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._ 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 section becomes
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: ]
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 had 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? And 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 yourself give the answer to your own question: 1, 2, 4, are
evidently in Geometrical Progression. What is the next number.
_I._ Eight.
_Sphere._ Exactly. The one Square produces a
_Something-which-you-do-not-as-yet-know-a-name-for-but-which-we-call-a-Cube_
with _eight_ terminal Points. Now are you convinced?
_I._ And has this Creature sides, as well as angles or what you call
“terminal Points?”
_Sphere._ Of course; and all according to Analogy. But, by the way, not
what _you_ call sides, but what _we_ call sides. You would call them
_solids_.
_I._ And how many solids or sides will appertain to this Being whom I am
to generate by the motion of my inside in an “upward” direction, and whom
you call a Cube?
_Sphere._ How can you ask? And you a mathematician! The side of anything
is always, if I may so say, one Dimension behind the thing. Consequently,
as there is no Dimension behind a Point, a Point has 0 sides; a Line, if
I may so say, has 2 sides (for the Points of a Line may be called by
courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression do
you call that?
_I._ Arithmetical.
_Sphere._ And what is the next number?
_I._ Six.
_Sphere._ Exactly. Then you see you have answered your own question. The
Cube which you will generate will be bounded by six sides, that is to
say, six of your insides. You see it all now, eh?
“Monster,” I shrieked, “be thou juggler, enchanter, dream, or devil, no
more will I endure thy mockeries. Either thou or I must perish.” And
saying these words I precipitated myself upon him.
§ 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 to nothing. Soon
there was a blank. But I still 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!”
§ 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
show you that which you have so 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: ]
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 pickpockets 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 beside 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.