THE FOCUS-APERTURE RATIO. 287 



But it is not always advisable to adhere to a theoretical limit ; 

 we ought to be satisfied with something less, and not try to work 

 up to a breaking-point. After half a century's almost constant 

 work with a microscope, and an acquaintance with an enormous 

 number of all kinds of objectives, I think the Club will not object 

 to my bringing to its notice a power-aperture curve, derived 

 from my own practical experience. First, let me point out that 

 a tabular list of objectives giving their foci, powers, numerical 

 apertures, and optical indices is of little value, for the mind 

 cannot take it in at a glance. A curve must be drawn from the 

 tabular data and exhibited to the eye for judgment. Now, it 

 wiU be perfectly clear to all our members that a curve, drawn 

 upon any plan at all, must be a smooth curve. For example, 

 the graph of the plan just dealt with, having an optical index 

 of 25, would with ordinates powers and abscissae N.A. be repre- 

 sented by a straight line, drawn at an angle to the base, with 

 a tangent of 04, or 22|°. This angle is small, because the plan 

 is based upon a high ratio ; if, however, we were content with 

 a lower ratio, say N.A. 0-2 for each additional 10 in initial 

 magnifying power — that is, optical index 20 — the straight line 

 would make the larger angle of 26|° to the base, and with optical 

 index 10 it would be 45°. We can then tell at once, from the angle 

 the line or curve makes with the base, whether the graph is that 

 of a wide or narrow angled scheme. (A straight Line making an 



tangents of small angles are nearly alike. With wide angles, however, 



an error would come in. The true value of x is ^^^ . . Example : — The 



2N.A. 



semiaperture angle of a lens //4 is tan 1/8 = 0-125, this is 7° 7' 30" ; but 



the true aperture of that lens should be measured by the sine or 0-124, 



etc. Now, as the rapidity of lenses is as the squares of their N.A.s, a 



lens of double the N.A. would have four times the speed ; but double 



sine 0-124, etc., is equivalent to an angle of 14° 21' 48", which represents 



//1-9525, and this is the true value of a lens having fovir times the speed 



of one of //4. By the photographer's scale, calculated from the tangent, 



we have //2, or tan 1/4, which is an angle of 14° 2' 10", and this is too 



small. Therefore, a lens //2 has not four times the speed of one of 



//4. The diameter of a photographic lens, 8 in. focus, having four times 



the speed of a lens //4, according to a photographer's scale, is //2, or 



4 in., whereas its true value is 4-1 in. The visual resolving power of a 



lens is 45287/a;, and its photographic 58494/.r lines to the inch. (See 



my table of correct apertures for U.S. numbers in The British Journal 



oj Photography , January 12th, 1891.) 



