# publicIvesvdf/flatland

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3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 3256 3257 3258 3259 3260 3261 3262 3263 3264 3265 3266 3267 3268 3269 3270 3271 3272 3273 3274 3275 3276 3277 3278 3279 3280 3281 3282 3283 3284 3285 3286 3287 3288 3289 3290 3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301 3302 3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320 3321 3322 3323 3324 3325 3326 3327 3328 3329 3330 3331 3332 3333 3334 3335 3336 3337 3338 3339 3340 \title{Flatland} \author{Edwin A. Abbott} \date{\today}\documentclass[12pt, a4paper, oneside]{memoir} \usepackage[english]{babel}\usepackage{graphicx} \usepackage[T1]{fontenc}\usepackage{hyperref}\let\footruleskip\relax % for compatibility of memoir and fancyhdr\let\rm\rmfamily % for compatibility of memoir\let\sl\emph % for compatibility of memoir\usepackage{fancyhdr}\pagestyle{fancy}\fancyhf{} % remove everything\renewcommand{\headrulewidth}{0pt} % remove lines as well%\fancyhead[CE, CO]{Flatland}%\fancyhead[CE, CO]{Flatland}%\fancyhead[LO]{\leftmark}%\fancyhead[RE]{\thetitle}\fancyfoot[CE,CO]{\thepage}\urlstyle{same}\begin{document} % \maketitle\thispagestyle{empty} \begin{vplace}[0.5]\includegraphics[trim=0mm 0mm 0mm 0mm,width=\linewidth]{flatland_cover}\end{vplace}\clearpage\frontmatterThis work is in the public domain in the United States because it waspublished before January 1, 1923.The author died in 1926, so this work is also in the public domain incountries and areas where the copyright term is the author's life plus 80years or less. This work may also be in the public domain in countries andareas with longer native copyright terms that apply the rule of the shorterterm to foreign works.\vspace{5mm}\noindent\emph{Written by} Edwin A. Abbott \\\emph{Illustrated by} Edwin A. Abbott \\\emph{Typeset by} Ives van der Flaas\\ \\\null\vfillFifth edition\chapter*{A Note From The Typesetter}Although Edwin A. Abbott's essay Flatland'' is readily available on theinternet, I failed to find a nicely typeset version. At this time,the rendering quality of internet browsers doesn't come anywhere near thequality of a nice book. For these reasons, as well as a healthy portion of boredom, I made the versionof Flatland'' you are currently reading, in \LaTeX. The \LaTeX\ source can befound on \url{https://github.com/Ivesvdf/flatland} (feel free to make issuesor send me pull requests when you find bugs or have improvements). It shouldalso be fairly easy --- a matter of minutes --- to produce a book-sizedversion, ready to send to a printer. \vspace{1cm}\hfill Ives van der Flaas , 2011-09-20\chapter*{Dedication}\begin{center}To\\The Inhabitants of SPACE IN GENERAL\\And H.C. IN PARTICULAR\\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\\\end{center}\chapter*{Preface}If my poor Flatland friend retained the vigour of mind which he enjoyed whenhe began to compose these Memoirs, I should not now need to represent him inthis preface, in which he desires, fully, to return his thanks to his readersand critics in Spaceland, whose appreciation has, with unexpected celerity,required a second edition of this work; secondly, to apologize for certainerrors and misprints (for which, however, he is not entirely responsible);and, thirdly, to explain on or two misconceptions. But he is not the Square heonce was. Years of imprisonment, and the still heavier burden of generalincredulity and mockery, have combined with the thoughts and notions, and muchalso of the terminology, which he acquired during his short stay in spaceland.He has, therefore, requested me to reply in his behalf to two specialobjections, one of an intellectual, the other of a moral nature.The first objection is, that a Flatlander, seeing a Line, sees something thatmust be thick to the eye as well as long to the eye (otherwise it would not bevisible, if it had not some thickness); and consequently he ought (it isargued) to acknowledge that his countrymen are not only long and broad, butalso (though doubtless to a very slight degree) thick or high. This objectionis 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'sanswer appears to me completely to meet it.I admit,'' said he --- when I mentioned to him this objection --- I admit thetruth of your critic's facts, but I deny his conclusions. It is true that wehave really in Flatland a Third unrecognized Dimension called height,' justas it also is true that you have really in Spaceland a Fourth unrecognizedDimension, 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 yourextra-height.' Even I --- who have been in Spaceland, and have had theprivilege of understanding for twenty-four hours the meaning of height' ---even I cannot now comprehend it, nor realize it by the sense of sight or byany process of reason; I can but apprehend it by faith.''The reason is obvious. Dimension implied direction, implies measurement,implies the more and the less. Now, all our lines are equally andinfinitesimally thick (or high, whichever you like); consequently, there isnothing in them to lead our minds to the conception of that Dimension. Nodelicate micrometer' --- as has been suggested by one too hasty Spacelandcritic --- 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 andbright; brightness, as well as length, is necessary to the existence of aLine; if the brightness vanishes, the Line is extinguished. Hence, all myFlatland friends --- when I talk to them about the unrecognized Dimension whichis somehow visible in a Line --- say, Ah, you mean brightness': and when Ireply, No, I mean a real Dimension,' they at once retort, Then measure it,or tell us in what direction it extends'; and this silences me, for I can doneither. Only yesterday, when the Chief Circle (in other words our HighPriest) 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?' Itried to prove to him that he was high', as well as long and broad, althoughhe did not know it. But what was his reply? You say I am high''; measure myhigh-ness'' and I will believe you.'. What could I do? How could I meet hischallenge? 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 tosay, Whenever you open your eyes, you see a Plane (which is of TwoDimensions) and you infer a Solid (which is of Three); but in reality you alsosee (though you do not recognize) a Fourth Dimension, which is not colour norbrightness nor anything of the kind, but a true Dimension, although I cannotpoint out to you its direction, nor can you possibly measure it.' What wouldyou say to such a visitor? Would not you have him locked up? Well, that is myfate: and it is as natural for us Flatlanders to lock up a Square forpreaching the Third Dimension, as it is for you Spacelanders to lock up a Cubefor preaching the Fourth. Alas, how strong a family likeness runs throughblind and persecuting humanity in all Dimensions! Points, Lines, Squares,Cubes, Extra-Cubes --- we are all liable to the same errors, all alike theSlavers of our respective Dimensional prejudices, as one of our Spacelandpoets has said --- One touch of Nature makes all worlds akin.' '' \footnote{TheAuthor desires me to add, that the misconceptions of some of his critics onthis matter has induced him to insert in his dialogue withthe Sphere, certain remarks which have a bearing on the point in question andwhich he had previously omitted as being tedious and unnecessary.}On this point the defence of the Square seems to me to be impregnable. I wishI could say that his answer to the second (or moral) objection was equallyclear and cogent. It has been objected that he is a woman-hater; and as thisobjection has been vehemently urged by those whom Nature's decree hasconstituted the somewhat larger half of the Spaceland race, I should like toremove it, so far as I can honestly do so. But the Square is so unaccustomedto the use of the moral terminology of Spaceland that I should be doing him aninjustice if I were literally to transcribe his defence against this charge.Acting, therefore, as his interpreter and summarizer, I gather that in thecourse of an imprisonment of seven years he has himself modified his ownpersonal views, both as regards Women and as regards the Isosceles or LowerClasses. Personally, he now inclines to the opinion of the Sphere that theStraight Lines are in many important respects superior to theCircles. But, writing as a Historian, he has identified himself (perhaps tooclosely) with the views generally adopted by Flatland, and (as he has beeninformed) even by Spaceland, Historians; in whose pages (until very recenttimes) the destinies of Women and of the masses of mankind have seldom beendeemed worthy of mention and never of careful consideration.In a still more obscure passage he now desires to disavow the Circular oraristocratic tendencies with which some critics have naturally credited him.While doing justice to the intellectual power with which a few Circles havefor many generations maintained their supremacy over immense multitudes oftheir countrymen, he believes that the facts of Flatland, speaking forthemselves without comment on his part, declare that Revolutions cannot alwaysbe suppressed by slaughter, and that Nature, in sentencing the Circles toinfecundity, has condemned them to ultimate failure --- and herein,'' he says,I see a fulfillment of the great Law of all worlds, that while the wisdom ofMan thinks it is working one thing, the wisdom of Nature constrains it to workanother, and quite a different and far better thing.'' For the rest, he begshis readers not to suppose that every minute detail in the daily life ofFlatland must needs correspond to some other detail in Spaceland; and yet hehopes that, taken as a whole, his work may prove suggestive as well asamusing, to those Spacelanders of moderate and modest minds who --- speaking ofthat which is of the highest importance, but lies beyond experience --- declineto say on the one hand, This can never be,'' and on the other hand, It mustneeds be precisely thus, and we know all about it.''\mainmatter\part{This World}\chapter{Of the Nature of Flatland} I call our world Flatland, not because wecall it so, but to make its nature clearer to you, my happy readers, who areprivileged 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 theirplaces, move freely about, on or in the surface, but without the power ofrising above or sinking below it, very much like shadows --- only hard withluminous edges --- and you will then have a pretty correct notion of my countryand countrymen. Alas, a few years ago, I should have said my universe'': butnow my mind has been opened to higher views of things.In such a country, you will perceive at once that it is impossible that thereshould be anything of what you call a solid'' kind; but I dare say you willsuppose that we could at least distinguish by sight the Triangles, Squares,and other figures, moving about as I have described them. On the contrary, wecould see nothing of the kind, not at least so as to distinguish one figurefrom another. Nothing was visible, nor could be visible, to us, exceptStraight 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 overit, look down upon it. It will appear a circle.But now, drawling back to the edge of the table, gradually lower your eye(thus bringing yourself more and more into the condition of the inhabitants ofFlatland), and you will find the penny becoming more and more oval to yourview, and at last when you have placed your eye exactly on the edge of thetable (so that you are, as it were, actually a Flatlander) the penny will thenhave ceased to appear oval at all, and will have become, so far as you cansee, a straight line.The same thing would happen if you were to treat in the same way a Triangle,or a Square, or any other figure cut out of pasteboard. As soon as you look atit with your eye on the edge of the table, you will find that it ceases toappear to you as a figure, and that it becomes in appearance a straight line.Take for example an equilateral Triangle --- who represents with us a Tradesmanof the respectable class. Figure 1 represents the Tradesman as you would seehim while you were bending over him from above; figures 2 and 3 represent theTradesman, as you would see him if your eye were close to the level, or allbut on the level of the table; and if your eye were quite on the level of thetable (and that is how we see him in Flatland) you would see nothing but astraight line.\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig1}When I was in Spaceland I heard that your sailors have very similarexperiences while they traverse your seas and discern some distant island orcoast lying on the horizon. The far-off land may have bays, forelands, anglesin and out to any number and extent; yet at a distance you see none of these(unless indeed your sun shines bright upon them revealing the projections andretirements by means of light and shade), nothing but a grey unbroken lineupon the water.Well, that is just what we see when one of our triangular or otheracquaintances comes towards us in Flatland. As there is neither sun with us,nor any light of such a kind as to make shadows, we have none of the helps tothe sight that you have in Spaceland. If our friend comes closer to us we seehis line becomes larger; if he leaves us it becomes smaller; but still helooks like a straight line; be he a Triangle, Square, Pentagon, Hexagon,Circle, what you will --- a straight Line he looks and nothing else.You may perhaps ask how under these disadvantageous circumstances we are ableto distinguish our friends from one another: but the answer to this verynatural question will be more fitly and easily given when I come to describethe inhabitants of Flatland. For the present let me defer this subject, andsay a word or two about the climate and houses in our country.\chapter{Of the Climate and Houses in Flatland}As with you, so also with us, there are four points of the compass North,South, East, and West.There being no sun nor other heavenly bodies,it is impossible for us to determine the North in the usual way; but we have amethod of our own. By a Law of Nature with us, there is a constant attractionto the South; and, although in temperate climates this is very slight --- sothat even a Woman in reasonable health can journey several furlongs northwardwithout much difficulty --- yet the hampering effect of the southward attractionis quite sufficient to serve as a compass in most parts of our earth.Moreover, the rain (which falls at stated intervals) coming always from theNorth, is an additional assistance; and in the towns we have the guidance ofthe houses, which of course have their side-walls running for the most partNorth and South, so that the roofs may keep off the rain from the North. Inthe country, where there are no houses, the trunks of the trees serve as somesort of guide. Altogether, we have not so much difficulty as might be expectedin determining our bearings.Yet in our more temperate regions, in which the southward attraction is hardlyfelt, walking sometimes in a perfectly desolate plain where there have been nohouses nor trees to guide me, I have been occasionally compelled to remainstationary for hours together, waiting till the rain came before continuing myjourney. On the weak and aged, and especially on delicate Females, the forceof attraction tells much more heavily than on the robust of the Male Sex, sothat it is a point of breeding, if you meet a Lady in the street, always togive her the North side of the way --- by no means an easy thing to do always atshort notice when you are in rude health and in a climate where it isdifficult to tell your North from your South.Windows there are none in our houses: for the light comes to us alike in ourhomes and out of them, by day and by night, equally at all times and in allplaces, whence we know not. It was in old days, with our learned men, aninteresting and oft-investigate question, What is the origin of light?'' andthe solution of it has been repeatedly attempted, with no other result than tocrowd our lunatic asylums with the would-be solvers. Hence, after fruitlessattempts to suppress such investigations indirectly by making them liable to aheavy tax, the Legislature, in comparatively recent times, absolutelyprohibited them. I --- alas, I alone in Flatland --- know now only too well thetrue solution of this mysterious problem; but my knowledge cannot be madeintelligible to a single one of my countrymen; and I am mocked at --- I, thesole possessor of the truths of Space and of the theory of the introduction ofLight from the world of three Dimensions --- as if I were the maddest of themad! But a truce to these painful digressions: let me return to our homes.The most common form for the construction of a house is five-sided orpentagonal, as in the annexed figure. The two Northern sides RO, OF,constitute the roof, and for the most part have no doors; on the East is asmall door for the Women; on the West a much larger one for the Men; the Southside or floor is usually doorless.\begin{center}\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig2}\end{center}Square and triangular houses are not allowed, and for this reason. The anglesof a Square (and still more those of an equilateral Triangle,) being much morepointed than those of a Pentagon, and the lines of inanimate objects (such ashouses) being dimmer than the lines of Men and Women, it follows that there isno little danger lest the points of a square of triangular house residencemight do serious injury to an inconsiderate or perhaps absentminded travellersuddenly running against them: and therefore, as early as the eleventh centuryof our era, triangular houses were universally forbidden by Law, the onlyexceptions being fortifications, powder-magazines, barracks, and other statebuildings, which is not desirable that the general public should approachwithout circumspection.At this period, square houses were still everywhere permitted, thoughdiscouraged by a special tax. But, about three centuries afterwards, the Lawdecided that in all towns containing a population above ten thousand, theangle of a Pentagon was the smallest house-angle that could be allowedconsistently with the public safety. The good sense of the community hasseconded the efforts of the Legislature; and now, even in the country, thepentagonal construction has superseded every other. It is only now and then insome very remote and backward agricultural district that an antiquarian maystill discover a square house.\chapter{Concerning the Inhabitants of Flatland}The greatest length or breadth of a full grown inhabitant of Flatland may beestimated at about eleven of your inches. Twelve inches may be regarded as amaximum.Our Women are Straight Lines.Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides,each about eleven inches long, and a base or third side so short (often notexceeding half an inch) that they form at their vertices a very sharp andformidable angle. Indeed when their bases are of the most degraded type (notmore than the eighth part of an inch in size), they can hardly bedistinguished from Straight lines or Women; so extremely pointed are theirvertices. With us, as with you, these Triangles are distinguished from othersby being called Isosceles; and by this name I shall refer to them in thefollowing pages.Our Middle Class consists of Equilateral or Equal-Sided Triangles.Our Professional Men and Gentlemen are Squares (to which class I myselfbelong) and Five-Sided Figures or Pentagons.Next above these come the Nobility, of whom there are several degrees,beginning at Six-Sided Figures, or Hexagons, and from thence rising in thenumber of their sides till they receive the honourable title of Polygonal, ormany-Sided. Finally when the number of the sides becomes so numerous, and thesides themselves so small, that the figure cannot be distinguished from acircle, he is included in the Circular or Priestly order; and this is thehighest class of all.It is a Law of Nature with us that a male child shall have one more side thanhis father, so that each generation shall rise (as a rule) one step in thescale of development and nobility. Thus the son of a Square is a Pentagon; theson of a Pentagon, a Hexagon; and so on.But this rule applies not always to the Tradesman, and still less often to theSoldiers, and to the Workmen; who indeed can hardly be said to deserve thename of human Figures, since they have not all their sides equal. With themtherefore the Law of Nature does not hold; and the son of an Isosceles (i.e. aTriangle with two sides equal) remains Isosceles still. Nevertheless, all hopeis not shut out, even from the Isosceles, that his posterity may ultimatelyrise above his degraded condition. For, after a long series of militarysuccesses, or diligent and skillful labours, it is generally found that themore intelligent among the Artisan and Soldier classes manifest a slightincrease of their third side or base, and a shrinkage of the two other sides.Intermarriages (arranged by the Priests) between the sons and daughters ofthese more intellectual members of the lower classes generally result in anoffspring approximating still more to the type of the Equal-Sided Triangle.Rarely --- in proportion to the vast numbers of Isosceles births --- is a genuineand certifiable Equal-Sided Triangle produced from Isosceles parents\footnote{ What need of a certificate?'' a Spaceland critic may ask: Is notthe procreation of a Square Son a certificate from Nature herself, proving theEqual-sidedness of the Father?'' I reply that no Lady of any position willmarry an uncertified Triangle. Square offspring has sometimes resulted from aslightly Irregular Triangle; but in almost every such case the Irregularity ofthe first generation is visited on the third; which either fails to attain thePentagonal rank, or relapses to the Triangular.