ellipse, and placing it in such a manner
that the vertex of the conjugate axis falls upon one of the reflectors,
and the vertex of the transverse axis upon the other.
The form of the pattern being determined, the next step is to select
an outline, and the colours which are to enter into its formation. In
order to do this to the greatest advantage, the differently coloured
worsteds which the manufacturer proposes to employ should be placed
upon a plane surface, either in the state of thread, or, what is
much better, when they are wrought into cloth. These differently
coloured pieces of carpet, which we may suppose to be blue, green, and
yellow, must then be placed at the distance of a few feet from the
Kaleidoscope, so that their image may, by means of the lens, be formed
at the end of the reflectors. In this state a very perfect pattern will
be created by the instrument, and the blue, green, and yellow colour
will predominate according as a greater or a lesser portion of these
colours happens to be opposite to the angular aperture. By shifting
the position of the Kaleidoscope, any one of the colours may be made
to predominate at pleasure; and the artist has it thus in his power,
not only to produce any kind of outline that he chooses, but regulate
the masses of colour by which it is to be filled up; and to try the
effects which will be produced by the juxtaposition of two colours, by
the separation of others, or by the transference of the separate or
combined masses to different parts of the design. It would be foreign
to our object to describe the apparatus by which these changes in the
quantities of colour, and in their relative position, may be most
easily and conveniently effected; the artist can have no difficulty in
constructing such an apparatus for himself, and by means of it he will
be enabled to obtain results from the Kaleidoscope which he would have
sought for in vain from any other method.
As the methods we have described of using the Kaleidoscope in
ornamental architecture, or ornamental painting, and in the manufacture
of carpets, will apply to the various other professions in which
the formation of symmetrical designs is a necessary part, I shall
merely state, that it will be found of the greatest advantage to the
jeweller in the arrangement of precious stones; to the bookbinder, the
wire-worker, the paper-stainer, and the artist who forms windows of
painted glass. In this last profession, in particular, the application
of the Kaleidoscope cannot fail to indicate combinations far superior
to anything that has yet been seen in this branch of art. From the
uniformity of tint in the separate pieces of glass which are to be
combined, the effect produced by the instrument from portions of the
very same glass that is to be used for the windows, may be considered
as a perfect fac-simile of the window when well executed on a large
scale.
CHAPTER XXI.
ON THE PHOTOGRAPHIC DELINEATION OF THE PICTURES
CREATED BY THE KALEIDOSCOPE.
In a preceding chapter we have referred to the delineation of the
pictures created by the Kaleidoscope, when they are received on the
ground glass of a camera obscura, or when a camera lucida is placed
at the eye-end of the instrument. Both of these methods are very
imperfect, and when the pictures have been copied for useful purposes,
we believe that they have generally been executed by a skilful
draughtsman, who delineated carefully one of the sectors of which the
figure is composed, and then repeated it so as to complete the picture.
[Illustration: FIG. 52.]
Since the invention of the Kaleidoscope, the discovery of the
photographic art—of the Daguerreotype and Talbotype processes, has
given a new value to the instrument. By means of a Kaleidoscope Camera,
the most complex figures can be almost instantaneously transferred to
paper, or to plated copper, and hundreds of designs offered to the
choice of the artist who is to employ them. The very same figure which
is obtained by one mode of illumination, may be altered indefinitely,
by merely changing the direction or the intensity of the light, all
the objects which give the figure remaining fixed in the object-box.
The optical arrangement by which these figures may be copied,
photographically, is substantially the same as that which is shown in
Fig. 52, where =C G D F= is the Kaleidoscope, =A O= the object-box,
and =L L= a small achromatic lens, of rock crystal or glass, a quarter
of an inch in diameter, placed in contact with the extremities of the
glass or metallic reflectors, and having its centre immediately behind
the small opening at =E=. When the rays of the sun, or any other strong
light, containing the actinic rays, are thrown obliquely upon the
object-box =A O=, an image of the Kaleidoscopic figure, produced by the
objects in the object-box, will be formed at =P P=, in the focus of the
lens =L L=. If the focal length of the lens =L L= is equal to one-half
=C G=, the length of the Kaleidoscope, the image will be formed behind
=L L=, at a distance equal to =C G=, and of the same size as the
object-box; but if the focal length of =L L= is greater than the half
of =C G=, the image will be formed at a greater distance than =C G=,
behind the lens, and the size of it will be greater than that of the
object-box, the distance and the size of the picture increasing as the
focal length of the lens increases; and when it becomes equal to =C G=,
the size and the distance of the picture will be infinitely great. When
the focal length of the lens is less than half =C G=, the figure will
be smaller than the object-box, and nearer the lens.
_The Kaleidoscope Camera._
Every camera employed for the purposes of photography may be readily
adapted for taking the pictures formed by the Kaleidoscope. We have
only to take out the lens or lenses which belong to it, and place the
Kaleidoscope furnished with its lens =L L=, Fig. 52, in the inner tube,
which is movable by means of the rack and pinion. If the picture can
be made distinct on the grey glass by the rack and pinion, a negative
or positive copy of it may be taken on collodion or paper, in the same
manner as other photographs. In some cameras the end of the box which
contains the grey glass is movable, backwards and forwards, so that the
adjustment for rendering the picture distinct may be effected, though
the Kaleidoscope is fixed in the front portion of the box.
When the camera is made for the express purpose of taking Kaleidoscope
pictures, it becomes a very simple instrument and may be constructed
easily and cheaply. It requires no lens excepting the small one =L
L= of rock crystal or of glass, a quarter of an inch in diameter.
The aperture required for this lens is so small that the spherical
and chromatic aberration cannot injure, in any sensible manner, the
distinctness of the picture. The difference between the chemical and
luminous focus, which cannot be made to coincide with a single lens,
may be easily determined by experiment, or in order to avoid this, the
small lens may be made achromatic.[16]
[16] A thin lens of rock crystal will transmit more of the actinic rays
than one rendered achromatic.
The form of the camera as fitted up with one of Mr. Bate’s
Kaleidoscopes, with metallic reflectors, is shown in the annexed
figure, where =M C D N= is the body of the camera, _m n o p_ the tube
which is moved out and in by a milled head attached to the pinion which
drives the rack on the tube _m n o p_. The Kaleidoscope =A B L L=,
exhibiting the figure which is to be copied, is temporarily fixed in
this tube by pieces of cork or wood, so that its axis, or the line of
junction of the reflectors, may be perpendicular to the surface of the
grey glass _p q_, which can be taken out, so as to allow the collodion
or paper slide to be placed in the same groove.
[Illustration: FIG. 53.]
