Kaleidoscope, an optical instrument, invented by Sir David Brewster, which, by a particular arrangement of mirrors, or reflecting surfaces, presents to the eye, placed in a certain position, symmetrical combinations of images, remarkable for their beauty and the infinite variations of which they are susceptible. The name is derived from the Greek words κάλος, beautiful, ἰδεῖν, a form or appearance, and εἶδος, to see.

The effect of combining two or more plane mirrors, so as to produce a multiplication of images, had long been known and described by writers on optics. Baptista the Younger Porta, in his Magia Naturalis, gives an account of the construction of an instrument, which he calls polyphaton, in which two rectangular specula are united by two of their sides, so that they may be opened or shut like a book, and the angles varied; and also of a polygonal speculum, consisting of several mirrors arranged in a polygon, for multiplying in different directions the images of ob- Kircher, also, in his *Ars Magna Lucis et Umbrae*, describes, as an invention of his own, the former of these constructions, and distinctly traces the relation between the angle of inclination of the mirrors and the number of images formed. The very same contrivance was afterwards adopted by Bradley, for the purpose of assisting in the designing of garden plots and fortifications; and he states that, "from the most trifling designs, we may, by this means, produce some thousands of good draughts. But the particular application of this principle in the case where the two reflectors are inclined to one another at a small angle, so as to form a series of symmetric images, distinctly visible only in a particular position of the eye, was a discovery reserved for Sir David Brewster. The first idea of this remarkable property occurred to him in the course of some experiments in which he was engaged on the polarization of light, during the year 1814. But the only circumstance which at that time attracted his attention, was 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, between which the light had undergone several successive reflections. In repeating, at a subsequent period, some experiments of M. Biot on the action of homogeneous fluids upon polarised light, and in extending them to other fluids which he had not tried, Sir David Brewster happened, for greater convenience, 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 placed 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 remarkably regular and symmetrical manner. Pursuing the hint thus obtained, and investigating the subject optically, he discovered the leading principles of the kaleidoscope, in as far as the inclination of the reflectors, the position of the object and that of the eye, were concerned. He then constructed an instrument in its simplest form, and showed it to some of the members of the Royal Society of Edinburgh, who were much struck with the beauty of its effects. Several very material improvements were subsequently made by the inventor, in the construction and application of the instrument, for which he then took out a patent. But, in consequence of one of these instruments having found its way to London, its properties became generally known before any number of the patent kaleidoscopes could be prepared for sale. It very quickly became popular, and the sensation it excited in London throughout all ranks of people was astonishing. Kaleidoscopes were manufactured in immense numbers, and were sold as rapidly as they could be made. The instrument was in every body's hands, and people were everywhere seen, even at the corners of streets, looking through the kaleidoscope. 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. Sir David Brewster states, that no fewer than two hundred thousand kaleidoscopes were sold in London and Paris in the space of three months; "and yet," says he, "out of this immense number, there is, perhaps, not one thousand constructed upon scientific principles, or capable of giving any thing like a correct idea of the power of the kaleidoscope; and of the millions who have witnessed its effects, there is perhaps not one hundred who have any idea of the principles upon which it is constructed, and of the mode in which those effects are produced." To convey a knowledge of these principles is the object of the present article.

It follows from the optical law of the equality of the optical angles which the incident and reflected rays make with principles a line perpendicular to the reflecting surface at the point on which of incidence, that rays which diverge from any object, and fall on a plane surface, will, after reflection, proceed in the same course as if they had immediately diverged from a point situated at the same distance behind the reflecting surface as the radiant point is before it. This point is called the virtual focus of those rays; and the eye receiving them will have the perception of a reversed image of the object in this situation. Thus the mirror AA' (Plate CCCXIX. fig. 1) will produce a reversed image of the object R, situated at the point S, in the line RP, perpendicular to the surface of the mirror; and this image will appear in the same place whatever be the situation of the eye, as E, provided the reflected rays rE meet it.

Since the course of the reflected rays is the same as if they had immediately proceeded from a real object of S, where its image is seen, this image will, with relation to another mirror, have all the effect of a real object; and a second reflection of the rays by a new mirror at BB', will produce, at the point T, equally distant from BB' as S is, but on the other side of it, an image of the first image, visible to the eye at E by the twice reflected rays rE. As the first image was reversed with respect to the object, so the second image will be reversed with respect to the first, and therefore direct when compared with the object. The second image may, it is evident, by a new reflection from the first mirror, give rise to a third, which will now again, like the first image, be reversed; and so on, in succession, may a series of images alternately reversed and direct be produced on each side, by two mirrors only, in consequence of multiplied reflections, provided the mirrors are of sufficient extent to admit of them, and provided the eye be so placed as to receive the rays which are last reflected.

