This excess will be _seen_ if we take an oil-immersion objective of, say 122° balsam angle, illuminating it so that the whole field is filled with the incident rays, and use it first on an object not mounted in balsam, but dry. We then have a _dry objective_ of nearly 180° angular aperture, for, as will be seen by reference to Fig. 36, the cover-glass is virtually the first surface of the objective, as the front lens, the immersion fluid, and the cover-glass are all approximately of the same index, and form, therefore, a front lens of extra thickness. When the object is close to the cover-glass the pencil radiating from it will be very nearly 180°, and the emergent pencil (observed by removing the eye-piece) will be seen to utilise as much of the back lens of the objective as is equal to twice the focal length, that is, the _inner_ of the two circles at the head of Fig. 35. If now balsam be run in beneath the cover-glass so that the angle of the pencil taken up by the objective is no longer 180°, but 122° only (that is, _smaller_), the diameter of the emergent pencil is _larger_ than it was before, when the angle of the pencil was 180° in air, and will be approximately represented by the _outer_ circle of Fig. 35. As the power remains the same in both cases, the larger diameter denotes the greater aperture of the immersion objective over a dry objective of even 180° angle, and the excess of aperture is made plainly visible. Having settled the principle, it is still necessary, however, to find a proper _notation_ for comparing apertures. The astronomer can compare the apertures of his various objectives by simply expressing them in inches, but this is obviously not available to the microscopist, who has to deal with the ratio of two varying quantities. In consequence of a discovery made by Professor Abbe in 1873, that a general relation existed between the pencil admitted into the front of the objective and that emerging from the back of the objective, he was able to show that the ratio of the semi-diameter of the emergent pencil to the focal length of the objective could be expressed by the formula _n_ Sin _u_, _i.e._, by the sine of half the angle of aperture (_u_) multiplied by the refractive index of the medium (_n_) in front of the objective (_n_ being 1·0 for air, 1·33 for water, and 1·52 for oil or balsam).