Flatland: A Romance of Many Dimensions
Edwin A. Abbott (1838-1926. English scholar, theologian, and writer.)
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| "O day and night, but this is wondrous strange" |
| ______ |
| / / /| ------ / /| /| / /-. |
| /---- / /__| / / /__| / | / / / |
| / /___ / | / /___ / | / |/ /__.-' |
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| No Dimensions One Dimension |
| . A ROMANCE OF MANY DIMENSIONS ----- |
| POINTLAND LINELAND |
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| Two Dimensions Three Dimensions |
| ___ __ |
| | | /__/| |
| |___| |__|/ |
| FLATLAND SPACELAND |
| "Fie, fie, how franticly I square my talk!" |
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With Illustrations by the Author, A SQUARE (Edwin A. Abbott)
To
The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most rare and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
Preface to the Second and Revised Edition, 1884.
By the Editor
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."
CONTENTS:
PART I: THIS WORLD
Section
1. Of the Nature of Flatland
2. Of the Climate and Houses in Flatland
3. Concerning the Inhabitants of Flatland
4. Concerning the Women
5. Of our Methods of Recognizing one another
6. Of Recognition by Sight
7. Concerning Irregular Figures
8. Of the Ancient Practice of Painting
9. Of the Universal Colour Bill
10. Of the Suppression of the Chromatic Sedition
11. Concerning our Priests
12. Of the Doctrine of our Priests
PART II: OTHER WORLDS
13. How I had a Vision of Lineland
14. How I vainly tried to explain the nature of Flatland
15. Concerning a Stranger from Spaceland
16. How the Stranger vainly endeavoured to reveal to me
in words the mysteries of Spaceland
17. How the Sphere, having in vain tried words,
resorted to deeds
18. How I came to Spaceland, and what I saw there
19. How, though the Sphere shewed me other mysteries
of Spaceland, I still desired more; and what came of it
20. How the Sphere encouraged me in a Vision
21. How I tried to teach the Theory of Three Dimensions
to my Grandson, and with what success
22. How I then tried to diffuse the Theory
of Three Dimensions by other means, and of the result
PART I: THIS WORLD
"Be patient, for the world is broad and wide."
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.
[Illustration 1]
[ASCII approximation follows]
(1) __________ (2) ___________ (3) _________
\ / --__ __-- ---
\ / -
\/
When I was in Spaceland I heard that your sailors have very similar
experiences while they traverse your seas and discern some distant
island or coast lying on the horizon. The far-off land may have bays,
forelands, angles in and out to any number and extent; yet at a
distance you see none of these (unless indeed your sun shines bright
upon them revealing the projections and retirements by means of light
and shade), nothing but a grey unbroken line upon the water.
Well, that is just what we see when one of our triangular or other
acquaintances comes toward us in Flatland. As there is neither sun
with us, nor any light of such a kind as to make shadows, we have none
of the helps to the sight that you have in Spaceland. If our friend
comes closer to us we see his line becomes larger; if he leaves us it
becomes smaller: but still he looks like a straight line; be he a
Triangle, Square, Pentagon, Hexagon, Circle, what you will--a straight
Line he looks and nothing else.
You may perhaps ask how under these disadvantageous circumstances we
are able to distinguish our friends from one another: but the answer to
this very natural question will be more fitly and easily given when I
come to describe the inhabitants of Flatland. For the present let me
defer this subject, and say a word or two about the climate and houses
in our country.