The pictures should always be taken, when it is possible, by sunlight
incident obliquely upon the object-box, or with the illuminators
already described. When this cannot be done, artificial light,
containing the actinic or chemical rays, may be thrown upon the object,
as shown in Fig. 52. The light of coal gas, which may be condensed for
any length of time upon the objects in the object-box, will be found
to answer the purpose; but if a more rapid process is required, we
may, as proposed by Mr. Moule, use the Bengal lights, which contain a
large quantity of the actinic rays.[17] These lights are composed of
_six_ parts of _nitre_, _two_ of _sulphur_, and _one_ of _the sulphuret
of antimony_. The powder is made up in the shape of a cone, which is
ignited at the top; but it would be better to spread it out widely, in
order that it may be condensed upon the object-box. The lime-ball light
may also be employed, but the Bude light, which is a coal gas flame,
rendered more intensely luminous by a stream of oxygen, is preferable
to any other artificial flame. In using it, or any other of the lights
already mentioned, for obtaining Kaleidoscope figures, it must either
be placed very near the object-box, or condensed upon it by a large
lens, on account of the small size of the aperture at the eye-end of
the Kaleidoscope.
[17] See the _Journal of the Photographic Society_, vol. iv. p. 137.
In the formation of the Kaleidoscope figures, it is necessary to pay
particular attention to the colour of the objects of which they are to
be composed, _red_, _orange_, and _dark yellow_ objects, which transmit
few, if any, actinic rays, should not be employed. Blue glasses, which
transmit no red rays, are actinically more luminous than colourless
glasses, and therefore pale blue, pale green, and pale yellow glasses
may be advantageously combined with colourless glasses as objects for
the object-box.
When we wish to copy photographically the pictures obtained from
external objects, we must use the Telescopic Kaleidoscope of Mr. Bates,
by which the images of these objects are formed at the extremity of
the reflectors. These objects may be flowers, plants, architectural
ornaments, or parts of them, paintings, photographs, statues, or loose
and irregularly placed materials of any kind. In the figures created
by the common Kaleidoscope, we must trust to chance for the formation
of the figures, and can only choose those which we like out of a
large number accidentally formed; but when we make use of external
objects, we can group them and illuminate them in any manner we choose,
selecting those forms and colours which, when combined, produce the
effect that we desire.
This power, of arranging the objects, when in contact with the ends of
the reflectors, may be obtained with the Kaleidoscope of Mr. Bates,
shown in Fig. 29, by removing the anterior half of the cone when the
objects are opaque, and placing the camera in a vertical position.
The objects being placed upon a horizontal plate, may, if opaque, be
illuminated, by reflected light, and the picture of them, when combined
by the Kaleidoscope, taken in the same manner as when the instrument
was in the position shown in Fig. 52. If the figure is not what we
like, we can shift any individual object, or remove it, or replace it
by another, till we are satisfied with the combination.
The very same power of altering the combination, may be obtained
for transparent objects. We have only to place these objects on a
horizontal plate of glass, not connected with the Kaleidoscope, but
capable of being removed from it, and again brought close to the ends
of the reflectors. When the objects on the glass plate have been put
into their proper place, so as to give the desired picture, the plate
is brought as near as possible to the ends of the reflectors, and
the objects are illuminated either by solar or other light reflected
upwards upon the object-box.
CHAPTER XXII.
ON THE ADVANTAGES OF THE KALEIDOSCOPE AS AN INSTRUMENT OF AMUSEMENT.
The splendid discoveries which have been made with the telescope and
microscope have invested them with a philosophical character which can
never be attached to any other instrument. It is only, however, in the
hands of the astronomer and naturalist that they are consecrated to the
great objects of science; their ordinary possessors employ them solely
as instruments of amusement, and it is singular to remark how soon they
lose their novelty and interest when devoted to this inferior purpose.
The solar microscope, the camera obscura, and the magic lantern, are
equally shortlived in their powers of entertainment; and even the
wonders of the electrical and galvanic apparatus are called forth, at
long intervals, for the occasional purposes of instructing the young,
or astonishing the ignorant. A serenity of sky very uncommon in our
northern climate, is absolutely necessary for displaying the powers
of some of the preceding instruments; and the effects of the rest can
only be exhibited after much previous preparation. From these causes,
but principally from a want of variety in their exhibitions, they have
constantly failed to excite, in ordinary minds, that intense and
continued interest which might have been expected from the ingenuity of
their construction and the splendour of their effects.
The pleasure which is derived from the use of musical instruments is
of a different kind, and far more intense in its effects, and more
general in its influence than that which is obtained from any of the
preceding instruments. There are, indeed, few minds that are not alive
to the soothing and exhilarating influence of musical sounds, or that
do not associate them with the dearest and most tender sympathies of
our nature. But the ear is not the only avenue to the heart; and though
sorrow and distress are represented by notes of a deep and solemn
character, and happiness and gaiety by more light and playful tones,
the same kind of feelings may also be excited by the exhibition of dark
and gloomy colours, and by the display of bright and aërial tints.
The association, indeed, is not so powerful in the one case as in the
other, for we have been taught from our infancy, in consequence of the
connexion of music and poetry, to associate particular sentiments with
particular sounds; but there can be no doubt that the association of
colour is naturally as powerful as that of sound, and that a person who
has never listened to any other music but that of nature, nor seen any
other colours but those of the material world, might have his feelings
as powerfully excited through the medium of the eye as through that of
the ear.
The first person who attempted to supply the organ of vision with the
luxuries of light and colour, was Father Castel, a learned Jesuit, who
had distinguished himself chiefly by his opposition to the splendid
optical discoveries of Newton. About the year 1725 or 1726, he
published in the _Mercure de Paris_, his first ideas of an organ, or
ocular harpsichord. A full account of this curious instrument was
afterwards published at Hamburgh, in 1739, by M. Tellemann, a German
musician, who had seen one of the harpsichords in the possession of the
inventor, when he was on a visit to Paris. This account was afterwards
translated into French, and printed at the end of Castel’s _L’Optique
des Couleurs_,[18] which appeared at Paris in 1740. The ocular
harpsichord is a common harpsichord, fitted up in such a manner, that
when a certain sound is produced by striking the keys, a colour related
to that sound is at the same instant exhibited to the eye in a box or
frame connected with the harpsichord; so that when a piece of music
is played for the gratification of the ear, the eye is simultaneously
delighted by the display of corresponding colours.
[18] This work is entitled, _L’Optique des Couleurs, fondée sur les
simples observations et tourné surtout à la pratique de la peinture,
de la teinture, et des autres arts coloristes_. Par le R. P. Castel,
Jesuite. Paris, 1740.
In adjusting the colours and the sounds, Castel lays down the following
six propositions:—
_1st_,—There is a fundamental and primitive sound in nature, which may
be called _ut_, and there is also an original and primitive colour,
which is the foundation of all other colours, namely, _blue_.
_2d_,—There are three chords, or essential sounds, which depend upon
the primitive sound _ut_, and which compose with it a primitive and
original accord, and these are _ut_, _mi_, _sol_. There are also three
original colours depending on the _blue_, which, while they are not
composed of any other colours, produce them all, namely, _blue_,
_yellow_, and _red_. The _blue_ is here the note of the tone, the _red_
is the fifth, and the _yellow_ is the third.
_3d_,—There are five tonic chords, _ut_, _re_, _mi_, _sol_, _la_, and
two semitonic chords, _fa_ and _si_. There are also five tonic colours,
to which all the rest are ordinarily related, namely, _blue_, _green_,
_yellow_, _red_, and _violet_, and two semitonic or equivocal colours,
namely, _aurora_ and _violant_ (related to the _orange_ and _indigo_ of
Newton).
