CATOPTRICS. General Laws of Reflection. 6. In order that the surface of a body may reflect the light regularly and give a distinct image of the luminous points by means of which it is rendered visible, it must be carefully polished, that is, its little inequalities must be removed, as far as possible, by friction. Such is the state to which glass, crystals, and metals, are reduced by art. We shall first consider the phenomena presented by surfaces thus prepared; and afterwards endeavour to show in what manner the polishing contributes to the effect in question. To proceed methodically in the study of the laws of reflection, we shall begin by determining what these laws are with respect to plane surfaces. It will be easy to extend them, when thus determined, to curved surfaces. For, in all the modifications which light undergoes, by the action of bodies of a sensible extent, the luminous rays may be considered, (at least, as far as they are perceptible to our senses) as mathematical straight lines; so that at each point of a curved surface, reflection takes place in exactly the same manner as if the ray fell upon a plane touching this point; and since we may always calculate the position of the tangent plane for all the points of a given surface, it follows that the reflection of light from any curved surface whatever will be a subject of pure calculation, when the laws according to which it takes place from plane surfaces are once known. 7. For the purposes of this inquiry, and with respect to optical experiments in general, it is indispensable that we should be provided with a room which receives the direct rays of the ure. sun during a part of the day, at least, and which is furnished with window-shutters made very tight, so that it may at any time be rendered completely dark. In one of these shutters an aperture is supposed to be made of any convenient size, and covered with a metallic plate, pierced with several holes of unequal diameters, and capable of being opened or shut at pleasOne of these holes being opened when the rays of the sun fall directly upon the shutter, the light will enter the room in the form of a beam or fine collection of rays, which become sensible by being reflected from the particles of dust floating in the atmosphere. We shall hereafter make known a more convenient and ingenious apparatus; the one just described will suffice for the present. Now if this beam be made to fall obliquely upon a polished, transparent plate of glass, placed horizontally, the following appearances will be exhibited; (1). The luminous beam is not wholly transmitted through the surface. A part is reflected upward, in a direction depending on its obliquity. If the eye be placed any where in this direction, a lively, brilliant image of the sun will be visible, which seems to come from beneath the glass in the direction of the floor. (2.) The point where the beam of light meets the plate is visible from every part of the room; but when thus viewed no regular image of the sun is presented, and the light is incomparably less brilliant than when the eye of the observer is placed in the direction of the reflected beam. (3.) A portion of the incident light escapes reflection at the first surface of the plate and passes to the interior. Arriving at the second surface, another partial reflection takes place, and the remainder passes into the air beneath the glass. If we now consider the phenomena presented by the first surface, we shall observe that there are three distinct operations. A part of the incident light is reflected regularly in a particular direction; another part is reflected indifferently in all directions, as if the body were not polished; and finally the remainder is transmitted without being reflected. To distinguish these two modes of reflection which take place in the same body, I shall call the first specular reflection, since it is this which gives a regular image, whatever be the reflecting surface. I shall call the other mode radiant reflection, because it scatters the light in all directions about the point of incidence, as if this point had by itself a radiating power. If we substitute for the plate of glass one of polished metal, the two first only of these phenomena will be observed. The polished metal reflects regularly one portion of the incident light, disperses another portion, and absorbs or extinguishes the rest; this remainder answers to what the transparent body transmits. By confining ourselves, therefore, for the present to the two first phenomena which constitute reflection, we shall in the first place inquire into this dissemination of light which renders the point of incidence visible from all parts of the dark room. If we repeat the experiment with different reflecting bodies, and with the same body having different degrees of polish, we shall soon perceive that the imperfection of the polish is the determining cause of the phenomenon. For, the greater this imperfection is, the nature of the reflecting body remaining the same, the more considerable is the portion of light thus dispersed, and the more feeble, on the contrary, is the specular reflection. To be convinced of this, we need only take a plate of glass, having a plane but unpolished surface, and to present it successively, under different angles of obliquity to a solar ray, introduced into a dark room. When the ray meets the plate in a direction nearly perpendicular to its surface, the portion of light reflected specularly