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Flatland: A Romance of Many Dimensions
Edwin A. Abbott (1838-1926. English scholar, theologian, and writer.)
-----------------------------------------------------------------
| "O day and night, but this is wondrous strange" |
| ______ |
| / / /| ------ / /| /| / /-. |
| /---- / /__| / / /__| / | / / / |
| / /___ / | / /___ / | / |/ /__.-' |
| |
| No Dimensions One Dimension |
| . A ROMANCE OF MANY DIMENSIONS ----- |
| POINTLAND LINELAND |
| |
| Two Dimensions Three Dimensions |
| ___ __ |
| | | /__/| |
| |___| |__|/ |
| FLATLAND SPACELAND |
| "Fie, fie, how franticly I square my talk!" |
-----------------------------------------------------------------
With Illustrations by the Author, A SQUARE (Edwin A. Abbott)
To
The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most rare and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
Preface to the Second and Revised Edition, 1884.
By the Editor
If my poor Flatland friend retained the vigour of mind which he
enjoyed when he began to compose these Memoirs, I should not now need
to represent him in this preface, in which he desires, firstly,
to return his thanks to his readers and critics in Spaceland,
whose appreciation has, with unexpected celerity, required a second
edition of his work; secondly, to apologize for certain errors
and misprints (for which, however, he is not entirely responsible);
and, thirdly, to explain one or two misconceptions. But he is not
the Square he once was. Years of imprisonment, and the still heavier
burden of general incredulity and mockery, have combined with
the natural decay of old age to erase from his mind many of
the thoughts and notions, and much also of the terminology,
which he acquired during his short stay in Spaceland. He has,
therefore, requested me to reply in his behalf to two special
objections, one of an intellectual, the other of a moral nature.
The first objection is, that a Flatlander, seeing a Line,
sees something that must be THICK to the eye as well as LONG
to the eye (otherwise it would not be visible, if it had not
some thickness); and consequently he ought (it is argued)
to acknowledge that his countrymen are not only long and broad,
but also (though doubtless in a very slight degree) THICK or HIGH.
This objection is plausible, and, to Spacelanders,
almost irresistible, so that, I confess, when I first heard it,
I knew not what to reply. But my poor old friend's answer
appears to me completely to meet it.
"I admit," said he -- when I mentioned to him this objection --
"I admit the truth of your critic's facts, but I deny his conclusions.
It is true that we have really in Flatland a Third
unrecognized Dimension called 'height', just as it is also true
that you have really in Spaceland a Fourth unrecognized Dimension,
called by no name at present, but which I will call 'extra-height'.
But we can no more take cognizance of our 'height' than you can
of your 'extra-height'. Even I -- who have been in Spaceland,
and have had the privilege of understanding for twenty-four hours
the meaning of 'height' -- even I cannot now comprehend it,
nor realize it by the sense of sight or by any process of reason;
I can but apprehend it by faith.
"The reason is obvious. Dimension implies direction,
implies measurement, implies the more and the less. Now,
all our lines are EQUALLY and INFINITESIMALLY thick (or high,
whichever you like); consequently, there is nothing in them
to lead our minds to the conception of that Dimension.
No 'delicate micrometer' -- as has been suggested by one too hasty
Spaceland critic -- would in the least avail us; for we should not
know WHAT TO MEASURE, NOR IN WHAT DIRECTION. When we see a Line,
we see something that is long and BRIGHT; BRIGHTNESS,
as well as length, is necessary to the existence of a Line;
if the brightness vanishes, the Line is extinguished. Hence,
all my Flatland friends -- when I talk to them about the unrecognized
Dimension which is somehow visible in a Line -- say, 'Ah,
you mean BRIGHTNESS': and when I reply, 'No, I mean
a real Dimension', they at once retort, 'Then measure it,
or tell us in what direction it extends'; and this silences me,
for I can do neither. Only yesterday, when the Chief Circle
(in other words our High Priest) came to inspect the State Prison
and paid me his seventh annual visit, and when for the seventh time
he put me the question, 'Was I any better?' I tried to prove to him
that he was 'high', as well as long and broad, although he did not
know it. But what was his reply? 'You say I am "high"; measure my
"high-ness" and I will believe you.' What could I do? How could I
meet his challenge? I was crushed; and he left the room triumphant.
"Does this still seem strange to you? Then put yourself in
a similar position. Suppose a person of the Fourth Dimension,
condescending to visit you, were to say, 'Whenever you open your eyes,
you see a Plane (which is of Two Dimensions) and you INFER
a Solid (which is of Three); but in reality you also see
(though you do not recognize) a Fourth Dimension, which is not colour
nor brightness nor anything of the kind, but a true Dimension,
although I cannot point out to you its direction, nor can you
possibly measure it.' What would you say to such a visitor?
Would not you have him locked up? Well, that is my fate:
and it is as natural for us Flatlanders to lock up a Square
for preaching the Third Dimension, as it is for you Spacelanders
to lock up a Cube for preaching the Fourth. Alas, how strong
a family likeness runs through blind and persecuting humanity
in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes --
we are all liable to the same errors, all alike the 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 desires me to add, that the misconception of some
of his critics on this matter has induced him to insert in his
dialogue with the Sphere, certain remarks which have a bearing
on the point in question, and which he had previously omitted
as being tedious and unnecessary.]
On this point the defence of the Square seems to me to be impregnable.
I wish I could say that his answer to the second (or moral) objection
was equally clear and cogent. It has been objected that he is
a woman-hater; and as this objection has been vehemently urged
by those whom Nature's decree has constituted the somewhat larger half
of the Spaceland race, I should like to remove it, so far as I can
honestly do so. But the Square is so unaccustomed to the use
of the moral terminology of Spaceland that I should be doing him
an injustice if I were literally to transcribe his defence against
this charge. Acting, therefore, as his interpreter and summarizer,
I gather that in the course of an imprisonment of seven years
he has himself modified his own personal views, both as regards Women
and as regards the Isosceles or Lower Classes. Personally,
he now inclines to the opinion of the Sphere that the Straight Lines
are in many important respects superior to the Circles.
But, writing as a Historian, he has identified himself
(perhaps too closely) with the views generally adopted by Flatland,
and (as he has been informed) even by Spaceland, Historians;
in whose pages (until very recent times) the destinies of Women
and of the masses of mankind have seldom been deemed worthy of mention
and never of careful consideration.
In a still more obscure passage he now desires to disavow the Circular
or aristocratic tendencies with which some critics have naturally
credited him. While doing justice to the intellectual power
with which a few Circles have for many generations maintained
their supremacy over immense multitudes of their countrymen,
he believes that the facts of Flatland, speaking for themselves
without comment on his part, declare that Revolutions cannot always
be suppressed by slaughter, and that Nature, in sentencing the Circles
to infecundity, has condemned them to ultimate failure --
"and herein," he says, "I see a fulfilment of the great Law
of all worlds, that while the wisdom of Man thinks it is working
one thing, the wisdom of Nature constrains it to work another,
and quite a different and far better thing." For the rest,
he begs his readers not to suppose that every minute detail
in the daily life of Flatland must needs correspond to
some other detail in Spaceland; and yet he hopes that,
taken as a whole, his work may prove suggestive as well as amusing,
to those Spacelanders of moderate and modest minds who --
speaking of that which is of the highest importance,
but lies beyond experience -- decline to say on the one hand,
"This can never be," and on the other hand, "It must needs be
precisely thus, and we know all about it."
CONTENTS:
PART I: THIS WORLD
Section
1. Of the Nature of Flatland
2. Of the Climate and Houses in Flatland
3. Concerning the Inhabitants of Flatland
4. Concerning the Women
5. Of our Methods of Recognizing one another
6. Of Recognition by Sight
7. Concerning Irregular Figures
8. Of the Ancient Practice of Painting
9. Of the Universal Colour Bill
10. Of the Suppression of the Chromatic Sedition
11. Concerning our Priests
12. Of the Doctrine of our Priests
PART II: OTHER WORLDS
13. How I had a Vision of Lineland
14. How I vainly tried to explain the nature of Flatland
15. Concerning a Stranger from Spaceland
16. How the Stranger vainly endeavoured to reveal to me
in words the mysteries of Spaceland
17. How the Sphere, having in vain tried words,
resorted to deeds
18. How I came to Spaceland, and what I saw there
19. How, though the Sphere shewed me other mysteries
of Spaceland, I still desired more; and what came of it
20. How the Sphere encouraged me in a Vision
21. How I tried to teach the Theory of Three Dimensions
to my Grandson, and with what success
22. How I then tried to diffuse the Theory
of Three Dimensions by other means, and of the result
PART I: THIS WORLD
"Be patient, for the world is broad and wide."
Section 1. Of the Nature of Flatland
I call our world Flatland, not because we call it so,
but to make its nature clearer to you, my happy readers,
who are privileged to live in Space.
Imagine a vast sheet of paper on which straight Lines, Triangles,
Squares, Pentagons, Hexagons, and other figures, instead of remaining
fixed in their places, move freely about, on or in the surface,
but without the power of rising above or sinking below it, very much
like shadows -- only hard and with luminous edges -- and you will then
have a pretty correct notion of my country and countrymen. Alas,
a few years ago, I should have said "my universe": but now my mind
has been opened to higher views of things.
In such a country, you will perceive at once that it is impossible
that there should be anything of what you call a "solid" kind;
but I dare say you will suppose that we could at least
distinguish by sight the Triangles, Squares, and other figures,
moving about as I have described them. On the contrary,
we could see nothing of the kind, not at least so as to distinguish
one figure from another. Nothing was visible, nor could be visible,
to us, except Straight Lines; and the necessity of this
I will speedily demonstrate.
Place a penny on the middle of one of your tables in Space;
and leaning over it, look down upon it. It will appear a circle.
But now, drawing back to the edge of the table, gradually lower
your eye (thus bringing yourself more and more into the condition of
the inhabitants of Flatland), and you will find the penny becoming
more and more oval to your view, and at last when you have placed
your eye exactly on the edge of the table (so that you are,
as it were, actually a Flatlander) the penny will then have ceased
to appear oval at all, and will have become, so far as you can see,
a straight line.
The same thing would happen if you were to treat in the same way
a Triangle, or Square, or any other figure cut out of pasteboard.
As soon as you look at it with your eye on the edge on the table,
you will find that it ceases to appear to you a figure,
and that it becomes in appearance a straight line. Take for example
an equilateral Triangle -- who represents with us a Tradesman
of the respectable class. Fig. 1 represents the Tradesman
as you would see him while you were bending over him from above;
figs. 2 and 3 represent the Tradesman, as you would see him
if your eye were close to the level, or all but on the level of
the table; and if your eye were quite on the level of the table
(and that is how we see him in Flatland) you would see nothing
but a straight line.
<>
<>
(1) __________ (2) ___________ (3) _________
\ / --__ __-- ---
\ / -
\/
When I was in Spaceland I heard that your sailors have very similar
experiences while they traverse your seas and discern some distant
island or coast lying on the horizon. The far-off land may have bays,
forelands, angles in and out to any number and extent;
yet at a distance you see none of these (unless indeed your sun shines
bright upon them revealing the projections and retirements by means of
light and shade), nothing but a grey unbroken line upon the water.
Well, that is just what we see when one of our triangular or other
acquaintances comes toward us in Flatland. As there is neither
sun with us, nor any light of such a kind as to make shadows,
we have none of the helps to the sight that you have in Spaceland.
If our friend comes closer to us we see his line becomes larger;
if he leaves us it becomes smaller: but still he looks like
a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle,
what you will -- a straight Line he looks and nothing else.
You may perhaps ask how under these disadvantageous circumstances
we are able to distinguish our friends from one another:
but the answer to this very natural question will be more fitly
and easily given when I come to describe the inhabitants of Flatland.
For the present let me defer this subject, and say a word or two
about the climate and houses in our country.
Section 2. Of the Climate and Houses in Flatland
As with you, so also with us, there are four points of the compass
North, South, East, and West.
There being no sun nor other heavenly bodies, it is impossible for us
to determine the North in the usual way; but we have a method of
our own. By a Law of Nature with us, there is a constant attraction
to the South; and, although in temperate climates this is very slight
-- so that even a Woman in reasonable health can journey
several furlongs northward without much difficulty --
yet the hampering effect of the southward attraction is
quite sufficient to serve as a compass in most parts of our earth.
Moreover, the rain (which falls at stated intervals) coming always
from the North, is an additional assistance; and in the towns we have
the guidance of the houses, which of course have their side-walls
running for the most part North and South, so that the roofs
may keep off the rain from the North. In the country, where there are
no houses, the trunks of the trees serve as some sort of guide.
Altogether, we have not so much difficulty as might be expected
in determining our bearings.
Yet in our more temperate regions, in which the southward attraction
is hardly felt, walking sometimes in a perfectly desolate plain
where there have been no houses nor trees to guide me, I have been
occasionally compelled to remain stationary for hours together,
waiting till the rain came before continuing my journey. On the weak
and aged, and especially on delicate Females, the force of attraction
tells much more heavily than on the robust of the Male Sex,
so that it is a point of breeding, if you meet a Lady in the street,
always to give her the North side of the way -- by no means
an easy thing to do always at short notice when you are in rude health
and in a climate where it is difficult to tell your North
from your South.
Windows there are none in our houses: for the light comes to us alike
in our homes and out of them, by day and by night, equally at
all times and in all places, whence we know not. It was in old days,
with our learned men, an interesting and oft-investigated question,
"What is the origin of light?" and the solution of it
has been repeatedly attempted, with no other result than to crowd
our lunatic asylums with the would-be solvers. Hence,
after fruitless attempts to suppress such investigations indirectly
by making them liable to a heavy tax, the Legislature,
in comparatively recent times, absolutely prohibited them.
I -- alas, I alone in Flatland -- know now only too well
the true solution of this mysterious problem; but my knowledge
cannot be made intelligible to a single one of my countrymen;
and I am mocked at -- I, the sole possessor of the truths of Space
and of the theory of the introduction of Light from the world
of three Dimensions -- as if I were the maddest of the mad!
But a truce to these painful digressions: let me return
to our houses.
The most common form for the construction of a house is five-sided
or pentagonal, as in the annexed figure. The two Northern sides RO,
OF, constitute the roof, and for the most part have no doors;
on the East is a small door for the Women; on the West a much
larger one for the Men; the South side or floor is usually doorless.
Square and triangular houses are not allowed, and for this reason.
The angles of a Square (and still more those of an equilateral
Triangle), being much more pointed than those of a Pentagon,
and the lines of inanimate objects (such as houses) being dimmer
than the lines of Men and Women, it follows that there is
no little danger lest the points of a square or triangular
house residence might do serious injury to an inconsiderate
or perhaps absent-minded traveller suddenly therefore,
running against them: and as early as the eleventh century
of our era, triangular houses were universally forbidden by Law,
the only exceptions being fortifications, powder-magazines, barracks,
and other state buildings, which it is not desirable that
the general public should approach without circumspection.
<>
<>
O
/\
/ \
/ \
/ \
/ \
R/ \F
\_ /
_/
Men's door _ Women's door
_ /
\____________/
A B
At this period, square houses were still everywhere permitted,
though discouraged by a special tax. But, about three centuries
afterwards, the Law decided that in all towns containing a population
above ten thousand, the angle of a Pentagon was the smallest
house-angle that could be allowed consistently with the public safety.
The good sense of the community has seconded the efforts
of the Legislature; and now, even in the country,
the pentagonal construction has superseded every other.
It is only now and then in some very remote and backward
agricultural district that an antiquarian may still discover
a square house.
Section 3. Concerning the Inhabitants of Flatland
The greatest length or breadth of a full grown inhabitant of Flatland
may be estimated at about eleven of your inches. Twelve inches may be
regarded as a maximum.
Our Women are Straight Lines.
Our Soldiers and Lowest Classes of Workmen are Triangles with two
equal sides, each about eleven inches long, and a base or third side
so short (often not exceeding half an inch) that they form
at their vertices a very sharp and formidable angle.
Indeed when their bases are of the most degraded type (not more than
the eighth part of an inch in size), they can hardly be distinguished
from Straight Lines or Women; so extremely pointed are their vertices.
With us, as with you, these Triangles are distinguished from others
by being called Isosceles; and by this name I shall refer to them
in the following pages.
Our Middle Class consists of Equilateral or Equal-Sided Triangles.
Our Professional Men and Gentlemen are Squares (to which class
I myself belong) and Five-Sided Figures or Pentagons.
Next above these come the Nobility, of whom there are several degrees,
beginning at Six-Sided Figures, or Hexagons, and from thence rising
in the number of their sides till they receive the honourable title
of Polygonal, or many-sided. Finally when the number of the sides
becomes so numerous, and the sides themselves so small,
that the figure cannot be distinguished from a circle,
he is included in the Circular or Priestly order; and this is
the highest class of all.
It is a Law of Nature with us that a male child shall have
one more side than his father, so that each generation shall rise
(as a rule) one step in the scale of development and nobility.
Thus the son of a Square is a Pentagon; the son of a Pentagon,
a Hexagon; and so on.
But this rule applies not always to the Tradesmen, and still
less often to the Soldiers, and to the Workmen; who indeed can hardly
be said to deserve the name of human Figures, since they have not
all their sides equal. With them therefore the Law of Nature
does not hold; and the son of an Isosceles (i.e. a Triangle with
two sides equal) remains Isosceles still. Nevertheless,
all hope is not shut out, even from the Isosceles, that his posterity
may ultimately rise above his degraded condition. For, after a long
series of military successes, or diligent and skilful labours,
it is generally found that the more intelligent among
the Artisan and Soldier classes manifest a slight increase
of their third side or base, and a shrinkage of the two other sides.
Intermarriages (arranged by the Priests) between the sons
and daughters of these more intellectual members of the lower classes
generally result in an offspring approximating still more to the type
of the Equal-Sided Triangle.
Rarely -- in proportion to the vast numbers of Isosceles births --
is a genuine and certifiable Equal-Sided Triangle produced
from Isosceles parents. [Note: "What need of a certificate?"
a Spaceland critic may ask: "Is not the procreation of a Square Son
a certificate from Nature herself, proving the Equal-sidedness
of the Father?" I reply that no Lady of any position will marry
an uncertified Triangle. Square offspring has sometimes resulted
from a slightly Irregular Triangle; but in almost every such case
the Irregularity of the first generation is visited on the third;
which either fails to attain the Pentagonal rank, or relapses to
the Triangular.] Such a birth requires, as its antecedents,
not only a series of carefully arranged intermarriages,
but also a long, continued exercise of frugality and self-control
on the part of the would-be ancestors of the coming Equilateral,
and a patient, systematic, and continuous development
of the Isosceles intellect through many generations.
The birth of a True Equilateral Triangle from Isosceles parents
is the subject of rejoicing in our country for many furlongs around.
After a strict examination conducted by the Sanitary and Social Board,
the infant, if certified as Regular, is with solemn ceremonial
admitted into the class of Equilaterals. He is then immediately
taken from his proud yet sorrowing parents and adopted by some
childless Equilateral, who is bound by oath never to permit the child
henceforth to enter his former home or so much as to look upon
his relations again, for fear lest the freshly developed organism may,
by force of unconscious imitation, fall back again into
his hereditary level.
The occasional emergence of an Equilateral from the ranks
of his serf-born ancestors is welcomed, not only by
the poor serfs themselves, as a gleam of light and hope shed upon
the monotonous squalor of their existence, but also by the Aristocracy
at large; for all the higher classes are well aware that
these rare phenomena, while they do little or nothing to vulgarize
their own privileges, serve as a most useful barrier against
revolution from below.
Had the acute-angled rabble been all, without exception,
absolutely destitute of hope and of ambition, they might have
found leaders in some of their many seditious outbreaks,
so able as to render their superior numbers and strength too much
even for the wisdom of the Circles. But a wise ordinance of Nature
has decreed that, in proportion as the working-classes increase
in intelligence, knowledge, and all virtue, in that same proportion
their acute angle (which makes them physically terrible)
shall increase also and approximate to the comparatively harmless
angle of the Equilateral Triangle. Thus, in the most brutal
and formidable of the soldier class -- creatures almost on a level
with women in their lack of intelligence -- it is found that,
as they wax in the mental ability necessary to employ
their tremendous penetrating power to advantage, so do they wane
in the power of penetration itself.
