surfaces of lenses to each other that exercises the art of the optician. So far we have got rid of this aberration in a single lens; it can also be done in the case of achromatic lenses. The foci of the two lenses in an achromatic combination must bear a certain relation to each other, and the curvatures of the surfaces must also have a certain relation for spherical aberration. In the achromatic lens there are four surfaces, _two of which can be altered for one aberration and two for the other_. For instance, in the case of the lens, Fig. 45, where the interior surfaces of the lenses are cemented together, although shown separate for clearness, we find that if the exterior surface of the crown double convex lens be of a curvature struck by a radius 672 units in length, and the exterior surface of the flint glass lens to a curvature due to a radius of 1,420 units, the lens will be corrected for spherical aberration, and these conditions leave the interior surfaces to be altered so that the relation between the powers of the lenses is such as to give achromatism. The flint is as useful in correcting the spherical aberration as the chromatic aberration; for although the relative thicknesses of the flint and crown are fixed in order to get achromatism, still we have by altering both the curvatures of each lens equally, and keeping the same foci, the power of altering the extent of spherical aberration; and it is in the applications of these two conditions that much of the higher art of our opticians is exercised. We have now therefore practically got rid of both aberrations in the modern object-glass, and hence it is that lenses of the large diameter of twenty-five and twenty-six inches are possible. The nearest approach to achromatism is known to be made when looking at a star of the first or second magnitude, the eyepiece being pushed out of focus towards the object-glass, the expanded disc has its margin of a claret colour. When the eyepiece is pushed beyond the focus outwards the margin of the expanded disc is of a light green colour. If the object-glass is well corrected for spherical aberration, the expanded discs both within and without the focus will be constituted of a series of rings equally dense with regard to light throughout, with the exception of the marginal ring, which will be a little stronger than the rest. * * * * * Having determined the radius of curvature of surface, both he who grinds the speculum, whether of speculum metal or glass, and he who grinds the object-glass, starts fair; only one has four times the work to do that the other has. The grinding is managed in a simple way, and the process of grinding or polishing an object-glass or speculum, either of glass or of metal, is the same.