 Part 1, sections 11 and 12 of Flatland. This LibriVox recording is in the public domain. Recording by Ruth Golding. Flatland are romance of many dimensions by Edwin Abbott Abbott. Part 1, section 11, concerning our priests. It is high time that I should pass from these brief and discursive notes about things in Flatland to the central event of this book, my initiation into the mysteries of space. That is my subject. All that has gone before is merely preface. For this reason I must admit many matters of which the explanation would not, I flatter myself, be without interest for my readers. As for example our method of propelling and stopping ourselves, although destitute of feet. The means by which we give fixity to structures of wood, stone or brick, although of course we have no hands, nor can we lay foundations as you can, nor avail ourselves of the lateral pressure of the earth. The manner in which the rain originates in the intervals between our various zones so that the northern regions do not intercept the moisture from falling on the southern. The nature of our hills and mines, our trees and vegetables, our seasons and harvests. Our alphabet and method of writing adapted to our linear tablets. These and a hundred other details of our physical existence I must pass over. Nor do I mention them now except to indicate to my readers that their omission proceeds not from forgetfulness on the part of the author, but from his regard for the time of the reader. Yet before I proceed to my legitimate subject some few final remarks will no doubt be expected by my readers upon those pillars and mainstays of the constitution of flatland, the controllers of our conduct and shapers of our destiny, the objects of universal homage and almost of adoration need I say that I mean our circles or priests. When I call them priests, let me not be understood as meaning no more than the term denotes with you. With us our priests are administrators of all business, art and science, directors of trade, commerce, generalship, architecture, engineering, education, statesmanship, legislature, morality, theology. Doing nothing themselves they are the causes of everything worth doing that is done by others. Although popularly every one called a circle is deemed a circle, yet among the better educated classes it is known that no circle is really a circle, but only a polygon with a very large number of very small sides. In proportion to the number of the sides the polygon approximates to a circle, and when the number is very great say for example three or four hundred it is extremely difficult for the most delicate touch to feel any polygonal angles. Let me say rather it would be difficult for as I have shown above recognition by feeling is unknown among the highest society and to feel a circle would be considered a most audacious insult. This habit of abstention from feeling in the best society enables a circle the more easily to sustain the veil of mystery in which from his earliest years he is wont to unwrap the exact nature of his perimeter or circumference. Three feet being the average perimeter it follows that in a polygon of three hundred sides each side will be no more than the hundredth part of a foot in length or little more than the tenth part of an inch, and in a polygon of six or seven hundred sides the sides are little larger than the diameter of a spaceland pinhead. It is always assumed by courtesy that the chief circle for the time being has ten thousand sides. The ascent of the posterity of the circles in the social scale is not restricted as it is among the lower regular classes by the law of nature which limits the increase of sides to one in each generation. If it were so the number of sides in a circle would be a mere question of pedigree and arithmetic, and the four hundred and ninety-seventh descendant of an equilateral triangle would necessarily be a polygon with five hundred sides. But this is not the case. Nature's law prescribes two antagonistic decrees affecting circular propagation. First, that as the race climbs higher in the scale of development so development shall proceed at an accelerated pace. Second, that in the same proportion the race shall become less fertile. Consequently, in the home of a polygon of four or five hundred sides it is rare to find a sun. More than one is never seen. On the other hand the sun of a five hundred sided polygon has been known to possess five hundred and fifty or even six hundred sides. Art also steps in to help the process of the higher evolution. Our physicians have discovered that the small and tender sides of an infant polygon of the higher class can be fractured and his whole frame reset with such exactness that a polygon of two or three hundred sides sometimes, by no means always, for the process is attended with serious risk, but sometimes overleaps two or three hundred generations, and as it were doubles at a stroke the number of his progenitors and the nobility of his descent. Many a promising child is sacrificed in this way. Scarcely one out of ten survives. Yet so strong is the parental ambition among those polygons who are, as it were, on the fringe of the circular class, that it is very rare to find a nobleman of that position in society who has neglected to place his first born son in the circular neotherapeutic gymnasium before he has attained the age of a month. One year determines success or failure. At the end of that time the child has, in all probability, added one more to the tombstones that crowd the neotherapeutic cemetery. But on rare occasions a glad procession bears back the little one to his exultant parents, no longer a polygon but a circle, at least by courtesy. And a single instance of so blessed a result induces multitudes of polygonal parents to submit to similar domestic sacrifices which have a dissimilar issue. Section 12 of The Doctrine of Our