328 THE MICROSCOPE AND ITS REVELATIONS. 



tnre, Microscopical Vision, and the Value of Wide-angled Immersion 

 Objectives;' contained in the "Journal of the Koyal Microscopical 

 Society," for April and June, 1881. 



It can be easily demonstrated mathematically, that the ' aperture ' of 

 a single lens used as a magnifying glass that is, its capacity for receiv- 

 ing, and bringing to a remote conjugate focus, the rays emanating from 

 the axial point of an object brought very near to itis determined by 

 thenitio between its absolute diameter (or clear ' opening') and its focal 

 length; while that of an ordinary Achromatic Objective, composed of 

 several lenses, is determined by the ratio of the diameter of its back lens 

 (so far as this is really utilized) to its focal length. This ratio is most 

 simply expressed, when the medium is the same, by the sine of its semi- 

 angle of aperture (sin u); and we hence see how different are the propor- 

 tionate ' apertures ' of different lenses from their proportionate ' angles 

 of aperture.' For as the sine of half 180, the largest possible theoreti- 

 cal angle, whose two boundaries lie in the same straight line, is equal 

 to radius, and as the sine of half 60 is equal to J- radius, it follows that 

 a lens having an angle of 60 has an aperture equal to half (instead of 

 being only one-third) of the theoretical maximum. And as the sines of 

 angles beyond 60 increase very slowly, an objective whose angle is 120 

 will have (instead of only two-thirds) as much as about 87-100ths of the 

 aperture given by the theoretical maximum. 



When, however, the medium in which the Objective works is not air, 

 but a liquid of higher refractive index such as water or oil an addi- 

 tional circumstance has to be taken into consideration; for we may now 

 have three angles of aperture expressed by the same number of degrees, 

 which yet denote quite different 'apertures.' For instance, an 'angle' 

 of 90 in oil will give a greater f aperture ' than one of 90 in water; and 

 the latter a greater aperture than 90 in air. For since, when light is 

 transmitted from any medium into another of greater refractive index 

 ( 1), its rays are bent towards the perpendicular, the rays forming a 

 pencil of given angular extension in air, will, when they pass into water 

 or oil, be closed-together or compressed; so that in comparing (for 

 instance) an object mounted in balsam with one mounted dry, the 

 balsam angle, though much reduced, may nevertheless contain all the 

 rays that were spread-out over the whole hemisphere when the object was 

 in the less dense medium. It follows, therefore, that a given ' angle ' 

 in oil or water represents an increase in ' aperture ' over the same angle 

 in air. The amount of this increase having been determined by Prof. 

 Abbe to be proportional, in each case, to the index of refraction of the 

 interposed medium, the comparative ' apertures ' of lenses working in 

 different media are in the compound ratio of two factors, the sines of 

 their respective semi-angles of aperture, and the refractive indices of 

 the interposed fluids. 



It is the product of these (n sin u) that gives what is termed by Prof. 

 Abbe the Numerical Aperture; which serves, therefore, as the standard 

 of comparison not only between ' immersion ' and ' dry ' objectives, but 

 also between objects of like kind. For, when the medium is the same, 

 the factor (w) which represents the refractive index may, of course, be 

 neglected; the 'numerical apertures' of such objectives then being 

 simply the sines of their respective semi-angles. 



Thus, taking as a standard of comparison a ' dry ' objective of the 

 maximum theoretical angle of 180, whose ' numerical aperture ' is the 



