"Fie, fie, how frantically I square my talk!"
1884 2025
This is an AI modernization of Flatland into contemporary English. The original is available on Project Gutenberg. A beautiful version of the original ebook is available from Standard Ebooks.
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 Introduced to the Mysteries
Of THREE Dimensions
Having previously known
ONLY TWO
So the Citizens of that Celestial Region
May aspire ever 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
"Be patient, for the world is broad and wide."
I call our world Flatland — not because that's what we call it, but to make its nature clearer to you, my happy readers, who have the privilege of living in Space.
Imagine an enormous sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other shapes — instead of staying fixed in place — move around freely, on or within the surface, but without any ability to rise above it or sink below it. Think of them as shadows, except solid, with glowing edges. If you can picture that, you'll have a pretty accurate idea of my country and my countrymen. A few years ago, I would have sadly said "my universe." But since then, my mind has been opened to higher views of things.
In such a country, you'll immediately realize that nothing "solid" — in the way you understand the word — can possibly exist. But I imagine you'd assume that we could at least tell the difference by sight between the Triangles, Squares, and other shapes I've described moving around. On the contrary — we can see nothing of the sort, or at least nothing that would let us tell one figure from another. The only thing visible to us, or that could ever be visible, is Straight Lines. I'll demonstrate why in just a moment.
Place a penny in the middle of one of your tables in Space, and lean over it to look straight down. It will appear as a circle.
Now, pull back to the edge of the table and gradually lower your eye — bringing yourself closer and closer to the condition of a Flatland inhabitant — and you'll see the penny becoming more and more oval. Finally, when you've placed your eye exactly at the edge of the table, so that you are, in effect, actually a Flatlander, the penny will no longer appear oval at all. As far as you can see, it will have become a straight line.
The same thing would happen if you tried this with a Triangle, or a Square, or any other shape cut out of cardboard. As soon as you look at it with your eye level with the table's edge, it stops appearing as a shape and becomes, to all appearances, a straight line. Take, for example, an equilateral Triangle — which represents, in our world, a Tradesman of the respectable class. Figure 1 shows the Tradesman as you would see him while bending over him from above. Figures 2 and 3 show the Tradesman as you would see him if your eye were close to the level of the table, or nearly at the level of the table. And if your eye were exactly at the level of the table — which is how we see him in Flatland — you would see nothing but a straight line.
When I was in Spaceland, I heard that your sailors have very similar experiences when they're crossing the open sea and spot some distant island or coastline on the horizon. The far-off land might have bays, headlands, and angles jutting in and out in any number and to any extent. Yet from a distance, you see none of these — unless the sun happens to be shining brightly on them, revealing the bumps and indentations through light and shadow. All you see is a gray, unbroken line on the water.
Well, that's exactly what we see when one of our triangular or otherwise-shaped acquaintances approaches us in Flatland. Since we have no sun and no light source that casts shadows, we don't have any of the visual aids that you enjoy in Spaceland. If a friend moves closer to us, we see his line getting larger. If he moves away, it gets smaller. But he still looks like a straight line. Whether he's a Triangle, Square, Pentagon, Hexagon, Circle, whatever you like — a straight Line is what he looks like, and nothing else.
You might be wondering how, under these rather challenging circumstances, we manage to tell our friends apart. But the answer to that perfectly natural question will be better and more easily given when I come to describe the inhabitants of Flatland. For now, let me set that subject aside and say a word or two about the climate and houses in our country.
Just like in your world, we have four points on the compass: North, South, East, and West.
Since we have no sun or other celestial bodies, we can't determine North the way you do. But we have our own method. There's a natural law in our world that creates a constant pull toward the South. In temperate climates, this pull is very slight -- even a Woman in reasonable health can walk several miles northward without much difficulty -- but the drag of that southward attraction is enough to serve as a reliable compass across most of our world. On top of that, the rain (which falls at regular intervals) always comes from the North, which helps. And in the towns, we have the houses themselves as guides, since their side-walls mostly run North to South so the roofs can keep the northern rain off. Out in the countryside, where there are no houses, the trunks of the trees provide a rough guide. All in all, we don't have nearly as much trouble getting our bearings as you might expect.
That said, in our more temperate regions where the southward pull is barely noticeable, I've occasionally found myself stuck in the middle of a perfectly flat, empty plain with no houses or trees to orient me. More than once, I've had to just stand there for hours, waiting for the rain to come before I could continue on my way. For the weak and elderly -- and especially for delicate Females -- the pull of gravity hits much harder than it does for robust Males. So it's considered good manners, if you meet a Lady on the street, to always give her the North side of the path. This is by no means easy to do on short notice when you're in good health and you're in a climate where it's hard to tell your North from your South.
We have no windows in our houses. Light reaches us the same way everywhere -- indoors and out, day and night, equally in all places. Where it comes from, we have no idea. In the old days, our scholars found this a fascinating question: "What is the origin of light?" But every attempt to answer it only succeeded in filling our asylums with the people who tried. After fruitless efforts to discourage such research indirectly -- by slapping a heavy tax on it -- the Legislature eventually banned it outright.
I alone in all of Flatland now know the true answer to this mystery. But my knowledge is impossible to explain to a single one of my countrymen. I am laughed at -- I, the sole possessor of the truths of Space and the theory of how Light enters our world from the realm of Three Dimensions -- as though I were the maddest of the mad! But enough of these painful digressions. Let me return to our homes.
The most common house design is five-sided, or pentagonal, as shown in the figure below. The two Northern sides, RO and OF, form the roof, and generally have no doors. On the East side is a small door for the Women. On the West, a much larger one for the Men. The South side, or floor, usually has no door at all.
Square and triangular houses are not allowed, and here's why. The angles of a Square -- and even more so those of an equilateral Triangle -- are much sharper than those of a Pentagon. And since the edges of inanimate objects like houses appear dimmer than the edges of living Men and Women, there's a real danger that the pointed corners of a square or triangular house could seriously injure some careless or maybe just absent-minded traveler who bumps into them. So as far back as the eleventh century of our era, triangular houses were universally banned by law. The only exceptions were fortifications, ammunition storage, barracks, and other government buildings -- places where you don't really want the general public wandering up without paying attention.
During that period, square houses were still allowed everywhere, though discouraged with a special tax. But about three centuries later, the Law determined that in any town with a population above ten thousand, the pentagon's angle was the smallest house-angle that could be permitted without putting public safety at risk. Common sense backed up the Legislature's efforts, and today, even in the countryside, pentagonal construction has replaced every other design. It's only now and then, in some very remote and backward farming district, that a history buff might still stumble across a square house.
The largest a full-grown inhabitant of Flatland gets, in terms of length or width, is about eleven of your inches. Twelve inches would be an extreme case.
Our Women are Straight Lines.
Our Soldiers and lowest class of Workers are Triangles with two equal sides, each about eleven inches long, and a base — the third side — so short (often no more than half an inch) that they come to an extremely sharp and dangerous point at the top. In fact, when their bases are at their most degraded (no more than an eighth of an inch across), they're practically indistinguishable from Straight Lines — that is, from Women. Their points are that narrow. Just as in your world, these Triangles are called Isosceles, and that's the name I'll use for them from here on.
Our Middle Class consists of Equilateral, or Equal-Sided, Triangles.
Our Professional Men and Gentlemen are Squares (the class I myself belong to) and Five-Sided Figures, or Pentagons.
Above them comes the Nobility, which has several ranks, starting with Six-Sided Figures, or Hexagons, and rising with each additional side until they earn the distinguished title of Polygonal — meaning many-sided. Finally, when the number of sides becomes so large, and the sides themselves so small, that the figure is indistinguishable from a Circle, he's inducted into the Circular or Priestly order — the highest class of all.
It is a Law of Nature in our world that a male child will have one more side than his father, so that each generation rises (as a rule) one step on the scale of development and nobility. The son of a Square is a Pentagon. The son of a Pentagon is a Hexagon. And so on.
But this rule doesn't always apply to Tradesmen, and applies even less often to Soldiers and Workers — who can hardly be said to deserve the name of human Figures at all, since their sides aren't all equal. For them, the Law of Nature doesn't hold, and the son of an Isosceles (that is, a Triangle with only two equal sides) remains Isosceles. Still, not all hope is lost, even for the Isosceles. There's a chance that his descendants may eventually rise above his degraded condition. After a long track record of military successes, or years of diligent and skillful labor, the more intelligent members of the Worker and Soldier classes tend to show a slight increase in the length of their third side — their base — along with a shrinkage of the other two sides. Intermarriages between the sons and daughters of these more intellectual lower-class members (arranged by the Priests) generally produce offspring that come even closer to the form of a true Equal-Sided Triangle.
Rarely — relative to the vast numbers of Isosceles births — is a genuine, certified Equal-Sided Triangle produced from Isosceles parents. (A reader from Spaceland might ask: "Why do you need a certificate? Isn't having a Square son proof enough from Nature herself that the father was Equal-Sided?" To that I'd reply: no Woman of any standing will marry an uncertified Triangle. Square offspring have occasionally resulted from slightly Irregular Triangles, but in almost every such case, the Irregularity of the first generation catches up with the third — which either fails to reach Pentagonal rank or falls back to Triangular.) Such a birth requires not only a series of carefully arranged intermarriages, but also a long, sustained practice of thrift and self-discipline on the part of the aspiring ancestors of the future Equilateral, and a patient, systematic, continuous development of the Isosceles intellect across many generations.
The birth of a true Equilateral Triangle from Isosceles parents is cause for celebration across our country for many miles around. After a rigorous examination by the Sanitary and Social Board, the infant — if certified as Regular — is admitted into the class of Equilaterals in a solemn ceremony. He is then immediately taken from his proud but heartbroken parents and adopted by some childless Equilateral, who is sworn by oath never to let the child return to his former home or so much as look at his old family again. The fear is that the newly developed organism might, through unconscious imitation, slip back down to 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 ray of light and hope breaking through the dull misery of their existence — but also by the upper classes at large. The higher ranks are well aware that these rare events, while doing little or nothing to cheapen their own privileges, serve as an almost indispensable barrier against revolution from below.
If the sharp-angled rabble had been entirely, without exception, stripped of all hope and ambition, they might have found leaders during their many uprisings capable enough to turn their superior numbers and strength against even the wisdom of the Circles. But a convenient ordinance of Nature has decreed that the more the working classes grow in intelligence, knowledge, and virtue, the more their sharp angle (which makes them physically dangerous) also grows wider, approaching the comparatively harmless angle of the Equilateral Triangle. So among the most brutal and fearsome of the Soldier class — creatures almost on the level of Women in their lack of intelligence — you find that as they develop the mental ability needed to use their tremendous piercing power effectively, that very piercing power diminishes.
What an admirable Law of Compensation! And what perfect proof of the natural fitness — I might almost say the divine origin — of the aristocratic constitution of the States of Flatland! Through a careful application of this Law of Nature, the Polygons and Circles are almost always able to crush rebellion in its cradle, exploiting the unstoppable and boundless hopefulness of the human mind. Art also comes to the aid of Law and Order. It's generally found possible — through a little artificial compression or expansion performed by the State physicians — to make some of the more intelligent rebel leaders perfectly Regular, and to admit them immediately into the privileged classes. A much larger number, still below the standard but tempted by the prospect of eventually being elevated, are persuaded to enter the State Hospitals, where they're kept in comfortable confinement for life. Only one or two of the most stubborn, foolish, and hopelessly Irregular are led to execution.
Then the wretched rabble of the Isosceles, without a plan and without leaders, are either run through without resistance by the small force of their fellow Isosceles that the Chief Circle keeps on payroll for exactly these emergencies — or, more often, through jealousies and suspicions skillfully stirred up among them by the Circular party, they turn on each other and die by one another's angles. No fewer than one hundred and twenty rebellions are recorded in our history, along with two hundred and thirty-five minor uprisings — and every single one has ended this way.
If our sharply pointed Triangles of the Soldier class are dangerous, it should be obvious that our Women are far more dangerous still. Because if a Soldier is a wedge, a Woman is a needle — she is, so to speak, all point, at least at both ends. Add to this her ability to make herself practically invisible at will, and you'll see that a Female in Flatland is a creature you do not want to mess with.
But here, perhaps, some of my younger readers might ask: how can a Woman in Flatland make herself invisible? This ought to be obvious without any explanation. Still, a few words will make it clear to even the most inattentive.
Place a needle on a table. Then, with your eye level with the tabletop, look at it from the side — and you see the whole length of it. But look at it from one end, and you see nothing but a point. It has become practically invisible. That's exactly how it works with one of our Women. When her side is turned toward us, we see her as a straight line. When the end containing her eye or mouth — for in our world, these two organs are one and the same — is the part facing us, then we see nothing but a highly luminous point. But when her back is presented to our view — being only faintly luminous, and in fact almost as dim as an inanimate object — her rear end serves as a kind of Invisible Cap.
The dangers we face from our Women should now be obvious to even the dullest intelligence in Spaceland. If even the angle of a respectable middle-class Triangle is not without its dangers; if bumping into a Working Man means getting gashed; if a collision with a military Officer means a serious wound; if a mere touch from the point of a common Soldier brings the danger of death — then what can it mean to run into a Woman, except absolute and immediate destruction? And when a Woman is invisible, or visible only as a faint, dimly glowing point, how difficult it must be, even for the most cautious among us, to always avoid collision!
Many laws have been passed at various times in the various States of Flatland to minimize this danger. In the Southern and less temperate climates, where the force of gravity is stronger and people are more prone to sudden, involuntary movements, the Laws concerning Women are naturally much more strict. But a general overview of the Code can be gathered from the following summary:
1. Every house shall have one entrance on the Eastern side, for the use of Females only. All women shall enter through it "in a becoming and respectful manner," and not through the Men's door on the Western side. (When I was in Spaceland, I learned that some of your Priestly Circles have the same sort of arrangement — a separate entrance for Villagers, Farmers, and Teachers of Board Schools, so that they may "approach in a becoming and respectful manner.")
2. No Female shall walk in any public place without continuously keeping up her Peace-cry, under penalty of death.
3. Any Female duly certified to be suffering from St. Vitus's Dance, epileptic fits, chronic cold accompanied by violent sneezing, or any disease causing involuntary movements, shall be instantly destroyed.
In some States there is an additional Law forbidding Females, under penalty of death, from walking or standing in any public place without constantly swinging their backs from right to left, to signal their presence to those behind them. Other States require a Woman, when traveling, to be followed by one of her sons, or servants, or by her husband. Still others confine Women entirely to their houses except during religious festivals. But the wisest of our Circles — our Statesmen — have found that piling up restrictions on Females tends not only to weaken and diminish the race, but also to increase domestic murders to such an extent that a State loses more than it gains from an overly restrictive Code.
Because whenever the temper of the Women is pushed to extremes by confinement at home or burdensome regulations outside it, they tend to take out their fury on their husbands and children. In the less temperate climates, the entire male population of a village has sometimes been wiped out in one or two hours during a simultaneous female rampage. So the Three Laws mentioned above are enough for the better-governed States, and can be taken as a rough summary of our Female Code.
In the end, our main safeguard lies not in legislation but in the self-interest of the Women themselves. Because although they can inflict instant death with a backward movement, unless they can immediately free their stinging end from the struggling body of their victim, their own fragile bodies are likely to shatter.
The power of Fashion is also on our side. I noted that in some less civilized States, no Female is allowed to stand in any public place without swaying her back from right to left. This practice has been universal among ladies of any claim to good breeding in all well-governed States, as far back as anyone can remember. It's considered a disgrace to any State that legislation should have to enforce what ought to be — and what is, in every respectable female — a natural instinct. The rhythmic and, if I may say so, elegantly modulated swaying of the back among our ladies of Circular rank is envied and imitated by the wife of a common Equilateral, who can manage nothing beyond a dull, monotonous swing, like the ticking of a pendulum. And the regular tick of the Equilateral's wife is no less admired and copied by the wife of the ambitious, upwardly mobile Isosceles, among whose women no "back-motion" of any kind has yet become a necessity of life. As a result, in every family of status and standing, "back motion" is as constant as time itself — and the husbands and sons in these households enjoy immunity from invisible attacks, at least.
Not that anyone should suppose for a moment that our Women are without affection. But unfortunately, the passion of the moment overwhelms every other consideration in the Weaker Sex. This is, of course, an unavoidable consequence of their unfortunate shape. Since they have no claim to an angle — being inferior in this respect to even the lowest of the Isosceles — they are consequently devoid of any brainpower whatsoever, and have neither reflection, judgment, nor forethought, and hardly any memory. So in their fits of fury, they remember no obligations and recognize no distinctions. I have actually known a case where a Woman exterminated her entire household, and half an hour later, when her rage had passed and the fragments had been swept away, asked what had become of her husband and children.
Obviously, then, a Woman must not be irritated as long as she's in a position where she can turn around. When you have them in their apartments — which are designed specifically to deny them that ability — you can say and do whatever you like, because they are then completely harmless. They won't remember, a few minutes from now, the thing they may be threatening you with death over at this very moment, nor the promises you may have found it necessary to make in order to calm them down.
On the whole, we get along fairly smoothly in our domestic lives, except in the lower ranks of the Military Classes. There, the husbands' lack of tact and discretion produces indescribable disasters from time to time. Relying too much on the offensive weapons of their sharp angles instead of the defensive tools of good sense and well-timed pretense, these reckless creatures too often neglect the required design of the women's apartments, or provoke their wives with thoughtless remarks in public, which they then refuse to take back. What's more, a blunt, stubborn devotion to literal truth makes them unwilling to offer the kind of lavish promises by which a wiser Circle can instantly calm his wife. The result is massacre — though not without its benefits, as it eliminates the more brutal and troublesome of the Isosceles. In fact, many of our Circles regard the destructive power of the Thinner Sex as one of several providential arrangements for suppressing excess population and nipping Revolution in the bud.
