commit aff2bc2e0174617a3bd13301e94764ff777d062e Author: Kenneth Odle Date: Sun Jun 18 11:38:27 2023 -0400 Initial commit diff --git a/README.MD b/README.MD new file mode 100755 index 0000000..288c8a4 --- /dev/null +++ b/README.MD @@ -0,0 +1,13 @@ +Flatland +======== +Although Edwin A. Abbott's essay "Flatland" is readily available on the +internet, I failed to find a nicely typeset version. At this time, +the rendering quality of internet browsers doesn't come anywhere near the +quality of a nice book. + +For these reasons, as well as a healthy portion of boredom, I made the version +of "Flatland" you are currently reading, in LaTeX. The LaTeX source can be +found on https://github.com/Ivesvdf/flatland (feel free to make issues +or send me pull requests when you find bugs or have improvements). It should +also be fairly easy --- a matter of minutes --- to produce a book-sized +version, ready to send to a printer. diff --git a/build/main.pdf b/build/main.pdf new file mode 100644 index 0000000..753f649 Binary files /dev/null and b/build/main.pdf differ diff --git a/fig1.png b/fig1.png new file mode 100755 index 0000000..34f782f Binary files /dev/null and b/fig1.png differ diff --git a/fig10.png b/fig10.png new file mode 100755 index 0000000..3f7a996 Binary files /dev/null and b/fig10.png differ diff --git a/fig11.png b/fig11.png new file mode 100755 index 0000000..5fe458d Binary files /dev/null and b/fig11.png differ diff --git a/fig2.png b/fig2.png new file mode 100755 index 0000000..0bc3447 Binary files /dev/null and b/fig2.png differ diff --git a/fig3.png b/fig3.png new file mode 100755 index 0000000..8181dbb Binary files /dev/null and b/fig3.png differ diff --git a/fig4.png b/fig4.png new file mode 100755 index 0000000..0da123e Binary files /dev/null and b/fig4.png differ diff --git a/fig5.png b/fig5.png new file mode 100755 index 0000000..4c40b65 Binary files /dev/null and b/fig5.png differ diff --git a/fig6.png b/fig6.png new file mode 100755 index 0000000..542f2c9 Binary files /dev/null and b/fig6.png differ diff --git a/fig7.png b/fig7.png new file mode 100755 index 0000000..e74d905 Binary files /dev/null and b/fig7.png differ diff --git a/fig8.png b/fig8.png new file mode 100755 index 0000000..1f1ff7d Binary files /dev/null and b/fig8.png differ diff --git a/fig9.png b/fig9.png new file mode 100755 index 0000000..f86e44f Binary files /dev/null and b/fig9.png differ diff --git a/flatland_cover.png b/flatland_cover.png new file mode 100755 index 0000000..4817c15 Binary files /dev/null and b/flatland_cover.png differ diff --git a/main.tex b/main.tex new file mode 100755 index 0000000..ef76b70 --- /dev/null +++ b/main.tex @@ -0,0 +1,3374 @@ +\title{Flatland} \author{Edwin A. Abbott} +\date{\today} + +\documentclass[10pt, twoside]{report} +\usepackage[english]{babel} +\usepackage{graphicx} +\usepackage[T1]{fontenc} +\usepackage{hyperref} +\usepackage[fulloldstylenums, light]{kpfonts} + + +\usepackage{geometry} +\geometry{ + paperheight=8.5in, + paperwidth=5.5in, +% heightrounded, + margin=0.5in +} + +% Adjust the top and bottom margins +% http://kb.mit.edu/confluence/pages/viewpage.action?pageId=3907057 +\addtolength{\topmargin}{0.4in} +\addtolength{\textheight}{-0.75in} + + +\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] +\vspace*{4cm} +\includegraphics[scale=0.45]{flatland_cover} +% \end{vplace} + +\clearpage + +% \frontmatter +This work is in the public domain in the United States because it was +published before January 1, 1923. +The author died in 1926, so this work is also in the public domain in +countries and areas where the copyright term is the author's life plus 80 +years or less. This work may also be in the public domain in countries and +areas with longer native copyright terms that apply the rule of the shorter +term 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 +\vfill +Fifth edition +\chapter*{A Note From The Typesetter} +Although Edwin A. Abbott's essay ``Flatland'' is readily available on the +internet, I failed to find a nicely typeset version. At this time, +the rendering quality of internet browsers doesn't come anywhere near the +quality of a nice book. + +For these reasons, as well as a healthy portion of boredom, I made the version +of ``Flatland'' you are currently reading, in \LaTeX. The \LaTeX\ source can be +found on \url{https://github.com/Ivesvdf/flatland} (feel free to make issues +or send me pull requests when you find bugs or have improvements). It should +also be fairly easy --- a matter of minutes --- to produce a book-sized +version, ready to send to a printer. + +\vspace{1cm} + +\hfill Ives van der Flaas , 2011-09-20 + +\vspace{1.5cm} + +\textbf{An additional note:} + +I found this on github, where it included an A4 version in pdf. I decided to update this to a letterhalf page size, which is more common in North America. I also changed to the \texttt{kpfont} font package, and reduced the size of the type from 12 points to 10 points. + +Where necessary, I also adjusted the scaling of the images. + +The source can be found on \url{https://git.kjodle.net/kjodle/Flatland} + +\begin{flushright} +\textit{Kenneth John Odle} +\end{flushright} + + +\newpage \ \newpage + + +\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 when +he began to compose these Memoirs, I should not now need to represent him in +this preface, in which he desires, fully, to return his thanks to his readers +and critics in Spaceland, whose appreciation has, with unexpected celerity, +required a second edition of this work; secondly, to apologize for certain +errors and misprints (for which, however, he is not entirely responsible); +and, thirdly, to explain on or two misconceptions. But he is not the Square he +once was. Years of imprisonment, and the still heavier burden of general +incredulity and mockery, have combined with the thoughts and notions, and much +also of the terminology, which he acquired during his short stay in spaceland. +He has, therefore, requested me to reply in his behalf to two special +objections, one of an intellectual, the other of a moral nature. + +The first objection is, that a Flatlander, seeing a Line, sees something that +must be thick to the eye as well as long to the eye (otherwise it would not be +visible, if it had not some thickness); and consequently he ought (it is +argued) to acknowledge that his countrymen are not only long and broad, but +also (though doubtless to a very slight degree) thick or high. This objection +is plausible, and, to Spacelanders, almost irresistible, so that, I confess, +when I first heard it, I knew not what to reply. But my poor old friend's +answer appears to me completely to meet it. + +``I admit,'' said he --- when I mentioned to him this objection --- ``I admit the +truth of your critic's facts, but I deny his conclusions. It is true that we +have really in Flatland a Third unrecognized Dimension called `height,' just +as it also is true that you have really in Spaceland a Fourth unrecognized +Dimension, called by no name at present, but which I will call `extra-height.' +But we can no more take cognizance of our `height' than you can of your +`extra-height.' Even I --- who have been in Spaceland, and have had the +privilege of understanding for twenty-four hours the meaning of `height' --- +even I cannot now comprehend it, nor realize it by the sense of sight or by +any 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 and +infinitesimally thick (or high, whichever you like); consequently, there is +nothing in them to lead our minds to the conception of that Dimension. No +`delicate micrometer' --- as has been suggested by one too hasty Spaceland +critic --- would in the least avail us; for we should not know what to measure, +nor in what direction. When we see a Line, we see something that is long and +bright; brightness, as well as length, is necessary to the existence of a +Line; if the brightness vanishes, the Line is extinguished. Hence, all my +Flatland friends --- when I talk to them about the unrecognized Dimension which +is somehow visible in a Line --- say, `Ah, you mean brightness': and when I +reply, `No, I mean a real Dimension,' they at once retort, `Then measure it, +or tell us in what direction it extends'; and this silences me, for I can do +neither. Only yesterday, when the Chief Circle (in other words our High +Priest) came to inspect the State Prison and paid me his seventh annual visit, +and when for the seventh time he put me the question, `Was I any better?' I +tried to prove to him that he was `high', as well as long and broad, although +he did not know it. But what was his reply? `You say I am ``high''; measure my +``high-ness'' and I will believe you.'. What could I do? How could I meet his +challenge? I was crushed; and he left the room triumphant.'' + +``Does this still seem strange to you? Then put yourself in a similar position. +Suppose a person of the Fourth Dimension, condescending to visit you, were to +say, `Whenever you open your eyes, you see a Plane (which is of Two +Dimensions) and you infer a Solid (which is of Three); but in reality you also +see (though you do not recognize) a Fourth Dimension, which is not colour nor +brightness nor anything of the kind, but a true Dimension, although I cannot +point out to you its direction, nor can you possibly measure it.' What would +you say to such a visitor? Would not you have him locked up? Well, that is my +fate: and it is as natural for us Flatlanders to lock up a Square for +preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube +for preaching the Fourth. Alas, how strong a family likeness runs through +blind and persecuting humanity in all Dimensions! Points, Lines, Squares, +Cubes, Extra-Cubes --- we are all liable to the same errors, all alike the +Slavers of our respective Dimensional prejudices, as one of our Spaceland +poets has said --- `One touch of Nature makes all worlds akin.' '' \footnote{The +Author desires me to add, that the misconceptions of some of his critics on +this matter has induced him to insert in his dialogue with +the Sphere, certain remarks which have a bearing on the point in question and +which he had previously omitted as being tedious and unnecessary.} + +On this point the defence of the Square seems to me to be impregnable. I wish +I could say that his answer to the second (or moral) objection was equally +clear and cogent. It has been objected that he is a woman-hater; and as this +objection has been vehemently urged by those whom Nature's decree has +constituted the somewhat larger half of the Spaceland race, I should like to +remove it, so far as I can honestly do so. But the Square is so unaccustomed +to the use of the moral terminology of Spaceland that I should be doing him an +injustice if I were literally to transcribe his defence against this charge. +Acting, therefore, as his interpreter and summarizer, I gather that in the +course of an imprisonment of seven years he has himself modified his own +personal views, both as regards Women and as regards the Isosceles or Lower +Classes. Personally, he now inclines to the opinion of the Sphere that the +Straight Lines are in many important respects superior to the +Circles. But, writing as a Historian, he has identified himself (perhaps too +closely) with the views generally adopted by Flatland, and (as he has been +informed) even by Spaceland, Historians; in whose pages (until very recent +times) the destinies of Women and of the masses of mankind have seldom been +deemed worthy of mention and never of careful consideration. + +In a still more obscure passage he now desires to disavow the Circular or +aristocratic tendencies with which some critics have naturally credited him. +While doing justice to the intellectual power with which a few Circles have +for many generations maintained their supremacy over immense multitudes of +their countrymen, he believes that the facts of Flatland, speaking for +themselves without comment on his part, declare that Revolutions cannot always +be suppressed by slaughter, and that Nature, in sentencing the Circles to +infecundity, has condemned them to ultimate failure --- ``and herein,'' he says, +``I see a fulfillment of the great Law of all worlds, that while the wisdom of +Man thinks it is working one thing, the wisdom of Nature constrains it to work +another, and quite a different and far better thing.'' For the rest, he begs +his readers not to suppose that every minute detail in the daily life of +Flatland must needs correspond to some other detail in Spaceland; and yet he +hopes that, taken as a whole, his work may prove suggestive as well as +amusing, to those Spacelanders of moderate and modest minds who --- speaking of +that which is of the highest importance, but lies beyond experience --- decline +to say on the one hand, ``This can never be,'' and on the other hand, ``It must +needs be precisely thus, and we know all about it.'' + + + +%b\mainmatter + +\part{This World} +\chapter{Of the Nature of Flatland} I call our world Flatland, not because we +call it so, but to make its nature clearer to you, my happy readers, who are +privileged to live in Space. + +Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, +Pentagons, Hexagons, and other figures, instead of remaining fixed in their +places, move freely about, on or in the surface, but without the power of +rising above or sinking below it, very much like shadows --- only hard with +luminous edges --- and you will then have a pretty correct notion of my country +and countrymen. Alas, a few years ago, I should have said ``my universe'': but +now my mind has been opened to higher views of things. + +In such a country, you will perceive at once that it is impossible that there +should be anything of what you call a ``solid'' kind; but I dare say you will +suppose that we could at least distinguish by sight the Triangles, Squares, +and other figures, moving about as I have described them. On the contrary, we +could see nothing of the kind, not at least so as to distinguish one figure +from another. Nothing was visible, nor could be visible, to us, except +Straight Lines; and the necessity of this I will speedily demonstrate. + +Place a penny on the middle of one of your tables in Space; and leaning over +it, look down upon it. It will appear a circle. + +But now, drawling back to the edge of the table, gradually lower your eye +(thus bringing yourself more and more into the condition of the inhabitants of +Flatland), and you will find the penny becoming more and more oval to your +view, and at last when you have placed your eye exactly on the edge of the +table (so that you are, as it were, actually a Flatlander) the penny will then +have ceased to appear oval at all, and will have become, so far as you can +see, a straight line. + +The same thing would happen if you were to treat in the same way a Triangle, +or a Square, or any other figure cut out of pasteboard. As soon as you look at +it with your eye on the edge of the table, you will find that it ceases to +appear to you as a figure, and that it becomes in appearance a straight line. +Take for example an equilateral Triangle --- who represents with us a Tradesman +of the respectable class. Figure 1 represents the Tradesman as you would see +him while you were bending over him from above; figures 2 and 3 represent the +Tradesman, as you would see him if your eye were close to the level, or all +but on the level of the table; and if your eye were quite on the level of the +table (and that is how we see him in Flatland) you would see nothing but a +straight line. + +\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig1} + +When I was in Spaceland I heard that your sailors have very similar +experiences while they traverse your seas and discern some distant island or +coast lying on the horizon. The far-off land may have bays, forelands, angles +in and out to any number and extent; yet at a distance you see none of these +(unless indeed your sun shines bright upon them revealing the projections and +retirements by means of light and shade), nothing but a grey unbroken line +upon the water. + +Well, that is just what we see when one of our triangular or other +acquaintances 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 to +the sight that you have in Spaceland. If our friend comes closer to us we see +his line becomes larger; if he leaves us it becomes smaller; but still he +looks 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 able +to distinguish our friends from one another: but the answer to this very +natural question will be more fitly and easily given when I come to describe +the inhabitants of Flatland. For the present let me defer this subject, and +say a word or two about the 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 a +method of our own. By a Law of Nature with us, there is a constant attraction +to the South; and, although in temperate climates this is very slight --- so +that even a Woman in reasonable health can journey several furlongs northward +without much difficulty --- yet the hampering effect of the southward attraction +is quite sufficient to serve as a compass in most parts of our earth. +Moreover, the rain (which falls at stated intervals) coming always from the +North, is an additional assistance; and in the towns we have the guidance of +the houses, which of course have their side-walls running for the most part +North and South, so that the roofs may keep off the rain from the North. In +the country, where there are no houses, the trunks of the trees serve as some +sort of guide. Altogether, we have not so much difficulty as might be expected +in determining our bearings. + +Yet in our more temperate regions, in which the southward attraction is hardly +felt, walking sometimes in a perfectly desolate plain where there have been no +houses nor trees to guide me, I have been occasionally compelled to remain +stationary for hours together, waiting till the rain came before continuing my +journey. On the weak and aged, and especially on delicate Females, the force +of attraction tells much more heavily than on the robust of the Male Sex, so +that it is a point of breeding, if you meet a Lady in the street, always to +give her the North side of the way --- by no means an easy thing to do always at +short notice when you are in rude health and in a climate where it is +difficult to tell your North from your South. + +Windows there are none in our houses: for the light comes to us alike in our +homes and out of them, by day and by night, equally at all times and in all +places, whence we know not. It was in old days, with our learned men, an +interesting and oft-investigate question, ``What is the origin of light?'' and +the solution of it has been repeatedly attempted, with no other result than to +crowd our lunatic asylums with the would-be solvers. Hence, after fruitless +attempts to suppress such investigations indirectly by making them liable to a +heavy tax, the Legislature, in comparatively recent times, absolutely +prohibited them. I --- alas, I alone in Flatland --- know now only too well the +true solution of this mysterious problem; but my knowledge cannot be made +intelligible to a single one of my countrymen; and I am mocked at --- I, the +sole possessor of the truths of Space and of the theory of the introduction of +Light from the world of three Dimensions --- as if I were the maddest of the +mad! 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 or +pentagonal, 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 a +small door for the Women; on the West a much larger one for the Men; the South +side 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 angles +of a Square (and still more those of an equilateral Triangle,) being much more +pointed than those of a Pentagon, and the lines of inanimate objects (such as +houses) being dimmer than the lines of Men and Women, it follows that there is +no little danger lest the points of a square of triangular house residence +might do serious injury to an inconsiderate or perhaps absentminded traveller +suddenly running against them: and therefore, as early as the eleventh century +of our era, triangular houses were universally forbidden by Law, the only +exceptions being fortifications, powder-magazines, barracks, and other state +buildings, which is not desirable that the general public should approach +without circumspection. + +At this period, square houses were still everywhere permitted, though +discouraged by a special tax. But, about three centuries afterwards, the Law +decided that in all towns containing a population above ten thousand, the +angle of a Pentagon was the smallest house-angle that could be allowed +consistently with the public safety. The good sense of the community has +seconded the efforts of the Legislature; and now, even in the country, the +pentagonal construction has superseded every other. It is only now and then in +some very remote and backward agricultural district that an antiquarian may +still discover a square house. + + + + + + + + + + + + + +\chapter{Concerning the Inhabitants of Flatland} +The greatest length or breadth of a full grown inhabitant of Flatland may be +estimated at about eleven of your inches. Twelve inches may be regarded as a +maximum. + +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 not +exceeding half an inch) that they form at their vertices a very sharp and +formidable angle. Indeed when their bases are of the most degraded type (not +more than the eighth part of an inch in size), they can hardly be +distinguished from Straight lines or Women; so extremely pointed are their +vertices. With us, as with you, these Triangles are distinguished from others +by being called Isosceles; and by this name I shall refer to them in the +following pages. + +Our Middle Class consists of Equilateral or Equal-Sided Triangles. + +Our Professional Men and Gentlemen are Squares (to which class I myself +belong) 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 the +number of their sides till they receive the honourable title of Polygonal, or +many-Sided. Finally when the number of the sides becomes so numerous, and the +sides themselves so small, that the figure cannot be distinguished from a +circle, he is included in the Circular or Priestly order; and this is the +highest class of all. + +It is a Law of Nature with us that a male child shall have one more side than +his father, so that each generation shall rise (as a rule) one step in the +scale of development and nobility. Thus the son of a Square is a Pentagon; the +son of a Pentagon, a Hexagon; and so on. + +But this rule applies not always to the Tradesman, and still less often to the +Soldiers, and to the Workmen; who indeed can hardly be said to deserve the +name of human Figures, since they have not all their sides equal. With them +therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a +Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope +is not shut out, even from the Isosceles, that his posterity may ultimately +rise above his degraded condition. For, after a long series of military +successes, or diligent and skillful labours, it is generally found that the +more intelligent among the Artisan and Soldier classes manifest a slight +increase of their third side or base, and a shrinkage of the two other sides. +Intermarriages (arranged by the Priests) between the sons and daughters of +these more intellectual members of the lower classes generally result in an +offspring approximating still more to the type of the Equal-Sided Triangle. + +Rarely --- in proportion to the vast numbers of Isosceles births --- is a genuine +and certifiable Equal-Sided Triangle produced from Isosceles parents +\footnote{ ``What need of a certificate?'' a Spaceland critic may ask: ``Is not +the procreation of a Square Son a certificate from Nature herself, proving the +Equal-sidedness of the Father?'' I reply that no Lady of any position will +marry an uncertified Triangle. Square offspring has sometimes resulted from a +slightly Irregular Triangle; but in almost every such case the Irregularity of +the first generation is visited on the third; which either fails to attain the +Pentagonal rank, or relapses to the Triangular.