If the mirrors be parallel to each other (see fig. 2), the images of the intervening objects, AAB'B', will be ranged in succession in a continued line on each side. If they be somewhat inclined to each other (as in fig. 3), the images will be disposed in the arch of a circle, having for its centre the point in which the directions of the mirror unite. If the mirrors be of sufficient length, or sufficiently inclined, so as actually to meet; and if, moreover, the angle they form be an even aliquot part of a circle, the images of all the objects situated in the space between them, ABC, fig. 4, will together occupy a circular field, and will be disposed in the form of sectors all round the circle.

This circular arrangement of the images, however legitimately it may have been deduced from the simplest law method of optics, appears to be so extraordinary an illusion of tracing the sense, as to call for somewhat further examination before we can feel perfectly assured that it is a necessary consequence of that law. Perhaps the most satisfactory method of prosecuting their examination is to investigate separately the mode in which each of the images results from the successive reflections by the two mirrors. A very simple and convenient rule may be laid down for enabling us to trace the whole course, however complex, of the rays which form these images; and this rule will be best understood by considering, as an example, its application to one of the remote images in the circular field. Thus, in the circular field AHL, fig. 5, divided into equal sectors by the radii CF, CG, CH, &c. let S be one of the remotest images of the object R, formed by four reflections from the mirrors AC, BC; and let E be the place of the eye. Draw the line ES, intersecting the radii al- ready mentioned, in P, Q, T, V; make Cg equal to CQ, and join Pg; make Ct equal to CT, and join gt; make Ce equal to CV, and join tv and vr. Then Rtv PE will be the real course of the rays, by which the image of R is seen at S by the eye at E; for it is sufficiently apparent, without the necessity of a formal demonstration, that by this construction, the equality of the angles of incidence and reflection is everywhere preserved. The different positions of the line PS, that is, PQ, QT, TV, and VS, are in fact the images of Pg, gt, tv, and vr respectively, which are so many portions of the real course of the reflected rays. It is evident that a similar construction will, in every other case, furnish us with the actual course of all the rays from which images result, through all their successive stages of reflection; and it has also the advantage of giving us the exact angles of incidence and reflection throughout the whole path.

We have hitherto, for the sake of perspicuity, supposed both the object and the eye, together with the path of the rays, to be in the same plane. But it is obvious that the same method of construction and of reasoning may be employed in tracing their course, if we suppose the mirrors to be prolonged in a direction perpendicular to the plane of the figure, and the eye raised above that plane. The space between the mirrors, instead of being the sector of a circle merely, is now the sector of a cylinder; which cylinder may be completed by supplying the other sectors which compose it, as is represented in fig. 8, where ACae and BCbe being the mirrors, the rest of the cylindrical space is occupied by complementary sectors. The course of the rays by which the eye at E will see the image S, for instance, of the object R, may readily be traced by drawing a straight line from E to S, which will pass through as many planes BCbe, &c., as the rays have suffered reflections. The portions of the lines ES, intercepted between these planes, may, as in the former case, be regarded as the images (either reversed or direct, as the case may be) of some portion of the actual path of the rays between the mirrors; A will occupy the same position with regard to the complementary sector it traverses, as the real path does in the original sector bounded by the mirrors. By drawing, in this sector, lines similarly situated with respect to its sides, as the several portions, PQ, QT, TS of the lines ES, are with respect to the sides of their respective sectors, we obtain the real course of the rays, RtvPE.

Symmetry appears to be the principal constituent of beauty in the forms given to the various works of art which have exercised the skill and ingenuity of man; and the richness of each individual ornament, as well as the pleasing effect of the whole assemblage, is generally in proportion as this principle has received a more perfect development. Even nature, in the multitude of forms with which she has invested the different tribes of the animal creation, has, with but few exceptions, followed the law of symmetry, in as far as respects the perfect similarity of the two sides of the body. In almost all the higher classes, or those which are comprehended under the great division of vertebrated animals, and in many of the inferior tribes, as in insects, one half of the animal form is the reflected image of the other half. A still higher degree of beauty, derived from a more extended symmetry of form, has been displayed in the structure of objects in the vegetable kingdom. Flowers, in particular, derive a peculiar beauty from their presenting to the eye a symmetrical combination of forms with reference to a common centre. This is also the general model followed in the structure of radiated animals, of which the star-fish and sea anemone are examples. In those works of art in which there is the greatest scope for the indulgence of fancy in the production of pleasing effects, the most perfect and successful kinds of ornament are those resulting from a symmetrical arrangement of parts, which is not confined to a single lateral repetition, but is extended in various directions in space, and is multiplied and alternated in different lines, and around different centres. It is the latter of these combinations, more especially, that is represented by the kaleidoscope, namely, the disposition of a certain number of pairs of images symmetrically disposed around one or more centres.

