3388 lines
194 KiB
TeX
Executable File
3388 lines
194 KiB
TeX
Executable File
\title{Flatland} \author{Edwin A. Abbott}
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\date{\today}
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\documentclass[9pt, twoside]{extreport}
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\usepackage[english]{babel}
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\usepackage{graphicx}
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\usepackage[T1]{fontenc}
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\usepackage{hyperref}
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\usepackage[fulloldstylenums]{kpfonts}
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\usepackage{extsizes}
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\usepackage{setspace}
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\onehalfspacing
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\usepackage[skip=0pt, indent=5mm]{parskip}
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\usepackage{geometry}
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\geometry{
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paperheight=8.5in,
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paperwidth=5.5in,
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% heightrounded,
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margin=0.5in
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}
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% Adjust the top and bottom margins
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% http://kb.mit.edu/confluence/pages/viewpage.action?pageId=3907057
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\addtolength{\topmargin}{0.4in}
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\addtolength{\textheight}{-0.75in}
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\let\footruleskip\relax % for compatibility of memoir and fancyhdr
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\let\rm\rmfamily % for compatibility of memoir
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\let\sl\emph % for compatibility of memoir
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\usepackage{fancyhdr}
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\pagestyle{fancy}
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\fancyhf{} % remove everything
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\renewcommand{\headrulewidth}{0pt} % remove lines as well
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\fancyhead[CE, CO]{Flatland}
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\fancyhead[CE, CO]{Flatland}
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%\fancyhead[LO]{\leftmark}
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%\fancyhead[RE]{\thetitle}
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\fancyfoot[CE,CO]{\thepage}
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\urlstyle{same}
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\begin{document} % \maketitle
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\thispagestyle{empty}
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\vspace*{2cm}
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\includegraphics[scale=0.45]{flatland_cover}
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\vspace*{2cm}
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\begin{center}
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\textit{``Fie, fie, how franticly I square my talk!''}
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\end{center}
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\clearpage
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% \frontmatter
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\noindent{} This work is in the public domain in the United States because it was
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published before January 1, 1923.
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The author died in 1926, so this work is also in the public domain in
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countries and areas where the copyright term is the author's life plus 80
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years or less. This work may also be in the public domain in countries and
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areas with longer native copyright terms that apply the rule of the shorter
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term to foreign works.
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\vspace{5mm}
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\noindent\emph{Written by} Edwin A. Abbott \\
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\emph{Illustrated by} Edwin A. Abbott \\
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\emph{Typeset by} Ives van der Flaas\\
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\emph{Additional typesetting by} Kenneth John Odle\\ \\
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\null
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\vfill
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Fifth edition
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\chapter*{A Note From The Typesetter}
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Although Edwin A. Abbott's essay ``Flatland'' is readily available on the
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internet, I failed to find a nicely typeset version. At this time,
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the rendering quality of internet browsers doesn't come anywhere near the
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quality of a nice book.
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For these reasons, as well as a healthy portion of boredom, I made the version
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of ``Flatland'' you are currently reading, in \LaTeX. The \LaTeX\ source can be
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found on \url{https://github.com/Ivesvdf/flatland} (feel free to make issues
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or send me pull requests when you find bugs or have improvements). It should
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also be fairly easy --- a matter of minutes --- to produce a book-sized
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version, ready to send to a printer.
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\vspace{1cm}
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\hfill Ives van der Flaas <ives.vdf@gmail.com>, 2011-09-20
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\vspace{1.5cm}
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\noindent{}\textbf{An additional note:}\\
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\noindent{}I found this at the location above, where it included an A4 version in pdf. I decided to update this to a letterhalf page size, which is more common in North America.
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Where necessary, I also made adjustments such as the scaling of the images, changing to the \texttt{kpfont} font package, reducing the size of the type from 12 points to 9 points, and adjusting the margins and line spacing.
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The source can be found on \texttt{\url{https://git.kjodle.net/kjodle/Flatland}}.
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\begin{flushright}
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\textit{Kenneth John Odle}
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\end{flushright}
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\newpage \ \newpage
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\chapter*{Dedication}
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\begin{center}
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To\\
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The Inhabitants of \textsc{Space in General}\\
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And \textsc{H.C. in Particular}\\
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This Work is Dedicated\\
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By a Humble Native of Flatland\\
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In the Hope that\\
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Even as he was Initiated into the Mysteries\\
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Of \textsc{Three Dimensions}\\
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Having been previously conversant\\
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With \textsc{Only Tow}\\
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So the Citizens of that Celestial Region\\
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May aspire yet higher and higher\\
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To the Secrets of \textsc{Four Five or Even Six} Dimensions\\
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Thereby contributing\\
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To the Enlargement of \textsc{the Imagination}\\
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And the possible Development\\
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Of that most rare and excellent Gift of \textsc{Modesty}\\
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Among the Superior Races\\
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Of \textsc{Solid Humanity}\\
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\end{center}
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\chapter*{Preface}
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If my poor Flatland friend retained the vigour of mind which he enjoyed when
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he began to compose these Memoirs, I should not now need to represent him in
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this preface, in which he desires, fully, to return his thanks to his readers
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and critics in Spaceland, whose appreciation has, with unexpected celerity,
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required a second edition of this work; secondly, to apologize for certain
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errors and misprints (for which, however, he is not entirely responsible);
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and, thirdly, to explain on or two misconceptions. But he is not the Square he
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once was. Years of imprisonment, and the still heavier burden of general
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incredulity and mockery, have combined with the thoughts and notions, and much
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also of the terminology, which he acquired during his short stay in spaceland.
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He has, therefore, requested me to reply in his behalf to two special
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objections, one of an intellectual, the other of a moral nature.
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The first objection is, that a Flatlander, seeing a Line, sees something that
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must be thick to the eye as well as long to the eye (otherwise it would not be
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visible, if it had not some thickness); and consequently he ought (it is
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argued) to acknowledge that his countrymen are not only long and broad, but
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also (though doubtless to a very slight degree) thick or high. This objection
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is plausible, and, to Spacelanders, almost irresistible, so that, I confess,
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when I first heard it, I knew not what to reply. But my poor old friend's
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answer appears to me completely to meet it.
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``I admit,'' said he --- when I mentioned to him this objection --- ``I admit the
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truth of your critic's facts, but I deny his conclusions. It is true that we
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have really in Flatland a Third unrecognized Dimension called `height,' just
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as it also is true that you have really in Spaceland a Fourth unrecognized
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Dimension, called by no name at present, but which I will call `extra-height.'
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But we can no more take cognizance of our `height' than you can of your
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`extra-height.' Even I --- who have been in Spaceland, and have had the
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privilege of understanding for twenty-four hours the meaning of `height' ---
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even I cannot now comprehend it, nor realize it by the sense of sight or by
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any process of reason; I can but apprehend it by faith.''
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``The reason is obvious. Dimension implied direction, implies measurement,
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implies the more and the less. Now, all our lines are equally and
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infinitesimally thick (or high, whichever you like); consequently, there is
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nothing in them to lead our minds to the conception of that Dimension. No
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`delicate micrometer' --- as has been suggested by one too hasty Spaceland
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critic --- would in the least avail us; for we should not know what to measure,
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nor in what direction. When we see a Line, we see something that is long and
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bright; brightness, as well as length, is necessary to the existence of a
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Line; if the brightness vanishes, the Line is extinguished. Hence, all my
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Flatland friends --- when I talk to them about the unrecognized Dimension which
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is somehow visible in a Line --- say, `Ah, you mean brightness': and when I
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reply, `No, I mean a real Dimension,' they at once retort, `Then measure it,
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or tell us in what direction it extends'; and this silences me, for I can do
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neither. Only yesterday, when the Chief Circle (in other words our High
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Priest) came to inspect the State Prison and paid me his seventh annual visit,
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and when for the seventh time he put me the question, `Was I any better?' I
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tried to prove to him that he was `high', as well as long and broad, although
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he did not know it. But what was his reply? `You say I am ``high''; measure my
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``high-ness'' and I will believe you.'. What could I do? How could I meet his
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challenge? I was crushed; and he left the room triumphant.''
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``Does this still seem strange to you? Then put yourself in a similar position.