}. Such a birth requires, as +its antecedents, not only a series of carefully arranged intermarriages, but +also a long-continued exercise of frugality and self-control on the part of +the would-be ancestors of the coming Equilateral, and a patient, systematic, +and continuous development of the Isosceles intellect through many +generations. + +The birth of a True Equilateral Triangle from Isosceles parents is the subject +of rejoicing in our country for many furlongs around. After a strict +examination conducted by the Sanitary and Social Board, the infant, if +certified as Regular, is with solemn ceremonial admitted into the class of +Equilaterals. He is then immediately taken from his proud yet sorrowing +parents and adopted by some childless Equilateral, who is bound by oath never +to permit the child henceforth to enter his former home or so much as to look +upon his relations again, for fear lest the freshly developed organism may, by +force of unconscious imitation, fall back again into his hereditary level. + +The occasional emergence of an Equilateral from the ranks of his serf-born +ancestors is welcomed, not only by the poor serfs themselves, as a gleam of +light and hope shed upon the monotonous squalor of their existence, but also +by the Aristocracy at large; for all the higher classes are well aware that +these rare phenomena, while they do little or nothing to vulgarize their own +privileges, serve as almost useful barrier against revolution from below. + +Had the acute-angled rabble been all, without exception, absolutely destitute +of hope and of ambition, they might have found leaders in some of their many +seditious outbreaks, so able as to render their superior numbers and strength +too much even for the wisdom of the Circles. But a wise ordinance of Nature +has decreed that, in proportion as the working-classes increase in +intelligence, knowledge, and all virtue, in that same proportion their acute +angle (which makes them physically terrible) shall increase also and +approximate to their comparatively harmless angle of the Equilateral Triangle. +Thus, in the most brutal and formidable off the soldier class --- creatures +almost 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 tremendous +penetrating power to advantage, so do they wane in the power of penetration +itself. + +How admirable is this Law of Compensation! And how perfect a proof of the +natural fitness and, I may almost say, the divine origin of the aristocratic +constitution of the States of Flatland! By a judicious use of this Law of +Nature, the Polygons and Circles are almost always able to stifle sedition in +its very cradle, taking advantage of the irrepressible and boundless +hopefulness of the human mind. Art also comes to the aid of Law and Order. It +is generally found possible --- by a little artificial compression or expansion +on the part of the State physicians --- to make some of the more intelligent +leaders of a rebellion perfectly Regular, and to admit them at once into the +privileged classes; a much larger number, who are still below the standard, +allured by the prospect of being ultimately ennobled, are induced to enter the +State Hospitals, where they are kept in honourable confinement for life; one +or two alone of the most obstinate, foolish, and hopelessly irregular are led +to execution. + +Then the wretched rabble of the Isosceles, planless and leaderless, are ether +transfixed without resistance by the small body of their brethren whom the +Chief Circle keeps in pay for emergencies of this kind; or else more often, by +means of jealousies and suspicious skillfully fomented among them by the +Circular party, they are stirred to mutual warfare, and perish by one +another's angles. No less than one hundred and twenty rebellions are recorded +in our annals, besides minor outbreaks numbered at two hundred and +thirty-five; and they have all ended thus. + + + + + + + + + + + + + + +\chapter{Concerning the Women} +If our highly pointed Triangles of the Soldier class are formidable, it may be +readily inferred that far more formidable are our Women. For, if a Soldier is +a wedge, a Woman is a needle; being, so to speak, all point, at least at the +two extremities. Add to this the power of making herself practically invisible +at will, and you will perceive that a Female, in Flatland, is a creature by no +means to be trifled with. + +But here, perhaps, some of my younger Readers may ask how a woman in Flatland +can make herself invisible. This ought, I think, to be apparent without any +explanation. However, a few words will make it clear to the most unreflecting. + +Place a needle on 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 it +end-ways, and you see nothing but a point, it has become practically +invisible. Just so is it with one of our Women. When her side is turned +towards us, we see her as a straight line; when the end containing her eye or +mouth --- for with us these two organs are identical --- is the part that meets +our eye, then we see nothing but a highly lustrous point; but when the back is +presented to our view, then --- being only sub-lustrous, and, indeed, almost as +dim as an inanimate object --- her hinder extremity serves her as a kind of +Invisible Cap. + +The dangers to which we are exposed from our Women must now be manifest to the +meanest capacity in Spaceland. If even the angle of a respectable Triangle in +the middle class is not without its dangers; if to run against a Working Man +involves a gash; if collision with an Officer of the military class +necessitates a serious wound; if a mere touch from the vertex of a Private +Soldier brings with it danger of death; --- what can it be to run against a +woman, except absolute and immediate destruction? And when a Woman is +invisible, or visible only as a dim sub-lustrous point, how difficult must it +be, even for the most cautious, always to avoid collision! + +Many are the enactments made at different times in the different States of +Flatland, in order to minimize this peril; and in the Southern and less +temperate climates, where the force of gravitation is greater, and human +beings more liable to casual and involuntary motions, the Laws concerning +Women are naturally much more stringent. But a general view of the Code may be +obtained from the following summary: --- + +Every house shall have one entrance on the Eastern side, for the use of +Females only; by which all females shall enter ``in a becoming and respectful +manner'' \footnote{ When I was in Spaceland I understood that some of your +Priestly circles have in the same way a separate entrance for Villagers, +Farmers and Teachers of Board Schools (Spectator, Sept. 1884, P. 1255) that +they may ``approach in a becoming and respectful manner.''} and not by the Men's +or Western door. No Female shall walk in any public place without continually +keeping up her Peace-cry, under penalty of death. Any Female, duly certified +to be suffering from St. Vitus's Dance, fits, chronic cold accompanied by +violent sneezing, or any disease necessitating involuntary motions, shall be +instantly destroyed. In some of the States there is an additional Law +forbidding Females, under penalty of death, from walking or standing in any +public place without moving their backs constantly from right to left so as to +indicate their presence to those behind them; others oblige a Woman, when +travelling, to be followed by one of her sons, or servants, or by her husband; +others confine Women altogether to their houses except during the religious +festivals. But it has been found by the wisest of our Circles or Statesmen +that the multiplication of restrictions on Females tends not only to the +debilitation and diminution of the race, but also to the increase of domestic +murders to such an extent that a State loses more than it gains by a too +prohibitive Code. + +For whenever the temper of the Women is thus exasperated by confinement at +home or hampering regulations abroad, they are apt to vent their spleen upon +their husbands and children; and in the less temperate climates the whole male +population of a village has been sometimes destroyed in one or two hours of +simultaneous female outbreak. Hence the Three Laws, mentioned above, suffice +for the better regulated States, and may be accepted as a rough +exemplification of our Female Code. + +After all, our principal safeguard is found, not in Legislature, but in the +interests of the Women themselves. For, although they can inflict +instantaneous death by a retrograde movement, yet unless they can at once +disengage their stinging extremity from the struggling body of their victim, +their own frail bodies are liable to be shattered. + +The power of Fashion is also on our side. I pointed +out that in some less civilized States no female is suffered to stand in any +public place without swaying her back from right to left. This practice has +been universal among ladies of any pretensions to breeding in all +well-governed States, as far back as the memory of Figures can reach. It is +considered a disgrace to any state that legislation should have to enforce +what ought to be, and is in every respectable female, a natural instinct. The +rhythmical and, if I may so say, well-modulated undulation of the back in our +ladies of Circular rank is envied and imitated by the wife of a common +Equilateral, who can achieve nothing beyond a mere monotonous swing, like the +ticking of a pendulum; and the regular tick of the Equilateral is no less +admired and copied by the wife of the progressive and aspiring Isosceles, in +the females of whose family no ``back-motion'' of any kind has become as yet a +necessity of life. Hence, in every family of position and consideration, ``back +motion'' is as prevalent as time itself; and the husbands and sons in these +households enjoy immunity at least from invisible attacks. + +Not that it must be for a moment supposed that our Women are destitute of +affection. But unfortunately the passion of the moment predominates, in the +Frail Sex, over every other consideration. This is, of course, a necessity +arising from their unfortunate conformation. For as they have no pretensions +to an angle, being inferior in this respect to the very lowest of the +Isosceles, they are consequently wholly devoid of brainpower, and have neither +reflection, judgment nor forethought, and hardly any memory. Hence, in their +fits of fury, they remember no claims and recognize no distinctions. I have +actually known a case where a Woman has exterminated her whole household, and +half an hour afterwards, when her rage was over and the fragments swept away, +has asked what has become of her husband and her children. + +Obviously then a Woman is not to be irritated as long as she is in a position +where she can turn round. When you have them in their apartments --- which are +constructed with a view to denying them that power --- you can say and do what +you like; for they are then wholly impotent for mischief, and will not +remember a few minutes hence the incident for which they may be at this moment +threatening you with death, nor the promises which you may have found it +necessary to make in order to pacify their fury. + +On the whole we got on pretty smoothly in our domestic +relations, except in the lower strata of the Military Classes. There the want +of tact and discretion on the part of the husbands produces at times +indescribable disasters. Relying too much on the offensive weapons of their +acute angles instead of the defensive organs of good sense and seasonable +simulations, these reckless creatures too often neglect the prescribed +construction of the women's apartments, or irritate their wives by ill-advised +expressions out of doors, which they refuse immediately to retract. Moreover a +blunt and stolid regard for literal truth indisposes them to make those lavish +promises by which the more judicious Circle can in a moment pacify his +consort. The result is massacre; not, however, without its advantages, as it +eliminates the more brutal and troublesome of the Isosceles; and by many of +our Circles the destructiveness of the Thinner Sex is regarded as one among +many providential arrangements for suppressing redundant population, and +nipping Revolution in the bud. + +Yet even in our best regulated and most approximately Circular families I +cannot say that the ideal of family life is so high as with you in Spaceland. +There is peace, in so far as the absence of slaughter may be called by that +name, but there is necessarily little harmony of tastes or pursuits; and the +cautious wisdom of the Circles has ensured safety at the cost of domestic +comfort. In every Circular or Polygonal household it has been a habit from +time immemorial --- and now has become a kind of instinct among the women of our +higher classes --- that the mothers and daughters should constantly keep their +eyes and mouths towards their husband and his male friends; and for a lady in +a family of distinction to turn her back upon her husband would be regarded as +a kind of portent, involving loss of status. But, as I shall soon shew, this +custom, 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 of +the Working Man or respectable Tradesman --- where the wife is allowed to turn +her back upon her husband, while pursuing her household avocations --- there are +at least intervals of quiet, when the wife is neither seen nor heard, except +for the humming sound of the continuous Peace-cry; but in the homes of the +upper classes there is too often no peace. There the voluble mouth and bright +penetrating eye are ever directed toward the Master of the household; and +light itself is not more persistent than the stream of Feminine discourse. The +tact and skill which suffice to avert a Woman's sting are unequal to the task +of 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 from +saying it, not a few cynics have been found to aver that they prefer the +danger of the death-dealing but inaudible sting to the safe sonorousness of a +Woman's other end. + +To my readers in Spaceland the condition of our Women may seen truly +deplorable, and so indeed it is. A Male of the lowest type of the Isosceles +may look forward to some improvement of his angle, and to the ultimate +elevation of the whole of his degraded caste; but no Woman can entertain such +hopes for her sex. ``Once a Woman, always a Woman'' is a Decree of Nature; and +the very Laws of Evolution seem suspended in her disfavour. Yet at least we +can admire the wise Prearrangement which has ordained that, as they have no +hopes, so they shall have no memory to recall, and no forethought to +anticipate, the miseries and humiliations which are at once a necessity of +their existence and the basis of the constitution of Flatland. + + + + + + + + +\chapter{Of our Methods of Recognizing one another} You, who are blessed with +shade as well as light, you, who are gifted with two eyes, endowed with a +knowledge of perspective, and charmed with the enjoyment of various colours, +you, who can actually see an angle, and contemplate the complete circumference +of a Circle in the happy region of the Three Dimensions --- how shall I make it +clear to you the extreme difficulty which we in Flatland experience in +recognizing 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 distinguished +from another, where all appear the same? + +The answer is threefold. The first means of recognition is the sense of +hearing; which with us is far more highly developed than with you, and which +enables us not only to distinguish by the voice our personal friends, but even +to discriminate between different classes, at least so far as concerns the +three lowest orders, the Equilateral, the Square, and the Pentagon --- for the +Isosceles I take no account. But as we ascend the social scale, the process of +discriminating and being discriminated by hearing increases in difficulty, +partly because voices are assimilated, partly because the faculty of +voice-discrimination is a plebeian virtue not much developed among the +Aristocracy. And wherever there is any danger of imposture we cannot trust to +this method. Amongst our lowest orders, the vocal organs are developed to a +degree more than correspondent with those of hearing, so that an Isosceles can +easily feign the voice of a Polygon, and, with some training, that of a Circle +himself. A second method is therefore more commonly resorted to. + +Feeling is, among our Women and lower classes --- about our upper classes I +shall speak presently --- the principal test of recognition, at all events +between strangers, and when the question is, not as to the individual, but as +to the class. What therefore ``introduction'' is among the higher classes in +Spaceland, that the process of ``feeling'' is with us. ``Permit me to ask you to +feel and be felt by my friend Mr. So-and-so'' --- is still, among the more +old-fashioned of our country gentlemen in districts remote from towns, the +customary formula for a Flatland introduction. But in the towns, and among men +of business, the words ``be felt by'' are omitted and the sentence is +abbreviated to, ``Let me ask you to feel Mr. So-and-so''; although it is +assumed, of course, that the ``feeling'' is to be reciprocal. Among our still +more modern and dashing young gentlemen --- who are extremely averse to +superfluous effort and supremely indifferent to the purity of their native +language --- the formula is still further curtailed by the use of ``to feel'' in a +technical sense, meaning, ``to +recommend-for-the-purposes-of-feeling-and-being-felt''; and at this moment the +``slang'' of polite or fast society in the upper classes sanctions such a +barbarism as ``Mr. Smith, permit me to feel Mr. Jones.''. + +Let not my Reader however suppose that ``feeling'' is with us the tedious +process that it would be with you, or that we find it necessary to feel right +round all the sides of every individual before we determine the class to which +he belongs. Long practice and training, begun in the schools and continued in +the experience of daily life, enable us to discriminate at once by the sense +of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; +and I need not say that the brainless vertex of an acute-angled Isosceles is +obvious to the dullest touch. It is therefore not necessary, as a rule, to do +more than feel a single angle of an individual; and this, once ascertained, +tells us the class of the person whom we are addressing, unless indeed he +belongs to the higher sections of the nobility. There the difficulty is much +greater. Even a Master of Arts in our University of Wentbridge has been known +to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a +Doctor of Science in or out of that famous University who could pretend to +decide promptly and unhesitatingly between a twenty-sided and a twenty-four +sided member of the Aristocracy. + +Those of my readers who recall the extracts I gave above from the Legislative +code concerning Women, will readily perceive that the process of introduction +by contact requires some care and discretion. Otherwise the angles might +inflict on the unwary Feeler irreparable injury. It is essential for the +safety of the Feeler that the Felt should stand perfectly still. A start, a +fidgety shifting of the position, yes, even a violent sneeze, has been known +before now to prove fatal to the incautious, and to nip in the bud many a +promising friendship. Especially is this true among the lower classes of the +Triangles. With them, the eye is situated so far from their vertex that they +can scarcely take cognizance of what goes on at that extremity of their frame. +They are, moreover, of a rough coarse nature, not sensitive to the delicate +touch of the highly organized Polygon. What wonder then if an involuntary toss +of the head has ere now deprived the State of a valuable life! + +I have heard that my excellent Grandfather --- one of the least irregular of his +unhappy Isosceles class, who indeed obtained, shortly before his decease, four +out of seven votes from the Sanitary and Social Board for passing him into the +class of the Equal-sided --- often deplored, with a tear in his venerable eye, a +miscarriage of this kind, which had occurred to his +great-great-great-Grandfather, a respectable Working Man with an angle or +brain of 59 degrees 30 minutes. According to his account, my unfortunate +Ancestor, being afflicted with rheumatism, and in the act of being felt by a +Polygon, by one sudden start accidentally transfixed the Great Man through the +diagonal and thereby, partly in consequence of his long imprisonment and +degradation, and partly because of the moral shock which pervaded the whole of +my Ancestor's relations, threw back our family a degree and a half in their +ascent towards better things. The result was that in the next generation the +family brain was registered at only 58 degrees, and not till the lapse of five +generations was the lost ground recovered, the full 60 degrees attained, and +the Ascent from the Isosceles finally achieved. And all this series of +calamities from one little accident in the process of Feeling. + +At this point I think I hear some of my better educated readers exclaim, ``How +could you in Flatland know anything about angles and degrees, or minutes? We +see an angle, because we, in the region of Space, can see two straight lines +inclined to one another; but you, who can see nothing but on straight line at +a time, or at all events only a number of bits of straight lines all in one +straight line, --- how can you ever discern any angle, and much less register +angles of different sizes?''