Those who are familiar with the literature of the subject know that it
has long been customary to make regarding space certain other
statements to which the plain man does not usually make serious
objection when he is introduced to them. Thus it is said:--

(1) The idea of space is _necessary_. We can think of objects in space
as annihilated, but we cannot conceive space to be annihilated. We can
clear space of things, but we cannot clear away space itself, even in
thought.

(2) Space must be _infinite_. We cannot conceive that we should come
to the end of space.

(3) Every space, however small, is _infinitely divisible_. That is to
say, even the most minute space must be composed of spaces. We cannot,
even theoretically, split a solid into mere surfaces, a surface into
mere lines, or a line into mere points.

Against such statements the plain man is not impelled to rise in
rebellion, for he can see that there seems to be some ground for making
them. He can conceive of any particular material object as
annihilated, and of the place which it occupied as standing empty; but
he cannot go on and conceive of the annihilation of this bit of empty
space. Its annihilation would not leave a gap, for a gap means a bit
of empty space; nor could it bring the surrounding spaces into
juxtaposition, for one cannot shift spaces, and, in any case, a
shifting that is not a shifting through space is an absurdity.

Again, he cannot conceive of any journey that would bring him to the
end of space. There is no more reason for stopping at one point than
at another; why not go on? What could end space?

As to the infinite divisibility of space, have we not, in addition to
the seeming reasonableness of the doctrine, the testimony of all the
mathematicians? Does any one of them ever dream of a line so short
that it cannot be divided into two shorter lines, or of an angle so
small that it cannot be bisected?

24. SPACE AS NECESSARY AND SPACE AS INFINITE.--That these statements
about space contain truth one should not be in haste to deny. It seems
silly to say that space can be annihilated, or that one can travel
"over the mountains of the moon" in the hope of reaching the end of it.
And certainly no prudent man wishes to quarrel with that coldly
rational creature the mathematician.

But it is well worth while to examine the statements carefully and to
see whether there is not some danger that they may be understood in
such