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Suppose a person of the Fourth Dimension, condescending to visit you, were to
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say, `Whenever you open your eyes, you see a Plane (which is of Two
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Dimensions) and you infer a Solid (which is of Three); but in reality you also
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see (though you do not recognize) a Fourth Dimension, which is not colour nor
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brightness nor anything of the kind, but a true Dimension, although I cannot
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point out to you its direction, nor can you possibly measure it.' What would
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you say to such a visitor? Would not you have him locked up? Well, that is my
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fate: and it is as natural for us Flatlanders to lock up a Square for
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preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube
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for preaching the Fourth. Alas, how strong a family likeness runs through
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blind and persecuting humanity in all Dimensions! Points, Lines, Squares,
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Cubes, Extra-Cubes --- we are all liable to the same errors, all alike the
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Slavers of our respective Dimensional prejudices, as one of our Spaceland
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poets has said --- `One touch of Nature makes all worlds akin.' '' \footnote{The
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Author desires me to add, that the misconceptions of some of his critics on
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this matter has induced him to insert in his dialogue with
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the Sphere, certain remarks which have a bearing on the point in question and
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which he had previously omitted as being tedious and unnecessary.}
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On this point the defence of the Square seems to me to be impregnable. I wish
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I could say that his answer to the second (or moral) objection was equally
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clear and cogent. It has been objected that he is a woman-hater; and as this
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objection has been vehemently urged by those whom Nature's decree has
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constituted the somewhat larger half of the Spaceland race, I should like to
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remove it, so far as I can honestly do so. But the Square is so unaccustomed
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to the use of the moral terminology of Spaceland that I should be doing him an
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injustice if I were literally to transcribe his defence against this charge.
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Acting, therefore, as his interpreter and summarizer, I gather that in the
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course of an imprisonment of seven years he has himself modified his own
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personal views, both as regards Women and as regards the Isosceles or Lower
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Classes. Personally, he now inclines to the opinion of the Sphere that the
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Straight Lines are in many important respects superior to the
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Circles. But, writing as a Historian, he has identified himself (perhaps too
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closely) with the views generally adopted by Flatland, and (as he has been
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informed) even by Spaceland, Historians; in whose pages (until very recent
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times) the destinies of Women and of the masses of mankind have seldom been
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deemed worthy of mention and never of careful consideration.
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In a still more obscure passage he now desires to disavow the Circular or
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aristocratic tendencies with which some critics have naturally credited him.
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While doing justice to the intellectual power with which a few Circles have
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for many generations maintained their supremacy over immense multitudes of
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their countrymen, he believes that the facts of Flatland, speaking for
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themselves without comment on his part, declare that Revolutions cannot always
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be suppressed by slaughter, and that Nature, in sentencing the Circles to
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infecundity, has condemned them to ultimate failure --- ``and herein,'' he says,
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``I see a fulfillment of the great Law of all worlds, that while the wisdom of
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Man thinks it is working one thing, the wisdom of Nature constrains it to work
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another, and quite a different and far better thing.'' For the rest, he begs
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his readers not to suppose that every minute detail in the daily life of
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Flatland must needs correspond to some other detail in Spaceland; and yet he
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hopes that, taken as a whole, his work may prove suggestive as well as
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amusing, to those Spacelanders of moderate and modest minds who --- speaking of
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that which is of the highest importance, but lies beyond experience --- decline
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to say on the one hand, ``This can never be,'' and on the other hand, ``It must
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needs be precisely thus, and we know all about it.''
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%b\mainmatter
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\part{This World}
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\chapter{Of the Nature of Flatland} I call our world Flatland, not because we
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call it so, but to make its nature clearer to you, my happy readers, who are
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privileged to live in Space.
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Imagine a vast sheet of paper on which straight Lines, Triangles, Squares,
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Pentagons, Hexagons, and other figures, instead of remaining fixed in their
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places, move freely about, on or in the surface, but without the power of
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rising above or sinking below it, very much like shadows --- only hard with
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luminous edges --- and you will then have a pretty correct notion of my country
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and countrymen. Alas, a few years ago, I should have said ``my universe'': but
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now my mind has been opened to higher views of things.
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In such a country, you will perceive at once that it is impossible that there
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should be anything of what you call a ``solid'' kind; but I dare say you will
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suppose that we could at least distinguish by sight the Triangles, Squares,
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and other figures, moving about as I have described them. On the contrary, we
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could see nothing of the kind, not at least so as to distinguish one figure
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from another. Nothing was visible, nor could be visible, to us, except
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Straight Lines; and the necessity of this I will speedily demonstrate.
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Place a penny on the middle of one of your tables in Space; and leaning over
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it, look down upon it. It will appear a circle.
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But now, drawling back to the edge of the table, gradually lower your eye
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(thus bringing yourself more and more into the condition of the inhabitants of
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Flatland), and you will find the penny becoming more and more oval to your
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view, and at last when you have placed your eye exactly on the edge of the
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table (so that you are, as it were, actually a Flatlander) the penny will then
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have ceased to appear oval at all, and will have become, so far as you can
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see, a straight line.
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The same thing would happen if you were to treat in the same way a Triangle,
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or a Square, or any other figure cut out of pasteboard. As soon as you look at
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it with your eye on the edge of the table, you will find that it ceases to
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appear to you as a figure, and that it becomes in appearance a straight line.
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Take for example an equilateral Triangle --- who represents with us a Tradesman
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of the respectable class. Figure 1 represents the Tradesman as you would see
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him while you were bending over him from above; figures 2 and 3 represent the
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Tradesman, as you would see him if your eye were close to the level, or all
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but on the level of the table; and if your eye were quite on the level of the
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table (and that is how we see him in Flatland) you would see nothing but a
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straight line.
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\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig1}
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When I was in Spaceland I heard that your sailors have very similar
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experiences while they traverse your seas and discern some distant island or
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coast lying on the horizon. The far-off land may have bays, forelands, angles
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in and out to any number and extent; yet at a distance you see none of these
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(unless indeed your sun shines bright upon them revealing the projections and
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retirements by means of light and shade), nothing but a grey unbroken line
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upon the water.
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Well, that is just what we see when one of our triangular or other
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acquaintances comes towards us in Flatland. As there is neither sun with us,
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nor any light of such a kind as to make shadows, we have none of the helps to
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the sight that you have in Spaceland. If our friend comes closer to us we see
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his line becomes larger; if he leaves us it becomes smaller; but still he
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looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon,
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Circle, what you will --- a straight Line he looks and nothing else.
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You may perhaps ask how under these disadvantageous circumstances we are able
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to distinguish our friends from one another: but the answer to this very
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natural question will be more fitly and easily given when I come to describe
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the inhabitants of Flatland. For the present let me defer this subject, and
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say a word or two about the climate and houses in our country.
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\chapter{Of the Climate and Houses in Flatland}
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As with you, so also with us, there are four points of the compass North,
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South, East, and West.
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There being no sun nor other heavenly bodies,
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it is impossible for us to determine the North in the usual way; but we have a
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method of our own. By a Law of Nature with us, there is a constant attraction
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to the South; and, although in temperate climates this is very slight --- so
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that even a Woman in reasonable health can journey several furlongs northward
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without much difficulty --- yet the hampering effect of the southward attraction
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is quite sufficient to serve as a compass in most parts of our earth.
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Moreover, the rain (which falls at stated intervals) coming always from the
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North, is an additional assistance; and in the towns we have the guidance of
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the houses, which of course have their side-walls running for the most part
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North and South, so that the roofs may keep off the rain from the North. In
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the country, where there are no houses, the trunks of the trees serve as some
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sort of guide. Altogether, we have not so much difficulty as might be expected
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in determining our bearings.
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Yet in our more temperate regions, in which the southward attraction is hardly
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felt, walking sometimes in a perfectly desolate plain where there have been no
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houses nor trees to guide me, I have been occasionally compelled to remain
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stationary for hours together, waiting till the rain came before continuing my
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journey. On the weak and aged, and especially on delicate Females, the force
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of attraction tells much more heavily than on the robust of the Male Sex, so
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that it is a point of breeding, if you meet a Lady in the street, always to
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give her the North side of the way --- by no means an easy thing to do always at
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short notice when you are in rude health and in a climate where it is
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difficult to tell your North from your South.
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Windows there are none in our houses: for the light comes to us alike in our
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homes and out of them, by day and by night, equally at all times and in all
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places, whence we know not. It was in old days, with our learned men, an
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interesting and oft-investigate question, ``What is the origin of light?'' and
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the solution of it has been repeatedly attempted, with no other result than to
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crowd our lunatic asylums with the would-be solvers. Hence, after fruitless
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attempts to suppress such investigations indirectly by making them liable to a
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heavy tax, the Legislature, in comparatively recent times, absolutely
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prohibited them. I --- alas, I alone in Flatland --- know now only too well the
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true solution of this mysterious problem; but my knowledge cannot be made
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intelligible to a single one of my countrymen; and I am mocked at --- I, the
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sole possessor of the truths of Space and of the theory of the introduction of
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Light from the world of three Dimensions --- as if I were the maddest of the
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mad! But a truce to these painful digressions: let me return to our homes.
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The most common form for the construction of a house is five-sided or
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pentagonal, as in the annexed figure. The two Northern sides RO, OF,
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constitute the roof, and for the most part have no doors; on the East is a
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small door for the Women; on the West a much larger one for the Men; the South
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side or floor is usually doorless.
