Flatland: A Romance of Many Dimensions [Pilot project. eBook available to selected US libraries only] 9781400866649

Table of contents :
CONTENTS
PREFACE TO THE SECOND AND REVISED EDITION
INTRODUCTION
Part I This World
1 Of the Nature of Flatland
2 Of the Climate and Houses in Flatland
3 Concerning the Inhabitants of Flatland
4 Concerning the Women
5 Of our Methods of Recognizing one another
6 Of Recognition by Sight
7 Concerning Irregular Figures
8 Of the Ancient Practice of Painting
9 Of the Universal Colour Bill
10 Of the Suppression of the Chromatic Sedition
11 Concerning our Priests
12 Of the Doctrine of our Priests
Part II Other Worlds
13 How I had a Vision of Lineland
14 How in my Vision I endeavoured to explain the nature of Flatland, but could not
15 Concerning a Stranger from Spaceland
16 How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland
17 How the Sphere, having in vain tried words, resorted to deeds
18 How I came to Spaceland and what I saw there
19 How, though the Sphere showed me other mysteries of Spaceland, I still desired more; and what came of it
20 How the Sphere encouraged me in a Vision
21 How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success
22 How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result

Citation preview

FLATLAND

“Fie, fie, how franticly I square my talk!”

SPACELAND

Flatland

A Romance of Many Dimensions EDWIN ABBOTT ABBOTT (1838–1926) With an introduction by THOMAS BANCHOFF

Princeton University Press Princeton and Oxford

Introduction copyright © 1991 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, NJ 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved New Princeton Science Library paperback printing, 2015 New Paperback ISBN 978-0-691-16555-4 The Library of Congress Control cataloged the 1991 edition as follows Abbott, Edwin Abbott, 1838–1926. Flatland : a romance of many dimensions / Edwin Abbott Abbott ; with a new introduction by Thomas Banchoff. p. cm.-(Princeton science library) Reprint. Originally published: 6th ed. New York : Dover Publications, 1953. ISBN: 978-0-691-12366-0 1. Fourth dimension. I. Title. II. Series. [QA699.A13 1991] 530.1'1-dc20 90–28266 British Library Cataloging-in-Publication Data is available This book has been composed in Adobe Garamond Pro Printed on acid-free paper. ∞ Printed in the United States of America 1 3 5 7 9 10 8 6 4 2

To The Inhabitants of Space in General And H. C. in Par tic u lar This Work is Dedicated By a Humble Native of Flatland In the Hope that Even as he was Initiated into the Mysteries Of Three Dimensions Having been previously conversant With Only Two So the Citizens of that Celestial Region May aspire yet higher and higher To the Secrets of Four Five or even Six Dimensions Thereby contributing To the Enlargement of the Imagination And the possible Development Of that most rare and excellent Gift of Modesty Among the Superior Races Of Solid Humanity

contents to Flatland: The Book contents to Flatland: The Book contents to Flatland: The Book preface to the second and revised edition

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introduction Part I This World Part I This World section Part I This World section 1 Of the Nature of Flatland 15 section Flatland 15 Flatland 16 21 Of Of the the Nature Climateofand Houses in the Nature of Flatland 3 Flatland 231 Of Climate and Houses in Concerning the Inhabitants 15 of Flatland 16 18 243 Concerning Of the Climate and Houses in Flatland 16 4 the Inhabitants of 18 Concerning the Women 20 345 Concerning the Inhabitants of Flatland 18 6 20 one another 24 Of our MethodsWomen of Recognizing 465 Concerning the Women 8 Of our Methods of Recognizing one another 24 Of Recognition by Sight 20 28 our Methods of Recognizing 24 675 Concerning Of Recognition by Sight 28 Irregular Figures 32one another 12 16 678 Concerning Of Recognition by Sight 28 Irregular Figures 32 Of the Ancient Practice of Painting 34 798 Concerning Irregular Figures 32 20 Of the Ancient Practice of Painting 34 Of the Universal Colour Bill 36 8 Ancient Practice of Painting 34 9 Of the Universal Colour Bill 36 22 10 Of the Suppression of the Chromatic Sedition 39 9 Of the Universal Colour Bill 36 24 10 Suppression of the 42 Chromatic Sedition 39 11 Concerning our Priests 10 Of of the Chromatic 39 11 Concerning our Priests 42 12 Of the the Suppression Doctrine of our Priests 44 Sedition 27 11 Concerning our Priests 42 12 Of the Doctrine of our Priests 44 30 12 Of the Doctrine of our Priests 44 32 Part II Other Worlds Part II Other Worlds 13 How I had a Vision of Lineland 51 Part II Other Worlds 13 How VisionI of Lineland to 51 explain 39 14 How Iinhad myaVision endeavoured 13 How I had a Vision of Lineland 51 explain 14 the nature in myofVision I endeavoured to Flatland, but could not 54 14 in myofVision I endeavoured to explain the nature but not 54 58 42 15 How Concerning aFlatland, Stranger fromcould Spaceland the nature of Flatland, but could not 15 Concerning a Stranger from Spaceland 58 46 16 How the Stranger vainly endeavoured to54reveal 15 Concerning a Stranger from Spaceland 58 61 16 How Stranger endeavoured to reveal to methe in words thevainly mysteries of Spaceland 16 How Stranger vainly endeavoured to reveal to methe in words mysteries of Spaceland 61 49 17 How the Sphere,the having in vain tried words, resorted to deeds 68 to methe in words mysteries of Spaceland 17 How Sphere,the having in vain tried words, 61 resorted to deeds 55 68 17 How the Sphere, having in vain tried words, resorted to deeds 68 18 I came to Spaceland and what I saw there 57 70 19 20 21 22

How, though the Sphere showed me other mysteries of Spaceland, I still desired more; and what came of it 61 73 How the Sphere encouraged me in a Vision 66 79 How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success 68 81 How I then tried to diff use the Theory of Three Dimensions by other means, and of the result 70 83

11 11 11

preface to the Second and Revised Edition By the Editor

If my poor Flatland friend retained the vigour of mind which he enjoyed when he began to compose these Memoirs, I should not now need to represent him in this Preface, in which he desires, firstly, to return his thanks to his readers and critics in Spaceland, whose appreciation has, with unexpected celerity, required a second edition of his work; secondly, to apologize for certain errors and misprints (for which, however, he is not entirely responsible); and, thirdly, to explain one or two misconceptions. But he is not the Square he once was. Years of imprisonment, and the still heavier burden of general incredulity and mockery, have combined with the natural decay of old age to erase from his mind many of the thoughts and notions, and much also of the terminology, which he acquired during his short stay in Spaceland. He has, therefore, requested me to reply in his behalf to two special objections, one of an intellectual, the other of a moral nature. The first objection is, that a Flatlander, seeing a Line, sees something that must be thick to the eye as well as long to the eye (otherwise it would not be visible, if it had not some thickness); and consequently he ought (it is argued) to acknowledge that his countrymen are not only long and broad, but also (though doubtless to a very slight degree) thick or high. This objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess, when I first heard it, I knew not what to reply. But my poor old friend’s answer appears to me completely to meet it. “I admit,” said he—when I mentioned to him this objection—“I admit the truth of your critic’s facts, but I deny his conclusions. It is true that we have really in Flatland a Third unrecognized Dimension called ‘height’, just as it is also true that you

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have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call ‘extra-height.’ But we can no more take cognizance of our ‘height’ than you can of your ‘extra-height.’ Even I—who have been in Spaceland, and have had the privilege of understanding for twenty-four hours the meaning of ‘height’—even I cannot now comprehend it, nor realise it by the sense of sight or by any process of reason: I can but apprehend it by faith. “The reason is obvious. Dimension implies direction, implies mea surement, implies the more and the less. Now, all our lines are equally and infinitestimally thick (or high, whichever you like); consequently, there is nothing in them to lead our minds to the conception of that Dimension. No ‘delicate micrometer’—as has been suggested by one too hasty Spaceland critic—would in the least avail us; for we should not know what to measure, nor in what direction. When we see a Line, we see something that is long and bright; brightness, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished. Hence, all my Flatland friends—when I talk to them about the unrecognized Dimension which is somehow visible in a Line—say, ‘Ah, you mean brightness’: and when I reply, ‘No, I mean a real Dimension,’ they at once retort, ‘Then mea sure it, or tell us in what direction it extends’: and this silences me, for I can do neither. Only yesterday, when the Chief Circle (in other words our High Priest) came to inspect the State Prison and paid me his seventh annual visit, and when for the seventh time he put me the question, ‘Was I any better?’ I tried to prove to him that he was ‘high,’ as well as long and broad, although he did not know it. But what was his reply? ‘You say I am “high”; mea sure my “high-ness” and I will believe you.’ What could I do? How could I meet his challenge? I was crushed; and he left the room triumphant. “Does this still seem strange to you? Then put yourself in a similar position. Suppose a person of the Fourth Dimension, condescending to visit you, were to say, ‘Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you infer a Solid (which is of Th ree); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly mea sure it! What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Th ird Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra- Cubes—we are all liable to the same errors,

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all alike the Slaves of our respective Dimensional prejudices, as one of our Spaceland poets has said—

‘One touch of Nature makes all worlds akin.’ ”* On this point the defense of the Square seems to me to be impregnable. I wish I could say that his answer to the second (or moral) objection was equally clear and cogent. It has been objected that he is a woman-hater, and as this objection has been vehemently urged by those whom Nature’s decree has constituted the somewhat larger half of the Spaceland race, I should like to remove it, so far as I can honestly do so. But the Square is so unaccustomed to the use of the moral terminology of Spaceland that I should be doing him an injustice if I were literally to transcribe his defense against this charge. Acting, therefore, as his interpreter and summarizer, I gather that in the course of an imprisonment of seven years he has himself modified his own personal views, both as regards Women and as regards the Isosceles or Lower Classes. Personally, he now inclines to the opinion of the Sphere that the Straight Lines are in many important respects superior to the Circles. But, writing as a Historian, he has identified himself (perhaps too closely) with the views generally adopted by Flatland, and (as he has been informed) even by Spaceland, Historians; in whose pages (until very recent times) the destinies of Women and of the masses of mankind have seldom been deemed worthy of mention and never of careful consideration. In a still more obscure passage he now desires to disavow the Circular or aristocratic tendencies with which some critics have naturally credited him. While doing justice to the intellectual power with which a few Circles have for many generations maintained their supremacy over immense multitudes of their countrymen, he believes that the facts of Flatland, speaking for themselves without comment on his part, declare that Revolutions cannot always be suppressed by slaughter, and that Nature, in sentencing the Circles to infecundity, has condemned them to ultimate failure—“and herein” he says, “I see a fulfi lment of the great Law of all worlds, that while the wisdom of Man thinks it is working one thing, the wisdom of Nature constrains it to work another, and quite a diff erent and far better, thing.” For the rest, he begs his readers not to suppose that every minute detail in

* The Author desires me to add, that the misconception of some of his critics on this matter has induced him to insert in his dialogue with the Sphere, certain remarks which have a bearing on the point in question, and which he had previously omitted as being tedious and unnecessary.

The Book

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the daily life of Flatland must needs correspond to some other detail in Spaceland; and yet he hopes that, taken as a whole, his work may prove suggestive, as well as amusing, to those Spacelanders of moderate and modest minds who—speaking of that which is of the highest importance, but lies beyond experience—decline to say on the one hand, “This can never be,” and on the other hand, “It must needs be precisely thus, and we know all about it.”

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F L ATL AND : an introduction Thomas F. Banchoff

Flatland first appeared over one hundred years ago and a dozen different editions have been published since then. Why now put out a new edition of this book in the Princeton Science Library? For each generation of readers, Edwin Abbott Abbott’s classic story of encounter between beings from different dimensions has had different significance. In 1884, the social satire of the limited perspective of Victorian England was as important as the comments on the use of analogy in treating higher dimensions, and both of these elements were clarified by the introduction to the second edition, purportedly written by an editor but actually written by Abbott himself. When the book was reissued in 1926, the main stimulus for considering higher dimensions was relativity theory, and a new introduction was written by William Garnett, a physicist student of Abbott’s. In 1952, when the first modern edition appeared, it was again a physicist, Banesh Hoffmann, who wrote the introduction, referring to the connetion between the dimensional analogy and the curvature of space. Other recent introductions have come from science-fiction and fantasy writers, a computer scientist, and a social historian. In our own day, there are new reasons to reconsider this book and its fundamental ideas. Abbott challenged his readers to imagine trying to understand the nature of phenomena in higher dimensions if all they could see directly were lower-dimensional slices. That is precisely the situation that radiologists face today as they analyze the slices produced by CAT scans or magnetic resonance imaging, in attempts to reconstruct the forms of objects in space by studying their planar cross-sections. But Abbott did not

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want his readers to stop at two-dimensional representations of the third dimension: What would happen if we were to apply the same techniques one dimension higher, and attempt to conceive objects from a spatial fourth dimension by considering threedimensional cross-sections? For mathematicians in Abbott’s day, that challenge was daunting. Their methods were adequate for visualizing surfaces in three-dimensional space, but it was only with great difficulty that these techniques could be applied to objects from higher dimensions. Frequently, researchers relied more and more on formal and abstract methods, leaving images behind. Now, however there is a new interest in visualization, primarily because of the emergence of modern computer graphics, which can literally bring these higher dimensions into view. We can manipulate objects in four dimensions and see their three-dimensional slices tumbling on the computer screen. But how do we interpret these images? There is no better introduction to the problem of dealing with higherdimensional slicing phenomena than this book, Flatland. It is not necessary to read an introduction to appreciate Flatland, and at this point the reader might be encouraged to go ahead and read the book, returning to the introduction only later on to see if some of the questions that arise are answered here. In this short commentary, we shall address the satirical elements in Flatland, sketch Abbott’s background and interests, provide overviews of the style and historical context of the work, and investigate Flatland ’s influence on contemporary approaches to new modes of visualization of higher dimensions, particularly computer graphics. Social Satire in Flatland When Edwin Abbott Abbott wrote his little masterpiece over one hundred years ago, he did it for several reasons. Some of these were obvious to his readers and remain obvious today. Others had to be explained by Abbott himself in the introduction to the second edition, which followed one month after the first, at the end of 1884. Still others were immediately clear to Victorian readers but need some clarification for readers in our day. Fortunately, there is enough evidence, direct and indirect, in this remarkable author’s forty-five books and other writings to give us answers to many questions, though not all, since the subject itself continues to encourage us to raise new ones. The first subject that has to be addressed is the treatment of women in Flatland. Abbott was a social reformer who criticized a great many aspects of the limitations of Victorian society. He was a firm believer in equality of educational opportunity, across social classes and in particular for women. He participated actively in the efforts to bring about changes, and the frustration he felt from the resistance of the educational establishment is mirrored in the satire of Flatland. This was the first generation in



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which women were permitted to attend classes at Oxford and Cambridge, but their access was still quite limited. Although there were many schools where a boy could be trained for the demanding university entrance examinations, there were few comparable opportunities for girls, and many of the young women who gained entrance to universities, like Abbott’s daughter, had received much of their education at home, often from private tutors. It was partially to aid in this effort that Abbott composed his Hints for Home Teaching, directed at parents who wished to help their children prepare for higher education. Abbott was also a vocal leader in the Teachers’ Training Syndicate, formed and primarily supported by the major female educators of Victorian England, who extensively praised Abbott for his efforts on behalf of education reform, in particular for proposing alternate ways of qualifying for entrance into university studies. The narrow-minded attitudes that blocked these efforts show up quite clearly in Flatland, where females are presented as incapable of comprehending the education given to males. Many people ignored one whole dimension of women’s existence, as symbolized by the representaiton of women in Flatland as straight line segments. Under the guise of protecting women, they kept them away from the means by which they might better their station. Abbott’s sentiments in this matter are clear by the end of Flatland, where the narrator comes to realize that the very (female) virtues his society has been putting down are ones that are to be most prized. Abbott was one of the first to recognize the implications of a “two cultures” society. The men in Flatland epitomize the rational, emphasizing the importance of that which can be measured empirically and described in precise scientific language. Qualitative properties not susceptible to such quantification are relegated to the world of women, who have an absolute corner on not only the intuitve aspects of knowledge but also the abstract concepts such as loyalty and love which are difficult to translate into a strictly utilitarian construct. A Square, the two-dimensional narrator of Flatland, considers himself enlightened when he propounds the view that the strain of maintaining two separate languages, one for conversation among men and one for talking with women, exacts too great a toll on young minds. There is always the danger that the language of men will be revealed to women, reminiscent of the prohibitions in earlier societies against teaching slaves to read. As a religious man with a well-developed traditional sense of morality, Abbott clearly did not subscribe to the prevailing scientistic view of knowledge, and he more than once pointed out the dangers of letting one side of the personality completely dominate the other. Abbott was a teacher who extolled blanace, and Flatland reduces to an absurdity the single-minded tendency of choosing either the totally rationalistic or the totally intuitive.

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Another aspect of the satire in Flatland is the treatment of those who did not fit in. In the rigid society of Victorian England, there was little tolerance for irregularity. It was often associated with criminal tendency, and some theories blamed deviant behavior on an abnormal shape in the frame or the scull. Frequently, the unusual were segregated from the rest of society in asylums. The rest of society maintained a fascination with the freakish element, and asylums often had viewing galleries so ordinary people could observe the activities and antics of the inmates. Abbott’s suggestion that irregulars be eliminated in a Swiftian exaggeration, especially when coupled with the cruel plan of keeping a number of these unfortunates available as object lessons, an expendable supply of individuals with no rights at all, to be studied by the regular Flatland pupils. Especially pointed are his remarks about the appearance of irregularities among the upper classes. His readers could certainly supply their own examples of men destined for high station who had failed to fulfill the prerequisites for completing a university education. Such people are not fit for lower employment, and so they simply cause trouble within society. Abbott’s solution is worthy of Jonathan Swift’s Modest Proposal: anyone who fails the final examinations of the university will be incarcerated for life or subjected to a painless death. No reader should miss the satirical intent of such a paragraph. Abbott definitely saw education as a means for students to transcend the social class into which they were born, and he was particularly censorious of the efforts of those who used education to perpetuate class distinctions. A basic liberal education should be provided to all students, so that some of them might go on to higher education and others might go into business from a more enlightened perspective. Abbott’s students at the City of London School learned practical science and art as well as theoretical subjects. Abbott resigned from the headmastership just seven years after the writing of Flatland, during a crisis over the splitting of the curriculum into seperate “modern” and classical sides. In Flatland, Abbott echoes this crisis as he satirically contrasts the “feeling” of the lower classes with the more refined “seeing” of the educated part of society. By preferring the more remote way of sensing, the higher classes built a barrier between themselves and the lower strata. Several of Abbott’s students who broke societal barriers were only too happy to thank their old master. Most famous was the Prime Minister of England, H. H. Asquith, Lord Oxford, a man from humble background who gained entrance to Oxford as a result of his classical education under Abbott. Bramwell Booth, the second director general of the Salvation Army, also thanked Abbott for his encouragement and for promoting the development of his self-worth. Sir Israel Gollancz went on from the City of London School to study at the University of London at a time when



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educational opportunities for Jewish students were meager. He always sent his Shakespearean volumes to his old teacher, and induced him to accept membership into the British Academy, which Gollancz had helped to found. It is evident that Abbott was proud of his former pupils and that he preferred the kind of education that allowed people to rise on the basis of merit rather than of the societal class into which they were born. The prevailing system is the target of his satire in Flatland. Abbott’s Background and Interests What sort of person was the writer of Flatland? Clearly Abbott was impressed by the power of language, and Flatland is about language at several levels. Throughout the book there are overt and hidden references to Abbott’s favorite author, Shakespeare. It is unlikely that his readers would recognize many of the allusions, since although Victorians did attend Shakespearean plays, they did not study them in the schools, except at the City of London School. There a tradition of prize examinations based on Shakespeare led the bright and highly competitive students to memorize large portions of plays, and to apply to Elizabethan English some of the linguistic analysis that was common with scholarship in classical languages. Abbott wrote his first major book, A Shakespearean Grammar, in 1870, and several new editions appeared in his lifetime, even though he lamented as late as the turn of the century that headmasters throughout England were unwilling to turn over to the study of English literature even a small portion of the time allotted to Latin and Greek. Abbott himself writes that in preparing a new edition of his work on Shakespeare, he reread all of the plays for a third time. It was a close reading, too. Abbott had been trained by his father, Edwin Abbott, headmaster of the Philological School in Marylebone, London, and compiler of the concordance of the works of Pope that is still the major reference for that subject. Abbott aided his father in this enterprise and wrote the introduction to his father’s life’s work. Abbott was somewhat of an intellectual radical, going so far as to suggest that the techniques of literary criticism developed by classical scholars should be used with the same vigor to examine the Scriptures. In his controversial article on the Gospels for the ninth edition of the Encyclopaedia Britannica, Abbott suggests linguistics tests for the dating and the authorship of the Gospels and Epistles. He refers to the “method of curves” as a way of analyzing style and structure, and he laments that the work involved is too much for a single person, perhaps anticipating our present day when computer searches routinely carry out the kinds of investigation Abbott could only dream of. Such suggestions were naturally threatening to those who wished to base their faith on less scientific grounds.

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Part of the rigidity of the educational system was due to the central place played by examinations, first for gaining entrance to the universities, and secondly for determining the sort of degree that would be awarded. Abbott himself did quite well under this system. By the time he had graduated from the City of London School, he had won prizes in just about all of the subjects in the curriculum. He had delivered prize declamations in English, French, German, Latin, and Greek, and he had even won a prize in mathematics. At Cambridge he was first in classics and Chancellor’s Medallist, the same year that the top mathematics prizes went to another graduate of the City of London School, Thomas Steadman Aldis, who was First Wrangler and Smith’s Prizeman. It was unusal for all four top prizes to go to boys from the same school, and unheard of that that school should be a day school, not one of the English public schools that were the traditional leaders in preparatory education. The Style of Flatland The narrative style of Flatland is somewhat different from some of the more familiar reports of visits to exotic lands, since the story is told not by the visitor but by the person visited. It is as if the story of Gulliver were told by the mayor of Lilliput or the advenutres of Alice by the White Rabbit. It is only in the latter half of the book that the narrator A Square can relate the changes that took place in his own perceptions as a result of the remarkable events caused by the visit of a being from a higher dimension. The reader has to stay with the story all the way through in order to appreciate the change that takes place in the storyteller. A similar thing happens with the narrator of another book written in exactly the same year, Huckleberry Finn. In the beginning of that book, the narrator shares the prejudices of his society with respect to slavery, and by the end of his journey down the river, his experiences have caused him to reject those views. Even the relatively enlightened ideas that the author expresses at the end of the book may seem inadequate in our present age, and they might not represent accurately the views of the author. Still, the novel itself is valuable for the tale it tells about the development of sensibility in a strange environment. That is what we have in Flatland as well. We have not erased racial prejudice from our world, nor do we treat women equally or provide for the needs of the handicapped. The social lessons of Flatland are still being learned. The Historical Context of Flatland Much of what Abbott wrote in Flatland came about in response to the mathematical and philosophical concerns of his day, when geometers attempted to visualize phenomena in the fourth and higher dimensions. They drew pictures and made models,

introduction xix and even attempted to use stereoscopic images in order to see what projections and slices of four-dimensional objects look like. They were often frustrated by their inability to see more than one incomplete image at a time. All that has changed with the advent of computer technology. We cannot answer all the questions raised by the investigators in the nineteenth century, but we can reconsider them in new ways. Abbott’s two-dimensional narrator led his readers through a good part of the exercises any citizen of Flatland would go through in the perception training, refining the sense of touch and sight so as to come to terms with all reasonable shapes. Similarly, we in three-space learn to interpret the sense data that come to us, gradually coming to the point where we can resolve ambiguities and draw correct inferences from the views presented to us. We can refine our own abilities to model and draw so as to construct the most effective images to communicate accurately about them. As the concepts become more and more complicated, our limitations become tested, and ultimately we experience frustration. This is especially true when an object or a configuration resists representation on the visual plane that is our most trusted means of receiving information. The problem may have to do with the intrinsic complexity of something in our own space, or it might be due to an entirely different reason, because some phenomenon comes from a space beyond our own, from a higher dimension. That is the analogue of the problem faced by A Square in Flatland. Abbott was not the first person to challenge his readers with the enigmas of perceptions in different dimensions. Plato had already used a powerful image that crossed dimensional boundaries when he posited a race of men whose only sense impressions came from the shadows they could see on the wall of a cave. How little of the solid reality of the third dimension would such limited creatures be able to appreciate, and how little of the true nature of the transcendent reality do we comprehend when all we can see are the solid shadows of ideal objects in the mind of God. In the nineteenth century in Germany and England, several writers seized on the idea of imagining the experience of beings confined to two dimensions. Gustav Fechner made up the story of shadow man cast on a wall by a slide projector. Karl Friedrich Gauss and Hermann von Helmholz both investigated the kind of geometry that would be discovered by an intelligent flatworm sliding about on the surface of a solid object. Helmholz’s ideas were well represented in British intellectual journals, which ran debates about the nature of higher dimensions and our ability to comprehend them. Charles Howard Hinton examined the physical properties of the life of flat creatures on a two-dimensional surface in the early 1880s, and it is likely that Abbott could have seen one or more of his articles before he wrote Flatland. Abbott’s best friend, Howard Candler, the “H.C.” to whom Flatland is dedicated, was mathematics master at the Uppingham School,