. + +I answer that though we cannot see angles, we can infer them, and this with +great precision. Our sense of touch, stimulated by necessity, and developed by +long training, enables us to distinguish angles far more accurately than your +sense of sight, when unaided by a rule or measure of angles. Nor must I omit +to explain that we have great natural helps. It is with us a Law of Nature +that the brain of the Isosceles class shall begin at half a degree, or thirty +minutes, and shall increase (if it increases at all) by half a degree in every +generation until the goal of 60 degrees is reached, when the condition of +serfdom is quitted, and the freeman enters the class of Regulars. + +Consequently, Nature herself supplies us with an ascending scale or Alphabet +of angles for half a degree up to 60 degrees, Specimen of which are placed in +every Elementary School throughout the land. Owing to occasional +retrogressions, to still more frequent moral and intellectual stagnation, and +to the extraordinary fecundity of the Criminal and Vagabond classes, there is +always a vast superfluity of individuals of the half degree and single degree +class, and a fair abundance of Specimens up to 10 degrees. These are +absolutely destitute of civic rights; and a great number of them, not having +even intelligence enough for the purposes of warfare, are devoted by the +States to the service of education. Fettered immovably so as to remove all +possibility of danger, they are placed in the classrooms of our Infant +Schools, and there they are utilized by the Board of Education for the purpose +of imparting to the offspring of the Middle Classes that tact and intelligence +of which these wretched creatures themselves are utterly devoid. + +In some States the Specimens are occasionally fed and suffered to exist for +several years; but in the more temperate and better regulated regions, it is +found in the long run more advantageous for the educational interests of the +young, to dispense with food, and to renew the Specimens every month --- which +is about the average duration of the foodless existence of the Criminal class. +In the cheaper schools, what is gained by the longer existence of the Specimen +is lost, partly in the expenditure for food, and partly in the diminished +accuracy of the angles, which are impaired after a few weeks of constant +``feeling''. Nor must we forget to add, in enumerating the advantages of the +more expensive system, that it tends, though slightly yet perceptibly, to the +diminution of the redundant Isosceles population --- an object which every +statesman in Flatland constantly keeps in view. On the whole therefore --- +although I am not ignorant that, in many popularly elected School Boards, +there is a reaction in favour of ``the cheap system'' as it is called --- I am +myself disposed to think that this is one of the many cases in which expense +is the truest economy. + +But I must not allow questions of School Board politics to divert me from my +subject. Enough has been said, I trust, to shew that Recognition by feeling is +not so tedious or indecisive a process as might have been supposed; and it is +obviously more trustworthy than Recognition by hearing. Still there remains, +as has been pointed out above, the objection that this method is not without +danger. For this reason many in the Middle and Lower classes, and all without +exception in the Polygonal and Circular orders, prefer a third method, the +description 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 that +all figures in Flatland present the appearance of a straight line; and it was +added or implied, that it is consequently impossible to distinguish by the +visual organ between individuals of different classes: yet now I am about to +explain to my Spaceland critics how we are able to recognize one another by +the sense of sight. + +If however the Reader will take the trouble to refer to the passage in which +Recognition by Feeling is stated to be universal, he will find this +qualification --- ``among the lower classes''. It is only among the higher classes +and in our more temperate climates that Sight Recognition is practised. + +That this power exists in any regions and for any classes is the result of +Fog; which prevails during the greater part of the year in all parts save the +torrid zones. That which is with you in Spaceland an unmixed evil, blotting +out the landscape, depressing the spirits, and enfeebling the health, is by us +recognized as a blessing scarcely inferior to air itself, and as the Nurse of +arts and Parent of sciences. But let me explain my meaning, without further +eulogies on this beneficent Element. + +If Fog were non-existent, all lines would appear equally and indistinguishably +clear; and this is actually the case in those unhappy countries in which the +atmosphere is perfectly dry and transparent. But wherever there is a rich +supply of Fog, objects that are at a distance, say of three feet, are +appreciably dimmer than those at the distance of two feet eleven inches; and +the result is that by careful and constant experimental observation of +comparative dimness and clearness, we are enabled to infer with great +exactness the configuration of the object observed. + +An instance will do more than a volume of generalities to make my meaning +clear. + +Suppose I see two individuals approaching whose rank I wish to ascertain. They +are, we will suppose, a Merchant and a Physician, or in other words, an +Equilateral 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 threshold +of Geometrical Studies, that, if I can bring my eye so that its glance may +bisect an angle (A) of the approaching stranger, my view will lie as it were +evenly between his two sides that are next to me (viz. CA and AB), so that I +shall contemplate the two impartially, and both will appear of the same size. + +Now in the case of (1) the Merchant, what shall I see? I shall see a straight +line DAE, in which the middle point (A) will be very bright because it is +nearest to me; but on either side the line will shade away rapidly to dimness, +because the sides AC and AB recede rapidly into the fog and what appear to me +as the Merchant's extremities, viz. D and E, will be very dim indeed. + +On the other hand in the case of (2) the Physician, though I shall here also +see a line (D'A'E') with a bright centre (A'), yet it will shade away less +rapidly to dimness, because the sides (A'C', A'B') recede less rapidly into +the fog: and what appear to me the Physician's extremities, viz. D' and E', +will not be not so dim as the extremities of the Merchant. + +The Reader will probably understand from these +two instances how --- after a very long training supplemented by constant +experience --- it is possible for the well-educated classes among us to +discriminate with fair accuracy between the middle and lowest orders, by the +sense of sight. If my Spaceland Patrons have grasped this general conception, +so far as to conceive the possibility of it and not to reject my account as +altogether incredible --- I shall have attained all I can reasonably expect. +Were I to attempt further details I should only perplex. Yet for the sake of +the young and inexperienced, who may perchance infer --- from the two simple +instances I have given above, of the manner in which I should recognize my +Father and my Sons --- that Recognition by sight is an easy affair, it may be +needful to point out that in actual life most of the problems of Sight +Recognition are far more subtle and complex. + +If for example, when my Father, the Triangle, approaches me, he happens to +present his side to me instead of his angle, then, until I have asked him to +rotate, or until I have edged my eye around him, I am for the moment doubtful +whether he may not be a Straight Line, or, in other words, a Woman. Again, +when I am in the company of one of my two hexagonal Grandsons, contemplating +one of his sides (AB) full front, it will be evident from the accompanying +diagram that I shall see one whole line (AB) in comparative brightness +(shading off hardly at all at the ends) and two smaller lines (CA and BD) dim +throughout and shading away into greater dimness towards the extremities C and +D. +\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 these +topics. The meanest mathematician in Spaceland will readily believe me when I +assert that the problems of life, which present themselves to the +well-educated --- when they are themselves in motion, rotating, advancing or +retreating, and at the same time attempting to discriminate by the sense of +sight between a number of Polygons of high rank moving in different +directions, as for example in a ball-room or conversazione --- must be of a +nature to task the angularity of the most intellectual, and amply justify the +rich endowments of the Learned Professors of Geometry, both Static and +Kinetic, in the illustrious University of Wentbridge, where the Science and +Art of Sight Recognition are regularly taught to large classes of the elite of +the States. + +It is only a few of the scions of our +noblest and wealthiest houses, who are able to give the time and money +necessary for the thorough prosecution of this noble and valuable Art. Even to +me, a Mathematician of no mean standing, and the Grandfather of two most +hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd +of rotating Polygons of the higher classes, is occasionally very perplexing. +And of course to a common Tradesman, or Serf, such a sight is almost as +unintelligible as it would be to you, my Reader, were you suddenly transported +to our country. + +In such a crowd you could see on all sides of you nothing but a Line, +apparently straight, but of which the parts would vary irregularly and +perpetually in brightness or dimness. Even if you had completed your third +year in the Pentagonal and Hexagonal classes in the University, and were +perfect in the theory of the subject, you would still find there was need of +many years of experience, before you could move in a fashionable crowd without +jostling against your betters, whom it is against etiquette to ask to ``feel'', +and who, by their superior culture and breeding, know all about your +movements, while you know very little or nothing about theirs. In a word, to +comport oneself with perfect propriety in Polygonal society, one ought to be a +Polygon oneself. Such at least is the painful teaching of my experience. + +It is astonishing how much the Art --- or I may almost +call it instinct --- of Sight Recognition is developed by the habitual practice +of it and by the avoidance of the custom of ``Feeling''. Just as, with you, the +deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet, +will never acquire the more difficult but far more valuable art of lip-speech +and lip-reading, so it is with us as regards ``Seeing'' and ``Feeling''. None who +in early life resort to ``Feeling'' will ever learn ``Seeing'' in perfection. + +For this reason, among our Higher Classes, ``Feeling'' is discouraged or +absolutely forbidden. From the cradle their children, instead of going to the +Public Elementary schools (where the art of Feeling is taught,) are sent to +higher Seminaries of an exclusive character; and at our illustrious +University, to ``feel'' is regarded as a most serious fault, involving +Rustication for the first offence, and Expulsion for the second. + +But among the lower classes the art of Sight Recognition is regarded as an +unattainable luxury. A common Tradesman cannot afford to let his son spend a +third of his life in abstract studies. The children of the poor are therefore +allowed to ``feel'' from their earliest years, and they gain thereby a precocity +and an early vivacity which contrast at first most favourably with the inert, +undeveloped, and listless behaviour of the half-instructed youths of the +Polygonal class; but when the latter have at last completed their University +course, and are prepared to put their theory into practice, the change that +comes over them may almost be described as a new birth, and in every art, +science, and social pursuit they rapidly overtake and distance their +Triangular competitors. + +Only a few of the Polygonal Class fail to pass the Final Test or Leaving +Examination at the University. The condition of the unsuccessful minority is +truly pitiable. Rejected from the higher class, they are also despised by the +lower. They have neither the matured and systematically trained powers of the +Polygonal Bachelors and Masters of Arts, nor yet the native precocity and +mercurial versatility of the youthful Tradesman. The professions, the public +services, are closed against them, and though in most States they are not +actually debarred from marriage, yet they have the greatest difficulty in +forming suitable alliances, as experience shews that the offspring of such +unfortunate and ill-endowed parents is generally itself unfortunate, if not +positively Irregular. + +It is from these specimens of the refuse of our Nobility that the great +Tumults and Seditions of past ages have generally derived their leaders; and +so great is the mischief thence arising that an increasing minority of our +more progressive Statesmen are of opinion that true mercy would dictate their +entire suppression, by enacting that all who fail to pass the Final +Examination of the University should be either imprisoned for life, or +extinguished by a painless death. + +But I find myself digressing into the subject of Irregularities, a matter of +such vital interest that it demands a separate section. + + + + + + +\chapter{Concerning Irregular Figures} +Throughout the previous pages I have been assuming --- what perhaps should have +been 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 of +regular construction. By this I mean that a Woman must not only be a line, but +a 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 a +humble member), four sides equal, and, generally, that in every Polygon, all +the 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 might +extend to a foot. As to the Males of every class, it may be roughly said that +the length of an adult's size, when added together, is two feet or a little +more. But the size of our sides is not under consideration. I am speaking of +the equality of sides, and it does not need much reflection to see that the +whole of the social life in Flatland rests upon the fundamental fact that +Nature wills all Figures to have their sides equal. + +If our sides were unequal our angles might be unequal. Instead of its being +sufficient to feel, or estimate by sight, a single angle in order to determine +the form of an individual, it would be necessary to ascertain each angle by +the experiment of Feeling. But life would be too short for such a tedious +groping. 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 would +become perilous or impossible; there would be an end to all confidence, all +forethought; no one would be safe in making the most simple social +arrangements; 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, must +convince every one that our social system is based upon Regularity, or +Equality of Angles. You meet, for example, two or three Tradesmen in the +street, whom your recognize at once to be Tradesman by a glance at their +angles and rapidly bedimmed sides, and you ask them to step into your house to +lunch. This you do at present with perfect confidence, because everyone knows +to an inch or two the area occupied by an adult Triangle: but imagine that +your Tradesman drags behind his regular and respectable vertex, a +parallelogram of twelve or thirteen inches in diagonal: --- what are you to do +with such a monster sticking fast in your house door? + +But I am insulting the intelligence of my Readers by accumulating details +which must be patent to everyone who enjoys the advantages of a Residence in +Spaceland. Obviously the measurements of a single angle would no longer be +sufficient under such portentous circumstances; one's whole life would be +taken up in feeling or surveying the perimeter of one's acquaintances. Already +the difficulties of avoiding a collision in a crowd are enough to tax the +sagacity of even a well-educated Square; but if no one could calculate the +Regularity of a single figure in the company, all would be chaos and +confusion, and the slightest panic would cause serious injuries, or --- if there +happened to be any Women or Soldiers present --- perhaps considerable loss of +life. + +Expediency therefore concurs with Nature in stamping the seal of its approval +upon Regularity of conformation: nor has the Law been backward in seconding +their efforts. ``Irregularity of Figure'' means with us the same as, or more +than, a combination of moral obliquity and criminality with you, and is +treated accordingly. There are not wanting, it is true, some promulgators of +paradoxes who maintain that there is no necessary connection between +geometrical and moral Irregularity. ``The Irregular,'' they say, ``is from his +birth scouted by his own parents, derided by his brothers and sisters, +neglected by the domestics, scorned and suspected by society, and excluded +from all posts of responsibility, trust, and useful activity. His every +movement is jealously watched by the police till he comes of age and presents +himself for inspection; then he is either destroyed, if he is found to exceed +the fixed margin of deviation, at an uninteresting occupation for a miserable +stipend; obliged to live and board at the office, and to take even his +vacation under close supervision; what wonder that human nature, even in the +best and purest, is embittered and perverted by such surroundings!'' + +All this very plausible reasoning does not convince me, as it has not +convinced the wisest of our Statesmen, that our ancestors erred in laying it +down as an axiom of policy that the toleration of Irregularity is incompatible +with the safety of the State. Doubtless, the life of an Irregular is hard; but +the interests of the Greater Number require that it shall be hard. If a man +with a triangular front and a polygonal back were allowed to exist and to +propagate a still more Irregular posterity, what would become of the arts of +life? Are the houses and doors and churches in Flatland to be altered in order +to accommodate such monsters? Are our ticket-collectors to be required to +measure every man's perimeter before they allow him to enter a theatre, or to +take his place in a lecture room? Is an Irregular to be exempted from the +militia? And if not, how is he to be prevented from carrying desolation into +the ranks of his comrades? Again, what irresistible temptations to fraudulent +impostures must needs beset such a creature! How easy for him to enter a shop +with his polygonal front foremost, and to order goods to any extent from a +confiding tradesman! Let the advocates of a falsely called Philanthropy plead +as they may for the abrogation of the Irregular Penal Laws, I for my part have +never known an Irregular who was not also what Nature evidently intended him +to be --- a hypocrite, a misanthropist, and, up to the limits of his power, a +perpetrator of all manner of mischief. + +Not that I should be disposed to recommend (at present) the extreme measures +adopted by some States, where an infant whose angle deviates by half a degree +from the correct angularity is summarily destroyed at birth. Some of our +highest and ablest men, men of real genius, have during their earliest days +laboured under deviations as great as, or even greater than forty-five +minutes: and the loss of their precious lives would have been an irreparable +injury to the State. The art of healing also has achieved some of its most +glorious triumphs in the compressions, extensions, trepannings, colligations, +and other surgical or diaetetic operations by which Irregularity has been +partly or wholly cured. Advocating therefore a Via Media, I would lay down no +fixed or absolute line of demarcation; but at the period when the frame is +just beginning to set, and when the Medical Board has reported that recovery +is improbably, I would suggest that the Irregular offspring be painlessly and +mercifully consumed. + + + + + + +\chapter{Of the Ancient Practice of Painting} +If my Readers have followed me with any attention up to this point, they will +not be surprised to hear that life is somewhat dull in Flatland. I do not, of +course, mean that there are not battles, conspiracies, tumults, factions, and +all those other phenomena which are supposed to make History interesting; nor +would I deny that the strange mixture of the problems of life and the problems +of Mathematics, continually inducing conjecture and giving an opportunity of +immediate verification, imparts to our existence a zest which you in Spaceland +can hardly comprehend. I speak now from the aesthetic and artistic point of +view 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 single +line, with no varieties except degrees of brightness and obscurity? + +It was not always thus. Colour, if Tradition speaks the truth, once for the +space of half a dozen centuries or more, threw a transient splendour over the +lives of our ancestors in the remotest ages. Some private individual --- a +Pentagon whose name is variously reported --- having casually discovered the +constituents of the simpler colours and a rudimentary method of painting, is +said to have begun by decorating first his house, then his slaves, then his +Father, his Sons, and Grandsons, lastly himself. The convenience as well as +the beauty of the results commended themselves to all. Wherever Chromatistes, +--- for by that name the most trustworthy authorities concur in calling him, --- +turned his variegated frame, there he at once excited attention, and attracted +respect. No one now needed to ``feel'' him; no one mistook his front for his +back; all his movements were readily ascertained by his neighbours without the +slightest strain on their powers of calculation; no one jostled him, or failed +to make way for him; his voice was saved the labour of that exhausting +utterance by which we colourless Squares and Pentagons are often forced to +proclaim our individuality when we move amid a crowd of ignorant Isosceles. + +The fashion spread like wildfire. Before a week was over, every Square and +Triangle in the district had copied the example of Chromatistes, and only a +few of the more conservative Pentagons still held out. A month or two found +even the Dodecagons infected with the innovation. A year had not elapsed +before the habit had spread to all but the very highest of the Nobility. +Needless to say, the custom soon made its way from the district of +Chromatistes to surrounding regions; and within two generations no one in all +Flatland was colourless except the Women and the Priests. + +Here Nature herself appeared to erect a barrier, and to plead against +extending the innovations to these two classes. Many-sidedness was almost +essential as a pretext for the Innovators. ``Distinction of sides is intended +by Nature to imply distinction of colours'' --- such was the sophism which in +those days flew from mouth to mouth, converting whole towns at a time to a new +culture. But manifestly to our Priests and Women this adage did not apply. The +latter had only one side, and therefore --- plurally and pedantically speaking --- +no sides. The former --- if at least they would assert their claim to be readily +and truly Circles, and not mere high-class Polygons, with an infinitely large +number of infinitesimally small sides --- were in the habit of boasting (what +Women confessed and deplored) that they also had no sides, being blessed with +a perimeter of only one line, or, in other words, a Circumference. Hence it +came to pass that these two Classes could see no force in the so-called axiom +about ``Distinction of Sides implying Distinction of Colour;'' and when all +others had succumbed to the fascinations of corporal decoration, the Priests +and the Women alone still remained pure from the pollution of paint. + +Immoral, licentious, anarchical, unscientific --- call them by what names you +will --- yet, from an aesthetic point of view, those ancient days of the Colour +Revolt were the glorious childhood of Art in Flatland --- a childhood, alas, +that never ripened into manhood, nor even reached the blossom of youth. To +live then in itself a delight, because living implied seeing. Even at a small +party, the company was a pleasure to behold; the richly varied hues of the +assembly in a church or theatre are said to have more than once proved too +distracting from our greatest teachers and actors; but most ravishing of all +is said to have been the unspeakable magnificence of a military review. + +The sight of a line of battle of twenty thousand +Isosceles suddenly facing about, and exchanging the sombre black of their +bases for the orange of the two sides including their acute angle; the militia +of the Equilateral Triangles tricoloured in red, white, and blue; the mauve, +ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly +rotating near their vermillion guns; the dashing and flashing of the +five-coloured and six-coloured Pentagons and Hexagons careering across the +field in their offices of surgeons, geometricians and aides-de-camp --- all +these may well have been sufficient to render credible the famous story how an +illustrious Circle, overcome by the artistic beauty of the forces under his +command, threw aside his marshal's baton and his royal crown, exclaiming that +he henceforth exchanged them for the artist's pencil. How great and glorious +the sensuous development of these days must have been is in part indicated by +the very language and vocabulary of the period. The commonest utterances of +the commonest citizens in the time of the Colour Revolt seem to have been +suffused with a richer tinge of word or thought; and to that era we are even +now indebted for our finest poetry and for whatever rhythm still remains in +the more scientific utterance of those modern days. + + +\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 neglect +even at our University. The inferior Art of Feeling speedily experienced the +same fate at our Elementary Schools. Then the Isosceles classes, asserting +that the Specimens were no longer used nor needed, and refusing to pay the +customary tribute from the Criminal classes to the service of Education, waxed +daily more numerous and more insolent on the strength of their immunity from +the old burden which had formerly exercised the twofold wholesome effect of at +once taming their brutal nature and thinning their