Hitherto we have considered the effects of combining two reflectors, by means of which the reflected images are arranged around one centre, either visible or invisible; but it must be obvious, from the principles already explained, that very singular effects will be obtained from the combination of three or more reflectors. As in instruments of this kind the reflected images are arranged round several centres, we have distinguished them by the name of _Polycentral_. As 90° is the greatest angle which is an even aliquot part of 360°, and as all regular polygons, with a greater number of sides than four, must have their interior angles greater than 90°, it follows, that symmetrical pictures cannot be created by any number of reflectors greater than four, arranged like the sides of a regular polygon. If the polygon is irregular, and consists of four sides, or more, then one of its angles must exceed 90°, and consequently it cannot give symmetrical patterns. In constructing Polycentral Kaleidoscopes, we are limited to combinations of four or three reflectors. The only modes in which we can combine four reflectors, are so as to form a hollow square, or a hollow rectangle; but though these combinations afford regular patterns, from their angles being even aliquot parts of 360°, yet these figures are composed merely of a great number of squares, or rectangles, the point where every four squares or rectangles meet being the centre of a pattern. Those, however, who may wish to construct such instruments, must make the plates as narrow as possible at the eye-end, so as to bring the eye, as much as can be done, into the plane of all the four reflectors. In combining three reflectors, the limitation is nearly as great; but the effect of the combination is highly pleasing. Since the angles at which the reflectors must be placed are even aliquot parts of 360°, such as 90°, 60°, 45°, 36°, 25-¹⁰/₁₄° 22½°, 20°, 18°, etc., which are the quotients of 360°, divided by the even numbers, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.; and since the reflectors are combined in the form of a prism, the section of which is everywhere a triangle, the sum of whose angles is 180°, we must select any three of the above even aliquot parts which amount to 180°; and when the reflectors are combined at these angles, they will afford forms perfectly symmetrical. Now, it is obvious, that these conditions will be complied with when the angles are— 90° + 45° + 45° = 180° 90° + 60° + 30° = 180° 60° + 60° + 60° = 180° The Polycentral Kaleidoscopes are therefore limited to _five_ different combinations, namely,— 1. Four reflectors of equal breadth, forming a square. 2. Four reflectors, two of which are broader than the other two, and form, a rectangle. 3. Three reflectors at angles of 90°, 45°, and 45°. 4. Three reflectors at angles of 90°, 60°, and 30°. 5. Three reflectors at angles of 60°, 60°, and 60°.