The Stranger Fails to Explain Spaceland’s Mysteries

As soon as the sound of the Peace-cry of my departing Wife had died away, I began to approach the Stranger with the intention of taking a nearer view and of bidding him be seated: but his appearance struck me dumb and motionless with astonishment. Without the slightest symptoms of angularity he nevertheless varied every instant with gradations of size and brightness scarcely possible for any Figure within the scope of my experience. The thought flashed across me that I might have before me a burglar or cut-throat, some monstrous Irregular Isosceles, who, by feigning the voice of a Circle, had obtained admission somehow into the house, and was now preparing to stab me with his acute angle.

As soon as my departing Wife’s Peace-cry faded away, I began to approach the Stranger. I intended to get a closer look and invite him to sit down. But his appearance left me speechless and frozen with astonishment. He showed no sign of having angles, yet his size and brightness changed from moment to moment. These shifts were scarcely possible for any Figure I had ever encountered. The thought flashed through my mind that he might be a burglar or murderer, some monstrous Irregular Isosceles. By imitating a Circle’s voice, he might somehow have gained entry to the house. Now he might be preparing to stab me with his sharp angle.

In a sitting-room, the absence of Fog (and the season happened to be remarkably dry), made it difficult for me to trust to Sight Recognition, especially at the short distance at which I was standing. Desperate with fear, I rushed forward with an unceremonious, "You must permit me, Sir—" and felt him. My Wife was right. There was not the trace of an angle, not the slightest roughness or inequality: never in my life had I met with a more perfect Circle. He remained motionless while I walked round him, beginning from his eye and returning to it again. Circular he was throughout, a perfectly satisfactory Circle; there could not be a doubt of it. Then followed a dialogue, which I will endeavour to set down as near as I can recollect it, omitting only some of my profuse apologies—for I was covered with shame and humiliation that I, a Square, should have been guilty of the impertinence of feeling a Circle. It was commenced by the Stranger with some impatience at the lengthiness of my introductory process.

In the sitting-room, the lack of Fog made it difficult for me to rely on Sight Recognition. The season happened to be unusually dry. Recognition was especially difficult at the short distance where I stood. Desperate with fear, I rushed forward without ceremony, saying, "You must allow me, Sir..." and felt him. My Wife was right. There was no trace of an angle, not the slightest roughness or unevenness. Never in my life had I met a more perfect Circle. He remained still while I walked around him, starting at his eye and returning to it. He was circular all the way around, a Circle perfect in every respect. There could be no doubt about it. A conversation followed, which I will try to record as closely as I can remember. I will leave out only some of my many apologies. I was overwhelmed with shame and humiliation that I, a Square, had been rude enough to feel a Circle. The Stranger began the conversation, somewhat impatient with how long my preliminary examination had taken.

STRANGER. Have you felt me enough by this time? Are you not introduced to me yet?

STRANGER. Have you felt me enough by now? Haven't you gotten to know me yet?

I. Most illustrious Sir, excuse my awkwardness, which arises not from ignorance of the usages of polite society, but from a little surprise and nervousness, consequent on this somewhat unexpected visit. And I beseech you to reveal my indiscretion to no one, and especially not to my Wife. But before your Lordship enters into further communications, would he deign to satisfy the curiosity of one who would gladly know whence his Visitor came?

I. Most distinguished Sir, please excuse my awkwardness. It comes not from ignorance of polite manners, but from a little surprise and nervousness at your somewhat unexpected visit. And I beg you not to tell anyone about my lapse, especially my Wife. But before your Lordship says more, would you kindly satisfy my curiosity and tell me where you came from?

STRANGER. From Space, from Space, Sir: whence else?

STRANGER. From Space, from Space, sir. Where else?

I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment?

I. Forgive me, my Lord, but aren't you and I, your humble servant, already in Space at this very moment?

STRANGER. Pooh! what do you know of Space? Define Space.

STRANGER. Nonsense! What do you know about Space? Define Space.

