358 ANNUAL RECORD OF SCIENCE AND INDUSTRY. 



cal aperture" all objectives, dry, water-, or oil-immersion, 

 can be directly compared. The apparatus cannot well be 

 described without a diagram. Suffice it that it is mainly a 

 semicircular disk of thick crown-glass, with polished edges. 

 On one of the faces of the disk two scales are engraved. 

 The inner one reads off the largest possible angle from air 

 into the medium (crown-glass) of which the disk is made, or 

 twice the " critical angle." The other reads off the corre- 

 sponding "numerical aperture," which is a number that is al- 

 ways the product of the index of refraction of the medium in 

 front of the objective multiplied by the sine of half the angle 

 of aperture. Knowing this number, and also the index of 

 the medium in front of the objective, we can from these get 

 the equivalent angle of aperture for that medium. The in- 

 ternal scale is graduated from in the middle to say 82 (or 

 double the critical angle for crown-glass) on either side ; and 

 this, as already said, reads off the air angle. The external 

 scale, concentric with, but outside of, the other, commences 

 with in the middle, and reads 1 on either side, coincident 

 with the 82-f of the inner scale; but the divisions extend 

 much beyond this say to 1.3 or 1.4. Suppose, now, the ob- 

 jective to be dry, and, as near as possible, 180 air angle, 

 using the apertometer, the angle would be read 82+ on the 

 inner scale i. e., 82 -f in glass, which is equivalent to 180 

 in air. On the outer scale we would read 1 (z=sine 90, half 

 the air angle). This is the equivalent numerical aperture ; 

 and since this is a number which is equal to the sine of the 

 semi-angle of aperture multiplied by the index of refraction 

 of the medium (in this case air = l nearly), we have l = sine 

 semi-angle x 1, or sine of semi-angle of aperture = 1 ; i. e., 

 semi-angle = 90, twice which = 180, or air angle, as before ; 

 and this is the utmost any objective could read icith only air 

 in front. Applying, now, a medium say water (whose index 

 is 1.33) the outer scale, with the same objective, would read 

 off say 1.1. This is the numerical aperture an aperture 

 which, if taken in air, would be imaginary i. e., surpass 180 ; 

 but to get water angle we proceed as before : sine i aperture 



= 0.827 = sine 55 30'; and twice this, 111, is the water 

 1.33 ' ' 



angle of the objective. If, instead of water, balsam (with 



index 1.5) had been used, and yet the scale had only read 



