C'D', cd', 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

eye R.

In like manner, by drawing LA, LB, we shall find that A'B' will represent the line AB, and a circle, whose

diameter is A'B', the base AEBG, as seen by the left eye. The summit, ceDg, of the frustum being in the plane of projection, will be represented by the circle ceDg. The representation of the frustum ABCD, therefore, upon a plane

surface, as seen by the left eye L, consists of two circles, whose diameters are AB, CD; and, as seen by the right eye, of other two circles, whose diameters are ab, CD, which, in Fig. 20, are represented by AB, CD, and ab, cd.

These

plane figures being also the representation of the solid on the retina of the two eyes, how comes it that we see the solid and not the plane pictures? When we look at the point B, Fig. 19, with both eyes, we converge upon it the optic axes IB, RB, and we therefore see the point single, and at the distances LB, RB from each eye. When we look at the point D, we withdraw the optic axes from B, and converge them upon D. We therefore see the point D single, and at the distances LD, RD from each eye; and in like manner the eyes run over the whole solid, seeing every point single and distinct upon which they converge their axes, and at the distance of the point of convergence from the observer. During this rapid survey of the object, the whole of it is seen distinctly as a solid, although every point of it is seen double and indistinct, excepting the point upon which the axes are for the instant converged.

From these observations it is obvious, that when we look with both eyes at any solid or body in relief, we see more of the right side of it by the right eye, and more of the left side

of it by the left eye. The right side of the frustum ABCD, Fig. 19, is represented by the line Db, as seen by the right eye, and by the shorter line DB', as seen by the left eye. In like manner, the left side AC is represented by ca', as seen by the left eye, and by the shorter line ca', as seen by the right eye.

When the body is hollow, like a wine glass, we see more of the right side with the left eye, and more of the left side with the right eye.

If we now separate, as in Fig. 20, the two projections shewn together on Fig. 19, we shall see that the two summits, CD, cd, of the frustum are farther from one another than the more distant bases, AB, ab, and it is true generally that in the two pictures of any solid in relief, the similar parts that are near the observer are more distant in the two pictures than the remoter parts, when the plane of perspective is beyond the object. In the binocular picture of the human face the distance between the two noses is greater than the distance between the two right or left eyes, and the distance between the two right or left eyes greater than the distance between the two remoter ears.

We are now in a condition to explain the process by which, with the eyes alone, we can see a solid in relief by uniting the right and left eye pictures of it,—or the theory ocular stereoscope. In order to obtain the proper relief we must place the right eye picture on the left side, and the left eye picture on the right side, as shewn in Fig. 21, by the pictures ABCD, abcd, of the frustum of a cone, as obtained from Fig. 19.

In order to unite these two dissimilar projections, we must converge the optical axes to a point nearer the ob