ZOOLOaY AND BOTANY, MICROSCOPy, ETC. 321 



Binall balsam-pencil of 82^, yet has some virtue that prevents the 

 latter being treated as its equivalent. We are entitled to ask ivhat 

 this virtue is, and to be shown that it is not a mere fancy. If it is 

 asserted that there must necessarily be a loss in passing from 180^ 

 air-angle to 82° balsam-angle (a large difference in angular extension), 

 surely this loss can be shown or deiined? At least, some intelli- 

 gible explanation can be given of its essence and existence — of the 

 optical theory on which it is based or the experiments by which it is 

 supported ? 



(8) Numerical Aperture — Having now shown that in whatever 

 way the matter may be regarded the expression of the degrees in 

 the angles is not a correct method of comparing apertures, and 

 that 180° in air is not a maximum, we are in a position to resume 

 the consideration of the notation to be adopted for the proper estima- 

 tion of aperture, and this it will be seen is essentially numerical 

 aperture, which is however supposed by the angular aperturist to be 

 a fanciful notation, not founded on any known natural phenomenon, 

 so that whether a person adopts it, or continues to use the expression 

 of " angular " aperture, is purely a question of taste, like the adoption 

 of the Fahrenheit or Centigrade scales for the thermometer. 



Aperture, in its true and legitimate meaning of " opening," 

 depends, as we have seen, on the ratio between the clear opening of the 

 objective and the power — a ratio which increases progressively from 

 the lowest angular aperture of a dry objective to the highest angular 

 aperture of an oil-immersion objective. The expression for the ratio 

 of the semi-diameter of the emergent pencil to the focal length is 

 n sin u, n being the refractive index of the medium and m the semi- 

 angle of aperture. It is simply this expression which is the numerical 

 aperture, and which is therefore the true measure of the relative 

 apertures of objectives of all kinds. 



We have also seen that whether we consider the amount of light 

 in the pencils, the number of rays in the plane angle, or the resolving 

 power, it is n sin u (or its square) which affords the only correct 

 comparison. 



The expression n sin u. 



We add here one of the forms of the deduction of the expression 

 n sin u which, though not the most strict, is one of the simplest. It 

 establishes two points. 1st. That the angle alone (2 w) can never 

 correctly define aperture, for equal apertures always require equal 

 values of n sin w, and different apertui'es different values ; so that 

 this expression (i. e. a function of the semi-angle compounded with 

 the refractive index of the medium, and not the angle itself) is neces- 

 sarily the adequate measure of aperture in general ; and 2nd, that a 

 dry objective cannot have so large an apertiu-e as a wide-angled 

 immersion objective. 



Let Fig. 63 represent any objective collecting a pencil exceeding 

 82° in balsam with any desired amplification of the image, and any 

 desired distance of the image (i. e. length of tube), let ?t be the semi- 

 angle of the admitted pencil within a medium of refractive index n 

 ( = 1 • 5 if it is oil), N the amplification of the image which is always 