_4th_,—Out of these five entire tones, and two semitones, is formed the
diatonic scale, _ut_, _re_, _mi_, _fa_, _sol_, _la_, and _si_; and,
in like manner, out of the five entire or tonic colours, and the two
demi-colours, are formed the gradation of colours, _blue_, _green_,
_yellow_, _aurora_, _red_, _violet_, and _violant_; for the _blue_
leads to _green_, which is _demi-blue_; the _yellow_ to _aurora_, which
is _gilded yellow_; the _aurora_ leads to _red_, the _red_ to _violet_,
which is two-thirds of _red_, and one-third of _blue_; and the _violet_
to _violant_, which has more _blue_ than _red_.
_5th_,— The entire tones divide themselves into semitones; and the
five entire tones of the scale or gamut, comprehending in this the
two natural semitones, make twelve semitones, viz., _ut_ natural,
_ut_ dieze, _re_, _re_ dieze, _mi_, _fa_, _fa_ dieze, _sol_, _sol_
dieze, _la_, _la_ dieze, and _si_. In like manner there are twelve
demi-colours, or demi-tints, and there can be neither more nor less,
according to the opinion of painters themselves, and as may be
demonstrated by other means. These colours are _blue_, _sea-green_,
_green_, _olive_, _yellow_, _aurora_, _orange_, _red_, _crimson_,
_violet_, _agathe_, and _violant_. _Blue_ leads to _sea-green_,
which is a _greenish blue_; _sea-green_ leads to _green_; _green_ to
_olive_, which is a _yellowish green_; _olive_ to _yellow_; _yellow_
to _aurora_; _aurora_ to _orange_; _orange_ to _red_, the colour
of fire; _red_ to _crimson_, which is _red_ mixed with a little
_blue_; _crimson_ to _violet_, which is still more _blue_; _violet_ to
_agathe_, or _bluish violet_; and _agathe_ to _violant_.
_6th_,—The progression of sounds is in a circle, setting out from _ut_,
and returning back: thus, _ut_, _mi_, _sol_, _ut_, or _ut_, _re_, _mi_,
_fa_, _sol_, _la_, _si_, _ut_. This is called an octave, when the last
_ut_ is one-half more acute than the first. The colours also have their
progression in a circle.
_7th_,—After an octave, _ut_, _re_, _mi_, _fa_, _sol_, _la_, _si_,
there recommences a new one, which is one-half more acute, and the
whole circle of music produces several octaves.
Such are the principles upon which the ocular harpsichord was founded;
but though the instrument, from its singularity, excited great
attention when it was first constructed, we have not been able to learn
that it was ever supposed to possess the power of affording pleasure to
the eye. It must be obvious, indeed, to any person who considers the
subject, that colour, independent of form, is incapable of yielding a
continued pleasure. Masses of rich and harmonious tints, following one
another in succession, or combined according to certain laws, would no
doubt give satisfaction to a person who had not been familiar with the
contemplation of colours; but this satisfaction would not be permanent,
and he would cease to admire them as soon as they ceased to be new.
Colour is a mere accident of light, which communicates richness and
variety to objects that are otherwise beautiful; but perfection of form
is a source of beauty, independent of all colours; and it is therefore
only from a combination of these two sources of beauty that a sensation
of pleasure can be excited.
Those who have been in the habit of using a correct Kaleidoscope,
furnished with proper objects, will have no hesitation in admitting
that this instrument realizes, in the fullest manner, the formerly
chimerical idea of an ocular harpsichord. The combination of fine
forms, and ever-varying tints, which it presents in succession to
the eye, have already been found, by experience, to communicate
to those who have a taste for this kind of beauty, a pleasure as
intense and as permanent as that which the finest ear derives from
musical sounds. An eye for admiring and appreciating the effect of
fine forms, seems, indeed, to be much more general than an ear for
music; and we have heard of many cases where the tedium of severe and
continued indisposition has been removed, and where many a dull and
solitary hour has been rendered cheerful, by the unceasing variety of
entertainment which the Kaleidoscope afforded. In one respect, indeed,
this instrument is superior to all others. When it is once properly
constructed, its effects are exhibited without either skill or labour;
and so numerous are its applications, and so inexhaustible its stores,
that the observer is constantly flattered with the belief that he has
obtained results which were never seen before, and that he has either
improved the instrument, or extended its power, by new applications.
Such are the advantages, as an instrument of amusement, which the
Kaleidoscope possesses, even in its present imperfect state. To what
degree of perfection it may yet arrive, it is not easy to anticipate;
but we may venture to predict, because we see the steps by which
the prediction is to be fulfilled, that combinations of forms and
colours may be made to succeed each other in such a manner as to
excite sentiments and ideas with as much vivacity as those which are
excited by musical composition. If it be true that there are harmonic
colours which inspire more pleasure by their combination than others;
that dull and gloomy masses, moving slowly before the eye, excite
feelings of sadness and distress; and that the aërial tracery of light
and evanescent forms, enriched with lively colours, are capable of
inspiring us with cheerfulness and gaiety; then it is unquestionable,
that, by a skilful combination of these passing visions, the mind may
derive a degree of pleasure far superior to that which arises from the
immediate impression which they make upon the organ of vision. A very
simple piece of machinery is alone necessary for introducing objects of
different forms and colours, for varying the direction of the motion
across the angular aperture, and for accommodating the velocity of
their motion to the effect which it is intended to produce.
These combinations of colours and forms may be adapted to a piece of
music, and their succession exhibited on a screen by means of the
electric, or lime-ball, or other lights to which we have already had
occasion to refer. The coloured objects might be fixed between long
stripes of glass, moved horizontally or obliquely across the ends of
the reflectors; and the effects thus obtained might be varied by the
occasional introduction of revolving object-boxes, containing objects
of various colours and forms, partly fixed and partly movable. Similar
forms in different colours, and in tints of varying intensity, losing
and resuming their peculiar character with different velocities, and in
different times, might exhibit a distinct relation between the optical
and the acoustic phenomena simultaneously presented to the senses.
Flashes of light, coloured and colourless, and clouds of different
depths of shadow, advancing into, or emerging from the centre of
symmetry, or passing across the radial lines of the figure at different
obliquities, would assist in marking more emphatically the gay or the
gloomy sounds with which they are accompanied.
A slight idea of the effects which might be expected from an ingenious
piece of mechanism for creating and combining the various optical
phenomena, and exhibiting them in connexion with musical sounds, may
be obtained by a single observer, who looks into a fine Kaleidoscope,
firmly fixed upon a stand, and produces with his two hands all the
variations in form and colour which he can effect by such inadequate
means, and which he considers appropriate to the musical piece that
accompanies them.
CHAPTER XXIII.
HISTORY OF THE COMBINATIONS OF PLANE MIRRORS WHICH HAVE
BEEN SUPPOSED TO RESEMBLE THE KALEIDOSCOPE.
