In order to find the height MN, or rather the depth of the cone in Fig. 22, let D, d, c, c, represent the same quantities as before, and we shall have

When D, C, d, d' have the same values as before, we shall have MN 18.7 feet!

When cd, MP will be infinite.

We have already explained how the two binocular pictures are combined or laid upon one another in the lenticular stereoscope. Let us now see how the relief is obtained. The two plane pictures abcd, ABCD, in Fig. 18, are, as we have already explained, combined or simply laid upon one another by the lenses LL, L'L', and in this state are shewn by the middle circles at AaBb, CcDd. The images of the bases AB, ab of the cone are accurately united in the double base AB, ab, but the summits of the conical frustum remain separate, as seen at c'D', c'd'. It is now the business of the eyes to unite these, or rather to make them appear as united. We have already seen how they are brought into relief when the summits are refracted so as to pass one another, as in Fig. 18. Let us therefore

summits CD, cd

take the case shewn in Fig. 20, where the are more distant than the bases AB, ab. The union of these figures is instantly effected, as shewn in Fig. 23, by converging the optic axes to points m and n successively, and thus uniting c and c and D and d, and making these points of the summit plane appear at m and n, the

points of convergence of the axes Lm, Rm, and Ln, Rn. In like manner, every pair of points in the summit

plane, and in the sides Am, Bn of the frustum, are converged to points corresponding to their distance from the base AB of the original solid frustum, from which the plane

pictures ABCD, abcd, were taken. We shall, therefore, see in relief the frustum of a cone whose section is Amn B. The theory of the stereoscope may be expressed and illustrated in the following manner, without

to binocular vision:

any reference

1. When a drawing of any object or series of objects is executed on a plane surface from one point of sight, according to the principles of geometrical perspective, every point of its surface that is visible from the point of sight will be represented on the plane.

2. If another drawing of the same object or series of objects is similarly executed on the same plane from a second point of sight, sufficiently distant from the first to make the two drawings separate without overlapping, every point of its surface visible from this second point of sight will also be represented on the plane, so that we shall have two different drawings of the object placed, at a short distance from each other, on the same plane.

3. Calling these different points of the object 1, 2, 3, 4, &c., it will be seen from Fig. 24, in which L, R are the

two points of sight, that the distances 1, 1, on the plane MN, of any pair of points in the two pictures representing the point 1 of the object, will be to the distance of any other pair 2, 2, representing the point 2, as the distances 1'P, 2'p of the points of the object from the plane MN, multiplied inversely by the distances of these points from the points of sight L, R, or the middle point o between them.

4. If the sculptor, therefore, or the architect, or the mechanist, or the surveyor, possesses two such pictures, either as drawn by a skilful artist or taken photographically, he can, by measuring the distances of every pair of points, obtain the relief or prominence of the original point, or its distance from the plane MN or AB; and without the use of the stereoscope, the sculptor may model the object from its plane picture, and the distances of every point from a given plane. In like manner, the other artists may determine distances in buildings, in machinery, and in the field.

5. If the distance of the points of sight is equal to the distance of the eyes L, R, the two plane pictures may be united and raised into relief by the stereoscope, and thus give the sculptor and other artists an accurate model, from which they will derive additional aid in the execution of their work.

6. In stereoscopic vision, therefore, when we join the points 1, 1 by converging the optic axes to l' in the line Po, and the points 2, 2 by converging them to 2' in the same line, we place these points at the distances o1, o2, and see the relief, or the various differences of distance which the sculptor and others obtained by the method which we have described.

7. Hence we infer, that if the stereoscopic vision of relief had never been thought of, the principles of the instrument are involved in the geometrical relief which is embodied in the two pictures of an object taken from two points of sight, and in the prominence of every part of it obtained geometrically.