PART II: OTHER WORLDS

"O brave new worlds, that have such people in them!"

Section 13. How I had a Vision of Lineland

It was the last day but one of the 1999th year of our era, and the

first day of the Long Vacation. Having amused myself till a late hour

with my favourite recreation of Geometry, I had retired to rest with an

unsolved problem in my mind. In the night I had a dream.

I saw before me a vast multitude of small Straight Lines (which I

naturally assumed to be Women) interspersed with other Beings still

smaller and of the nature of lustrous points--all moving to and fro in

one and the same Straight Line, and, as nearly as I could judge, with

the same velocity.

A noise of confused, multitudinous chirping or twittering issued from

them at intervals as long as they were moving; but sometimes they

ceased from motion, and then all was silence.

Approaching one of the largest of what I thought to be Women, I

accosted her, but received no answer. A second and a third appeal on

my part were equally ineffectual. Losing patience at what appeared to

me intolerable rudeness, I brought my mouth into a position full in

front of her mouth so as to intercept her motion, and loudly repeated

my question, "Woman, what signifies this concourse, and this strange

and confused chirping, and this monotonous motion to and fro in one and

the same Straight Line?"

[Illustration 6]

[ASCII approximation follows]

My view of Lineland

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| |

| Myself|

| |

My eye o--------

Women A boy Men The KING Men A boy Women

+ + + + - --- -- -- -- -- (>----<) -- -- -- -- --- - + + + +

^ ^

The KING'S eyes

much larger than the reality

shewing that HIS MAJESTY

could see nothing but a point.

"I am no Woman," replied the small Line. "I am the Monarch of the

world. But thou, whence intrudest thou into my realm of Lineland?"

Receiving this abrupt reply, I begged pardon if I had in any way

startled or molested his Royal Highness; and describing myself as a

stranger I besought the King to give me some account of his dominions.

But I had the greatest possible difficulty in obtaining any information

on points that really interested me; for the Monarch could not refrain

from constantly assuming that whatever was familiar to him must also be

known to me and that I was simulating ignorance in jest. However, by

persevering questions I elicited the following facts:

It seemed that this poor ignorant Monarch--as he called himself--was

persuaded that the Straight Line which he called his Kingdom, and in

which he passed his existence, constituted the whole of the world, and

indeed the whole of Space. Not being able either to move or to see,

save in his Straight Line, he had no conception of anything out of it.

Though he had heard my voice when I first addressed him, the sounds had

come to him in a manner so contrary to his experience that he had made

no answer, "seeing no man", as he expressed it, "and hearing a voice as

it were from my own intestines." Until the moment when I placed my

mouth in his World, he had neither seen me, nor heard anything except

confused sounds beating against--what I called his side, but what he

called his INSIDE or STOMACH; nor had he even now the least conception

of the region from which I had come. Outside his World, or Line, all

was a blank to him; nay, not even a blank, for a blank implies Space;

say, rather, all was non-existent.

His subjects--of whom the small Lines were men and the Points

Women--were all alike confined in motion and eye-sight to that single

Straight Line, which was their World. It need scarcely be added that

the whole of their horizon was limited to a Point; nor could any one

ever see anything but a Point. Man, woman, child, thing--each was a

Point to the eye of a Linelander. Only by the sound of the voice could

sex or age be distinguished. Moreover, as each individual occupied the

whole of the narrow path, so to speak, which constituted his Universe,

and no one could move to the right or left to make way for passers by,

it followed that no Linelander could ever pass another. Once

neighbours, always neighbours. Neighbourhood with them was like

marriage with us. Neighbours remained neighbours till death did them

part.

Such a life, with all vision limited to a Point, and all motion to a

Straight Line, seemed to me inexpressibly dreary; and I was surprised

to note the vivacity and cheerfulness of the King. Wondering whether

it was possible, amid circumstances so unfavourable to domestic

relations, to enjoy the pleasures of conjugal union, I hesitated for

some time to question his Royal Highness on so delicate a subject; but

at last I plunged into it by abruptly inquiring as to the health of his

family. "My wives and children," he replied, "are well and happy."

Staggered at this answer--for in the immediate proximity of the Monarch

(as I had noted in my dream before I entered Lineland) there were none

but Men--I ventured to reply, "Pardon me, but I cannot imagine how your

Royal Highness can at any time either see or approach their Majesties,

when there are at least half a dozen intervening individuals, whom you

can neither see through, nor pass by? Is it possible that in Lineland

proximity is not necessary for marriage and for the generation of

children?"

"How can you ask so absurd a question?" replied the Monarch. "If it

were indeed as you suggest, the Universe would soon be depopulated.

No, no; neighbourhood is needless for the union of hearts; and the

birth of children is too important a matter to have been allowed to

depend upon such an accident as proximity. You cannot be ignorant of

this. Yet since you are pleased to affect ignorance, I will instruct

you as if you were the veriest baby in Lineland. Know, then, that

marriages are consummated by means of the faculty of sound and the

sense of hearing.

"You are of course aware that every Man has two mouths or voices--as

well as two eyes--a bass at one and a tenor at the other of his

extremities. I should not mention this, but that I have been unable to

distinguish your tenor in the course of our conversation." I replied

that I had but one voice, and that I had not been aware that his Royal

Highness had two. "That confirms my impression," said the King, "that

you are not a Man, but a feminine Monstrosity with a bass voice, and an

utterly uneducated ear. But to continue.

"Nature having herself ordained that every Man should wed two wives--"

"Why two?" asked I. "You carry your affected simplicity too far", he

cried. "How can there be a completely harmonious union without the

combination of the Four in One, viz. the Bass and Tenor of the Man and

the Soprano and Contralto of the two Women?" "But supposing," said I,

"that a man should prefer one wife or three?" "It is impossible," he

said; "it is as inconceivable as that two and one should make five, or

that the human eye should see a Straight Line." I would have

interrupted him; but he proceeded as follows:

"Once in the middle of each week a Law of Nature compels us to move to

and fro with a rhythmic motion of more than usual violence, which

continues for the time you would take to count a hundred and one. In

the midst of this choral dance, at the fifty-first pulsation, the

inhabitants of the Universe pause in full career, and each individual

sends forth his richest, fullest, sweetest strain. It is in this

decisive moment that all our marriages are made. So exquisite is the

adaptation of Bass to Treble, of Tenor to Contralto, that oftentimes

the Loved Ones, though twenty thousand leagues away, recognize at once

the responsive note of their destined Lover; and, penetrating the

paltry obstacles of distance, Love unites the three. The marriage in

that instant consummated results in a threefold Male and Female

offspring which takes its place in Lineland."

"What! Always threefold?" said I. "Must one wife then always have

twins?"

"Bass-voiced Monstrosity! yes," replied the King. "How else could the

balance of the Sexes be maintained, if two girls were not born for

every boy? Would you ignore the very Alphabet of Nature?" He ceased,

speechless for fury; and some time elapsed before I could induce him to

resume his narrative.

"You will not, of course, suppose that every bachelor among us finds

his mates at the first wooing in this universal Marriage Chorus. On

the contrary, the process is by most of us many times repeated. Few

are the hearts whose happy lot it is at once to recognize in each

other's voices the partner intended for them by Providence, and to fly

into a reciprocal and perfectly harmonious embrace. With most of us

the courtship is of long duration. The Wooer's voices may perhaps

accord with one of the future wives, but not with both; or not, at

first, with either; or the Soprano and Contralto may not quite

harmonize. In such cases Nature has provided that every weekly Chorus

shall bring the three Lovers into closer harmony. Each trial of voice,

each fresh discovery of discord, almost imperceptibly induces the less

perfect to modify his or her vocal utterance so as to approximate to

the more perfect. And after many trials and many approximations, the

result is at last achieved. There comes a day at last, when, while the

wonted Marriage Chorus goes forth from universal Lineland, the three

far-off Lovers suddenly find themselves in exact harmony, and, before

they are awake, the wedded Triplet is rapt vocally into a duplicate

embrace; and Nature rejoices over one more marriage and over three more

births."

Section 14. How I vainly tried to explain the nature of Flatland

Thinking that it was time to bring down the Monarch from his raptures

to the level of common sense, I determined to endeavour to open up to

him some glimpses of the truth, that is to say of the nature of things

in Flatland. So I began thus: "How does your Royal Highness

distinguish the shapes and positions of his subjects? I for my part

noticed by the sense of sight, before I entered your Kingdom, that some

of your people are Lines and others Points, and that some of the Lines

are larger--" "You speak of an impossibility," interrupted the King;

"you must have seen a vision; for to detect the difference between a

Line and a Point by the sense of sight is, as every one knows, in the

nature of things, impossible; but it can be detected by the sense of

hearing, and by the same means my shape can be exactly ascertained.

Behold me--I am a Line, the longest in Lineland, over six inches of

Space--" "Of Length", I ventured to suggest. "Fool," said he, "Space

is Length. Interrupt me again, and I have done."

I apologized; but he continued scornfully, "Since you are impervious to

argument, you shall hear with your ears how by means of my two voices I

reveal my shape to my Wives, who are at this moment six thousand miles

seventy yards two feet eight inches away, the one to the North, the

other to the South. Listen, I call to them."

He chirruped, and then complacently continued: "My wives at this

moment receiving the sound of one of my voices, closely followed by the

other, and perceiving that the latter reaches them after an interval in

which sound can traverse 6.457 inches, infer that one of my mouths is

6.457 inches further from them than the other, and accordingly know my

shape to be 6.457 inches. But you will of course understand that my

wives do not make this calculation every time they hear my two voices.

They made it, once for all, before we were married. But they COULD

make it at any time. And in the same way I can estimate the shape of

any of my Male subjects by the sense of sound."

