inclining to the belief that he must be no Circle
at all, but some extremely clever juggler; or else that the old wives'
tales were true, and that after all there were such people as
Enchanters and Magicians.

After a long pause he muttered to himself, "One resource alone remains,
if I am not to resort to action. I must try the method of Analogy."
Then followed a still longer silence, after which he continued our
dialogue.

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. This is all in accordance with Analogy; surely it must be clear
to you.

Restraining my impatience--for I was now under a strong temptation to
rush blindly at my Visitor and to precipitate him into Space, or out of
Flatland, anywhere, so that I could get rid of him--I replied:--

"And what may be the nature of the Figure which I am to shape out by
this motion which you are pleased to denote by the word 'upward'? I
presume it is describable in the language of Flatland."

Sphere. Oh, certainly. It is all plain and simple, and in strict
accordance with Analogy--only, by the way, you must not speak of the
result as