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where Hinton spent several years as science master. Hinton wrote that his purpose was to examine scientific principles, while the purpose of Abbott’s book was not science but rather “his philosophy and his lessons.” Abbott was one of the first writers to realize the full power of the dimensional analogy for investigating contact among beings from different dimensions. In his later theological works, he developed this idea further as an analogy for the all-seeing power of God, who makes revelations to a lower world and interacts with creatures there. Abbott was also one of the first to exploit the notion of slicing as a means of gaining information about solid objects by studying planar cross-sections. He was familiar with the process of calculating the volume of a solid ball by keeping track of the areas of circular slices, and he probed the psychological process of trying to appreciate a form by keeping track of its different cross-sections. It is difficult for A Square to comprehend the nature of a sphere from the third dimension. It would be even more difficult for him to comprehend the surface of a doughnut. The challenge for us is apparent. Just as a sphere penetrating Flatland is viewed by A Square as a circle growing and then shrinking in time, so also if we were visited by a hypersphere from a space of four dimensions, we might see a sphere growing and then shrinking in time. The ability to treat such a sequence of impressions as the gradual revelation of an entity from a higher dimension is the first exercise for anyone who wishes to accept the challenge of Flatland. More difficult is the exercise implied in the book concerning the analogue of a square in three and higher dimensions. In the text, the number of corners of analogues of cubes in different dimensions can be suggested by arguments based on the characteristics of various formulas. It is not so easy to imagine the ways in which such cubes and hypercubes appear when they are put through a slicing plane or hyperplane at different angles. Flatland and New Modes of Visualization Today the major reason for our interest in Flatland is that for the first time we can achieve some of the dreams of our ancestors a century ago and obtain direct visual experience of phenomena in a dimension higher than our own. The breakthrough that brings this about is modern computer graphics. Images moving on a computer screen can give us a view of complicated two-dimensional phenomena like networks and patterns, and they can also enable us investigate complex objects in three dimensions, like the plans of a building or a machine part. As we watch, the turn of the dial makes the image spin around on the screen, revealing new aspects of the structure and new relationships as different parts of the object come into view. The building and the machine part are three-dimensional, with every point speci-

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fied by three coordinates. Taking those coordinates two at a time gives us the different elevations, front, side, and top views. Computers can calculate quickly enough to display mixed views from any desired angle. By showing a sequence of views, thirty views every second, each slightly different from the previous one, the computer can create an animated film that reveals the structure of a three-dimensional object much better than any single picture. If such animations are valuable for studying three-dimensional objects, they are even more crucial in giving us insight into the phenomena in four or more dimensions. The computer techniques from architecture and machine design help us approach collections where each point is specified by four or more coordinates. Again, we show the coordinates two or three at a time, and use our imaginations to gain an appreciation of the overall relationships so we can more effectively predict what will come next. Dramatic evidence of the power of such techniques can be found in the field of radiology, where CAT scans and magnetic resonance imaging has produced data sets that can monitor the development of a tumor over the course of weeks, with all information available to be investigated on a computer graphics screen. Precisely the same kinds of techniques enable geologists to study the data on global warming trends over tens of thousands of years, or the economics of a large urban area, with each ward contributing numerical observations in a dozen different categories. The use of computer graphics in statistics has changed that subject in dramatic ways, especially in the area of exploratory data analysis. time and the fourth dimension “The fourth dimension? That’s time, isn’t it?” It isn’t hard to imagine A Square’s rejoinder to such a question. He would turn the question back on the poser and remind him that in Flatland, the comparable response to the introduction to the concept of the third dimension is, “The third dimension? That’s time, isn’t it?” For beings whose universe has a spatial fourth dimension, the fifth dimension would be the one assigned to time, and in general we can expect that time will come in right after the dimensions that are reserved for space. For some modern physicists, there are ten dimensions which act like space, and the eleventh is time. The trouble is that when we treat time as a dimension, we do not expect it to act like the homogeneous interchangeable dimensions of space. There is a perfectly fine mathematics of space-time, but it is not one that Abbott is pointing to in Flatland. There we are asked to contemplate a fourth spatial dimension equivalent in all ways to the first three. As it happens, in order to introduce the geometry of space-time, almost all physics writers lower the dimension

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to that of one or two dimensions of space to go with one dimension of time, thereby reducing the total number of dimensions to two or three, allowing easier visualization. Relativity in Flatland helps us to understand relativistic effects in our own universe. Flatland was written a full generation before Albert Einstein developed his theories of relativity, and most of the people who thought about dimensions in Abbott’s day were concerned with the nature of space rather than space-time. (A good reference for the place of geometry in Victorian England is the recent book Mathematical Visions by Joan Richards.) There were several writers who recognized that time could be treated as a dimension similar to the other three, but with certain differences. Hermann Minkowski had already worked out some of the consequences of measuring space in a way that treated one direction differently from the others, in the same way that time measurement is different from the measurement of space. Edmond Laguerre in France had developed a geometry of spheres that was four-dimensional in that each sphere could be specified by three dimensions for the center and one more dimension for the radius. But to tell whether or not two spheres intersected, it was necessary to use formulas in which the radius coordinate behaved like time rather than space. Another important geometry involving four variables was the geometry of lines in space. We can specify all non-horizontal lines by indicating the points where the line intersects a pair of horizontal planes, describing each of these points by two coordinates. Such four-dimensional geometries were well understood by mathematicians as abstract constructs, but resistance remained to the concept of a physical fourth dimension equivalent in every way to the accepted three dimensions. It is the challenge of visualizing such a homogeneous four-dimensional space that formed the basis of Flatland. higher dimensions It is important to note that Flatland is not specifically about the fourth dimension. One of the problems of thinking of the fourth dimension as time is that it tends to stop things at that stage. One is not likely to go on to a fifth dimension in order to find a fourth dimension of space (although that is precisely what happens in Madeleine L’Engle’s book A Wrinkle in Time, where the author takes space-time for granted and has her powerful beings taking shortcuts through a five-dimensional void). In Flatland, there is to be no limit on the number of dimensions, and indeed the point of the story is that the Sphere, at first an omniscent observer, is revealed as subject to his own dimensional prejudices, unable to conceive of a dimension beyond the third, as a four-dimensional being might begin by doubting the reality of anything of dimension five or higher. All of us are slaves to the prejudices of our own dimension. Even the frontispiece of the book invites readers to contemplate worlds of up to ten dimensions

introduction xxiii in the mammoth cloud around the perspective drawing of the name of the book. Curiously enough, it is a universe of ten spatial dimensions which has recently gained quite a bit of popularity in quantum physics. There is little attempt to visualize so many dimensions at once, but the framework of higher dimensions still provides a way of keeping track of a large number of variables, some so small that they can pass unnoticed when we concentrate on the usual three dimensions of space. Another important model employs twenty-six dimensions, even more beyond our powers of visualization. Even though the eventual aim of the physicists is to deal with spaces of these high dimensions, any popular exposition of the subject begins by describing analogues in two, three, and four dimensions, more often than not referring to the analogies presented in Flatland. Another area in which higher dimensions arise in physics is in the curvature of space. In almost any treatment of this concept, either Flatland is cited directly, or the writer or speaker will refer to Sphereland, the sequel written in 1965 by the Dutch physics teacher Dionys Burger. In that book, decendants of A Square set out on long journeys and discover that their world is not flat after all, but curved into the surface of a large sphere. In the same way we can imagine space explorers travelling so far that they determine that our own supposedly flat three-dimensional universe is in reality finite like the surface of a sphere. At such a moment we may be sure that the Flatland analogy will be invoked to help people come to terms with this new worldview. Although strictly speaking the concept of curvature of space does not necessarily involve a higher dimensional space that our space is curved into, this is undoubtedly the most natural way to begin thinking about this idea. A more thorough treatment of these concepts is found in The Shape of Space, by Jeff Weeks, which uses Flatland as a starting point and describes two-dimensional worlds of varying topological complexity, shaped like doughnuts or twisted Möbius bands or Klein bottles. Understanding such twodimensional examples is crucial for appreciating the current developments in the shape of three- and four-dimensional space. For further reading on these and other topics using the dimensional analogy, see The Fourth Dimension and Geometry, Relativity, and the Fourth Dimension by Rudolf Rucker, The Planiverse by A. K. Dewdney and my own volume, Beyond the Third Dimension. Other scientists who have profited by recent developments in visualization of data sets in high dimensions are the statisticians, especially those engaged in the field of exploratory data analysis. Researchers quite often have to analyze a mass of data from a sociological survey or a set of physical or biological experiments. The numbers of parameters in such investigations can be very large indeed, with a single observation leading to a string of a dozen or more numbers. Searching for patterns in such data col-

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lections can reveal hidden relationships, allowing scientists to predict the size of certain variables once others are known, and suggesting trends that can lead to predictions of short- or long-range behavior. A recent book on data visualization by Edward Tufte, Envisioning Information, has as its first chapter “Escaping from Flatland,” and at the centenary conference on Flatland at Brown University in 1984, John and Paul Tukey introduced a full range of techniques in exploratory data analysis by starting with the world of data visualization accessible to A Square. Software for computers of moderate size now enables researchers to manipulate multidimensional data sets with great freedom, projecting point clouds into three or four dimensions and then analyzing the patterns on a graphics screen as the collection of points is rotated in space, or as the data set is sliced to reveal the points lying in a thin slab. As such a test slab proceeds through the data set, we gain important information about the distribution of points in the whole collection. The slicing technique from Flatland still remains one of the most powerful tools for dealing with aggregates in higher dimensions. The technique of slicing is also at work when a cartographer produces a contour map of a mountainous region, shading in the points that correspond to the region between two different levels. This technique does not have to stop with the analysis of three-dimensional data using two-dimensional slices since it works as well in higher dimensions. If each point in space is given a fourth coordinate indicating its temperature, then by indicating the points with temperature between two given values, we obtain an analogue of a contour map in more than three dimensions. These same approaches work to help mathematicians analyze the structure of abstract functions of three or more variables, which in turn can serve as models for the analysis of more and more complicated physical phenomena. Although such methods have been theoretically possible for many years, it is only in recent times that computers have become powerful enough to carry out the procedures effectively. Now even relatively small computers produce contour maps, and it will not be long before it will be easy to apply the same approach to the direct study of objects in higher dimensions. The dream of mathematicians of the last century, used so effectively by Abbott in his treatment of the ways we comprehend and the ways we communicate, will finally be within our grasp. We do not know all of the ways that this new way of seeing will change our approach to geometry and its applications, but one thing is sure-anyone who wants to make good use of these developments in visualization will begin by reading the book you now hold, Flatland.

Part I

This World

“Be patient, for the world is broad and wide.”

§I Of the Nature of Flatland

§ 1: Of the Nature of Flatland I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows—only hard and with luminous edges—and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said “my universe”: but now my mind has been opened to higher views of things. In such a country, you will perceive at once that it is impossible that there should be anything of what you call a “solid” kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate. Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view; and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line. The same thing would happen if you were to treat in the same way a Triangle, or Square, or any other figure cut out of pasteboard. As soon as you look at it with your eye on the edge of the table, you will find that it ceases to appear to you a figure, and that it becomes in appearance a straight line. Take for example an equilateral Triangle-who a Tradesman of the respectable class. Fig.Fig. 1 represents Triangle—whorepresents representswith withusus a Tradesman of the respectable class. 1 reprethe Tradesman as you would him while youwhile were you bending him from figs. sents the Tradesman as yousee would see him wereover bending over above; him from 2above; and 3firepresent Tradesman, as you wouldas see if your eye were close the gs. 2 and the 3 represent the Tradesman, youhim would see him if your eyetowere level, or the all but onorthe of the table; your and eye were quite thequite levelon of the close to level, alllevel but on level ofand theiftable; if your eyeon were level of thethat table (andwe that howinwe see himyou in Flatland) would seea straight nothingline. but table (and is how seeishim Flatland) would seeyou nothing but a straight line.

15

When I was in Spaceland I heard that your sailors have very similar experiences while 4 they traverse your seas and discern some distant island or coast lying on the horizon. The far-off land may have bays, forelands, angles in and out to any number and When in Spaceland I heard that your(unless sailorsindeed have very experiences extent; yet Iatwas a distance you see none of these yoursimilar sun shines bright while them, they traverse your and discern distant island or coast lyingand on the houpon revealing theseas projections andsome retirements by means of light shade), rizon. Thbut e faroff land may have nothing a grey unbroken linebays, uponforelands, the water.angles in and out to any number and extent; at aisdistance these (unless indeed or your sunacquaintances shines bright Well,yetthat just whatyou wesee seenone whenofone of our triangular other upon them, revealing the projections and retirements by means of any lightlight and of shade), comes towards us in Flatland. As there is neither sun with us, nor such a grey shadows, unbrokenwe linehave upon the of water. anothing kind asbut to make none the helps to the sight that you have in Well, that justfriend what comes we see when oneusofwe oursee triangular other acquaintances Spaceland. If isour close to his line orbecomes larger; if he comes us towards us in smaller: Flatland.but Asstill there neither with us,line; nor be anyhelight of such leaves it becomes heislooks likesun a straight a Triangle, a kind asPentagon, to make shadows, have none thewill—a helps tostraight the sight thatheyou haveand in Square, Hexagon,weCircle, what ofyou Line looks Spaceland. If our friend comes close to us we see his line becomes larger; if he nothing else. leaves it becomes but still he looks like a straight line; be he aweTriangle, Youusmay perhaps smaller: ask how under these disadvantageous circumstances are able Square, Pentagon, Hexagon, what you will—a straight looksquesand to distinguish our friends fromCircle, one another: but the answer to thisLine very he natural nothing else. tion will be more fitly and easily given when I come to describe the inhabitants of You may ask how under these are able Flatland. Forperhaps the present let me defer thisdisadvantageous subject, and saycircumstances a word or twowe about the to distinguish our friends one another: but the answer to this very natural quesclimate and houses in our from country. tion will be more fitly and easily given when I come to describe the inhabitants of Flatland. For the present let me defer this subject, and say a word or two about the and houses in our country. §climate 2

Of the climate and houses in Flatland § 2: Of the climate and houses in Flatland §2 Of with the climate As you, so and also with us, there are four points of the compass North, South, East, houses in Flatland and West. There being no sun nor other heavenly bodies, it is impossible for us to determine As with you, withway; us, but therewearehave foura points of of theour compass North, South, East, the North in so thealso usual method own. By a Law of Nature and West. with us, there is a constant attraction to the South; and, although in temperate cliThthis ere being sun nor other heavenly bodies, is impossible forcan us to determine mates is verynoslight—so that even a Woman in itreasonable health journey sevthe North in northward the usual way; but we havediffi a method of our By a Law of Nature eral furlongs without much culty—yet theown. hampering effect of the with us, there is a constant South; although in temperate clisouthward attraction is quiteattraction sufficienttotothe serve as aand, compass in most parts of our mates this is very the slight—so that even Woman reasonable healthalways can journey earth. Moreover rain (which fallsa at statedinintervals) coming from sevthe eral furlongs northward assistance; without much culty—yet thehave hampering effect of the North, is an additional and diffi in the towns we the guidance southward attraction is quite sufficient to serve as a compass in most of and our houses, which of course have their side-walls running for the most partparts North earth. Moreover rain (which fallstheatrain stated intervals) coming from the South, so that thethe roofs may keep off from the North. In thealways country, where North, additional assistance; the as towns have the guidance of the there areisnoanhouses, the trunks of theand treesinserve somewe sort of guide. Altogether, we houses, which of course have their sidewalls running for the most part North and have not so much difficulty as might be expected in determining our bearings. South, the roofs may keep off theinrain fromthe thesouthward North. Inattraction the country, where Yet so in that our more temperate regions, which is hardly therewalking are no houses, the trunks of the trees serve as some sort of guide. Altogether, we felt, sometimes in a perfectly desolate plain where there have been no houses have not so much diffi culty as might be expected in determining our bearings. nor treesflatland to guide me, I have been occasionally compelled to remain stationary for 16 hours together, waiting till the rain came before continuing my journey. On the weak and aged, and especially on delicate Females, the force of attraction tells much more 16 flatland heavily than on the robust of the Male Sex, so that it is a point of breeding, if you meet a Lady in the street always to give her the North side of the way—by no means

Yet in our more temperate regions, in which the southward attraction is hardly felt, walking sometimes in a perfectly desolate plain where there have been no houses5 nor trees to guide me, I have been occasionally compelled to remain stationary for hours waiting till the rain cameinbefore my journey. On the weak Yettogether, in our more temperate regions, whichcontinuing the southward attraction is hardly and andsometimes especiallyin ona delicate force of attraction more felt, aged, walking perfectlyFemales, desolatethe plain where there havetells beenmuch no houses heavily than on the robust of the Male Sex, so that it is a point of breeding, if nor trees to guide me, I have been occasionally compelled to remain stationaryyou for meet Lady in waiting the streettillalways to give the North side of way—by hoursatogether, the rain cameher before continuing mythe journey. Onno themeans weak an thing do always short notice when rude health in more a cliandeasy aged, and to especially onatdelicate Females, theyou forceareofinattraction tellsand much mate where is diffi to tell your North youritSouth. heavily thaniton the cult robust of the Male Sex,from so that is a point of breeding, if you Windows are none in to ourgive houses: forNorth the light comes to us alike our meet a Lady inthere the street always her the side of the way—by no in means homes outtoofdothem, byatday andnotice by night, allrude timeshealth and in allinplaces, an easyand thing always short whenequally you areat in and a cliwhence we know not. It to was old North days, with learned mate where it is diffi cult tellinyour fromour your South.men, an interesting and oft-investigated question, “What is the originforofthe light?” the tosolution it our has Windows there are none in our houses: lightand comes us alikeof in been repeatedly attempted, with no other result than to crowd our lunatic asylums homes and out of them, by day and by night, equally at all times and in all places, with thewe wouldbenot. solvers. Hence, fruitless attempts to men, suppress such investigawhence know It was in oldafter days, with our learned an interesting and tions indirectly by making them liable to a heavy tax, the Legislature, in comparaoft-investigated question, “What is the origin of light?” and the solution of it has tively recent times, absolutely prohibited alone our in Flatland—know been repeatedly attempted, with no otherthem. result I—alas than to Icrowd lunatic asylums now too well the trueHence, solution of this mysterious problem; but such my knowledge with only the wouldbe solvers. after fruitless attempts to suppress investigabe made intelligible to a single one of my cannot be made intelligible to a single one of my countrymen; and I am tions indirectly by making them liable to a heavy tax, the Legislature, in comparacountrymen; and ampossessor mocked the sole mocked at—I,times, the Isole ofat-I, the truths of I—alas Space and of theintheory of tively recent absolutely prohibited them. I alone Flatland—know possessor thewell truths of Space andworld ofofthe the of the Light from the oftheory Th ree Dimensions—as if Imy were nowintroduction only of too true solution this mysterious problem; but knowledge the maddest of the mad! But a truce to these painful digressions: let me reof the introduction of Light from the world of cannot be made intelligible to a single one of my countrymen; and I am turn toDimensions-as our houses. Three I were of thethe maddest of Space and of the theory of mocked at—I, the sole if possessor truths of e most for theworld construction a house is five-if sided or the Th mad! But common a truce toform these painful digressions: introduction of Light from the of Three of Dimensions—as I were pentagonal, as in the annexed fi gure. Th e two Northern sides RO, OF, conlet me return to our houses. the maddest of the mad! But a truce to these painful digressions: let me restitute and forform the most part have no doors; on the East is a small most common for the construction turnThe tothe ourroof, houses. door for theisWomen; onform West much larger the is Men; South of aTh house five-sided orthepentagonal, as in the one e most common for thea construction of afor house five-the sided or side or fl oor is usually doorless. annexed figure. The two Northern sides RO, OF, pentagonal, as in the annexed figure. The two Northern sides RO, OF, conSquare androof, triangular are allowed, for the this reason. of a constitute forhouses the part have noand doors; on East the East is ae angles small door stitute the the roof, andand for the mostmost partnot have no doors; on is a Th small Square (and stillon more ana equilateral being for theonthose West aofmuch largerlarger oneTriangle) for themuch South side pointed or floor doorthe forWomen; the Women; the West much onethe forMen; the Men; the more South side or than those of a Pentagon, and the lines of inanimate objects (such as houses) being is usually doorless. floor is usually doorless. dimmer than lines of houses Men and it follows is no danger Square andthe triangular are Women, not allowed, and forthat thisthere reason. Thlittle e angles of a lest the points of amore square or triangular house residence might seriousmore injury to an Square (and still those of an equilateral Triangle) beingdomuch pointed inconsiderate perhaps absentminded suddenly running them:being and than those of or a Pentagon, and the linestraveller of inanimate objects (suchagainst as houses) therefore, as early as the eleventh century of our era, triangular houses were univerdimmer than the lines of Men and Women, it follows that there is no little danger sally forbidden exceptions fortifi cations, powder-injury magazines, lest the points ofbya Law, squarethe or only triangular housebeing residence might do serious to an barracks, and or other stateabsentminded buildings, which it is not desirable that the general public inconsiderate perhaps traveller suddenly running against them: and should approach without circumspection. therefore, as early as the eleventh century of our era, triangular houses were universally forbidden by Law, the only exceptions being fortifications, powder-magazines, barracks, and other state buildings, which it is not desirable that the general public The Book 17 should approach without circumspection.

The Book

17

At this period, square houses were still everywhere permitted, though discouraged by a special tax. But, about three centuries afterwards, the Law decided that in all towns containing a population above ten thousand, the angle of a Pentagon was 6 the smallest house-angle that could be allowed consistently with the public safety. TheAt good of the community has still seconded the efforts of the though Legislature; and thissense period, square houses were everywhere permitted, discournow, even in the country, pentagonal construction has superseded every other. It aged by a special tax. But,the about three centuries afterwards, the Law decided that in is now and then ainpopulation some very remote andthousand, backwardthe agricultural thatwas an allonly towns containing above ten angle of adistrict Pentagon antiquarian stillangle discover squarebehouse. the smallest may housethat acould allowed consistently with the public safety. The good sense of the community has seconded the efforts of the Legislature; and now, even in the country, the pentagonal construction has superseded every other. It § is 3only now and then in some very remote and backward agricultural district that an Concerning antiquarian may still discover a square house.

the Inhabitants of Flatland § 3: Concerning the Inhabitants of Flatland §3 Th e greatest length or breadth of a full-grown inhabitant of Flatland may be estimated Concerning at eleven of your inches. Twelve inches may be regarded as a maximum. theabout Inhabitants Our Women are Straight Lines. of Flatland

Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each about eleven long,ofand a base or third side so (often The greatest lengthinches or breadth a fullgrown inhabitant of short Flatland maynot be exceeding estimated half an inch) that they form at their vertices a very sharp and formidable angle. Inat about eleven of your inches. Twelve inches may be regarded as a maximum. deedOur when their bases are of the most degraded type (not more than the eighth part Women are Straight Lines. of an inch in size)and they can hardly distinguished Straight or Women; Our Soldiers Lowest Class be of Workmen are from Triangles withLines two equal sides, so extremely pointed are their vertices. With us, as with you, these Triangles are diseach about eleven inches long, and a base or third side so short (often not exceeding tinguished from by being called Isosceles; by and this formidable name I shall referInto half an inch) thatothers they form at their vertices a veryand sharp angle. them in thetheir following deed when bases pages. are of the most degraded type (not more than the eighth part Our Middle Class of Equilateral or EqualsidedStraight Triangles. of an inch in size) theyconsists can hardly be distinguished from Lines or Women; Our Professional Mentheir andvertices. Gentlemen I myself beso extremely pointed are Withare us,Squares as with (to you,which these class Triangles are dislong) and Fivefigures or Pentagons. tinguished fromsided others by being called Isosceles; and by this name I shall refer to Next above these come the Nobility, of whom there are several degrees, beginning them in the following pages. at Sixsided Figures, Hexagons, and from thence rising in the number of their sides Our Middle Classorconsists of Equilateral or Equalsided Triangles. till Our they Professional receive the honourable title of Polygonal, or manysided. class Finally when bethe Men and Gentlemen are Squares (to which I myself number the sides soPentagons. numerous, and the sides themselves so small, that the long) andofFivesidedbecomes figures or figure cannot distinguished from a circle, he is there included in the Circular or Priestly Next abovebethese come the Nobility, of whom are several degrees, beginning order; and this is theorhighest class and of all. at Six-sided Figures, Hexagons, from thence rising in the number of their sides is areceive Law ofthe Nature with ustitle thatofa male child shall havesided. one more sidewhen than the his till It they honourable Polygonal, or manyFinally father, so that each generation shall rise (as a rule) one step in the scale of developnumber of the sides becomes so numerous, and the sides themselves so small, that the ment nobility. Thus the son of aa Square is is a Pentagon; of a Pentagon, figureand cannot be distinguished from circle, he included inthe theson Circular or Priestlya Hexagon; order; andand thisso is on. the highest class of all. It is this a Law of applies, Nature not withalways us thattoathe male child shalland havestill oneless more side his But rule Tradesmen, often to than the Sol18 father, soflatland that each generation shall rise (as rule) one scalethe of developdiers, and to the Workmen; who indeed cana hardly be step said in to the deserve name of ment and nobility. Thus thehave son of a Pentagon; the son of atherefore Pentagon, human Figures, since they nota Square all theirissides equal. With them thea Hexagon; and sodoes on. not hold; and the son of an Isosceles (i.e. a Triangle with two Law of Nature sides equal) remains Isosceles still. Nevertheless, all hope is not shut out, even from 18 flatland the Isosceles, that his posterity may ultimately rise above his degraded condition. For, after a long series of military successes, or diligent and skilful labours, it is generally

But this rule applies, not always to the Tradesmen, and still less often to the Sol 7 diers, and to the Workmen; who indeed can hardly be said to deserve the name of human Figures, since they have not all their sides equal. With them therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope is not shut out, even from the Isosceles, that his posterity may ultimately rise above his degraded condition. For, after a long series of military successes, or diligent and skilful labours, it is generally found that the more intelligent among the Artisan and Soldier classes manifest a slight increase of their third side or base, and a shrinkage of the two other sides. Intermarriages (arranged by the Priests) between the sons and daughters of these more intellectual members of the lower classes generally result in an offspring approximating still more to the type of the Equal-sided Triangle. Rarely—in proportion to the vast numbers of Isosceles births—is a genuine and certifiable Equal-sided Triangle produced from Isosceles parents.* Such a birth requires, as its antecedents, not only a series of carefully arranged intermarriages, but also a long-continued exercise of frugality and self-control on the part of the wouldbe ancestors of the coming Equilateral, and a patient, systematic, and continuous development of the Isosceles intellect through many generations. The birth of a True Equilateral Triangle from Isosceles parents is the subject of rejoicing in our country for many furlongs round. After a strict examination conducted by the Sanitary and Social Board, the infant, if certified as Regular, is with solemn ceremonial admitted into the class of Equilaterals. He is then immediately taken from his proud yet sorrowing parents and adopted by some childless Equilateral, who is bound by oath never to permit the child henceforth to enter his former home or so much as to look upon his relations again, for fear lest the freshly developed organism may, by force of unconscious imitation, fall back again into his hereditary level. The occasional emergence of an Isosceles from the ranks of his serf-born ancestors, is welcomed not only by the poor serfs themselves, as a gleam of light and hope shed upon the monotonous squalor of their existence, but also by the Aristocracy at large; for all the higher classes are well aware that these rare phenomena, while they do little or nothing to vulgarise their own privileges, serve as a most useful barrier against revolution from below.

