246 Tur Microscope. 
placed. The proportions of the slit should be such that the 
middle of the knife should rest on the edges of the cut while ue 
cutting edge does not reach the bottom. 
APERTURE AND WORKING DISTANCE. 
BY E. GUNDLACH. 
shee aperture of a lens magnifier or objective, is its actual 
diameter; but, only the proportion of the diameter to the 
focal length is optically of importance. This proportion is given 
by the so-called angular aperture of the lens; the radius of the 
actual aperture being the szne of one-half the angle of aperture 
if the focal length be the radius of said angle. Working dis- 
tance is the distance between the object and the lens when in 
focus. If the lens were without thickness, then its focal length 
would be its working distance. But this can only be admitted 
to be practically true for glasses of comparatively small angular 
aperture. A lens of ten inches focal length and one inch 
diameter will form an image of a remote object at ten inches 
distance from the lens. So its working distance is ten inches or 
equal to its focal length. Now, suppose the lens is desired to be 
of 90° angular aperture. To make this plain we will suppose 
further, that the lens is without thickness and plane; then its 
working distance would not be reduced by its increased diame- 
ter, which would then have to be 20 inches. If we take a step 
further and increase the angular aperture of the hypothetical 
lens to 170° it would then require a diameter of 228.6 inches, or, 
twice the tangent of one half the angle of aperture, the focal 
length of the lens being the radius of the angle. But, fortu- 
nately the proper form of the equivalent of an objective is not 
a plane, but a segment of a sphere, having its center in the 
focus of the objective. Consequently, the radius of its clear 
aperture is equal to the sine of one-half the angle of aperture, 
the focal length being the radius. Thus the diameter of a ten 
inch lens of 170° aperture, spherically formed, will be only 19.9 
inches. But this spherical shape reduces the working distance 
considerably, because it brings the edge of the lens so much 
nearer the plane of the object. Hence, a lens or objective will 
have a proportionately shorter working distance the greater the 
