LIGHT
complement of fig. 1, gives therefore the effect of four equal sources of light shining on a wall through a circular hole. And it is evident that, with the change of a word here and there, the previous reasoning may be applied to this case also. The umbra in the former case becomes the fully illuminated portion, and vice versa. The penumbra remains the penumbra, but it is now darkest where before it was brightest, and vice versa. For further information we subjoin the complement (fig. 4) of the second case above, – the same four sources, but the smaller hole. Here we have four equally bright, separate images – one belonging to each of the sources. Thus we see how,
Images by small hole.
when a small hole is cut in the window-shutter of a dark room, a picture of the sun, and bright clouds about it, is formed on the opposite wall. This picture is obviously inverted, and also perverted, for not only are objects depicted lower the higher they are, but also objects seen to the right are depicted to the left, &c. But it will be seen unperverted (though still inverted) if it be received on a sheet of ground glass and looked at from behind. The smaller the hole (so far at least as Geometrical Optics is concerned) the less confused will the picture be. As the hole is made larger the illuminated portions from different sources gradually overlap; and when the hole becomes a window we have no indications of such a picture except from a body (like the sun) much brighter than the other external objects. Here the picture has ceased to be one of the sun, it is now a picture of the window. But if the wall could be placed 100 miles off, the picture would be one of the sun. To prevent this overlapping of images, and yet to admit a good deal of light, is one main object of the lens which usually forms part of the camera obscura.
The formation of pictures of the sun in this way is well seen on a calm sunny day under trees, where the sunlight penetrating through small chinks forms elliptic spots on the ground. During a partial eclipse these pictures have, of course, a crescent form. When detached clouds are drifting rapidly across the sun, we often seethe shadows of the bars of the window on the walls or floor suddenly shifted by an inch or two, and for a moment very much more sharply defined. They are, in fact, shadows cast by a small portion of the sun s limb, from opposite sides alternately. Another beautiful illustration is easily ob tained by cutting with a sharp knife a very small T aperture in a piece of note paper. Place this close to the eye, and an inch or so behind it place another piece of paper with a fine needle-hole in it. The light of the sky passing through the needle-hole forms a bright picture of the T on the retina. The eye perceives this picture, which gives the impression of the T much magnified, but turned upside down. Another curious phenomenon may fitly be referred to in this connexion, viz., the phantoms which are seen when we look at two parallel sets of palisades or railings, one behind the other, or look through two parallel sides of a meat-safe formed of perforated zinc. The appearance pre sented is that of a magnified set of bars or apertures which appear to move rapidly as we slowly walk past. Their origin is the fact that where the bars appear nearly to coincide the apparent gaps bear the greatest ratio to the dark spaces; i.e., these parts of the field are the most highly illuminated. The exact determination of the appearances in any given case is a mere problem of convergents to a continued fraction. But the fact that the Shadow image. PI pali- sa les apparent rapidity of motion of this phantom may exceed in any ratio that of the spectator is of importance, enabling us to see how velocities, apparently of impossible magnitude, may be accounted for by the mere running along of the condition of visibility among a group of objects no one of which is moving at an extravagant rate. (b) Another important consequence of this law is that Illumi- if the medium l>e transparent the intensity of illumination nation. which a luminous point can produce on a white surface directly exposed to it is inversely as the square of the distance. The word transparent implies that no light is absorbed or stopped. Whatever, therefore, leaves the source of light must in succession pass through each of a series of spherical surfaces described round the source as centre. The same amount of light falls perpendicularly on all these surfaces in succession. The amount received in a given time by a unit of surface on each is therefore inversely as the number of such units in each. But the surfaces of spheres are as the squares of their radii, whence the proposition. (We assume here that the velocity of light is constant in the medium, and that the source gives out its light uniformly and not by fits and starts. ) When the rays fall otherwise than perpendicularly on the surface, the illumination pro duced is proportional to the cosine of the obliquity ; for the area seen under a given spherical angle increases as the secant of the obliquity, the distance remaining the same. As a corollary to this we have the further proposition Bright- that the apparent brightness of a luminous surface (seen ness at through a transparent homogeneous medium) is the same at different 7 v , distances. all distances. The word brightness is here taken as a measure of the amount of light falling on the pupil per unit of spherical angle subtended by the luminous surface. The spherical angle subtended by any small surface whose plane is at right angles to the line of sight is inversely as the square of the distance. So also is the light received from it. Hence the brightness is the same at all distances. The word brightness is often used (even scientifically) in another sense from that just defined. Thus we speak of a bright star, of the question When is Venus at its brightest] ifec. Strictly, such expressions are not defen sible except for sources of light which (like a star) have no apparent surface, so that we cannot tell from what amount of spherical angle their light appears to come. In that case the spherical angle is, for want of knowledge, assumed to be the same for all, and therefore the bright ness of each is now estimated in terms of the whole quantity of light we receive from it. It is in this sense Maxi- only that we use the word when we speak of Venus at its im . mi brightest : for if we take the former definition of bright- brigbt- 11GSS Oi ness the solution of this once celebrated problem would be y enus . very different from that usually given. As the question, however, is an interesting one both in itself and histori cally, we give an approximate solution of it. The approxi mation assumes what is certainly not true, that the illuminated portion of Venus always appears uniformly bright, and of the same degree of brightness in all aspects. Let a be tlic radius of the earth s orbit, b that of the orbit of Venus, 5 the distance between the planets when Venus is brightest. Then if be the apparent angular distance of the earth from the sun as seen from Venus, the illuminated part of the disk of Venus as seen from the earth is 1 -f cos 6 of the whole disk. Hence 1 + cos e 2S 2 is a maximum, with the obvious trigonometrical relation