I. Space, my Lord, is height and breadth indefinitely prolonged.

I. Space, my Lord, is height and width extending without limit.

STRANGER. Exactly: you see you do not even know what Space is. You think it is of Two Dimensions only; but I have come to announce to you a Third—height, breadth, and length.

STRANGER. Exactly. You see, you don’t even know what Space is. You think Space has only two dimensions. But I’ve come to tell you about a third: height, breadth, and length.

I. Your Lordship is pleased to be merry. We also speak of length and height, or breadth and thickness, thus denoting Two Dimensions by four names.

I. Your Lordship must be joking. We also speak of length and height, or breadth and thickness. So we use four names for Two Dimensions.

STRANGER. But I mean not only three names, but Three Dimensions.

STRANGER. But I don't just mean three names. I mean Three Dimensions.

I. Would your Lordship indicate or explain to me in what direction is the Third Dimension, unknown to me?

I. Could your Lordship show or explain to me which direction the Third Dimension lies in? I know nothing about it.

STRANGER. I came from it. It is up above and down below.

STRANGER. I came from there. It is above and below.

I. My Lord means seemingly that it is Northward and Southward.

I. My Lord seems to mean that it is northward and southward.

STRANGER. I mean nothing of the kind. I mean a direction in which you cannot look, because you have no eye in your side.

STRANGER. I mean nothing like that. I mean a direction you cannot look, because you have no eye on your side.

I. Pardon me, my Lord, a moment's inspection will convince your Lordship that I have a perfect luminary at the juncture of two of my sides.

I. Forgive me, my Lord. A quick look will convince you that I have a perfect luminary where two of my sides meet.

STRANGER. Yes: but in order to see into Space you ought to have an eye, not on your Perimeter, but on your side, that is, on what you would probably call your inside; but we in Spaceland should call it your side.

STRANGER. Yes, but to see into Space, you need an eye on your side, not on your Perimeter. You would probably call that side your inside. But we in Spaceland would call it your side.

I. An eye in my inside! An eye in my stomach! Your Lordship jests.

I. An eye inside me! An eye in my stomach! You're joking, Your Lordship.

STRANGER. I am in no jesting humour. I tell you that I come from Space, or, since you will not understand what Space means, from the Land of Three Dimensions whence I but lately looked down upon your Plane which you call Space forsooth. From that position of advantage I discerned all that you speak of as SOLID (by which you mean "enclosed on four sides"), your houses, your churches, your very chests and safes, yes even your insides and stomachs, all lying open and exposed to my view.

STRANGER. I am not joking. I tell you, I come from Space. Or, since you will not understand what Space means, I come from the Land of Three Dimensions. From there, I recently looked down on your Plane, which you actually call Space! From that better viewpoint, I could see everything you call SOLID (by which you mean "enclosed on four sides"). Your houses, your churches, your very chests and safes lay open and exposed to my view. Yes, even your insides and stomachs were fully visible to me.

I. Such assertions are easily made, my Lord.

I. Claims like that are easy to make, my Lord.

STRANGER. But not easily proved, you mean. But I mean to prove mine.

STRANGER. You mean it's not easy to prove. But I intend to prove mine.

When I descended here, I saw your four Sons, the Pentagons, each in his apartment, and your two Grandsons the Hexagons; I saw your youngest Hexagon remain a while with you and then retire to his room, leaving you and your Wife alone. I saw your Isosceles servants, three in number, in the kitchen at supper, and the little Page in the scullery. Then I came here, and how do you think I came?

When I came down here, I saw your four Sons, the Pentagons, each in his own room. I also saw your two Grandsons, the Hexagons. I saw your youngest Hexagon stay with you for a while and then go to his room. He left you and your Wife alone. I saw your three Isosceles servants eating supper in the kitchen, and the little Page in the scullery. Then I came here. How do you think I came?

I. Through the roof, I suppose.

I. Through the roof, I suppose.

STRANGER. Not so. Your roof, as you know very well, has been recently repaired, and has no aperture by which even a Woman could penetrate. I tell you I come from Space. Are you not convinced by what I have told you of your children and household?