How admirable is this Law of Compensation! And how perfect a proof
of the natural fitness and, I may almost say, the divine origin
of the aristocratic constitution of the States in Flatland!
By a judicious use of this Law of Nature, the Polygons and Circles
are almost always able to stifle sedition in its very cradle,
taking advantage of the irrepressible and boundless hopefulness
of the human mind. Art also comes to the aid of Law and Order.
It is generally found possible -- by a little artificial
compression or expansion on the part of the State physicians --
to make some of the more intelligent leaders of a rebellion
perfectly Regular, and to admit them at once into
the privileged classes; a much larger number, who are still below
the standard, allured by the prospect of being ultimately ennobled,
are induced to enter the State Hospitals, where they are kept
in honourable confinement for life; one or two alone
of the more obstinate, foolish, and hopelessly irregular are led
to execution.
Then the wretched rabble of the Isosceles, planless and leaderless,
are either transfixed without resistance by the small body
of their brethren whom the Chief Circle keeps in pay
for emergencies of this kind; or else more often, by means of
jealousies and suspicions skilfully fomented among them
by the Circular party, they are stirred to mutual warfare,
and perish by one another's angles. No less than one hundred
and twenty rebellions are recorded in our annals, besides minor
outbreaks numbered at two hundred and thirty-five;
and they have all ended thus.
Section 4. Concerning the Women
If our highly pointed Triangles of the Soldier class are formidable,
it may be readily inferred that far more formidable are our Women.
For if a Soldier is a wedge, a Woman is a needle; being, so to speak,
ALL point, at least at the two extremities. Add to this the power
of making herself practically invisible at will, and you will perceive
that a Female, in Flatland, is a creature by no means
to be trifled with.
But here, perhaps, some of my younger Readers may ask HOW a woman
in Flatland can make herself invisible. This ought, I think,
to be apparent without any explanation. However, a few words
will make it clear to the most unreflecting.
Place a needle on a table. Then, with your eye on the level of
the table, look at it side-ways, and you see the whole length of it;
but look at it end-ways, and you see nothing but a point,
it has become practically invisible. Just so is it with one
of our Women. When her side is turned towards us, we see her
as a straight line; when the end containing her eye or mouth --
for with us these two organs are identical -- is the part that meets
our eye, then we see nothing but a highly lustrous point;
but when the back is presented to our view, then -- being only
sub-lustrous, and, indeed, almost as dim as an inanimate object --
her hinder extremity serves her as a kind of Invisible Cap.
The dangers to which we are exposed from our Women must now be
manifest to the meanest capacity in Spaceland. If even the angle
of a respectable Triangle in the middle class is not without
its dangers; if to run against a Working Man involves a gash;
if collision with an officer of the military class necessitates
a serious wound; if a mere touch from the vertex of a Private Soldier
brings with it danger of death; -- what can it be to run against
a Woman, except absolute and immediate destruction? And when a Woman
is invisible, or visible only as a dim sub-lustrous point,
how difficult must it be, even for the most cautious,
always to avoid collision!
Many are the enactments made at different times in the different
States of Flatland, in order to minimize this peril;
and in the Southern and less temperate climates where
the force of gravitation is greater, and human beings more liable to
casual and involuntary motions, the Laws concerning Women
are naturally much more stringent. But a general view of the Code
may be obtained from the following summary: --
1. Every house shall have one entrance in the Eastern side,
for the use of Females only; by which all females shall enter
"in a becoming and respectful manner" and not by the Men's
or Western door. [Note: When I was in Spaceland I understood that
some of your Priestly circles have in the same way a separate entrance
for Villagers, Farmers and Teachers of Board Schools (`Spectator',
Sept. 1884, p. 1255) that they may "approach in a becoming
and respectful manner."]
2. No Female shall walk in any public place without continually
keeping up her Peace-cry, under penalty of death.
3. Any Female, duly certified to be suffering from St. Vitus's Dance,
fits, chronic cold accompanied by violent sneezing, or any disease
necessitating involuntary motions, shall be instantly destroyed.
In some of the States there is an additional Law forbidding Females,
under penalty of death, from walking or standing in any public place
without moving their backs constantly from right to left
so as to indicate their presence to those behind them;
others oblige a Woman, when travelling, to be followed by one
of her sons, or servants, or by her husband; others confine Women
altogether to their houses except during the religious festivals.
But it has been found by the wisest of our Circles or Statesmen
that the multiplication of restrictions on Females tends not only
to the debilitation and diminution of the race, but also to
the increase of domestic murders to such an extent that a State loses
more than it gains by a too prohibitive Code.
For whenever the temper of the Women is thus exasperated
by confinement at home or hampering regulations abroad,
they are apt to vent their spleen upon their husbands and children;
and in the less temperate climates the whole male population
of a village has been sometimes destroyed in one or two hours
of simultaneous female outbreak. Hence the Three Laws,
mentioned above, suffice for the better regulated States,
and may be accepted as a rough exemplification of our Female Code.
After all, our principal safeguard is found, not in Legislature,
but in the interests of the Women themselves. For, although they can
inflict instantaneous death by a retrograde movement,
yet unless they can at once disengage their stinging extremity
from the struggling body of their victim, their own frail bodies
are liable to be shattered.
The power of Fashion is also on our side. I pointed out that in some
less civilized States no female is suffered to stand
in any public place without swaying her back from right to left.
This practice has been universal among ladies of any pretensions
to breeding in all well-governed States, as far back as the memory
of Figures can reach. It is considered a disgrace to any State
that legislation should have to enforce what ought to be,
and is in every respectable female, a natural instinct.
The rhythmical and, if I may so say, well-modulated undulation
of the back in our ladies of Circular rank is envied and imitated
by the wife of a common Equilateral, who can achieve nothing beyond
a mere monotonous swing, like the ticking of a pendulum;
and the regular tick of the Equilateral is no less admired and copied
by the wife of the progressive and aspiring Isosceles,
in the females of whose family no "back-motion" of any kind
has become as yet a necessity of life. Hence, in every family
of position and consideration, "back motion" is as prevalent
as time itself; and the husbands and sons in these households
enjoy immunity at least from invisible attacks.
Not that it must be for a moment supposed that our Women are
destitute of affection. But unfortunately the passion of the moment
predominates, in the Frail Sex, over every other consideration.
This is, of course, a necessity arising from their
unfortunate conformation. For as they have no pretensions
to an angle, being inferior in this respect to the very lowest
of the Isosceles, they are consequently wholly devoid of brain-power,
and have neither reflection, judgment nor forethought,
and hardly any memory. Hence, in their fits of fury, they remember
no claims and recognize no distinctions. I have actually known a case
where a Woman has exterminated her whole household,
and half an hour afterwards, when her rage was over and the fragments
swept away, has asked what has become of her husband and her children.
Obviously then a Woman is not to be irritated as long as she is in
a position where she can turn round. When you have them
in their apartments -- which are constructed with a view
to denying them that power -- you can say and do what you like;
for they are then wholly impotent for mischief, and will not remember
a few minutes hence the incident for which they may be at this moment
threatening you with death, nor the promises which you may have
found it necessary to make in order to pacify their fury.
On the whole we get on pretty smoothly in our domestic relations,
except in the lower strata of the Military Classes. There the want
of tact and discretion on the part of the husbands produces at times
indescribable disasters. Relying too much on the offensive weapons
of their acute angles instead of the defensive organs of good sense
and seasonable simulation, these reckless creatures too often neglect
the prescribed construction of the women's apartments,
or irritate their wives by ill-advised expressions out of doors,
which they refuse immediately to retract. Moreover a blunt and stolid
regard for literal truth indisposes them to make those lavish promises
by which the more judicious Circle can in a moment pacify his consort.
The result is massacre; not, however, without its advantages,
as it eliminates the more brutal and troublesome of the Isosceles;
and by many of our Circles the destructiveness of the Thinner Sex
is regarded as one among many providential arrangements for
suppressing redundant population, and nipping Revolution in the bud.
Yet even in our best regulated and most approximately Circular
families I cannot say that the ideal of family life is so high
as with you in Spaceland. There is peace, in so far as the absence
of slaughter may be called by that name, but there is necessarily
little harmony of tastes or pursuits; and the cautious wisdom
of the Circles has ensured safety at the cost of domestic comfort.
In every Circular or Polygonal household it has been a habit
from time immemorial -- and now has become a kind of instinct among
the women of our higher classes -- that the mothers and daughters
should constantly keep their eyes and mouths towards their husband
and his male friends; and for a lady in a family of distinction
to turn her back upon her husband would be regarded as a kind
of portent, involving loss of STATUS. But, as I shall soon shew,
this custom, though it has the advantage of safety,
is not without its disadvantages.
In the house of the Working Man or respectable Tradesman --
where the wife is allowed to turn her back upon her husband,
while pursuing her household avocations -- there are at least
intervals of quiet, when the wife is neither seen nor heard,
except for the humming sound of the continuous Peace-cry;
but in the homes of the upper classes there is too often no peace.
There the voluble mouth and bright penetrating eye are ever directed
towards the Master of the household; and light itself is not
more persistent than the stream of feminine discourse.
The tact and skill which suffice to avert a Woman's sting are unequal
to the task of stopping a Woman's mouth; and as the wife
has absolutely nothing to say, and absolutely no constraint of wit,
sense, or conscience to prevent her from saying it,
not a few cynics have been found to aver that they prefer the danger
of the death-dealing but inaudible sting to the safe sonorousness
of a Woman's other end.
To my readers in Spaceland the condition of our Women may seem
truly deplorable, and so indeed it is. A Male of the lowest type
of the Isosceles may look forward to some improvement of his angle,
and to the ultimate elevation of the whole of his degraded caste;
but no Woman can entertain such hopes for her sex. "Once a Woman,
always a Woman" is a Decree of Nature; and the very Laws of Evolution
seem suspended in her disfavour. Yet at least we can
admire the wise Prearrangement which has ordained that,
as they have no hopes, so they shall have no memory to recall,
and no forethought to anticipate, the miseries and humiliations
which are at once a necessity of their existence and the basis of
the constitution of Flatland.
Section 5. Of our Methods of Recognizing one another
You, who are blessed with shade as well as light, you,
who are gifted with two eyes, endowed with a knowledge of perspective,
and charmed with the enjoyment of various colours, you,
who can actually SEE an angle, and contemplate the complete
circumference of a circle in the happy region of the Three Dimensions
-- how shall I make clear to you the extreme difficulty which we
in Flatland experience in recognizing one another's configuration?
Recall what I told you above. All beings in Flatland,
animate or inanimate, no matter what their form, present TO OUR VIEW
the same, or nearly the same, appearance, viz. that of
a straight Line. How then can one be distinguished from another,
where all appear the same?
The answer is threefold. The first means of recognition
is the sense of hearing; which with us is far more highly developed
than with you, and which enables us not only to distinguish
by the voice our personal friends, but even to discriminate
between different classes, at least so far as concerns
the three lowest orders, the Equilateral, the Square, and the Pentagon
-- for of the Isosceles I take no account. But as we ascend
in the social scale, the process of discriminating and being
discriminated by hearing increases in difficulty, partly because
voices are assimilated, partly because the faculty of
voice-discrimination is a plebeian virtue not much developed among
the Aristocracy. And wherever there is any danger of imposture
we cannot trust to this method. Amongst our lowest orders,
the vocal organs are developed to a degree more than correspondent
with those of hearing, so that an Isosceles can easily feign the voice
of a Polygon, and, with some training, that of a Circle himself.
A second method is therefore more commonly resorted to.
FEELING is, among our Women and lower classes -- about our
upper classes I shall speak presently -- the principal test
of recognition, at all events between strangers, and when
the question is, not as to the individual, but as to the class.
What therefore "introduction" is among the higher classes
in Spaceland, that the process of "feeling" is with us.
"Permit me to ask you to feel and be felt by my friend Mr. So-and-so"
-- is still, among the more old-fashioned of our country gentlemen
in districts remote from towns, the customary formula for
a Flatland introduction. But in the towns, and among men of business,
the words "be felt by" are omitted and the sentence is abbreviated to,
"Let me ask you to feel Mr. So-and-so"; although it is assumed,
of course, that the "feeling" is to be reciprocal.
Among our still more modern and dashing young gentlemen -- who are
extremely averse to superfluous effort and supremely indifferent
to the purity of their native language -- the formula is still
further curtailed by the use of "to feel" in a technical sense,
meaning, "to recommend-for-the-purposes-of-feeling-and-being-felt";
and at this moment the "slang" of polite or fast society
in the upper classes sanctions such a barbarism as "Mr. Smith,
permit me to feel Mr. Jones."
Let not my Reader however suppose that "feeling" is with us
the tedious process that it would be with you, or that we find it
necessary to feel right round all the sides of every individual
before we determine the class to which he belongs. Long practice
and training, begun in the schools and continued in the experience
of daily life, enable us to discriminate at once by
the sense of touch, between the angles of an equal-sided Triangle,
Square, and Pentagon; and I need not say that the brainless vertex
of an acute-angled Isosceles is obvious to the dullest touch.
It is therefore not necessary, as a rule, to do more than feel
a single angle of an individual; and this, once ascertained,
tells us the class of the person whom we are addressing,
unless indeed he belongs to the higher sections of the nobility.
There the difficulty is much greater. Even a Master of Arts
in our University of Wentbridge has been known to confuse a ten-sided
with a twelve-sided Polygon; and there is hardly a Doctor of Science
in or out of that famous University who could pretend
to decide promptly and unhesitatingly between a twenty-sided
and a twenty-four sided member of the Aristocracy.
Those of my readers who recall the extracts I gave above
from the Legislative code concerning Women, will readily perceive
that the process of introduction by contact requires
some care and discretion. Otherwise the angles might inflict
on the unwary Feeler irreparable injury. It is essential
for the safety of the Feeler that the Felt should stand
perfectly still. A start, a fidgety shifting of the position, yes,
even a violent sneeze, has been known before now to prove fatal
to the incautious, and to nip in the bud many a promising friendship.
Especially is this true among the lower classes of the Triangles.
With them, the eye is situated so far from their vertex that they
can scarcely take cognizance of what goes on at that extremity
of their frame. They are, moreover, of a rough coarse nature,
not sensitive to the delicate touch of the highly organized Polygon.
What wonder then if an involuntary toss of the head has ere now
deprived the State of a valuable life!
I have heard that my excellent Grandfather -- one of the least
irregular of his unhappy Isosceles class, who indeed obtained,
shortly before his decease, four out of seven votes from the Sanitary
and Social Board for passing him into the class of the Equal-sided --
often deplored, with a tear in his venerable eye, a miscarriage
of this kind, which had occured to his great-great-great-Grandfather,
a respectable Working Man with an angle or brain of 59 degrees
30 minutes. According to his account, my unfortunate Ancestor,
being afflicted with rheumatism, and in the act of being felt
by a Polygon, by one sudden start accidentally transfixed
the Great Man through the diagonal; and thereby, partly in consequence
of his long imprisonment and degradation, and partly because of
the moral shock which pervaded the whole of my Ancestor's relations,
threw back our family a degree and a half in their ascent
towards better things. The result was that in the next generation
the family brain was registered at only 58 degrees, and not till
the lapse of five generations was the lost ground recovered,
the full 60 degrees attained, and the Ascent from the Isosceles
finally achieved. And all this series of calamities from one
little accident in the process of Feeling.
At this point I think I hear some of my better educated
readers exclaim, "How could you in Flatland know anything about
angles and degrees, or minutes? We can SEE an angle, because we,
in the region of Space, can see two straight lines inclined
to one another; but you, who can see nothing but one straight line
at a time, or at all events only a number of bits of straight lines
all in one straight line -- how can you ever discern any angle,
and much less register angles of different sizes?"
I answer that though we cannot SEE angles, we can INFER them,
and this with great precision. Our sense of touch,
stimulated by necessity, and developed by long training,
enables us to distinguish angles far more accurately than your
sense of sight, when unaided by a rule or measure of angles.
Nor must I omit to explain that we have great natural helps.
It is with us a Law of Nature that the brain of the Isosceles class
shall begin at half a degree, or thirty minutes, and shall increase
(if it increases at all) by half a degree in every generation;
until the goal of 60 degrees is reached, when the condition of serfdom
is quitted, and the freeman enters the class of Regulars.
Consequently, Nature herself supplies us with an ascending scale
or Alphabet of angles for half a degree up to 60 degrees,
Specimens of which are placed in every Elementary School
throughout the land. Owing to occasional retrogressions,
to still more frequent moral and intellectual stagnation, and to
the extraordinary fecundity of the Criminal and Vagabond Classes,
there is always a vast superfluity of individuals of the half degree
and single degree class, and a fair abundance of Specimens
up to 10 degrees. These are absolutely destitute of civic rights;
and a great number of them, not having even intelligence enough
for the purposes of warfare, are devoted by the States to the service
of education. Fettered immovably so as to remove all possibility
of danger, they are placed in the class rooms of our Infant Schools,
and there they are utilized by the Board of Education for the purpose
of imparting to the offspring of the Middle Classes that tact
and intelligence of which these wretched creatures themselves
are utterly devoid.
In some States the Specimens are occasionally fed and suffered
to exist for several years; but in the more temperate
and better regulated regions, it is found in the long run
more advantageous for the educational interests of the young,
to dispense with food, and to renew the Specimens every month --
which is about the average duration of the foodless existence
of the Criminal class. In the cheaper schools, what is gained
by the longer existence of the Specimen is lost, partly in
the expenditure for food, and partly in the diminished accuracy
of the angles, which are impaired after a few weeks
of constant "feeling". Nor must we forget to add, in enumerating
the advantages of the more expensive system, that it tends,
though slightly yet perceptibly, to the diminution of the redundant
Isosceles population -- an object which every statesman in Flatland
constantly keeps in view. On the whole therefore --
although I am not ignorant that, in many popularly elected
School Boards, there is a reaction in favour of "the cheap system"
as it is called -- I am myself disposed to think that this is one
of the many cases in which expense is the truest economy.
But I must not allow questions of School Board politics to divert me
from my subject. Enough has been said, I trust, to shew
that Recognition by Feeling is not so tedious or indecisive a process
as might have been supposed; and it is obviously more trustworthy
than Recognition by hearing. Still there remains, as has been
pointed out above, the objection that this method is not
without danger. For this reason many in the Middle and Lower classes,
and all without exception in the Polygonal and Circular orders,
prefer a third method, the description of which shall be reserved
for the next section.
Section
6. Of Recognition by Sight
I am about to appear very inconsistent. In previous sections
I have said that all figures in Flatland present the appearance
of a straight line; and it was added or implied, that it is
consequently impossible to distinguish by the visual organ
between individuals of different classes: yet now I am about
to explain to my Spaceland critics how we are able to recognize
one another by the sense of sight.
If however the Reader will take the trouble to refer to the passage
in which Recognition by Feeling is stated to be universal,
he will find this qualification -- "among the lower classes".
It is only among the higher classes and in our temperate climates
that Sight Recognition is practised.
That this power exists in any regions and for any classes
is the result of Fog; which prevails during the greater part
of the year in all parts save the torrid zones. That which is
with you in Spaceland an unmixed evil, blotting out the landscape,
depressing the spirits, and enfeebling the health, is by us recognized
as a blessing scarcely inferior to air itself, and as the Nurse
of arts and Parent of sciences. But let me explain my meaning,
without further eulogies on this beneficent Element.
If Fog were non-existent, all lines would appear equally
and indistinguishably clear; and this is actually the case
in those unhappy countries in which the atmosphere is perfectly dry
and transparent. But wherever there is a rich supply of Fog
objects that are at a distance, say of three feet, are appreciably
dimmer than those at a distance of two feet eleven inches;
and the result is that by careful and constant experimental
observation of comparative dimness and clearness, we are enabled to
infer with great exactness the configuration of the object observed.
An instance will do more than a volume of generalities to make
my meaning clear.
Suppose I see two individuals approaching whose rank I wish
to ascertain. They are, we will suppose, a Merchant and a Physician,
or in other words, an Equilateral Triangle and a Pentagon:
how am I to distinguish them?