On examining the subject more minutely, we find that the first element of this symmetry consists in the union of any particular form, or of its direct image with its reversed image, by which a new form is created, composed of two simple forms similar to each other, and similarly situated with respect to a given line. If a succession of these compound forms be now arranged around a centre, they will combine into a perfect whole, in which all the similar parts are brought into union, and which must thus afford pleasure, by enabling the mind readily to take in and comprehend every part at a single glance. The operation of the kaleidoscope is, in this way, to create regularity and symmetry out of every form that is presented to it, however irregular in itself that form may be. Thus, out of the few simple lines contained in fig. 9, the appearance presented in fig. 6 is created by the instrument. It is scarcely necessary to observe, that the original lines, which occupy the sector between the mirrors, are seen by direct vision, and that their appearance unites itself on each side with their images seen by reflection. We shall in future designate the whole of the appearance thus produced by the kaleidoscope by the term spectrum.

If we examine the effect produced by each elementary portion of the compound figure of the spectrum, we shall find that any straight line reaching directly across the sector, as fg (fig. 9), is formed by the kaleidoscope into a regular polygon, having as many sides as the numbers into which the circular field is divided; if it be at right angles to either of the sides, the polygon will have only half the number of sides. A line, as mn, crossing the field between the mirrors in an oblique direction, is converted by the instrument into a polygon of the same number of sides as the former, but with salient and re-entering angles; that is, into the form of a star, with a number of rays equal to half the number of sectors. Another line crossing the field in an opposite direction gives another star, having its rays intermediate to those of the former. Curved lines form by their union a multitude of beautiful and elegant figures, of which the variety is inexhaustible. Each group, taken separately, possesses its peculiar and intrinsic beauty; but the effect of the whole assemblage is considerably heightened by the combination, and by the regularity of the relations that each part bears to all the others.

Having thus given an account of the general principles upon which the kaleidoscope is constructed, and of the conditions in which it acts, we are now prepared to direct our attention to the conditions which are required for the perfect performance of its functions.

If the mirrors of the kaleidoscope could reflect the whole extent of the light which falls upon them, the images would possess the same degree of brilliancy as the objects from which they are derived; and their number would be limited only by the more or less favourable position of the mirrors, and of the eye with relation to the objects. But as a very large portion of the incident light is, in most cases, destroyed by reflection, it follows that each successive image will be fainter than that which preceded it; and that in the progress of the reflections we must very soon arrive at a limit beyond which they become no longer visible. It is found, from experiment, that the quantity of light lost by reflection is in all cases greatest when the rays fall perpendicularly on the mirror, and least when they fall with the greatest est obliquity. The difference is more considerable in the case of glass than in that of metallic surfaces. Thus, in a common looking-glass, the images of objects seen by holding it directly opposite to them are produced wholly by the surface of the quicksilver, those reflected by the glass being too faint to occasion any interference. If the glass be placed obliquely, so that the angles of incidence and reflection be large, a greater proportion of light will be reflected from the glass, and the images formed by it will be bright enough to be seen, and will mix themselves with the images from the quicksilver. At a certain angle, both sets of images will appear of equal brightness; and by still further increasing the obliquity, those produced by the quicksilver will gradually fade away, and vanish, leaving the images produced by the glass perfectly distinct, and nearly as brilliant as the objects themselves.

The following table, abridged from one given by Sir David Brewster, and founded on the experiments of Bouguer, shows the number of rays reflected from plate-glass at various angles of incidence, the number of incident rays being supposed to be 1000.

| Complement of the Angles of Incidence | Rays reflected out of 1000 | Complement of the Angles of Incidence | Rays reflected out of 1000 | |---------------------------------------|--------------------------|---------------------------------------|--------------------------| | 2½° | 584 | 30° | 112 | | 5 | 543 | 35 | 79 | | 7½° | 474 | 40 | 57 | | 10 | 412 | 50 | 34 | | 12½° | 356 | 60 | 27 | | 15 | 299 | 70 | 25 | | 20 | 222 | 80 | 25 | | 25 | 157 | 90 | 25 |

With the help of this table, and the method above explained of tracing the course of the rays, and on investigating the angles of incidence, the degree of illumination of any part of the spectrum might be calculated, were it not for a new condition, termed polarisation, with which the rays of light are affected by reflection, and which may also contribute to the further loss of light, when the reflection is repeated at certain angles, and in certain positions of the plane in which it takes place; a circumstance which is not without its influence in the case of the kaleidoscope, especially in those constructed with glass mirrors.

As the effect which the kaleidoscope is intended to produce is to be the result of repeated reflections, it is an object of essential importance, in order that as little light may be lost as possible, that all these reflections should take place with the greatest obliquity. With this view, the mirrors should be of considerable length, and the eye should be raised above the field of view, and brought as near as possible to the planes of the mirrors; that is, as near as possible to the remote end of the line of their intersection. From this situation the remoter sectors will be seen by a greater quantity of light than from any other, and consequently the illumination of the spectrum will be more equal in every part. This position of the eye affords the further advantage of giving to the spectrum a circular appearance; for it is obvious, that if viewed from any other and more oblique situation, it would, from the laws of perspective, appear more or less elliptical. It is scarcely necessary to remark, that the eye cannot be mathematically in the line of junction of the mirrors, for no light would in that case reach it by reflection from them.

