Lastly, it should also be noted that it is numerical and not angular aperture which measures the quantity of light admitted to the objective by different pencils. [Illustration: Fig. 38.] [Illustration: Fig. 38_a_.] First take the case of the medium being the same. The popular notion of a pencil of light may be illustrated by Fig. 38, which assumes that there is equal intensity of emission in all directions, so that the quantity of light contained in any given pencils may be compared by simply comparing the contents of the solid cones. The Bouguer-Lambert law, however, shows that the quantity of light emitted by any bright point varies with the obliquity of the direction of emission, being _greater_ in a perpendicular than in an oblique direction. The rays are less intense in proportion as they are more inclined to the surface which emits them, so that a pencil is not correctly represented by Fig. 38, but by Fig. 38_a_, the density of the rays decreasing continuously from the vertical to the horizontal, and the squares of the sines of the semi-angles (_i.e._, of the numerical aperture) constituting the true measure of the quantity of light contained in any solid pencil. If, again, the media are of different refractive indices, as air (1·0), water (1·33), and oil (1·52), the total amount of light emitted over the whole 180° from radiant points in these media under a given illumination is not the same, but is _greater_ in the case of the media of greater refractive indices in the ratio of the squares of those indices (_i.e._, as 1·0, 1·77 and 2·25). The quantity of light in pencils of different angle and in different media must therefore be compared by squaring the product of the sines and the refractive indices, _i.e._ (_n_ Sin _u_^2), for the square of the numerical aperture. The fact is therefore made clear that the aperture of a dry objective of 180° does not represent, as was supposed, a maximum, but that aperture increases with the increase in the refractive index of the immersion fluid; and it should be borne in mind that this result has been arrived at in strict accordance with the ordinary propositions of geometrical optics, and without any reference to or deductions from the diffraction theory of Professor Abbe.