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 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?