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 showed 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 guard. “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.
§ 19.—_How, though the Sphere showed 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: ]
Once more we ascended into space. “Hitherto,” said the Sphere, “I have
shown 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 show 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 shown 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._ Your Lordship tempts his servant to see whether he remembers the
revelations imparted to him. Trifle not with me, my Lord; I crave, I
thirst, for more knowledge. Doubtless we cannot _see_ that other higher
Spaceland now, because we have no eye in our stomachs. But, just as there
_was_ the realm of Flatland, though that poor puny Lineland Monarch could
neither turn to left nor right to discern it, and just as there _was_
close at hand, and touching my frame, the land of Three Dimensions,
though I, blind senseless wretch, had no power to touch it, no eye in my
interior to discern it, so of a surety there is a Fourth Dimension, which
my Lord perceives with the inner eye of thought. And that it must exist
my Lord himself has taught me. Or can he have forgotten what he himself
imparted to his servant?
In One Dimension, did not a moving Point produce a Line with _two_
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 commands 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.
§ 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 know 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 blue-bottles, 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, showed 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, showing 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.
§ 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 interchanged 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 3³ 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.