flatland(弗莱特兰)-第15章
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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。
Strange。 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 of the
neighbourhood possessing your Lordship's ample means of 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
examplehow 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 of 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
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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 oughtbesides inferring her breadthto see her length； and to
SEE what we call her HEIGHT； although the 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 the 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 understand 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； or in； the top of which you and your
countrymen move about； without rising above 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
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Flatland
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 aSECTION which you； very rightly； call a Circle。 For even a
Spherewhich is my proper name in my own countryif he manifest
himself at all to an inhabitant of Flatland must needs manifest himself as
a Circle。
Do you not rememberfor 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 orSECTION 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 orSECTION 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 mySECTIONs； 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 mySECTIONs 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。
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
voiceclose 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
mySECTION 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
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Flatland
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 re…
appeared 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。
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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 youfor you are a Square and will serve the
purpose of my illustrationevery 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 impatiencefor 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 himI 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 Analogyonly； 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 coursebeing itself a Point
has only ONE terminal Point。
One Point produces a Line with TWO terminal Points。
One Line produces a Square with FOUR t