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\begin{center}
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\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig2}
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\end{center}
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Square and triangular houses are not allowed, and for this reason. The angles
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of a Square (and still more those of an equilateral Triangle,) being much more
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pointed than those of a Pentagon, and the lines of inanimate objects (such as
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houses) being dimmer than the lines of Men and Women, it follows that there is
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no little danger lest the points of a square of triangular house residence
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might do serious injury to an inconsiderate or perhaps absentminded traveller
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suddenly running against them: and therefore, as early as the eleventh century
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of our era, triangular houses were universally forbidden by Law, the only
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exceptions being fortifications, powder-magazines, barracks, and other state
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buildings, which is not desirable that the general public should approach
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without circumspection.
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At this period, square houses were still everywhere permitted, though
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discouraged by a special tax. But, about three centuries afterwards, the Law
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decided that in all towns containing a population above ten thousand, the
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angle of a Pentagon was the smallest house-angle that could be allowed
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consistently with the public safety. The good sense of the community has
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seconded the efforts of the Legislature; and now, even in the country, the
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pentagonal construction has superseded every other. It is only now and then in
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some very remote and backward agricultural district that an antiquarian may
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still discover a square house.
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\chapter{Concerning the Inhabitants of Flatland}
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The greatest length or breadth of a full grown inhabitant of Flatland may be
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estimated at about eleven of your inches. Twelve inches may be regarded as a
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maximum.
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Our Women are Straight Lines.
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Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides,
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each about eleven inches long, and a base or third side so short (often not
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exceeding half an inch) that they form at their vertices a very sharp and
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formidable angle. Indeed when their bases are of the most degraded type (not
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more than the eighth part of an inch in size), they can hardly be
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distinguished from Straight lines or Women; so extremely pointed are their
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vertices. With us, as with you, these Triangles are distinguished from others
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by being called Isosceles; and by this name I shall refer to them in the
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following pages.
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Our Middle Class consists of Equilateral or Equal-Sided Triangles.
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Our Professional Men and Gentlemen are Squares (to which class I myself
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belong) and Five-Sided Figures or Pentagons.
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Next above these come the Nobility, of whom there are several degrees,
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beginning at Six-Sided Figures, or Hexagons, and from thence rising in the
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number of their sides till they receive the honourable title of Polygonal, or
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many-Sided. Finally when the number of the sides becomes so numerous, and the
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sides themselves so small, that the figure cannot be distinguished from a
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circle, he is included in the Circular or Priestly order; and this is the
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highest class of all.
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It is a Law of Nature with us that a male child shall have one more side than
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his father, so that each generation shall rise (as a rule) one step in the
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scale of development and nobility. Thus the son of a Square is a Pentagon; the
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son of a Pentagon, a Hexagon; and so on.
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But this rule applies not always to the Tradesman, and still less often to the
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Soldiers, and to the Workmen; who indeed can hardly be said to deserve the
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name of human Figures, since they have not all their sides equal. With them
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therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a
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Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope
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is not shut out, even from the Isosceles, that his posterity may ultimately
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rise above his degraded condition. For, after a long series of military
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successes, or diligent and skillful labours, it is generally found that the
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more intelligent among the Artisan and Soldier classes manifest a slight
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increase of their third side or base, and a shrinkage of the two other sides.
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Intermarriages (arranged by the Priests) between the sons and daughters of
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these more intellectual members of the lower classes generally result in an
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offspring approximating still more to the type of the Equal-Sided Triangle.
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Rarely --- in proportion to the vast numbers of Isosceles births --- is a genuine
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and certifiable Equal-Sided Triangle produced from Isosceles parents
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\footnote{ ``What need of a certificate?'' a Spaceland critic may ask: ``Is not
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the procreation of a Square Son a certificate from Nature herself, proving the
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Equal-sidedness of the Father?'' I reply that no Lady of any position will
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marry an uncertified Triangle. Square offspring has sometimes resulted from a
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slightly Irregular Triangle; but in almost every such case the Irregularity of
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the first generation is visited on the third; which either fails to attain the
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Pentagonal rank, or relapses to the Triangular.}. Such a birth requires, as
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its antecedents, not only a series of carefully arranged intermarriages, but
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also a long-continued exercise of frugality and self-control on the part of
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the would-be ancestors of the coming Equilateral, and a patient, systematic,
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and continuous development of the Isosceles intellect through many
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generations.
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The birth of a True Equilateral Triangle from Isosceles parents is the subject
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of rejoicing in our country for many furlongs around. After a strict
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examination conducted by the Sanitary and Social Board, the infant, if
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certified as Regular, is with solemn ceremonial admitted into the class of
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Equilaterals. He is then immediately taken from his proud yet sorrowing
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parents and adopted by some childless Equilateral, who is bound by oath never
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to permit the child henceforth to enter his former home or so much as to look
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upon his relations again, for fear lest the freshly developed organism may, by
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force of unconscious imitation, fall back again into his hereditary level.
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The occasional emergence of an Equilateral from the ranks of his serf-born
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ancestors is welcomed, not only by the poor serfs themselves, as a gleam of
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light and hope shed upon the monotonous squalor of their existence, but also
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by the Aristocracy at large; for all the higher classes are well aware that
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these rare phenomena, while they do little or nothing to vulgarize their own
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privileges, serve as almost useful barrier against revolution from below.
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Had the acute-angled rabble been all, without exception, absolutely destitute
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of hope and of ambition, they might have found leaders in some of their many
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seditious outbreaks, so able as to render their superior numbers and strength
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too much even for the wisdom of the Circles. But a wise ordinance of Nature
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has decreed that, in proportion as the working-classes increase in
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intelligence, knowledge, and all virtue, in that same proportion their acute
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angle (which makes them physically terrible) shall increase also and
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approximate to their comparatively harmless angle of the Equilateral Triangle.
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Thus, in the most brutal and formidable of the soldier class --- creatures
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almost on a level with women in their lack of intelligence --- it is found that,
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as they wax in the mental ability necessary to employ their tremendous
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penetrating power to advantage, so do they wane in the power of penetration
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itself.
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How admirable is this Law of Compensation! And how perfect a proof of the
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natural fitness and, I may almost say, the divine origin of the aristocratic
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constitution of the States of Flatland! By a judicious use of this Law of
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Nature, the Polygons and Circles are almost always able to stifle sedition in
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its very cradle, taking advantage of the irrepressible and boundless
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hopefulness of the human mind. Art also comes to the aid of Law and Order. It
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is generally found possible --- by a little artificial compression or expansion
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on the part of the State physicians --- to make some of the more intelligent
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leaders of a rebellion perfectly Regular, and to admit them at once into the
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privileged classes; a much larger number, who are still below the standard,
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allured by the prospect of being ultimately ennobled, are induced to enter the
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State Hospitals, where they are kept in honourable confinement for life; one
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or two alone of the most obstinate, foolish, and hopelessly irregular are led
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to execution.
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Then the wretched rabble of the Isosceles, planless and leaderless, are ether
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transfixed without resistance by the small body of their brethren whom the
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Chief Circle keeps in pay for emergencies of this kind; or else more often, by
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means of jealousies and suspicious skillfully fomented among them by the
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Circular party, they are stirred to mutual warfare, and perish by one
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another's angles. No less than one hundred and twenty rebellions are recorded
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in our annals, besides minor outbreaks numbered at two hundred and
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thirty-five; and they have all ended thus.
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\chapter{Concerning the Women}
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If our highly pointed Triangles of the Soldier class are formidable, it may be
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readily inferred that far more formidable are our Women. For, if a Soldier is
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a wedge, a Woman is a needle; being, so to speak, all point, at least at the
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two extremities. Add to this the power of making herself practically invisible
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at will, and you will perceive that a Female, in Flatland, is a creature by no
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means to be trifled with.
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But here, perhaps, some of my younger Readers may ask how a woman in Flatland
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can make herself invisible. This ought, I think, to be apparent without any
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explanation. However, a few words will make it clear to the most unreflecting.
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Place a needle on the table. Then, with your eye on the level of the table,
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look at it side-ways, and you see the whole length of it; but look at it
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end-ways, and you see nothing but a point, it has become practically
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invisible. Just so is it with one of our Women. When her side is turned
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towards us, we see her as a straight line; when the end containing her eye or
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mouth --- for with us these two organs are identical --- is the part that meets
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our eye, then we see nothing but a highly lustrous point; but when the back is
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presented to our view, then --- being only sub-lustrous, and, indeed, almost as
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dim as an inanimate object --- her hinder extremity serves her as a kind of
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Invisible Cap.
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The dangers to which we are exposed from our Women must now be manifest to the
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meanest capacity in Spaceland. If even the angle of a respectable Triangle in
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the middle class is not without its dangers; if to run against a Working Man
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involves a gash; if collision with an Officer of the military class
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necessitates a serious wound; if a mere touch from the vertex of a Private
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Soldier brings with it danger of death; --- what can it be to run against a
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woman, except absolute and immediate destruction? And when a Woman is
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invisible, or visible only as a dim sub-lustrous point, how difficult must it
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be, even for the most cautious, always to avoid collision!
