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07 - Flatland- A Romance of Many Dimensions - Edwin Abbott Abbott · Classic Sci Fi Books - Daily

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Part two, Section fifteen, concerning a stranger from Spaceland. From dreams,
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I proceed to facts. It was the last day of the nineteen hundred and
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ninety ninth year of our era. The pattering of the rain had long ago
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announced nightfall, and I was sitting in the company of my wife, musing
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on the events of the past and the prospects of the coming year, the
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coming century, the coming millennium. Footnote. When I say sitting, of
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course, I do not mean any change of attitude, such as you in
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Spaceland signify by that word. For as we have no feat, we can
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no more sit nor stand in your sense of the word than one of your
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souls or flounders. Nevertheless, we perfectly well recognize the different mental states of
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volition implied in lying, sitting, and standing, which are to some extent
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indicated to a beholder by a slight increase of luster corresponding to the increase of
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volition. But on this and a thousand other kindred subjects, time forbids me
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to dwell. End of footnote. My four sons and two orphan grandchildren had
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retired to their several apartments, and my wife alone remained with me to see
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the old millennium out and the new one in. I was wrapt in thought,
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pondering in my mind some words that had casually issued from the mouth of
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my youngest grandson, a most promising young hexagon of unusual brilliancy and perfect angularity.
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His uncles and I had been giving him his usual practical lesson in sight
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recognition, turning ourselves upon our centers, now rapidly, now more slowly,
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and questioning him as to our positions. And his answers had been so satisfactory
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that I had been induced to reward him by giving him a few hints on
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arithmetic as applied to geometry. Taking nine squares each an inch every way,
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I had put them together so as to make one large square with a side
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of three inches. And I had hence proved to my little grandson that though
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it was impossible for us to see the inside of the square, yet we
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might ascertain the number of square inches in a square by simply squaring the number
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of inches in the side. And thus said I, we know that three
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squared or nine represents the number of square inches in a square whose side is
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three inches long. The little hexagon meditated on this a while and then said
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to me, but you have been teaching me to raise numbers to the third
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power. I suppose three cubed must mean something in geometry. What does it
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mean? Nothing at all? Replied I, not at least in geometry,
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for geometry has only two dimensions. And then I began to show the boy
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how a point, by moving through a length of three inches, makes a
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line of three inches, which may be represented by three, and how a
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line of three inches moving parallel to itself through a length of three inches makes
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a square of three inches every way, which may be represented by three squared.
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Upon this, my grandson, again returning to his former suggestion, took
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me up rather suddenly and exclaimed, well, then, if a point,
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by moving three inches makes a line of three inches represented by three, and
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if a straight line of three inches moving parallel to itself makes a square of
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three inches every way presented by three squared, it must be that a square
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of three inches every way, moving somehow parallel to itself. But I don't
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see how must make something else. But I don't see what of three inches
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every way? And this must be represented by three cubed. Go to bed,
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said I, a little ruffled by his interruption. If you would talk
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less nonsense, you would remember more sense. So my grandson had disappeared in
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disgrace, and there I sat by my wife's side, endeavoring to form a
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retrospect of the year nineteen ninety nine and of the possibilities of the year two
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thousand, but not quite able to shake off the thoughts suggested by the prattle
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of my bright little hexagon. Only a few sands now remained in the half
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hour glass. Rousing myself from my reverie, I turned the glo northward for
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the last time in the old millennium, and in the act I exclaimed aloud,
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the boy is a fool. Straightway, I became conscious of a presence
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in the room, and a chilling breath thrilled through my very being. He
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is no such thing, cried my wife. And you are breaking the commandments
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in thus dishonoring your own grandson. But I took no notice of her.
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Looking round in every direction, I could see nothing. Yet still I felt
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a presence and shivered. As the cold whisper came again. I started up.
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What is the matter, said my wife? There is no draft.
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What are you looking for? There is nothing? There was nothing and I
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resumed my seat again, exclaiming, the boy is a fool. I say,
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three cubed can have no meaning in geometry, But once there came a
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distinctly audible reply, the boy is not a fool, and three cubed has
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an obvious geometrical meaning. My wife, as well as myself heard the words,
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although she did not understand their meaning, and both of us sprang forward
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in the direction of the sound. What was our horror when we saw before
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us a figure. At the first glance, it appeared to be a woman
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seen sideways, But a moment's observation showed me that the extremities passed into dimness
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too rapidly to represent one of the female sex. And I should have thought
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it a circle, only that it seemed to change its size in a manner
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impossible for a circle or for any regular figure of which I had had experience.