<>
<>
C (1)
|\ - _ D
| \ ||- _
| \ || - _
| <--- >|| -----------+(> Eye-glance
___C' (2) | / A|| _ -
___--- \ - _D' | / ||_ -
__--- \ || - _ |/ _ - E
| \ || - _ B
| \ || - _
| Eye-glance \ || - _
| <----------- A'>|| ------------------------+(>
| / || _ -
| / || _ -
|__ / || _ -
---___ / || _ -
---___/ _ -E'
B'
It will be obvious, to every child in Spaceland who has touched
the threshold of Geometrical Studies, that, if I can bring my eye so
that its glance may bisect an angle (A) of the approaching stranger,
my view will lie as it were evenly between his two sides that are
next to me (viz. CA and AB), so that I shall contemplate
the two impartially, and both will appear of the same size.
Now in the case of (1) the Merchant, what shall I see? I shall see
a straight line DAE, in which the middle point (A) will be very bright
because it is nearest to me; but on either side the line will
shade away RAPIDLY INTO DIMNESS, because the sides AC and AB
RECEDE RAPIDLY INTO THE FOG and what appear to me as
the Merchant's extremities, viz. D and E, will be VERY DIM INDEED.
On the other hand in the case of (2) the Physician, though I shall
here also see a line (D'A'E') with a bright centre (A'),
yet it will shade away LESS RAPIDLY into dimness, because the sides
(A'C', A'B') RECEDE LESS RAPIDLY INTO THE FOG: and what appear
to me the Physician's extremities, viz. D' and E', will not be
NOT SO DIM as the extremities of the Merchant.
The Reader will probably understand from these two instances how --
after a very long training supplemented by constant experience --
it is possible for the well-educated classes among us to discriminate
with fair accuracy between the middle and lowest orders,
by the sense of sight. If my Spaceland Patrons have grasped
this general conception, so far as to conceive the possibility of it
and not to reject my account as altogether incredible --
I shall have attained all I can reasonably expect. Were I to attempt
further details I should only perplex. Yet for the sake of the young
and inexperienced, who may perchance infer -- from the two simple
instances I have given above, of the manner in which I should
recognize my Father and my Sons -- that Recognition by sight
is an easy affair, it may be needful to point out that in actual life
most of the problems of Sight Recognition are far more
subtle and complex.
If for example, when my Father, the Triangle, approaches me,
he happens to present his side to me instead of his angle, then,
until I have asked him to rotate, or until I have edged my eye
round him, I am for the moment doubtful whether he may not be
a Straight Line, or, in other words, a Woman. Again, when I am
in the company of one of my two hexagonal Grandsons, contemplating one
of his sides (AB) full front, it will be evident from
the accompanying diagram that I shall see one whole line (AB)
in comparative brightness (shading off hardly at all at the ends)
and two smaller lines (CA and BD) dim throughout and shading away
into greater dimness towards the extremities C and D.
<>
<>
/\ - _ C
/ \ || _
/ \ || - _
/ \|| - _
| A || - _
| || -+(> (Eye)
| B || _ -
\ /|| _ -
\ / || _ -
\ / || -
\/ _ - D
But I must not give way to the temptation of enlarging on
these topics. The meanest mathematician in Spaceland will readily
believe me when I assert that the problems of life, which present
themselves to the well-educated -- when they are themselves in motion,
rotating, advancing or retreating, and at the same time attempting to
discriminate by the sense of sight between a number of Polygons
of high rank moving in different directions, as for example in
a ball-room or conversazione -- must be of a nature to task
the angularity of the most intellectual, and amply justify
the rich endowments of the Learned Professors of Geometry,
both Static and Kinetic, in the illustrious University of Wentbridge,
where the Science and Art of Sight Recognition are regularly taught
to large classes of the ELITE of the States.
It is only a few of the scions of our noblest and wealthiest houses,
who are able to give the time and money necessary for the thorough
prosecution of this noble and valuable Art. Even to me,
a Mathematician of no mean standing, and the Grandfather of two
most hopeful and perfectly regular Hexagons, to find myself
in the midst of a crowd of rotating Polygons of the higher classes,
is occasionally very perplexing. And of course to a common Tradesman,
or Serf, such a sight is almost as unintelligible as it would be
to you, my Reader, were you suddenly transported into our country.
In such a crowd you could see on all sides of you nothing but a Line,
apparently straight, but of which the parts would vary
irregularly and perpetually in brightness or dimness. Even if you
had completed your third year in the Pentagonal and Hexagonal classes
in the University, and were perfect in the theory of the subject,
you would still find that there was need of many years of experience,
before you could move in a fashionable crowd without jostling against
your betters, whom it is against etiquette to ask to "feel", and who,
by their superior culture and breeding, know all about your movements,
while you know very little or nothing about theirs. In a word,
to comport oneself with perfect propriety in Polygonal society,
one ought to be a Polygon oneself. Such at least is
the painful teaching of my experience.
It is astonishing how much the Art -- or I may almost call it instinct
-- of Sight Recognition is developed by the habitual practice of it
and by the avoidance of the custom of "Feeling". Just as, with you,
the deaf and dumb, if once allowed to gesticulate and to use
the hand-alphabet, will never acquire the more difficult
but far more valuable art of lipspeech and lip-reading, so it is
with us as regards "Seeing" and "Feeling". None who in early life
resort to "Feeling" will ever learn "Seeing" in perfection.
For this reason, among our Higher Classes, "Feeling" is discouraged
or absolutely forbidden. From the cradle their children,
instead of going to the Public Elementary schools (where the art
of Feeling is taught), are sent to higher Seminaries
of an exclusive character; and at our illustrious University,
to "feel" is regarded as a most serious fault, involving Rustication
for the first offence, and Expulsion for the second.
But among the lower classes the art of Sight Recognition is regarded
as an unattainable luxury. A common Tradesman cannot afford
to let his son spend a third of his life in abstract studies.
The children of the poor are therefore allowed to "feel"
from their earliest years, and they gain thereby a precocity
and an early vivacity which contrast at first most favourably with
the inert, undeveloped, and listless behaviour of the half-instructed
youths of the Polygonal class; but when the latter have at last
completed their University course, and are prepared to put
their theory into practice, the change that comes over them
may almost be described as a new birth, and in every art, science,
and social pursuit they rapidly overtake and distance
their Triangular competitors.
Only a few of the Polygonal Class fail to pass the Final Test
or Leaving Examination at the University. The condition of
the unsuccessful minority is truly pitiable. Rejected from
the higher class, they are also despised by the lower.
They have neither the matured and systematically trained powers
of the Polygonal Bachelors and Masters of Arts, nor yet the native
precocity and mercurial versatility of the youthful Tradesman.
The professions, the public services, are closed against them;
and though in most States they are not actually debarred
from marriage, yet they have the greatest difficulty in forming
suitable alliances, as experience shews that the offspring of such
unfortunate and ill-endowed parents is generally itself unfortunate,
if not positively Irregular.
It is from these specimens of the refuse of our Nobility
that the great Tumults and Seditions of past ages have generally
derived their leaders; and so great is the mischief thence arising
that an increasing minority of our more progressive Statesmen
are of opinion that true mercy would dictate their entire suppression,
by enacting that all who fail to pass the Final Examination
of the University should be either imprisoned for life,
or extinguished by a painless death.
But I find myself digressing into the subject of Irregularities,
a matter of such vital interest that it demands a separate section.
Section
7. Concerning Irregular Figures
Throughout the previous pages I have been assuming --
what perhaps should have been laid down at the beginning as a distinct
and fundamental proposition -- that every human being in Flatland
is a Regular Figure, that is to say of regular construction.
By this I mean that a Woman must not only be a line,
but a straight line; that an Artisan or Soldier must have
two of his sides equal; that Tradesmen must have three sides equal;
Lawyers (of which class I am a humble member), four sides equal,
and generally, that in every Polygon, all the sides must be equal.
The size of the sides would of course depend upon the age of
the individual. A Female at birth would be about an inch long,
while a tall adult Woman might extend to a foot. As to the Males
of every class, it may be roughly said that the length of
an adult's sides, when added together, is two feet or a little more.
But the size of our sides is not under consideration.
I am speaking of the EQUALITY of sides, and it does not need
much reflection to see that the whole of the social life in Flatland
rests upon the fundamental fact that Nature wills all Figures
to have their sides equal.
If our sides were unequal our angles might be unequal.
Instead of its being sufficient to feel, or estimate by sight,
a single angle in order to determine the form of an individual,
it would be necessary to ascertain each angle by the experiment
of Feeling. But life would be too short for such a tedious grouping.
The whole science and art of Sight Recognition would at once perish;
Feeling, so far as it is an art, would not long survive;
intercourse would become perilous or impossible; there would be
an end to all confidence, all forethought; no one would be safe
in making the most simple social arrangements; in a word,
civilization would relapse into barbarism.
Am I going too fast to carry my Readers with me to these
obvious conclusions? Surely a moment's reflection, and a single
instance from common life, must convince every one that our whole
social system is based upon Regularity, or Equality of Angles.
You meet, for example, two or three Tradesmen in the street,
whom you recognize at once to be Tradesmen by a glance at their angles
and rapidly bedimmed sides, and you ask them to step into your house
to lunch. This you do at present with perfect confidence,
because everyone knows to an inch or two the area occupied
by an adult Triangle: but imagine that your Tradesman drags
behind his regular and respectable vertex, a parallelogram
of twelve or thirteen inches in diagonal: -- what are you to do
with such a monster sticking fast in your house door?
But I am insulting the intelligence of my Readers by accumulating
details which must be patent to everyone who enjoys the advantages of
a Residence in Spaceland. Obviously the measurements of
a single angle would no longer be sufficient under such
portentous circumstances; one's whole life would be taken up
in feeling or surveying the perimeter of one's acquaintances.
Already the difficulties of avoiding a collision in a crowd are enough
to tax the sagacity of even a well-educated Square; but if no one
could calculate the Regularity of a single figure in the company,
all would be chaos and confusion, and the slightest panic
would cause serious injuries, or -- if there happened to be
any Women or Soldiers present -- perhaps considerable loss of life.
Expediency therefore concurs with Nature in stamping the seal
of its approval upon Regularity of conformation: nor has the Law
been backward in seconding their efforts. "Irregularity of Figure"
means with us the same as, or more than, a combination of
moral obliquity and criminality with you, and is treated accordingly.
There are not wanting, it is true, some promulgators of paradoxes
who maintain that there is no necessary connection between
geometrical and moral Irregularity. "The Irregular", they say,
"is from his birth scouted by his own parents, derided by
his brothers and sisters, neglected by the domestics,
scorned and suspected by society, and excluded from all posts
of responsibility, trust, and useful activity. His every movement
is jealously watched by the police till he comes of age
and presents himself for inspection; then he is either destroyed,
if he is found to exceed the fixed margin of deviation,
or else immured in a Government Office as a clerk of
the seventh class; prevented from marriage; forced to drudge
at an uninteresting occupation for a miserable stipend;
obliged to live and board at the office, and to take even his vacation
under close supervision; what wonder that human nature,
even in the best and purest, is embittered and perverted
by such surroundings!"
All this very plausible reasoning does not convince me, as it has not
convinced the wisest of our Statesmen, that our ancestors erred
in laying it down as an axiom of policy that the toleration
of Irregularity is incompatible with the safety of the State.
Doubtless, the life of an Irregular is hard; but the interests of
the Greater Number require that it shall be hard. If a man with
a triangular front and a polygonal back were allowed to exist
and to propagate a still more Irregular posterity, what would become
of the arts of life? Are the houses and doors and churches
in Flatland to be altered in order to accommodate such monsters?
Are our ticket-collectors to be required to measure every man's
perimeter before they allow him to enter a theatre or to take
his place in a lecture room? Is an Irregular to be exempted
from the militia? And if not, how is he to be prevented from
carrying desolation into the ranks of his comrades? Again,
what irresistible temptations to fraudulent impostures must
needs beset such a creature! How easy for him to enter a shop
with his polygonal front foremost, and to order goods
to any extent from a confiding tradesman! Let the advocates of
a falsely called Philanthropy plead as they may for the abrogation
of the Irregular Penal Laws, I for my part have never known
an Irregular who was not also what Nature evidently intended him to be
-- a hypocrite, a misanthropist, and, up to the limits of his power,
a perpetrator of all manner of mischief.
Not that I should be disposed to recommend (at present)
the extreme measures adopted by some States, where an infant
whose angle deviates by half a degree from the correct angularity
is summarily destroyed at birth. Some of our highest and ablest men,
men of real genius, have during their earliest days laboured under
deviations as great as, or even greater than, forty-five minutes:
and the loss of their precious lives would have been an irreparable
injury to the State. The art of healing also has achieved
some of its most glorious triumphs in the compressions, extensions,
trepannings, colligations, and other surgical or diaetetic operations
by which Irregularity has been partly or wholly cured.
Advocating therefore a VIA MEDIA, I would lay down no fixed
or absolute line of demarcation; but at the period when the frame
is just beginning to set, and when the Medical Board has reported that
recovery is improbable, I would suggest that the Irregular offspring
be painlessly and mercifully consumed.
Section
8. Of the Ancient Practice of Painting
If my Readers have followed me with any attention up to this point,
they will not be surprised to hear that life is somewhat dull
in Flatland. I do not, of course, mean that there are not battles,
conspiracies, tumults, factions, and all those other phenomena which
are supposed to make History interesting; nor would I deny
that the strange mixture of the problems of life and the problems
of Mathematics, continually inducing conjecture and giving
the opportunity of immediate verification, imparts to our existence
a zest which you in Spaceland can hardly comprehend. I speak now
from the aesthetic and artistic point of view when I say that life
with us is dull; aesthetically and artistically, very dull indeed.
How can it be otherwise, when all one's prospect, all one's
landscapes, historical pieces, portraits, flowers, still life,
are nothing but a single line, with no varieties except degrees of
brightness and obscurity?
It was not always thus. Colour, if Tradition speaks the truth,
once for the space of half a dozen centuries or more,
threw a transient splendour over the lives of our ancestors
in the remotest ages. Some private individual -- a Pentagon
whose name is variously reported -- having casually discovered
the constituents of the simpler colours and a rudimentary method
of painting, is said to have begun decorating first his house,
then his slaves, then his Father, his Sons, and Grandsons,
lastly himself. The convenience as well as the beauty of the results
commended themselves to all. Wherever Chromatistes, --
for by that name the most trustworthy authorities concur
in calling him, -- turned his variegated frame, there he at once
excited attention, and attracted respect. No one now needed
to "feel" him; no one mistook his front for his back;
all his movements were readily ascertained by his neighbours
without the slightest strain on their powers of calculation;
no one jostled him, or failed to make way for him; his voice was saved
the labour of that exhausting utterance by which we colourless Squares
and Pentagons are often forced to proclaim our individuality
when we move amid a crowd of ignorant Isosceles.
The fashion spread like wildfire. Before a week was over,
every Square and Triangle in the district had copied the example
of Chromatistes, and only a few of the more conservative Pentagons
still held out. A month or two found even the Dodecagons
infected with the innovation. A year had not elapsed before
the habit had spread to all but the very highest of the Nobility.
Needless to say, the custom soon made its way from the district of
Chromatistes to surrounding regions; and within two generations no one
in all Flatland was colourless except the Women and the Priests.
Here Nature herself appeared to erect a barrier, and to plead
against extending the innovation to these two classes.
Many-sidedness was almost essential as a pretext for the Innovators.
"Distinction of sides is intended by Nature to imply distinction
of colours" -- such was the sophism which in those days
flew from mouth to mouth, converting whole towns at a time
to the new culture. But manifestly to our Priests and Women
this adage did not apply. The latter had only one side,
and therefore -- plurally and pedantically speaking -- NO SIDES.
The former -- if at least they would assert their claim to be
really and truly Circles, and not mere high-class Polygons
with an infinitely large number of infinitesimally small sides --
were in the habit of boasting (what Women confessed and deplored)
that they also had no sides, being blessed with a perimeter of
one line, or, in other words, a Circumference. Hence it came to pass
that these two Classes could see no force in the so-called axiom about
"Distinction of Sides implying Distinction of Colour"; and when
all others had succumbed to the fascinations of corporal decoration,
the Priests and the Women alone still remained pure from
the pollution of paint.
Immoral, licentious, anarchical, unscientific -- call them
by what names you will -- yet, from an aesthetic point of view,
those ancient days of the Colour Revolt were the glorious childhood of
Art in Flatland -- a childhood, alas, that never ripened into manhood,
nor even reached the blossom of youth. To live was then in itself
a delight, because living implied seeing. Even at a small party,
the company was a pleasure to behold; the richly varied hues
of the assembly in a church or theatre are said to have more than once
proved too distracting for our greatest teachers and actors;
but most ravishing of all is said to have been the unspeakable
magnificence of a military review.
The sight of a line of battle of twenty thousand Isosceles suddenly
facing about, and exchanging the sombre black of their bases for
the orange and purple of the two sides including their acute angle;
the militia of the Equilateral Triangles tricoloured in red, white,
and blue; the mauve, ultra-marine, gamboge, and burnt umber
of the Square artillerymen rapidly rotating near their vermilion guns;
the dashing and flashing of the five-coloured and six-coloured
Pentagons and Hexagons careering across the field in their offices
of surgeons, geometricians and aides-de-camp -- all these may well
have been sufficient to render credible the famous story
how an illustrious Circle, overcome by the artistic beauty
of the forces under his command, threw aside his marshal's baton
and his royal crown, exclaiming that he henceforth exchanged them
for the artist's pencil. How great and glorious the sensuous
development of these days must have been is in part
indicated by the very language and vocabulary of the period.
The commonest utterances of the commonest citizens in the time
of the Colour Revolt seem to have been suffused with a richer tinge
of word or thought; and to that era we are even now indebted for
our finest poetry and for whatever rhythm still remains
in the more scientific utterance of these modern days.
Section
9. Of the Universal Colour Bill
But meanwhile the intellectual Arts were fast decaying.
The Art of Sight Recognition, being no longer needed,
was no longer practised; and the studies of Geometry, Statics,
Kinetics, and other kindred subjects, came soon to be
considered superfluous, and fell into disrespect and neglect even at
our University. The inferior Art of Feeling speedily experienced
the same fate at our Elementary Schools. Then the Isosceles classes,
asserting that the Specimens were no longer used nor needed,
and refusing to pay the customary tribute from the Criminal classes
to the service of Education, waxed daily more numerous
and more insolent on the strength of their immunity from
the old burden which had formerly exercised the twofold
wholesome effect of at once taming their brutal nature and thinning
their excessive numbers.
Year by year the Soldiers and Artisans began more vehemently to assert
-- and with increasing truth -- that there was no great difference
between them and the very highest class of Polygons, now that they
were raised to an equality with the latter, and enabled to grapple
with all the difficulties and solve all the problems of life,
whether Statical or Kinetical, by the simple process
of Colour Recognition. Not content with the natural neglect
into which Sight Recognition was falling, they began boldly to demand
the legal prohibition of all "monopolizing and aristocratic Arts"
and the consequent abolition of all endowments for the studies of
Sight Recognition, Mathematics, and Feeling. Soon, they began
to insist that inasmuch as Colour, which was a second Nature,
had destroyed the need of aristocratic distinctions, the Law
should follow in the same path, and that henceforth all individuals
and all classes should be recognized as absolutely equal and entitled
to equal rights.
Finding the higher Orders wavering and undecided, the leaders
of the Revolution advanced still further in their requirements,
and at last demanded that all classes alike, the Priests and the Women
not excepted, should do homage to Colour by submitting to be painted.
When it was objected that Priests and Women had no sides,
they retorted that Nature and Expediency concurred in dictating
that the front half of every human being (that is to say,
the half containing his eye and mouth) should be distinguishable
from his hinder half. They therefore brought before a general
and extraordinary Assembly of all the States of Flatland
a Bill proposing that in every Woman the half containing
the eye and mouth should be coloured red, and the other half green.
The Priests were to be painted in the same way, red being applied
to that semicircle in which the eye and mouth formed the middle point;
while the other or hinder semicircle was to be coloured green.
There was no little cunning in this proposal, which indeed emanated
not from any Isosceles -- for no being so degraded would have had
angularity enough to appreciate, much less to devise, such a model
of state-craft -- but from an Irregular Circle who, instead of being
destroyed in his childhood, was reserved by a foolish indulgence
to bring desolation on his country and destruction on
myriads of his followers.
On the one hand the proposition was calculated to bring
the Women in all classes over to the side of the Chromatic Innovation.
For by assigning to the Women the same two colours as were assigned
to the Priests, the Revolutionists thereby ensured that,
in certain positions, every Woman would appear like a Priest,
and be treated with corresponding respect and deference --
a prospect that could not fail to attract the Female Sex in a mass.