* “What need of a certificate?” a Spaceland critic may ask: “Is not the procreation of a Square Son a certificate from Nature herself, proving the Equal-sidedness of the Father?” I reply that no Lady of any position will marry an uncertified Triangle. Square offspring has sometimes resulted from a slightly Irregular Triangle: but in almost every such case the Irregularity of the first generation is visited on the third; which either fails to attain the Pentagonal rank, or relapses to the Triangular.

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8 Had the acute-angled rabble been all, without exception, absolutely destitute of hope and of ambition, they might have found leaders in some of their many seditious outbreaks, able as to render their superior numbers and strength too much for the Had thesoacuteangled rabble been all, without exception, absolutely destitute of wisdom of the Circles. But a have wise ordinance of Nature decreed thatseditious in prohope andeven of ambition, they might found leaders in somehas of their many portion as the workingclasses their increase in intelligence, knowledge, all virtue, in outbreaks, so able as to render superior numbers and strength and too much for the that sameeven proportion their acute (which makes them physically wisdom of the Circles. But aangle wise ordinance of Nature has decreedterrible) that in shall proincrease approximate to the harmless angle knowledge, of the Equilateral Triangle. portion asalso theand workingclasses increase in intelligence, and all virtue, in Th us,same in theproportion most brutal andacute formidable of the soldier almost on a shall level that their angle (which makesclass themcreatures physically terrible) with women their lack of intelligence—it is found that, wax in the mental increase also inand approximate to the harmless angle of as thethey Equilateral Triangle. ability employ tremendous to advantage, do Thus, innecessary the mosttobrutal andtheir formidable of thepenetrating soldier classpower creatures almost on asolevel they in in thetheir power itself. is found that, as they wax in the mental with wane women lackofofpenetration intelligence—it How admirable the Law of Compensation! And howpower perfecttoa advantage, proof of thesonatability necessary to isemploy their tremendous penetrating do ural tnessin and, may almost say, the divine they fiwane theIpower of penetration itself. origin of the aristocratic constitution of the How Statesadmirable of Flatland! By aLaw judicious use of this Law of how Nature, the Polygons is the of Compensation! And perfect a proof ofand the Cirnatcles almost to stifl sedition its very taking advantage of the uralare fitness and,always I mayable almost say,e the divineinorigin ofcradle, the aristocratic constitution of irrepressible boundless hopefulness of this the Law human mind. Art comes toCirthe the States of and Flatland! By a judicious use of of Nature, the also Polygons and aid and always Order. able It is generally found possible—by a little artificial compression clesof areLaw almost to stifle sedition in its very cradle, taking advantage of the or expansionand on the part of hopefulness the State physicians—to make some the comes more intelliirrepressible boundless of the human mind. Artofalso to the gent of aOrder. rebellion and to admitathem once the priviaid ofleaders Law and It isperfectly generallyRegular, found possible—by little at artifi cialinto compression leged classes; aonmuch larger are still below thesome standard, by the or expansion the part of number, the Statewho physicians—to make of theallured more intelliprospect of of being ultimately ennobled, areand induced to them enter atthe State gent leaders a rebellion perfectly Regular, to admit once intoHospitals, the priviwhere they area kept honourable confi nement forbelow life; one two alone of theby more leged classes; muchinlarger number, who are still theor standard, allured the obstinate, hopelessly irregularare areinduced led to execution. prospect offoolish, being and ultimately ennobled, to enter the State Hospitals, Then theare wretched of theconfi Isosceles, planless are either where they kept in rabble honourable nement for life;and oneleaderless, or two alone of thetransmore fiobstinate, xed without re sis tance by the small body of their brethren whom the Chief Circle foolish, and hopelessly irregular are led to execution. keeps emergencies kind; or else, moreand often, by means jealousies Thin en pay the for wretched rabble of of this the Isosceles, planless leaderless, are of either transand among them by the Circularwhom party,the they are stirred fixedsuspicions without reskilfully sistancefomented by the small body of their brethren Chief Circle to mutual warfare, and perishofbythis onekind; another’s angles. less than one hundred and keeps in pay for emergencies or else, moreNo often, by means of jealousies twenty rebellions are recorded in among our annals, outbreaks numbered at and suspicions skilfully fomented thembesides by the minor Circular party, they are stirred two hundred and thirtyfive; and theyanother’s have all angles. ended thus. to mutual warfare, and perish by one No less than one hundred and twenty rebellions are recorded in our annals, besides minor outbreaks numbered at two hundred and thirty-five; and they have all ended thus.

§4 Concerning the § 4Women § 4: Concerning the Women Concerning If highly pointed Triangles of the Soldier class are formidable, it may be readily theour Women inferred that far more formidable are our Women. For, if a Soldier is a wedge, a If our highly pointedbeing, Triangles the Soldier class may be readily Woman is a needle; so toofspeak, all point, at are leastformidable, at the two itextremities. Add inferred that far more formidable are our Women. For, if a Soldier is a wedge, to this the power of making herself practically invisible at will, and you will perceivea 20 flatland that a Female in Flatland, is a creature by no means to be trifled with. But here, perhaps, some of my younger Readers may ask how a woman in Flatland 20 can makeflatland herself invisible. This ought, I think, to be apparent without any explanation. However, a few words will make it clear to the most unreflecting.



9

Woman is a needle; being, so to speak, all point, at least at the two extremities. Add to this the power of making herself practically invisible at will, and you will perceive that a Female in Flatland, is a creature by no means to be trifled with. But here, perhaps, some of my younger Readers may ask how a woman in Flatland can make herself invisible. This ought, I think, to be apparent without any explanation. However, a few words will make it clear to the most unreflecting. Place a needle on a table. Then, with your eye on the level of the table, look at it side-ways, and you see the whole length of it; but look at it end-ways, and you see nothing but a point: it has become practically invisible. Just so is it with one of our Women. When her side is turned towards us, we see her as a straight line; when the end containing her eye or mouth—for with us these two organs are identical—is the part that meets our eye, then we see nothing but a highly lustrous point; but when the back is presented to our view, then—being only sub-lustrous, and, indeed, almost as dim as an inanimate object—her hinder extremity serves her as a kind of Invisible Cap. The dangers to which we are exposed from our Women must now be manifest to the meanest capacity in Spaceland. If even the angle of a respectable Triangle in the middle class is not without its dangers; if to run against a Working Man involves a gash; if collision with an Officer of the military class necessitates a serious wound; if a mere touch from the vertex of a Private Soldier brings with it danger of death;— what can it be to run against a Woman, except absolute and immediate destruction? And when a Woman is invisible, or visible only as a dim sub-lustrous point, how difficult must it be, even for the most cautious, always to avoid collision! Many are the enactments made at different times in the different States of Flatland, in order to minimize this peril; and in the Southern and less temperate climates, where the force of gravitation is greater, and human beings more liable to casual and involuntary motions, the Laws concerning Women are naturally much more stringent. But a general view of the Code may be obtained from the following summary:— 1. Every house shall have one entrance in the Eastern side, for the use of Females only; by which all females shall enter “in a becoming and respectful manner”* and not by the Men’s or Western door. 2. No Female shall walk in any public place without continually keeping up her Peace-cry, under penalty of death.

* When I was in Spaceland I understood that some of your Priestly Circles have in the same way a separate entrance for Villagers, Farmers, and Teachers of Board Schools (Spectator, Sept. 1884, p. 1255) that they may “approach in a becoming and respectful manner.”

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10 3. Any Female, duly certified to be suffering from St. Vitus’s Dance, fits, chronic cold accompanied by violent sneezing, or any disease necessitating involuntary motions, shall be instantly destroyed. In some of the States there is an additional Law forbidding Females, under penalty of death, from walking or standing in any public place without moving their backs constantly from right to left so as to indicate their presence to those behind them; others oblige a Woman, when travelling, to be followed by one of her sons, or servants, or by her husband; others confine Women altogether in their houses except during the religious festivals. But it has been found by the wisest of our Circles or Statesmen that the multiplication of restrictions on Females tends not only to the debilitation and diminution of the race, but also to the increase of domestic murders to such an extent that a State loses more than it gains by a too prohibitive Code. For whenever the temper of the Women is thus exasperated by confinement at home or hampering regulations abroad, they are apt to vent their spleen upon their husbands and children; and in the less temperate climates the whole male population of a village has been sometimes destroyed in one or two hours of a simultaneous female outbreak. Hence the Three Laws, mentioned above, suffice for the better regulated States, and may be accepted as a rough exemplification of our Female Code. After all, our principal safeguard is found, not in Legislature, but in the interests of the Women themselves. For, although they can inflict instantaneous death by a retrograde movement, yet unless they can at once disengage their stinging extremity from the struggling body of their victim, their own frail bodies are liable to be shattered. The power of Fashion is also on our side. I pointed out that in some less civilised States no female is suffered to stand in any public place without swaying her back from right to left. This practice has been universal among ladies of any pretensions to breeding in all well-governed States, as far back as the memory of Figures can reach. It is considered a disgrace to any State that legislation should have to enforce what ought to be, and is in every respectable female, a natural instinct. The rhythmical and, if I may so say, well-modulated undulation of the back in our ladies of Circular rank is envied and imitated by the wife of a common Equilateral, who can achieve nothing beyond a mere monotonous swing, like the ticking of a pendulum; and the regular tick of the Equilateral is no less admired and copied by the wife of the progressive and aspiring Isosceles, in the females of whose family no “backmotion” of any kind has become as yet a necessity of life. Hence, in every family of position and consideration, “back motion” is as prevalent as time itself; and the husbands and sons in these households enjoy immunity at least from invisible attacks.

22

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Not that it must be for a moment supposed that our Women are destitute of affection. But unfortunately the passion of the moment predominates, in the Frail Sex, over every other consideration. This is, of course, a necessity arising from their unfortunate conformation. For as they have no pretensions to an angle, being inferior in this respect to the very lowest of the Isosceles, they are consequently wholly devoid of brainpower, and have neither reflection, judgment nor forethought, and hardly any memory. Hence, in their fits of fury, they remember no claims and recognise no distinctions. I have actually known a case where a Woman has exterminated her whole household, and half an hour afterwards, when her rage was over and the fragments swept away, has asked what has become of her husband and children! Obviously then a Woman is not to be irritated as long as she is in a position where she can turn round. When you have them in their apartments—which are constructed with a view to denying them that power—you can say and do what you like; for they are then wholly impotent for mischief, and will not remember a few minutes hence the incident for which they may be at this moment threatening you with death, nor the promises which you may have found it necessary to make in order to pacify their fury. On the whole we got on pretty smoothly in our domestic relations, except in the lower strata of the Military Classes. There the want of tact and discretion on the part of the husbands produces at times indescribable disasters. Relying too much on the offensive weapons of their acute angles instead of the defensive organs of good sense and seasonable simulations, these reckless creatures too often neglect the prescribed construction of the Women’s apartments, or irritate their wives by ill-advised expressions out of doors, which they refuse immediately to retract. Moreover a blunt and stolid regard for literal truth indisposes them to make those lavish promises by which the more judicious Circle can in a moment pacify his consort. The result is massacre; not however without its advantages, as it eliminates the more brutal and troublesome of the Isosceles; and by many of our Circles the destructiveness of the Thinner Sex is regarded as one among many providential arrangements for suppressing redundant population, and nipping Revolution in the bud. Yet even in our best regulated and most approximately circular families I cannot say that the ideal of family life is so high as with you in Spaceland. There is peace, in so far as the absence of slaughter may be called by that name, but there is necessarily little harmony of tastes or pursuits; and the cautious wisdom of the Circles has ensured safety at the cost of domestic comfort. In every Circular or Polygonal household it has been a habit from time immemorial—and has now become a kind of instinct among the women of our higher classes—that the mothers and daughters should constantly keep their eyes and mouths towards their husband and

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his male friends; and for a lady in a family of distinction to turn her back upon her husband would be regarded as a kind of portent, involving loss of status. But, as I 12 shall soon shew, this custom, though it has the advantage of safety, is not without disadvantages. his male friends; and for a lady in a family of distinction to turn her back upon her In thewould house of Working respectable Tradesman—where the wife husband be the regarded as aMan kindorof portent, involving loss of status. But,isasal-I lowedsoon to turn herthis backcustom, upon herthough husband, while herofhouse holdis avocations— shall shew, it has thepursuing advantage safety, not without there are at least intervals of quiet, when the wife is neither seen nor heard, except for disadvantages. the In humming the continuous Peace-cry;Tradesman—where but in the homes the of the the housesound of the of Working Man or respectable wifeupper is alclassesto there tooback oftenupon no peace. There the voluble mouthher and bright penetrating eye lowed turnisher her husband, while pursuing house hold avocations— are ever towards the Master of the and light is not except more perthere aredirected at least intervals of quiet, when thehouse wifehold; is neither seen itself nor heard, for sis tent than the stream of feminine discourse. Th e tact and skill which suffi ce to avert the humming sound of the continuous Peace-cry; but in the homes of the upper a Woman’s unequal to theTh task Woman’s and as the wife classes theresting is tooare often no peace. ere of thestopping voluble amouth and mouth; bright penetrating eye hasever absolutely nothing to the say,Master and absolutely no constraint wit,itself sense, or conscience are directed towards of the house hold; and of light is not more pertotent prevent saying it, not a few cynics have beenand found aver that sis thanher thefrom stream of feminine discourse. The tact skilltowhich suffithey ce toprefer avert the danger of the deathdealing but inaudible sting to the safe sonorousness of a Woma Woman’s sting are unequal to the task of stopping a Woman’s mouth; and as the wife an’sabsolutely other end.nothing to say, and absolutely no constraint of wit, sense, or conscience has To my her readers Spaceland condition of our Women seem to prevent frominsaying it, notthe a few cynics have been foundmay to aver thattruly theydeplorprefer able, and so indeed it is. A Male of the lowest type of the Isosceles may look the danger of the death-dealing but inaudible sting to the safe sonorousness of aforward Womto some of his angle, and to the ultimate elevation of the whole of his an’s otherimprovement end. degraded caste; but no Woman hopes for her sex.seem “Once a Woman, To my readers in Spaceland can the entertain conditionsuch of our Women may truly deploralways a Woman” is a Decree of Nature; and the very Laws of Evolution susable, and so indeed it is. A Male of the lowest type of the Isosceles may lookseem forward pended her disfavour. Yet angle, at leastand we to can theelevation wise Prearrangement to some in improvement of his theadmire ultimate of the whole which of his has ordained that, as they have no hopes, so they shall have no memory to recall, and degraded caste; but no Woman can entertain such hopes for her sex. “Once a Woman, no forethought to anticipate, the miseries and humiliations which are at once a necesalways a Woman” is a Decree of Nature; and the very Laws of Evolution seem sussity of their existence andYet theatbasis thecan constitution Flatland. pended in her disfavour. leastofwe admire theofwise Prearrangement which has ordained that, as they have no hopes, so they shall have no memory to recall, and no forethought to anticipate, the miseries and humiliations which are at once a neces§ 5 of their existence and the basis of the constitution of Flatland. sity

Of our methods of recognizing §§one 5 another 5: Of our methods of recognizing one another

Of our methods You, who are blessed with shade as well as light, you who are gifted with two eyes, enof recognizing dowed with a knowledge of perspective, and charmed with the enjoyment of various one another

colours, you, who can actually see an angle, and contemplate the complete circumference who of a are Circle in the Thyou ree who Dimensions—how shalleyes, I make You, blessed withhappy shaderegion as wellofas the light, are gifted with two enclear towith you athe extreme of diffi culty which wecharmed in Flatland in recognizing dowed knowledge perspective, and withexperience the enjoyment of various one another’s confican guration? colours, you, who actually see an angle, and contemplate the complete circumferwhat Iintold above. All of beings in ree Flatland, animate or inanimate, no enceRecall of a Circle theyou happy region the Th Dimensions—how shall I make mattertowhat form, present to our view we thein same, or nearly the same, clear you their the extreme difficulty which Flatland experience in appearance, recognizing 24 another’s viz. that flatland of a straight Line. How then can one be distinguished from another, where one configuration? all appear the same? The answer is threefold. The first means of recognition is the sense of hearing; which with us is far more highly developed than with you, and which enables us not 24 flatland only to distinguish by the voice of our personal friends, but even to discriminate be-

Recall what I told you above. All beings in Flatland, animate or inanimate, 13 no matter what their form, present to our view the same, or nearly the same, appearance, viz. that of a straight Line. How then can one be distinguished from another, where all appear the same? The answer is threefold. The first means of recognition is the sense of hearing; which with us is far more highly developed than with you, and which enables us not only to distinguish by the voice of our personal friends, but even to discriminate between different classes, at least so far as concerns the three lowest orders, the Equilateral, the Square, and the Pentagon—for of the Isosceles I take no account. But as we ascend in the social scale, the process of discriminating and being discriminated by hearing increases in difficulty, partly because voices are assimilated, partly because the faculty of voice-discrimination is a plebeian virtue not much developed among the Aristocracy. And wherever there is any danger of imposture we cannot trust to this method. Amongst our lowest orders, the vocal organs are developed to a degree more than correspondent with those of hearing, so that an Isosceles can easily feign the voice of a Polygon, and, with some training, that of a Circle himself. A second method is therefore more commonly resorted to. Feeling is, among our Women and lower classes—about our upper classes I shall speak presently—the principal test of recognition, at all events between strangers, and when the question is, not as to the individual, but as to the class. What therefore “introduction” is among the higher classes in Spaceland, that the process of “feeling” is with us. “Permit me to ask you to feel and be felt by my friend Mr. So-and-so”—is still, among the more old-fashioned of our country gentlemen in districts remote from towns, the customary formula for a Flatland introduction. But in the towns, and among men of business, the words “be felt by” are omitted and the sentence is abbreviated to, “Let me ask you to feel Mr. So-and-so”; although it is assumed, of course, that the “feeling” is to be reciprocal. Among our still more modern and dashing young gentlemen—who are extremely averse to superfluous effort and supremely indifferent to the purity of their native language—the formula is still further curtailed by the use of “to feel” in a technical sense, meaning, “to recommend-for-thepurposes-of-feeling-and being-felt”; and at this moment the “slang” of polite or fast society in the upper classes sanctions such a barbarism as “Mr. Smith, permit me to feel you Mr. Jones.” Let not my Reader however suppose that “feeling” is with us the tedious process that it would be with you, or that we find it necessary to feel right round all the sides of every individual before we determine the class to which he belongs. Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of any individual; and this,The onceBook ascertained, 25 tells us the class of the person whom we are addressing, unless indeed he belongs to the higher sections of the nobility. There the difficulty is much greater. Even a Master of Arts in our University of Wentbridge has been known to confuse a ten-sided with a

14

Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of any individual; and this, once ascertained, tells us the class of the person whom we are addressing, unless indeed he belongs to the higher sections of the nobility. There the difficulty is much greater. Even a Master of Arts in our University of Wentbridge has been known to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a Doctor of Science in or out of that famous University who could pretend to decide promptly and unhesitatingly between a twenty-sided and a twenty-four sided member of the Aristocracy. Those of my readers who recall the extracts I gave above from the Legislative code concerning Women, will readily perceive that the process of introduction by contact requires some care and discretion. Otherwise the angles might inflict on the unwary Feeler irreparable injury. It is essential for the safety of the Feeler that the Felt should stand perfectly still. A start, a fidgety shifting of the position, yes, even a violent sneeze, has been known before now to prove fatal to the incautious, and to nip in the bud many a promising friendship. Especially is this true among the lower classes of the Triangles. With them, the eye is situated so far from their vertex that they can scarcely take cognizance of what goes on at that extremity of their frame. They are moreover of a rough coarse nature, not sensitive to the delicate touch of the highly organized Polygon. What wonder then if an involuntary toss of the head has ere now deprived the State of a valuable life! I have heard that my excellent Grandfather—one of the least irregular of his unhappy Isosceles class, who indeed obtained, shortly before his decease, four out of seven votes from the Sanitary and Social Board for passing him into the class of the Equal-sided—often deplored, with a tear in his venerable eye, a miscarriage of this kind, which had occurred to his great-great-great-Grandfather, a respectable Working Man with an angle or brain of 59° 30’. According to his account, my unfortunate Ancestor, being afflicted with rheumatism, and in the act of being felt by a Polygon, by one sudden start accidentally transfixed the Great Man through the diagonal; and thereby, partly in consequence of his long imprisonment and degradation, and partly because of the moral shock which pervaded the whole of my Ancestor’s relations, threw back our family a degree and a half in their ascent towards better things. The result was that in the next generation the family brain was registered at only 58°, and not till the lapse of five generations was the lost ground recovered, the full 60° attained, and the Ascent from the Isosceles finally achieved. And all this series of calamities from one little accident in the process of Feeling. At this point I think I hear some of my better educated readers exclaim, “How could you in Flatland know anything about angles and degrees, or minutes? We can see an angle, because we, in the region of Space, can see two straight lines inclined to one another; but you, who can see nothing but one straight line at a time, or at all events only a number of bits of straight lines all in one straight line,—how can you 26 flatland ever discern any angle, and much less register angles of different sizes?” I answer that though we cannot see angles, we can infer them, and this with great



15

see an angle, because we, in the region of Space, can see two straight lines inclined to one another; but you, who can see nothing but one straight line at a time, or at all events only a number of bits of straight lines all in one straight line,—how can you ever discern any angle, and much less register angles of different sizes?” I answer that though we cannot see angles, we can infer them, and this with great precision. Our sense of touch, stimulated by necessity, and developed by long training, enables us to distinguish angles far more accurately than your sense of sight, when unaided by a rule or measure of angles. Nor must I omit to explain that we have great natural helps. It is with us a Law of Nature that the brain of the Isosceles class shall begin at half a degree, or thirty minutes, and shall increase (if it increases at all) by half a degree in every generation; until the goal of 60° is reached, when the condition of serfdom is quitted, and the freeman enters the class of Regulars. Consequently, Nature herself supplies us with an ascending scale or Alphabet of angles for half a degree up to 60°, specimens of which are placed in every Elementary School throughout the land. Owing to occasional retrogressions, to still more frequent moral and intellectual stagnation, and to the extraordinary fecundity of the Criminal and Vagabond Classes, there is always a vast superfluity of individuals of the half degree and single degree class, and a fair abundance of Specimens up to 10°. These are absolutely destitute of civil rights; and a great number of them, not having even intelligence enough for the purposes of warfare, are devoted by the States to the service of education. Fettered immovably so as to remove all possibility of danger, they are placed in the class rooms of our Infant Schools, and there they are utilized by the Board of Education for the purpose of imparting to the offspring of the Middle Classes that tact and intelligence of which these wretched creatures themselves are utterly devoid. In some states the Specimens are occasionally fed and suffered to exist for several years; but in the more temperate and better regulated regions, it is found in the long run more advantageous for the educational interests of the young, to dispense with food, and to renew the Specimens every month,—which is about the average duration of the foodless existence of the Criminal class. In the cheaper schools, what is gained by the longer existence of the Specimens is lost, partly in the expenditure for food, and partly in the diminished accuracy of the angles, which are impaired after a few weeks of constant “feeling.” Nor must we forget to add, in enumerating the advantages of the more expensive system, that it tends, though slightly yet perceptibly, to the diminution of the redundant Isosceles population—an object which every statesman in Flatland constantly keeps in view. On the whole therefore—although I am not ignorant that, in many popularly elected School Boards, there is a reaction in favour of “the cheap system,” as it is called— I am myself disposed to think that this is one of the many cases in which expense is the truest economy. But I must not allow questions of School Board politics to divert me from my subject. Enough has been said, I trust, to shew that Recognition by Feeling is not so tedious or indecisive a process as might have been supposed; and it The is obviously Book more 27 trustworthy than Recognition by hearing. Still there remains, as has been pointed out above, the objection that this method is not without danger. For this reason

subject. Enough has been said, I trust, to shew that Recognition by Feeling is not so tedious or indecisive a process as might have been supposed; and it is obviously more trustworthy than Recognition by hearing. Still there remains, as has been pointed 16 out But above, the not objection that this method not without Forme this reason I must allow questions of Schoolis Board politicsdanger. to divert from my many inEnough the Middle and Lower classes,toand without exception the Polygonal subject. has been said, I trust, shewallthat Recognition by in Feeling is not so and Circular orders, prefer a third method, description which be reserved tedious or indecisive a process as might havethe been supposed;ofand it isshall obviously more for the next section. trustworthy than Recognition by hearing. Still there remains, as has been pointed out above, the objection that this method is not without danger. For this reason many in the Middle and Lower classes, and all without exception in the Polygonal § 6 Circular orders, prefer a third method, the description of which shall be reserved and Of the Recognition for next section.