Yet even in our best-regulated and most nearly Circular families, I can't say that the ideal of family life is as high as yours in Spaceland. There is peace, insofar as the absence of slaughter can be called peace, but there is necessarily little harmony of tastes or interests. The cautious wisdom of the Circles has purchased safety at the cost of domestic comfort. In every Circular or Polygonal household, it has been the custom since time immemorial — and has now become a kind of instinct among the women of our upper classes — that mothers and daughters should constantly keep their eyes and mouths turned toward their husband and his male friends. For a lady of distinguished family to turn her back on her husband would be regarded as a kind of omen, involving loss of status. But, as I will soon show, this custom — though it has the advantage of safety — is not without its drawbacks.
In the house of a Working Man or respectable Tradesman, where the wife is allowed to turn her back on her husband while doing housework, there are at least intervals of quiet, when the wife is neither seen nor heard except for the humming sound of the constant Peace-cry. But in the homes of the upper classes, there is too often no peace at all. There, the nonstop mouth and bright, piercing eye are always directed at the Master of the household, and light itself is not more constant than the stream of Feminine conversation. The tact and skill that are enough to avoid a Woman's sting are no match for the task of stopping a Woman's mouth. And since the wife has absolutely nothing to say, and absolutely no restraint of wit, sense, or conscience to prevent her from saying it, more than a few cynics have been known to declare that they'd prefer the danger of a silent, death-dealing sting to the safe but relentless noise of a Woman's other end.
To my readers in Spaceland, the condition of our Women may seem truly deplorable — and indeed it is. A Male of the lowest type of Isosceles may look forward to some improvement in his angle, and to the eventual elevation of his entire degraded caste. But no Woman can hold out any such hope for her sex. "Once a Woman, always a Woman" is a Decree of Nature, and the very Laws of Evolution seem suspended to her disadvantage. Yet at least we can admire the wise Design that has ordained that, since they have no hopes, they shall also have no memory to recall, and no forethought to anticipate, the miseries and humiliations that are at once a necessity of their existence and the foundation of the constitution of Flatland.
You, who are blessed with shade as well as light -- you, who have the gift of two eyes, an understanding of perspective, and the pleasure of seeing in full color -- you, who can actually see an angle and take in the complete circumference of a Circle in the happy realm of Three Dimensions -- how can I possibly convey to you how incredibly difficult it is for us in Flatland to recognize one another's shape?
Think back to what I told you earlier. Every being in Flatland, living or otherwise, no matter what their form, looks to us the same, or very nearly the same: a straight Line. So how can one person be distinguished from another, when everyone looks identical?
The answer comes in three parts. The first method of recognition is hearing. Our sense of hearing is far more developed than yours, and it allows us not only to recognize the voices of our personal friends, but even to tell different classes apart -- at least when it comes to the three lowest ranks: the Equilateral, the Square, and the Pentagon. (I'm not counting the Isosceles here.) But as we climb the social ladder, telling people apart by sound gets harder and harder, partly because voices become more similar at higher ranks, and partly because the ability to distinguish voices is considered a common skill, not one the Aristocracy bothers to develop. And whenever there's any risk of someone impersonating a higher class, we can't rely on this method at all. Among our lowest orders, the vocal organs are developed to a degree far exceeding their sense of hearing, so an Isosceles can easily fake the voice of a Polygon, and with some practice, even that of a Circle. A second method is therefore more commonly used.
Feeling is, among our Women and lower classes -- I'll get to the upper classes in a moment -- the main way of identifying someone, at least between strangers, and when the question isn't who someone is, but what class they belong to. What "introductions" are among the higher classes in Spaceland, "Feeling" is for us. "Allow me to ask you to feel and be felt by my friend Mr. So-and-so" -- this is still, among the more old-fashioned of our country gentlemen in districts far from the cities, the standard formula for a Flatland introduction. But in the cities, and among businessmen, the words "be felt by" are dropped and the phrase is shortened to: "Let me ask you to feel Mr. So-and-so" -- although it's assumed, of course, that the Feeling will go both ways. Among our even more modern and fashionable young gentlemen -- who hate unnecessary effort and couldn't care less about the purity of their native language -- the formula is shortened still further by using "to feel" in a technical sense, meaning "to recommend-for-the-purposes-of-feeling-and-being-felt." And so the current slang among polite and fast society in the upper classes allows such a mangling of language as "Mr. Smith, allow me to feel Mr. Jones."
Don't assume, though, that Feeling is the tedious process for us that it would be for you, or that we need to feel our way around every single side of a person before we can determine what class they belong to. Years of practice and training, starting in school and continuing through everyday life, let us instantly tell the difference, just by touch, between the angles of an Equilateral Triangle, a Square, and a Pentagon. And I hardly need to mention that the pointy, brainless vertex of an acute-angled Isosceles is obvious to even the clumsiest touch. So it's not usually necessary to feel more than a single angle of a person, and that one angle tells us the class of whoever we're dealing with -- unless they happen to belong to the higher ranks of the nobility. There, the difficulty gets much greater. Even a Master of Arts at our University of Wentbridge has been known to confuse a ten-sided Polygon with a twelve-sided one. And there's hardly a Doctor of Science, inside or outside that famous University, who could claim to quickly and confidently tell the difference between a twenty-sided and a twenty-four-sided member of the Aristocracy.
Those of you who remember the excerpts I shared earlier from the legal code regarding Women will immediately see that the process of introduction by touch requires some care and caution. Otherwise, the sharp angles involved could inflict serious -- even irreparable -- injury on the unwary Feeler. It's absolutely essential for the safety of the Feeler that the person being Felt should stand perfectly still. A sudden start, a fidgety shift in position -- even a violent sneeze -- has been known to prove fatal to the careless, cutting short many a promising friendship before it could begin. This is especially true among the lower classes of the Triangles. Their eye is positioned so far from their vertex that they can barely keep track of what's happening at that sharp end of their body. They're also rough and coarse by nature, not sensitive to the delicate touch of a highly refined Polygon. So is it any wonder that an involuntary toss of the head has, before now, robbed the State of a valuable life?
I've heard that my excellent Grandfather -- one of the least irregular of his unfortunate Isosceles class, who actually received, shortly before his death, four out of seven votes from the Sanitary and Social Board to be promoted into the class of Equal-sided -- often lamented, with a tear in his aged eye, a tragedy of exactly this kind. It had happened to his great-great-great-Grandfather, a respectable Working Man with an angle (or brain) of 59 degrees and 30 minutes. According to his account, my unfortunate Ancestor was suffering from rheumatism and was in the process of being Felt by a Polygon when, with one sudden jolt, he accidentally ran the Great Man through on his diagonal. The consequences were devastating: partly because of his long imprisonment and disgrace, and partly because of the moral shock that rippled through our entire family, this single accident set us back a degree and a half in our ascent toward better things. The result was that in the next generation, the family brain was registered at only 58 degrees, and it took five full generations to recover the lost ground, reach the full 60 degrees, and finally achieve the Ascent out of the Isosceles class. And all this cascade of disasters from one little accident during Feeling.
At this point, I think I can hear some of my better-educated readers protesting: "How could you in Flatland know anything about angles and degrees, or arc minutes? We see an angle because we, living in the realm of Space, can see two straight lines leaning toward each other. But you, who can see nothing but one straight line at a time -- or at best, a bunch of little segments all in one straight line -- how could you ever perceive an angle, let alone measure angles of different sizes?"
My answer is that though we can't see angles, we can infer them, and we do so with great precision. Our sense of touch, sharpened by necessity and refined through long training, lets us distinguish angles far more accurately than your sense of sight can when it's not aided by a protractor or other measuring tool. And I should also explain that we have some major natural advantages. In Flatland, it's a Law of Nature that the brain of the Isosceles class starts at half a degree (or thirty minutes) and increases -- if it increases at all -- by half a degree in every generation, until the goal of 60 degrees is reached. At that point, the condition of serfdom is left behind, and the freed individual enters the class of Regulars.
As a result, Nature herself provides us with a built-in scale -- an Alphabet of angles, you might say -- running from half a degree all the way up to 60 degrees. Specimens of these angles are placed in every Elementary School across the land. Thanks to occasional backsliding, even more frequent moral and intellectual stagnation, and the extraordinary birth rate among the Criminal and Vagabond classes, there's always an enormous surplus of individuals at the half-degree and single-degree level, and a fair supply of Specimens up to 10 degrees. These individuals have absolutely no civil rights, and a great many of them, not even having enough intelligence for military service, are assigned by the State to serve education. Chained down so securely that there's no possibility of danger, they're placed in the classrooms of our Infant Schools, where they're used by the Board of Education to teach the children of the Middle Classes the tactile skills and intelligence that these wretched creatures themselves completely lack.
In some States, the Specimens are occasionally fed and allowed to survive for several years. But in the more moderate and better-run regions, it's been found, over time, more beneficial for the educational interests of the young to skip the feeding altogether and replace the Specimens every month -- which is roughly how long a member of the Criminal class can survive without food. In the cheaper schools, whatever is gained by keeping a Specimen alive longer is lost, partly in the cost of food, and partly in the declining accuracy of the angles, which deteriorate after a few weeks of constant Feeling. And we shouldn't forget to mention, when listing the advantages of the more expensive system, that it also contributes -- slightly but noticeably -- to reducing the surplus Isosceles population, a goal that every statesman in Flatland keeps constantly in mind. So on the whole -- although I'm well aware that in many popularly elected School Boards, there's a push to go back to "the cheap system," as they call it -- I'm personally inclined to think that this is one of those many cases where spending more actually saves more in the long run.
But I shouldn't let questions of School Board politics distract me from my subject. I trust I've said enough to show that Recognition by Feeling is not nearly as tedious or unreliable a process as you might have imagined. And it's clearly more trustworthy than Recognition by hearing. Still, as I pointed out earlier, there remains the objection that this method is not without its dangers. For this reason, many in the Middle and Lower classes -- and absolutely everyone in the Polygonal and Circular orders -- prefer a third method, which I'll describe in the next section.
I'm about to seem very inconsistent. In the previous sections, I said that all figures in Flatland look like straight lines, and I implied -- or outright stated -- that it's impossible to tell different classes apart by sight alone. Yet here I am, about to explain to my Spaceland readers how we actually can recognize one another visually.
If you go back and reread the passage where I described Recognition by Feeling as universal, though, you'll find a key qualifier: "among the lower classes." It's only among the higher classes, and in our more temperate climates, that Sight Recognition is practiced.
The fact that this ability exists at all, in any region or for any class, is thanks to Fog. Fog blankets our world for the greater part of the year everywhere except the torrid zones. What is for you in Spaceland a pure nuisance -- blotting out the landscape, dampening the spirits, undermining your health -- is for us a blessing almost as essential as air itself, the nursemaid of arts and the parent of sciences. But let me explain what I mean without heaping any more praise on this wonderful element.
If Fog didn't exist, all lines would appear equally and indistinguishably clear -- and this is actually the case in those unfortunate countries where the atmosphere is perfectly dry and transparent. But wherever there's a rich supply of Fog, objects at a distance of, say, three feet are noticeably dimmer than those at a distance of two feet and eleven inches. The result is that through careful, constant observation of comparative dimness and clearness, we can figure out the shape of any object we're looking at with remarkable precision.
A single example will make this clearer than a whole volume of generalities.
Say I see two individuals approaching and I want to figure out their rank. Let's suppose they're a Merchant and a Physician -- in other words, an Equilateral Triangle and a Pentagon. How do I tell them apart?
It should be obvious to any child in Spaceland who has taken even a basic geometry class that if I position my eye so that my line of sight bisects one angle (A) of the approaching stranger, my view will fall evenly between the two sides nearest to me (namely CA and AB). I'll be looking at both sides equally, and both will appear the same size.
Now, in the case of (1) the Merchant, what will I see? I'll see a straight line DAE, in which the middle point (A) will be very bright because it's closest to me. But on either side, the line will fade rapidly into dimness, because sides AC and AB recede quickly into the Fog. And what appear to me as the Merchant's outer edges -- namely D and E -- will be very dim indeed.
On the other hand, in the case of (2) the Physician, I'll also see a line (D'A'E') with a bright center (A'), but it will fade less rapidly into dimness, because the sides (A'C', A'B') recede less quickly into the Fog. And what appear to me as the Physician's outer edges -- namely D' and E' -- will be not as dim as the Merchant's.
From these two examples, you can probably see how -- after a very long period of training, backed up by constant real-world experience -- it becomes possible for the well-educated classes among us to distinguish with reasonable accuracy between the middle and lowest ranks, just by looking. If my Spaceland readers have grasped the general concept well enough to see that it's at least possible, and haven't written off my account as totally unbelievable, I'll have achieved all I can reasonably expect. If I tried to go into further detail, I'd only confuse things. Still, for the sake of the young and inexperienced -- who might conclude from the two simple examples I just gave of how I'd recognize my Father and my Sons that Sight Recognition is easy -- I should point out that in actual practice, the problems of Sight Recognition are far more subtle and complicated.
For instance, if my Father, the Triangle, happens to approach me showing his side rather than his angle, then until I've asked him to rotate or until I've shifted my eye around him, I'm momentarily unsure whether he might actually be a Straight Line -- in other words, a Woman. And when I'm in the company of one of my two hexagonal Grandsons, looking straight at one of his sides (AB), it should be clear from the accompanying diagram that I'll see one full line (AB) in comparative brightness (hardly fading at all at the ends) and two shorter lines (CA and BD) dim throughout and fading into even greater dimness toward the far points C and D.
But I shouldn't give in to the temptation of going on and on about these details. Even the most average mathematician in Spaceland will easily believe me when I say that the challenges of everyday life that present themselves to the well-educated -- when they're moving, rotating, advancing, or retreating, all while trying to visually distinguish between a crowd of high-ranking Polygons moving in different directions (as you might encounter at, say, a formal ball or an evening reception) -- are enough to tax the sharpest intellect. This more than justifies the generous endowments of the Learned Professors of Geometry, both Static and Kinetic, at the distinguished University of Wentbridge, where the Science and Art of Sight Recognition are regularly taught to large classes of the elite of the States.
Only a few scions of our noblest and wealthiest families can actually afford the time and money necessary for a thorough education in this noble and valuable Art. Even for me -- a Mathematician of no small standing and the Grandfather of two wonderfully promising and perfectly regular Hexagons -- finding myself in the middle of a crowd of rotating Polygons from the higher classes is occasionally very disorienting. And for a common Tradesman or Serf, of course, such a scene is almost as incomprehensible as it would be to you, my Reader, if you were suddenly dropped into my country.
In such a crowd, you'd see nothing on all sides but a Line, apparently straight, whose parts would flicker irregularly and constantly between brightness and dimness. Even if you'd completed your third year in the Pentagonal and Hexagonal courses at the University and had mastered the theory completely, you'd still find that you needed many years of hands-on experience before you could navigate a fashionable crowd without bumping into your social superiors -- people it would be a breach of etiquette to ask to "feel," and who, thanks to their superior education and breeding, know all about your movements while you know little or nothing about theirs. In short, to conduct yourself with perfect grace in Polygonal society, you really need to be a Polygon yourself. That, at least, is the painful lesson of my own experience.
It's amazing how much the Art -- or I might almost call it the instinct -- of Sight Recognition is sharpened by habitual practice and by avoiding the habit of Feeling. Just as in your world, deaf people who are allowed to rely on sign language and hand-spelling will never fully master the harder but far more valuable skills of lip-reading and spoken language, the same applies to us when it comes to Seeing versus Feeling. Anyone who falls back on Feeling in their early years will never learn Seeing to its fullest potential.
For this reason, among our Higher Classes, Feeling is discouraged or outright forbidden. From the cradle, their children are sent not to the Public Elementary schools (where the art of Feeling is taught) but to exclusive private academies. And at our distinguished University, to "feel" is considered a serious offense, punishable by suspension for the first violation and expulsion for the second.
But among the lower classes, the art of Sight Recognition is seen as an unattainable luxury. A common Tradesman can't afford to let his son spend a third of his life on abstract studies. The children of the poor are therefore allowed to "feel" from their earliest years, and they gain from this a quickness and early liveliness that at first compares very favorably with the sluggish, undeveloped, and listless behavior of the half-educated young Polygons. But when those Polygons have finally completed their University training and are ready to put theory into practice, the transformation that comes over them can almost be described as a rebirth. In every art, science, and social pursuit, they rapidly overtake and leave behind their Triangular rivals.
Only a few of the Polygonal Class fail to pass the Final Test or Leaving Examination at the University. The situation of this unsuccessful minority is truly pitiful. Rejected by the higher class, they're also despised by the lower. They have neither the mature, systematically trained abilities of the Polygonal Bachelors and Masters of Arts, nor the natural quickness and lively adaptability of the young Tradesman. The professions and public services are closed to them, and though in most States they aren't actually forbidden from marrying, they have enormous difficulty forming suitable matches, since experience shows that the children of such unfortunate and poorly equipped parents are generally unfortunate themselves -- if not outright Irregular.
It's from these specimens of our Nobility's rejects that the great Uprisings and Rebellions of past ages have generally drawn their leaders. And the trouble they cause is so great that a growing minority of our more progressive Statesmen believe that true mercy would require their complete suppression -- by decreeing that all who fail to pass the University's Final Examination should be either imprisoned for life or put to a painless death.
