s millions of times that of light,
we might at length find ourselves approaching the earth from a
direction the opposite of that in which we started. Our straight-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 "flat-land,"
instead of being spherical, is saddle-shaped. Apparently straight
parallel lines drawn upon it would then diverge, as supposed by Bolyai.
We cannot, however, imagine such a surface extended indefinitely
without losing its properties. The analogy is not so clearly marked as
in the other case.

To explain hypergeometry proper we must first set forth what a fourth
dimension of space means, and show how natural the way is by which it
may be approached. We continue our analogy from "flat-land" In this
supposed land let us make a cross--two straight line