Flatland, by Edwin A. Abbott
Part 2 out of 2
and fills up many of your deficiencies. But permit me to point out
that your life in Lineland must be deplorably dull. To see nothing
but a Point! Not even to be able to contemplate a Straight Line!
Nay, not even to know what a Straight Line is! To see, yet to be cut
off from those Linear prospects which are vouchsafed to us in Flatland!
Better surely to have no sense of sight at all than to see so little!
I grant you I have not your discriminative faculty of hearing;
for the concert of all Lineland which gives you such intense pleasure,
is to me no better than a multitudinous twittering or chirping.
But at least I can discern, by sight, a Line from a Point.
And let me prove it. Just before I came into your kingdom,
I saw you dancing from left to right, and then from right to left,
with Seven Men and a Woman in your immediate proximity on the left,
and eight Men and two Women on your right. Is not this correct?"
"It is correct," said the King, "so far as the numbers and sexes
are 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 clearer to you.
But surely you cannot be ignorant of so simple a distinction.
King. I do not in the least understand you.
I. Alas! How shall I make it clear? When you move straight on,
does it not sometimes occur to you that you COULD move in some other way,
turning your eye round so as to look in the direction towards which your
side is now fronting? In other words, instead of always moving
in the direction of one of your extremities, do you never feel
a desire to move in the direction, so to speak, of your side?
King. Never. And what do you mean? How can a man's inside "front"
in any direction? Or how can a man move in the direction of his inside?
I. Well then, since words cannot explain the matter, I will try deeds,
and will move gradually out of Lineland in the direction which I desire
to indicate to you.
At the word I began to move my body out of Lineland. As long
as any part of me remained in his dominion and in his view, the King
kept exclaiming, "I see you, I see you still; you are not moving."
But when I had at last moved myself out of his Line, he cried in his
shrillest voice, "She is vanished; she is dead." "I am not dead,"
replied I; "I am simply out of Lineland, that is to say, out of the
Straight Line which you call Space, and in the true Space, where I can
see things as they are. And at this moment I can see your Line,
or side--or inside as you are pleased to call it; and I can see also
the Men and Women on the North and South of you, whom I will now enumerate,
describing their order, their size, and the interval between each."
When I had done this at great length, I cried triumphantly,
"Does that at last convince you?" And, with that, I once more
entered Lineland, taking up the same position as before.
But the Monarch replied, "If you were a Man of sense--though, as
you appear to have only one voice I have little doubt you are not a
Man but a Woman--but, if you had a particle of sense, you would
listen to reason. You ask me to believe that there is another Line
besides that which my senses indicate, and another motion besides that
of which I am daily conscious. I, in return, ask you to describe
in words or indicate by motion that other Line of which you speak.
Instead of moving, you merely exercise some magic art of vanishing
and returning to sight; and instead of any lucid description of your
new World, you simply tell me the numbers and sizes of some forty
of my retinue, facts known to any child in my capital. Can anything
be more irrational or audacious? Acknowledge your folly or depart
from my dominions."
Furious at his perversity, and especially indignant that he professed
to be ignorant of my sex, I retorted in no measured terms, "Besotted Being!
You think yourself the perfection of existence, while you are in reality
the most imperfect and imbecile. You profess to see, whereas you see
nothing but a Point! You plume yourself on inferring the existence
of a Straight Line; but I CAN SEE Straight Lines, and infer the existence
of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles.
Why waste more words? Suffice it that I am the completion of your
incomplete self. You are a Line, but I am a Line of Lines called
in my country a Square: and even I, infinitely superior though
I am to you, am of little account among the great nobles of Flatland,
whence I have come to visit you, in the hope of 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 movement
there arose from myriads of his subjects a multitudinous war-cry,
increasing in vehemence till at last methought it rivalled the roar
of an army of a hundred thousand Isosceles, and the artillery
of a thousand Pentagons. Spell-bound and motionless, I could
neither speak nor move to avert the impending destruction;
and still the noise grew louder, and the King came closer,
when I awoke to find the breakfast-bell recalling me
to the realities of Flatland.
SECTION 15 Concerning a Stranger from Spaceland
From dreams I proceed to facts.
It was the last day of our 1999th year of our era.
The patterning of the rain had long ago announced nightfall;
and I was sitting (footnote 3) in the company of my wife,
musing on the events of the past and the prospects of the coming year,
the coming century, the coming Millennium.
My four Sons and two orphan Grandchildren had retired to their
several apartments; and my wife alone remained with me to see
the old Millennium out and the new one in.
I was rapt in thought, pondering in my mind some words that had
casually issued from the mouth of my youngest Grandson, a most
promising young Hexagon of unusual brilliancy and perfect angularity.
His uncles and I had been giving him his usual practical lesson in
Sight Recognition, turning ourselves upon our centres, now rapidly,
now more slowly, and questioning him as to our positions; and his
answers had been so satisfactory that I had been induced to reward him
by giving him a few hints on Arithmetic, as applied to Geometry.
Taking nine Squares, each an inch every way, I had put them together
so as to make one large Square, with a side of three inches,
and I had hence proved to my little Grandson that--though it
was impossible for us to SEE the inside of the Square--
yet we might ascertain the number of square inches in a Square
by simply squaring the number of inches in the side: "and thus,"
said I, "we know that three-to-the-second, or nine, represents the
number of square inches in a Square whose side is three inches long."
The little Hexagon meditated on this a while and then said to me;
"But you have been teaching me to raise numbers to the third power:
I suppose three-to-the-third must mean something in Geometry; what does
it mean?" "Nothing at all," replied I, "not at least in Geometry;
for Geometry has only Two Dimensions." And then I began to shew the boy
how a Point by moving through a length of three inches makes a Line of
three inches, which may be represented by three; and how a Line of three
inches, moving parallel to itself through a length of three inches,
makes a Square of three inches every way, which may be represented
Upon this, my Grandson, again returning to his former suggestion,
took me up rather suddenly and exclaimed, "Well, then, if a Point by
moving three inches, makes a Line of three inches represented by three;
and if a straight Line of three inches, moving parallel to itself,
makes a Square of three inches every way, represented by three-to-the-second;
it must be that a Square of three inches every way, moving somehow parallel
to itself (but I don't see how) must make Something else (but I don't see what)
of three inches every way--and this must be represented by three-to-the-third."
"Go to bed," said I, a little ruffled by this interruption:
"if you would talk less nonsense, you would remember more sense."
So my Grandson had disappeared in disgrace; and there I sat by my
Wife's side, endeavouring to form a retrospect of the year 1999 and of
the possibilities of the year 2000; but not quite able to shake of the
thoughts suggested by the prattle of my bright little Hexagon. Only a
few sands now remained in the half-hour glass. Rousing myself from my
reverie I turned the glass Northward for the last time in the old
Millennium; and in the act, I exclaimed aloud, "The boy is a fool."
Straightway I became conscious of a Presence in the room, and a
chilling breath thrilled through my very being. "He is no such thing,"
cried my Wife, "and you are breaking the Commandments in thus
dishonouring your own Grandson." But I took no notice of her.
Looking around in every direction I could see nothing; yet still
I FELT a Presence, and shivered as the cold whisper came again.
I started up. "What is the matter?" said my Wife, "there is no draught;
what are you looking for? There is nothing." There was nothing;
and I resumed my seat, again exclaiming, "The boy is a fool, I say;
three- to-the-third can have no meaning in Geometry."