But I find myself drifting into the subject of Irregularities, a matter of such vital importance that it demands its own section.
Throughout everything I've written so far, I've been assuming something that probably should have been stated up front as a basic, fundamental principle: that every human being in Flatland is a Regular Figure — meaning, a figure with regular construction. By this I mean that a Woman must not only be a line, but a perfectly straight line. An Artisan or Soldier must have two of his sides equal. Tradesmen must have three sides equal. Lawyers (a class of which I am a humble member) must have four sides equal. And in general, for every Polygon, all sides must be equal.
The sizes of those sides naturally depend on the individual's age. A female at birth would be about an inch long, while a tall adult Woman might extend to a foot. As for the Males of every class, you can roughly say that when an adult's sides are added together, the total comes to two feet or a little more. But the size of our sides isn't what I'm talking about here. I'm talking about the equality of sides — and it shouldn't take much thought to see that the entire fabric of social life in Flatland rests on the fundamental fact that Nature intends all Figures to have their sides equal.
If our sides were unequal, our angles could be unequal too. Instead of only needing to feel or visually estimate a single angle in order to identify someone, it would be necessary to verify each angle individually through the painstaking process of Feeling. But life would be too short for that kind of tedious groping around. The whole science and art of Sight Recognition would immediately collapse. Feeling, as far as it's an art, wouldn't survive much longer. Social interaction would become dangerous or impossible. All trust, all forward planning — gone. No one would be safe making even the simplest social arrangements. In a word, civilization might slide right back into barbarism.
Am I going too fast for my Readers to follow me to these obvious conclusions? Surely a moment's thought, and a single example from everyday life, is enough to convince anyone that our social system is built on Regularity — that is, on Equality of Angles. Say you run into two or three Tradesmen on the street, and you recognize them immediately as Tradesmen by a glance at their angles and quickly fading sides. You ask them to come to your house for lunch. Right now, you do this with complete confidence, because everyone knows within an inch or two the area that an adult Triangle takes up. But imagine that your Tradesman is dragging behind his regular, respectable front vertex a parallelogram twelve or thirteen inches across. What exactly are you supposed to do with such a monster jammed in your doorway?
But I'm insulting the intelligence of my Readers by piling up details that must be obvious to everyone who enjoys the advantage of living in Spaceland. Clearly, measuring a single angle would no longer be enough under such alarming circumstances. Your entire life would be spent feeling or surveying the perimeter of your acquaintances. Already, the difficulty of avoiding a collision in a crowd is enough to test the skill of even a well-educated Square. But if no one could calculate the Regularity of a single figure in the room, the result would be total chaos and confusion. The slightest panic would cause serious injuries — or, if any Women or Soldiers happened to be present, quite possibly significant loss of life.
Practical necessity, then, agrees with Nature in putting its stamp of approval on Regularity. And the Law has not been slow in backing up their joint efforts. "Irregularity of Figure" means for us the same thing as — or actually worse than — what you would understand by a combination of moral corruption and outright criminality. It's treated accordingly. There are, it's true, certain advocates of contrarian views who insist there is no necessary connection between geometrical and moral Irregularity. "The Irregular," they say, "is from birth shunned by his own parents, mocked by his brothers and sisters, neglected by the household servants, scorned and distrusted by society, and shut out of every position of responsibility, trust, and meaningful work. His every move is jealously watched by the police until he comes of age and presents himself for inspection. At that point, he is either destroyed — if he's found to exceed the allowable margin of deviation — or assigned a dead-end job for miserable pay, forced to live and eat at his workplace, and required to take even his holidays under close supervision. Is it any wonder that human nature, even in the best and purest of us, is warped and twisted by such conditions?"
All of this very reasonable-sounding argument doesn't convince me, just as it has never convinced the wisest of our Statesmen, that our ancestors were wrong in laying down as a core principle of policy that tolerating Irregularity is incompatible with the safety of the State. Yes, the life of an Irregular is hard. But the interests of the Greater Number require that it be hard. If a man with a triangular front and a polygonal back were allowed to exist and to produce an even more Irregular next generation, what would become of the basic functioning of our society? Are we supposed to redesign all the houses, doors, and churches in Flatland to accommodate such monsters? Should our ticket-collectors be required to measure every man's perimeter before letting him into a theater or taking his seat in a lecture hall? Should an Irregular be exempt from military service? And if not, how do you stop him from causing devastation in the ranks of his fellow soldiers? And then there's the irresistible temptation to fraud that such a creature would face! How easy for him to walk into a shop with his polygonal front leading, and order goods to any extent from a trusting Tradesman! Let the advocates of a falsely named Philanthropy plead all they want for the repeal of the Irregular Penal Laws. I, for my part, have never known an Irregular who was not also exactly what Nature clearly intended him to be — a hypocrite, a misanthrope, and, up to the limits of his power, a perpetrator of every kind of harm.
Not that I would recommend — at the present time — the extreme measures adopted by some States, where an infant whose angle deviates by as little as half a degree from the correct angularity is immediately destroyed at birth. Some of our greatest and most capable men, men of genuine genius, struggled during their earliest days with deviations as large as, or even larger than, forty-five minutes of arc. The loss of their valuable lives would have been an irreparable blow to the State. The art of medicine has also achieved some of its most glorious triumphs in the compressions, extensions, bone-settings, bindings, and other surgical or therapeutic procedures by which Irregularity has been partly or fully cured. Advocating, therefore, a Via Media — a middle path — I would not draw any fixed or absolute line of demarcation. But at the point when the body is just beginning to set into its final form, and when the Medical Board has reported that recovery is unlikely, I would suggest that the Irregular offspring be painlessly and mercifully put to death.
If you've been following along with any attention up to this point, you won't be surprised to hear that life is somewhat dull in Flatland. I don't mean, of course, that we lack battles, conspiracies, upheavals, factions, and all the other events that are supposed to make History interesting. Nor would I deny that the strange overlap between the problems of everyday life and the problems of Mathematics — constantly sparking new theories and offering immediate chances to test them — gives our existence a kind of intellectual thrill that you in Spaceland can hardly comprehend. What I mean is that from an aesthetic and artistic point of view, life here is dull. Aesthetically and artistically, very dull indeed.
How could it be otherwise, when everything you see — every landscape, every historical painting, every portrait, every flower arrangement, every still life — is nothing but a single line, with no variety except degrees of brightness and darkness?
It wasn't always this way. Color, if tradition can be trusted, once threw a brief and glorious splendor over the lives of our ancestors for half a dozen centuries or more, back in the most remote ages. A certain private citizen — a Pentagon whose name is reported differently depending on the source — apparently stumbled upon the ingredients of the simpler colors and a basic method of painting. He's said to have started by decorating his house, then his slaves, then his Father, his Sons, and Grandsons, and finally himself. Both the convenience and the beauty of the results impressed everyone. Wherever Chromatistes — for that's the name the most reliable authorities agree on — turned his multicolored frame, he immediately attracted attention and commanded respect. No one needed to "Feel" him anymore. No one mistook his front for his back. All his movements could be easily tracked by his neighbors without the slightest strain on their powers of calculation. No one jostled him or failed to get out of his way. His voice was spared the exhausting effort of constantly announcing himself — something we colorless Squares and Pentagons are often forced to do when moving through a crowd of clueless Isosceles.
The fashion spread like wildfire. Within a week, every Square and Triangle in the district had copied Chromatistes's example, and only a few of the more conservative Pentagons were still holding out. A month or two later, even the Dodecagons had caught the bug. Within a year, the habit had spread to everyone except the very highest ranks of the Nobility. Needless to say, the custom quickly traveled from Chromatistes's home district to the surrounding regions, and within two generations no one in all of Flatland was colorless except the Women and the Priests.
Here, Nature herself seemed to put up a barrier and argue against extending the trend to these two classes. Having many sides was almost essential as a justification for the innovators. "Distinction of sides is meant by Nature to imply distinction of colors" — that was the slogan that flew from mouth to mouth in those days, converting entire towns to the new culture overnight. But this catchphrase obviously didn't apply to our Priests and Women. Women had only one side, and therefore — to be technical and pedantic about it — no sides. The Priests, meanwhile — at least if they were going to insist on their claim to be true Circles rather than merely high-ranking Polygons with an infinitely large number of infinitely small sides — had long boasted (as Women openly admitted and lamented) that they too had no sides, being blessed with a perimeter consisting of only one line, or in other words, a Circumference. And so it came about that these two Classes saw no force in the so-called axiom about "Distinction of Sides Implying Distinction of Color." When everyone else had given in to the allure of decorating their bodies, the Priests and the Women alone remained free from the pollution of paint.
Immoral, licentious, anarchical, unscientific — call them whatever you like — but from an aesthetic point of view, those ancient days of the Colour Revolt were the glorious childhood of Art in Flatland. A childhood that, sadly, never ripened into adulthood, or even reached the bloom of youth. To live then was itself a delight, because living meant seeing. Even at a small party, the company was a pleasure to look at. The richly varied hues of a congregation in a church or an audience in a theatre are said to have more than once proved too distracting for even our greatest teachers and actors. But the most dazzling spectacle of all, by every account, was the unspeakable magnificence of a military review.
Picture it: a battle line of twenty thousand Isosceles suddenly wheeling around, swapping the somber black of their bases for the bright orange of the two sides flanking their acute angle. The militia of Equilateral Triangles, tricolored in red, white, and blue. The mauve, ultramarine, gamboge, and burnt umber of the Square artillerymen rapidly rotating beside their vermillion guns. The dashing and flashing of the five-colored and six-colored Pentagons and Hexagons streaking across the field in their roles as surgeons, geometricians, and aides-de-camp — all of this was apparently enough to make believable the famous story of 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, declaring that from now on he was trading them for the artist's pencil. How magnificent the sensory and artistic development of those days must have been is partly revealed by the very language and vocabulary of the period. Even the most ordinary remarks of the most ordinary citizens during the Colour Revolt seem to have been infused with a richer tone of word and thought. And to that era we still owe our finest poetry and whatever sense of rhythm still survives in the more scientific language of our modern days.
But meanwhile, the intellectual arts were rapidly declining.
The Art of Sight Recognition, no longer needed, was no longer practiced. The study of Geometry, Statics, Kinetics, and other related subjects soon came to be seen as pointless and fell into disrepute, even at our University. The simpler Art of Feeling quickly met the same fate in our elementary schools. Then the Isosceles classes, claiming that the Specimens were no longer used or needed and refusing to pay the traditional tribute from the Criminal classes toward the cost of Education, grew more numerous and more brazen by the day. They were emboldened by their freedom from the old burden — a burden that had previously served the double benefit of taming their violent nature and thinning their excessive numbers.
Year by year the Soldiers and Artisans began to argue more forcefully — and with increasing justification — that there was no real difference between them and the very highest class of Polygons, now that they'd been raised to the same level. After all, they could now tackle every difficulty and solve every problem of life, whether fixed or in motion, through the simple process of Colour Recognition. Not satisfied with the natural decline of Sight Recognition, they boldly demanded the legal prohibition of all "monopolizing and aristocratic Arts" and the resulting elimination of all funding for the study of Sight Recognition, Mathematics, and Feeling. Before long, they insisted that since Colour — which had become second nature — had destroyed the need for aristocratic distinctions, the Law should follow the same path. From now on, they argued, all individuals and all classes should be recognized as absolutely equal and entitled to equal rights.
Seeing the higher Orders hesitating and unable to make up their minds, the leaders of the Revolution pushed their demands even further. At last, they demanded that all classes — even the Priests and Women — should pay homage to Colour by agreeing to be painted. When it was objected that Priests and Women had no sides, they shot back that Nature and common sense both dictated that the front half of every human being (that is, the half containing the eye and mouth) should be distinguishable from the back half. They therefore brought before a general and extraordinary Assembly of all the States of Flatland a Bill proposing that every Woman's front half — the one containing the eye and mouth — should be coloured red, and the back half green. The Priests were to be painted the same way: red applied to the semicircle where the eye and mouth formed the middle point, and green to the other, rear semicircle.
There was no small amount of cunning in this proposal. And it didn't actually come from any Isosceles — no being that simple would have had the mental sharpness to appreciate, let alone design, such a masterpiece of political strategy. Instead, it came from an Irregular Circle who, rather than being destroyed in childhood as he should have been, had been spared by a foolish act of mercy — only to bring devastation to his country and destruction to countless followers.
On one hand, the proposal was designed to win over Women of every class to the side of the Chromatic Innovation. By assigning Women the same two colours as the Priests, the Revolutionists guaranteed that, from certain angles, every Woman would look exactly like a Priest and be treated with the same respect and deference. You can imagine what an irresistible prospect that was for the Female Sex as a whole.
But some of my readers may not see how Priests and Women could look identical under this new law. If so, let me spell it out.
Imagine a Woman properly decorated according to the new Code: the front half (containing the eye and mouth) red, and the back half green. Look at her from one side. Obviously, you'll see a straight line — half red, half green.
Now imagine a Priest whose mouth is at M, and whose front semicircle (AMB) is therefore coloured red while his back semicircle is green, so that the diameter AB divides the green from the red. If you look at this great man with your eye positioned along the same line as his dividing diameter (AB), what you'll see is a straight line (CBD). One half (CB) will be red, and the other (BD) green. The whole line (CD) will be somewhat shorter than that of a full-sized Woman and will fade more quickly toward its ends. But the identical colours would immediately give you the impression that they were the same class, making you overlook the other details. Keep in mind that Sight Recognition was decaying rapidly during the time of the Colour Revolt. Add to that the certainty that Women would quickly learn to shade off their extremities to imitate the Circles, and it must be obvious to you, my dear reader, that the Colour Bill put us in serious danger of mistaking a Priest for a young Woman.
You can easily imagine how thrilling this prospect was for the Frail Sex. They looked forward with delight to the confusion that would follow. At home, they could overhear political and religious secrets meant not for them but for their husbands and brothers, and might even issue commands in the name of a priestly Circle. Out on the street, the striking combination of red and green — without any other colours — would inevitably lead ordinary people into endless mix-ups. Women would gain whatever respect the Circles lost in the eyes of passersby. As for the scandal that would fall upon the Circular Class if the frivolous and improper behavior of Women were blamed on them, and as for the resulting threat to the Constitution — well, the Female Sex couldn't be expected to spare a thought for such concerns. Even in the households of the Circles themselves, the Women were all in favor of the Universal Colour Bill.
The second goal of the Bill was the gradual weakening of the Circles themselves. Amid the general intellectual decline, they had still maintained their original clarity and strength of mind. From their earliest childhood, growing up in Circular households where Colour was entirely absent, the Nobles alone had preserved the Sacred Art of Sight Recognition — along with all the benefits that come from such excellent training of the intellect. So up to the time the Universal Colour Bill was introduced, the Circles had not only held their own but had actually widened their advantage over the other classes by refusing to follow the popular trend.
And so the cunning Irregular I described above — the true architect of this diabolical Bill — aimed to accomplish two things in a single stroke. First, to lower the standing of the Priesthood by forcing them to submit to the contamination of Colour. Second, to destroy their home advantage in training for Sight Recognition, thereby weakening their minds by stripping away their pure and colourless households. Once the stain of colour was introduced, every parent and every child in a Circular family would undermine each other. The only challenge left for a Circular infant's intellect would be telling Father from Mother — and even that would often be corrupted by a mother's impersonations, shaking the child's faith in all logical reasoning. Gradually, the intellectual brilliance of the Priestly Order would fade, and the path would lie wide open for the total destruction of all aristocratic government and the overthrow of our Privileged Classes.
The campaign for the Universal Colour Bill raged on for three years, and right up until the very end, it looked like anarchy was going to win.
An entire army of Polygons who had enlisted to fight as common soldiers was completely wiped out by a larger force of Isosceles Triangles — while the Squares and Pentagons stood by and did nothing.
Even worse, some of the most distinguished Circles fell victim to trouble at home. Wives in many noble households, driven to fury by the political conflict, badgered their husbands relentlessly to drop their opposition to the Colour Bill. Some of them, when their pleas went nowhere, snapped — killing their own innocent children and husbands, and dying themselves in the violence. The records show that during those three years of unrest, no fewer than twenty-three Circles perished in domestic disputes.
The danger was very real. It seemed like the Priests had no choice left — either give in or be wiped out. And then, suddenly, the entire course of events was transformed by one of those dramatic incidents that statesmen should never overlook, should often anticipate, and should sometimes perhaps even engineer — because of the absurdly outsized power such events have to sway public sympathy.
Here's what happened. An Isosceles of the lowest type — with a brain angle barely above four degrees — accidentally got his hands on some paint from a Tradesman's shop he'd robbed. He painted himself (or had himself painted — accounts differ) in the twelve colors of a Dodecagon. Then he walked into the Market Place and, disguising his voice, approached a young woman — the orphan daughter of a noble Polygon whose affection he had unsuccessfully pursued in the past. Through a series of deceptions — aided on one side by a string of lucky coincidences too long to detail here, and on the other by an almost unbelievable carelessness on the part of the bride's relatives — he managed to carry out the marriage. The poor girl killed herself when she discovered the fraud.
When news of this disaster spread from State to State, the Women were shaken to their core. Sympathy for the victim, combined with the terrifying thought that they themselves — or their sisters, or their daughters — could be deceived the same way, made them see the Colour Bill in an entirely new light. More than a few openly declared themselves opposed to it. The rest only needed a small push to do the same. Seizing this opportunity, the Circles quickly called an emergency Assembly of the States. In addition to the usual guard of Convicts, they made sure a large number of reactionary Women were in attendance.
