NUMERICAL APERTURE. 75 



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 micro- 

 scopist, 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 I'O 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 I'O (since %=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, sup- 

 pose we desire to compare the apertures of threo 

 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 " 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 recognized, and it is seen, for instance, 

 that the aperture of the water immersion is some- 