"But how," said I, "if a Man feigns a Woman's voice with one of his two

voices, or so disguises his Southern voice that it cannot be recognized

as the echo of the Northern? May not such deceptions cause great

inconvenience? And have you no means of checking frauds of this kind

by commanding your neighbouring subjects to feel one another?" This of

course was a very stupid question, for feeling could not have answered

the purpose; but I asked with the view of irritating the Monarch, and I

succeeded perfectly.

"What!" cried he in horror, "explain your meaning." "Feel, touch, come

into contact," I replied. "If you mean by FEELING," said the King,

"approaching so close as to leave no space between two individuals,

know, Stranger, that this offence is punishable in my dominions by

death. And the reason is obvious. The frail form of a Woman, being

liable to be shattered by such an approximation, must be preserved by

the State; but since Women cannot be distinguished by the sense of

sight from Men, the Law ordains universally that neither Man nor Woman

shall be approached so closely as to destroy the interval between the

approximator and the approximated.

"And indeed what possible purpose would be served by this illegal and

unnatural excess of approximation which you call TOUCHING, when all the

ends of so brutal and coarse a process are attained at once more easily

and more exactly by the sense of hearing? As to your suggested danger

of deception, it is non-existent: for the Voice, being the essence of

one's Being, cannot be thus changed at will. But come, suppose that I

had the power of passing through solid things, so that I could

penetrate my subjects, one after another, even to the number of a

billion, verifying the size and distance of each by the sense of

FEELING: how much time and energy would be wasted in this clumsy and

inaccurate method! Whereas now, in one moment of audition, I take as

it were the census and statistics, local, corporeal, mental and

spiritual, of every living being in Lineland. Hark, only hark!"

So saying he paused and listened, as if in an ecstasy, to a sound which

seemed to me no better than a tiny chirping from an innumerable

multitude of lilliputian grasshoppers.

"Truly," replied I, "your sense of hearing serves you in good stead,

and fills up many of your deficiencies. But permit me to point out

that your life in Lineland must be deplorably dull. To see nothing but

a Point! Not even to be able to contemplate a Straight Line! Nay, not

even to know what a Straight Line is! To see, yet be cut off from

those Linear prospects which are vouchsafed to us in Flatland! Better

surely to have no sense of sight at all than to see so little! I grant

you I have not your discriminative faculty of hearing; for the concert

of all Lineland which gives you such intense pleasure, is to me no

better than a multitudinous twittering or chirping. But at least I can

discern, by sight, a Line from a Point. And let me prove it. Just

before I came into your kingdom, I saw you dancing from left to right,

and then from right to left, with Seven Men and a Woman in your

immediate proximity on the left, and eight Men and two Women on your

right. Is not this correct?"

"It is correct," said the King, "so far as the numbers and sexes are

concerned, though I know not what you mean by 'right' and 'left'. But

I deny that you saw these things. For how could you see the Line, that

is to say the inside, of any Man? But you must have heard these

things, and then dreamed that you saw them. And let me ask what you

mean by those words 'left' and 'right'. I suppose it is your way of

saying Northward and Southward."

"Not so," replied I; "besides your motion of Northward and Southward,

there is another motion which I call from right to left."

KING. Exhibit to me, if you please, this motion from left to right.

I. Nay, that I cannot do, unless you could step out of your Line

altogether.

KING. Out of my Line? Do you mean out of the world? Out of Space?

I. Well, yes. Out of YOUR World. Out of YOUR Space. For your Space

is not the true Space. True Space is a Plane; but your Space is only a

Line.

KING. If you cannot indicate this motion from left to right by

yourself moving in it, then I beg you to describe it to me in words.

I. If you cannot tell your right side from your left, I fear that no

words of mine can make my meaning clear to you. But surely you cannot

be ignorant of so simple a distinction.

KING. I do not in the least understand you.

I. Alas! How shall I make it clear? When you move straight on, does

it not sometimes occur to you that you COULD move in some other way,

turning your eye round so as to look in the direction towards which

your side is now fronting? In other words, instead of always moving in

the direction of one of your extremities, do you never feel a desire to

move in the direction, so to speak, of your side?

KING. Never. And what do you mean? How can a man's inside "front" in

any direction? Or how can a man move in the direction of his inside?

I. Well then, since words cannot explain the matter, I will try deeds,

and will move gradually out of Lineland in the direction which I desire

to indicate to you.

At the word I began to move my body out of Lineland. As long as any

part of me remained in his dominion and in his view, the King kept

exclaiming, "I see you, I see you still; you are not moving." But when

I had at last moved myself out of his Line, he cried in his shrillest

voice, "She is vanished; she is dead." "I am not dead," replied I; "I

am simply out of Lineland, that is to say, out of the Straight Line

which you call Space, and in the true Space, where I can see things as

they are. And at this moment I can see your Line, or side--or inside

as you are pleased to call it; and I can see also the Men and Women on

the North and South of you, whom I will now enumerate, describing their

order, their size, and the interval between each."

[Illustration 7]

[ASCII approximation follows]

My body just before I disappeared

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|\ \ \ \ \|

|\ \ \ \ \|

|\ \ \ \ \|

Lineland ----> |\ \ \ \ \| The King

--------------------+---------+--------------========

When I had done this at great length, I cried triumphantly, "Does that

at last convince you?" And, with that, I once more entered Lineland,

taking up the same position as before.

But the Monarch replied, "If you were a Man of sense--though, as you

appear to have only one voice I have little doubt you are not a Man but

a Woman--but, if you had a particle of sense, you would listen to

reason. You ask me to believe that there is another Line besides that

which my senses indicate, and another motion besides that of which I am

daily conscious. I, in return, ask you to describe in words or

indicate by motion that other Line of which you speak. Instead of

moving, you merely exercise some magic art of vanishing and returning

to sight; and instead of any lucid description of your new World, you

simply tell me the numbers and sizes of some forty of my retinue, facts

known to any child in my capital. Can anything be more irrational or

audacious? Acknowledge your folly or depart from my dominions."

Furious at his perversity, and especially indignant that he professed

to be ignorant of my sex, I retorted in no measured terms, "Besotted

Being! You think yourself the perfection of existence, while you are

in reality the most imperfect and imbecile. You profess to see,

whereas you can see nothing but a Point! You plume yourself on

inferring the existence of a Straight Line; but I CAN SEE Straight

Lines, and infer the existence of Angles, Triangles, Squares,

Pentagons, Hexagons, and even Circles. Why waste more words? Suffice

it that I am the completion of your incomplete self. You are a Line,

but I am a Line of Lines, called in my country a Square: and even I,

infinitely superior though I am to you, am of little account among the

great nobles of Flatland, whence I have come to visit you, in the hope

of enlightening your ignorance."

Hearing these words the King advanced towards me with a menacing cry as

if to pierce me through the diagonal; and in that same moment there

arose from myriads of his subjects a multitudinous war-cry, increasing

in vehemence till at last methought it rivalled the roar of an army of

a hundred thousand Isosceles, and the artillery of a thousand

Pentagons. Spell-bound and motionless, I could neither speak nor move

to avert the impending destruction; and still the noise grew louder,

and the King came closer, when I awoke to find the breakfast-bell

recalling me to the realities of Flatland.

Section 15. Concerning a Stranger from Spaceland

From dreams I proceed to facts.

It was the last day of the 1999th year of our era. The pattering of

the rain had long ago announced nightfall; and I was sitting in the

company of my wife, musing on the events of the past and the prospects

of the coming year, the coming century, the coming Millennium.

[Note: When I say "sitting", of course I do not mean any change of

attitude such as you in Spaceland signify by that word; for as we have

no feet, we can no more "sit" nor "stand" (in your sense of the word)

than one of your soles or flounders.

Nevertheless, we perfectly well recognize the different mental states

of volition implied in "lying", "sitting", and "standing", which are to

some extent indicated to a beholder by a slight increase of lustre

corresponding to the increase of volition.

But on this, and a thousand other kindred subjects, time forbids me to

dwell.]

My four Sons and two orphan Grandchildren had retired to their several

apartments; and my wife alone remained with me to see the old

Millennium out and the new one in.

I was rapt in thought, pondering in my mind some words that had

casually issued from the mouth of my youngest Grandson, a most

promising young Hexagon of unusual brilliancy and perfect angularity.

His uncles and I had been giving him his usual practical lesson in

Sight Recognition, turning ourselves upon our centres, now rapidly, now

more slowly, and questioning him as to our positions; and his answers

had been so satisfactory that I had been induced to reward him by

giving him a few hints on Arithmetic, as applied to Geometry.

Taking nine Squares, each an inch every way, I had put them together so

as to make one large Square, with a side of three inches, and I had

hence proved to my little Grandson that--though it was impossible for

us to SEE the inside of the Square--yet we might ascertain the number

of square inches in a Square by simply squaring the number of inches in

the side: "and thus," said I, "we know that 3^2, or 9, represents the

number of square inches in a Square whose side is 3 inches long."

The little Hexagon meditated on this a while and then said to me; "But

you have been teaching me to raise numbers to the third power: I

suppose 3^3 must mean something in Geometry; what does it mean?"

"Nothing at all," replied I, "not at least in Geometry; for Geometry

has only Two Dimensions." And then I began to shew the boy how a Point

by moving through a length of three inches makes a Line of three

inches, which may be represented by 3; and how a Line of three inches,

moving parallel to itself through a length of three inches, makes a

Square of three inches every way, which may be represented by 3^2.

Upon this, my Grandson, again returning to his former suggestion, took

me up rather suddenly and exclaimed, "Well, then, if a Point by moving

three inches, makes a Line of three inches represented by 3; and if a

straight Line of three inches, moving parallel to itself, makes a

Square of three inches every way, represented by 3^2; it must be that a

Square of three inches every way, moving somehow parallel to itself

(but I don't see how) must make Something else (but I don't see what)

of three inches every way--and this must be represented by 3^3."

"Go to bed," said I, a little ruffled by this interruption: "if you

would talk less nonsense, you would remember more sense."