Before an unprecedented crowd, the Chief Circle of that era — a man named Pantocyclus — rose to speak and immediately found himself hissed and booed by a hundred and twenty thousand Isosceles. But he silenced them by announcing that from now on, the Circles would pursue a policy of Concession. Bowing to the will of the majority, they would accept the Colour Bill. The jeering instantly turned to cheering. He then invited Chromatistes, the leader of the Sedition, to the center of the hall to receive, on behalf of his followers, the Hierarchy's formal surrender. What followed was a speech — a masterpiece of persuasion — that took nearly an entire day to deliver, and that no summary could possibly do justice to.
With a convincing appearance of fairness, Pantocyclus declared that since they were now committing themselves once and for all to Reform and Innovation, it would be wise to take one final look at the whole picture — the drawbacks as well as the benefits. Gradually introducing the subject of dangers facing the Tradesmen, the Professional Classes, and the Gentlemen, he quieted the rising grumbling from the Isosceles by reminding them that despite all these drawbacks, he was still willing to accept the Bill if the majority approved. But it was clear that everyone except the Isosceles had been moved by his words and were either neutral or opposed.
Turning to the Workmen, he insisted that their interests must not be ignored, and that if they intended to support the Colour Bill, they should at least do so with their eyes wide open to the consequences. Many of them, he pointed out, were on the verge of being promoted to the class of Regular Triangles. Others expected their children to achieve a status they could never reach themselves. That honorable ambition would have to be thrown away. With the universal adoption of Colour, all distinctions would vanish. Regularity would be confused with Irregularity. Progress would give way to regression. Within a few generations, the Workman would be degraded to the level of the Military, or even the Convict Class. Political power would end up in the hands of the greatest number — which is to say, the Criminal Classes — who already outnumbered the Workmen and would soon outnumber every other class combined, once the usual balancing Laws of Nature were thrown out of order.
A quiet murmur of agreement rippled through the ranks of the Artisans. Chromatistes, alarmed, tried to step forward and address them. But he found himself surrounded by guards and forced to stay silent while the Chief Circle, in a few passionate words, made his final appeal to the Women. He cried out that if the Colour Bill passed, no marriage would ever be safe again, no woman's honor secure. Fraud, deception, and hypocrisy would infect every household. Domestic happiness would share the fate of the Constitution and be destroyed. "Better death than this!" he shouted.
Those words were the prearranged signal for action.
The Isosceles Convicts rushed forward and killed the unfortunate Chromatistes on the spot. The Regular Classes opened their ranks to let through a band of Women who, under the direction of the Circles, advanced rear-end first — invisible and unerring — toward the unsuspecting soldiers. The Artisans, following the example of their social betters, also opened their ranks. Meanwhile, squads of Convicts sealed off every exit with an impenetrable wall of bodies.
The battle — or rather, the slaughter — was over quickly. Under the Circles' expert command, nearly every charge by a Woman was fatal, and many managed to pull their stingers out unharmed, ready for a second strike. But no second strike was needed. The Isosceles mob did the rest of the work themselves. Caught off guard, leaderless, attacked from the front by invisible enemies, and finding every way out blocked by Convicts behind them, they immediately — as was their nature — lost all composure and raised the cry of "Treachery!" That sealed their fate. Every Isosceles now saw an enemy in every other Isosceles. Within half an hour, not one of that enormous crowd was still alive, and the shattered remains of a hundred and forty thousand members of the Criminal Class — killed by each other's angles — stood as proof of the triumph of Order.
The Circles wasted no time pressing their victory to its fullest extent. They spared the Working Men but decimated them. The Militia of the Equilaterals was called up immediately, and every Triangle suspected on reasonable grounds of Irregularity was executed by Court Martial, without the formality of precise measurement by the Social Board. The homes of the Military and Artisan classes were subjected to inspections that stretched over more than a year. During that time, every town, village, and hamlet was systematically purged of the surplus of lower-order citizens that had built up through the neglect of sending Criminals to the Schools and University, and through the violation of other natural Laws of the Constitution of Flatland. And so the balance of classes was restored.
It goes without saying that from then on, the use of Colour was abolished and its possession made illegal. Even speaking any word referring to Colour — unless you were a Circle or a qualified scientific teacher — was punished with a severe penalty. Only at our University, in some of the very highest and most exclusive classes — which I myself have never had the privilege of attending — is it understood that the limited use of Colour is still permitted for the purpose of illustrating some of the deeper problems of mathematics. But on that point, I can only speak from hearsay.
Everywhere else in Flatland, Colour is now nonexistent. The art of making it is known to only one living person: the current Chief Circle. On his deathbed, he passes the secret to no one but his Successor. A single factory produces it, and to prevent the secret from leaking out, the workers are killed every year and replaced with new ones.
That is how deep the terror runs — even now — when our Aristocracy looks back to those distant days of the campaign for the Universal Colour Bill.
It's high time I moved past these brief, scattered observations about life in Flatland and got to the central event of this book: my initiation into the mysteries of Space. That is my real subject. Everything before this has been merely preface.
For that reason, I have to skip over many topics whose explanation would not, I flatter myself, be without interest to my readers. For example: how we propel and stop ourselves, despite having no feet. How we anchor structures made of wood, stone, or brick, even though we obviously have no hands, can't lay foundations the way you can, and can't rely on the lateral pressure of the earth. How the rain originates in the gaps between our various zones, so that the northern regions don't intercept the moisture meant for the south. 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 — and I mention them now only so my readers know that these omissions aren't due to forgetfulness on my part, but out of respect for your time.
Still, before I move on to my real subject, a few final remarks are no doubt expected about those pillars and mainstays of the Flatland Constitution, the controllers of our conduct and shapers of our destiny, the objects of universal reverence and near-worship. Do I need to say that I mean our Circles — our Priests?
When I call them Priests, don't misunderstand me as meaning only what that term means to you. In our world, our Priests are administrators of all business, art, and science. They are the directors of trade, commerce, military leadership, architecture, engineering, education, government, legislature, morality, and theology. They do nothing themselves, yet they are the cause of everything worth doing that gets done by others.
Although in common usage everyone called a Circle is considered a Circle, the better-educated classes know that no Circle is really a Circle. They're all just Polygons with a very large number of very small sides. As the number of sides increases, a Polygon gets closer and closer to being a Circle. And when the number gets very large indeed — say, three or four hundred — it's extremely difficult for even the most delicate touch to detect any polygonal angles. Actually, let me rephrase that: it would be difficult. As I've explained earlier, Recognition by Feeling is unheard of among the highest ranks of society, and to feel a Circle would be considered a deeply audacious insult. This custom of not touching in polite society makes it all the easier for a Circle to maintain the veil of mystery that, from his earliest years, he keeps wrapped around the exact nature of his Perimeter or Circumference. Since three feet is the average Perimeter, it follows that in a Polygon of three hundred sides, each side will be no more than a hundredth of a foot in length — a little over a tenth of an inch. And in a Polygon of six or seven hundred sides, each side is barely larger than the diameter of a Spaceland pinhead. It is always assumed, as a matter of courtesy, that the Chief Circle of any given era has ten thousand sides.
The upward climb of the Circles' descendants on the social scale isn't restricted the way it is among the lower Regular classes, where the Law of Nature limits the increase to just one additional side per generation. If it were, the number of sides in a Circle would be a simple matter of pedigree and arithmetic — the four hundred and ninety-seventh descendant of an Equilateral Triangle would necessarily be a Polygon with five hundred sides. But that's not how it works. Nature's Law lays down two opposing decrees when it comes to Circular reproduction. First: as the race climbs higher on the scale of development, that development accelerates. Second: in the same proportion, the race becomes less fertile. As a result, in the home of a Polygon with four or five hundred sides, it's rare to find even one son. More than one is never seen. On the other hand, the son of a five-hundred-sided Polygon has been known to have five hundred and fifty, or even six hundred sides.
Medicine also steps in to help the process of higher evolution. Our physicians have discovered that the small, delicate sides of an infant Polygon from the upper classes can be fractured and his entire frame reset with such precision that a Polygon of two or three hundred sides can sometimes — by no means always, since the procedure carries serious risk — but sometimes leap ahead by two or three hundred generations, effectively doubling the number of his ancestors and the nobility of his descent in a single stroke.
Many a promising child is sacrificed this way. Barely one in ten survives. Yet parental ambition runs so strong among those Polygons who sit, as it were, on the fringe of the Circular class, that you'll rarely find a Nobleman of that social standing who has failed to place his firstborn in the Circular Neo-Therapeutic Gymnasium before the child is even a month old.
One year determines success or failure. At the end of that time, the child has, in all likelihood, added one more tombstone to the crowded Neo-Therapeutic Cemetery. But on rare occasions, a joyful procession carries the little one back to his overjoyed parents — no longer a Polygon, but a Circle, at least by courtesy. And a single instance of such a blessed outcome is enough to convince multitudes of Polygonal parents to submit their own children to the same sacrifice, though their results are rarely so fortunate.
The entire doctrine of the Circles can be boiled down to a single motto: "Pay attention to your Configuration." Whether the subject is politics, religion, or morality, everything they teach is aimed at improving individual and collective Configuration -- with special emphasis, naturally, on the Configuration of the Circles themselves, to which all other goals take a backseat.
To their credit, the Circles have effectively stamped out those old heresies that led people to waste their energy and compassion on the foolish belief that behavior depends on willpower, effort, training, encouragement, praise, or anything other than Configuration. It was Pantocyclus -- the famous Circle I mentioned earlier, the one who crushed the Colour Revolt -- who first convinced the public that Configuration makes the man. If, for example, you're born an Isosceles with two uneven sides, you will inevitably go wrong unless you get them corrected -- which means a trip to the Isosceles Hospital. Likewise, if you're a Triangle, Square, or even a Polygon born with any Irregularity, you need to be admitted to one of the Regular Hospitals to have your condition treated. Otherwise, you'll end your days in the State Prison, or at the angle of the State Executioner.
All faults and defects, from the most minor misbehavior to the most outrageous crime, Pantocyclus traced back to some deviation from perfect Regularity in the body. These deviations might be caused (if they weren't present from birth) by getting bumped in a crowd, by not getting enough exercise, or by getting too much of it, or even by a sudden temperature change that caused some vulnerable part of the body to shrink or expand. Therefore, that distinguished Philosopher concluded, neither good behavior nor bad behavior deserves any real praise or blame. After all, why should you praise a Square for faithfully defending his client's interests when what you should really admire is the precise accuracy of his right angles? And why blame a lying, thieving Isosceles when you should really be mourning the hopeless inequality of his sides?
In theory, this doctrine is beyond question. But in practice, it has some awkward drawbacks. When you're dealing with an Isosceles, and the scoundrel claims he can't help stealing because of his unevenness, you just reply that for that very reason -- because he can't help being a menace to his neighbors -- you, the Magistrate, can't help sentencing him to be consumed. End of discussion. But in minor domestic squabbles, where the penalty of consumption -- that is, death -- is off the table, this theory of Configuration can get pretty uncomfortable. I'll confess that when one of my own Hexagonal Grandsons uses as his excuse for disobedience that a sudden change in temperature has been too much for his Perimeter, and that I really ought to blame not him but his Configuration -- which, he assures me, can only be strengthened by generous helpings of the finest sweets -- I can neither find a logical way to reject his argument nor a practical way to accept his conclusions.
For my part, I find it best to assume that a good firm scolding or punishment has some hidden, strengthening effect on my Grandson's Configuration, though I admit I have no real basis for thinking so. At any rate, I'm not alone in taking this way out of the dilemma. I've noticed that many of the highest Circles, when sitting as judges in the courts, freely use praise and blame toward both Regular and Irregular Figures. And in their own homes, I know from personal experience that when scolding their children, they talk about "right" and "wrong" just as passionately and vehemently as if they truly believed those words referred to real things, and that a person is actually capable of choosing between them.
In keeping with their constant policy of making Configuration the central idea in every mind, the Circles have actually reversed that Commandment which in Spaceland governs the relationship between parents and children. In your world, children are taught to honor their parents. In ours -- after the Circles, who are the primary object of universal reverence -- a man is taught to honor his Grandson, if he has one, or failing that, his Son. By "honor," though, they definitely don't mean "spoil." They mean a deep respect for their highest potential. The Circles teach that fathers have a duty to put their own interests below those of their descendants, and in doing so, they advance the welfare of the whole State as well as their own immediate family line.
The weak point in the Circles' system -- if a humble Square may dare to suggest that anything Circular has a weak point -- seems to me to lie in their treatment of Women.
Since it's critically important for Society that Irregular births be discouraged, it follows that no Woman with any Irregularities in her family history is a suitable wife for a man who wants his descendants to rise steadily through the social ranks.
Now, the Irregularity of a Male is something you can measure. But since all Women are straight lines -- and therefore all visibly Regular, so to speak -- you need some other way to determine what I might call their invisible Irregularity: that is, their potential for producing Irregular offspring. This is handled through carefully maintained pedigrees, which are kept and overseen by the State. Without a certified pedigree, no Woman is allowed to marry.
You might expect that a Circle -- proud of his ancestry and protective of a lineage that might someday produce a Chief Circle -- would be the most careful of all in choosing a wife with a spotless family record. But it's actually the opposite. The care taken in choosing a Regular wife seems to decrease as you go up the social ladder. Nothing would persuade an ambitious Isosceles, who dreams of fathering an Equilateral Son, to marry a wife with even a single Irregularity among her ancestors. A Square or Pentagon, confident that his family is on the rise, doesn't bother checking beyond five hundred generations back. A Hexagon or Dodecagon is even more careless about his wife's pedigree. But a Circle has been known to deliberately marry a wife whose Great-Grandfather was Irregular, and all because of some slight superiority in her complexion's shine, or because of the charms of a soft voice -- which in our world, even more than in yours, is considered "an excellent thing in a Woman."
These poorly judged marriages are, as you'd expect, childless -- when they don't actually produce Irregularity or a loss of sides. But none of these consequences have so far proven to be a strong enough deterrent. The loss of a few sides in a highly developed Polygon isn't easily noticed, and can sometimes be fixed by a successful procedure at the Neo-Therapeutic Gymnasium, as I described earlier. And the Circles are too inclined to accept infertility as simply a natural law of their superior development. Yet if this trend isn't stopped, the gradual shrinking of the Circular class could soon accelerate. The day may not be far off when, with the race no longer able to produce a Chief Circle, the Constitution of Flatland itself must collapse.
One more word of warning comes to mind, though I can't as easily suggest a remedy. This too has to do with our treatment of Women. About three hundred years ago, the Chief Circle decreed that since women are lacking in Reason but overflowing with Emotion, they should no longer be treated as rational beings, nor receive any mental education whatsoever. The result was that they were no longer taught to read, or even to learn enough arithmetic to count the angles of their husband or children. And so, generation by generation, they steadily declined in intellectual ability. This system of female non-education -- this enforced ignorance -- is still in effect today.
My fear is that, however well-intentioned, this policy has been pushed so far that it's actually backfiring on the Male Sex.
Here's why. As things stand now, we Males have to lead a kind of bilingual -- and I might almost say bimental -- existence. With Women, we talk about "love," "duty," "right," "wrong," "pity," "hope," and other irrational, emotional concepts that have no real existence. The whole fiction has no purpose except to keep feminine impulses in check. But among ourselves, and in our books, we use an entirely different vocabulary -- and really, a different way of thinking altogether. "Love" becomes "the anticipation of benefits." "Duty" becomes "necessity" or "fitness." And all other words get similarly transformed. What's more, when speaking with Women, we use language that implies the deepest respect for their Sex. They genuinely believe that the Chief Circle Himself is not more devoutly worshipped by us than they are. But behind their backs, they are both regarded and spoken of -- by everyone except the very young -- as little better than "mindless organisms."
Our theology in the Women's quarters is also completely different from our theology everywhere else.
Now, my humble concern is that this double training -- in language as well as in thought -- places too heavy a burden on the young. Especially when, at the age of three, they're taken from their mother's care and taught to unlearn the old language (except for the purpose of repeating it in the presence of Mothers and Nurses) and to learn the vocabulary and way of thinking used in science. Already I think I can detect a weakness in the grasp of mathematical truth today compared to the more powerful intellects of our ancestors three hundred years ago. I'll say nothing of the potential danger if a Woman should ever secretly learn to read and share with her Sex what she discovers in even a single popular book. Nor of the possibility that some disobedient or careless young boy might accidentally reveal to his Mother the secrets of our logical language. On the simple grounds that male intellect is being weakened, I rest this humble appeal to the highest Authorities to reconsider the rules governing Female education.
PART II OTHER WORLDS
"O brave new worlds, That have such people in them!"
"O brave new worlds,
That have such people in them!"
It was the second-to-last day of the year 1999 in our era, and the first day of the Long Vacation. I'd been amusing myself well into the late hours with my favorite pastime -- Geometry -- and had gone to bed with an unsolved problem rattling around in my mind. That night, I had a dream.
I saw before me a vast crowd of small Straight Lines (which I naturally assumed to be Women) mixed in with other beings even smaller -- tiny, glowing points -- all moving back and forth along one and the same Straight Line, and, as far as I could tell, all at the same speed.
A noise of confused, high-pitched chirping or twittering came from them at intervals, as long as they were moving. But sometimes they stopped, and then everything went silent.
I approached one of the largest of what I took to be Women and spoke to her, but got no answer. A second and third attempt on my part were equally useless. Losing patience at what struck me as unbearable rudeness, I moved my mouth to a position directly in front of her mouth so as to block her path, and loudly repeated my question: "Woman, what is this gathering about, and what is all this strange, confused chirping, and this monotonous back-and-forth movement along one and the same Straight Line?"