It has always been the fate of new inventions to have their origin
referred to some remote period; and those who labour to enlarge the
boundaries of science, or to multiply the means of improvement, are
destined to learn, at a very early period of their career, that the
desire of doing justice to the living is a much less powerful principle
than that of being generous to the dead. This mode of distributing
fame, injurious as it is to the progress of science, by taking away one
of the strongest excitements of early genius, has yet the advantage of
erring on the side of generosity; and there are few persons who would
reclaim against a decision invested with such a character, were it
pronounced by the grave historian of science, who had understood and
studied the subject to which it referred.
The apparent simplicity, both of the theory and the construction of the
Kaleidoscope, has deceived very well-meaning persons into the belief
that they understood its mode of operation; and it was only those that
possessed more than a moderate share of optical knowledge, who saw that
it was not only more difficult to understand, but also more difficult
to execute, than most of the philosophical instruments now in use. The
persons who considered the Kaleidoscope as an instrument consisting
of two reflectors, which multiplied objects, wherever these objects
were placed, and whatever was the position of the eye, provided that
it received only the reflected rays, were at no loss to find numerous
candidates for the invention. All those, indeed, who had observed the
multiplication and circular arrangement of a fire blazing between two
polished plates of brass or steel; who had dressed themselves by the
aid of a pair of looking-glasses, or who had observed the effects of
two mirrors placed upon the rectangular sides of a drawing-room, were
entitled, upon such a definition, to be constituted inventors of the
Kaleidoscope. The same claim might be urged for every jeweller who had
erected in his window two perpendicular mirrors, and placed his wares
before them, in order to be multiplied and exhibited to advantage;
and for every Dutch toy-maker, and dealer in optical wonders, who had
manufactured show-boxes, for the purpose of heaping together, in some
sort of order, a crowd of images of the same object, of different
intensities, seen under different angles, and presenting different
sides to the eye. This mode of grouping images, dissimilar in their
degree of light, dissimilar in their magnitude, and dissimilar in
their very outlines, produced such a poor effect, that the reflecting
show-boxes have for a long series of years disappeared from among the
number of philosophical toys.
From these causes, the candidates for the merit of inventing the
Kaleidoscope have been so numerous, that they have started up in every
part of the world; and many individuals, who are scarcely acquainted
with the equality of the angles of incidence and reflexion, have not
scrupled to favour the world with an account of the improvements which
they fancy they have made upon the instrument.[19]
[19] It would be an easy matter to amuse the reader with an account
of these improvements. One of the most notable of them consists
in covering the back of the reflectors with white paint, for the
purpose of increasing the light of the circular field. This scheme
is identically the same as if the author had proposed to improve the
magnificent telescopes of Herschel, when rendered dark with a high
magnifying power, by _white-washing_ the interior of the tube.
The earliest writer who appears to have described the use of two plane
mirrors, was Baptista Porta, who has given an account of several
experiments which he performed with them, in the second chapter of the
seventh book of his _Magia Naturalis_.
As the combination of plane mirrors which he there describes has been
represented as the same as the Kaleidoscope, we shall give the passage
at full length:—
_Speculum è planis multividum construere._
“Speculum construitur, quod _polyphaton_ id est multorum
visibilium dicitur, illud enim aperiendo et claudendo solius
digiti viginti et plura demonstrat simulacra. Sic igitur id
parabis. Ærea duo specula vel crystallina rectangula super
basim eandem erigantur, sintque in hemiolia proportione,
vel alia, et secundum longitudinis latus unum simul colligentur,
ut libri instar apte claudi et aperiri possint, et anguli
diversentur, qualia Venetiis factitari solent: faciem
enim unam objiciens, in utroque plura cernes ora, et hoc
quanto arctius clauseris, minorique fuerint angulo: aperiendo
autem minuentur; et obtusiori cernes angulo, pauciora numero
conspicientur. Sic digitum ostendens, non nisi digitos
cernes, dextra insuper dextra, et sinistra sinistra convisuntur;
quod speculis contrarium est: mutuaque id evenit
reflexione, et pulsatione, unde imaginum vicissitudo.”
—_Edit. Amstelod_, 1664.
The following is an exact and literal translation of this passage:—
_How to construct a multiplying speculum out of plane ones._
A speculum is constructed, called _polyphaton_, that is,
_which shows many objects_, for by opening and shutting it,
it exhibits twenty and more images of the finger alone.
You will therefore prepare it in the following manner:—Let
two rectangular specula of brass or crystal be erected
upon the same base, and let their length be one and a half
times their width, or in any other proportion; and let two
of their sides be placed together, so that they may be opened
and shut like a book, and the angles varied, _as they are
generally made at Venice_. For by presenting your face,
you will see in both more faces the more they are shut, and
the less that the angle is; but they will be diminished by
opening it, and you will see fewer as you observe with a
more obtuse angle. If you exhibit your finger, you will
see only fingers, the right fingers being seen on the right
side, and the left on the left side, which is contrary to
what happens in looking-glasses, and this arises from the
mutual reflexion and repulsion which produce a change of
the images.[20]
[20] We request that the reader will take the trouble of comparing with
the original the following translation of Baptista Porta’s description,
which was published in London, and copied into all the foreign
newspapers, etc. We hope the translator of it had no improper motive
in altering the obvious meaning of the original; yet it is singular,
that in the journal where this translation appeared, the specification
of the patent was published under the title of Directions for making
the Kaleidoscope, purporting to be an original communication to that
journal, the name of the patentee, and the technical parts of the
specification having been left out, apparently to promote the belief
that there was no patent, and that every person might make them with
impunity. The following is the translation alluded to:—
“In the following manner we may construct a mirror for seeing a
multitude of objects on a plain surface. This kind of mirror,
when constructed, is what is called _polyphaton_, that is to say,
multiplying, for by opening and shutting, it shows twenty and more
images of one single finger. If, therefore, you wish to prepare it, let
two brazen or crystal rectangular mirrors be erected on the same base,
and let the proportion of length be one and a half of the width, or any
other proportion; and let each side for the whole of its length be so
connected together that they may easily be shut and opened like a book,
and that the angles may be varied, as they are usually constructed at
Venice; for if you place _one object opposite to the face of each_,
you will see _several figures_; and this in proportion as you shut it
closer, and the angle shall be less. But, by opening, _the objects_
will be reduced in number, and the more obtuse the angle under which
you see it, the _fewer objects_ will be seen. So if you exhibit your
finger _as the object_, you will see nothing but fingers. The right
fingers will be seen on the right side, and the left on the left side,
which is contrary to the usual custom with looking-glasses; but this
happens from the mutual reflexion and repulsion which produce a change
of the images.”