At once there came a distinctly audible reply,
"The boy is not a fool; and three-to-the-third
has an obvious Geometrical meaning."
My Wife as well as myself heard the words, although she did not
understand their meaning, and both of us sprang forward in the direction
of the sound. What was our horror when we saw before us a Figure!
At the first glance it appeared to be a Woman, seen sideways;
but a moment's observation shewed me that the extremities passed
into dimness too rapidly to represent one of the Female Sex;
and I should have thought it a Circle, only that it seemed
to change its size in a manner impossible for a Circle
or for any regular Figure of which I had had experience.
But my Wife had not my experience, nor the coolness necessary
to note these characteristics. With the usual hastiness
and unreasoning jealousy of her Sex, she flew at once
to the conclusion that a Woman had entered the house
through some small aperture. "How comes this person here?"
she exclaimed, "you promised me, my dear, that there should
be no ventilators in our new house." "Nor are they any," said I;
"but what makes you think that the stranger is a Woman?
I see by my power of Sight Recognition --"
"Oh, I have no patience with your Sight Recognition," replied she,
"`Feeling is believing' and `A Straight Line to the touch is worth
a Circle to the sight'"--two Proverbs, very common with the Frailer
Sex in Flatland.
"Well," said I, for I was afraid of irritating her, "if it must be so,
demand an introduction." Assuming her most gracious manner, my Wife
advanced towards the Stranger, "Permit me, Madam to feel and be felt by--"
then, suddenly recoiling, "Oh! it is not a Woman, and there are no angles
either, not a trace of one. Can it be that I have so misbehaved
to a perfect Circle?"
"I am indeed, in a certain sense a Circle," replied the Voice,
"and a more perfect Circle than any in Flatland; but to speak more
accurately, I am many Circles in one." Then he added more mildly,
"I have a message, dear Madam, to your husband, which I must not
deliver in your presence; and, if you would suffer us to retire
for a few minutes --" But my wife would not listen to the proposal
that our august Visitor should so incommode himself, and assuring
the Circle that the hour of her own retirement had long passed,
with many reiterated apologies for her recent indiscretion,
she at last retreated to her apartment.
I glanced at the half-hour glass. The last sands had fallen.
The third Millennium had begun.
Footnote 3. When I say "sitting," of course I do not mean any change
of attitude such as you in Spaceland signify by that word; for as we
have no feet, we can no more "sit" nor "stand" (in your sense of the
word) than one of your soles or flounders.
Nevertheless, we perfectly well recognize the different mental
states of volition implied by "lying," "sitting," and "standing,"
which are to some extent indicated to a beholder by a slight increase
of lustre corresponding to the increase of volition.
But on this, and a thousand other kindred subjects, time forbids me to dwell.
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 graduations
of size and brightness scarcely possible for any Figure within the scope
of my experience. The thought flashed across me that I might have before
me a burglar or cut-throat, some monstrous Irregular Isosceles, who,
by feigning the voice of a Circle, had obtained admission somehow
into the house, and was now preparing to stab me with his acute angle.
In a sitting-room, the absence of Fog (and the season happened
to be remarkably dry), made it difficult for me to trust to Sight
Recognition, especially at the short distance at which I was standing.
Desperate with fear, I rushed forward with an unceremonious, "You must
permit me, Sir --" and felt him. My Wife was right. There was not
the trace of an angle, not the slightest roughness or inequality:
never in my life had I met with a more perfect Circle. He remained
motionless while I walked around him, beginning from his eye
and returning to it again. Circular he was throughout,
a perfectly satisfactory Circle; there could not be a doubt of it.
Then followed a dialogue, which I will endeavour to set down as near
as I can recollect it, omitting only some of my profuse apologies--
for I was covered with shame and humiliation that I, a Square,
should have been guilty of the impertinence of feeling a Circle.
It was commenced by the Stranger with some impatience at the
lengthiness of my introductory process.
Stranger. Have you felt me enough by this time? Are you not
introduced to me yet?
I. Most illustrious Sir, excuse my awkwardness, which arises not
from ignorance of the usages of polite society, but from a little
surprise and nervousness, consequent on this somewhat unexpected visit.
And I beseech you to reveal my indiscretion to no one, and especially
not to my Wife. But before your Lordship enters into further
communications, would he deign to satisfy the curiosity
of one who would gladly know whence his visitor came?
Stranger. From Space, from Space, Sir: whence else?
I. Pardon me, my Lord, but is not your Lordship already in Space,
your Lordship and his humble servant, even at this moment?
Stranger. Pooh! what do you know of Space? Define Space.
I. Space, my Lord, is height and breadth indefinitely prolonged.
Stranger. Exactly: you see you do not even know what Space is.
You think it is of Two Dimensions only; but I have come to announce
to you a Third--height, breadth, and length.
I. Your Lordship is pleased to be merry. We also speak of length
and height, or breadth and thickness, thus denoting Two Dimensions
by four names.
Stranger. But I mean not only three names, but Three Dimensions.
I. Would your Lordship indicate or explain to me
in what direction is the Third Dimension, unknown to me?
Stranger. I came from it. It is up above and down below.
I. My Lord means seemingly that it is Northward and Southward.
Stranger. I mean nothing of the kind. I mean a direction
in which you cannot look, because you have no eye in your side.
I. Pardon me, my Lord, a moment's inspection will convince your Lordship
that I have a perfectly luminary at the juncture of my two sides.
Stranger: Yes: but in order to see into Space you ought to have an eye,
not on your Perimeter, but on your side, that is, on what you would probably
call your inside; but we in Spaceland should call it your side.
I. An eye in my inside! An eye in my stomach! Your Lordship jests.
Stranger. I am in no jesting humour. I tell you that I come from Space,
or, since you will not understand what Space means, from the Land
of Three Dimensions whence I but lately looked down upon your Plane
which you call Space forsooth. From that position of advantage
I discerned all that you speak of as SOLID (by which you mean
"enclosed on four sides"), your houses, your churches,
your very chests and safes, yes even your insides and stomachs,
all lying open and exposed to my view.
I. Such assertions are easily made, my Lord.
Stranger. But not easily proved, you mean. But I mean to prove mine.
When I descended here, I saw your four Sons, the Pentagons,
each in his apartment, and your two Grandsons the Hexagons;
I saw your youngest Hexagon remain a while with you and then
retire to his room, leaving you and your Wife alone.
I saw your Isosceles servants, three in number,
in the kitchen at supper, and the little Page
in the scullery. Then I came here, and how do you think I came?
I. Through the roof, I suppose.
Strange. Not so. Your roof, as you know very well, has been
recently repaired, and has no aperture by which even a Woman
could penetrate. I tell you I come from Space. Are you not convinced
by what I have told you of your children and household?
I. Your Lordship must be aware that such facts touching
the belongings of his humble servant might be easily ascertained
by any one of the neighbourhood possessing your Lordship's
ample means of information.
Stranger. (TO HIMSELF.) What must I do? Stay; one more
argument suggests itself to me. When you see a Straight Line--
your wife, for example--how many Dimensions do you attribute to her?
I. Your Lordship would treat me as if I were one of the vulgar who,
being ignorant of Mathematics, suppose that a Woman is really a Straight Line,
and only of One Dimension. No, no, my Lord; we Squares are better advised,
and are as well aware of your Lordship that a Woman, though popularly
called a Straight Line, is, really and scientifically,
a very thin Parallelogram, possessing Two Dimensions,
like the rest of us, viz., length and breadth (or thickness).
