07 - Flatland- A Romance of Many Dimensions - Edwin Abbott Abbott

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Episode Transcript

Speaker 1 (00:00):
Part two, Section fifteen, concerning a stranger from Spaceland. From dreams,
I proceed to facts. It was the last day of
the nineteen hundred and ninety ninth year of our era.
The pattering of the rain had long ago announced nightfall,

(00:21):
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. Footnote.
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 feat, we

(00:44):
can no more sit nor stand in your sense of
the word than one of your souls 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

(01:04):
luster corresponding to the increase of volition. But on this
and a thousand other kindred subjects, time forbids me to dwell.
End of footnote. 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

(01:26):
the new one in. I was wrapt 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

(01:48):
sight recognition, turning ourselves upon our centers, 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

(02:10):
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

(02:32):
the number of inches in the side. And thus said I,
we know that three squared or nine represents the number
of square inches in a square whose side is three
inches long. The little hexagon meditated on this a while
and then said to me, but you have been teaching

(02:52):
me to raise numbers to the third power. I suppose
three cubed 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 show the boy how a point, by moving through

(03:14):
a length of three inches, makes a line of three inches,
which may be represented by three, and how a line
of three inches moving parallel to itself through a length
of three inches makes a square of three inches every way,
which may be represented by three squared. Upon this, my grandson,

(03:38):
again returning to his former suggestion, took me up rather
suddenly and exclaimed, well, then, if a point, by moving
three inches makes a line of three inches represented by three,
and if a straight line of three inches moving parallel
to itself makes a square of three inches every way

(04:00):
presented by three squared, it must be that a square
of three inches every way, moving somehow parallel to itself.
But I don't see how must make something else. But
I don't see what of three inches every way? And
this must be represented by three cubed. Go to bed,

(04:23):
said I, a little ruffled by his 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, endeavoring to form a retrospect of
the year nineteen ninety nine and of the possibilities of

(04:43):
the year two thousand, 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 glo northward for the last time in the old millennium,
and in the act I exclaimed aloud, the boy is

(05:06):
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 dishonoring your own grandson.
But I took no notice of her. Looking round in

(05:29):
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 draft. What are you looking for? There
is nothing? There was nothing and I resumed my seat again, exclaiming,

(05:54):
the boy is a fool. I say, three cubed can
have no meaning in geometry, But once there came a
distinctly audible reply, the boy is not a fool, and
three cubed has an obvious geometrical meaning. My wife, as
well as myself heard the words, although she did not

(06:17):
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 showed me that the extremities passed into dimness

(06:37):
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

(07:01):
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,

(07:21):
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,

(07:44):
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 to be felt by, then suddenly recoiling. Oh, it
is not a woman, and there are no angles either.

(08:07):
What a trait of one? Can it be that I
have so misbehaved to a perfect circle. I am, indeed,
in a sense a circle, replied the voice, and a
more perfect circle than any in flat land. But to
speak more accurately, I am many circles in one. Then

(08:27):
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 for her own retirement

(08:49):
had long passed. With many reiterated apologies for her recent indiscretion,
she at last retreated to her apartment. I glanced at
the half the last sands had fallen, the second millennium
had begun. Section sixteen. How the stranger vainly endeavored to

(09:14):
reveal to me in word as the mysteries of spaceland.
As soon as the sound of my wife's retreating footsteps
had died away, I began to approach the stranger, with
the intention of taking a nearer view, and of bidding
him be seated. That his appearance struck me dumb and

(09:34):
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

(09:55):
me a burglar or cutthroat, some monstrous, irregular eye soscely 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,

(10:16):
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

(10:36):
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.

(10:57):
There could not be a doubt of it. Then followed
a dialog, which I will endeavor 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

(11:19):
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 rises not from ignorance of the
usages of polite society, but from a little surprise and

(11:42):
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 curious of
one who would gladly know whence his visitor came stranger

(12:06):
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, pool,
what do you know of space? Define space? I? Space,

(12:31):
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

(12:52):
and length. I your Lordship is pleased to be merry.
We also speak of length and high or breadth and thickness,
thus denoting two dimensions by four names. Stranger, But I
mean not only three names, but three dimensions. I would

(13:15):
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 northwood and southward. Stranger,

(13:37):
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,

(14:00):
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 I in my inside, an I in my stomach,

(14:24):
your lordship jests, Stranger. I am in no jesting humor.
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

(14:44):
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.

(15:08):
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.

(15:29):
I saw your youngest Hexagon remain awhile 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?

(15:49):
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

(16:10):
and household. I, your lordship, must be aware that such
facts touching the belongings of his humble servant might be
easily ascertained by anyone in the neighborhood possessing your Lordship's
ample means of obtaining information. Stranger, how shall I convince him?
Surely A plain statement of facts followed by ocular demonstration

(16:34):
ought to suffice. Now, sir, listen to me. You are
living on a plane. What you style flat land 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

(16:57):
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.

