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When the lines cross each other at a right angle, the salient angles of the opening polygon can never touch the salient angles of the stationary polygon, but always its re-entering angles. If the lines, however, form a less angle than the complement of the angle formed by the mirrors, then the salient angles of the opening polygon may touch the salient angles of the stationary polygon, by placing the mirrors so as to form re-entering angles in the polygon. When the lines form an angle between 90° and the complement of the angle formed by the mirrors, the salient angle of the opening polygon may be made to touch the salient angles of the stationary one, but in this case the stationary polygon can have no re-entering angles. The preceding effects are finely exemplified by the use of a Kaleidoscope with a draw-tube and lens described in Chapter X., and by employing the vertical and horizontal bars of a window, which may be set at different angles, by viewing them in perspective.

CHAPTER IV.

ON THE EFFECTS PRODUCED UPON THE SYMMETRY OF THE PICTURE BY VARYING THE POSITION OF THE EYE.

IT has been taken for granted in the preceding chapters, not only that the object seen by direct vision is in a state of perfect junction with the images of it formed by reflection; but that the object and its images have the same apparent magnitude, and nearly the same intensity of light. As these conditions are absolutely necessary to the produc tion of symmetrical and beautiful forms, and may be all effected by particular methods of construction, we shall proceed to investigate the principles upon which these methods are founded, in so far as the position of the eye is concerned.

When any object is made to touch a common lookingglass in one or more points, the reflected image does not touch the object in these points, but is always separated from it by a space equal to the thickness of the glass, in consequence of the reflexion being performed by the posterior surface of the mirror. The image and the object must therefore be always disunited; and as the interval of separation must be interposed between all the reflected images, there cannot possibly exist that union of forms which constitutes the very essence of symmetry. In mirror glass

there is a series of images reflected from the first surface, which unite perfectly with the object, and with one another. When the angles of incidence are not great, this series of images is very faint, and does not much interfere with the more brilliant images formed by the metallic surface. As the angles of incidence increase, the one series of images destroys the effect of the other, from their overlapping or imperfect coincidence-an effect which increases with the thickness of the glass; but when the reflexions are made at very oblique incidences, the images formed by the metallic surface become almost invisible, while those formed by the first surface are as brilliant and nearly as perfect as if the effect of the posterior metallic surface had been entirely removed. In the following observations, therefore, it is understood that the images are reflected either from a polished metallic surface, or from the first surface of glass. In order to explain the effects produced upon the sym

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metry of the picture by a variation in the position of the eye, we must suppose the object to be placed at a small distance from the end of the mirror. This position is re

presented in Fig. 13, where A E is a section of the mirror in the direction of its length; M N O P an object placed at a distance from the extremity A of the mirror, and m n o p, its image seen by an eye to the right hand of E, and which, by the principles of catoptrics, will be similar to the object and similarly situated with respect to the mirror A E Now, if the eye is placed at , it will see distinctly the whole object M N O P, but it will only see the portion nrs o of the image cut off by drawing the line Ar through the extremity of the mirror, so that there cannot be a symmetrical form produced by observing at the same time the object M N O P and this portion of its image; and the deviation from symmetry will be still greater, if M N O P is brought nearer the line B A, for the image m n o p will be entirely included between the lines A r and A B, so that no part whatever of the image will be visible to an eye at ɛ. As the eye of the observer moves from to e, the line ɛ A will move into the position e Ax, and when it has reached the point e, the whole of the image m n o p will be visible. The symmetry, therefore, arising from the simultaneous contemplation of the object and its image will be improved; but it will still be imperfect, as the image will appear to be distant from the plane of the mirror, only by the space m x, while the distance of the object is м x. As the eye moves from e to E, the line e Ax will move into E A B, and the object and its image will seem to be placed at the equal distances м в, m в from the plane of the mirror, and will therefore form a symmetrical combination. When the object is moved, and arrives at B A the image will touch the object, and they will form one perfect and united whole, whatever be the shape of the line M P. Hence we con

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clude, that when an object is placed at a little distance from the extremity of a plain mirror, its image formed by reflection from the mirror cannot unite with the object in forming a conjoined and symmetrical picture, unless the eye is in the plane of the mirror.

When two mirrors, therefore, are combined, as in Fig. 14, the eye must be in the plane of both, in order that the object and its image may have a symmetrical coincidence, and therefore it must be at the point E where the two planes cut each other. The necessity of this position, and the effects of any considerable deviation from it, will be understood from Fig. 14, where AOB is the angle formed by the mirrors, and M N the place of the object.

FIG. 14.

Then if

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the eye is placed at , the aperture A O B will be projected into a bw upon a plane passing through M N and at right angles to E o'; but the orthographic projection of A BO upon the same plane is A' B' o', or, what is the same thing, the reflecting surfaces of which A 0, B O are sections, will, when prolonged, cut the plane passing through M N in the lines A' O', B' o'; hence, rays from the objects situated

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