escope we have to deal with a linear
measure of aperture and an angular limit of resolution, whereas in the
case of the microscope the limit of resolution is linear, and it is
expressed in terms of angular aperture.

It must be understood that the above argument distinctly assumes that
the different parts of the object are self-luminous, or at least that
the light proceeding from the various points is without phase
relations. As has been emphasized by G. J. Stoney, the restriction is
often, perhaps usually, violated in the microscope. A different
treatment is then necessary, and for some of the problems which arise
under this head the method of Abbe is convenient.

The importance of the general conclusions above formulated, as
imposing a limit upon our powers of direct observation, can hardly be
overestimated; but there has been in some quarters a tendency to
ascribe to it a more precise character than it can bear, or even to
mistake its meaning altogether. A few words of further explanation may
therefore be desirable. The first point to be emphasized is that
nothing whatever is said as to the smallness of a single object that
may be made visible. The eye, unaided or armed with a telescope, is
able to see, as points of light, stars subtending no sensible angle.
The visibility of a star is a question of brightness simply, and has
nothing to do with resolving power. The latter element enters only
when it is a question of recognizing the duplicity of a double star,
or of distinguishing detail upon the surface of a planet. So in the
microscope there is nothing except lack of light to hinder the
visibility of an object however small. But if its dimensions be much
less than the half wave-length, it can only be seen as a whole, and
its parts cannot be distinctly separated, although in cases near the
border line some inference may be possible, founded upon experience of
what appearances are presented in various cases. Interesting
observations upon particles, _ultra-microscopic_ in the above sense,
have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy
(_Drude's Ann._, 1903, 10, p. 1).

In a somewhat similar way a dark linear interruption in a bright
ground may be visible, although its actual width is much inferior to
the half wave-length. In illustration of this fact a simple experiment
may be mentioned. In front of the naked eye was held a piece o