So my Grandson had disappeared in disgrace; and there I sat by my

Wife's side, endeavouring to form a retrospect of the year 1999 and of

the possibilities of the year 2000, but not quite able to shake off the

thoughts suggested by the prattle of my bright little Hexagon. Only a

few sands now remained in the half-hour glass. Rousing myself from my

reverie I turned the glass Northward for the last time in the old

Millennium; and in the act, I exclaimed aloud, "The boy is a fool."

Straightway I became conscious of a Presence in the room, and a

chilling breath thrilled through my very being. "He is no such thing,"

cried my Wife, "and you are breaking the Commandments in thus

dishonouring your own Grandson." But I took no notice of her. Looking

round in every direction I could see nothing; yet still I FELT a

Presence, and shivered as the cold whisper came again. I started up.

"What is the matter?" said my Wife, "there is no draught; what are you

looking for? There is nothing." There was nothing; and I resumed my

seat, again exclaiming, "The boy is a fool, I say; 3^3 can have no

meaning in Geometry." At once there came a distinctly audible reply,

"The boy is not a fool; and 3^3 has an obvious Geometrical meaning."

My Wife as well as myself heard the words, although she did not

understand their meaning, and both of us sprang forward in the

direction of the sound. What was our horror when we saw before us a

Figure! At the first glance it appeared to be a Woman, seen sideways;

but a moment's observation shewed me that the extremities passed into

dimness too rapidly to represent one of the Female Sex; and I should

have thought it a Circle, only that it seemed to change its size in a

manner impossible for a Circle or for any regular Figure of which I had

had experience.

But my Wife had not my experience, nor the coolness necessary to note

these characteristics. With the usual hastiness and unreasoning

jealousy of her Sex, she flew at once to the conclusion that a Woman

had entered the house through some small aperture. "How comes this

person here?" she exclaimed, "you promised me, my dear, that there

should be no ventilators in our new house." "Nor are there any," said

I; "but what makes you think that the stranger is a Woman? I see by my

power of Sight Recognition----" "Oh, I have no patience with your Sight

Recognition," replied she, "'Feeling is believing' and 'A Straight Line

to the touch is worth a Circle to the sight'"--two Proverbs, very

common with the Frailer Sex in Flatland.

"Well," said I, for I was afraid of irritating her, "if it must be so,

demand an introduction." Assuming her most gracious manner, my Wife

advanced towards the Stranger, "Permit me, Madam, to feel and be felt

by----" then, suddenly recoiling, "Oh! it is not a Woman, and there

are no angles either, not a trace of one. Can it be that I have so

misbehaved to a perfect Circle?"

"I am indeed, in a certain sense a Circle," replied the Voice, "and a

more perfect Circle than any in Flatland; but to speak more accurately,

I am many Circles in one." Then he added more mildly, "I have a

message, dear Madam, to your husband, which I must not deliver in your

presence; and, if you would suffer us to retire for a few minutes----"

But my Wife would not listen to the proposal that our august Visitor

should so incommode himself, and assuring the Circle that the hour of

her own retirement had long passed, with many reiterated apologies for

her recent indiscretion, she at last retreated to her apartment.

I glanced at the half-hour glass. The last sands had fallen. The

third Millennium had begun.

Section 16. How the Stranger vainly endeavoured to reveal to me

in words the mysteries of Spaceland

As soon as the sound of the Peace-cry of my departing Wife had died

away, I began to approach the Stranger with the intention of taking a

nearer view and of bidding him be seated: but his appearance struck me

dumb and motionless with astonishment. Without the slightest symptoms

of angularity he nevertheless varied every instant with gradations of

size and brightness scarcely possible for any Figure within the scope

of my experience. The thought flashed across me that I might have

before me a burglar or cut-throat, some monstrous Irregular Isosceles,

who, by feigning the voice of a Circle, had obtained admission somehow

into the house, and was now preparing to stab me with his acute angle.

In a sitting-room, the absence of Fog (and the season happened to be

remarkably dry), made it difficult for me to trust to Sight

Recognition, especially at the short distance at which I was standing.

Desperate with fear, I rushed forward with an unceremonious, "You must

permit me, Sir--" and felt him. My Wife was right. There was not the

trace of an angle, not the slightest roughness or inequality: never in

my life had I met with a more perfect Circle. He remained motionless

while I walked round him, beginning from his eye and returning to it

again. Circular he was throughout, a perfectly satisfactory Circle;

there could not be a doubt of it. Then followed a dialogue, which I

will endeavour to set down as near as I can recollect it, omitting only

some of my profuse apologies--for I was covered with shame and

humiliation that I, a Square, should have been guilty of the

impertinence of feeling a Circle. It was commenced by the Stranger

with some impatience at the lengthiness of my introductory process.

STRANGER. Have you felt me enough by this time? Are you not

introduced to me yet?

I. Most illustrious Sir, excuse my awkwardness, which arises not from

ignorance of the usages of polite society, but from a little surprise

and nervousness, consequent on this somewhat unexpected visit. And I

beseech you to reveal my indiscretion to no one, and especially not to

my Wife. But before your Lordship enters into further communications,

would he deign to satisfy the curiosity of one who would gladly know

whence his Visitor came?

STRANGER. From Space, from Space, Sir: whence else?

I. Pardon me, my Lord, but is not your Lordship already in Space, your

Lordship and his humble servant, even at this moment?

STRANGER. Pooh! what do you know of Space? Define Space.

I. Space, my Lord, is height and breadth indefinitely prolonged.

STRANGER. Exactly: you see you do not even know what Space is. You

think it is of Two Dimensions only; but I have come to announce to you

a Third--height, breadth, and length.

I. Your Lordship is pleased to be merry. We also speak of length and

height, or breadth and thickness, thus denoting Two Dimensions by four

names.

STRANGER. But I mean not only three names, but Three Dimensions.

I. Would your Lordship indicate or explain to me in what direction is

the Third Dimension, unknown to me?

STRANGER. I came from it. It is up above and down below.

I. My Lord means seemingly that it is Northward and Southward.

STRANGER. I mean nothing of the kind. I mean a direction in which you

cannot look, because you have no eye in your side.

I. Pardon me, my Lord, a moment's inspection will convince your

Lordship that I have a perfect luminary at the juncture of two of my

sides.

STRANGER. Yes: but in order to see into Space you ought to have an

eye, not on your Perimeter, but on your side, that is, on what you

would probably call your inside; but we in Spaceland should call it

your side.

I. An eye in my inside! An eye in my stomach! Your Lordship jests.

STRANGER. I am in no jesting humour. I tell you that I come from

Space, or, since you will not understand what Space means, from the

Land of Three Dimensions whence I but lately looked down upon your

Plane which you call Space forsooth. From that position of advantage I

discerned all that you speak of as SOLID (by which you mean "enclosed

on four sides"), your houses, your churches, your very chests and

safes, yes even your insides and stomachs, all lying open and exposed

to my view.

I. Such assertions are easily made, my Lord.

STRANGER. But not easily proved, you mean. But I mean to prove mine.

When I descended here, I saw your four Sons, the Pentagons, each in his

apartment, and your two Grandsons the Hexagons; I saw your youngest

Hexagon remain a while with you and then retire to his room, leaving

you and your Wife alone. I saw your Isosceles servants, three in

number, in the kitchen at supper, and the little Page in the scullery.

Then I came here, and how do you think I came?

I. Through the roof, I suppose.

STRANGER. Not so. Your roof, as you know very well, has been recently

repaired, and has no aperture by which even a Woman could penetrate. I

tell you I come from Space. Are you not convinced by what I have told

you of your children and household?

I. Your Lordship must be aware that such facts touching the belongings

of his humble servant might be easily ascertained by any one in the

neighbourhood possessing your Lordship's ample means of obtaining

information.

STRANGER. (TO HIMSELF.) What must I do? Stay; one more argument

suggests itself to me. When you see a Straight Line--your wife, for

example--how many Dimensions do you attribute to her?

I. Your Lordship would treat me as if I were one of the vulgar who,

being ignorant of Mathematics, suppose that a Woman is really a

Straight Line, and only of One Dimension. No, no, my Lord; we Squares

are better advised, and are as well aware as your Lordship that a

Woman, though popularly called a Straight Line, is, really and

scientifically, a very thin Parallelogram, possessing Two Dimensions,

like the rest of us, viz., length and breadth (or thickness).

STRANGER. But the very fact that a Line is visible implies that it

possesses yet another Dimension.

I. My Lord, I have just acknowledged that a Woman is broad as well as

long. We see her length, we infer her breadth; which, though very

slight, is capable of measurement.

STRANGER. You do not understand me. I mean that when you see a Woman,

you ought--besides inferring her breadth--to see her length, and to SEE

what we call her HEIGHT; although that last Dimension is infinitesimal

in your country. If a Line were mere length without "height", it would

cease to occupy Space and would become invisible. Surely you must

recognize this?

I. I must indeed confess that I do not in the least understand your

Lordship. When we in Flatland see a Line, we see length and

BRIGHTNESS. If the brightness disappears, the Line is extinguished,

and, as you say, ceases to occupy Space. But am I to suppose that your

Lordship gives to brightness the title of a Dimension, and that what we

call "bright" you call "high"?

STRANGER. No, indeed. By "height" I mean a Dimension like your

length: only, with you, "height" is not so easily perceptible, being

extremely small.

I. My Lord, your assertion is easily put to the test. You say I have

a Third Dimension, which you call "height". Now, Dimension implies

direction and measurement. Do but measure my "height", or merely

indicate to me the direction in which my "height" extends, and I will

become your convert. Otherwise, your Lordship's own understanding must

hold me excused.

STRANGER. (TO HIMSELF.) I can do neither. How shall I convince him?

Surely a plain statement of facts followed by ocular demonstration

ought to suffice. --Now, Sir; listen to me.

You are living on a Plane. What you style Flatland is the vast level

surface of what I may call a fluid, on, or in, the top of which you and

your countrymen move about, without rising above it or falling below it.

