All the intermediate apertures from 0'^ were, of course, neces- 

 sarily calculated in obtaining these. But the insertion of them 

 does not seem to be practically of use, and might serve to confuse, 

 perhaps, rather than to assist where general results are being 

 looked for. The index for glass is taken at the common value for 

 crown glass, i. e. 1'525 ; for water, 1'336, the relative index being 

 therefore 1*141. And in the tabular form it is necessary to 

 observe that the total emitted light for the full cone of 180° is 

 taken, for convenience of fractions, not as unity, but 1000. The 

 nearest whole numbers only are given. On this scale, then, of 

 1000 we have the amount of light actually in work for any angle, 

 and can compare it either with a higher angle or with the same 

 angle under different conditions. Above 170° the figures are 

 suppressed intentionally, because for such extreme apertures the 

 measurements are doubtful and uncertain in the highest degree, if 

 indeed such apertures can be said to have a real existence at all. 

 It was, in feet, in attempting to form a judgment of this that I 

 was first led to this calculation ; and in continuing it on above 170^ 

 we are led to see by theory what may be verified easily in practice, 

 the worthlessness of all professions of exactness in such extreme 

 apertures. They are purely paper angles. And it is foolish and 

 absurd in the highest degree for opticians to profess this extreme 

 exactness in their lists, although they can scarcely be aware of the 

 extent to which the absurdity reaches. One optician has on his list, 

 let us say, the highest angle as 170° ; another has 175°; and then, 

 when we read in some other list 176°, we think, of course, that this firm 

 has beaten the last — very little indeed, — by a head and neck only ; — 

 but still beaten them. Now what is the fact ? Suppose a ray at 

 this obliquity could be got at ; how much of it could we finally use ? 

 In such a case it must pass necessarily through a cover ; and if we 

 try, by continuing this table, how much we arrive at as efiective, or 

 actually transmitted into the front — what " percentage " e. g. of the 