by Sight

§ 6: Of Recognition by Sight

§I am 6 about to appear very inconsistent. In the previous sections I have said that all figures in Flatland present the appearance of a straight line; and it was added or imOf Recognition plied, that it is consequently impossible to distinguish by the visual organ between by Sight individuals of different classes: yet now I am about to explain to my Spaceland critics are able to recognize one anotherInbythe theprevious sense of sections sight. I have said that all Ihow am we about to appear very inconsistent. If however the Reader the trouble refer toline; the passage in which figures in Flatland presentwill thetake appearance of atostraight and it was addedRecogor imnition that by Feeling is stated to beimpossible universal,to hedistinguish will find this cation—“among the plied, it is consequently byqualifi the visual organ between lower classes.” It iserent only classes: among yet thenow higher classes our more temperate climates individuals of diff I am aboutand to in explain to my Spaceland critics that Sight is practised. how we areRecognition able to recognize one another by the sense of sight. Thhowever at this power existswill in take any regions andtofor anytoclasses, is the Fog; If the Reader the trouble refer the passage in result which of Recogwhich by prevails during thetogreater part of he thewill year in this all parts the torrid zones. nition Feeling is stated be universal, find qualifisave cation—“among the That which is Itwith youamong in Spaceland an unmixed blotting out the landscape, lower classes.” is only the higher classes andevil, in our more temperate climates depressing the spirits, and enfeebling the health, is by us recognized as a blessing that Sight Recognition is practised. scarcely to air itself,inand the Nurse Parentis ofthesciences. ButFog; let That inferior this power exists anyasregions and offorarts anyand classes, result of me explain my during meaning, benefisave centthe Element. which prevails thewithout greater further part of eulogies the yearon in this all parts torrid zones. Fog were non-you existent, all lines an would appear equally andout indistinguishably ThatIf which is with in Spaceland unmixed evil, blotting the landscape, clear; and this is actually case in those unhappy countries in which atmodepressing the spirits, andthe enfeebling the health, is by us recognized as the a blessing sphere isinferior perfectlytodry transparent. wherever a richofsupply of Fog, scarcely air and itself, and as theBut Nurse of artsthere and is Parent sciences. Butoblet jectsexplain that are a distance, say of further three feet, are appreciably dimmer than those at a me myatmeaning, without eulogies on this benefi cent Element. distance two feet inches; and would the result is that by careful and constant exIf Fogofwere non-eleven existent, all lines appear equally and indistinguishably perimental observation of comparative dimness and clearness, weinarewhich enabled infer clear; and this is actually the case in those unhappy countries thetoatmowith great exactness confi guration of thewherever object observed. sphere is perfectly drythe and transparent. But there is a rich supply of Fog, obinstance do more a volume generalitiesdimmer to makethan mythose meaning jectsAn that are at awill distance, say than of three feet, areofappreciably at a clear. of two feet eleven inches; and the result is that by careful and constant exdistance Suppose observation I see two individuals approaching whose I wishwe toare ascertain. are, perimental of comparative dimness and rank clearness, enabledThey to infer we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle with great exactness the configuration of the object observed. and a Pentagon: how to distinguish them? of generalities to make my meaning instance will am do Imore than a volume 28 An flatland It will be obvious, to every child in Spaceland who has touched the threshold of clear. Geometrical Studies, that, if I can bring my eye so that its glance may bisect an angle 28

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(A) of the approaching stranger, myapproaching view will lie as it were evenly between his two sides Suppose I see two individuals whose rank I wish to ascerthat to me and ab),a soMerchant that I shall contemplate theor two and tain.are Thnext ey are, we(viz. willca suppose, and a Physician, inimpartially, other both will appear of the same size. words, an Equilateral Triangle and a Pentagon: how am I to distinguish Now in the case of (i) the Merchant, what shall I see? I shall see a straight line dae, them? in which point (a) willchild be very bright because is nearest me; but on It willthe bemiddle obvious, to every in Spaceland who ithas touchedtothe either side of theGeometrical line will shade away rapidly dimness, ac and threshold Studies, that, ifinto I can bringbecause my eyesides so that its ab recede rapidly fog;an and what(A) appear me as the Merchant’s extremities, viz. d and e, glance into may the bisect angle of thetoapproaching stranger, my will veryliedim viewbewill as indeed. it were evenly between his two sides that are next to me (viz. ca and ab), so that I shall contemplate the two impartially, and both will appear of the same size. Now in the case of (1) the Merchant, what shall I see? I shall see a straight line dae, in which the middle point (a) will be very bright because it is nearest to me; but on either side the line will shade away rapidly into dimness, because the sides ac and ab recede rapidly into the fog; and what appear to me as the Merchant’s extremities, viz. d and e, will be very dim indeed. On the other hand in the case of (2) the Physician, though I shall here also see a line (d´a´e´) with a bright centre (a´), yet it will shade away less rapidly into dimness, because the sides (a´c´, a´b´) recede less rapidly into the fog; and what appear to me the Physician’s extremities, viz. d´ and e´, will be not so dim as the extremities of the Merchant. The Reader will probably understand from these two instances how—after a very long training supplemented by constant experience—it is possible for the welleducated classes among us to discriminate with fair accuracy between the middle and lowest orders, by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible—I shall have attained all I can reasonably expect. Were I to attempt further details I should only perplex. Yet for the sake of the young and inexperienced, who may perchance infer—from the two simple instances I have given above, of the manner in which I should recognize my Father and my Sons—that Recognition by sight is an easy affair, it may be needful to point out that in actual life most of the problems of Sight Recognition are far more subtle and complex. If for example, when my Father, the Triangle, approaches me, he happens to present his side to me instead of his angle, then, until I have asked him to rotate, or until I have edged my eye around him, I am for the moment doubtful whether he may not be a Straight Line, or, in other words, a Woman. Again, when I am in the company of one of my two hexagonal Grandsons, contemplating one of his sides (ab) full front,

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18 it will be evident from the accompanying diagram that I shall see one whole line (ab) in comparative brightness (shading off hardly at all at the ends) and two smaller lines (ca and bd) dim throughout and shading away into greater dimness toward the extremities c and d. But I must not give way to the temptation of enlarging on these topics. The meanest mathematician in Spaceland will readily believe me when I assert that the prob-

lems of life, which present themselves to the well-educated—when they are themselves in motion, rotating, advancing or retreating, and at the same time attempting to discriminate by the sense of sight between a number of Polygons of high rank moving in different directions, as for example in a ball-room or conversazione—must be of a nature to task the angularity of the most intellectual, and amply justify the rich endowments of the Learned Professors of Geometry, both Static and Kinetic, in the illustrious University of Wentbridge, where the Science and Art of Sight Recognition are regularly taught to large classes of the élite of the States. It is only a few of the scions of our noblest and wealthiest houses, who are able to give the time and money necessary for the thorough prosecution of this noble and valuable Art. Even to me, a Mathematician of no mean standing, and the Grandfather of two most hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd of rotating Polygons of the higher classes, is occasionally very perplexing. And of course to a common Tradesman, or Serf, such a sight is almost as unintelligible as it would be to you, my Reader, were you suddenly transported into our country. In such a crowd you could see on all sides of you nothing but a Line, apparently straight, but of which the parts would vary irregularly and perpetually in brightness or dimness. Even if you had completed your third year in the Pentagonal and Hexagonal classes in the University, and were perfect in the theory of the subject, you would still find that there was need of many years of experience, before you could move in a fashionable crowd without jostling against your betters, whom it is against etiquette to ask to “feel,” and who, by their superior culture and breeding, know all about your movements, while you know very little or nothing about theirs. In a word, to comport oneself with perfect propriety in Polygonal society, one ought to be a Polygon oneself. Such at least is the painful teaching of my experience.

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It is astonishing how much the Art—or I may almost call it instinct—of Sight Recognition is developed by the habitual practice of it and by the avoidance of the custom of “Feeling.” Just as, with you, the deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet, will never acquire the more difficult but far more valuable art of lip-speech and lip-reading, so it is with us as regards “Seeing” and “Feeling.” None who in early life resort to “Feeling” will ever learn “Seeing” in perfection. For this reason, among our Higher Classes, “Feeling” is discouraged or absolutely forbidden. From the cradle their children, instead of going to the Public Elementary schools (where the art of Feeling is taught,) are sent to higher Seminaries of an exclusive character; and at our illustrious University, to “feel” is regarded as a most serious fault, involving Rustication for the first offence, and Expulsion for the second. But among the lower classes the art of Sight Recognition is regarded as an unattainable luxury. A common Tradesman cannot afford to let his son spend a third of his life in abstract studies. The children of the poor are therefore allowed to “feel” from their earliest years, and they gain thereby a precocity and an early vivacity which contrast at first most favourably with the inert, undeveloped, and listless behaviour of the half-instructed youths of the Polygonal class; but when the latter have at last completed their University course, and are prepared to put their theory into practice, the change that comes over them may almost be described as a new birth, and in every art, science, and social pursuit they rapidly overtake and distance their Triangular competitors. Only a few of the Polygonal Class fail to pass the Final Test or Leaving Examination at the University. The condition of the unsuccessful minority is truly pitiable. Rejected from the higher class, they are also despised by the lower. They have neither the matured and systematically trained powers of the Polygonal Bachelors and Masters of Arts, nor yet the native precocity and mercurial versatility of the youthful Tradesman. The professions, the public ser vices, are closed against them; and though in most States they are not actually debarred from marriage, yet they have the greatest difficulty in forming suitable alliances, as experience shows that the offspring of such unfortunate and ill-endowed parents is generally itself unfortunate, if not positively Irregular. It is from these specimens of the refuse of our Nobility that the great Tumults and Seditions of past ages have generally derived their leaders; and so great is the mischief thence arising that an increasing minority of our more progressive Statesmen are of opinion that true mercy would dictate their entire suppression, by enacting that all

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who fail to pass the Final Examination of the University should be either imprisoned 20 for life, or extinguished by a painless death. find myself digressing into the of subject of Irregularities, such vital whoBut failIto pass the Final Examination the University should abematter eitherof imprisoned interest that it demands a separate section. for life, or extinguished by a painless death. But I find myself digressing into the subject of Irregularities, a matter of such vital interest that it demands a separate section.

§7 Of Irregular Figures § 7: Concerning Irregular Figures §7 Throughout previous pages I have been assuming—what perhaps should have Of Irregular the Figures

been laid down at the beginning as a distinct and fundamental proposition—that everyroughout human being in Flatland is aI Regular Figure, that is to sayperhaps of regular construcTh the previous pages have been assuming—what should have tion. By this I mean that a Woman must not only be a line, but a straight line; been laid down at the beginning as a distinct and fundamental proposition—thatthat evan Artisan Soldier must have of his sides equal; Tradesmen have ery human or being in Flatland is a two Regular Figure, that is that to say of regularmust constructhree By sides Lawyers whichmust class not I amonly a humble member), four sides tion. thisequal; I mean that a (of Woman be a line, but a straight line;equal, that and, generally, that in every Polygon, all the sides must be equal. an Artisan or Soldier must have two of his equal; that Tradesmen must have Thsides e sizes of the sides would of course upon the age of the A three equal; Lawyers (of which class Idepend am a humble member), fourindividual. sides equal, Female at birththat would be about an inch whilemust a tallbeadult Woman might extend and, generally, in every Polygon, alllong, the sides equal. to aTh foot. As to of every may be upon roughly theindividual. length of an e sizes of the Males sides would of class, courseit depend thesaid age that of the A adult’s sides, when added is twolong, feet while or a little more. the size of our sides Female at birth would be together, about an inch a tall adultBut Woman might extend is not under speaking the be equality of sides, andthe it does notof need to a foot. As consideration. to the Males ofI am every class, itofmay roughly said that length an much refl ection to see that the whole of the social life in Flatland rests upon the funadult’s sides, when added together, is two feet or a little more. But the size of our sides damental fact that Nature wills Figuresoftothe have their of sides equal. is not under consideration. I am all speaking equality sides, and it does not need If our sides were angles might belife unequal. Instead its being sufmuch reflection to seeunequal that theour whole of the social in Flatland restsofupon the funfi cient to feel, or estimate by sight, a single angle in order to determine the form of damental fact that Nature wills all Figures to have their sides equal. an individual, it would be necessary to ascertain each angle by the experiment of If our sides were unequal our angles might be unequal. Instead of its being sufwould beby too shorta single for such a tedious groping. The whole science fiFeeling. cient toBut feel,life or estimate sight, angle in order to determine the form of and art of Sight Recognition would at once perish; Feeling, so far as it is an art, an individual, it would be necessary to ascertain each angle by the experiment of would not would orTh impossible; there Feeling. Butlong life survive; would beintercourse too short for suchbecome a tediousperilous groping. e whole science would end Recognition to all confidence, allatforethought; one would be safe and artbeofanSight would once perish;noFeeling, so far as itinis making an art, the most social arrangements; in a become word, civilization relapsethere into would notsimple long survive; intercourse would perilous ormight impossible; barbarism. would be an end to all confidence, all forethought; no one would be safe in making fast to carry my Readers with me to these obvious the Am mostI going simpletoo social arrangements; in a word, civilization might conclusions? relapse into Surely a moment’s reflection, and a single instance from common life, must convince barbarism. every that our is based upon or Equality of AnAmone I going toowhole fast tosocial carrysystem my Readers with meRegularity, to these obvious conclusions? gles. You meet, forrefl example, three instance Tradesmen in common the street,life, whom recogSurely a moment’s ection, two and or a single from mustyou convince nize atone once to our be Tradesmen bysystem a glanceis at theirupon anglesRegularity, and rapidly every that whole social based orbedimmed Equality ofsides, Anand you ask them to step into your house to lunch. Th is you do at present with gles. You meet, for example, two or three Tradesmen in the street, whom you recogperfect confito dence, because everyone knows to an inch and or two the area occupied by nize at once be Tradesmen by a glance at their angles rapidly bedimmed sides, an adult your Tradesman drags hisatregular and reand you Triangle: ask thembut to imagine step intothat your house to lunch. Thisbehind you do present with spectableflatland vertex, a parallelogram of twelve or thirteen inches in diagonal:—what are 32 you to do with such a monster sticking fast in your house door? am insulting the intelligence of my Readers by accumulating details which 32 But Iflatland must be patent to everyone who enjoys the advantages of a Residence in Spaceland.

perfect confidence, because everyone knows to an inch or two the area occupied 21 by an adult Triangle: but imagine that your Tradesman drags behind his regular and respectable vertex, a parallelogram of twelve or thirteen inches in diagonal:—what are you to do with such a monster sticking fast in your house door? But I am insulting the intelligence of my Readers by accumulating details which must be patent to everyone who enjoys the advantages of a Residence in Spaceland. Obviously the measurements of a single angle would no longer be sufficient under such portentous circumstances; one’s whole life would be taken up in feeling or surveying the perimeter of one’s acquaintances. Already the difficulties of avoiding a collision in a crowd are enough to tax the sagacity of even a well-educated Square; but if no one could calculate the Regularity of a single figure in the company, all would be chaos and confusion, and the slightest panic would cause serious injuries, or—if there happened to be any Women or Soldiers present—perhaps considerable loss of life. Expediency therefore concurs with Nature in stamping the seal of its approval upon Regularity of conformation: nor has the Law been backward in seconding their efforts. “Irregularity of Figure” means with us the same as, or more than, a combination of moral obliquity and criminality with you, and is treated accordingly. There are not wanting, it is true, some promulgators of paradoxes who maintain that there is no necessary connection between geometrical and moral Irregularity. “The Irregular,” they say, “is from his birth scouted by his own parents, derided by his brothers and sisters, neglected by the domestics, scorned and suspected by society, and excluded from all posts of responsibility, trust, and useful activity. His every movement is jealously watched by the police till he comes of age and presents himself for inspection; then he is either destroyed, if he is found to exceed the fi xed margin of deviation, or else immured in a Government Office as a clerk of the seventh class; prevented from marriage; forced to drudge at an uninteresting occupation for a miserable stipend; obliged to live and board at the office, and to take even his vacation under close supervision; what wonder that human nature, even in the best and purest, is embittered and perverted by such surroundings!” All this very plausible reasoning does not convince me, as it has not convinced the wisest of our Statesmen, that our ancestors erred in laying it down as an axiom of policy that the toleration of Irregularity is incompatible with the safety of the State. Doubtless, the life of an Irregular is hard; but the interests of the Greater Number require that it shall be hard. If a man with a triangular front and a polygonal back were allowed to exist and to propagate a still more Irregular posterity, what would become of the arts of life? Are the houses and doors and churches in Flatland to be altered in order to accommodate such monsters? Are our ticket-collectors to be required to measure every man’s perimeter before they allow him to enter a theatre, or to take his place in a lecture room? Is an Irregular to be exempted from the militia? And if not, how is he to be prevented from carrying desolation into the ranksThe of his comrades? Book 33 Again, what irresistible temptations to fraudulent impostures must needs beset such a creature! How easy for him to enter a shop with his polygonal front foremost, and to order goods to any extent from a confiding Tradesman! Let the advocates of a

place in a lecture room? Is an Irregular to be exempted from the militia? And if not, how is he to be prevented from carrying desolation into the ranks of his comrades? measurewhat everyirresistible man’s perimeter beforetothey allow him to enter amust theatre, or beset to takesuch his Again, temptations fraudulent impostures needs 22 in a lecture room? an to Irregular be exempted from the front militia? And if not, aplace creature! How easy for Ishim enter a to shop with his polygonal foremost, and how is hegoods to be to prevented from carry desolation into the ranks his comrades? to order any extent from a ing confi ding Tradesman! Let theof advocates of a Again,called what irresistible temptations to fraudulent besetPenal such falsely Philanthropy plead as they may for theimpostures abrogationmust of theneeds Irregular a creature! How easy fornever him to enter an a shop with who his polygonal front foremost, and Laws, I for my part have known Irregular was not also what Nature evito order goods to from a confi ding Tradesman! advocates dently intended himany to extent be—a hypocrite, a misanthropist, and,Let up the to the limits ofofhisa falsely called Philanthropy as they may for the abrogation of the Irregular Penal power—a perpetrator of allplead manner of mischief. Laws, part have never known an Irregular was not also whatmea Nature NotI for thatmy I should be disposed to recommend (atwho present) the extreme sures eviaddentlyin intended him to be—a a misanthropist, up to the limits of the his opted some States, where anhypocrite, infant whose angle deviatesand, by half a degree from power—a perpetrator of all manner of mischief. correct angularity is summarily destroyed at birth. Some of our highest and ablest Not that should be disposed to recommend (at days present) the extreme measures admen, men ofIreal genius, have during their earliest laboured under deviations as optedas, in or some States, where anfortyinfantfive whose angleand deviates by half a degree fromlives the great even greater than, minutes: the loss of their precious correcthave angularity summarilyinjury destroyed birth. our highest ablest would been anisirreparable to the at State. TheSome art ofofhealing also hasand achieved men, of men realglorious genius,triumphs have during their earliest daysextensions, laboured under deviations as some its of most in the compressions, trepannings, colligreat as, and or even greater than, five minutes: loss Irregularity of their precious lives gations, other surgical or fortydiaetetic operationsand bythe which has been would or have beencured. an irreparable injurytherefore to the State. e art ofI would healinglay also has achieved partly wholly Advocating, a ViaTh Media, down no fixed some of its most triumphs collior absolute line ofglorious demarcation; butinatthe thecompressions, period when extensions, the frame istrepannings, just beginning to gations, and other surgical Board or diaetetic operations whichisIrregularity been set, and when the Medical has reported that by recovery improbable,has I would partly orthat wholly cured. Advocating, a Via I would lay down no fixed suggest the Irregular offspring betherefore painlessly andMedia, mercifully consumed. or absolute line of demarcation; but at the period when the frame is just beginning to set, and when the Medical Board has reported that recovery is improbable, I would that the Irregular offspring be painlessly and mercifully consumed. §suggest 8

Of the Ancient Practice of Painting § 8: Of the Ancient Practice of Painting §8 Ofmy theReaders Ancienthave followed me with any attention up to this point, they will not be If Practice oftoPainting surprised hear that life is somewhat dull in Flatland. I do not, of course, mean that there are not battles, conspiracies, tumults, factions, and all those other phenomena If my Readers have followed with any attentionnor upwould to thisIpoint, they the willstrange not be which are supposed to make me History interesting; deny that surprisedoftothe hear that lifeofis life somewhat in Flatland. I do not, ofcontinually course, mean that mixture problems and thedull problems of Mathematics, inducthereconjecture are not battles, conspiracies, tumults,offactions, and verifi all those other phenomena ing and giving the opportunity immediate cation, imparts to our which areasupposed to you make interesting; norcomprehend. would I deny that the strange existence zest which in History Spaceland can hardly I speak now from mixture of theand problems life and the problems of that Mathematics, continually inducthe aesthetic artistic of point of view when I say life with us is dull; aesthetiing conjecture and giving the opportunity of immediate verification, imparts to our cally and artistically, very dull indeed. existence a zest which you in when Spaceland canprospect, hardly comprehend. I speak now from How can it be otherwise, all one’s all one’s landscapes, historical the aesthetic andflartistic pointlife, of are viewnothing when Ibut say athat lifeline, withwith us isnodull; aesthetipieces, portraits, owers, still single varieties excallydegrees and artistically, very dull indeed. cept of brightness and obscurity? 34 It wasflatland not always thus. Colour, if Tradition speaks the truth, once for the space of half a dozen centuries or more, threw a transient splendour over the lives of our ancestors in remote ages. Some private individual—a Pentagon whose name is variously 34 flatlandcasually discovered the constituents of the simpler colours and a reported—having rudimentary method of painting, is said to have begun by decorating first his house, then his slaves, then his Father, his Sons and Grandsons, lastly himself. The convenience as well as the beauty of the results commended themselves to all. Wherever

cept degrees of brightness and obscurity? It was not always thus. Colour, if Tradition speaks the truth, once for the space of half a dozen centuries or more, threw a transient splendour over the lives of our an 23 cestors in remote ages. Some private individual—a Pentagon whose name is variously reported—having casually discovered the constituents of the simpler colours and a rudimentary method of painting, is said to have begun by decorating first his house, then his slaves, then his Father, his Sons and Grandsons, lastly himself. The convenience as well as the beauty of the results commended themselves to all. Wherever Chromatistes,—for by that name the most trustworthy authorities concur in calling him,—turned his variegated frame, there he at once excited attention, and attracted respect. No one now needed to “feel” him; no one mistook his front for his back; all his movements were readily ascertained by his neighbours without the slightest strain on their powers of calculation; no one jostled him, or failed to make way for him; his voice was saved the labour of that exhausting utterance by which we colourless Squares and Pentagons are often forced to proclaim our individuality when we move amid a crowd of ignorant Isosceles. The fashion spread like wildfire. Before a week was over, every Square and Triangle in the district had copied the example of Chromatistes, and only a few of the more conservative Pentagons still held out. A month or two found even the Dodecagons infected with the innovation. A year had not elapsed before the habit had spread to all but the very highest of the Nobility. Needless to say, the custom soon made its way from the district of Chromatistes to surrounding regions; and within two generations no one in all Flatland was colourless except the Women and the Priests. Here Nature herself appeared to erect a barrier, and to plead against extending the innovations to these two classes. Many-sidedness was almost essential as a pretext for the Innovators. “Distinction of sides is intended by Nature to imply distinction of colours”—such was the sophism which in those days flew from mouth to mouth, converting whole towns at a time to the new culture. But manifestly to our Priests and Women this adage did not apply. The latter had only one side, and therefore— plurally and pedantically speaking—no sides. The former—if at least they would assert their claim to be really and truly Circles, and not mere high-class Polygons with an infinitely large number of infinitesimally small sides—were in the habit of boasting (what Women confessed and deplored) that they also had no sides, being blessed with a perimeter of one line or, in other words, a Circumference. Hence it came to pass that these two Classes could see no force in the so-called axiom about “Distinction of Sides implying Distinction of Colour”; and when all others had succumbed to the fascinations of corporal decoration, the Priests and the Women alone still remained pure from the pollution of paint. 35 Immoral, licentious, anarchical, unscientific—call them by The whatBook names you will—yet, from an aesthetic point of view, those ancient days of the Colour Revolt were the glorious childhood of Art in Flatland—a childhood, alas, that never ripened into manhood, nor even reached the blossom of youth. To live was then in itself a delight, because living implied seeing. Even at a small party, the company was a pleasure to behold; the richly varied hues of the assembly in a church or theatre are said to have more than once proved too distracting from our greatest teachers and actors; but most ravishing of all is said to have been the unspeakable magnificence of a mili-

anarchical, unscientifi c—callTothem by then whatinnames intoImmoral, manhood,licentious, nor even reached the blossom of youth. live was itself ayou dewill—yet, from an aesthetic point of view, those ancient days of the Colour light, because living implied seeing. Even at a small party, the company was aRevolt pleawere thebehold; gloriousthe childhood of Arthues in Flatland—a childhood, alas, that never ripened sure to richly varied of the assembly in a church or theatre are said 24 into manhood, nor even reached the blossom of youth. To live was then in itself a deto have more than once proved too distracting from our greatest teachers and actors; light, because living implied seeing. Even at a small party, the company was a but most ravishing of all is said to have been the unspeakable magnificence of a pleamilisure to behold; the richly varied hues of the assembly in a church or theatre are said tary review. to have onceofproved tootwenty distracting from Isosceles our greatest teachers and actors; The more sight than of a line battle of thousand suddenly facing about, but most ravishing of all is said to have been the unspeakable magnifi cence of aofmiliand exchanging the sombre black of their bases for the orange and purple the tary two review. sides including their acute angle; the militia of the Equilateral Triangles tricoThein sight a lineand of battle of twenty Isosceles suddenly about, loured red,ofwhite, blue; the mauve,thousand ultramarine, gamboge, and facing burnt umber and exchanging the sombre black of their bases for the orange and purple of the of the Square artillerymen rapidly rotating near their vermilion guns; the dashing two including acute angle; thecoloured militia ofPentagons the Equilateral Trianglescareertricoand sides flashing of the fitheir ve-coloured and sixand Hexagons loured in red, blue; the of mauve, ultramarine, gamboge, and burnt umber ing across the white, field inand their offices surgeons, geometricians and aidesde-camp— of Square artillerymen rapidly rotating near their vermilion guns;story the dashing all the these may well have been sufficient to render credible the famous how an and flashing of theovercome five-coloured six-coloured Hexagons careerillustrious Circle, by theand artistic beauty ofPentagons the forcesand under his command, ing across in theirbaton offices of his surgeons, geometricians andthat aidescamp— threw asidethe hisfield marshal’s and royal crown, exclaiming hedehenceforth all these may wellfor have suffi cient How to render the famous story how an exchanged them the been artist’s pencil. greatcredible and glorious the sensuous develillustrious overcome by the artistic of the forces under command, opment of Circle, these days must have been is inbeauty part indicated by the veryhis language and threw aside of histhe marshal’s and his royal crown, exclaiming that he henceforth vocabulary period.baton The commonest utterances of the commonest citizens in exchanged for the artist’s theasensuous develthe time ofthem the Colour Revolt pencil. seem toHow havegreat beenand suffglorious used with richer tinge of opment of these days must have been is in part indicated by the very language and word or thought; and to that era we are even now indebted for our finest poetry vocabulary ofever the period. e commonest the commonest in and for what rhythmTh still remains inutterances the more of scientifi c utterancecitizens of those the time of the Colour Revolt seem to have been suff used with a richer tinge of modern days. word or thought; and to that era we are even now indebted for our finest poetry and for whatever rhythm still remains in the more scientific utterance of those modern days. §9