But by some of my Readers the possibility of the identical appearance
of Priests and Women, under the new Legislation, may not
be recognized; if so, a word or two will make it obvious.
Imagine a woman duly decorated, according to the new Code;
with the front half (i.e. the half containing eye and mouth) red,
and with the hinder half green. Look at her from one side.
Obviously you will see a straight line, HALF RED, HALF GREEN.
<>
<>
<>
M
_____
/ \ - C_
/ \|| - _
| || - _
A|- - - - - - -||B- - - - - -_-+(> (Eye)
| || _ -
\ /||_ -
\ _____ / - D
Now imagine a Priest, whose mouth is at M, and whose front semicircle
(AMB) is consequently coloured red, while his hinder semicircle
is green; so that the diameter AB divides the green from the red.
If you contemplate the Great Man so as to have your eye in the same
straight line as his dividing diameter (AB), what you will see will be
a straight line (CBD), of which ONE HALF (CB) WILL BE RED,
AND THE OTHER (BD) GREEN. The whole line (CD) will be
rather shorter perhaps than that of a full-sized Woman,
and will shade off more rapidly towards its extremities;
but the identity of the colours would give you an immediate impression
of identity of Class, making you neglectful of other details.
Bear in mind the decay of Sight Recognition which threatened society
at the time of the Colour Revolt; add too the certainty that Women
would speedily learn to shade off their extremities so as to imitate
the Circles; it must then be surely obvious to you, my dear Reader,
that the Colour Bill placed us under a great danger of confounding
a Priest with a young Woman.
How attractive this prospect must have been to the Frail Sex may
readily be imagined. They anticipated with delight the confusion that
would ensue. At home they might hear political and ecclesiastical
secrets intended not for them but for their husbands and brothers,
and might even issue commands in the name of a priestly Circle;
out of doors the striking combination of red and green,
without addition of any other colours, would be sure to lead
the common people into endless mistakes, and the Women would gain
whatever the Circles lost, in the deference of the passers by.
As for the scandal that would befall the Circular Class if
the frivolous and unseemly conduct of the Women were imputed to them,
and as to the consequent subversion of the Constitution,
the Female Sex could not be expected to give a thought
to these considerations. Even in the households of the Circles,
the Women were all in favour of the Universal Colour Bill.
The second object aimed at by the Bill was the gradual demoralization
of the Circles themselves. In the general intellectual decay
they still preserved their pristine clearness and strength
of understanding. From their earliest childhood, familiarized in
their Circular households with the total absence of Colour,
the Nobles alone preserved the Sacred Art of Sight Recognition,
with all the advantages that result from that admirable training
of the intellect. Hence, up to the date of the introduction
of the Universal Colour Bill, the Circles had not only held their own,
but even increased their lead of the other classes by abstinence from
the popular fashion.
Now therefore the artful Irregular whom I described above
as the real author of this diabolical Bill, determined at one blow
to lower the status of the Hierarchy by forcing them to submit to
the pollution of Colour, and at the same time to destroy their
domestic opportunities of training in the Art of Sight Recognition,
so as to enfeeble their intellects by depriving them of their pure
and colourless homes. Once subjected to the chromatic taint,
every parental and every childish Circle would demoralize each other.
Only in discerning between the Father and the Mother would
the Circular infant find problems for the exercise of
its understanding -- problems too often likely to be corrupted by
maternal impostures with the result of shaking the child's faith
in all logical conclusions. Thus by degrees the intellectual lustre
of the Priestly Order would wane, and the road would then lie open
for a total destruction of all Aristocratic Legislature
and for the subversion of our Privileged Classes.
Section
10. Of the Suppression of the Chromatic Sedition
The agitation for the Universal Colour Bill continued for three years;
and up to the last moment of that period it seemed as though Anarchy
were destined to triumph.
A whole army of Polygons, who turned out to fight as private soldiers,
was utterly annihilated by a superior force of Isosceles Triangles --
the Squares and Pentagons meanwhile remaining neutral.
Worse than all, some of the ablest Circles fell a prey to
conjugal fury. Infuriated by political animosity, the wives
in many a noble household wearied their lords with prayers
to give up their opposition to the Colour Bill; and some,
finding their entreaties fruitless, fell on and slaughtered
their innocent children and husband, perishing themselves in the act
of carnage. It is recorded that during that triennial agitation
no less than twenty-three Circles perished in domestic discord.
Great indeed was the peril. It seemed as though the Priests
had no choice between submission and extermination; when suddenly
the course of events was completely changed by one of those
picturesque incidents which Statesmen ought never to neglect,
often to anticipate, and sometimes perhaps to originate,
because of the absurdly disproportionate power with which they appeal
to the sympathies of the populace.
It happened that an Isosceles of a low type, with a brain little
if at all above four degrees -- accidentally dabbling in the colours
of some Tradesman whose shop he had plundered -- painted himself,
or caused himself to be painted (for the story varies)
with the twelve colours of a Dodecagon. Going into the Market Place
he accosted in a feigned voice a maiden, the orphan daughter
of a noble Polygon, whose affection in former days he had sought
in vain; and by a series of deceptions -- aided, on the one side,
by a string of lucky accidents too long to relate, and on the other,
by an almost inconceivable fatuity and neglect of ordinary precautions
on the part of the relations of the bride -- he succeeded in
consummating the marriage. The unhappy girl committed suicide
on discovering the fraud to which she had been subjected.
When the news of this catastrophe spread from State to State
the minds of the Women were violently agitated. Sympathy with
the miserable victim and anticipations of similar deceptions
for themselves, their sisters, and their daughters, made them
now regard the Colour Bill in an entirely new aspect.
Not a few openly avowed themselves converted to antagonism;
the rest needed only a slight stimulus to make a similar avowal.
Seizing this favourable opportunity, the Circles hastily convened
an extraordinary Assembly of the States; and besides the usual
guard of Convicts, they secured the attendance of a large number
of reactionary Women.
Amidst an unprecedented concourse, the Chief Circle of those days
-- by name Pantocyclus -- arose to find himself hissed and hooted
by a hundred and twenty thousand Isosceles. But he secured silence
by declaring that henceforth the Circles would enter on a policy
of Concession; yielding to the wishes of the majority,
they would accept the Colour Bill. The uproar being at once converted
to applause, he invited Chromatistes, the leader of the Sedition,
into the centre of the hall, to receive in the name of his followers
the submission of the Hierarchy. Then followed a speech,
a masterpiece of rhetoric, which occupied nearly a day
in the delivery, and to which no summary can do justice.
With a grave appearance of impartiality he declared that as
they were now finally committing themselves to Reform or Innovation,
it was desirable that they should take one last view of the perimeter
of the whole subject, its defects as well as its advantages.
Gradually introducing the mention of the dangers to the Tradesmen,
the Professional Classes and the Gentlemen, he silenced
the rising murmurs of the Isosceles by reminding them that,
in spite of all these defects, he was willing to accept the Bill
if it was approved by the majority. But it was manifest that all,
except the Isosceles, were moved by his words and were either
neutral or averse to the Bill.
Turning now to the Workmen he asserted that their interests must not
be neglected, and that, if they intended to accept the Colour Bill,
they ought at least to do so with full view of the consequences.
Many of them, he said, were on the point of being admitted to
the class of the Regular Triangles; others anticipated
for their children a distinction they could not hope for themselves.
That honourable ambition would now have to be sacrificed.
With the universal adoption of Colour, all distinctions would cease;
Regularity would be confused with Irregularity; development would
give place to retrogression; the Workman would in a few generations
be degraded to the level of the Military, or even the Convict Class;
political power would be in the hands of the greatest number,
that is to say the Criminal Classes, who were already more numerous
than the Workmen, and would soon out-number all the other Classes
put together when the usual Compensative Laws of Nature were violated.
A subdued murmur of assent ran through the ranks of the Artisans,
and Chromatistes, in alarm, attempted to step forward
and address them. But he found himself encompassed with guards
and forced to remain silent while the Chief Circle in a few
impassioned words made a final appeal to the Women, exclaiming that,
if the Colour Bill passed, no marriage would henceforth be safe,
no woman's honour secure; fraud, deception, hypocrisy would pervade
every household; domestic bliss would share the fate
of the Constitution and pass to speedy perdition. "Sooner than this,"
he cried, "Come death."
At these words, which were the preconcerted signal for action,
the Isosceles Convicts fell on and transfixed the wretched
Chromatistes; the Regular Classes, opening their ranks,
made way for a band of Women who, under direction of the Circles,
moved, back foremost, invisibly and unerringly upon
the unconscious soldiers; the Artisans, imitating the example
of their betters, also opened their ranks. Meantime bands of Convicts
occupied every entrance with an impenetrable phalanx.
The battle, or rather carnage, was of short duration.
Under the skillful generalship of the Circles almost every Woman's
charge was fatal and very many extracted their sting uninjured,
ready for a second slaughter. But no second blow was needed;
the rabble of the Isosceles did the rest of the business
for themselves. Surprised, leader-less, attacked in front
by invisible foes, and finding egress cut off by the Convicts
behind them, they at once -- after their manner -- lost all presence
of mind, and raised the cry of "treachery". This sealed their fate.
Every Isosceles now saw and felt a foe in every other.
In half an hour not one of that vast multitude was living;
and the fragments of seven score thousand of the Criminal Class
slain by one another's angles attested the triumph of Order.
The Circles delayed not to push their victory to the uttermost.
The Working Men they spared but decimated. The Militia of
the Equilaterals was at once called out; and every Triangle
suspected of Irregularity on reasonable grounds, was destroyed
by Court Martial, without the formality of exact measurement
by the Social Board. The homes of the Military and Artisan classes
were inspected in a course of visitations extending through
upwards of a year; and during that period every town, village,
and hamlet was systematically purged of that excess of
the lower orders which had been brought about by the neglect to pay
the tribute of Criminals to the Schools and University,
and by the violation of the other natural Laws of the Constitution
of Flatland. Thus the balance of classes was again restored.
Needless to say that henceforth the use of Colour was abolished,
and its possession prohibited. Even the utterance of any word
denoting Colour, except by the Circles or by qualified
scientific teachers, was punished by a severe penalty. Only at
our University in some of the very highest and most esoteric classes
-- which I myself have never been privileged to attend --
it is understood that the sparing use of Colour is still sanctioned
for the purpose of illustrating some of the deeper problems
of mathematics. But of this I can only speak from hearsay.
Elsewhere in Flatland, Colour is now non-existent. The art
of making it is known to only one living person, the Chief Circle
for the time being; and by him it is handed down on his death-bed
to none but his Successor. One manufactory alone produces it; and,
lest the secret should be betrayed, the Workmen are annually consumed,
and fresh ones introduced. So great is the terror with which even now
our Aristocracy looks back to the far-distant days of the agitation
for the Universal Colour Bill.
Section
11. Concerning our Priests
It is high time that I should pass from these brief and discursive
notes about things in Flatland to the central event of this book,
my initiation into the mysteries of Space. THAT is my subject;
all that has gone before is merely preface.
For this reason I must omit many matters of which the explanation
would not, I flatter myself, be without interest for my Readers:
as for example, our method of propelling and stopping ourselves,
although destitute of feet; the means by which we give fixity
to structures of wood, stone, or brick, although of course
we have no hands, nor can we lay foundations as you can,
nor avail ourselves of the lateral pressure of the earth;
the manner in which the rain originates in the intervals between
our various zones, so that the northern regions do not intercept
the moisture from falling on the southern; the nature of our
hills and mines, our trees and vegetables, our seasons and harvests;
our Alphabet and method of writing, adapted to our linear tablets;
these and a hundred other details of our physical existence I must
pass over, nor do I mention them now except to indicate to my readers
that their omission proceeds not from forgetfulness on the part of
the author, but from his regard for the time of the Reader.
Yet before I proceed to my legitimate subject some few
final remarks will no doubt be expected by my Readers upon those
pillars and mainstays of the Constitution of Flatland,
the controllers of our conduct and shapers of our destiny,
the objects of universal homage and almost of adoration:
need I say that I mean our Circles or Priests?
When I call them Priests, let me not be understood as meaning
no more than the term denotes with you. With us, our Priests
are Administrators of all Business, Art, and Science;
Directors of Trade, Commerce, Generalship, Architecture, Engineering,
Education, Statesmanship, Legislature, Morality, Theology;
doing nothing themselves, they are the Causes of everything
worth doing, that is done by others.
Although popularly everyone called a Circle is deemed a Circle,
yet among the better educated Classes it is known that no Circle
is really a Circle, but only a Polygon with a very large number
of very small sides. As the number of the sides increases,
a Polygon approximates to a Circle; and, when the number
is very great indeed, say for example three or four hundred,
it is extremely difficult for the most delicate touch to feel
any polygonal angles. Let me say rather, it WOULD be difficult:
for, as I have shown above, Recognition by Feeling is unknown
among the highest society, and to FEEL a Circle would be considered
a most audacious insult. This habit of abstention from Feeling
in the best society enables a Circle the more easily to sustain
the veil of mystery in which, from his earliest years, he is wont
to enwrap the exact nature of his Perimeter or Circumference.
Three feet being the average Perimeter it follows that,
in a Polygon of three hundred sides each side will be no more than
the hundredth part of a foot in length, or little more than the tenth
part of an inch; and in a Polygon of six or seven hundred sides
the sides are little larger than the diameter of a Spaceland pin-head.
It is always assumed, by courtesy, that the Chief Circle
for the time being has ten thousand sides.
The ascent of the posterity of the Circles in the social scale
is not restricted, as it is among the lower Regular classes,
by the Law of Nature which limits the increase of sides to one
in each generation. If it were so, the number of sides in a Circle
would be a mere question of pedigree and arithmetic,
and the four hundred and ninety-seventh descendant of
an Equilateral Triangle would necessarily be a Polygon with
five hundred sides. But this is not the case. Nature's Law
prescribes two antagonistic decrees affecting Circular propagation;
first, that as the race climbs higher in the scale of development,
so development shall proceed at an accelerated pace; second,
that in the same proportion, the race shall become less fertile.
Consequently in the home of a Polygon of four or five hundred sides
it is rare to find a son; more than one is never seen.
On the other hand the son of a five-hundred-sided Polygon has been
known to possess five hundred and fifty, or even six hundred sides.
Art also steps in to help the process of the higher Evolution.
Our physicians have discovered that the small and tender sides
of an infant Polygon of the higher class can be fractured,
and his whole frame re-set, with such exactness that a Polygon
of two or three hundred sides sometimes -- by no means always,
for the process is attended with serious risk -- but sometimes
overleaps two or three hundred generations, and as it were doubles
at a stroke, the number of his progenitors and the nobility
of his descent.
Many a promising child is sacrificed in this way. Scarcely one
out of ten survives. Yet so strong is the parental ambition
among those Polygons who are, as it were, on the fringe of
the Circular class, that it is very rare to find a Nobleman
of that position in society, who has neglected to place his first-born
in the Circular Neo-Therapeutic Gymnasium before he has attained
the age of a month.
One year determines success or failure. At the end of that time
the child has, in all probability, added one more to the tombstones
that crowd the Neo-Therapeutic Cemetery; but on rare occasions
a glad procession bears back the little one to his exultant parents,
no longer a Polygon, but a Circle, at least by courtesy:
and a single instance of so blessed a result induces multitudes
of Polygonal parents to submit to similar domestic sacrifices,
which have a dissimilar issue.
Section
12. Of the Doctrine of our Priests
As to the doctrine of the Circles it may briefly be summed up
in a single maxim, "Attend to your Configuration." Whether political,
ecclesiastical, or moral, all their teaching has for its object
the improvement of individual and collective Configuration --
with special reference of course to the Configuration of the Circles,
to which all other objects are subordinated.
It is the merit of the Circles that they have effectually suppressed
those ancient heresies which led men to waste energy and sympathy
in the vain belief that conduct depends upon will, effort, training,
encouragement, praise, or anything else but Configuration.
It was Pantocyclus -- the illustrious Circle mentioned above,
as the queller of the Colour Revolt -- who first convinced mankind
that Configuration makes the man; that if, for example, you are born
an Isosceles with two uneven sides, you will assuredly go wrong
unless you have them made even -- for which purpose you must go
to the Isosceles Hospital; similarly, if you are a Triangle,
or Square, or even a Polygon, born with any Irregularity,
you must be taken to one of the Regular Hospitals to have your
disease cured; otherwise you will end your days in the State Prison
or by the angle of the State Executioner.
All faults or defects, from the slightest misconduct to the most
flagitious crime, Pantocyclus attributed to some deviation from
perfect Regularity in the bodily figure, caused perhaps
(if not congenital) by some collision in a crowd; by neglect
to take exercise, or by taking too much of it; or even by a sudden
change of temperature, resulting in a shrinkage or expansion
in some too susceptible part of the frame. Therefore,
concluded that illustrious Philosopher, neither good conduct
nor bad conduct is a fit subject, in any sober estimation,
for either praise or blame. For why should you praise, for example,
the integrity of a Square who faithfully defends the interests
of his client, when you ought in reality rather to admire
the exact precision of his right angles? Or again, why blame a lying,
thievish Isosceles when you ought rather to deplore the incurable
inequality of his sides?
Theoretically, this doctrine is unquestionable; but it has
practical drawbacks. In dealing with an Isosceles, if a rascal pleads
that he cannot help stealing because of his unevenness,
you reply that for that very reason, because he cannot help being
a nuisance to his neighbours, you, the Magistrate, cannot help
sentencing him to be consumed -- and there's an end of the matter.
But in little domestic difficulties, where the penalty of consumption,
or death, is out of the question, this theory of Configuration
sometimes comes in awkwardly; and I must confess that occasionally
when one of my own Hexagonal Grandsons pleads as an excuse
for his disobedience that a sudden change of the temperature has been
too much for his Perimeter, and that I ought to lay the blame
not on him but on his Configuration, which can only be strengthened
by abundance of the choicest sweetmeats, I neither see my way
logically to reject, nor practically to accept, his conclusions.
For my own part, I find it best to assume that a good sound scolding
or castigation has some latent and strengthening influence on
my Grandson's Configuration; though I own that I have no grounds
for thinking so. At all events I am not alone in my way
of extricating myself from this dilemma; for I find that many
of the highest Circles, sitting as Judges in law courts,
use praise and blame towards Regular and Irregular Figures;
and in their homes I know by experience that, when scolding
their children, they speak about "right" or "wrong" as vehemently
and passionately as if they believed that these names represented
real existences, and that a human Figure is really capable
of choosing between them.
Constantly carrying out their policy of making Configuration
the leading idea in every mind, the Circles reverse the nature
of that Commandment which in Spaceland regulates the relations
between parents and children. With you, children are taught
to honour their parents; with us -- next to the Circles,
who are the chief object of universal homage -- a man is taught
to honour his Grandson, if he has one; or, if not, his Son.
By "honour", however, is by no means meant "indulgence",
but a reverent regard for their highest interests: and the Circles
teach that the duty of fathers is to subordinate their own interests
to those of posterity, thereby advancing the welfare of
the whole State as well as that of their own immediate descendants.
The weak point in the system of the Circles -- if a humble Square
may venture to speak of anything Circular as containing
any element of weakness -- appears to me to be found
in their relations with Women.
As it is of the utmost importance for Society that Irregular births
should be discouraged, it follows that no Woman who has
any Irregularities in her ancestry is a fit partner for one
who desires that his posterity should rise by regular degrees
in the social scale.
Now the Irregularity of a Male is a matter of measurement;
but as all Women are straight, and therefore visibly Regular
so to speak, one has to devise some other means of ascertaining
what I may call their invisible Irregularity, that is to say
their potential Irregularities as regards possible offspring.
This is effected by carefully-kept pedigrees, which are preserved
and supervised by the State; and without a certified pedigree
no Woman is allowed to marry.
Now it might have been supposed that a Circle -- proud of his ancestry
and regardful for a posterity which might possibly issue hereafter
in a Chief Circle -- would be more careful than any other to choose
a wife who had no blot on her escutcheon. But it is not so.
