Edwin A. Abbot
Flatland: A Romance of Many Dimensions, Illustrated
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Table of contents
Preface
PART I: THIS WORLD
Section 1. Of the Nature of Flatland
Section 2. Of the Climate and Houses in Flatland
Section 3. Concerning the Inhabitants of Flatland
Section 4. Concerning the Women
Section 5. Of our Methods of Recognizing one another
Section 6. Of Recognition by Sight
Section 7. Concerning Irregular Figures
Section 8. Of the Ancient Practice of Painting
Section 9. Of the Universal Colour Bill
Section 10. Of the Suppression of the Chromatic Sedition
Section 11. Concerning our Priests
Section 12. Of the Doctrine of our Priests
PART II: OTHER WORLDS
Section 13. How I had a Vision of Lineland
Section 14. How I vainly tried to explain the nature of Flatland
Section 15. Concerning a Stranger from Spaceland
Section 16. How the Stranger vainly endeavoured to reveal to mein words the mysteries of Spaceland
Section 17. How the Sphere, having in vain tried words,resorted to deeds
Section 18. How I came to Spaceland, and what I saw there
Section 19. How, though the Sphere shewed me other mysteriesof Spaceland, I still desired more; and what came of it
Section 20. How the Sphere encouraged me in a Vision
Section 21. How I tried to teach the Theory of Three Dimensionsto my Grandson, and with what success
Section 22. How I then tried to diffuse the Theoryof Three Dimensions by other means, and of the result
Preface
Preface
If
my poor Flatland friend retained the vigour of mind which he
enjoyed
when he began to compose these Memoirs, I should not now need to
represent him in this preface, in which he desires, firstly, to
return his thanks to his readers and critics in Spaceland, whose
appreciation has, with unexpected celerity, required a second
edition
of his work; secondly, to apologize for certain errors and
misprints
(for which, however, he is not entirely responsible); and, thirdly,
to explain one or two misconceptions. But he is not the Square he
once was. Years of imprisonment, and the still heavier burden of
general incredulity and mockery, have combined with the natural
decay
of old age to erase from his mind many of the thoughts and notions,
and much also of the terminology, which he acquired during his
short
stay in Spaceland. He has, therefore, requested me to reply in his
behalf to two special objections, one of an intellectual, the other
of a moral nature.The
first objection is, that a Flatlander, seeing a Line, sees
something
that must be THICK to the eye as well as LONG to the eye (otherwise
it would not be visible, if it had not some thickness); and
consequently he ought (it is argued) to acknowledge that his
countrymen are not only long and broad, but also (though doubtless
in
a very slight degree) THICK or HIGH. This objection is plausible,
and, to Spacelanders, almost irresistible, so that, I confess, when
I
first heard it, I knew not what to reply. But my poor old friend's
answer appears to me completely to meet it."I
admit," said he—when I mentioned to him this objection—"I
admit the truth of your critic's facts, but I deny his conclusions.
It is true that we have really in Flatland a Third unrecognized
Dimension called 'height', just as it is also true that you have
really in Spaceland a Fourth unrecognized Dimension, called by no
name at present, but which I will call 'extra-height'. But we can
no
more take cognizance of our 'height' than you can of your
'extra-height'. Even I—who have been in Spaceland, and have had the
privilege of understanding for twenty-four hours the meaning of
'height'—even I cannot now comprehend it, nor realize it by the
sense of sight or by any process of reason; I can but apprehend it
by
faith."The
reason is obvious. Dimension implies direction, implies
measurement,
implies the more and the less. Now, all our lines are EQUALLY and
INFINITESIMALLY thick (or high, whichever you like); consequently,
there is nothing in them to lead our minds to the conception of
that
Dimension. No 'delicate micrometer'—as has been suggested by one
too hasty Spaceland critic—would in the least avail us; for we
should not know WHAT TO MEASURE, NOR IN WHAT DIRECTION. When we see
a
Line, we see something that is long and BRIGHT; BRIGHTNESS, as well
as length, is necessary to the existence of a Line; if the
brightness
vanishes, the Line is extinguished. Hence, all my Flatland
friends—when I talk to them about the unrecognized Dimension which
is somehow visible in a Line—say, 'Ah, you mean BRIGHTNESS': and
when I reply, 'No, I mean a real Dimension', they at once retort,
'Then measure it, or tell us in what direction it extends'; and
this
silences me, for I can do neither. Only yesterday, when the Chief
Circle (in other words our High Priest) came to inspect the State
Prison and paid me his seventh annual visit, and when for the
seventh
time he put me the question, 'Was I any better?' I tried to prove
to
him that he was 'high', as well as long and broad, although he did
not know it. But what was his reply? 'You say I am "high";
measure my "high-ness" and I will believe you.' What could
I do? How could I meet his challenge? I was crushed; and he left
the
room triumphant."Does
this still seem strange to you? Then put yourself in a similar
position. Suppose a person of the Fourth Dimension, condescending
to
visit you, were to say, 'Whenever you open your eyes, you see a
Plane
(which is of Two Dimensions) and you INFER a Solid (which is of
Three); but in reality you also see (though you do not recognize) a
Fourth Dimension, which is not colour nor brightness nor anything
of
the kind, but a true Dimension, although I cannot point out to you
its direction, nor can you possibly measure it.' What would you say
to such a visitor? Would not you have him locked up? Well, that is
my
fate: and it is as natural for us Flatlanders to lock up a Square
for
preaching the Third Dimension, as it is for you Spacelanders to
lock
up a Cube for preaching the Fourth. Alas, how strong a family
likeness runs through blind and persecuting humanity in all
Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes—we are all
liable to the same errors, all alike the Slaves of our respective
Dimensional prejudices, as one of your Spaceland poets has
said—'One
touch of Nature makes all worlds akin'."