Stranger. But the very fact that a Line is visible implies
that it possesses yet another Dimension.
I. My Lord, I have just 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 the last Dimension
is infinitesimal in your country. If a Line were mere length
without "height," it would cease to occupy Space and would become invisible.
Surely you must recognize this?
I. I must indeed confess that I do not in the least understand
your Lordship. When we in Flatland see a Line, we see length
and BRIGHTNESS. If the brightness disappears, the Line is extinguished,
and, as you say, ceases to occupy Space. But am I to suppose that
your Lordship gives the brightness the title of a Dimension,
and that what we call "bright" you call "high"?
Stranger. No, indeed. By "height" I mean a Dimension like your length:
only, with you, "height" is not so easily perceptible, being extremely small.
I. My Lord, your assertion is easily put to the test. You say
I have a Third Dimension, which you call "height." Now, Dimension
implies direction and measurement. Do but measure my "height,"
or merely indicate to me the direction in which my "height" extends,
and I will become your convert. Otherwise, your Lordship's
own understand must hold me excused.
Stranger. (TO HIMSELF.) I can do neither. How shall I convince him?
Surely a plain statement of facts followed by ocular demonstration ought
to suffice. --Now, Sir; listen to me.
You are living on a Plane. What you style Flatland is the vast level
surface of what I may call a fluid, or in, the top of which you and your
countrymen move about, without rising above or falling below it.
I am not a plane Figure, but a Solid. You call me a Circle; but in
reality I am not a Circle, but an infinite number of Circles, of size
varying from a Point to a Circle of thirteen inches in diameter,
one placed on the top of the other. When I cut through your plane as
I am now doing, I make in your plane a section which you, very rightly,
call a Circle. For even a Sphere--which is my proper name in my own
country--if he manifest himself at all to an inhabitant of Flatland--
must needs manifest himself as a Circle.
Do you not remember--for I, who see all things, discerned last
night the phantasmal vision of Lineland written upon your brain--
do you not remember, I say, how when you entered the realm of Lineland,
you were compelled to manifest yourself to the King, not as a Square,
but as a Line, because that Linear Realm had not Dimensions enough
to represent the whole of you, but only a slice or section of you?
In precisely the same way, your country of Two Dimensions is not spacious
enough to represent me, a being of Three, but can only exhibit a slice
or section of me, which is what you call a Circle.
The diminished brightness of your eye indicates incredulity.
But now prepare to receive proof positive of the truth of my assertions.
You cannot indeed see more than one of my sections, or Circles, at a time;
for you have no power to raise your eye out of the plane of Flatland;
but you can at least see that, as I rise in Space, so my sections
become smaller. See now, I will rise; and the effect upon your eye
will be that my Circle will become smaller and smaller till it dwindles
to a point and finally vanishes.
There was no "rising" that I could see; but he diminished and
finally vanished. I winked once or twice to make sure that
I was not dreaming. But it was no dream. For from the depths
of nowhere came forth a hollow voice--close to my heart it seemed--
"Am I quite gone? Are you convinced now? Well, now I will gradually
return to Flatland and you shall see my section become larger and larger."
Every reader in Spaceland will easily understand that my mysterious Guest
was speaking the language of truth and even of simplicity. But to me,
proficient though I was in Flatland Mathematics, it was by no means
a simple matter. The rough diagram given above will make it clear
to any Spaceland child that the Sphere, ascending in the three positions
indicated there, must needs have manifested himself to me, or to any
Flatlander, as a Circle, at first of full size, then small, and at last
very small indeed, approaching to a Point. But to me, although I saw
the facts before me, the causes were as dark as ever. All that I could
comprehend was, that the Circle had made himself smaller and vanished,
and that he had now re-appeared and was rapidly making himself larger.
When he regained his original size, he heaved a deep sigh; for he
perceived by my silence that I had altogether failed to comprehend
him. And indeed I was now inclining to the belief that he must be
no Circle at all, but some extremely clever juggler; or else that
the old wives' tales were true, and that after all there were such
people as Enchanters and Magicians.
After a long pause he muttered to himself, "One resource alone remains,
if I am not to resort to action. I must try the method of Analogy."
Then followed a still longer silence, after which he continued our dialogue.
Sphere. Tell me, Mr. Mathematician; if a Point moves Northward,
and leaves a luminous wake, what name would you give to the wake?
I. A straight Line.
Sphere. And a straight Line has how many extremities?
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?
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
Sphere. Of course; and all according to Analogy. But, by the way,
not what YOU call sides, but what WE call sides. You would call them SOLIDS.
I. And how many solids or sides will appertain to this Being whom
I am to generate by the motion of my inside in an "upward" direction,
and whom you call a Cube?
Sphere. How can you ask? And you a mathematician! The side of anything
is always, if I may so say, one Dimension behind the thing. Consequently,
as there is no Dimension behind a Point, a Point has 0 sides; a Line,
if I may so say, has 2 sides (for the points of a Line may be called
by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression
do you call that?
Sphere. And what is the next number?
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; not edging to the right nor to the left, but moving somehow
out of the world, and vanishing into nothing. Soon there was a blank.
But still I heard the Intruder's voice.
Sphere. Why will you refuse to listen to reason? I had hoped to find
in you--as being a man of sense and an accomplished mathematician--
a fit apostle for the Gospel of the Three Dimensions, which I am allowed
to preach once only in a thousand years: but now I know not how
to convince you. Stay, I have it. Deeds, and not words,
shall proclaim the truth. Listen, my friend.
I have told you I can see from my position in Space the inside
of all things that you consider closed. For example, I see in yonder
cupboard near which you are standing, several of what you call boxes
(but like everything else in Flatland, they have no tops or bottom)
full of money; I see also two tablets of accounts. I am about
to descend into that cupboard and to bring you one of those tablets.
I saw you lock the cupboard half an hour ago, and I know you have
the key in your possession. But I descend from Space; the doors, you see,
remain unmoved. Now I am in the cupboard and am taking the tablet.
Now I have it. Now I ascent with it.
I rushed to the closet and dashed the door open. One of the tablets
was gone. With a mocking laugh, the Stranger appeared in the other
corner of the room, and at the same time the tablet appeared upon the floor.
I took it up. There could be no doubt--it was the missing tablet.
I groaned with horror, doubting whether I was not out of my sense;
but the Stranger continued: "Surely you must now see that my explanation,
and no other, suits the phenomena. What you call Solid things are really
superficial; what you call Space is really nothing but a great Plane.
I am in Space, and look down upon the insides of the things of which
you only see the outsides. You could leave the Plane yourself,
if you could but summon up the necessary volition. A slight upward
or downward motion would enable you to see all that I can see.
"The higher I mount, and the further I go from your Plane,
the more I can see, though of course I see it on a smaller scale.
For example, I am ascending; now I can see your neighbour the Hexagon
and his family in their several apartments; now I see the inside of
the Theatre, ten doors off, from which the audience is only just departing;
and on the other side a Circle in his study, sitting at his books.
Now I shall come back to you. And, as a crowning proof, what do
you say to my giving you a touch, just the least touch, in your stomach?
It will not seriously injure you, and the slight pain you may suffer
cannot be compared with the mental benefit you will receive."
