376 T>r. Roget on the Kaleidoscope. [May, 



of the images at the extreme parts will not coalesce, but will 

 overlap and confuse each other. This will be apparent by consi- 

 dering what takes place when the angle is 120°, 72°, 40 , &c. 

 which are the third, fifth, ninth, &c. parts of the whole circum- 

 ference. The field is, indeed, regularly divided into this number 

 of sectors ; but the images of the objects near the edges of the 

 mirrors, occurring in pairs, will not coalesce when followed 

 round the circle. If MM, for example, 

 be the edges of the mirrors seen in the 

 direction of the line of their intersection, 

 the images of a and b, if the divisions of 

 the circle be an odd number, as five, will 

 occur together on both sides of the ra- 

 dius R opposite to the interval between 

 the mirrors. On the other hand, when 

 this number is even, as six, similar 

 images coalesce, and the optical illusion 

 is perfect.* 



in the polygonal kaleidoscopes, or 

 those in which a number of plane mirrors 

 are disposed along the sides of a polygon, 

 so as to form a hollow prism, which 

 repeat the reflections in every direction, 

 and present the appearance of an ex- 

 tended plane instead of a circular field 

 of view, we are restricted by the above 

 condition to aveiy limited number of arrangements. It excludes, 

 in the first place, all angles above 90°, and, therefore, all poly- 

 gons having more than four sides. The square and the rectangle 

 are the only four-sided figures which will aflbrd regular appear- 

 ances. Triangles, therefore, alone remain ; and the particular 

 triangles can only be such as are formed with angles 

 of 90°, 60°, 45°, or 30°, which are the quotients of 

 180° divided by two, three, four, and six ; other ali- 

 quot parts of the semi-circle being excluded by the 

 necessary condition that the sum of the three angles 

 must be equal to 180°. We are, therefore, limited to 

 the three following species of triangles, which are 

 represented in the margin. The first has all the angles 

 equal to 60° ; the second has one of 90°, and the 

 two others of 45° each ; and the third has angles of 

 90°, 60°, and 30°. Let us now inquire into the 

 effects resulting from each of these combinations. 



The square kaleidoscope, composed of four mirrors, pro- 

 duces by no means so pleasing an effect as the others ; because 

 the regularity of form is in general most apparent in one 



» The above-mentioned condition also results from the mathematical formulas 

 for calculating the number of images of an object situated between two plane 

 mirrors inclined to each other at a given angle. (See Wood's Elements of Optics, 

 Prop, xiv.) 



