Flatland: A Romance of Many Dimensions
[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