Before I could utter a word of remonstrance, I felt a shooting pain
in my inside, and a demoniacal laugh seemed to issue from within me.
A moment afterwards the sharp agony had ceased, leaving nothing but
a dull ache behind, and the Stranger began to reappear, saying,
as he gradually increased in size, "There, I have not hurt you much,
have I? If you are not convinced now, I don't know what will convince you.
What say you?"
My resolution was taken. It seemed intolerable that I should endure
existence subject to the arbitrary visitations of a Magician who could
thus play tricks with one's very stomach. If only I could in any way
manage to pin him against the wall till help came!
Once more I dashed my hardest angle against him, at the same time
alarming the whole household by my cries for aid. I believe,
at the moment of my onset, the Stranger had sunk below our Plane,
and really found difficulty in rising. In any case he remained motionless,
while I, hearing, as I thought, the sound of some help approaching,
pressed against him with redoubled vigor, and continued to shout
A convulsive shudder ran through the Sphere. "This must not be,"
I thought I heard him say: "either he must listen to reason,
or I must have recourse to the last resource of civilization."
Then, addressing me in a louder tone, he hurriedly exclaimed, "Listen:
no stranger must witness what you have witnessed. Send your Wife back
at once, before she enters the apartment. The Gospel of Three Dimensions
must not be thus frustrated. Not thus must the fruits of one thousand
years of waiting be thrown away. I hear her coming. Back! back!
Away from me, or you must go with me--wither you know not--into
the Land of Three Dimensions!"
"Fool! Madman! Irregular!" I exclaimed; "never will I release thee;
thou shalt pay the penalty of thine impostures."
"Ha! Is it come to this?" thundered the Stranger: "then meet your fate:
out of your Plane you go. Once, twice, thrice! `Tis done!"
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 loud in agony, "Either this is madness
or it is Hell." "It is neither, calmly replied the voice of the Sphere,
"it is Knowledge; it is Three Dimensions: open your eye once again
and try to look steadily."
I looked, and, behold, a new world! There stood before me,
visibly incorporate, all that I had before inferred, conjectured,
dreamed, of perfect Circular beauty. What seemed the centre
of the Stranger's form lay open to my view: yet I could see no heart,
lungs, nor arteries, only a beautiful harmonious Something--
for which I had no words; but you, my Readers in Spaceland,
would call it the surface of the Sphere.
Prostrating myself mentally before my Guide, I cried, "How 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 giving to you, nor to any other Being, to behold
my internal parts. I am of a different order of Beings
from those in Flatland. 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 represents
the appearance of a Circle."
Bewildered though I was by my Teacher's enigmatic utterance,
I no longer chafed against it, but worshipped him in silent adoration.
He continued, with more mildness in his voice. "Distress not yourself
if you cannot at first understand the deeper mysteries of Spaceland.
By degrees they will dawn upon you. Let us begin by casting back
a glance at the region whence you came. Return with me a while to
the plains of Flatland and I will shew you that which you have often
reasoned and thought about, but never seen with the sense of sight--
a visible angle." "Impossible!" I cried; but, the Sphere leading the way,
I followed as if in a dream, till once more his voice arrested me:
"Look yonder, and behold your own Pentagonal house, and all its inmates."
I looked below, and saw with my physical eye all that domestic
individuality which I had hitherto merely inferred with
the understanding. And how poor and shadowy was the inferred conjecture
in comparison with the reality which I now behold! My four Sons
calmly asleep in the North-Western rooms, my two orphan Grandsons
to the South; the Servants, the Butler, my Daughter, all in their
several apartments. Only my affection Wife, alarmed by my continued
absence, had quitter her room and was roving up and down in the Hall,
anxiously awaiting my return. Also the Page, aroused by my cries,
had left his room, and under pretext of ascertaining whether I had
fallen somewhere in a faint, was prying into the cabinet in my study.
All this I could now SEE, not merely infer; and as we came nearer
and nearer, I could discern even the contents of my cabinet, and the
two chests of gold, and the tablets of which the Sphere had made mention.
Touched by my Wife's distress, I would have sprung downward
to reassure her, but I found myself incapable of motion.
"Trouble not yourself about your Wife," said my Guide:
"she will not be long left in anxiety; meantime,
let us take a survey of Flatland."
Once more I felt myself rising through space. It was even as
the Sphere had said. The further we receded from the object we beheld,
the larger became the field of vision. My native city, with
the interior of every house and every creature therein, lay open
to my view in miniature. We mounted higher, and lo, the secrets
of the earth, the depths of the mines and inmost caverns of the hills,
were bared before me.
Awestruck at the sight of the mysteries of the earth, thus unveiled
before my unworthy eye, I said to my Companion, "Behold, I am become
as a God. For the wise men in our country say that to see all things,
or as they express it, OMNIVIDENCE, is the attribute of God alone."
There was something of scorn in the voice of my Teacher as he made answer:
"it is so indeed? Then the very pick-pockets and cut-throats
of my country are to be worshipped by your wise men as being Gods:
for there is not one of them that does not see as much as you see now.
But trust me, your wise men are wrong."
I. Then is omnividence the attribute of others besides Gods?
Sphere. I do not know. But, if a pick-pocket or a cut-throat
of our country can see everything that is in your country, surely
that is no reason why the pick-pocket or cut-throat should be accepted
by you as a God. This omnividence, as you call it--it is not a common word
in Spaceland--does it make you more just, more merciful, less selfish,
more loving? Not in the least. Then how does it make you more divine?
I. "More merciful, more loving!" But these are the qualities of women!
And we know that a Circle is a higher Being than a Straight Line,
in so far as knowledge and wisdom are more to be esteemed than mere affection.
Sphere. It is not for me to classify human faculties according to merit.
Yet many of the best and wisest in Spaceland think more of the affections
than of the understand, more of your despised Straight Lines than of your
belauded Circles. But enough of this. Look yonder. Do you know
I looked, and afar off I saw an immense Polygonal structure,
in which I recognized the General Assembly Hall of the States
of Flatland, surrounded by dense lines of Pentagonal buildings
at right angles to each other, which I knew to be streets;
and I perceived that I was approaching the great Metropolis.
"Here we descend," said my Guide. It was now morning, the first
hour of the first day of the two thousandth year of our era.
Acting, as was their wont, in strict accordance with precedent,
the highest Circles of the realm were meeting in solemn conclave,
as they had met on the first hour of the first day of the year 1000,
and also on the first hour of the first day of the year 0.
The minutes of the previous meetings were now read by one whom
I at once recognized as my brother, a perfectly Symmetrical Square,
and the Chief Clerk of the High Council. It was found recorded on
each occasion that: "Whereas the States had been troubled by divers
ill-intentioned persons pretending to have received revelations
from another World, and professing to produce demonstrations whereby
they had instigated to frenzy both themselves and others, it had been
for this cause unanimously resolved by the Grand Council that on the
first day of each millenary, special injunctions be sent to the Prefects
in the several districts of Flatland, to make strict search for such
misguided persons, and without formality of mathematical examination,
to destroy all such as were Isosceles of any degree, to scourge
and imprison any regular Triangle, to cause any Square or Pentagon
to be sent to the district Asylum, and to arrest any one of higher rank,
sending him straightway to the Capital to be examined and judged
by the Council."
"You hear your fate," said the Sphere to me, while the Council
was passing for the third time the formal resolution. "Death or
imprisonment awaits the Apostle of the Gospel of Three Dimensions."