It is quite obvious, from the preceding passage, that the multiplying
speculum described by Porta was not an invention of his own, but had
been long made at Venice. In the very next chapter, indeed, where he
describes a _speculum theatrale_ or _amphitheatrale_ (the _show-box_
of Harris and other modern authors), he expressly states, that a
multiplying speculum was invented by the ancients.—“Speculum autem
è planis compactum, cui si unum spectabile demonstrabitur, plura
illius rei simulachra demonstrabit, _prudens invenit vetustas_; ut ex
quibusdam Ptolemæi scriptis quæ circumferuntur percipitur;” that is, “a
speculum consisting of plane ones, which, when one object is presented
to it, will exhibit several images of it, was invented by skilful
antiquity, as appears from some of the writings of Ptolemy.” This
speculum, which consists of several mirrors arranged in a polygon, with
the object within it, is characterized by Porta as puerile, and much
less wonderful and agreeable than one of his own, which he proceeds to
describe. This new speculum consists of ten mirrors, placed within a
box, in a sort of polygonal form, with one of the sides of the polygon
open. Architectural columns, pictures, gems, pearls, coloured birds,
etc., are all placed within the box, and their images are seen heaped
together in inextricable disorder, as a whole, but so as to astonish
the spectator by their number, and by the arrangement of individual
groups. Porta speaks of this effect as so beautiful, “_ut nil jucundius
nil certe admirabilius oculis occurset nostris_,”—that nothing more
agreeable, and certainly nothing more admirable, was ever presented to
our eyes.
It would be an insult to the capacities of the most ordinary readers
to show that the instruments here described by Baptista Porta have no
farther connexion with the Kaleidoscope than that they are composed of
plane mirrors.
The sole purpose of these instruments was to _multiply_ objects by
reflexion; and so little did the idea of producing a symmetrical
picture enter into Baptista Porta’s contemplation, that he directs the
mirrors to be placed at any angle, because the multiplying property
of the mirrors is equally developed, whatever be the angle of their
inclination.
The show-box of which Porta speaks with such admiration, has so many
mirrors, and these are placed at such angles, that not one of the
effects of the Kaleidoscope can be produced from them. Its beauty is
entirely derived from the accumulation of individual images.
The next competitor for the invention of the Kaleidoscope is the
celebrated Kircher, who describes, as an invention of his own, the
construction of two mirrors which can be opened and shut like the
leaves of a book. This instrument is represented in Fig. 54, and is
described in the following passage:—
[Illustration: FIG. 54.]
PARASTASIS I.
_Specula plana multiplicativa sunt specierum unius rei._
Vide Fig. 54.
Mira quædam et a nemine, quod sciam observata proprietas elucescit in
duobus speculis ita constructis, ut ad instar libri claudi et aperiri
possint; ponantur illa in plano quopiam, in quo semicirculum in gradus
suos descriptum habeas. Si enim punctum, in quo specula committuntur,
in centro semicirculi statuas, ita ut utrumque speculi latus diametro
insistat, semel tantam videbitur rei imago, apparebuntque duæ res, una
extra speculum vera, altera intra, phantastica: Si vero specula ita
posueris, ut divaricatio laterum 120 gradus intercipiat, videbis rei
intra latera est, quia angulus reflexionis et incidentiæ tantus est,
quantus est angulus interceptus a lateribus, videlicet 120 grad. qui
cum obtusius sit, non nisi binam imaginem causare potest, ut in Propos.
v. fol. 848, ostensum est. Si vero specula interceperint angulum 90
graduum, videbis in plano circulum in quatuor partes divisum, in
quibus totidem simulacra rei positæ comparabunt, tria phantastica,
et unum verum; cum enim reflectio fiat ad angulos rectos utrumque
latus reflectens formam causabit intra se alias duas formas, unde et
consequentur pro multiplicatione laterum formæ multiplicabuntur, quæ
et in reflectione laterum normam servabunt uti in Propos. v. fol.
848, ostendimus. Porro si speculorum latera interceperint angulum 72
graduum, videbis in plano horizontali efformari perfectum et regulare
pentagonum, in quo totidem formæ apparebunt, item, si sexaginta
graduum interceperint angulum, videbis hexagonum totidemque formas
quinque nimirum phantasticas unam veram. Ita, si speculorum angulus
interceperit 51 gradus cum ³/₇ comparebit perfectum heptagonum,
cum totidem rei intra specula collocatæ formis; non secus angulus
speculorum 45 graduum dabit octogonum; 40 graduum dabit enneagonum;
36 graduum decagonum; 32 graduum angulus cum ⁸/₁₁ dabit endecagonum
et denique angulus 30 graduum referet dodecagonum cum totidem formis,
et sic in infinitum; ita ut semper tot laterum sit futurum polygonum
anacampticum totidemque formarum, quot polygonum cuivis latus
speculorum intercipit divaricatio, latera habuerit: quorum omnium
rationes dependent a Propos. v. precedentis distinctionis. —Kircheri,
_Ars Magna Lucis et Umbræ_. Rom. 1646, p. 890.
The following is a translation of the preceding passage:—
PARASTASIS I.
_Plane specula may be made to multiply the images of one object._ See
Fig. 54.
A wonderful, and, so far as I know, a new property, is exhibited by
two specula, so constructed that they may be opened and shut like a
book. If they are placed upon any plane, in which there is a semicircle
divided into degrees, in such a manner that the point where the
specula meet is in the centre of the circle, and the edge of each
speculum stands upon the diameter of the semicircle, one image only
will be visible, and there will appear two things, namely, a real one,
without the specula, and another formed by reflexion behind them,[21]
and so on, as in the following table, where the first column shows
the inclination of the specula, and the second the figure which is
produced:—
[21] I have thrown the rest of the passage into a tabular form, that
the reader may see, more readily, the effect produced by the variation
of the angle.
_Angle of Specula._ _Effect produced._
180° one image and one object,
*120° two images and one object,
90° four images,
*72° a pentagon and five forms,
60° a hexagon and six forms,
*51³/₇° a heptagon and seven forms,
45° an octagon and eight forms,
*40° an enneagon and nine forms,
36° a decagon and ten forms,
*32⁸/₁₁° an endecagon and eleven forms,
30° a dodecagon and twelve forms,
and so on, _ad infinitum_, the polygon formed by reflexion having
always as many sides as the number of times that the angle of the
specula is contained in 360°.
The combination of plane mirrors, which Kircher describes in the
preceding extract, is precisely the same as that which is given by
Baptista Porta. The latter, indeed, only mentions, that the number of
images increases by the diminution of the angle, whereas Kircher gives
the number of images produced at different angles, and enumerates the
_regular polygons_ which are thus formed.
It must be quite obvious to any person who attends to Kircher’s
description, that the idea never once occurred to him of producing
beautiful and symmetrical forms by means of plane mirrors. His _sole
object_ was to multiply a given regular form a certain number of times;
and he never imagined that, when the mirrors were placed at the angles
marked with an asterisk, there could be no symmetry in the figure, and
no union of the two last reflected images, unless in the case where
a regular object was placed, either by design or by accident, in a
position symmetrically related to both the reflectors.
In Kircher’s mirrors the eye was placed in front of them. The object
therefore was much nearer the eye than the images, and the light of
the different reflected images was not only extremely unequal, but
the difference in their angular magnitude was such that they could
not possibly be united into a symmetrical whole. From the accidental
circumstance of his using _lines upon paper_ as an object, the
distortion of the pictures arising from the erroneous position of the
eye was prevented; but if the same combination of mirrors were applied
to the object-plates of the Kaleidoscope, it would be found utterly
incapable of producing any of the fine forms which are peculiar to that
instrument.
