USEFUL AND EMPTY MAGNIFICATION IN THE MICROSCOPE. 
47 
USEFUL AND EMPTY MAGNIFICATION IN THE MICROSCOPE. 
Experience has shown that the normal eye is able to resolve details in an 
object which are separated by slightly less than i' of arc. If, therefore, 
the angular distance between two points in the image is 2' or more, they are 
clearly distinguishable; if this distance be much less than 2' they do not 
appear distinctly separate and the limit of useful efficiency of the micro- 
scope has been reached. The angular size 5 of the image details depends 
directly on the magnification by the microscope. If an object e be viewed 
through the microscope, its apparent size 5 is then 
IT 
= e.V= 
(10) 
where V is the magnifying power of the instrument and A T the magnification 
of the instrument for the distance of normal vision / as defined on page 41. 
Combining equations (9) and (10) we find 
V\ N\ 
2dl 
or 
2a 
.. ... 
V = and N = 
X 
(12) 
From equation (12) it is evident that the limit of normal magnification 
for the microscope varies with the numerical aperture a of the objective 
and inversely with the wave-length of light used. In table i* the 
size e of the details distinguishable for the different numerical apertures 
and the limits of useful magnification are listed on the assumption that 
5 = 2' and X = 0.00055 Him. =0.55/1; /' is the equivalent focal length (E. F.) 
of the microscope. 
TABLE i. 
a = n sin 
e 
N 
f 
M 
mm. 
0. 10 
2-75 
53 
4 72 
0.30 
0.92 159 1.58 
0.60 
0.46 317 0.79 
0.90 
0.31 476 0.52 
I .20 0.23 635 0.39 
I .40 
o. 19 
74' 
o 34 
I .60 
0.17 
847 
0.30 
The higher the magnifying power used, therefore, the greater must be the 
numerical aperture of the objective for satisfactory definition. Low-power 
objectives have low numerical apertures, but high-power objectives must 
have correspondingly higher numerical apertures. As a rule it may be 
stated that with each increase of 50 in magnifying power the numerical 
aperture of the objective should be increased about o . i . 
As indicated on page 41, equation (23), the magnification of the micro- 
scope for the distance of distinct vision may be considered to be due to two 
*S. Czapski. Theorie d. Opt. Inst.. ad ed., 355. 1904. 
