ON  E___<_—_—_<__£_ ea a ee eee ee 
THE THEORY OF APERTURE IN THE MICROSCOPE, 327 
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of the full moon, and that of a planet through a telescope, to have 
greatly increased brilliancy from the centre to the circumference, 
because in any equal small apparent areas of their surfaces the 
number of real surface-elements emitting light must increase 
rapidly as the areas recede from the centre, and enormously as 
they approach the circumference. This is not the case ; and the 
reason why this increase of brightness from the centre does not 
take place is to be found in the decrease of intensity of the emitted 
rays in the ratio of the cosine of their obliquity from the line of 
perpendicular emission. This principle is illustrated in Fig. 37, 
where the two theories are compared with reference to pencils 
containing equal quantities of light emitted from a luminous hemis- 
phere, such as that of the moon, ina parallel direction towards 
the eye. It will be observed that the Jalse theory of angular 
equivalents of light gives rise to pencils decreasing in diameter 
towards the sides of the hemisphere, and therefore of greater in- 
tensity in that direction ; whilst the ¢rue theory of angular equiva- 
lents results in equal pencils of therefore equal intensity. 
Angular aperture is thus shown not to bea scientific method 
of formulating vea/ aperture, and we are left to seek it in a more 
correct notation, bearing the expression of the relations of the 
sines of angles to each other. This expression we find in the 
system of “numerical aperture” tabulated by the Royal Micros- 
copical Society, and now published on a fly-leaf of their Journal, 
together with other interesting matter connected with it. In this 
system, vadius, or the sine of the semi-angle of 180°, is taken as 
unity or 1.0; so that all apertures in air must be less than 1.0 in 
proportion as the sines of their semi-angles are less than radius. 
For instance, the angle of 140° in air has a semi-angle of 70°, the 
sine of which is .9397, or .94 nearly, which is the xumerical aperture 
of 140° in air. A useful table of natural sines, which will enable 
anyone to convert angular into numerical aperture almost at a 
glance, will be found in Mr. G. E. Davis’s “ Practical Microscopy,” 
chapter IX., and in chapter V. there is a copy of the numerical 
aperture table of the Royal Microscopical Society. 
Those who wish to know something of the controversy which 
has taken place between the old school of angular aperturists ” 
and the new school of “ numerical aperturists,” especially if they 
have any lingering scepticism as to the superiority of the new 
system of notation, should read the very able article in the Journal 
of the R. M.S., for 1881, p. 303, written by Mr. Frank Crisp, 
B,A., LL.B., Hon. Sec. R.M.S., from notes by Prof. Abbe and 
others in possession of the Society, and forming a most convincing 
argument in support of the new theory and in refutation of the 
old one. 
So far we have considered the case of angles in air only, ze, 