If it were necessary to prove that Kircher and his pupils were entirely
ignorant of the positions of the eye and the object which are necessary
to the production of a picture, symmetrical in all its parts, and
uniformly illuminated, and that they went no farther than the mere
multiplication of forms that were previously regular and symmetrical,
we would refer the reader to Schottus’ _Magia Universalis Naturæ et
Artis_, printed at Wurtzbourg in 1657, where he repeats, almost word
for word, the description of Kircher, and adds the following curious
observation:—“But it is not only the objects placed in the semicircle
in the angle of the glasses that are seen and _multiplied_, but also
those which are more distant; for example, a wall, with its windows,
and in this case the multiplication produced by the mirrors will create
an immense public place, adorned with edifices and palaces.” This
passage shows, in the clearest manner, not only that the multiplication
of an object, independent of the union of the multiplied objects into
a symmetrical whole, was all that Kircher and his followers proposed
to accomplish; but also that they were entirely unacquainted with
the effects produced by varying the distance of the object from the
mirrors. If any person should doubt the accuracy of this observation,
we would request him to take Kircher’s two mirrors, to direct them to
a “wall with its windows,” either by Kircher’s method, or even by any
other way that he chooses, and to contemplate “the public place adorned
with edifices and with palaces.” He will see heaps of windows and of
walls, some of the heaps being much larger than others; and some being
farther from, and others nearer to, the centre; and some being dark,
and others luminous; while all of them are disunited. Let him now take
a Telescopic Kaleidoscope, and direct it to the same object; he will
instantly perceive the most perfect order arise out of confusion, and
he will not scruple to acknowledge, that no two things in nature can
be more different than the effects which are produced by these two
combinations of mirrors.
[Illustration: FIG. 55.]
[Illustration: FIG. 56.]
We come now to consider the claims of Mr. Bradley, Professor of
Botany in the University of Cambridge. In a work, entitled, _New
Improvements in Planting and Gardening_, published in 1717, this
author has drawn and described Kircher’s apparatus as an invention of
his own; and, instead of having in any respect improved it, he has
actually deteriorated it, in so far as he has made the breadth of the
mirrors greater than their height. An exact copy of the mirrors used
by Bradley is shown in Fig. 55, from which it will be at once seen,
that it is precisely the same as Kircher’s, shown in Fig. 54. We are
far from saying that Bradley stole the invention from Kircher, or that
Kircher stole it from Baptista Porta, or that Baptista Porta stole it
from the ancients. There is reason, on the contrary, to think that the
apparatus had been entirely forgotten, in the long intervals which
elapsed between these different authors, and there can be no doubt that
each of them added some little improvement to the instrument of their
predecessors. Baptista Porta saw the superiority of two mirrors, as a
multiplying machine, to a greater number used by the ancients. Kircher
showed the relation between the number of images and the inclination
of the mirrors; and Bradley, though he rather injured the apparatus,
yet he had the merit of noticing, that figures upon paper, which had a
certain degree of irregularity, like those in Fig. 56, could still form
a regular figure.
In order that the reader may fully understand Bradley’s method of using
the mirrors, we shall give it in his own words:—
“We must choose two pieces of looking-glass (says he), of equal
bigness, of the figure of a long square, _five_ inches in length, and
_four_ in breadth; they must be covered on the back with paper or silk,
to prevent rubbing off the silver, which would else be too apt to crack
off by frequent use. This covering for the back of the glasses must be
so put on, that nothing of it may appear about the edges on the bright
side.
“The glasses being thus prepared, they must be laid face to face, and
hinged together, so that they may be made to open and shut at pleasure,
like the leaves of a book; and now the glasses being thus fitted for
our purpose, I shall proceed to _explain the use of them_.
“Draw a large circle upon paper; divide it into three, four, five, six,
seven, or eight equal parts; which being done, we may draw in every one
of the divisions a figure, at our pleasure, either for garden-plats or
fortifications; as, for example, in Fig. 55, we see a circle divided
into six parts, and upon the division marked =F= is drawn part of a
design for a garden. Now, to see that design entire, which is yet
confused, we must place our glasses upon the paper, and open them to
the sixth part of the circle, _i.e._, one of them must stand upon the
line _b_, to the centre, and the other must be opened exactly to the
point _c_; so shall we discover an entire garden-plot in a circular
form (if we look into the glasses), divided into six parts, with as
many walks leading to the centre, where we shall find a basin of an
hexagonal figure.
“The line =A=, where the glasses join, stands immediately over the
centre of the circle, the glass =B= stands upon the line drawn from the
centre to the point =C=, and the glass =D= stands upon the line leading
from the centre to the point =E=: the glasses being thus placed, cannot
fail to produce the complete figure we look for; and so whatever equal
part of a circle you mark out, let the line =A= stand always upon the
centre, and open your glasses to the division you have made with your
compasses. If, instead of a circle, you would have the figure of a
hexagon, draw a straight line with a pen from the point _c_ to the
point _b_, and, by placing the glasses as before, you will have the
figure desired.
“So likewise a pentagon may be perfectly represented, by finding the
fifth part of a circle, and placing the glasses upon the outlines of
it; and the fourth part of a circle will likewise produce a square, by
means of the glasses, or, by the same rule, will give us any figure
of equal sides. I easily suppose that a curious person, by a little
practice with these glasses, may make many improvements with them,
which, perhaps, I may not have yet discovered, or have, for brevity
sake, omitted to describe.
“It next follows that I explain how, by these glasses, we may, from the
figure of a circle, drawn upon paper, make an oval; and also, by the
same rule, represent a long square from a perfect square. To do this,
open the glasses, and fix them to an exact square; place them over a
circle, and move them to and fro till you see the representation of
the oval figure you like best; and so, having the glasses fixed, in
like manner move them over a square piece of work till you find the
figure you desire of a long square. In these trials you will meet with
many varieties of designs. As for instance, Fig. 56, although it seems
to contain but a confused representation, may be varied into above
two hundred different representations, by moving the glasses over it,
which are opened and fixed to an exact square. In a word, from the most
trifling designs, we may, by this means, produce some thousands of good
draughts.
“But that Fig. 56 may yet be more intelligible and useful, I have drawn
on every side of it a scale, divided into equal parts, by which means
we may ascertain the just proportion of any design we shall meet within
it.
“I have also marked every side of it with a letter, as =A=, =B=, =C=,
=D=, the better to inform my reader of the use of the invention, and
put him in the way to find out every design contained in that figure.
“Example I.—Turn the side =A= to any certain point, either to the
north, or to the window of your room; and when you have opened your
glasses to an exact square, set one of them on the line of the side
=D=, and the other on the line of the side =C=, you will then have a
square figure four times as big as the engraved design in the plate:
but if that representation should not be agreeable, move the glasses
(still open to a square) to the number 5 of the side =D=, so will one
of them be parallel to =D=, and the other stand upon the line of the
side =C=, your first design will then be varied; and so by moving your
glasses, in like manner, from point to point, the draughts will differ
by every variation of the glasses, till you have discovered at least
fifty plans, differing from one another.
