Sphere. Tell me, Mr. Mathematician; if a Point moves Northward,
and leaves a luminous wake, what name would you give to the wake?
I. A straight Line.
Sphere. And a straight Line has how many extremities?
I. Two.
Sphere. Now conceive the Northward straight Line moving parallel
to itself, East and West, so that every point in it leaves behind
it the wake of a straight Line. What name will you give to the Figure
thereby formed? We will suppose that it moves through a distance equal
to the original straight line. --What name, I say?
I. A square.
Sphere. And how many sides has a Square? How many angles?
I. Four sides and four angles.
Sphere. Now stretch your imagination a little, and conceive a Square
in Flatland, moving parallel to itself upward.
I. What? Northward?
Sphere. No, not Northward; upward; out of Flatland altogether.
If it moved Northward, the Southern points in the Square
would have to move through the positions previously occupied
by the Northern points. But that is not my meaning.
I mean that every Point in you--for you are a Square and will serve
the purpose of my illustration--every Point in you, that is to say
in what you call your inside, is to pass upwards through Space
in such a way that no Point shall pass through the position previously
occupied by any other Point; but each Point shall describe a straight
Line of its own.
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