will be insensible; the radiant portion, on the contrary, will be very strong, and will render the point of incidence dazzling; but by inclining the reflecting surface to the ray, we shall find that this portion becomes less, and the specular reflection beginning to take place, gives at first a feeble reddish light; soon this light increases, and finally, when the ray becomes nearly parallel to the surface of the plate, it will be almost as strong and as white, as it would be from glass highly polished. Analogous results are observed, if instead of examining the reflection of a solar ray, we undertake to observe, by reflection from an unpolished plate, the images of very bright objects, for example, that of a building illuminated by the sun; for while the incident rays make large angles with the unpolished surface, the images of the objects are not distinctly formed. But soon they will Fig. 1. begin to appear when the rays begin to be inclined to the reflecting surface; and in the end they will become perfectly bright and distinct with the smallest inclinations. Now the direction of the incident ray in relation to the reflecting surface being given, let us inquire what will be the direction. of that part of the light which is reflected specularly. For this purpose we may make use of the instrument represented in figure 1, which we shall hereafter have occasion to use. It consists, in the first place, of a circular plane AZB, placed vertically upon a firm stand capable of being levelled. The circumference AZB is graduated, and carries too metallic indices S, O, turning on the same centre and having two small holes S', O', at equal distances from the plane of the circle. At the centre C is placed a plate of polished glass, CG, which, by means of screws, is fixed perpendicularly to the plane of the circle; it consequently takes a horizontal position when the circle is vertical. In order to give it this adjustment the plane of the circle is first rendered vertical by being directed to the vertical bars of a window frame or other vertical object; a spirit level is then placed upon the glass plate, and the plate is moved by means of its adjusting screws till this level indicates a horizontal position. Finally, above this glass and before the centre of the circle is fixed a metallic plate, or thin wedge, the rectilineal edge of which CL forms a straight line, proceeding from the centre C perpendicularly to the plane of the circle; and upon this line is made a light stroke C', at the same distance from this plane with the hole in the indices; so that the three points, S, O, C', are always in the same plane parallel to the graduated circle. The instrument is now to be placed before an open window, so that the light from abroad may enter the hole S, and fall upon the glass under different angles. The index O is moved backward or forward till the image of the hole S is seen through the opening at O exactly upon the edge CL of the central plate. It is shown by experiment that this is possible; and the point of incidence is always found to fall precisely upon the stroke C', therefore, the incident ray and the reflected ray are comprehended in the same plane perpendicular to the reflecting surface. This is the first fundamental law of reflection. Moreover, the two rays meeting at the axis CC' of the graduated circle, their respective inclina tions to the reflecting surface are measured by the arcs BS, AO, which may be read off upon the graduated circle, the divisions commencing at the horizontal diameter AB. This being done, it will be found that for all possible positions of the indices, the incident and reflected rays make the same angle with the reflecting surface. This is the second general law of reflection, and being taken in connexion with the preceding, it determines all the circumstances of this phenomenon. 8. Through the point of incidence C' suppose a line C'Z', drawn perpendicularly to the reflecting surface; the angle S'C'Z', formed with this perpendicular by the incident ray is commonly called the angle of incidence, or simply the incidence, and O'C'Z' is called the angle of reflection. The second general law of reflection, therefore, deduced from the preceding observations, is, that the angle of reflection is always equal to the angle of incidence. Of the Plane Mirror. 9. THE laws of reflection being known, it is easy to deduce from them the appearances which are observed when the reflecting surface is a plane. Let S be a radiant point, O the eye, and AB the reflecting Fig. 2. plane, which I shall suppose for the present of indefinite extent. Among all the luminous rays which proceed from S, there will be one, as SI, which, after being reflected, will go to meet the eye at O, in the direction IO. Then the angles SIA, OIB, will be equal, according to the second law of reflection. Now draw from the point S, a perpendicular SA, meeting the reflecting surface in A, and produce this line on the opposite side of the mirror to D, making AD equal to AS. From the point D draw the straight line DO to the eye; DO, it is manifest, will be the direction of the reflected ray, and the point S, where it cuts the surface of the mirror, will be the point of incidence. Moreover, if the luminous object and the eye are considered as mathematical points, without sensible extent, the ray, thus determined, is the only one which can be reflected to the eye. But the pupil or opening through which the light is admitted into the eye is |