The care in choosing a Regular wife appears to diminish as one rises
in the social scale. Nothing would induce an aspiring Isosceles,
who had hopes of generating an Equilateral Son, to take a wife
who reckoned a single Irregularity among her Ancestors;
a Square or Pentagon, who is confident that his family is steadily
on the rise, does not inquire above the five-hundredth generation;
a Hexagon or Dodecagon is even more careless of the wife's pedigree;
but a Circle has been known deliberately to take a wife
who has had an Irregular Great-Grandfather, and all because
of some slight superiority of lustre, or because of the charms
of a low voice -- which, with us, even more than you,
is thought "an excellent thing in Woman".
Such ill-judged marriages are, as might be expected, barren,
if they do not result in positive Irregularity or in
diminution of sides; but none of these evils have hitherto proved
sufficiently deterrent. The loss of a few sides in a highly-developed
Polygon is not easily noticed, and is sometimes compensated
by a successful operation in the Neo-Therapeutic Gymnasium,
as I have described above; and the Circles are too much disposed
to acquiesce in infecundity as a Law of the superior development.
Yet, if this evil be not arrested, the gradual diminution
of the Circular class may soon become more rapid, and the time
may be not far distant when, the race being no longer able to produce
a Chief Circle, the Constitution of Flatland must fall.
One other word of warning suggests itself to me, though I cannot
so easily mention a remedy; and this also refers to our relations
with Women. About three hundred years ago, it was decreed by
the Chief Circle that, since women are deficient in Reason
but abundant in Emotion, they ought no longer to be treated
as rational, nor receive any mental education. The consequence
was that they were no longer taught to read, nor even to master
Arithmetic enough to enable them to count the angles of their husband
or children; and hence they sensibly declined during each generation
in intellectual power. And this system of female non-education
or quietism still prevails.
My fear is that, with the best intentions, this policy has been
carried so far as to react injuriously on the Male Sex.
For the consequence is that, as things now are, we Males have to lead
a kind of bi-lingual, and I may almost say bi-mental, existence.
With Women, we speak of "love", "duty", "right", "wrong", "pity",
"hope", and other irrational and emotional conceptions,
which have no existence, and the fiction of which has no object
except to control feminine exuberances; but among ourselves,
and in our books, we have an entirely different vocabulary
and I may almost say, idiom. "Love" then becomes "the anticipation
of benefits"; "duty" becomes "necessity" or "fitness"; and other words
are correspondingly transmuted. Moreover, among Women,
we use language implying the utmost deference for their Sex;
and they fully believe that the Chief Circle Himself is not more
devoutly adored by us than they are: but behind their backs they are
both regarded and spoken of -- by all except the very young --
as being little better than "mindless organisms".
Our Theology also in the Women's chambers is entirely different from
our Theology elsewhere.
Now my humble fear is that this double training, in language as well
as in thought, imposes somewhat too heavy a burden upon the young,
especially when, at the age of three years old, they are taken
from the maternal care and taught to unlearn the old language --
except for the purpose of repeating it in the presence of
their Mothers and Nurses -- and to learn the vocabulary and idiom
of science. Already methinks I discern a weakness in the grasp of
mathematical truth at the present time as compared with
the more robust intellect of our ancestors three hundred years ago.
I say nothing of the possible danger if a Woman should ever
surreptitiously learn to read and convey to her Sex the result
of her perusal of a single popular volume; nor of the possibility
that the indiscretion or disobedience of some infant Male
might reveal to a Mother the secrets of the logical dialect.
On the simple ground of the enfeebling of the Male intellect,
I rest this humble appeal to the highest Authorities to reconsider
the regulations of Female education.
PART II: OTHER WORLDS
"O brave new worlds, that have such people in them!"
Section 13. How I had a Vision of Lineland
It was the last day but one of the 1999th year of our era,
and the first day of the Long Vacation. Having amused myself
till a late hour with my favourite recreation of Geometry,
I had retired to rest with an unsolved problem in my mind.
In the night I had a dream.
I saw before me a vast multitude of small Straight Lines
(which I naturally assumed to be Women) interspersed with other Beings
still smaller and of the nature of lustrous points -- all moving
to and fro in one and the same Straight Line, and, as nearly as I
could judge, with the same velocity.
A noise of confused, multitudinous chirping or twittering
issued from them at intervals as long as they were moving;
but sometimes they ceased from motion, and then all was silence.
Approaching one of the largest of what I thought to be Women,
I accosted her, but received no answer. A second and a third appeal
on my part were equally ineffectual. Losing patience at what
appeared to me intolerable rudeness, I brought my mouth
into a position full in front of her mouth so as to intercept
her motion, and loudly repeated my question, "Woman, what signifies
this concourse, and this strange and confused chirping,
and this monotonous motion to and fro in one and the same
Straight Line?"
<>
<>
My view of Lineland
---------
| |
| Myself|
| |
My eye o--------
Women A boy Men The KING Men A boy Women
+ + + + - --- -- -- -- -- (>----<) -- -- -- -- --- - + + + +
^ ^
The KING'S eyes
much larger than the reality
shewing that HIS MAJESTY
could see nothing but a point.
"I am no Woman," replied the small Line. "I am the Monarch
of the world. But thou, whence intrudest thou into my realm
of Lineland?" Receiving this abrupt reply, I begged pardon
if I had in any way startled or molested his Royal Highness;
and describing myself as a stranger I besought the King to give me
some account of his dominions. But I had the greatest possible
difficulty in obtaining any information on points that really
interested me; for the Monarch could not refrain from constantly
assuming that whatever was familiar to him must also be known to me
and that I was simulating ignorance in jest. However,
by persevering questions I elicited the following facts:
It seemed that this poor ignorant Monarch -- as he called himself --
was persuaded that the Straight Line which he called his Kingdom,
and in which he passed his existence, constituted the whole
of the world, and indeed the whole of Space. Not being able either
to move or to see, save in his Straight Line, he had no conception
of anything out of it. Though he had heard my voice when I first
addressed him, the sounds had come to him in a manner so contrary
to his experience that he had made no answer, "seeing no man",
as he expressed it, "and hearing a voice as it were from
my own intestines." Until the moment when I placed my mouth
in his World, he had neither seen me, nor heard anything except
confused sounds beating against -- what I called his side,
but what he called his INSIDE or STOMACH; nor had he even now
the least conception of the region from which I had come.
Outside his World, or Line, all was a blank to him; nay,
not even a blank, for a blank implies Space; say, rather,
all was non-existent.
His subjects -- of whom the small Lines were men and the Points Women
-- were all alike confined in motion and eye-sight to that single
Straight Line, which was their World. It need scarcely be added that
the whole of their horizon was limited to a Point; nor could any one
ever see anything but a Point. Man, woman, child, thing -- each was
a Point to the eye of a Linelander. Only by the sound of the voice
could sex or age be distinguished. Moreover, as each individual
occupied the whole of the narrow path, so to speak, which constituted
his Universe, and no one could move to the right or left
to make way for passers by, it followed that no Linelander
could ever pass another. Once neighbours, always neighbours.
Neighbourhood with them was like marriage with us.
Neighbours remained neighbours till death did them part.
Such a life, with all vision limited to a Point, and all motion
to a Straight Line, seemed to me inexpressibly dreary; and I was
surprised to note the vivacity and cheerfulness of the King.
Wondering whether it was possible, amid circumstances so unfavourable
to domestic relations, to enjoy the pleasures of conjugal union,
I hesitated for some time to question his Royal Highness
on so delicate a subject; but at last I plunged into it
by abruptly inquiring as to the health of his family.
"My wives and children," he replied, "are well and happy."
Staggered at this answer -- for in the immediate proximity
of the Monarch (as I had noted in my dream before I entered Lineland)
there were none but Men -- I ventured to reply, "Pardon me,
but I cannot imagine how your Royal Highness can at any time either
see or approach their Majesties, when there are at least half a dozen
intervening individuals, whom you can neither see through,
nor pass by? Is it possible that in Lineland proximity is not
necessary for marriage and for the generation of children?"
"How can you ask so absurd a question?" replied the Monarch.
"If it were indeed as you suggest, the Universe would soon
be depopulated. No, no; neighbourhood is needless for the union
of hearts; and the birth of children is too important a matter
to have been allowed to depend upon such an accident as proximity.
You cannot be ignorant of this. Yet since you are pleased
to affect ignorance, I will instruct you as if you were the veriest
baby in Lineland. Know, then, that marriages are consummated
by means of the faculty of sound and the sense of hearing.
"You are of course aware that every Man has two mouths or voices
-- as well as two eyes -- a bass at one and a tenor at the other
of his extremities. I should not mention this, but that I have been
unable to distinguish your tenor in the course of our conversation."
I replied that I had but one voice, and that I had not been aware
that his Royal Highness had two. "That confirms my impression,"
said the King, "that you are not a Man, but a feminine Monstrosity
with a bass voice, and an utterly uneducated ear. But to continue.
"Nature having herself ordained that every Man should wed two wives --"
"Why two?" asked I. "You carry your affected simplicity too far",
he cried. "How can there be a completely harmonious union
without the combination of the Four in One, viz. the Bass and Tenor
of the Man and the Soprano and Contralto of the two Women?"
"But supposing," said I, "that a man should prefer one wife or three?"
"It is impossible," he said; "it is as inconceivable as that
two and one should make five, or that the human eye should see
a Straight Line." I would have interrupted him; but he proceeded
as follows:
"Once in the middle of each week a Law of Nature compels us
to move to and fro with a rhythmic motion of more than usual violence,
which continues for the time you would take to count
a hundred and one. In the midst of this choral dance,
at the fifty-first pulsation, the inhabitants of the Universe
pause in full career, and each individual sends forth his richest,
fullest, sweetest strain. It is in this decisive moment
that all our marriages are made. So exquisite is the adaptation
of Bass to Treble, of Tenor to Contralto, that oftentimes
the Loved Ones, though twenty thousand leagues away,
recognize at once the responsive note of their destined Lover; and,
penetrating the paltry obstacles of distance, Love unites the three.
The marriage in that instant consummated results in a threefold
Male and Female offspring which takes its place in Lineland."
"What! Always threefold?" said I. "Must one wife then
always have 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 ceased, speechless for fury; and some time elapsed before
I could induce him to resume his narrative.
"You will not, of course, suppose that every bachelor among us
finds his mates at the first wooing in this universal Marriage Chorus.
On the contrary, the process is by most of us many times repeated.
Few are the hearts whose happy lot it is at once to recognize
in each other's voices the partner intended for them by Providence,
and to fly into a reciprocal and perfectly harmonious embrace.
With most of us the courtship is of long duration. The Wooer's voices
may perhaps accord with one of the future wives, but not with both;
or not, at first, with either; or the Soprano and Contralto
may not quite harmonize. In such cases Nature has provided that
every weekly Chorus shall bring the three Lovers into closer harmony.
Each trial of voice, each fresh discovery of discord,
almost imperceptibly induces the less perfect to modify
his or her vocal utterance so as to approximate to the more perfect.
And after many trials and many approximations, the result is
at last achieved. There comes a day at last, when, while the wonted
Marriage Chorus goes forth from universal Lineland, the three
far-off Lovers suddenly find themselves in exact harmony, and,
before they are awake, the wedded Triplet is rapt vocally
into a duplicate embrace; and Nature rejoices over one more marriage
and over three more births."
Section
14. How I vainly tried to explain the nature of Flatland
Thinking that it was time to bring down the Monarch from his raptures
to the level of common sense, I determined to endeavour to
open up to him some glimpses of the truth, that is to say
of the nature of things in Flatland. So I began thus:
"How does your Royal Highness distinguish the shapes and positions
of his subjects? I for my part noticed by the sense of sight,
before I entered your Kingdom, that some of your people are Lines
and others Points, and that some of the Lines are larger --"
"You speak of an impossibility," interrupted the King;
"you must have seen a vision; for to detect the difference between
a Line and a Point by the sense of sight is, as every one knows,
in the nature of things, impossible; but it can be detected by
the sense of hearing, and by the same means my shape can be
exactly ascertained. Behold me -- I am a Line, the longest
in Lineland, over six inches of Space --" "Of Length",
I ventured to suggest. "Fool," said he, "Space is Length.
Interrupt me again, and I have done."
I apologized; but he continued scornfully, "Since you are impervious
to argument, you shall hear with your ears how by means of
my two voices I reveal my shape to my Wives, who are at this moment
six thousand miles seventy yards two feet eight inches away, the one
to the North, the other to the South. Listen, I call to them."
He chirruped, and then complacently continued: "My wives at this
moment receiving the sound of one of my voices, closely followed by
the other, and perceiving that the latter reaches them after
an interval in which sound can traverse 6.457 inches, infer that one
of my mouths is 6.457 inches further from them than the other,
and accordingly know my shape to be 6.457 inches. But you will
of course understand that my wives do not make this calculation
every time they hear my two voices. They made it, once for all,
before we were married. But they COULD make it at any time.
And in the same way I can estimate the shape of any of
my Male subjects by the sense of sound."
"But how," said I, "if a Man feigns a Woman's voice with one of
his two voices, or so disguises his Southern voice that it cannot
be recognized as the echo of the Northern? May not such deceptions
cause great inconvenience? And have you no means of checking frauds
of this kind by commanding your neighbouring subjects to feel
one another?" This of course was a very stupid question,
for feeling could not have answered the purpose; but I asked
with the view of irritating the Monarch, and I succeeded perfectly.
"What!" cried he in horror, "explain your meaning." "Feel, touch,
come into contact," I replied. "If you mean by FEELING,"
said the King, "approaching so close as to leave no space
between two individuals, know, Stranger, that this offence
is punishable in my dominions by death. And the reason is obvious.
The frail form of a Woman, being liable to be shattered
by such an approximation, must be preserved by the State;
but since Women cannot be distinguished by the sense of sight
from Men, the Law ordains universally that neither Man nor Woman
shall be approached so closely as to destroy the interval
between the approximator and the approximated.
"And indeed what possible purpose would be served by this illegal
and unnatural excess of approximation which you call TOUCHING,
when all the ends of so brutal and coarse a process are attained
at once more easily and more exactly by the sense of hearing?
As to your suggested danger of deception, it is non-existent:
for the Voice, being the essence of one's Being, cannot be thus
changed at will. But come, suppose that I had the power of passing
through solid things, so that I could penetrate my subjects,
one after another, even to the number of a billion, verifying the size
and distance of each by the sense of FEELING: how much time
and energy would be wasted in this clumsy and inaccurate method!
Whereas now, in one moment of audition, I take as it were the census
and statistics, local, corporeal, mental and spiritual,
of every living being in Lineland. Hark, only hark!"
So saying he paused and listened, as if in an ecstasy,
to a sound which seemed to me no better than a tiny chirping
from an innumerable multitude of lilliputian grasshoppers.
"Truly," replied I, "your sense of hearing serves you in good stead,
and fills up many of your deficiencies. But permit me to point out
that your life in Lineland must be deplorably dull. To see nothing
but a Point! Not even to be able to contemplate a Straight Line!
Nay, not even to know what a Straight Line is! To see, yet be cut off
from those Linear prospects which are vouchsafed to us in Flatland!
Better surely to have no sense of sight at all than to see so little!
I grant you I have not your discriminative faculty of hearing;
for the concert of all Lineland which gives you such intense pleasure,
is to me no better than a multitudinous twittering or chirping.
But at least I can discern, by sight, a Line from a Point.
And let me prove it. Just before I came into your kingdom,
I saw you dancing from left to right, and then from right to left,
with Seven Men and a Woman in your immediate proximity on the left,
and eight Men and two Women on your right. Is not this correct?"
"It is correct," said the King, "so far as the numbers and sexes
are concerned, though I know not what you mean by 'right' and 'left'.
But I deny that you saw these things. For how could you see the Line,
that is to say the inside, of any Man? But you must have
heard these things, and then dreamed that you saw them.
And let me ask what you mean by those words 'left' and 'right'.
I suppose it is your way of saying Northward and Southward."
"Not so," replied I; "besides your motion of Northward and Southward,
there is another motion which I call from right to left."
KING. Exhibit to me, if you please, this motion from left to right.
I. Nay, that I cannot do, unless you could step out
of your Line altogether.
KING. Out of my Line? Do you mean out of the world? Out of Space?
I. Well, yes. Out of YOUR World. Out of YOUR Space.
For your Space is not the true Space. True Space is a Plane;
but your Space is only a Line.
KING. If you cannot indicate this motion from left to right by
yourself moving in it, then I beg you to describe it to me in words.
I. If you cannot tell your right side from your left,
I fear that no words of mine can make my meaning clear to you.
But surely you cannot be ignorant of so simple a distinction.
KING. I do not in the least understand you.
I. Alas! How shall I make it clear? When you move straight on,
does it not sometimes occur to you that you COULD move
in some other way, turning your eye round so as to look
in the direction towards which your side is now fronting?
In other words, instead of always moving in the direction
of one of your extremities, do you never feel a desire to move
in the direction, so to speak, of your side?
KING. Never. And what do you mean? How can a man's inside
"front" in any direction? Or how can a man move in the direction
of his inside?
I. Well then, since words cannot explain the matter,
I will try deeds, and will move gradually out of Lineland
in the direction which I desire to indicate to you.
At the word I began to move my body out of Lineland.
As long as any part of me remained in his dominion and in his view,
the King kept exclaiming, "I see you, I see you still;
you are not moving." But when I had at last moved myself
out of his Line, he cried in his shrillest voice, "She is vanished;
she is dead." "I am not dead," replied I; "I am simply
out of Lineland, that is to say, out of the Straight Line
which you call Space, and in the true Space, where I can see things
as they are. And at this moment I can see your Line, or side --
or inside as you are pleased to call it; and I can see also the Men
and Women on the North and South of you, whom I will now enumerate,
describing their order, their size, and the interval between each."
<>
<>
My body just before I disappeared
+---------+
|\ \ \ \ \|
|\ \ \ \ \|
|\ \ \ \ \|
Lineland ----> |\ \ \ \ \| The King
--------------------+---------+--------------========
When I had done this at great length, I cried triumphantly,
"Does that at last convince you?" And, with that, I once more
entered Lineland, taking up the same position as before.
But the Monarch replied, "If you were a Man of sense -- though,
as you appear to have only one voice I have little doubt
you are not a Man but a Woman -- but, if you had a particle of sense,
you would listen to reason. You ask me to believe that there is
another Line besides that which my senses indicate, and another motion
besides that of which I am daily conscious. I, in return,
ask you to describe in words or indicate by motion that other Line
of which you speak. Instead of moving, you merely exercise
some magic art of vanishing and returning to sight; and instead of
any lucid description of your new World, you simply tell me
the numbers and sizes of some forty of my retinue, facts known
to any child in my capital. Can anything be more irrational
or audacious? Acknowledge your folly or depart from my dominions."
Furious at his perversity, and especially indignant that he professed
to be ignorant of my sex, I retorted in no measured terms,
"Besotted Being! You think yourself the perfection of existence,
while you are in reality the most imperfect and imbecile.
You profess to see, whereas you can see nothing but a Point!
You plume yourself on inferring the existence of a Straight Line;
but I CAN SEE Straight Lines, and infer the existence of Angles,
Triangles, Squares, Pentagons, Hexagons, and even Circles.
Why waste more words? Suffice it that I am the completion
of your incomplete self. You are a Line, but I am a Line of Lines,
called in my country a Square: and even I, infinitely superior
though I am to you, am of little account among the great nobles
of Flatland, whence I have come to visit you, in the hope of
enlightening your ignorance."
Hearing these words the King advanced towards me with a menacing cry
as if to pierce me through the diagonal; and in that same moment
there arose from myriads of his subjects a multitudinous war-cry,
increasing in vehemence till at last methought it rivalled
the roar of an army of a hundred thousand Isosceles, and the artillery
of a thousand Pentagons. Spell-bound and motionless,
I could neither speak nor move to avert the impending destruction;
and still the noise grew louder, and the King came closer,
when I awoke to find the breakfast-bell recalling me to
the realities of Flatland.
Section
15. Concerning a Stranger from Spaceland
From dreams I proceed to facts.
It was the last day of the 1999th year of our era.
The pattering of the rain had long ago announced nightfall;
and I was sitting in the company of my wife, musing on the events
of the past and the prospects of the coming year, the coming century,
the coming Millennium.
[Note: When I say "sitting", of course I do not mean
any change of attitude such as you in Spaceland signify by that word;
for as we have no feet, we can no more "sit" nor "stand"
(in your sense of the word) than one of your soles or flounders.