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Many are the enactments made at different times in the different States of
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Flatland, in order to minimize this peril; and in the Southern and less
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temperate climates, where the force of gravitation is greater, and human
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beings more liable to casual and involuntary motions, the Laws concerning
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Women are naturally much more stringent. But a general view of the Code may be
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obtained from the following summary: ---
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Every house shall have one entrance on the Eastern side, for the use of
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Females only; by which all females shall enter ``in a becoming and respectful
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manner'' \footnote{ When I was in Spaceland I understood that some of your
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Priestly circles have in the same way a separate entrance for Villagers,
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Farmers and Teachers of Board Schools (Spectator, Sept. 1884, P. 1255) that
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they may ``approach in a becoming and respectful manner.''} and not by the Men's
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or Western door. No Female shall walk in any public place without continually
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keeping up her Peace-cry, under penalty of death. Any Female, duly certified
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to be suffering from St. Vitus's Dance, fits, chronic cold accompanied by
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violent sneezing, or any disease necessitating involuntary motions, shall be
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instantly destroyed. In some of the States there is an additional Law
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forbidding Females, under penalty of death, from walking or standing in any
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public place without moving their backs constantly from right to left so as to
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indicate their presence to those behind them; others oblige a Woman, when
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travelling, to be followed by one of her sons, or servants, or by her husband;
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others confine Women altogether to their houses except during the religious
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festivals. But it has been found by the wisest of our Circles or Statesmen
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that the multiplication of restrictions on Females tends not only to the
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debilitation and diminution of the race, but also to the increase of domestic
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murders to such an extent that a State loses more than it gains by a too
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prohibitive Code.
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For whenever the temper of the Women is thus exasperated by confinement at
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home or hampering regulations abroad, they are apt to vent their spleen upon
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their husbands and children; and in the less temperate climates the whole male
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population of a village has been sometimes destroyed in one or two hours of
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simultaneous female outbreak. Hence the Three Laws, mentioned above, suffice
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for the better regulated States, and may be accepted as a rough
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exemplification of our Female Code.
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After all, our principal safeguard is found, not in Legislature, but in the
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interests of the Women themselves. For, although they can inflict
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instantaneous death by a retrograde movement, yet unless they can at once
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disengage their stinging extremity from the struggling body of their victim,
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their own frail bodies are liable to be shattered.
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The power of Fashion is also on our side. I pointed
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out that in some less civilized States no female is suffered to stand in any
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public place without swaying her back from right to left. This practice has
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been universal among ladies of any pretensions to breeding in all
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well-governed States, as far back as the memory of Figures can reach. It is
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considered a disgrace to any state that legislation should have to enforce
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what ought to be, and is in every respectable female, a natural instinct. The
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rhythmical and, if I may so say, well-modulated undulation of the back in our
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ladies of Circular rank is envied and imitated by the wife of a common
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Equilateral, who can achieve nothing beyond a mere monotonous swing, like the
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ticking of a pendulum; and the regular tick of the Equilateral is no less
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admired and copied by the wife of the progressive and aspiring Isosceles, in
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the females of whose family no ``back-motion'' of any kind has become as yet a
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necessity of life. Hence, in every family of position and consideration, ``back
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motion'' is as prevalent as time itself; and the husbands and sons in these
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households enjoy immunity at least from invisible attacks.
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Not that it must be for a moment supposed that our Women are destitute of
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affection. But unfortunately the passion of the moment predominates, in the
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Frail Sex, over every other consideration. This is, of course, a necessity
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arising from their unfortunate conformation. For as they have no pretensions
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to an angle, being inferior in this respect to the very lowest of the
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Isosceles, they are consequently wholly devoid of brainpower, and have neither
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reflection, judgment nor forethought, and hardly any memory. Hence, in their
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fits of fury, they remember no claims and recognize no distinctions. I have
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actually known a case where a Woman has exterminated her whole household, and
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half an hour afterwards, when her rage was over and the fragments swept away,
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has asked what has become of her husband and her children.
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Obviously then a Woman is not to be irritated as long as she is in a position
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where she can turn round. When you have them in their apartments --- which are
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constructed with a view to denying them that power --- you can say and do what
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you like; for they are then wholly impotent for mischief, and will not
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remember a few minutes hence the incident for which they may be at this moment
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threatening you with death, nor the promises which you may have found it
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necessary to make in order to pacify their fury.
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On the whole we got on pretty smoothly in our domestic
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relations, except in the lower strata of the Military Classes. There the want
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of tact and discretion on the part of the husbands produces at times
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indescribable disasters. Relying too much on the offensive weapons of their
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acute angles instead of the defensive organs of good sense and seasonable
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simulations, these reckless creatures too often neglect the prescribed
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construction of the women's apartments, or irritate their wives by ill-advised
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expressions out of doors, which they refuse immediately to retract. Moreover a
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blunt and stolid regard for literal truth indisposes them to make those lavish
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promises by which the more judicious Circle can in a moment pacify his
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consort. The result is massacre; not, however, without its advantages, as it
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eliminates the more brutal and troublesome of the Isosceles; and by many of
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our Circles the destructiveness of the Thinner Sex is regarded as one among
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many providential arrangements for suppressing redundant population, and
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nipping Revolution in the bud.
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Yet even in our best regulated and most approximately Circular families I
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cannot say that the ideal of family life is so high as with you in Spaceland.
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There is peace, in so far as the absence of slaughter may be called by that
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name, but there is necessarily little harmony of tastes or pursuits; and the
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cautious wisdom of the Circles has ensured safety at the cost of domestic
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comfort. In every Circular or Polygonal household it has been a habit from
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time immemorial --- and now has become a kind of instinct among the women of our
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higher classes --- that the mothers and daughters should constantly keep their
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eyes and mouths towards their husband and his male friends; and for a lady in
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a family of distinction to turn her back upon her husband would be regarded as
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a kind of portent, involving loss of status. But, as I shall soon shew, this
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custom, though it has the advantage of safety, is not without disadvantages.
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Moving with ease and smoothness in uttering words; of rapid speech;
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nimble in speaking; glib; as, a flippant, voluble, tongue. In the house of
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the Working Man or respectable Tradesman --- where the wife is allowed to turn
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her back upon her husband, while pursuing her household avocations --- there are
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at least intervals of quiet, when the wife is neither seen nor heard, except
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for the humming sound of the continuous Peace-cry; but in the homes of the
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upper classes there is too often no peace. There the voluble mouth and bright
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penetrating eye are ever directed toward the Master of the household; and
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light itself is not more persistent than the stream of Feminine discourse. The
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tact and skill which suffice to avert a Woman's sting are unequal to the task
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of stopping a Woman's mouth; and as the wife has absolutely nothing to say,
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and absolutely no constraint of wit, sense, or conscience to prevent her from
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saying it, not a few cynics have been found to aver that they prefer the
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danger of the death-dealing but inaudible sting to the safe sonorousness of a
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Woman's other end.
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To my readers in Spaceland the condition of our Women may seen truly
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deplorable, and so indeed it is. A Male of the lowest type of the Isosceles
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may look forward to some improvement of his angle, and to the ultimate
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elevation of the whole of his degraded caste; but no Woman can entertain such
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hopes for her sex. ``Once a Woman, always a Woman'' is a Decree of Nature; and
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the very Laws of Evolution seem suspended in her disfavour. Yet at least we
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can admire the wise Prearrangement which has ordained that, as they have no
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hopes, so they shall have no memory to recall, and no forethought to
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anticipate, the miseries and humiliations which are at once a necessity of
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their existence and the basis of the constitution of Flatland.
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\chapter{Of our Methods of Recognizing one another} You, who are blessed with
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shade as well as light, you, who are gifted with two eyes, endowed with a
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knowledge of perspective, and charmed with the enjoyment of various colours,
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you, who can actually see an angle, and contemplate the complete circumference
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of a Circle in the happy region of the Three Dimensions --- how shall I make it
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clear to you the extreme difficulty which we in Flatland experience in
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recognizing one another's configuration?
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Recall what I told you above. All beings in Flatland, animate and inanimate,
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no matter what their form, present to our view the same, or nearly the same,
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appearance, viz. that of a straight Line. How then can one be distinguished
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from another, where all appear the same?
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The answer is threefold. The first means of recognition is the sense of
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hearing; which with us is far more highly developed than with you, and which
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enables us not only to distinguish by the voice our personal friends, but even
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to discriminate between different classes, at least so far as concerns the
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three lowest orders, the Equilateral, the Square, and the Pentagon --- for the
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Isosceles I take no account. But as we ascend the social scale, the process of
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discriminating and being discriminated by hearing increases in difficulty,
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partly because voices are assimilated, partly because the faculty of
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voice-discrimination is a plebeian virtue not much developed among the
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Aristocracy. And wherever there is any danger of imposture we cannot trust to
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this method. Amongst our lowest orders, the vocal organs are developed to a
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degree more than correspondent with those of hearing, so that an Isosceles can
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easily feign the voice of a Polygon, and, with some training, that of a Circle
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himself. A second method is therefore more commonly resorted to.
