CHAPTER II.
ON THE PRINCIPLES OF THE KALEIDOSCOPE, AND THEFORMATION OF SYMMETRICAL PICTURES BY THECOMBINATION 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,simpleorirregularforms, andcompoundorregularforms. 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 inFigs. 4and5the effects produced by the multiplication of single and compound forms. The linea 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. 5represents 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 forma b c.
Fig. 4.Fig. 5.
Fig. 4.
Fig. 4.
Fig. 5.
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 inFig. 5, we must consider that the simple formm n,Fig. 2, is seen by direct vision through the open sectorA O B, and that the imagen o, of the objectm n, formed by one reflexion in the sectorB Oa, is necessarily an inverted image. But since the imageo p, in the sectoraO α, is a reflected and consequently an inverted image of theinverted image,m t, in the sectorA Ob, it follows, that the wholen o pis an inverted image of the wholen m t. Hence the imagen owill unite with the imageo p, in the same manner asm nunites withm 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 halfβ Oeof the last sectorβ O αwas formerly shown to be an image of the half sectoraOs, the lineq vwill also be an image of the lineo z, and for the same reason the linev pwill be an image oft y. But the imagev pforms the same angle withB Oorn qthatt ydoes, and is equal and similar tot y; andq vforms the same angle withA Othato zdoes, and is equal and similar too z. Hence,Oo=o q, andOy= Ov, and thereforeq vandv pwill form one straight line, equal and similar tot q, and similarly situated with respect toB O. The figurem 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 apertureA O B, whether simple or compound, and to every angleA O B, which is an even aliquot part of a circle, it follows,—
1. That when the inclination of the mirror is anevenaliquot 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 angleA O Bis 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 anoddaliquot part of 360°, such as ⅕th, as shown inFig. 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 ofFig. 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.
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 sectoraOe,Fig. 7, by the mirrorB O, will not join with the image formed in the last sectorbOe, by the mirrorA 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 inFig. 7. LetA O,B O, be the reflecting planes, andm na line,inclined to the radius which bisects the angleA O B, so thato m>o n; thenm nʹ,n mʹ,will be the images formed by the first reflexion fromA OandB O, andnʹ mʺ,mʹ nʺ, the images formed by the second reflexion; but by the principles of catoptrics,Om= Omʹ= Omʺ, andOn= Onʹ= Onʺ, consequently sinceOmis by hypothesis greater thanOn, we shall haveOmʺgreater thanOnʺ; that is, the imagesmʹ nʺ,nʹ mʺ, will not coincide. AsOnapproaches to an equality withOm,Onʺapproaches to an equality withOmʺ, and whenOm= On, we haveOnʺ= Omʺ, and at this limit the images are symmetrically arranged, which is the case of the straight line 1, 2 inFig. 6.By tracing the images of the other lines, as is done inFig. 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.
Fig. 7.Fig. 8.
Fig. 7.
Fig. 7.
Fig. 8.
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 anoddaliquot 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 halfthe 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., andvice 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:—
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 anodd 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 sectorβ O α,Fig. 2, is greater thanA O B, the partq vin one half must be the image of more thano z, andv pthe image of more thant y, andvice versa, whenβ O αis less thanA 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 imagesq vandv p. The anglesOq vandOo pare obviously equal, and also the anglesOp v,Op o; but since the angleβ O α, orqOp, is by hypothesis greater or less thanpOo, it follows that the angles of the triangleqOpare either greater or less than two right angles, because they are greater or less than the three angles of the trianglepOo. But as this is absurd, the linesq 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 angleβ O αis greater thanpOo, orA O B, the linesq v,v p, will form a re-entering angle towardsO, and when it is less thanA O B, the same lines will form a salient angle towardsO.