I am not a plane Figure, but a Solid. You call me a Circle; but in

reality I am not a Circle, but an infinite number of Circles, of size

varying from a Point to a Circle of thirteen inches in diameter, one

placed on the top of the other. When I cut through your plane as I am

now doing, I make in your plane a section which you, very rightly, call

a Circle. For even a Sphere--which is my proper name in my own

country--if he manifest himself at all to an inhabitant of

Flatland--must needs manifest himself as a Circle.

Do you not remember--for I, who see all things, discerned last night

the phantasmal vision of Lineland written upon your brain--do you not

remember, I say, how, when you entered the realm of Lineland, you were

compelled to manifest yourself to the King, not as a Square, but as a

Line, because that Linear Realm had not Dimensions enough to represent

the whole of you, but only a slice or section of you? In precisely the

same way, your country of Two Dimensions is not spacious enough to

represent me, a being of Three, but can only exhibit a slice or section

of me, which is what you call a Circle.

The diminished brightness of your eye indicates incredulity. But now

prepare to receive proof positive of the truth of my assertions. You

cannot indeed see more than one of my sections, or Circles, at a time;

for you have no power to raise your eye out of the plane of Flatland;

but you can at least see that, as I rise in Space, so my sections

become smaller. See now, I will rise; and the effect upon your eye

will be that my Circle will become smaller and smaller till it dwindles

to a point and finally vanishes.

[Illustration 8]

[ASCII approximation follows]

The Sphere on the

point of vanishing

(2) __-----__

The Sphere with The Sphere rising / \ (3)

his section __-----__ / \

at full size / \ | |

__-----__ / \ | |

/ \ | | | |

/ __ - __ \ | | \ / My

| -- -- | | __ --- __ | \ __ __ / Eye

--|-----------------|----\--__-------__--/------------===----------+(>

| -- __ __ -- | \ __ --- __ /

\ - / -----

\ __ __ /

-----

There was no "rising" that I could see; but he diminished and finally

vanished. I winked once or twice to make sure that I was not dreaming.

But it was no dream. For from the depths of nowhere came forth a

hollow voice--close to my heart it seemed--"Am I quite gone? Are you

convinced now? Well, now I will gradually return to Flatland and you

shall see my section become larger and larger."

Every reader in Spaceland will easily understand that my mysterious

Guest was speaking the language of truth and even of simplicity. But

to me, proficient though I was in Flatland Mathematics, it was by no

means a simple matter. The rough diagram given above will make it

clear to any Spaceland child that the Sphere, ascending in the three

positions indicated there, must needs have manifested himself to me, or

to any Flatlander, as a Circle, at first of full size, then small, and

at last very small indeed, approaching to a Point. But to me, although

I saw the facts before me, the causes were as dark as ever. All that I

could comprehend was, that the Circle had made himself smaller and

vanished, and that he had now reappeared and was rapidly making himself

larger.

When he regained his original size, he heaved a deep sigh; for he

perceived by my silence that I had altogether failed to comprehend him.

And indeed I was now inclining to the belief that he must be no Circle

at all, but some extremely clever juggler; or else that the old wives'

tales were true, and that after all there were such people as

Enchanters and Magicians.

After a long pause he muttered to himself, "One resource alone remains,

if I am not to resort to action. I must try the method of Analogy."

Then followed a still longer silence, after which he continued our

dialogue.

SPHERE. Tell me, Mr. Mathematician; if a Point moves Northward, and

leaves a luminous wake, what name would you give to the wake?

I. A straight Line.

SPHERE. And a straight Line has how many extremities?

I. Two.

SPHERE. Now conceive the Northward straight Line moving parallel to

itself, East and West, so that every point in it leaves behind it the

wake of a straight Line. What name will you give to the Figure thereby

formed? We will suppose that it moves through a distance equal to the

original straight Line. --What name, I say?

I. A Square.

SPHERE. And how many sides has a Square? How many angles?

I. Four sides and four angles.

SPHERE. Now stretch your imagination a little, and conceive a Square

in Flatland, moving parallel to itself upward.

I. What? Northward?

SPHERE. No, not Northward; upward; out of Flatland altogether.

If it moved Northward, the Southern points in the Square would have to

move through the positions previously occupied by the Northern points.

But that is not my meaning.

I mean that every Point in you--for you are a Square and will serve the

purpose of my illustration--every Point in you, that is to say in what

you call your inside, is to pass upwards through Space in such a way

that no Point shall pass through the position previously occupied by

any other Point; but each Point shall describe a straight Line of its

own. This is all in accordance with Analogy; surely it must be clear

to you.

Restraining my impatience--for I was now under a strong temptation to

rush blindly at my Visitor and to precipitate him into Space, or out of

Flatland, anywhere, so that I could get rid of him--I replied:--

"And what may be the nature of the Figure which I am to shape out by

this motion which you are pleased to denote by the word 'upward'? I

presume it is describable in the language of Flatland."

SPHERE. Oh, certainly. It is all plain and simple, and in strict

accordance with Analogy--only, by the way, you must not speak of the

result as being a Figure, but as a Solid. But I will describe it to

you. Or rather not I, but Analogy.

We began with a single Point, which of course--being itself a

Point--has only ONE terminal Point.

One Point produces a Line with TWO terminal Points.

One Line produces a Square with FOUR terminal Points.

Now you can give yourself the answer to your own question: 1, 2, 4,

are evidently in Geometrical Progression. What is the next number?

I. Eight.

SPHERE. Exactly. The one Square produces a SOMETHING-WHICH-

YOU-DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE with EIGHT

terminal Points. Now are you convinced?

I. And has this Creature sides, as well as angles or what you call

"terminal Points"?

SPHERE. Of course; and all according to Analogy. But, by the way, not

what YOU call sides, but what WE call sides. You would call them

SOLIDS.

I. And how many solids or sides will appertain to this Being whom I am

to generate by the motion of my inside in an "upward" direction, and

whom you call a Cube?

SPHERE. How can you ask? And you a mathematician! The side of

anything is always, if I may so say, one Dimension behind the thing.

Consequently, as there is no Dimension behind a Point, a Point has 0

sides; a Line, if I may say, has 2 sides (for the Points of a Line may

be called by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what

Progression do you call that?

I. Arithmetical.

SPHERE. And what is the next number?

I. Six.

SPHERE. Exactly. Then you see you have answered your own question.

The Cube which you will generate will be bounded by six sides, that is

to say, six of your insides. You see it all now, eh?

"Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no

more will I endure thy mockeries. Either thou or I must perish." And

saying these words I precipitated myself upon him.

Section 17. How the Sphere, having in vain tried words,

resorted to deeds

It was in vain. I brought my hardest right angle into violent

collision with the Stranger, pressing on him with a force sufficient to

have destroyed any ordinary Circle: but I could feel him slowly and

unarrestably slipping from my contact; no edging to the right nor to

the left, but moving somehow out of the world, and vanishing to

nothing. Soon there was a blank. But still I heard the Intruder's

voice.

SPHERE. Why will you refuse to listen to reason? I had hoped to find

in you--as being a man of sense and an accomplished mathematician--a

fit apostle for the Gospel of the Three Dimensions, which I am allowed

to preach once only in a thousand years: but now I know not how to

convince you. Stay, I have it. Deeds, and not words, shall proclaim

the truth. Listen, my friend.

I have told you I can see from my position in Space the inside of all

things that you consider closed. For example, I see in yonder cupboard

near which you are standing, several of what you call boxes (but like

everything else in Flatland, they have no tops nor bottoms) full of

money; I see also two tablets of accounts. I am about to descend into

that cupboard and to bring you one of those tablets. I saw you lock

the cupboard half an hour ago, and I know you have the key in your

possession. But I descend from Space; the doors, you see, remain

unmoved. Now I am in the cupboard and am taking the tablet. Now I

have it. Now I ascend with it.

I rushed to the closet and dashed the door open. One of the tablets

was gone. With a mocking laugh, the Stranger appeared in the other

corner of the room, and at the same time the tablet appeared upon the

floor. I took it up. There could be no doubt--it was the missing

tablet.

I groaned with horror, doubting whether I was not out of my senses; but

the Stranger continued: "Surely you must now see that my explanation,

and no other, suits the phenomena. What you call Solid things are

really superficial; what you call Space is really nothing but a great

Plane. I am in Space, and look down upon the insides of the things of

which you only see the outsides. You could leave this Plane yourself,

if you could but summon up the necessary volition. A slight upward or

downward motion would enable you to see all that I can see.

"The higher I mount, and the further I go from your Plane, the more I

can see, though of course I see it on a smaller scale. For example, I

am ascending; now I can see your neighbour the Hexagon and his family

in their several apartments; now I see the inside of the Theatre, ten

doors off, from which the audience is only just departing; and on the

other side a Circle in his study, sitting at his books. Now I shall

come back to you. And, as a crowning proof, what do you say to my

giving you a touch, just the least touch, in your stomach? It will not

seriously injure you, and the slight pain you may suffer cannot be

compared with the mental benefit you will receive."

Before I could utter a word of remonstrance, I felt a shooting pain in

my inside, and a demoniacal laugh seemed to issue from within me. A

moment afterwards the sharp agony had ceased, leaving nothing but a

dull ache behind, and the Stranger began to reappear, saying, as he

gradually increased in size, "There, I have not hurt you much, have I?

If you are not convinced now, I don't know what will convince you.

What say you?"

My resolution was taken. It seemed intolerable that I should endure

existence subject to the arbitrary visitations of a Magician who could

thus play tricks with one's very stomach. If only I could in any way

manage to pin him against the wall till help came!

Once more I dashed my hardest angle against him, at the same time

alarming the whole household by my cries for aid. I believe, at the

moment of my onset, the Stranger had sunk below our Plane, and really

found difficulty in rising. In any case he remained motionless, while

I, hearing, as I thought, the sound of some help approaching, pressed

against him with redoubled vigour, and continued to shout for

assistance.