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Feeling is, among our Women and lower classes --- about our upper classes I
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shall speak presently --- the principal test of recognition, at all events
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between strangers, and when the question is, not as to the individual, but as
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to the class. What therefore ``introduction'' is among the higher classes in
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Spaceland, that the process of ``feeling'' is with us. ``Permit me to ask you to
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feel and be felt by my friend Mr. So-and-so'' --- is still, among the more
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old-fashioned of our country gentlemen in districts remote from towns, the
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customary formula for a Flatland introduction. But in the towns, and among men
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of business, the words ``be felt by'' are omitted and the sentence is
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abbreviated to, ``Let me ask you to feel Mr. So-and-so''; although it is
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assumed, of course, that the ``feeling'' is to be reciprocal. Among our still
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more modern and dashing young gentlemen --- who are extremely averse to
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superfluous effort and supremely indifferent to the purity of their native
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language --- the formula is still further curtailed by the use of ``to feel'' in a
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technical sense, meaning, ``to
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recommend-for-the-purposes-of-feeling-and-being-felt''; and at this moment the
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``slang'' of polite or fast society in the upper classes sanctions such a
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barbarism as ``Mr. Smith, permit me to feel Mr. Jones.''.
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Let not my Reader however suppose that ``feeling'' is with us the tedious
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process that it would be with you, or that we find it necessary to feel right
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round all the sides of every individual before we determine the class to which
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he belongs. Long practice and training, begun in the schools and continued in
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the experience of daily life, enable us to discriminate at once by the sense
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of touch, between the angles of an equal-sided Triangle, Square, and Pentagon;
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and I need not say that the brainless vertex of an acute-angled Isosceles is
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obvious to the dullest touch. It is therefore not necessary, as a rule, to do
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more than feel a single angle of an individual; and this, once ascertained,
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tells us the class of the person whom we are addressing, unless indeed he
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belongs to the higher sections of the nobility. There the difficulty is much
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greater. Even a Master of Arts in our University of Wentbridge has been known
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to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a
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Doctor of Science in or out of that famous University who could pretend to
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decide promptly and unhesitatingly between a twenty-sided and a twenty-four
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sided member of the Aristocracy.
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Those of my readers who recall the extracts I gave above from the Legislative
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code concerning Women, will readily perceive that the process of introduction
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by contact requires some care and discretion. Otherwise the angles might
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inflict on the unwary Feeler irreparable injury. It is essential for the
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safety of the Feeler that the Felt should stand perfectly still. A start, a
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fidgety shifting of the position, yes, even a violent sneeze, has been known
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before now to prove fatal to the incautious, and to nip in the bud many a
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promising friendship. Especially is this true among the lower classes of the
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Triangles. With them, the eye is situated so far from their vertex that they
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can scarcely take cognizance of what goes on at that extremity of their frame.
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They are, moreover, of a rough coarse nature, not sensitive to the delicate
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touch of the highly organized Polygon. What wonder then if an involuntary toss
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of the head has ere now deprived the State of a valuable life!
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I have heard that my excellent Grandfather --- one of the least irregular of his
|
|
unhappy Isosceles class, who indeed obtained, shortly before his decease, four
|
|
out of seven votes from the Sanitary and Social Board for passing him into the
|
|
class of the Equal-sided --- often deplored, with a tear in his venerable eye, a
|
|
miscarriage of this kind, which had occurred to his
|
|
great-great-great-Grandfather, a respectable Working Man with an angle or
|
|
brain of 59 degrees 30 minutes. According to his account, my unfortunate
|
|
Ancestor, being afflicted with rheumatism, and in the act of being felt by a
|
|
Polygon, by one sudden start accidentally transfixed the Great Man through the
|
|
diagonal and thereby, partly in consequence of his long imprisonment and
|
|
degradation, and partly because of the moral shock which pervaded the whole of
|
|
my Ancestor's relations, threw back our family a degree and a half in their
|
|
ascent towards better things. The result was that in the next generation the
|
|
family brain was registered at only 58 degrees, and not till the lapse of five
|
|
generations was the lost ground recovered, the full 60 degrees attained, and
|
|
the Ascent from the Isosceles finally achieved. And all this series of
|
|
calamities from one little accident in the process of Feeling.
|
|
|
|
At this point I think I hear some of my better educated readers exclaim, ``How
|
|
could you in Flatland know anything about angles and degrees, or minutes? We
|
|
see an angle, because we, in the region of Space, can see two straight lines
|
|
inclined to one another; but you, who can see nothing but on straight line at
|
|
a time, or at all events only a number of bits of straight lines all in one
|
|
straight line, --- how can you ever discern any angle, and much less register
|
|
angles of different sizes?''.
|
|
|
|
I answer that though we cannot see angles, we can infer them, and this with
|
|
great precision. Our sense of touch, stimulated by necessity, and developed by
|
|
long training, enables us to distinguish angles far more accurately than your
|
|
sense of sight, when unaided by a rule or measure of angles. Nor must I omit
|
|
to explain that we have great natural helps. It is with us a Law of Nature
|
|
that the brain of the Isosceles class shall begin at half a degree, or thirty
|
|
minutes, and shall increase (if it increases at all) by half a degree in every
|
|
generation until the goal of 60 degrees is reached, when the condition of
|
|
serfdom is quitted, and the freeman enters the class of Regulars.
|
|
|
|
Consequently, Nature herself supplies us with an ascending scale or Alphabet
|
|
of angles for half a degree up to 60 degrees, Specimens of which are placed in
|
|
every Elementary School throughout the land. Owing to occasional
|
|
retrogressions, to still more frequent moral and intellectual stagnation, and
|
|
to the extraordinary fecundity of the Criminal and Vagabond classes, there is
|
|
always a vast superfluity of individuals of the half degree and single degree
|
|
class, and a fair abundance of Specimens up to 10 degrees. These are
|
|
absolutely destitute of civic rights; and a great number of them, not having
|
|
even intelligence enough for the purposes of warfare, are devoted by the
|
|
States to the service of education. Fettered immovably so as to remove all
|
|
possibility of danger, they are placed in the classrooms of our Infant
|
|
Schools, and there they are utilized by the Board of Education for the purpose
|
|
of imparting to the offspring of the Middle Classes that tact and intelligence
|
|
of which these wretched creatures themselves are utterly devoid.
|
|
|
|
In some States the Specimens are occasionally fed and suffered to exist for
|
|
several years; but in the more temperate and better regulated regions, it is
|
|
found in the long run more advantageous for the educational interests of the
|
|
young, to dispense with food, and to renew the Specimens every month --- which
|
|
is about the average duration of the foodless existence of the Criminal class.
|
|
In the cheaper schools, what is gained by the longer existence of the Specimen
|
|
is lost, partly in the expenditure for food, and partly in the diminished
|
|
accuracy of the angles, which are impaired after a few weeks of constant
|
|
``feeling''. Nor must we forget to add, in enumerating the advantages of the
|
|
more expensive system, that it tends, though slightly yet perceptibly, to the
|
|
diminution of the redundant Isosceles population --- an object which every
|
|
statesman in Flatland constantly keeps in view. On the whole therefore ---
|
|
although I am not ignorant that, in many popularly elected School Boards,
|
|
there is a reaction in favour of ``the cheap system'' as it is called --- I am
|
|
myself disposed to think that this is one of the many cases in which expense
|
|
is the truest economy.
|
|
|
|
But I must not allow questions of School Board politics to divert me from my
|
|
subject. Enough has been said, I trust, to shew that Recognition by feeling is
|
|
not so tedious or indecisive a process as might have been supposed; and it is
|
|
obviously more trustworthy than Recognition by hearing. Still there remains,
|
|
as has been pointed out above, the objection that this method is not without
|
|
danger. For this reason many in the Middle and Lower classes, and all without
|
|
exception in the Polygonal and Circular orders, prefer a third method, the
|
|
description of which shall be reserved for the next section.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{Of Recognition by Sight}
|
|
I am about to appear very inconsistent. In previous sections I have said that
|
|
all figures in Flatland present the appearance of a straight line; and it was
|
|
added or implied, that it is consequently impossible to distinguish by the
|
|
visual organ between individuals of different classes: yet now I am about to
|
|
explain to my Spaceland critics how we are able to recognize one another by
|
|
the sense of sight.
|
|
|
|
If however the Reader will take the trouble to refer to the passage in which
|
|
Recognition by Feeling is stated to be universal, he will find this
|
|
qualification --- ``among the lower classes''. It is only among the higher classes
|
|
and in our more temperate climates that Sight Recognition is practised.