Priests As to the doctrine of the circles it may briefly be summed up in a single maxim, attend to your configuration. Whether political, ecclesiastical or moral all their teaching has for its object the improvement of individual and collective configuration, with special reference of course to the configuration of the circles to which all other objects are subordinated. It is the merit of the circles that they have effectively suppressed those ancient heresies which led men to waste energy and sympathy in the vain belief that conduct depends upon will, effort, training, encouragement, praise or anything else but configuration. It was Pantocyclus, the illustrious circle mentioned above as the Queller of the Colour of Oat, who first convinced mankind that configuration makes the man, that if for example you are born anisosceles with two uneven sides, you will assuredly go wrong unless you have them made even, for which purpose you must go to the isosceles hospital. Similarly, if you are a triangle or square or even a polygon born with any irregularity, you must be taken to one of the regular hospitals to have your disease cured, otherwise you will end your days in the state prison or by the angle of the state executioner. All faults or defects, from the slightest misconduct to the most logitious crime, Pantocyclus attributed to some deviation from perfect regularity in the bodily figure, caused perhaps, if not congenital, by some collision in a crowd, by neglect to take exercise or by taking too much of it, or even by a sudden change of temperature resulting in a shrinkage or expansion in some too susceptible part of the frame. Therefore, concluded that illustrious philosopher, neither good conduct nor bad conduct is a fit subject in any sober estimation for either praise or blame. For why should you praise, for example, the integrity of a square who faithfully defends the interests of his client, when you ought, in reality, rather to admire the exact precision of his rectangles? Or again, why blame a lying, thievish isosceles when you ought rather to deplore the incurable inequality of his sides? Theoretically, this doctrine is unquestionable, but it has practical drawbacks. In dealing with an isosceles, if a rascal pleads that he cannot help stealing because of his unevenness, you reply that for that very reason, because he cannot help being a nuisance to his neighbours, you, the magistrate, cannot help sentencing him to be consumed, and there's an end of the matter. But in little domestic difficulties, whether penalty of consumption or death is out of the question, this theory of configuration sometimes comes in awkwardly. And I must confess, that occasionally, when one of my own hexagonal grandsons pleads, as an excuse for his disobedience, that a sudden change of the temperature has been too much for his perimeter, and that I ought to lay the blame not on him, but on his configuration, which can only be strengthened by abundance of the choicest sweet-meats, I neither see my way logically to reject, nor practically to accept, his conclusions. For my own part, I find it best to assume that a good sound scolding or castigation has some latent and strengthening influence on my grandsons' configuration, though I own that I have no grounds for thinking so. At all events I am not alone in my way of extricating myself from this dilemma, for I find that many of the highest circles, sitting as judges in law courts, use praise and blame towards regular and irregular figures. And in their homes, I know by experience that, when scolding their children, they speak about right or wrong, as vehemently and passionately, as if they believed that these names represented real existences, and that a human figure is really capable of choosing between them. Simply carrying out their policy of making configuration the leading idea in every mind, the circles reverse the nature of that commandment which in space-land regulates the relations between parents and children. With you children are taught to honour their parents. With us, next to the circles who are the chief object of universal homage, a man is taught to honour his grandson if he has one, or if not his son. By honour, however, is by no means meant indulgence, but a reverent regard for their highest interests. And the circles teach that the duty of fathers is to subordinate their own interests to those of posterity, thereby advancing the welfare of the whole state, as well as that of their own immediate descendants. The weak point in the system of the circles, if a humble square may venture to speak of anything circular as containing any element of weakness, appears to me to be found in their relations with women. As it is of the utmost importance for society that irregular births should be discouraged, it follows that no woman who has any irregularities in her ancestry is a fit partner for one who desires that his posterity should rise by regular degrees in the social scale. Now the irregularity of a male is a matter of measurement, but as all women are straight and therefore visibly regular, so to speak, one has to devise some other means of ascertaining what I may call their invisible irregularity. That is to say, their potential irregularities as regards possible offspring. This is effected by carefully kept pedigrees, which are preserved and supervised by the state, and without a certified pedigree no woman is allowed to marry. Now it might have been supposed that a circle, proud of his ancestry and regardful for a posterity which might possibly issue her after in a chief circle, would be more careful than any other to choose a wife who had no blot on her escutcheon. But it is not so. The care in choosing a regular wife appears to diminish as one rises in the social scale. Nothing would induce an aspiring isosceles who had hopes of generating an equilateral son to take a wife