I apologized; but he continued scornfully, “Since you are impervious to argument, you shall hear with your ears how by means of my two voices I reveal my shape to my Wives, who are at this moment six thousand miles seventy yards two feet eight inches away, the one to the North, the other to the South. Listen, I call to them.”
He chirruped, and then complacently continued: “My wives at this moment receiving the sound of one of my voices, closely followed by the other, and perceiving that the latter reaches them after an interval in which sound can traverse 6.457 inches, infer that one of my mouths is 6.457 inches further from them than the other, and accordingly know my shape to be 6.457 inches. But you will of course understand that my wives do not make this calculation every time they hear my two voices. They made it, once for all, before we were married. But they could make it at any time. And in the same way I can estimate the shape of any of my Male subjects by the sense of sound.”
“But how,” said I, “if a Man feigns a Woman’s voice with one of his two voices, or so disguises his Southern voice that it cannot be recognized as the echo of the Northern? May not such deceptions cause great inconvenience? And have you no means of checking frauds of this kind by commanding your neighbouring subjects to feel one another?” This of course was a very stupid question, for feeling could not have answered the purpose; but I asked with the view of irritating the Monarch, and I succeeded perfectly.
“What!” cried he in horror, “explain your meaning.” “Feel, touch, come into contact,” I replied. “If you mean by feeling,” said the King, “approaching so close as to leave no space between two individuals, know, Stranger, that this offence is punishable in my dominions by death. And the reason is obvious. The frail form of a Woman, being liable to be shattered by such an approximation, must be preserved by the State; but since Women cannot be distinguished by the sense of sight from Men, the Law ordains universally that neither Man nor Woman shall be approached so closely as to destroy the interval between the approximator and the approximated.
“And indeed what possible purpose would be served by this illegal and unnatural excess of approximation which you call touching, when all the ends of so brutal and coarse a process are attained at once more easily and more exactly by the sense of hearing? As to your suggested danger of deception, it is non-existent: for the Voice, being the essence of one’s Being, cannot be thus changed at will. But come, suppose that I had the power of passing through solid things, so that I could penetrate my subjects, one after another, even to the number of a billion, verifying the size and distance of each by the sense of feeling: how much time and energy would be wasted in this clumsy and inaccurate method! Whereas now, in one moment of audition, I take as it were the census and statistics, local, corporeal, mental and spiritual, of every living being in Lineland. Hark, only hark!”
So saying he paused and listened, as if in an ecstasy, to a sound which seemed to me no better than a tiny chirping from an innumerable multitude of lilliputian grasshoppers.
“Truly,” replied I, “your sense of hearing serves you in good stead, and fills up many of your deficiencies. But permit me to point out that your life in Lineland must be deplorably dull. To see nothing but a Point! Not even to be able to contemplate a Straight Line! Nay, not even to know what a Straight Line is! To see, yet be cut off from those Linear prospects which are vouchsafed to us in Flatland! Better surely to have no sense of sight at all than to see so little! I grant you I have not your discriminative faculty of hearing; for the concert of all Lineland which gives you such intense pleasure, is to me no better than a multitudinous twittering or chirping. But at least I can discern, by sight, a Line from a Point. And let me prove it. Just before I came into your kingdom, I saw you dancing from left to right, and then from right to left, with Seven Men and a Woman in your immediate proximity on the left, and eight Men and two Women on your right. Is not this correct?”
“It is correct,” said the King, “so far as the numbers and sexes are concerned, though I know not what you mean by ‘right’ and ‘left.’ But I deny that you saw these things. For how could you see the Line, that is to say the inside, of any Man? But you must have heard these things, and then dreamed that you saw them. And let me ask what you mean by those words ‘left’ and ‘right.’ I suppose it is your way of saying Northward and Southward.”
“Not so,” replied I; “besides your motion of Northward and Southward, there is another motion which I call from right to left.”
King. Exhibit to me, if you please, this motion from left to right.
I. Nay, that I cannot do, unless you could step out of your Line altogether.
King. Out of my Line? Do you mean out of the world? Out of Space?
I. Well, yes. Out of your World. Out of your Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line.
King. If you cannot indicate this motion from left to right by yourself moving in it, then I beg you to describe it to me in words.
I. If you cannot tell your right side from your left, I fear that no words of mine can make my meaning clear to you. But surely you cannot be ignorant of so simple a distinction.
King. I do not in the least understand you.
I. Alas! How shall I make it clear? When you move straight on, does it not sometimes occur to you that you could move in some other way, turning your eye round so as to look in the direction towards which your side is now fronting? In other words, instead of always moving in the direction of one of your extremities, do you never feel a desire to move in the direction, so to speak, of your side?
King. Never. And what do you mean? How can a man’s inside “front” in any direction? Or how can a man move in the direction of his inside?
I. Well then, since words cannot explain the matter, I will try deeds, and will move gradually out of Lineland in the direction which I desire to indicate to you.
At the word I began to move my body out of Lineland. As long as any part of me remained in his dominion and in his view, the King kept exclaiming, “I see you, I see you still; you are not moving.” But when I had at last moved myself out of his Line, he cried in his shrillest voice, “She is vanished; she is dead.” “I am not dead,” replied I; “I am simply out of Lineland, that is to say, out of the Straight Line which you call Space, and in the true Space, where I can see things as they are. And at this moment I can see your Line, or side—or inside as you are pleased to call it; and I can see also the Men and Women on the North and South of you, whom I will now enumerate, describing their order, their size, and the interval between each.”
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 sex, 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 exist
ence 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 nobles 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 multitudinous warcry, 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.
15.—Concerning a Stranger from Spaceland
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.
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 32, 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 33 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 32.
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 32; 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 33.”
“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 glass. Rousing myself from my reverie I turned the glass 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; 33 can have no meaning in Geometry.” At once there came a distinctly audible reply, “The boy is not a fool; and 33 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 passed into dimness too rapidly to represent one of the Female Sex; 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 Sex, 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 Sex in Flatland.
“Well,” said I, for I was afraid of irritating her, “if it must be so, demand an introduction.” Assuming 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 assuring the Circle that the hour of her own retirement had long passed, with many reiterated apologies for her recent indiscretion, she at last retreated to her apartment.
I glanced at the half-hour glass. The last sands had fallen. The third Millennium had begun.
16.—How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland
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 standi
ng. 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 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?
Stranger. From Space, from Space, Sir: whence else?
I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment?
Stranger. Pooh! what do you know of Space? Define Space.
I. Space, my Lord, is height and breadth indefinitely prolonged.
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.
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