Determination of the Images produced by Diverging Lenses. Use of these Lenses in correcting shortsightedness. 49. We shall first apply the method to diverging lenses. Let Fig. 48. MAM be such a lens, of which A is the centre of figure; and place the object SS' before its surface, at any distance whatever, provided it be such that the angles of incidence shall not exceed the limits supposed in our approximations. If from the extremity S of the object, we draw the line SA to the centre of figure of the lens, the cone of incident rays proceeding from the point S, will have for its axis SA, and its focus will be found somewhere in this straight line, and on the same side of the lens (since it is diverging), for example, in f. The focus will be found in the same way in S'A as at f'; and these two foci comprehending between them all the others, ff' will be the image of the object. It will always be erect and smaller than the object, since it is comprehended between the sides of the angle SAS', and nearer its vertex A. Moreover, the absolute value of its distance from the lens will always be less than the principal focal distance AF, and so much less according as the object itself is nearer the glass. According to this construction, when the luminous rays, proceeding from the same point S or S' of the object, have passed through the lens, their course is exactly the same as if they had set out from the corresponding point for f' of the image. There in figure 47. This always happens, for instance, in converging lenses, when ▲ is infinite, which renders nothing. For then, ac 1 Δ cording to figures 36, 37, 38, the ray r of the anterior surface becomes negative; and whether the other lens be convex or concave, 1 will remain a negative quantity. We shall then have this is precisely the rule enunciated in the text for the calculation of the principal focal distance AF. fore, if a spectator had his eye situated at OO on the other side of the lens so as to receive all these rays or only a part of them, he would see neither the points S, S', nor the points between them, but their image ff'; and his eye would be affected in the same way as if the object had really become smaller, and had been transferred to the place where the foci are formed. He will thus see this imaginary object erect, diminished and brought nearer. But although these may be in fact the only elements. of the sensation produced in the eye, yet we do not estimate correctly the distance and magnitude of the image, because our judgment is affected at the same time by other considerations, altogether independent of the direction of the luminous rays. To convince ourselves of this singular fact, we may take any diverging glass, for example, that nearest the eye in an opera glass, which is usually double concave; and look through it at objects which we will suppose to be very distant compared with the focal distance of the glass. When the eye is placed at a proper distance from the posterior surface, we shall see a very distinct image of these objects. It will appear erect like them and smaller, but instead of supposing it near the glass and in the focus ƒƒ', where it is really formed, it will seem to us more distant than the object itself. This is because the sensation of the visual angle, and that of the greater or less divergency of the luminous rays which reach us, are not the only elements from which we estimate distances. We add to these, without being sensible of it, the impressions which we may have of the absolute dimensions of objects, A man seen successively at the distances 20, 40, and 60 yards, appears always of the same absolute magnitude. Nevertheless the rays of light which render him visible at these different distances, cross each other on entering the eye under very different angles, since they are to each other nearly as the numbers 1,,; so that were we to judge only from the openings of these angles, the apparent magnitudes would seem to us to decrease in the same ratio. 50. This habit of connecting the idea of absolute magnitude with the sensation of the visual angle in judging of the distance of objects, is derived from the experience of our whole life, and it becomes as rapid as sensation itself; or rather, the sensation which is transmitted to the mind when we look at an external object, is the compound result of these different elements. But the involuntary application which we make of them, deceives us when we look through a diverging lens; for the objects which we had just viewed with the naked eye, of the absolute magnitude and distance of which we were consequently able to form a pretty accurate judgment, being suddenly presented to the eye with much smaller dimensions, we do not hence conclude simply that their images are smaller, but that they are further removed from us; and no reasoning can prevent this conclusion being formed even when we know theoretically that it is false. With the exception of this case, in which our senses are but poor judges, observation perfectly confirms the results indicated by theory; this we may verify not only when the object is at a great distance, as we have just now supposed, but also when it is gradually brought near. If it be brought, however, very near the lens, it must be very small, and the refracting surfaces must be held nearly perpendicular to the rays which proceed from it; otherwise we should exceed the limits of incidence and emergence supposed in our approximations. In performing these experiments, we find that in order to see the image distinctly, we must place the eye at a certain distance from the lens, and this distance varies with different eyes. If the eye be brought nearer, the image becomes larger and