Nevertheless, we perfectly well recognize the different mental states
of volition implied in "lying", "sitting", and "standing",
which are to some extent indicated to a beholder by a slight
increase of lustre corresponding to the increase of volition.
But on this, and a thousand other kindred subjects, time forbids me
to dwell.]
My four Sons and two orphan Grandchildren had retired
to their several apartments; and my wife alone remained with me
to see the old Millennium out and the new one in.
I was rapt in thought, pondering in my mind some words that had
casually issued from the mouth of my youngest Grandson,
a most promising young Hexagon of unusual brilliancy
and perfect angularity. His uncles and I had been giving him
his usual practical lesson in Sight Recognition, turning ourselves
upon our centres, now rapidly, now more slowly, and questioning him
as to our positions; and his answers had been so satisfactory
that I had been induced to reward him by giving him a few hints
on Arithmetic, as applied to Geometry.
Taking nine Squares, each an inch every way, I had put them together
so as to make one large Square, with a side of three inches,
and I had hence proved to my little Grandson that -- though it was
impossible for us to SEE the inside of the Square --
yet we might ascertain the number of square inches in a Square
by simply squaring the number of inches in the side: "and thus,"
said I, "we know that 3^2, or 9, represents the number
of square inches in a Square whose side is 3 inches long."
The little Hexagon meditated on this a while and then said to me;
"But you have been teaching me to raise numbers to the third power:
I suppose 3^3 must mean something in Geometry; what does it mean?"
"Nothing at all," replied I, "not at least in Geometry;
for Geometry has only Two Dimensions." And then I began
to shew the boy how a Point by moving through a length of three inches
makes a Line of three inches, which may be represented by 3;
and how a Line of three inches, moving parallel to itself through
a length of three inches, makes a Square of three inches every way,
which may be represented by 3^2.
Upon this, my Grandson, again returning to his former suggestion,
took me up rather suddenly and exclaimed, "Well, then,
if a Point by moving three inches, makes a Line of three inches
represented by 3; and if a straight Line of three inches,
moving parallel to itself, makes a Square of three inches every way,
represented by 3^2; 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 3^3."
"Go to bed," said I, a little ruffled by this interruption:
"if you would talk less nonsense, you would remember more sense."
So my Grandson had disappeared in disgrace; and there I sat
by my Wife's side, endeavouring to form a retrospect of the year 1999
and of the possibilities of the year 2000, but not quite able
to shake off the thoughts suggested by the prattle of my bright
little Hexagon. Only a few sands now remained in the half-hour glass.
Rousing myself from my reverie I turned the glass Northward
for the last time in the old Millennium; and in the act,
I exclaimed aloud, "The boy is a fool."
Straightway I became conscious of a Presence in the room,
and a chilling breath thrilled through my very being.
"He is no such thing," cried my Wife, "and you are breaking
the Commandments in thus dishonouring your own Grandson."
But I took no notice of her. Looking round in every direction
I could see nothing; yet still I FELT a Presence, and shivered
as the cold whisper came again. I started up. "What is the matter?"
said my Wife, "there is no draught; what are you looking for?
There is nothing." There was nothing; and I resumed my seat,
again exclaiming, "The boy is a fool, I say; 3^3 can have no meaning
in Geometry." At once there came a distinctly audible reply,
"The boy is not a fool; and 3^3 has an obvious Geometrical meaning."
My Wife as well as myself heard the words, although she did not
understand their meaning, and both of us sprang forward
in the direction of the sound. What was our horror when we saw
before us a Figure! At the first glance it appeared to be a Woman,
seen sideways; but a moment's observation shewed me that
the extremities passed into dimness too rapidly to represent
one of the Female Sex; and I should have thought it a Circle,
only that it seemed to change its size in a manner impossible
for a Circle or for any regular Figure of which I had had experience.
But my Wife had not my experience, nor the coolness necessary to note
these characteristics. With the usual hastiness and unreasoning
jealousy of her Sex, she flew at once to the conclusion
that a Woman had entered the house through some small aperture.
"How comes this person here?" she exclaimed, "you promised me,
my dear, that there should be no ventilators in our new house."
"Nor are there any," said I; "but what makes you think that
the stranger is a Woman? I see by my power of Sight Recognition ----"
"Oh, I have no patience with your Sight Recognition," replied she,
"'Feeling is believing' and 'A Straight Line to the touch is worth
a Circle to the sight'" -- two Proverbs, very common
with the Frailer Sex in Flatland.
"Well," said I, for I was afraid of irritating her, "if it must be so,
demand an introduction." Assuming her most gracious manner,
my Wife advanced towards the Stranger, "Permit me, Madam,
to feel and be felt by ----" then, suddenly recoiling, "Oh!
it is not a Woman, and there are no angles either, not a trace of one.
Can it be that I have so misbehaved to a perfect Circle?"
"I am indeed, in a certain sense a Circle," replied the Voice,
"and a more perfect Circle than any in Flatland; but to speak
more accurately, I am many Circles in one." Then he added
more mildly, "I have a message, dear Madam, to your husband,
which I must not deliver in your presence; and, if you would suffer us
to retire for a few minutes ----" But my Wife would not listen
to the proposal that our august Visitor should so incommode himself,
and assuring the Circle that the hour of her own retirement
had long passed, with many reiterated apologies for her
recent indiscretion, she at last retreated to her apartment.
I glanced at the half-hour glass. The last sands had fallen.
The third Millennium had begun.
Section
16. How the Stranger vainly endeavoured to reveal to me
in words the mysteries of Spaceland
As soon as the sound of the Peace-cry of my departing Wife
had died away, I began to approach the Stranger with the intention
of taking a nearer view and of bidding him be seated:
but his appearance struck me dumb and motionless with astonishment.
Without the slightest symptoms of angularity he nevertheless varied
every instant with gradations of size and brightness scarcely possible
for any Figure within the scope of my experience. The thought
flashed across me that I might have before me a burglar or cut-throat,
some monstrous Irregular Isosceles, who, by feigning the voice
of a Circle, had obtained admission somehow into the house,
and was now preparing to stab me with his acute angle.
In a sitting-room, the absence of Fog (and the season happened
to be remarkably dry), made it difficult for me to trust to
Sight Recognition, especially at the short distance at which
I was standing. Desperate with fear, I rushed forward
with an unceremonious, "You must permit me, Sir --" and felt him.
My Wife was right. There was not the trace of an angle,
not the slightest roughness or inequality: never in my life had I met
with a more perfect Circle. He remained motionless while I walked
round him, beginning from his eye and returning to it again.
Circular he was throughout, a perfectly satisfactory Circle;
there could not be a doubt of it. Then followed a dialogue,
which I will endeavour to set down as near as I can recollect it,
omitting only some of my profuse apologies -- for I was covered
with shame and humiliation that I, a Square, should have been guilty
of the impertinence of feeling a Circle. It was commenced
by the Stranger with some impatience at the lengthiness
of my introductory process.
STRANGER. Have you felt me enough by this time? Are you not
introduced to me yet?
I. Most illustrious Sir, excuse my awkwardness, which arises not
from ignorance of the usages of polite society, but from a little
surprise and nervousness, consequent on this somewhat
unexpected visit. And I beseech you to reveal my indiscretion
to no one, and especially not to my Wife. But before your Lordship
enters into further communications, would he deign to satisfy
the curiosity of one who would gladly know whence his Visitor came?
STRANGER. From Space, from Space, Sir: whence else?
I. Pardon me, my Lord, but is not your Lordship already in Space,
your Lordship and his humble servant, even at this moment?
STRANGER. Pooh! what do you know of Space? Define Space.
I. Space, my Lord, is height and breadth indefinitely prolonged.
STRANGER. Exactly: you see you do not even know what Space is.
You think it is of Two Dimensions only; but I have come
to announce to you a Third -- height, breadth, and length.
I. Your Lordship is pleased to be merry. We also speak
of length and height, or breadth and thickness, thus denoting
Two Dimensions by four names.
STRANGER. But I mean not only three names, but Three Dimensions.
I. Would your Lordship indicate or explain to me in what direction
is the Third Dimension, unknown to me?
STRANGER. I came from it. It is up above and down below.
I. My Lord means seemingly that it is Northward and Southward.
STRANGER. I mean nothing of the kind. I mean a direction in which
you cannot look, because you have no eye in your side.
I. Pardon me, my Lord, a moment's inspection will convince
your Lordship that I have a perfect luminary at the juncture of two
of my sides.
STRANGER. Yes: but in order to see into Space you ought to have
an eye, not on your Perimeter, but on your side, that is,
on what you would probably call your inside; but we in Spaceland
should call it your side.
I. An eye in my inside! An eye in my stomach! Your Lordship jests.
STRANGER. I am in no jesting humour. I tell you that
I come from Space, or, since you will not understand what Space means,
from the Land of Three Dimensions whence I but lately looked down
upon your Plane which you call Space forsooth. From that position
of advantage I discerned all that you speak of as SOLID
(by which you mean "enclosed on four sides"), your houses,
your churches, your very chests and safes, yes even your insides
and stomachs, all lying open and exposed to my view.
I. Such assertions are easily made, my Lord.
STRANGER. But not easily proved, you mean. But I mean to prove mine.
When I descended here, I saw your four Sons, the Pentagons,
each in his apartment, and your two Grandsons the Hexagons;
I saw your youngest Hexagon remain a while with you and then
retire to his room, leaving you and your Wife alone.
I saw your Isosceles servants, three in number, in the kitchen
at supper, and the little Page in the scullery. Then I came here,
and how do you think I came?
I. Through the roof, I suppose.
STRANGER. Not so. Your roof, as you know very well,
has been recently repaired, and has no aperture by which even a Woman
could penetrate. I tell you I come from Space. Are you not convinced
by what I have told you of your children and household?
I. Your Lordship must be aware that such facts touching
the belongings of his humble servant might be easily ascertained
by any one in the neighbourhood possessing your Lordship's
ample means of obtaining information.
STRANGER. (TO HIMSELF.) What must I do? Stay; one more argument
suggests itself to me. When you see a Straight Line -- your wife,
for example -- how many Dimensions do you attribute to her?
I. Your Lordship would treat me as if I were one of the vulgar who,
being ignorant of Mathematics, suppose that a Woman is really
a Straight Line, and only of One Dimension. No, no, my Lord;
we Squares are better advised, and are as well aware as your Lordship
that a Woman, though popularly called a Straight Line, is,
really and scientifically, a very thin Parallelogram,
possessing Two Dimensions, like the rest of us, viz.,
length and breadth (or thickness).
STRANGER. But the very fact that a Line is visible implies
that it possesses yet another Dimension.
I. My Lord, I have just acknowledged that a Woman is broad
as well as long. We see her length, we infer her breadth;
which, though very slight, is capable of measurement.
STRANGER. You do not understand me. I mean that when you see
a Woman, you ought -- besides inferring her breadth --
to see her length, and to SEE what we call her HEIGHT;
although that last Dimension is infinitesimal in your country.
If a Line were mere length without "height", it would cease to
occupy Space and would become invisible. Surely you must
recognize this?
I. I must indeed confess that I do not in the least
understand your Lordship. When we in Flatland see a Line,
we see length and BRIGHTNESS. If the brightness disappears,
the Line is extinguished, and, as you say, ceases to occupy Space.
But am I to suppose that your Lordship gives to brightness the title
of a Dimension, and that what we call "bright" you call "high"?
STRANGER. No, indeed. By "height" I mean a Dimension like
your length: only, with you, "height" is not so easily perceptible,
being extremely small.
I. My Lord, your assertion is easily put to the test.
You say I have a Third Dimension, which you call "height".
Now, Dimension implies direction and measurement. Do but measure
my "height", or merely indicate to me the direction in which
my "height" extends, and I will become your convert. Otherwise,
your Lordship's own understanding must hold me excused.
STRANGER. (TO HIMSELF.) I can do neither. How shall I
convince him? Surely a plain statement of facts followed by
ocular demonstration ought to suffice. -- Now, Sir; listen to me.
You are living on a Plane. What you style Flatland is
the vast level surface of what I may call a fluid, on, or in,
the top of which you and your countrymen move about,
without rising above it or falling below it.
I am not a plane Figure, but a Solid. You call me a Circle;
but in reality I am not a Circle, but an infinite number of Circles,
of size varying from a Point to a Circle of thirteen inches
in diameter, one placed on the top of the other. When I cut through
your plane as I am now doing, I make in your plane a section
which you, very rightly, call a Circle. For even a Sphere --
which is my proper name in my own country -- if he manifest himself
at all to an inhabitant of Flatland -- must needs manifest himself
as a Circle.
Do you not remember -- for I, who see all things, discerned last night
the phantasmal vision of Lineland written upon your brain --
do you not remember, I say, how, when you entered the realm
of Lineland, you were compelled to manifest yourself to the King,
not as a Square, but as a Line, because that Linear Realm had not
Dimensions enough to represent the whole of you, but only a slice
or section of you? In precisely the same way, your country
of Two Dimensions is not spacious enough to represent me,
a being of Three, but can only exhibit a slice or section of me,
which is what you call a Circle.
The diminished brightness of your eye indicates incredulity. But now
prepare to receive proof positive of the truth of my assertions.
You cannot indeed see more than one of my sections, or Circles,
at a time; for you have no power to raise your eye out of the plane
of Flatland; but you can at least see that, as I rise in Space,
so my sections become smaller. See now, I will rise; and the effect
upon your eye will be that my Circle will become smaller and smaller
till it dwindles to a point and finally vanishes.
<>
<>
The Sphere on the
point of vanishing
(2) __-----__
The Sphere with The Sphere rising / \ (3)
his section __-----__ / \
at full size / \ | |
__-----__ / \ | |
/ \ | | | |
/ __ - __ \ | | \ / My
| -- -- | | __ --- __ | \ __ __ / Eye
--|-----------------|----\--__-------__--/------------===----------+(>
| -- __ __ -- | \ __ --- __ /
\ - / -----
\ __ __ /
-----
There was no "rising" that I could see; but he diminished
and finally vanished. I winked once or twice to make sure
that I was not dreaming. But it was no dream. For from the depths
of nowhere came forth a hollow voice -- close to my heart it seemed --
"Am I quite gone? Are you convinced now? Well, now I will
gradually return to Flatland and you shall see my section become
larger and larger."
Every reader in Spaceland will easily understand that
my mysterious Guest was speaking the language of truth
and even of simplicity. But to me, proficient though I was
in Flatland Mathematics, it was by no means a simple matter.
The rough diagram given above will make it clear to any
Spaceland child that the Sphere, ascending in the three positions
indicated there, must needs have manifested himself to me,
or to any Flatlander, as a Circle, at first of full size, then small,
and at last very small indeed, approaching to a Point. But to me,
although I saw the facts before me, the causes were as dark as ever.
All that I could comprehend was, that the Circle had made himself
smaller and vanished, and that he had now reappeared and was rapidly
making himself larger.
When he regained his original size, he heaved a deep sigh;
for he perceived by my silence that I had altogether failed
to comprehend him. And indeed I was now inclining to the belief
that he must be no Circle at all, but some extremely clever juggler;
or else that the old wives' tales were true, and that after all
there were such people as Enchanters and Magicians.
After a long pause he muttered to himself, "One resource alone remains,
if I am not to resort to action. I must try the method of Analogy."
Then followed a still longer silence, after which he continued
our dialogue.
SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward,
and leaves a luminous wake, what name would you give to the wake?
I. A straight Line.
SPHERE. And a straight Line has how many extremities?
I. Two.
SPHERE. Now conceive the Northward straight Line moving parallel
to itself, East and West, so that every point in it leaves behind it
the wake of a straight Line. What name will you give to the Figure
thereby formed? We will suppose that it moves through a distance
equal to the original straight Line. -- What name, I say?
I. A Square.
SPHERE. And how many sides has a Square? How many angles?
I. Four sides and four angles.
SPHERE. Now stretch your imagination a little, and conceive
a Square in Flatland, moving parallel to itself upward.
I. What? Northward?
SPHERE. No, not Northward; upward; out of Flatland altogether.
If it moved Northward, the Southern points in the Square would have to
move through the positions previously occupied by the Northern points.
But that is not my meaning.
I mean that every Point in you -- for you are a Square and will serve
the purpose of my illustration -- every Point in you, that is to say
in what you call your inside, is to pass upwards through Space
in such a way that no Point shall pass through the position
previously occupied by any other Point; but each Point shall describe
a straight Line of its own. This is all in accordance with Analogy;
surely it must be clear to you.
Restraining my impatience -- for I was now under a strong temptation
to rush blindly at my Visitor and to precipitate him into Space,
or out of Flatland, anywhere, so that I could get rid of him --
I replied: --
"And what may be the nature of the Figure which I am to shape out
by this motion which you are pleased to denote by the word 'upward'?
I presume it is describable in the language of Flatland."
SPHERE. Oh, certainly. It is all plain and simple,
and in strict accordance with Analogy -- only, by the way,
you must not speak of the result as being a Figure, but as a Solid.
But I will describe it to you. Or rather not I, but Analogy.
We began with a single Point, which of course -- being itself a Point
-- has only ONE terminal Point.
One Point produces a Line with TWO terminal Points.
One Line produces a Square with FOUR terminal Points.
Now you can give yourself the answer to your own question: 1, 2, 4,
are evidently in Geometrical Progression. What is the next number?
I. Eight.
SPHERE. Exactly. The one Square produces a SOMETHING-WHICH-
YOU-DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE
with EIGHT terminal Points. Now are you convinced?
I. And has this Creature sides, as well as angles or what you call
"terminal Points"?
SPHERE. Of course; and all according to Analogy. But, by the way,
not what YOU call sides, but what WE call sides.
You would call them SOLIDS.
I. And how many solids or sides will appertain to this Being whom
I am to generate by the motion of my inside in an "upward" direction,
and whom you call a Cube?
SPHERE. How can you ask? And you a mathematician!
The side of anything is always, if I may so say, one Dimension behind
the thing. Consequently, as there is no Dimension behind a Point,
a Point has 0 sides; a Line, if I may say, has 2 sides
(for the Points of a Line may be called by courtesy, its sides);
a Square has 4 sides; 0, 2, 4; what Progression do you call that?
I. Arithmetical.
SPHERE. And what is the next number?
I. Six.
SPHERE. Exactly. Then you see you have answered your own question.
The Cube which you will generate will be bounded by six sides,
that is to say, six of your insides. You see it all now, eh?
"Monster," I shrieked, "be thou juggler, enchanter, dream, or devil,
no more will I endure thy mockeries. Either thou or I must perish."
And saying these words I precipitated myself upon him.
Section 17. How the Sphere, having in vain tried words,
resorted to deeds
It was in vain. I brought my hardest right angle into violent
collision with the Stranger, pressing on him with a force sufficient
to have destroyed any ordinary Circle: but I could feel him
slowly and unarrestably slipping from my contact; no edging to
the right nor to the left, but moving somehow out of the world,
and vanishing to nothing. Soon there was a blank. But still I heard
the Intruder's voice.
SPHERE. Why will you refuse to listen to reason?
I had hoped to find in you -- as being a man of sense
and an accomplished mathematician -- a fit apostle for the Gospel
of the Three Dimensions, which I am allowed to preach once only
in a thousand years: but now I know not how to convince you.
Stay, I have it. Deeds, and not words, shall proclaim the truth.
Listen, my friend.
I have told you I can see from my position in Space the inside
of all things that you consider closed. For example,
I see in yonder cupboard near which you are standing,
several of what you call boxes (but like everything else in Flatland,
they have no tops nor bottoms) full of money; I see also
two tablets of accounts. I am about to descend into that cupboard
and to bring you one of those tablets. I saw you lock the cupboard
half an hour ago, and I know you have the key in your possession.
But I descend from Space; the doors, you see, remain unmoved.
Now I am in the cupboard and am taking the tablet. Now I have it.
Now I ascend with it.
I rushed to the closet and dashed the door open. One of the tablets
was gone. With a mocking laugh, the Stranger appeared
in the other corner of the room, and at the same time the tablet
appeared upon the floor. I took it up. There could be no doubt --
it was the missing tablet.
I groaned with horror, doubting whether I was not out of my senses;
but the Stranger continued: "Surely you must now see
that my explanation, and no other, suits the phenomena. What you call
Solid things are really superficial; what you call Space is really
nothing but a great Plane. I am in Space, and look down upon
the insides of the things of which you only see the outsides.
You could leave this Plane yourself, if you could but summon up
the necessary volition. A slight upward or downward motion
would enable you to see all that I can see.
