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Title: Flatland: A Romance of Many Dimensions (Illustrated)
Author: Abbott, Edwin Abbott
Language: English
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*** Start of this LibraryBlog Digital Book "Flatland: A Romance of Many Dimensions (Illustrated)" ***
Flatland: A Romance of Many Dimensions
Edwin A. Abbott (1838-1926. English scholar, theologian, and writer.)
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| "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!" |
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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.
[Illustration 1]
[ASCII approximation follows]
(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.
[Illustration 2]
[ASCII approximation follows]
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?
[Illustration 3]
[ASCII approximation follows]
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.
[Illustration 4]
[ASCII approximation follows]
/\ - _ 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.
[Illustration 5]
[ASCII approximation follows]
[for simplicity's sake, the circle is approximated as an octogon]
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?"
[Illustration 6]
[ASCII approximation follows]
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."
[Illustration 7]
[ASCII approximation follows]
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.
[Illustration 8]
[ASCII approximation follows]
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.
[Illustration 9]
[ASCII approximation follows]
/\
/ |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."
[Illustration 10]
[ASCII approximation follows]
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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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