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 reflection 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 abcd, 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

B

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

FIG. 4

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 abc.

FIG. 5

a

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 mn, 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 op, in the sector a o a, 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 no will unite with the image op, in the same manner as mn 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 Boe of the last sector Bo a 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

Hence, o o

=

v p will form similar to t q, and similarly The figure m n o p q t, thereobject, and several reflected

angle with B O or n q that t y does, and is equal and similar to ty; and q v forms the same angle with Ao that o z does, and is equal and similar to o z. o q, and o y =ov, and therefore q v and one straight line, equal and situated with respect to B 0. fore, composed of one direct 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 th, 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