}. Such a birth requires, asits antecedents, not only a series of carefully arranged intermarriages, butalso a long-continued exercise of frugality and self-control on the part ofthe would-be ancestors of the coming Equilateral, and a patient, systematic,and continuous development of the Isosceles intellect through manygenerations.The birth of a True Equilateral Triangle from Isosceles parents is the subjectof rejoicing in our country for many furlongs around. After a strictexamination conducted by the Sanitary and Social Board, the infant, ifcertified as Regular, is with solemn ceremonial admitted into the class ofEquilaterals. He is then immediately taken from his proud yet sorrowingparents and adopted by some childless Equilateral, who is bound by oath neverto permit the child henceforth to enter his former home or so much as to lookupon his relations again, for fear lest the freshly developed organism may, byforce of unconscious imitation, fall back again into his hereditary level.The occasional emergence of an Equilateral from the ranks of his serf-bornancestors is welcomed, not only by the poor serfs themselves, as a gleam oflight and hope shed upon the monotonous squalor of their existence, but alsoby the Aristocracy at large; for all the higher classes are well aware thatthese rare phenomena, while they do little or nothing to vulgarize their ownprivileges, serve as almost useful barrier against revolution from below.Had the acute-angled rabble been all, without exception, absolutely destituteof hope and of ambition, they might have found leaders in some of their manyseditious outbreaks, so able as to render their superior numbers and strengthtoo much even for the wisdom of the Circles. But a wise ordinance of Naturehas decreed that, in proportion as the working-classes increase inintelligence, knowledge, and all virtue, in that same proportion their acuteangle (which makes them physically terrible) shall increase also andapproximate to their comparatively harmless angle of the Equilateral Triangle.Thus, in the most brutal and formidable off the soldier class --- creaturesalmost on a level with women in their lack of intelligence --- it is found that,as they wax in the mental ability necessary to employ their tremendouspenetrating power to advantage, so do they wane in the power of penetrationitself.How admirable is this Law of Compensation! And how perfect a proof of thenatural fitness and, I may almost say, the divine origin of the aristocraticconstitution of the States of Flatland! By a judicious use of this Law ofNature, the Polygons and Circles are almost always able to stifle sedition inits very cradle, taking advantage of the irrepressible and boundlesshopefulness of the human mind. Art also comes to the aid of Law and Order. Itis generally found possible --- by a little artificial compression or expansionon the part of the State physicians --- to make some of the more intelligentleaders of a rebellion perfectly Regular, and to admit them at once into theprivileged classes; a much larger number, who are still below the standard,allured by the prospect of being ultimately ennobled, are induced to enter theState Hospitals, where they are kept in honourable confinement for life; oneor two alone of the most obstinate, foolish, and hopelessly irregular are ledto execution.Then the wretched rabble of the Isosceles, planless and leaderless, are ethertransfixed without resistance by the small body of their brethren whom theChief Circle keeps in pay for emergencies of this kind; or else more often, bymeans of jealousies and suspicious skillfully fomented among them by theCircular party, they are stirred to mutual warfare, and perish by oneanother's angles. No less than one hundred and twenty rebellions are recordedin our annals, besides minor outbreaks numbered at two hundred andthirty-five; and they have all ended thus.\chapter{Concerning the Women}If our highly pointed Triangles of the Soldier class are formidable, it may bereadily inferred that far more formidable are our Women. For, if a Soldier isa wedge, a Woman is a needle; being, so to speak, all point, at least at thetwo extremities. Add to this the power of making herself practically invisibleat will, and you will perceive that a Female, in Flatland, is a creature by nomeans to be trifled with.But here, perhaps, some of my younger Readers may ask how a woman in Flatlandcan make herself invisible. This ought, I think, to be apparent without anyexplanation. However, a few words will make it clear to the most unreflecting.Place a needle on the 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 itend-ways, and you see nothing but a point, it has become practicallyinvisible. Just so is it with one of our Women. When her side is turnedtowards us, we see her as a straight line; when the end containing her eye ormouth --- for with us these two organs are identical --- is the part that meetsour eye, then we see nothing but a highly lustrous point; but when the back ispresented to our view, then --- being only sub-lustrous, and, indeed, almost asdim as an inanimate object --- her hinder extremity serves her as a kind ofInvisible Cap.The dangers to which we are exposed from our Women must now be manifest to themeanest capacity in Spaceland. If even the angle of a respectable Triangle inthe middle class is not without its dangers; if to run against a Working Maninvolves a gash; if collision with an Officer of the military classnecessitates a serious wound; if a mere touch from the vertex of a PrivateSoldier brings with it danger of death; --- what can it be to run against awoman, except absolute and immediate destruction? And when a Woman isinvisible, or visible only as a dim sub-lustrous point, how difficult must itbe, even for the most cautious, always to avoid collision!Many are the enactments made at different times in the different States ofFlatland, in order to minimize this peril; and in the Southern and lesstemperate climates, where the force of gravitation is greater, and humanbeings more liable to casual and involuntary motions, the Laws concerningWomen are naturally much more stringent. But a general view of the Code may beobtained from the following summary: ---Every house shall have one entrance on the Eastern side, for the use ofFemales only; by which all females shall enter in a becoming and respectfulmanner'' \footnote{ When I was in Spaceland I understood that some of yourPriestly circles have in the same way a separate entrance for Villagers,Farmers and Teachers of Board Schools (Spectator, Sept. 1884, P. 1255) thatthey may approach in a becoming and respectful manner.''} and not by the Men'sor Western door. No Female shall walk in any public place without continuallykeeping up her Peace-cry, under penalty of death. Any Female, duly certifiedto be suffering from St. Vitus's Dance, fits, chronic cold accompanied byviolent sneezing, or any disease necessitating involuntary motions, shall beinstantly destroyed. In some of the States there is an additional Lawforbidding Females, under penalty of death, from walking or standing in anypublic place without moving their backs constantly from right to left so as toindicate their presence to those behind them; others oblige a Woman, whentravelling, to be followed by one of her sons, or servants, or by her husband;others confine Women altogether to their houses except during the religiousfestivals. But it has been found by the wisest of our Circles or Statesmenthat the multiplication of restrictions on Females tends not only to thedebilitation and diminution of the race, but also to the increase of domesticmurders to such an extent that a State loses more than it gains by a tooprohibitive Code.For whenever the temper of the Women is thus exasperated by confinement athome or hampering regulations abroad, they are apt to vent their spleen upontheir husbands and children; and in the less temperate climates the whole malepopulation of a village has been sometimes destroyed in one or two hours ofsimultaneous female outbreak. Hence the Three Laws, mentioned above, sufficefor the better regulated States, and may be accepted as a roughexemplification of our Female Code.After all, our principal safeguard is found, not in Legislature, but in theinterests of the Women themselves. For, although they can inflictinstantaneous death by a retrograde movement, yet unless they can at oncedisengage 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 pointedout that in some less civilized States no female is suffered to stand in anypublic place without swaying her back from right to left. This practice hasbeen universal among ladies of any pretensions to breeding in allwell-governed States, as far back as the memory of Figures can reach. It isconsidered a disgrace to any state that legislation should have to enforcewhat ought to be, and is in every respectable female, a natural instinct. Therhythmical and, if I may so say, well-modulated undulation of the back in ourladies of Circular rank is envied and imitated by the wife of a commonEquilateral, who can achieve nothing beyond a mere monotonous swing, like theticking of a pendulum; and the regular tick of the Equilateral is no lessadmired and copied by the wife of the progressive and aspiring Isosceles, inthe females of whose family no back-motion'' of any kind has become as yet anecessity of life. Hence, in every family of position and consideration, backmotion'' is as prevalent as time itself; and the husbands and sons in thesehouseholds enjoy immunity at least from invisible attacks.Not that it must be for a moment supposed that our Women are destitute ofaffection. But unfortunately the passion of the moment predominates, in theFrail Sex, over every other consideration. This is, of course, a necessityarising from their unfortunate conformation. For as they have no pretensionsto an angle, being inferior in this respect to the very lowest of theIsosceles, they are consequently wholly devoid of brainpower, and have neitherreflection, judgment nor forethought, and hardly any memory. Hence, in theirfits of fury, they remember no claims and recognize no distinctions. I haveactually known a case where a Woman has exterminated her whole household, andhalf an hour afterwards, when her rage was over and the fragments swept away,has asked what has become of her husband and her children.Obviously then a Woman is not to be irritated as long as she is in a positionwhere she can turn round. When you have them in their apartments --- which areconstructed with a view to denying them that power --- you can say and do whatyou like; for they are then wholly impotent for mischief, and will notremember a few minutes hence the incident for which they may be at this momentthreatening you with death, nor the promises which you may have found itnecessary to make in order to pacify their fury.On the whole we got on pretty smoothly in our domesticrelations, except in the lower strata of the Military Classes. There the wantof tact and discretion on the part of the husbands produces at timesindescribable disasters. Relying too much on the offensive weapons of theiracute angles instead of the defensive organs of good sense and seasonablesimulations, these reckless creatures too often neglect the prescribedconstruction of the women's apartments, or irritate their wives by ill-advisedexpressions out of doors, which they refuse immediately to retract. Moreover ablunt and stolid regard for literal truth indisposes them to make those lavishpromises by which the more judicious Circle can in a moment pacify hisconsort. The result is massacre; not, however, without its advantages, as iteliminates the more brutal and troublesome of the Isosceles; and by many ofour Circles the destructiveness of the Thinner Sex is regarded as one amongmany providential arrangements for suppressing redundant population, andnipping Revolution in the bud.Yet even in our best regulated and most approximately Circular families Icannot 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 thatname, but there is necessarily little harmony of tastes or pursuits; and thecautious wisdom of the Circles has ensured safety at the cost of domesticcomfort. In every Circular or Polygonal household it has been a habit fromtime immemorial --- and now has become a kind of instinct among the women of ourhigher classes --- that the mothers and daughters should constantly keep theireyes and mouths towards their husband and his male friends; and for a lady ina family of distinction to turn her back upon her husband would be regarded asa kind of portent, involving loss of status. But, as I shall soon shew, thiscustom, though it has the advantage of safety, is not without disadvantages.Moving with ease and smoothness in uttering words; of rapid speech;nimble in speaking; glib; as, a flippant, voluble, tongue. In the house ofthe Working Man or respectable Tradesman --- where the wife is allowed to turnher back upon her husband, while pursuing her household avocations --- there areat least intervals of quiet, when the wife is neither seen nor heard, exceptfor the humming sound of the continuous Peace-cry; but in the homes of theupper classes there is too often no peace. There the voluble mouth and brightpenetrating eye are ever directed toward the Master of the household; andlight itself is not more persistent than the stream of Feminine discourse. Thetact and skill which suffice to avert a Woman's sting are unequal to the taskof stopping a Woman's mouth; and as the wife has absolutely nothing to say,and absolutely no constraint of wit, sense, or conscience to prevent her fromsaying it, not a few cynics have been found to aver that they prefer thedanger of the death-dealing but inaudible sting to the safe sonorousness of aWoman's other end.To my readers in Spaceland the condition of our Women may seen trulydeplorable, and so indeed it is. A Male of the lowest type of the Isoscelesmay look forward to some improvement of his angle, and to the ultimateelevation of the whole of his degraded caste; but no Woman can entertain suchhopes for her sex. Once a Woman, always a Woman'' is a Decree of Nature; andthe very Laws of Evolution seem suspended in her disfavour. Yet at least wecan admire the wise Prearrangement which has ordained that, as they have nohopes, so they shall have no memory to recall, and no forethought toanticipate, the miseries and humiliations which are at once a necessity oftheir existence and the basis of the constitution of Flatland.\chapter{Of our Methods of Recognizing one another} You, who are blessed withshade as well as light, you, who are gifted with two eyes, endowed with aknowledge of perspective, and charmed with the enjoyment of various colours,you, who can actually see an angle, and contemplate the complete circumferenceof a Circle in the happy region of the Three Dimensions --- how shall I make itclear to you the extreme difficulty which we in Flatland experience inrecognizing one another's configuration?Recall what I told you above. All beings in Flatland, animate and inanimate,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 distinguishedfrom another, where all appear the same?The answer is threefold. The first means of recognition is the sense ofhearing; which with us is far more highly developed than with you, and whichenables us not only to distinguish by the voice our personal friends, but evento discriminate between different classes, at least so far as concerns thethree lowest orders, the Equilateral, the Square, and the Pentagon --- for theIsosceles I take no account. But as we ascend the social scale, the process ofdiscriminating and being discriminated by hearing increases in difficulty,partly because voices are assimilated, partly because the faculty ofvoice-discrimination is a plebeian virtue not much developed among theAristocracy. And wherever there is any danger of imposture we cannot trust tothis method. Amongst our lowest orders, the vocal organs are developed to adegree more than correspondent with those of hearing, so that an Isosceles caneasily feign the voice of a Polygon, and, with some training, that of a Circlehimself. A second method is therefore more commonly resorted to.Feeling is, among our Women and lower classes --- about our upper classes Ishall speak presently --- the principal test of recognition, at all eventsbetween strangers, and when the question is, not as to the individual, but asto the class. What therefore introduction'' is among the higher classes inSpaceland, that the process of feeling'' is with us. Permit me to ask you tofeel and be felt by my friend Mr. So-and-so'' --- is still, among the moreold-fashioned of our country gentlemen in districts remote from towns, thecustomary formula for a Flatland introduction. But in the towns, and among menof business, the words be felt by'' are omitted and the sentence isabbreviated to, Let me ask you to feel Mr. So-and-so''; although it isassumed, of course, that the feeling'' is to be reciprocal. Among our stillmore modern and dashing young gentlemen --- who are extremely averse tosuperfluous effort and supremely indifferent to the purity of their nativelanguage --- the formula is still further curtailed by the use of to feel'' in atechnical sense, meaning, torecommend-for-the-purposes-of-feeling-and-being-felt''; and at this moment theslang'' of polite or fast society in the upper classes sanctions such abarbarism as Mr. Smith, permit me to feel Mr. Jones.''.Let not my Reader however suppose that feeling'' is with us the tediousprocess that it would be with you, or that we find it necessary to feel rightround all the sides of every individual before we determine the class to whichhe belongs. Long practice and training, begun in the schools and continued inthe experience of daily life, enable us to discriminate at once by the senseof 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 isobvious to the dullest touch. It is therefore not necessary, as a rule, to domore than feel a single angle of an individual; and this, once ascertained,tells us the class of the person whom we are addressing, unless indeed hebelongs to the higher sections of the nobility. There the difficulty is muchgreater. Even a Master of Arts in our University of Wentbridge has been knownto confuse a ten-sided with a twelve-sided Polygon; and there is hardly aDoctor of Science in or out of that famous University who could pretend todecide promptly and unhesitatingly between a twenty-sided and a twenty-foursided member of the Aristocracy.Those of my readers who recall the extracts I gave above from the Legislativecode concerning Women, will readily perceive that the process of introductionby contact requires some care and discretion. Otherwise the angles mightinflict on the unwary Feeler irreparable injury. It is essential for thesafety of the Feeler that the Felt should stand perfectly still. A start, afidgety shifting of the position, yes, even a violent sneeze, has been knownbefore now to prove fatal to the incautious, and to nip in the bud many apromising friendship. Especially is this true among the lower classes of theTriangles. With them, the eye is situated so far from their vertex that theycan 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 delicatetouch of the highly organized Polygon. What wonder then if an involuntary tossof 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 hisunhappy Isosceles class, who indeed obtained, shortly before his decease, fourout of seven votes from the Sanitary and Social Board for passing him into theclass of the Equal-sided --- often deplored, with a tear in his venerable eye, amiscarriage of this kind, which had occurred to hisgreat-great-great-Grandfather, a respectable Working Man with an angle orbrain of 59 degrees 30 minutes. According to his account, my unfortunateAncestor, being afflicted with rheumatism, and in the act of being felt by aPolygon, by one sudden start accidentally transfixed the Great Man through thediagonal and thereby, partly in consequence of his long imprisonment anddegradation, and partly because of the moral shock which pervaded the whole ofmy Ancestor's relations, threw back our family a degree and a half in theirascent towards better things. The result was that in the next generation thefamily brain was registered at only 58 degrees, and not till the lapse of fivegenerations was the lost ground recovered, the full 60 degrees attained, andthe Ascent from the Isosceles finally achieved. And all this series ofcalamities from one little accident in the process of Feeling.At this point I think I hear some of my better educated readers exclaim, Howcould you in Flatland know anything about angles and degrees, or minutes? Wesee an angle, because we, in the region of Space, can see two straight linesinclined to one another; but you, who can see nothing but on straight line ata time, or at all events only a number of bits of straight lines all in onestraight line, --- how can you ever discern any angle, and much less registerangles of different sizes?''