|
|
|
|
That this power exists in any regions and for any classes is the result of
|
|
Fog; which prevails during the greater part of the year in all parts save the
|
|
torrid zones. That which is with you in Spaceland an unmixed evil, blotting
|
|
out the landscape, depressing the spirits, and enfeebling the health, is by us
|
|
recognized as a blessing scarcely inferior to air itself, and as the Nurse of
|
|
arts and Parent of sciences. But let me explain my meaning, without further
|
|
eulogies on this beneficent Element.
|
|
|
|
If Fog were non-existent, all lines would appear equally and indistinguishably
|
|
clear; and this is actually the case in those unhappy countries in which the
|
|
atmosphere is perfectly dry and transparent. But wherever there is a rich
|
|
supply of Fog, objects that are at a distance, say of three feet, are
|
|
appreciably dimmer than those at the distance of two feet eleven inches; and
|
|
the result is that by careful and constant experimental observation of
|
|
comparative dimness and clearness, we are enabled to infer with great
|
|
exactness the configuration of the object observed.
|
|
|
|
An instance will do more than a volume of generalities to make my meaning
|
|
clear.
|
|
|
|
Suppose I see two individuals approaching whose rank I wish to ascertain. They
|
|
are, we will suppose, a Merchant and a Physician, or in other words, an
|
|
Equilateral Triangle and a Pentagon; how am I to distinguish them?
|
|
|
|
\includegraphics[trim=20mm 0mm 0mm 0mm, width=\linewidth]{fig3}
|
|
|
|
It will be obvious, to every child in Spaceland who has touched the threshold
|
|
of Geometrical Studies, that, if I can bring my eye so that its glance may
|
|
bisect an angle (A) of the approaching stranger, my view will lie as it were
|
|
evenly between his two sides that are next to me (viz. CA and AB), so that I
|
|
shall contemplate the two impartially, and both will appear of the same size.
|
|
|
|
Now in the case of (1) the Merchant, what shall I see? I shall see a straight
|
|
line DAE, in which the middle point (A) will be very bright because it is
|
|
nearest to me; but on either side the line will shade away rapidly to dimness,
|
|
because the sides AC and AB recede rapidly into the fog and what appear to me
|
|
as the Merchant's extremities, viz. D and E, will be very dim indeed.
|
|
|
|
On the other hand in the case of (2) the Physician, though I shall here also
|
|
see a line (D'A'E') with a bright centre (A'), yet it will shade away less
|
|
rapidly to dimness, because the sides (A'C', A'B') recede less rapidly into
|
|
the fog: and what appear to me the Physician's extremities, viz. D' and E',
|
|
will not be not so dim as the extremities of the Merchant.
|
|
|
|
The Reader will probably understand from these
|
|
two instances how --- after a very long training supplemented by constant
|
|
experience --- it is possible for the well-educated classes among us to
|
|
discriminate with fair accuracy between the middle and lowest orders, by the
|
|
sense of sight. If my Spaceland Patrons have grasped this general conception,
|
|
so far as to conceive the possibility of it and not to reject my account as
|
|
altogether incredible --- I shall have attained all I can reasonably expect.
|
|
Were I to attempt further details I should only perplex. Yet for the sake of
|
|
the young and inexperienced, who may perchance infer --- from the two simple
|
|
instances I have given above, of the manner in which I should recognize my
|
|
Father and my Sons --- that Recognition by sight is an easy affair, it may be
|
|
needful to point out that in actual life most of the problems of Sight
|
|
Recognition are far more subtle and complex.
|
|
|
|
If for example, when my Father, the Triangle, approaches me, he happens to
|
|
present his side to me instead of his angle, then, until I have asked him to
|
|
rotate, or until I have edged my eye around him, I am for the moment doubtful
|
|
whether he may not be a Straight Line, or, in other words, a Woman. Again,
|
|
when I am in the company of one of my two hexagonal Grandsons, contemplating
|
|
one of his sides (AB) full front, it will be evident from the accompanying
|
|
diagram that I shall see one whole line (AB) in comparative brightness
|
|
(shading off hardly at all at the ends) and two smaller lines (CA and BD) dim
|
|
throughout and shading away into greater dimness towards the extremities C and
|
|
D.
|
|
\begin{center}
|
|
\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig4}
|
|
\end{center}
|
|
|
|
|
|
But I must not give way to the temptation of enlarging on these
|
|
topics. The meanest mathematician in Spaceland will readily believe me when I
|
|
assert that the problems of life, which present themselves to the
|
|
well-educated --- when they are themselves in motion, rotating, advancing or
|
|
retreating, and at the same time attempting to discriminate by the sense of
|
|
sight between a number of Polygons of high rank moving in different
|
|
directions, as for example in a ball-room or conversazione --- must be of a
|
|
nature to task the angularity of the most intellectual, and amply justify the
|
|
rich endowments of the Learned Professors of Geometry, both Static and
|
|
Kinetic, in the illustrious University of Wentbridge, where the Science and
|
|
Art of Sight Recognition are regularly taught to large classes of the elite of
|
|
the States.
|
|
|
|
It is only a few of the scions of our
|
|
noblest and wealthiest houses, who are able to give the time and money
|
|
necessary for the thorough prosecution of this noble and valuable Art. Even to
|
|
me, a Mathematician of no mean standing, and the Grandfather of two most
|
|
hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd
|
|
of rotating Polygons of the higher classes, is occasionally very perplexing.
|
|
And of course to a common Tradesman, or Serf, such a sight is almost as
|
|
unintelligible as it would be to you, my Reader, were you suddenly transported
|
|
to our country.
|
|
|
|
In such a crowd you could see on all sides of you nothing but a Line,
|
|
apparently straight, but of which the parts would vary irregularly and
|
|
perpetually in brightness or dimness. Even if you had completed your third
|
|
year in the Pentagonal and Hexagonal classes in the University, and were
|
|
perfect in the theory of the subject, you would still find there was need of
|
|
many years of experience, before you could move in a fashionable crowd without
|
|
jostling against your betters, whom it is against etiquette to ask to ``feel'',
|
|
and who, by their superior culture and breeding, know all about your
|
|
movements, while you know very little or nothing about theirs. In a word, to
|
|
comport oneself with perfect propriety in Polygonal society, one ought to be a
|
|
Polygon oneself. Such at least is the painful teaching of my experience.
|
|
|
|
It is astonishing how much the Art --- or I may almost
|
|
call it instinct --- of Sight Recognition is developed by the habitual practice
|
|
of it and by the avoidance of the custom of ``Feeling''. Just as, with you, the
|
|
deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet,
|
|
will never acquire the more difficult but far more valuable art of lip-speech
|
|
and lip-reading, so it is with us as regards ``Seeing'' and ``Feeling''. None who
|
|
in early life resort to ``Feeling'' will ever learn ``Seeing'' in perfection.
|
|
|
|
For this reason, among our Higher Classes, ``Feeling'' is discouraged or
|
|
absolutely forbidden. From the cradle their children, instead of going to the
|
|
Public Elementary schools (where the art of Feeling is taught,) are sent to
|
|
higher Seminaries of an exclusive character; and at our illustrious
|
|
University, to ``feel'' is regarded as a most serious fault, involving
|
|
Rustication for the first offence, and Expulsion for the second.
|
|
|
|
But among the lower classes the art of Sight Recognition is regarded as an
|
|
unattainable luxury. A common Tradesman cannot afford to let his son spend a
|
|
third of his life in abstract studies. The children of the poor are therefore
|
|
allowed to ``feel'' from their earliest years, and they gain thereby a precocity
|
|
and an early vivacity which contrast at first most favourably with the inert,
|
|
undeveloped, and listless behaviour of the half-instructed youths of the
|
|
Polygonal class; but when the latter have at last completed their University
|
|
course, and are prepared to put their theory into practice, the change that
|
|
comes over them may almost be described as a new birth, and in every art,
|
|
science, and social pursuit they rapidly overtake and distance their
|
|
Triangular competitors.
|
|
|
|
Only a few of the Polygonal Class fail to pass the Final Test or Leaving
|
|
Examination at the University. The condition of the unsuccessful minority is
|
|
truly pitiable. Rejected from the higher class, they are also despised by the
|
|
lower. They have neither the matured and systematically trained powers of the
|
|
Polygonal Bachelors and Masters of Arts, nor yet the native precocity and
|
|
mercurial versatility of the youthful Tradesman. The professions, the public
|
|
services, are closed against them, and though in most States they are not
|
|
actually debarred from marriage, yet they have the greatest difficulty in
|
|
forming suitable alliances, as experience shews that the offspring of such
|
|
unfortunate and ill-endowed parents is generally itself unfortunate, if not
|
|
positively Irregular.
|
|
|
|
It is from these specimens of the refuse of our Nobility that the great
|
|
Tumults and Seditions of past ages have generally derived their leaders; and
|
|
so great is the mischief thence arising that an increasing minority of our
|
|
more progressive Statesmen are of opinion that true mercy would dictate their
|
|
entire suppression, by enacting that all who fail to pass the Final
|
|
Examination of the University should be either imprisoned for life, or
|
|
extinguished by a painless death.
