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 2½ 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 AB will be transferred by refraction to A′B′, and thus united with _a′b′_. [Illustration: FIG. 18] The pictures _ab_, ABB 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 _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′.