"I am no Woman," replied the small Line. "I am the Monarch of the world. But you -- where have you come from, intruding into my realm of Lineland?" Taken aback by this blunt reply, I begged his pardon if I had in any way startled or bothered his Royal Highness. Describing myself as a stranger, I asked the King to tell me something about his dominions. But I had the hardest time getting any information about the things that actually interested me. The Monarch couldn't stop assuming that whatever was familiar to him must also be known to me, and that I was just pretending to be ignorant as a joke. Still, by persisting with my questions, I managed to piece together the following facts:
It seemed that this poor, ignorant Monarch -- as he called himself -- was completely convinced that the Straight Line he called his Kingdom, the one in which he spent his entire existence, made up the whole of the world, and in fact the whole of Space. Unable to either move or see in any direction except along his Straight Line, he had no concept of anything outside of it. Although he'd heard my voice when I first spoke to him, the sounds had reached him in a way so contrary to his experience that he hadn't answered, "seeing no man," as he put it, "and hearing a voice as if from my own intestines." Until the moment I placed my mouth inside his World, he had neither seen me nor heard anything except confused sounds beating against what I would call his side, but what he called his inside or stomach. Even now, he had absolutely no concept of the region I'd come from. Outside his World -- his Line -- everything was a blank to him. No, not even a blank, because a blank implies Space. Better to say: everything outside was simply nonexistent.
His subjects -- the small Lines being men and the Points being Women -- were all equally confined in motion and vision to that single Straight Line, their entire World. It hardly needs saying that their whole horizon was limited to a Point; nobody could ever see anything but a Point. Man, woman, child, thing -- each appeared as a Point to the eye of a Linelander. Only by the sound of someone's voice could sex or age be told apart. What's more, since each individual occupied the whole of the narrow path (so to speak) that made up his Universe, and no one could move to the right or left to make way for anyone else, it followed that no Linelander could ever pass another. Once neighbors, always neighbors. Neighborhood with them was like marriage with us. Neighbors remained neighbors till death did them part.
Such a life, with all vision limited to a Point and all movement limited to a Straight Line, seemed to me impossibly dreary. I was surprised to notice the King's vivacity and good cheer. Wondering whether it was even possible, under circumstances so unfavorable to domestic relations, to enjoy the pleasures of married life, I hesitated for some time to question his Royal Highness on such a delicate subject. But at last I dove in by abruptly asking about the health of his family. "My wives and children," he replied, "are well and happy."
I was staggered by this answer -- because in the immediate vicinity of the Monarch (as I'd noticed in my dream before entering Lineland) there were none but Men. I ventured to reply, "Pardon me, but I can't imagine how your Royal Highness can at any time either see or approach their Majesties, when there are at least half a dozen people between you and them, whom you can neither see through nor get past. Is it possible that in Lineland, physical closeness isn't necessary for marriage and for having children?"
"How can you ask such an absurd question?" replied the Monarch. "If it were indeed as you suggest, the Universe would soon be depopulated. No, no -- proximity is not needed for the union of hearts, and the birth of children is far too important a matter to have been left to depend on such an accident as physical closeness. You can't possibly be ignorant of this. Yet since you insist on pretending you are, I will instruct you as though you were the most clueless baby in Lineland. Know, then, that marriages are consummated through the faculty of sound and the sense of hearing.
"You are of course aware that every Man has two mouths or voices -- just as he has two eyes -- a bass at one end and a tenor at the other of his extremities. I shouldn't need to mention this, but I have been unable to make out your tenor in the course of our conversation." I replied that I had only one voice, and that I hadn't been aware his Royal Highness had two. "That confirms my impression," said the King, "that you are not a Man, but a feminine Monstrosity with a bass voice and an utterly uneducated ear. But to continue.
"Nature having herself ordained that every Man should wed two wives --" "Why two?" I asked. "You carry your fake simplicity too far," he cried. "How can there be a completely harmonious union without the combination of the Four in One -- namely, the Bass and Tenor of the Man and the Soprano and Contralto of the two Women?" "But suppose," I said, "that a man preferred one wife or three?" "It is impossible," he said. "It is as inconceivable as two and one making five, or the human eye seeing a Straight Line." I would have interrupted him, but he went on:
"Once in the middle of each week, a Law of Nature compels us to move back and forth with a rhythmic motion of more than usual force, which continues for about the time it would take you to count to a hundred and one. In the midst of this choral dance, at the fifty-first beat, the inhabitants of the Universe pause in mid-stride, and each individual sends out his richest, fullest, sweetest note. It is in this decisive moment that all our marriages are made. So perfectly tuned is the match of Bass to Treble, of Tenor to Contralto, that the Loved Ones -- though twenty thousand leagues away -- often recognize at once the answering note of their destined Lover. And, cutting through the trivial obstacles of distance, Love unites the three. The marriage thus consummated results in a threefold offspring -- Male and Female -- which takes its place in Lineland."
"What! Always threefold?" I said. "Must one wife then always have twins?"
"Bass-voiced 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 stopped, speechless with fury, and some time passed before I could coax him into resuming his story.
"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 repeated many times by most of us. Few are the hearts whose happy lot is to immediately recognize in each other's voice the partner intended for them by Providence, and to fly into a mutual and perfectly harmonious embrace. For most of us, the courtship is long. The Wooer's voices may perhaps match one of his future wives, but not both; or at first, not 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 nudges the less perfect partner to adjust his or her singing so as to come closer to the more perfect. And after many trials and many adjustments, the result is at last achieved. There comes a day when, while the usual Marriage Chorus rings out from all of Lineland, the three far-off Lovers suddenly find themselves in exact harmony, and before they know it, the wedded Triplet is swept vocally into a double embrace -- and Nature rejoices over one more marriage and three more births."
Figuring it was time to bring the Monarch back down from his raptures to something resembling common sense, I decided to try opening his eyes to a few glimpses of the truth -- that is, the nature of things as they exist in Flatland. So I began:
"How does your Royal Highness distinguish the shapes and positions of his subjects? Before I entered your Kingdom, I could see -- with my own sense of sight -- that some of your people are Lines and others are Points, and that some of the Lines are larger--"
"You're describing something impossible," the King interrupted. "You must have been seeing things. Everyone knows that telling the difference between a Line and a Point by sight is simply impossible -- it's in the very nature of things. But it can be detected by hearing, and by the same method my own shape can be determined exactly. Look at me -- I am a Line, the longest in Lineland, over six inches of Space--"
"Of Length," I ventured to suggest.
"Fool," he said. "Space is Length. Interrupt me again and I'm done talking."
I apologized, but he went on scornfully: "Since you're impervious to argument, you can hear with your own ears how I reveal my shape to my Wives using my two voices. They are at this moment six thousand miles, seventy yards, two feet, and eight inches away -- one to the North, the other to the South. Listen -- I'll call to them."
He chirped, and then went on, quite pleased with himself: "My Wives are at this moment receiving the sound of one of my voices, closely followed by the other. They can tell that the second sound reaches them after a slight delay -- specifically, the time it takes sound to travel 6.457 inches. From this, they know that one of my mouths is 6.457 inches farther from them than the other, and therefore my shape measures 6.457 inches. But of course you understand that my Wives don't run this calculation every single time they hear my two voices. They worked it out once and for all before we were married. But they could redo it at any time. And in the same way, I can determine the shape of any of my Male subjects by the sense of sound."
"But what happens," I asked, "if a Man fakes a Woman's voice with one of his two voices, or disguises his Southern voice so it can't be recognized as the echo of the Northern one? Couldn't such deceptions cause serious problems? And don't you have any way to catch these kinds of frauds -- say, by having your neighboring subjects feel one another?"
This was, of course, a very stupid question, since feeling wouldn't have served the purpose at all. But I asked it deliberately to annoy the Monarch, and I succeeded perfectly.
"What?!" he cried in horror. "Explain what you mean."
"Feel. Touch. Come into physical contact," I replied.
"If by feeling you mean," said the King, "getting so close that there is no space left between two individuals -- know this, Stranger: that offense is punishable by death in my dominions. And the reason is obvious. The fragile form of a Woman, which could be shattered by such close contact, must be protected by the State. But since Women cannot be told apart from Men by sight, the Law decrees universally that neither Man nor Woman shall be approached so closely as to eliminate the gap between the one approaching and the one being approached.
"And really, what possible purpose could be served by this illegal and unnatural act you call touching, when everything such a brutal and crude process could achieve is already accomplished far more easily and precisely by the sense of hearing? As for your supposed danger of deception -- it doesn't exist. A person's Voice is the essence of their Being and cannot simply be changed at will. But come now -- suppose I had the power to pass through solid things, so that I could penetrate one subject after another, even a billion of them, verifying each one's size and distance by the sense of feeling. How much time and energy would be wasted on such a clumsy and inaccurate method! As it stands, in a single moment of listening, I take in what amounts to the census and statistics -- local, physical, mental, and spiritual -- of every living being in Lineland. Listen -- just listen!"
With that he paused, listening in what seemed like pure ecstasy to a sound that, to me, was nothing more than a tiny chirping from an innumerable swarm of miniature grasshoppers.
"I'll grant you," I replied, "your sense of hearing serves you remarkably well and makes up for many of your deficiencies. But let me point out that your life in Lineland must be depressingly dull. To see nothing but a Point! To not even be able to look at a Straight Line! No, not even to know what a Straight Line is! To have the power of sight, yet to be cut off from the kind of Linear views that we take for granted in Flatland! Honestly, it would be better to have no sense of sight at all than to see so little! I'll admit I don't have your incredible ability to discriminate sounds -- the great concert of all Lineland that gives you such intense pleasure is, to me, nothing more than a mass of twittering and chirping. But at least I can tell, by sight, the difference between a Line and a Point. And let me prove it. Just before I entered your Kingdom, I watched you dancing from left to right and then from right to left, with Seven Men and a Woman close by on the left, and Eight Men and Two Women on the right. Am I wrong?"
"That is correct," said the King, "as far as the numbers and sexes go, though I have no idea what you mean by 'right' and 'left.' But I deny that you saw these things. How could you possibly see a Line -- that is, the inside of any Man? You must have heard them and then dreamed you saw them. And tell me -- what do you mean by these words 'left' and 'right'? I assume it's your way of saying Northward and Southward."
"Not at all," I replied. "Besides your motion Northward and Southward, there is another kind of motion, which I call from right to left."
King. Show me, if you would, this motion from left to right.
I. I can't do that unless you could step out of your Line entirely.
King. Out of my Line? You mean out of the world? Out of Space?
I. Well, yes. Out of your world. Out of your Space. Because your Space is not true Space. True Space is a Plane; your Space is only a Line.
King. If you can't demonstrate this motion from left to right by actually performing it yourself, then I ask you to describe it to me in words.
I. If you can't tell your right side from your left, I'm afraid no words of mine can make my meaning any clearer. But surely you can't be ignorant of such a simple distinction.
King. I don't understand you in the slightest.
I. This is hopeless! How can I make it clear? When you move straight ahead, doesn't it ever occur to you that you could move in some other direction -- turning so that your eye looks toward the direction your side currently faces? In other words, instead of always moving toward one of your endpoints, don't you ever feel a desire to move in the direction of your side?
King. Never. And what do you mean? How can a person's inside "face" any direction? And how can anyone move in the direction of their own inside?
I. Well then, since words can't explain the matter, I'll let actions speak instead. I'm going to gradually move out of Lineland in the direction I've been trying to describe to you.
As I said this, I began to move my body out of Lineland. As long as any part of me remained within his domain and within his view, the King kept exclaiming, "I see you, I still see you -- you're not moving." But when I had finally moved myself entirely out of his Line, he cried in his shrillest voice, "She has vanished! She is dead!"
"I'm not dead," I called back. "I've simply moved out of Lineland -- that is, out of the Straight Line that you call Space -- and into true Space, where I can see things as they actually are. And right now I can see your Line, or side -- or 'inside,' as you prefer to call it. And I can also see the Men and Women to the North and South of you, whom I will now list out, describing their order, their size, and the distance between each."
When I had done this at considerable length, I cried triumphantly, "Does that finally convince you?" And with that, I dropped back into Lineland, taking up my former position.
But the Monarch replied, "If you were a Man of sense -- though, since you appear to have only one voice, I have little doubt you are not a Man at all but a Woman -- but if you had a shred of sense, you would listen to reason. You're asking me to believe that there is another Line besides the one my senses reveal to me, and another kind of motion besides the one I experience every day. In return, I ask you to describe in words, or demonstrate by motion, this other Line you're talking about. Instead of moving, you simply perform some magic trick of vanishing and reappearing. And instead of any clear description of your supposed new World, you just rattle off the numbers and sizes of some forty of my retinue -- facts that any child in my capital already knows. Could anything be more irrational or arrogant? Either admit your foolishness or leave my dominions."
Furious at his stubbornness -- and especially outraged that he claimed not to know my sex -- I shot back in the harshest possible terms: "You deluded creature! You think you're the pinnacle of existence, when in reality you are the most imperfect and dim-witted being imaginable. You claim to see, yet all you see is a Point! You pride yourself on deducing the existence of a Straight Line, but I can see Straight Lines and deduce the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? It's enough to say 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 to you as I am, count for very little among the great nobles of Flatland, from where I've come to visit you in the hope of enlightening your ignorance."
Hearing these words, the King charged toward me with a menacing cry, as if to pierce me straight through my diagonal. At that same moment, a massive war-cry rose from thousands upon thousands of his subjects, building in ferocity until I could have sworn it rivaled the roar of an army of a hundred thousand Isosceles and the artillery of a thousand Pentagons. Frozen in place, I could neither speak nor move to escape the oncoming destruction. And still the noise grew louder, and the King came closer -- when I woke up to find the breakfast bell calling me back to the realities of Flatland.
From dreams, I move on to facts.
It was the last day of our 1,999th year. The patter of the rain had announced nightfall long ago, and I was sitting in the company of my Wife, reflecting on the events of the past year and the prospects of the coming one -- the coming century, the coming Millennium.
(When I say "sitting," of course I don't mean any change of posture like you in Spaceland would picture. Since we have no feet, we can no more "sit" or "stand" -- in your sense of those words -- than one of your soles or flounders could. Still, we perfectly well recognize the different mental states of intention behind "lying," "sitting," and "standing," which are to some degree visible to an observer as a slight increase in brightness corresponding to an increase in effort of will. But on this, and a thousand related subjects, time doesn't allow me to dwell.)
My four Sons and two orphan Grandchildren had retired to their separate rooms, and my Wife alone remained with me to see the old Millennium out and the new one in.
I was deep in thought, mulling over some words that had casually come from the mouth of my youngest Grandson -- a most promising young Hexagon of unusual brilliance and perfect angularity. His uncles and I had been giving him his usual hands-on lesson in Sight Recognition, spinning ourselves on our centers, sometimes quickly, sometimes more slowly, and quizzing him on our positions. His answers had been so good that I'd been moved to reward him with a few hints on Arithmetic as applied to Geometry.
Taking nine Squares, each one inch on every side, I'd put them together to make one large Square with a side of three inches. And from this I'd proved to my little Grandson that -- though it was impossible for us to see the inside of the Square -- we could still figure out the number of square inches in a Square by simply squaring the number of inches in the side. "And so," I told him, "we know that 3 squared, or 9, represents the number of square inches in a Square whose side is 3 inches long."
The little Hexagon thought about this for a while and then said to me, "But you've been teaching me to raise numbers to the third power. I suppose 3 cubed must mean something in Geometry. What does it mean?"
"Nothing at all," I replied, "not in Geometry, at any rate. Geometry has only Two Dimensions." And then I started showing the boy how a Point, by moving through a length of three inches, makes a Line of three inches, which can be represented by 3. And how a Line of three inches, moving parallel to itself through a length of three inches, makes a Square of three inches on every side, which can be represented by 3 squared.
At this, my Grandson returned to his earlier idea and cut in rather abruptly. "Well then," he exclaimed, "if a Point moving three inches makes a Line of three inches, represented by 3 -- and if a straight Line of three inches, moving parallel to itself, makes a Square of three inches every way, represented by 3 squared -- then 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 3 cubed."
"Go to bed," I said, a little annoyed by this interruption. "If you talked less nonsense, you'd remember more sense."
So my Grandson disappeared in disgrace. And there I sat by my Wife's side, trying to put together a mental review of the year 1999 and the possibilities of the year 2000 -- but not quite able to shake off the thoughts sparked by my bright little Hexagon's chatter. Only a few grains of sand remained in the half-hour glass. Rousing myself from my daydream, I turned the glass Northward for the last time in the old Millennium. And as I did it, I exclaimed aloud, "The boy is a fool."
Instantly I became aware of a Presence in the room, and a chilling breath ran through my entire being. "He is no such thing," cried my Wife, "and you are breaking the Commandments by disrespecting your own Grandson." But I paid no attention to her. Looking around in every direction, I could see nothing -- and yet I felt a Presence, and shivered as the cold whisper came again. I jumped to my feet. "What's the matter?" said my Wife. "There's no draft. What are you looking for? There's nothing." There was nothing. And I sat back down, exclaiming again, "The boy is a fool, I say. 3 cubed can have no meaning in Geometry." At once there came a distinctly audible reply: "The boy is not a fool, and 3 cubed has an obvious Geometrical meaning."
