326 SUMMARY OF CURRENT RESEARCHES RELATING TO 



0-60, 0*90, and 1-30, are given, and it is seen also by how much 

 they fall short of, or exceed 1 (= 180^ angular aperture in air). 

 Compare with these figures those denoting the angular aperture ! 



" Numerical aperture," therefore, so far from being a fanciful 

 arbitrary notation, expresses the plain fact that we want to under- 

 stand, viz. how does the relative " opening " of any given objective — 

 i. 6. its aperture or the capacity of the objective for receiving rays 

 from an object — stand in relation to that of another objective (whether 

 dry, water-immersion, or oil-immersion) ? Has it the same opening, 

 or a larger or smaller one ? 



II. Angular-Aperture Fallacies. 



We have now dealt with the insufficiency of the angular-aperture 

 theory from all the points on which it has been attempted to base it : 

 showing, 1st, the fallacy of the photometrical test and of the supposed 

 identity of the hemispheres in different media, so that 180° in air 

 does not represent the " whole " contended for ; 2nd, that i-esolving 

 power is not proportional to the angles nor attains a maximum with 

 180° in air ; 3rd, that there is no virtue in greater or less "angular 

 grip " ; and dth, that numerical aperture is the only scientific nota- 

 tion for the comparison of the " apertures " of objectives, using the 

 term in its true sense as the capacity of an objective for admitting 

 rays from an object. 



We propose now to deal with some special fallacies of angular 

 aperture which could not be conveniently included in the preceding 

 pages, and the following notes meet some of the difficulties which 

 we have from time to time met with in discussing the aj^erture 

 question. They are none of them imaginary, but all have actually 

 occurred, and not only so, but have been really felt to be difficulties. 

 It is easy to be surprised, after the explanation is given, that they 

 could have appeared as " difficulties " ; but as they have occurred more 

 than once already, they are very likely to occur again, and we do not 

 think therefore that the space devoted to them is wasted. 



(1) The Hemisphere Puzzles.— 



(a) The Convex Hemispliere. 



This puzzle is now hung up in the Society's Library, and will no 

 doubt be a source of wonder to microscopists of the future that their 

 forefathers could ever have been puzzled by it. It has been sug- 

 gested in various forms, the most notable of which is the following. 



The cardinal fallacy of the angular aperturist has always been the 

 idea that when an object is mounted in balsam the aperture of the dry 

 objective is " cut down." Thus the aperture of a dry objective of say 

 a + inch of 90° (air-angle) used upon an object in air would, it was 

 thought, be largely " cut down" if the object were placed in balsam, 

 which would reduce the angle at the radiant to say 55°, the air above 

 the cover-glass preventing part of the rays formerly emitted from 

 emerging.* The aperture of a ;^-inch immersion objective of 90° 



* Cf. Figs. 61 aud 02. 