excessive numbers. + +Year by year the Soldiers and Artisans began more vehemently to assert --- and +with increasing truth --- that there was no great difference between them and +the very highest class of Polygons, now that they were raised to an equality +with the latter, and enabled to grapple with all the difficulties and solve +all the problems of life, whether Statical or Kinetical, by the simple process +of Colour Recognition. Not content with the natural neglect into which Sight +Recognition was falling, they began boldly to demand the legal prohibition of +all ``monopolizing and aristocratic Arts'' and the consequent abolition of all +endowments for the studies of Sight Recognition, Mathematics, and Feeling. +Soon, they began to insist that inasmuch as Colour, which was a second Nature, +had destroyed the need of aristocratic distinctions, the Law should follow in +the same path, and that henceforth all individuals and all classes should be +recognized as absolutely equal and entitled to equal rights. + +Finding the higher Orders wavering and undecided, the leaders of the +Revolution advanced still further in their requirements, and at last demanded +that all classes alike, the Priests and the Women not excepted, should do +homage to Colour by submitting to be painted. When it was objected that +Priests and Women had no sides, they retorted that Nature and Expediency +concurred in dictating that the front half of every human being (that is to +say, the half containing his eye and mouth) should be distinguishable from his +hinder half. They therefore brought before a general and extraordinary +Assembly of all the States of Flatland a Bill proposing that in every Woman +the half containing the eye and mouth should be coloured red, and the other +half green. The Priests were to be painted in the same way, red being applied +to that semicircle in which the eye and mouth formed the middle point; while +the other or hinder semicircle was to be coloured green. + +There was no little cunning in this proposal, which indeed emanated not from +any Isosceles --- for no being so degraded would have angularity enough to +appreciate, much less to devise, such a model of state-craft --- but from an +Irregular Circle who, instead of being destroyed in his childhood, was +reserved by a foolish indulgence to bring desolation on his country and +destruction on myriads of followers. + +On the one hand the proposition was calculated to bring the Women in all +classes over to the side of the Chromatic Innovation. For by assigning to the +Women the same two colours as were assigned to the Priests, the Revolutionists +thereby ensured that, in certain positions, every Woman would appear as a +Priest, and be treated with corresponding respect and deference --- a prospect +that could not fail to attract the Female Sex in a mass. + +But by some of my Readers the possibility of the identical appearance of +Priests and Women, under a new Legislation, may not be recognized; if so, a +word or two will make it obvious. + +Imagine a woman duly decorated, according to the new Code; with the front half +(i.e., the half containing the eye and mouth) red, and with the hinder half +green. Look at her from one side. Obviously you will see a straight line, half +red, half green. +\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) is +consequently coloured red, while his hinder semicircle is green; so that the +diameter AB divides the green from the red. If you contemplate the Great Man +so as to have your eye in the same straight line as his dividing diameter +(AB), what you will see will be a straight line (CBD), of which one half (CB) +will be red, and the other (BD) green. The whole line (CD) will be rather +shorter perhaps than that of a full-sized Woman, and will shade off more +rapidly towards its extremities; but the identity of the colours would give +you an immediate impression of identity in Class, making you neglectful of +other details. Bear in mind the decay of Sight Recognition which threatened +society at the time of the Colour revolt; add too the certainty that Woman +would speedily learn to shade off their extremities so as to imitate the +Circles; it must then be surely obvious to you, my dear Reader, that the +Colour Bill placed us under a great danger of confounding a Priest with a +young Woman. + +How attractive this prospect must have been to the Frail Sex may readily be +imagined. They anticipated with delight the confusion that would ensue. At +home they might hear political and ecclesiastical secrets intended not for +them but for their husbands and brothers, and might even issue some commands +in the name of a priestly Circle; out of doors the striking combination of red +and green without addition of any other colours, would be sure to lead the +common people into endless mistakes, and the Woman would gain whatever the +Circles lost, in the deference of the passers by. As for the scandal that +would befall the Circular Class if the frivolous and unseemly conduct of the +Women were imputed to them, and as to the consequent subversion of the +Constitution, the Female Sex could not be expected to give a thought to these +considerations. Even in the households of the Circles, the Women were all in +favour of the Universal Colour Bill. + +The second object aimed at by the Bill was the gradual demoralization of the +Circles themselves. In the general intellectual decay they still preserved +their pristine clearness and strength of understanding. From their earliest +childhood, familiarized in their Circular households with the total absence of +Colour, the Nobles alone preserved the Sacred Art of Sight Recognition, with +all the advantages that result from that admirable training of the intellect. +Hence, up to the date of the introduction of the Universal Colour Bill, the +Circles had not only held their own, but even increased their lead of the +other classes by abstinence from the popular fashion. + +Now therefore the artful Irregular whom I described above as the real author +of this diabolical Bill, determined at one blow to lower the status of the +Hierarchy by forcing them to submit to the pollution of Colour, and at the +same time to destroy their domestic opportunities of training in the Art of +Sight Recognition, so as to enfeeble their intellects by depriving them of +their pure and colourless homes. Once subjected to the chromatic taint, every +parental and every childish Circle would demoralize each other. Only in +discerning between the Father and the Mother would the Circular infant find +problems for the exercise of his understanding --- problems too often likely to +be corrupted by maternal impostures with the result of shaking the child's +faith in all logical conclusions. Thus by degrees the intellectual lustre of +the Priestly Order would wane, and the road would then lie open for a total +destruction of all Aristocratic Legislature and for the subversion of our +Privileged Classes. + + +\chapter{Of the Suppression of the Chromatic Sedition} +The agitation for the Universal Colour Bill continued for three years; and up +to the last moment of that period it seemed as though Anarchy were destined to +triumph. + +A whole army of Polygons, who turned out to fight as private soldiers, was +utterly annihilated by a superior force of Isosceles Triangles --- the Squares +and Pentagons meanwhile remaining neutral. Worse than all, some of the ablest +Circles fell a prey to conjugal fury. Infuriated by political animosity, the +wives in many a noble household wearied their lords with prayers to give up +their opposition to the Colour Bill; and some, finding their entreaties +fruitless, fell on and slaughtered their innocent children and husband, +perishing themselves in the act of carnage. It is recorded that during that +triennial agitation no less than twenty-three Circles perished in domestic +discord. + +Great indeed was the peril. It seemed as though the Priests had no choice +between submission and extermination; when suddenly the course of events was +completely changed by one of those picturesque incidents which Statesmen ought +never to neglect, often to anticipate, and sometimes perhaps to originate, +because of the absurdly disproportionate power with which they appeal to the +sympathies of the populace. + +It happened that an Isosceles of a low type, with a brain little if at all +above four degrees --- accidentally dabbling in the colours of some Tradesman +whose shop he had plundered --- painted himself, or caused himself to be painted +(for the story varies) with the twelve colours of a Dodecagon. Going into the +Market Place he accosted in a feigned voice a maiden, the orphan daughter of a +noble Polygon, whose affection in former days he had sought in vain; and by a +series of deceptions --- aided, on the one side, by a string of lucky accidents +too long to relate, and, on the other, by an almost inconceivable fatuity and +neglect of ordinary precautions on the part of the relations of the bride --- he +succeeded in consummating the marriage. The unhappy girl committed suicide on +discovering the fraud to which she had been subjected. + +When the news of this catastrophe spread from State to State the minds of the +Women were violently agitated. Sympathy with the miserable victim and +anticipations of similar deceptions for themselves, their sisters, and their +daughters, made them now regard the Colour Bill in an entirely new aspect. Not +a few openly avowed themselves converted to antagonism; the rest needed only a +slight stimulus to make a similar avowal. Seizing this favourable opportunity, +the Circles hastily convened an extraordinary Assembly of the States; and +besides the usual guard of Convicts, they secured the attendance of a large +number of reactionary Women. + +Amidst an unprecedented concourse, the Chief Circle of those days --- by name +Pantocyclus --- arose to find himself hissed and hooted by a hundred and twenty +thousand Isosceles. But he secured silence by declaring that henceforth the +Circles would enter on a policy of Concession; yielding to the wishes of the +majority, they would accept the Colour Bill. The uproar being at once +converted to applause, he invited Chromatistes, the leader of the Sedition, +into the centre of the hall, to receive in the name of his followers the +submission of the Hierarchy. Then followed a speech, a masterpiece of +rhetoric, which occupied nearly a day in the delivery, and to which no summary +can do justice. + +With a grave appearance of impartiality he declared that as they were now +finally committing themselves to Reform or Innovation, it was desirable that +they should take one last view of the perimeter of the whole subject, its +defects as well as its advantages. Gradually introduction the mention of the +dangers to the Tradesmen, the Professional Classes and the Gentlemen, he +silenced the rising murmurs of the Isosceles by reminding them that, in spite +of all these defects, he was willing to accept the Bill if it was approved by +the majority. But it was manifest that all, except the Isosceles, were moved +by his words and were either neutral or averse to the Bill. + +Turning now to the Workmen he asserted that their interests must not be +neglected, and that, if they intended to accept the Colour Bill, they ought at +least to do so with full view of the consequences. Many of them, he said, were +on the point of being admitted to the class of the Regular Triangles; others +anticipated for their children a distinction they could not hope for +themselves. That honourable ambition would now have to be sacrificed. With the +universal adoption of Colour, all distinctions would cease; Regularity would +be confused with Irregularity; development would give place to retrogression; +the Workman would in a few generations be degraded to the level of the +Military, or even the Convict Class; political power would be in the hands of +the greatest number, that is to say the Criminal Classes, who were already +more numerous than the Workmen, and would soon out-number all the other +Classes put together when the usual Compensative Laws of Nature were violated. + +A subdued murmur of assent ran through the ranks of the Artisans, and +Chromatistes, in alarm, attempted to step forward and address them. But he +found himself encompassed with guards and forced to remain silent while the +Chief Circle in a few impassioned words made a final appeal to the Women, +exclaiming that, if the Colour Bill passed, no marriage would henceforth be +safe, no woman's honour secure; fraud, deception, hypocrisy would pervade +every household; domestic bliss would share the fate of the Constitution and +pass to speedy perdition. ``Sooner than this'', he cried, ``Come death''. + +At these words, which were the preconcerted signal for action, the Isosceles +Convicts fell on and transfixed the wretched Chromatistes; the Regular +Classes, opening their ranks, made way for a band of Women who, under +direction of the Circles, moved back foremost, invisibly and unerringly upon +the unconscious soldiers; the Artisans, imitating the example of their +betters, also opened their ranks. Meantime bands of Convicts occupied every +entrance with an impenetrable phalanx. + +The battle, or rather carnage, was of short duration. Under the skillful +generalship of the Circles almost every Woman's charge was fatal and very many +extracted their sting uninjured, ready for a second slaughter. But no second +blow was needed; the rabble of the Isosceles did the rest of the business for +themselves. Surprised, leader-less, attacked in front by invisible foes, and +finding egress cut off by the Convicts behind them, they at once --- after their +manner --- lost all presence of mind, and raised the cry of ``treachery''. This +sealed their fate. Every Isosceles now saw and felt a foe in every other. In +half an hour not one of that vast multitude was living; and the fragments of +seven score thousand of the Criminal Class slain by one another's angles +attested the triumph of Order. + +The Circles delayed not to push their victory to the uttermost. The Working +Men they spared but decimated. The Militia of the Equilaterals was at once +called out, and every Triangle suspected of Irregularity on reasonable +grounds, was destroyed by Court Martial, without the formality of exact +measurement by the Social Board. The homes of the Military and Artisan classes +were inspected in a course of visitation extending through upwards of a year; +and during that period every town, village, and hamlet was systematically +purged of that excess of the lower orders which had been brought about by the +neglect to pay the tribute of Criminals to the Schools and University, and by +the violation of other natural Laws of the Constitution of Flatland. Thus the +balance of classes was again restored. + +Needless to say that henceforth the use of Colour was abolished, and its +possession prohibited. Even the utterance of any word denoting Colour, except +by the Circles or by qualified scientific teachers, was punished by a severe +penalty. Only at our University in some of the very highest and most esoteric +classes --- which I myself have never been privileged to attend --- it is +understood that the sparing use of Colour is still sanctioned for the purpose +of illustrating some of the deeper problems of mathematics. But of this I can +only speak from hearsay. + +Elsewhere in Flatland, Colour is now non-existent. The art of making it is +known to only one living person, the Chief Circle for the time being; and by +him it is handed down on his death-bed to none but his Successor. One +manufactory alone produces it; and, lest the secret should be betrayed, the +Workmen are annually consumed, and fresh ones introduced. So great is the +terror with which even now our Aristocracy looks back to the far-distant days +of 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 about +things in Flatland to the central event of this book, my initiation into the +mysteries of Space. That is my subject; all that has gone before is merely +preface. + +For this reason I must omit many matters of which the explanation would not, I +flatter myself, be without interest for my Readers: as for example, our method +of propelling and stopping ourselves, although destitute of feet; the means by +which we give fixity to structures of wood, stone, or brick, although of +course we have no hands, nor can we lay foundations as you can, nor avail +ourselves of the lateral pressure of the earth; the manner in which the rain +originates in the intervals between our various zones, so that the northern +regions do not intercept the moisture falling on the southern; the nature of +our hills and mines, our trees and vegetables, our seasons and harvests; our +Alphabet and method of writing, adapted to our linear tablets; these and a +hundred other details of our physical existence I must pass over, nor do I +mention them now except to indicate to my readers that their omission proceeds +not from forgetfulness on the part of the author, but from his regard for the +time of the Reader. + +Yet before I proceed to my legitimate subject some few final remarks will no +doubt be expected by my Readers upon these pillars and mainstays of the +Constitution of Flatland, the controllers of our conduct and shapers of our +destiny, the objects of universal homage and almost of adoration: need I say +that I mean our Circles or Priests? + +When I call them Priests, let me not be understood as meaning no more than the +term denotes with you. With us, our Priests are Administrators of all +Business, Art, and Science; Directors of Trade, Commerce, Generalship, +Architecture, Engineering, Education, Statesmanship, Legislature, Morality, +Theology; doing nothing themselves, they are the Causes of everything worth +doing, that is done by others. + +Although popularly everyone called a Circle is deemed a Circle, yet among the +better educated Classes it is known that no Circle is really a Circle, but +only a Polygon with a very large number of very small sides. As the number of +the sides increases, a Polygon approximates to a Circle; and, when the number +is very great indeed, say for example three or four hundred, it is extremely +difficult for the most delicate touch to feel any polygonal angles. Let me say +rather it would be difficult: for, as I have shown above, Recognition by +Feeling is unknown among the highest society, and to feel a Circle would be +considered a most audacious insult. This habit of abstention from Feeling in +the best society enables a Circle the more easily to sustain the veil of +mystery in which, from his earliest years, he is wont to enwrap the exact +nature of his Perimeter or Circumference. Three feet being the average +Perimeter it follows that, in a Polygon of three hundred sides each side will +be no more than the hundredth part of a foot in length, or little more than +the tenth part of an inch; and in a Polygon of six or seven hundred sides the +sides are little larger than the diameter of a Spaceland pin-head. It is +always assumed, by courtesy, that the Chief Circle for the time being has ten +thousand sides. + +The ascent of the posterity of the Circles in the social scale is not +restricted, as it is among the lower Regular classes, by the Law of Nature +which limits the increase of sides to one in each generation. If it were so, +the number of sides in the Circle would be a mere question of pedigree and +arithmetic, and the four hundred and ninety-seventh descendant of an +Equilateral Triangle would necessarily be a polygon With five hundred sides. +But this is not the case. Nature's Law prescribes two antagonistic decrees +affecting Circular propagation; first, that as the race climbs higher in the +scale of development, so development shall proceed at an accelerated pace; +second, that in the same proportion, the race shall become less fertile. +Consequently in the home of a Polygon of four or five hundred sides it is rare +to find a son; more than one is never seen. On the other hand the son of a +five-hundred-sided Polygon has been known to possess five hundred and fifty, +or even six hundred sides. + +Art also steps in to help the process of higher Evolution. Our physicians have +discovered that the small and tender sides of an infant Polygon of the higher +class can be fractured, and his whole frame re-set, with such exactness that a +Polygon of two or three hundred sides sometimes --- by no means always, for the +process is attended with serious risk --- but sometimes overleaps two or three +hundred generations, and as it were double at a stroke, the number of his +progenitors and the nobility of his descent. + +Many a promising child is sacrificed in this way. Scarcely one out of ten +survives. Yet so strong is the parental ambition among those Polygons who are, +as it were, on the fringe of the Circular class, that it is very rare to find +the Nobleman of that position in society, who has neglected to place his +first-born in the Circular Neo-Therapeutic Gymnasium before he has attained +the age of a month. + +One year determines success or failure. At the end of that time the child has, +in all probability, added one more to the tombstones that crowd the +Neo-Therapeutic Cemetery; but on rare occasional a glad procession bares back +the little one to his exultant parents, no longer a Polygon, but a Circle, at +least by courtesy: and a single instance of so blessed a result induces +multitudes of Polygonal parents to submit to similar domestic sacrifice, which +have 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 single +maxim, ``Attend to your Configuration''. Whether political, ecclesiastical, or +moral, all their teaching has for its object the improvement of individual and +collective Configuration --- with special reference of course to the +Configuration of the Circles, to which all other objects are subordinated. + +It is the merit of the Circles that they have effectually suppressed those +ancient heresies which led men to waste energy and sympathy in the vain belief +that conduct depends upon will, effort, training, encouragement, praise, or +anything else but Configuration. It was Pantocyclus --- the illustrious Circle +mentioned above, as the queller of the Colour Revolt --- who first convinced +mankind that Configuration makes the man; that if, for example, you are born +an Isosceles with two uneven sides, you will assuredly go wrong unless you +have them made even --- for which purpose you must go to the Isosceles Hospital; +similarly, if you are a Triangle, or Square, or even a Polygon, born with any +Irregularity, you must be taken to one of the Regular Hospitals to have your +disease cured; otherwise you will end your days in the State Prison or by the +angle of the State Executioner. + +All faults or defects, from the slightest misconduct to the most flagitious +crime, Pantocyclus attributed to some deviation from perfect Regularity in the +bodily figure, caused perhaps (if not congenital by some collision in a crowd; +by neglect to take exercise, or by taking too much of it; or even by a sudden +change of temperature, resulting in a shrinkage or expansion in some too +susceptible part of the frame. Therefore, concluded that illustrious +Philosopher, neither good conduct nor bad conduct is a fit subject, in any +sober estimation, for either praise or blame. For why should you praise, for +example, the integrity of a Square who faithfully defends the interests of his +client, when you ought in reality rather to admire the exact precision of his +right angles? Or again, why blame a lying, thievish Isosceles, when you ought +rather to deplore the incurable inequality of his sides? + +Theoretically, this doctrine is unquestionable; but it has practical +drawbacks. In dealing with an Isosceles, if a rascal pleads that he cannot +help stealing because of his unevenness, you reply that for that very reason, +because he cannot help being a nuisance to his neighbours, you, the +Magistrate, cannot help sentencing him to be consumed --- and there's an end of +the matter. But in little domestic difficulties, when the penalty of +consumption, or