.I answer that though we cannot see angles, we can infer them, and this withgreat precision. Our sense of touch, stimulated by necessity, and developed bylong training, enables us to distinguish angles far more accurately than yoursense of sight, when unaided by a rule or measure of angles. Nor must I omitto explain that we have great natural helps. It is with us a Law of Naturethat the brain of the Isosceles class shall begin at half a degree, or thirtyminutes, and shall increase (if it increases at all) by half a degree in everygeneration until the goal of 60 degrees is reached, when the condition ofserfdom is quitted, and the freeman enters the class of Regulars.Consequently, Nature herself supplies us with an ascending scale or Alphabetof angles for half a degree up to 60 degrees, Specimen of which are placed inevery Elementary School throughout the land. Owing to occasionalretrogressions, to still more frequent moral and intellectual stagnation, andto the extraordinary fecundity of the Criminal and Vagabond classes, there isalways a vast superfluity of individuals of the half degree and single degreeclass, and a fair abundance of Specimens up to 10 degrees. These areabsolutely destitute of civic rights; and a great number of them, not havingeven intelligence enough for the purposes of warfare, are devoted by theStates to the service of education. Fettered immovably so as to remove allpossibility of danger, they are placed in the classrooms of our InfantSchools, and there they are utilized by the Board of Education for the purposeof imparting to the offspring of the Middle Classes that tact and intelligenceof which these wretched creatures themselves are utterly devoid.In some States the Specimens are occasionally fed and suffered to exist forseveral years; but in the more temperate and better regulated regions, it isfound in the long run more advantageous for the educational interests of theyoung, to dispense with food, and to renew the Specimens every month --- whichis 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 Specimenis lost, partly in the expenditure for food, and partly in the diminishedaccuracy of the angles, which are impaired after a few weeks of constantfeeling''. Nor must we forget to add, in enumerating the advantages of themore expensive system, that it tends, though slightly yet perceptibly, to thediminution of the redundant Isosceles population --- an object which everystatesman 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 ammyself disposed to think that this is one of the many cases in which expenseis the truest economy.But I must not allow questions of School Board politics to divert me from mysubject. Enough has been said, I trust, to shew that Recognition by feeling isnot so tedious or indecisive a process as might have been supposed; and it isobviously more trustworthy than Recognition by hearing. Still there remains,as has been pointed out above, the objection that this method is not withoutdanger. For this reason many in the Middle and Lower classes, and all withoutexception in the Polygonal and Circular orders, prefer a third method, thedescription of which shall be reserved for the next section.\chapter{Of Recognition by Sight} I am about to appear very inconsistent. In previous sections I have said thatall figures in Flatland present the appearance of a straight line; and it wasadded or implied, that it is consequently impossible to distinguish by thevisual organ between individuals of different classes: yet now I am about toexplain to my Spaceland critics how we are able to recognize one another bythe sense of sight.If however the Reader will take the trouble to refer to the passage in whichRecognition by Feeling is stated to be universal, he will find thisqualification --- among the lower classes''. It is only among the higher classesand in our more temperate climates that Sight Recognition is practised.That this power exists in any regions and for any classes is the result ofFog; which prevails during the greater part of the year in all parts save thetorrid zones. That which is with you in Spaceland an unmixed evil, blottingout the landscape, depressing the spirits, and enfeebling the health, is by usrecognized as a blessing scarcely inferior to air itself, and as the Nurse ofarts and Parent of sciences. But let me explain my meaning, without furthereulogies on this beneficent Element.If Fog were non-existent, all lines would appear equally and indistinguishablyclear; and this is actually the case in those unhappy countries in which theatmosphere is perfectly dry and transparent. But wherever there is a richsupply of Fog, objects that are at a distance, say of three feet, areappreciably dimmer than those at the distance of two feet eleven inches; andthe result is that by careful and constant experimental observation ofcomparative dimness and clearness, we are enabled to infer with greatexactness the configuration of the object observed.An instance will do more than a volume of generalities to make my meaningclear.Suppose I see two individuals approaching whose rank I wish to ascertain. Theyare, we will suppose, a Merchant and a Physician, or in other words, anEquilateral Triangle and a Pentagon; how am I to distinguish them?\includegraphics[trim=20mm 0mm 0mm 0mm, width=\linewidth]{fig3}It will be obvious, to every child in Spaceland who has touched the thresholdof Geometrical Studies, that, if I can bring my eye so that its glance maybisect an angle (A) of the approaching stranger, my view will lie as it wereevenly between his two sides that are next to me (viz. CA and AB), so that Ishall 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 straightline DAE, in which the middle point (A) will be very bright because it isnearest to me; but on either side the line will shade away rapidly to dimness,because the sides AC and AB recede rapidly into the fog and what appear to meas 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 alsosee a line (D'A'E') with a bright centre (A'), yet it will shade away lessrapidly to dimness, because the sides (A'C', A'B') recede less rapidly intothe fog: and what appear to me the Physician's extremities, viz. D' and E',will not be not so dim as the extremities of the Merchant.The Reader will probably understand from thesetwo instances how --- after a very long training supplemented by constantexperience --- it is possible for the well-educated classes among us todiscriminate with fair accuracy between the middle and lowest orders, by thesense 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 asaltogether incredible --- I shall have attained all I can reasonably expect.Were I to attempt further details I should only perplex. Yet for the sake ofthe young and inexperienced, who may perchance infer --- from the two simpleinstances I have given above, of the manner in which I should recognize myFather and my Sons --- that Recognition by sight is an easy affair, it may beneedful to point out that in actual life most of the problems of SightRecognition are far more subtle and complex.If for example, when my Father, the Triangle, approaches me, he happens topresent his side to me instead of his angle, then, until I have asked him torotate, or until I have edged my eye around him, I am for the moment doubtfulwhether 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, contemplatingone of his sides (AB) full front, it will be evident from the accompanyingdiagram 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) dimthroughout and shading away into greater dimness towards the extremities C andD. \begin{center}\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig4}\end{center}But I must not give way to the temptation of enlarging on thesetopics. The meanest mathematician in Spaceland will readily believe me when Iassert that the problems of life, which present themselves to thewell-educated --- when they are themselves in motion, rotating, advancing orretreating, and at the same time attempting to discriminate by the sense ofsight between a number of Polygons of high rank moving in differentdirections, as for example in a ball-room or conversazione --- must be of anature to task the angularity of the most intellectual, and amply justify therich endowments of the Learned Professors of Geometry, both Static andKinetic, in the illustrious University of Wentbridge, where the Science andArt of Sight Recognition are regularly taught to large classes of the elite ofthe States.It is only a few of the scions of ournoblest and wealthiest houses, who are able to give the time and moneynecessary for the thorough prosecution of this noble and valuable Art. Even tome, a Mathematician of no mean standing, and the Grandfather of two mosthopeful and perfectly regular Hexagons, to find myself in the midst of a crowdof rotating Polygons of the higher classes, is occasionally very perplexing.And of course to a common Tradesman, or Serf, such a sight is almost asunintelligible as it would be to you, my Reader, were you suddenly transportedto 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 andperpetually in brightness or dimness. Even if you had completed your thirdyear in the Pentagonal and Hexagonal classes in the University, and wereperfect in the theory of the subject, you would still find there was need ofmany years of experience, before you could move in a fashionable crowd withoutjostling against your betters, whom it is against etiquette to ask to feel'',and who, by their superior culture and breeding, know all about yourmovements, while you know very little or nothing about theirs. In a word, tocomport oneself with perfect propriety in Polygonal society, one ought to be aPolygon oneself. Such at least is the painful teaching of my experience.It is astonishing how much the Art --- or I may almostcall it instinct --- of Sight Recognition is developed by the habitual practiceof it and by the avoidance of the custom of Feeling''. Just as, with you, thedeaf 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-speechand lip-reading, so it is with us as regards Seeing'' and Feeling''. None whoin early life resort to Feeling'' will ever learn Seeing'' in perfection.For this reason, among our Higher Classes, Feeling'' is discouraged orabsolutely forbidden. From the cradle their children, instead of going to thePublic Elementary schools (where the art of Feeling is taught,) are sent tohigher Seminaries of an exclusive character; and at our illustriousUniversity, to feel'' is regarded as a most serious fault, involvingRustication for the first offence, and Expulsion for the second.But among the lower classes the art of Sight Recognition is regarded as anunattainable luxury. A common Tradesman cannot afford to let his son spend athird of his life in abstract studies. The children of the poor are thereforeallowed to feel'' from their earliest years, and they gain thereby a precocityand an early vivacity which contrast at first most favourably with the inert,undeveloped, and listless behaviour of the half-instructed youths of thePolygonal class; but when the latter have at last completed their Universitycourse, and are prepared to put their theory into practice, the change thatcomes over them may almost be described as a new birth, and in every art,science, and social pursuit they rapidly overtake and distance theirTriangular competitors.Only a few of the Polygonal Class fail to pass the Final Test or LeavingExamination at the University. The condition of the unsuccessful minority istruly pitiable. Rejected from the higher class, they are also despised by thelower. They have neither the matured and systematically trained powers of thePolygonal Bachelors and Masters of Arts, nor yet the native precocity andmercurial versatility of the youthful Tradesman. The professions, the publicservices, are closed against them, and though in most States they are notactually debarred from marriage, yet they have the greatest difficulty informing suitable alliances, as experience shews that the offspring of suchunfortunate and ill-endowed parents is generally itself unfortunate, if notpositively Irregular.It is from these specimens of the refuse of our Nobility that the greatTumults and Seditions of past ages have generally derived their leaders; andso great is the mischief thence arising that an increasing minority of ourmore progressive Statesmen are of opinion that true mercy would dictate theirentire suppression, by enacting that all who fail to pass the FinalExamination of the University should be either imprisoned for life, orextinguished by a painless death.But I find myself digressing into the subject of Irregularities, a matter ofsuch vital interest that it demands a separate section.\chapter{Concerning Irregular Figures}Throughout the previous pages I have been assuming --- what perhaps should havebeen laid down at the beginning as a distinct and fundamental proposition ---that every human being in Flatland is a Regular Figure, that is to say ofregular construction. By this I mean that a Woman must not only be a line, buta straight line; that an Artisan or Soldier must have two of his sides equal;that Tradesmen must have three sides equal; Lawyers (of which class I am ahumble member), four sides equal, and, generally, that in every Polygon, allthe sides must be equal.The sizes of the sides would of course depend upon the age of the individual.A Female at birth would be about an inch long, while a tall adult Woman mightextend to a foot. As to the Males of every class, it may be roughly said thatthe length of an adult's size, when added together, is two feet or a littlemore. But the size of our sides is not under consideration. I am speaking ofthe equality of sides, and it does not need much reflection to see that thewhole of the social life in Flatland rests upon the fundamental fact thatNature wills all Figures to have their sides equal.If our sides were unequal our angles might be unequal. Instead of its beingsufficient to feel, or estimate by sight, a single angle in order to determinethe form of an individual, it would be necessary to ascertain each angle bythe experiment of Feeling. But life would be too short for such a tediousgroping. The whole science and art of Sight Recognition would at once perish;Feeling, so far as it is an art, would not long survive; intercourse wouldbecome perilous or impossible; there would be an end to all confidence, allforethought; no one would be safe in making the most simple socialarrangements; in a word, civilization might relapse into barbarism.Am I going too fast to carry my Readers with me to these obvious conclusions?Surely a moment's reflection, and a single instance from common life, mustconvince every one that our social system is based upon Regularity, orEquality of Angles. You meet, for example, two or three Tradesmen in thestreet, whom your recognize at once to be Tradesman by a glance at theirangles and rapidly bedimmed sides, and you ask them to step into your house tolunch. This you do at present with perfect confidence, because everyone knowsto an inch or two the area occupied by an adult Triangle: but imagine thatyour Tradesman drags behind his regular and respectable vertex, aparallelogram of twelve or thirteen inches in diagonal: --- what are you to dowith such a monster sticking fast in your house door?But I am insulting the intelligence of my Readers by accumulating detailswhich must be patent to everyone who enjoys the advantages of a Residence inSpaceland. Obviously the measurements of a single angle would no longer besufficient under such portentous circumstances; one's whole life would betaken up in feeling or surveying the perimeter of one's acquaintances. Alreadythe difficulties of avoiding a collision in a crowd are enough to tax thesagacity of even a well-educated Square; but if no one could calculate theRegularity of a single figure in the company, all would be chaos andconfusion, and the slightest panic would cause serious injuries, or --- if therehappened to be any Women or Soldiers present --- perhaps considerable loss oflife.Expediency therefore concurs with Nature in stamping the seal of its approvalupon Regularity of conformation: nor has the Law been backward in secondingtheir efforts. Irregularity of Figure'' means with us the same as, or morethan, a combination of moral obliquity and criminality with you, and istreated accordingly. There are not wanting, it is true, some promulgators ofparadoxes who maintain that there is no necessary connection betweengeometrical and moral Irregularity. The Irregular,'' they say, is from hisbirth scouted by his own parents, derided by his brothers and sisters,neglected by the domestics, scorned and suspected by society, and excludedfrom all posts of responsibility, trust, and useful activity. His everymovement is jealously watched by the police till he comes of age and presentshimself for inspection; then he is either destroyed, if he is found to exceedthe fixed margin of deviation, at an uninteresting occupation for a miserablestipend; obliged to live and board at the office, and to take even hisvacation under close supervision; what wonder that human nature, even in thebest and purest, is embittered and perverted by such surroundings!''All this very plausible reasoning does not convince me, as it has notconvinced the wisest of our Statesmen, that our ancestors erred in laying itdown as an axiom of policy that the toleration of Irregularity is incompatiblewith the safety of the State. Doubtless, the life of an Irregular is hard; butthe interests of the Greater Number require that it shall be hard. If a manwith a triangular front and a polygonal back were allowed to exist and topropagate a still more Irregular posterity, what would become of the arts oflife? Are the houses and doors and churches in Flatland to be altered in orderto accommodate such monsters? Are our ticket-collectors to be required tomeasure every man's perimeter before they allow him to enter a theatre, or totake his place in a lecture room? Is an Irregular to be exempted from themilitia? And if not, how is he to be prevented from carrying desolation intothe ranks of his comrades? Again, what irresistible temptations to fraudulentimpostures must needs beset such a creature! How easy for him to enter a shopwith his polygonal front foremost, and to order goods to any extent from aconfiding tradesman! Let the advocates of a falsely called Philanthropy pleadas they may for the abrogation of the Irregular Penal Laws, I for my part havenever known an Irregular who was not also what Nature evidently intended himto be --- a hypocrite, a misanthropist, and, up to the limits of his power, aperpetrator of all manner of mischief.Not that I should be disposed to recommend (at present) the extreme measuresadopted by some States, where an infant whose angle deviates by half a degreefrom the correct angularity is summarily destroyed at birth. Some of ourhighest and ablest men, men of real genius, have during their earliest dayslaboured under deviations as great as, or even greater than forty-fiveminutes: and the loss of their precious lives would have been an irreparableinjury to the State. The art of healing also has achieved some of its mostglorious triumphs in the compressions, extensions, trepannings, colligations,and other surgical or diaetetic operations by which Irregularity has beenpartly or wholly cured. Advocating therefore a Via Media, I would lay down nofixed or absolute line of demarcation; but at the period when the frame isjust beginning to set, and when the Medical Board has reported that recoveryis improbably, I would suggest that the Irregular offspring be painlessly andmercifully consumed.\chapter{Of the Ancient Practice of Painting}If my Readers have followed me with any attention up to this point, they willnot be surprised to hear that life is somewhat dull in Flatland. I do not, ofcourse, mean that there are not battles, conspiracies, tumults, factions, andall those other phenomena which are supposed to make History interesting; norwould I deny that the strange mixture of the problems of life and the problemsof Mathematics, continually inducing conjecture and giving an opportunity ofimmediate verification, imparts to our existence a zest which you in Spacelandcan hardly comprehend. I speak now from the aesthetic and artistic point ofview when I say that life with us is dull; aesthetically and artistically,very dull indeed.How can it be otherwise, when all one's prospect, all one's landscapes,historical pieces, portraits, flowers, still life, are nothing but a singleline, with no varieties except degrees of brightness and obscurity?It was not always thus. Colour, if Tradition speaks the truth, once for thespace of half a dozen centuries or more, threw a transient splendour over thelives of our ancestors in the remotest ages. Some private individual --- aPentagon whose name is variously reported --- having casually discovered theconstituents of the simpler colours and a rudimentary method of painting, issaid to have begun by decorating first his house, then his slaves, then hisFather, his Sons, and Grandsons, lastly himself. The convenience as well asthe 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 attractedrespect. No one now needed to feel'' him; no one mistook his front for hisback; all his movements were readily ascertained by his neighbours without theslightest strain on their powers of calculation; no one jostled him, or failedto make way for him; his voice was saved the labour of that exhaustingutterance by which we colourless Squares and Pentagons are often forced toproclaim our individuality when we move amid a crowd of ignorant Isosceles.The fashion spread like wildfire. Before a week was over, every Square andTriangle in the district had copied the example of Chromatistes, and only afew of the more conservative Pentagons still held out. A month or two foundeven the Dodecagons infected with the innovation. A year had not elapsedbefore 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 ofChromatistes to surrounding regions; and within two generations no one in allFlatland was colourless except the Women and the Priests.Here Nature herself appeared to erect a barrier, and to plead againstextending the innovations to these two classes. Many-sidedness was almostessential as a pretext for the Innovators. Distinction of sides is intendedby Nature to imply distinction of colours'' --- such was the sophism which inthose days flew from mouth to mouth, converting whole towns at a time to a newculture. But manifestly to our Priests and Women this adage did not apply. Thelatter had only one side, and therefore --- plurally and pedantically speaking ---no sides. The former --- if at least they would assert their claim to be readilyand truly Circles, and not mere high-class Polygons, with an infinitely largenumber of infinitesimally small sides --- were in the habit of boasting (whatWomen confessed and deplored) that they also had no sides, being blessed witha perimeter of only one line, or, in other words, a Circumference. Hence itcame to pass that these two Classes could see no force in the so-called axiomabout Distinction of Sides implying Distinction of Colour;'' and when allothers had succumbed to the fascinations of corporal decoration, the Priestsand the Women alone still remained pure from the pollution of paint.Immoral, licentious, anarchical, unscientific --- call them by what names youwill --- yet, from an aesthetic point of view, those ancient days of the ColourRevolt were the glorious childhood of Art in Flatland --- a childhood, alas,that never ripened into manhood, nor even reached the blossom of youth. Tolive then in itself a delight, because living implied seeing. Even at a smallparty, the company was a pleasure to behold; the richly varied hues of theassembly in a church or theatre are said to have more than once proved toodistracting from our greatest teachers and actors; but most ravishing of allis said to have been the unspeakable magnificence of a military review.The sight of a line of battle of twenty thousandIsosceles suddenly facing about, and exchanging the sombre black of theirbases for the orange of the two sides including their acute angle; the militiaof the Equilateral Triangles tricoloured in red, white, and blue; the mauve,ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidlyrotating near their vermillion guns; the dashing and flashing of thefive-coloured and six-coloured Pentagons and Hexagons careering across thefield in their offices of surgeons, geometricians and aides-de-camp --- allthese may well have been sufficient to render credible the famous story how anillustrious Circle, overcome by the artistic beauty of the forces under hiscommand, threw aside his marshal's baton and his royal crown, exclaiming thathe henceforth exchanged them for the artist's pencil. How great and gloriousthe sensuous development of these days must have been is in part indicated bythe very language and vocabulary of the period. The commonest utterances ofthe commonest citizens in the time of the Colour Revolt seem to have beensuffused with a richer tinge of word or thought; and to that era we are evennow indebted for our finest poetry and for whatever rhythm still remains inthe more scientific utterance of those modern days.