CHAPTER I.
PRELIMINARY PRINCIPLES RESPECTING THEEFFECTS 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 puttogether at an angle of 60°, or the sixth part of a circle, as shown inFig. 1, and the eye placed at the narrow endE, it will observe the openingA O Bmultiplied six times, and arranged round the centreO, as shown inFig. 2.
Fig. 1.Fig. 2.
Fig. 1.
Fig. 2.
In order to understand how this effect is produced, let us take a small sector of white paper of the shapeA O B,Fig. 2, and having laid it on a black ground, let the extremityA Oof one of the reflectors be placed upon the edgeA Oof the sector. It is then obvious that an imageA Obof this white sector of paper will be formed behind the mirrorA O, and will have the same magnitude and the same situation behind the mirror as the sectorA O Bhad before it. In like manner, if we place the edgeB Oof the other reflector upon the other sideB Oof the paper sector, a similar imageB Oawill be formed behind it. The origin of three of the sectors seen roundOis therefore explained: the first,A O B, is the white paper sector seen by direct vision; the second,A Ob, is an image of the first formed byonereflexion from the mirrorA O; and the third is another image of the first formed byonereflexionfrom the other mirrorB 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 imageA Obbeing, as it were, a new object placed before the mirrorB O, will have an imageaO αof itself formed behindB O; and for the same reason the imageB Oawill have an imagebO βof itself, formed behind the mirrorA 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 thexviith Prop. of his Optics, has evaded this difficulty, and given a false demonstration of the proposition. He remarks, that the last sector,α O βis produced “by the reflexion of the rays forming either of the two last images” (namely,bO βandaO α); but this is clearly absurd, for the sectorα O βwould 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 sectorα O β, we must recollect that the lineO Eis the line of junction of the mirrors, and that the eye is placed any where in the plane passing throughO Eand bisectingA O B. Now, if the mirror,B O, had extended as far asO β, the sectorα O βwould have been the image of the sectorbO β, reflected fromB O; and in like manner, if the mirrorA Ohad extended as far asO α, the sectorα O βwould have been the image of the sectoraO αreflected fromA O; but as this overlapping or extension of the mirrors isimpossible, and as they must necessarily join at the lineO E, it follows, that an imageα Oe, of only half the sectorbO β, viz.,bOr, can be seen by reflexion from the mirrorB O; and that an imageβ Oe, of only half the sectoraO α, viz.,aOs, can be seen by reflexion from the mirrorA 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 oβ,a oα, but is composed of the images of two half sectors, one of which is formed by the mirrorA O, and the other by the mirrorB 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 imagesD,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:—“Whena(the angle of the mirrors) is a measure of 180°two images coincide,” and it is demonstrated, thatsince two images of any objectX(Fig. 2)must be formed, viz., one by each mirror, and since these two images must be formed at 180° from the objectX, placed between the mirrors, that is, at the same pointx, 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 imagexis seen by the last reflexion from the mirrorB O E, and another imagewould be seen at x, if the mirrorA O Ehad extended as far asx; but as this is impossible, without covering the part of the mirrorB O E, which gives the first imagex, there can be only one image seen atx. When the objectXis equidistant fromAandB, then one-half of the last reflected imagexwill be formed by the last reflexion from the mirrorB O, and the other half by the last reflexion from the mirrorA O, and these two half images will join each other, and form a whole image ate, as perfect as any of the rest. In this last case, when the angleA O Bis a little different from an even aliquot part of 360°, the eye atEwill perceive atean appearance of two incoincident images; but this arises from the pupil of the eye being partly on one side ofEand partly on the other; and, therefore, the apparent duplication of the image is removed by looking through a very small aperture atE. 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 whenA O Bis ¼, ⅙, ⅛, ⅒, ¹/₁₂, etc., of a circle, the number of reflected images of any objectX, is 4 - 1, 6 - 1, 8 - 1, 10 - 1, 12 - 1.
2. That whenXis 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 whichXis 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 nearestX, than from the mirror farthest from it.
3. That whenXis equidistant fromA OandB O, the number of images which reaches the eye from each mirror is equal, and is always
which are fractional values, showing that the last image is composed of two half images.
When the inclination of the mirrors, or the angleA O B,Fig. 3, is anoddaliquot 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 ofreflected sectorsmust in this case always beeven, the lineO E, where the mirrors join, will separate the two last reflected sectors,bOe,aOe. Hence it follows,—
Fig. 3.
Fig. 3.
1. That whenA O Bis 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
which are always even numbers.
Hitherto we have supposed the inclination of the mirrors to beexactlyeither 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 angleA O B,Fig. 2, is made to increase from being anevenaliquot part of a circle, such as ⅙th, till it becomes anoddaliquot part, such as ⅐th, the last reflected imageβ O α, composed of the two halvesβ Oe,α Oe, will gradually increase, in consequence of each of the halves increasing; and whenA O Bbecomes ⅐th of the circle, the sectorβ O αwill become double ofA O B, andα Oe,β Oewill become each complete sectors, or equal toA O B.
If the angleA O Bis made to vary from ⅙th to ⅕th of a circle, the last sectorβ O αwill gradually diminish, in consequence of each of its halves,β Oe,α Oe, diminishing; and just when the angle becomes ⅙th of a circle, the sectorβ O αwill have become infinitely small, and the two sectors,bO β,aO α, will join each other exactly at the lineOe, as inFig. 3.