When I had done this at great length, I cried triumphantly, "Does that at last convince you?" And, with that, I once more entered Lineland, taking up the same position as before.
But the Monarch replied, "If you were a Man of sense -- though, as you appear to have only one voice I have little doubt you are not a Man but a Woman -- but, if you had a particle of sense, you would listen to reason. You ask me to believe that there is another Line besides that which my senses indicate, and another motion besides that of which I am daily conscious. I, in return, ask you to describe in words or indicate by motion that other Line of which you speak. Instead of moving, you merely exercise some magic art of vanishing and returning to sight; and instead of any lucid description of your new World, you simply tell me the numbers and sizes of some forty of my retinue, facts known to any child in my capital. Can anything be more irrational or audacious? Acknowledge your folly or depart from my dominions."
Furious at his perversity, and especially indignant that he professed to be ignorant of my s.e.x, I retorted in no measured terms, "Besotted Being! You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I CAN SEE Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines, called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great n.o.bles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance."
Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal; and in that same moment there arose from myriads of his subjects a mult.i.tudinous war-cry, increasing in vehemence till at last methought it rivalled the roar of an army of a hundred thousand Isosceles, and the artillery of a thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to avert the impending destruction; and still the noise grew louder, and the King came closer, when I awoke to find the breakfast-bell recalling me to the realities of Flatland.
Section 15. Concerning a Stranger from s.p.a.celand From dreams I proceed to facts.
It was the last day of the 1999th year of our era. The pattering of the rain had long ago announced nightfall; and I was sitting in the company of my wife, musing on the events of the past and the prospects of the coming year, the coming century, the coming Millennium.
[Note: When I say "sitting", of course I do not mean any change of att.i.tude such as you in s.p.a.celand signify by that word; for as we have no feet, we can no more "sit" nor "stand" (in your sense of the word) than one of your soles or flounders.
Nevertheless, we perfectly well recognize the different mental states of volition implied in "lying", "sitting", and "standing", which are to some extent indicated to a beholder by a slight increase of l.u.s.tre corresponding to the increase of volition.
But on this, and a thousand other kindred subjects, time forbids me to dwell.]
My four Sons and two orphan Grandchildren had retired to their several apartments; and my wife alone remained with me to see the old Millennium out and the new one in.
I was rapt in thought, pondering in my mind some words that had casually issued from the mouth of my youngest Grandson, a most promising young Hexagon of unusual brilliancy and perfect angularity. His uncles and I had been giving him his usual practical lesson in Sight Recognition, turning ourselves upon our centres, now rapidly, now more slowly, and questioning him as to our positions; and his answers had been so satisfactory that I had been induced to reward him by giving him a few hints on Arithmetic, as applied to Geometry.
Taking nine Squares, each an inch every way, I had put them together so as to make one large Square, with a side of three inches, and I had hence proved to my little Grandson that -- though it was impossible for us to SEE the inside of the Square -- yet we might ascertain the number of square inches in a Square by simply squaring the number of inches in the side: "and thus," said I, "we know that 3^2, or 9, represents the number of square inches in a Square whose side is 3 inches long."
The little Hexagon meditated on this a while and then said to me; "But you have been teaching me to raise numbers to the third power: I suppose 3^3 must mean something in Geometry; what does it mean?" "Nothing at all," replied I, "not at least in Geometry; for Geometry has only Two Dimensions." And then I began to shew the boy how a Point by moving through a length of three inches makes a Line of three inches, which may be represented by 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 3^2.
Upon this, my Grandson, again returning to his former suggestion, took me up rather suddenly and exclaimed, "Well, then, if a Point by moving three inches, makes a Line of three inches represented by 3; and if a straight Line of three inches, moving parallel to itself, makes a Square of three inches every way, represented by 3^2; it must be that a Square of three inches every way, moving somehow parallel to itself (but I don"t see how) must make Something else (but I don"t see what) of three inches every way -- and this must be represented by 3^3."