\chapter{Of the Universal Colour Bill}But meanwhile the intellectual Arts were fast decaying.The 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 to be considered superfluous, and fell into disrespect and neglecteven at our University. The inferior Art of Feeling speedily experienced thesame fate at our Elementary Schools. Then the Isosceles classes, assertingthat the Specimens were no longer used nor needed, and refusing to pay thecustomary tribute from the Criminal classes to the service of Education, waxeddaily more numerous and more insolent on the strength of their immunity fromthe old burden which had formerly exercised the twofold wholesome effect of atonce taming their brutal nature and thinning their excessive numbers.Year by year the Soldiers and Artisans began more vehemently to assert --- andwith increasing truth --- that there was no great difference between them andthe very highest class of Polygons, now that they were raised to an equalitywith the latter, and enabled to grapple with all the difficulties and solveall the problems of life, whether Statical or Kinetical, by the simple processof Colour Recognition. Not content with the natural neglect into which SightRecognition was falling, they began boldly to demand the legal prohibition ofall monopolizing and aristocratic Arts'' and the consequent abolition of allendowments 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 inthe same path, and that henceforth all individuals and all classes should berecognized as absolutely equal and entitled to equal rights.Finding the higher Orders wavering and undecided, the leaders of theRevolution advanced still further in their requirements, and at last demandedthat all classes alike, the Priests and the Women not excepted, should dohomage to Colour by submitting to be painted. When it was objected thatPriests and Women had no sides, they retorted that Nature and Expediencyconcurred in dictating that the front half of every human being (that is tosay, the half containing his eye and mouth) should be distinguishable from hishinder half. They therefore brought before a general and extraordinaryAssembly of all the States of Flatland a Bill proposing that in every Womanthe half containing the eye and mouth should be coloured red, and the otherhalf green. The Priests were to be painted in the same way, red being appliedto that semicircle in which the eye and mouth formed the middle point; whilethe other or hinder semicircle was to be coloured green.There was no little cunning in this proposal, which indeed emanated not fromany Isosceles --- for no being so degraded would have angularity enough toappreciate, much less to devise, such a model of state-craft --- but from anIrregular Circle who, instead of being destroyed in his childhood, wasreserved by a foolish indulgence to bring desolation on his country anddestruction on myriads of followers.On the one hand the proposition was calculated to bring the Women in allclasses over to the side of the Chromatic Innovation. For by assigning to theWomen the same two colours as were assigned to the Priests, the Revolutioniststhereby ensured that, in certain positions, every Woman would appear as aPriest, and be treated with corresponding respect and deference --- a prospectthat could not fail to attract the Female Sex in a mass.But by some of my Readers the possibility of the identical appearance ofPriests and Women, under a new Legislation, may not be recognized; if so, aword 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 the eye and mouth) red, and with the hinder halfgreen. Look at her from one side. Obviously you will see a straight line, halfred, half green. \begin{center}\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig5}\end{center}Now imagine a Priest, whose mouth is at M, and whose front semicircle (AMB) isconsequently coloured red, while his hinder semicircle is green; so that thediameter AB divides the green from the red. If you contemplate the Great Manso 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 rathershorter perhaps than that of a full-sized Woman, and will shade off morerapidly towards its extremities; but the identity of the colours would giveyou an immediate impression of identity in Class, making you neglectful ofother details. Bear in mind the decay of Sight Recognition which threatenedsociety at the time of the Colour revolt; add too the certainty that Womanwould speedily learn to shade off their extremities so as to imitate theCircles; it must then be surely obvious to you, my dear Reader, that theColour Bill placed us under a great danger of confounding a Priest with ayoung Woman.How attractive this prospect must have been to the Frail Sex may readily beimagined. They anticipated with delight the confusion that would ensue. Athome they might hear political and ecclesiastical secrets intended not forthem but for their husbands and brothers, and might even issue some commandsin the name of a priestly Circle; out of doors the striking combination of redand green without addition of any other colours, would be sure to lead thecommon people into endless mistakes, and the Woman would gain whatever theCircles lost, in the deference of the passers by. As for the scandal thatwould befall the Circular Class if the frivolous and unseemly conduct of theWomen were imputed to them, and as to the consequent subversion of theConstitution, the Female Sex could not be expected to give a thought to theseconsiderations. Even in the households of the Circles, the Women were all infavour of the Universal Colour Bill.The second object aimed at by the Bill was the gradual demoralization of theCircles themselves. In the general intellectual decay they still preservedtheir pristine clearness and strength of understanding. From their earliestchildhood, familiarized in their Circular households with the total absence ofColour, the Nobles alone preserved the Sacred Art of Sight Recognition, withall the advantages that result from that admirable training of the intellect.Hence, up to the date of the introduction of the Universal Colour Bill, theCircles had not only held their own, but even increased their lead of theother classes by abstinence from the popular fashion.Now therefore the artful Irregular whom I described above as the real authorof this diabolical Bill, determined at one blow to lower the status of theHierarchy by forcing them to submit to the pollution of Colour, and at thesame time to destroy their domestic opportunities of training in the Art ofSight Recognition, so as to enfeeble their intellects by depriving them oftheir pure and colourless homes. Once subjected to the chromatic taint, everyparental and every childish Circle would demoralize each other. Only indiscerning between the Father and the Mother would the Circular infant findproblems for the exercise of his understanding --- problems too often likely tobe corrupted by maternal impostures with the result of shaking the child'sfaith in all logical conclusions. Thus by degrees the intellectual lustre ofthe Priestly Order would wane, and the road would then lie open for a totaldestruction of all Aristocratic Legislature and for the subversion of ourPrivileged Classes.\chapter{Of the Suppression of the Chromatic Sedition}The agitation for the Universal Colour Bill continued for three years; and upto the last moment of that period it seemed as though Anarchy were destined totriumph.A whole army of Polygons, who turned out to fight as private soldiers, wasutterly annihilated by a superior force of Isosceles Triangles --- the Squaresand Pentagons meanwhile remaining neutral. Worse than all, some of the ablestCircles fell a prey to conjugal fury. Infuriated by political animosity, thewives in many a noble household wearied their lords with prayers to give uptheir opposition to the Colour Bill; and some, finding their entreatiesfruitless, fell on and slaughtered their innocent children and husband,perishing themselves in the act of carnage. It is recorded that during thattriennial agitation no less than twenty-three Circles perished in domesticdiscord.Great indeed was the peril. It seemed as though the Priests had no choicebetween submission and extermination; when suddenly the course of events wascompletely changed by one of those picturesque incidents which Statesmen oughtnever to neglect, often to anticipate, and sometimes perhaps to originate,because of the absurdly disproportionate power with which they appeal to thesympathies of the populace.It happened that an Isosceles of a low type, with a brain little if at allabove four degrees --- accidentally dabbling in the colours of some Tradesmanwhose 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 theMarket Place he accosted in a feigned voice a maiden, the orphan daughter of anoble Polygon, whose affection in former days he had sought in vain; and by aseries of deceptions --- aided, on the one side, by a string of lucky accidentstoo long to relate, and, on the other, by an almost inconceivable fatuity andneglect of ordinary precautions on the part of the relations of the bride --- hesucceeded in consummating the marriage. The unhappy girl committed suicide ondiscovering the fraud to which she had been subjected.When the news of this catastrophe spread from State to State the minds of theWomen were violently agitated. Sympathy with the miserable victim andanticipations of similar deceptions for themselves, their sisters, and theirdaughters, made them now regard the Colour Bill in an entirely new aspect. Nota few openly avowed themselves converted to antagonism; the rest needed only aslight stimulus to make a similar avowal. Seizing this favourable opportunity,the Circles hastily convened an extraordinary Assembly of the States; andbesides the usual guard of Convicts, they secured the attendance of a largenumber of reactionary Women.Amidst an unprecedented concourse, the Chief Circle of those days --- by namePantocyclus --- arose to find himself hissed and hooted by a hundred and twentythousand Isosceles. But he secured silence by declaring that henceforth theCircles would enter on a policy of Concession; yielding to the wishes of themajority, they would accept the Colour Bill. The uproar being at onceconverted to applause, he invited Chromatistes, the leader of the Sedition,into the centre of the hall, to receive in the name of his followers thesubmission of the Hierarchy. Then followed a speech, a masterpiece ofrhetoric, which occupied nearly a day in the delivery, and to which no summarycan do justice.With a grave appearance of impartiality he declared that as they were nowfinally committing themselves to Reform or Innovation, it was desirable thatthey should take one last view of the perimeter of the whole subject, itsdefects as well as its advantages. Gradually introduction the mention of thedangers to the Tradesmen, the Professional Classes and the Gentlemen, hesilenced the rising murmurs of the Isosceles by reminding them that, in spiteof all these defects, he was willing to accept the Bill if it was approved bythe majority. But it was manifest that all, except the Isosceles, were movedby his words and were either neutral or averse to the Bill.Turning now to the Workmen he asserted that their interests must not beneglected, and that, if they intended to accept the Colour Bill, they ought atleast to do so with full view of the consequences. Many of them, he said, wereon the point of being admitted to the class of the Regular Triangles; othersanticipated for their children a distinction they could not hope forthemselves. That honourable ambition would now have to be sacrificed. With theuniversal adoption of Colour, all distinctions would cease; Regularity wouldbe confused with Irregularity; development would give place to retrogression;the Workman would in a few generations be degraded to the level of theMilitary, or even the Convict Class; political power would be in the hands ofthe greatest number, that is to say the Criminal Classes, who were alreadymore numerous than the Workmen, and would soon out-number all the otherClasses put together when the usual Compensative Laws of Nature were violated.A subdued murmur of assent ran through the ranks of the Artisans, andChromatistes, in alarm, attempted to step forward and address them. But hefound himself encompassed with guards and forced to remain silent while theChief Circle in a few impassioned words made a final appeal to the Women,exclaiming that, if the Colour Bill passed, no marriage would henceforth besafe, no woman's honour secure; fraud, deception, hypocrisy would pervadeevery household; domestic bliss would share the fate of the Constitution andpass to speedy perdition. Sooner than this'', he cried, Come death''.At these words, which were the preconcerted signal for action, the IsoscelesConvicts fell on and transfixed the wretched Chromatistes; the RegularClasses, opening their ranks, made way for a band of Women who, underdirection of the Circles, moved back foremost, invisibly and unerringly uponthe unconscious soldiers; the Artisans, imitating the example of theirbetters, also opened their ranks. Meantime bands of Convicts occupied everyentrance with an impenetrable phalanx.The battle, or rather carnage, was of short duration. Under the skillfulgeneralship of the Circles almost every Woman's charge was fatal and very manyextracted their sting uninjured, ready for a second slaughter. But no secondblow was needed; the rabble of the Isosceles did the rest of the business forthemselves. Surprised, leader-less, attacked in front by invisible foes, andfinding egress cut off by the Convicts behind them, they at once --- after theirmanner --- lost all presence of mind, and raised the cry of treachery''. Thissealed their fate. Every Isosceles now saw and felt a foe in every other. Inhalf an hour not one of that vast multitude was living; and the fragments ofseven score thousand of the Criminal Class slain by one another's anglesattested the triumph of Order.The Circles delayed not to push their victory to the uttermost. The WorkingMen they spared but decimated. The Militia of the Equilaterals was at oncecalled out, and every Triangle suspected of Irregularity on reasonablegrounds, was destroyed by Court Martial, without the formality of exactmeasurement by the Social Board. The homes of the Military and Artisan classeswere inspected in a course of visitation extending through upwards of a year;and during that period every town, village, and hamlet was systematicallypurged of that excess of the lower orders which had been brought about by theneglect to pay the tribute of Criminals to the Schools and University, and bythe violation of other natural Laws of the Constitution of Flatland. Thus thebalance of classes was again restored.Needless to say that henceforth the use of Colour was abolished, and itspossession prohibited. Even the utterance of any word denoting Colour, exceptby the Circles or by qualified scientific teachers, was punished by a severepenalty. Only at our University in some of the very highest and most esotericclasses --- which I myself have never been privileged to attend --- it isunderstood that the sparing use of Colour is still sanctioned for the purposeof illustrating some of the deeper problems of mathematics. But of this I canonly speak from hearsay.Elsewhere in Flatland, Colour is now non-existent. The art of making it isknown to only one living person, the Chief Circle for the time being; and byhim it is handed down on his death-bed to none but his Successor. Onemanufactory alone produces it; and, lest the secret should be betrayed, theWorkmen are annually consumed, and fresh ones introduced. So great is theterror with which even now our Aristocracy looks back to the far-distant daysof the agitation for the Universal Colour Bill.\chapter{Concerning our Priests}It is high time that I should pass from these brief and discursive notes aboutthings in Flatland to the central event of this book, my initiation into themysteries of Space. That is my subject; all that has gone before is merelypreface.For this reason I must omit many matters of which the explanation would not, Iflatter myself, be without interest for my Readers: as for example, our methodof propelling and stopping ourselves, although destitute of feet; the means bywhich we give fixity to structures of wood, stone, or brick, although ofcourse we have no hands, nor can we lay foundations as you can, nor availourselves of the lateral pressure of the earth; the manner in which the rainoriginates in the intervals between our various zones, so that the northernregions do not intercept the moisture falling on the southern; the nature ofour hills and mines, our trees and vegetables, our seasons and harvests; ourAlphabet and method of writing, adapted to our linear tablets; these and ahundred other details of our physical existence I must pass over, nor do Imention them now except to indicate to my readers that their omission proceedsnot from forgetfulness on the part of the author, but from his regard for thetime of the Reader.Yet before I proceed to my legitimate subject some few final remarks will nodoubt be expected by my Readers upon these pillars and mainstays of theConstitution of Flatland, the controllers of our conduct and shapers of ourdestiny, the objects of universal homage and almost of adoration: need I saythat I mean our Circles or Priests?When I call them Priests, let me not be understood as meaning no more than theterm denotes with you. With us, our Priests are Administrators of allBusiness, Art, and Science; Directors of Trade, Commerce, Generalship,Architecture, Engineering, Education, Statesmanship, Legislature, Morality,Theology; doing nothing themselves, they are the Causes of everything worthdoing, that is done by others.Although popularly everyone called a Circle is deemed a Circle, yet among thebetter educated Classes it is known that no Circle is really a Circle, butonly a Polygon with a very large number of very small sides. As the number ofthe sides increases, a Polygon approximates to a Circle; and, when the numberis very great indeed, say for example three or four hundred, it is extremelydifficult for the most delicate touch to feel any polygonal angles. Let me sayrather it would be difficult: for, as I have shown above, Recognition byFeeling is unknown among the highest society, and to feel a Circle would beconsidered a most audacious insult. This habit of abstention from Feeling inthe best society enables a Circle the more easily to sustain the veil ofmystery in which, from his earliest years, he is wont to enwrap the exactnature of his Perimeter or Circumference. Three feet being the averagePerimeter it follows that, in a Polygon of three hundred sides each side willbe no more than the hundredth part of a foot in length, or little more thanthe tenth part of an inch; and in a Polygon of six or seven hundred sides thesides are little larger than the diameter of a Spaceland pin-head. It isalways assumed, by courtesy, that the Chief Circle for the time being has tenthousand sides.The ascent of the posterity of the Circles in the social scale is notrestricted, as it is among the lower Regular classes, by the Law of Naturewhich limits the increase of sides to one in each generation. If it were so,the number of sides in the Circle would be a mere question of pedigree andarithmetic, and the four hundred and ninety-seventh descendant of anEquilateral Triangle would necessarily be a polygon With five hundred sides.But this is not the case. Nature's Law prescribes two antagonistic decreesaffecting Circular propagation; first, that as the race climbs higher in thescale of development, so development shall proceed at an accelerated pace;second, that in the same proportion, the race shall become less fertile.Consequently in the home of a Polygon of four or five hundred sides it is rareto find a son; more than one is never seen. On the other hand the son of afive-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 havediscovered that the small and tender sides of an infant Polygon of the higherclass can be fractured, and his whole frame re-set, with such exactness that aPolygon of two or three hundred sides sometimes --- by no means always, for theprocess is attended with serious risk --- but sometimes overleaps two or threehundred generations, and as it were double at a stroke, the number of hisprogenitors and the nobility of his descent.Many a promising child is sacrificed in this way. Scarcely one out of tensurvives. 