STRANGER. No. As you know very well, your roof was recently repaired. It has no opening that even a Woman could get through. I tell you, I come from Space. Doesn't what I've told you about your children and household convince you?

I. Your Lordship must be aware that such facts touching the belongings of his humble servant might be easily ascertained by any one in the neighbourhood possessing your Lordship's ample means of obtaining information.

I. Your Lordship must know that these facts about your humble servant's belongings could easily be discovered. Anyone nearby with your Lordship's extensive resources for gathering information could discover them.

STRANGER. (TO HIMSELF.) What must I do? Stay; one more argument suggests itself to me. When you see a Straight Line—your wife, for example—how many Dimensions do you attribute to her?

STRANGER. (TO HIMSELF.) What should I do? Wait, I can think of one more argument. Take a Straight Line like your wife, for example. When you see her, how many Dimensions do you think she has?

I. Your Lordship would treat me as if I were one of the vulgar who, being ignorant of Mathematics, suppose that a Woman is really a Straight Line, and only of One Dimension. No, no, my Lord; we Squares are better advised, and are as well aware as your Lordship that a Woman, though popularly called a Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness).

I. Your Lordship would treat me as if I were one of the ignorant common people who know no Mathematics. They suppose that a Woman really is a Straight Line, with only One Dimension. No, no, my Lord. We Squares know better. We know as well as your Lordship that, though commonly called a Straight Line, a Woman is scientifically a very thin Parallelogram. Like the rest of us, she has Two Dimensions: length and breadth (or thickness).

STRANGER. But the very fact that a Line is visible implies that it possesses yet another Dimension.

STRANGER. But the very fact that a Line is visible means it has another Dimension.

I. My Lord, I have just acknowledged that a Woman is broad as well as long. We see her length, we infer her breadth; which, though very slight, is capable of measurement.

I. My Lord, I have just admitted that a Woman has width as well as length. We see her length and infer her width. Although her width is very small, it can be measured.

STRANGER. You do not understand me. I mean that when you see a Woman, you ought—besides inferring her breadth—to see her length, and to SEE what we call her HEIGHT; although that last Dimension is infinitesimal in your country. If a Line were mere length without "height", it would cease to occupy Space and would become invisible. Surely you must recognize this?

STRANGER. You do not understand me. I mean that when you see a Woman, you should not just infer her breadth. You should also see her length and SEE what we call her HEIGHT. That last Dimension, however, is infinitesimally small in your country. If a Line had only length without "height", it would no longer take up Space and would become invisible. Surely you must recognize that?

I. I must indeed confess that I do not in the least understand your Lordship. When we in Flatland see a Line, we see length and BRIGHTNESS. If the brightness disappears, the Line is extinguished, and, as you say, ceases to occupy Space. But am I to suppose that your Lordship gives to brightness the title of a Dimension, and that what we call "bright" you call "high"?

I. I must admit that I do not understand your Lordship at all. When we in Flatland see a Line, we see length and BRIGHTNESS. If the brightness disappears, the Line goes out and, as you say, no longer occupies Space. But am I to suppose that your Lordship calls brightness a Dimension? And that what we call "bright" you call "high"?

STRANGER. No, indeed. By "height" I mean a Dimension like your length: only, with you, "height" is not so easily perceptible, being extremely small.

STRANGER. No, not at all. By "height" I mean a dimension like your length. But for you, "height" is harder to notice because it is extremely small.

I. My Lord, your assertion is easily put to the test. You say I have a Third Dimension, which you call "height". Now, Dimension implies direction and measurement. Do but measure my "height", or merely indicate to me the direction in which my "height" extends, and I will become your convert. Otherwise, your Lordship's own understanding must hold me excused.

I. My Lord, your claim is easy to test. You say I have a Third Dimension, which you call "height". Now, a dimension involves direction and measurement. Just measure my "height", or simply show me the direction in which my "height" extends, and I will believe you. Otherwise, your Lordship must understand why I remain unconvinced.

