Presuming the refractive and dispersive properties of the glass not known, it will be necessary to have a small bit of glass of the same kind to experiment with. That the optician may make no mistake in this important matter, some glass manufacturers make the discs with projecting pieces to be cut off; these the object-glass maker works into prisms to determine the exact refraction and dispersion, including the position in the spectrum of the Fraunhofer lines C and G, for both the crown and flint glass. With these numbers and the desired focal length he has all the necessary data for the mathematical operation of calculating the _powers_ to be given to the two lenses—flint and crown, and the radii of curvature of the four surfaces in order that the object-glass may be aplanatic or free from aberration both spherical and chromatic. The problem is what mathematicians call an indeterminate one, as an infinite number of different curvatures is possible. Assume, however, the radius of curvature of one surface, and all the rest are limited. In assuming the radius of curvature on one of the crown-glass surfaces, it is well to avoid deep ones. It is better to divide the refraction of the four surfaces as equally as the nature of the problem will admit, as any little deviation from a true spherical figure in the polishing will produce less effect in injuring the performance of the object-glass from surfaces so arranged than if the curves were deep. But whatever curves he chooses he goes to work so that the spherical aberration of the compound lens shall be eliminated as far as possible, and the chromatism in one lens shall be corrected by the other, or in other words, that what is called the _secondary spectrum_ shall be as small as possible; and it is to be feared that this will never be abolished.[6] [Illustration: FIG. 64.—Images of planet produced by short and long focus lenses of the same aperture giving images of different size, but with the same amount of colour round the edges. ] This matter requires a somewhat detailed treatment in order that it may be seen how the four surfaces to which reference has been made are determined.