so MICROSCOPE AND ACCESSORIES \_CH.I 



the rays actually passing from the back combination of the objectives to form 

 the real image ; he thus takes into account the medium in front of the objec- 

 tive as well as the angular aperture. The term "Numerical Aperture," 

 (N. A.) was introduced by Abbe to indicate the capacity of an optical instru- 

 ment " for receiving rays from the object and transmitting them to the image". 



\ 36. Numerical Aperture (abbreviated N. A.), as now employed for 

 microscope objectives, is the ratio of the semi-diameter of the emergent pencil 

 to the focal length of the lens. Or as the factors are more readily obtainable 

 it is simpler to utilize the relationship shown in the La Grange-Helmholtz- 

 Abbe formula, and indicate the aperture by the expression: N. A.=n sin u. 

 In this formula n is the index of refraction of the medium in front of the 

 objective (air, water or homogeneous liquid), sin u is thesine of half the angle 

 of aperture (Fig. 32, D B A). For the mathematical discussion showing that 

 the expressions 

 semi-diameter of emergent pencil =n s . ^ student is referred to the Jour . 



focal length of the lens 

 nal of the Royal Microscopical Society \ 1881, pp. 392-395, 1898, p. 363. 



§ 37. Comparison of Dry and Immersion Objectives. — For example, take 

 three objectives each of 3 mm. equivalent focus, one being a dry, one a water 

 immersion, and one a homogeneous immersion. Suppose that the dry objec- 

 tive has an angular aperture of 106 , the water immersion of 94 and the homo- 

 geneous immersion of 90 . Simply compared as to their angular aperture, 

 without regard to the medium in front of the objective, it would look as if the 

 dry objective would actually take in and transmit a wider pencil of light than 

 either of the others. However, if the medium in front of the objective is 

 considered, that is to say, if the numerical instead of the angular apertures are 

 compared, the results would be as follows : Numerical Aperture of a dry ob- 

 jective of 106 , N. A.=ra sin u. In the case of dry objectives the medium in 

 front of the objective being air, the index of refraction is unity, whence «=1. 

 Half the angular aperature is i££°=53°. By consulting a table of natural 

 sines it will be found that the sine of 53° is 0.799, whence N. K.=n or 1 X sin 

 u or 0.799=0.799.* 



*{S 38. Interpolation. — In practice, as in solving problems similar to those 

 on the following pages and those in refraction if one cannot find a sine exactly 

 corresponding to a given angle ; or if one has an angle which does not corres- 

 pond to any sine or angle given in the table, the sine or angle may be closely 

 approximated by the method of interpolation, as follows : Find the sine in the 

 table nearest the sine whose angle is to be determined. Get the difference of 

 the sines of the angles greater and less than the sine whose angle is to be de- 

 termined. That will give the increase of sine for that region of the arc for 15 

 minutes. Divide this increase by 15 and it will give with approximate accur- 

 acy the increase for 1 minute. Now get the difference between the sine whose 

 angle is to be determined and the sine just below it in value. Divide this 

 difference by the amount found necessary for an increase in angle of 1 minute 

 and the quotient will give the number of minutes the sine is greater than the; 