A convulsive shudder ran through the Sphere. "This must not be," I

thought I heard him say: "either he must listen to reason, or I must

have recourse to the last resource of civilization." Then, addressing

me in a louder tone, he hurriedly exclaimed, "Listen: no stranger must

witness what you have witnessed. Send your Wife back at once, before

she enters the apartment. The Gospel of Three Dimensions must not be

thus frustrated. Not thus must the fruits of one thousand years of

waiting be thrown away. I hear her coming. Back! back! Away from me,

or you must go with me--whither you know not--into the Land of Three

Dimensions!"

"Fool! Madman! Irregular!" I exclaimed; "never will I release thee;

thou shalt pay the penalty of thine impostures."

"Ha! Is it come to this?" thundered the Stranger: "then meet your

fate: out of your Plane you go. Once, twice, thrice! 'Tis done!"

Section 18. How I came to Spaceland, and what I saw there

An unspeakable horror seized me. There was a darkness; then a dizzy,

sickening sensation of sight that was not like seeing; I saw a Line

that was no Line; Space that was not Space: I was myself, and not

myself. When I could find voice, I shrieked aloud in agony, "Either

this is madness or it is Hell." "It is neither," calmly replied the

voice of the Sphere, "it is Knowledge; it is Three Dimensions: open

your eye once again and try to look steadily."

I looked, and, behold, a new world! There stood before me, visibly

incorporate, all that I had before inferred, conjectured, dreamed, of

perfect Circular beauty. What seemed the centre of the Stranger's form

lay open to my view: yet I could see no heart, nor lungs, nor

arteries, only a beautiful harmonious Something--for which I had no

words; but you, my Readers in Spaceland, would call it the surface of

the Sphere.

Prostrating myself mentally before my Guide, I cried, "How is it, O

divine ideal of consummate loveliness and wisdom that I see thy inside,

and yet cannot discern thy heart, thy lungs, thy arteries, thy liver?"

"What you think you see, you see not," he replied; "it is not given to

you, nor to any other Being to behold my internal parts. I am of a

different order of Beings from those in Flatland. Were I a Circle, you

could discern my intestines, but I am a Being, composed as I told you

before, of many Circles, the Many in the One, called in this country a

Sphere. And, just as the outside of a Cube is a Square, so the outside

of a Sphere presents the appearance of a Circle."

Bewildered though I was by my Teacher's enigmatic utterance, I no

longer chafed against it, but worshipped him in silent adoration. He

continued, with more mildness in his voice. "Distress not yourself if

you cannot at first understand the deeper mysteries of Spaceland. By

degrees they will dawn upon you. Let us begin by casting back a glance

at the region whence you came. Return with me a while to the plains of

Flatland, and I will shew you that which you have often reasoned and

thought about, but never seen with the sense of sight--a visible

angle." "Impossible!" I cried; but, the Sphere leading the way, I

followed as if in a dream, till once more his voice arrested me: "Look

yonder, and behold your own Pentagonal house, and all its inmates."

I looked below, and saw with my physical eye all that domestic

individuality which I had hitherto merely inferred with the

understanding. And how poor and shadowy was the inferred conjecture in

comparison with the reality which I now beheld! My four Sons calmly

asleep in the North-Western rooms, my two orphan Grandsons to the

South; the Servants, the Butler, my Daughter, all in their several

apartments. Only my affectionate Wife, alarmed by my continued

absence, had quitted her room and was roving up and down in the Hall,

anxiously awaiting my return. Also the Page, aroused by my cries, had

left his room, and under pretext of ascertaining whether I had fallen

somewhere in a faint, was prying into the cabinet in my study. All

this I could now SEE, not merely infer; and as we came nearer and

nearer, I could discern even the contents of my cabinet, and the two

chests of gold, and the tablets of which the Sphere had made mention.

[Illustration 9]

[ASCII approximation follows]

/\

/ |My \

/ <> |Study \

/______ | ___ \

/ <> My Sons\ \|The \

/______/ \ Page / \

N / <> \ / My \

^ /______/ THE HALL \ Bedroom \

| \ <> My\ /

| \____| /\Wife's /

W--+--E \ My Wife / Apartment/

| ------- /\ --- \ WOMEN'S DOOR

| MEN'S DOOR \My Daughter

| /\ --== \ / The Scullion

S \ My Grandsons \ -==# \/ The Footman

\___ ___ _ _/ \-=#|/ The Butler

\ <> | <> | |THE CELLAR \ /

\____|____|_|____________/

###===--- ---===###

Policeman Policeman

Touched by my Wife's distress, I would have sprung downward to reassure

her, but I found myself incapable of motion. "Trouble not yourself

about your Wife," said my Guide: "she will not be long left in anxiety;

meantime, let us take a survey of Flatland."

Once more I felt myself rising through space. It was even as the

Sphere had said. The further we receded from the object we beheld, the

larger became the field of vision. My native city, with the interior

of every house and every creature therein, lay open to my view in

miniature. We mounted higher, and lo, the secrets of the earth, the

depths of mines and inmost caverns of the hills, were bared before me.

Awestruck at the sight of the mysteries of the earth, thus unveiled

before my unworthy eye, I said to my Companion, "Behold, I am become as

a God. For the wise men in our country say that to see all things, or

as they express it, OMNIVIDENCE, is the attribute of God alone." There

was something of scorn in the voice of my Teacher as he made answer:

"Is it so indeed? Then the very pick-pockets and cut-throats of my

country are to be worshipped by your wise men as being Gods: for there

is not one of them that does not see as much as you see now. But trust

me, your wise men are wrong."

I. Then is omnividence the attribute of others besides Gods?

SPHERE. I do not know. But, if a pick-pocket or a cut-throat of our

country can see everything that is in your country, surely that is no

reason why the pick-pocket or cut-throat should be accepted by you as a

God. This omnividence, as you call it--it is not a common word in

Spaceland--does it make you more just, more merciful, less selfish,

more loving? Not in the least. Then how does it make you more divine?

I. "More merciful, more loving!" But these are the qualities of

women! And we know that a Circle is a higher Being than a Straight

Line, in so far as knowledge and wisdom are more to be esteemed than

mere affection.

SPHERE. It is not for me to classify human faculties according to

merit. Yet many of the best and wisest in Spaceland think more of the

affections than of the understanding, more of your despised Straight

Lines than of your belauded Circles. But enough of this. Look yonder.

Do you know that building?

I looked, and afar off I saw an immense Polygonal structure, in which I

recognized the General Assembly Hall of the States of Flatland,

surrounded by dense lines of Pentagonal buildings at right angles to

each other, which I knew to be streets; and I perceived that I was

approaching the great Metropolis.

"Here we descend," said my Guide. It was now morning, the first hour

of the first day of the two thousandth year of our era. Acting, as was

their wont, in strict accordance with precedent, the highest Circles of

the realm were meeting in solemn conclave, as they had met on the first

hour of the first day of the year 1000, and also on the first hour of

the first day of the year 0.

The minutes of the previous meetings were now read by one whom I at

once recognized as my brother, a perfectly Symmetrical Square, and the

Chief Clerk of the High Council. It was found recorded on each

occasion that: "Whereas the States had been troubled by divers

ill-intentioned persons pretending to have received revelations from

another World, and professing to produce demonstrations whereby they

had instigated to frenzy both themselves and others, it had been for

this cause unanimously resolved by the Grand Council that on the first

day of each millenary, special injunctions be sent to the Prefects in

the several districts of Flatland, to make strict search for such

misguided persons, and without formality of mathematical examination,

to destroy all such as were Isosceles of any degree, to scourge and

imprison any regular Triangle, to cause any Square or Pentagon to be

sent to the district Asylum, and to arrest any one of higher rank,

sending him straightway to the Capital to be examined and judged by the

Council."

"You hear your fate," said the Sphere to me, while the Council was

passing for the third time the formal resolution. "Death or

imprisonment awaits the Apostle of the Gospel of Three Dimensions."

"Not so," replied I, "the matter is now so clear to me, the nature of

real space so palpable, that methinks I could make a child understand

it. Permit me but to descend at this moment and enlighten them." "Not

yet," said my Guide, "the time will come for that. Meantime I must

perform my mission. Stay thou there in thy place." Saying these

words, he leaped with great dexterity into the sea (if I may so call

it) of Flatland, right in the midst of the ring of Counsellors. "I

come," cried he, "to proclaim that there is a land of Three Dimensions."

I could see many of the younger Counsellors start back in manifest

horror, as the Sphere's circular section widened before them. But on a

sign from the presiding Circle--who shewed not the slightest alarm or

surprise--six Isosceles of a low type from six different quarters

rushed upon the Sphere. "We have him," they cried; "No; yes; we have

him still! he's going! he's gone!"

"My Lords," said the President to the Junior Circles of the Council,

"there is not the slightest need for surprise; the secret archives, to

which I alone have access, tell me that a similar occurrence happened

on the last two millennial commencements. You will, of course, say

nothing of these trifles outside the Cabinet."

Raising his voice, he now summoned the guards. "Arrest the policemen;

gag them. You know your duty." After he had consigned to their fate

the wretched policemen--ill-fated and unwilling witnesses of a

State-secret which they were not to be permitted to reveal--he again

addressed the Counsellors. "My Lords, the business of the Council

being concluded, I have only to wish you a happy New Year." Before

departing, he expressed, at some length, to the Clerk, my excellent but

most unfortunate brother, his sincere regret that, in accordance with

precedent and for the sake of secrecy, he must condemn him to perpetual

imprisonment, but added his satisfaction that, unless some mention were

made by him of that day's incident, his life would be spared.

