Page:Flatland, a Romance of Many Dimensions (1963, fifth edition).djvu/96

 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 so say, has 2 sides (for the Points of a Line may be called by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that?

I. Arithmetical.