“Example II.—Turn the side marked =B=, of Fig. 56, to the same point
where =A= was before, and by moving your glasses as you did in the
former example, you will discover as great a variety of designs as
had been observed in the foregoing experiment; then turn the side
=C= to the place of =B=, and, managing the glasses in the manner I
have directed in the first example, you may have a great variety of
different plans, which were not in the former trials; and the fourth
side, =D=, must be managed in the same manner with the others; so that
from one plan alone, not exceeding the bigness of a man’s hand, we
may vary the figure at least two hundred times; and so, consequently,
from _five_ figures of the like nature, we might show about a thousand
several sorts of garden-plats; and if it should happen that the reader
has any number of plans for parterres or wilderness-works by him, he
may, by this method, alter them at his pleasure, and produce such
innumerable varieties, that it is not possible the most able designer
could ever have contrived.”
In reading the preceding description, the following conclusions cannot
fail to be drawn by every person who understands it.
1. Dr. Bradley, like Kircher, considers his mirrors as applicable to
regular figures, such as are represented in Fig. 55, and was entirely
unacquainted with the fact, that the inclination of the mirrors must
be an _even_ aliquot part of a circle. This is obvious, from his
stating that the mirrors may be set at the _third_, fourth, _fifth_,
sixth, _seventh_, or eighth part of a circle; for if he had tried to
set an irregular object between the mirrors when placed at the _third_,
_fifth_, or _seventh_ part of 360°, he would have found that a complete
figure could not possibly be produced.
2. From the erroneous position of the eye in front of the mirrors,
there is such an inequality of light in the reflected sectors, that the
last is scarcely visible, and therefore cannot be united into a uniform
picture with the real objects.
3. As the place of the eye in Bradley’s instrument is in front, and
therefore much nearer the object, or sector, seen by direct vision,
the angular magnitudes of all the different sectors are different,
and hence they cannot unite into a symmetrical figure. This is so
unavoidable a result of the erroneous position of the eye—a position
too, rendered necessary from the absurd form of the mirrors—that
Bradley actually employs his mirrors to convert a circle into an
ellipse, and a square into a rectangle!
4. In Bradley’s instrument, the sectors thus unequally shaped, and
unequally illuminated, are all separated from one another, by a space
equal to the thickness of the glass plates; and from the same cause,
the images reflected from the first surface interfere with those
reflected from the second, and produce a confusion or overlapping of
images entirely incompatible with a symmetrical picture.
These results are deduced upon the supposition that the object
consists of lines drawn upon the surface of paper, the only purpose
to which Bradley ever applied his mirrors; but when we attempt to use
these mirrors as a Kaleidoscope, and apply them to the objects which
are usually adapted to that instrument, we shall fail entirely, as
the mirrors are utterly incapable of producing the beautiful and
symmetrical forms which belong to that instrument.
We come now to consider the method of applying the mirrors to the
object shown in Fig. 55, which is an exact, but reduced copy of
Bradley’s figure. Because the angles of this figure are 90°, he directs
that the mirrors “be opened and fixed to an exact square; and that the
edge of one of the mirrors must always be placed so as to coincide
exactly with one of the sides of the rectangle, and carefully kept
in this line when the position of the mirrors is changed.” Hence it
is manifest, that Bradley was entirely ignorant of the fundamental
principle of the Kaleidoscope, namely, that if the inclination of the
mirrors is an even aliquot part of a circle, and if they are _set in
any position upon any object or set of objects, however confused,
or distorted, or irregular, they will create the most perfect and
symmetrical designs_. But even if Bradley had been aware of this
principle in theory, neither he nor any other person was acquainted
with the mode of constructing and fitting up the reflectors, in order
to render them capable of producing the effect. The position of the
eye, for symmetry of light, and for symmetry of form;—the position of
the mirrors, to produce a perfect junction of the last sectors;—the
position of the object necessary to produce an equality and perfect
junction between the object and the reflected images;—and the method
of accomplishing this for objects at all distances, were fundamental
points, in the combination of plane mirrors, which, so far as I
know, have never been investigated by any author. But even though a
knowledge of these theoretical points had been obtained, numerous
difficulties in the practical construction of the instrument remained
to be surmounted; and it required no slight degree of labour and
attention to enable the instrument to develop, in the most simple and
efficacious manner, the various effects which, in theory, it was found
susceptible of producing.
As the combination of mirrors, described by Kircher and Bradley, had
been long known to opticians, and had excited so little attention that
they had even ceased to be noticed in works on optical instruments, it
became necessary to discover some other origin for the Kaleidoscope.
The thirteenth and fourteenth Propositions of Wood’s _Optics_, and
some analogous propositions in Harris’s _Optics_, were therefore
presumed to be an anticipation of the invention. Professor Wood gives a
mathematical investigation of the number and arrangement of the images
formed by two reflectors, either inclined or parallel to each other.
These theorems assign no position to the eye, or to the object, and do
not include the principle of inversion, which is absolutely necessary
to the production of symmetrical forms. The theorems, indeed, which
have no connexion whatever with any instrument, are true, whatever
be the position of the eye or the object; and Mr. Wood has frankly
acknowledged, “that the effects produced by the Kaleidoscope were never
in his contemplation.”[22]
[22] See Appendix, p. 185.
The propositions in Harris’s _Optics_ relate, like Professor Wood’s,
merely to the multiplication and circular arrangement of the apertures
or sectors formed by the inclined mirrors, and to the progress of a
ray of light reflected between two inclined or parallel mirrors; and
no allusion whatever is made, in the propositions themselves, to any
instrument. In the propositions respecting the multiplication of the
sectors, the eye of the observer is never once mentioned; and the
proposition is true, if the eye has an infinite number of positions;
whereas, in the Kaleidoscope, the eye can only have one position. In
the other proposition (Prop. XVII.) respecting the progress of the
rays, the eye and the object are actually stated to be placed _between
the reflectors_; and even if the eye had been placed without the
reflectors, as in the Kaleidoscope, the position assigned it, at a
great distance from the angular point, is a demonstration that Harris
was _entirely ignorant of the positions of symmetry, either for the
object or the eye_, and could not have combined two reflectors so as to
form a Kaleidoscope for producing beautiful or symmetrical forms.[23]
It is important also to remark, that all Harris’s propositions relate
either to sectors or to small circular objects; and that he supposes
the very same effects to be produced when the inclination of the
mirrors is an _odd_, as when it is an _even_, aliquot part of a
circle. It is clear, therefore, that he was neither acquainted with
the fundamental point in the theory of the Kaleidoscope, nor with
any of its practical effects. The _only practical part_ of Harris’s
propositions is the fifth and sixth scholia to Prop. XVII. In the
fifth scholium he proposes a sort of catoptric box, or cistula,
known long before his time, composed of four mirrors, arranged in
a most unscientific manner, and containing opaque objects _between
the speculums_. “Whatever they are,” says he, when speaking of the
objects, “the upright figures between the speculums should be slender,
and not too many in number, otherwise they will too much _obstruct the
reflected rays from coming to the eye_.” This shows, in a most decisive
manner, that Harris knew nothing of the Kaleidoscope, and that he has
not even improved the common catoptric cistula, which had been known
long before. The principle of inversion, and the positions of symmetry,
were entirely unknown to him. In the sixth scholium, he speaks of rooms
lined with looking-glasses, and of luminous amphitheatres, which were
known even to the ancients, and have been described and figured by all
the old writers on optics.