death, is out of the question, this theory of Configuration +sometimes comes in awkwardly; and I must confess that occasionally when one of +my own Hexagonal Grandsons pleads as an excuse for his disobedience that a +sudden change of temperature has been too much for his Perimeter, and that I +ought to lay the blame not on him but on his Configuration, which can only be +strengthened by abundance of the choicest sweetmeats, I neither see my way +logically to reject, nor practically to accept, his conclusions. + +For my own part, I find it best to assume that a good sound scolding or +castigation has some latent and strengthening influence on my Grandson's +Configuration; though I own that I have no grounds for thinking so. At all +events I am not alone in my way of extricating myself from this dilemma; for I +find that many of the highest Circles, sitting as Judges in law courts, use +praise and blame towards Regular and Irregular Figures; and in their homes I +know by experience that, when scolding their children, they speak about +``right'' and ``wrong'' as vehemently and passionately as if they believe that +these names represented real existence, and that a human Figure is really +capable of choosing between them. + +Constantly carrying out their policy of making Configuration the leading idea +in every mind, the Circles reverse the nature of that Commandment which in +Spaceland regulates the relations between parents and children. With you, +children are taught to honour their parents; with us --- next to the Circles, +who are the chief object of universal homage --- a man is taught to honour his +Grandson, if he has one; or, if not, his Son. By ``honour'', however, is by no +means mean ``indulgence'', but a reverent regard for their highest interests: +and the Circles teach that the duty of fathers is to subordinate their own +interests to those of posterity, thereby advancing the welfare of the whole +State as well as that of their own immediate descendants. + +The weak point in the system of the Circles --- if a humble Square may venture +to speak of anything Circular as containing any element of weakness --- appears +to me to be found in their relations with Women. + +As it is of the utmost importance for Society that Irregular births should be +discouraged, it follows that no Woman who has any Irregularities in her +ancestry is a fit partner for one who desires that his posterity should rise +by regular degrees in the social scale. + +Now the Irregularity of a Male is a matter of measurement; but as all Women +are straight, and therefore visibly Regular so to speak, one has to device +some other means of ascertaining what I may call their invisible Irregularity, +that is to say their potential Irregularities as regards possible offspring. +This is effected by carefully-kept pedigrees, which are preserved and +supervised by the State; and without a certified pedigree no Woman is allowed +to marry. + +Now it might have been supposed the a Circle --- proud of his ancestry and +regardful for a posterity which might possibly issue hereafter in a Chief +Circle --- would be more careful than any other to choose a wife who had no blot +on her escutcheon. But it is not so. The care in choosing a Regular wife +appears to diminish as one rises in the social scale. Nothing would induce an +aspiring Isosceles, who has hopes of generating an Equilateral Son, to take a +wife who reckoned a single Irregularity among her Ancestors; a Square or +Pentagon, who is confident that his family is steadily on the rise, does not +inquire above the five-hundredth generation; a Hexagon or Dodecagon is even +more careless of the wife's pedigree; but a Circle has been known deliberately +to take a wife who has had an Irregular Great-Grandfather, and all because of +some slight superiority of lustre, or because of the charms of a low voice --- +which, with us, even more than with you, is thought ``an excellent thing in a +Woman''. + +Such ill-judged marriages are, as might be expected, barren, if they do not +result in positive Irregularity or in diminution of sides; but none of these +evils have hitherto provided sufficiently deterrent. The loss of a few sides +in a highly-developed Polygon is not easily noticed, and is sometimes +compensated by a successful operation in the Neo-Therapeutic Gymnasium, as I +have described above; and the Circles are too much disposed to acquiesce in +infecundity as a law of the superior development. Yet, if this evil be not +arrested, the gradual diminution of the Circular class may soon become more +rapid, and the time may not be far distant when, the race being no longer able +to produce a Chief Circle, the Constitution of Flatland must fall. + +One other word of warning suggest itself to me, though I cannot so easily +mention a remedy; and this also refers to our relations with Women. About +three hundred years ago, it was decreed by the Chief Circle that, since women +are deficient in Reason but abundant in Emotion, they ought no longer to be +treated as rational, nor receive any mental education. The consequence was +that they were no longer taught to read, nor even to master Arithmetic enough +to enable them to count the angles of their husband or children; and hence +they sensibly declined during each generation in intellectual power. And this +system of female non-education or quietism still prevails. + +My fear is that, with the best intentions, this policy has been carried so far +as to react injuriously on the Male Sex. + +For the consequence is that, as things now are, we Males have to lead a kind +of bi-lingual, and I may almost say bimental, existence. With Women, we speak +of ``love'', ``duty'', ``right'', ``wrong'', ``pity'', ``hope'', and other irrational and +emotional conceptions, which have no existence, and the fiction of which has +no object except to control feminine exuberances; but among ourselves, and in +our books, we have an entirely different vocabulary and I may also say, idiom. +``Love'' them becomes ``the anticipation of benefits''; ``duty'' becomes ``necessity'' +or ``fitness''; and other words are correspondingly transmuted. Moreover, among +Women, we use language implying the utmost deference for their Sex; and they +fully believe that the Chief Circle Himself is not more devoutly adored by us +than they are: but behind their backs they are both regarded and spoken of --- +by all but the very young --- as being little better than ``mindless organisms''. + +Our Theology also in the Women's chambers is entirely different from our +Theology elsewhere. + +Now my humble fear is that this double training, in language as well as in +thought, imposes somewhat too heavy a burden upon the young, especially when, +at the age of three years old, they are taken from the maternal care and +taught to unlearn the old language --- except for the purpose of repeating it in +the presence of the Mothers and Nurses --- and to learn the vocabulary and idiom +of science. Already methinks I discern a weakness in the grasp of mathematical +truth at the present time as compared with the more robust intellect of our +ancestors three hundred years ago. I say nothing of the possible danger if a +Woman should ever surreptitiously learn to read and convey to her Sex the +result of her perusal of a single popular volume; nor of the possibility that +the indiscretion or disobedience of some infant Male might reveal to a Mother +the secrets of the logical dialect. On the simple ground of the enfeebling of +the male intellect, I rest this humble appeal to the highest Authorities to +reconsider the regulations of Female education. + + + + +\part{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 day +of the Long Vacation. Having amused myself till a late hour with my favourite +recreation of Geometry, I had retired to rest with an unsolved problem in my +mind. In the night I had a dream. + +I saw before me a vast multitude of small Straight Lines (which I naturally +assumed to be Women) interspersed with other Beings still smaller and of the +nature of lustrous points --- all moving to and fro in one and the same Straight +Line, and, as nearly as I could judge, with the same velocity. + +A noise of confused, multitudinous chirping or twittering issued from them at +intervals as long as they were moving; but sometimes they ceased from motion, +and then all was silence. + +Approaching one of the largest of what I thought to be Women, I accosted her, +but received no answer. A second and third appeal on my part were equally +ineffectual. Losing patience at what appeared to me intolerable rudeness, I +brought my mouth to a position full in front of her mouth so as to intercept +her motion, and loudly repeated my question, ``Woman, what signifies this +concourse, and this strange and confused chirping, and this monotonous motion +to and fro in one and the same Straight Line?'' + +``I am no Woman'', replied the small Line: ``I am the Monarch of the world. But +thou, whence intrudest thou into my realm of Lineland?'' Receiving this abrupt +reply, I begged pardon if I had in any way startled or molested his Royal +Highness; and describing myself as a stranger I besought the King to give me +some account of his dominions. But I had the greatest possible difficulty in +obtaining any information on points that really interested me; for the Monarch +could not refrain from constantly assuming that whatever was familiar to him +must also be known to me and that I was simulating ignorance in jest. However, +by preserving questions I elicited the following facts: + +\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth, scale=0.4]{fig6} + + +It seemed that this poor ignorant Monarch --- as he called himself --- was +persuaded that the Straight Line which he called his Kingdom, and in which he +passed his existence, constituted the whole of the world, and indeed the whole +of Space. Not being able either to move or to see, save in his Straight Line, +he had no conception of anything out of it. Though he had heard my voice when +I first addressed him, the sounds had come to him in a manner so contrary to +his experience that he had made no answer, ``seeing no man,'' as he expressed +it, ``and hearing a voice as it were from my own intestines.'' Until the moment +when I placed my mouth in his World, he had neither seen me, nor heard +anything except confused sounds beating against, what I called his side, but +what he called his inside or stomach; nor had he even now the least conception +of the region from which I had come. Outside his World, or Line, all was a +blank to him; nay, not even a blank, for a blank implies Space; say, rather, +all was non-existent. + +His subjects --- of whom the small Lines were men and the Points Women --- were +all alike confined in motion and eyesight to that single Straight Line, which +was their World. It need scarcely be added that the whole of their horizon was +limited to a Point; nor could any one ever see anything but a Point. Man, +woman, child, thing --- each as a Point to the eye of a Linelander. Only by the +sound of the voice could sex or age be distinguished. Moreover, as each +individual occupied the whole of the narrow path, so to speak, which +constituted his Universe, and no one could move to the right or left to make +way for passers by, it followed that no Linlander could ever pass another. +Once neighbours, always neighbours. Neighbourhood with them was like marriage +with us. Neighbours remained neighbours till death did them part. + +Such a life, with all vision limited to a Point, and all motion to a Straight +Line, seemed to me inexpressibly dreary; and I was surprised to note that +vivacity and cheerfulness of the King. Wondering whether it was possible, amid +circumstances so unfavourable to domestic relations, to enjoy the pleasures of +conjugal union, I hesitated for some time to question his Royal Highness on so +delicate a subject; but at last I plunged into it by abruptly inquiring as to +the health of his family. ``My wives and children,'' he replied, ``are well and +happy.'' + +Staggered at this answer --- for in the immediate proximity of the Monarch (as I +had noted in my dream before I entered Lineland) there were none but Men --- I +ventured to reply, ``Pardon me, but I cannot imagine how your Royal Highness +can at any time either See or approach their Majesties, when there at least +half a dozen intervening individuals, whom you can neither see through, nor +pass by? Is it possible that in Lineland proximity is not necessary for +marriage and for the generation of children?'' + +``How can you ask so absurd a question?'' replied the Monarch. ``If it were +indeed as you suggest, the Universe would soon be depopulated. No, no; +neighbourhood is needless for the union of hearts; and the birth of children +is too important a matter to have been allowed to depend upon such an accident +as proximity. You cannot be ignorant of this. Yet since you are pleased to +affect ignorance, I will instruct you as if you were the veriest baby in +Lineland. Know, then, that marriages are consummated by means of the faculty +of sound and the sense of hearing. + +``You are of course aware that every Man has two mouths or voices --- as well as +two eyes --- a bass at one and a tenor at the other of his extremities. I should +not mention this, but that I have been unable to distinguish your tenor in the +course of our conversation.'' I replied that I had but one voice, and that I +had not been aware that his Royal Highness had two. ``That confirms by +impression,'' said the King, ``that you are not a Man, but a feminine +Monstrosity with a bass voice, and an utterly uneducated ear. But to continue. + +``Nature having herself ordained that every Man should wed two wives ---'' ``Why +two?'' asked I. ``You carry your affected simplicity too far,'' he cried. ``How +can there be a completely harmonious union without the combination of the Four +in One, viz. the Bass and Tenor of the Man and the Soprano and Contralto of +the two Women?'' ``But supposing,'' said I, ``that a man should prefer one wife or +three?'' ``It is impossible,'' he said; ``it is as inconceivable as that two and +one should make five, or that the human eye should see a Straight Line.'' I +would have interrupted him; but he proceeded as follows: + +``Once in the middle of each week a Law of Nature compels us to move to and fro +with a rhythmic motion of more than usual violence, which continues for the +time you would take to count a hundred and one. In the midst of this choral +dance, at the fifty-first pulsation, the inhabitants of the Universe pause in +full career, and each individual sends forth his richest, fullest, sweetest +strain. It is in this decisive moment that all our marriages are made. So +exquisite is the adaptation of Bass and Treble, of Tenor to Contralto, that +oftentimes the Loved Ones, though twenty thousand leagues away, recognize at +once the responsive note of their destined Lover; and, penetrating the paltry +obstacles of distance, Love unites the three. The marriage in that instance +consummated results in a threefold Male and Female offspring which takes its +place in Lineland.'' + +``What! Always threefold?'' said I. ``Must one wife then always have twins?'' + +``Bass-voice Monstrosity! yes,'' replied the King. ``How else could the balance +of the Sexes be maintained, if two girls were not born for every boy? Would +you ignore the very Alphabet of Nature?'' He ceased, speechless for fury; and +some time elapsed before I could induce him to resume his narrative. + +``You will not, of course, suppose that every bachelor among us finds his mates +at the first wooing in this universal Marriage Chorus. On the contrary, the +process is by most of us many times repeated. Few are the hearts whose happy +lot is at once to recognize in each other's voice the partner intended for +them by Providence, and to fly into a reciprocal and perfectly harmonious +embrace. With most of us the courtship is of long duration. The Wooer's voices +may perhaps accord with one of the future wives, but not with both; or not, at +first, with either; or the Soprano and Contralto may not quite harmonize. In +such cases Nature has provided that every weekly Chorus shall bring the three +Lovers into closer harmony. Each trial of voice, each fresh discovery of +discord, almost imperceptibly induces the less perfect to modify his or her +vocal utterance so as to approximate to the more perfect. And after many +trials and many approximations, the result is at last achieved. There comes a +day at last when, while the wonted Marriage Chorus goes forth from universal +Lineland, the three far-off Lovers suddenly find themselves in exact harmony, +and, before they are aware, the wedded Triplet is rapt vocally into a +duplicate embrace; and Nature rejoices over one more marriage and over three +more 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 the +level of common sense, I determined to endeavour to open up to him some +glimpses of the truth, that is to say of the nature of things in Flatland. So +I began thus: ``How does your Royal Highness distinguish the shapes and +positions of his subjects? I for my part noticed by the sense of sight, before +I entered your Kingdom, that some of your people are lines and others Points; +and that some of the lines are larger ---'' ``You speak of an impossibility,'' +interrupted the King; ``you must have seen a vision; for to detect the +difference between a Line and a Point by the sense of sight is, as every one +knows, in the nature of things, impossible; but it can be detected by the +snese 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 to +argument, you shall hear with your ears how by means of my two voices I reveal +my shape to my Wives, who are at this moment six thousand miles seventy yards +two feet eight inches away, the one to the North, the other to the South. +Listen, I call to them.'' + +He chirruped, and then complacently continued: ``My wives at this moment +receiving the sound of one of my voice, closely followed by the other, and +perceiving that the latter reaches them after an interval in which sound can +traverse 6.457 inches, infer that one of my mouths is 6.457 inches further +from them than the other, and accordingly know my shape to be 6.457 inches. +But you will of course understand that my wives do not make this calculation +every time they hear my two voices. They made it, once for all, before we were +married. But they could make it at any time. And in the same way I can +estimate the shape of any of my Male subjects by the sense of sound.'' + +``But how,'' said I, ``if a Man feigns a Woman's voice with one of his two +voices, or so disguises his Southern voice that it cannot be recognized as the +echo of the Northern? May not such deceptions cause great inconvenience? And +have you no means of checking frauds of this kind by commanding your +neighbouring subjects to feel one another?'' This of course was a very stupid +question, for feeling could not have answered the purpose; but I asked with +the view of irritating the Monarch, and I succeeded perfectly. + +``What!'' cried he in horror, ``explain your meaning.'' ``Feel, touch, come into +contact,'' I replied.. ``If you mean by feeling,`` said the King, ``approaching so +close as to leave no space between two individuals, know, Stranger, that this +offence is punishable in my dominions by death. And the reason is obvious. The +frail form of a Woman, being liable to be shattered by such an approximation, +must be preserved by the State; but since Women cannot be distinguished by the +sense of sight from Men, the Law ordains universally that neither Man nor +Woman shall be approached so closely as to destroy the interval between the +approximator and the approximated. + +``And indeed what possible purpose would be served by this illegal and +unnatural excess of approximation which you call touching, when all the ends +of so brutal and course a process are attained at once more easily and more +exactly by the sense of hearing? As to your suggested danger of deception, it +is non-existent: for the Voice, being the essence of one's Being, cannot be +thus changed at will. But come, suppose that I had the power of passing +through solid things, so that I could penetrate my subjects, one after +another, even to the number of a billion, verifying the size and distance of +each by the sense of feeling: How much time and energy would be wasted in this +clumsy and inaccurate method! Whereas now, in one moment of audition, I take +as it were the census and statistics, local, corporeal, mental and spiritual, +of every living being in Lineland. Hark, only hark!'' + +So saying he paused and listened, as if in an ecstasy, to a sound which seemed +to me no better than a tiny chirping from an innumerable multitude of +lilliputian grasshoppers. + +``Truly,'' replied I, ``your sense of hearing serves you in good stead, and fills +up many of your deficiencies. But permit me to point out that your life in +Lineland must be deplorably dull. To see nothing but a Point! Not even to be +able to contemplate a Straight Line! Nay, not even to know what a Straight +Line is! To see, yet to be cut off from those Linear prospects which are +vouchsafed to us in Flatland! Better surely to have no sense of sight at all +than to see so little! I grant you I have not your discriminative faculty of +hearing; for the concert of all Lineland which gives you such intense +pleasure, is to me no better than a multitudinous twittering or chirping. But +at least I can discern, by sight, a Line from a Point. And let me prove it. +Just before I came into your kingdom, I saw you dancing from left to right, +and then from right to left, with Seven Men and a Woman in your immediate +proximity on the left, and eight Men and two Women on your right. Is not this +correct?'' + +``It is correct,'' said the King, ``so far as the numbers and sexes are +cocnerned, though I know not what you mean by `right' and `left.' But I deny +that you saw these things. For how could you see the Line, that is to say the +inside, of any Man? But you must have heard these things, and then dreamed +that you saw them. And let me ask what you mean by those words `left' and +`right'. I suppose it is your way of saying Northward and Southward.'' + +``Not so,'' replied I; ``besides your motion of Northward and Southward, there is +another motion which I call from right to left.'' + +King. Exhibit to me, if you please, this motion from left to right. + +I. Nay, that I cannot do, unless you could 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 the +true Space. True Space is a Plane; but your Space is only a Line. + +King. If you cannot indicate this motion from left to right by yourself moving +in it, then I beg you to describe it to me in words. + +I. If you cannot tell your right side from your left, I fear that no words of +mine can make my meaning clearer to you. But surely you cannot be ignorant of +so simple a distinction. + +King. I do not in the least understand you. + +I. Alas! How shall I make it clear? When you move straight on, does it not +sometimes occur to you that you could move in some other way, turning your eye +round so as to look in the direction towards which your side is now fronting? +In other words, instead of always moving in the direction of one of your +extremities, do you never feel a desire to move in the direction, so to speak, +of your side? + +King. Never. And what do you mean? How can a man's inside ``front'' in any +direction? Or how can a man move in the direction of his inside? + +I. Well then, since words cannot explain the matter, I will try deeds, and +will move gradually out of Lineland in the direction which I desire to +indicate to you. + +At the word I began to move my body out of Lineland. As long as any part of me +remained in his dominion and in his view, the King kept exclaiming, ``I see +you, I see you still; you are not moving.'' But when I had at last moved myself +out of his Line, he cried in his shrillest voice, ``She is vanished; she is +dead.'' ``I am not dead,'' replied I; ``I am simply out of Lineland, that is to +say, out of the Straight Line which you call Space, and in the true Space, +where I can see things as they are. And at this moment I can see your Line, or +side --- or inside as you are pleased to call it; and I can see also the Men and +Women on the North and South of you, whom I will now enumerate, describing +their order, their size, and the interval between each.'' + +\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig7} + +When I had done this at great length, I cried triumphantly, ``Does that at last +convince you?'' And, with that, I once more entered Lineland, taking up the +same position as before. + +But the Monarch replied, ``If you were a Man of sense --- though, as you appear +to have only one voice I have little doubt you are not a Man but a Woman --- +but, if you had a particle of sense, you would listen to reason. You ask me to +believe that there is another Line besides that which my senses indicate, and +another motion besides that of which I am daily conscious. I, in return, ask +you to describe in words or indicate by motion that other Line of which you +speak. Instead of moving, you merely exercise some magic art of vanishing and +returning to sight; and instead of any lucid description of your new World, +you simply tell me the numbers and sizes of some forty of my retinue, facts +known to any child in my capital. Can anything be more irrational or +audacious? Acknowledge your folly or depart from my dominions.'' + +Furious at his perversity, and especially indignant that he professed to be +ignorant of my sex, I retorted in no measured terms, ``Besotted Being! You +think yourself the perfection of existence, while you are in reality the most +imperfect and imbecile. You profess to see, whereas you see nothing but a +Point! You plume yourself on inferring the existence of a Straight Line; but I +can see Straight Lines, and infer the existence of Angles, Triangles, Squares, +Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I +am the completion of your incomplete self. You are a Line, but I am a Line of +Lines called in my country a Square: and even I, infinitely superior though I +am to you, am of little account among the great nobles of Flatland, whence I +have come to visit you, in the hope of enightening your ignorance.'' + +Hearing these words the King advanced towards me with a menacing cry as if to +pierce me through the diagonal; and in that same movement there arose from +myriads of his subjects a multitudinous war-cry, increasing in vehemence till +at last methought it rivalled the roar of an army of a hundred thousand +Isosceles, and the artillery of a thousand Pentagons. Spell-bound and +motionless, I could neither speak nor move to avert the impending destruction; +and still the noise grew louder, and the King came closer, when I awoke to +find the breakfast-bell recalling me to the realities of Flatland. + + + + + + + +\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 rain +had long ago announced nightfall; and I was sitting \footnote{ When I say +``sitting,'' of course I do not mean any change of attitude such as you in +Spaceland signify by that word; for as we have no feet, we can no more ``sit'' +nor ``stand'' (in your sense of the word) than one of your soles or flounders. + +Nevertheless, we perfectly well recognize the different mental states of +volition implied by ``lying,'' ``sitting,'' and ``standing,'' which are to some +extent indicated to a beholder by a slight increase of lustre corresponding to +the increase of volition. + +But on this, and a thousand other kindred subjects, time forbids me to dwell.