The task before us is difficult. We have to make up from the outside what the appearances of a higher space existence are to us in our space, and then we have to look at the facts of nature and see if they correspond to these appearances.
Let us take a few isolated points and look at them patiently.
To a being standing on the rim of a plane world a straight line absolutely shuts out the prospect before him. If the straight line is infinite it cuts his world in two; he can never hope to get beyond it.
It is to him what an infinite plane would be to us, stretching impassably in front of us, cutting us off from all that lies on the other side.
But we know that a point can move round this line. It can revolve round it by going out of the plane, and coming down again into the plane on the other side of the line.
This movement would be inconceivable to a plane being; for he can only conceive it possible to get to the other side of the line by going to the end of it and coming back along the other side of the line.
Now take a piece of paper and put a dot right in the middle and suppose that it has no means of passing through the paper. We can only conceive the dot getting to the other side of the paper by passing round the edge and coming back again to the position underneath where it was.
But by a four-dimensional movement it can slip round the paper without going to the edge.
A set of words may help. In a plane a body rotates round a point—rotation takes place round a point. In space rotation is always round a line—the axis. In four-dimensional space rotation takes place round a plane.
To take a farther consideration of this point—a plane being can see one side or the opposite of a straight line. He can only see it in one direction or in the reverse direction. But we can look at a straight line from a direction at right angles to that in which a plane being looks at it. We can look at a straight line from points which go all round it.
Similarly, a being in four-dimensional space can look at a plane from a direction at right angles to that in which we look at it. If we try to think of this we shall imagine ourselves looking at the thin edge. But this is not what a four-dimensional being would mean. He would see the plane exactly as we see it, but it would be from a direction at right angles to that in which we look.