Latest / Daily Sci-Fi Books / 07 - Flatland- A Romance of Many Dimensions - Edwin Abbott Abbott
Transcript
- Part two, Section fifteen, concerning a stranger from Spaceland. From dreams, I proceed to facts. It was the last day of the nineteen hundred and ninety ninth 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. Footnote. When I say sitting, of course, I do not mean any change of attitude, such as you in Spaceland signify by that word. For as we have no feat, we can no more sit nor stand in your sense of the word than one of your souls 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 luster corresponding to the increase of volition. But on this and a thousand other kindred subjects, time forbids me to dwell. End of footnote. 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 wrapt 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 centers, 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 three squared or nine represents the number of square inches in a square whose side is three 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 three cubed 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 show the boy how a point, by moving through a length of three inches, makes a line of three inches, which may be represented by three, 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 three squared. 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 three, and if a straight line of three inches moving parallel to itself makes a square of three inches every way presented by three squared, 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 three cubed. Go to bed, said I, a little ruffled by his 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, endeavoring to form a retrospect of the year nineteen ninety nine and of the possibilities of the year two thousand, 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 glo 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 dishonoring 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 draft. 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, three cubed can have no meaning in geometry, But once there came a distinctly audible reply, the boy is not a fool, and three cubed 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 showed 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 to be felt by, then suddenly recoiling. Oh, it is not a woman, and there are no angles either. What a trait of one? Can it be that I have so misbehaved to a perfect circle. I am, indeed, in a sense a circle, replied the voice, and a more perfect circle than any in flat land. 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 for 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 the last sands had fallen, the second millennium had begun. Section sixteen. How the stranger vainly endeavored to reveal to me in word as the mysteries of spaceland. As soon as the sound of my wife's retreating footsteps had died away, I began to approach the stranger, with the intention of taking a nearer view, and of bidding him be seated. That 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 cutthroat, some monstrous, irregular eye soscely 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 dialog, which I will endeavor 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 rises 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 curious 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, pool, 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. I your Lordship is pleased to be merry. We also speak of length and high 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 northwood 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 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. I an I in my inside, an I in my stomach, your lordship jests, Stranger. I am in no jesting humor. 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. I such assertions 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 awhile 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 space. 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 anyone in the neighborhood possessing your Lordship's ample means of obtaining information. Stranger, 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 flat land 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 Limeland written upon your brain. Do you not remember? I say, how, when you entered the realm of Limeland, you are 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. 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 section becomes 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. Reader's note. The following paragraph makes reference to a diagram. The diagram shows a horizontal line cutting the line. There are three spheres. The first, or leftmost, is half above and half below the line, with its equator drawn at the level of the line to indicate the sphere's solid nature. Over it is written the sphere with his section at full size. The second, or central sphere is positioned about five sixths above the line and one sixth below, with a circumference drawn at the level of the line. This is labeled two, the sphere rising. The third, or the right hand sphere, is nearly all above the line, with just a sliver below, the circumference being drawn at the level of the line. This is labeled three, the sphere on the point of vanishing. At the right hand end of the line. There is an eye marked my eye looking towards the left end of Reader's note. 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 flat land, and you shall see my section become larger and larger. Every reader in Spaceland will easy 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 a 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 had 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 analogy. Then followed a still longer silence, after which he continued, our dialog sphere. Tell me, mister 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? And how many angles? I? Four sides and four angles sphere? Now stretch your imagination a little and conceive a square in flat land moving parallel to itself upward. I what northward sphere, No, not northward upward out of flatland altogether. If it moved northward, the southern point in the square would have to move through the positions previously occupied by the northern points. But that is not my meaning. I mean that every point in you, for you are a square, and will serve the purpose of my illustration. Every point in you, that is to say, in what you call your inside, is to pass upwards through space in such a way that no point shall pass through the position previously occupied by any other point, but each point shall describe a straight line of its own. This is all in accordance with analogy. Surely it must be clear to you. Restraining my impatience, for I was now under a strong temptation to rush blindly at my visitor and to precipitate him into space or out of flat land, 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 upwards. I presume it is describable in the language of flat land sphere. Oh, certainly, it is all plain and simple and in strict accordance with analogy. Only, by the way, you must not speak of the result as being a figure, but as a solid. But I will describe it to you, or rather not I, but analogy. We began with a single point, which, of course, being itself a point, has only one terminal point. One point produces a line with two terminal points. One line produces a square with four terminal points. Now you can yourself give the answer to your own question. One, two, four are evidently in geometrical progression. What is the next number? I? Eight? Sphere? Exactly? The one square produces a something which you do not as yet know a name for, but which we call a cube with eight terminal points. Now are you convinced I? And has this creature sides as well as angles or what you call terminal points? Sphere of course, and all according to analogy, But by the way, not what you call sides, but what we call sides you would call them solids. I And how many solids or so 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, A 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 no sides. A line, if I may so say, has two sides. For the points of a line may be called by courtesy its sides. A square has four sides, not two four. 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 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 seventeen. 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, not 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 I still 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 space 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 are have no tops nor bottoms full of money. I also see 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 space. 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 space is really nothing but a great plane. I am in space 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. The higher I mount, and the further I go from your plane, the more I can see, though of course I see it on a smaller scale. For example, I am ascending. Now I can see your neighbor, the hexagon, and his family in their several apartments. Now I see the inside of the theater, ten doors off from which the audience is only just departing, and on the other side a circle in his study, sitting at his books. Now I shall come back to you, And as a crowning proof, what do you say to my giving you a touch, just the least touch in your stomach, it will not seriously injure you, and the slight pain you may suffer cannot be compared with the mental benefit you will receive. Before I could utter a word of remonstrance, I felt a shooting pain in my inside, and a demoniacal laugh seemed to issue from within me. A moment afterwards, the sharp agony had ceased, leaving nothing but a dull ache behind, and the stranger began to reappear, saying, as he gradually increased in size. There I had not hurt you much, have I? If you are not convinced now, I don't know what will convince you. What say you? My resolution was taken. It seemed intolerable that I should endure existence subject to the arbitrary visitations of a magician who could thus play tricks with one's very stomach. If only I could, in any way manage to pin him against the wall till help came. Once more, I dashed, by hardest angle against him, at the same time alarming the whole household by my cries for aid. I believe at the moment of my onset the stranger had sunk below our plane and really found difficulty in rising. In any case, he remained motionless while I, hearing as I thought the sound of some help approaching, pressed against him with redoubled figure, and continued to shout for assistance. Convulsive shudder ran through the sphere. This must not be, I thought. I heard him say, either he must listen to reason, or I must have recourse to the last resource of civilization. Then, addressing me in a louder tone, he hurriedly exclaimed, listen, no stranger must witness what you have witnessed. Send your wife back at once before she enters the apartment. The Gospel of three dimensions must not be thus frustrated. Not thus must the fruits of one thousand years of waiting be thrown away. I hear her coming back back away from me, or you must go with me whither you know not, into the land of three dimensions fall Madman irregular, I exclaimed, Never will I release thee. Thou shalt pay the penalty of thine impostures. Ha Is it come to this, thundered the stranger. Then meet your fate. Out of your plane you go once twice thrice tis done. End of Section seventeen.