speculorum 45 graduum dabit octogonum; 40 graduum dabit enneagonum; 36 graduum decagonum; 32 graduum angulus cum ⁸/₁₁ dabit endecagonum et denique angulus 30 graduum referet dodecagonum cum totidem formis, et sic in infinitum; ita ut semper tot laterum sit futurum polygonum anacampticum totidemque formarum, quot polygonum cuivis latus speculorum intercipit divaricatio, latera habuerit: quorum omnium rationes dependent a Propos. v. precedentis distinctionis. —Kircheri, _Ars Magna Lucis et Umbræ_. Rom. 1646, p. 890. The following is a translation of the preceding passage:— PARASTASIS I. _Plane specula may be made to multiply the images of one object._ See Fig. 54. A wonderful, and, so far as I know, a new property, is exhibited by two specula, so constructed that they may be opened and shut like a book. If they are placed upon any plane, in which there is a semicircle divided into degrees, in such a manner that the point where the specula meet is in the centre of the circle, and the edge of each speculum stands upon the diameter of the semicircle, one image only will be visible, and there will appear two things, namely, a real one, without the specula, and another formed by reflexion behind them,[21] and so on, as in the following table, where the first column shows the inclination of the specula, and the second the figure which is produced:— [21] I have thrown the rest of the passage into a tabular form, that the reader may see, more readily, the effect produced by the variation of the angle. _Angle of Specula._ _Effect produced._ 180° one image and one object, *120° two images and one object, 90° four images, *72° a pentagon and five forms, 60° a hexagon and six forms, *51³/₇° a heptagon and seven forms, 45° an octagon and eight forms, *40° an enneagon and nine forms, 36° a decagon and ten forms, *32⁸/₁₁° an endecagon and eleven forms, 30° a dodecagon and twelve forms, and so on, _ad infinitum_, the polygon formed by reflexion having always as many sides as the number of times that the angle of the specula is contained in 360°. The combination of plane mirrors, which Kircher describes in the preceding extract, is precisely the same as that which is given by Baptista Porta. The latter, indeed, only mentions, that the number of images increases by the diminution of the angle, whereas Kircher gives the number of images produced at different angles, and enumerates the _regular polygons_ which are thus formed.