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 findthe Nobleman of that position in society, who has neglected to place hisfirst-born in the Circular Neo-Therapeutic Gymnasium before he has attainedthe age of a month.One year determines success or failure. At the end of that time the child has,in all probability, added one more to the tombstones that crowd theNeo-Therapeutic Cemetery; but on rare occasional a glad procession bares backthe little one to his exultant parents, no longer a Polygon, but a Circle, atleast by courtesy: and a single instance of so blessed a result inducesmultitudes of Polygonal parents to submit to similar domestic sacrifice, whichhave a dissimilar issue.\chapter{Of the Doctrine of our Priests}As to the doctrine of the Circles it may briefly be summed up in a singlemaxim, Attend to your Configuration''. Whether political, ecclesiastical, ormoral, all their teaching has for its object the improvement of individual andcollective Configuration --- with special reference of course to theConfiguration of the Circles, to which all other objects are subordinated.It is the merit of the Circles that they have effectually suppressed thoseancient heresies which led men to waste energy and sympathy in the vain beliefthat conduct depends upon will, effort, training, encouragement, praise, oranything else but Configuration. It was Pantocyclus --- the illustrious Circlementioned above, as the queller of the Colour Revolt --- who first convincedmankind that Configuration makes the man; that if, for example, you are bornan Isosceles with two uneven sides, you will assuredly go wrong unless youhave 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 anyIrregularity, you must be taken to one of the Regular Hospitals to have yourdisease cured; otherwise you will end your days in the State Prison or by theangle of the State Executioner.All faults or defects, from the slightest misconduct to the most flagitiouscrime, Pantocyclus attributed to some deviation from perfect Regularity in thebodily 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 suddenchange of temperature, resulting in a shrinkage or expansion in some toosusceptible part of the frame. Therefore, concluded that illustriousPhilosopher, neither good conduct nor bad conduct is a fit subject, in anysober estimation, for either praise or blame. For why should you praise, forexample, the integrity of a Square who faithfully defends the interests of hisclient, when you ought in reality rather to admire the exact precision of hisright angles? Or again, why blame a lying, thievish Isosceles, when you oughtrather to deplore the incurable inequality of his sides?Theoretically, this doctrine is unquestionable; but it has practicaldrawbacks. In dealing with an Isosceles, if a rascal pleads that he cannothelp stealing because of his unevenness, you reply that for that very reason,because he cannot help being a nuisance to his neighbours, you, theMagistrate, cannot help sentencing him to be consumed --- and there's an end ofthe matter. But in little domestic difficulties, when the penalty ofconsumption, or death, is out of the question, this theory of Configurationsometimes comes in awkwardly; and I must confess that occasionally when one ofmy own Hexagonal Grandsons pleads as an excuse for his disobedience that asudden change of temperature has been too much for his Perimeter, and that Iought to lay the blame not on him but on his Configuration, which can only bestrengthened by abundance of the choicest sweetmeats, I neither see my waylogically to reject, nor practically to accept, his conclusions.For my own part, I find it best to assume that a good sound scolding orcastigation has some latent and strengthening influence on my Grandson'sConfiguration; though I own that I have no grounds for thinking so. At allevents I am not alone in my way of extricating myself from this dilemma; for Ifind that many of the highest Circles, sitting as Judges in law courts, usepraise and blame towards Regular and Irregular Figures; and in their homes Iknow by experience that, when scolding their children, they speak aboutright'' and wrong'' as vehemently and passionately as if they believe thatthese names represented real existence, and that a human Figure is reallycapable of choosing between them.Constantly carrying out their policy of making Configuration the leading ideain every mind, the Circles reverse the nature of that Commandment which inSpaceland 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 hisGrandson, if he has one; or, if not, his Son. By honour'', however, is by nomeans mean indulgence'', but a reverent regard for their highest interests:and the Circles teach that the duty of fathers is to subordinate their owninterests to those of posterity, thereby advancing the welfare of the wholeState as well as that of their own immediate descendants.The weak point in the system of the Circles --- if a humble Square may ventureto speak of anything Circular as containing any element of weakness --- appearsto me to be found in their relations with Women.As it is of the utmost importance for Society that Irregular births should bediscouraged, it follows that no Woman who has any Irregularities in herancestry is a fit partner for one who desires that his posterity should riseby regular degrees in the social scale.Now the Irregularity of a Male is a matter of measurement; but as all Womenare straight, and therefore visibly Regular so to speak, one has to devicesome 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 andsupervised by the State; and without a certified pedigree no Woman is allowedto marry.Now it might have been supposed the a Circle --- proud of his ancestry andregardful for a posterity which might possibly issue hereafter in a ChiefCircle --- would be more careful than any other to choose a wife who had no bloton her escutcheon. But it is not so. The care in choosing a Regular wifeappears to diminish as one rises in the social scale. Nothing would induce anaspiring Isosceles, who has hopes of generating an Equilateral Son, to take awife who reckoned a single Irregularity among her Ancestors; a Square orPentagon, who is confident that his family is steadily on the rise, does notinquire above the five-hundredth generation; a Hexagon or Dodecagon is evenmore careless of the wife's pedigree; but a Circle has been known deliberatelyto take a wife who has had an Irregular Great-Grandfather, and all because ofsome 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 aWoman''.Such ill-judged marriages are, as might be expected, barren, if they do notresult in positive Irregularity or in diminution of sides; but none of theseevils have hitherto provided sufficiently deterrent. The loss of a few sidesin a highly-developed Polygon is not easily noticed, and is sometimescompensated by a successful operation in the Neo-Therapeutic Gymnasium, as Ihave described above; and the Circles are too much disposed to acquiesce ininfecundity as a law of the superior development. Yet, if this evil be notarrested, the gradual diminution of the Circular class may soon become morerapid, and the time may not be far distant when, the race being no longer ableto produce a Chief Circle, the Constitution of Flatland must fall.One other word of warning suggest itself to me, though I cannot so easilymention a remedy; and this also refers to our relations with Women. Aboutthree hundred years ago, it was decreed by the Chief Circle that, since womenare deficient in Reason but abundant in Emotion, they ought no longer to betreated as rational, nor receive any mental education. The consequence wasthat they were no longer taught to read, nor even to master Arithmetic enoughto enable them to count the angles of their husband or children; and hencethey sensibly declined during each generation in intellectual power. And thissystem of female non-education or quietism still prevails.My fear is that, with the best intentions, this policy has been carried so faras to react injuriously on the Male Sex.For the consequence is that, as things now are, we Males have to lead a kindof bi-lingual, and I may almost say bimental, existence. With Women, we speakof love'', duty'', right'', wrong'', pity'', hope'', and other irrational andemotional conceptions, which have no existence, and the fiction of which hasno object except to control feminine exuberances; but among ourselves, and inour books, we have an entirely different vocabulary and I may also say, idiom.Love'' them becomes the anticipation of benefits''; duty'' becomes necessity''or fitness''; and other words are correspondingly transmuted. Moreover, amongWomen, we use language implying the utmost deference for their Sex; and theyfully believe that the Chief Circle Himself is not more devoutly adored by usthan they are: but behind their backs they are both regarded and spoken of ---by all but the very young --- as being little better than mindless organisms''.Our Theology also in the Women's chambers is entirely different from ourTheology elsewhere.Now my humble fear is that this double training, in language as well as inthought, 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 andtaught to unlearn the old language --- except for the purpose of repeating it inthe presence of the Mothers and Nurses --- and to learn the vocabulary and idiomof science. Already methinks I discern a weakness in the grasp of mathematicaltruth at the present time as compared with the more robust intellect of ourancestors three hundred years ago. I say nothing of the possible danger if aWoman should ever surreptitiously learn to read and convey to her Sex theresult of her perusal of a single popular volume; nor of the possibility thatthe indiscretion or disobedience of some infant Male might reveal to a Motherthe secrets of the logical dialect. On the simple ground of the enfeebling ofthe male intellect, I rest this humble appeal to the highest Authorities toreconsider the regulations of Female education.\part{Other Worlds}\chapter{How I had a Vision of Lineland}It was the last day but one of the 1999th year of our era, and the first dayof the Long Vacation. Having amused myself till a late hour with my favouriterecreation of Geometry, I had retired to rest with an unsolved problem in mymind. In the night I had a dream.I saw before me a vast multitude of small Straight Lines (which I naturallyassumed to be Women) interspersed with other Beings still smaller and of thenature of lustrous points --- all moving to and fro in one and the same StraightLine, and, as nearly as I could judge, with the same velocity.A noise of confused, multitudinous chirping or twittering issued from them atintervals 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 equallyineffectual. Losing patience at what appeared to me intolerable rudeness, Ibrought my mouth to a position full in front of her mouth so as to intercepther motion, and loudly repeated my question, Woman, what signifies thisconcourse, and this strange and confused chirping, and this monotonous motionto and fro in one and the same Straight Line?''I am no Woman'', replied the small Line: I am the Monarch of the world. Butthou, whence intrudest thou into my realm of Lineland?'' Receiving this abruptreply, I begged pardon if I had in any way startled or molested his RoyalHighness; and describing myself as a stranger I besought the King to give mesome account of his dominions. But I had the greatest possible difficulty inobtaining any information on points that really interested me; for the Monarchcould not refrain from constantly assuming that whatever was familiar to himmust also be known to me and that I was simulating ignorance in jest. However,by preserving questions I elicited the following facts:\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig6}It seemed that this poor ignorant Monarch --- as he called himself --- waspersuaded that the Straight Line which he called his Kingdom, and in which hepassed his existence, constituted the whole of the world, and indeed the wholeof 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 whenI first addressed him, the sounds had come to him in a manner so contrary tohis experience that he had made no answer, seeing no man,'' as he expressedit, and hearing a voice as it were from my own intestines.'' Until the momentwhen I placed my mouth in his World, he had neither seen me, nor heardanything except confused sounds beating against, what I called his side, butwhat he called his inside or stomach; nor had he even now the least conceptionof the region from which I had come. Outside his World, or Line, all was ablank 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 --- wereall alike confined in motion and eyesight to that single Straight Line, whichwas their World. It need scarcely be added that the whole of their horizon waslimited to a Point; nor could any one ever see anything but a Point. Man,woman, child, thing --- each as a Point to the eye of a Linelander. Only by thesound of the voice could sex or age be distinguished. Moreover, as eachindividual occupied the whole of the narrow path, so to speak, whichconstituted his Universe, and no one could move to the right or left to makeway for passers by, it followed that no Linlander could ever pass another.Once neighbours, always neighbours. Neighbourhood with them was like marriagewith us. Neighbours remained neighbours till death did them part.Such a life, with all vision limited to a Point, and all motion to a StraightLine, seemed to me inexpressibly dreary; and I was surprised to note thatvivacity and cheerfulness of the King. Wondering whether it was possible, amidcircumstances so unfavourable to domestic relations, to enjoy the pleasures ofconjugal union, I hesitated for some time to question his Royal Highness on sodelicate a subject; but at last I plunged into it by abruptly inquiring as tothe health of his family. My wives and children,'' he replied, are well andhappy.''Staggered at this answer --- for in the immediate proximity of the Monarch (as Ihad noted in my dream before I entered Lineland) there were none but Men --- Iventured to reply, Pardon me, but I cannot imagine how your Royal Highnesscan at any time either See or approach their Majesties, when there at leasthalf a dozen intervening individuals, whom you can neither see through, norpass by? Is it possible that in Lineland proximity is not necessary formarriage and for the generation of children?''How can you ask so absurd a question?'' replied the Monarch. If it wereindeed as you suggest, the Universe would soon be depopulated. No, no;neighbourhood is needless for the union of hearts; and the birth of childrenis too important a matter to have been allowed to depend upon such an accidentas proximity. You cannot be ignorant of this. Yet since you are pleased toaffect ignorance, I will instruct you as if you were the veriest baby inLineland. Know, then, that marriages are consummated by means of the facultyof sound and the sense of hearing.You are of course aware that every Man has two mouths or voices --- as well astwo eyes --- a bass at one and a tenor at the other of his extremities. I shouldnot mention this, but that I have been unable to distinguish your tenor in thecourse of our conversation.'' I replied that I had but one voice, and that Ihad not been aware that his Royal Highness had two. That confirms byimpression,'' said the King, that you are not a Man, but a feminineMonstrosity with a bass voice, and an utterly uneducated ear. But to continue.Nature having herself ordained that every Man should wed two wives ---'' Whytwo?'' asked I. You carry your affected simplicity too far,'' he cried. Howcan there be a completely harmonious union without the combination of the Fourin One, viz. the Bass and Tenor of the Man and the Soprano and Contralto ofthe two Women?'' But supposing,'' said I, that a man should prefer one wife orthree?'' It is impossible,'' he said; it is as inconceivable as that two andone should make five, or that the human eye should see a Straight Line.'' Iwould 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 frowith a rhythmic motion of more than usual violence, which continues for thetime you would take to count a hundred and one. In the midst of this choraldance, at the fifty-first pulsation, the inhabitants of the Universe pause infull career, and each individual sends forth his richest, fullest, sweeteststrain. It is in this decisive moment that all our marriages are made. Soexquisite is the adaptation of Bass and Treble, of Tenor to Contralto, thatoftentimes the Loved Ones, though twenty thousand leagues away, recognize atonce the responsive note of their destined Lover; and, penetrating the paltryobstacles of distance, Love unites the three. The marriage in that instanceconsummated results in a threefold Male and Female offspring which takes itsplace in Lineland.''What! Always threefold?'' said I. Must one wife then always have twins?''Bass-voice Monstrosity! yes,'' replied the King. How else could the balanceof the Sexes be maintained, if two girls were not born for every boy? Wouldyou ignore the very Alphabet of Nature?'' He ceased, speechless for fury; andsome time elapsed before I could induce him to resume his narrative.You will not, of course, suppose that every bachelor among us finds his matesat the first wooing in this universal Marriage Chorus. On the contrary, theprocess is by most of us many times repeated. Few are the hearts whose happylot is at once to recognize in each other's voice the partner intended forthem by Providence, and to fly into a reciprocal and perfectly harmoniousembrace. With most of us the courtship is of long duration. The Wooer's voicesmay perhaps accord with one of the future wives, but not with both; or not, atfirst, with either; or the Soprano and Contralto may not quite harmonize. Insuch cases Nature has provided that every weekly Chorus shall bring the threeLovers into closer harmony. Each trial of voice, each fresh discovery ofdiscord, almost imperceptibly induces the less perfect to modify his or hervocal utterance so as to approximate to the more perfect. And after manytrials and many approximations, the result is at last achieved. There comes aday at last when, while the wonted Marriage Chorus goes forth from universalLineland, the three far-off Lovers suddenly find themselves in exact harmony,and, before they are aware, the wedded Triplet is rapt vocally into aduplicate embrace; and Nature rejoices over one more marriage and over threemore births.''\chapter{How I vainly tried to explain the nature of Flatland}Thinking that it was time to bring down the Monarch from his raptures to thelevel of common sense, I determined to endeavour to open up to him someglimpses of the truth, that is to say of the nature of things in Flatland. SoI began thus: How does your Royal Highness distinguish the shapes andpositions of his subjects? I for my part noticed by the sense of sight, beforeI entered your Kingdom, that some of your people are lines and others Points;and that some of the lines are larger ---'' You speak of an impossibility,''interrupted the King; you must have seen a vision; for to detect thedifference between a Line and a Point by the sense of sight is, as every oneknows, in the nature of things, impossible; but it can be detected by thesnese of hearing, and by the same means my shape can be exactly ascertained.Behold me --- I am 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.''I apologized; but he continued scornfully, Since you are impervious toargument, you shall hear with your ears how by means of my two voices I revealmy shape to my Wives, who are at this moment six thousand miles seventy yardstwo 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 momentreceiving the sound of one of my voice, closely followed by the other, andperceiving that the latter reaches them after an interval in which sound cantraverse 6.457 inches, infer that one of my mouths is 6.457 inches furtherfrom 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 calculationevery time they hear my two voices. They made it, once for all, before we weremarried. But they could make it at any time. And in the same way I canestimate 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 twovoices, or so disguises his Southern voice that it cannot be recognized as theecho of the Northern? May not such deceptions cause great inconvenience? Andhave you no means of checking frauds of this kind by commanding yourneighbouring subjects to feel one another?'' This of course was a very stupidquestion, for feeling could not have answered the purpose; but I asked withthe view of irritating the Monarch, and I succeeded perfectly.What!'' cried he in horror, explain your meaning.'' Feel, touch, come intocontact,'' I replied.. If you mean by feeling, said the King, approaching soclose as to leave no space between two individuals, know, Stranger, that thisoffence is punishable in my dominions by death. And the reason is obvious. Thefrail 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 thesense of sight from Men, the Law ordains universally that neither Man norWoman shall be approached so closely as to destroy the interval between theapproximator and the approximated.And indeed what possible purpose would be served by this illegal andunnatural excess of approximation which you call touching, when all the endsof so brutal and course a process are attained at once more easily and moreexactly by the sense of hearing? As to your suggested danger of deception, itis non-existent: for the Voice, being the essence of one's Being, cannot bethus changed at will. But come, suppose that I had the power of passingthrough solid things, so that I could penetrate my subjects, one afteranother, even to the number of a billion, verifying the size and distance ofeach by the sense of feeling: How much time and energy would be wasted in thisclumsy and inaccurate method! Whereas now, in one moment of audition, I takeas 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 seemedto me no better than a tiny chirping from an innumerable multitude oflilliputian grasshoppers.Truly,'' replied I, your sense of hearing serves you in good stead, and fillsup many of your deficiencies. But permit me to point out that your life inLineland must be deplorably dull. To see nothing but a Point! Not even to beable to contemplate a Straight Line! Nay, not even to know what a StraightLine is! To see, yet to be cut off from those Linear prospects which arevouchsafed to us in Flatland! Better surely to have no sense of sight at allthan to see so little! I grant you I have not your discriminative faculty ofhearing; for the concert of all Lineland which gives you such intensepleasure, is to me no better than a multitudinous twittering or chirping. Butat 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 immediateproximity on the left, and eight Men and two Women on your right. Is not thiscorrect?''It is correct,'' said the King, so far as the numbers and sexes arecocnerned, though I know not what you mean by right' and left.' But I denythat you saw these things. For how could you see the Line, that is to say theinside, of any Man? But you must have heard these things, and then dreamedthat you saw them. And let me ask what you mean by those words left' andright'. I suppose it is your way of saying Northward and Southward.''Not so,'' replied I; besides your motion of Northward and Southward, there isanother 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 setp 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 thetrue 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 movingin it, then I beg you to describe it to me in words.I. If you cannot tell your right side from your left, I fear that no words ofmine can make my meaning clearer to you. But surely you cannot be ignorant ofso 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 notsometimes occur to you that you could move in some other way, turning your eyeround 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 yourextremities, 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 anydirection? 