The essential parts of the kaleidoscope, then, are the two mirrors ACE and BCE (fig. 10), which should be from six to ten inches in length, and from one inch to an inch and a half in breadth at the object end C, while they are made narrower at the other end E. They are kept apart at their upper edges, and united along their lower edges CE, so as to form an angle which must be an even aliquot part of a circle. The angles 36°, 30°, 25°, 22½°, 20°, or 18°, which divide the circumference of the circle respectively into 10, 12, 14, 16, 18, and 20 equal parts, are the only angles which can conveniently be employed with glass mirrors. The objects to be viewed must occupy the space ABC, between the ends of the mirrors, and must be situated in the plane formed by these lines. They are to be viewed from the opposite or narrow end e; the eye being placed near to the angular point E, formed by the junction of the ends of the mirrors. It should, however, be a little above this point, in order that a sufficient quantity of light may enter through the pupil. By trial, the proper distance at which the maximum of illumination is obtained will easily be found.

It is of considerable importance that the junction of the Accuracy mirrors be a perfectly straight line, free from roughness, in their and from particles of dust. Any irregularity in this line will interfere with the perfection of the image at that part most remote from the object. The projection of this line of junction of the mirrors on the field of view is a line CD, fig. 4, immediately opposite to the middle of the space between the mirrors. If tolerable pains have been taken to apply a straight and smooth edge of one mirror upon the surface of the other, and to preserve them clean, this line will scarcely be seen, more especially as the greater part of it is placed much nearer than the objects contemplated, and lies, therefore, within the distance to which the refractive powers of the eye are adapted; it is, therefore, seen only indistinctly.

Any deviation in the angle formed by the mirrors from Angle of that which accurately divides the circle into an even number of sectors, is quickly perceived by the eye, from the consequent irregularity which takes place in the compound figure of the spectrum at the part most remote from the object. This is illustrated by fig. 7, where the last ray of the star is seen to be imperfect, from the want of correspondence in the images which meet in the remote sector. If the angle be too small, the image is redundant, from a reduplication of one portion; if too large, the image presents a deficiency. But, in consequence of the aperture of the pupil being of sufficient size to admit portions of the images from both mirrors, reflected from the parts immediately adjacent to the line of their junction, these two images will be, for a certain space, seen in the same direction, and will consequently overlap and interfere with each other. As soon as the angle of the mirrors is rendered correct, the double images coalesce into one, and perfect symmetry is restored to the spectrum. It is necessary to observe, that the angle must be an even aliquot part of a circle; that is, must divide it into an even number of equal parts. If the division were into an odd number of parts, as in fig. 7, the discordance of the adjacent images at the remote sectors would be the greatest possible. This will appear from considering that the images in the successive sectors on each side, being alternately reversed and direct, those in the sectors immediately adjacent to the radius most remote from the mirrors, would both be of the same kind; the one, therefore, could not be the reverse of the other, a relation which, as we have already seen, is the elementary condition of symmetry in each pair of images. The corresponding parts of each, indeed, instead of being adjacent, would then be the most remote from one another. This circumstance, namely, the necessity of the angle of the mirrors being the even aliquot part of a circle, although it be an essential condition of the instrument, is not mentioned in the specification of Sir David Brewster's patent. It was first noticed by the author of this article in the Annals of Philosophy.

If we investigate the proportion of light distributed over the field of view, by considering the degrees of obliquity with which the rays impinge upon the mirrors, and also the number of reflections which they sustain, we shall find that it diminishes nearly in the same proportion as we recede from the edge of the sector bounded by the mirrors, and is least in the remotest sector. The line of equal illumination in each individual sector, or the isophotal line, as Sir David Brewster has termed it, is parallel to that radius of the sector which is nearest to the mirror on that side. It follows as a consequence, that the light will diminish in each sector in proportion as we recede from the angular point or common centre of the field. This last circumstance limits us to the magnitude which it would be proper to allow to the field of view, and therefore restricts us in the breadth of the mirrors when they are of a given length. In general, their breadth should not exceed one sixth of their length, and the angle subtended by the circular field will then be about 19°. The proper proportion, however, varies according to the angle at which the mirrors are inclined. The larger this angle, the greater latitude may be allowed in extending the field of view; while, if the angle be small, the number of reflections for completing the remote parts of the spectrum will be great; the light will become too faint to allow the eye to distinguish the parts at the circumference; and the diameter of the field must be contracted by lessening the breadth of the mirrors.

What has now been said relates only to the proportional length and breadth of the mirrors. With regard to their absolute size, we must be guided in our choice by other circumstances of convenience. As the length of the instrument determines the distance of the eye from the field, it should be such as to admit of the distinct vision of every part of the spectrum. This may be effected, if requisite, by interposing a convex lens, of the proper focal distance, between the eye and the narrow end of the mirrors.