} +in the company of my wife, musing on the events of the past and the prospects +of the coming year, the coming century, the coming Millennium. + +My four Sons and two orphan Grandchildren had retired to their several +apartments; and my wife alone remained with me to see the old Millennium out +and the new one in. + +I was rapt in thought, pondering in my mind some words that had casually +issued from the mouth of my youngest Grandson, a most promising young Hexagon +of unusual brilliancy and perfect angularity. His uncles and I had been giving +him his usual practical lesson in Sight Recognition, turning ourselves upon +our centres, now rapidly, now more slowly, and questioning him as to our +positions; and his answers had been so satisfactory that I had been induced to +reward him by giving him a few hints on Arithmetic, as applied to Geometry. + +Taking nine Squares, each an inch every way, I had put them together so as to +make one large Square, with a side of three inches, and I had hence proved to +my little Grandson that --- though it was impossible for us to see the inside of +the Square --- yet we might ascertain the number of square inches in a Square by +simply squaring the number of inches in the side: ``and thus,'' said I, ``we know +that three-to-the-second, or nine, represents the number of square inches in a +Square whose side is three inches long.'' + +The little Hexagon meditated on this a while and then said to me; ``But you +have been teaching me to raise numbers to the third power: I suppose +three-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 only +Two Dimensions.'' And then I began to shew the boy how a Point by moving +through a length of three inches makes a Line of three inches, which may be +represented by three; and how a Line of three inches, moving parallel to +itself through a length of three inches, makes a Square of three inches every +way, which may be represented by three-to-the-second. + +Upon this, my Grandson, again returning to his former suggestion, took me up +rather suddenly and exclaimed, ``Well, then, if a Point by moving three inches, +makes a Line of three inches represented by three; and if a straight Line of +three inches, moving parallel to itself, makes a Square of three inches every +way, represented by three-to-the-second; it must be that a Square of three +inches every way, moving somehow parallel to itself (but I don't see how) must +make Something else (but I don't see what) of three inches every way --- and +this must be represented by three-to-the-third.'' + +``Go to bed,'' said I, a little ruffled by this interruption: ``if you would talk +less nonsense, you would remember more sense.'' + +So my Grandson had disappeared in disgrace; and there I sat by my Wife's side, +endeavouring to form a retrospect of the year 1999 and of the possibilities of +the year 2000; but not quite able to shake of the thoughts suggested by the +prattle of my bright little Hexagon. Only a few sands now remained in the +half-hour glass. Rousing myself from my reverie I turned the glass Northward +for the last time in the old Millennium; and in the act, I exclaimed aloud, +``The boy is a fool.'' + +Straightway I became conscious of a Presence in the room, and a chilling +breath thrilled through my very being. ``He is no such thing,'' cried my Wife, +``and you are breaking the Commandments in thus dishonouring your own +Grandson.'' But I took no notice of her. Looking around in every direction I +could see nothing; yet still I felt a Presence, and shivered as the cold +whisper came again. I started up. ``What is the matter?'' said my Wife, ``there +is no draught; what are you looking for? There is nothing.'' There was nothing; +and I resumed my seat, again exclaiming, ``The boy is a fool, I say; +three-to-the-third can have no meaning in Geometry.'' At once there came a +distinctly audible reply, ``The boy is not a fool; and three-to-the-third has +an obvious Geometrical meaning.'' + +My Wife as well as myself heard the words, although she did not understand +their meaning, and both of us sprang forward in the direction of the sound. +What was our horror when we saw before us a Figure! At the first glance it +appeared to be a Woman, seen sideways; but a moment's observation shewed me +that the extremities passed into dimness too rapidly to represent one of the +Female Sex; and I should have thought it a Circle, only that it seemed to +change its size in a manner impossible for a Circle or for any regular Figure +of which I had had experience. + +But my Wife had not my experience, nor the coolness necessary to note these +characteristics. With the usual hastiness and unreasoning jealousy of her Sex, +she flew at once to the conclusion that a Woman had entered the house through +some small aperture. ``How comes this person here?'' she exclaimed, ``you +promised me, my dear, that there should be no ventilators in our new house.'' +``Nor are they any,'' said I; ``but what makes you think that the stranger is a +Woman? I see by my power of Sight Recognition ---'' ``Oh, I have no patience with +your Sight Recognition,'' replied she, ```Feeling is believing' and `A Straight +Line to the touch is worth a Circle to the sight''' --- two Proverbs, very common +with the Frailer Sex in Flatland. + +``Well,'' said I, for I was afraid of irritating her, ``if it must be so, demand +an introduction.'' Assuming her most gracious manner, my Wife advanced towards +the Stranger, ``Permit me, Madam to feel and be felt by ---'' then, suddenly +recoiling, ``Oh! it is not a Woman, and there are no angles either, not a trace +of one. Can it be that I have so misbehaved to a perfect Circle?'' + +``I am indeed, in a certain sense a Circle,'' replied the Voice, ``and a more +perfect Circle than any in Flatland; but to speak more accurately, I am many +Circles in one.'' Then he added more mildly, ``I have a message, dear Madam, to +your husband, which I must not deliver in your presence; and, if you would +suffer us to retire for a few minutes ---'' But my wife would not listen to the +proposal that our august Visitor should so incommode himself, and assuring the +Circle that the hour of her own retirement had long passed, with many +reiterated apologies for her recent indiscretion, she at last retreated to her +apartment. + +I glanced at the half-hour glass. The last sands had fallen. The third +Millennium 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, I +began to approach the Stranger with the intention of taking a nearer view and +of bidding him be seated: but his appearance struck me dumb and motionless +with astonishment. Without the slightest symptoms of angularity he +nevertheless varied every instant with graduations of size and brightness +scarcely possible for any Figure within the scope of my experience. The +thought flashed across me that I might have before me a burglar or cut-throat, +some monstrous Irregular Isosceles, who, by feigning the voice of a Circle, +had obtained admission somehow into the house, and was now preparing to stab +me with his acute angle. + +In a sitting-room, the absence of Fog (and the season happened to be +remarkably dry), made it difficult for me to trust to Sight Recognition, +especially at the short distance at which I was standing. Desperate with fear, +I rushed forward with an unceremonious, ``You must permit me, Sir ---'' and felt +him. My Wife was right. There was not the trace of an angle, not the slightest +roughness or inequality: never in my life had I met with a more perfect +Circle. He remained motionless while I walked around him, beginning from his +eye and returning to it again. Circular he was throughout, a perfectly +satisfactory Circle; there could not be a doubt of it. Then followed a +dialogue, which I will endeavour to set down as near as I can recollect it, +omitting only some of my profuse apologies --- for I was covered with shame and +humiliation that I, a Square, should have been guilty of the impertinence of +feeling a Circle. It was commenced by the Stranger with some impatience at the +lengthiness of my introductory process. + +STRANGER. Have you felt me enough by this time? Are you not introduced to me +yet? + +I. Most illustrious Sir, excuse my awkwardness, which arises not from +ignorance of the usages of polite society, but from a little surprise and +nervousness, consequent on this somewhat unexpected visit. And I beseech you +to reveal my indiscretion to no one, and especially not to my Wife. But before +your Lordship enters into further communications, would he deign to satisfy +the curiosity of one who would gladly know whence his visitor came? + +STRANGER. From Space, from Space, Sir: whence else? + +I. Pardon me, my Lord, but is not your Lordship already in Space, your +Lordship and his humble servant, even at this moment? + +STRANGER. Pooh! what do you know of Space? Define Space. + +I. Space, my Lord, is height and breadth indefinitely prolonged. + +STRANGER. Exactly: you see you do not even know what Space is. You think it is +of Two Dimensions only; but I have come to announce to you a Third --- height, +breadth, and length. + +I. Your Lordship is pleased to be merry. We also speak of length and height, +or breadth and thickness, thus denoting Two Dimensions by four names. + +STRANGER. But I mean not only three names, but Three Dimensions. + +I. Would your Lordship indicate or explain to me in what direction is the +Third Dimension, unknown to me? + +STRANGER. I came from it. It is up above and down below. + +I. My Lord means seemingly that it is Northward and Southward. + +STRANGER. I mean nothing of the kind. I mean a direction in which you cannot +look, because you have no eye in your side. + +I. Pardon me, my Lord, a moment's inspection will convince your Lordship that +I have a 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 on +your Perimeter, but on your side, that is, on what you would probably call +your inside; but we in Spaceland should call it your side. + +I. An eye in my inside! An eye in my stomach! Your Lordship jests. + +STRANGER. I am in no jesting humour. I tell you that I come from Space, or, +since you will not understand what Space means, from the Land of Three +Dimensions whence I but lately looked down upon your Plane which you call +Space forsooth. From that position of advantage I discerned all that you speak +of as solid (by which you mean ``enclosed on four sides''), your houses, your +churches, your very chests and safes, yes even your insides and stomachs, all +lying open and exposed to my view. + +I. Such assertions are easily made, my Lord. + +STRANGER. But not easily proved, you mean. But I mean to prove mine. + +When I descended here, I saw your four Sons, the Pentagons, each in his +apartment, and your two Grandsons the Hexagons; I saw your youngest Hexagon +remain a while with you and then retire to his room, leaving you and your Wife +alone. I saw your Isosceles servants, three in number, in the kitchen at +supper, and the little Page in the scullery. Then I came here, and how do you +think I came? + +I. Through the roof, I suppose. + +STRANGER. Not so. Your roof, as you know very well, has been recently +repaired, and has no aperture by which even a Woman could penetrate. I tell +you I come from Space. Are you not convinced by what I have told you of your +children and household? + +I. Your Lordship must be aware that such facts touching the belongings of his +humble servant might be easily ascertained by any one of the neighbourhood +possessing your Lordship's ample means of information. + +STRANGER. (To himself.) What must I do? Stay; one more argument suggests +itself to me. When you see a Straight Line --- your wife, for example --- how many +Dimensions do you attribute to her? + +I. Your Lordship would treat me as if I were one of the vulgar who, being +ignorant of Mathematics, suppose that a Woman is really a Straight Line, and +only of One Dimension. No, no, my Lord; we Squares are better advised, and are +as well aware of your Lordship that a Woman, though popularly called a +Straight Line, is, really and scientifically, a very thin Parallelogram, +possessing Two Dimensions, like the rest of us, viz., length and breadth (or +thickness). + +STRANGER. But the very fact that a Line is visible implies that it possesses +yet another Dimension. + +I. My Lord, I have just acknowledge that a Woman is broad as well as long. We +see her length, we infer her breadth; which, though very slight, is capable of +measurement. + +STRANGER. You do not understand me. I mean that when you see a Woman, you +ought --- besides inferring her breadth --- to see her length, and to see what we +call her height; although the last Dimension is infinitesimal in your country. +If a Line were mere length without ``height,'' it would cease to occupy Space +and 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 the +brightness disappears, the Line is extinguished, and, as you say, ceases to +occupy Space. But am I to suppose that your Lordship gives the brightness the +title of a Dimension, and that what we call ``bright'' you call ``high''? + +STRANGER. No, indeed. By ``height'' I mean a Dimension like your length: only, +with you, ``height'' is not so easily perceptible, being extremely small. + +I. My Lord, your assertion is easily put to the test. You say I have a Third +Dimension, which you call ``height.'' Now, Dimension implies direction and +measurement. Do but measure my ``height,'' or merely indicate to me the +direction in which my ``height'' extends, and I will become your convert. +Otherwise, your Lordship's own understanding must hold me excused. + +STRANGER. (To himself.) I can do neither. How shall I convince him? Surely a +plain statement of facts followed by ocular demonstration ought to suffice. --- +Now, Sir; listen to me. + +You are living on a Plane. What you style Flatland is the vast level surface +of what I may call a fluid, or in, the top of which you and your countrymen +move 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 I +am not a Circle, but an infinite number of Circles, of size varying from a +Point to a Circle of thirteen inches in diameter, one placed on the top of the +other. When I cut through your plane as I am now doing, I make in your plane a +section which you, very rightly, call a Circle. For even a Sphere --- which is +my proper name in my own country --- if he manifest himself at all to an +inhabitant of Flatland --- must needs manifest himself as a Circle. + +Do you not remember --- for I, who see all things, discerned last night the +phantasmal vision of Lineland written upon your brain --- do you not remember, I +say, how when you entered the realm of Lineland, you were compelled to +manifest yourself to the King, not as a Square, but as a Line, because that +Linear Realm had not Dimensions enough to represent the whole of you, but only +a slice or section of you? In precisely the same way, your country of Two +Dimensions is not spacious enough to represent me, a being of Three, but can +only exhibit a slice or section of me, which is what you call a Circle. + +The diminished brightness of your eye indicates incredulity. But now prepare +to receive proof positive of the truth of my assertions. You cannot indeed see +more than one of my sections, or Circles, at a time; for you have no power to +raise your eye out of the plane of Flatland; but you can at least see that, as +I rise in Space, so my sections become smaller. See now, I will rise; and the +effect upon your eye will be that my Circle will become smaller and smaller +till it dwindles to a point and finally vanishes. + +\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig8} + +There was no ``rising'' that I could see; but he diminished and finally +vanished. I winked once or twice to make sure that I was not dreaming. But it +was no dream. For from the depths of nowhere came forth a hollow voice --- close +to my heart it seemed --- ``Am I quite gone? Are you convinced now? Well, now I +will gradually return to Flatland and you shall see my section become larger +and larger.'' + +Every reader in Spaceland will easily understand that my mysterious Guest was +speaking the language of truth and even of simplicity. But to me, proficient +though I was in Flatland Mathematics, it was by no means a simple matter. The +rough diagram given above will make it clear to any Spaceland child that the +Sphere, ascending in the three positions indicated there, must needs have +manifested himself to me, or to any Flatlander, as a Circle, at first of full +size, then small, and at last very small indeed, approaching to a Point. But +to me, although I saw the facts before me, the causes were as dark as ever. +All that I could comprehend was, that the Circle had made himself smaller and +vanished, and that he had now re-appeared and was rapidly making himself +larger. + +When he regained his original size, he heaved a deep sigh; for he perceived by +my silence that I had altogether failed to comprehend him. And indeed I was +now inclining to the belief that he must be no Circle at all, but some +extremely clever juggler; or else that the old wives' tales were true, and +that after all there were such people as Enchanters and Magicians. + +After a long pause he muttered to himself, ``One resource alone remains, if I +am not to resort to action. I must try the method of Analogy.'' Then followed a +still longer silence, after which he continued our dialogue. + +SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a +luminous wake, what name would you give to the wake? + +I. A straight Line. + +SPHERE. And a straight Line has how many extremities? + +I. Two. + +SPHERE. Now conceive the Northward straight Line moving parallel to itself, +East and West, so that every point in it leaves behind it the wake of a +straight Line. What name will you give to the Figure thereby formed? We will +suppose that it moves through a distance equal to the original straight line. +--- What name, I say? + +I. A square. + +SPHERE. And how many sides has a Square? How many angles? + +I. Four sides and four angles. + +SPHERE. Now stretch your imagination a little, and conceive a Square in +Flatland, moving parallel to itself upward. + +I. What? Northward? + +SPHERE. No, not Northward; upward; out of Flatland altogether. + +If it moved Northward, the Southern points in the Square would have to move +through the positions previously occupied by the Northern points. But that is +not my meaning. + +I mean that every Point in you --- for you are a Square and will serve the +purpose of my illustration --- every Point in you, that is to say in what you +call your inside, is to pass upwards through Space in such a way that no Point +shall pass through the position previously occupied by any other Point; but +each Point shall describe a straight Line of its own. This is all in +accordance with Analogy; surely it must be clear to you. + +Restraining my impatience --- for I was now under a strong temptation to rush +blindly at my Visitor and to precipitate him into Space, or out of Flatland, +anywhere, so that I could get rid of him --- I replied: --- + +``And what may be the nature of the Figure which I am to shape out by this +motion which you are pleased to denote by the word `upward'? I presume it is +describable in the language of Flatland.'' + +SPHERE. Oh, certainly. It is all plain and simple, and in strict accordance +with Analogy --- only, by the way, you must not speak of the result as being a +Figure, but as a Solid. But I will describe it to you. Or rather not I, but +Analogy. + +We began with a single Point, which of course --- being itself a Point --- has +only one terminal Point. + +One Point produces a Line with two terminal Points. + +One Line produces a Square with four terminal Points. + +Now you can give yourself the answer to your own question: 1, 2, 4, are +evidently in Geometrical Progression. What is the next number? + +I. Eight. + +SPHERE. Exactly. The one Square produces a +Something-which-you-do-not-as-yet-know-a-name-for-but-which-we-call-a-Cube +with eight terminal Points. Now are you convinced? + +I. And has this Creature sides, as well as Angles or what you call ``terminal +Points''? + +SPHERE. Of course; and all according to Analogy. But, by the way, not what you +call sides, but what we call sides. You would call them solids. + +I. And how many solids or sides will appertain to this Being whom I am to +generate by the motion of my inside in an ``upward'' direction, and whom you +call a Cube? + +SPHERE. How can you ask? And you a mathematician! The side of anything is +always, if I may so say, one