who reckoned a single irregularity among her ancestors. A square or pentagon, who is confident that his family is steadily on the rise, does not inquire above the five hundredth generation. A hexagon, or dodecahedron, is even more careless of the wife's pedigree. But a circle has been known deliberately to take a wife who has had an irregular great-grandfather, and all because of some slight superiority of lustre, or because of the charms of the low voice, which with us, even more than with you, is thought an excellent thing in woman. Such ill-judged marriages are, as might be expected, barren if they do not result in positive irregularity or in diminution of sides. But none of these evils have hitherto proved sufficiently deterrent. The loss of a few sides in a highly developed polygon is not easily noticed, and is sometimes compensated by a successful operation in the neotherapeutic gymnasium, as I have described above. And the circles are too much disposed to acquiesce in infecundity as a law of the superior development. Yet, it's as evil be not arrested, the gradual diminution of the circular class may soon become more rapid, and the time may not be far distant when, the race being no longer able to produce a chief circle, the constitution of flatland must fall. One other word of warning suggests itself to me, though I cannot so easily mention a remedy, and this also refers to our relations with women. About three hundred years ago it was decreed by the chief circle that, since women are deficient in reason but abundant in emotion, they ought no longer to be treated as rational nor receive any mental education. The consequence was that they were no longer taught to read, nor even to master arithmetic enough to enable them to count the angles of their husband or children, and hence they sensibly declined during each generation in intellectual power, and this system of female non-education or quietism still prevails. My fear is that, with the best intentions, this policy has been carried so far as to react injuriously on the male sex. For the consequence is that, as things now are, we males have to lead a kind of bilingual, and I may almost say, bimental existence. With the women we speak of love, duty, right, wrong, pity, hope, and other irrational and emotional conceptions which have no existence, and the fiction of which has no object except to control feminine exuberances. But among ourselves and in our books we have an entirely different vocabulary, and I may almost say, idiom. Love then becomes the anticipation of benefits, duty becomes necessity or fitness, and other words are correspondingly transmuted. Moreover, among women we use language implying the utmost deference for their sex, and they fully believe that the chief circle himself is not more devoutly adored by us than they are. But behind their backs they are both regarded and spoken of by all except the very young as being little better than mindless organisms. Our theology also in the women's chambers is entirely different from our theology elsewhere. Now, my humble fear is that this double training in language as well as in thought imposes somewhat too heavy a burden upon the young, especially when at the age of three years old they are taken from the maternal care and taught to unlearn the old language except for the purpose of repeating it in the presence of their mothers and nurses, and to learn the vocabulary and idiom of science. Already me thinks I discern a weakness in the grasp of mathematical truth at the present time as compared with the more robust intellect of our ancestors three hundred years ago. I say nothing of the possible danger if a woman should ever surreptitiously learn to read and convey to her sex the result of her perusal of a single popular volume, nor of the possibility that the indiscretion or disobedience of some infant male might reveal to a mother the secrets of the logical dialect. On the simple ground of the enfeebling of the male intellect I rest this humble appeal to the highest authorities to reconsider the regulations of female education. End of Section 12 and of Part 1. Recording by Ruth Golding. Part 2. Section 13 and 14 of Flatland. This LibriVox recording is in the public domain, recording by Ruth Golding. Flatland, a romance of many dimensions by Edwin Abbott Abbott. Part 2. Other Worlds. O brave new worlds that have such people in them. Section 13. How I Had a Vision of Lineland. It was the last day but one of the nineteen-hundred and ninety-ninth year of our era and the first day of the long vacation. Having amused myself till a late hour with my favourite recreation of geometry I had retired to rest with an unsolved problem in my mind. In the night I had a dream. Readers note. The following paragraph makes reference to a diagram. The diagram is headed My View of Lineland. Beneath this title and centrally placed is a square labelled myself. Under that, from left to right on a horizontal plane, are four dots labelled women, a short dash labelled a boy, six longer dashes marked men, then directly below myself a thick dash labelled the king, with an eye looking out from either end. Under the eyes is written the king's eyes much larger than the reality, showing that his majesty could see nothing but a point. The horizontal line then continues towards the right, with seven dashes marked men, one dash marked a boy, and seven dots marked women. End of Readers note. I saw before me a vast multitude of small straight lines, which I naturally assumed to be women, interspersed with other beings still smaller and of the nature of lustrous points, all moving to and fro in one and the same straight line, and as nearly as I could judge with the same velocity. A noise of confused, multitudinous chirping or twittering issued from them at intervals as long as they were moving, but sometimes they ceased from motion, and then