The last circumstance we shall notice as essential to the perfect operation of the instrument, is, that the objects must be situated as nearly as possible in the plane ABC, fig. 10, formed by the ends of the mirrors. All deviations from this position are productive of irregularity in the spectrum. If the eye, indeed, were a mere mathematical point, and were it possible for it to receive the rays while placed at the very point of the angle E, the distance of the object from the mirrors would, in strictness, produce no deviation from symmetry. Let the plane MN be taken at a little distance from the ends of the mirrors, and the planes of the mirrors produced till they meet it in the lines ae and be. It is evident that the space comprehended between these lines, is the only situation in that plane from whence rays can proceed so as to fall upon the mirrors; no object, therefore, which is not within that space, can have its image formed by reflection from the mirrors. The lines ae and be are the projections of AC and BC as viewed from the point E. But if the eye be raised to e, it will be apparent that a space below the former, and bounded by the lines de and fe, which will now be the projections of AC and BC, will come into view. The objects situated in this space will have no corresponding images, and their introduction into the field of view will produce confusion in every part of the spectrum. The magnitude of this additional space, measured by the interval er, which is unrepresented by the instrument, and which may be termed the aberration, is dependent upon and proportional to two separate causes, namely, the distance of the eye from the angular point, and the distance of the object from the mirrors. The deviation from regularity which it produces in the spectrum increases as the object approaches to the centre. An eye accustomed to observe and admire the symmetry of the combined images will instantly perceive it to be violated, even when the distance of the object Cc is less than the twentieth part of an inch. When the object is very distant, the defect of symmetry is so enormous, that, although the object is seen by direct vision, and also in some of the sectors, it is entirely invisible in the rest. If the object, on the other hand, be placed within the reflectors, a symmetrical spectrum will indeed be formed; but the centre of this spectrum will not coincide with the centre of the circular field of view, and its effect in producing a symmetrical picture is thereby entirely destroyed. In order to insure perfect mathematical symmetry, the objects should, strictly speaking, be limited to lines lying in the same plane, which plane must be exactly in contact with the ends of the mirrors.

We have hitherto considered the effects resulting from the combination of only two mirrors, in which case the field of view is necessarily limited to a circle. But on the very same principles we may, by employing a greater number of mirrors, obtain an extension of this field in all directions, and produce groups of images around several centres, which shall be repeated in perpetual succession on every side. Kaleidoscopes of this description have on that account been called Polycentral, and, when properly constructed, their effects are exceedingly beautiful. With respect, also, to their utility, as applicable to the arts, they very far excel the simple kaleidoscope, inasmuch as the occasions requiring an ornamental design for a flat extended surface are of much more ordinary occurrence than those in which we are limited to a circular space. The principles upon which polycentral kaleidoscopes should be constructed, and the conditions to which they are limited, were first pointed out by the author of this article, in the Annals of Philosophy (vol. xi. p. 375), soon after the common instrument became known in London.

It is evident that, by joining together a number of mirrors, so as to compose the sides of a prism, we might obtain a succession of images in every possible direction. But we must recollect that, for the production of symmetrical combinations of images, we are restricted in our choice of a base for the required prism, to such angles only as will divide the circle into an even number of aliquot parts. This condition confines us to a very limited range. It excludes, in the first place, all angles above 90°; and, therefore, all polygons having more than four sides. Of four-sided polygons, the square and the rectangle, where all the angles are right angles, are the only figures that can give symmetrical combinations. After these, there remain only triangles; and, among all the possible varieties of triangles, we can take only such as are formed with angles of 90°, 60°, 45°, or 30°, which are the quotients of 360°, divided by 4, 6, 8, and 12; all the other even aliquot parts of the circle being excluded by the necessary condition that the sum of those angles must be equal to 180°. We are, therefore, limited to the three following species of triangles, represented in figures 15, 16, and 17:

The first having all its angles equal to 60°, 60°, and 60°; The second its angles respectively equal to 90°, 45°, and 45°; And the third its angles respectively equal to 90°, 60°, and 30°;

The sum of these angles, in each case, being 180°.

Let us now inquire into the effects resulting from each of these combinations.

The comparative effects of these four species of polycentral kaleidoscopes are illustrated by figures 14, 15, 16, and 17, where A, in each case, represents the sections of the mirrors, or the base of each prism; B, the elements of each pattern; and C, the pattern itself, resulting from the series of reflected images.

It will be seen that the square polycentral kaleidoscope, fig. 14, produces a less pleasing effect than the others, because the attention being more particularly directed to the repetition of the same set of images in one direction only, the whole pattern appears composed of an alternation of longitudinal stripes. The direction of the stripes is determined by the general direction of the lines, in the elementary pattern approaching more to one of the sides of the base than to the other side. It is scarcely neces- sary to observe, that the spectrum produced by a rect- angular is quite similar to that of a square kaleidoscope, only that it is more extended in one direction.