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But my wife had not my experience nor the coolness necessary to note these
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characteristics. With the usual hastiness and unreasoning jealousy of her sex, she flew
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at once to the conclusion that a woman had entered the house through some small
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aperture. How comes this person here, she exclaimed, you promised me,
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my dear, that there should be no ventilators in our new house, nor
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are there any, said I, But what makes you think that the stranger
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is a woman? I see by my power of sight recognition. Oh,
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I have no patience with your sight recognition, replied she feeling is believing,
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and a straight line to the touch is worth a circle to the sight,
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two proverbs very common with the frailer sex in flatland. Well said I,
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For I was afraid of irritating her. If it must be so, demand
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an introduction, Assuming her most gracious manner, My wife advanced towards the stranger.
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Permit me, madam, to feel to be felt by, then suddenly
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recoiling. Oh, it is not a woman, and there are no angles
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either. What a trait of one? Can it be that I have so
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misbehaved to a perfect circle. I am, indeed, in a sense a
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circle, replied the voice, and a more perfect circle than any in flat
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land. But to speak more accurately, I am many circles in one.
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Then he added, more mildly, I have a message, dear madam,
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to your husband, which I must not deliver in your presence, and if
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you would suffer us to retire for a few minutes. But my wife would
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not listen to the proposal that our August visitor should so incommode himself. And
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assuring the circle that the hour for her own retirement had long passed. With
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many reiterated apologies for her recent indiscretion, she at last retreated to her apartment.
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I glanced at the half the last sands had fallen, the second millennium
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had begun. Section sixteen. How the stranger vainly endeavored to reveal to me
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in word as the mysteries of spaceland. As soon as the sound of my
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wife's retreating footsteps had died away, I began to approach the stranger, with
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the intention of taking a nearer view, and of bidding him be seated.
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That his appearance struck me dumb and motionless with astonishment. Without the slightest symptoms
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of angularity, he nevertheless varied every instant with gradations of size and brightness,
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scarcely possible for any figure within the scope of my experience. The thought flashed
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across me that I might have before me a burglar or cutthroat, some monstrous,
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irregular eye soscely who, by feigning the voice of a circle had obtained
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admission somehow into the house and was now preparing to stab me with his acute
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angle in a sitting room. The absence of fog and the season happened to
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be remarkably dry, made it difficult for me to trust to sight recognition,
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especially at the short distance at which I was standing. Desperate with fear,
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I rushed forward with an unceremonious you must permit me, sir, and felt
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him. My wife was right. There was not the trace of an angle,
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not the slightest roughness or inequality. Never in my life had I met
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with a more perfect circle. He remained motionless while I walked round him,
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beginning from his eye and returning to it again circular. He was throughout a
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perfectly satisfactory circle. There could not be a doubt of it. Then followed
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a dialog, which I will endeavor to set down as near as I can
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recollect it, omitting only some of my profuse apologies, for I was covered
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with shame and humiliation that I a square should have been guilty of the impertinence
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of feeling a circle. It was commenced by the stranger with some impatience at
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the lengthiness of my introductory process. Stranger, have you felt me enough by
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this time? Are you not introduced to me yet? I most illustrious,
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sir, excuse my awkwardness, which rises not from ignorance of the usages of
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polite society, but from a little surprise and nervousness consequent on this somewhat unexpected
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visit. And I beseech you to reveal my indiscretion to no one, and
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especially not to my wife. But before your Lordship enters into further communications,
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would he deign to satisfy the curious of one who would gladly know whence his
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visitor came stranger from space? From space? Sir? Whence else? I
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pardon me, my lord, but is not your lordship already in space?
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Your lordship and his humble servant, even at this moment, Stranger, pool,
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what do you know of space? Define space? I? Space,
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my Lord, is height and breadth, indefinitely prolonged. Stranger, exactly you
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see, you do not even know what space is. You think it is
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of two dimensions only. But I have come to announce to you a third
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height, breadth and length. I your Lordship is pleased to be merry.