|
|
|
|
But I find myself digressing into the subject of Irregularities, a matter of
|
|
such vital interest that it demands a separate section.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{Concerning Irregular Figures}
|
|
Throughout the previous pages I have been assuming --- what perhaps should have
|
|
been laid down at the beginning as a distinct and fundamental proposition ---
|
|
that every human being in Flatland is a Regular Figure, that is to say of
|
|
regular construction. By this I mean that a Woman must not only be a line, but
|
|
a straight line; that an Artisan or Soldier must have two of his sides equal;
|
|
that Tradesmen must have three sides equal; Lawyers (of which class I am a
|
|
humble member), four sides equal, and, generally, that in every Polygon, all
|
|
the sides must be equal.
|
|
|
|
The sizes of the sides would of course depend upon the age of the individual.
|
|
A Female at birth would be about an inch long, while a tall adult Woman might
|
|
extend to a foot. As to the Males of every class, it may be roughly said that
|
|
the length of an adult's size, when added together, is two feet or a little
|
|
more. But the size of our sides is not under consideration. I am speaking of
|
|
the equality of sides, and it does not need much reflection to see that the
|
|
whole of the social life in Flatland rests upon the fundamental fact that
|
|
Nature wills all Figures to have their sides equal.
|
|
|
|
If our sides were unequal our angles might be unequal. Instead of its being
|
|
sufficient to feel, or estimate by sight, a single angle in order to determine
|
|
the form of an individual, it would be necessary to ascertain each angle by
|
|
the experiment of Feeling. But life would be too short for such a tedious
|
|
groping. The whole science and art of Sight Recognition would at once perish;
|
|
Feeling, so far as it is an art, would not long survive; intercourse would
|
|
become perilous or impossible; there would be an end to all confidence, all
|
|
forethought; no one would be safe in making the most simple social
|
|
arrangements; in a word, civilization might relapse into barbarism.
|
|
|
|
Am I going too fast to carry my Readers with me to these obvious conclusions?
|
|
Surely a moment's reflection, and a single instance from common life, must
|
|
convince every one that our social system is based upon Regularity, or
|
|
Equality of Angles. You meet, for example, two or three Tradesmen in the
|
|
street, whom your recognize at once to be Tradesman by a glance at their
|
|
angles and rapidly bedimmed sides, and you ask them to step into your house to
|
|
lunch. This you do at present with perfect confidence, because everyone knows
|
|
to an inch or two the area occupied by an adult Triangle: but imagine that
|
|
your Tradesman drags behind his regular and respectable vertex, a
|
|
parallelogram of twelve or thirteen inches in diagonal: --- what are you to do
|
|
with such a monster sticking fast in your house door?
|
|
|
|
But I am insulting the intelligence of my Readers by accumulating details
|
|
which must be patent to everyone who enjoys the advantages of a Residence in
|
|
Spaceland. Obviously the measurements of a single angle would no longer be
|
|
sufficient under such portentous circumstances; one's whole life would be
|
|
taken up in feeling or surveying the perimeter of one's acquaintances. Already
|
|
the difficulties of avoiding a collision in a crowd are enough to tax the
|
|
sagacity of even a well-educated Square; but if no one could calculate the
|
|
Regularity of a single figure in the company, all would be chaos and
|
|
confusion, and the slightest panic would cause serious injuries, or --- if there
|
|
happened to be any Women or Soldiers present --- perhaps considerable loss of
|
|
life.
|
|
|
|
Expediency therefore concurs with Nature in stamping the seal of its approval
|
|
upon Regularity of conformation: nor has the Law been backward in seconding
|
|
their efforts. ``Irregularity of Figure'' means with us the same as, or more
|
|
than, a combination of moral obliquity and criminality with you, and is
|
|
treated accordingly. There are not wanting, it is true, some promulgators of
|
|
paradoxes who maintain that there is no necessary connection between
|
|
geometrical and moral Irregularity. ``The Irregular,'' they say, ``is from his
|
|
birth scouted by his own parents, derided by his brothers and sisters,
|
|
neglected by the domestics, scorned and suspected by society, and excluded
|
|
from all posts of responsibility, trust, and useful activity. His every
|
|
movement is jealously watched by the police till he comes of age and presents
|
|
himself for inspection; then he is either destroyed, if he is found to exceed
|
|
the fixed margin of deviation, at an uninteresting occupation for a miserable
|
|
stipend; obliged to live and board at the office, and to take even his
|
|
vacation under close supervision; what wonder that human nature, even in the
|
|
best and purest, is embittered and perverted by such surroundings!''
|
|
|
|
All this very plausible reasoning does not convince me, as it has not
|
|
convinced the wisest of our Statesmen, that our ancestors erred in laying it
|
|
down as an axiom of policy that the toleration of Irregularity is incompatible
|
|
with the safety of the State. Doubtless, the life of an Irregular is hard; but
|
|
the interests of the Greater Number require that it shall be hard. If a man
|
|
with a triangular front and a polygonal back were allowed to exist and to
|
|
propagate a still more Irregular posterity, what would become of the arts of
|
|
life? Are the houses and doors and churches in Flatland to be altered in order
|
|
to accommodate such monsters? Are our ticket-collectors to be required to
|
|
measure every man's perimeter before they allow him to enter a theatre, or to
|
|
take his place in a lecture room? Is an Irregular to be exempted from the
|
|
militia? And if not, how is he to be prevented from carrying desolation into
|
|
the ranks of his comrades? Again, what irresistible temptations to fraudulent
|
|
impostures must needs beset such a creature! How easy for him to enter a shop
|
|
with his polygonal front foremost, and to order goods to any extent from a
|
|
confiding tradesman! Let the advocates of a falsely called Philanthropy plead
|
|
as they may for the abrogation of the Irregular Penal Laws, I for my part have
|
|
never known an Irregular who was not also what Nature evidently intended him
|
|
to be --- a hypocrite, a misanthropist, and, up to the limits of his power, a
|
|
perpetrator of all manner of mischief.
|
|
|
|
Not that I should be disposed to recommend (at present) the extreme measures
|
|
adopted by some States, where an infant whose angle deviates by half a degree
|
|
from the correct angularity is summarily destroyed at birth. Some of our
|
|
highest and ablest men, men of real genius, have during their earliest days
|
|
laboured under deviations as great as, or even greater than forty-five
|
|
minutes: and the loss of their precious lives would have been an irreparable
|
|
injury to the State. The art of healing also has achieved some of its most
|
|
glorious triumphs in the compressions, extensions, trepannings, colligations,
|
|
and other surgical or diaetetic operations by which Irregularity has been
|
|
partly or wholly cured. Advocating therefore a Via Media, I would lay down no
|
|
fixed or absolute line of demarcation; but at the period when the frame is
|
|
just beginning to set, and when the Medical Board has reported that recovery
|
|
is improbably, I would suggest that the Irregular offspring be painlessly and
|
|
mercifully consumed.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{Of the Ancient Practice of Painting}
|
|
If my Readers have followed me with any attention up to this point, they will
|
|
not be surprised to hear that life is somewhat dull in Flatland. I do not, of
|
|
course, mean that there are not battles, conspiracies, tumults, factions, and
|
|
all those other phenomena which are supposed to make History interesting; nor
|
|
would I deny that the strange mixture of the problems of life and the problems
|
|
of Mathematics, continually inducing conjecture and giving an opportunity of
|
|
immediate verification, imparts to our existence a zest which you in Spaceland
|
|
can hardly comprehend. I speak now from the aesthetic and artistic point of
|
|
view when I say that life with us is dull; aesthetically and artistically,
|
|
very dull indeed.
|
|
|
|
How can it be otherwise, when all one's prospect, all one's landscapes,
|
|
historical pieces, portraits, flowers, still life, are nothing but a single
|
|
line, with no varieties except degrees of brightness and obscurity?
|
|
|
|
It was not always thus. Colour, if Tradition speaks the truth, once for the
|
|
space of half a dozen centuries or more, threw a transient splendour over the
|
|
lives of our ancestors in the remotest ages. Some private individual --- a
|
|
Pentagon whose name is variously reported --- having casually discovered the
|
|
constituents of the simpler colours and a rudimentary method of painting, is
|
|
said to have begun by decorating first his house, then his slaves, then his
|
|
Father, his Sons, and Grandsons, lastly himself. The convenience as well as
|
|
the beauty of the results commended themselves to all. Wherever Chromatistes,
|
|
--- for by that name the most trustworthy authorities concur in calling him, ---
|
|
turned his variegated frame, there he at once excited attention, and attracted
|
|
respect. No one now needed to ``feel'' him; no one mistook his front for his
|
|
back; all his movements were readily ascertained by his neighbours without the
|
|
slightest strain on their powers of calculation; no one jostled him, or failed
|
|
to make way for him; his voice was saved the labour of that exhausting
|
|
utterance by which we colourless Squares and Pentagons are often forced to
|
|
proclaim our individuality when we move amid a crowd of ignorant Isosceles.