"Go to bed," said I, a little ruffled by this interruption: "if you would talk less nonsense, you would remember more sense."
So my Grandson had disappeared in disgrace; and there I sat by my Wife"s side, endeavouring to form a retrospect of the year 1999 and of the possibilities of the year 2000, but not quite able to shake off the thoughts suggested by the prattle of my bright little Hexagon. Only a few sands now remained in the half-hour gla.s.s. Rousing myself from my reverie I turned the gla.s.s Northward for the last time in the old Millennium; and in the act, I exclaimed aloud, "The boy is a fool."
Straightway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being. "He is no such thing," cried my Wife, "and you are breaking the Commandments in thus dishonouring your own Grandson." But I took no notice of her. Looking round in every direction I could see nothing; yet still I FELT a Presence, and shivered as the cold whisper came again. I started up. "What is the matter?" said my Wife, "there is no draught; what are you looking for? There is nothing." There was nothing; and I resumed my seat, again exclaiming, "The boy is a fool, I say; 3^3 can have no meaning in Geometry." At once there came a distinctly audible reply, "The boy is not a fool; and 3^3 has an obvious Geometrical meaning."
My Wife as well as myself heard the words, although she did not understand their meaning, and both of us sprang forward in the direction of the sound. What was our horror when we saw before us a Figure! At the first glance it appeared to be a Woman, seen sideways; but a moment"s observation shewed me that the extremities pa.s.sed into dimness too rapidly to represent one of the Female s.e.x; and I should have thought it a Circle, only that it seemed to change its size in a manner impossible for a Circle or for any regular Figure of which I had had experience.
But my Wife had not my experience, nor the coolness necessary to note these characteristics. With the usual hastiness and unreasoning jealousy of her s.e.x, she flew at once to the conclusion that a Woman had entered the house through some small aperture. "How comes this person here?" she exclaimed, "you promised me, my dear, that there should be no ventilators in our new house." "Nor are there any," said I; "but what makes you think that the stranger is a Woman? I see by my power of Sight Recognition ----" "Oh, I have no patience with your Sight Recognition," replied she, ""Feeling is believing" and "A Straight Line to the touch is worth a Circle to the sight"" -- two Proverbs, very common with the Frailer s.e.x in Flatland.
"Well," said I, for I was afraid of irritating her, "if it must be so, demand an introduction." a.s.suming her most gracious manner, my Wife advanced towards the Stranger, "Permit me, Madam, to feel and be felt by ----" then, suddenly recoiling, "Oh! it is not a Woman, and there are no angles either, not a trace of one. Can it be that I have so misbehaved to a perfect Circle?"
"I am indeed, in a certain sense a Circle," replied the Voice, "and a more perfect Circle than any in Flatland; but to speak more accurately, I am many Circles in one." Then he added more mildly, "I have a message, dear Madam, to your husband, which I must not deliver in your presence; and, if you would suffer us to retire for a few minutes ----" But my Wife would not listen to the proposal that our august Visitor should so incommode himself, and a.s.suring the Circle that the hour of her own retirement had long pa.s.sed, with many reiterated apologies for her recent indiscretion, she at last retreated to her apartment.
I glanced at the half-hour gla.s.s. The last sands had fallen. The third Millennium had begun.
Section 16. How the Stranger vainly endeavoured to reveal to me in words the mysteries of s.p.a.celand 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.
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.
STRANGER. Have you felt me enough by this time? Are you not introduced to me yet?
I. Most ill.u.s.trious 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?
STRANGER. From s.p.a.ce, from s.p.a.ce, Sir: whence else?
I. Pardon me, my Lord, but is not your Lordship already in s.p.a.ce, your Lordship and his humble servant, even at this moment?
STRANGER. Pooh! what do you know of s.p.a.ce? Define s.p.a.ce.
I. s.p.a.ce, my Lord, is height and breadth indefinitely prolonged.
STRANGER. Exactly: you see you do not even know what s.p.a.ce is. You think it is of Two Dimensions only; but I have come to announce to you 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.
