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.
Pages:
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223