Fundamental Propositions in Optics. 473 



so placed that the emergence is more nearly grazing than the 

 incidence, is readily traced*. 



There is, of course, no limit, either in telescopes or in 

 microscopes, to the magnifying which may be obtained by 

 sufficiently diminishing the section of the emergent pencil ; 

 but, as is now well known, the resolving power cannot thus 

 be indefinitely augmented. It is interesting to note that 

 Smith was aware of the fact, though he could have no know- 

 ledge of the reasons for it. He points out that in examining 

 objects of great intrinsic brightness, there would appear to be 

 advantage in diminishing the aperture of the object-glass. 

 For by this means the disturbing influence of aberration, both 

 spherical and chromatic, would be mitigated. " But f in 

 reality it is quite otherwise ; and that for two reasons. First 

 because the minute parts . . . may be better discerned when 

 all the light remains in the telescope than when it is reduced 

 to T lo part, though not in the same proportion. The other 

 reason is that when the aperture is too much contracted the 

 outlines that circumscribe the pictures in the eye become con- 

 fused, which is carefully to be minded, and also what are the 

 limits of this confusion. This is certain, that as the aperture 

 is contracted the slender pencils or cylinders of rays that 

 emerge from the eye-glass into the eye are also contracted in 

 the same proportion. Now if the breadth of one of these 

 pencils be less . . . than -^ or ^ part of an inch, the out- 

 lines of the pictures are spoiled, for some unknown reason in 

 the make of the eye . . . For by looking through a hole in 

 a thin plate, narrower than -J- or -J of a line, the edges of 

 objects begin to appear confused, and so much the more as 

 the hole is made narrower." If we assume that a given ap- 

 parent definition requires a given diameter of emergent pencil, 

 it follows that the resolving power of telescopes is propor- 

 tional to aperture. 



The theory of the microscopic limit has been much dis- 

 cussed in recent years, and has been placed upon a satisfactory 

 basis by Abbe, and by v. Helmholtz, who treats the subject in 

 his usual masterly style. But I think that due credit has 

 not been given to Frauenhofer, whose argument appears 

 substantially correct. In his discussion on gratings Frauen- 

 hofer X remarks: — "Es istnicht wohl denkbar, das die Politur, 

 welche wir durch Kunst auf Grlas etc. hervorbringen konnen, 

 mathematisch vollkommen sey. Besteht diese Politur aus 



* See " Investigations in Optics," Phil. Mag. January 1880. 

 f Book 2, Oh. 7, p. 144. Smith is here expounding the views of 

 Huyghens (Dioptricd). 

 X Gilbert, Ann. 1823, p. 337. 