Section 19. How, though the Sphere shewed me other mysteries

of Spaceland, I still desired more; and what came of it

When I saw my poor brother led away to imprisonment, I attempted to

leap down into the Council Chamber, desiring to intercede on his

behalf, or at least bid him farewell. But I found that I had no motion

of my own. I absolutely depended on the volition of my Guide, who said

in gloomy tones, "Heed not thy brother; haply thou shalt have ample

time hereafter to condole with him. Follow me."

[Illustration 10]

[ASCII approximation follows]

(1) (2)

__________ __________

|\ |\ | \

| \ | \ | \

| \ ____|____\ | \

| | | | | |

|_____|____| | | |

\ | \ | \ |

\ | \ | \ |

\|_________\| \ __________|

Once more we ascended into space. "Hitherto," said the Sphere, "I have

shewn you naught save Plane Figures and their interiors. Now I must

introduce you to Solids, and reveal to you the plan upon which they are

constructed. Behold this multitude of moveable square cards. See, I

put one on another, not, as you supposed, Northward of the other, but

ON the other. Now a second, now a third. See, I am building up a

Solid by a multitude of Squares parallel to one another. Now the Solid

is complete, being as high as it is long and broad, and we call it a

Cube."

"Pardon me, my Lord," replied I; "but to my eye the appearance is as of

an Irregular Figure whose inside is laid open to the view; in other

words, methinks I see no Solid, but a Plane such as we infer in

Flatland; only of an Irregularity which betokens some monstrous

criminal, so that the very sight of it is painful to my eyes."

"True," said the Sphere, "it appears to you a Plane, because you are

not accustomed to light and shade and perspective; just as in Flatland

a Hexagon would appear a Straight Line to one who has not the Art of

Sight Recognition. But in reality it is a Solid, as you shall learn by

the sense of Feeling."

He then introduced me to the Cube, and I found that this marvellous

Being was indeed no Plane, but a Solid; and that he was endowed with

six plane sides and eight terminal points called solid angles; and I

remembered the saying of the Sphere that just such a Creature as this

would be formed by a Square moving, in Space, parallel to himself: and

I rejoiced to think that so insignificant a Creature as I could in some

sense be called the Progenitor of so illustrious an offspring.

But still I could not fully understand the meaning of what my Teacher

had told me concerning "light" and "shade" and "perspective"; and I did

not hesitate to put my difficulties before him.

Were I to give the Sphere's explanation of these matters, succinct and

clear though it was, it would be tedious to an inhabitant of Space, who

knows these things already. Suffice it, that by his lucid statements,

and by changing the position of objects and lights, and by allowing me

to feel the several objects and even his own sacred Person, he at last

made all things clear to me, so that I could now readily distinguish

between a Circle and a Sphere, a Plane Figure and a Solid.

This was the Climax, the Paradise, of my strange eventful History.

Henceforth I have to relate the story of my miserable Fall:--most

miserable, yet surely most undeserved! For why should the thirst for

knowledge be aroused, only to be disappointed and punished? My

volition shrinks from the painful task of recalling my humiliation;

yet, like a second Prometheus, I will endure this and worse, if by any

means I may arouse in the interiors of Plane and Solid Humanity a

spirit of rebellion against the Conceit which would limit our

Dimensions to Two or Three or any number short of Infinity. Away then

with all personal considerations! Let me continue to the end, as I

began, without further digressions or anticipations, pursuing the plain

path of dispassionate History. The exact facts, the exact words,--and

they are burnt in upon my brain,--shall be set down without alteration

of an iota; and let my Readers judge between me and Destiny.

The Sphere would willingly have continued his lessons by indoctrinating

me in the conformation of all regular Solids, Cylinders, Cones,

Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I

ventured to interrupt him. Not that I was wearied of knowledge. On

the contrary, I thirsted for yet deeper and fuller draughts than he was

offering to me.

"Pardon me," said I, "O Thou Whom I must no longer address as the

Perfection of all Beauty; but let me beg thee to vouchsafe thy servant

a sight of thine interior."

SPHERE. My what?

I. Thine interior: thy stomach, thy intestines.

SPHERE. Whence this ill-timed impertinent request? And what mean you

by saying that I am no longer the Perfection of all Beauty?

I. My Lord, your own wisdom has taught me to aspire to One even more

great, more beautiful, and more closely approximate to Perfection than

yourself. As you yourself, superior to all Flatland forms, combine

many Circles in One, so doubtless there is One above you who combines

many Spheres in One Supreme Existence, surpassing even the Solids of

Spaceland. And even as we, who are now in Space, look down on Flatland

and see the insides of all things, so of a certainty there is yet above

us some higher, purer region, whither thou dost surely purpose to lead

me--O Thou Whom I shall always call, everywhere and in all Dimensions,

my Priest, Philosopher, and Friend--some yet more spacious Space, some

more dimensionable Dimensionality, from the vantage-ground of which we

shall look down together upon the revealed insides of Solid things, and

where thine own intestines, and those of thy kindred Spheres, will lie

exposed to the view of the poor wandering exile from Flatland, to whom

so much has already been vouchsafed.

SPHERE. Pooh! Stuff! Enough of this trifling! The time is short,

and much remains to be done before you are fit to proclaim the Gospel

of Three Dimensions to your blind benighted countrymen in Flatland.

I. Nay, gracious Teacher, deny me not what I know it is in thy power

to perform. Grant me but one glimpse of thine interior, and I am

satisfied for ever, remaining henceforth thy docile pupil, thy

unemancipable slave, ready to receive all thy teachings and to feed

upon the words that fall from thy lips.

SPHERE. Well, then, to content and silence you, let me say at once, I

would shew you what you wish if I could; but I cannot. Would you have

me turn my stomach inside out to oblige you?

I. But my Lord has shewn me the intestines of all my countrymen in the

Land of Two Dimensions by taking me with him into the Land of Three.

What therefore more easy than now to take his servant on a second

journey into the blessed region of the Fourth Dimension, where I shall

look down with him once more upon this land of Three Dimensions, and

see the inside of every three-dimensioned house, the secrets of the

solid earth, the treasures of the mines in Spaceland, and the

intestines of every solid living creature, even of the noble and

adorable Spheres.

SPHERE. But where is this land of Four Dimensions?

I. I know not: but doubtless my Teacher knows.

SPHERE. Not I. There is no such land. The very idea of it is utterly

inconceivable.

I. Not inconceivable, my Lord, to me, and therefore still less

inconceivable to my Master. Nay, I despair not that, even here, in

this region of Three Dimensions, your Lordship's art may make the

Fourth Dimension visible to me; just as in the Land of Two Dimensions

my Teacher's skill would fain have opened the eyes of his blind servant

to the invisible presence of a Third Dimension, though I saw it not.

Let me recall the past. Was I not taught below that when I saw a Line

and inferred a Plane, I in reality saw a Third unrecognized Dimension,

not the same as brightness, called "height"? And does it not now

follow that, in this region, when I see a Plane and infer a Solid, I

really see a Fourth unrecognized Dimension, not the same as colour, but

existent, though infinitesimal and incapable of measurement?

And besides this, there is the Argument from Analogy of Figures.

SPHERE. Analogy! Nonsense: what analogy?

I. Your Lordship tempts his servant to see whether he remembers the

revelations imparted to him. Trifle not with me, my Lord; I crave, I

thirst, for more knowledge. Doubtless we cannot SEE that other higher

Spaceland now, because we we have no eye in our stomachs. But, just as

there WAS the realm of Flatland, though that poor puny Lineland Monarch

could neither turn to left nor right to discern it, and just as there

WAS close at hand, and touching my frame, the land of Three Dimensions,

though I, blind senseless wretch, had no power to touch it, no eye in

my interior to discern it, so of a surety there is a Fourth Dimension,

which my Lord perceives with the inner eye of thought. And that it

must exist my Lord himself has taught me. Or can he have forgotten

what he himself imparted to his servant?

In One Dimension, did not a moving Point produce a Line with TWO

terminal points?

In Two Dimensions, did not a moving Line produce a Square with FOUR

terminal points?

In Three Dimensions, did not a moving Square produce--did not this eye

of mine behold it--that blessed Being, a Cube, with EIGHT terminal

points?

And in Four Dimensions shall not a moving Cube--alas, for Analogy, and

alas for the Progress of Truth, if it be not so--shall not, I say, the

motion of a divine Cube result in a still more divine Organization with

SIXTEEN terminal points?

Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not

this a Geometrical Progression? Is not this--if I might quote my

Lord's own words--"strictly according to Analogy"?

Again, was I not taught by my Lord that as in a Line there are TWO

bounding Points, and in a Square there are FOUR bounding Lines, so in a

Cube there must be SIX bounding Squares? Behold once more the

confirming Series, 2, 4, 6: is not this an Arithmetical Progression?

And consequently does it not of necessity follow that the more divine

offspring of the divine Cube in the Land of Four Dimensions, must have

8 bounding Cubes: and is not this also, as my Lord has taught me to

believe, "strictly according to Analogy"?

O, my Lord, my Lord, behold, I cast myself in faith upon conjecture,

not knowing the facts; and I appeal to your Lordship to confirm or deny

my logical anticipations. If I am wrong, I yield, and will no longer

demand a fourth Dimension; but, if I am right, my Lord will listen to

reason.

I ask therefore, is it, or is it not, the fact, that ere now your

countrymen also have witnessed the descent of Beings of a higher order

than their own, entering closed rooms, even as your Lordship entered

mine, without the opening of doors or windows, and appearing and

vanishing at will? On the reply to this question I am ready to stake

everything. Deny it, and I am henceforth silent. Only vouchsafe an

answer.

SPHERE. (AFTER A PAUSE). It is reported so. But men are divided in

opinion as to the facts. And even granting the facts, they explain

them in different ways. And in any case, however great may be the

number of different explanations, no one has adopted or suggested the

theory of a Fourth Dimension. Therefore, pray have done with this

trifling, and let us return to business.

