"I see, And another line would make a third dimension."
"Don't go too fast. Your third line is no use. If I want to show the position
of any point on the paper, I can do it by reference to these two lines only.
Make a point, and I'll show you."
She obeyed.
"Now, I draw lines from your point to make right angles with my lines. I
say your point is so far east of the central point, and so far north. You see? I
determine the position by only two measurements."
"But if I made my point right in the air here?"
"Exactly. We need a third line, but it must be at right angles to the other
two; sticking straight up, as you might say. Then we can measure in three
directions, and determine the point. It is so far east, so far south, and so high."
"Yes."
"Now I'll go over that again in another way. [56]
"Here is a point, not long nor broad nor thick: no dimension.
"Here is a line, long but neither broad nor thick one dimension.
Here is a surface, long and broad, but not thick: two dimensions.
"Here is a solid, long and broad and thick: three dimensions."
"Now I quite understand. But you said: four dimensions.
"I will say it presently. But just now I am going to hammer at two.
"Observe: I make a triangle. All the sides are equal. Now I draw a line
through it from one angle to the middle of the other side. I have two triangles.
They are exactly alike, as you see; same size, same shape. But — they point
in opposite directions. Now we will cut them out with scissors.
He did so.
"Slide them about, so that one lies exactly to cover the other!"
She tried and failed: then, with a laugh, turned one over, when it easily
fitted.
"Ah, you cheated. I said, 'slide them.'"
"I'm sorry."
"On the contrary, you have acted divinely, in the best sense! You took the
thing that wouldn't fit out of its world of two into the world of three, put it
back, and they all lived happy ever after!
"The next thing is this. Everything that exists — everything material — has
these three dimensions. These points and lines and surfaces have all a minute
extension in some other dimension, or they would be merely things in our
imagination. The surface of water, for instance, is merely the boundary