CHAPTER IV.
ON THE EFFECTS PRODUCED UPON THESYMMETRY OF THE PICTURE BY VARYINGTHE 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 reflexion; 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 production 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 looking-glass 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 ofsymmetry. In mirror-glass there is a series of images reflected from thefirstsurface, 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.
Fig. 13.
Fig. 13.
In order to explain the effects produced upon the symmetry 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 represented inFig. 13, whereA Eis a section of the mirror in the direction of its length;M N O Pan object placed at a distance from the extremityAof the mirror, andm n o p, its image seen by an eye to the right hand ofE, and which, by the principles of catoptrics, will be similar to the object and similarly situated with respect to the mirrorA E. Now, if the eye is placed at ε, it will see distinctly the whole objectM N O P, but it will only see the portionn r s oof the image cut off by drawing the lineε Arthrough the extremity of the mirror, so that there cannot be a symmetrical form produced by observing at the same time the objectM N O Pand this portion of its image; and the deviation from symmetry will be still greater, ifM N O Pis brought nearer the lineB A, for the imagem n o pwill be entirely included between the linesArandA B, so that no part whatever of the image will be visible to an eye atε. As the eye of the observer moves from ε toe, the lineε Arwill move into the positioneAx, and when it has reached the pointe, the whole of the imagem n o pwill 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 spacem x, while the distance of the object isMx. As the eye moves frometoE, the lineeAxwill move intoE A B, and the object and its image will seem to be placed at the equal distancesM B,mBfrom the plane of the mirror, and will therefore form a symmetrical combination. When the object is moved, and arrives atB Athe image will touch the object, and they will form one perfect and united whole, whatever be the shape of the lineM P.Hence we conclude,that when an object is placed at a little distance from the extremity of a plain mirror, its image formed by reflexion 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.
Fig. 14.
Fig. 14.
When two mirrors, therefore, are combined, as inFig. 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 pointEwhere the two planes cut each other. The necessity of this position, and the effects of any considerable deviation from it, will be understood fromFig. 14, whereA O Bis the angle formed by the mirrors, andM Nthe place of the object. Then if the eye is placed at ε, the apertureA O Bwill be projected intoa bωupon a plane passing throughM Nand at right angles toE Oʹ; but the orthographic projection ofA B Oupon the same plane isAʹ Bʹ Oʹ, or, what is the same thing, the reflecting surfaces of whichA O,B Oare sections, will, when prolonged, cut the plane passing throughM Nin the linesAʹ Oʹ,Bʹ Oʹ; hence, rays from the objects situated betweenAʹ Oʹ Bʹandaωbcannot fall upon the mirrorsA O E,B O E, or images of these objects cannot be formed by the mirrors. The images, therefore, in the different sectors formed by reflexion roundOas a centre, cannot include any objects withoutAʹ Oʹ Bʹ; and since the eye at ε sees all the objects betweenAʹ Oʹ Bʹandaωb, there can be no symmetry and uniformity in the picture formed by the combination of such an object with the images in the sectors. When the eye descends toe, the apertureA O Bis projected intoaʹ oʹ bʹ, which approaches nearer toA O B; but for the reasons already assigned, the symmetry of the picture is still imperfect. As the eye descends, the linesaʹ oʹ,bʹ oʹapproach toAʹ Oʹ,Bʹ Oʹ, and when the eye arrives atE, a point in the plane of both the reflecting surfaces, the projection of the apertureA O Bwill beAʹ Oʹ Bʹ, and the images in all the sectors will be exactly similar to the object presented to the aperture. Hence we conclude in general,that when an object is placed at any distance before two mirrors inclined at an angle, which is an even aliquot part of 360°, the symmetry of the picture is perfect, when the eye, considered as a mathematical point, is placed atE, and thatthe deviation from symmetry increases as the eye recedes fromEtowardsε.
If the object were a mathematical surface, all the parts of which were in contact with the extremitiesA O,B Oof the mirrors, then it is easy to see that the symmetry of the picture will not be affected by the deviation of the eye from the pointE, and, in consequence of the enlargement of the sector, seen by direct vision. The symmetry of the picture, is, however, affected in another way, by the deviation of the eye from the pointE.