I. I was certain of it. I was certain that my anticipations would be

fulfilled. And now have patience with me and answer me yet one more

question, best of Teachers! Those who have thus appeared--no one knows

whence--and have returned--no one knows whither--have they also

contracted their sections and vanished somehow into that more Spacious

Space, whither I now entreat you to conduct me?

SPHERE (MOODILY). They have vanished, certainly--if they ever

appeared. But most people say that these visions arose from the

thought--you will not understand me--from the brain; from the perturbed

angularity of the Seer.

I. Say they so? Oh, believe them not. Or if it indeed be so, that

this other Space is really Thoughtland, then take me to that blessed

Region where I in Thought shall see the insides of all solid things.

There, before my ravished eye, a Cube, moving in some altogether new

direction, but strictly according to Analogy, so as to make every

particle of his interior pass through a new kind of Space, with a wake

of its own--shall create a still more perfect perfection than himself,

with sixteen terminal Extra-solid angles, and Eight solid Cubes for his

Perimeter. And once there, shall we stay our upward course? In that

blessed region of Four Dimensions, shall we linger on the threshold of

the Fifth, and not enter therein? Ah, no! Let us rather resolve that

our ambition shall soar with our corporal ascent. Then, yielding to

our intellectual onset, the gates of the Sixth Dimension shall fly

open; after that a Seventh, and then an Eighth-- How long I should have

continued I know not. In vain did the Sphere, in his voice of thunder,

reiterate his command of silence, and threaten me with the direst

penalties if I persisted. Nothing could stem the flood of my ecstatic

aspirations. Perhaps I was to blame; but indeed I was intoxicated with

the recent draughts of Truth to which he himself had introduced me.

However, the end was not long in coming. My words were cut short by a

crash outside, and a simultaneous crash inside me, which impelled me

through space with a velocity that precluded speech. Down! down! down!

I was rapidly descending; and I knew that return to Flatland was my

doom. One glimpse, one last and never-to-be-forgotten glimpse I had of

that dull level wilderness--which was now to become my Universe

again--spread out before my eye. Then a darkness. Then a final,

all-consummating thunder-peal; and, when I came to myself, I was once

more a common creeping Square, in my Study at home, listening to the

Peace-Cry of my approaching Wife.

Section 20. How the Sphere encouraged me in a Vision

Although I had less than a minute for reflection, I felt, by a kind of

instinct, that I must conceal my experiences from my Wife. Not that I

apprehended, at the moment, any danger from her divulging my secret,

but I knew that to any Woman in Flatland the narrative of my adventures

must needs be unintelligible. So I endeavoured to reassure her by some

story, invented for the occasion, that I had accidentally fallen

through the trap-door of the cellar, and had there lain stunned.

The Southward attraction in our country is so slight that even to a

Woman my tale necessarily appeared extraordinary and well-nigh

incredible; but my Wife, whose good sense far exceeds that of the

average of her Sex, and who perceived that I was unusually excited, did

not argue with me on the subject, but insisted that I was ill and

required repose. I was glad of an excuse for retiring to my chamber to

think quietly over what had happened. When I was at last by myself, a

drowsy sensation fell on me; but before my eyes closed I endeavoured to

reproduce the Third Dimension, and especially the process by which a

Cube is constructed through the motion of a Square. It was not so

clear as I could have wished; but I remembered that it must be "Upward,

and yet not Northward", and I determined steadfastly to retain these

words as the clue which, if firmly grasped, could not fail to guide me

to the solution. So mechanically repeating, like a charm, the words,

"Upward, yet not Northward", I fell into a sound refreshing sleep.

During my slumber I had a dream. I thought I was once more by the side

of the Sphere, whose lustrous hue betokened that he had exchanged his

wrath against me for perfect placability. We were moving together

towards a bright but infinitesimally small Point, to which my Master

directed my attention. As we approached, methought there issued from

it a slight humming noise as from one of your Spaceland bluebottles,

only less resonant by far, so slight indeed that even in the perfect

stillness of the Vacuum through which we soared, the sound reached not

our ears till we checked our flight at a distance from it of something

under twenty human diagonals.

"Look yonder," said my Guide, "in Flatland thou hast lived; of Lineland

thou hast received a vision; thou hast soared with me to the heights of

Spaceland; now, in order to complete the range of thy experience, I

conduct thee downward to the lowest depth of existence, even to the

realm of Pointland, the Abyss of No dimensions.

"Behold yon miserable creature. That Point is a Being like ourselves,

but confined to the non-dimensional Gulf. He is himself his own World,

his own Universe; of any other than himself he can form no conception;

he knows not Length, nor Breadth, nor Height, for he has had no

experience of them; he has no cognizance even of the number Two; nor

has he a thought of Plurality; for he is himself his One and All, being

really Nothing. Yet mark his perfect self-contentment, and hence learn

this lesson, that to be self-contented is to be vile and ignorant, and

that to aspire is better than to be blindly and impotently happy. Now

listen."

He ceased; and there arose from the little buzzing creature a tiny,

low, monotonous, but distinct tinkling, as from one of your Spaceland

phonographs, from which I caught these words, "Infinite beatitude of

existence! It is; and there is none else beside It."

"What," said I, "does the puny creature mean by 'it'?" "He means

himself," said the Sphere: "have you not noticed before now, that

babies and babyish people who cannot distinguish themselves from the

world, speak of themselves in the Third Person? But hush!"

"It fills all Space," continued the little soliloquizing Creature, "and

what It fills, It is. What It thinks, that It utters; and what It

utters, that It hears; and It itself is Thinker, Utterer, Hearer,

Thought, Word, Audition; it is the One, and yet the All in All. Ah,

the happiness ah, the happiness of Being!"

"Can you not startle the little thing out of its complacency?" said I.

"Tell it what it really is, as you told me; reveal to it the narrow

limitations of Pointland, and lead it up to something higher." "That

is no easy task," said my Master; "try you."

Hereon, raising my voice to the uttermost, I addressed the Point as

follows:

"Silence, silence, contemptible Creature. You call yourself the All in

All, but you are the Nothing: your so-called Universe is a mere speck

in a Line, and a Line is a mere shadow as compared with--" "Hush,

hush, you have said enough," interrupted the Sphere, "now listen, and

mark the effect of your harangue on the King of Pointland."

The lustre of the Monarch, who beamed more brightly than ever upon

hearing my words, shewed clearly that he retained his complacency; and

I had hardly ceased when he took up his strain again. "Ah, the joy,

ah, the joy of Thought! What can It not achieve by thinking! Its own

Thought coming to Itself, suggestive of Its disparagement, thereby to

enhance Its happiness! Sweet rebellion stirred up to result in triumph!

Ah, the divine creative power of the All in One! Ah, the joy, the joy

of Being!"

"You see," said my Teacher, "how little your words have done. So far

as the Monarch understands them at all, he accepts them as his own--for

he cannot conceive of any other except himself--and plumes himself upon

the variety of 'Its Thought' as an instance of creative Power. Let us

leave this God of Pointland to the ignorant fruition of his

omnipresence and omniscience: nothing that you or I can do can rescue

him from his self-satisfaction."

After this, as we floated gently back to Flatland, I could hear the

mild voice of my Companion pointing the moral of my vision, and

stimulating me to aspire, and to teach others to aspire. He had been

angered at first--he confessed--by my ambition to soar to Dimensions

above the Third; but, since then, he had received fresh insight, and he

was not too proud to acknowledge his error to a Pupil. Then he

proceeded to initiate me into mysteries yet higher than those I had

witnessed, shewing me how to construct Extra-Solids by the motion of

Solids, and Double Extra-Solids by the motion of Extra-Solids, and all

"strictly according to Analogy", all by methods so simple, so easy, as

to be patent even to the Female Sex.

Section 21. How I tried to teach the Theory of Three Dimensions

to my Grandson, and with what success

I awoke rejoicing, and began to reflect on the glorious career before

me. I would go forth, methought, at once, and evangelize the whole of

Flatland. Even to Women and Soldiers should the Gospel of Three

Dimensions be proclaimed. I would begin with my Wife.

Just as I had decided on the plan of my operations, I heard the sound

of many voices in the street commanding silence. Then followed a

louder voice. It was a herald's proclamation. Listening attentively,

I recognized the words of the Resolution of the Council, enjoining the

arrest, imprisonment, or execution of any one who should pervert the

minds of the people by delusions, and by professing to have received

revelations from another World.

I reflected. This danger was not to be trifled with. It would be

better to avoid it by omitting all mention of my Revelation, and by

proceeding on the path of Demonstration--which after all, seemed so

simple and so conclusive that nothing would be lost by discarding the

former means. "Upward, not Northward"--was the clue to the whole

proof. It had seemed to me fairly clear before I fell asleep; and when

I first awoke, fresh from my dream, it had appeared as patent as

Arithmetic; but somehow it did not seem to me quite so obvious now.

Though my Wife entered the room opportunely just at that moment, I

decided, after we had exchanged a few words of commonplace

conversation, not to begin with her.

My Pentagonal Sons were men of character and standing, and physicians

of no mean reputation, but not great in mathematics, and, in that

respect, unfit for my purpose. But it occurred to me that a young and

docile Hexagon, with a mathematical turn, would be a most suitable

pupil. Why therefore not make my first experiment with my little

precocious Grandson, whose casual remarks on the meaning of 3^3 had met

with the approval of the Sphere? Discussing the matter with him, a

mere boy, I should be in perfect safety; for he would know nothing of

the Proclamation of the Council; whereas I could not feel sure that my

Sons--so greatly did their patriotism and reverence for the Circles

predominate over mere blind affection--might not feel compelled to hand

me over to the Prefect, if they found me seriously maintaining the

seditious heresy of the Third Dimension.

