line circle would be complete.

Another result of the theory is that, if it be true, space, though
still unbounded, is not infinite, just as the surface of a sphere,
though without any edge or boundary, has only a limited extent of
surface. Space would then have only a certain volume--a volume
which, though perhaps greater than that of all the atoms in the
material universe, would still be capable of being expressed in
cubic miles. If we imagine our earth to grow larger and larger in
every direction without limit, and with a speed similar to that we
have described, so that to-morrow it was large enough to extend to
the nearest fixed stars, the day after to yet farther stars, and
so on, and we, living upon it, looked out for the result, we
should, in time, see the other side of the earth above us, coming
down upon us? as it were. The space intervening would grow
smaller, at last being filled up. The earth would then be so
expanded as to fill all existing space.

This, although to us the most interesting form of the non-
Euclidian geometry, is not the only one. The idea which
Lobatchewsky worked out was that through a point more than one
parallel to a given line could be drawn; that is to say, if
through the point P we have already supposed another line were
drawn making ever so small an angle with CD, this line also would
never meet the line AB. It might approach the latter at first, but
would eventually diverge. The two lines AB and CD, starting
parallel, would eventually, perhaps at distances greater than that
of the fixed stars, gradually diverge from each other. This system
does not admit of being shown by analogy so easily as the other,
but an idea of it may be had by supposing that the surface of