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tance as the object, and the image formed there will be of the same magnitude, but inverted. And hence the advantage of such lenses to painters, and such as desire the images they would copy, as large as the objects.

CAUSTICS BY REFRACTION.

In all that we have hitherto said concerning the focus of rays, we have confined ourselves to such as fall near to the vertex of the lens or to its axis. More distant rays are united in different points in the axis, still nearer to the vertex, the more distant that the point of incidence is from it. The more diverging rays will succes. sively intersect the less diverging, in different points before they arrive at the axis; and these points of intersection, gradually diverging from the principal focus, will form luminous curves, denominated caustics by refraction. When the light of a candle is refracted through a globe or decanter on the table, and falls upon a piece of white paper laid parallel to the axis, the luminous space which receives the refracted rays will be bounded by two bright curves, one on each side of the axis; which accede towards each other as they approach to the principal focus, where they form a sharp point. As these curves are formed by the successive intersections of every ray with the next to it in order from the axis to the outermost, the brightness of the paper within the curves, and its darkness without them, is caused by the multiplicity of these intersections within the curves. Every ray crosses the next to it, in a point of the caustic, before it meets the axis; those which pass nearest to the axis, proceeding farthest before their intersections, and thereby intersecting nearest to the principal focus; and those that pass farthest from the axis intersect each other soonest and nearest to the globe.

As the candle approaches to the globe, the rays that pass nearest to the axis will soonest become parallel, and afterwards diverge, while more distant rays still converge, until by the continued approach of the candle they also will become parallel and afterwards diverge. Consequently if the emergent rays, when they begin to diverge, were produced backwards, they would intersect the axis beyond the candle, and each contiguous couple of rays would afterwards intersect each other, and these imaginary intersections would form an imaginary caustic, with an acute angle at the radiant point, and diverging from thence. And while some of the rays that pass nearest to the axis are diverging, others are still converging, as they pass at a greater distance from the axis, and the intersections of these will form a caus tic beyond the focus.

A lens in the same manner forms a caustic between the principal focus and the lens. Hence the exterior rays, that fall farthest from the axis of the lens are too much refracted to belong to the principal focus, and therefore will cross the axis nearer to the lens. This is called the aberration of the rays, by the spherical figure of the lens. A spherical surface therefore having the same degree of curvature every where, cannot refract the rays of different pencils to the same point ar focus.

To accomplish this design it must grow flatter towards the edges, that the concurrence of the exterior rays may be prolonged. Nor will a plano-convex lens answer the purpose, for although it occasions less aberration than the double convex lens, yet the lens must be spherical on both sides near the center, to shorten the concurrence of the middle rays, and gradually become concave towards the edges, to lengthen the con

currence of the exterior pencils, with the axis. Nevertheless, the middle pencils are so closely crowded together in the focus, and the exterior rays are scattered so thinly over a plane supposed to pass perpendicularly through the focus, especially towards the verges of the picture formed there, that the confusion thereby occasioned is seldom very sensible, provided the lens have a considerable aperture. It is very little in comparison with the confusion occasioned by the different refrangibility of the rays of different colours.

IMAGES OF OBJECTS BY REFRACTION.

WHEN a pencil of rays diverges from any point of an object, and falls upon a convex lens, the middle ray that passes through the center of the lens suffers no refraction, and all the other rays of that pencil will be refracted to their proper focus in it, where they will consequently form an image of that point; and as this is true of every point of the object, there must be a complete image of the object formed in the focus; the image of each point of the object being formed in the foci of their diverging pencils. As the middle ray of each pencil passes through the center of the lens, they must all cross one another there, so that the image will be inverted, with respect to the object; the rays, that proceed from the points above the axis, having their foci below it.

For the same reason, the image and object will always be viewed under the same angle, from the center of the lens; as the axis of each pencil crosses the other in the center of the lens: and hence the diameters of the object and image will be to each other, as their respective distances from the lens; when the object is parallel to the lens. But as any inclination of the object alters its appa

rent diameter, by diminishing the angle under which it is seen from the lens, the image will sustain a similar diminution, and still their apparent diameters will be in the same proportion.

If you would receive, on a white screen, in a darkened room, the images of exterior objects, by suffering the rays from them to pass through a convex lens, placed in a window shutter, for the purpose, you must be careful to exclude all other rays of light, which otherwise falling upon the picture would so weaken and dilute the shadings and colourings of it, as to make it obscure or even to disappear entirely. The screen must also be placed where the several pencils unite in their respective foci, and where the distinct images are formed. As the object recedes from the lens, the image approaches to it, but can never come nearer than the principal focus, because that is the place of the image, when the distance of the object becomes infinite. Hence the distinct images of objects placed at different distances from the lens, can never be received at the same time on the screen. Because, while a nearer position of the screen is necessary to render the images of distant objects distinct, the images of nearer objects will thereby become obscure, until the screen be again removed to the place, where the several pencils of rays from them unite. As the images will be more bright and pleasing, the more strongly the objects are illuminated, the lens should be placed in a window opposite to the sun, that the side of the objects next to it may thereby be more copiously enlightened.

As the diameter of an image depends upon the diameter of the object at a given distance, and decreases as the distance of the object increases, the diameter of an image must be directly as the diameter of the object

and inversely as its distance. I=

dist'

And the area of

the image being as the square of its diameter directly, it must be also as the square of the distance of the object inversely, or the square of its own distance from the lens directly; when the diameter of the object is

given. a=I'==d2. a being the area of the image, I= D2

its diameter, d=its distance, O=the diameter of the object, and Dits distance.

The distinctness of a picture depends upon the several pencils of rays being accurately collected into their respective foci upon the screen; but its brightness, on the quantity of light transmitted through the lens: which let be B. Now as the brightness of an image depends upon the quantity of light thrown together in the place of the image, it must be different, according to the distance of the object, the area of the lens, and its focal distance, or the area of the image. The greater that the area of the lens is, the more light it transmits in the same proportion, therefore B=L, L being the area of the lens. The quantity of light that falls upon a given surface, from different distances will be inversely as the

1

squares of these distances; and therefore B=D2, when the area of the lens and its focal distance are given. The brightness of an image will also be inversely as the area of the image or the square of its distance from the lens, the other circumstances being equal, and therefore B=

1 1

a

=

de

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Because, when the same quantity of light is dif

fused over a larger space, it must enlighten it in a less degree, than when it is spread over a less space, and kept closer together. Hence, when neither the distance

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