The first of the triangular polycentral kaleidoscopes (fig. 15), which has for the base of its prism an equilate- ral triangle, affords very regular combinations of images, disposed in three different directions, which cross each other at angles of 60° and 120°; thus presenting what may be called a triangular symmetry. The circumstance of each pair of images being combined in groups of three together in every part of the spectrum, has suggested the name of Triassiccope for this species of triangular polycen- tral kaleidoscope.

The second species of triangle (fig. 16), which may be made the base of the prism, is that composed of two con- tiguous sides, together with the connecting diagonal of a square; or, in other words, of a right-angled isosceles tri- angle. The result of this construction is to produce a division of the field of view into regular square compart- ments, having the base of the above-mentioned triangle for their sides. The very perfect symmetry which re- sults from this construction is the source of remarkably beautiful designs; the predominant character of which is an arrangement of forms grouped together by fours at a time, and symmetrically disposed in squares. Such an instrument may, on this account, be called a Tetra- scope.

The last species of triangular polycentric kaleidoscope, or that which takes for its base the half of an equilateral triangle (fig. 17), resulting from its division by a perpen- dicular drawn from an angle to the opposite side, affords also appearances of very considerable beauty. Here the predominant form is the hexagon, from the circumstance that the smallest of the angles, which is that of 30°, pro- ducing the greatest number of repetitions of the same image around one centre, the symmetry is most conspi- cuous with reference to that centre, and the attention of the spectator is immediately directed to the hexagonal compartments into which the field is thereby divided.

As the pairs of images in these leading objects (such as the stars in the figure, which, it will be observed, have each six rays) are six in number, we shall, following the analogy of the other names, denominate this variety of the instrument a Hexascope. These names, derived from the circumstance which gives the chief character of sym- metry to the extended spectrum, will perhaps be consi- dered as sufficiently appropriate. They will, at all events, recommend themselves by their brevity, when we consi- der the very compound epithets which would otherwise be required in order to designate correctly the equi- angular, triangular, polycentral kaleidoscope; the rect- angular, isosceles, triangular, polycentral kaleidoscope; and the semi, equilateral, triangular, polycentral kaleido- scope.

As a plane surface of indefinite extent admits of sub- division, by regular polygons of the same kind, only in three ways, namely, by triangles, by squares, and by hex- agons, so each of these modes of division is the result of a separate arrangement of three plane mirrors, namely, that of the triascope, the tetrascope, and the hexascope.

Of these, the last two appear to be those more especially calculated to afford assistance to artists in the invention of ornamental patterns.

It is evident, that the principal advantage which the polycentral kaleidoscopes have over the simple ones, is the greater extension they give to the field of view. This field might, in theory, appear to be infinite; but in prac- tice it soon becomes limited, from the great loss of light attendant on repeated reflections. The effects of polar- ization, in further diminishing the light, is also greater in them than in the simple kaleidoscope. On both these accounts, metallic are preferable to glass mirrors for their construction. The number of reflections required, in or- der to obtain any extent of spectrum, being greater than in the ordinary kinds of simple kaleidoscopes, the instru- ment must be of greater length comparatively with the breadth of the mirrors, as in this way the course of the rays will be more oblique with respect to the mirrors, and a larger portion of light will reach the eye. A greater obliquity is also obtained, with the same proportion be- tween the length and breadth of the mirrors, by making them taper at the end next the eye. The instrument will then, see fig. 18, have the form of a truncated pyra- mid instead of a prism; ABC being the triangular base, to which the objects are to be applied, and abe the nar- rower end at which the eye is applied. It is true, that, in mathematical strictness, this construction is incorrect: for the mirrors in that case having necessarily a degree of inclination to the base, the spectrum will be composed of portions, not of a plane, but of a spherical surface, which does not admit of the same divisions; but the field really visible to the eye is too limited to render this inaccuracy of any consequence.

After the detailed explanation which has been given of Construc- the principles on which kaleidoscopes act, it will not be ne- cessary to enter into any minute account of the me- thods of constructing them. A few practical directions may, however, be useful for the guidance of such as wish to provide themselves with this source of innocent amuse- ment. In order to construct the simple kaleidoscope, two slips of plate-glass, about six or eight inches long, and about an inch or an inch and a half in breadth, must be procured. The best form for these plates is that repre- sented in fig. 10, where one end of them is only half the breadth of the other. The newest plate-glass should be employed, as that which is old has frequently scratches and imperfections on its surface, which occasion a great loss of light. They should have been skilfully cut with a diamond, so that at least one of the edges may be per- fectly smooth, and free from chips. If this be not the case, one of the edges must be made quite straight, and freed from all imperfections, by grinding it with very fine emery upon a flat surface, such as another piece of plate- glass. The posterior surfaces of each of the plates are now to be covered with a black varnish, or with black sealing-wax, so as to remove its reflective power. When this has been done, and the varnish is dry, take the plate of which the edge has been rendered perfect, and apply this edge against the surface of the other plate, as near as convenient to the edge of this latter plate, and keep the edges so applied in contact, by means of a strip of black silk or cloth glued along the back of the plates, so as to serve the purpose of a hinge, allowing of their opening and closing to a certain extent, like the leaves of a book. They are now to be adjusted to the proper angle, which may be done with the greatest accuracy, by directing the mirrors, placed as in fig. 10, to any line, or the straight edge of any object in contact with the broad ends, and very ob- liquely situated with respect to the edge of either of the mirrors; then, looking from the other end, open or shut the plates till the figure of a star appears, having six, se- ven, eight, or any other number of rays which may be thought desirable, and observing that the images of the rays in the spectrum most remote from the object perfectly coalesce. The mirrors must now be fixed in their position by small arches of wood or brass, extending across the open ends of the plates AB in two or three places. These may at first be attached temporarily by means of sealing-wax; but they should afterwards be fastened more securely by other pieces glued to the plates in several places along the edges Aa, Be. The clearness of the effect of the instrument is much promoted by excluding all light, except what comes from the field of view; and this is best accomplished by laying a strip of black velvet, previous to the fixing of the pieces just described, all along the upper side of the instrument, so as to line the whole of the space between the upper edges of the mirrors. All reflection of light from that quarter is thus effectually precluded.