Dimension behind the thing. Consequently, as +there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may so +say, has 2 sides (for the points of a Line may be called by courtesy, its +sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that? + +I. Arithmetical. + +SPHERE. And what is the next number? + +I. Six. + +SPHERE. Exactly. Then you see you have answered your own question. The Cube +which you will generate will be bounded by six sides, that is to say, six of +your insides. You see it all now, eh? + +``Monster,'' I shrieked, ``be thou juggler, enchanter, dream, or devil, no more +will I endure thy mockeries. Either thou or I must perish.'' And saying these +words I precipitated myself upon him. + + + +\chapter{How the Sphere, having in vain +tried words, resorted to deeds} +It was in vain. I brought my hardest right angle into violent collision with +the Stranger, pressing on him with a force sufficient to have destroyed any +ordinary Circle: but I could feel him slowly and unarrestably slipping from my +contact; not edging to the right nor to the left, but moving somehow out of +the world, and vanishing into nothing. Soon there was a blank. But still I +heard the Intruder's voice. + +SPHERE. Why will you refuse to listen to reason? I had hoped to find in you --- +as being a man of sense and an accomplished mathematician --- a fit apostle for +the Gospel of the Three Dimensions, which I am allowed to preach once only in +a thousand years: but now I know not how to convince you. Stay, I have it. +Deeds, and not words, shall proclaim the truth. Listen, my friend. + +I have told you I can see from my position in Space the inside of all things +that you consider closed. For example, I see in yonder cupboard near which you +are standing, several of what you call boxes (but like everything else in +Flatland, they have no tops or bottom) full of money; I see also two tablets +of accounts. I am about to descend into that cupboard and to bring you one of +those tablets. I saw you lock the cupboard half an hour ago, and I know you +have the key in your possession. But I descend from Space; the doors, you see, +remain unmoved. Now I am in the cupboard and am taking the tablet. Now I have +it. Now I 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. There +could be no doubt --- it was the missing tablet. + +I groaned with horror, doubting whether I was not out of my sense; but the +Stranger continued: ``Surely you must now see that my explanation, and no +other, suits the phenomena. What you call Solid things are really superficial; +what you call Space is really nothing but a great Plane. I am in Space, and +look down upon the insides of the things of which you only see the outsides. +You could leave the Plane yourself, if you could but summon up the necessary +volition. A slight upward or downward motion would enable you to see all that +I can see. + +``The higher I mount, and the further I go from your Plane, the more I can see, +though of course I see it on a smaller scale. For example, I am ascending; now +I can see your neighbour the Hexagon and his family in their several +apartments; now I see the inside of the Theatre, ten doors off, from which the +audience is only just departing; and on the other side a Circle in his study, +sitting at his books. Now I shall come back to you. And, as a crowning proof, +what do you say to my giving you a touch, just the least touch, in your +stomach? It will not seriously injure you, and the slight pain you may suffer +cannot be compared with the mental benefit you will receive.'' + +Before I could utter a word of remonstrance, I felt a shooting pain in my +inside, and a demoniacal laugh seemed to issue from within me. A moment +afterwards the sharp agony had ceased, leaving nothing but a dull ache behind, +and the Stranger began to reappear, saying, as he gradually increased in size, +``There, I have not hurt you much, have I? If you are not convinced now, I +don't know what will convince you. What say you?'' + +My resolution was taken. It seemed intolerable that I should endure existence +subject to the arbitrary visitations of a Magician who could thus play tricks +with one's very stomach. If only I could in any way manage to pin him against +the wall till help came! + +Once more I dashed my hardest angle against him, at the same time alarming the +whole household by my cries for aid. I believe, at the moment of my onset, the +Stranger had sunk below our Plane, and really found difficulty in rising. In +any case he remained motionless, while I, hearing, as I thought, the sound of +some help approaching, pressed against him with redoubled vigor, and continued +to shout for assistance. + +A convulsive shudder ran through the Sphere. ``This must not be,'' I thought I +heard him say: ``either he must listen to reason, or I must have recourse to +the last resource of civilization.'' Then, addressing me in a louder tone, he +hurriedly exclaimed, ``Listen: no stranger must witness what you have +witnessed. Send your Wife back at once, before she enters the apartment. The +Gospel of Three Dimensions must not be thus frustrated. Not thus must the +fruits of one thousand years of waiting be thrown away. I hear her coming. +Back! back! Away from me, or you must go with me --- wither you know not --- into +the Land of Three Dimensions!'' + +``Fool! Madman! Irregular!'' I exclaimed; ``never will I release thee; thou shalt +pay the penalty of thine impostures.'' + +``Ha! Is it come to this?'' thundered the Stranger: ``then meet your fate: out of +your Plane you go. Once, twice, thrice! 'Tis done!'' + + + +\chapter{How I came to Spaceland, and what I saw there} +An unspeakable horror seized me. There was a darkness; then a dizzy, sickening +sensation of sight that was not like seeing; I saw a Line that was no Line; +Space that was not Space: I was myself, and not myself. When I could find +voice, I shrieked loud in agony, ``Either this is madness or it is Hell.'' ``It +is neither, calmly replied the voice of the Sphere, ``it is Knowledge; it is +Three Dimensions: open your eye once again and try to look steadily.'' + +I looked, and, behold, a new world! There stood before me, visibly +incorporate, all that I had before inferred, conjectured, dreamed, of perfect +Circular beauty. What seemed the centre of the Stranger's form lay open to my +view: yet I could see no heart, lungs, nor arteries, only a beautiful +harmonious Something --- for which I had no words; but you, my Readers in +Spaceland, would call it the surface of the Sphere. + +Prostrating myself mentally before my Guide, I cried, ``How it is, O divine +ideal of consummate loveliness and wisdom that I see thy inside, and yet +cannot discern thy heart, thy lungs, thy arteries, thy liver?'' ``What you think +you see, you see not,'' he replied; ``it is not giving to you, nor to any other +Being, to behold my internal parts. I am of a different order of Beings from +those in Flatland. Where I a Circle, you could discern my intestines, but I am +a Being, composed as I told you before, of many Circles, the Many in the One, +called in this country a Sphere. And, just as the outside of a Cube is a +Square, so the outside of a Sphere represents the appearance of a Circle.'' + +Bewildered though I was by my Teacher's enigmatic utterance, I no longer +chafed against it, but worshipped him in silent adoration. He continued, with +more mildness in his voice. ``Distress not yourself if you cannot at first +understand the deeper mysteries of Spaceland. By degrees they will dawn upon +you. Let us begin by casting back a glance at the region whence you came. +Return with me a while to the plains of Flatland and I will shew you that +which you have often reasoned and thought about, but never seen with the sense +of sight --- a visible angle.'' ``Impossible!'' I cried; but, the Sphere leading +the way, I followed as if in a dream, till once more his voice arrested me: +``Look yonder, and behold your own Pentagonal house, and all its inmates.'' +\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 individuality +which I had hitherto merely inferred with the understanding. And how poor and +shadowy was the inferred conjecture in comparison with the reality which I now +behold! My four Sons calmly asleep in the North-Western rooms, my two orphan +Grandsons to the South; the Servants, the Butler, my Daughter, all in their +several apartments. Only my affectionate Wife, alarmed by my continued +absence, had quitted her room and was roving up and down in the Hall, +anxiously awaiting my return. Also the Page, aroused by my cries, had left his +room, and under pretext of ascertaining whether I had fallen somewhere in a +faint, was prying into the cabinet in my study. All this I could now see, not +merely infer; and as we came nearer and nearer, I could discern even the +contents of my cabinet, and the two chests of gold, and the tablets of which +the Sphere had made mention. + +Touched by my Wife's distress, I would have sprung downward to reassure her, +but I found myself incapable of motion. ``Trouble not yourself about your +Wife,'' said my Guide: ``she will not be long left in anxiety; meantime, let us +take a survey of Flatland.'' + +Once more I felt myself rising through space. It was even as the Sphere had +said. The further we receded from the object we beheld, the larger became the +field of vision. My native city, with the interior of every house and every +creature therein, lay open to my view in miniature. We mounted higher, and lo, +the secrets of the earth, the depths of the mines and inmost caverns of the +hills, were bared before me. + +Awestruck at the sight of the mysteries of the earth, thus unveiled before my +unworthy eye, I said to my Companion, ``Behold, I am become as a God. For the +wise men in our country say that to see all things, or as they express it, +omnividence, is the attribute of God alone.'' There was something of scorn in +the voice of my Teacher as he made answer: ``it is so indeed? Then the very +pick-pockets and cut-throats of my country are to be worshipped by your wise +men as being Gods: for there is not one of them that does not see as much as +you see now. But trust me, your wise men are wrong.'' + +I. Then is omnividence the attribute of +others besides Gods? + +SPHERE. I do not know. But, if a pick-pocket or a cut-throat of our country +can see everything that is in your country, surely that is no reason why the +pick-pocket or cut-throat should be accepted by you as a God. This +omnividence, as you call it --- it is not a common word in Spaceland --- does it +make you more just, more merciful, less selfish, more loving? Not in the +least. Then how does it make you more divine? + +I. ``More merciful, more loving!'' But these are the qualities of women! And we +know that a Circle is a higher Being than a Straight Line, in so far as +knowledge and wisdom are more to be esteemed than mere affection. + +SPHERE. It is not for me to classify human faculties according to merit. Yet +many of the best and wisest in Spaceland think more of the affections than of +the understanding, more of your despised Straight Lines than of your belauded +Circles. But enough of this. Look yonder. Do you know that building? + +I looked, and afar off I saw an immense Polygonal structure, in which I +recognized the General Assembly Hall of the States of Flatland, surrounded by +dense lines of Pentagonal buildings at right angles to each other, which I +knew to be streets; and I perceived that I was approaching the great +Metropolis. + +``Here we descend,'' said my Guide. It was now morning, the first hour of the +first day of the two thousandth year of our era. Acting, as was their wont, in +strict accordance with precedent, the highest Circles of the realm were +meeting in solemn conclave, as they had met on the first hour of the first day +of the year 1000, and also on the first hour of the first day of the year 0. + +The minutes of the previous meetings were now read by one whom I at once +recognized as my brother, a perfectly Symmetrical Square, and the Chief Clerk +of the High Council. It was found recorded on each occasion that: ``Whereas the +States had been troubled by divers ill-intentioned persons pretending to have +received revelations from another World, and professing to produce +demonstrations whereby they had instigated to frenzy both themselves and +others, it had been for this cause unanimously resolved by the Grand Council +that on the first day of each millenary, special injunctions be sent to the +Prefects in the several districts of Flatland, to make strict search for such +misguided persons, and without formality of mathematical examination, to +destroy all such as were Isosceles of any degree, to scourge and imprison any +regular Triangle, to cause any Square or Pentagon to be sent to the district +Asylum, and to arrest any one of higher rank, sending him straightway to the +Capital to be examined and judged by the Council.'' + +``You hear your fate,'' said the Sphere to me, while the Council was passing for +the third time the formal resolution. ``Death or imprisonment awaits the +Apostle of the Gospel of Three Dimensions.'' ``Not so,'' replied I, ``the matter +is now so clear to me, the nature of real space so palpable, that methinks I +could make a child understand it. Permit me but to descend at this moment and +enlighten them.'' ``Not yet,'' said my Guide, ``the time will come for that. +Meantime I must perform my mission. Stay thou there in thy place.'' Saying +these words, he leaped with great dexterity into the sea (if I may so call it) +of Flatland, right in the midst of the ring of Counsellors. ``I come,'' said he, +``to proclaim that there is a land of Three Dimensions.'' + +I could see many of the younger Counsellors start back in manifest horror, as +the Sphere's circular section widened before them. But on a sign from the +presiding Circle --- who shewed not the slightest alarm or surprise --- six +Isosceles 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's +gone!'' + +``My Lords,'' said the President to the Junior Circles of the Council, ``there is +not the slightest need for surprise; the secret archives, to which I alone +have access, tell me that a similar occurrence happened on the last two +millennial commencements. You will, of course, say nothing of these trifles +outside the Cabinet.'' + +Raising his voice, he now summoned the guards. ``Arrest the policemen; gag +them. You know your duty.'' After he had consigned to their fate the wretched +policemen --- ill-fated and unwilling witnesses of a State-secret which they +were not to be permitted to reveal --- he again addressed the Counsellors. ``My +Lords, the business of the Council being concluded, I have only to wish you a +happy New Year.'' Before departing, he expressed, at some length, to the Clerk, +my excellent but most unfortunate brother, his sincere regret that, in +accordance with precedent and for the sake of secrecy, he must condemn him to +perpetual imprisonment, but added his satisfaction that, unless some mention +were made by him of that day's incident, his life would be spared. + + + + + + +\chapter{How, though the Sphere +showed me other mysteries of Spaceland, I still desire more; and what came of +it} +When I saw my poor brother led away to imprisonment, I attempted to leap down +into the Council Chamber, desiring to intercede on his behalf, or at least bid +him farewell. But I found that I had no motion of my own. I absolutely +depended on the volition of my Guide, who said in gloomy tones, ``Heed not thy +brother; haply thou shalt have ample time hereafter to condole with him. +Follow me.'' +\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 shewn +you naught save Plane Figures and their interiors. Now I must introduce you to +Solids, and reveal to you the plan upon which they are constructed. Behold +this multitude of moveable square cards. See, I put one on another, not, as +you supposed, Northward of the other, but on the other. Now a second, now a +third. See, I am building up a Solid by a multitude of Squares parallel to one +another. Now the Solid is complete, being as high as it is long and broad, and +we call it a Cube.'' + +``Pardon me, my Lord,'' replied I; ``but to my eye the appearance is as of an +Irregular Figure whose inside is laid open to view; in other words, methinks I +see no Solid, but a Plane such as we infer in Flatland; only of an +Irregularity which betokens some monstrous criminal, so that the very sight of +it is painful to my eyes.'' + +``True,'' said the Sphere; ``it appears to you a Plane, because you are not +accustomed to light and shade and perspective; just as in Flatland a Hexagon +would appear a Straight Line to one who has not the Art of Sight Recognition. +But in reality it is a Solid, as you shall learn by the sense of Feeling.'' + +He then introduced me to the Cube, and I found that this marvellous Being was +indeed no Plane, but a Solid; and that he was endowed with six plane sides and +eight terminal points called solid angles; and I remembered the saying of the +Sphere that just such a Creature as this would be formed by the Square moving, +in Space, parallel to himself: and I rejoiced to think that so insignificant a +Creature as I could in some sense be called the Progenitor of so illustrious +an offspring. + +But still I could not fully understand the meaning of what my Teacher had told +me concerning ``light'' and ``shade'' and ``perspective''; and I did not hesitate to +put my difficulties before him. + +Were I to give the Sphere's explanation of these matters, succinct and clear +though it was, it would be tedious to an inhabitant of Space, who knows these +things already. Suffice it, that by his lucid statements, and by changing the +position of objects and lights, and by allowing me to feel the several objects +and even his own sacred Person, he at last made all things clear to me, so +that I could now readily distinguish between a Circle and a Sphere, a Plane +Figure and a Solid. + +This was the Climax, the Paradise, of my strange eventful History. Henceforth +I have to relate the story of my miserable Fall: --- most miserable, yet surely +most undeserved! For why should the thirst for knowledge be aroused, only to +be disappointed and punished? My volition shrinks from the painful task of +recalling my humiliation; yet, like a second Prometheus, I will endure this +and worse, if by any means I may arouse in the interiors of Plane and Solid +Humanity a spirit of rebellion against the Conceit which would limit our +Dimensions to Two or Three or any number short of Infinity. Away then with all +personal considerations! Let me continue to the end, as I began, without +further digressions or anticipations, pursuing the plain path of dispassionate +History. The exact facts, the exact words, --- and they are burnt in upon my +brain, --- shall be set down without alteration of an iota; and let my Readers +judge between me and Destiny. + +The Sphere would willingly have continued his lessons by indoctrinating me in +the conformation of all regular Solids, Cylinders, Cones, Pyramids, +Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to +interrupt him. Not that I was wearied of knowledge. On the contrary, I +thirsted 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 Perfection +of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine +interior.'' + +SPHERE. My what? + +I. Thine interior: thy stomach, thy intestines. + +SPHERE. Whence this ill-timed impertinent request? And what mean you by saying +that I am no longer the Perfection of all Beauty? + +I. My Lord, your own wisdom has taught me to aspire to One even more great, +more beautiful, and more closely approximate to Perfection than yourself. As +you yourself, superior to all Flatland forms, combine many Circles in One, so +doubtless there is One above you who combines many Spheres in One Supreme +Existence, surpassing even the Solids of Spaceland. And even as we, who are +now in Space, look down on Flatland and see the insides of all things, so of a +certainty there is yet above us some higher, purer region, whither thou dost +surely purpose to lead me --- O Thou Whom I shall always call, everywhere and in +all Dimensions, my Priest, Philosopher, and Friend --- some yet more spacious +Space, some more dimensionable Dimensionality, from the vantage-ground of +which we shall look down together upon the revealed insides of Solid things, +and where thine own intestines, and those of thy kindred Spheres, will lie +exposed to the view of the poor wandering exile from Flatland, to whom so much +has already been vouchsafed. + +SPHERE. Pooh! Stuff! Enough of this trifling! The time is short, and much +remains to be done before you are fit to proclaim the Gospel of Three +Dimensions to your blind benighted countrymen in Flatland. + +I. Nay, gracious Teacher, deny me not what I know it is in thy power to +perform. Grant me but one glimpse of thine interior, and I am satisfied for +ever, remaining henceforth thy docile pupil, thy unemacipable slave, ready to +receive all thy teachings and to feed upon the words that fall from thy lips. + +SPHERE. Well, then, to content and silence you, let me say at once, I would +shew you what you wish if I could; but I cannot. Would you have me turn my +stomach inside out to oblige you? + +I. But my Lord has shewn me the intestines of all my countrymen in the Land of +Two Dimensions by taking me with him into the Land of Three. What therefore +more easy than now to take his servant on a second journey into the blessed +region of the Fourth Dimension, where I shall look down with him once more +upon this land of Three Dimensions, and see the inside of every +three-dimensioned house, the secrets of the solid earth, the treasures of the +mines of Spaceland, and the intestines of every solid living creature, even +the noble and adorable Spheres. + +SPHERE. But where is this land of Four Dimensions? + +I. I know not: but doubtless my Teacher knows. + +SPHERE. Not I. There is no such land. The very idea of it is utterly +inconceivable. + +I. Not inconceivable, my Lord, to me, and therefore still less inconceivable +to my Master. Nay, I despair not that, even here, in this region of Three +Dimensions, your Lordship's art may make the Fourth Dimension visible to me; +just as in the Land of Two Dimensions my Teacher's skill would fain have +opened the eyes of his blind servant to the invisible presence of a Third +Dimension, though I saw it not. + +Let me recall the past. Was I not taught below that when I saw a Line and +inferred a Plane, I in reality saw a Third unrecognized Dimension, not the +same as brightness, called ``height''? And does it not now follow that, in this +region, when I see a Plane and infer a Solid, I really see a Fourth +unrecognized Dimension, not the same as colour, but existent, though +infinitesimal and incapable of 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 the +revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst, +for more knowledge. Doubtless we cannot see that other higher Spaceland now, +because we have no eye in our stomachs. But, just as there was the realm of +Flatland, though that poor puny Lineland Monarch could neither turn to left +nor right to discern it, and just as there was close at hand, and touching my +frame, the land of Three Dimensions, though I, blind senseless wretch, had no +power to touch it, no eye in my interior to discern it, so of a surety there +is a Fourth Dimension, which my Lord perceives with the inner eye of thought. +And that it must exist my Lord himself has taught me. Or can he have forgotten +what he himself imparted to his servant? + +In One Dimension, did not a moving Point produce a Line with two terminal +points? + +In Two Dimensions, did not a moving Line produce a Square with four terminal +points? + +In Three Dimensions, did not a moving Square produce --- did not this eye of +mine behold it --- that blessed Being, a Cube, with eight terminal points? + +And in Four Dimensions shall not a moving Cube --- alas, for Analogy, and alas +for the Progress of Truth, if it be not so --- shall not, I say, the motion of a +divine Cube