My Wife heard the words as well as I did, though she didn't understand what they meant, and both of us sprang forward in the direction of the sound. Imagine our horror when we saw before us a Figure! At first glance it appeared to be a Woman, seen sideways. But a moment's observation showed me that its edges faded into dimness too rapidly to be one of the Female Sex. And I would have thought it was a Circle, except that it seemed to change its size in a way that was impossible for a Circle -- or for any regular Figure I had ever encountered.
But my Wife didn't have my experience, or the composure needed to notice these details. With the typical quickness to judgment and irrational jealousy of her Sex, she leapt straight to the conclusion that a Woman had entered the house through some small opening. "How did this person get in here?" she exclaimed. "You promised me, my dear, that there would be no ventilators in our new house."
"There aren't any," I said, "but what makes you think the Stranger is a Woman? I can see by my power of Sight Recognition--"
"Oh, I have no patience with your Sight Recognition," she replied. "'Feeling is believing' and 'A Straight Line to the touch is worth a Circle to the sight'" -- two proverbs very popular with the Frailer Sex in Flatland.
"Well," I said, since I was afraid of setting her off, "if it must be so, ask for an introduction." Putting on her most gracious manner, my Wife advanced toward 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 toward 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 gently, "I have a message, dear Madam, for your husband, which I must not deliver in your presence. If you would allow us to step aside for a few minutes--" But my Wife would not hear of our distinguished Visitor putting himself out like that. Assuring the Circle that the hour of her own bedtime had long since passed, and with many repeated apologies for her recent blunder, she at last retreated to her room.
I glanced at the half-hour glass. The last grains of sand had fallen. The third Millennium had begun.
As soon as the sound of my departing Wife's Peace-cry had faded away, I moved toward the Stranger, intending to get a closer look and invite him to sit down. But his appearance struck me dumb. I stood there, motionless with astonishment. Without the slightest hint of angularity, he was nevertheless shifting every instant through gradations of size and brightness that I'd never seen in any Figure within my experience. A thought flashed through my mind: what if this was a burglar or a cutthroat — some monstrous Irregular Isosceles who had faked the voice of a Circle to gain entry to the house, and was now preparing to stab me with his sharp angle?
In a sitting-room, the absence of Fog (and the season happened to be remarkably dry) made it hard for me to rely on Sight Recognition, especially at such a short distance. Desperate with fear, I rushed forward with an unceremonious "You must allow me, Sir —" and felt him. My Wife was right. There was no trace of an angle, not the slightest roughness or irregularity. I had never in my life encountered a more perfect Circle. He stayed motionless while I walked all the way around him, starting from his eye and returning to it again. Circular he was, through and through — a perfectly satisfactory Circle. There could be no doubt about it. What followed was a dialogue that I'll try to set down as closely as I can remember it, leaving out most of my groveling apologies — for I was covered with shame and humiliation that I, a Square, had been guilty of the rudeness of feeling a Circle. It was the Stranger who started things off, with some impatience at how long my introductory process was taking.
Stranger. Have you felt me enough by this time? Aren't you introduced to me yet?
I. Most illustrious Sir, please excuse my awkwardness, which comes not from ignorance of polite society, but from a little surprise and nervousness brought on by this somewhat unexpected visit. And I beg you not to reveal my rudeness to anyone, especially not to my Wife. But before your Lordship goes any further, would he be so kind as to satisfy the curiosity of one who would dearly like to know where his visitor came from?
Stranger. From Space, from Space, Sir — where else?
I. Forgive me, my Lord, but isn't your Lordship already in Space? Both your Lordship and his humble servant, even at this very moment?
Stranger. Pah! What do you know of Space? Define Space.
I. Space, my Lord, is height and width extended indefinitely.
Stranger. Exactly — you see, you don't even know what Space is. You think it has only Two Dimensions. But I've come to announce a Third: height, width, and depth.
I. Your Lordship must be joking. We also use the words "length" and "height," or "width" and "thickness," giving four names to describe just Two Dimensions.
Stranger. But I don't mean just three names. I mean Three Dimensions.
I. Would your Lordship point out or explain to me the direction in which this Third Dimension runs — the one I know nothing about?
Stranger. I came from it. It is up above and down below.
I. My Lord seems to be saying Northward and Southward.
Stranger. I mean nothing of the kind. I mean a direction in which you cannot look, because you have no eye on your side.
I. Forgive me, my Lord, but a moment's inspection will show your Lordship that I have a perfectly good eye at the junction of my two sides.
Stranger. Yes — but in order to see into Space, you'd need an eye not on your Perimeter, but on your side. That is, on what you'd probably call your inside. But we in Spaceland would call it your side.
I. An eye on my inside! An eye in my stomach! Your Lordship must be joking.
Stranger. I am in no joking mood. I'm telling you that I come from Space, or — since you refuse to understand what Space means — from the Land of Three Dimensions, from which I recently looked down upon your Plane, the thing you call Space. From that vantage point, I could see everything you think 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 claims are easy to make, my Lord.
Stranger. But not easy to prove, you mean. Well, I intend to prove mine.
When I came down here, I saw your four Sons, the Pentagons, each in his own room, and your two Grandsons the Hexagons. I saw your youngest Hexagon stay a while with you and then go back to his room, leaving you and your Wife alone. I saw your Isosceles servants, three of them, in the kitchen having supper, and the little Page in the scullery. Then I came here. And how do you think I got in?
I. Through the roof, I suppose.
Stranger. No. Your roof, as you know perfectly well, has been recently repaired and has no opening that even a Woman could slip through. I'm telling you: I came from Space. Aren't you convinced by what I've told you about your children and household?
I. Your Lordship must realize that facts about his humble servant's household could easily be gathered by anyone in the neighborhood who had your Lordship's considerable means of gathering information.
Stranger. (to himself.) What am I supposed to do? Wait — one more argument occurs 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 common people who, being ignorant of Mathematics, imagine that a Woman is actually a Straight Line and has only One Dimension. No, no, my Lord — we Squares know better, and are just as aware as your Lordship that a Woman, though commonly called a Straight Line, is really and scientifically a very thin Parallelogram, possessing Two Dimensions like the rest of us: namely, length and width (or thickness).
Stranger. But the very fact that a Line is visible implies that it must have yet another Dimension.
I. My Lord, I just acknowledged that a Woman has width as well as length. We see her length; we infer her width, which, though very slight, can still be measured.
Stranger. You're not understanding me. I mean that when you see a Woman, you should — besides inferring her width — be able to see her length, and to see what we call her height, although that last Dimension is infinitely small in your country. If a Line were nothing but length without "height," it would stop occupying Space and become invisible. Surely you can see that?
I. I have to admit that I don't understand your Lordship in the least. When we in Flatland see a Line, we see length and brightness. If the brightness vanishes, the Line goes out, and, as you say, stops occupying Space. But am I to understand that your Lordship is giving brightness the title of a Dimension — that what we call "bright" you call "high"?
Stranger. No, not at all. By "height" I mean a Dimension like your length — it's just that in your world, "height" isn't easily perceptible, being extremely small.
I. My Lord, your claim is easy enough to test. You say I have a Third Dimension, which you call "height." Now, a Dimension implies direction and measurement. Just measure my "height," or even just point out the direction in which my "height" extends, and I'll become your convert. Otherwise, your Lordship's own good judgment must excuse me.
Stranger. (to himself.) I can do neither. How am I going to convince him? Surely a clear statement of facts followed by a visual demonstration should be enough. — Now, Sir, listen to me.
You are living on a Plane. What you call Flatland is the vast level surface of what I might describe as a fluid, on the top of which you and your fellow citizens move around, without ever rising above it or sinking below it.
I am not a plane Figure — I am a Solid. You call me a Circle, but in reality I'm not a Circle. I'm an infinite number of Circles, ranging in size from a Point to a Circle thirteen inches in diameter, stacked one on top of the other. When I cut through your plane, as I'm doing right now, I create a cross-section that you quite rightly call a Circle. Because even a Sphere — which is my proper name in my own country — if he shows himself at all to an inhabitant of Flatland, can only show himself as a Circle.
Don't you remember — for I, who see all things, noticed last night the dream-vision of Lineland imprinted on your brain — don't you remember, I say, how when you entered the realm of Lineland, you were forced to appear to the King not as a Square, but as a Line, because that Linear Realm didn't have enough Dimensions to show all of you, only a slice or cross-section? In exactly the same way, your country of Two Dimensions doesn't have enough room to show me, a being of Three, and can only display a slice or cross-section of me — which is what you call a Circle.
The dimming of your eye tells me you're skeptical. But now get ready for definitive proof of what I'm saying. You can't see more than one of my cross-sections, or Circles, at a time, because you have no ability to raise your eye out of the plane of Flatland. But you can at least see that, as I rise in Space, my cross-sections get smaller. Watch now — I'll rise, and what you'll see is my Circle growing smaller and smaller until it shrinks to a point and finally disappears.
There was no "rising" that I could see. But he shrank and finally vanished. I blinked once or twice to make sure I wasn't dreaming. But it was no dream. From the depths of nowhere came a hollow voice — it seemed to come from right next to my heart — "Am I completely gone? Are you convinced now? Well, now I'll gradually return to Flatland, and you'll see my cross-section grow larger and larger."
Every reader in Spaceland will easily understand that my mysterious Guest was speaking the plain and simple truth. But to me — even though I was well versed in Flatland Mathematics — it was anything but simple. The rough diagram above will make it obvious to any child in Spaceland that the Sphere, rising through the three positions shown, would have had to appear to me, or to any Flatlander, as a Circle — first full-sized, then small, and finally very small indeed, approaching a Point. But for me, even though I saw it all happening right in front of me, the causes remained completely dark. All I could grasp was that the Circle had made himself smaller and vanished, and that he had now reappeared and was rapidly making himself bigger again.
When he returned to his original size, he let out a deep sigh, because my silence told him that I had completely failed to understand. And indeed, I was now leaning toward the belief that he couldn't be a Circle at all, but rather some extraordinarily clever illusionist — or else that the old wives' tales were true, and there really were such things as Enchanters and Magicians.
After a long pause, he muttered to himself, "One option remains, if I'm not going to resort to action. I have to try the method of Analogy." Then came an even longer silence, after which he resumed our dialogue.
Sphere. Tell me, Mr. Mathematician — if a Point moves Northward and leaves a glowing trail behind it, what would you call that trail?
I. A straight Line.
Sphere. And a straight Line has how many endpoints?
I. Two.
Sphere. Now imagine that Northward straight Line moving parallel to itself, East and West, so that every point on it leaves behind a trail in the shape of a straight Line. What would you call the Figure that's formed? We'll say it moves a distance equal to the length of the original straight Line. What name, I say?
I. A Square.
Sphere. And how many sides does a Square have? How many angles?
I. Four sides and four angles.
Sphere. Now stretch your imagination a little and picture 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 of the Square would have to pass through the positions already occupied by the Northern points. But that's not what I mean.
I mean that every Point in you — because you are a Square, and you'll serve nicely for my example — every Point in you, meaning every Point in what you call your inside, is to move upward through Space in such a way that no Point passes through the position already occupied by any other Point. Instead, each Point traces out a straight Line of its own. This is all perfectly consistent with Analogy — surely it must be clear to you.
Restraining my impatience — for I was now powerfully tempted to throw myself blindly at my Visitor and hurl him into Space, or out of Flatland, anywhere, just to be rid of him — I replied:
"And what exactly would be the nature of the Figure I'd create through this motion you're pleased to call 'upward'? I assume it can be described in the language of Flatland."
Sphere. Oh, absolutely. It's all plain and simple, and in perfect keeping with Analogy — only, by the way, you must not call the result a Figure. You must call it a Solid. But let me describe it to you. Or rather, not me — Analogy will do it.
We started 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 answer your own question: 1, 2, 4 — that's clearly a Geometric Progression. What's 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 does this Creature have 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 — what we call sides. You would call them solids.
I. And how many solids, or sides, will belong to this Being that I'm supposed to generate by moving my inside in an "upward" direction — this thing you call a Cube?
Sphere. How can you even ask? And you a mathematician! The side of anything is always, if I may put it this way, one Dimension behind the thing itself. So: since there's no Dimension behind a Point, a Point has 0 sides. A Line, if I may say so, has 2 sides (since the points of a Line can be called, as a courtesy, its sides). A Square has 4 sides. 0, 2, 4 — what kind of Progression would you call that?
I. Arithmetic.
Sphere. And what's the next number?
I. Six.
Sphere. Exactly. So you see, you've answered your own question. The Cube that you'll generate will be bounded by six sides — that is to say, six of your insides. You see it all now, don't you?
"Monster," I shrieked, "be you juggler, enchanter, dream, or devil, I will endure your mockeries no longer. Either you or I must die." And with those words I hurled myself at him.
It was no use. I slammed my sharpest right angle into the Sphere with all my might, pressing against him with enough force to have destroyed any ordinary Circle. But I could feel him slowly and unstoppably sliding away from me — not edging to the right or to the left, but somehow moving out of the world entirely, vanishing into nothing. A moment later, there was just... blank space. But I could still hear his voice.
Sphere. Why won't you listen to reason? I had hoped to find in you — a man of intelligence and an accomplished mathematician — the perfect apostle for the Gospel of the Three Dimensions, which I'm only allowed to preach once every thousand years. But now I don't know how to convince you. Wait — I've got it. Actions, not words, will prove the truth. Listen, my friend.
I've told you that from my position in Space, I can see the inside of everything you consider sealed shut. For example, I can see into that cupboard near where you're standing. There are several of what you call boxes in there (though like everything else in Flatland, they have no tops or bottoms), full of money. I can also see two account tablets. I'm about to descend into that cupboard and bring you one of those tablets. I watched you lock it half an hour ago, and I know you have the key on you. But I'm descending from Space — the doors, you'll notice, haven't moved. Now I'm inside the cupboard and picking up the tablet. Now I have it. Now I'm rising back up with it.
I rushed to the closet and threw the door open. One of the tablets was gone. With a mocking laugh, the Sphere appeared in the far corner of the room, and at the same instant the tablet appeared on the floor. I picked it up. There was no doubt — it was the missing tablet.
I groaned in horror, wondering if I'd lost my mind. But the Sphere continued: "Surely you can see now that my explanation, and no other, fits what just happened. What you call Solid things are really just surfaces. What you call Space is really nothing more than a vast Plane. I am in Space, and I look down upon the insides of things that you can only see the outsides of. You could leave the Plane yourself, if you could just summon the willpower to do it. A slight upward or downward motion would let you see everything I can see.
"The higher I rise, and the further I go from your Plane, the more I can see — though of course it all looks smaller from up there. For example, I'm ascending right now. Now I can see your neighbor the Hexagon and his family in their separate rooms. Now I can see inside the Theatre, ten doors down, where the audience is just filing out. And on the other side, a Circle in his study, sitting at his books. Now I'll come back down to you. And as a final, crowning proof — what would you say to my giving you a little touch, just the slightest touch, in your stomach? It won't seriously hurt you, and the small pain you might feel is nothing compared to the mental benefit you'll get from it."
Before I could say a word of protest, I felt a sharp, shooting pain in my insides, and a demonic laugh seemed to come from within me. A moment later the sharp agony faded, leaving nothing but a dull ache, and the Sphere began to reappear, saying as he gradually grew larger, "There — I haven't hurt you much, have I? If you're not convinced now, I don't know what will convince you. What do you say?"
My mind was made up. It seemed unbearable that I should go on living at the mercy of some Magician who could play tricks with my very stomach whenever he pleased. If only I could somehow pin him against the wall until help arrived!
Once more I charged my sharpest angle at him, and at the same time I raised the alarm across the entire household with my cries for help. I believe that at the moment of my attack, the Sphere had sunk below our Plane and was actually having difficulty rising back up. In any case, he stayed motionless while I — hearing what I thought was the sound of help approaching — pressed against him with twice the force, continuing to shout for assistance.
A convulsive shudder ran through the Sphere. "This can't go on," I thought I heard him say. "Either he listens to reason, or I'll have to use the last resort of civilization." Then, calling out to me in a louder voice, he exclaimed urgently, "Listen: no one else can witness what you've witnessed. Send your Wife back at once, before she enters the room. The Gospel of Three Dimensions must not be thwarted like this. The fruits of a thousand years of waiting must not be thrown away. I can hear her coming. Back! Back! Get away from me, or you'll have to come with me — to a place you can't imagine — into the Land of Three Dimensions!"
"Fool! Madman! Irregular!" I shouted. "I'll never let you go! You'll pay the price for your tricks!"
"So it's come to this?" the Sphere thundered. "Then face your fate — out of your Plane you go. Once, twice, three times! It's done!"
An unspeakable horror seized me. There was darkness, and 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 finally find my voice, I shrieked in agony: "Either this is madness or it is Hell."
"It is neither," the Sphere replied calmly. "It is Knowledge. It is Three Dimensions. Open your eye once more and try to look steadily."
I looked — and behold, a new world! There stood before me, visibly real, everything I had previously only inferred, guessed at, dreamed of — perfect Circular beauty in the flesh. What seemed to be the center of the Stranger's form lay open to my view. And yet I couldn't see any heart, lungs, or 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.
Mentally prostrating myself before my Guide, I cried, "How is it, O divine ideal of perfect loveliness and wisdom, that I can see your inside, and yet I can't make out your heart, your lungs, your arteries, your liver?"
"What you think you see, you do not actually see," he replied. "It is not given to you, or to any other Being, to look at my internal parts. I am a different order of Being from those in Flatland. If I were a Circle, you could see 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, the outside of a Sphere presents the appearance of a Circle."