The plates thus prepared are to be placed in a tube, as represented in fig. 11, so that the broad ends of the mirrors shall barely project beyond the end of the tube; while the narrow end is placed so that the angle formed by the junction of the mirrors shall be a little below the middle of that end of the tube. The plates must then be kept in this position by pieces of cork or wood wedged in between them and the tube; taking care, however, that they press but lightly on the mirrors, for a very slight force is capable of bending and altering the figure, even of very thick plates of glass. A cover, with a circular aperture in the centre, is then to be fitted to the end abc, which should, in general, be furnished with a convex lens, whose focal length is an inch or two greater than the length of the mirrors, in order to allow the eye to see every part of the spectrum with perfect distinctness. Persons who are short-sighted will of course not require this lens; but it will still be expedient to close the end abc of the mirrors with a piece of plane glass, as a security against the introduction of dust.

In constructing polycentral kaleidoscopes, where three mirrors are employed, the third mirror occupies the place of the black velvet and connecting pieces already described. Great care should be taken to have three very perfect edges for the junctions of the plates with each other; and considerable attention should be paid to their being fixed at the exact angles required by the construction; and when once placed correctly, they are to be retained in their relative position by effectual securities. Similar remarks apply to the construction of square and rectangular polycentral kaleidoscopes.

The instrument, when so far completed, is now ready to be applied to the objects which are to form the spectrum. A case for holding these objects, and for communicating to them a revolving motion, is fitted to the object end of the tube. The best construction for such a case is the following: Upon the end of the tube abcd, fig. 12 (corresponding to the end of the mirrors ABC, fig. 11), is placed a ring of brass, mn, which moves easily upon the tube, and is kept in its place by a shoulder of brass on each side of it. A brass cell, MN, is then made to slip tightly upon the moveable ring mn, so that when the cell is turned round by means of the milled end at MN, the ring mn may move freely upon the tube. The objects are to be placed in a small box, consisting of two glasses, one transparent, and the other ground, kept at the distance of one eighth or one tenth of an inch by a brass rim. This brass rim should consist of two pieces which should screw into one another, so that the box can be opened by unscrewing it, and the objects changed at pleasure. This object box is placed at the bottom of the cell MN, as shown at OP; and the depth of the cell is such as to allow the side O to touch the end of the mirrors when the cell is slipped upon the ring mn. The instrument, when used, is to be held in one hand, with the angular point E downwards, and the cell is turned round with the other, so as to present the objects in succession before the aperture ACB, fig. 11.

The objects best fitted for producing pleasing effects are small fragments of coloured glass, of sufficient size to occupy a certain portion of the interval between the mirrors, but not so large as to engross the greater part, or to interfere with each other's motions, as they are made to fall in succession into the field of view, by the revolution of the case which holds them. Wires of glass, both span and twisted, and of different colours and shades of colours, and of various shapes, both curvilinear and angular, may be intermixed with the larger masses of coloured glass, together with one or two beads, bugles, fine needles, bent metallic wires, small pieces of lace, and fragments of fine sea-weed. Looped curves like the figure 8, double curves like the letter S or the figure 3, circles, ovals, spirals, triangles, or lines bent into angles like the letter W or Z, have generally a good effect, either alone or in combination with other objects. Care should, however, be taken not to crowd the case with too many objects at a time, as an excess in this respect produces a degree of complexity totally inconsistent with beauty. In order to obtain a greater variety in the styles of patterns produced, a number of different cases, with objects, may be provided, so as to fit on occasionally, and be changed at pleasure.

