ard; 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 being a Figure, but as a Solid. But I will describe it to
you. Or rather not I, but Analogy.

We began with a single Point, which of course--being itself a
Point--has only ONE terminal Point.

One Point produces a Line with TWO terminal Points.

One Line produces a Square with FOUR terminal Points.

Now you can give yourself the answer to your own question: 1, 2, 4,
are evidently in Geometrical Progression. What is the next number?

I. Eight.

SPHERE. Exactly. The one Square produces a SOMETHING-WHICH-
YOU-DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE with EIGHT
terminal Points. Now are you convinced?

I. And has this Creature sides, as well as angles or what you call
"terminal Points"?

SPHERE. Of course; and all according to Analogy. But, by the way, not
what YOU call sides, but what WE call sides. You would call them
SOLIDS.

I. And how many solids or sides will appertain to this Being whom I am
to generate by the motion of my inside in an "upward" direction, and
whom you call a Cube?

SPHERE. How can you ask? And you a mathematician! The side of
anything is always, if I may so say, one Dimension behind the thing.
Consequently, as there is no Dimension behind a Point, a Point has 0
sides; a Line, if I may say, h