all was silence. Approaching one of the largest of what I thought to be women, I accosted her, but received no answer. A second and third appeal on my part were equally ineffectual. Losing patience at what appeared to me intolerable rudeness, I brought my mouth into a position full in front of her mouth, so as to intercept her motion, and loudly repeated my question. Woman, what signifies this concourse, and this strange and confused chirping, and this monotonous motion to and fro in one and the same straight line? I am no woman," replied the small line, I am the monarch of the world, but thou wence intrudous thou into my realm of line-land. Receiving this abrupt reply, I begged pardon if I had in any way startled or molested his royal highness, and, describing myself as a stranger, I besought the king to give me some account of his dominions, but I had the greatest possible difficulty in obtaining any information on points that really interested me, for the monarch could not refrain from constantly assuming that whatever was familiar to him must also be known to me, and that I was simulating ignorance in jest. However, by persevering questions, I elicited the following facts. It seemed that this poor ignorant monarch, as he called himself, was persuaded that the straight line which he called his kingdom, and in which he passed his existence, constituted the whole of the world, and indeed the whole of space. Not being able either to move or to see, save in his straight line, he had no conception of anything out of it. Though he had heard my voice when I first addressed him, the sounds had come to him in a manner so contrary to his experience that he had made no answer, seeing no man, as he expressed it, and hearing a voice, as it were, from my own intestines. Until the moment when I placed my mouth in his world, he had neither seen me, nor heard anything except confused sounds, beating against what I called his side, but what he called his inside or stomach, nor had he even now the least conception of the region from which I had come. Inside his world, or line, all was a blank to him. Nay, not even a blank, for a blank implies space, say, rather, all was non-existent. His subjects, of whom the small lines were men and the points women, were all alike confined in motion and eyesight to that single straight line, which was their world. It needs scarcely be added that the whole of their horizon was limited to a point. Nor could any one ever see anything but a point, man, woman, child, thing, each was a point to the eye of a line-lander. Only by the sound of the voice could sex or age be distinguished. Moreover, as each individual occupied the whole of the narrow path, so to speak, which constituted his universe, and no one could move to the right or left to make way for passers-by, it followed that no line-lander could ever pass another. Once neighbours, always neighbours. Neighbourhood with them was like marriage with us. Neighbours remained neighbours till death did them part. Such a life, with all vision limited to a point, and all motion to a straight line, seemed to me inexpressibly dreary, and I was surprised to note the vivacity and cheerfulness of the king. Wondering whether it was possible, amid circumstances so unfavourable to domestic relations, to enjoy the pleasures of conjugal union, I hesitated for some time to question his royal highness on so delicate a subject. But at last I plunged into it, by abruptly inquiring as to the health of his family. My wives and children, he replied, are well and happy. Staggered at this answer, for in the immediate proximity of the monarch, as I had noted in my dream before I entered line-land, there were none but men. I ventured to reply, pardon me, but I cannot imagine how your royal highness can at any time either see or approach their majesties when there are at least half a dozen intervening individuals whom you can neither see through nor pass by. Is it possible that in line-land proximity is not necessary for marriage and for the generation of children? How can you ask so absurd a question? replied the monarch. If it were indeed, as you suggest, the universe would soon be depopulated. No, no, neighbourhood is needless for the union of hearts, and the birth of children is too important a matter to have been allowed to depend on such an accident as proximity. You cannot be ignorant of this. Yet, since you are pleased to affect ignorance, I will instruct you as if you were the various baby in line-land. Know then that marriages are consummated by means of the faculty of sound and the sense of hearing. You are, of course, aware that every man has two mouths or voices, as well as two eyes, a base at one and a tenor at the other of his extremities. I should not mention this, but that I have been unable to distinguish your tenor in the course of our conversation. I replied that I had but one voice, and that I had not been aware that his Royal Highness had two. That confirms my impression, said the king, that you are not a man, but a feminine monstrosity with a base voice and an utterly uneducated ear. But to continue. Nature herself having ordained that every man should wed two wives. Why, too, asked I. You carry your affected simplicity too far, he cried. How can there be a completely harmonious union without the combination of the four in one? Fizz the base and tenor of the man, and the soprano and contralto of the two women. But supposing, said I, that a man should prefer one wife or three. It is impossible, he said. It is as inconceivable as that two and one should make five, or that the human eye should see a straight line. I would have interrupted him, but he proceeded as follows. Once in the middle of each week a law of nature compels us to move two and fro with the rhythmic motion of more than usual violence, which continues for the time you would take to count a hundred