[23] See Chap. I. pp. 11, 12, where we have shown that Harris was not
even acquainted with the way in which the last sector is formed by
reflexion.
APPENDIX.
Although I have no doubt that the observations contained in the
preceding Chapters are sufficient to satisfy every candid and
intelligent person respecting the true nature of all the combinations
of plane mirrors that preceded the invention of the Kaleidoscope, yet
as there are many who are incapable, from want of optical knowledge,
to understand the comparison which has been made between them, I shall
here present the opinions of four of the most eminent mathematicians
and natural philosophers.
The first of these is contained in a note from the late Professor
Wood, Master of St. John’s College, Cambridge, written in reply to a
letter, in which I requested him to say, if he had any idea of the
effects of the Kaleidoscope when he wrote the thirteenth and fourteenth
Propositions of his works on Optics.
“ST. JOHN’S, _May 19, 1818_.
“SIR,—The propositions I have given relating to the
number of images formed by plane reflectors, inclined to
each other, contain merely the mathematical calculation of
their number and arrangement. _The effects produced by
the Kaleidoscope were never in my contemplation._ My
attention has for some years been turned to other subjects,
and I regret that I have not time to read your Optical
Treatise, which I am sure would give me great pleasure.—I
am, Sir, your obedient humble servant,
“J. WOOD.”
The following is the opinion of the late celebrated Mr. James Watt:—
“It has been said here,” says Mr. Watt, “that you
took the idea of the Kaleidoscope from an old book on
gardening. My friend, the Rev. Mr. Corrie, has procured
me a sight of the book. It is Bradley’s Improvements of
Planting and Gardening. London, 1731, Part II. Chap. I.
It consists of two pieces of looking-glass, of equal bigness,
of the figure of a long square, five inches long, and four
inches high, hinged together upon one of the narrow sides,
so as to open and shut like the leaves of a book, which,
being set upon their edges upon a drawing, will show it
multiplied by repeated reflexions. This instrument I have
seen in my father’s possession seventy years ago, and frequently
since, but what has become of it I know not. In
my opinion, the application of the principle is very different
from that of your Kaleidoscope.”
The following is the opinion of the late Mr. Playfair, Professor of
Natural Philosophy in the University of Edinburgh:—
“EDINBURGH, _May 11, 1818_.
“I have examined the Kaleidoscope invented by Dr.
Brewster, and compared it with the description of an
instrument which it has been said to resemble, constructed
by Bradley in 1717. I have also compared its effect with
an experiment to which it may be thought to have some
analogy, described by Mr. Wood in his _Optics_, Prop. XIII.
and XIV.
“From both these contrivances, and from every optical
instrument with which I am acquainted, the Kaleidoscope
appears to differ essentially, both in its effect and in the
principles of its construction.
“As to the effect, the thing produced by the Kaleidoscope
is a series of figures presented with the most perfect
symmetry, so as always to compose a whole, in which
nothing is wanting and nothing redundant. It matters not
what the object be to which the instrument is directed; if
it only be in its proper place, the effect just described is
sure to take place, and with an endless variety. In this
respect the Kaleidoscope appears to be quite singular among
optical instruments. Neither the instrument of Bradley,
nor the experiment or theorem in Wood’s book, have any
resemblance to this; they go no further than the multiplication
of the figure.
“Next, as to the principle of construction, Dr. Brewster’s
instrument requires _a particular position of the eye of the
observer, and of the object looked at_, in order to produce its
effect. If either of these is wanting, the symmetry vanishes,
and the figures are irregular and disunited. In the other
two cases, no particular position, either for the eye or the
object, is required.
“For these reasons, Dr. Brewster’s invention seems to
me quite unlike the other two. Indeed, as far as I know,
it is quite singular among optical instruments; and it will
be matter of sincere regret, if any imaginary or vague analogy,
between it and other optical instruments, should be
the means of depriving the Doctor of any part of the reward
to which his skill, ingenuity, and perseverance, entitle him
so well.
“JOHN PLAYFAIR,
_Professor of Natural Philosophy in
the University of Edinburgh_.”
“_P.S._—Granting that there were a resemblance between
the Kaleidoscope and Bradley’s instrument, in any of the
particulars mentioned above, the introduction of coloured
and movable objects, at the end of the reflectors, is quite
peculiar to Dr. Brewster’s instrument. Besides this, a circumstance
highly deserving of attention, is the use of two
lenses and a draw-tube; so that the action of the Kaleidoscope
is extended to objects of all sizes, and at all distances
from the observer, and united, by that means, to the
advantages of the telescope.
“J. P.”
Professor Pictet’s opinion is stated in the following letter:—
“SIR,—Among your friends, I have not been one of the
least painfully affected by the shameful invasion of your
rights as an inventor, which I have been a witness of lately
in London. Not only none of the allegations of the invaders
of your patent, grounded on a pretended similarity
between your Kaleidoscope and Bradley’s instrument, or
such as Wood’s or Harris’s theories might have suggested,
appear to me to have any real foundation; but I can affirm,
that, neither in any of the French, German, or Italian
authors, who, to my knowledge, have treated of optics, nor
in Professor Charles’s justly-celebrated and most complete
collection of optical instruments at Paris, have I read or
seen anything resembling your ingenious apparatus, which,
from its numberless applications, and the pleasure it affords,
and will continue to afford to millions of beholders of its
matchless effects, may be ranked among the most happy
inventions that science ever presented to the lovers of
rational enjoyment.
“M. A. PICTET,
_Professor of Natural Philosophy in
the Academy of Geneva_.”
Those who wish to examine farther the ancient combinations of plane
mirrors, and other subjects connected with the Kaleidoscope, are
referred to the following works:—
Baptista Porta’s _Magia Naturalis_. Kircher’s _Ars Magna Lucis et
Umbræ_. Schottus’s _Magia Universalis Naturæ et Artis_. Bradley’s
_Treatise on Planting and Gardening_. Harris’s _Treatise on Optics_.
Wood’s _Optics_. Dr. Roget on the _Kaleidoscope_, in the _Annals
of Philosophy_, vol. xi. p. 375. _Encyclopædia Britannica_, Art.
KALEIDOSCOPE, _by_ Dr. Roget; and the _Compte Rendu des Travaux de
L’Académie de Dijon_, pour 1818, pp. 108-117.
Works by SIR DAVID BREWSTER.
MORE WORLDS THAN ONE: the Creed of the Philosopher, the
Hope of the Christian. Cr. 8vo., 4s. 6d.
THE MARTYRS OF SCIENCE: Galileo, Tycho Brahe, and
Kepler. Cr. 8vo., 4s. 6d.
THE KALEIDOSCOPE Practically Described. Cr. 8vo.,
_with Illustrations_, 4s. 6d.
THE STEREOSCOPE Practically Described. Cr. 8vo.,
_with Illustrations_, 4s. 6d.
LONDON:
JOHN CAMDEN HOTTEN, 74, PICCADILLY.
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