ERSCHEL was correct; it is for his
successors to look for the special cases and limitations, to answer the
question, At a certain distance from us, what are the variations which
actually take place in the brilliancy and the sizes of stars? The answer
to this question is to be found in the study of the clusters of regular
forms, where we _know_ the stars to be all at the same distance from us.

_Researches on Light and Heat, Etc._

Frequently in the course of his astronomical work, HERSCHEL found
himself confronted by questions of physics which could not be
immediately answered in the state of the science at that time. In his
efforts to find a method for determining the dimensions of the stellar
universe, he was finally led, as has been shown, to regard the
brightness of a star as, in general, the best attainable measure of its
distance from us. His work, however, was done with telescopes of various
dimensions and powers, and it was therefore necessary to find some law
for comparing the different results among themselves as well as with
those given by observations with an unassisted eye. This necessity
prompted an investigation, published in 1800, in which, after drawing
the distinction between absolute and intrinsic brightness, HERSCHEL gave
an expression for the _space-penetrating power_ of a telescope. The
reasoning at the base of this conception was as follows.

The ratio of the light entering the eye when directed toward a star, to
the whole light given out by the star, would be as the area of the pupil
of the eye to the area of the whole sphere having the star as a centre
and our distance from the star as a radius. If the eye is assisted by a
telescope, the ratio is quite different. In that case the ratio of the
light which enters the eye to the whole light, would be as the area of
the mirror or object-glass to the area of the whole sphere having the
star as a centre and its distance as a radius. Thus the light received
by the _eye_ in the two cases would be as the area of the pupil is to
the area of the object-glass. For instance, if the pupil has a diameter
of two-fifths of an inch, and the mirror a diameter of four inches, then
a hundred times as much light would enter the eye when assisted by the
telescope as when unarmed, since the _area_ of the pupil is
one-hundredth the _area_ of the objective.

If a particular star is just visible to the naked eye, it will be quite
bright if viewed