confused. If, on the contrary, it be removed, the image becomes smaller and more difficult to be seen distinctly. This results from the fact, that the eye is itself an optical instrument which cannot concentrate the rays with sufficient exactness for perfect vision, except when they fall upon its surface within certain limits of incidence. Suppose, for example, that the luminous point S, forms its Fig. 49. image in F, and that this image appears distinct when the eye is placed at 00; then vision is produced by a cone of rays FOO which has for its base the surface 00 of the pupil, and for its vertex the point F. If the eye be brought nearer, suppose to O'O', then the pupil intercepts a more diverging cone of rays, and consequently the rays FO' which form the exterior surface of this cone, fall upon the eye with a greater angle of incidence. If this angle become so great that the eye cannot concentrate all the rays upon its retina, vision is necessarily confused; and this in fact takes place when we bring the eye too near the lens, and consequently too near, the focus F, the common centre of the emerging rays. If, on the contrary, after having found the point where vision is most perfect, we remove the eye further from the lens, the image, which remains always in the same place, is at a greater distance from the eye. It ought, therefore, to appear smaller and less distinct in its outline, like other objects when removed from us; and this actually happens. 51. The least distance at which objects are seen differs with different eyes. Those persons who are shortsighted are obliged to approach the focus F; those who are longsighted, on the contrary, find it necessary to go to a greater distance from it. The reason of this will be readily understood when we remember that this focus, from which the rays diverge, is to the eye as it were an object situated in F; and thus each one, in order to see it distinctly, is obliged to place his eye at the distance at which he would see a common object most distinctly. This distance is commonly from eight to ten inches, when we wish to distinguish the minute parts of small objects; but some persons cannot distinguish objects placed at so small a distance; and, on the contrary, there are others with respect to whom this distance is only two or three inches. The latter are called myopes, or shortsighted persons, and the former presbytes or longsighted. the 52. It is not to be understood that distinct vision is, in every case, strictly confined to a particular distance. On the contrary, eye is endued with a power by which it is enabled to adapt itself, within certain limits, to the different distances of objects. But beyond these the image is confused, and vision becomes imperfect. Thus persons possessing the longest sight, cease to distinguish the minute parts of objects when placed at a considerable distance. But these parts disappear to shortsighted persons, even when the objects are very much nearer. The defect of shortsightedness may be corrected by placing a diverging lens between the object and the eye, as represented in figure 49. For by means of such a lens, there is substituted instead of the real object the image formed at its focus, and we have only to give to the lens a focal distance equal to that of distinct vision of the organ for which it is designed; then by bringing it near the eye, the shortsighted person will see distant objects as distinctly as if they were situated near him, although he refers them to their true places. But he cannot use the same glasses, at least, when brought close to the eye, for the purpose of seeing very near objects; because the foci of the rays which proceed from them being formed still nearer the glass, the image of the objects would be too near the eye to be seen distinctly without fatiguing it. It is necessary, therefore, to employ lenses of a greater focal distance; or, which is better, to dispense with them entirely, since it is these near objects which shortsighted persons perceive most distinctly with the naked eye, and to reserve the use of diverging glasses for distant objects. The principal focal distance ought, moreover, to exceed a little that at which small objects are seen most distinctly; for the eye would be soon fatigued by the too near approach of the image. Such is the purpose and operation of those spectacles by which shortsighted persons remedy the imperfection of their sight. It is plain that the glasses in question would be useless and worse than useless to the longsighted who cannot perceive near objects distinctly, on account of their being too close to the eye; for, the foci of diverging lenses being always nearer than the object itself, the evil would only be increased. We must, in this case, therefore, seek some means of removing the image beyond the object which produces it; and this purpose is effected by means of converging glasses, as we shall soon see when we have studied their properties. Fig. 50. Method of determining the Images produced by Converging Lenses. Persons. 53. LET MAM be a lens which we now suppose to be converging; and let SS' be the object placed beyond the focal distance of parallel rays. Through its upper extremity S draw the ray S to the centre of the lens. This ray will be the axis of the pencil of rays proceeding from the point S; and the lens, being supposed to be very thin, the focus of this pencil will be situated somewhere beyond the lens with respect to the point S, as at f, for example, in SA produced, as appears from our general construction. All the emerging rays which come from the point S will meet in this point from which they will pass on and diverge as if from a real object situated in the same place. If |