Of the Universal Colour Bill § 9: Of the Universal Colour Bill §9 Of Universalthe intellectual Arts were fast decaying. Butthe meanwhile Colour The Bill Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon But intellectual Arts decaying. to bemeanwhile consideredthe superfl uous, and fellwere intofast disrepute and neglect even at our Univerof Sight beingexperienced no longer needed, was noatlonger practised; sity.Th TheeArt inferior Art Recognition, of Feeling speedily the same fate our Elementary and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon Schools. Th en the Isosceles classes, asserting that the Specimens were no longer used 36 flatland nor needed, and refusing to pay the customary tribute from the Criminal classes to the ser vice of Education, waxed daily more numerous and more insolent on the 36 strength flatland of their immunity from the old burden which had formerly exercised the twofold wholesome effect of at once taming their brutal nature and thinning their excessive numbers. Year by year the Soldiers and Artisans began more vehemently to assert—and with increasing truth—that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to

twofold wholesome effect of at once taming their brutal nature and thinning their excessive numbers. Year by year the Soldiers and Artisans began more vehemently to assert—and 25 with increasing truth—that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to demand the legal prohibition of all “monopolising and aristocratic Arts” and the consequent abolition of all endowments for the studies of Sight Recognition, Mathematics, and Feeling. Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights. Finding the higher Orders wavering and undecided, the leaders of the Revolution advanced still further in their requirements, and at last demanded that all classes alike, the Priests and the Women not excepted, should do homage to Colour by submitting to be painted. When it was objected that Priests and Women had no sides, they retorted that Nature and Expediency concurred in dictating that the front half of every human being (that is to say, the half containing his eye and mouth) should be distinguishable from his hinder half. They therefore brought before a general and extraordinary Assembly of all the States of Flatland a Bill proposing that in every Woman the half containing the eye and mouth should be coloured red, and the other half green. The Priests were to be painted in the same way, red being applied to that semicircle in which the eye and mouth formed the middle point; while the other or hinder semicircle was to be coloured green. There was no little cunning in this proposal, which indeed emanated, not from any Isosceles–for no being so degraded would have had angularity enough to appreciate, much less to devise, such a model of state-craft—but from an Irregular Circle who, instead of being destroyed in his childhood, was reserved by a foolish indulgence to bring desolation on his country and destruction on myriads of his followers. On the one hand the proposition was calculated to bring the Women in all classes over to the side of the Chromatic Innovation. For by assigning to the Women the same two colours as were assigned The Book 37 to the Priests, the Revolutionists thereby ensured that, in certain positions, every Woman would appear like a Priest, and be treated with corresponding respect and deference-a prospect that could not fail to attract the Female Sex in a mass.

to the Priests, the Revolutionists thereby ensured that, in certain positions, every Woman would appear like a Priest, and be treated with corresponding respect and deference—a prospect that could not fail 26 to attract the Female Sex in a mass. But by some of my Readers the possibility of the identical appearance of Priests and Women, under the new Legislation, may not be recognized; if so, a word or two will make it obvious. Imagine a woman duly decorated, according to the new Code; with the front half (i.e., the half containing eye and mouth) red, and with the hinder half green. Look at her from one side. Obviously you will see a straight line, half red, half green. Now imagine a Priest, whose mouth is at m, and whose front semicircle (amb) is consequently coloured red, while his hinder semicircle is green; so that the diameter ab divides the green from the red. If you contemplate the Great Man so as to have your eye in the same straight line as his dividing diameter (ab), what you will see will be a straight line (cbd), of which one half (cb) will be red, and the other (bd) green. The whole line (cd) will be rather shorter perhaps than that of a full-sized Woman, and will shade off more rapidly towards its extremities; but the identity of the colours would give you an immediate impression of identity of Class, making you neglectful of other details. Bear in mind the decay of Sight Recognition which threatened society at the time of the Colour revolt; add too the certainty that Women would speedily learn to shade off their extremities so as to imitate the Circles; it must then be surely obvious to you, my dear Reader, that the Colour Bill placed us under a great danger of confounding a Priest with a young Woman. How attractive this prospect must have been to the Frail Sex may readily be imagined. They anticipated with delight the confusion that would ensue. At home they might hear political and ecclesiastical secrets intended not for them but for their husbands and brothers, and might even issue commands in the name of a priestly Circle; out of doors the striking combination of red and green, without addition of any other colours, would be sure to lead the common people into endless mistakes, and the Women would gain whatever the Circles lost, in the deference of the passers by. As for the scandal that would befall the Circular Class if the frivolous and unseemly conduct of the Women were imputed to them, and as to the consequent subversion of the Constitution, the Female Sex could not be expected to give a thought to these considerations. Even in the households of the Circles, the Women were all in favour of the Universal Colour Bill. The second object aimed at by the Bill was the gradual demoralization of the Circles themselves. In the general intellectual decay they still preserved their pristine clearness and strength of understanding. From their earliest childhood, familiarized 38 flatland in their Circular households with the total absence of Colour, the Nobles alone preserved the Sacred Art of Sight Recognition, with all the advantages that result from that admirable training of the intellect. Hence, up to the date of the introduction of the Universal Colour Bill, the Circles had not only held their own, but even increased their lead of other classes by abstinence from the popular fashion. Now therefore the artful Irregular whom I described above as the real author of this diabolical Bill, determined at one blow to lower the status of the Hierarchy by forcing them to submit to the pollution of Colour, and at the same time to destroy

servedlead theofSacred of Sight Recognition, all uthe their other Art classes by abstinence fromwith the pop laradvantages fashion. that result from thatNow admirable training of theIrregular intellect.whom Hence, up to the above date ofasthe therefore the artful I described theintroduction real author of the Universal Colour Bill, the Circles had not only held their own, but even increased this diabolical Bill, determined at one blow to lower the status of the by Hierarchy 27 their lead of other classestobythe abstinence the popand ularatfashion. forcing them to submit pollutionfrom of Colour, the same time to destroy therefore the artful of Irregular described above as the realsoauthor of theirNow domestic opportunities trainingwhom in theI Art of Sight Recognition, as to enthis diabolical Bill, determined at one blow to lower status of the homes. Hierarchy by feeble their intellects by depriving them of their purethe and colourless Once forcing them to chromatic submit to the pollution of Colour, at childish the sameCircle time would to destroy subjected to the taint, every parental andand every detheir domestic opportunities training in the Artthe of Sight so as would to enmoralize each other. Only inofdiscerning between FatherRecognition, and the Mother feeble their intellects depriving of theirofpure and colourless homes. Once the Circular infant findbyproblems forthem the exercise its understanding—problems too subjected to to thebechromatic every parental and every would the deoften likely corruptedtaint, by maternal impostures with childish the resultCircle of shaking moralize eachinother. Onlyconclusions. in discerning the Father and the Mother child’s faith all logical Thbetween us by degrees the intellectual lustre would of the the Circular infant find problems for road the exercise of its lie understanding—problems too Priestly Order would wane, and the would then open for a total destruction often likely to be Legislature corrupted by imposturesofwith the result Classes. of shaking the of all Aristocratic andmaternal for the subversion our Privileged child’s faith in all logical conclusions. Thus by degrees the intellectual lustre of the Priestly Order would wane, and the road would then lie open for a total destruction all Aristocratic Legislature and for the subversion of our Privileged Classes. §of 10

Of the Suppression of the Chromatic § 10 Sedition § 10: Of the Suppression of the Chromatic Sedition Of the Suppression of the Chromatic Th e agitation for the Universal Colour Bill continued for three years; and up to the Sedition last moment of that period it seemed as though Anarchy were destined to triumph. A whole army of Polygons, who turned out to fight as private soldiers, was utterly The agitationbyfora the Universal Bill continued for three years;and andPentagons up to the annihilated superior force Colour of Isosceles Triangles—the Squares last momentremaining of that period it seemed though Anarchy destined triumph. meanwhile neutral. Worse as than all, some of thewere ablest Circlestofell a prey to A whole army of Polygons, who out tothe fight as private utterly conjugal fury. Infuriated by politi calturned animosity, wives in manysoldiers, a noblewas house hold annihilated a superior force of Triangles—the Squares and Pentagons wearied theirbylords with prayers to Isosceles give up their opposition to the Colour Bill; and meanwhile remaining neutral.fruitless, Worse than all,and some of the ablest Circles fell achildren prey to some, finding their entreaties fell on slaughtered their innocent conjugal fury. Infuriated political in animosity, wives inItmany a noblethat house hold and husbands, perishing by themselves the act ofthe carnage. is recorded during wearied their lords with prayers give twentyup theirthree opposition the Colour Bill; and that triennial agitation no less tothan Circlestoperished in domestic some, finding their entreaties fruitless, fell on and slaughtered their innocent children discord. indeedperishing was the peril. It seemed as though the Priests no choice andGreat husbands, themselves in the act of carnage. It ishad recorded thatbetween during submission andagitation extermination; the course of events wasincompletely that triennial no lesswhen thansuddenly twenty-three Circles perished domestic changed never to nediscord. by one of those picturesque incidents which Statesmen ought The Book 39 glect, often to anticipate, and sometimes perhaps to originate, because of the absurdly disproportionate power with which they appeal to the sympathies of the populace. It happened that an Isosceles of a low type, with a brain little ifThe at all above four Book 39 degrees—accidentally dabbling in the colours of some Tradesman whose shop he had plundered—painted himself, or caused himself to be painted (for the story varies) with the twelve colours of a Dodecagon. Going into the Market Place he accosted in a feigned voice a maiden, the orphan daughter of a noble Polygon, whose affection in former days he had sought in vain; and by a series of deceptions—aided, on the one side, by a string of lucky accidents too long to relate, and, on the other, by an almost inconceivable fatuity and neglect of ordinary precautions on the part of the relations of the bride—he succeeded in consummating the marriage. The unhappy girl committed suicide on discovering the fraud to which she had been subjected.

with the twelve colours of a Dodecagon. Going into the Market Place he accosted in a feigned voice a maiden, the orphan daughter of a noble Polygon, whose affection in former days he had sought in vain; and by a series of deceptions—aided, on the one 28 side, by a string of lucky accidents too long to relate, and, on the other, by an almost inconceivable fatuity and neglect of ordinary precautions on the part of the relations of the bride—he succeeded in consummating the marriage. The unhappy girl committed suicide on discovering the fraud to which she had been subjected. When the news of this catastrophe spread from State to State the minds of the Women were violently agitated. Sympathy with the miserable victim and anticipations of similar deceptions for themselves, their sisters, and their daughters, made them now regard the Colour Bill in an entirely new aspect. Not a few openly avowed themselves converted to antagonism; the rest needed only a slight stimulus to make a similar avowal. Seizing this favourable opportunity, the Circles hastily convened an extraordinary Assembly of the States; and besides the usual guard of Convicts, they secured the attendance of a large number of reactionary Women. Amidst an unprecedented concourse, the Chief Circle of those days—by name Pantocyclus—arose to find himself hissed and hooted by a hundred and twenty thousand Isosceles. But he secured silence by declaring that henceforth the Circles would enter on a policy of Concession; yielding to the wishes of the majority, they would accept the Colour Bill. The uproar being at once converted to applause, he invited Chromatistes, the leader of the Sedition, into the centre of the hall, to receive in the name of his followers the submission of the Hierarchy. Then followed a speech, a masterpiece of rhetoric, which occupied nearly a day in the delivery, and to which no summary can do justice. With a grave appearance of impartiality he declared that, as they were now finally committing themselves to Reform or Innovation, it was desirable that they should take one last view of the perimeter of the whole subject, its defects as well as its advantages. Gradually introducing the mention of the dangers to the Tradesmen, the Professional Classes and the Gentlemen, he silenced the rising murmurs of the Isosceles by reminding them that, in spite of all these defects, he was willing to accept the Bill if it was approved by the majority. But it was manifest that all, except the Isosceles, were moved by his words and were either neutral or averse to the Bill. 40 Turning flatland now to the Workmen he asserted that their interests must not be neglected, and that, if they intended to accept the Colour Bill, they ought at least to do so with a full view of the consequences. Many of them, he said, were on the point of being admitted to the class of the Regular Triangles; others anticipated for their children a distinction they could not hope for themselves. That honourable ambition would now have to be sacrificed. With the universal adoption of Colour, all distinctions would cease; Regularity would be confused with Irregularity; development would give place to retrogression; the Workman would in a few generations be degraded to the level of the Military, or even the Convict Class; political power would be in the hands of the greatest number, that is to say the Criminal Classes, who were already more numerous than the Workmen, and would soon out-number all the other Classes put together when the usual Compensative Laws of Nature were violated. A subdued murmur of assent ran through the ranks of the Artisans, and Chroma-

tions would cease; Regularity would be confused with Irregularity; development would give place to retrogression; the Workman would in a few generations be degraded to the level of the Military, or even the Convict Class; political power would29 be in the hands of the greatest number, that is to say the Criminal Classes, who were already more numerous than the Workmen, and would soon out-number all the other Classes put together when the usual Compensative Laws of Nature were violated. A subdued murmur of assent ran through the ranks of the Artisans, and Chromatistes, in alarm, attempted to step forward and address them. But he found himself encompassed with guards and forced to remain silent while the Chief Circle in a few impassioned words made a final appeal to the Women, exclaiming that, if the Colour Bill passed, no marriage would henceforth be safe, no woman’s honour secure; fraud, deception, hypocrisy would pervade every household; domestic bliss would share the fate of the Constitution and pass to speedy perdition. “Sooner than this,” he cried, “Come death.” At these words, which were the preconcerted signal for action, the Isosceles Convicts fell on and transfixed the wretched Chromatistes; the Regular Classes, opening their ranks, made way for a band of Women who, under direction of the Circles, moved, back foremost, invisibly and unerringly upon the unconscious Soldiers; the Artisans, imitating the example of their betters, also opened their ranks. Meantime bands of Convicts occupied every entrance with an impenetrable phalanx. The battle, or rather carnage, was of short duration. Under the skilful generalship of the Circles almost every Woman’s charge was fatal, and very many extracted their sting uninjured, ready for a second slaughter. But no second blow was needed; the rabble of the Isosceles did the rest of the business for themselves. Surprised, leaderless, attacked in front by invisible foes, and finding egress cut off by the Convicts behind them, they at once—after their manner—lost all presence of mind, and raised the cry of “treachery.” This sealed their fate. Every Isosceles now saw and felt a foe in every other. In half an hour not one of that vast multitude was living; and the fragments of seven score thousand of the Criminal Class slain by one another’s angles attested the triumph of Order. The Circles delayed not to push their victory to the uttermost. The Working Men they spared but decimated. The Militia of the Equilaterals was at once called out; and The Book 41 every Triangle suspected of Irregularity on reasonable grounds, was destroyed by Court Martial, without the formality of exact measurement by the Social Board. The homes of the Military and Artisan classes were inspected in a course of visitations extending through upwards of a year; and during that period every town, village, and hamlet was systematically purged of that excess of the lower orders which had been brought about by the neglect to pay the Tribute of Criminals to the Schools and University, and by the violation of the other natural Laws of the Constitution of Flatland. Thus the balance of classes was again restored. Needless to say that henceforth the use of Colour was abolished, and its possession prohibited. Even the utterance of any word denoting Colour, except by the Circles or by qualified scientific teachers, was punished by a severe penalty. Only at our University in some of the very highest and most esoteric classes—which I myself have never been privileged to attend—it is understood that the sparing use of Colour is still sanctioned for the purpose of illustrating some of the deeper problems of mathe-

versity, and by theEven violation of the other natural of theColour, Constitution sion prohibited. the utterance of any wordLaws denoting exceptofbyFlatland. the CirTh of scientifi classes was again restored. clesusorthe bybalance qualified c teachers, was punished by a severe penalty. Only at our Needlessinto say of that theand usemost of Colour abolished, and its possesUniversity some thehenceforth very highest esotericwas classes—which I myself have 30 sion prohibited. Even the utterance of any word denoting Colour, except by the Cir-is never been privileged to attend—it is understood that the sparing use of Colour cles by qualififor ed the scientifi c teachers, was punished a severe Only at our still or sanctioned purpose of illustrating some ofbythe deeperpenalty. problems of matheUniversity the only very speak highestfrom andhearsay. most esoteric classes—which I myself have matics. Butinofsome this of I can never been privileged to attend—it is understood thatTh the sparing use ofitColour is Elsewhere in Flatland, Colour is now non-existent. e art of making is known still sanctioned forperson, the purpose of illustrating of the deeper matheto only one living the Chief Circle for some the time being; andproblems by him it of is handed matics. But this Ibed can to only speak hearsay. One manufactory alone produces down on hisofdeathnone butfrom his Successor. Elsewhere Flatland, Colour is now nonexistent. Thare e art of making it is known it; and, lest theinsecret should be betrayed, the Workmen annually consumed, and to onlyones one introduced. living person, Chief Circle for the time being; andnow by him is handed fresh Sothe great is the terror with which even our itAristocracy down on histodeathbed to none One alone Bill. produces looks back the fardistant daysbut of his the Successor. agitation for themanufactory Universal Colour it; and, lest the secret should be betrayed, the Workmen are annually consumed, and fresh ones introduced. So great is the terror with which even now our Aristocracy looks § 11 back to the far-distant days of the agitation for the Universal Colour Bill.

Concerning our Priests § 11: Concerning our Priests § 11 Concerning It is high time that I should pass from these brief and discursive notes about things in our Prieststo the central event of this book, my initiation into the mysteries of Space. Flatland That is my subject; all that has gone before is merely preface. It isFor highthis time that IIshould pass from briefofand discursive notes about thingsnot, in reason must omit manythese matters which the explanation would Flatland to the central event of this book, my initiation into the mysteries of Space. I flatter myself, be without interest for my Readers: as for example, our method of Th at is my subject; all that has gonealthough before isdestitute merely preface. propelling and stopping ourselves, of feet; the means by which we thistoreason I mustofomit many matters of which the explanation would giveFor fixity structures wood, stone, or brick, although of course we havenot, no Ihands, flatternor myself, be without interest for my Readers: as for example, our method of can we lay foundations as you can, nor avail ourselves of the lateral prespropelling stopping ourselves, although of feet;inthethe means by which we sure of theand earth; the manner in which the destitute rain originates intervals between give fixity tozones, structures wood, stone,regions or brick, although of the course we have no our various so thatofthe northern do not intercept moisture falling hands, nor can wethe laynature foundations as you can, nor avail ourselves of the lateral on the southern; of our hills and mines, our trees and vegetables, ourpresseasons harvests; our Alphabet, suited to our linear tablets; our eyes adapted to our 42 andflatland linear sides; 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 42 proceeds,flatland 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 everyone 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 Poly-

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, 31 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 everyone 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. As the number of the sides increases, a Polygon approximates to a Circle; and, when the number is very great indeed, 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 enwrap 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 pin-head. 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 The in the scale of deBook 43 velopment, 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 son; more than one is never seen. On the other hand the son 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 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 re-set, 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.

descent. with serious risk—but sometimes overleaps two or three hundred generations, and as Many a promising child isthe sacrifi ced inofthis Scarcely and one out ten survives. it were doubles at a stroke, number hisway. progenitors the of nobility of his Yet so strong is the parental ambition among those Polygons who are, as it were, on descent. 32 the Many fringe aofpromising the Circular class, that ced it isinvery find a Nobleman, of that posichild is sacrifi thisrare way.toScarcely one out of ten survives. tion society, whoparental has neglected to among place his first-Polygons born sonwho in the NeoYet soinstrong is the ambition those are,Circular as it were, on Th he has attained thetoage of a Nobleman, month. theerapeutic fringe ofGymnasium the Circularbefore class, that it is very rare find of that posiyear determines or failure. thefiend of thatson time has, Neoin all tionOne in society, who has success neglected to placeAthis rst-born in the the child Circular probability, added one more crowd the Neo-Therapeutic Therapeutic Gymnasium beforetohethe hastombstones attained thethat age of a month. Cemetery; but on rare occasions a glad proAt cession bears backtime the the littlechild one has, to his One year determines success or failure. the end of that in exall ultant parents, no longer a Polygon, a Circle, atthat least crowd by courtesy: andTh a single inprobability, added one more to thebuttombstones the Neoerapeutic stance of sobut blessed result induces Polygonal parents to submit to Cemetery; on rarea occasions a gladmultitudes processionofbears back the little one to his exsimilar domestic ces, which have ultant parents, nosacrifi longer a Polygon, butaadissimilar Circle, at issue. 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.

§ 12 Of the Doctrine of our Priests § 12 § 12: Of the Doctrine of our Priests Of the Doctrine As to the doctrine of the Circles it may briefly be summed up in a single maxim, “Atof our Priests tend to your Configuration.” Whether political, ecclesiastical, or moral, all their teaching for its object improvement of individual and collective Confimaxim, guration— As to thehas doctrine of thethe Circles it may briefl y be summed up in a single “Atwith of course to the Confi of the Circles,ortomoral, which all other tend special to yourreference Configuration.” Whether poguration litical, ecclesiastical, their objects teachingare hassubordinated. for its object the improvement of individual and collective Configuration— is the merit of the Circlestothat they have effectually suppressed thoseall ancient withIt special reference of course the Confi guration of the Circles, to which other heresies which led men to waste energy and sympathy in the vain belief that conduct objects are subordinated. 44 flatland depends upon will, effort, training, encouragement, praise, or anything else but Configuration. It was Pantocyclus—the illustrious Circle mentioned above, as the queller of Revolt—who first convinced mankind that Configuration makes the 44 the Colour flatland man; that if, for example, you are born an Isosceles 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 flagitious 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 right angles? 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

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 33 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 right angles? 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, where the 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 sweetmeats, 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 Grandson’s 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 The passionately as 45 if Book they believed that these names represented real existences, and that a human Figure is really capable of choosing between them. Consistently carrying out their policy of making Configuration the leading idea in every mind, the Circles reverse the nature of that Commandment which in Spaceland 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

speak of anything Circular as containing any element of weakness—appears to me to be found in their relations with Women. 34 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 hereafter 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 has 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 enquire above the five-hundredth generation; a Hexagon or Dodecagon 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 a low voice—which, with us, even more than with you, is thought “an excellent thing in Woman.” ill-judged marriages are, as might be expected, barren, if they do not result 46 Suchflatland 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 Neo-Therapeutic 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, if this evil be not arrested, the gradual diminution of the Circular class may soon become more rapid, and the time may be not 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 bi-

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 35 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 also 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 ofThe Science. Book Already 47 methinks 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.

Part II

Other Worlds

“O brave new worlds, That have such people in them!”

§ 13 How I had a Vision Lineland §of13: How I had a Vision of Lineland It was the last day but one of the 1999th 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. 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 to 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, whence intrudest thou into my realm of Lineland?” 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 51 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 follow-

I begged pardon if I had in any way startled or molested his Royal Highness; and de40

scribing 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. Outside 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 eye-sight to that single Straight Line, which was their World. It need 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 Linelander. 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 Linelander 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 Lineland) there were none but Men—I ventured to reply, “Pardon me, but I cannot imagine how your Royal Highness can 52 any time flatland at 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 Lineland proximity is not necessary for marriage and for the generation of children?”

