The ellipse and the hyperbola are forms of lenses in which the curvature diminishes from the central ray, or axis, to the circumference _b_; and mathematicians have shown that spherical aberration may be practically got rid of by employing lenses whose sections are ellipses or hyperbolas. The remarkable discovery of these forms of lenses is attributed to Descartes, who mathematically demonstrated the fact.
If _a l_, _a l′_, for example (Fig. 16) be part of an ellipse whose greater axis is to the distance between its foci _f f_ as the index of refraction is to unity, then parallel rays _r l′_, _r′′ l_ incident upon the elliptical surface _l′ a l_, will be refracted by the single action of that surface into lines which would meet exactly in the farther focus _f_, if there were no second surface intervening between _l a l′_ and _f_. But as every useful lens must have two surfaces, we have only to describe a circle _l a′ l′_ round _f_ as a centre, for the second surface of the lens _l′ l_.
[Illustration: Fig. 16.--Converging Meniscus.]
As all the rays refracted at the surface _l a l′_ converge accurately to _f_, and as the circular surface _l a′ l′_ is perpendicular to every one of the refracted rays, all these rays will go on to _f_ without suffering any refraction at the circular surface. Hence it should follow, that a meniscus whose convex surface is part of an ellipsoid, and whose concave surface is part of any spherical surface whose centre is in the farther focus, will have no appreciable spherical aberration, and will refract parallel rays incident on its convex surface to the farther focus.
[Illustration: Fig. 17.--Aplanatic Doublet.]
The spherical form of lens is that most generally used in the construction of the microscope. If a true elliptical or hyperbolic curve could be ground, lenses would very nearly approach perfection, and spherical aberration would be considerably reduced. Even this defect can be further reduced in practice by observing a certain ratio between the radii of the anterior and posterior surfaces of lenses; thus the spherical aberration of a lens, the radius of one surface of which is six or seven times greater than that of the other, will be much reduced when its more convex surface is turned forward to receive parallel rays, than when its less convex surface is turned forwards. It should be borne in mind that in lenses having curvatures of the kind the object would only be correctly seen in focus at one point--the mathematical or geometrical axis of the lens.