and one. In the midst of this choral dance, at the fifty-first pulsation, the inhabitants of the universe pause in full career, and each individual sends forth his richest, fullest, sweetest strain. It is in this decisive moment that all our marriages are made. So exquisite is the adaptation of base to treble, of tenor to contralto, that oftentimes the loved ones, though twenty thousand leagues away, recognise at once the responsive note of their destined lover, and penetrating the paltry obstacles of distance, love unites the three. The marriage, in that instant consummated, results in a threefold male and female offspring, which takes its place in line-land. What, always threefold, said I, must one wife then always have twins? Base voice monstrosity, yes, replied the King, how else could the balance of the sexes be maintained, if two girls were not born for every boy? Would you ignore the very alphabet of nature? He ceased, speechless for fury, and some time elapsed before I could induce him to resume his narrative. You will not, of course, suppose that every bachelor among us finds his mates at the first wooing in this universal marriage-chorus. On the contrary, the process is by most of us many times repeated. You are the hearts whose happy lot it is at once to recognise in each other's voices the partner intended for them by providence, and to fly into a reciprocal and perfectly harmonious embrace. With most of us the courtship is of long duration. The wooer's voices may perhaps accord with one of the future wives, but not with both, or not at first with either, or the soprano and contralto may not quite harmonise. In such cases, nature has provided that every weekly chorus shall bring the three lovers into closer harmony. Each trial of voice, each fresh discovery of discord, almost imperceptibly induces the less perfect to modify his or her vocal utterance, so as to approximate to the more perfect. And after many trials, and many approximations, the result is at last achieved. There comes a day at last when, while the won'ted marriage-chorus goes forth from universal line-land, the three far-off lovers suddenly find themselves in exact harmony. And before they are aware, the wedded triplet is wrapped vocally into a duplicate embrace. And nature rejoices over one more marriage and over three more births. Section 14 How I Veinly Tried to Explain the Nature of Flatland Thinking that it was time to bring down the monarch from his raptures to the level of common sense, I determined to endeavour to open up to him some glimpses of the truth, that is to say, of the nature of things in flatland. So I began thus. How does your royal highness distinguish the shapes and positions of his subjects? I, for my part, noticed by the sense of sight, before I entered your kingdom, that some of your people are lines and others points, and that some of the lines are larger. You speak of an impossibility, interrupted the king. You must have seen a vision. For to detect the difference between a line and a point by the sense of sight is, as everyone knows, in the nature of things, impossible. But it can be detected by the sense of hearing, and by the same means my shape can be exactly as attained. Behold me, I am a line, the longest in line-land, over six inches of space—of length, I ventured to suggest. Fool, said he, space is length, interrupt me again, and I have done. I apologised, 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 six point four five seven inches, in fur that one of my mouths is six point four five seven inches further from them than the other, and accordingly know my shape to be six point four five seven 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 recognised 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, no 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 man, 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 coarser 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 senses and statistics, local, corporal, mental and spiritual, of every living being in Lionland, 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 lily-pution 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 Lionland must be deplorably dull, to see nothing but a point, not even to be able to contemplate a straight line. No, not even to know what a straight line is. To see, yet to 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 Lionland 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. But 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 my 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 to 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 line-land in the direction which I desire to indicate to you. Readers note, the following paragraph makes reference to a diagram. The diagram shows a horizontal line. At the left is marked line-land, with an arrow pointing rightward. At the right, on the line, is a broad dash labelled the King. In the centre, on the line, is a horizontally shaded square over which is written my body just before I disappeared. End of Readers note. At the word, I began to move my body out of line-land. 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's vanished, she's dead. I'm not dead, replied I. I am simply out of line-land, 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 also see 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 this at last convince you? And with that I once more entered line-land, 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, this sotted 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 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 war cry, increasing environments till at last me thought it rivaled the roar of an army of a hundred thousand isosceles and the artillery of a thousand pentagons. Spellbound 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. End of Section 14. Recording by Ruth Golding. Part 2. Sections 15 to 17 of Flatland. This LibriVox recording is in the public domain. Recording by Ruth Golding. Flatland, a romance of many dimensions, by Edwin Abbott Abbott. Part 2. Section 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. 