We have already seen, that, in order to possess perfect symmetry, an object must consist of two parts in complete contact, one of which is an inverted image of the other. But in order that an object possessing perfect symmetry may appear perfectly symmetrical, four conditions are required. The two halves of the object must be so placed with respect to the eye of the observer, that no part of the one half shall conceal any part of the other; that whatever parts of the one half are seen, the corresponding parts of the other must also be seen; and that the corresponding parts of both halves, and both halves themselves, must subtend the same angle at the eye. When we stand before a looking-glass, and hold out one hand so as to touch it, the hand will be found to conceal various parts of its image; and, in some positions of the eye, the whole image will be concealed, so that a symmetrical picture cannot possibly be formed by the union of the two. If the eye is placed so obliquely to the looking-glass, that the hand no longer interferes with its image, it will still be seen, that parts of the hand which are not directly visible, are visible in its reflected image, and therefore that a symmetrical picture cannot be created by the union of two parts apparently dissimilar. If the eye of the observer is placed near his hand, so that he can see distinctly both the hand and its image, the angular magnitude of his hand is much greater than that of its image; and therefore, when the two are united, they cannot form a symmetrical object. This will be better understood fromFig. 15. When the eye is placed at ε, the objectM N O Pis obviously nearer than its imagem n o p, and must therefore appear larger; and this difference in their apparent magnitudes willincrease as the eye rises above the plane of the mirrorA E. As the eye approaches toE, the distances of the object and its image approach to an equality; and when the eye is atE, the objectM N O P, and its imagem n o p, are situated at exactly the same distance from the eye, and therefore have the same angular magnitude. Hence it follows, that when they are united, they will form a perfectly symmetrical combination.
Fig. 15.
Fig. 15.
When the eye is placed in the plane of both the mirrors, the field of view arising from the multiplication of the sectorA O B,Fig. 14, will be perfectly circular; but as the eye rises above the plane of both the mirrors, this circle will become a sort of ellipse, becoming more and more eccentric as the eye comes in front of the mirrors, or rises in the directionE ε. If the observer were infinitely distant, these figures would be correct ellipses; but as the eye, particularly when the mirrors are broad, must be nearly twice as far from the last reflected sector as from the sector seen by directvision, the field of view, and consequently every pattern which it contains, must be distorted and destitute of beauty, from this cause alone.
Hitherto we have alluded only to symmetry of form, but it is manifest that before the union of two similar forms can give pleasure to the eye, there must be also a symmetry of light. If the objectM N O P,Fig. 15, is white, and its imagem n o pblack, they cannot possibly form, by their combination, an agreeable picture. As any considerable difference in the intensity of the light will destroy the beauty of the patterns, it becomes a matter of indispensable importance to determine the position of the eye, which will give the greatest possible uniformity to the different images of which the picture is composed.
It has been ascertained by the accurate experiments of Bouguer, that when light is reflected perpendicularly from good plate glass, only 25 rays are reflected out of 1000; that is, the intensity of the light of any object seen perpendicularly in plate glass, is to the intensity of the light of its image as 1000 to 25, or as 40 is to 1. When the angle of incidence is 60°, the number of reflected rays is 112, and the intensities are nearly as 9 to 1.—When the angle of incidence is 87½°, the number of reflected rays is 584, and the intensities are nearly as 17 to 10, so that the luminousness of the object and its image approach rapidly to an equality. It is therefore clear, that, in order to have the greatest uniformity of light in the different images which compose the figure, the eye must be placed as nearly as possible in the plane of both the mirrors, that is, as near as possible to the angular point.—But as it is impracticable to have the eye exactly in the plane of both mirrors, the images formed by one reflexion must always be lessbright than the direct object, even at the part nearly in contact with the object. The second and third reflexions, etc., where the rays fall with less obliquity, will be still darker than the first; though this difference will not be very perceptible when the inclination of the mirrors is 30° or upwards, and the eye placed in the position already described.
It is a curious circumstance, that the positions of the eye which are necessary to effect a complete union of the images—to represent similar parts of the object and its images—to observe the object and its image under the same angular magnitude, and to give a maximum intensity of light to the reflected images—should all unite in the same point. Had this not been the case, the construction of the Kaleidoscope would have been impracticable, and hence it will be seen how vain is the attempt to produce beautiful and symmetrical forms from any combination of plain mirrors in which this position of the eye is not a radical and essential principle.