"The higher I mount, and the further I go from your Plane,
the more I can see, though of course I see it on a smaller scale.
For example, I am ascending; now I can see your neighbour the Hexagon
and his family in their several apartments; now I see
the inside of the Theatre, ten doors off, from which the audience
is only just departing; and on the other side a Circle in his study,
sitting at his books. Now I shall come back to you.
And, as a crowning proof, what do you say to my giving you a touch,
just the least touch, in your stomach? It will not seriously
injure you, and the slight pain you may suffer cannot be compared with
the mental benefit you will receive."
Before I could utter a word of remonstrance, I felt a shooting pain
in my inside, and a demoniacal laugh seemed to issue from within me.
A moment afterwards the sharp agony had ceased, leaving nothing but
a dull ache behind, and the Stranger began to reappear, saying,
as he gradually increased in size, "There, I have not hurt you much,
have I? If you are not convinced now, I don't know what will
convince you. What say you?"
My resolution was taken. It seemed intolerable that I should endure
existence subject to the arbitrary visitations of a Magician who could
thus play tricks with one's very stomach. If only I could in any way
manage to pin him against the wall till help came!
Once more I dashed my hardest angle against him, at the same time
alarming the whole household by my cries for aid. I believe,
at the moment of my onset, the Stranger had sunk below our Plane,
and really found difficulty in rising. In any case
he remained motionless, while I, hearing, as I thought,
the sound of some help approaching, pressed against him
with redoubled vigour, and continued to shout for assistance.
A convulsive shudder ran through the Sphere. "This must not be,"
I thought I heard him say: "either he must listen to reason,
or I must have recourse to the last resource of civilization."
Then, addressing me in a louder tone, he hurriedly exclaimed,
"Listen: no stranger must witness what you have witnessed.
Send your Wife back at once, before she enters the apartment.
The Gospel of Three Dimensions must not be thus frustrated.
Not thus must the fruits of one thousand years of waiting
be thrown away. I hear her coming. Back! back! Away from me,
or you must go with me -- whither you know not -- into the Land
of Three Dimensions!"
"Fool! Madman! Irregular!" I exclaimed; "never will I release thee;
thou shalt pay the penalty of thine impostures."
"Ha! Is it come to this?" thundered the Stranger: "then meet
your fate: out of your Plane you go. Once, twice, thrice!
'Tis done!"
Section 18. How I came to Spaceland, and what I saw there
An unspeakable horror seized me. There was a darkness;
then a dizzy, sickening sensation of sight that was not like seeing;
I saw a Line that was no Line; Space that was not Space:
I was myself, and not myself. When I could find voice,
I shrieked aloud in agony, "Either this is madness or it is Hell."
"It is neither," calmly replied the voice of the Sphere,
"it is Knowledge; it is Three Dimensions: open your eye once again
and try to look steadily."
I looked, and, behold, a new world! There stood before me,
visibly incorporate, all that I had before inferred, conjectured,
dreamed, of perfect Circular beauty. What seemed the centre
of the Stranger's form lay open to my view: yet I could see no heart,
nor lungs, nor arteries, only a beautiful harmonious Something --
for which I had no words; but you, my Readers in Spaceland,
would call it the surface of the Sphere.
Prostrating myself mentally before my Guide, I cried, "How is it,
O divine ideal of consummate loveliness and wisdom that I see
thy inside, and yet cannot discern thy heart, thy lungs, thy arteries,
thy liver?" "What you think you see, you see not," he replied;
"it is not given to you, nor to any other Being to behold
my internal parts. I am of a different order of Beings from those
in Flatland. Were I a Circle, you could discern my intestines,
but I am a Being, composed as I told you before, of many Circles,
the Many in the One, called in this country a Sphere. And,
just as the outside of a Cube is a Square, so the outside of a Sphere
presents the appearance of a Circle."
Bewildered though I was by my Teacher's enigmatic utterance,
I no longer chafed against it, but worshipped him in silent adoration.
He continued, with more mildness in his voice. "Distress not yourself
if you cannot at first understand the deeper mysteries of Spaceland.
By degrees they will dawn upon you. Let us begin by casting back
a glance at the region whence you came. Return with me a while
to the plains of Flatland, and I will shew you that which
you have often reasoned and thought about, but never seen
with the sense of sight -- a visible angle." "Impossible!" I cried;
but, the Sphere leading the way, I followed as if in a dream,
till once more his voice arrested me: "Look yonder,
and behold your own Pentagonal house, and all its inmates."
I looked below, and saw with my physical eye all that
domestic individuality which I had hitherto merely inferred
with the understanding. And how poor and shadowy was the inferred
conjecture in comparison with the reality which I now beheld!
My four Sons calmly asleep in the North-Western rooms,
my two orphan Grandsons to the South; the Servants, the Butler,
my Daughter, all in their several apartments. Only my
affectionate Wife, alarmed by my continued absence, had quitted
her room and was roving up and down in the Hall, anxiously awaiting
my return. Also the Page, aroused by my cries, had left his room,
and under pretext of ascertaining whether I had fallen
somewhere in a faint, was prying into the cabinet in my study.
All this I could now SEE, not merely infer; and as we came
nearer and nearer, I could discern even the contents of my cabinet,
and the two chests of gold, and the tablets of which the Sphere
had made mention.
<>
<>
/\
/ |My \
/ <> |Study \
/______ | ___ \
/ <> My Sons\ \|The \
/______/ \ Page / \
N / <> \ / My \
^ /______/ THE HALL \ Bedroom \
| \ <> My\ /
| \____| /\Wife's /
W--+--E \ My Wife / Apartment/
| ------- /\ --- \ WOMEN'S DOOR
| MEN'S DOOR \My Daughter
| /\ --== \ / The Scullion
S \ My Grandsons \ -==# \/ The Footman
\___ ___ _ _/ \-=#|/ The Butler
\ <> | <> | |THE CELLAR \ /
\____|____|_|____________/
###===--- ---===###
Policeman Policeman
Touched by my Wife's distress, I would have sprung downward
to reassure her, but I found myself incapable of motion.
"Trouble not yourself about your Wife," said my Guide:
"she will not be long left in anxiety; meantime, let us take
a survey of Flatland."
Once more I felt myself rising through space. It was even as
the Sphere had said. The further we receded from the object
we beheld, the larger became the field of vision. My native city,
with the interior of every house and every creature therein,
lay open to my view in miniature. We mounted higher, and lo,
the secrets of the earth, the depths of mines and inmost caverns
of the hills, were bared before me.
Awestruck at the sight of the mysteries of the earth,
thus unveiled before my unworthy eye, I said to my Companion,
"Behold, I am become as a God. For the wise men in our country say
that to see all things, or as they express it, OMNIVIDENCE,
is the attribute of God alone." There was something of scorn
in the voice of my Teacher as he made answer: "Is it so indeed?
Then the very pick-pockets and cut-throats of my country
are to be worshipped by your wise men as being Gods:
for there is not one of them that does not see as much as you see now.
But trust me, your wise men are wrong."
I. Then is omnividence the attribute of others besides Gods?
SPHERE. I do not know. But, if a pick-pocket or a cut-throat
of our country can see everything that is in your country,
surely that is no reason why the pick-pocket or cut-throat should be
accepted by you as a God. This omnividence, as you call it --
it is not a common word in Spaceland -- does it make you more just,
more merciful, less selfish, more loving? Not in the least.
Then how does it make you more divine?
I. "More merciful, more loving!" But these are the qualities
of women! And we know that a Circle is a higher Being
than a Straight Line, in so far as knowledge and wisdom
are more to be esteemed than mere affection.
SPHERE. It is not for me to classify human faculties according
to merit. Yet many of the best and wisest in Spaceland think more
of the affections than of the understanding, more of your despised
Straight Lines than of your belauded Circles. But enough of this.
Look yonder. Do you know that building?
I looked, and afar off I saw an immense Polygonal structure, in which
I recognized the General Assembly Hall of the States of Flatland,
surrounded by dense lines of Pentagonal buildings at right angles
to each other, which I knew to be streets; and I perceived that
I was approaching the great Metropolis.
"Here we descend," said my Guide. It was now morning,
the first hour of the first day of the two thousandth year of our era.
Acting, as was their wont, in strict accordance with precedent,
the highest Circles of the realm were meeting in solemn conclave,
as they had met on the first hour of the first day of the year 1000,
and also on the first hour of the first day of the year 0.
The minutes of the previous meetings were now read by one whom I
at once recognized as my brother, a perfectly Symmetrical Square,
and the Chief Clerk of the High Council. It was found recorded
on each occasion that: "Whereas the States had been troubled
by divers ill-intentioned persons pretending to have received
revelations from another World, and professing to produce
demonstrations whereby they had instigated to frenzy both themselves
and others, it had been for this cause unanimously resolved
by the Grand Council that on the first day of each millenary,
special injunctions be sent to the Prefects in the several districts
of Flatland, to make strict search for such misguided persons,
and without formality of mathematical examination, to destroy all such
as were Isosceles of any degree, to scourge and imprison
any regular Triangle, to cause any Square or Pentagon to be sent
to the district Asylum, and to arrest any one of higher rank,
sending him straightway to the Capital to be examined and judged
by the Council."
"You hear your fate," said the Sphere to me, while the Council
was passing for the third time the formal resolution.
"Death or imprisonment awaits the Apostle of the Gospel
of Three Dimensions." "Not so," replied I, "the matter is now
so clear to me, the nature of real space so palpable, that methinks
I could make a child understand it. Permit me but to descend
at this moment and enlighten them." "Not yet," said my Guide,
"the time will come for that. Meantime I must perform my mission.
Stay thou there in thy place." Saying these words,
he leaped with great dexterity into the sea (if I may so call it)
of Flatland, right in the midst of the ring of Counsellors. "I come,"
cried he, "to proclaim that there is a land of Three Dimensions."
I could see many of the younger Counsellors start back
in manifest horror, as the Sphere's circular section widened
before them. But on a sign from the presiding Circle
-- who shewed not the slightest alarm or surprise -- six Isosceles
of a low type from six different quarters rushed upon the Sphere.
"We have him," they cried; "No; yes; we have him still! he's going!
he's gone!"
"My Lords," said the President to the Junior Circles of the Council,
"there is not the slightest need for surprise; the secret archives,
to which I alone have access, tell me that a similar occurrence
happened on the last two millennial commencements. You will,
of course, say nothing of these trifles outside the Cabinet."
Raising his voice, he now summoned the guards. "Arrest the policemen;
gag them. You know your duty." After he had consigned to their fate
the wretched policemen -- ill-fated and unwilling witnesses
of a State-secret which they were not to be permitted to reveal --
he again addressed the Counsellors. "My Lords, the business
of the Council being concluded, I have only to wish you
a happy New Year." Before departing, he expressed, at some length,
to the Clerk, my excellent but most unfortunate brother,
his sincere regret that, in accordance with precedent and for the sake
of secrecy, he must condemn him to perpetual imprisonment,
but added his satisfaction that, unless some mention were made by him
of that day's incident, his life would be spared.
Section
19. How, though the Sphere shewed me other mysteries
of Spaceland, I still desired more; and what came of it
When I saw my poor brother led away to imprisonment, I attempted
to leap down into the Council Chamber, desiring to intercede
on his behalf, or at least bid him farewell. But I found that
I had no motion of my own. I absolutely depended on the volition
of my Guide, who said in gloomy tones, "Heed not thy brother;
haply thou shalt have ample time hereafter to condole with him.
Follow me."
<>
<>
(1) (2)
__________ __________
|\ |\ | \
| \ | \ | \
| \ ____|____\ | \
| | | | | |
|_____|____| | | |
\ | \ | \ |
\ | \ | \ |
\|_________\| \ __________|
Once more we ascended into space. "Hitherto," said the Sphere,
"I have shewn you naught save Plane Figures and their interiors.
Now I must introduce you to Solids, and reveal to you the plan
upon which they are constructed. Behold this multitude
of moveable square cards. See, I put one on another, not,
as you supposed, Northward of the other, but ON the other.
Now a second, now a third. See, I am building up a Solid
by a multitude of Squares parallel to one another. Now the Solid
is complete, being as high as it is long and broad,
and we call it a Cube."
"Pardon me, my Lord," replied I; "but to my eye the appearance is as
of an Irregular Figure whose inside is laid open to the view;
in other words, methinks I see no Solid, but a Plane such as
we infer in Flatland; only of an Irregularity which betokens
some monstrous criminal, so that the very sight of it is painful
to my eyes."
"True," said the Sphere, "it appears to you a Plane,
because you are not accustomed to light and shade and perspective;
just as in Flatland a Hexagon would appear a Straight Line to one
who has not the Art of Sight Recognition. But in reality
it is a Solid, as you shall learn by the sense of Feeling."
He then introduced me to the Cube, and I found that this
marvellous Being was indeed no Plane, but a Solid; and that he was
endowed with six plane sides and eight terminal points
called solid angles; and I remembered the saying of the Sphere
that just such a Creature as this would be formed by a Square moving,
in Space, parallel to himself: and I rejoiced to think
that so insignificant a Creature as I could in some sense be called
the Progenitor of so illustrious an offspring.
But still I could not fully understand the meaning of what my Teacher
had told me concerning "light" and "shade" and "perspective";
and I did not hesitate to put my difficulties before him.
Were I to give the Sphere's explanation of these matters,
succinct and clear though it was, it would be tedious to an inhabitant
of Space, who knows these things already. Suffice it, that by his
lucid statements, and by changing the position of objects and lights,
and by allowing me to feel the several objects and even his own
sacred Person, he at last made all things clear to me,
so that I could now readily distinguish between a Circle and a Sphere,
a Plane Figure and a Solid.
This was the Climax, the Paradise, of my strange eventful History.
Henceforth I have to relate the story of my miserable Fall: --
most miserable, yet surely most undeserved! For why should the thirst
for knowledge be aroused, only to be disappointed and punished?
My volition shrinks from the painful task of recalling my humiliation;
yet, like a second Prometheus, I will endure this and worse,
if by any means I may arouse in the interiors of Plane and Solid
Humanity a spirit of rebellion against the Conceit which would limit
our Dimensions to Two or Three or any number short of Infinity.
Away then with all personal considerations! Let me continue
to the end, as I began, without further digressions or anticipations,
pursuing the plain path of dispassionate History. The exact facts,
the exact words, -- and they are burnt in upon my brain, --
shall be set down without alteration of an iota; and let my Readers
judge between me and Destiny.
The Sphere would willingly have continued his lessons
by indoctrinating me in the conformation of all regular Solids,
Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons,
and Spheres: but I ventured to interrupt him. Not that I was
wearied of knowledge. On the contrary, I thirsted for yet deeper
and fuller draughts than he was offering to me.
"Pardon me," said I, "O Thou Whom I must no longer address
as the Perfection of all Beauty; but let me beg thee to vouchsafe
thy servant a sight of thine interior."
SPHERE. My what?
I. Thine interior: thy stomach, thy intestines.
SPHERE. Whence this ill-timed impertinent request? And what
mean you by saying that I am no longer the Perfection of all Beauty?
I. My Lord, your own wisdom has taught me to aspire to One
even more great, more beautiful, and more closely approximate
to Perfection than yourself. As you yourself, superior to all
Flatland forms, combine many Circles in One, so doubtless there is One
above you who combines many Spheres in One Supreme Existence,
surpassing even the Solids of Spaceland. And even as we,
who are now in Space, look down on Flatland and see the insides
of all things, so of a certainty there is yet above us some higher,
purer region, whither thou dost surely purpose to lead me --
O Thou Whom I shall always call, everywhere and in all Dimensions,
my Priest, Philosopher, and Friend -- some yet more spacious Space,
some more dimensionable Dimensionality, from the vantage-ground
of which we shall look down together upon the revealed insides
of Solid things, and where thine own intestines, and those of thy
kindred Spheres, will lie exposed to the view of the poor wandering
exile from Flatland, to whom so much has already been vouchsafed.
SPHERE. Pooh! Stuff! Enough of this trifling! The time is short,
and much remains to be done before you are fit to proclaim the Gospel
of Three Dimensions to your blind benighted countrymen in Flatland.
I. Nay, gracious Teacher, deny me not what I know it is
in thy power to perform. Grant me but one glimpse of thine interior,
and I am satisfied for ever, remaining henceforth thy docile pupil,
thy unemancipable slave, ready to receive all thy teachings
and to feed upon the words that fall from thy lips.
SPHERE. Well, then, to content and silence you, let me say at once,
I would shew you what you wish if I could; but I cannot.
Would you have me turn my stomach inside out to oblige you?
I. But my Lord has shewn me the intestines of all my countrymen
in the Land of Two Dimensions by taking me with him
into the Land of Three. What therefore more easy than now
to take his servant on a second journey into the blessed region
of the Fourth Dimension, where I shall look down with him once more
upon this land of Three Dimensions, and see the inside
of every three-dimensioned house, the secrets of the solid earth,
the treasures of the mines in Spaceland, and the intestines of every
solid living creature, even of the noble and adorable Spheres.
SPHERE. But where is this land of Four Dimensions?
I. I know not: but doubtless my Teacher knows.
SPHERE. Not I. There is no such land. The very idea of it
is utterly inconceivable.
I. Not inconceivable, my Lord, to me, and therefore still less
inconceivable to my Master. Nay, I despair not that, even here,
in this region of Three Dimensions, your Lordship's art
may make the Fourth Dimension visible to me; just as in the Land
of Two Dimensions my Teacher's skill would fain have opened the eyes
of his blind servant to the invisible presence of a Third Dimension,
though I saw it not.
Let me recall the past. Was I not taught below that when I saw a Line
and inferred a Plane, I in reality saw a Third unrecognized Dimension,
not the same as brightness, called "height"? And does it not now
follow that, in this region, when I see a Plane and infer a Solid,
I really see a Fourth unrecognized Dimension, not the same as colour,
but existent, though infinitesimal and incapable of measurement?
And besides this, there is the Argument from Analogy of Figures.
SPHERE. Analogy! Nonsense: what analogy?
I. Your Lordship tempts his servant to see whether he remembers
the revelations imparted to him. Trifle not with me, my Lord;
I crave, I thirst, for more knowledge. Doubtless we cannot SEE
that other higher Spaceland now, because we we have no eye
in our stomachs. But, just as there WAS the realm of Flatland,
though that poor puny Lineland Monarch could neither turn to left
nor right to discern it, and just as there WAS close at hand,
and touching my frame, the land of Three Dimensions,
though I, blind senseless wretch, had no power to touch it,
no eye in my interior to discern it, so of a surety there is
a Fourth Dimension, which my Lord perceives with the inner eye
of thought. And that it must exist my Lord himself has taught me.
Or can he have forgotten what he himself imparted to his servant?
In One Dimension, did not a moving Point produce a Line
with TWO terminal points?
In Two Dimensions, did not a moving Line produce a Square
with FOUR terminal points?
In Three Dimensions, did not a moving Square produce --
did not this eye of mine behold it -- that blessed Being, a Cube,
with EIGHT terminal points?
And in Four Dimensions shall not a moving Cube -- alas, for Analogy,
and alas for the Progress of Truth, if it be not so -- shall not,
I say, the motion of a divine Cube result in a still more divine
Organization with SIXTEEN terminal points?
Behold the infallible confirmation of the Series, 2, 4, 8, 16:
is not this a Geometrical Progression? Is not this -- if I might
quote my Lord's own words -- "strictly according to Analogy"?
Again, was I not taught by my Lord that as in a Line there are
TWO bounding Points, and in a Square there are FOUR
bounding Lines, so in a Cube there must be SIX bounding Squares?
Behold once more the confirming Series, 2, 4, 6: is not this
an Arithmetical Progression? And consequently does it not
of necessity follow that the more divine offspring of the divine Cube
in the Land of Four Dimensions, must have 8 bounding Cubes:
and is not this also, as my Lord has taught me to believe,
"strictly according to Analogy"?
O, my Lord, my Lord, behold, I cast myself in faith upon conjecture,
not knowing the facts; and I appeal to your Lordship to confirm
or deny my logical anticipations. If I am wrong, I yield,
and will no longer demand a fourth Dimension; but, if I am right,
my Lord will listen to reason.