(17:21):
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

(17:41):
manifest himself as a circle. Do you not remember, for I,
who see all things, discerned last night the phantasmal vision
of Limeland written upon your brain. Do you not remember?
I say, how, when you entered the realm of Limeland,
you are compelled to manifest yourself to the King not

(18:03):
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

(18:26):
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

(18:48):
a time, for you have no power to raise your
eye out of the plane of flatland. But you can
at least see that as I rise in space, so
my section becomes smaller. See now I will rise, and
the effect upon your eye will be that my circle
will become smaller and smaller till it dwindles. To a

(19:10):
point and finally vanishes. Reader's note. The following paragraph makes
reference to a diagram. The diagram shows a horizontal line
cutting the line. There are three spheres. The first, or leftmost,
is half above and half below the line, with its

(19:31):
equator drawn at the level of the line to indicate
the sphere's solid nature. Over it is written the sphere
with his section at full size. The second, or central
sphere is positioned about five sixths above the line and
one sixth below, with a circumference drawn at the level

(19:52):
of the line. This is labeled two, the sphere rising.
The third, or the right hand sphere, is nearly all
above the line, with just a sliver below, the circumference
being drawn at the level of the line. This is
labeled three, the sphere on the point of vanishing. At

(20:15):
the right hand end of the line. There is an
eye marked my eye looking towards the left end of
Reader's note. 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

(20:39):
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 flat land, and you shall
see my section become larger and larger. Every reader in
Spaceland will easy understand that my mysterious guest was speaking

(21:03):
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

(21:26):
have manifested himself to me, or to any Flatlander, as
a circle, at first a 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

(21:47):
comprehend was that the circle had made himself smaller and vanished,
and that he had now reappeared and was rapidly making
himself larger. When he had regained his original size. He
heaved a deep sigh, for he perceived, by my silence
that I had altogether failed to comprehend him, And indeed

(22:10):
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

(22:32):
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 dialog sphere.
Tell me, mister mathematician, if a point moves northward and

(22:54):
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,

(23:19):
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

(23:41):
many sides has a square? And how many angles? I?
Four sides and four angles sphere? Now stretch your imagination
a little and conceive a square in flat land moving
parallel to itself upward. I what northward sphere, No, not

(24:08):
northward upward out of flatland altogether. If it moved northward,
the southern point 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

(24:30):
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

(24:51):
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 flat land, anywhere, so that I could
get rid of him, I replied, And what may be

(25:14):
the nature of the figure which I am to shape
out by this motion, which you are pleased to denote
by the word upwards. I presume it is describable in
the language of flat land sphere. Oh, certainly, it is
all plain and simple and in strict accordance with analogy. Only,

(25:35):
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

(25:56):
produces a line with two terminal points. One line produces
a square with four terminal points. Now you can yourself
give the answer to your own question. One, two, four
are evidently in geometrical progression. What is the next number? I? Eight? Sphere? Exactly?

(26:22):
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

(26:45):
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 so will
appertain to this being whom I am to generate by
the motion of my inside in an upward direction, and

(27:07):
whom you call a cube? Sphere. How can you ask?
And you a mathematician, A 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 no sides. A line, if I may so say,

(27:32):
has two sides. For the points of a line may
be called by courtesy its sides. A square has four sides,
not two four. What progression do you call that? I
arithmetical sphere? And what is the next number? I? Six sphere? Exactly?

(27:58):
Then you see you have answered your 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

(28:21):
endure thy mockeries. Either thou or I must perish. And
saying these words, I precipitated myself upon him. Section seventeen.
How the sphere, having in vain tried words, resorted to deeds.

(28:41):
It was in vain I brought my hardest right angle
into violent collision with the stranger, pressing on him with
a force sufficient to have destroyed any ordinary circle. But
I could feel him slowly and unarrestably slipping from my contact,
not edging to the right, nor to the left, but

(29:02):
moving somehow out of the world and vanishing to nothing.
Soon there was a blank, but I still heard the
intruder's voice, sphere, Why will you refuse to listen to reason?
I had hoped to find in you as being a
man of sense and an accomplished mathematician, a fit apostle

(29:24):
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

(29:44):
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 are have no
tops nor bottoms full of money. I also see two

(30:05):
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.

(30:28):
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.

(30:49):
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

(31:12):
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

(31:36):
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 neighbor,

(31:56):
the hexagon, and his family in their several apartments. Now
I see the inside of the theater, 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

(32:19):
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

(32:42):
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 had not hurt
you much, have I? If you are not convinced now,

(33:04):
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

(33:26):
against the wall till help came. Once more, I dashed,
by 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

(33:46):
any case, he remained motionless while I, hearing as I
thought the sound of some help approaching, pressed against him
with redoubled figure, and continued to shout for assistance. Convulsive
shudder ran through the sphere. This must not be, I thought.
I heard him say, either he must listen to reason,

(34:10):
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

(34:33):
thus must the fruits of one thousand years of waiting
be thrown away.

Speaker 2 (34:37):
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 fall Madman irregular, I exclaimed,
Never will.

Speaker 1 (34:53):
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. End of Section seventeen.