162 NUMERICAL APERTURE. 



representing its angular aperture, it is evident that its angular 



aperture is less than that of A B. This shows that focal length is 



an element in the calculation of aperture. 



If a lamp with a circular ground glass globe is looked at, the 



part perpendicular to the eye is seen to be brighter than the edges ; 



but if the sun or moon is looked at they seem equally bright all 



over, and this in spite of the fact that equal visual angles take in 



more of the surface radiating light at the edges than perpendicular 



to the eye : the investigation of these phenomena, and of light 



emitted from an infinitesimal bright surface element has given rise 



to the Lambert law, which shows that the amount of light given 



out does not depend on the ratios of the angles, but on the sines 



of their inclination to the perpendicular. For exam]:)le, take an 



angle of 60"^ and an angle of 120" round the perpendicular. 



1 2 o *^ 2 

 Then-—- =— and the amount of light is not as 2= : i ^ 4 : i ; 



but as -, = I -^ 4 = ^- , or light from 120'^ : light from 



sm- 30 I 



60^ :: 3 : I, 



All these points show that the angle of aperture is insufficient 

 of itself to give a correct notion of the performance of a lens, but 

 that some function of this angle is required. 



The word aperture simply means opening, and what we want 

 to get is the lineal opening of a lens, which lets the light through, 

 so as to be able to compare lenses together. The size of this 

 opening has already been shown to vary with the focal length; that 

 is to say, the longer the focal length the smaller the aperture for 

 the same diameter of lens, so that, if 

 = this opening, 

 / = the focal length, 

 we want to find the ratio of to/ 



The next question is. Where shall we measure this lineal 

 opening? If we measure the front lens, some of the light may 

 be cut off by diaphragms or in other ways ; that is to say, the 

 light might come in, but never get out again. The only useful 

 rays are those that get out, and as they get out into air we get rid 

 of refractive indexes. Therefore, the rays emerging from the 

 back lens are the ones to measure. There is another advantage, 