result in a still more divine Organization with sixteen terminal +points? + +Behold the infallible confirmation of the Series, 2, 4, +8, 16: is not this a Geometrical Progression? Is not this --- if I might quote +my Lord's own words --- ``strictly according to Analogy''? + +Again, was I not taught by my Lord that as in a Line there are two bounding +Points, and in a Square there are four bounding Lines, so in a Cube there must +be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is +not this an Arithmetical Progression? And consequently does it not of +necessity follow that the more divine offspring of the divine Cube in the Land +of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my +Lord has taught me to believe, ``strictly according to Analogy''? + +O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not +knowing the facts; and I appeal to your Lordship to confirm or deny my logical +anticipations. If I am wrong, I yield, and will no longer demand a Fourth +Dimension; but, if I am right, my Lord will listen to reason. + +I ask therefore, is it, or is it not, the fact, that ere now your countrymen +also have witnessed the descent of Beings of a higher order than their own, +entering closed rooms, even as your Lordship entered mine, without the opening +of doors or windows, and appearing and vanishing at will? On the reply to this +question I am ready to stake everything. Deny it, and I am henceforth silent. +Only vouchsafe an answer. + +Sphere (after a pause). It is reported so. But men are divided in opinion as +to the facts. And even granting the facts, they explain them in different +ways. And in any case, however great may be the number of different +explanations, no one has adopted or suggested the theory of a Fourth +Dimension. Therefore, pray have done with this trifling, and let us return to +business. + +I. I was certain of it. I was certain that my anticipations would be +fulfilled. And now have patience with me and answer me yet one more question, +best of Teachers! Those who have thus appeared --- no one knows whence --- and +have returned --- no one knows whither --- have they also contracted their +sections and vanished somehow into that more Spacious Space, whither I now +entreat you to conduct me? + +Sphere (moodily). They have vanished, certainly --- if they ever appeared. But +most people say that these visions arose from the thought --- you will not +understand 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 other +Space is really Thoughtland, then take me to that blessed Region where I in +Thought shall see the insides of all solid things. There, before my ravished +eye, a Cube moving in some altogether new direction, but strictly according to +Analogy, so as to make every particle of his interior pass through a new kind +of Space, with a wake of its own --- shall create a still more perfect +perfection than himself, with sixteen terminal Extra-solid angles, and Eight +solid Cubes for his Perimeter. And once there, shall we stay our upward +course? In that blessed region of Four Dimensions, shall we linger at the +threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve +that our ambition shall soar with our corporal ascent. Then, yielding to our +intellectual onset, the gates of the Six Dimension shall fly open; after that +a Seventh, and then an Eighth --- + +How long I should have continued I know not. In vain did the Sphere, in his +voice of thunder, reiterate his command of silence, and threaten me with the +direst penalties if I persisted. Nothing could stem the flood of my ecstatic +aspirations. Perhaps I was to blame; but indeed I was intoxicated with the +recent draughts of Truth to which he himself had introduced me. However, the +end was not long in coming. My words were cut short by a crash outside, and a +simultaneous crash inside me, which impelled me through space with a velocity +that precluded speech. Down! down! down! I was rapidly descending; and I knew +that return to Flatland was my doom. One glimpse, one last and +never-to-be-forgotten glimpse I had of that dull level wilderness --- which was +now to become my Universe again --- spread out before my eye. Then a darkness. +Then a final, all-consummating thunder-peal; and, when I came to myself, I was +once more a common creeping Square, in my Study at home, listening to the +Peace-Cry of my approaching Wife. + + + + + + + +\chapter{How the Sphere encouraged me in a Vision} +Although I had less than a minute for reflection, I felt, by a kind of +instinct, that I must conceal my experiences from my Wife. Not that I +apprehended, at the moment, any danger from her divulging my secret, but I +knew that to any Woman in Flatland the narrative of my adventures must needs +be unintelligible. So I endeavoured to reassure her by some story, invented +for the occasion, that I had accidentally fallen through the trap-door of the +cellar, and had there lain stunned. + +The Southward attraction in our country is so slight that even to a Woman my +tale necessarily appeared extraordinary and well-nigh incredible; but my Wife, +whose good sense far exceeds that of the average of her Sex, and who perceived +that I was unusually excited, did not argue with me on the subject, but +insisted that I was ill and required repose. I was glad of an excuse for +retiring to my chamber to think quietly over what had happened. When I was at +last by myself, a drowsy sensation fell on me; but before my eyes closed I +endeavoured to reproduce the Third Dimension, and especially the process by +which a Cube is constructed through the motion of a Square. It was not so +clear as I could have wished; but I remembered that it must be ``Upward, and +yet not Northward,'' and I determined steadfastly to retain these words as the +clue which, if firmly grasped, could not fail to guide me to the solution. So +mechanically repeating, like a charm, the words, ``Upward, yet not Northward,'' +I fell into a sound refreshing sleep. + +During my slumber I had a dream. I thought I was once more by the side of the +Sphere, whose lustrous hue betokened that he had exchanged his wrath against +me for perfectly placability. We were moving together towards a bright but +infinitesimally small Point, to which my Master directed my attention. As we +approached, methought there issued from it a slight humming noise as from one +of your Spaceland bluebottles, only less resonant by far, so slight indeed +that even in the perfect stillness of the Vacuum through which we soared, the +sound reached not our ears till we checked our flight at a distant from it of +something under twenty human diagonals. + +``Look yonder,'' said my Guide, ``in Flatland thou hast lived; of Lineland thou +hast received a vision; thou hast soared with me to the heights of Spaceland; +now, in order to complete the range of thy experience, I conduct thee downward +to the lowest depth of existence, even to the realm of Pointland, the Abyss of +No dimensions. + +``Behold yon miserable creature. That Point is a Being like ourselves, but +confined to the non-dimensional Gulf. He is himself his own World, his own +Universe; of any other than himself he can form no conception; he knows not +Length, nor Breadth, nor Height, for he has had no experience of them; he has +no cognizance even of the number Two; nor has he a thought of Plurality; for +he is himself his One and All, being really Nothing. Yet mark his perfect +self-contentment, and hence learn his lesson, that to be self-contented is to +be vile and ignorant, and that to aspire is better than to be blindly and +impotently happy. Now listen.'' + +He ceased; and there arose from the little buzzing creature a tiny, low, +monotonous, but distinct tinkling, as from one of your Spaceland phonographs, +from which I caught these words, ``Infinite beatitude of existence! It is; and +there is nothing else beside It.'' + +``What,'' said I, ``does the puny creature mean by `it'?'' ``He means himself,'' +said the Sphere: ``have you not noticed before now, that babies and babyish +people who cannot distinguish themselves from the world, speak of themselves +in the Third Person? But hush!'' + +``It fills all Space,'' continued the little soliloquizing Creature, ``and what +It fills, It is. What It thinks, that It utters; and what It utters, that It +hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it +is the One, and yet the All in All. Ah, the happiness, ah, the happiness of +Being!'' + +``Can you not startle the little thing out of its complacency?'' said I. ``Tell +it what it really is, as you told me; reveal to it the narrow limitations of +Pointland, and lead it up to something higher.'' ``That is no easy task,'' said +my Master; ``try you.'' + +Hereon, raising by voice to the uttermost, I addressed the Point as follows: + +``Silence, silence, contemptible Creature. You call yourself the All in All, +but you are the Nothing: your so-called Universe is a mere speck in a Line, +and a Line is a mere shadow as compared with ---'' ``Hush, hush, you have said +enough,'' interrupted the Sphere, ``now listen, and mark the effect of your +harangue on the King of Pointland.'' + +The lustre of the Monarch, who beamed more brightly than ever upon hearing my +words, shewed clearly that he retained his complacency; and I had hardly +ceased when he took up his strain again. ``Ah, the joy, ah, the joy of Thought! +What can It not achieve by thinking! Its own Thought coming to Itself, +suggestive of its disparagement, thereby to enhance Its happiness! Sweet +rebellion stirred up to result in triumph! Ah, the divine creative power of +the All in One! Ah, the joy, the joy of Being!'' + +``You see,'' said my Teacher, ``how little your words have done. So far as the +Monarch understand them at all, he accepts them as his own --- for he cannot +conceive of any other except himself --- and plumes himself upon the variety of +`Its Thought' as an instance of creative Power. Let us leave this God of +Pointland 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 voice +of 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 had +received fresh insight, and he was not too proud to acknowledge his error to a +Pupil. Then he proceeded to initiate me into mysteries yet higher than those I +had witnessed, shewing me how to construct Extra-Solids by the motion of +Solids, and Double Extra-Solids by the motion of Extra-Solids, and all +``strictly according to Analogy,'' all by methods so simple, so easy, as to be +patent even to the Female Sex.? + + + +\chapter{How I tried to teach the +Theory of Three Dimensions to my Grandson, and with what success} +I awoke rejoicing, and began to reflect on the glorious career before me. I +would go forth, methought, at once, and evangelize the whole of Flatland. Even +to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I +would begin with my Wife. + +Just as I had decided on the plan of my operations, I heard the sound of many +voices in the street commanding silence. Then followed a louder voice. It was +a herald's proclamation. Listening attentively, I recognized the words of the +Resolution of the Council, enjoining the arrest, imprisonment, or execution of +any one who should pervert the minds of people by delusions, and by professing +to have received revelations from another World. + +I reflected. This danger was not to be trifled with. It would be better to +avoid it by omitting all mention of my Revelation, and by proceeding on the +path of Demonstration --- which after all, seemed so simple and so conclusive +that nothing would be lost by discarding the former means. ``Upward, not +Northward'' --- was the clue to the whole proof. It had seemed to me fairly clear +before I fell asleep; and when I first awoke, fresh from my dream, it had +appeared as patent as Arithmetic; but somehow it did not seem to me quite so +obvious now. Though my Wife entered the room opportunely at just that moment, +I decided, after we had exchanged a few words of commonplace conversation, not +to begin with her. + +My Pentagonal Sons were men of character and standing, and physicians of no +mean reputation, but not great in mathematics, and, in that respect, unfit for +my purpose. But it occurred to me that a young and docile Hexagon, with a +mathematical turn, would be a most suitable pupil. Why therefore not make my +first experiment with my little precocious Grandson, whose casual remarks on +the meaning of 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; for +he would know nothing of the Proclamation of the Council; whereas I could not +feel sure that my Sons --- so greatly did their patriotism and reverence for the +Circles predominate over mere blind affection --- might not feel compelled to +hand me over to the Prefect, if they found me seriously maintaining the +seditious heresy of the Third Dimension. + +But the first thing to be done was to satisfy in some way the curiosity of my +Wife, who naturally wished to know something of the reasons for which the +Circle had desired that mysterious interview, and of the means by which he had +entered the house. Without entering into the details of the elaborate account +I gave her, --- an account, I fear, not quite so consistent with truth as my +Readers in Spaceland might desire, --- I must be content with saying that I +succeeded at last in persuading her to return quitely to her household duties +without eliciting from me any reference to the World of Three Dimensions. This +done, I immediately sent for my Grandson; for, to confess the truth, I felt +that all that I had seen and heard was in some strange way slipping away from +me, like the image of a half- grasped, tantalizing dream, and I longed to +essay my skill in making a first disciple. + +When my Grandson entered the room I carefully secured the door. Then, sitting +down by his side and taking our mathematical tablets, --- or, as you would call +them, Lines --- I told him we would resume the lesson of yesterday. I taught him +once more how a Point by motion in One Dimension produces a Line, and how a +straight Line in Two Dimensions produces a Square. After this, forcing a +laugh, I said, ``And now, you scamp, you wanted to make believe that a Square +may in the same way by motion `Upward, not Northward' produce another figure, +a sort of extra square in Three Dimensions. Say that again, you young rascal.'' + +At this moment we heard once more the herald's ``O yes! O yes!'' outside in the +street proclaiming the Resolution of the Council. Young though he was, my +Grandson --- who was unusually intelligent for his age, and bred up in perfect +reverence for the authority of the Circles --- took in the situation with an +acuteness for which I was quite unprepared. He remained silent till the last +words of the Proclamation had died away, and then, bursting into tears, ``Dear +Grandpapa,'' he said, ``that was only my fun, and of course I meant nothing at +all by it; and we did not know anything then about the new Law; and I don't +think I said anything about the Third Dimension; and I am sure I did not say +one word about `Upward, not Northward,' for that would be such nonsense, you +know. How could a thing move Upward, and not Northward? Upward and not +Northward! Even if I were a baby, I could not be so absurd as that. How silly +it is! Ha! ha! ha!'' + +``Not at all silly,'' said I, losing my temper; ``here for example, I take this +Square,'' and, at the word, I grasped a moveable Square, which was lying at +hand --- ``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, but +somehow ---'' Here I brought my sentence to an inane conclusion, shaking the +Square about in a purposeless manner, much to the amusement of my Grandson, +who burst out laughing louder than ever, and declared that I was not teaching +him, but joking with him; and so saying he unlocked the door and ran out of +the room. Thus ended my first attempt to convert a pupil to the Gospel of +Three Dimensions. + + + + +\chapter{How I then tried to diffuse the Theory of Three Dimensions by other +means, and of the result} +My failure with my Grandson did not encourage me to communicate my secret to +others 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, not +Northward,'' but must rather endeavour to seek a demonstration by setting +before the public a clear view of the whole subject; and for this purpose it +seemed necessary to resort to writing. + +So I devoted several months in privacy to the composition of a treatise on the +mysteries of Three Dimensions. Only, with the view of evading the Law, if +possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in +theory, a Figure could look down upon Flatland and see simultaneously the +insides of all things, and where it was possible that there might be supposed +to exist a Figure environed, as it were, with six Squares, and containing +eight terminal Points. But in writing this book I found myself sadly hampered +by the impossibility of drawing such diagrams as were necessary for my +purpose: for of course, in our country of Flatland, there are no tablets but +Lines, and no diagrams but Lines, all in one straight Line and only +distinguishable by difference of size and brightness; so that, when I had +finished my treatise (which I entitled, ``Through Flatland to Thoughtland'') I +could not feel certain that many would understand my meaning. + +Meanwhile my life was under a cloud. All pleasures palled upon me; all sights +tantalized and tempted me to outspoken treason, because I could not but +compare what I saw in Two Dimensions with what it really was if seen in Three, +and could hardly refrain from making my comparisons aloud. I neglected my +clients and my own business to give myself to the contemplation of the +mysteries which I had once beheld, yet which I could impart to no one, and +found daily more difficult to reproduce even before my own mental vision. + +One day, about eleven months after my return from Spaceland, I tried to see a +Cube with my eye closed, but failed; and though I succeeded afterwards, I was +not then quite certain (nor have I been ever afterwards) that I had exactly +realized the original. This made me more melancholy than before, and +determined me to take some step; yet what, I knew not. I felt that I would +have been willing to sacrifice my life for the Cause, if thereby I could have +produced conviction. But if I could not convince my Grandson, how could I +convince the highest and most developed Circles in the land? + +And yet at times my spirit was too strong for me, and I gave vent to +dangerous utterances. Already I was considered heterodox if not treasonable, +and I was keenly alive to the danger of my position; nevertheless I could not +at times refrain from bursting out into suspicious or half-seditious +utterances, even among the highest Polygonal or Circular society. When, for +example, the question arose about the treatment of those lunatics who said +that they had received the power of seeing the insides of things, I would +quote the saying of an ancient Circle, who declared that prophets and inspired +people are always considered by the majority to be mad; and I could not help +occasionally dropping such expressions as ``the eye that discerns the interiors +of things,'' and ``the all-seeing land''; once or twice I even let fall the +forbidden terms ``the Third and Fourth Dimensions.'' At last, to complete a +series of minor indiscretions, at a meeting of our Local Speculative Society +held at the palace of the Prefect himself, --- some extremely silly person +having read an elaborate paper exhibiting the precise reasons why Providence +has limited the number of Dimensions to Two, and why the attribute of +omnividence is assigned to the Supreme alone --- I so far forgot myself as to +give an exact account of the whole of my voyage with the Sphere into Space, +and to the Assembly Hall in our Metropolis, and then to Space again, and of my +return home, and of everything that I had seen and heard in fact or vision. At +first, indeed, I pretended that I was describing the imaginary experiences of +a fictitious person; but my enthusiasm soon forced me to throw off all +disguise, and finally, in a fervent peroration, I exhorted all my hearers to +divest 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 the +Sphere had stood in my company, I was allowed to begin and to continue my +narration unquestioned and uninterrupted. But from the first I foresaw my +fate; for the President, noting that a guard of the better sort of Policemen +was in attendance, of angularity little, if at all, under 55 degrees, ordered +them to be relieved before I began my defence, by an inferior class of 2 or 3 +degrees. I knew only too well what that meant. I was to be executed or +imprisoned, and my story was to be kept secret from the world by the +simultaneous destruction of the officials who had heard it; and, this being +the case, the President desired to substitute the cheaper for the more +expensive victims. + +After I had concluded my defence, the President, perhaps perceiving that some +of the junior Circles had been moved by evident earnestness, asked me two +questions: --- + +\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 I +could not do better. I must be sentenced to perpetual imprisonment; but if the +Truth intended that I should emerge from prison and evangelize the world, the +Truth might be trusted to bring that result to pass. Meanwhile I should be +subjected to no discomfort that was not necessary to preclude escape, and, +unless I forfeited the privilege by misconduct, I should be occasionally +permitted to see my brother who had preceded me to my prison. + +Seven years have elapsed and I am still a prisoner, and --- if I except the +occasional visits of my brother --- debarred from all companionship save that of +my jailers. My brother is one of the best of Squares, just sensible, cheerful, +and not without fraternal affection; yet I confess that my weekly interviews, +at least in one respect, cause me the bitterest pain. He was present when the +Sphere manifested himself in the Council Chamber; he saw the Sphere's changing +sections; he heard the explanation of the phenomena then give to the Circles. +Since that time, scarcely a week has passed during seven whole years, without +his hearing from me a repetition of the part I played in that manifestation, +together with ample descriptions of all the phenomena in Spaceland, and the +arguments for the existence of Solid things derivable from Analogy. Yet --- I +take shame to be forced to confess it --- my brother has not yet grasped the +nature of Three Dimensions, and frankly avows his disbelief in the existence +of a Sphere. + +Hence I am absolutely destitute of converts, and, for +aught that I can see, the millennial Revelation has been made to me for +nothing. Prometheus up in Spaceland was bound for bringing down fire for +mortals, but I --- poor Flatland Prometheus --- lie here in prison for bringing +down nothing to my countrymen. Yet I exist in the hope that these memoirs, in +some manner, I know not how, may find their way to the minds of humanity in +Some Dimension, and may stir up a race of rebels who shall refuse to be +confined to limited Dimensionality. + +That is the hope of my brighter moments. Alas, it is +not always so. Heavily weights on me at times the burdensome reflection that I +cannot honestly say I am confident as to the exact shape of the once-seen, +oft-regretted Cube; and in my nightly visions the mysterious precept, ``Upward, +not Northward,'' haunts me like a soul-devouring Sphinx. It is part of the +martyrdom which I endure for the cause of Truth that there are seasons of +mental weakness, when Cubes and Spheres flit away into the background of +scarce-possible existences; when the Land of Three Dimensions seems almost as +visionary as the Land of One or None; nay, when even this hard wall that bars +me from my freedom, these very tablets on which I am writing, and all the +substantial realities of Flatland itself, appear no better than the offspring +of a diseased imagination, or the baseless fabric of a dream. + +\begin{center} +\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig11} +\end{center} + +\end{document} + +