"Not so," replied I, "the matter is now so clear to me, the nature of real
space so palpable, that methinks I could make a child understand it.
Permit me but to descend at this moment and enlighten them."
"Not yet," said my Guide, "the time will come for that.
Meantime I must perform my mission. Stay thou there in thy place."
Saying these words, he leaped with great dexterity into the sea
(if I may so call it) of Flatland, right in the midst of the ring
of Counsellors. "I come," said he, "to proclaim that there is a land
of Three Dimensions."
I could see many of the younger Counsellors start back in manifest horror,
as the Sphere's circular section widened before them. But on a sign from
the presiding Circle--who shewed not the slightest alarm or surprise--
six Isosceles of a low type from six different quarters rushed upon the Sphere.
"We have him," they cried; "No; yes; we have him still! he's going! he's gone!"
"My Lords," said the President to the Junior Circles of the Council,
"there is not the slightest need for surprise; the secret archives,
to which I alone have access, tell me that a similar occurrence
happened on the last two millennial commencements. You will,
of course, say nothing of these trifles outside the Cabinet."
Raising his voice, he now summoned the guards. "Arrest the policemen;
gag them. You know your duty." After he had consigned to their fate
the wretched policemen--ill-fated and unwilling witnesses
of a State-secret which they were not to be permitted to reveal--
he again addressed the Counsellors. "My Lords, the business of the
Council being concluded, I have only to wish you a happy New Year."
Before departing, he expressed, at some length, to the Clerk,
my excellent but most unfortunate brother, his sincere regret that,
in accordance with precedent and for the sake of secrecy, he must condemn
him to perpetual imprisonment, but added his satisfaction that,
unless some mention were made by him of that day's incident,
his life would be spared.
SECTION 19 How, though the Sphere shewed me other mysteries of
Spaceland, I still desire more; and what came of it
When I saw my poor brother led away to imprisonment, I attempted to leap
down into the Council Chamber, desiring to intercede on his behalf,
or at least bid him farewell. But I found that I had no motion of my own.
I absolutely depended on the volition of my Guide, who said in gloomy tones,
"Heed not thy brother; haply thou shalt have ample time hereafter
to condole with him. Follow me."
Once more we ascended into space. "Hitherto," said the Sphere,
"I have shewn you naught save Plane Figures and their interiors.
Now I must introduce you to Solids, and reveal to you the plan upon which
they are constructed. Behold this multitude of moveable square cards.
See, I put one on another, not, as you supposed, Northward of the other,
but ON the other. Now a second, now a third. See, I am building up
a Solid by a multitude of Squares parallel to one another.
Now the Solid is complete, being as high as it is long and broad,
and we call it a Cube."
"Pardon me, my Lord," replied I; "but to my eye the appearance
is as of an Irregular Figure whose inside is laid open to view;
in other words, methinks I see no Solid, but a Plane such as we
infer in Flatland; only of an Irregularity which betokens some
monstrous criminal, so that the very sight of it is painful to my eyes."
"True," said the Sphere; "it appears to you a Plane, because you
are not accustomed to light and shade and perspective; just as in
Flatland a Hexagon would appear a Straight Line to one who has not
the Art of Sight Recognition. But in reality it is a Solid,
as you shall learn by the sense of Feeling."
He then introduced me to the Cube, and I found that this marvellous
Being was indeed no Plane, but a Solid; and that he was endowed with
six plane sides and eight terminal points called solid angles;
and I remembered the saying of the Sphere that just such a Creature
as this would be formed by the Square moving, in Space, parallel to himself:
and I rejoiced to think that so insignificant a Creature as I could
in some sense be called the Progenitor of so illustrious an offspring.
But still I could not fully understand the meaning of what my Teacher
had told me concerning "light" and "shade" and "perspective";
and I did not hesitate to put my difficulties before him.
Were I to give the Sphere's explanation of these matters, succinct
and clear though it was, it would be tedious to an inhabitant of Space,
who knows these things already. Suffice it, that by his lucid statements,
and by changing the position of objects and lights, and by allowing me
to feel the several objects and even his own sacred Person, he at last
made all things clear to me, so that I could now readily distinguish
between a Circle and a Sphere, a Plane Figure and a Solid.
This was the Climax, the Paradise, of my strange eventful History.
Henceforth I have to relate the story of my miserable Fall:--most miserable,
yet surely most undeserved! For why should the thirst for knowledge
be aroused, only to be disappointed and punished? My volition shrinks
from the painful task of recalling my humiliation; yet, like a second
Prometheus, I will endure this and worse, if by any means I may arouse
in the interiors of Plane and Solid Humanity a spirit of rebellion against
the Conceit which would limit our Dimensions to Two or Three or any number
short of Infinity. Away then with all personal considerations!
Let me continue to the end, as I began, without further digressions
or anticipations, pursuing the plain path of dispassionate History.
The exact facts, the exact words,--and they are burnt in upon my brain,
--shall be set down without alteration of an iota; and let my Readers
judge between me and Destiny.
The Sphere would willingly have continued his lessons by
indoctrinating me in the conformation of all regular Solids,
Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons,
and Spheres: but I ventured to interrupt him. Not that I was wearied
of knowledge. On the contrary, I thirsted for yet deeper and fuller
draughts than he was offering to me.
"Pardon me," said I, "O Thou Whom I must no longer address as the Perfection
of all Beauty; but let me beg thee to vouchsafe thy servant a slight
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 reform. Grant me but one glimpse of thine interior,
and I am satisfied for ever, remaining henceforth thy docile pupil,
thy unemacipable slave, ready to receive all thy teachings and to feed
upon the words that fall from thy lips.
Sphere. Well, then, to content and silence you, let me say at once,
I would shew you what you wish if I could; but I cannot. Would you
have me turn my stomach inside out to oblige you?
I. But my Lord has shewn me the intestines of all my countrymen in
the Land of Two Dimensions by taking me with him into the Land of Three.
What therefore more easy than now to take his servant on a second journey
into the blessed region of the Fourth Dimension, where I shall look down
with him once more upon this land of Three Dimensions, and see the inside
of every three-dimensioned house, the secrets of the solid earth,
the treasures of the mines of Spaceland, and the intestines of every
solid living creature, even the noble and adorable Spheres.
Sphere. But where is this land of Four Dimensions?
I. I know not: but doubtless my Teacher knows.
Sphere. Not I. There is no such land. The very idea of it is
I. Not inconceivable, my Lord, to me, and therefore still less
inconceivable to my Master. Nay, I despair not that, even here,
in this region of Three Dimensions, your Lordship's art may make
the Fourth Dimension visible to me; just as in the Land of
Two Dimensions my Teacher's skill would fain have opened the eyes
of his blind servant to the invisible presence of a Third Dimension,
though I saw it not.
Let me recall the past. Was I not taught below that when I saw a Line
and inferred a Plane, I in reality saw a Third unrecognized Dimension,
not the same as brightness, called "height"? And does it not
now follow that, in this region, when I see a Plane and infer a Solid,
I really see a Fourth unrecognized Dimension, not the same as colour,
but existent, though infinitesimal and incapable of measurement?
And besides this, there is the Argument from Analogy of Figures.
Sphere. Analogy! Nonsense: what analogy?