By Sir David Brewster's very ingenious contrivance of substituting for the case of objects above described, and which is applied in contact with the ends of the mirrors, a convex lens placed at a certain distance from them, the images of distant objects may be brought to occupy the exact place adapted for their reflection by the kaleidoscope, and may thus afford a still greater variety of symmetric combinations. This operation of the lens is illustrated by fig. 13, where the lens L forms an image of the object R at F, the space between the ends of the mirrors, which image is multiplied by the reflecting powers of the instrument, and forms a symmetric spectrum, precisely in the same way as if a real object of that size had occupied its place. The lens may be fitted to the end of a separate tube, external to that of the instrument, and capable of being drawn out upon it to the proper distance, which is known by observing when the spectrum appears perfectly symmetrical. The instrument in this form has been called the Telescopic or Compound Kaleidoscope; and is applicable to distant objects of every description, and equally so to those in motion as to those at rest. All their movements are represented with singular effect in the spectrum. A blazing fire viewed by it gives the appearance of beautiful fire-works, at one time rushing with great rapidity towards the centre, and at another issuing from it towards the circumference, or darting in splendid stariform corrucations over the field of vision. These varieties in the spectrum are occasioned both by turning the instrument round its axis, and by moving it forwards in any direction.

The compound kaleidoscope has thus a much more extended range than the common kind; and it has this further advantage, that it admits of the symmetry of the spectrum being rendered perfectly correct, since the images may be brought exactly to the ends of the mirrors; a condition which can never be completely obtained when the objects are confined in a glass case, as they must then always be separated from the mirrors by at least the thickness of the glass.

The focal length of the lens should always be much less than the length of the outer tube, and should in general be such as to be capable of forming an image at the end of the mirrors, when the object is four or five inches from the lens. Its diameter should be such as that, when it is at its greatest distance from the mirrors, it shall still occupy the whole of the field of view which is seen by direct vision; or, in other words, that the eye shall not see any part of its edge.

The exhibition of the effects of the kaleidoscope to a number of spectators at the same time, by throwing the images on a wall, after the manner of the magic lantern, or solar microscope, might be easily accomplished, if sufficient light could be procured for the illumination of the objects. The form of an instrument for this purpose is represented in fig. 20, where L is the lamp, the light from which being augmented by the reflector M, and concentrated by the very convex lens N, upon the transparent objects at the end of the kaleidoscope K, is formed into an image on the opposite wall by refraction through the lens P, the focal distance of which is somewhat shorter than the length of the tube. The brilliant light produced during combustion carried on by means of a stream of oxygen gas is peculiarly fitted for the exhibition of these effects, as was very successfully shown by Mr William Allan, in his lectures at Guy's Hospital, London.

By a contrivance on a similar principle, the patterns formed by the kaleidoscope may be copied, if thought necessary, by receiving them in a camera obscura. The readiest mode of tracing them, however, is by the use of a camera lucida, applied to the instrument at the end next the eye. The kaleidoscope might also be applied to the microscope, if it were worth while to multiply these applications; for which, however, considering the infinite variety of designs which the simpler instruments afford, there appears not to be the least necessity.

Instead of employing the exterior surfaces of glass as the reflectors, we may employ the interior surfaces of a prism of solid glass for that same purpose; and we may obtain in this way, as was shown by Sir David Brewster, a total instead of a partial reflection of light. This solid form of the instrument is peculiarly fitted for polycentral kaleidoscopes; but it is liable to the objection of its being extremely difficult to procure a piece of glass of sufficient size entirely free from veins, and also to obtain the perfect junction of the two reflecting planes.

Simple kaleidoscopes have been variously constructed with reference to the angles of inclination of the mirrors, allowing the mirrors to move on their connecting edges as on a hinge, so as to open or close at pleasure by means of a screw. Others admit of the mirrors entirely separating, so as gradually to become parallel to each other, and thus give rectilinear or annular patterns, as is seen in figures 2 and 3. But there is no occasion to dwell more particularly on these subjects, as the circumstances of their construction and effects must be sufficiently obvious from what has been already said; and there is probably more ingenuity than utility in devising these variations. We shall therefore conclude, by merely noticing a convenient mode of uniting several of these instruments, which was suggested by the author of this article, with a view to compare the effects of the simple and polycentric kaleidoscopes, applied to the very same set of objects. Fig. 20 shows a section of that instrument, in which mn are the mirrors of a simple kaleidoscope in the middle of the tube t, and which might be set to any angle; the mirrors def on one side forming a tetrascope, and gpr on the other, a hexascope. The whole was enclosed in the tube t, at the eye end of which were three separate apertures, in order to allow the observer to look through each in succession. The other end was fitted with a case of moveable objects, as in figure 12; and was also provided with an additional tube for the reception of a lens instead of the case, and capable of being drawn, so as to convert the whole into a telescopic instrument. The effect of the whole combination was very striking.

See Sir David Brewster's Treatise on the Kaleidoscope, Edinb. 1819; Harris's Treatise on Optics; Wood's Optics; Dr Roget on the Kaleidoscope, in the Annals of Philosophy, vol. xi. p. 375; and the Compte rendu des Travaux de l'Académie de Dijon, pour 1818, p. 108-117.