The fundamental principle, therefore, of the Kaleidoscope is, that it produces symmetrical and beautiful pictures, by converting simple into compound or beautiful forms, and arranging them, by successive reflexions, into one perfect whole. This principle, it will be readily seen, cannot be discovered by any examination of the luminous sectors which compose the circular field of the Kaleidoscope, and is not even alluded to in any of the propositions given by Mr. Harris and Mr. Wood. In looking at the circular field composed of an even and an odd number of reflexions, the arrangement of the sectors is perfect in both cases; but when the number is odd, and the form of the object simple, and when the object is not similarly placed with regard to the two mirrors, a symmetrical and united picture cannot possibly be produced. Hence it is manifest, that neither the principles nor the effects of the Kaleidoscope could possibly be deduced from any practical knowledge respecting the luminous sectors. In order to explain the formation of the symmetrical picture shown in Fig. 5, we must consider that the simple form _m n_, Fig. 2, is seen by direct vision through the open sector =A O B=, and that the image _n o_, of the object _m n_, formed by one reflexion in the sector =B O _a_=, is necessarily an inverted image. But since the image _o p_, in the sector =_a_ O α=, is a reflected and consequently an inverted image of the _inverted image_, _m t_, in the sector =A O _b_=, it follows, that the whole _n o p_ is an inverted image of the whole _n m t_. Hence the image _n o_ will unite with the image _o p_, in the same manner as _m n_ unites with _m t_. But as these two last unite into a regular form, the two first will also unite into a regular or compound form. Now, since the half =β O _e_= of the last sector =β O α= was formerly shown to be an image of the half sector =_a_ O _s_=, the line _q v_ will also be an image of the line _o z_, and for the same reason the line _v p_ will be an image of _t y_. But the image _v p_ forms the same angle with =B O= or _n q_ that _t y_ does, and is equal and similar to _t y_; and _q v_ forms the same angle with =A O= that _o z_ does, and is equal and similar to _o z_. Hence, =O _o_ = _o q_=, and =O _y_ O _v_=, and therefore _q v_ and _v p_ will form one straight line, equal and similar to _t q_, and similarly situated with respect to =B O=. The figure _m n o p q t_, therefore, composed of one direct object, and several reflected images of that object, will be symmetrical. As the same reasoning is applicable to every object extending across the aperture =A O B=, whether simple or compound, and to every angle =A O B=, which is an even aliquot part of a circle, it follows,—