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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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Flatland
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 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
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Flatland
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 so 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。
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Flatland
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； not edging to the right nor to the
left； but moving somehow out of the world； and vanishing into 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
youas 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