STRANGER. (TO HIMSELF.) I can do neither. How shall I convince him? Surely a plain statement of facts followed by ocular demonstration ought to suffice. —Now, Sir; listen to me.

STRANGER. (TO HIMSELF.) I can do neither. How can I convince him? Surely a clear statement of the facts, followed by a demonstration he can see, should be enough. Now, sir, listen to me.

You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, on, or in, the top of which you and your countrymen move about, without rising above it or falling below it.

You live on a Plane. What you call Flatland is the vast, level surface of what I might call a fluid. You and your countrymen move on or within the top of this fluid, never rising above or falling below it.

I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere—which is my proper name in my own country—if he manifest himself at all to an inhabitant of Flatland—must needs manifest himself as a Circle.

I am not a flat Figure, but a Solid. You call me a Circle, but I am not really a Circle. I am an infinite number of Circles stacked on top of one another. Their sizes range from a Point to a Circle thirteen inches across. When I pass through your plane, as I am doing now, I create a cross-section in your plane. You quite rightly call that cross-section a Circle. For even a Sphere must appear as a Circle if he appears at all to an inhabitant of Flatland. Sphere is my proper name in my own country.

Do you not remember—for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain—do you not remember, I say, how, when you entered the realm of Lineland, you were compelled to manifest yourself to the King, not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or section of you? In precisely the same way, your country of Two Dimensions is not spacious enough to represent me, a being of Three, but can only exhibit a slice or section of me, which is what you call a Circle.

Do you not remember? For I, who see all things, saw the ghostly vision of Lineland written on your brain last night. Do you not remember, I ask, how you had to appear to the King when you entered Lineland? You had to appear not as a Square, but as a Line. This was because that Linear Realm lacked enough Dimensions to show all of you, only a slice or section. In precisely the same way, your country of Two Dimensions lacks enough space to show me, a being of Three. Your country can only show a slice or section of me, which you call a Circle.

The diminished brightness of your eye indicates incredulity. But now prepare to receive proof positive of the truth of my assertions. You cannot indeed see more than one of my sections, or Circles, at a time; for you have no power to raise your eye out of the plane of Flatland; but you can at least see that, as I rise in Space, so my sections become smaller. See now, I will rise; and the effect upon your eye will be that my Circle will become smaller and smaller till it dwindles to a point and finally vanishes.

Your eye has grown dimmer, showing your disbelief. But now prepare for definite proof that my claims are true. You can only see one of my cross-sections, or Circles, at a time. This is because you cannot raise your eye out of the plane of Flatland. But you can at least see that as I rise in Space, my cross-sections grow smaller. Watch now as I rise. To your eye, my Circle will grow smaller and smaller until it shrinks to a point and finally disappears.

There was no "rising" that I could see; but he diminished and finally vanished. I winked once or twice to make sure that I was not dreaming. But it 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."

I could see no "rising", but he grew smaller and finally vanished. I blinked once or twice to make sure I was not dreaming. But it was no dream. A hollow voice came from the depths of nowhere, seeming close to my heart. "Am I completely gone? Are you convinced now? Well, now I will gradually return to Flatland. You will see my cross-section grow 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.

Every reader in Spaceland will easily understand that my mysterious Guest was speaking truthfully and even simply. But although I was skilled in Flatland Mathematics, I found his explanation far from simple. The rough diagram above will make the explanation clear to any child in Spaceland. As the Sphere rose through the three positions shown, he must have appeared to me as a Circle. Any other Flatlander would have seen the same thing. At first, the Circle was full size, then small, and finally very small indeed, almost a Point. But although I saw what happened, the causes remained as mysterious as ever. All I could understand was that the Circle had made himself smaller and vanished. Now he had 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.

When he returned to his original size, he let out a deep sigh. My silence showed him that I had completely failed to understand him. And indeed, I was beginning to believe he must not be a Circle at all, but an extremely clever juggler. Or perhaps the old wives' tales were true, and people such as Enchanters and Magicians really did exist after all.

