286 America ?i Quarterly Microscopical Journal . 



vertically. At the center of the semicircle, on the face of the 

 disc, a silvered coverglass is cemented, having a clear, central hole. 

 On the upper face are two engraved scales; the inner one indicat- 

 ing air angle; the outer one numerical aperture. To use the 

 instrument, the microscope is placed upright, and the apertometer 

 laid upon the stage with the clear spot of the cover glass in focus. 

 The measurement of the angle is made by moving two index points, 

 which slide along the curved edge, until their images are seen at 

 the margins of the field of view. The angle is then read off from 

 the graduations. An examining lens, furnished with the instru- 

 ment, may be used in this operation, and is necessary with high 

 powers. 



The numerical aperture depends upon the index of refraction of 

 the medium and the semi-angular aperture of the objective. If 

 we designate the index of refraction by ii, the semi-angular aper- 

 ture, determined by experiment, as 7V, while a represents numeri- 

 cal aperture, then a^^n-sin. w and sin. 70=--. For water ;;=: 



1.33 ; for balsam, ;;=i.5o. For three objectives, having respect- 

 ively numerical apertures i.oo, i.io and 1.25, as read directly 

 from the apertometer of Abbe, the angular apertures are therefore: 

 i--^=sin. 70=48" 45' and 2 7o=g'j° 30' 

 ^]^=:sin. a'=55° 45' and 2 7ci:=iii° 30' 

 ^]^z=sin. 7C'=^6° 30' and 2 ti>=^ii^° 

 The first two are water angles ; the last is for cedar oil. 

 The full value of numerical aperture does not appear to be 

 generally understood. Its application is far more general and 

 scientific than angular aperture or " interior angle." The 

 former will vary with every change in the immersion fluid ; the nu- 

 merical aperture is constant. Prof. Abbe says :* " But since the 

 ' numerical ' equivalent of the angular aperture (the measure which 

 determines the number of rays taken in by the objective) is pro- 

 portional not only to the sine of half the angle of aperture, but 

 also to the refractive indices of the respective media employed, 

 and since all the functions of the angle of aperture, and especially 

 the resolving power of the microscope, are regulated by this 

 numerical equivalent, it follows that, according to theory, the 

 capacity of the new objective, compared with that of ordinary 

 immersion lenses, is increased in the proportion of 1.50 to 1.33, and, 

 as compared with the highest dry objectives, as 1.50 to i." The ratio 



* Jotn-iial of the Royal Microscopical Society. 