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, andwill move gradually out of Lineland in the direction which I desire toindicate to you.At the word I began to move my body out of Lineland. As long as any part of meremained in his dominion and in his view, the King kept exclaiming, I seeyou, I see you still; you are not moving.'' But when I had at last moved myselfout of his Line, he cried in his shrillest voice, She is vanished; she isdead.'' I am not dead,'' replied I; I am simply out of Lineland, that is tosay, out of the Straight Line which you call Space, and in the true Space,where I can see things as they are. And at this moment I can see your Line, orside --- or inside as you are pleased to call it; and I can see also the Men andWomen on the North and South of you, whom I will now enumerate, describingtheir order, their size, and the interval between each.''\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig7}When I had done this at great length, I cried triumphantly, Does that at lastconvince you?'' And, with that, I once more entered Lineland, taking up thesame position as before.But the Monarch replied, If you were a Man of sense --- though, as you appearto 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 tobelieve that there is another Line besides that which my senses indicate, andanother motion besides that of which I am daily conscious. I, in return, askyou to describe in words or indicate by motion that other Line of which youspeak. Instead of moving, you merely exercise some magic art of vanishing andreturning 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, factsknown to any child in my capital. Can anything be more irrational oraudacious? Acknowledge your folly or depart from my dominions.''Furious at his perversity, and especially indignant that he professed to beignorant of my sex, I retorted in no measured terms, Besotted Being! Youthink yourself the perfection of existence, while you are in reality the mostimperfect and imbecile. You profess to see, whereas you see nothing but aPoint! You plume yourself on inferring the existence of a Straight Line; but Ican see Straight Lines, and infer the existence of Angles, Triangles, Squares,Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that Iam the completion of your incomplete self. You are a Line, but I am a Line ofLines called in my country a Square: and even I, infinitely superior though Iam to you, am of little account among the great nobles of Flatland, whence Ihave come to visit you, in the hope of enightening your ignorance.''Hearing these words the King advanced towards me with a menacing cry as if topierce me through the diagonal; and in that same movement there arose frommyriads of his subjects a multitudinous war-cry, increasing in vehemence tillat last methought it rivalled the roar of an army of a hundred thousandIsosceles, and the artillery of a thousand Pentagons. Spell-bound andmotionless, I could neither speak nor move to avert the impending destruction;and still the noise grew louder, and the King came closer, when I awoke tofind the breakfast-bell recalling me to the realities of Flatland.\chapter{Concerning a Stranger from Spaceland}From dreams I proceed to facts.It was the last day of our 1999th year of our era. The patterning of the rainhad long ago announced nightfall; and I was sitting \footnote{ When I saysitting,'' of course I do not mean any change of attitude such as you inSpaceland signify by that word; for as we have no feet, we can no more sit''nor stand'' (in your sense of the word) than one of your soles or flounders.Nevertheless, we perfectly well recognize the different mental states ofvolition implied by lying,'' sitting,'' and standing,'' which are to someextent indicated to a beholder by a slight increase of lustre corresponding tothe increase of volition.But on this, and a thousand other kindred subjects, time forbids me to dwell.}in the company of my wife, musing on the events of the past and the prospectsof the coming year, the coming century, the coming Millennium.My four Sons and two orphan Grandchildren had retired to their severalapartments; and my wife alone remained with me to see the old Millennium outand the new one in.I was rapt in thought, pondering in my mind some words that had casuallyissued from the mouth of my youngest Grandson, a most promising young Hexagonof unusual brilliancy and perfect angularity. His uncles and I had been givinghim his usual practical lesson in Sight Recognition, turning ourselves uponour centres, now rapidly, now more slowly, and questioning him as to ourpositions; and his answers had been so satisfactory that I had been induced toreward 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 tomake one large Square, with a side of three inches, and I had hence proved tomy little Grandson that --- though it was impossible for us to see the inside ofthe Square --- yet we might ascertain the number of square inches in a Square bysimply squaring the number of inches in the side: and thus,'' said I, we knowthat three-to-the-second, or nine, represents the number of square inches in aSquare whose side is three inches long.''The little Hexagon meditated on this a while and then said to me; But youhave been teaching me to raise numbers to the third power: I supposethree-to-the-third must mean something in Geometry; what does it mean?''Nothing at all,'' replied I, not at least in Geometry; for Geometry has onlyTwo Dimensions.'' And then I began to shew the boy how a Point by movingthrough a length of three inches makes a Line of three inches, which may berepresented by three; and how a Line of three inches, moving parallel toitself through a length of three inches, makes a Square of three inches everyway, which may be represented by three-to-the-second.Upon this, my Grandson, again returning to his former suggestion, took me uprather suddenly and exclaimed, Well, then, if a Point by moving three inches,makes a Line of three inches represented by three; and if a straight Line ofthree inches, moving parallel to itself, makes a Square of three inches everyway, represented by three-to-the-second; it must be that a Square of threeinches every way, moving somehow parallel to itself (but I don't see how) mustmake Something else (but I don't see what) of three inches every way --- andthis must be represented by three-to-the-third.''Go to bed,'' said I, a little ruffled by this interruption: if you would talkless 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 ofthe year 2000; but not quite able to shake of the thoughts suggested by theprattle of my bright little Hexagon. Only a few sands now remained in thehalf-hour glass. Rousing myself from my reverie I turned the glass Northwardfor the last time in the old Millennium; and in the act, I exclaimed aloud,The boy is a fool.''Straightway I became conscious of a Presence in the room, and a chillingbreath thrilled through my very being. He is no such thing,'' cried my Wife,and you are breaking the Commandments in thus dishonouring your ownGrandson.'' But I took no notice of her. Looking around in every direction Icould see nothing; yet still I felt a Presence, and shivered as the coldwhisper came again. I started up. What is the matter?'' said my Wife, thereis 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;three-to-the-third can have no meaning in Geometry.'' At once there came adistinctly audible reply, The boy is not a fool; and three-to-the-third hasan obvious Geometrical meaning.''My Wife as well as myself heard the words, although she did not understandtheir 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 itappeared to be a Woman, seen sideways; but a moment's observation shewed methat the extremities passed into dimness too rapidly to represent one of theFemale Sex; and I should have thought it a Circle, only that it seemed tochange its size in a manner impossible for a Circle or for any regular Figureof which I had had experience.But my Wife had not my experience, nor the coolness necessary to note thesecharacteristics. With the usual hastiness and unreasoning jealousy of her Sex,she flew at once to the conclusion that a Woman had entered the house throughsome small aperture. How comes this person here?'' she exclaimed, youpromised me, my dear, that there should be no ventilators in our new house.''Nor are they any,'' said I; but what makes you think that the stranger is aWoman? I see by my power of Sight Recognition ---'' Oh, I have no patience withyour Sight Recognition,'' replied she, Feeling is believing' and A StraightLine to the touch is worth a Circle to the sight''' --- two Proverbs, very commonwith the Frailer Sex in Flatland.Well,'' said I, for I was afraid of irritating her, if it must be so, demandan introduction.'' Assuming her most gracious manner, my Wife advanced towardsthe Stranger, Permit me, Madam to feel and be felt by ---'' then, suddenlyrecoiling, Oh! it is not a Woman, and there are no angles either, not a traceof 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 moreperfect Circle than any in Flatland; but to speak more accurately, I am manyCircles in one.'' Then he added more mildly, I have a message, dear Madam, toyour husband, which I must not deliver in your presence; and, if you wouldsuffer us to retire for a few minutes ---'' But my wife would not listen to theproposal that our august Visitor should so incommode himself, and assuring theCircle that the hour of her own retirement had long passed, with manyreiterated apologies for her recent indiscretion, she at last retreated to herapartment.I glanced at the half-hour glass. The last sands had fallen. The thirdMillennium had begun.\chapter{How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland}As soon as the sound of the Peace-cry of my departing Wife had died away, Ibegan to approach the Stranger with the intention of taking a nearer view andof bidding him be seated: but his appearance struck me dumb and motionlesswith astonishment. Without the slightest symptoms of angularity henevertheless varied every instant with graduations of size and brightnessscarcely possible for any Figure within the scope of my experience. Thethought 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 stabme with his acute angle.In a sitting-room, the absence of Fog (and the season happened to beremarkably 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 felthim. My Wife was right. There was not the trace of an angle, not the slightestroughness or inequality: never in my life had I met with a more perfectCircle. He remained motionless while I walked around him, beginning from hiseye and returning to it again. Circular he was throughout, a perfectlysatisfactory Circle; there could not be a doubt of it. Then followed adialogue, 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 andhumiliation that I, a Square, should have been guilty of the impertinence offeeling a Circle. It was commenced by the Stranger with some impatience at thelengthiness of my introductory process.STRANGER. Have you felt me enough by this time? Are you not introduced to meyet?I. Most illustrious Sir, excuse my awkwardness, which arises not fromignorance of the usages of polite society, but from a little surprise andnervousness, consequent on this somewhat unexpected visit. And I beseech youto reveal my indiscretion to no one, and especially not to my Wife. But beforeyour Lordship enters into further communications, would he deign to satisfythe 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, yourLordship 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 isof 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 theThird 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 cannotlook, because you have no eye in your side.I. Pardon me, my Lord, a moment's inspection will convince your Lordship thatI have a perfectly luminary at the juncture of my two sides.Stranger: Yes: but in order to see into Space you ought to have an eye, not onyour Perimeter, but on your side, that is, on what you would probably callyour 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 ThreeDimensions whence I but lately looked down upon your Plane which you callSpace forsooth. From that position of advantage I discerned all that you speakof as solid (by which you mean enclosed on four sides''), your houses, yourchurches, your very chests and safes, yes even your insides and stomachs, alllying 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 hisapartment, and your two Grandsons the Hexagons; I saw your youngest Hexagonremain a while with you and then retire to his room, leaving you and your Wifealone. I saw your Isosceles servants, three in number, in the kitchen atsupper, and the little Page in the scullery. Then I came here, and how do youthink I came?I. Through the roof, I suppose.STRANGER. Not so. Your roof, as you know very well, has been recentlyrepaired, and has no aperture by which even a Woman could penetrate. I tellyou I come from Space. Are you not convinced by what I have told you of yourchildren and household?I. Your Lordship must be aware that such facts touching the belongings of hishumble servant might be easily ascertained by any one of the neighbourhoodpossessing your Lordship's ample means of information.STRANGER. (To himself.) What must I do? Stay; one more argument suggestsitself to me. When you see a Straight Line --- your wife, for example --- how manyDimensions do you attribute to her?I. Your Lordship would treat me as if I were one of the vulgar who, beingignorant of Mathematics, suppose that a Woman is really a Straight Line, andonly of One Dimension. No, no, my Lord; we Squares are better advised, and areas well aware of your Lordship that a Woman, though popularly called aStraight Line, is, really and scientifically, a very thin Parallelogram,possessing Two Dimensions, like the rest of us, viz., length and breadth (orthickness).STRANGER. But the very fact that a Line is visible implies that it possessesyet another Dimension.I. My Lord, I have just acknowledge that a Woman is broad as well as long. Wesee her length, we infer her breadth; which, though very slight, is capable ofmeasurement.STRANGER. You do not understand me. I mean that when you see a Woman, youought --- besides inferring her breadth --- to see her length, and to see what wecall her height; although the last Dimension is infinitesimal in your country.If a Line were mere length without height,'' it would cease to occupy Spaceand would become invisible. Surely you must recognize 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 thebrightness disappears, the Line is extinguished, and, as you say, ceases tooccupy Space. But am I to suppose that your Lordship gives the brightness thetitle 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 ThirdDimension, which you call height.'' Now, Dimension implies direction andmeasurement. Do but measure my height,'' or merely indicate to me thedirection in which my height'' extends, and I will become your convert.Otherwise, your Lordship's own understanding must hold me excused.STRANGER. (To himself.) I can do neither. How shall I convince him? Surely aplain 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 surfaceof what I may call a fluid, or in, the top of which you and your countrymenmove about, without rising above or falling below it.I am not a plane Figure, but a Solid. You call me a Circle; but in reality Iam not a Circle, but an infinite number of Circles, of size varying from aPoint to a Circle of thirteen inches in diameter, one placed on the top of theother. When I cut through your plane as I am now doing, I make in your plane asection which you, very rightly, call a Circle. For even a Sphere --- which ismy proper name in my own country --- if he manifest himself at all to aninhabitant of Flatland --- must needs manifest himself as a Circle.Do you not remember --- for I, who see all things, discerned last night thephantasmal vision of Lineland written upon your brain --- do you not remember, Isay, how when you entered the realm of Lineland, you were compelled tomanifest yourself to the King, not as a Square, but as a Line, because thatLinear Realm had not Dimensions enough to represent the whole of you, but onlya slice or section of you? In precisely the same way, your country of TwoDimensions is not spacious enough to represent me, a being of Three, but canonly exhibit a slice or section of me, which is what you call a Circle.The diminished brightness of your eye indicates incredulity. But now prepareto receive proof positive of the truth of my assertions. You cannot indeed seemore than one of my sections, or Circles, at a time; for you have no power toraise your eye out of the plane of Flatland; but you can at least see that, asI rise in Space, so my sections become smaller. See now, I will rise; and theeffect upon your eye will be that my Circle will become smaller and smallertill it dwindles to a point and finally vanishes.\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig8}There was no rising'' that I could see; but he diminished and finallyvanished. I winked once or twice to make sure that I was not dreaming. But itwas no dream. For from the depths of nowhere came forth a hollow voice --- closeto my heart it seemed --- Am I quite gone? Are you convinced now? Well, now Iwill gradually return to Flatland and you shall see my section become largerand larger.''Every reader in Spaceland will easily understand that my mysterious Guest wasspeaking the language of truth and even of simplicity. But to me, proficientthough I was in Flatland Mathematics, it was by no means a simple matter. Therough diagram given above will make it clear to any Spaceland child that theSphere, ascending in the three positions indicated there, must needs havemanifested himself to me, or to any Flatlander, as a Circle, at first of fullsize, then small, and at last very small indeed, approaching to a Point. Butto 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 andvanished, and that he had now re-appeared and was rapidly making himselflarger.When he regained his original size, he heaved a deep sigh; for he perceived bymy silence that I had altogether failed to comprehend him. And indeed I wasnow inclining to the belief that he must be no Circle at all, but someextremely clever juggler; or else that the old wives' tales were true, andthat after all there were such people as Enchanters and Magicians.After a long pause he muttered to himself, One resource alone remains, if Iam not to resort to action. I must try the method of Analogy.'' Then followed astill longer silence, after which he continued our dialogue.SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves aluminous 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 astraight Line. What name will you give to the Figure thereby formed? We willsuppose 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? How many angles?I. Four sides and four angles.SPHERE. Now stretch your imagination a little, and conceive a Square inFlatland, 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 movethrough the positions previously occupied by the Northern points. But that isnot my meaning.I mean that every Point in you --- for you are a Square and will serve thepurpose of my illustration --- every Point in you, that is to say in what youcall your inside, is to pass upwards through Space in such a way that no Pointshall pass through the position previously occupied by any other Point; buteach Point shall describe a straight Line of its own. This is all inaccordance with Analogy; surely it must be clear to you.Restraining my impatience --- for I was now under a strong temptation to rushblindly 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 thismotion which you are pleased to denote by the word upward'? I presume it isdescribable in the language of Flatland.''SPHERE. Oh, certainly. It is all plain and simple, and in strict accordancewith Analogy --- only, by the way, you must not speak of the result as being aFigure, but as a Solid. But I will describe it to you. Or rather not I, butAnalogy.We began with a single Point, which of course --- being itself a Point --- hasonly one terminal Point.One Point produces a Line with two terminal Points.One Line produces a Square with four terminal Points.Now you can give yourself the answer to your own question: 1, 2, 4, areevidently in Geometrical Progression. What is the next number?I. Eight.SPHERE. Exactly. The one Square produces aSomething-which-you-do-not-as-yet-know-a-name-for-but-which-we-call-a-Cubewith eight terminal Points. Now are you convinced?I. And has this Creature sides, as well as Angles or what you call terminalPoints''?SPHERE. Of course; and all according to Analogy. But, by the way, not what youcall 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 togenerate by the motion of my inside in an upward'' direction, and whom youcall a Cube?SPHERE. How can you ask? And you a mathematician! The side of anything isalways, if I may so say, one Dimension behind the thing. Consequently, asthere is no Dimension behind a Point, a Point has 0 sides; a Line, if I may sosay, has 2 sides (for the points of a Line may be called by courtesy, itssides); 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 Cubewhich you will generate will be bounded by six sides, that is to say, six ofyour insides. You see it all now, eh?Monster,'' I shrieked, be thou juggler, enchanter, dream, or devil, no morewill I endure thy mockeries. Either thou or I must perish.'' And saying thesewords I precipitated myself upon him.