Fig. 3. not a mathematical point, but a circle of sensible extent which in man is about 0,08 of an inch in diameter, and which may be represented by LL. All the reflected rays which can enter this aperture will reach the retina and contribute to vision. Now each of these rays is determined by the construction just given; whence it is evident that they will form a pencil or cone with a circular base, the vertex of which will be D, and the base LL. It is a fact, that when the eye estimates freely the distances of luminous points, it supposes them placed at the point from which the rays that enter the eye diverge. Thus to the eye situated at O, the luminous point, seen by reflection, will appear at the point D, that is, as far behind the mirror as it really is before it. If the radiant object has a definite extent, each of the radiant points, presented by it, will produce its own particular image, according to the laws which we have just explained, and the assemblage of these images will compose the image of the object. Fig. 4. Suppose, for example, that this object is an arrow SS'; the point S of the arrow will have its image at D, the image of the other extremity will appear at D', and the intermediate points will be seen in the straight line DD. Thus the entire image will be comprehended between the extreme reflected pencils DO, D'O; its absolute magnitude DD' will be equal to that of the arrow itself; but it will appear inverted from right to left. According to what we have just now said respecting figure 3, each point of this image is regarded by the eye as situated where it is actually fixed by our geometrical construction. If we substitute, therefore, in the place of this image a real and similar object, that is, an arrow in every respect equal to SS', placing its point at D, and its base at D', and if we look at it from the point O, the mirror being removed, we shall see it precisely as we see the image DD', having the same magnitude, the same visual angles, and the same degree of brightness, since all the luminous points which compose the real object would transmit to the eye cones of luminous rays exactly similar to those which seem to proceed from the image. It hence follows that objects are seen in the plane mirror with the same forms, the same brightness, and at the same distance, as they would be if viewed through the medium of the air, provided that the quantity of light is the same in the two cases; for we must make allowance for the portion of light which is transmitted when the mirror is transparent, or absorbed when it is opaque. 10. What has now been said is sufficient for the resolution of all problems which may occur respecting reflection and vision by plane mirrors. As the reflection of light takes place rigorously according to the law we have just demonstrated, it may be employed with great advantage in measuring the angles formed by two plane and polished surfaces. As this measurement is necessary in a great number of physical inquiries, and as no other method has ever been employed sufficiently nice and accurate for this purpose, I shall give some simple examples of the process. When we operate upon bodies having faces of a sensible extent, for example, on prisms cut and polished by art, like those which are employed in optical experiments, we make use of the instrument which has already been applied to determine the laws of reflection. For this purpose, instead of bringing the glass plate GG in contact with the edge CL, it should be placed at the distance of about of an inch, being always made per- Fig. 5. pendicular to the plane of the circle, by means of the adjusting screws with which the instrument is provided. In order to give it this position without other aid than that which the instrument affords, we move the index O nearly to the highest part of the circle, and placing the eye behind it, we regulate the glass so that the reflected image of the hole O' and that of the eye shall return through this hole from the fixed mark C' upon the edge CL. When this takes place we are sure that the ray OC, parallel to the circle, is at the same time perpendicular to the glass, and reciprocally that the glass is perpendicular to the plane of the circle. We then place upon the glass one of the faces of the prism whose angles we wish to measure. The edge of the prism is now made to slide under CL, by turning it in such a manner that the upper surface shall also be perpendicular to the plane of the circle; and this condition is fulfilled when, the index S being placed in any point of the circumference, its image, as seen through the other index, moved to a corresponding point, appears on the mark C'. Let S and O be the positions of the two indices answering to this condition. Then it is evident that N, the centre of the arc SO (a point easily determined by means of the divisions upon the arc) is in the direction of the line rais ed from the centre C perpendicularly to the reflecting surface. Moreover, CZ is perpendicular to the other face; therefore the angle ZCN, measured by the arc ZN, is the angle comprehended between the two surfaces of the prism, and which it was proposed to determine. This ingenious method, which was invented by M. Cauchoix, requires but little time and is very accurate in its results. 