That light moves in straight lines is a statement which is true only when the media through which it passes are uniform; for it is easily proved that when light passes from one medium to another, a change of direction takes place at the common surface of the media in all rays that meet this surface otherwise than perpendicularly. As a consequence of this, it really is possible to see round a corner, as the reader may convince himself by performing the following easy experiment. Having procured a cup or basin, Fig. [202], let him, by means of a little bees’-wax or tallow, attach to the bottom of the vessel, at R, a small coin. If he now places the cup so that its edge just conceals the coin from view, and maintains his eye steadily in the same position as at I, he will, when water is poured into the cup, perceive the coin apparently above the edge of the vessel in the direction I R´, that is, the bottom of the cup will appear to have risen higher. Since it is known that in each medium the rays pass in straight lines, the bending which renders the coin visible can therefore only take place at the common junction of the media, or, in other words, the ray, R O, passing from the object in a straight line through the water, is bent abruptly aside as it passes out at the surface of the water, A B, and enters the air, in which it again pursues a straight course, reaching the eye at I, where it gives the spectator an impression of an object at R´. This experiment is also an illustration of the cause of the well-known tendency we have to under-estimate the depth of water when we can see the bottom. The broken appearance presented by an oar plunged into clear water is due to precisely the same cause. The curious exaggerated sizes and distorted shapes of the gold-fish seen in a transparent globe have their origin in the same bending aside of the rays. This deviation which light undergoes in passing obliquely from one medium into another is known by the name of refraction, and it is essential for the understanding of the sequel that the reader should be acquainted with some of the laws of this phenomenon, although their discovery by Snell dates two centuries and a half anterior to the present time. Let T O, Fig. [203], be a ray of light which falls obliquely upon a plane surface, A B, common to two different media, one of which is represented by the shaded portion of the figure, A B C D, of which C D represents another plane surface, parallel to the former. If the ray, T O, suffered no refraction, it would pursue its course in a straight line to R´; but as a matter of fact it is found that such a ray is always bent aside at O, if the medium A B C D is more or less dense than the other. If, for example, A B C D is water, and the medium above it glass, then the ray entering at O will take the course O R; but if A B C D is a plate of glass with water above and below it, the ray will take the course T O, O R, R B, suffering refraction on entering the glass, and again on leaving it, so that R B will emerge from the glass parallel to its original direction at T O. If through the point of incidence, O, we suppose a line, O P, to be drawn perpendicular to the surface, A B, then we may say that the ray in passing from the rarer medium (water, air, &c.) into the denser medium (glass, &c.) is bent towards the perpendicular, or normal, as at O; but that on leaving the denser to enter the rarer medium, as at R, it is bent away from the perpendicular. In other words, the angle b O a is less than the angle m O T, and O R forms a less angle with R P´ than R B´ does. It is also a law of ordinary refraction that the normal, O P, at the point of incidence, the incident ray, T O, and the refracted ray, O R, are all in the same plane. Besides, there is the important and interesting law discovered by Snell and by Descartes, which may thus be explained with reference to Fig. [203]. On the incident and refracted rays, T O and O R, let us suppose that any equal distances, O d and O b, are measured off from O, and that from each of the points a and b, perpendiculars, a m and b n, are drawn to the normal, P P, which passes through O; then it is found that, whatever may be the angle of incidence, T O P, or however it is made to vary, the length of the line a m bears always the same proportion to the line b n for the same two media. Thus, if A B C D be water, and T O enters it out of the air, the length of the line a m divided by the length of the line a b will always (whatever slope T O may have) give the quotient 1·33. This number is, therefore, a constant quantity for air and water, and is called the index of refraction for air into water. The law just explained is expressed by the language of mathematics thus: For two given media the ratio of the sines of the angles of incidence and of refraction is constant.
Fig. 202.
Fig. 203.
It is an axiom in optical science that a ray of light when sent in the opposite direction will pursue the same path. Thus in Fig. [203] the direction of the light is represented as from T towards B´; but if we suppose B´ R to be an incident ray, it would pursue the path B´ R, R O, O T, and in passing out of the denser medium, A B C D at O, its direction is farther from the normal, P P, or O T, as the law of sines, a m will be always longer than n b, and will bear a constant ratio to it. Suppose the angle R O P to increase, then P O B will become a right angle; that is, the emergent ray, O T, will just graze the surface, A B, when the angle R O P has some definite value. If this last angle be further increased, no light at all will pass out of the medium A B C D, but the ray R O will be totally reflected at O back into the medium, A B C D, according to the laws of reflection. The angle which R O forms with O P when O T just skims the surface, A B, is termed the limiting angle, or the critical angle, and its value varies with the media. The reader may easily see the total reflection in an aquarium, or even in a tumbler of water, when he looks up through the glass at the surface of the water, which has then all the properties of a perfect mirror.
The power of lenses to form images of objects is entirely due to these laws of refraction. The ordinary double-convex lens, for example, having its surfaces formed of portions of spheres, refracts the rays so that all the rays which from one luminous point fall upon the lens, meet together again at a point on the other side, the said point being termed their focus. It is thus that images of luminous bodies are formed by lenses. An explanation of the construction and theory of lenses cannot, however, be entered into in this place.
One important remark remains to be made—namely, that in the above statement of the laws of reflection and refraction, certain limitations and conditions under which they are true and perfectly general have not been expressed; for the mention of a number of particulars, which the reader would probably not be in a condition to understand, would only tend to confuse, and the explanation of them would lead us beyond our limits. Some of these conditions belong to the phenomena we have to describe, and are named in connection with them, and others, which are not in immediate relation to our subject, we leave the reader to find for himself in any good treatise on optics.
DOUBLE REFRACTION AND POLARIZATION.
About two hundred years ago, a traveller, returning from Iceland, brought to Copenhagen some crystals, which he had obtained from the Bay of Roërford, in that island. These crystals, which are remarkable for their size and transparency, were sent by the traveller to his friend, Erasmus Bartholinus, a medical man of great learning, who examined them with great interest, and was much surprised by finding that all objects viewed through them appeared double. He published an account of this singular circumstance in 1669, and by the discovery of this property of Iceland spar, it became evident that the theory of refraction, the laws of which had been studied by Snell and by Huyghens a few years before, required some modification, for these laws required only one refracted ray, and Iceland spar gave two. Huyghens studied the subject afresh, and was able, by a geometrical conception, to bring the new phenomena within the general theory of light. Iceland spar is chemically carbonate of lime (calcium carbonate), and hence is also called calc spar, and, from the shape of the crystals, it has also been termed rhombohedral spar. The form in which the crystals actually present themselves is seen in Fig. [204], which also represents the phenomenon of double refraction. Iceland spar splits up very readily, but only along certain definite directions, and from such a piece as that represented in Fig. [204] a perfect rhombohedron, such as that shown in Fig. [206], is readily obtained by cleavage; and then we have a solid having six lozenge-shaped sides, each lozenge or side having two obtuse angles of 101° 55´, and two acute angles of 78° 5´. Of the eight solid corners, such as A B C, &c., six are produced by the meeting of one obtuse and two acute angles, and the remaining two solid corners are formed by the meeting of three obtuse angles. Let us imagine that a line is drawn from one of these angles to the other: the diagonal so drawn forms the optic axis of the crystal, and a plane passing through the optic axis, A B, Fig. [205], and through the bisectors of the angles, E A D and F B G, marks a certain definite direction in the crystal, to which also belong all planes parallel to that just indicated. Any one of such planes forms what is termed a “principal section,” to which we shall presently refer.