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.
Fig. 6.