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We also speak of length and high or breadth and thickness, thus denoting two
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dimensions by four names. Stranger, But I mean not only three names,
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but three dimensions. I would your lordship indicate or explain to me in what
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direction is the third dimension? Unknown to me, Stranger, I came from
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it. It is up above and down below, I, my lord,
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means seemingly that it is northwood and southward. Stranger, I mean nothing of
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the kind. I mean a direction in which you cannot look because you have
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no eye in your side. I pardon me, my lord. A moment's
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inspection will convince your lordship that I have a perfect luminary at the juncture of
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two of my sides. Stranger. Yes, but in order to see into
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space, you ought to have an eye, not on your perimeter, but
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on your side. That is, on what you would probably call your inside,
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but we in Spaceland should call it your side. I an I in
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my inside, an I in my stomach, your lordship jests, Stranger.
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I am in no jesting humor. I tell you that I come from space,
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or since you will not understand what space means, from the land of
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three dimensions, whence I, but lately looked down upon your plane, which
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you call space. Forsooth, from that position of advantage, I discerned all
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that you speak of as solid, by which you mean enclosed on four sides,
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your houses, your churches, your very chests and safes, yes,
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even your insides and stomachs, all lying open and exposed to my view.
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I such assertions are easily made, my Lord Stranger, but not easily proved,
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you mean, But I mean to prove mine. When I descended here,
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I saw your four sons, the Pentagons, each in his apartment,
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and your two grandsons, the Hexagons. I saw your youngest Hexagon remain awhile
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with you, and then retire to his room, leaving you and your wife
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alone. I saw your Isosceles servants three in number in the kitchen at supper,
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and the little page in the scullery. Then I came here, And
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how do you think I came? I through the roof, I suppose,
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Stranger. Not so. Your roof, as you know very well, has
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been recently repaired, and has no aperture by which even a woman could penetrate.
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I tell you I come from space. Are you not convinced by what
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I have told you? Of your children and household. I, your lordship,
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must be aware that such facts touching the belongings of his humble servant might
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be easily ascertained by anyone in the neighborhood possessing your Lordship's ample means of obtaining
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information. Stranger, how shall I convince him? Surely A plain statement of
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facts followed by ocular demonstration ought to suffice. Now, sir, listen to
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me. You are living on a plane. What you style flat land is
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the vast level surface of what I may call a fluid on or in the
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top of which you and your countrymen move about without rising above it or falling
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below it. I am not a plane figure, but a solid. You
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call me a circle, But in reality I am not a circle, but
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an infinite number of circles, of size varying from a point to a circle
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of thirteen inches in diameter, one placed on the top of the other.
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When I cut through your plane, as I am now doing, I make
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in your plane a section which you very rightly call a circle, for even
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a sphere, which is my proper name in my own country. If he
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manifest himself at all to an inhabitant of flatland, must needs manifest himself as
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a circle. Do you not remember, for I, who see all things,
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discerned last night the phantasmal vision of Limeland written upon your brain. Do
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you not remember? I say, how, when you entered the realm of
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Limeland, you are compelled to manifest yourself to the King not as a square,
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but as a line, because that linear realm had not dimensions enough to
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represent the whole of you, but only a slice or section of you.
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In precisely the same way, your country of two dimensions is not spacious enough
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to represent me, a being of three, but can only exhibit a slice
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or section of me, which is what you call a circle. The diminished
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brightness of your eye indicates incredulity. But now prepare to receive proof positive of
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the truth of my assertions. You cannot indeed see more than one of my
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sections or circles at a time, for you have no power to raise your
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eye out of the plane of flatland. But you can at least see that
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as I rise in space, so my section becomes smaller. See now I
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will rise, and the effect upon your eye will be that my circle will
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become smaller and smaller till it dwindles. To a point and finally vanishes.
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Reader's note. The following paragraph makes reference to a diagram. The diagram shows
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a horizontal line cutting the line. There are three spheres. The first,
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or leftmost, is half above and half below the line, with its equator
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drawn at the level of the line to indicate the sphere's solid nature. Over
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it is written the sphere with his section at full size. The second,
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or central sphere is positioned about five sixths above the line and one sixth below,
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with a circumference drawn at the level of the line. This is labeled
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two, the sphere rising. The third, or the right hand sphere,
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is nearly all above the line, with just a sliver below, the circumference
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being drawn at the level of the line. This is labeled three, the
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sphere on the point of vanishing. At the right hand end of the line.