I. Your Lordship tempts his servant to see whether he remembers
the revelations imparted to him. Trifle not with me, my Lord;
I crave, I thirst, for more knowledge. Doubtless we cannot SEE that
other higher Spaceland now, because we have no eye in our stomachs.
But, just as there WAS the realm of Flatland, though that poor puny
Lineland Monarch could neither turn to left nor right to discern it,
and just as there WAS close at hand, and touching my frame, the land
of Three Dimensions, though I, blind senseless wretch, had no power
to touch it, no eye in my interior to discern it, so of a surety there
is a Fourth Dimension, which my Lord perceives with the inner eye
of thought. And that it must exist my Lord himself has taught me.
Or can he have forgotten what he himself imparted to his servant?
In One Dimension, did not a moving Point produce a Line with TWO
In Two Dimensions, did not a moving Line produce a Square with FOUR
In Three Dimensions, did not a moving Square produce--did not
this eye of mine behold it--that blessed Being, a Cube, with EIGHT
And in Four Dimensions shall not a moving Cube--alas, for Analogy,
and alas for the Progress of Truth, if it be not so--shall not,
I say, the motion of a divine Cube result in a still more divine
Organization with SIXTEEN terminal points?
Behold the infallible confirmation of the Series, 2, 4, 8, 16: is
not this a Geometrical Progression? Is not this--if I might quote
my Lord's own words--"strictly according to Analogy"?
Again, was I not taught by my Lord that as in a Line there are TWO
bounding Points, and in a Square there are FOUR bounding Lines,
so in a Cube there must be SIX bounding Squares? Behold once more
the confirming Series, 2, 4, 6: is not this an Arithmetical Progression?
And consequently does it not of necessity follow that the more divine
offspring of the divine Cube in the Land of Four Dimensions,
must have 8 bounding Cubes: and is not this also, as my Lord
has taught me to believe, "strictly according to Analogy"?
O, my Lord, my Lord, behold, I cast myself in faith upon conjecture,
not knowing the facts; and I appeal to your Lordship to confirm
or deny my logical anticipations. If I am wrong, I yield,
and will no longer demand a Fourth Dimension; but,
if I am right, my Lord will listen to reason.
I ask therefore, is it, or is it not, the fact, that ere now your
countrymen also have witnessed the descent of Beings of a higher order
than their own, entering closed rooms, even as your Lordship entered mine,
without the opening of doors or windows, and appearing and vanishing at will?
On the reply to this question I am ready to stake everything. Deny it,
and I am henceforth silent. Only vouchsafe an answer.
Sphere (AFTER A PAUSE). It is reported so. But men are divided
in opinion as to the facts. And even granting the facts, they explain
them in different ways. And in any case, however great may be the
number of different explanations, no one has adopted or suggested
the theory of a Fourth Dimension. Therefore, pray have done with
this trifling, and let us return to business.
I. I was certain of it. I was certain that my anticipations
would be fulfilled. And now have patience with me and answer me
yet one more question, best of Teachers! Those who have thus appeared--
no one knows whence--and have returned--no one knows whither--
have they also contracted their sections and vanished somehow into
that more Spacious Space, whither I now entreat you to conduct me?
Sphere (MOODILY). They have vanished, certainly--if they ever appeared.
But most people say that these visions arose from the thought--you will not
understand me--from the brain; from the perturbed angularity of the Seer.
I. Say they so? Oh, believe them not. Or if it indeed be so,
that this other SPace is really Thoughtland, then take me to that
blessed Region where I in Thought shall see the insides of all solid
things. There, before my ravished eye, a Cube moving in some
altogether new direction, but strictly according to Analogy, so as to
make every particle of his interior pass through a new kind of Space,
with a wake of its own--shall create a still more perfect perfection
than himself, with sixteen terminal Extra-solid angles, and Eight
solid Cubes for his Perimeter. And once there, shall we stay our
upward course? In that blessed region of Four Dimensions, shall we
linger at the threshold of the Fifth, and not enter therein? Ah, no!
Let us rather resolve that our ambition shall soar with our corporal
ascent. Then, yielding to our intellectual onset, the gates of the
Six Dimension shall fly open; after that a Seventh, and then an Eighth--
How long I should have continued I know not. In vain did the Sphere,
in his voice of thunder, reiterate his command of silence,
and threaten me with the direst penalties if I persisted.
Nothing could stem the flood of my ecstatic aspirations.
Perhaps I was to blame; but indeed I was intoxicated with
the recent draughts of Truth to which he himself had introduced me.
However, the end was not long in coming. My words were cut short
by a crash outside, and a simultaneous crash inside me,
which impelled me through space with a velocity that precluded speech.
Down! down! down! I was rapidly descending; and I knew that return
to Flatland was my doom. One glimpse, one last and never-to-be-forgotten
glimpse I had of that dull level wilderness--which was now to become
my Universe again-- spread out before my eye. Then a darkness.
Then a final, all- consummating thunder-peal; and, when I came to myself,
I was once more a common creeping Square, in my Study at home,
listening to the Peace- Cry of my approaching Wife.
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 will
and required repose. I was glad of an excuse for retiring to my chamber
to think quietly over what had happened. When I was at last by myself,
a drowsy sensation fell on me; but before my eyes closed I endeavoured
to reproduce the Third Dimension, and especially the process by which
a Cube is constructed through the motion of a Square. It was not
so clear as I could have wished; but I remembered that it must be
"Upward, and yet not Northward," and I determined steadfastly
to retain these words as the clue which, if firmly grasped,
could not fail to guide me to the solution. So mechanically
repeating, like a charm, the words, "Upward, yet not Northward,"
I fell into a sound refreshing sleep.
During my slumber I had a dream. I thought I was once more
by the side of the Sphere, whose lustrous hue betokened that
he had exchanged his wrath against me for perfectly placability.
We were moving together towards a bright but infinitesimally small Point,
to which my Master directed my attention. As we approached, methought
there issued from it a slight humming noise as from one of your Spaceland
bluebottles, only less resonant by far, so slight indeed that even
in the perfect stillness of the Vacuum through which we soared,
the sound reached not our ears till we checked our flight
at a distant from it of something under twenty human diagonals.
"Look yonder," said my Guide, "in Flatland thou hast lived;
of Lineland thou hast received a vision; thou hast soared with me
to the heights of Spaceland; now,, in order to complete the range
of thy experience, I conduct thee downward to the lowest depth of existence,
even to the realm of Pointland, the Abyss of No dimensions.
"Behold yon miserable creature. That Point is a Being like ourselves,
but confined to the non-dimensional Gulf. He is himself his own World,
his own Universe; of any other than himself he can form no conception;
he knows not Length, nor Breadth, nor Height, for he has had no
experience of them; he has no cognizance even of the number Two;
nor has he a thought of Plurality; for he is himself his One and All,
being really Nothing. Yet mark his perfect self-contentment,
and hence learn his lesson, that to be self-contented is to be vile
and ignorant, and that to aspire is better than to be blindly
and impotently happy. Now listen."
He ceased; and there arose from the little buzzing creature a tiny,
low, monotonous, but distinct tinkling, as from one of your Spaceland
phonographs, from which I caught these words, "Infinite beatitude
of existence! It is; and there is nothing else beside It."
"What," said I, "does the puny creature mean by `it'?" "He means
himself," said the Sphere: "have you not noticed before now,
that babies and babyish people who cannot distinguish themselves
from the world, speak of themselves in the Third Person? But hush!"