But the first thing to be done was to satisfy in some way the curiosity

of my Wife, who naturally wished to know something of the reasons for

which the Circle had desired that mysterious interview, and of the

means by which he had entered the house. Without entering into the

details of the elaborate account I gave her,--an account, I fear, not

quite so consistent with truth as my Readers in Spaceland might

desire,--I must be content with saying that I succeeded at last in

persuading her to return quietly to her household duties without

eliciting from me any reference to the World of Three Dimensions. This

done, I immediately sent for my Grandson; for, to confess the truth, I

felt that all that I had seen and heard was in some strange way

slipping away from me, like the image of a half-grasped, tantalizing

dream, and I longed to essay my skill in making a first disciple.

When my Grandson entered the room I carefully secured the door. Then,

sitting down by his side and taking our mathematical tablets,--or, as

you would call them, Lines--I told him we would resume the lesson of

yesterday. I taught him once more how a Point by motion in One

Dimension produces a Line, and how a straight Line in Two Dimensions

produces a Square. After this, forcing a laugh, I said, "And now, you

scamp, you wanted to make me believe that a Square may in the same way

by motion 'Upward, not Northward' produce another figure, a sort of

extra Square in Three Dimensions. Say that again, you young rascal."

At this moment we heard once more the herald's "O yes! O yes!" outside

in the street proclaiming the Resolution of the Council. Young though

he was, my Grandson--who was unusually intelligent for his age, and

bred up in perfect reverence for the authority of the Circles--took in

the situation with an acuteness for which I was quite unprepared. He

remained silent till the last words of the Proclamation had died away,

and then, bursting into tears, "Dear Grandpapa," he said, "that was

only my fun, and of course I meant nothing at all by it; and we did not

know anything then about the new Law; and I don't think I said anything

about the Third Dimension; and I am sure I did not say one word about

'Upward, not Northward', for that would be such nonsense, you know.

How could a thing move Upward, and not Northward? Upward and not

Northward! Even if I were a baby, I could not be so absurd as that.

How silly it is! Ha! ha! ha!"

"Not at all silly," said I, losing my temper; "here for example, I take

this Square," and, at the word, I grasped a moveable Square, which was

lying at hand--"and I move it, you see, not Northward but--yes, I move

it Upward--that is to say, not Northward, but I move it somewhere--not

exactly like this, but somehow--" Here I brought my sentence to an

inane conclusion, shaking the Square about in a purposeless manner,

much to the amusement of my Grandson, who burst out laughing louder

than ever, and declared that I was not teaching him, but joking with

him; and so saying he unlocked the door and ran out of the room. Thus

ended my first attempt to convert a pupil to the Gospel of Three

Dimensions.

Section 22. How I then tried to diffuse the Theory

of Three Dimensions by other means, and of the result

My failure with my Grandson did not encourage me to communicate my

secret to others of my household; yet neither was I led by it to

despair of success. Only I saw that I must not wholly rely on the

catch-phrase, "Upward, not Northward", but must rather endeavour to

seek a demonstration by setting before the public a clear view of the

whole subject; and for this purpose it seemed necessary to resort to

writing.

So I devoted several months in privacy to the composition of a treatise

on the mysteries of Three Dimensions. Only, with the view of evading

the Law, if possible, I spoke not of a physical Dimension, but of a

Thoughtland whence, in theory, a Figure could look down upon Flatland

and see simultaneously the insides of all things, and where it was

possible that there might be supposed to exist a Figure environed, as

it were, with six Squares, and containing eight terminal Points. But

in writing this book I found myself sadly hampered by the impossibility

of drawing such diagrams as were necessary for my purpose; for of

course, in our country of Flatland, there are no tablets but Lines, and

no diagrams but Lines, all in one straight Line and only

distinguishable by difference of size and brightness; so that, when I

had finished my treatise (which I entitled, "Through Flatland to

Thoughtland") I could not feel certain that many would understand my

meaning.

Meanwhile my life was under a cloud. All pleasures palled upon me; all

sights tantalized and tempted me to outspoken treason, because I could

not but compare what I saw in Two Dimensions with what it really was if

seen in Three, and could hardly refrain from making my comparisons

aloud. I neglected my clients and my own business to give myself to

the contemplation of the mysteries which I had once beheld, yet which I

could impart to no one, and found daily more difficult to reproduce

even before my own mental vision.

One day, about eleven months after my return from Spaceland, I tried to

see a Cube with my eye closed, but failed; and though I succeeded

afterwards, I was not then quite certain (nor have I been ever

afterwards) that I had exactly realized the original. This made me

more melancholy than before, and determined me to take some step; yet

what, I knew not. I felt that I would have been willing to sacrifice

my life for the Cause, if thereby I could have produced conviction.

But if I could not convince my Grandson, how could I convince the

highest and most developed Circles in the land?

And yet at times my spirit was too strong for me, and I gave vent to

dangerous utterances. Already I was considered heterodox if not

treasonable, and I was keenly alive to the danger of my position;

nevertheless I could not at times refrain from bursting out into

suspicious or half-seditious utterances, even among the highest

Polygonal and Circular society. When, for example, the question arose

about the treatment of those lunatics who said that they had received

the power of seeing the insides of things, I would quote the saying of

an ancient Circle, who declared that prophets and inspired people are

always considered by the majority to be mad; and I could not help

occasionally dropping such expressions as "the eye that discerns the

interiors of things", and "the all-seeing land"; once or twice I even

let fall the forbidden terms "the Third and Fourth Dimensions". At

last, to complete a series of minor indiscretions, at a meeting of our

Local Speculative Society held at the palace of the Prefect

himself,--some extremely silly person having read an elaborate paper

exhibiting the precise reasons why Providence has limited the number of

Dimensions to Two, and why the attribute of omnividence is assigned to

the Supreme alone--I so far forgot myself as to give an exact account

of the whole of my voyage with the Sphere into Space, and to the

Assembly Hall in our Metropolis, and then to Space again, and of my

return home, and of everything that I had seen and heard in fact or

vision. At first, indeed, I pretended that I was describing the

imaginary experiences of a fictitious person; but my enthusiasm soon

forced me to throw off all disguise, and finally, in a fervent

peroration, I exhorted all my hearers to divest themselves of prejudice

and to become believers in the Third Dimension.

Need I say that I was at once arrested and taken before the Council?

Next morning, standing in the very place where but a very few months

ago the Sphere had stood in my company, I was allowed to begin and to

continue my narration unquestioned and uninterrupted. But from the

first I foresaw my fate; for the President, noting that a guard of the

better sort of Policemen was in attendance, of angularity little, if at

all, under 55 degrees, ordered them to be relieved before I began my

defence, by an inferior class of 2 or 3 degrees. I knew only too well

what that meant. I was to be executed or imprisoned, and my story was

to be kept secret from the world by the simultaneous destruction of the

officials who had heard it; and, this being the case, the President

desired to substitute the cheaper for the more expensive victims.

After I had concluded my defence, the President, perhaps perceiving

that some of the junior Circles had been moved by my evident

earnestness, asked me two questions:--

1. Whether I could indicate the direction which I meant when I used

the words "Upward, not Northward"?

2. Whether I could by any diagrams or descriptions (other than the

enumeration of imaginary sides and angles) indicate the Figure I was

pleased to call a Cube?

I declared that I could say nothing more, and that I must commit myself

to the Truth, whose cause would surely prevail in the end.

The President replied that he quite concurred in my sentiment, and that

I could not do better. I must be sentenced to perpetual imprisonment;

but if the Truth intended that I should emerge from prison and

evangelize the world, the Truth might be trusted to bring that result

to pass. Meanwhile I should be subjected to no discomfort that was not

necessary to preclude escape, and, unless I forfeited the privilege by

misconduct, I should be occasionally permitted to see my brother who

had preceded me to my prison.

Seven years have elapsed and I am still a prisoner, and--if I except

the occasional visits of my brother--debarred from all companionship

save that of my jailers. My brother is one of the best of Squares,

just, sensible, cheerful, and not without fraternal affection; yet I

confess that my weekly interviews, at least in one respect, cause me

the bitterest pain. He was present when the Sphere manifested himself

in the Council Chamber; he saw the Sphere's changing sections; he heard

the explanation of the phenomena then given to the Circles. Since that

time, scarcely a week has passed during seven whole years, without his

hearing from me a repetition of the part I played in that

manifestation, together with ample descriptions of all the phenomena in

Spaceland, and the arguments for the existence of Solid things

derivable from Analogy. Yet--I take shame to be forced to confess

it--my brother has not yet grasped the nature of the Third Dimension,

and frankly avows his disbelief in the existence of a Sphere.

Hence I am absolutely destitute of converts, and, for aught that I can

see, the millennial Revelation has been made to me for nothing.

Prometheus up in Spaceland was bound for bringing down fire for

mortals, but I--poor Flatland Prometheus--lie here in prison for

bringing down nothing to my countrymen. Yet I exist in the hope that

these memoirs, in some manner, I know not how, may find their way to

the minds of humanity in Some Dimension, and may stir up a race of

rebels who shall refuse to be confined to limited Dimensionality.

That is the hope of my brighter moments. Alas, it is not always so.

Heavily weighs on me at times the burdensome reflection that I cannot

honestly say I am confident as to the exact shape of the once-seen,

oft-regretted Cube; and in my nightly visions the mysterious precept,

"Upward, not Northward", haunts me like a soul-devouring Sphinx. It is

part of the martyrdom which I endure for the cause of the Truth that

there are seasons of mental weakness, when Cubes and Spheres flit away

into the background of scarce-possible existences; when the Land of

Three Dimensions seems almost as visionary as the Land of One or None;

nay, when even this hard wall that bars me from my freedom, these very

tablets on which I am writing, and all the substantial realities of

Flatland itself, appear no better than the offspring of a diseased

imagination, or the baseless fabric of a dream.

THE END of FLATLAND

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| THE END of |

| ______ |

| / / /| ------ / /| /| / /-. |

| /---- / /__| / / /__| / | / / / |

| / /___ / | / /___ / | / |/ /__.-' |

| |

| The baseless fabric of my vision |

| Melted into air into thin air |

| Such stuff as dreams are made of |

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