measures are made with the
spectroscope. Unfortunately, they can be best made only on the brighter
stars--becoming very difficult in the case of stars not plainly visible
to the naked eye. Still the motions of several hundreds have been
measured and the number is constantly increasing.

A general result of all these measures and of other estimates may be
summed up by saying that there is a certain average speed with which
the individual stars move in space; and that this average is about
twenty miles per second. We are also able to form an estimate as to
what proportion of the stars move with each rate of speed from the
lowest up to a limit which is probably as high as 150 miles per second.
Knowing these proportions we have, by observation of the proper motions
of the stars, another method of estimating how thickly they are
scattered in space; in other words, what is the volume of space which,
on the average, contains a single star. This method gives a thickness
of the stars greater by about twenty-five per cent, than that derived
from the measures of parallax. That is to say, a sphere like the second
we have proposed, having a radius 800,000 times the distance of the
sun, and therefore a diameter 1,600,000 times this distance, would,
judging by the proper motions, have ten or twelve stars contained
within it, while the measures of parallax only show eight stars within
the sphere of this diameter having the sun as its centre. The
probabilities are in favor of the result giving the greater thickness
of the stars. But, after all, the discrepancy does not change the
general conclusion as to the limits of the visible universe. If we
cannot estimate its extent with the same certainty that we can
determine the size of the earth, we can still form a general idea of it.

The estimates we have made are based on the supposition that the stars
are equally scattered in space. We have good reason to believe that
this is true of all the stars except those of the Milky Way. But, after
all, the latter probably includes half the whole number of stars
visible with a telescope, and the question may arise whether our
results are seriously wrong from this cause. This question can best be
solved by yet another method of estimating the average distance of
certain classes of stars.

The parallaxes of which we have heretofore spoken consist in the change
in the direction of a star produced by the swing of the earth from one
side of its orbit