"It fills all Space," continued the little soliloquizing Creature,
"and what It fills, It is. What It thinks, that It utters;
and what It utters, that It hears; and It itself is Thinker, Utterer,
Hearer, THought, Word, Audition; it is the One, and yet the All in All.
Ah, the happiness, ah, the happiness of Being!"
"Can you not startle the little thing out of its complacency?" said I.
"Tell it what it really is, as you told me; reveal to it the narrow
limitations of Pointland, and lead it up to something higher."
"That is no easy task," said my Master; "try you."
Hereon, raising by voice to the uttermost, I addressed the Point as follows:
"Silence, silence, contemptible Creature. You call yourself the
All in All, but you are the Nothing: your so-called Universe is a
mere speck in a Line, and a Line is a mere shadow as compared with--"
"Hush, hush, you have said enough," interrupted the Sphere, "now listen,
and mark the effect of your harangue on the King of Pointland."
The lustre of the Monarch, who beamed more brightly than ever upon
hearing my words, shewed clearly that he retained his complacency;
and I had hardly ceased when he took up his strain again. "Ah,
the joy, ah, the joy of Thought1 What can It not achieve by thinking!
Its own Thought coming to Itself, suggestive of its disparagement,
thereby to enhance Its happiness! Sweet rebellion stirred up to result
in triumph! Ah, the divine creative power of the All in One!
Ah, the joy, the joy of Being!"
"You see," said my Teacher, "how little your words have done.
So far as the Monarch understand them at all, he accepts them as his own--
for he cannot conceive of any other except himself--and plumes himself
upon the variety of `Its Thought' as an instance of creative Power.
Let us leave this God of Pointland to the ignorant fruition of his
omnipresence and omniscience: nothing that you or I can do can rescue
him from his self-satisfaction."
After this, as we floated gently back to Flatland, I could hear
the mild voice of my Companion pointing the moral of my vision,
and stimulating me to aspire, and to teach others to aspire.
He had been angered at first--he confessed--by my ambition to soar
to Dimensions above the Third; but, since then, he had received fresh
insight, and he was not too proud to acknowledge his error to a Pupil.
Then he proceeded to initiate me into mysteries yet higher than those
I had witnessed, shewing me how to construct Extra-Solids by the motion
of Solids, and Double Extra-Solids by the motion of Extra-Solids,
and all "strictly according to Analogy," all by methods so simple,
so easy, as to be patent even to the Female Sex.
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 people by delusions,
and by professing to have received revelations from another World.
I reflected. This danger was not to be trifled with. It would
be better to avoid it by omitting all mention of my Revelation,
and by proceeding on the path of Demonstration--which after all,
seemed so simple and so conclusive that nothing would be lost by
discarding the former means. "Upward, not Northward"--was the clue to
the whole proof. It had seemed to me fairly clear before I fell asleep;
and when I first awoke, fresh from my dream, it had appeared as patent
as Arithmetic; but somehow it did not seem to me quite so obvious now.
Though my Wife entered the room opportunely at just that moment,
I decided, after we had exchanged a few words of commonplace conversation,
not to begin with her.
My Pentagonal Sons were men of character and standing, and physicians
of no mean reputation, but not great in mathematics, and, in that respect,
unfit for my purpose. But it occurred to me that a young and docile Hexagon,
with a mathematical turn, would be a most suitable pupil. Why therefore
not make my first experiment with my little precocious Grandson,
whose casual remarks on the meaning of three-to-the-third had met
with the approval of the Sphere? Discussing the matter with him,
a mere boy, I should be in perfect safety; for he would know nothing
of the Proclamation of the Council; whereas I could not feel sure
that my Sons--so greatly did their patriotism and reverence
for the Circles predominate over mere blind affection--
might not feel compelled to hand me over to the Prefect,
if they found me seriously maintaining the seditious
heresy of the Third Dimension.
But the first thing to be done was to satisfy in some way the curiosity
of my Wife, who naturally wished to know something of the reasons
for which the Circle had desired that mysterious interview,
and of the means by which he had entered the house. Without entering
into the details of the elaborate account I gave her,--an account,
I fear, not quite so consistent with truth as my Readers in Spaceland
might desire,--I must be content with saying that I succeeded
at last in persuading her to return 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 believe that a Square may in
the same way by motion `Upward, not Northward' produce another figure,
a sort of extra square in Three Dimensions. Say that again, you young rascal."
At this moment we heard once more the herald's "O yes! O yes!"
outside in the street proclaiming the REsolution of the Council.
Young though he was, my Grandson--who was unusually intelligent
for his age, and bred up in perfect reverence for the authority
of the Circles--took in the situation with an acuteness for which
I was quite unprepared. He remained silent till the last words
of the Proclamation had died away, and then, bursting into tears,
"Dear Grandpapa," he said, "that was only my fun, and of course I meant
nothing at all by it; and we did not know anything then about the new Law;
and I don't think I said anything about the Third Dimension; and I am sure
I did not say one word about `Upward, not Northward,' for that would be
such nonsense, you know. How could a thing move Upward, and not Northward?
Upward and not Northward! Even if I were a baby, I could not be so absurd
as that. How silly it is! Ha! ha! ha!"
"Not at all silly," said I, losing my temper; "here for example,
I take this Square," and, at the word, I grasped a moveable Square,
which was lying at hand--"and I move it, you see, not Northward but
--yes, I move it Upward--that is to say, Northward but I move it
somewhere--not exactly like this, but somehow --" Here I brought
my sentence to an inane conclusion, shaking the Square about in
a purposeless manner, much to the amusement of my Grandson, who burst
out laughing louder than ever, and declared that I was not teaching
him, but joking with him; and so saying he unlocked the door and ran
out of the room. Thus ended my first attempt to convert a pupil to
the Gospel of Three Dimensions.
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 wife was under a cloud. All pleasures palled upon me;
all sights tantalized and tempted me to outspoken treason, because
I could not compare what I saw in Two Dimensions with what it really
was if seen in Three, and could hardly refrain from making my comparisons
aloud. I neglected my clients and my own business to give myself
to the contemplation of the mysteries which I had once beheld,
yet which I could impart to no one, and found daily more difficult
to reproduce even before my own mental vision.
One day, about eleven months after my return from Spaceland, I tried
to see a Cube with my eye closed, but failed; and though I succeeded
afterwards, I was not then quite certain (nor have I been ever afterwards)
that I had exactly realized the original. This made me more melancholy
than before, and determined me to take some step; yet what, I knew not.
I felt that I would have been willing to sacrifice my life for the Cause,
if thereby I could have produced conviction. But if I could not convince
my Grandson, how could I convince the highest and most developed Circles
in the land?
And yet at times my spirit was too strong for me, and I gave vent
to dangerous utterances. Already I was considered heterodox if not
treasonable, and I was keenly alive to the danger of my position;
nevertheless I could not at times refrain from bursting out into
suspicious or half-seditious utterances, even among the highest
Polygonal or Circular society. When, for example, the question arose
about the treatment of those lunatics who said that they had received
the power of seeing the insides of things, I would quote the saying
of an ancient Circle, who declared that prophets and inspired people
are always considered by the majority to be mad; and I could not help
occasionally dropping such expressions as "the eye that discerns
the interiors of things," and "the all-seeing land"; once or twice
I even let fall the forbidden terms "the Third and Fourth Dimensions."
