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racy as, in ordinary binocular vision, we see the same object in relief by uniting on the retina two pictures exactly the same as the binocular ones, the mere statement of this fact has been regarded as the theory of the stereoscope. We shall see, however, that it is not, and that it remains to be explained, more minutely than we have done in Chapter III., both how we see objects in relief in ordinary binocular vision, and how we see them in the same relief by uniting ocularly, or in the stereoscope, two dissimilar images of them.

Before proceeding, however, to this subject, we must explain the manner in which half and quarter lenses unite the two dissimilar pictures.

In Fig. 17 is shewn a semi-lens MN, with its section

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M'N.' If we look at any object successively through the portions A A'A" in the semi-lens MN, corresponding to aa'a" in the section M'N', which is the same as in a quarter-lens, the object will be magnified equally in all of them, but it will be more displaced, or more refracted, towards N, by looking through A' or a than through A or a, and most of all by looking through A" or a", the refraction being greatest at A" or a", less at A' or a', and still less at A or a. By means of a semi-lens, or a quarter of a lens of the size of MN, we can,

with an aperture of the size of A, obtain three different degrees of displacement or refraction, without any change of the magnifying power.

If we use a thicker lens, as shewn at M'N'nm, keeping the curvature of the surface the same, we increase the refracting angle at its margin N'n, we can produce any degree of displacement we require, either for the purposes of experiment, or for the duplication of large binocular pictures.

When two half or quarter lenses are used as a stereoscope, the displacement of the two pictures is produced in the manner shewn in Fig. 18, where LL is the lens for the left eye E, and L'L' that for the right eye E', placed so that the middle points, no, n'o', of each are 2 inches distant, like the two eyes. The two binocular pictures which are to be united are shewn at ab, AB, and placed at nearly the same distance. The pictures being fixed in the focus of the lenses, the pencils ano, A'n'o', bno, B'n'o', will be refracted at the points n, o, n',o', and at their points of incidence on the second surface, so as to enter the eyes, E, E', in parallel directions, though not shewn in the Figure. The points a, A, of one of the pictures, will therefore be seen distinctly in the direction of the refracted ray—that is, the pencils an, ao, issuing from a', will be seen as if they came from a', and the pencils bn, bo, as if they came from b', so that ab will be transferred by refraction to a'b'. In like manner, the picture A B will be transferred by refraction to A'B', and thus united with a'b'.

The pictures ab, AB thus united are merely circles, and will therefore be seen as a single circle at A'B'. But if we suppose ab to be the base of the frustum of a cone, and cd its summit, as seen by the left eye, and the circles AB, CD

to represent the base and summit of the same solid as seen by the right eye, then it is obvious that when the pictures of

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cd and CD are similarly displaced or refracted by the lenses LL L'L', so that cc' is equal to a A' and DD' to BB', the circles will not be united, but will overlap one another as at

C'D', c'd', in consequence of being carried beyond their place of union. The eyes, however, will instantly unite them into one by converging their axes to a remoter point, and the united circles will rise from the paper, or from the base A'B', and place the single circle at the point of convergence, as the summit of the frustum of a hollow cone whose base is A'B'. If cd, CD had been farther from one another than ab, AB, as in Figs. 20 and 21, they would still have overlapped though not carried up to their place of union. The eyes, however, will instantly unite them by converging their axes to a nearer point, and the united circles will rise from the paper, or from the base AB, and form the summit of the frustum of a raised cone whose base is A'B'.

In the preceding illustration we have supposed the solid to consist only of a base and a summit, or of parts at two different distances from the eye; but what is true of two distances is true of any number, and the instant that the two pictures are combined by the lenses they will exhibit in relief the body which they represent. If the pictures are refracted too little, or if they are refracted too much, so as not to be united, their tendency to unite is so great, that they are soon brought together by the increased or diminished convergency of the optic axes, and the stereoscopic effect is produced. Whenever two pictures are seen, no relief is visible; when only one picture is distinctly seen, the relief must be complete.

In the preceding diagram we have not shewn the refraction at the second surface of the lenses, nor the parallelism of the rays when they enter the eye,-facts well known in elementary optics.

CHAPTER V.

ON THE THEORY OF STEREOSCOPIC VISION.

HAVING, in the preceding chapter, described the ocular, the reflecting, and the lenticular stereoscopes, and explained the manner in which the two binocular pictures are combined or laid upon one another in the last of these instruments, we shall now proceed to consider the theory of stereoscopic vision.

In order to understand how the two pictures, when placed the one above the other, rise into relief, we must first explain the manner in which a solid object itself is, in ordinary vision, seen in relief, and we shall then shew how this process takes place in the two forms of the ocular stereoscope, and in the lenticular stereoscope. For this purpose, let ABCD, Fig. 19, be a section of the frustum of a cone, that is, a cone with its top cut off by a plane ceDg, and having AEBG for its base. In order that the figure may not be complicated, it will be sufficient to consider how we see, with two eyes, L and R, the cone as projected upon a plane passing through its summit ceDg. The points L, R being the points of sight, draw the lines RA, RB, which will cut the plane on which the projection is to be made in the points a, b, so that ab will represent the line AB, and a circle, whose diameter is ab, will represent the base of the cone, as seen by the right

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