
THE THEORY OF APERTURE IN THE MICROSCOPE. 329 

necessarily follows that the interposition of the immersion fluid 
between the front lens and the object has increased the quantity of 
light entering any given semi-diameter of the front surface of the 
lens in the proportion of the refractive index of water, 1.33 ; because, 
if water had not been interposed, the light from the object would 
have spread over the larger angle E O D, equal to B O A in air; 
and as the areas of circles are to each other as the squares of their 
radii, so the amount of light received by the whole surface of the 
front lens of a water-immersion objective must bear to that received 
by the surface of a dry objective of similar angular capacity the 
ratio of the square of 1.33, or 1.77 to 1.0. Similarly in the case of 
oil as the immersion medium, with an index of refraction = 1.52, 
the semi-diameter of the front surface of the lens will receive 1.52 
times the quantity of light that a similar portion of a lens of the 
same capacity will receive in air; and the illuminating power of 
the two lenses will be in the ratio of the square of 1.52, or 2.31 to 
1.0, A correct system of notation for apertures will, therefore, 
require that, when the light passes through different immersion 
media, the numerical expressions for aperture shall be multiples of 
the sines of the semi-angles with the refractive indices of those 
media. ‘This principle is carried out in the system of numerical 
aperture, where radius, or the sine of half the angle of 180°, 
being taken as = 1.0 in air, it becomes = 1.33 in water, and 1.52 
in oil. 
We have seen that in the diagram, Fig. 38, the ray of light A O, 
falling upon the water at O, is refracted to C. In like manner a 
ray of light C O, passing from water into air, will be refracted in 
the direction O A, the angle of incidence in this case being R, and 
the angle of refraction I. Now, if we enlarge the angle of incidence 
until it is nearly 484°, we shall find that a ray of light E O will be 
refracted in such an oblique direction O F, that any increase in the 
angle of incidence will result in no light passing out of the water, 
but in the whole of it being totally reflected at O in the direction 
of O G; and the angle E O D is, therefore, called the critical 
angle, which for water is 485°, and for crown glass or oil 41°. If the 
system of notation in numerical aperture is correct, therefore, we 
ought to find that 1.0, = 180° in air, is also equal to twice the 
critical angles in water and in oil; and we accordingly find that 
radius divided by the refractive index in each case is equal to the 
sine of the critical angle, and that 180° in air is consequently the 
same thing as 97° in water and 82° in oil. It will be obvious that 
any excess over these apertures in water and oil must represent 
greater apertures than 18o° in air. 
It may be asked whether we can see these apertures in excess of 
180° in air through the microscope ; and, if they really exist, what 
is their practical utility, since dry objectives are made that will take 