At last, to complete a series of minor indiscretions, at a meeting of
our Local Speculative Society held at the palace of the Prefect himself,
--some extremely silly person having read an elaborate paper exhibiting
the precise reasons why Providence has limited the number of Dimensions
to Two, and why the attribute of omnividence is assigned to the Supreme
alone--I so far forgot myself as to give an exact account of the whole
of my voyage with the Sphere into Space, and to the Assembly Hall
in our Metropolis, and then to Space again, and of my return home,
and of everything that I had seen and heard in fact or vision.
At first, indeed, I pretended that I was describing the imaginary
experiences of a fictitious person; but my enthusiasm soon forced me
to throw off all disguise, and finally, in a fervent peroration,
I exhorted all my hearers to divest themselves of prejudice
and to become believers in the Third Dimension.
Need I say that I was at once arrested and taken before the Council?
Next morning, standing in the very place where but a very few months ago
the Sphere had stood in my company, I was allowed to begin and to continue
my narration unquestioned and uninterrupted. But from the first I foresaw
my fate; for the President, noting that a guard of the better sort
of Policemen was in attendance, of angularity little, if at all,
under 55 degrees, ordered them to be relieved before I began my defence,
by an inferior class of 2 or 3 degrees. I knew only too well what that meant.
I was to be executed or imprisoned, and my story was to be kept secret
from the world by the simultaneous destruction of the officials
who had heard it; and, this being the case, the President desired
to substitute the cheaper for the more expensive victims.
After I had concluded my defence, the President, perhapsperceiving
that some of the junior Circles had been moved by 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 give
to the Circles. Since that time, scarcely a week has passed during
seven whole years, without his hearing from me a repetition of the part
I played in that manifestation, together with ample descriptions of all
the phenomena in Spaceland, and the arguments for the existence of Solid
things derivable from Analogy. Yet--I take shame to be forced to confess it--
my brother has not yet grasped the nature of Three Dimensions, and frankly
avows his disbelief in the existence of a Sphere.
Hence I am absolutely destitute of converts, and, for aught that
I can see, the millennial Revelation has been made to me for nothing.
Prometheus up in Spaceland was bound for bringing down fire
for mortals, but I--poor Flatland Prometheus--lie here in prison
for bringing down nothing to my countrymen. Yet I existing the hope
that these memoirs, in some manner, I know not how, may find their way
to the minds of humanity in Some Dimension, and may stir up a race
of rebels who shall refuse to be confined to limited Dimensionality.
That is the hope of my brighter moments. Alas, it is not always so.
Heavily weights on me at times the burdensome reflection that I cannot
honestly say I am confident as to the exact shape of the once-seen,
oft-regretted Cube; and in my nightly visions the mysterious precept,
"Upward, not Northward," haunts me like a soul-devouring Sphinx.
It is part of the martyrdom which I endure for the cause of Truth
that there are seasons of mental weakness, when Cubes and Spheres
flit away into the background of scarce-possible existences;
when the Land of Three Dimensions seems almost as visionary
as the Land of One or None; nay, when even this hard wall that bars
me from my freedom, these very tablets on which I am writing,
and all the substantial realities of Flatland itself,
appear no better than the offspring of a diseased imagination,
or the baseless fabric of a dream.
PREFACE TO THE
SECOND AND REVISED
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, fully, to return his thanks
to his readers and critics in Spaceland, whose appreciation has,
with unexpected celerity, required a second edition of this work;
secondly, to apologize for certain errors and misprints (for which,
however, he is not entirely responsible); and, thirdly, to explain
on or two misconceptions. But he is not the Square he once was.
Years of imprisonment, and the still heavier burden of general
incredulity and mockery, have combined with the thoughts and notions,
and much also of the terminology, which he acquired during his
short stay in spaceland. He has, therefore, requested me to reply
in his behalf to two special objections, one of an intellectual,
the other of a moral nature.
The first objection is, that a Flatlander, seeing a Line,
sees something that must be THICK to the eye as well as LONG
to the eye (otherwise it would not be visible, if it had not
some thickness); and consequently he ought (it is argued)
to acknowledge that his countrymen are not only long and broad,
but also (though doubtless to a very slight degree) THICK or HIGH.
This objection is plausible, and, to Spacelanders, almost irresistible,
so that, I confess, when I first heard it, I knew not what to reply.
But my poor old friend's answer appears to me completely to meet it.
"I admit," said he--when I mentioned to him this objection--
"I admit the truth of your critic's facts, but I deny his conclusions.
It is true that we have really in Flatland a Third unrecognized Dimension
called `height,' just as it also is true that you have really in Spaceland
a Fourth unrecognized Dimension, called by no name at present, but which
I will call `extra-height.' But we can no more take cognizance of our
`height' than you can of your `extra-height.' Even I--who have been in
Spaceland, and have had the privilege of understanding for twenty-four hours
the meaning of `height'--even I cannot now comprehend it, nor realize it
by the sense of sight or by any process of reason; I can but apprehend
it by faith.
"The reason is obvious. Dimension implied direction, implies
measurement, implies the more and the less. Now, all our lines
are EQUALLY and INFINITESIMALLY thick (or high, whichever you like);
consequently, there is nothing in them to lead our minds to the
conception of that Dimension. No `delicate micrometer'--as has been
suggested by one too hasty Spaceland critic--would in the least
avail us; for we should not know WHAT TO MEASURE, NOR IN WHAT DIRECTION.
When we see a Line, we see something that is long and BRIGHT;
BRIGHTNESS, as well as length, is necessary to the existence of a Line;
if the brightness vanishes, the Line is extinguished. Hence, all my
Flatland friends--when I talk to them about the unrecognized Dimension
which is somehow visible in a Line--say, `Ah, you mean BRIGHTNESS':
and when I reply, `No, I mean a real Dimension,' they at once retort,
`Then measure it, or tell us in what direction it extends'; and this
silences me, for I can do neither. Only yesterday, when the Chief Circle
(in other words our High Priest) came to inspect the State Prison
and paid me his seventh annual visit, and when for the seventh time
he put me the question, `Was I any better?' I tried to prove to him
that he was `high,' as well as long and broad, although he did not know it.
But what was his reply? `You say I am "high"; measure my "high-ness"
and I will believe you.' What could I do? How could I meet his challenge?
I was crushed; and he left the room triumphant.
"Does this still seem strange to you? Then put yourself
in asimilar position. Suppose a person of the Fourth Dimension,
condescending to visit you, were to say, `Whenever you open your eyes,
you see a Plane (which is of Two Dimensions) and you INFER a Solid
(which is of Three); but in reality you also see (though you do
not recognize) a Fourth Dimension, which is not colour nor brightness
nor anything of the kind, but a true Dimension, although I cannot
point out to you its direction, nor can you possibly measure it.'
What would you say to such a visitor? Would not you have him locked up?
Well, that is my fate: and it is as natural for us Flatlanders
to lock up a Square for preaching the Third Dimension,
as it is for you Spacelanders to lock up a Cube for preaching the Fourth.
Alas, how strong a family likeness runs through blind and persecuting
humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes--
we are all liable to the same errors, all alike the Slavers of our
respective Dimensional prejudices, as one of our Spaceland poets has said--
`One touch of Nature makes all worlds akin.'" (footnote 1)
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 (see page 86) 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."
Footnote 1. The Author desires me to add, that the misconceptions of
some of his critics on this matter has induced him to insert (on pp.
74 and 92) 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.
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