I was bewildered by my Teacher's puzzling words, but I no longer resisted them. Instead, I worshipped him in silent adoration. He continued, his voice growing gentler. "Don't distress yourself if you can't immediately understand the deeper mysteries of Spaceland. They'll dawn on you gradually. Let's begin by casting a glance back at the region you came from. Come back with me for a moment to the plains of Flatland, and I'll show you something you've often reasoned and thought about but never actually seen with your eyes — a visible angle."
"Impossible!" I cried. But with the Sphere leading the way, I followed as if in a dream, until his voice stopped me once more: "Look down there, and see your own Pentagonal house and everyone in it."
I looked below, and saw with my physical eye all the domestic detail that I had up to now only inferred with my understanding. And how poor and shadowy those inferences were compared to the reality I now beheld! My four Sons calmly asleep in the Northwestern rooms. My two orphan Grandsons to the South. The Servants, the Butler, my Daughter — all in their separate rooms. Only my loving Wife, alarmed by my long absence, had left her room and was pacing up and down the Hall, anxiously waiting for me to come home. The Page, too, had been woken by my cries and had left his room. Under the pretense of checking whether I'd fainted somewhere, he was actually snooping through the cabinet in my study. All this I could now see, not merely deduce. And as we came closer and closer, I could make out even the contents of my cabinet — the two chests of gold and the tablets that the Sphere had mentioned.
Moved by my Wife's distress, I would have sprung downward to reassure her, but I found myself unable to move. "Don't worry about your Wife," said my Guide. "She won't be left anxious for long. In the meantime, let's take a survey of Flatland."
Once more I felt myself rising through space. It was exactly as the Sphere had said. The farther we moved from the object we were looking at, the larger our field of vision became. My native city, with the interior of every house and every creature in it, lay open to my view in miniature. We rose higher, and there — the secrets of the earth, the depths of the mines and the innermost caverns of the hills — all laid bare before me.
Awestruck at the sight of the earth's mysteries unveiled before my unworthy eye, I said to my Companion, "I have become like a God! The wise men in our country say that seeing all things — or as they put it, omnividence — is the attribute of God alone."
There was a note of scorn in my Teacher's voice as he answered: "Is that so? Then the very pickpockets and cutthroats of my country deserve to be worshipped by your wise men as Gods, because every single one of them can see just as much as you're seeing now. But trust me — your wise men are wrong."
I. Then is omnividence the attribute of others besides Gods?
Sphere. I don't know. But if a pickpocket or a cutthroat in our country can see everything that's in your country, surely that's no reason why you should accept the pickpocket or cutthroat as a God. This omnividence, as you call it — it's not a common word in Spaceland, by the way — does it make you more just? More merciful? Less selfish? More loving? Not in the least. So how does it make you more divine?
I. "More merciful, more loving!" But those are the qualities of women! And we know that a Circle is a higher Being than a Straight Line, insofar as knowledge and wisdom are more valuable than mere affection.
Sphere. It's not for me to rank human faculties by merit. Yet many of the best and wisest in Spaceland value the affections more than the intellect — your despised Straight Lines more than your celebrated Circles. But enough of that. Look down there. Do you recognize that building?
I looked, and far off I saw an immense Polygonal structure that I recognized as the General Assembly Hall of the States of Flatland. It was surrounded by dense lines of Pentagonal buildings at right angles to each other, which I knew to be streets. I realized 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 habit, in strict accordance with precedent, the highest Circles of the realm were meeting in solemn assembly, just as they had met at the first hour of the first day of the year 1000, and also at the first hour of the first day of the year 0.
The minutes of the previous meetings were now read by someone I immediately recognized as my brother — a perfectly Symmetrical Square and the Chief Clerk of the High Council. It was recorded on each occasion that: "Whereas the States had been troubled by various ill-intentioned persons claiming to have received revelations from another World, and claiming to produce demonstrations that had driven both themselves and others into a frenzy, it had been for this reason unanimously resolved by the Grand Council that on the first day of each millennium, special orders be sent to the Prefects in all districts of Flatland to conduct a strict search for such misguided persons; and without the formality of mathematical examination, to destroy all such as were Isosceles of any degree; to flog and imprison any regular Triangle; to have any Square or Pentagon sent to the district Asylum; and to arrest anyone of higher rank, sending him straight 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 the formal resolution for the third time. "Death or imprisonment awaits the Apostle of the Gospel of Three Dimensions."
"Not so," I replied. "The matter is so clear to me now, the nature of real space so obvious, that I think I could make a child understand it. Just let me go down there right now and enlighten them."
"Not yet," said my Guide. "The time will come for that. In the meantime, I must carry out my mission. Stay where you are." With those words, he leaped with great agility into the sea (if I can call it that) of Flatland, right into the middle of the ring of Counsellors. "I come," he said, "to proclaim that there is a land of Three Dimensions."
I could see many of the younger Counsellors recoil in obvious horror as the Sphere's circular cross-section widened before them. But at a signal from the presiding Circle — who showed not the slightest alarm or surprise — six Isosceles of a low type rushed in from six different directions and threw themselves at the Sphere. "We have him," they cried. "No — yes — we still have him! 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 alarm. The secret archives, to which I alone have access, tell me that a similar event occurred at the last two millennial gatherings. You will, of course, say nothing of these trifles outside the Cabinet."
Raising his voice, he summoned the guards. "Arrest the policemen. Gag them. You know your duty." After he had consigned the wretched policemen to their fate — those unlucky and unwilling witnesses of a state secret they would never be allowed to reveal — he addressed the Counsellors again. "My Lords, the business of the Council being concluded, I have only to wish you a Happy New Year." Before leaving, he expressed at some length to the Clerk — my excellent but deeply unfortunate brother — his sincere regret that, in accordance with precedent and for the sake of secrecy, he must condemn him to life in prison. He added, however, his satisfaction that unless my brother mentioned anything about the day's incident, his life would be spared.
When I saw my poor brother led away to prison, I tried to leap down into the Council Chamber, hoping to speak up on his behalf, or at least say goodbye. But I found I had no ability to move on my own. I was completely dependent on the will of my Guide, who said in gloomy tones, "Don't worry about your brother. Perhaps you'll have plenty of time later to mourn with him. Follow me."
Once more we rose into space. "So far," said the Sphere, "I've shown you nothing but Plane Figures and their interiors. Now I need to introduce you to Solids, and reveal to you the principle on which they're constructed. Look at this stack of moveable square cards. See, I place one on another — not, as you assumed, Northward of the other, but on the other. Now a second, now a third. See, I'm building up a Solid by stacking many 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."
"Forgive me, my Lord," I replied, "but to my eye it looks like an Irregular Figure whose inside is open to view. In other words, I think I see no Solid, but a Plane like the ones we have in Flatland — only with an Irregularity that suggests some monstrous criminal, so that the very sight of it is painful to my eyes."
"True," said the Sphere. "It appears to you as a Plane, because you aren't accustomed to light and shade and perspective — just as in Flatland a Hexagon would appear as a Straight Line to someone without the Art of Sight Recognition. But in reality it is a Solid, as you'll learn through the sense of Feeling."
He then introduced me to the Cube, and I found that this marvelous Being was indeed no Plane, but a Solid — endowed with six plane sides and eight terminal points called solid angles. I remembered the Sphere saying that just such a Creature would be formed by a Square moving, in Space, parallel to itself. And I rejoiced to think that so insignificant a Creature as I could, in some sense, be called the ancestor of so illustrious an offspring.
But still I couldn't fully understand what my Teacher had told me about "light" and "shade" and "perspective," and I didn't hesitate to raise my difficulties with him.
If I were to reproduce the Sphere's explanation of these matters, clear and concise though it was, it would be tedious for an inhabitant of Space, who already knows these things. Suffice it to say that through his lucid explanations, by changing the position of objects and lights, and by allowing me to feel the various objects and even his own sacred Person, he finally made everything clear to me, so that I could now easily distinguish between a Circle and a Sphere, a Plane Figure and a Solid.
This was the Climax, the Paradise, of my strange eventful History. From here on I have to tell 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 will 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 can arouse in the hearts of Plane and Solid Humanity a spirit of rebellion against the Conceit that 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, following the plain path of dispassionate History. The exact facts, the exact words — and they are burned into my brain — shall be set down without changing a single detail. Let my Readers judge between me and Destiny.
The Sphere would gladly have continued his lessons by teaching me the structure of all regular Solids — Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres. But I dared to interrupt him. Not because I was tired of learning. On the contrary, I thirsted for even deeper and fuller draughts of knowledge than he was offering me.
"Forgive me," said I, "O You Whom I must no longer address as the Perfection of all Beauty — but let me beg you to grant your servant a look at your interior."
Sphere. My what?
I. Your interior: your stomach, your intestines.
Sphere. Where did this ill-timed, impertinent request come from? And what do you mean by saying I'm no longer the Perfection of all Beauty?
I. My Lord, your own wisdom has taught me to aspire to One even greater, more beautiful, and more closely approaching Perfection than yourself. Just as you, superior to all Flatland forms, combine many Circles in One, so surely there must be One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland. And just as we, who are now in Space, look down on Flatland and see the insides of all things, so there must certainly be some higher, purer region above us, where you surely intend to lead me — O You 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 whose vantage point we could look down together upon the revealed insides of Solid things, and where your own intestines, and those of your fellow Spheres, would lie exposed to the view of the poor wandering exile from Flatland, to whom so much has already been granted.
Sphere. Bah! Nonsense! Enough of this foolishness! Time is short, and much remains to be done before you're fit to proclaim the Gospel of Three Dimensions to your blind, benighted countrymen in Flatland.
I. No, gracious Teacher, don't deny me what I know it's in your power to do. Grant me just one glimpse of your interior, and I'll be satisfied forever, remaining from now on your obedient pupil, your devoted slave, ready to receive all your teachings and to hang on every word that falls from your lips.
Sphere. Well, then, to satisfy and silence you, let me say at once: I would show you what you wish if I could, but I cannot. Would you have me turn my stomach inside out to oblige you?
I. But my Lord has shown me the intestines of all my countrymen in the Land of Two Dimensions by taking me with him into the Land of Three. What could be easier, then, than to take his servant on a second journey into the blessed region of the Fourth Dimension, where I could look down with him once more upon this land of Three Dimensions, and see the inside of every three-dimensional 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 don't know. But surely my Teacher knows.
Sphere. Not I. There is no such land. The very idea of it is utterly inconceivable.
I. Not inconceivable, my Lord — not to me, and therefore still less inconceivable to my Master. No, I don't despair that, even here in this region of Three Dimensions, your Lordship's skill could make the Fourth Dimension visible to me — just as in the Land of Two Dimensions my Teacher would gladly have opened the eyes of his blind servant to the invisible presence of a Third Dimension, though I couldn't see it.
Let me recall the past. Wasn't I taught down below that when I saw a Line and inferred a Plane, I was actually seeing a Third unrecognized Dimension — not the same as brightness, called "height"? And doesn't it follow now that, in this region, when I see a Plane and infer a Solid, I'm really seeing a Fourth unrecognized Dimension — not the same as color, but real, though infinitesimal and impossible to measure?
And besides this, there is the Argument from Analogy of Figures.
Sphere. Analogy! Nonsense — what analogy?
I. Your Lordship is testing his servant to see whether he remembers the revelations given to him. Don't toy with me, my Lord. I crave, I thirst, for more knowledge. Of course we can't 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 little Lineland Monarch could neither turn left nor right to see it — and just as there was, close at hand and touching my very body, the land of Three Dimensions, though I, blind senseless wretch, had no power to touch it, no eye in my interior to perceive it — so surely 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 explained 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 doesn't it necessarily 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 before now your countrymen have also witnessed the descent of Beings of a higher order than their own, entering closed rooms, just as your Lordship entered mine, without the opening of doors or windows, appearing and vanishing at will? On the reply to this question I am ready to stake everything. Deny it, and I am from now on silent. Only grant me an answer.
Sphere. (after a pause). It is reported so. But people are divided in their opinion as to the facts. And even granting the facts, they explain them in different ways. And in any case, however great the number of different explanations, no one has adopted or suggested the theory of a Fourth Dimension. Therefore, please stop this foolishness, 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 one more question, best of Teachers! Those who have appeared like this — no one knows from where — and have returned — no one knows to where — have they also contracted their cross-sections and vanished somehow into that more Spacious Space, where I now beg you to take me?
Sphere. (moodily). They have vanished, certainly — if they ever appeared. But most people say that these visions arose from the thought — you won't understand me — from the brain, from the disturbed mental state of the Seer.
I. They say that? Oh, don't believe them. Or if it really is so — if 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 enraptured eye, a Cube moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of its interior pass through a new kind of Space, with a wake of its own — shall create a still more perfect perfection than itself, with sixteen terminal Extra-solid angles and Eight solid Cubes for its Perimeter. And once there, shall we stop our upward course? In that blessed region of Four Dimensions, shall we linger at the threshold of the Fifth, and not enter? Ah, no! Let us rather resolve that our ambition shall soar with our bodily ascent. Then, yielding to our intellectual advance, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth —
How long I would have continued I don't know. In vain did the Sphere, in his voice of thunder, repeat his command of silence and threaten me with the most terrible penalties if I persisted. Nothing could stop the flood of my ecstatic aspirations. Perhaps I was to blame. But truly 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 hurled me through space with a velocity that made speech impossible. 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-consuming 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.
Even though I had less than a minute to think, I felt by a kind of instinct that I had to hide my experiences from my Wife. It wasn't that I was worried, in the moment, about her revealing my secret — but I knew that to any Woman in Flatland, the story of my adventures would be completely incomprehensible. So I tried to reassure her with a story I made up on the spot: that I'd accidentally fallen through the trapdoor to the cellar and had been lying down there, unconscious.
The Southward pull of gravity in our country is so slight that even to a Woman, my tale must have sounded far-fetched and nearly unbelievable. But my Wife, whose good sense far exceeds the average for her Sex, and who could tell I was unusually agitated, didn't argue with me about it. She simply insisted I was ill and needed rest. I was grateful for the excuse to retreat to my room and think quietly about everything that had happened. When I was finally alone, a drowsy feeling came over me. But before my eyes closed, I tried to reconstruct the Third Dimension in my mind — especially the process by which a Cube is built through the motion of a Square. It wasn't as clear as I would have liked, but I remembered that it had to be "Upward, and yet not Northward," and I resolved to hold firmly to those words as the key that, if I never let go of it, would surely guide me to the answer. So, mechanically repeating the words like a magic charm — "Upward, yet not Northward" — I fell into a deep, refreshing sleep.
During my sleep, I had a dream. I thought I was once again beside the Sphere, whose shining glow indicated that he had traded his anger toward me for perfect calm. We were moving together toward a bright but infinitesimally small Point, to which my Master directed my attention. As we got closer, I thought I could hear a faint humming noise coming from it — like one of your Spaceland bluebottle flies, only far less resonant. It was so faint, in fact, that even in the perfect silence of the Vacuum through which we were soaring, the sound didn't reach our ears until we slowed our flight to a distance of something under twenty human diagonals.
"Look there," said my Guide. "In Flatland you have lived. Of Lineland you have received a vision. You have soared with me to the heights of Spaceland. Now, to complete the range of your experience, I'm taking you downward to the very lowest depth of existence — all the way to the realm of Pointland, the Abyss of No Dimensions.
"Look at that miserable creature. That Point is a Being like us, but confined to the non-dimensional Gulf. He is his own World, his own Universe. He can form no concept of anything other than himself. He knows nothing of Length, or Width, or Height, because he has never experienced them. He doesn't even grasp the number Two. He has no concept of Plurality — because he himself is his One and All, while really being Nothing. Yet notice his perfect self-satisfaction, and learn this lesson from it: that to be self-satisfied is to be worthless and ignorant, and that to aspire is better than to be blindly and helplessly happy. Now listen."
He stopped speaking, and from the little buzzing creature there arose a tiny, low, monotonous but distinct tinkling — like one of your Spaceland phonographs — from which I caught these words: "Infinite bliss of existence! It is; and there is nothing else beside It."
"What," I said, "does the tiny creature mean by 'it'?" "He means himself," said the Sphere. "Haven't you noticed that babies and immature people who can't distinguish themselves from the world always talk about themselves in the Third Person? But hush!"
"It fills all Space," continued the little self-talking Creature, "and what It fills, It is. What It thinks, that It speaks; and what It speaks, that It hears; and It itself is Thinker, Speaker, Hearer, Thought, Word, Sound. It is the One, and yet the All in All. Ah, the happiness — ah, the happiness of Being!"
"Can't you startle the little thing out of its smugness?" I said. "Tell it what it really is, the way you told me. Show it the narrow limitations of Pointland and lead it up to something higher." "That's no easy task," said my Master. "You try."
At that, raising my voice as loud as I possibly could, I addressed the Point as follows:
"Silence, silence, you pathetic Creature. You call yourself the All in All, but you are the Nothing. Your so-called Universe is a mere speck on a Line, and a Line is a mere shadow compared with —" "Hush, hush, you've said enough," interrupted the Sphere. "Now listen, and watch the effect of your speech on the King of Pointland."
The glow of the Monarch, who beamed more brightly than ever upon hearing my words, showed clearly that he had kept his smugness intact. I'd barely finished when he picked up his refrain again. "Ah, the joy — ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming back to Itself, hinting at Its own criticism, only to make Its happiness greater! Sweet rebellion stirred up just to end 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 accomplished. To whatever extent the Monarch understood them at all, he accepted them as his own — because he can't conceive of anyone except himself — and congratulated himself on the variety of 'Its Thought' as proof of creative Power. Let us leave this God of Pointland to the ignorant enjoyment of his omnipresence and omniscience. Nothing you or I can do will rescue him from his self-satisfaction."