11. The same principle differently applied, has led to a great number of instruments analogous to this, which are called goniometers, that is, instruments for measuring angles. They will be easily understood after what has now been said, for the means of observation and verification are essentially the same in all. I shall nevertheless describe that invented by Dr Wollaston, because it is particularly applicable to mineralogy• This instrument, represented by figure 7, is composed of a vertical circle of copper, graduated on its edge, and turning about a horizontal axis AA. It is itself supported upon a vertitical foot CP. Within the axis AA, made hollow for the purpose, turns the axis a a, the projecting extremity of which carries several pieces having rectangular movements, upon which are fixed, with pincers or wax, the crystal whose angles are to be measured. To make use of this goniometer, we stand before a building having several horizontal lines parallel to each other; we then place the base of the instrument on some horizontal plane, so that the limb shall become vertical and perpendicular, or nearly perpendicular, to the lines which are to be employed as sights. The first condition is easily fulfilled by arranging the plane of the limb along some of the vertical lines presented by the building. This being done, we place the eye very near the crystal, and looking at the building by reflection from one of its faces, we turn this in such a manner, that one of the horizontal lines the most elevated, thus seen, shall coincide with one of the inferior lines seen directly. I shall mention soon how we may arrive at this condition. When we have obtained it, we turn the interior axis a a, until the same coincidence shall be observed likewise on the other face, whose inclination to the first we wish to measure. We arrive at this by several trials. Now when this coincidence can be thus obtained successively upon the two faces without changing the place of the eye, without touching the crystal, and by the rotation merely of the axis a a, we are sure that the intersection of the two surfaces is exactly horizontal, and consequently parallel to the axis a a. Then we do not touch the crystal, but setting out from one of its positions, in which the reflection is observed on one of the two surfaces, we turn the limb until the reflexion and the coincidence are observed likewise on the other. This motion takes place by means of the great axis AA, which turns with the axis a a, the crystal and the limb. The arc by which this has turned, is measured by the division traced upon the limb, and it is evidently equal to the supplement of the angle formed by the two surfaces. But the division traced upon the limb is written in a manner to indicate the angle itself, at least when we first put the index on the point zero. In order that the application of this method may be easy and sure, it is necessary that the dimensions of the crystal and its distance from the eye should be infinitely small, compared with the distance of the objects used as sights. For if this is observed, the fixed position of the eye is no longer a necessary condition any more than it is in observations made at sea with instruments of reflection. Thus by placing the eye very near the crystal, the approximation of the lines of sight may be considered as indefinite. Dr Wollaston was accustomed to place the instru ment in a chamber at some distance from the window, the bars of which were used as sights, and of which the upright parts served to place the instrument in a vertical plane. But without the address of this practised observer, sights so near could not be safely employed; for the liability to error increases with their nearness. We may in general make use of the horizontal and vertical lines of an edifice sixty or eighty yards distant; then, to render the first face of the crystal perpendicular to the limb, we first direct the shaft to parallel to its surface; then, Fig. 8. without taking it from this direction, we turn it upon its axis until the reflected image of one of the horizontal lines becomes parallel to the direct image, and a coincidence may be effected by the mere motion of the axis a a. We then see if the same condition is fulfilled for the other face of the crystal, and as in general we find it is not, we effect it by turning the branch b c about the point c without touching the shaft to; this motion being perpendicular to the plane of the limb cannot alter the perpendicularity of the first face; but for greater security we turn the axis a a to effect the coincidence; and as this is almost always necessary, if it have been subjected to any deviation, the correction is applied by the single rotatory motion of the shaft to upon its axis. After one or two verifications of this kind, the coincidence upon the two faces will take place exactly, and their inclination may be observed. One advantage peculiar to this goniometer, is, that it enables us to measure the angles of the smallest fragments of a crystal, to which it is exclusively and exactly applicable; and this property is so much the more valuable, as the small crystals seem to be the only ones in which a perfect regularity of internal construction is to be looked for. Of Curved Mirrors. 12. In order to discover and determine generally the apparent place, form, and magnitude of images, reflected by curved mirrors, whatever be their figure, it is sufficient to consider the reflection of each luminous ray as taking place upon a plane, tangent to the surface at the point of incidence. We can thus extend to all surfaces by the calculus, the results which we have obtained for the plane mirror. But it is unnecessary to make our calculations so general, for in practice we employ only concave and convex spherical mirrors, these being the only forms which can be accurately made and polished. Indeed, we cannot obtain from these perfect images except when the luminous rays fall upon the surface nearly in a perpendicular direction. This, therefore, will be the only case which we shall need to examine. In order to determine with exactness the circumstances to be considered, let us imagine, in space, a luminous point sending forth rays upon the different parts of any spherical surface either concave or convex; and considering one of the rays separately, let us endeavour to determine the direction which it takes after reflection. We must remember that reflection takes place always in the plane which passes through the luminous point and the perpendicular erected at the point of incidence. This perpendicular is the radius of the sphere; so that reflection |