STRANGER. But I mean not only three names, but Three Dimensions.
I. Would your Lordship indicate or explain to me in what direction is the Third Dimension, unknown to me?
STRANGER. I came from it. It is up above and down below.
I. My Lord means seemingly 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.
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.
STRANGER. Yes: but in order to see into s.p.a.ce 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 s.p.a.celand should call it your side.
I. An eye in my inside! An eye in my stomach! Your Lordship jests.
STRANGER. I am in no jesting humour. I tell you that I come from s.p.a.ce, or, since you will not understand what s.p.a.ce means, from the Land of Three Dimensions whence I but lately looked down upon your Plane which you call s.p.a.ce 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.
I. Such a.s.sertions are easily made, my Lord.
STRANGER. But not easily proved, you mean. But I mean 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?
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 s.p.a.ce. Are you not convinced by what I have told you of your children and household?
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.
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?
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).
STRANGER. But the very fact that a Line is visible implies that it possesses yet 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.
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 s.p.a.ce and would become invisible. Surely you must recognize this?
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 s.p.a.ce. But am I to suppose that your Lordship gives to brightness the t.i.tle of 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.
I. My Lord, your a.s.sertion 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.
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.
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.
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.
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 s.p.a.cious 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.
The diminished brightness of your eye indicates incredulity. But now prepare to receive proof positive of the truth of my a.s.sertions. 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 s.p.a.ce, 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.
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."
Every reader in s.p.a.celand 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 s.p.a.celand 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 a.n.a.logy." 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; 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 ill.u.s.tration -- every Point in you, that is to say in what you call your inside, is to pa.s.s upwards through s.p.a.ce in such a way that no Point shall pa.s.s 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 a.n.a.logy; 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 s.p.a.ce, 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 a.n.a.logy -- 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 a.n.a.logy.
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 a.n.a.logy. 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, 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.
SPHERE. And what is the next number?
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?
"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.
Section 17. How the Sphere, having in vain tried words, resorted to deeds It was in vain. I brought my hardest right angle into violent collision with the Stranger, pressing on him with a force sufficient to have destroyed any ordinary Circle: but I could feel him slowly and unarrestably slipping from my contact; no edging to the right nor to the left, but moving somehow out of the world, and vanishing to nothing. Soon there was a blank. But still I heard the Intruder"s voice.
SPHERE. Why will you refuse to listen to reason? I had hoped to find in you -- as being a man of sense and an accomplished mathematician -- a fit apostle for the Gospel of the Three Dimensions, which I am allowed to preach once only in a thousand years: but now I know not how to convince you. Stay, I have it. Deeds, and not words, shall proclaim the truth. Listen, my friend.
I have told you I can see from my position in s.p.a.ce the inside of all things that you consider closed. For example, I see in yonder cupboard near which you are standing, several of what you call boxes (but like everything else in Flatland, they have no tops nor bottoms) full of money; I see also two tablets of accounts. I am about to descend into that cupboard and to bring you one of those tablets. I saw you lock the cupboard half an hour ago, and I know you have the key in your possession. But I descend from s.p.a.ce; the doors, you see, remain unmoved. Now I am in the cupboard and am taking the tablet. Now I have it. Now I ascend with it.
I rushed to the closet and dashed the door open. One of the tablets was gone. With a mocking laugh, the Stranger appeared in the other corner of the room, and at the same time the tablet appeared upon the floor. I took it up. There could be no doubt -- it was the missing tablet.
I groaned with horror, doubting whether I was not out of my senses; but the Stranger continued: "Surely you must now see that my explanation, and no other, suits the phenomena. What you call Solid things are really superficial; what you call s.p.a.ce is really nothing but a great Plane. I am in s.p.a.ce, and look down upon the insides of the things of which you only see the outsides. You could leave this Plane yourself, if you could but summon up the necessary volition. A slight upward or downward motion would enable you to see all that I can see.