The Helmholtz Theory of the Microscope. By J. W. Gordon. 415 



diminishing the numerical aperture and increasing the diameter of 

 the antipoint in the centre of the field in reciprocal proportion, hut 

 restoring the uniformity of illumination which the first operation 

 had disturbed. 



Between these two methods of measuring resolving power there 

 cannot therefore be, even for a moment, any question as to which 

 is the more scientific. The apertometer is wholly misleading and 

 incurably bad, for it cannot be applied under working conditions. 

 Helmholtz' method, if properly applied with due precautions, tells 

 us precisely what we want to know, namely what is the com- 

 parative capacity of a given objective for producing a finely resolved 

 ima^e under the conditions of actual use. 



The Ultimate Limit of Resolving Power. 



Coming back now to Helmholtz' paper, our author, having 

 thrown out this practical suggestion as to the rating of objectives, 

 proceeds to discuss the theoretical question, " What is the ultimate 

 limit of resolving power ? " Speaking off-hand one might be dis- 

 posed to say that since the picture formed by a lens is built up of 

 antipoints, no object smaller than an antipoint could be delineated 

 within it. And if an antipoint were a material object like the 

 grain of a photograph, so that one antipoint could not penetrate 

 another, this would be a perfectly sound conclusion. But an 

 antipoint is a kind of diffusion disc, and there is no reason in the 

 world why any number of antipoints should not occupy the same 

 space interpenetrating one another like — what in truth they are — 

 beams from contiguous lamps. 



But although Helmholtz was quite well aware that you could 

 picture, by means of antipoints, objects smaller than the antipoints 

 themselves, he thought that objects would cease to be separately 

 discernible in the picture if they were smaller than one-half tha 

 diameter of the false disc of the antipoint. 



The diagram (fig. 98, plate VI.) will illustrate his view. Hera 

 four pairs of antipoints are shown in varying degrees of propin* 

 quity, the false discs only being taken into account. Case a 

 exhibits separate discs and presents no difficulty. Obviously these 

 represent distinct objects in the picture. Case o shows the two 

 antipoints overlapping to the extent of one-fourth of the diameter 

 of each. Here experiment shows that the eye distinguishes. In 

 c and cl this overlapping of adjacent antipoints is carried farther, 

 and we may assume that in d it has been carried so far that the 

 points in the object from which they start are no longer distin- 

 guishable by the eye. They are like irresoluble double stars in 

 the heavens, At c we assume that they are in the critical position 

 and may be described as just distinguishable or just indis- 



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