Staggered at this answer—for in the immediate proximity of the Monarch (as I had noted in my dream before I entered Lineland) there were none but Men—I 41 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 Lineland 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 upon 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 veriest baby in Lineland. 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 bass 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 bass voice and an utterly uneducated ear. But to continue. “Nature herself having ordained that every Man should wed two wives——” “Why two?” 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, viz. the Bass 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 to and fro with a 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 Bass 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 Lineland. “What! Always threefold?” said I. “Must one wife then always have The twins?” Book 53 “Bass-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.

the “What! paltry obstacles of distance,said Love unitesone the wife three. Thealways marriage that instant Always threefold?” I. “Must then haveintwins?” consummated in a threefold Male and Female“How offspring whichthe takes its place “Bass-voice results Monstrosity! yes,” replied the King. else could balance of in Lineland. the Sexes be maintained, if two girls were not born for every boy? Would you ignore 42 Alwaysofthreefold?” saidceased, I. “Must one wifefor then always the “What! very Alphabet Nature?” He speechless fury; and have sometwins?” time elapsed “Bassvoiceinduce Monstrosity! replied the King. “How else could the balance of before I could him to yes,” resume his narrative. the “You Sexeswill be maintained, if two girls that wereevery not born for every boy? you ignore not, of course, suppose bachelor among usWould finds his mates at Alphabet Nature?” ceased, speechless forthe fury; and some timecess elapsed the fivery rst wooing in of this universalHeMarriage Chorus. On contrary, the pro is by beforeofI us could induce to resume most many timeshim repeated. Fewhis arenarrative. the hearts whose happy lot it is at once to “You will not, other’s of course, suppose that every bachelor amongbyusProvidence, finds his mates at recognise in each voices the partner intended for them and to first awooing in this Chorus. On the contrary, cesscourtis by flthe y into reciprocal anduniversal perfectlyMarriage harmonious embrace. With most the of uspro the mostisofofuslong many times repeated. Few voices are themay hearts whoseaccord happywith lot itone is at to ship duration. The Wooer’s perhaps of once the furecognise each voices intended fororthem by Providence, and to ture wives,inbut notother’s with both; or the not,partner at first, with either; the Soprano and Contralto fly into reciprocal and perfectly harmonious Withthat most of us the courtmay notaquite harmonise. In such cases Nature embrace. has provided every weekly Choshipshall is ofbring long duration. e Wooer’s voicesharmony. may perhaps with oneeach of the furus the three Th Lovers into closer Eachaccord trial of voice, fresh ture wives, but not with both; or not, at fi rst, with either; or the Soprano and Contralto discovery of discord, almost imperceptibly induces the less perfect to modify his or mayvocal not quite harmonise. such cases Nature provided Choher utterance so as toInapproximate to the has more perfect.that Andevery afterweekly many trials rus shall the three Lovers into closer harmony. Each of voice, eachat fresh and manybring approximations, the result is at last achieved. Thtrial ere comes a day last, discovery of discord, almost imperceptibly thefrom less perfect to modify histhe or when, while the wonted Marriage Chorus induces goes forth universal Lineland, her vocal utterance so as to approximate to the more harmony, perfect. And many three far-off Lovers suddenly find themselves in exact and,after before theytrials are and many result is atinto last aachieved. ere comes at last, aware, the approximations, wedded Triplet isthe rapt vocally duplicateTh embrace; anda day Nature rewhen,over while wonted Marriage Chorus from universal Lineland, the joices onethe more marriage and over threegoes moreforth births.” three far-off Lovers suddenly find themselves in exact harmony, and, before they are aware, the wedded Triplet is rapt vocally into a duplicate embrace; and Nature re§joices 14 over one more marriage and over three more births.”

How I vainly tried to explain the nature § 14: 14 of Flatland How in my Vision I endeavoured to explain the nature of Flatland, but could not How I vainly tried to inking explainthat the itnature Th was time to bring down the Monarch from his raptures to the level Flatland sense, I determined to endeavour to open up to him some glimpses of the of common truth, that is to say of the nature of things in Flatland. So I began thus: “How does Thinking it was time to bringthe down the and Monarch fromofhis to Ithe your Royalthat Highness distinguish shapes positions hisraptures subjects? forlevel my of common determined to before endeavour to open upKingdom, to him some the part noticedsense, by theI sense of sight, I entered your thatglimpses some ofofyour truth, that is to say the nature things Flatland. I began thus: “How“You does people are Lines andofothers Points,ofand thatinsome of theSo Lines are larger——” your Royal Highness distinguish the shapes and“you positions of hisseen subjects? I for speak of an impossibility,” interrupted the King; must have a vision; formy to detectnoticed the diffbyerence between a Line and Ia entered Point byyour the sense of sight as every one part the sense of sight, before Kingdom, thatis,some of your knows, are in the nature things, impossible; butsome it can bylarger——” the sense of hearpeople Lines and of others Points, and that ofbe thedetected Lines are “You ing, and by impossibility,” the same meansinterrupted my shape can be exactly Behold me—I speak of flatland an the King; “youascertained. must have seen a vision; foram to 54 a Line, the longest in Lineland, over six inches of Space——” “Of Length,” I ventured to suggest. “Fool,” said he, “Space is Length. Interrupt me again, and I have done.” flatland 54 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 re-

a Line, the longest in Lineland, over six inches of Space——” “Of Length,” I ventured to suggest. “Fool,” said he, “Space is Length. Interrupt me again, and I have done.” 43 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 6.457 inches, infer that one of my mouths is 6.457 inches further from them than the other, and accordingly know my shape to be 6.457 inches. But you will of course understand that my Wives do not make this calculation every time they hear my two voices. They made it, once for all, before we were married. But they could make it at any time. And in the same way I can estimate the shape of any of my Male subjects by the sense of sound.” “But how,” said I, “if a Man feigns a Woman’s voice with one of his two voices, or so disguises his Southern voice that it cannot be 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, know, Stranger, that this offence is punishable in my dominions by death. And the reason is obvious. The frail form of a Woman, being liable to be shattered by such an approximation, must be preserved by the State; but since Women cannot be distinguished by the sense of sight from Men, the Law ordains universally that neither Man nor Woman shall be approached so closely as to destroy the interval between the approximator and the approximated. “And indeed what possible purpose would be served by this illegal and unnatural excess of approximation which you call touching, when all the ends of so brutal and coarse a process are attained at once more easily and more exactly by the sense of hearing. As to your suggested danger of deception, it is non-existent: for the Voice, being the essence of one’s Being, cannot be thus changed at will. But come, suppose that I had the power of passing through solid things, so that I could my The penetrate Book 55 subjects, one after another, even to the number of a billion, verifying the size and distance of each by the sense of feeling: how much time and energy would be wasted in this clumsy and inaccurate method! Whereas now, in one moment of audition, I take as it were the census and statistics, local, corporeal, mental and spiritual, of every living being in Lineland. Hark, only hark!” So saying he paused and listened, as if in an ecstasy, to a sound which seemed to me no better than a tiny chirping from an innumerable multitude of lilliputian grasshoppers.

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 44 as it were the census and statistics, local, corporeal, mental and spiritual, of every living being in Lineland. Hark, only hark!” So saying he paused and listened, as if in an ecstasy, to a sound which seemed to me no better than a tiny chirping from an innumerable multitude of lilliputian grasshoppers. “Truly,” replied I, “your sense of hearing serves you in good stead, and fi lls up many of your deficiencies. But permit me to point out that your life in Lineland must be deplorably dull. To see nothing but a Point! Not even to be able to contemplate a Straight Line! Nay, not even to know what a Straight Line is! To see, yet 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 Lineland which gives you such intense plea sure, is to me no better than a multitudinous twittering or chirping. But at least I can discern, by sight, a Line from a Point. And let me prove it. Just before I came into your kingdom, I saw you dancing from left to right, and then from right to left, with seven Men and a Woman in your immediate proximity on the left, and eight Men and two Women on your right. Is not this correct?” “It is correct,” said the King, “so far as the numbers and sexes are concerned, though I know not what you mean by ‘right’ and ‘left.’ But I deny that you saw these things. For how could you see the Line, that is to say the inside, of any Man? But you must have heard these things, and then dreamed that you saw them. And let me ask what you mean by those words ‘left’ and ‘right.’ I suppose it is your way of saying Northward and Southward.” “Not so,” replied I; “besides your motion of Northward and Southward, there is another motion which I call from right to left.” King. Exhibit to me, if you please, this motion from left to right. I. Nay, that I cannot do, unless you could step out of your Line altogether. King. Out of my Line? Do you mean out of the world? Out of Space? I. Well, yes. Out of your world. Out of your Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line. King. If you cannot indicate this motion from left to right by yourself moving in it, then I beg you to describe it to me in words. 56 I. If you flatland cannot tell your right side from your left, I fear that no words of mine can make my meaning clear to you. But surely you cannot be ignorant of so simple a distinction. King. I do not in the least understand you. I. Alas! How shall I make it clear? When you move straight on, does it not sometimes occur to you that you could move in some other way, turning your eye round so as to look in the direction towards which your side is now fronting? In other words, instead of always moving in the direction of one of your extremities,

I. If you cannot tell your right side from your left, I fear that no words of mine can make my meaning clear to you. But surely you cannot be ignorant of so simcan make my meaning clear to you. But surely you cannot be ignorant of so simple a distinction. ple a distinction. 45 King. I do not in the least understand you. 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 someI. 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 times occur to you that you could move in some other way, turning your eye round so as to look in the direction towards which your side is now fronting? In round so as to look in the direction towards which your side is now fronting? In other words, instead of always moving in the direction of one of your extremities, 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? 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 diKing. 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? rection? 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 I. Well then, since words cannot explain the matter, I will try deeds, and will move gradually out of Lineland in the direction which I desire to indicate to you. gradually out of Lineland in the direction which I desire to indicate to you. At the word I began to move my body out of Lineland. As long as any part of me At the word I began to move my body out of Lineland. As long as any part of me remained in his dominion and in his view, the King kept exclaiming, “I see you, I see 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, you still; you are not moving.” But when I had at last moved myself out of his Line, he cried in his shrillest voice, “She is vanished; she is dead.” “I am not dead,” replied he cried in his shrillest voice, “She is vanished; she is dead.” “I am not dead,” replied I; “I am simply out of Lineland, that is to say, out of the Straight Line which you call I; “I am simply out of Lineland, that is to say, out of the Straight Line which you call Space, and in the true Space, where I can see things as they are. And at this moment Space, and in the true Space, where I can see things as they are. And at this moment I can see your Line, or side—or inside as you are pleased to call it; and I can see also I can see your Line, or side—or inside as you are pleased to call it; and I can see also the Men and Women on the North and South of you, whom I will now enumerate, 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.” 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 conWhen I had done this at great length, I cried triumphantly, “Does this at last convince you?” And, with that, I once more entered Lineland, taking up the same posivince you?” And, with that, I once more entered Lineland, taking up the same position as before. tion as before. But the Monarch replied, “If you were a Man of sense—though, as you appear to 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 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 The Book 57 of which I am daily conscious. I, in return, ask you to describe in The words or indicate Book 57 by motion that other Line of which you speak. Instead of moving, you merely exercise some magic art of vanishing and returning to sight; and instead of any lucid description of your new World, you simply tell me the numbers and sizes of some forty of my retinue, facts known to any child in my capital. Can anything be more irrational or audacious? Acknowledge your folly or depart from my dominions.” Furious at his perversity, and especially indignant that he professed to be ignorant of my Sex, I retorted in no measured terms, “Besotted Being! You think your-

cise some Imagic art of vanishingI,and returning sight; and instead of any deof which am daily conscious. in return, ask to you to describe in words orlucid indicate scription yourother new Line World, simply me Instead the numbers and sizes some forty by motionofthat of you which you tell speak. of moving, youofmerely exerof retinue, facts to anyand child in my capital. Can anything morelucid irratiocisemy some magic art known of vanishing returning to sight; and instead be of any de46 nal or audacious? Acknowledge follytell or me depart my dominions.” scription of your new World, youyour simply the from numbers and sizes of some forty Furious at of hissense, perversity, indignant thatme he to professed to bethere ignohad a particle you would listen to my reason. YouCan ask believe that is of my retinue, facts known toand any especially child in capital. anything be more irratiorant of my Sex, I retorted in no mea sured terms, “Besotted Being! You think youranother Line besides that which my senses indicate, and another motion besides that nal or audacious? Acknowledge your folly or depart from my dominions.” selfwhich the perfection of existence, you ask are you in reality the most imperfect and of I am daily conscious. I, while in return, to that describe in words or be indicate Furious at his perversity, and especially indignant he professed to ignoimbecile. to see, whereas you can see nothing but a Point! You plume by motion that other Line of no which speak. Instead of moving, you exerrant of myYou Sex,profess I retorted in meayou sured terms, “Besotted Being! You merely think youryourself on inferring existence ofreturning a you Straight Line; but I instead canmost see of Straight Lines, cise magic art ofthe vanishing and sight; andthe any lucid deself some the perfection existence, while aretoin reality imperfect and and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even scription your new World, simply tellcan mesee thenothing numbersbut anda sizes of You someplume forty imbecile.ofYou profess to see, you whereas you Point! Circles. Why waste Suffi ce aitinStraight that amLine; theCan completion of be your incomplete of my retinue, facts more known to any child my Icapital. more irratioyourself on inferring thewords? existence of butanything I can see Straight Lines, self. You arethe a Line, but IofamAngles, a Line offolly Linesorcalled in from my country aHexagons, Square: and even I, nal audacious? Acknowledge your depart my dominions.” andor infer existence Triangles, Squares, Pentagons, and even infi nitely superior though I am to you, am of little account among the great Nobles of Furious his perversity, andSuffi especially that he professed be ignoCircles. Whyatwaste more words? ce it thatindignant I am the completion of yourto incomplete Flatland, whence have visit in terms, the hope of enlightening your ignorance.” rant of my Sex, IIretorted no mea sured “Besotted Being! You think yourself. You are a Line, but Icome aminato Line ofyou, Lines called in my country a Square: and even I, Hearing these words the King advanced towards me with a menacing cry as if to self the perfection of existence, while you are in reality the most imperfect and infinitely superior though I am to you, am of little account among the great Nobles of pierce mewhence through the diagonal; and you, in you that same movement there imbecile. You profess tocome see, to whereas see nothing but aarose Point! Youmyriads plume Flatland, I have visit in can the hope of enlightening yourfrom ignorance.” of his subjects a multitudinous war-advanced cry, in vehemence tillseeatStraight last methought yourself on inferring thethe existence of aincreasing Straight Line; Lines, Hearing these words King towards mebut withI can a menacing cry as if to it rivalled the roar of an army of a hundred thousand Isosceles, and the artillery of a and infer existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even pierce me through the diagonal; and in that same movement there arose from myriads thousand Pentagons. Spellbound and motionless, I could neither speak nor move to Circles. Why waste more words?warSuffi ce itincreasing that I aminthe completion incomplete of his subjects a multitudinous cry, vehemence tillofatyour last methought avert theare impending destruction; still the noise louder,and theartillery King came self. You but a Line of Lines called in grew myIsosceles, country aand Square: and even it rivalled thea Line, roar of anI am army of aand hundred thousand the of I,a closer, when I awoke to fi nd the breakfastbell recalling me to the realities of Flatland. infi nitely superior though I am to you, am of little account among the great Nobles thousand Pentagons. Spell-bound and motionless, I could neither speak nor move of to Flatland, I have come to visit thenoise hope grew of enlightening avert the whence impending destruction; andyou, stillinthe louder, andyour theignorance.” King came Hearing words King advanced a menacing as if to closer, when these I awoke to fithe nd the breakfastbelltowards recallingmemewith to the realities ofcry Flatland. § 15 me through the diagonal; and in that same movement there arose from myriads pierce Concerning of his subjectsa aStranger multitudinous war-cry, increasing in vehemence till at last methought from Spaceland it rivalled the roar an army of Spaceland a hundred thousand Isosceles, and the artillery of a § 15 § 15: Concerning a of Stranger from thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to Concerning a Stranger From dreams I proceed to facts. and still the noise grew louder, and the King came avert impending destruction; from the Spaceland It was theI last daytooffithe 1999th year ofbell ourrecalling era. Theme pattering of the rain had long closer, when awoke nd the breakfastto the realities of Flatland. ago announced nightfall; From dreams I proceed to and facts.I was sitting* in the company of my wife, musing on the It events of the andthethe prospects of our the coming the coming century, the was the lastpast day of 1999th year of era. The year, pattering of the rain had long coming Millennium. § 15 announced ago nightfall; and I was sitting* in the company of my wife, musing on * When I say “sitting,” of course I do not mean any change of attitude such as you in Spaceland sigMy four Sons Grandchildren to your theirsense several Concerning a Stranger nify by that word; for and as wetwo have orphan no feet, we can no more “sit”had nor retired “stand” (in of theapartword) than one of your or flalone ounders. ments; and mysoles Wife remained with me to see the old Millennium out and the from Spaceland * When I say “sitting,” of course I do not mean any change of attitude such as you in Spaceland signew one in. nify by that word; for as we have no feet, we can no more “sit” nor “stand” (in your sense of the word) was thought, in my mind some words that had casually issued From dreams Isoles proceed topondering facts. thanI one ofrapt yourin or flounders. 58 It the from mouth myofyoun Grandson, a most promising young Hexagon unwasflatland the lastofday the gest 1999th year of our era. Th e pattering of the rain hadoflong usual brilliancy and perfect angularity. His uncles and I had been giving him ago announced nightfall; and I was sitting* in the company of my wife, musing his on usual lesson in Sight Recognition, turning ourselves upon our centres, now 58 practical flatland rapidly, now more slowly, and questioning him as to our positions; and his answers * When I say “sitting,” of course I do not mean any change of attitude such as you in Spaceland sighad been so satisfactory that I had been induced to reward him by giving him a few nify by that word; for as we have no feet, we can no more “sit” nor “stand” (in your sense of the word) hintsone onofArithmetic applied to Geometry. than your soles or as flounders. 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— 58 flatland yet we might ascertain the number of square inches in a Square by simply squaring

I was rapt in thought, pondering in my mind some words that had casually issued from the mouth of my youngest Grandson, a most promising young Hexagon of unusual brilliancy and perfect angularity. His uncles and I had been giving him 47 his usual practical lesson in Sight Recognition, turning ourselves upon our centres, now rapidly, now more slowly, and questioning him as to our positions; and his answers had been so satisfactory that I had been induced to reward him by giving him a few hints on Arithmetic as applied to Geometry. Taking nine Squares, each an inch every way, I had put them together so as to make one large Square, with a side of three inches, and I had hence proved to my little Grandson that—though it was impossible for us to see the inside of the Square— yet we might ascertain the number of square inches in a Square by simply squaring the number of inches in the side: “and thus,” said I, “we know that 32, or 9, represents the number of square inches in a Square whose side is 3 inches long.” The little Hexagon meditated on this a while and then said to me: “But you have been teaching me to raise numbers to the third power; I suppose 33 must mean something in Geometry; what does it mean?” “Nothing at all,” replied I, “not at least in Geometry; for Geometry has only Two Dimensions.” And then I began to 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 3; and how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches every way, which may be represented by 32. Upon this, my Grandson, again returning to his former suggestion, took me up rather suddenly and exclaimed, “Well, then, if a Point by moving three inches, makes a Line of three inches represented by 3; and if a straight Line of three inches, moving parallel to itself, makes a Square of three inches every way, represented by 32; it must be that a Square of three inches every way, moving somehow parallel to itself (but I don’t see how) must make Something else (but I don’t see what) of three inches every way—and this must be represented by 33.” “Go to bed,” said I, a little ruffled by this interruption; “if you would talk less nonsense, you would remember more sense.” So my Grandson had disappeared in disgrace; and there I sat by my Wife’s side, endeavouring to form a retrospect of the year 1999 and of the possibilities of the year 2000, but not quite able to shake off the thoughts suggested by the prattle of my bright little Hexagon. Only a few sands now remained in the half-hour glass. Rousing myself from my reverie I turned the glass Northward for the last time in the old Millennium; and in the act, I exclaimed aloud, “The boy is a fool.” The Book 59 Straightway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being. “He is no such thing,” cried my Wife, “and you are breaking the Commandments in thus dishonouring your own Grandson.” But I took no notice of her. Looking around in every direction I could see nothing; yet still I felt a Presence, and shivered as the cold whisper came again. I started up. “What is the matter?” said my Wife, “there is no draught; what are you looking for? There is nothing.” There was nothing; and I resumed my seat, again exclaiming, “The boy is a fool, I say; 33 can have no meaning in Geometry.” At once there came a distinctly audible reply, “The boy is not a fool; and 33 has an obvious Geometrical meaning.” My Wife as well as myself heard the words, although she did not understand their

breaking the Commandments in thus dishonouring your own Grandson.” But I took no notice of her. Looking around 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 48 matter?” said my Wife, “there is no draught; what are you looking for? There is nothing.” There was nothing; and I resumed my seat, again exclaiming, “The boy is a fool, I say; 33 can have no meaning in Geometry.” At once there came a distinctly audible reply, “The boy is not a fool; and 33 has an obvious Geometrical meaning.” My Wife as well as myself heard the words, although she did not understand their meaning, and both of us sprang forward in the direction of the sound. What was our horror when we saw before us a Figure! At the first glance it appeared to be a Woman, seen sideways; but a moment’s observation shewed me that the extremities passed into dimness too rapidly to represent one of the Female Sex; and I should have thought it a Circle, only that it seemed to change its size in a manner impossible for a Circle or for any Regular Figure of which I had had experience. But my Wife had not my experience, nor the coolness necessary to note these characteristics. With the usual hastiness and unreasoning jealousy of her Sex, she flew at once to the conclusion that a Woman had entered the house through some small aperture. “How comes this person here?” she exclaimed, “you promised me, my dear, that there should be no ventilators in our new house.” “Nor are there any,” said I; “but what makes you think that the stranger is a Woman? I see by my power of Sight Recognition——” “Oh, I have no patience with your Sight Recognition,” replied she, “ ‘Feeling is believing’ and ‘A Straight Line to the touch is worth a Circle to the sight’ ”—two Proverbs, very common with the Frailer Sex in Flatland. “Well,” said I, for I was afraid of irritating her, “if it must be so, demand an introduction.” Assuming her most gracious manner, my Wife advanced towards the Stranger, “Permit me, Madam, to feel and be felt by——” then, suddenly recoiling, “Oh! it is not a Woman, and there are no angles either, not a trace of one. Can it be that I have so misbehaved to a perfect Circle?” “I am indeed, in a certain sense a Circle,” replied the Voice, “and a more perfect Circle than any in Flatland; but to speak more accurately, I am many Circles in one.” Then he added more mildly, “I have a message, dear Madam, to your husband, which I must not deliver in your presence; and, if you would suffer us to retire for a few minutes——” But my Wife would not listen to the proposal that our august Visitor should so incommode himself, and assuring the Circle that the hour of her own retirement had long passed, with many reiterated apologies for her recent indiscretion, she at last retreated to her apartment. 60 flatland I glanced at the half-hour glass. The last sands had fallen. The third Millennium had begun. Nevertheless, we perfectly well recognise 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.

§ 16 How the Stranger

How the Stranger vainly endeavoured to reveal to me words §in16: Howthe themysteries Stranger vainly endeavoured to reveal to me in words of Spaceland the mysteries of Spaceland

49

As soon as the sound of the Peace-cry of my departing Wife had died away, I began to approach the Stranger with the intention of taking a nearer view and of bidding him be seated: but his appearance struck me dumb and motionless with astonishment. Without the slightest symptoms of angularity he nevertheless varied every instant with gradations of size and brightness scarcely possible for any Figure within the scope of my experience. The thought flashed across me that I might have before me a burglar or cut-throat, some monstrous Irregular Isosceles, who, by feigning the voice of a Circle, had obtained admission somehow into the house, and was now preparing to stab me with his acute angle. In a sitting-room, the absence of Fog (and the season happened to be remarkably dry), made it difficult for me to trust to Sight Recognition, especially at the short distance at which I was standing. Desperate with fear, I rushed forward with an unceremonious “You must permit me, Sir—” and felt him. My Wife was right. There was not the trace of an angle, not the slightest roughness or inequality: never in my life had I met with a more perfect Circle. He remained motionless while I walked around 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. The Book to me 61 Stranger. Have you felt me enough by this time? Are you not introduced yet? I. Most illustrious Sir, excuse my awkwardness, which arises not from ignorance of the usages of polite society, but from a little surprise and ner vousness, consequent on this somewhat unexpected visit. And I beseech you to reveal my indiscretion to no one, and especially not to my Wife. But before your Lordship enters into further communications, would he deign to satisfy the curiosity of one who would gladly know whence his visitor came? Stranger. From Space, from Space, Sir: whence else? I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment? Stranger. Pooh! what do you know of Space? Define Space. I. Space, my Lord, is height and breadth indefinitely prolonged. Stranger. Exactly: you see you do not even know what Space is. You think it is of Two Dimensions only; but I have come to announce to you a Third—height,

I. Pardon me, my Lord, but is not your Lordship already in Space, your Lordship and his humble servant, even at this moment?

50

Stranger. Pooh! what do you know of Space? Define Space.

I. Space, my Lord, is height and breadth indefinitely prolonged. Stranger. Exactly: you see you do not even know what Space is. You think it is of Two Dimensions only; but I have come to announce to you a Third—height, breadth, and length. 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 forsooth. From that 62 position flatland 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

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.