[Note:
The Author desires me to add, that the misconception of some of his
critics on this matter has induced him to insert in his dialogue
with
the Sphere, certain remarks which have a bearing on the point in
question, and which he had previously omitted as being tedious and
unnecessary.]On
this point the defence of the Square seems to me to be impregnable.
I
wish I could say that his answer to the second (or moral) objection
was equally clear and cogent. It has been objected that he is a
woman-hater; and as this objection has been vehemently urged by
those
whom Nature's decree has constituted the somewhat larger half of
the
Spaceland race, I should like to remove it, so far as I can
honestly
do so. But the Square is so unaccustomed to the use of the moral
terminology of Spaceland that I should be doing him an injustice if
I
were literally to transcribe his defence against this charge.
Acting,
therefore, as his interpreter and summarizer, I gather that in the
course of an imprisonment of seven years he has himself modified
his
own personal views, both as regards Women and as regards the
Isosceles or Lower Classes. Personally, he now inclines to the
opinion of the Sphere that the Straight Lines are in many important
respects superior to the Circles. But, writing as a Historian, he
has
identified himself (perhaps too closely) with the views generally
adopted by Flatland, and (as he has been informed) even by
Spaceland,
Historians; in whose pages (until very recent times) the destinies
of
Women and of the masses of mankind have seldom been deemed worthy
of
mention and never of careful consideration.In
a still more obscure passage he now desires to disavow the Circular
or aristocratic tendencies with which some critics have naturally
credited him. While doing justice to the intellectual power with
which a few Circles have for many generations maintained their
supremacy over immense multitudes of their countrymen, he believes
that the facts of Flatland, speaking for themselves without comment
on his part, declare that Revolutions cannot always be suppressed
by
slaughter, and that Nature, in sentencing the Circles to
infecundity,
has condemned them to ultimate failure—"and herein," he
says, "I see a fulfilment of the great Law of all worlds, that
while the wisdom of Man thinks it is working one thing, the wisdom
of
Nature constrains it to work another, and quite a different and far
better thing." For the rest, he begs his readers not to suppose
that every minute detail in the daily life of Flatland must needs
correspond to some other detail in Spaceland; and yet he hopes
that,
taken as a whole, his work may prove suggestive as well as amusing,
to those Spacelanders of moderate and modest minds who—speaking of
that which is of the highest importance, but lies beyond
experience—decline to say on the one hand, "This can never
be," and on the other hand, "It must needs be precisely
thus, and we know all about it."
PART I: THIS WORLD
Section 1. Of the Nature of Flatland
I
call our world Flatland, not because we call it so, but to make its
nature clearer to you, my happy readers, who are privileged to live
in Space.Imagine
a vast sheet of paper on which straight Lines, Triangles, Squares,
Pentagons, Hexagons, and other figures, instead of remaining fixed
in
their places, move freely about, on or in the surface, but without
the power of rising above or sinking below it, very much like
shadows—only hard and with luminous edges—and you will then have
a pretty correct notion of my country and countrymen. Alas, a few
years ago, I should have said "my universe": but now my
mind has been opened to higher views of things.In
such a country, you will perceive at once that it is impossible
that
there should be anything of what you call a "solid" kind;
but I dare say you will suppose that we could at least distinguish
by
sight the Triangles, Squares, and other figures, moving about as I
have described them. On the contrary, we could see nothing of the
kind, not at least so as to distinguish one figure from another.
Nothing was visible, nor could be visible, to us, except Straight
Lines; and the necessity of this I will speedily
demonstrate.Place
a penny on the middle of one of your tables in Space; and leaning
over it, look down upon it. It will appear a circle.But
now, drawing back to the edge of the table, gradually lower your
eye
(thus bringing yourself more and more into the condition of the
inhabitants of Flatland), and you will find the penny becoming more
and more oval to your view, and at last when you have placed your
eye
exactly on the edge of the table (so that you are, as it were,
actually a Flatlander) the penny will then have ceased to appear
oval
at all, and will have become, so far as you can see, a straight
line.The
same thing would happen if you were to treat in the same way a
Triangle, or Square, or any other figure cut out of pasteboard. As
soon as you look at it with your eye on the edge on the table, you
will find that it ceases to appear to you a figure, and that it
becomes in appearance a straight line. Take for example an
equilateral Triangle—who represents with us a Tradesman of the
respectable class. Fig. 1 represents the Tradesman as you would see
him while you were bending over him from above; figs. 2 and 3
represent the Tradesman, as you would see him if your eye were
close
to the level, or all but on the level of the table; and if your eye
were quite on the level of the table (and that is how we see him in
Flatland) you would see nothing but a straight line.
