Professor Ahles Ajpertometer, &c. By Carl Zeiss. 21 
tlie indices are adjusted, as before described, so that their sharp 
points just touch the margin of the circular field limited by the 
contour of the back lens of, or any diaphragm in x. If the aperture 
is great enough, the indices should be brought to the disk in such a 
position as to touch the margin from within. 
The position of the straight edges of the indices on the internal 
scale of the disk will give the semi air-angle of the objective aj, as 
stated before. The external scale will give another definition of 
the aperture which is more abstract, and may be applied to those 
immersion lenses the angular aperture of which, taken in air, would 
surpass 180° ; i. e. would be imaginary. This external scale will 
give the value of the product a = . sin. w, n denoting the 
refractive index of any medium in front of the objective, and w 
the angle of semi-aperture belonging to the same medium. This 
quantity a, which Professor Abbe calls “ numerical aperture,” 
gives an absolute definition of aperture, which will not depend on 
the nature of the medium, supposed in front of the lens — air, water, 
or balsam; and by which lenses of every kind are directly com- 
parable. This value, taken note of as above described, the middle of 
the readings of both the indices considered, will afibrd the angular 
semi-aperture w of the lens for any definite medium, by the 
formula 
a 
sin, w = - •> 
n 
a denoting the number observed, 7i the index of the supposed 
medium. 
For instance, the immersion lenses of Zeiss will give approxi- 
mately, 
a = 1,1; 
calculated for water (n = 1 • 333), for balsam = 1 ' 50), 
1,1 
w = 55° 30' 
2w = 111 ° 
= water angle. 
sin. 10 = ^ = O' 733 
1 '50 
w = 47° 15' 
2x0 = 94° 30' 
= balsam angle. 
The internal scale gives the semi air-angle from 5-5 degrees, 
the external scale the value of a from 5-5 units of the second 
decimal ; by estimate the single degrees and the units of the second 
decimal of a are easily deduced. 
The exactness of the observation does not depend either on a 
very exact focussing to, or an exact centering of the hole in the 
silvered cover (the centre of which forms the geometric centre of 
the scales). It is sufficient if any point whatever of the hole is in 