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There is an eye marked my eye looking towards the left end of Reader's
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note. There was no rising that I could see, but he diminished and
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finally vanished. I winked once or twice to make sure that I was not
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dreaming, but it was no dream, for from the depths of nowhere came
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forth a hollow voice close to my heart, it seemed, Am I quite
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gone? Are you convinced now? Well? Now I will gradually return to
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flat land, and you shall see my section become larger and larger. Every
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reader in Spaceland will easy understand that my mysterious guest was speaking the language of
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truth and even of simplicity. But to me, proficient though I was in
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Flatland mathematics, it was by no means a simple matter. The rough diagram
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given above will make it clear to any Spaceland child that the sphere ascending in
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the three positions indicated there must needs have manifested himself to me, or to
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any Flatlander, as a circle, at first a full size, then small,
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and at last very small, indeed approaching to a point. But to
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me, although I saw the facts before me, the causes were as dark
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as ever. All that I could comprehend was that the circle had made himself
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smaller and vanished, and that he had now reappeared and was rapidly making himself
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larger. When he had regained his original size. He heaved a deep sigh,
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for he perceived, by my silence that I had altogether failed to comprehend
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him, And indeed I was now inclining to the belief that he must be
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no circle at all, but some extremely clever juggler, or else that the
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old wives tales were true, and that after all, there were such people
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as enchanters and magicians. After a long pause, he muttered to himself,
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one resource alone remains. If I am not to resort to action, I
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must try the method of analogy. Then followed a still longer silence, after
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which he continued, our dialog sphere. Tell me, mister mathematician, if
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a point moves northward and leaves a luminous wake, what name would you give
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to the wake? I a straight line sphere, and a straight line has
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how many extremities I two sphere? Now conceive the northward straight line moving parallel
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to itself east and west, so that every point in it leaves behind it
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the wake of a straight line. What name will you give to the figure
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thereby formed? We will suppose that it moves through a distance equal to the
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original straight line. What name I say? I a square sphere? And
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how many sides has a square? And how many angles? I? Four
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sides and four angles sphere? Now stretch your imagination a little and conceive a
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square in flat land moving parallel to itself upward. I what northward sphere,
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No, not northward upward out of flatland altogether. If it moved northward,
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the southern point in the square would have to move through the positions previously occupied
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by the northern points. But that is not my meaning. I mean that
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every point in you, for you are a square, and will serve the
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purpose of my illustration. Every point in you, that is to say,
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in what you call your inside, is to pass upwards through space in such
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a way that no point shall pass through the position previously occupied by any other
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point, but each point shall describe a straight line of its own. This
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is all in accordance with analogy. Surely it must be clear to you.
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Restraining my impatience, for I was now under a strong temptation to rush blindly
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at my visitor and to precipitate him into space or out of flat land,
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anywhere, so that I could get rid of him, I replied, And
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what may be the nature of the figure which I am to shape out by
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this motion, which you are pleased to denote by the word upwards. I
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presume it is describable in the language of flat land sphere. Oh, certainly,
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it is all plain and simple and in strict accordance with analogy. Only,
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by the way, you must not speak of the result as being a
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figure, but as a solid. But I will describe it to you,
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or rather not I, but analogy. We began with a single point,
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which, of course, being itself a point, has only one terminal point.
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One point produces a line with two terminal points. One line produces a
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square with four terminal points. Now you can yourself give the answer to your
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own question. One, two, four are evidently in geometrical progression. What
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is the next number? I? Eight? Sphere? Exactly? The one
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square produces a something which you do not as yet know a name for,
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but which we call a cube with eight terminal points. Now are you convinced
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I? And has this creature sides as well as angles or what you call
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terminal points? Sphere of course, and all according to analogy, But by
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the way, not what you call sides, but what we call sides you
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would call them solids. I And how many solids or so will appertain to
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this being whom I am to generate by the motion of my inside in an
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upward direction, and whom you call a cube? Sphere. How can you
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ask? And you a mathematician, A side of anything is always, if
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I may so, say, one dimension behind the thing. Consequently, as
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there is no dimension behind a point, a point has no sides. A
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line, if I may so say, has two sides. For the points
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of a line may be called by courtesy its sides. A square has four
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sides, not two four. What progression do you call that? I arithmetical
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sphere? And what is the next number? I? Six sphere? Exactly?