\chapter{How the Sphere, having in vaintried words, resorted to deeds}It was in vain. I brought my hardest right angle into violent collision withthe Stranger, pressing on him with a force sufficient to have destroyed anyordinary Circle: but I could feel him slowly and unarrestably slipping from mycontact; not edging to the right nor to the left, but moving somehow out ofthe world, and vanishing into nothing. Soon there was a blank. But still Iheard 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 forthe Gospel of the Three Dimensions, which I am allowed to preach once only ina 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 thingsthat you consider closed. For example, I see in yonder cupboard near which youare standing, several of what you call boxes (but like everything else inFlatland, they have no tops or bottom) full of money; I see also two tabletsof accounts. I am about to descend into that cupboard and to bring you one ofthose tablets. I saw you lock the cupboard half an hour ago, and I know youhave 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 haveit. Now I ascent 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. Therecould be no doubt --- it was the missing tablet.I groaned with horror, doubting whether I was not out of my sense; but theStranger continued: Surely you must now see that my explanation, and noother, 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, andlook down upon the insides of the things of which you only see the outsides.You could leave the Plane yourself, if you could but summon up the necessaryvolition. A slight upward or downward motion would enable you to see all thatI 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; nowI can see your neighbour the Hexagon and his family in their severalapartments; now I see the inside of the Theatre, ten doors off, from which theaudience is only just departing; and on the other side a Circle in his study,sitting at his books. Now I shall come back to you. And, as a crowning proof,what do you say to my giving you a touch, just the least touch, in yourstomach? It will not seriously injure you, and the slight pain you may suffercannot be compared with the mental benefit you will receive.''Before I could utter a word of remonstrance, I felt a shooting pain in myinside, and a demoniacal laugh seemed to issue from within me. A momentafterwards 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, Idon't know what will convince you. What say you?''My resolution was taken. It seemed intolerable that I should endure existencesubject to the arbitrary visitations of a Magician who could thus play trickswith one's very stomach. If only I could in any way manage to pin him againstthe wall till help came!Once more I dashed my hardest angle against him, at the same time alarming thewhole household by my cries for aid. I believe, at the moment of my onset, theStranger had sunk below our Plane, and really found difficulty in rising. Inany case he remained motionless, while I, hearing, as I thought, the sound ofsome help approaching, pressed against him with redoubled vigor, and continuedto shout for assistance.A convulsive shudder ran through the Sphere. This must not be,'' I thought Iheard him say: either he must listen to reason, or I must have recourse tothe last resource of civilization.'' Then, addressing me in a louder tone, hehurriedly exclaimed, Listen: no stranger must witness what you havewitnessed. Send your Wife back at once, before she enters the apartment. TheGospel of Three Dimensions must not be thus frustrated. Not thus must thefruits 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 --- intothe Land of Three Dimensions!''Fool! Madman! Irregular!'' I exclaimed; never will I release thee; thou shaltpay the penalty of thine impostures.''Ha! Is it come to this?'' thundered the Stranger: then meet your fate: out ofyour Plane you go. Once, twice, thrice! 'Tis done!''\chapter{How I came to Spaceland, and what I saw there}An unspeakable horror seized me. There was a darkness; then a dizzy, sickeningsensation 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 findvoice, I shrieked loud in agony, Either this is madness or it is Hell.'' Itis neither, calmly replied the voice of the Sphere, it is Knowledge; it isThree Dimensions: open your eye once again and try to look steadily.''I looked, and, behold, a new world! There stood before me, visiblyincorporate, all that I had before inferred, conjectured, dreamed, of perfectCircular beauty. What seemed the centre of the Stranger's form lay open to myview: yet I could see no heart, lungs, nor arteries, only a beautifulharmonious Something --- for which I had no words; but you, my Readers inSpaceland, would call it the surface of the Sphere.Prostrating myself mentally before my Guide, I cried, How it is, O divineideal of consummate loveliness and wisdom that I see thy inside, and yetcannot discern thy heart, thy lungs, thy arteries, thy liver?'' What you thinkyou see, you see not,'' he replied; it is not giving to you, nor to any otherBeing, to behold my internal parts. I am of a different order of Beings fromthose in Flatland. Where I a Circle, you could discern my intestines, but I ama 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 aSquare, so the outside of a Sphere represents the appearance of a Circle.''Bewildered though I was by my Teacher's enigmatic utterance, I no longerchafed against it, but worshipped him in silent adoration. He continued, withmore mildness in his voice. Distress not yourself if you cannot at firstunderstand the deeper mysteries of Spaceland. By degrees they will dawn uponyou. 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 shew you thatwhich you have often reasoned and thought about, but never seen with the senseof sight --- a visible angle.'' Impossible!'' I cried; but, the Sphere leadingthe 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.''\begin{center}\includegraphics[trim=20mm 0mm 20mm 0mm,scale=0.7]{fig9}\end{center}I looked below, and saw with my physical eye all that domestic individualitywhich I had hitherto merely inferred with the understanding. And how poor andshadowy was the inferred conjecture in comparison with the reality which I nowbehold! My four Sons calmly asleep in the North-Western rooms, my two orphanGrandsons to the South; the Servants, the Butler, my Daughter, all in theirseveral apartments. Only my affectionate Wife, alarmed by my continuedabsence, 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 hisroom, and under pretext of ascertaining whether I had fallen somewhere in afaint, was prying into the cabinet in my study. All this I could now see, notmerely infer; and as we came nearer and nearer, I could discern even thecontents of my cabinet, and the two chests of gold, and the tablets of whichthe 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 yourWife,'' said my Guide: she will not be long left in anxiety; meantime, let ustake a survey of Flatland.''Once more I felt myself rising through space. It was even as the Sphere hadsaid. The further we receded from the object we beheld, the larger became thefield of vision. My native city, with the interior of every house and everycreature 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 thehills, were bared before me.Awestruck at the sight of the mysteries of the earth, thus unveiled before myunworthy eye, I said to my Companion, Behold, I am become as a God. For thewise 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 inthe voice of my Teacher as he made answer: it is so indeed? Then the verypick-pockets and cut-throats of my country are to be worshipped by your wisemen as being Gods: for there is not one of them that does not see as much asyou see now. But trust me, your wise men are wrong.''I. Then is omnividence the attribute ofothers besides Gods?SPHERE. I do not know. But, if a pick-pocket or a cut-throat of our countrycan see everything that is in your country, surely that is no reason why thepick-pocket or cut-throat should be accepted by you as a God. Thisomnividence, as you call it --- it is not a common word in Spaceland --- does itmake you more just, more merciful, less selfish, more loving? Not in theleast. Then how does it make you more divine?I. More merciful, more loving!'' But these are the qualities of women! And weknow that a Circle is a higher Being than a Straight Line, in so far asknowledge and wisdom are more to be esteemed than mere affection.SPHERE. It is not for me to classify human faculties according to merit. Yetmany of the best and wisest in Spaceland think more of the affections than ofthe understanding, more of your despised Straight Lines than of your belaudedCircles. But enough of this. Look yonder. Do you know that building?I looked, and afar off I saw an immense Polygonal structure, in which Irecognized the General Assembly Hall of the States of Flatland, surrounded bydense lines of Pentagonal buildings at right angles to each other, which Iknew to be streets; and I perceived that I was approaching the greatMetropolis.Here we descend,'' said my Guide. It was now morning, the first hour of thefirst day of the two thousandth year of our era. Acting, as was their wont, instrict accordance with precedent, the highest Circles of the realm weremeeting in solemn conclave, as they had met on the first hour of the first dayof 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 oncerecognized as my brother, a perfectly Symmetrical Square, and the Chief Clerkof the High Council. It was found recorded on each occasion that: Whereas theStates had been troubled by divers ill-intentioned persons pretending to havereceived revelations from another World, and professing to producedemonstrations whereby they had instigated to frenzy both themselves andothers, it had been for this cause unanimously resolved by the Grand Councilthat on the first day of each millenary, special injunctions be sent to thePrefects in the several districts of Flatland, to make strict search for suchmisguided persons, and without formality of mathematical examination, todestroy all such as were Isosceles of any degree, to scourge and imprison anyregular Triangle, to cause any Square or Pentagon to be sent to the districtAsylum, and to arrest any one of higher rank, sending him straightway to theCapital to be examined and judged by the Council.''You hear your fate,'' said the Sphere to me, while the Council was passing forthe third time the formal resolution. Death or imprisonment awaits theApostle of the Gospel of Three Dimensions.'' Not so,'' replied I, the matteris now so clear to me, the nature of real space so palpable, that methinks Icould make a child understand it. Permit me but to descend at this moment andenlighten them.'' Not yet,'' said my Guide, the time will come for that.Meantime I must perform my mission. Stay thou there in thy place.'' Sayingthese 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, asthe Sphere's circular section widened before them. But on a sign from thepresiding Circle --- who shewed not the slightest alarm or surprise --- sixIsosceles of a low type from six different quarters rushed upon the Sphere.We have him,'' they cried; No; yes; we have him still! he's going! he'sgone!''My Lords,'' said the President to the Junior Circles of the Council, there isnot the slightest need for surprise; the secret archives, to which I alonehave access, tell me that a similar occurrence happened on the last twomillennial commencements. You will, of course, say nothing of these triflesoutside the Cabinet.''Raising his voice, he now summoned the guards. Arrest the policemen; gagthem. You know your duty.'' After he had consigned to their fate the wretchedpolicemen --- ill-fated and unwilling witnesses of a State-secret which theywere not to be permitted to reveal --- he again addressed the Counsellors. MyLords, the business of the Council being concluded, I have only to wish you ahappy New Year.'' Before departing, he expressed, at some length, to the Clerk,my excellent but most unfortunate brother, his sincere regret that, inaccordance with precedent and for the sake of secrecy, he must condemn him toperpetual imprisonment, but added his satisfaction that, unless some mentionwere made by him of that day's incident, his life would be spared.\chapter{How, though the Sphereshowed me other mysteries of Spaceland, I still desire more; and what came ofit} When I saw my poor brother led away to imprisonment, I attempted to leap downinto the Council Chamber, desiring to intercede on his behalf, or at least bidhim farewell. But I found that I had no motion of my own. I absolutelydepended on the volition of my Guide, who said in gloomy tones, Heed not thybrother; haply thou shalt have ample time hereafter to condole with him.Follow me.'' \begin{center}\includegraphics[trim=20mm 0mm 20mm 0mm,scale=0.6]{fig10}\end{center}Once more we ascended into space. Hitherto,'' said the Sphere, I have shewnyou naught save Plane Figures and their interiors. Now I must introduce you toSolids, and reveal to you the plan upon which they are constructed. Beholdthis multitude of moveable square cards. See, I put one on another, not, asyou supposed, Northward of the other, but on the other. Now a second, now athird. See, I am building up a Solid by a multitude of Squares parallel to oneanother. Now the Solid is complete, being as high as it is long and broad, andwe call it a Cube.''Pardon me, my Lord,'' replied I; but to my eye the appearance is as of anIrregular Figure whose inside is laid open to view; in other words, methinks Isee no Solid, but a Plane such as we infer in Flatland; only of anIrregularity which betokens some monstrous criminal, so that the very sight ofit is painful to my eyes.''True,'' said the Sphere; it appears to you a Plane, because you are notaccustomed to light and shade and perspective; just as in Flatland a Hexagonwould 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 wasindeed no Plane, but a Solid; and that he was endowed with six plane sides andeight terminal points called solid angles; and I remembered the saying of theSphere 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 aCreature as I could in some sense be called the Progenitor of so illustriousan offspring.But still I could not fully understand the meaning of what my Teacher had toldme concerning light'' and shade'' and perspective''; and I did not hesitate toput my difficulties before him.Were I to give the Sphere's explanation of these matters, succinct and clearthough it was, it would be tedious to an inhabitant of Space, who knows thesethings already. Suffice it, that by his lucid statements, and by changing theposition of objects and lights, and by allowing me to feel the several objectsand even his own sacred Person, he at last made all things clear to me, sothat I could now readily distinguish between a Circle and a Sphere, a PlaneFigure and a Solid.This was the Climax, the Paradise, of my strange eventful History. HenceforthI have to relate the story of my miserable Fall: --- most miserable, yet surelymost undeserved! For why should the thirst for knowledge be aroused, only tobe disappointed and punished? My volition shrinks from the painful task ofrecalling my humiliation; yet, like a second Prometheus, I will endure thisand worse, if by any means I may arouse in the interiors of Plane and SolidHumanity a spirit of rebellion against the Conceit which would limit ourDimensions to Two or Three or any number short of Infinity. Away then with allpersonal considerations! Let me continue to the end, as I began, withoutfurther digressions or anticipations, pursuing the plain path of dispassionateHistory. The exact facts, the exact words, --- and they are burnt in upon mybrain, --- shall be set down without alteration of an iota; and let my Readersjudge between me and Destiny.The Sphere would willingly have continued his lessons by indoctrinating me inthe conformation of all regular Solids, Cylinders, Cones, Pyramids,Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured tointerrupt him. Not that I was wearied of knowledge. On the contrary, Ithirsted for yet deeper and fuller draughts than he was offering to me.Pardon me,'' said I, O Thou Whom I must no longer address as the Perfectionof all Beauty; but let me beg thee to vouchsafe thy servant a sight of thineinterior.''SPHERE. My what?I. Thine interior: thy stomach, thy intestines.SPHERE. Whence this ill-timed impertinent request? And what mean you by sayingthat 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. Asyou yourself, superior to all Flatland forms, combine many Circles in One, sodoubtless there is One above you who combines many Spheres in One SupremeExistence, surpassing even the Solids of Spaceland. And even as we, who arenow in Space, look down on Flatland and see the insides of all things, so of acertainty there is yet above us some higher, purer region, whither thou dostsurely purpose to lead me --- O Thou Whom I shall always call, everywhere and inall Dimensions, my Priest, Philosopher, and Friend --- some yet more spaciousSpace, some more dimensionable Dimensionality, from the vantage-ground ofwhich we shall look down together upon the revealed insides of Solid things,and where thine own intestines, and those of thy kindred Spheres, will lieexposed to the view of the poor wandering exile from Flatland, to whom so muchhas already been vouchsafed.SPHERE. Pooh! Stuff! Enough of this trifling! The time is short, and muchremains to be done before you are fit to proclaim the Gospel of ThreeDimensions to your blind benighted countrymen in Flatland.I. Nay, gracious Teacher, deny me not what I know it is in thy power toperform. Grant me but one glimpse of thine interior, and I am satisfied forever, remaining henceforth thy docile pupil, thy unemacipable slave, ready toreceive 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 wouldshew you what you wish if I could; but I cannot. Would you have me turn mystomach inside out to oblige you?I. But my Lord has shewn me the intestines of all my countrymen in the Land ofTwo Dimensions by taking me with him into the Land of Three. What thereforemore easy than now to take his servant on a second journey into the blessedregion of the Fourth Dimension, where I shall look down with him once moreupon this land of Three Dimensions, and see the inside of everythree-dimensioned house, the secrets of the solid earth, the treasures of themines of Spaceland, and the intestines of every solid living creature, eventhe 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 utterlyinconceivable.I. Not inconceivable, my Lord, to me, and therefore still less inconceivableto my Master. Nay, I despair not that, even here, in this region of ThreeDimensions, 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 haveopened the eyes of his blind servant to the invisible presence of a ThirdDimension, though I saw it not.Let me recall the past. Was I not taught below that when I saw a Line andinferred a Plane, I in reality saw a Third unrecognized Dimension, not thesame as brightness, called height''? And does it not now follow that, in thisregion, when I see a Plane and infer a Solid, I really see a Fourthunrecognized Dimension, not the same as colour, but existent, thoughinfinitesimal and incapable of measurement?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 therevelations 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 ofFlatland, though that poor puny Lineland Monarch could neither turn to leftnor right to discern it, and just as there was close at hand, and touching myframe, the land of Three Dimensions, though I, blind senseless wretch, had nopower to touch it, no eye in my interior to discern it, so of a surety thereis 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 forgottenwhat he himself imparted to his servant?In One Dimension, did not a moving Point produce a Line with two terminalpoints?In Two Dimensions, did not a moving Line produce a Square with four terminalpoints?In Three Dimensions, did not a moving Square produce --- did not this eye ofmine behold it --- that blessed Being, a Cube, with eight terminal points?And in Four Dimensions shall not a moving Cube --- alas, for Analogy, and alasfor the Progress of Truth, if it be not so --- shall not, I say, the motion of adivine Cube result in a still more divine Organization with sixteen terminalpoints?Behold the infallible confirmation of the Series, 2, 4,8, 16: is not this a Geometrical Progression? Is not this --- if I might quotemy Lord's own words --- strictly according to Analogy''?Again, was I not taught by my Lord that as in a Line there are two boundingPoints, and in a Square there are four bounding Lines, so in a Cube there mustbe six bounding Squares? Behold once more the confirming Series, 2, 4, 6: isnot this an Arithmetical Progression? And consequently does it not ofnecessity follow that the more divine offspring of the divine Cube in the Landof Four Dimensions, must have 8 bounding Cubes: and is not this also, as myLord has taught me to believe, strictly according to Analogy''?O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, notknowing the facts; and I appeal to your Lordship to confirm or deny my logicalanticipations. If I am wrong, I yield, and will no longer demand a FourthDimension; 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 countrymenalso have witnessed the descent of Beings of a higher order than their own,entering closed rooms, even as your Lordship entered mine, without the openingof doors or windows, and appearing and vanishing at will? On the reply to thisquestion 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 asto the facts. And even granting the facts, they explain them in differentways. And in any case, however great may be the number of differentexplanations, no one has adopted or suggested the theory of a FourthDimension. Therefore, pray have done with this trifling, and let us return tobusiness.I. I was certain of it. I was certain that my anticipations would befulfilled. 