A being who was on the line might not perceive the fact of this bending, and it might not affect the measurements he made. But if the line ran into itself again, and he found that he was moving on what we should call a circle, this would in no way affect his idea of space. He would recognize that what he called space, namely, his line, was not space, but a curved thing in 2 space.
Similarly, taking a plane—this is by definition not curved in any way, known or unknown, and it can only be conceived to be bent by ampler space being conceived, and its being imagined as having force applied to it so as to become a bent thing in this ampler space. In this case the term “plane” is not the correct name.
And so about our three-dimensional space; we cannot be robbed of that idea, although it might conceivably be proved that our earth and our whole universe were on a curved thing in 4 space.
We will then keep the term “space” for the ordinary conception; and call it 1, 2, 3, 4 space, according to the number of supposed independent directions.
A curved line or surface or solid we will call a 1, 2, or 3 thing, according to the number of dimensions in it.
A straight line is a 1 thing possible in 1 space. A circle is a 1 thing possible in 2 space. At any point of it a being in it is limited to motion in one direction, while the circle itself involves two dimensions. The surface of a sphere is a 2 thing possible in 3 space. The rind of an orange, or the orange itself, is a 3 thing possible in 3 space.
It will be observed that the surface of the sphere, although only a 2 thing, involves the conception of 3 space, and cannot be understood without the use of the idea of 3 space. It is a 2 thing because at any point of the surface a being can only move in two independent directions. A crooked line drawn on the surface of a sphere is a 1 thing in a 2 thing in 3 space.
Another very common misconception is occasioned by the use of a figure of this kind
to represent a “knot” in 2 space.