PROCEEDINGS OF THE SOCIETY, 167 



it has misled him into assuming that equal angles denote optical 

 equality, and that having 180° of radiation in air, that must be the 

 whole, and that there can be nothing more. There is not one point of 

 view, however, from which apertures can be compared quantitatively 

 by means of the angles, because there is not one function in the per- 

 formance of the Microscope in which twice the angle represents a 

 double effect, or three times the angle a triple effect, except when the 

 angles do not exceed a few degrees. Neither the quantity of light, 

 nor the resolving power, nor any other performance connected with 

 the aperture is increased in the ratio of 1 : 2 if the angle is increased 

 from 60° to 120°. 



" Thus, if a dry, a water-immersion, and a homogeneous-immersion 

 objective are respectively denoted as of 60°, 53°, and 48° angular 

 aperture, a further calculation is necessary for the appreciation of 

 their true relations inter se. 



" Aperture is only correctly defined when the various refractive 

 indices of the media are taken into account as one of the factors, and, 

 further, as two objectives with the same medium cannot be compared by 

 the degrees, it is readily seen that the sine and not the number of the 

 degrees must be the other factor, and the complete formula becomes 

 n sin w = a, where n is the refractive index of the medium, w the 

 sine of half the angle, and a the numerical aperture. Angular aper- 

 ture was an insufficient expression when only dry objectives existed, 

 and it has continued to be used since the introduction of water- and 

 oil-immersion objectives simply from force of habit. If, however, all 

 three classes of objectives had always co-existed, is it conceivable that 

 any one would have established a notation ichich denoted things as the 

 same when they were different (60° in air and 60° in oil), and different 

 when they were the same (180° in air and 82° in oil) ? 



" Although it has already been so fully done by Professor Abbe in 

 the paper to which I have already referred, as well as in the further 

 paper now awaiting publication,* I may summarize (entirely from these 

 papers) the advantages of the expression of numerical over angular 

 aperture. 



"1st. It introduces an absolute measure of aperturs, the unity of 

 which is based on a fundamental phenomenon. This is the radiation 

 of a luminous point in a medium of the refractive index = 1, the rays 

 in which are embraced by an entire hemisphere. The capacity of an 

 objective for collecting the total hemisphere in air is the unit of aper- 

 ture with which, by the value of a, every individual aperture may be 

 compared. 



" 2nd. It is the only one which affords a correct idea of the true 

 relation of different apertures in the same medium; whilst, on the 

 contrary, the angles inevitably lead to a false conception of this 

 relation. 



" 3rd. It is the only way in lohich apertures pertaining to different 

 worJcing media can be compared. A comparison by the angles is 



* The paper of Professor Abbe, which the Council have decided to publish in 

 a separate volume when complete, constitutes the most exhaustive, and indeed the 

 only, scientific exposition of the function of aperture in microscopical vision which 

 has been attempted in this or any other country. 