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 feet 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 lustre 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 orphaned 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 wrapped 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 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 represented 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 a 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, 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! Straight away 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 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 ended 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 frailess ex 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 to 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 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 for her own retirement had long passed, with many reiterated apologies for her recent indiscretion, she had last retreated to her apartment. I glanced at the half hour-glass. The last sands had fallen. The second millennium had begun. Section 16 How the stranger vainly endeavored to reveal to me in words the mysteries of space-land. 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, 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 cite 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 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 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, what do you know of space, defined 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 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 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 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 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 for sooth. 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 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 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 neighbourhood 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 plain. 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 plain 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 13 inches in diameter, one placed on the top of the other. When I cut through your plain as I am now doing, I make in your plain a section which you very rightly call a circle. For even a steer, which is my proper name in my own country, if he manifest himself at all to an inhabitant of flat land, must needs manifest himself as a circle. Do you not remember? For I, who see all things discerned last night the fantasmal vision of limeland written upon your brain, do you not remember, I say, how, when you entered the realm of limeland, you were compelled to manifest yourself to the king, not as a square, but as a line? Because that linear realm had not dimensions enough to represent the whole of you, but only a slice or section of you? In precisely the same way, your country of two dimensions is not spacious enough to represent me, a being of three, but can only exhibit a slice or section of me, which is what you call a circle. 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 plain of flat land, 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. Readers 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 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 Readers 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 Flatland and you shall see my section become larger and larger. Every Reader in Spaceland will easily understand that my mysterious guest was speaking the language of truth and even of simplicity. But to me, proficient though I was in Flatland mathematics, it was by no means a simple matter. The rough diagram given above will make it clear to any Spaceland child that the sphere ascending in the three positions indicated there must needs have manifested himself to me, or to any Flatlander, as a circle, at first 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 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, and 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, all together. 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 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 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 of 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 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? 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 naught 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, naught, two, four. What progression do you call that? I, arosmetical. 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 chocolate, 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, 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 have no tops nor bottoms, full of money. I also see two tablets of accounts. I'm 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 I'm 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 evolution. 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 neighbour the hexagon and his family in their several apartments. Now I see the inside of the theatre 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 have 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 vigor and continued to shout for assistance. A 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 civilisation. 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, wither you know not, into the land of three dimensions. Fool! Madman! Irregular! I exclaimed. Never will I release thee. Thou shalt pay the penalty of thine impostures. Ah! Is it come to this? Thundered the stranger. Then meet your fate. Out of your plane you go. Once, twice, thrice. Tears done!