In a single lens, the rays admitted within one meridional plane evidently increase as the diameter of the lens (all other circumstances remaining the same), and in the microscope we have, at the back of the lens, the same conditions to deal with as are in front in the case of the telescope; the larger or smaller number of emergent rays will therefore be measured by the clear diameter, and as no rays can emerge that have not first been admitted, this will give the measure of the admitted rays under similar circumstances. If the lenses compared have different focal lengths but the same clear “openings,” they will transmit the same number of rays to equal areas of an image at a definite distance, because they would admit the same number if an object were substituted for the image; that is, if the lens were used as a telescope-objective. But as the focal lengths are different, the amplification of the images is different also, and equal areas of these images correspond to different areas of the object from which the rays are collected. Therefore, the higher power lens with the same opening as the lower power, will admit a _greater_ number of rays in all from the same object, because it admits the _same_ number as the latter from a _smaller_ portion of the object. Thus, if the focal lengths of two lenses are as 2 : 1, and the first amplifies N diameters, the second will amplify 2 N with the same distance of the image, so that the rays which are collected _to_ a given field of 1 mm. diameter of the image are admitted _from_ a field of 1/N mm. in the first case, and of 1/(2N) mm. in the second. As the “opening” of the objective is estimated by the diameter (and not by the area) the higher power lens admits _twice_ as many rays as the lower power, because it admits the same number from a field of half the diameter, and, in general, the admission of rays by the same opening, but different powers, must be in the inverse ratio of the focal lengths. In the case of the single lens, therefore, its aperture is determined by the ratio between the clear opening and the focal length. The same considerations apply to the case of a compound objective, substituting, however, for the clear opening of the single lens the diameter of the pencil at its emergence from the objective, that is, the clear utilised diameter of the back lens. All equally holds good whether the medium in which the objective is placed is the same in the case of the two objectives or different, as an alteration of the medium makes no difference in the power. [Illustration: 180° Oil Angle. (Numerical Aperture 1·52.) Illustration: 180° Water Angle. (Numerical Aperture 1·33.) Illustration: 180° Air Angle. 96° Water Angle. 82° Oil Angle. (Numerical Aperture 1·00.) Illustration: 97° Air Angle. (Numerical Aperture ·75.) Illustration: 60° Air Angle. (Numerical Aperture ·50.)