I ask therefore, is it, or is it not, the fact, that ere now
your countrymen also have witnessed the descent of Beings
of a higher order than their own, entering closed rooms,
even as your Lordship entered mine, without the opening of doors
or windows, and appearing and vanishing at will? On the reply
to this question I am ready to stake everything. Deny it,
and I am henceforth silent. Only vouchsafe an answer.
SPHERE. (AFTER A PAUSE). It is reported so. But men are divided
in opinion as to the facts. And even granting the facts,
they explain them in different ways. And in any case,
however great may be the number of different explanations,
no one has adopted or suggested the theory of a Fourth Dimension.
Therefore, pray have done with this trifling, and let us return
to business.
I. I was certain of it. I was certain that my anticipations
would be fulfilled. And now have patience with me and answer me yet
one more question, best of Teachers! Those who have thus appeared --
no one knows whence -- and have returned -- no one knows whither --
have they also contracted their sections and vanished somehow into
that more Spacious Space, whither I now entreat you to conduct me?
SPHERE (MOODILY). They have vanished, certainly --
if they ever appeared. But most people say that these visions arose
from the thought -- you will not understand me -- from the brain;
from the perturbed angularity of the Seer.
I. Say they so? Oh, believe them not. Or if it indeed be so,
that this other Space is really Thoughtland, then take me to
that blessed Region where I in Thought shall see the insides
of all solid things. There, before my ravished eye, a Cube,
moving in some altogether new direction, but strictly according
to Analogy, so as to make every particle of his interior pass through
a new kind of Space, with a wake of its own -- shall create
a still more perfect perfection than himself, with sixteen terminal
Extra-solid angles, and Eight solid Cubes for his Perimeter.
And once there, shall we stay our upward course? In that blessed
region of Four Dimensions, shall we linger on the threshold
of the Fifth, and not enter therein? Ah, no! Let us rather resolve
that our ambition shall soar with our corporal ascent. Then,
yielding to our intellectual onset, the gates of the Sixth Dimension
shall fly open; after that a Seventh, and then an Eighth --
How long I should have continued I know not. In vain did the Sphere,
in his voice of thunder, reiterate his command of silence,
and threaten me with the direst penalties if I persisted.
Nothing could stem the flood of my ecstatic aspirations.
Perhaps I was to blame; but indeed I was intoxicated with
the recent draughts of Truth to which he himself had introduced me.
However, the end was not long in coming. My words were cut short
by a crash outside, and a simultaneous crash inside me,
which impelled me through space with a velocity that precluded speech.
Down! down! down! I was rapidly descending; and I knew
that return to Flatland was my doom. One glimpse, one last
and never-to-be-forgotten glimpse I had of that dull
level wilderness -- which was now to become my Universe again --
spread out before my eye. Then a darkness. Then a final,
all-consummating thunder-peal; and, when I came to myself,
I was once more a common creeping Square, in my Study at home,
listening to the Peace-Cry of my approaching Wife.
Section 20. How the Sphere encouraged me in a Vision
Although I had less than a minute for reflection, I felt, by a kind
of instinct, that I must conceal my experiences from my Wife.
Not that I apprehended, at the moment, any danger from her
divulging my secret, but I knew that to any Woman in Flatland
the narrative of my adventures must needs be unintelligible.
So I endeavoured to reassure her by some story, invented for
the occasion, that I had accidentally fallen through
the trap-door of the cellar, and had there lain stunned.
The Southward attraction in our country is so slight
that even to a Woman my tale necessarily appeared extraordinary
and well-nigh incredible; but my Wife, whose good sense far exceeds
that of the average of her Sex, and who perceived that I was
unusually excited, did not argue with me on the subject,
but insisted that I was ill and required repose. I was glad
of an excuse for retiring to my chamber to think quietly over
what had happened. When I was at last by myself, a drowsy sensation
fell on me; but before my eyes closed I endeavoured to reproduce
the Third Dimension, and especially the process by which a Cube
is constructed through the motion of a Square. It was not so clear
as I could have wished; but I remembered that it must be "Upward,
and yet not Northward", and I determined steadfastly to retain
these words as the clue which, if firmly grasped, could not fail
to guide me to the solution. So mechanically repeating,
like a charm, the words, "Upward, yet not Northward",
I fell into a sound refreshing sleep.
During my slumber I had a dream. I thought I was once more
by the side of the Sphere, whose lustrous hue betokened that he
had exchanged his wrath against me for perfect placability. We were
moving together towards a bright but infinitesimally small Point,
to which my Master directed my attention. As we approached,
methought there issued from it a slight humming noise as from one
of your Spaceland bluebottles, only less resonant by far,
so slight indeed that even in the perfect stillness of the Vacuum
through which we soared, the sound reached not our ears
till we checked our flight at a distance from it of something under
twenty human diagonals.
"Look yonder," said my Guide, "in Flatland thou hast lived;
of Lineland thou hast received a vision; thou hast soared with me
to the heights of Spaceland; now, in order to complete the range
of thy experience, I conduct thee downward to the lowest depth
of existence, even to the realm of Pointland, the Abyss of
No dimensions.
"Behold yon miserable creature. That Point is a Being like ourselves,
but confined to the non-dimensional Gulf. He is himself
his own World, his own Universe; of any other than himself he can form
no conception; he knows not Length, nor Breadth, nor Height,
for he has had no experience of them; he has no cognizance even
of the number Two; nor has he a thought of Plurality;
for he is himself his One and All, being really Nothing.
Yet mark his perfect self-contentment, and hence learn this lesson,
that to be self-contented is to be vile and ignorant,
and that to aspire is better than to be blindly and impotently happy.
Now listen."
He ceased; and there arose from the little buzzing creature a tiny,
low, monotonous, but distinct tinkling, as from one
of your Spaceland phonographs, from which I caught these words,
"Infinite beatitude of existence! It is; and there is none else
beside It."
"What," said I, "does the puny creature mean by 'it'?"
"He means himself," said the Sphere: "have you not noticed
before now, that babies and babyish people who cannot distinguish
themselves from the world, speak of themselves in the Third Person?
But hush!"
"It fills all Space," continued the little soliloquizing Creature,
"and what It fills, It is. What It thinks, that It utters;
and what It utters, that It hears; and It itself is Thinker, Utterer,
Hearer, Thought, Word, Audition; it is the One, and yet
the All in All. Ah, the happiness ah, the happiness of Being!"
"Can you not startle the little thing out of its complacency?" said I.
"Tell it what it really is, as you told me; reveal to it
the narrow limitations of Pointland, and lead it up to
something higher." "That is no easy task," said my Master; "try you."
Hereon, raising my voice to the uttermost, I addressed the Point
as follows:
"Silence, silence, contemptible Creature. You call yourself
the All in All, but you are the Nothing: your so-called Universe
is a mere speck in a Line, and a Line is a mere shadow
as compared with --" "Hush, hush, you have said enough,"
interrupted the Sphere, "now listen, and mark the effect
of your harangue on the King of Pointland."
The lustre of the Monarch, who beamed more brightly than ever upon
hearing my words, shewed clearly that he retained his complacency;
and I had hardly ceased when he took up his strain again.
"Ah, the joy, ah, the joy of Thought! What can It not achieve
by thinking! Its own Thought coming to Itself, suggestive of
Its disparagement, thereby to enhance Its happiness! Sweet rebellion
stirred up to result in triumph! Ah, the divine creative power
of the All in One! Ah, the joy, the joy of Being!"
"You see," said my Teacher, "how little your words have done. So far
as the Monarch understands them at all, he accepts them as his own --
for he cannot conceive of any other except himself --
and plumes himself upon the variety of 'Its Thought' as an instance
of creative Power. Let us leave this God of Pointland to the ignorant
fruition of his omnipresence and omniscience: nothing that you or I
can do can rescue him from his self-satisfaction."
After this, as we floated gently back to Flatland, I could hear
the mild voice of my Companion pointing the moral of my vision,
and stimulating me to aspire, and to teach others to aspire.
He had been angered at first -- he confessed -- by my ambition to soar
to Dimensions above the Third; but, since then, he had received
fresh insight, and he was not too proud to acknowledge his error
to a Pupil. Then he proceeded to initiate me into mysteries
yet higher than those I had witnessed, shewing me how
to construct Extra-Solids by the motion of Solids,
and Double Extra-Solids by the motion of Extra-Solids,
and all "strictly according to Analogy", all by methods so simple,
so easy, as to be patent even to the Female Sex.
Section
21. How I tried to teach the Theory of Three Dimensions
to my Grandson, and with what success
I awoke rejoicing, and began to reflect on the glorious career
before me. I would go forth, methought, at once, and evangelize
the whole of Flatland. Even to Women and Soldiers should the Gospel
of Three Dimensions be proclaimed. I would begin with my Wife.
Just as I had decided on the plan of my operations, I heard
the sound of many voices in the street commanding silence.
Then followed a louder voice. It was a herald's proclamation.
Listening attentively, I recognized the words of the Resolution
of the Council, enjoining the arrest, imprisonment, or execution
of any one who should pervert the minds of the people by delusions,
and by professing to have received revelations from another World.
I reflected. This danger was not to be trifled with. It would be
better to avoid it by omitting all mention of my Revelation,
and by proceeding on the path of Demonstration -- which after all,
seemed so simple and so conclusive that nothing would be lost
by discarding the former means. "Upward, not Northward" --
was the clue to the whole proof. It had seemed to me fairly clear
before I fell asleep; and when I first awoke, fresh from my dream,
it had appeared as patent as Arithmetic; but somehow it did not
seem to me quite so obvious now. Though my Wife entered the room
opportunely just at that moment, I decided, after we had exchanged
a few words of commonplace conversation, not to begin with her.
My Pentagonal Sons were men of character and standing,
and physicians of no mean reputation, but not great in mathematics,
and, in that respect, unfit for my purpose. But it occurred to me
that a young and docile Hexagon, with a mathematical turn,
would be a most suitable pupil. Why therefore not make
my first experiment with my little precocious Grandson,
whose casual remarks on the meaning of 3^3 had met with the approval
of the Sphere? Discussing the matter with him, a mere boy,
I should be in perfect safety; for he would know nothing
of the Proclamation of the Council; whereas I could not feel sure
that my Sons -- so greatly did their patriotism and reverence
for the Circles predominate over mere blind affection --
might not feel compelled to hand me over to the Prefect,
if they found me seriously maintaining the seditious heresy
of the Third Dimension.
But the first thing to be done was to satisfy in some way
the curiosity of my Wife, who naturally wished to know
something of the reasons for which the Circle had desired
that mysterious interview, and of the means by which he had
entered the house. Without entering into the details
of the elaborate account I gave her, -- an account, I fear,
not quite so consistent with truth as my Readers in Spaceland
might desire, -- I must be content with saying that I succeeded
at last in persuading her to return quietly to her household duties
without eliciting from me any reference to the World
of Three Dimensions. This done, I immediately sent for my Grandson;
for, to confess the truth, I felt that all that I had seen and heard
was in some strange way slipping away from me, like the image
of a half-grasped, tantalizing dream, and I longed to essay my skill
in making a first disciple.
When my Grandson entered the room I carefully secured the door.
Then, sitting down by his side and taking our mathematical tablets,
-- or, as you would call them, Lines -- I told him we would resume
the lesson of yesterday. I taught him once more how a Point by motion
in One Dimension produces a Line, and how a straight Line
in Two Dimensions produces a Square. After this, forcing a laugh,
I said, "And now, you scamp, you wanted to make me believe
that a Square may in the same way by motion 'Upward, not Northward'
produce another figure, a sort of extra Square in Three Dimensions.
Say that again, you young rascal."
At this moment we heard once more the herald's "O yes! O yes!"
outside in the street proclaiming the Resolution of the Council.
Young though he was, my Grandson -- who was unusually intelligent
for his age, and bred up in perfect reverence for the authority
of the Circles -- took in the situation with an acuteness for which
I was quite unprepared. He remained silent till the last words
of the Proclamation had died away, and then, bursting into tears,
"Dear Grandpapa," he said, "that was only my fun, and of course
I meant nothing at all by it; and we did not know anything then
about the new Law; and I don't think I said anything about
the Third Dimension; and I am sure I did not say one word about
'Upward, not Northward', for that would be such nonsense,
you know. How could a thing move Upward, and not Northward?
Upward and not Northward! Even if I were a baby, I could not be
so absurd as that. How silly it is! Ha! ha! ha!"
"Not at all silly," said I, losing my temper; "here for example,
I take this Square," and, at the word, I grasped a moveable Square,
which was lying at hand -- "and I move it, you see, not Northward but
-- yes, I move it Upward -- that is to say, not Northward,
but I move it somewhere -- not exactly like this, but somehow --"
Here I brought my sentence to an inane conclusion, shaking the Square
about in a purposeless manner, much to the amusement of my Grandson,
who burst out laughing louder than ever, and declared that I was not
teaching him, but joking with him; and so saying he unlocked the door
and ran out of the room. Thus ended my first attempt to convert
a pupil to the Gospel of Three Dimensions.
Section 22. How I then tried to diffuse the Theory
of Three Dimensions by other means, and of the result
My failure with my Grandson did not encourage me to communicate
my secret to others of my household; yet neither was I led by it
to despair of success. Only I saw that I must not wholly rely
on the catch-phrase, "Upward, not Northward", but must rather
endeavour to seek a demonstration by setting before the public
a clear view of the whole subject; and for this purpose
it seemed necessary to resort to writing.
So I devoted several months in privacy to the composition
of a treatise on the mysteries of Three Dimensions. Only,
with the view of evading the Law, if possible, I spoke not
of a physical Dimension, but of a Thoughtland whence, in theory,
a Figure could look down upon Flatland and see simultaneously
the insides of all things, and where it was possible that there might
be supposed to exist a Figure environed, as it were, with six Squares,
and containing eight terminal Points. But in writing this book
I found myself sadly hampered by the impossibility of drawing
such diagrams as were necessary for my purpose; for of course,
in our country of Flatland, there are no tablets but Lines,
and no diagrams but Lines, all in one straight Line
and only distinguishable by difference of size and brightness;
so that, when I had finished my treatise (which I entitled,
"Through Flatland to Thoughtland") I could not feel certain
that many would understand my meaning.
Meanwhile my life was under a cloud. All pleasures palled upon me;
all sights tantalized and tempted me to outspoken treason,
because I could not but compare what I saw in Two Dimensions
with what it really was if seen in Three, and could hardly refrain
from making my comparisons aloud. I neglected my clients
and my own business to give myself to the contemplation
of the mysteries which I had once beheld, yet which I could impart
to no one, and found daily more difficult to reproduce even before
my own mental vision.
One day, about eleven months after my return from Spaceland,
I tried to see a Cube with my eye closed, but failed;
and though I succeeded afterwards, I was not then quite certain
(nor have I been ever afterwards) that I had exactly realized
the original. This made me more melancholy than before,
and determined me to take some step; yet what, I knew not.
I felt that I would have been willing to sacrifice my life
for the Cause, if thereby I could have produced conviction.
But if I could not convince my Grandson, how could I convince
the highest and most developed Circles in the land?
And yet at times my spirit was too strong for me, and I gave vent
to dangerous utterances. Already I was considered heterodox
if not treasonable, and I was keenly alive to the danger
of my position; nevertheless I could not at times refrain
from bursting out into suspicious or half-seditious utterances,
even among the highest Polygonal and Circular society. When,
for example, the question arose about the treatment of those lunatics
who said that they had received the power of seeing the insides
of things, I would quote the saying of an ancient Circle,
who declared that prophets and inspired people are always considered
by the majority to be mad; and I could not help occasionally dropping
such expressions as "the eye that discerns the interiors of things",
and "the all-seeing land"; once or twice I even let fall
the forbidden terms "the Third and Fourth Dimensions". At last,
to complete a series of minor indiscretions, at a meeting of our
Local Speculative Society held at the palace of the Prefect himself,
-- some extremely silly person having read an elaborate paper
exhibiting the precise reasons why Providence has limited
the number of Dimensions to Two, and why the attribute of omnividence
is assigned to the Supreme alone -- I so far forgot myself as to give
an exact account of the whole of my voyage with the Sphere into Space,
and to the Assembly Hall in our Metropolis, and then to Space again,
and of my return home, and of everything that I had seen and heard
in fact or vision. At first, indeed, I pretended that I was
describing the imaginary experiences of a fictitious person;
but my enthusiasm soon forced me to throw off all disguise,
and finally, in a fervent peroration, I exhorted all my hearers
to divest themselves of prejudice and to become believers
in the Third Dimension.
Need I say that I was at once arrested and taken before the Council?
Next morning, standing in the very place where but a very few
months ago the Sphere had stood in my company, I was allowed to begin
and to continue my narration unquestioned and uninterrupted.
But from the first I foresaw my fate; for the President,
noting that a guard of the better sort of Policemen was in attendance,
of angularity little, if at all, under 55 degrees, ordered them
to be relieved before I began my defence, by an inferior class
of 2 or 3 degrees. I knew only too well what that meant.
I was to be executed or imprisoned, and my story was to be kept secret
from the world by the simultaneous destruction of the officials
who had heard it; and, this being the case, the President desired
to substitute the cheaper for the more expensive victims.
After I had concluded my defence, the President, perhaps perceiving
that some of the junior Circles had been moved by my
evident earnestness, asked me two questions: --
1. Whether I could indicate the direction which I meant
when I used the words "Upward, not Northward"?
2. Whether I could by any diagrams or descriptions (other than
the enumeration of imaginary sides and angles) indicate the Figure
I was pleased to call a Cube?
I declared that I could say nothing more, and that I must
commit myself to the Truth, whose cause would surely prevail
in the end.
The President replied that he quite concurred in my sentiment,
and that I could not do better. I must be sentenced to
perpetual imprisonment; but if the Truth intended that I should emerge
from prison and evangelize the world, the Truth might be trusted
to bring that result to pass. Meanwhile I should be subjected
to no discomfort that was not necessary to preclude escape, and,
unless I forfeited the privilege by misconduct, I should be
occasionally permitted to see my brother who had preceded me
to my prison.
Seven years have elapsed and I am still a prisoner, and
-- if I except the occasional visits of my brother --
debarred from all companionship save that of my jailers.
My brother is one of the best of Squares, just, sensible,
cheerful, and not without fraternal affection; yet I confess
that my weekly interviews, at least in one respect, cause me
the bitterest pain. He was present when the Sphere manifested himself
in the Council Chamber; he saw the Sphere's changing sections;
he heard the explanation of the phenomena then given to the Circles.
Since that time, scarcely a week has passed during seven whole years,
without his hearing from me a repetition of the part I played
in that manifestation, together with ample descriptions
of all the phenomena in Spaceland, and the arguments for the existence
of Solid things derivable from Analogy. Yet -- I take shame
to be forced to confess it -- my brother has not yet grasped
the nature of the Third Dimension, and frankly avows his disbelief
in the existence of a Sphere.
Hence I am absolutely destitute of converts, and, for aught that
I can see, the millennial Revelation has been made to me for nothing.
Prometheus up in Spaceland was bound for bringing down fire
for mortals, but I -- poor Flatland Prometheus -- lie here in prison
for bringing down nothing to my countrymen. Yet I exist in the hope
that these memoirs, in some manner, I know not how, may find their way
to the minds of humanity in Some Dimension, and may stir up a race
of rebels who shall refuse to be confined to limited Dimensionality.
That is the hope of my brighter moments. Alas, it is not always so.
Heavily weighs on me at times the burdensome reflection that I cannot
honestly say I am confident as to the exact shape of the once-seen,
oft-regretted Cube; and in my nightly visions the mysterious precept,
"Upward, not Northward", haunts me like a soul-devouring Sphinx.
It is part of the martyrdom which I endure for the cause of the Truth
that there are seasons of mental weakness, when Cubes and Spheres
flit away into the background of scarce-possible existences;
when the Land of Three Dimensions seems almost as visionary
as the Land of One or None; nay, when even this hard wall that bars me
from my freedom, these very tablets on which I am writing,
and all the substantial realities of Flatland itself, appear no better
than the offspring of a diseased imagination, or the baseless fabric
of a dream.
THE END of FLATLAND
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| THE END of |
| ______ |
| / / /| ------ / /| /| / /-. |
| /---- / /__| / / /__| / | / / / |
| / /___ / | / /___ / | / |/ /__.-' |
| |
| The baseless fabric of my vision |
| Melted into air into thin air |
| Such stuff as dreams are made of |
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End of this Project Gutenberg Etext of Flatland