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).
When, then, the values in any given cases of the expression _n_ Sin _u_ (which is known as the “numerical aperture”) has been ascertained, the objectives are instantly compared as regards their aperture, and, moreover, as 180° in air is equal to 1·0 (since _n_ = 1·0 and the sine of half 180° = 1·0) we see, with equal readiness, whether the aperture is smaller or larger than that corresponding to 180° in air. Thus, suppose we desire to compare the apertures of three objectives, one a dry objective, the second a water immersion, and the third an oil immersion; these would be compared on the angular aperture view as, say 74° air angle, 85° water angle, and 118° oil angle, so that a calculation must be worked out to arrive at the actual relation between them. Applying, however, the _numerical_[15] notation, which gives ·60 for the dry objective, ·90 for the water immersion, and 1·30 for the oil immersion, their relative apertures are immediately recognised, and it is seen, for instance, that the aperture of the water immersion is somewhat less than that of a dry objective of 180°, and that the aperture of the oil immersion exceeds that of the latter by 30%.
The advantage of immersion, in comparison with dry objectives, becomes at once apparent. Instead of consisting merely in a diminution of the loss of light by reflection or increased working distance, it is seen that a wide-angled immersion objective has a larger aperture than a dry objective of maximum angle, so that for any of the purposes for which aperture is essential an immersion must necessarily be preferred to a dry objective.
That pencils of identical angular extension but in different media are different physically, will cease to appear in any way paradoxical if we recall the simple optical fact that rays, which in air are spread out over the whole hemisphere, are in a medium of higher refractive index such as oil _compressed_ into a cone of 82° round the perpendicular, _i.e._, twice the critical angle. A cone exceeding twice the critical angle of the medium will therefore embrace a _surplus_ of rays which do not exist even in the hemisphere when the object is in air.