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 --- andhave returned --- no one knows whither --- have they also contracted theirsections and vanished somehow into that more Spacious Space, whither I nowentreat you to conduct me?Sphere (moodily). They have vanished, certainly --- if they ever appeared. Butmost people say that these visions arose from the thought --- you will notunderstand me --- from the brain; from the perturbed angularity of the Seer.I. Say they so? Oh, believe them not. Or if it indeed be so, that this otherSpace is really Thoughtland, then take me to that blessed Region where I inThought shall see the insides of all solid things. There, before my ravishedeye, a Cube moving in some altogether new direction, but strictly according toAnalogy, so as to make every particle of his interior pass through a new kindof Space, with a wake of its own --- shall create a still more perfectperfection than himself, with sixteen terminal Extra-solid angles, and Eightsolid Cubes for his Perimeter. And once there, shall we stay our upwardcourse? In that blessed region of Four Dimensions, shall we linger at thethreshold of the Fifth, and not enter therein? Ah, no! Let us rather resolvethat our ambition shall soar with our corporal ascent. Then, yielding to ourintellectual onset, the gates of the Six Dimension shall fly open; after thata Seventh, and then an Eighth ---How long I should have continued I know not. In vain did the Sphere, in hisvoice of thunder, reiterate his command of silence, and threaten me with thedirest penalties if I persisted. Nothing could stem the flood of my ecstaticaspirations. Perhaps I was to blame; but indeed I was intoxicated with therecent draughts of Truth to which he himself had introduced me. However, theend was not long in coming. My words were cut short by a crash outside, and asimultaneous crash inside me, which impelled me through space with a velocitythat precluded speech. Down! down! down! I was rapidly descending; and I knewthat return to Flatland was my doom. One glimpse, one last andnever-to-be-forgotten glimpse I had of that dull level wilderness --- which wasnow to become my Universe again --- spread out before my eye. Then a darkness.Then a final, all-consummating thunder-peal; and, when I came to myself, I wasonce more a common creeping Square, in my Study at home, listening to thePeace-Cry of my approaching Wife.\chapter{How the Sphere encouraged me in a Vision}Although I had less than a minute for reflection, I felt, by a kind ofinstinct, that I must conceal my experiences from my Wife. Not that Iapprehended, at the moment, any danger from her divulging my secret, but Iknew that to any Woman in Flatland the narrative of my adventures must needsbe unintelligible. So I endeavoured to reassure her by some story, inventedfor the occasion, that I had accidentally fallen through the trap-door of thecellar, and had there lain stunned.The Southward attraction in our country is so slight that even to a Woman mytale necessarily appeared extraordinary and well-nigh incredible; but my Wife,whose good sense far exceeds that of the average of her Sex, and who perceivedthat I was unusually excited, did not argue with me on the subject, butinsisted that I was ill and required repose. I was glad of an excuse forretiring to my chamber to think quietly over what had happened. When I was atlast by myself, a drowsy sensation fell on me; but before my eyes closed Iendeavoured to reproduce the Third Dimension, and especially the process bywhich a Cube is constructed through the motion of a Square. It was not soclear as I could have wished; but I remembered that it must be Upward, andyet not Northward,'' and I determined steadfastly to retain these words as theclue which, if firmly grasped, could not fail to guide me to the solution. Somechanically 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 theSphere, whose lustrous hue betokened that he had exchanged his wrath againstme for perfectly placability. We were moving together towards a bright butinfinitesimally small Point, to which my Master directed my attention. As weapproached, methought there issued from it a slight humming noise as from oneof your Spaceland bluebottles, only less resonant by far, so slight indeedthat even in the perfect stillness of the Vacuum through which we soared, thesound reached not our ears till we checked our flight at a distant from it ofsomething under twenty human diagonals.Look yonder,'' said my Guide, in Flatland thou hast lived; of Lineland thouhast 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 downwardto the lowest depth of existence, even to the realm of Pointland, the Abyss ofNo dimensions.Behold yon miserable creature. That Point is a Being like ourselves, butconfined to the non-dimensional Gulf. He is himself his own World, his ownUniverse; of any other than himself he can form no conception; he knows notLength, nor Breadth, nor Height, for he has had no experience of them; he hasno cognizance even of the number Two; nor has he a thought of Plurality; forhe is himself his One and All, being really Nothing. Yet mark his perfectself-contentment, and hence learn his lesson, that to be self-contented is tobe vile and ignorant, and that to aspire is better than to be blindly andimpotently 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; andthere 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 babyishpeople who cannot distinguish themselves from the world, speak of themselvesin the Third Person? But hush!''It fills all Space,'' continued the little soliloquizing Creature, and whatIt fills, It is. What It thinks, that It utters; and what It utters, that Ithears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; itis the One, and yet the All in All. Ah, the happiness, ah, the happiness ofBeing!''Can you not startle the little thing out of its complacency?'' said I. Tellit what it really is, as you told me; reveal to it the narrow limitations ofPointland, and lead it up to something higher.'' That is no easy task,'' saidmy 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 saidenough,'' interrupted the Sphere, now listen, and mark the effect of yourharangue on the King of Pointland.''The lustre of the Monarch, who beamed more brightly than ever upon hearing mywords, shewed clearly that he retained his complacency; and I had hardlyceased 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! Sweetrebellion stirred up to result in triumph! Ah, the divine creative power ofthe All in One! Ah, the joy, the joy of Being!''You see,'' said my Teacher, how little your words have done. So far as theMonarch understand them at all, he accepts them as his own --- for he cannotconceive of any other except himself --- and plumes himself upon the variety ofIts Thought' as an instance of creative Power. Let us leave this God ofPointland to the ignorant fruition of his omnipresence and omniscience:nothing that you or I can do can rescue him from his self-satisfaction.''After this, as we floated gently back to Flatland, I could hear the mild voiceof my Companion pointing the moral of my vision, and stimulating me to aspire,and to teach others to aspire. He had been angered at first --- he confessed ---by my ambition to soar to Dimensions above the Third; but, since then, he hadreceived fresh insight, and he was not too proud to acknowledge his error to aPupil. Then he proceeded to initiate me into mysteries yet higher than those Ihad witnessed, shewing me how to construct Extra-Solids by the motion ofSolids, and Double Extra-Solids by the motion of Extra-Solids, and allstrictly according to Analogy,'' all by methods so simple, so easy, as to bepatent even to the Female Sex.?\chapter{How I tried to teach theTheory of Three Dimensions to my Grandson, and with what success}I awoke rejoicing, and began to reflect on the glorious career before me. Iwould go forth, methought, at once, and evangelize the whole of Flatland. Evento Women and Soldiers should the Gospel of Three Dimensions be proclaimed. Iwould begin with my Wife.Just as I had decided on the plan of my operations, I heard the sound of manyvoices in the street commanding silence. Then followed a louder voice. It wasa herald's proclamation. Listening attentively, I recognized the words of theResolution of the Council, enjoining the arrest, imprisonment, or execution ofany one who should pervert the minds of people by delusions, and by professingto have received revelations from another World.I reflected. This danger was not to be trifled with. It would be better toavoid it by omitting all mention of my Revelation, and by proceeding on thepath of Demonstration --- which after all, seemed so simple and so conclusivethat nothing would be lost by discarding the former means. Upward, notNorthward'' --- was the clue to the whole proof. It had seemed to me fairly clearbefore I fell asleep; and when I first awoke, fresh from my dream, it hadappeared as patent as Arithmetic; but somehow it did not seem to me quite soobvious now. Though my Wife entered the room opportunely at just that moment,I decided, after we had exchanged a few words of commonplace conversation, notto begin with her.My Pentagonal Sons were men of character and standing, and physicians of nomean reputation, but not great in mathematics, and, in that respect, unfit formy purpose. But it occurred to me that a young and docile Hexagon, with amathematical turn, would be a most suitable pupil. Why therefore not make myfirst experiment with my little precocious Grandson, whose casual remarks onthe meaning of three-to-the-third had met with the approval of the Sphere?Discussing the matter with him, a mere boy, I should be in perfect safety; forhe would know nothing of the Proclamation of the Council; whereas I could notfeel sure that my Sons --- so greatly did their patriotism and reverence for theCircles predominate over mere blind affection --- might not feel compelled tohand me over to the Prefect, if they found me seriously maintaining theseditious heresy of the Third Dimension.But the first thing to be done was to satisfy in some way the curiosity of myWife, who naturally wished to know something of the reasons for which theCircle had desired that mysterious interview, and of the means by which he hadentered the house. Without entering into the details of the elaborate accountI gave her, --- an account, I fear, not quite so consistent with truth as myReaders in Spaceland might desire, --- I must be content with saying that Isucceeded at last in persuading her to return quitely to her household dutieswithout eliciting from me any reference to the World of Three Dimensions. Thisdone, I immediately sent for my Grandson; for, to confess the truth, I feltthat all that I had seen and heard was in some strange way slipping away fromme, like the image of a half- grasped, tantalizing dream, and I longed toessay my skill in making a first disciple.When my Grandson entered the room I carefully secured the door. Then, sittingdown by his side and taking our mathematical tablets, --- or, as you would callthem, Lines --- I told him we would resume the lesson of yesterday. I taught himonce more how a Point by motion in One Dimension produces a Line, and how astraight Line in Two Dimensions produces a Square. After this, forcing alaugh, I said, And now, you scamp, you wanted to make believe that a Squaremay 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 thestreet proclaiming the Resolution of the Council. Young though he was, myGrandson --- who was unusually intelligent for his age, and bred up in perfectreverence for the authority of the Circles --- took in the situation with anacuteness for which I was quite unprepared. He remained silent till the lastwords of the Proclamation had died away, and then, bursting into tears, DearGrandpapa,'' he said, that was only my fun, and of course I meant nothing atall by it; and we did not know anything then about the new Law; and I don'tthink I said anything about the Third Dimension; and I am sure I did not sayone word about Upward, not Northward,' for that would be such nonsense, youknow. How could a thing move Upward, and not Northward? Upward and notNorthward! Even if I were a baby, I could not be so absurd as that. How sillyit is! Ha! ha! ha!''Not at all silly,'' said I, losing my temper; here for example, I take thisSquare,'' and, at the word, I grasped a moveable Square, which was lying athand --- and I move it, you see, not Northward but --- yes, I move it Upward ---that is to say, Northward but I move it somewhere --- not exactly like this, butsomehow ---'' Here I brought my sentence to an inane conclusion, shaking theSquare about in a purposeless manner, much to the amusement of my Grandson,who burst out laughing louder than ever, and declared that I was not teachinghim, but joking with him; and so saying he unlocked the door and ran out ofthe room. Thus ended my first attempt to convert a pupil to the Gospel ofThree Dimensions.\chapter{How I then tried to diffuse the Theory of Three Dimensions by othermeans, and of the result} My failure with my Grandson did not encourage me to communicate my secret toothers of my household; yet neither was I led by it to despair of success.Only I saw that I must not wholly rely on the catch-phrase, Upward, notNorthward,'' but must rather endeavour to seek a demonstration by settingbefore the public a clear view of the whole subject; and for this purpose itseemed necessary to resort to writing.So I devoted several months in privacy to the composition of a treatise on themysteries of Three Dimensions. Only, with the view of evading the Law, ifpossible, I spoke not of a physical Dimension, but of a Thoughtland whence, intheory, a Figure could look down upon Flatland and see simultaneously theinsides of all things, and where it was possible that there might be supposedto exist a Figure environed, as it were, with six Squares, and containingeight terminal Points. But in writing this book I found myself sadly hamperedby the impossibility of drawing such diagrams as were necessary for mypurpose: for of course, in our country of Flatland, there are no tablets butLines, and no diagrams but Lines, all in one straight Line and onlydistinguishable by difference of size and brightness; so that, when I hadfinished my treatise (which I entitled, Through Flatland to Thoughtland'') Icould not feel certain that many would understand my meaning.Meanwhile my life was under a cloud. All pleasures palled upon me; all sightstantalized and tempted me to outspoken treason, because I could not butcompare 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 myclients and my own business to give myself to the contemplation of themysteries which I had once beheld, yet which I could impart to no one, andfound 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 aCube with my eye closed, but failed; and though I succeeded afterwards, I wasnot then quite certain (nor have I been ever afterwards) that I had exactlyrealized the original. This made me more melancholy than before, anddetermined me to take some step; yet what, I knew not. I felt that I wouldhave been willing to sacrifice my life for the Cause, if thereby I could haveproduced conviction. But if I could not convince my Grandson, how could Iconvince the highest and most developed Circles in the land?And yet at times my spirit was too strong for me, and I gave vent todangerous utterances. Already I was considered heterodox if not treasonable,and I was keenly alive to the danger of my position; nevertheless I could notat times refrain from bursting out into suspicious or half-seditiousutterances, even among the highest Polygonal or Circular society. When, forexample, the question arose about the treatment of those lunatics who saidthat they had received the power of seeing the insides of things, I wouldquote the saying of an ancient Circle, who declared that prophets and inspiredpeople are always considered by the majority to be mad; and I could not helpoccasionally dropping such expressions as the eye that discerns the interiorsof things,'' and the all-seeing land''; once or twice I even let fall theforbidden terms the Third and Fourth Dimensions.'' At last, to complete aseries of minor indiscretions, at a meeting of our Local Speculative Societyheld at the palace of the Prefect himself, --- some extremely silly personhaving read an elaborate paper exhibiting the precise reasons why Providencehas limited the number of Dimensions to Two, and why the attribute ofomnividence is assigned to the Supreme alone --- I so far forgot myself as togive 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 myreturn home, and of everything that I had seen and heard in fact or vision. Atfirst, indeed, I pretended that I was describing the imaginary experiences ofa fictitious person; but my enthusiasm soon forced me to throw off alldisguise, and finally, in a fervent peroration, I exhorted all my hearers todivest themselves of prejudice and to become believers in the Third Dimension.Need I say that I was at once arrested and taken before the Council?Next morning, standing in the very place where but a very few months ago theSphere had stood in my company, I was allowed to begin and to continue mynarration unquestioned and uninterrupted. But from the first I foresaw myfate; for the President, noting that a guard of the better sort of Policemenwas in attendance, of angularity little, if at all, under 55 degrees, orderedthem to be relieved before I began my defence, by an inferior class of 2 or 3degrees. I knew only too well what that meant. I was to be executed orimprisoned, and my story was to be kept secret from the world by thesimultaneous destruction of the officials who had heard it; and, this beingthe case, the President desired to substitute the cheaper for the moreexpensive victims.After I had concluded my defence, the President, perhaps perceiving that someof the junior Circles had been moved by evident earnestness, asked me twoquestions: ---\begin{enumerate} \item Whether I could indicate the direction which I meant when I used the words Upward, not Northward''? \item 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.\end{enumerate}The President replied that he quite concurred in my sentiment, and that Icould not do better. I must be sentenced to perpetual imprisonment; but if theTruth intended that I should emerge from prison and evangelize the world, theTruth might be trusted to bring that result to pass. Meanwhile I should besubjected to no discomfort that was not necessary to preclude escape, and,unless I forfeited the privilege by misconduct, I should be occasionallypermitted 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 theoccasional visits of my brother --- debarred from all companionship save that ofmy 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 theSphere manifested himself in the Council Chamber; he saw the Sphere's changingsections; he heard the explanation of the phenomena then give to the Circles.Since that time, scarcely a week has passed during seven whole years, withouthis hearing from me a repetition of the part I played in that manifestation,together with ample descriptions of all the phenomena in Spaceland, and thearguments for the existence of Solid things derivable from Analogy. Yet --- Itake shame to be forced to confess it --- my brother has not yet grasped thenature of Three Dimensions, and frankly avows his disbelief in the existenceof a Sphere.Hence I am absolutely destitute of converts, and, foraught that I can see, the millennial Revelation has been made to me fornothing. Prometheus up in Spaceland was bound for bringing down fire formortals, but I --- poor Flatland Prometheus --- lie here in prison for bringingdown nothing to my countrymen. Yet I exist in the hope that these memoirs, insome manner, I know not how, may find their way to the minds of humanity inSome Dimension, and may stir up a race of rebels who shall refuse to beconfined to limited Dimensionality.That is the hope of my brighter moments. Alas, it isnot always so. Heavily weights on me at times the burdensome reflection that Icannot 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 themartyrdom which I endure for the cause of Truth that there are seasons ofmental weakness, when Cubes and Spheres flit away into the background ofscarce-possible existences; when the Land of Three Dimensions seems almost asvisionary as the Land of One or None; nay, when even this hard wall that barsme from my freedom, these very tablets on which I am writing, and all thesubstantial realities of Flatland itself, appear no better than the offspringof a diseased imagination, or the baseless fabric of a dream.\begin{center}\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig11}\end{center}\end{document}`
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