51

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 any one of the neighbourhood possessing your Lordship’s ample means of obtaining information. Stranger. (To himself ). What must I do? Stay; one more argument suggests itself to me. When you see a Straight Line—your wife, for example—how many Dimensions do you attribute to her? I. Your Lordship would treat me as if I were one of the vulgar who, being ignorant of Mathematics, suppose that a Woman is really a Straight Line, and only of One Dimension. No, no, my Lord; we Squares are better advised, and are as well aware as your Lordship that a Woman, though popularly called a Straight Line, is, really and scientifically, a very thin Parallelogram, possessing Two Dimensions, like the rest of us, viz., length and breadth (or thickness). Stranger. But the very fact that a Line is visible implies that it possesses yet another Dimension. I. My Lord, I have just acknowledged that a Woman is broad as well as long. We see her length, we infer her breadth; which, though very slight, is capable of measurement. Stranger. You do not understand me. I mean that when you see a Woman, you ought—besides inferring her breadth—to see her length, and to see what we call her height; although that last Dimension is infinitesimal in your country. If a Line were mere length without “height,” it would cease to occupy space would beTheand Book 63 come invisible. Surely you must recognise this? I. I must indeed confess that I do not in the least understand your Lordship. When we in Flatland see a Line, we see length and brightness. If the brightness disappears, the line is extinguished, and, as you say, ceases to occupy space. But am I to suppose that your Lordship gives to brightness the title of a Dimension, and that what we call “bright” you call “high”? Stranger. No, indeed. By “height” I mean a Dimension like your length: only, with you, “height” is not so easily perceptible, being extremely small. I. My Lord, your assertion is easily put to the test. You say I have a Third Dimension, which you call “height.” Now, Dimension implies direction and measurement. Do but measure my “height,” or merely indicate to me the direction in which my “height” extends, and I will become your convert. Otherwise, your Lordship’s own understand must hold me excused. Stranger. (To himself ). I can do neither. How shall I convince him? Surely a plain statement of facts followed by ocular demonstration ought to suffice.—Now, Sir;

Stranger. No, indeed. By “height” I mean a Dimension like your length: only, with you, “height” is not so easily perceptible, being extremely small. I. My Lord, your assertion is easily put to the test. You say I have a Third Dimen52 sion, which you call “height.” Now, Dimension implies direction and measurement. Do but measure my “height,” or merely indicate to me the direction in which my “height” extends, and I will become your convert. Otherwise, your Lordship’s own understand must hold me excused.

Stranger. (To himself ). I can do neither. How shall I convince him? Surely a plain statement of facts followed by ocular demonstration ought to suffice.—Now, Sir; listen to me. You are living on a Plane. What you style Flatland is the vast level surface of what I may call a fluid, on, or in, the top of which you and your countrymen move about, without rising above it or falling below it. I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere—which is my proper name in my own country—if he manifest himself at all to an inhabitant of Flatland— must needs manifest himself as a Circle. Do you not remember—for I, who see all things, discerned last night the phantasmal vision of Lineland written upon your brain—do you not remember, I say, how when you entered the realm of Lineland, you were compelled to manifest yourself to the King not as a Square, but as a Line, because that Linear Realm had not Dimensions enough to represent the whole of you, but only a slice or section enough to precisely representthe me,same a being Three, but can only Dimensions exhibit a slice of of you? In way,ofyour country of Two is or notsection spacious me, which is what you a Circle. enough to represent me,call a being of Three, but can only exhibit a slice or section of e diminished brightness of your eye indicates incredulity. But now prepare me,Th which is what you call a Circle. to receive proof positive of the truth of my You cannotBut indeed more The diminished brightness of your eye assertions. indicates incredulity. nowsee prepare than one of my sections, Circles, time; for you You havecannot no power to raise your receive proof positive oforthe truth at of amy assertions. indeed see more 64 to flatland eye out of Flatland; but at you can atfor least that, I riseto inraise Space, so than oneofofthe myplane sections, or Circles, a time; yousee have noaspower your my section becomes smaller. See but now,you I will and see thethat, effectasupon eye will eye out of the plane of Flatland; canrise; at least I riseyour in Space, so be that my becomes Circle will become dwindles a point and my section smaller. Seesmaller now, I and will smaller rise; andtill theit eff ect upontoyour eye will finally be thatvanishes. my Circle will become smaller and smaller till it dwindles to a point and finally vanishes.

There was no “rising” that I could see; but he diminished and finally vanished. I winked or “rising” twice to that makeI could sure that was he notdiminished dreaming. and But it was no dream. There once was no see;I but finally vanished. from once the depths came hollow voice—close to my it IFor winked or twiceoftonowhere make sure thatforth I wasa not dreaming. But it was no heart dream.



53

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 of full size, then small, and at last very small indeed, approaching to a Point. But to me, although I saw the facts before me, the causes were as dark as ever. All that I could comprehend was, that the Circle had made himself smaller and vanished, and that he had now reappeared and was rapidly making himself larger. When he 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. The Book Sphere. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves65 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 altogether.

I. A square. Sphere. And how many sides has a Square? And how many angles?

54

I. Four sides and four angles.

Sphere. Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward. I. What? Northward? Sphere. No, not Northward; upward; out of Flatland altogether. If it moved Northward, the Southern points in the Square would have to move through the positions previously occupied by the Northern points. But that is not my meaning. I mean that every Point in you—for you are a Square and will serve the purpose of my 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. 66 flatland 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: 1, 2, 4, are evidently in Geometrical Progression. What is the next number? I. Eight. Sphere. Exactly. The one Square produces a Something-which-you-do-not-as-yetknow-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 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?

Points”? Sphere. Of course; and all according to Analogy. But, by the way, not what you 55 call sides, but what we call sides. You would call them solids.



I. And how many solids or sides will appertain to this Being whom I am to generate by the motion of my inside in an “upward” direction, and whom you call a Cube? Sphere. How can you ask? And you a mathematician! The side of anything is always, if I may so say, one Dimension behind the thing. Consequently, as there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may so say, has 2 sides (for the Points of a Line may be called by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that? I. Arithmetical. Sphere. And what is the next number? I. Six. Sphere. Exactly. Then you see you have answered your own question. The Cube which you will generate will be bounded by six sides, that is to say, six of your insides. You see it all now, eh?

§ 17“Monster,” I shrieked, “be thou juggler, enchanter, dream, or devil, no more will I endure mockeries. Either thou or I must perish.” And saying these words I preHow the thy Sphere, cipitated myself having in vain upon him. tried words, resorted to deeds

§ 17: How the Sphere, having in vain tried words, resorted to deeds

The Bookwith the 67 It was in vain. I brought my hardest right angle into violent collision 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 still I heard the Intruder’s voice. Sphere. Why will you refuse to listen to reason? I had hoped to find in you—as being a man of sense and an accomplished mathematician—a fit apostle for the Gospel of the Three Dimensions, which I am allowed to preach once only in a thousand years: but now I know not how to convince you. Stay, I have it. Deeds, and not words, shall proclaim the truth. Listen, my friend. I have told you I can see from my position in 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 bottom) full of money; I see also two tablets of accounts. I am about to descend into that cupboard and to bring you one of those tablets. I saw you lock the cupboard half an hour ago, and I know you have the key in your possession. But I descend from 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

ing, several of what you call boxes (but like everything else in Flatland, they have no tops nor bottom) full of money; I see also two tablets of accounts. I am about to descend into that cupboard and to bring you one of those tablets. I saw you lock the 56 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 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 to you. And, as a crowning proof, what do you say to my giving you a 68 back flatland 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 my 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 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

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 57 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 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—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.” “Ha! Is it come to this?” thundered the Stranger: “then meet your fate: out of your Plane you go. Once, twice, thrice! ’Tis done!”

§ 18 How I came to Spaceland, and I sawI there §what 18: How came to Spaceland, and what I saw there An unspeakable horror seized me. There was a darkness; then a dizzy, senThesickening Book 69 sation of sight that was not like seeing; I saw a Line that was no Line; Space that was not Space; I was myself, and not myself. When I could find voice, I shrieked aloud in agony, “Either this is madness or it is Hell.” “It is neither,” calmly replied the voice of the Sphere, “it is Knowledge; it is Three Dimensions: open your eye once again and try to look steadily.” I looked, and, behold, a new world! There stood before me, visibly incorporate, all that I had before inferred, conjectured, dreamed, of perfect Circular beauty. What seemed the centre of the Stranger’s form lay open to my view: yet I could see no heart, lungs, nor arteries, only a beautiful harmonious Something—for which I had no words; but you, my Readers in Spaceland, would call it the surface of the Sphere. Prostrating myself mentally before my Guide, I cried, “How is it, O divine ideal of consummate loveliness and wisdom, that I see thy inside, and yet cannot discern thy heart, thy lungs, thy arteries, thy liver?” “What you think you see, you see not,” he replied; “it is not given to you, nor to any other Being, to behold my internal parts. I am of a different order of Beings from those in Flatland. Were I a Circle, you could discern my intestines, but I am a Being composed, as I told you before, of many Circles, the Many in the One, called in this country a Sphere. And, just as the outside of a Cube is a Square, so the outside of a Sphere represents the appearance of a Circle.” Bewildered though I was by my Teacher’s enigmatic utterance, I no longer chafed against it, but worshipped him in silent adoration. He continued, with more mildness in his voice. “Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland. By degrees they will dawn upon you. Let us begin by casting back a glance at the region whence you came. Return with me a while to the plains of

many Circles, the Many in the One, called in this country a Sphere. And, just as the outside of a Cube is a Square, so the outside of a Sphere represents the appearance of a Circle.” 58 Bewildered though I was by my Teacher’s enigmatic utterance, I no longer chafed against it, but worshipped him in silent adoration. He continued, with more mildness in his voice. “Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland. By degrees they will dawn upon you. Let us begin by casting back a glance at the region whence you came. Return with me a while to the plains of Flatland, and I will show you that which you have so often reasoned and thought about, but never seen with the sense of sight—a visible angle.” “Impossible!” I cried; but, the Sphere leading the way, I followed as if in a dream, till once more his voice arrested me: “Look yonder, and behold your own Pentagonal house, and all its inmates.” I looked below, and saw with my physical eye all that domestic individuality which I had hitherto merely inferred with the understanding. And how poor and shadowy was the inferred conjecture in comparison with the reality which I now beheld! My four Sons calmly asleep in the North-Western rooms, my two orphan Grandsons to the South; the Servants, the Butler, my Daughter, all in their several 70 flatland apartments. Only my affectionate Wife, alarmed by my continued absence, had quitted her room and was roving up and down the Hall, anxiously awaiting my return. Also the Page, aroused by my cries, had left his room, and under pretext of ascertaining whether I had fallen somewhere in a faint, was prying into the cabinet in my study. All this I could now see, not merely infer; and as we came nearer and nearer, I could discern even the contents of my cabinet, and the two chests of gold, and the tablets of which the Sphere had made mention. beheld! My four Sons calmly asleep in the NorthWestern rooms, my two orphan Grandsons to the South; the Servants, the Butler, my Daughter, all in their several apartments. Only my affectionate Wife, alarmed by my continued absence, had quitted her room and was roving up and down in the Hall, anxiously awaiting my return. Also the Page, aroused by my cries, had left his room, and under pretext of ascertaining whether I had fallen somewhere in a faint, was prying into the cabinet in my study. All this I could now see, not merely infer; and as we came nearer and nearer, I could discern even the contents of my cabinet, and the two chests of gold, and the tablets of which the Sphere had made mention. Touched by my Wife’s distress, I would have sprung downward to reassure her, but I found myself incapable of motion. “Trouble not yourself about your Wife,” said my Guide: “she will not be long left in anxiety; meantime, let us take a survey of Flatland.” Once more I felt myself rising through space. It was even as the Sphere had said. The further we receded from the object we beheld, the larger became the field of vision. My native city, with the interior of every house and every creature therein, lay

Touched by my Wife’s distress, I would have sprung downward to reassure her, but I found myself incapable of motion. “Trouble not yourself about your Wife,” said my Guide: “she will not be long left in anxiety; meantime, let us take a survey59 of Flatland.” Once more I felt myself rising through space. It was even as the Sphere had said. The further we receded from the object we beheld, the larger became the field of vision. My native city, with the interior of every house and every creature therein, lay open to my view in miniature. We mounted higher, and lo, the secrets of the earth, the depths of the mines and inmost caverns of the hills, were bared before me. Awestruck at the sight of the mysteries of the earth, thus unveiled before my unworthy eye, I said to my Companion, “Behold, I am become as a God. For the wise men in our country say that to see all things, or as they express it, omnividence, is the attribute of God alone.” There was something of scorn in the voice of my Teacher as he made answer: “Is it so indeed? Then the very pick-pockets and cut-throats of my country are to be worshipped by your wise men as being Gods: for there is not one of them that does not see as much as you see now. But trust me, your wise men are wrong.” I. Then is omnividence the attribute of others besides Gods? Sphere. I do not know. But, if a pick-pocket or a cut-throat of our country can see everything that is in your country, surely that is no reason why the pick-pocket or cut-throat should be accepted by you as a God. This omnividence, as you call it— it is not a common word in Spaceland—does it make you more just, more merciful, less selfish, more loving? Not in the least. Then how does it make you more divine? I. “More merciful, more loving!” But these are the qualities of women! And we know that a Circle is a higher Being than a Straight Line, in so far as knowledge The Book 71 and wisdom are more to be esteemed than mere affection. Sphere. It is not for me to classify human faculties according to merit. Yet many of the best and wisest in Spaceland think more of the affections than of the understanding, more of your despised Straight Lines than of your belauded Circles. But enough of this. Look yonder. Do you know that building? I looked, and afar off I saw an immense Polygonal structure, in which I recognized the General Assembly Hall of the States of Flatland, surrounded by dense lines of Pentagonal buildings at right angles to each other, which I knew to be streets; and I perceived that I was approaching the great Metropolis. “Here we descend,” said my Guide. It was now morning, the first hour of the first day of the two thousandth year of our era. Acting, as was their wont, in strict accordance with precedent, the highest Circles of the realm were meeting in solemn conclave, as they had met on the first hour of the first day of the year 1000, and also on the first hour of the first day of the year 0. The minutes of the previous meetings were now read by one whom I at once recognised as my brother, a perfectly Symmetrical Square, and the Chief Clerk of the High Council. It was found recorded on each occasion that: “Whereas the States had been

day of the two thousandth year of our era. Acting, as was their wont, in strict accordance with precedent, the highest Circles of the realm were meeting in solemn conclave, as they had met on the first hour of the first day of the year 1000, and also on 60 the first hour of the first day of the year 0. The minutes of the previous meetings were now read by one whom I at once recognised as my brother, a perfectly Symmetrical Square, and the Chief Clerk of the High Council. It was found recorded on each occasion that: “Whereas the States had been troubled by divers ill-intentioned persons pretending to have received revelations from another World, and professing to produce demonstrations whereby they had instigated to frenzy both themselves and others, it had been for this cause unanimously resolved by the Grand Council that on the first day of each millenary, special injunctions be sent to the Prefects in the several districts of Flatland, to make strict search for such misguided persons, and without formality of mathematical examination, to destroy all such as were Isosceles of any degree, to scourge and imprison any regular Triangle, to cause any Square or Pentagon to be sent to the district Asylum, and to arrest any one of higher rank, sending him straightway to the Capital to be examined and judged by the Council.” “You hear your fate,” said the Sphere to me, while the Council was passing for the third time the formal resolution. “Death or imprisonment awaits the Apostle of the Gospel of Three Dimensions.” “Not so,” replied I, “the matter is now so clear to me, the nature of real space so palpable, that methinks I could make a child understand it. Permit me but to descend at this moment and enlighten them.” “Not yet,” said my Guide, “the time will come for that. Meantime I must perform my mission. Stay thou there in thy place.” Saying these words, he leaped with great dexterity into the sea (if I may so call it) of Flatland, right in the midst of the ring of Counsellors. “I come,” said he, “to proclaim that there is a land of Three Dimensions.” I could see many of the younger Counsellors start back in manifest horror, as the Sphere’s circular section widened before them. But on a sign from the presiding Circles—who showed not the slightest alarm or surprise—six Isosceles of a low type 72 flatland from six different quarters rushed upon the Sphere. “We have him,” they cried; “No; yes; we have him still! he’s going! he’s gone!” “My Lords,” said the President to the Junior Circles of the Council, “there is not the slightest need for surprise; the secret archives, to which I alone have access, tell me that a similar occurrence happened on the last two millennial commencements. You will, of course, say nothing of these trifles outside the Cabinet.” Raising his voice, he now summoned the guards. “Arrest the policemen; gag them. You know your duty.” After he had consigned to their fate the wretched policemen— ill-fated and unwilling witnesses of a State-secret which they were not to be permitted to reveal—he again addressed the Counsellors. “My Lords, the business of the Council being concluded, I have only to wish you a happy New Year.” Before departing, he expressed, at some length, to the Clerk, my excellent but most unfortunate brother, his sincere regret that, in accordance with precedent and for the sake of secrecy, he must condemn him to perpetual imprisonment, but added his satisfaction that, unless some mention were made by him of that day’s incident, his life would be spared.

§ 19

§ 19 How, though the Sphere showed me other mysteries of Spaceland, I still desired § 19: How, though the Sphere showed me other mysteries of Spaceland, more; and what came of it

61

I still desired more; and what came of it

When I saw my poor brother led away to imprisonment, I attempted to leap down into the Council Chamber, desiring to intercede on his behalf, or at least bid him farewell. But I found that I had no motion of my own. I absolutely depended on the volition of my Guide, who said in gloomy tones, “Heed not thy brother; haply thou shalt have ample time hereafter to condole with him. Follow me.” Once more we ascended into space. “Hitherto,” said the Sphere, “I have shown you naught save Plane Figures and their interiors. Now I must introduce you to Solids, and reveal to you the plan upon which they are constructed. Behold this multitude of moveable square cards. See, I put one on another, not, as you supposed, Northward of the other, but on the other. Now a second, now a third. See, I am building up a Solid by a multitude of Squares parallel to one another. Now the Solid is complete, being as high as it is long and broad, and we call it a Cube.”

The Book

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“Pardon me, my Lord,” replied I; “but to my eye the appearance is as of an Irregular Figure whose inside is laid open to the view; in other words, methinks I see no Solid, but a Plane such as we infer in Flatland; only of an Irregularity which betokens some monstrous criminal, so that the very sight of it is painful to my eyes.” “True,” said the Sphere; “it appears to you a Plane, because you are not accustomed to light and shade and perspective; just as in Flatland a Hexagon would appear a Straight Line to one who has not the Art of Sight Recognition. But in reality it is a Solid, as you shall learn by the sense of Feeling.” He then introduced me to the Cube, and I found that this marvellous Being was indeed no Plane, but a Solid; and that he was endowed with six plane sides and eight terminal points called solid angles; and I remembered the saying of the Sphere that just such a Creature as this would be formed by the Square moving, in Space, parallel to himself: and I rejoiced to think that so insignificant a Creature as I could in some sense be called the Progenitor of so illustrious an offspring. But still I could not fully understand the meaning of what my Teacher had told me concerning “light” and “shade” and “perspective”; and I did not hesitate to put my difficulties before him.

terminal points called solid angles; and I remembered the saying of the Sphere that just such a Creature as this would be formed by the Square moving, in Space, parallel to himself: and I rejoiced to think that so insignificant a Creature as I could in some 62 sense be called the Progenitor of so illustrious an offspring. But still I could not fully understand the meaning of what my Teacher had told me concerning “light” and “shade” and “perspective”; and I did not hesitate to put my difficulties before him. Were I to give the Sphere’s explanation of these matters, succinct and clear though it was, it would be tedious to an inhabitant of Space, who knows these things already. Suffice it, that by his lucid statements, and by changing the position of objects and lights, and by allowing me to feel the several objects and even his own sacred Person, he at last made all things clear to me, so that I could now readily distinguish between a Circle and a Sphere, a Plane Figure and a Solid. This was the Climax, the Paradise, of my strange eventful History. Henceforth I have to relate the story of my miserable Fall:—most miserable, yet surely most undeserved! For why should the thirst for knowledge be aroused, only to be disappointed and punished! My volition shrinks from the painful task of recalling my humiliation; yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity. Away then with all personal considerations! Let me continue to the end, as I began, without further digressions or anticipations, pursuing the plain path of dispassionate History. The exact facts, the exact words,—and they are burnt in upon my brain,—shall be set down without alteration of an iota; and let my Readers judge between me and Destiny. The Sphere would willingly have continued his lessons by indoctrinating me in the conformation of all regular Solids, Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons and Spheres: but I ventured to interrupt him. Not that I was wearied of knowledge.and OnSpheres: the contrary, thirsted for deeper and Hexahedrons, Dodecahedrons but II ventured toyet interrupt him.fuller Not draughts he was offering to me. that I wasthan wearied of knowledge. On the contrary, I thirsted for yet deeper and fuller 74 flatland “Pardon me,” saidoff I, ering “O Th Whom I must no longer address as the Perfecdraughts than he was toou me. tion“Pardon of all Beauty; but let me beg thee toIvouchsafe thy servant a sight thine me,” said I, “O Thou Whom must no longer address as theofPerfecinterior.” tion of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine interior.” Sphere. My what? Sphere. what?thy stomach, thy intestines. I. ThineMy interior: I. ThineWhence interior:this thyillstomach, thy intestines. Sphere. timed impertinent request? And what mean you by saying that I am no longer the Perfection of all Beauty? Sphere. Whence this ill-timed impertinent request? And what mean you by saying that no your longer thewisdom Perfection all Beauty? I. MyI am Lord, own hasoftaught me to aspire to One even more great, more and more closelyhas approximate than even yourself. you I. Mybeautiful, Lord, your own wisdom taught metotoPerfection aspire to One moreAsgreat, yourself, superior to all Flatland forms, combine many Circles in One, so doubtless more beautiful, and more closely approximate to Perfection than yourself. As you there is One abovetoyou who combines many Spheres One in Supreme yourself, superior all Flatland forms, combine many in Circles One, soExistence, doubtless surpassing even the you Solids of combines Spaceland.many And Spheres even as in we,One whoSupreme are nowExistence, in Space, there is One above who look down on Flatland and see the insides of all things, so of a certainty there is yet surpassing even the Solids of Spaceland. And even as we, who are now in Space, above us some higher, purer region, whither thou dost surely purpose to lead me— look down on Flatland and see the insides of all things, so of a certainty there is yet O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, me—

that I am no longer the Perfection of all Beauty? I. My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself. As you 63 yourself, superior to all Flatland forms, combine many Circles in One, so doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And even as we, who are now in Space, look down on Flatland and see the insides of all things, so of a certainty there is yet above us some higher, purer region, whither thou dost surely purpose to lead me— O Thou Whom I shall always call, everywhere and in all Dimensions, my Priest, Philosopher, and Friend—some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of Solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been vouchsafed. Sphere. Pooh! Stuff ! Enough of this trifling! The time is short, and much remains to be done before you are fit to proclaim the Gospel of Three Dimensions to your blind benighted countrymen in Flatland. I. Nay, gracious Teacher, deny me not what I know it is in thy power to reform. Grant me but one glimpse of thine interior, and I am satisfied for ever, remaining henceforth thy docile pupil, thy unemancipable slave, ready to receive all thy teachings and to feed upon the words that fall from thy lips. Sphere. Well, then, to content and silence you, let me say at once, I would show you what you wish if I could; but I cannot. Would you have me turn my stomach inside out to oblige you? I. But my Lord has shown me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What therefore more easy than now to take his servant on a second journey into the blessed region of the Fourth Dimension, where I shall look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensioned house, the secrets of the solid earth, the treasures of the mines of and TheSpaceland, Book 75 the intestines of every solid living creature, even the noble and adorable Spheres. Sphere. But where is this land of Four Dimensions? I. I know not: but doubtless my Teacher knows. Sphere. Not I. There is no such land. The very idea of it is utterly inconceivable. I. Not inconceivable, my Lord, to me, and therefore still less inconceivable to my Master. Nay, I despair not that, even here, in this region of Th ree Dimensions, your Lordship’s art may make the Fourth Dimension visible to me; just as in the Land of Two Dimensions my Teacher’s skill would fain have opened the eyes of his blind servant to the invisible presence of a Th ird Dimension, though I saw it not. Let me recall the past. Was I not taught below that when I saw a Line and inferred a Plane, I in reality saw a Th ird unrecognized Dimension, not the same as brightness, called “height”? And does it not now follow that, in this region,

your Lordship’s art may make the Fourth Dimension visible to me; just as in the Land of Two Dimensions my Teacher’s skill would fain have opened the eyes of his blind servant to the invisible presence of a Th ird Dimension, though I saw 64 it not. Let me recall the past. Was I not taught below that when I saw a Line and inferred a Plane, I in reality saw a Th ird unrecognized Dimension, not the same as brightness, called “height”? And does it not now follow that, in this region, when I see a Plane and infer a Solid, I really see a Fourth unrecognized Dimension, not the same as colour, but existent, though infinitesimal and incapable of mea surement? And besides this, there is the Argument from Analogy of Figures. Sphere. Analogy! Nonsense: what analogy? I. Your Lordship tempts his servant to see whether he remembers the revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, for more knowledge. Doubtless we cannot see that other higher Spaceland now, because we have no eye in our stomachs. But, just as there was the realm of Flatland, though that poor puny Lineland Monarch could neither turn to left nor right to discern it, and just as there was close at hand, and touching my frame, the land of Th ree Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to discern it, so of a surety there is a Fourth Dimension, which my Lord perceives with the inner eye of thought. And that it must exist my Lord himself has taught me. Or can he have forgotten what he himself imparted to his servant? In One Dimension, did not a moving Point produce a Line with two terminal points? In Two Dimensions, did not a moving Line produce a Square with four terminal points? In Three Dimensions, did not a moving Square produce—did not this eye of behold it—that blessed Being, a Cube, with eight terminal points? 76 mineflatland And in Four Dimensions shall not a moving Cube—alas, for Analogy, and alas for the Progress of Truth, if it be not so—shall not, I say, the motion of a divine Cube result in a still more divine Organization with sixteen terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this—if I might quote my Lord’s own words— “strictly according to Analogy”? Again, was I not taught by my Lord that as in a Line there are two bounding Points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, “strictly according to Analogy”? O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a Fourth Dimen-

an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught65 me to believe, “strictly according to Analogy”? O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts; and I appeal to your Lordship to confirm or deny my logical anticipations. If I am wrong, I yield, and will no longer demand a Fourth Dimension; but, if I am right, my Lord will listen to reason. I ask therefore, is it, or is it not, the fact, that ere now your countrymen also have witnessed the descent of Beings of a higher order than their own, entering closed rooms, even as your Lordship entered mine, without the opening of doors or windows, and appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am henceforth silent. Only vouchsafe an answer. Sphere (after a pause). It is reported so. But men are divided in opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great may be the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, pray have done with this trifling, and let us return to business. I. I was certain of it. I was certain that my anticipations would be fulfi lled. And now have patience with me and answer me yet one more question, best of Teachers! Those who have thus appeared—no one knows whence—and have returned—no one knows whither—have they also contracted their sections and vanished somehow into into that that more more Spacious Spacious Space, Space, whither whither II now now entreat entreat you you to to vanished somehow conduct me? conduct me? Sphere (moodily). (moodily). Th They ey have have vanished, vanished, certainly—if certainly—if they they ever ever appeared. appeared. But But most most Sphere 77 people say say that that these these visions visions arose arose from from the the thought—you thought—you will willThe notBook understand people not understand me—from the brain; from the perturbed angularity of the Seer. me—from the brain; from the perturbed angularity of the Seer. I. Say Say they they so? so? Oh, Oh, believe believe them them not. not. Or Or if if it it indeed indeed be be so, so, that that this this other other Space Space is is I. really Th Thoughtland, oughtland, then then take take me me to to that that blessed blessed Region Region where where II in in Th Thought ought shall shall really see the the insides insides of of all all solid solid things. things. Th There, ere, before before my my ravished ravished eye, eye, aa Cube, Cube, moving moving see in some some altogether altogether new new direction, direction, but but strictly strictly according according to to Analogy, Analogy, so so as as to to make make in every particle particle of of his his interior interior pass pass through through aa new new kind kind of of Space Space with with aa wake wake of of its its every own—shall create create aa still still more more perfect perfect perfection perfection than than himself, himself, with with sixteen sixteen termitermiown—shall nal ExtraExtra-solid solid angles, angles, and and Eight Eight solid solid Cubes Cubes for for his his Perimeter. Perimeter. And And once once there, there, nal shall we we stay stay our our upward upward course? course? In In that that blessed blessed region region of of Four Four Dimensions, Dimensions, shall shall shall we linger linger at at the the threshold threshold of of the the Fifth, Fifth, and and not not enter enter therein? therein? Ah, Ah, no! no! Let Let us us rather rather we resolve that that our our ambition ambition shall shall soar soar with with our our corporal corporal ascent. ascent. Th Then, en, yielding yielding to to resolve our intellectual intellectual onset, onset, the the gates gates of of the the Sixth Sixth Dimension Dimension shall shall fl flyy open; open; after after that that aa our Seventh, and and then then an an Eighth—— Eighth—— Seventh, How long long II should should have have continued continued II know know not. not. In In vain vain did did the the Sphere, Sphere, in in his his How voice of thunder, reiterate his commands of silence, and threaten me with the dirvoice of thunder, reiterate his commands of silence, and threaten me with the direst penalties penalties if if II persisted. persisted. Nothing Nothing could could stem stem the the fl flood ood of of my my ecstatic ecstatic aspiraaspiraest tions. Perhaps Perhaps II was was to to blame; blame; but but indeed indeed II was was intoxicated intoxicated with with the the recent recent tions.