After this, as we floated gently back to Flatland, I could hear the calm voice of my Companion drawing the moral of my vision and urging me to aspire — and to teach others to aspire. He had been angry at first, he admitted, at my ambition to soar to Dimensions above the Third. But since then, he had gained fresh insight, and he was not too proud to acknowledge his error to a Pupil. Then he went on to introduce me to mysteries even higher than those I had already witnessed, showing me how to construct Extra-Solids by the motion of Solids, and Double Extra-Solids by the motion of Extra-Solids — all "strictly according to Analogy," and all by methods so simple and easy that even the Female Sex could grasp them.
I woke up overjoyed, and immediately started thinking about the glorious mission ahead of me. I would go out at once, I thought, and spread the word across all of Flatland. Even Women and Soldiers would hear the Gospel of Three Dimensions. I'd start with my Wife.
But just as I'd settled on a plan of action, I heard the sound of many voices in the street calling for silence. Then came a louder voice — a herald's proclamation. Listening carefully, I recognized the words of the Council's Resolution, ordering the arrest, imprisonment, or execution of anyone who might corrupt the public mind with delusions, or who claimed to have received revelations from another World.
I paused to think. This was not a danger to take lightly. It would be smarter to avoid it entirely by skipping any mention of my Revelation, and instead relying on the path of Demonstration — which, after all, had seemed so simple and so conclusive that nothing would be lost by dropping the more direct approach. "Upward, not Northward" — that was the key to the whole proof. It had seemed fairly clear to me before I fell asleep, and when I first woke up, still fresh from my dream, the logic had seemed as obvious as basic arithmetic. But somehow it didn't feel quite so obvious now. My Wife happened to walk into the room at just that moment, and after we'd exchanged a few words of ordinary conversation, I decided not to start with her after all.
My Pentagonal sons were men of good character and standing, and physicians with solid reputations, but they weren't particularly strong in mathematics — which made them unsuitable for what I had in mind. Then it occurred to me: a young, teachable Hexagon with a knack for math would make the ideal student. So why not try my first experiment on my precocious little Grandson, whose offhand remarks about the meaning of 3 cubed had earned the Sphere's approval? If I discussed the matter with him — just a boy — I'd be perfectly safe. He wouldn't know anything about the Council's Proclamation. My Sons, on the other hand — their patriotism and reverence for the Circles ran so deep, far outweighing mere family loyalty — I couldn't be sure they wouldn't feel obligated to hand me over to the Prefect if they found me seriously pushing the seditious heresy of the Third Dimension.
But first things first: I had to deal with my Wife's curiosity. She naturally wanted to know why the Circle had requested that mysterious private meeting, and how he'd gotten into the house in the first place. Without going into all the details of the elaborate story I gave her — a story that, I'm afraid, wasn't entirely consistent with the truth, not as consistent as my readers in Spaceland might hope for — let me just say that I eventually managed to persuade her to go back to her household duties without getting me to say a single word about the World of Three Dimensions. That settled, I immediately sent for my Grandson. Because the truth was, I could feel that everything I'd seen and heard was somehow slipping away from me, like the image of a half-remembered, tantalizingly vivid dream, and I was desperate to test my ability to make a first disciple.
When my Grandson came in, I carefully locked the door behind him. Then, sitting down beside him with our mathematical tablets — or, as you would call them, Lines — I told him we'd pick up where we left off in yesterday's lesson. I walked him through it again: how a Point, by moving in One Dimension, produces a Line, and how a straight Line, moving in Two Dimensions, produces a Square. After that, forcing a laugh, I said, "And now, you little rascal, remember how you tried to claim that a Square could do the same thing — that by moving 'Upward, not Northward,' it could produce some other figure, a sort of extra Square in Three Dimensions? Go on, say that again, you troublemaker."
At that exact moment, we heard the herald's "Hear ye! Hear ye!" ringing out again in the street, proclaiming the Council's Resolution. Young as he was, my Grandson — who was unusually smart for his age, and raised in absolute reverence for the authority of the Circles — sized up the situation with a sharpness I wasn't at all prepared for. He stayed silent until the last words of the Proclamation had faded away. Then he burst into tears.
"Dear Grandpapa," he said, "that was only a joke, and of course I didn't mean anything by it at all. And we didn't know anything about the new Law back then. And I don't think I said anything about the Third Dimension. And I'm absolutely sure I never said anything about 'Upward, not Northward,' because that would be such nonsense, you know. How could something move Upward and not Northward? Upward and not Northward! Even if I were a baby, I couldn't say anything that ridiculous. How silly! Ha! Ha! Ha!"
"Not silly at all," I said, losing my temper. "Look — for example, I'll take this Square" — and as I said it, I grabbed a moveable Square that happened to be lying nearby — "and I'll move it, you see, not Northward but — yes, I'll move it Upward — that is to say, not Northward, but I'll move it somewhere — not exactly like this, but somehow —" Here my sentence trailed off into incoherence as I shook the Square around aimlessly, much to the amusement of my Grandson, who burst out laughing louder than ever and declared that I wasn't teaching him — I was just messing with him. And with that, he unlocked the door and ran out of the room.
So ended my first attempt to convert a student to the Gospel of Three Dimensions.
My failure with my Grandson didn't encourage me to share my secret with anyone else in my household. But it didn't drive me to despair, either. I simply realized that I couldn't rely entirely on the catchphrase "Upward, not Northward." Instead, I needed to seek a proper demonstration by laying the whole subject out clearly before the public. And for that, I decided I would have to write.
So I spent several months in private composing a treatise on the mysteries of Three Dimensions. To try to stay on the right side of the Law, I didn't write about a physical Dimension. Instead, I wrote about a "Thoughtland" — a theoretical realm from which a Figure could look down on Flatland and see the insides of all things simultaneously, a place where a Figure might exist surrounded by six Squares and containing eight terminal Points. But writing this book, I found myself badly hampered by the impossibility of drawing the diagrams I needed. In our country of Flatland, after all, the only tablets are Lines, and the only diagrams are Lines — all arranged in a single straight Line, distinguishable only by differences in size and brightness. So by the time I'd finished my treatise (which I titled "Through Flatland to Thoughtland"), I couldn't feel confident that many readers would understand what I meant.
Meanwhile, my wife was living under a cloud. All pleasures had lost their appeal for me. Every sight tantalized and tempted me toward outright treason, because I couldn't help comparing what I saw in Two Dimensions with what it really was when seen in Three — and I could barely stop myself from making those comparisons out loud. I neglected my clients and my own business so I could devote myself to contemplating the mysteries I had once witnessed but could share with no one, mysteries I found harder to reproduce with each passing day, even in my own mind's eye. One day, about eleven months after my return from Spaceland, I tried to picture a Cube with my eyes closed and failed. I managed it later, but I wasn't entirely certain then — and I've never been entirely certain since — that I'd gotten it exactly right. This made me more melancholy than before, and I resolved to take some kind of action. But what, I had no idea. I felt I would have been willing to sacrifice my life for the Cause, if that would have convinced anyone. But if I couldn't convince my own Grandson, how could I possibly convince the highest and most advanced Circles in the land?
And yet there were times when my spirit was too strong for me, and I let dangerous words slip out. I was already considered unorthodox, if not outright treasonous, and I was painfully aware of how precarious my position was. Still, I couldn't always stop myself from bursting out with suspicious or half-seditious remarks, even in the highest Polygonal or Circular society. When, for example, the question came up about how to treat those so-called lunatics who claimed they could see the insides of things, I would quote an ancient Circle who once declared that prophets and inspired people are always considered mad by the majority. I couldn't help occasionally letting slip expressions like "the eye that sees into the interiors of things" and "the all-seeing land." Once or twice I even dropped the forbidden terms "the Third and Fourth Dimensions."
Finally, to cap off a series of minor indiscretions, I went too far at a meeting of our Local Speculative Society, held at the palace of the Prefect himself. Some extremely foolish person had just read an elaborate paper laying out the precise reasons why Providence had limited the number of Dimensions to Two, and why the attribute of all-seeing vision is reserved for the Supreme alone. I so completely forgot myself that I gave an exact account of my entire voyage with the Sphere into Space, and to the Assembly Hall in our Metropolis, and then back to Space again, and of my return home, and of everything I had seen and heard in fact or vision. At first, I'll admit, I pretended I was describing the imaginary experiences of a fictional character. But my enthusiasm soon forced me to drop all pretense, and finally, in a fervent conclusion, I urged all my listeners to cast aside their prejudice and become believers in the Third Dimension.
Do I need to tell you that I was immediately arrested and brought before the Council?
The next morning, standing in the very place where just a few months earlier the Sphere had stood beside me, I was allowed to begin my account and continue it without questions or interruptions. But from the start, I could see my fate. The President, noticing that a guard of the better class of Policemen was in attendance — officers of angularity no less than 55 degrees — ordered them to be replaced before I began my defense by an inferior class of just 2 or 3 degrees. I knew all too well what that meant. I was going to be executed or imprisoned, and my story was going to be kept secret from the world by the simultaneous destruction of the officials who had heard it. That being the case, the President preferred to substitute the cheaper victims for the more expensive ones.
After I had finished my defense, the President — perhaps noticing that some of the junior Circles had been moved by my obvious earnestness — asked me two questions:
1. Could I point in the direction I meant when I used the words "Upward, not Northward"?
2. Could I, by any diagram or description (other than listing imaginary sides and angles), show the Figure I was pleased to call a Cube?
I declared that I could say nothing more, and that I must commit myself to the Truth, whose cause would surely prevail in the end.
The President replied that he quite agreed with my sentiment, and that I could do no better. I must be sentenced to perpetual imprisonment. But if the Truth intended for me to emerge from prison and spread my message to the world, then the Truth could be trusted to bring that about. In the meantime, I would be subjected to no discomfort beyond what was necessary to prevent escape, and — unless I forfeited the privilege through misconduct — I would be occasionally permitted to see my brother, who had been sent to prison before me.
Seven years have passed, and I am still a prisoner. Apart from the occasional visits of my brother, I am cut off from all companionship except that of my jailers. My brother is one of the best of Squares — fair, sensible, cheerful, and not without brotherly affection. Yet I confess that my weekly visits with him cause me, in at least one respect, the bitterest pain. He was there when the Sphere appeared in the Council Chamber. He saw the Sphere's changing cross-sections. He heard the explanation of those phenomena given to the Circles at the time. Since then, hardly a week has passed across seven long years without him hearing me retell the part I played in that event, along with detailed descriptions of everything in Spaceland and the arguments for the existence of Solid objects derived from Analogy. Yet — and I am ashamed to admit it — my brother has still not grasped the nature of Three Dimensions, and he openly admits he doesn't believe a Sphere exists.
So I am utterly without converts. And as far as I can tell, this once-in-a-millennium Revelation was given to me for nothing. Prometheus up in Spaceland was bound and punished for bringing fire down to mortals. But I — poor Flatland Prometheus — lie here in prison for bringing down nothing to my countrymen. Yet I still hold onto the hope that these memoirs might somehow, in some way I can't foresee, find their way to the minds of humanity in Some Dimension, and stir up a race of rebels who will refuse to be confined to limited Dimensionality.
That is the hope of my brighter moments. But it is not always so. There are times when a crushing thought weighs heavily on me: that I cannot honestly say I am confident about the exact shape of that once-seen, deeply-regretted Cube. And in my nightly visions, the mysterious command "Upward, not Northward" haunts me like a soul-devouring Sphinx. It is part of the martyrdom I endure for the cause of Truth that there are seasons of mental weakness — when Cubes and Spheres fade into the background of barely-possible existences, when the Land of Three Dimensions seems almost as imaginary as the Land of One or None. In fact, there are moments when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the solid realities of Flatland itself, seem no better than the offspring of a diseased imagination, or the baseless fabric of a dream.
If my poor Flatland friend still had the sharpness of mind he enjoyed when he first began writing these memoirs, I wouldn't need to speak for him in this preface. He wants, first, to offer his full thanks to his readers and critics in Spaceland, whose appreciation has — with unexpected speed — required a second edition of this work; second, to apologize for certain errors and misprints (though he's not entirely to blame for those); and third, to clear up one or two misconceptions. But he is not the Square he once was. Years of imprisonment, combined with the even heavier burden of widespread disbelief and mockery, have taken their toll — as have the thoughts, ideas, and much of the terminology he picked up during his brief stay in Spaceland. He has therefore asked me to respond on his behalf to two specific objections — one intellectual, the other moral.
The first objection goes like this: a Flatlander, seeing a Line, must be seeing something that has some thickness as well as length (otherwise it wouldn't be visible at all, if it had no thickness). And therefore — so the argument goes — he ought to acknowledge that his countrymen are not only long and broad, but also (however slightly) thick or high. This is a plausible objection, and to Spacelanders it feels nearly unanswerable. I'll confess that when I first heard it, I had no idea how to respond. But my poor old friend's answer seems to me to address it completely.
"I'll admit," he said — when I brought this objection to his attention — "I admit the truth of your critic's facts, but I reject his conclusions. It's true that we really do have in Flatland a Third unrecognized Dimension called 'height,' just as it's true that you really have in Spaceland a Fourth unrecognized Dimension — one that has no name yet, but which I'll call 'extra-height.' But we can no more perceive our 'height' than you can perceive your 'extra-height.' Even I — who have been to Spaceland, who had the privilege of understanding what 'height' means for twenty-four hours — even I can't comprehend it now, can't grasp it through sight or through any process of reasoning. I can only accept it on faith.
"The reason is obvious. Dimension implies direction, implies measurement, implies more and less. Now, all our lines are equally and infinitesimally thick (or high, whichever you prefer). As a result, there's nothing about them that could lead our minds toward the concept of that Dimension. No 'delicate micrometer' — as one overeager Spaceland critic has suggested — would help us in the slightest, because we wouldn't know what to measure, or in what direction. When we see a Line, we see something that is long and bright. Brightness, just as much as length, is essential to a Line's existence — if the brightness vanishes, the Line disappears. So all my Flatland friends — when I try to tell them about this unrecognized Dimension that is somehow visible in a Line — just say, 'Ah, you mean brightness.' And when I reply, 'No, I mean a real Dimension,' they immediately shoot back, 'Then measure it, or tell us which direction it extends in.' And that shuts me up, because I can do neither. Just 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 for the seventh time asked me, 'Are you any better?' — I tried to prove to him that he was 'high,' as well as long and broad, even though he didn't know it. But what was his response? 'You say I am "high." Measure my "highness" and I'll believe you.' What could I do? How could I meet that challenge? I was crushed. And he left the room triumphant.
"Does this still seem strange to you? Then put yourself in the same position. Suppose a being from the Fourth Dimension condescended to visit you and said, 'Every time you open your eyes, you see a Plane (which has Two Dimensions), and you infer a Solid (which has Three). But in reality, you also see — though you don't recognize it — a Fourth Dimension, which is not color, not brightness, not anything of the kind, but a true Dimension, even though I can't point out its direction to you, and you can't possibly measure it.' What would you say to such a visitor? Wouldn't you have him locked up? Well, that is my fate. And it's just 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. How strong a family resemblance runs through blind and persecuting humanity across all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes — we are all prone to the same errors, all equally slaves of our respective dimensional prejudices, as one of your Spaceland poets has said —
'One touch of Nature makes all worlds akin.'"
[Note: The Author asked me to add that the misconceptions of some of his critics on this point led him to insert, in his dialogue with the Sphere, certain remarks that bear on the question — remarks he had previously left out as tedious and unnecessary.]
On this point, the Square's defense seems to me rock-solid. I wish I could say that his answer to the second — the moral — objection was equally clear and convincing. It has been objected that he is a woman-hater. And since this objection has been pressed with particular force by those whom Nature has made the somewhat larger half of the Spaceland population, I'd like to address it as honestly as I can. But the Square is so unused to Spaceland's moral vocabulary that I'd be doing him an injustice if I simply transcribed his defense word for word. So, acting as his interpreter and summarizer, I gather that over the course of seven years in prison, he has changed his own personal views — both about Women and about the Isosceles, or Lower Classes. Personally, he now leans toward the Sphere's opinion (expressed earlier in these pages) that Straight Lines are in many important respects superior to the Circles. But writing as a historian, he identified himself — perhaps too closely — with the views generally held by Flatland, and (as he's been told) even by Spaceland, historians, in whose pages, until very recent times, the lives of women and of ordinary people have rarely been considered worth mentioning, and never worth studying carefully.
In an even more obscure passage, he now wishes to distance himself from the pro-Circle, aristocratic leanings that some critics have naturally read into his work. While giving full credit to the intellectual power with which a few Circles have maintained their dominance over vast numbers of their countrymen for many generations, he believes that the facts of Flatland, speaking for themselves without his commentary, show that revolutions can't always be put down by slaughter, and that Nature, by sentencing the Circles to infertility, has condemned them to eventual extinction. "And in this," he says, "I see a fulfillment of the great Law of all worlds: that while the wisdom of Man thinks it's achieving one thing, the wisdom of Nature steers it toward something else entirely — something different and far better." Beyond that, he asks his readers not to assume that every tiny detail of daily life in Flatland must have a corresponding detail in Spaceland. And yet he hopes that, taken as a whole, his work may prove thought-provoking as well as entertaining to those Spacelanders of moderate and modest minds who — when speaking of things that are of the highest importance but lie beyond experience — are willing to say neither "This can never be" on the one hand, nor "It must be exactly like this, and we know everything about it" on the other.