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].
Fig. 21.
In order to unite these two dissimilar projections, we must converge the optical axes to a point nearer the observer, or look at some point about m. Both pictures will immediately be doubled. An image of the figure ab will advance towards p, and an image of ab will likewise advance towards p; and the instant these images are united, the frustum of a cone, which they represent, will appear in relief at mn, the place where the optic axes meet or cross each other. At first the solid figure will appear in the middle, between the two pictures from which it is formed and of the same size, but after some practice it will appear smaller and nearer the eye. Its smallness is an optical illusion, as it has the same angle of apparent magnitude as the plane figures, namely, mnl = abl; but its position at mn is a reality, for if we look at the point of our finger held beyond m the solid figure will be seen nearer the eye. The difficulty which we experience in seeing it of the size and in the position shewn in [Fig. 21], arises from its being seen along with its two plane representations, as we shall prove experimentally when we treat in a future chapter of the union of similar figures by the eye.
The two images being thus superimposed, or united, we shall now see that the combined images are seen in relief in the very same way that in ordinary vision we saw the real solid, abcd,[ Fig. 19], in relief, by the union of the two pictures of it on the retina. From the points a, b, c, d, a, b, c, d, draw lines to l and r, the centres of visible direction of each eye, and it will be seen that the circles ab, ab, representing the base of the cone, can be united by converging the optical axes to points in the line mn, and that the circles cd, cd, which are more distant, can be united only by converging the optic axes to points in the line op. The points a, a, for example, united by converging the axes to m, are seen at that point single; the points b, b at n single, the points c, c at o single, the points d, d at p single, the centres s, s of the base at m single, and the centres s′, s′ of the summit plane at n single. Hence the eyes l and r see the combined pictures at mn in relief, exactly in the same manner as they saw in relief the original solid mn in [Fig. 19].
In order to find the height mn of the conical frustum thus seen, let d = distance op; d = ss, the distance of the two points united at m; d′ = s′s′, the distance of the two points united at n; and c = lr = 2½ inches, the distance of the eyes. Then we have—
| MP = | Dd |
| C + d |
| NP = | Dd′ | , | and |
| C + d′ | |||
| MN = | Dd | - | Dd′ |
| C + d | C + d′ | ||
| If D = | 9.24 inches, |
| C = | 2.50, then |
| d = | 2.14 |
| d′ = | 2.42, and |
| MN = | 0.283, the height of the cone. |