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resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth——

How long I should have continued I know not. In vain did the Sphere, in his voice of thunder, reiterate his commands of silence, and threaten me with the direst penalties if I persisted. Nothing could stem the flood of my ecstatic aspirations. Perhaps I was to blame; but indeed I was intoxicated with the recent draughts of Truth to which he himself had introduced me. However, the end was not long in coming. My words were cut short by a crash outside, and a simultaneous crash inside me, which impelled me through Space with a velocity that precluded speech. Down! down! down! I was rapidly descending; and I knew that return to Flatland was my doom. One glimpse, one last and never-to-be-forgotten glimpse I had of that dull level wilderness—which was now to become my Universe again—spread out before my eye. Then a darkness. Then a final, allconsummating thunder-peal; and, when I came to myself, I was once more a common creeping Square, in my Study at home, listening to the Peace- Cry of my approaching Wife.

§ 20 How the Sphere encouraged me a vision §in20: How the Sphere encouraged me in a vision Although I had less than a minute for reflection, I felt, by a kind of instinct, that I must conceal my experiences from my Wife. Not that I apprehended, at the moment, any danger from her divulging my secret, but I know that to any Woman in Flatland 78 flatland the narrative of my adventures must needs be unintelligible. So I endeavoured to reassure her by some story, invented for the occasion, that I had accidentally fallen through the trap-door of the cellar, and had there lain stunned. The Southward attraction in our country is so slight that even to a Woman my tale necessarily appeared extraordinary and well-nigh incredible; but my Wife, whose good sense far exceeds that of the average of her Sex, and who perceived that I was unusually excited, did not argue with me on the subject, but insisted that I was ill and required repose. I was glad of an excuse for retiring to my chamber to think quietly over what had happened. When I was at last by myself, a drowsy sensation fell on me; but before my eyes closed I endeavoured to reproduce the Third Dimension, and especially the process by which a Cube is constructed through the motion of a Square. It was not so clear as I could have wished; but I remembered that it must be “Upward, and yet not Northward,” and I determined steadfastly to retain these words as the clue which, if firmly grasped, could not fail to guide me to the solution. So mechanically repeating, like a charm, the words, “Upward, yet not Northward,” I fell into a sound refreshing sleep. During my slumber I had a dream. I thought I was once more by the side of the Sphere, whose lustrous hue betokened that he had exchanged his wrath against me for perfect placability. We were moving together towards a bright but infinitesimally small Point, to which my Master directed my attention. As we approached, methought

not Northward,” and I determined steadfastly to retain these words as the clue which, if firmly grasped, could not fail to guide me to the solution. So mechanically repeating, like a charm, the words, “Upward, yet not Northward,” I fell into a sound refresh 67 ing sleep. During my slumber I had a dream. I thought I was once more by the side of the Sphere, whose lustrous hue betokened that he had exchanged his wrath against me for perfect placability. We were moving together towards a bright but infinitesimally small Point, to which my Master directed my attention. As we approached, methought there issued from it a slight humming noise as from one of your Spaceland bluebottles, only less resonant by far, so slight indeed that even in the perfect stillness of the Vacuum through which we soared, the sound reached not our ears till we checked our flight at a distance from it of something under twenty human diagonals. “Look yonder,” said my Guide, “in Flatland thou hast lived; of Lineland thou hast received a vision; thou hast soared with me to the heights of Spaceland; now, in order to complete the range of thy experience, I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No Dimensions. “Behold yon miserable creature. That Point is a Being like ourselves, but confined to the non-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself hisThe OneBook and All, be79 ing really Nothing. Yet mark his perfect self-contentment, and hence learn his lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy. Now listen.” He ceased; and there arose from the little buzzing creature a tiny, low, monotonous, but distinct tinkling, as from one of your Spaceland phonographs, from which I caught these words, “Infinite beatitude of existence! It is; and there is nothing else beside It.” “What,” said I, “does the puny creature mean by ‘it’?” “He means himself,” said the Sphere: “have you not noticed before now, that babies and babyish people who cannot distinguish themselves from the world, speak of themselves in the Third Person? But hush!” “It fills all Space,” continued the little soliloquizing Creature, “and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness, ah, the happiness of Being!” “Can you not startle the little thing out of its complacency?” said I. “Tell it what it really is, as you told me; reveal to it the narrow limitations of Pointland, and lead it up to something higher.” “That is no easy task,” said my Master; “try you.” Hereon, raising by voice to the uttermost, I addressed the Point as follows: “Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with—” “Hush, hush, you have said enough,” interrupted the Sphere, “now listen, and mark the effect of your harangue on the King of Pointland.” The lustre of the Monarch, who beamed more brightly than ever upon hearing

up to something higher.” “That is no easy task,” said my Master; “try you.” Hereon, raising by voice to the uttermost, I addressed the Point as follows: 68 “Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with—” “Hush, hush, you have said enough,” interrupted the Sphere, “now listen, and mark the effect of your harangue on the King of Pointland.” The lustre of the Monarch, who beamed more brightly than ever upon hearing my words, showed clearly that he retained his complacency; and I had hardly ceased when he took up his strain again. “Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming to Itself, suggestive of Its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of instance Being!” of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence omniscience: nothing you or done. I can do canasrescue him “You see,” said myand Teacher, “how little your that words have So far the Monfrom his self-satisfaction.” arch understands them at all, he accepts them as his own—for he cannot conceive of as we floated gently back himself to Flatland, could hear of the‘Its mild voice ofasmy anyAfter otherthis, except himself—and plumes uponI the variety Thought’ an Companion pointing the moral of my vision, and stimulating me to aspire, and to instance of creative Power. Let us leave this God of Pointland to the ignorant fruition teach others to aspire. He had been angered at fi rst—he confessed—by my ambition of his omnipresence and omniscience: nothing that you or I can do can rescue him to soarhistoselfDimensions above the Third; but, since then, he had received fresh insight, from satisfaction.” and he was notastoo acknowledge his error Itocould a Pupil. enmild he proceeded to 80 After flatland this, we proud floatedtogently back to Flatland, hearTh the voice of my initiate me into mysteries yet higher than those I had witnessed, showing me how to Companion pointing the moral of my vision, and stimulating me to aspire, and construct ExtraSolids He by the of Solids, Double Extra-Solidsmy byambition the moteach others to aspire. hadmotion been angered at fiand rst—he confessed—by tion of ExtraSolids, and all “strictly according to Analogy,” all by methods so insight, simple, to soar to Dimensions above the Third; but, since then, he had received fresh so easy, as to be patent even to the Female Sex. and he was not too proud to acknowledge his error to a Pupil. Then he proceeded to initiate me into mysteries yet higher than those I had witnessed, showing me how to construct Extra-Solids by the motion of Solids, and Double Extra-Solids by the mo§ 21 of Extra-Solids, and all “strictly according to Analogy,” all by methods so simple, tion How I tried so easy, as to be patent even to the Female Sex.

to teach the theory of Three Dimensions to 21 my Grandson, and §§ 21: How I tried to teach the theory of Three Dimensions to my Grandson, with what success How I tried and with what success to teach the theory Iofawoke and began to reflect on the glorious career before me. I would go Three rejoicing, Dimensions forth, at once, and evangelize the whole of Flatland. Even to Women and to my methought, Grandson, and Soldiers should the Gospel of Three Dimensions be proclaimed. I would begin with with what success my Wife. Just as I had decided on the plan I heard the me. sound of many I awoke rejoicing, and began to refl ectofonmy theoperations, glorious career before I would go voices in the street commanding silence. Th en followed a louder voice. It was a herforth, methought, at once, and evangelize the whole of Flatland. Even to Women and ald’s proclamation. I recognised the words of the Resolution of Soldiers should the Listening Gospel of attentively, Three Dimensions be proclaimed. I would begin with the Wife. Council, enjoining the arrest, imprisonment, or execution of any one who should my pervert of the people delusions, and by professing havesound received reveJust the as Iminds had decided on thebyplan of my operations, I heardtothe of many lations from another World. voices in the street commanding silence. Then followed a louder voice. It was a herreflected. This Listening danger was not to beI trifl ed with.the It would better to avoid of it ald’sI proclamation. attentively, recognised words be of the Resolution by omitting all mention of my imprisonment, Revelation, and by proceeding of the Council, enjoining the arrest, or execution of any on onethe whopath should

Just as I had decided on the plan of my operations, I heard the sound of many voices in the street commanding silence. Then followed a louder voice. It was a herald’s proclamation. Listening attentively, I recognised the words of the Resolution69 of the Council, enjoining the arrest, imprisonment, or execution of any one who should pervert the minds of the people by delusions, and by professing to have received revelations from another World. I reflected. This danger was not to be trifled with. It would be better to avoid it by omitting all mention of my Revelation, and by proceeding on the path of Demonstration—which after all, seemed so simple and so conclusive that nothing would be lost by discarding the former means. “Upward, not Northward”—was the clue to the whole proof. It had seemed to me fairly clear before I fell asleep; and when I first awoke, fresh from my dream, it had appeared as patent as Arithmetic; but somehow it did not seem to me quite so obvious now. Though my Wife entered the room opportunely at just that moment, I decided, after we had interchanged a few words of commonplace conversation, not to begin with her. Book of no 81 My Pentagonal Sons were men of character and standing, andThe physicians mean reputation, but not great in mathematics, and, in that respect, unfit for my purpose. But it occurred to me that a young and docile Hexagon, with a mathematical turn, would be a most suitable pupil. Why therefore not make my first experiment with my little precocious Grandson, whose casual remarks on the meaning of 33 had met with the approval of the Sphere? Discussing the matter with him, a mere boy, I should be in perfect safety; for he would know nothing of the Proclamation of the Council; whereas I could not feel sure that my Sons—so greatly did their patriotism and reverence for the Circles predominate over mere blind affection—might not feel compelled to hand me over to the Prefect, if they found me seriously maintaining the seditious heresy of the Third Dimension. But the first thing to be done was to satisfy in some way the curiosity of my Wife, who naturally wished to know something of the reasons for which the Circle had desired that mysterious interview, and of the means by which he had entered our house. Without entering into the details of the elaborate account I gave her,—an account, I fear, not quite so consistent with truth as my Readers in Spaceland might desire,— I must be content with saying that I succeeded at last in persuading her to return quietly to her household duties without eliciting from me any reference to the World of Three Dimensions. This done, I immediately sent for my Grandson; for, to confess the truth, I felt that all that I had seen and heard was in some strange way slipping away from me, like the image of a half-grasped, tantalizing dream, and I longed to essay my skill in making a first disciple. When my Grandson entered the room I carefully secured the door. Then, sitting down by his side and taking our mathematical tablets—or, as you would call them, Lines—I told him we would resume the lesson of yesterday. I taught him once more how a Point by motion in One Dimension produces a Line, and how a straight Line in Two Dimensions produces a Square. After this, forcing a laugh, I said, “And now, you scamp, you wanted to make me believe that a Square may in the same way by motion ‘Upward, not Northward’ produce another figure, a sort of extra square in Three Dimensions. Say that again, you young rascal.” At this moment we heard once more the herald’s “O yes! O yes!” outside in the

down by his side and taking our mathematical tablets—or, as you would call them, Lines—I told him we would resume the lesson of yesterday. I taught him once more how a Point by motion in One Dimension produces a Line, and how a straight Line 70 in Two Dimensions produces a Square. After this, forcing a laugh, I said, “And now, you scamp, you wanted to make me believe that a Square may in the same way by motion ‘Upward, not Northward’ produce another figure, a sort of extra square in Three Dimensions. Say that again, you young rascal.” hadAt died and then, bursting “Dear Grandpapa,” he said, “that thisaway, moment we heard once into moretears, the herald’s “O yes! O yes!” outside in was the only fun, and ofthe course I meant nothing at all by it; and we did notheknow streetmy proclaiming Resolution of the Council. Young though was, anymy thing then about the Law; and I don’t think Third DiGrandson—who wasnew unusually intelligent for hisI said age, anything and bred about up in the perfect revermension; and I am sure I did not say one in word ‘Upward, Northward,’ for ence for the authority of the Circles—took the about situation with annot acuteness for which that would be such nonsense, you know. How could a thing move Upward, and not I was quite unprepared. He remained silent till the last words of the Proclamation Northward? Upward and not Northward! Even“Dear if I were a baby, I could not “that be so was abhad died away, and then, bursting into tears, Grandpapa,” he said, surd it is! Ha! ha! ha!” only as mythat. fun,How and silly of course I meant nothing at all by it; and we did not know any“Not at about all silly,” losingand myItemper; “hereI for thing then thesaid newI,Law; don’t think saidexample, anythingI take aboutthis theSquare,”— Third Di82 flatland and, at the word, grasped moveable which was lying hand—”andforI mension; and I amIsure I didanot say oneSquare, word about ‘Upward, notatNorthward,’ move it, youbesee, notnonsense, Northward, I move it Upward—that is to say,and Norththat would such youbut—yes, know. How could a thing move Upward, not ward but I move it somewhere—not exactly like this, but somehow—” Here I brought Northward? Upward and not Northward! Even if I were a baby, I could not be so abmy an silly inaneit conclusion, shaking the Square about in a purposeless mansurdsentence as that. to How is! Ha! ha! ha!” ner,“Not muchatto amusement of my Grandson, who out laughing than allthe silly,” said I, losing my temper; “here forburst example, I take thislouder Square,”— ever, and declared that I was not teaching him, but joking with him. So saying he un-I and, at the word, I grasped a moveable Square, which was lying at hand—”and locked door out of thebut—yes, room; and thus ended my first attempt to convert move it,the you see,and not ran Northward, I move it Upward—that is to say, Northaward pupil toIthe Gospel of Three Dimensions. but move it somewhere—not exactly like this, but somehow—” Here I brought my sentence to an inane conclusion, shaking the Square about in a purposeless manner, much to the amusement of my Grandson, who burst out laughing louder than §ever, 22 and declared that I was not teaching him, but joking with him. So saying he unHow thendoor triedand ran out of the room; and thus ended my first attempt to convert lockedI the to diffuse eory of Three Dimensions. a pupil to the the Th Gospel

of Three Dimensions by other means, § 22: How I then tried to diffuse the Theory of Three Dimensions by other means, and § 22of the result and of the result How I then tried My failure with my Grandson did not encourage me to communicate my secret to to diff use the Theory others of my house hold; yet neither was I led by it to despair of success. Only I saw of Three Dimensions that I must not wholly rely on the catchphrase, “Upward, not Northward,” but must by other means, rather and ofendeavour the result to seek a demonstration by setting before the public a clear view of

the whole subject; and for this purpose it seemed necessary to resort to writing. I devoted several monthsdid in not privacy to the me composition of a treatise on the MySo failure with my Grandson encourage to communicate my secret to mysteries of Th ree Dimensions. Only,was with theby view evadingofthe Law, ifOnly possible, others of my house hold; yet neither I led it toofdespair success. I sawI spoke a physical but of a Thoughtland in theory,but a Figure that I not mustofnot wholly Dimension, rely on the catchphrase, “Upward,whence, not Northward,” must could down upon and see simultaneously the the insides of all things, rather look endeavour to seekFlatland a demonstration by setting before public a clear viewand of where it was possible might be supposed to exist to a Figure environed, the whole subject; andthat for there this purpose it seemed necessary resort to writing. as it were, six Squares, containing eight Points. But writing So with I devoted several and months in privacy to terminal the composition of a in treatise on this the book I found sadly hampered thethe impossibility of drawing such diagramsI mysteries of Thmyself ree Dimensions. Only,by with view of evading the Law, if possible, as werenot necessary for myDimension, purpose; for in our country of Flatland, are spoke of a physical butofofcourse, a Thoughtland whence, in theory,there a Figure could look down upon Flatland and see simultaneously the insides of all things, and

mysteries of Three Dimensions. Only, with the view of evading the Law, if possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in theory, a Figure could look down upon Flatland and see simultaneously the insides of all things, and 71 where it was possible that there might be supposed to exist a Figure environed, as it were, with six Squares, and containing eight terminal Points. But in writing this book I found myself sadly hampered by the impossibility of drawing such diagrams as were necessary for my purpose; for of course, in our country of Flatland, there are no tablets but Lines, and no diagrams but Lines, all in one straight Line and only distinguishable by difference of size and brightness; so that, when I had finished my treatise (which I entitled, “Through Flatland to Thoughtland”) I could not feel cerThe Book 83 tain that many would understand my meaning. Meanwhile my wife was under a cloud. All pleasures palled upon me; all sights tantalized and tempted me to outspoken treason, because I could not but compare what I saw in Two Dimensions with what it really was if seen in Three, and could hardly refrain from making my comparisons aloud. I neglected my clients and my own business to give myself to the contemplation of the mysteries which I had once beheld, yet which I could impart to no one, and found daily more difficult to reproduce even before my own mental vision. One day, about eleven months after my return from Spaceland, I tried to see a Cube with my eye closed, but failed; and though I succeeded afterwards, I was not then quite certain (nor have I been ever afterwards) that I had exactly realized the original. This made me more melancholy than before, and determined me to take some step; yet what, I knew not. I felt that I would have been willing to sacrifice my life for the Cause, if thereby I could have produced conviction. But if I could not convince my Grandson, how could I convince the highest and most developed Circles in the land? And yet at times my spirit was too strong for me, and I gave vent to dangerous utterances. Already I was considered heterodox if not treasonable, and I was keenly alive to the dangers of my position; nevertheless I could not at times refrain from bursting out into suspicious or half-seditious utterances, even among the highest Polygonal and Circular society. When, for example, the question arose about the treatment of those lunatics who said that they had received the power of seeing the insides of things, I would quote the saying of an ancient Circle, who declared that prophets and inspired people are always considered by the majority to be mad; and I could not help occasionally dropping such expressions as “the eye that discerns the interiors of things,” and “the all-seeing land”: once or twice I even let fall the forbidden terms “the Third and Fourth Dimensions.” At last, to complete a series of minor indiscretions, at a meeting of our Local Speculative Society held at the palace of the Prefect himself,—some extremely silly person having read an elaborate paper exhibiting the precise reasons why Providence has limited the number of Dimensions to Two, and why the attribute of omnividence is assigned to the Supreme alone—I so far forgot myself as to give an exact account of the whole of my voyage with the Sphere into Space, and to the Assembly Hall in our Metropolis, and then to Space again, and of my return home, and of everything that I had seen and heard in fact or vision. At first, indeed, I pretended that I was describing the imaginary experiences of a fictitious person; but my enthusiasm soon forced me to throw off all disguise, and

ing the precise reasons why Providence has limited the number of Dimensions to Two, and why the attribute of omnividence is assigned to the Supreme alone—I so far forgot myself as to give an exact account of the whole of my voyage with the 72 Sphere into Space, and to the Assembly Hall in our Metropolis, and then to Space again, and of my return home, and of everything that I had seen and heard in fact or vision. At first, indeed, I pretended that I was describing the imaginary experiences of a fictitious person; but my enthusiasm soon forced me to throw off all disguise, and finally, in a fervent peroration, I exhorted all my hearers to divest themselves of prejudice and to become believers in the Third Dimension. I say that I was at once arrested and taken before the Council? 84 Needflatland Next morning, standing in the very place where but a very few months ago the Sphere had stood in my company, I was allowed to begin and to continue my narration unquestioned and uninterrupted. But from the first I foresaw my fate; for the President, noting that a guard of the better sort of Policemen was in attendance, of angularity little, if at all, under 55°, ordered them to be relieved before I began my defence, by an inferior class of 2° or 3°. I knew only too well what that meant. I was to be executed or imprisoned, and my story was to be kept secret from the world by the simultaneous destruction of the officials who had heard it; and, this being the case, the President desired to substitute the cheaper for the more expensive victims. After I had concluded my defence, the President, perhaps perceiving that some of the junior Circles had been moved by evident earnestness, asked me two questions:— 1. Whether I could indicate the direction which I meant when I used the words “Upward, not Northward”? 2. Whether I could by any diagrams or descriptions (other than the enumeration of imaginary sides and angles) indicate the Figure I was pleased to call a Cube? I declared that I could say nothing more, and that I must commit myself to the Truth, whose cause would surely prevail in the end. The President replied that he quite concurred in my sentiment, and that I could not do better. I must be sentenced to perpetual imprisonment; but if the Truth intended that I should emerge from prison and evangelize the world, the Truth might be trusted to bring that result to pass. Meanwhile I should be subjected to no discomfort that was not necessary to preclude escape, and, unless I forfeited the privilege by misconduct, I should be occasionally permitted to see my brother, who had preceded me to my prison. Seven years have elapsed and I am still a prisoner, and—if I except the occasional visits of my brother—debarred from all companionship save that of my jailers. My brother is one of the best of Squares, just, sensible, cheerful, and not without fraternal affection; yet I confess that my weekly interviews, at least in one respect, cause me the bitterest pain. He was present when the Sphere manifested himself in the Council Chamber; he saw the Sphere’s changing sections; he heard the explanation of the phenomena then given to the Circles. Since that time, scarcely a week has passed during seven whole years, without his hearing from me a repetition of the part I played in that manifestation, together with ample descriptions of all the phenomena in Spaceland, and the arguments for the existence of Solid things derivable from Analogy. Yet—I take shame to be forced to confess it—my brother has not yet grasped the

bitterest pain. He was present when the Sphere manifested himself in the Council Chamber; he saw the Sphere’s changing sections; he heard the explanation of the phenomena then given to the Circles. Since that time, scarcely a week has passed during 73 seven whole years, without his hearing from me a repetition of the part I played in that manifestation, together with ample descriptions of all the phenomena in Spaceland, and the arguments for the existence of Solid things derivable from Analogy. Yet—I take shame to be forced to confess it—my brother has not yet grasped the nature of the Third Dimension, and frankly avows his disbelief in the existence of a Sphere. Hence I am absolutely destitute of converts, and, for aught that The I canBook see, the mil85 lennial Revelation has been made to me for nothing. Prometheus up in Spaceland was bound for bringing down fire for mortals, but I—poor Flatland Prometheus—lie here in prison for bringing down nothing to my countrymen. Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality. That is the hope of my brighter moments. Alas, it is not always so. Heavily weighs on me at times the burdensome reflection that I cannot honestly say I am confident as to the exact shape of the once-seen, oft-regretted Cube; and in my nightly visions the mysterious precept, “Upward, not Northward,” haunts me like a soul-devouring Sphinx. It is part of the martyrdom which I endure for the cause of the Truth that there are seasons of mental weakness, when Cubes and Spheres flit away into the background of scarce-possible existences; when the Land of Three Dimensions seems almost as visionary as the Land of One or None; nay, when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the offspring of a diseased imagination, or the baseless fabric of a dream.