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