t was no dream. For from the depths of nowhere came forth a
hollow voice--close to my heart it seemed--"Am I quite gone? Are you
convinced now? Well, now I will gradually return to Flatland and you
shall see my section become larger and larger."

Every reader in Spaceland will easily understand that my mysterious
Guest was speaking the language of truth and even of simplicity. But
to me, proficient though I was in Flatland Mathematics, it was by no
means a simple matter. The rough diagram given above will make it
clear to any Spaceland child that the Sphere, ascending in the three
positions indicated there, must needs have manifested himself to me, or
to any Flatlander, as a Circle, at first of full size, then small, and
at last very small indeed, approaching to a Point. But to me, although
I saw the facts before me, the causes were as dark as ever. All that I
could comprehend was, that the Circle had made himself smaller and
vanished, and that he had now reappeared and was rapidly making himself
larger.

When he regained his original size, he heaved a deep sigh; for he
perceived by my silence that I had altogether failed to comprehend him.
And indeed I was now 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; upw