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Then you see you have answered your your own question. The cube which
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you will generate will be bounded by six sides, that is to say,
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six of your insides. You see it all now, eh, monster,
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I shrieked, be thou juggler, enchanter, dream or devil. No more
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will I endure thy mockeries. Either thou or I must perish. And saying
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these words, I precipitated myself upon him. Section seventeen. How the sphere,
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having in vain tried words, resorted to deeds. It was in vain
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I brought my hardest right angle into violent collision with the stranger, pressing on
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him with a force sufficient to have destroyed any ordinary circle. But I could
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feel him slowly and unarrestably slipping from my contact, not edging to the right,
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nor to the left, but moving somehow out of the world and vanishing
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to nothing. Soon there was a blank, but I still heard the intruder's
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voice, sphere, Why will you refuse to listen to reason? I had
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hoped to find in you as being a man of sense and an accomplished mathematician,
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a fit apostle for the Gospel of the Three Dimensions, which I am
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allowed to preach once only in a thousand years. But now I know not
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how to convince you stay. I have it deeds and not words shall proclaim
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the truth. Listen, my friend, I have told you I can see
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from my position in space the inside of all things that you consider closed.
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For example, I see in yonder cupboard near which you are standing, several
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of what you call boxes, But like everything else in flatland, they are
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have no tops nor bottoms full of money. I also see two tablets of
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accounts. I am about to descend into that cupboard and to bring you one
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of those tablets. I saw you lock the cupboard half an hour ago,
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and I know you have the key in your possession. But I descend from
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space. The doors you see remain unmoved. Now I am in the cupboard
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and am taking the tablet. Now I have it. Now I ascend with
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it. I rushed to the closet and dashed the door open. One of
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the tablets was gone. With a mocking laugh, the stranger appeared in the
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other corner of the room, and at the same time the tablet appeared upon
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the floor. I took it up. There could be no doubt it was
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the missing tablet. I groaned with horror, doubting whether I was not out
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of my senses. But the stranger continued, Surely you must now see that
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my explanation, and no other suits the phenomena. What you call solid things
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are really superficial. What you call space is really nothing but a great plane.
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I am in space and look down upon the insides of the things of
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which you only see the outsides. You could leave this plane yourself, if
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you could but summon up the necessary volition. A slight upward or downward motion
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would enable you to see all that I can see. The higher I mount,
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and the further I go from your plane, the more I can see,
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though of course I see it on a smaller scale. For example,
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I am ascending. Now I can see your neighbor, the hexagon, and
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his family in their several apartments. Now I see the inside of the theater,
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ten doors off from which the audience is only just departing, and on
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the other side a circle in his study, sitting at his books. Now
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I shall come back to you, And as a crowning proof, what do
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you say to my giving you a touch, just the least touch in your
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stomach, it will not seriously injure you, and the slight pain you may
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suffer cannot be compared with the mental benefit you will receive. Before I could
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utter a word of remonstrance, I felt a shooting pain in my inside,
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and a demoniacal laugh seemed to issue from within me. A moment afterwards,
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the sharp agony had ceased, leaving nothing but a dull ache behind, and
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the stranger began to reappear, saying, as he gradually increased in size.
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There I had not hurt you much, have I? If you are not
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convinced now, I don't know what will convince you. What say you?
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My resolution was taken. It seemed intolerable that I should endure existence subject to
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the arbitrary visitations of a magician who could thus play tricks with one's very stomach.
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If only I could, in any way manage to pin him against the
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wall till help came. Once more, I dashed, by hardest angle against
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him, at the same time alarming the whole household by my cries for aid.
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I believe at the moment of my onset the stranger had sunk below our
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plane and really found difficulty in rising. In any case, he remained motionless
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while I, hearing as I thought the sound of some help approaching, pressed
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against him with redoubled figure, and continued to shout for assistance. Convulsive shudder
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ran through the sphere. This must not be, I thought. I heard
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him say, either he must listen to reason, or I must have recourse
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to the last resource of civilization. Then, addressing me in a louder tone,
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he hurriedly exclaimed, listen, no stranger must witness what you have witnessed.
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Send your wife back at once before she enters the apartment. The Gospel
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of three dimensions must not be thus frustrated. Not thus must the fruits of
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one thousand years of waiting be thrown away. I hear her coming back back
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away from me, or you must go with me whither you know not,
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into the land of three dimensions fall Madman irregular, I exclaimed, Never will
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I release thee. Thou shalt pay the penalty of thine impostures. Ha Is
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it come to this, thundered the stranger. Then meet your fate. Out
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of your plane you go once twice thrice tis done. End of Section seventeen.