108 



THE AMERICAN MONTHLY 



[June, 



eliminated which have enveloped the 

 consideration of the admitted pencil, 

 which may be in air, water, oil, or 

 other substances of various refractive 

 indices." 



The subject of aperture is not, in 

 reality, a difficult one, and any intri- 

 cacy in which it may seem to be in- 

 volved Avill be found to arise from the 

 necessity of clearing away some 

 of the old entanglements, such as the 

 curious mistake involved in the "hemi- 

 sphere puzzle " and similar matters. 

 Looked at de novo, there are two sim- 

 ple stages in the aperture question. 



(i ) To appreciate that, in using the 

 term "aperture," we use it not in any 

 artificial sense, but as meaning open- 

 ing and nothing else, — defining, sim- 

 ply, the capacity of an objective for 

 receiving rays from the object and 

 transmitting them to the image. 



(2) That the aperture (as so defin- 

 ed) of an objective is determined by 

 the ratio between the diameter of the 

 emergent beam and the focal length 

 of the objective. According as this 

 ratio is greater or less, so the object- 

 ive will receive and transmit a larger 

 or smaller portion of the total quanti- 

 ty of rays presented to it. 



The emergent beam of an air-ob- 

 jective of 180° angle cannot exceed 

 in diameter twice the focal length ; 

 that of a similar water-immersion ob- 

 jective may be one-third larger, and 

 of an oil-immersion half as large 

 again, and the relative capacities of 

 such objectives (with equal angles) to 

 receive and transmit rays will always 

 be as I, lYz and i^. 



It cannot be too carefully borne in 

 mind that it is not a question of this 

 or that theory, but the ordinary laws 

 of geometrical optics which determine 

 that, all other things being equal, one 

 objective will receive and transmit a 

 greater quantity of light than another, 

 and therefore has the larger or small- 

 er aperture, according as the diame- 

 ter of the beam emerging from it is 

 greater or smaller. 



As Fellows of this Society we may, 

 I think, be proud of the able commu- 



nications relating to this subject, 

 which were published in the April and 

 June numbers of the journal. 



Numerical Aperture. — The 

 abandonment of the angular notation 

 for aperture necessarily follows, as 

 soon as the correct view of aperture 

 is appreciated ; for when we know 

 that the apertures of three objectives 

 are, for instance, as 98, 126, and 138, 

 no one would insist that they should 

 be designated 157°, 142'', and 130^. 

 A notation can have no title to be 

 considered a scientific one, which 

 denotes things as the same when they 

 are really different (60'' in air and 

 oil) or different when they are the 

 same (180"^ in air and 82'' in oil). 



Until, however, the " law of apla- ■ 

 natic convergence " had been demon- 

 strated by Professor Abbe, no prin- 

 ciple had been established by which 

 the ratio between emergent beam and 

 focal length, could be conveniently 

 denoted. 



It would not be possible for me to 

 condense, without a sacrifice of intel- 

 ligibility, the steps by which he sub- 

 sequently showed, in a very beautiful 

 manner, that the ratio in question 

 can be expressed by the product of 

 the refractive index of the medium 

 in front of the objective, and the sine 

 of half the angle of aperture, that is, 

 by n sin. u. 



Taking for our unit the capacity of 

 an objective for collecting the whole 

 hemisphere of rays from an object in 

 air {i. e. the case of a dry objective 

 of 180° angle) we obtain the " numeri- 

 cal " notation, which, commencing 

 with the lowest numbers advances as 

 far as 1.52 with oil-immersion ob- 

 jectives, and by the use of which not 

 only are apertures compared in the 

 same medium, but in different media 

 also, and we see whether they are 

 smaller or larger than the maximum 

 of a dry objective. 



It is gratifying to find that reproach 

 hitherto attaching to microscopists 

 for the use of a misleading notation, 

 is, thanks to the efforts of this Socie- 

 ty, being rapidly removed, and that 



