APPENDIX. 327 



APPENDIX. 



'NUMERICAL APERTURE' AND 'ANGULAR APERTURE.' 



THE introduction of the ' immersion system ' has rendered necessary a 

 considerable modification in the mode of determining the real ' Aper- 

 tures ' of Achromatic Objectives; which were formerly estimated entirely 

 by their respective ' angles of aperture/ such angles being (as formerly 

 explained, 10), those contained, in each case, between the most 

 diverging of the rays issuing from the axial point of an object, that can 

 enter the lens and take part in the formation of an image. A careful 

 investigation of the whole subject of ( Aperture,' both theoretically and 

 practically, has of late been carried out with the greatest ability by Prof. 

 Abbe, of Jena; of whose important discovery of the dependence of ' re- 

 solving power' upon dif fraction not refraction an account has been 

 already given ( 157). This investigation has enabled him to place the 

 question on an exact basis; and not only to clear up a great deal that was 

 formerly obscure, but to formulate a definite principle for the compari- 

 son of ' immersion ' with ' dry ' or ' air ' objectives, which shows that the 

 advantages obtainable from the use of the former are much greater than 

 had been previously conceived. 



Prof. Abbe has also made an important contribution to the practical 

 part of this inquiry, by the invention of an ' Apertometer ' for the pre- 

 cise measurement of angular apertures, 1 by which more exact and definite 

 results can be obtained than by any of the methods previously in use: 

 and he has further shown that a comparison of ' dry ' and of ' immer- 

 sion ' lenses by their respective ' angles ' alone is so completely fallacious, 

 as to necessitate the introduction of a new scale of ( numerical apertures/ 

 to which, as to a common standard, both could be referred. It is the 

 object of this Addendum, in the first place, to explain to the readers of 

 this treatise the precise meaning of Prof. Abbe's term; and then to put 

 before them the new views in regard to the capacities of f immersion ' 

 Objectives, to which his investigations have led him. As (for obvious 

 reasons) conclusions only can be here stated, those who desire to master 

 the train of reasoning by which those conclusions have been worked-out, 

 are recommended to study the two most recent expositions of the doc- 

 trine; one given by Prof. Abbe himself in his Paper ' On the Estimation 

 of Aperture,' and the other by his disciple, Mr. Frank Crisp (one of the 

 Secretaries of the Eoyal Microscopical Society) in his ' Notes on Aper- 



1 " Journ. of Roy. Microsc. Soc.," Vol. i. (1878), p. 19. Another method devised 

 by Prof . Hamilton Smith (Op. cit., Vol. ii., 1879, p. 775), gives nearly the same 

 results as that of Prof. Abbe. And yet another has been proposed by Mr. Tolles 

 (Op. cit., Vol. iii., 1880, p. 887), who does not, however, give any reason to ques- 

 tion the accuracy of Prof. Abbe's instrument. 