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.

After a long pause, he muttered to himself, "I have only one option left if I am to avoid taking action. I must try using Analogy." An even longer silence followed. Then he continued our conversation.

SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a luminous wake, what name would you give to the wake?

SPHERE. Tell me, Mr. Mathematician. If a Point moves north and leaves a glowing trail, what would you call the trail?

I. A straight Line.

I. A straight line.

SPHERE. And a straight Line has how many extremities?

SPHERE. And how many endpoints does a straight line have?

I. Two.

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?

SPHERE. Now imagine the northward straight line moving east and west, staying parallel to its original position. As the line moves, each point on it traces a straight line. What name would you give to the shape formed this way? Suppose the line moves a distance equal to its original length. What name, I ask?

I. A Square.

I. A Square.

SPHERE. And how many sides has a Square? How many angles?

SPHERE. And how many sides does a Square have? How many angles?

I. Four sides and four 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.

SPHERE. Now stretch your imagination a little. Imagine a Square in Flatland moving upward while staying parallel to its original position.

I. What? Northward?

I. What? North?

SPHERE. No, not Northward; upward; out of Flatland altogether.

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.

If it moved north, the Square's southern points would have to pass through places its northern points previously occupied. But that is not what I mean.

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.

I mean that every Point in you is to move upwards through Space. You are a Square, so you will serve as my example. By every Point in you, I mean every Point in what you call your inside. No Point is to pass through a position previously occupied by any other Point. Instead, each Point is to trace its own straight Line. All of this follows Analogy. Surely the idea 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:—

I held back my impatience. I was now strongly tempted to charge blindly at my Visitor and hurl him into Space, or out of Flatland. Anywhere would do, so long as I could get rid of him. But 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."

"And what kind of Figure am I to form by moving in the direction you choose to call 'upward'? I assume the Figure can be described 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.

SPHERE. Oh, certainly. It is all clear and simple, and follows Analogy exactly. But by the way, you must not call the result a Figure. You must call it a Solid. But I will describe it to you. Or rather, Analogy will, not I.

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

We began with a single Point. Of course, because it is itself a Point, it has only ONE endpoint.

One Point produces a Line with TWO terminal Points.

One Point creates a Line with TWO endpoints.

One Line produces a Square with FOUR terminal Points.

One Line creates a Square with FOUR corners.

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

Now you can answer your own question: 1, 2, 4 clearly form a geometric sequence. What is the next number?

I. Eight.

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?

SPHERE. Exactly. The single Square produces a SOMETHING-WHICH-YOU-DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE with EIGHT corner points. Are you convinced now?

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

I. And does this creature have 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.

SPHERE. Of course, and all following the same analogy. But, by the way, I mean what WE call sides, not what YOU 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?

I. And how many solids or sides will this Being have? I am to create this Being by moving my inside in an "upward" direction. You call this Being 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, 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?

SPHERE. How can you ask? And you a mathematician! The side of anything is always one dimension below the thing itself, if I may put it that way. So, since no dimension is below a Point, a Point has 0 sides. A Line has 2 sides, if I may say so (we may loosely call the Line's Points its sides). A Square has 4 sides. 0, 2, 4: what kind of progression do you call that?

I. Arithmetical.

I. Arithmetic.

SPHERE. And what is the next number?

SPHERE. And what's the next number?

I. Six.

I. Six.

SPHERE. Exactly. Then you see you have answered your own question. The Cube which you will generate will be bounded by six sides, that is to say, six of your insides. You see it all now, eh?

SPHERE. Exactly. So you see, you have answered your own question. The Cube you create will be bounded by six sides, meaning six of your insides. You see it all now, eh?

"Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. Either thou or I must perish." And saying these words I precipitated myself upon him.

"Monster," I shrieked, "whether you're a juggler, magician, dream, or devil, I won't put up with your mockery any longer. Either you or I must die." And with those words, I threw myself at him.