|
|
|
|
The fashion spread like wildfire. Before a week was over, every Square and
|
|
Triangle in the district had copied the example of Chromatistes, and only a
|
|
few of the more conservative Pentagons still held out. A month or two found
|
|
even the Dodecagons infected with the innovation. A year had not elapsed
|
|
before the habit had spread to all but the very highest of the Nobility.
|
|
Needless to say, the custom soon made its way from the district of
|
|
Chromatistes to surrounding regions; and within two generations no one in all
|
|
Flatland was colourless except the Women and the Priests.
|
|
|
|
Here Nature herself appeared to erect a barrier, and to plead against
|
|
extending the innovations to these two classes. Many-sidedness was almost
|
|
essential as a pretext for the Innovators. ``Distinction of sides is intended
|
|
by Nature to imply distinction of colours'' --- such was the sophism which in
|
|
those days flew from mouth to mouth, converting whole towns at a time to a new
|
|
culture. But manifestly to our Priests and Women this adage did not apply. The
|
|
latter had only one side, and therefore --- plurally and pedantically speaking ---
|
|
no sides. The former --- if at least they would assert their claim to be readily
|
|
and truly Circles, and not mere high-class Polygons, with an infinitely large
|
|
number of infinitesimally small sides --- were in the habit of boasting (what
|
|
Women confessed and deplored) that they also had no sides, being blessed with
|
|
a perimeter of only one line, or, in other words, a Circumference. Hence it
|
|
came to pass that these two Classes could see no force in the so-called axiom
|
|
about ``Distinction of Sides implying Distinction of Colour;'' and when all
|
|
others had succumbed to the fascinations of corporal decoration, the Priests
|
|
and the Women alone still remained pure from the pollution of paint.
|
|
|
|
Immoral, licentious, anarchical, unscientific --- call them by what names you
|
|
will --- yet, from an aesthetic point of view, those ancient days of the Colour
|
|
Revolt were the glorious childhood of Art in Flatland --- a childhood, alas,
|
|
that never ripened into manhood, nor even reached the blossom of youth. To
|
|
live then in itself a delight, because living implied seeing. Even at a small
|
|
party, the company was a pleasure to behold; the richly varied hues of the
|
|
assembly in a church or theatre are said to have more than once proved too
|
|
distracting from our greatest teachers and actors; but most ravishing of all
|
|
is said to have been the unspeakable magnificence of a military review.
|
|
|
|
The sight of a line of battle of twenty thousand
|
|
Isosceles suddenly facing about, and exchanging the sombre black of their
|
|
bases for the orange of the two sides including their acute angle; the militia
|
|
of the Equilateral Triangles tricoloured in red, white, and blue; the mauve,
|
|
ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly
|
|
rotating near their vermillion guns; the dashing and flashing of the
|
|
five-coloured and six-coloured Pentagons and Hexagons careering across the
|
|
field in their offices of surgeons, geometricians and aides-de-camp --- all
|
|
these may well have been sufficient to render credible the famous story how an
|
|
illustrious Circle, overcome by the artistic beauty of the forces under his
|
|
command, threw aside his marshal's baton and his royal crown, exclaiming that
|
|
he henceforth exchanged them for the artist's pencil. How great and glorious
|
|
the sensuous development of these days must have been is in part indicated by
|
|
the very language and vocabulary of the period. The commonest utterances of
|
|
the commonest citizens in the time of the Colour Revolt seem to have been
|
|
suffused with a richer tinge of word or thought; and to that era we are even
|
|
now indebted for our finest poetry and for whatever rhythm still remains in
|
|
the more scientific utterance of those modern days.
|
|
|
|
|
|
\chapter{Of the Universal Colour Bill}
|
|
But meanwhile the intellectual Arts were fast decaying.
|
|
|
|
The Art of Sight Recognition, being no longer needed, was no longer practised;
|
|
and the studies of Geometry, Statics, Kinetics, and other kindred subjects,
|
|
came soon to be considered superfluous, and fell into disrespect and neglect
|
|
even at our University. The inferior Art of Feeling speedily experienced the
|
|
same fate at our Elementary Schools. Then the Isosceles classes, asserting
|
|
that the Specimens were no longer used nor needed, and refusing to pay the
|
|
customary tribute from the Criminal classes to the service of Education, waxed
|
|
daily more numerous and more insolent on the strength of their immunity from
|
|
the old burden which had formerly exercised the twofold wholesome effect of at
|
|
once taming their brutal nature and thinning their excessive numbers.
|
|
|
|
Year by year the Soldiers and Artisans began more vehemently to assert --- and
|
|
with increasing truth --- that there was no great difference between them and
|
|
the very highest class of Polygons, now that they were raised to an equality
|
|
with the latter, and enabled to grapple with all the difficulties and solve
|
|
all the problems of life, whether Statical or Kinetical, by the simple process
|
|
of Colour Recognition. Not content with the natural neglect into which Sight
|
|
Recognition was falling, they began boldly to demand the legal prohibition of
|
|
all ``monopolizing and aristocratic Arts'' and the consequent abolition of all
|
|
endowments for the studies of Sight Recognition, Mathematics, and Feeling.
|
|
Soon, they began to insist that inasmuch as Colour, which was a second Nature,
|
|
had destroyed the need of aristocratic distinctions, the Law should follow in
|
|
the same path, and that henceforth all individuals and all classes should be
|
|
recognized as absolutely equal and entitled to equal rights.
|
|
|
|
Finding the higher Orders wavering and undecided, the leaders of the
|
|
Revolution advanced still further in their requirements, and at last demanded
|
|
that all classes alike, the Priests and the Women not excepted, should do
|
|
homage to Colour by submitting to be painted. When it was objected that
|
|
Priests and Women had no sides, they retorted that Nature and Expediency
|
|
concurred in dictating that the front half of every human being (that is to
|
|
say, the half containing his eye and mouth) should be distinguishable from his
|
|
hinder half. They therefore brought before a general and extraordinary
|
|
Assembly of all the States of Flatland a Bill proposing that in every Woman
|
|
the half containing the eye and mouth should be coloured red, and the other
|
|
half green. The Priests were to be painted in the same way, red being applied
|
|
to that semicircle in which the eye and mouth formed the middle point; while
|
|
the other or hinder semicircle was to be coloured green.
|
|
|
|
There was no little cunning in this proposal, which indeed emanated not from
|
|
any Isosceles --- for no being so degraded would have angularity enough to
|
|
appreciate, much less to devise, such a model of state-craft --- but from an
|
|
Irregular Circle who, instead of being destroyed in his childhood, was
|
|
reserved by a foolish indulgence to bring desolation on his country and
|
|
destruction on myriads of followers.
|
|
|
|
On the one hand the proposition was calculated to bring the Women in all
|
|
classes over to the side of the Chromatic Innovation. For by assigning to the
|
|
Women the same two colours as were assigned to the Priests, the Revolutionists
|
|
thereby ensured that, in certain positions, every Woman would appear as a
|
|
Priest, and be treated with corresponding respect and deference --- a prospect
|
|
that could not fail to attract the Female Sex in a mass.
|
|
|
|
But by some of my Readers the possibility of the identical appearance of
|
|
Priests and Women, under a new Legislation, may not be recognized; if so, a
|
|
word or two will make it obvious.
|
|
|
|
Imagine a woman duly decorated, according to the new Code; with the front half
|
|
(i.e., the half containing the eye and mouth) red, and with the hinder half
|
|
green. Look at her from one side. Obviously you will see a straight line, half
|
|
red, half green.
|
|
\begin{center}
|
|
\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig5}
|
|
\end{center}
|
|
|
|
Now imagine a Priest, whose mouth is at M, and whose front semicircle (AMB) is
|
|
consequently coloured red, while his hinder semicircle is green; so that the
|
|
diameter AB divides the green from the red. If you contemplate the Great Man
|
|
so as to have your eye in the same straight line as his dividing diameter
|
|
(AB), what you will see will be a straight line (CBD), of which one half (CB)
|
|
will be red, and the other (BD) green. The whole line (CD) will be rather
|
|
shorter perhaps than that of a full-sized Woman, and will shade off more
|
|
rapidly towards its extremities; but the identity of the colours would give
|
|
you an immediate impression of identity in Class, making you neglectful of
|
|
other details. Bear in mind the decay of Sight Recognition which threatened
|
|
society at the time of the Colour revolt; add too the certainty that Woman
|
|
would speedily learn to shade off their extremities so as to imitate the
|
|
Circles; it must then be surely obvious to you, my dear Reader, that the
|
|
Colour Bill placed us under a great danger of confounding a Priest with a
|
|
young Woman.
|
|
|
|
How attractive this prospect must have been to the Frail Sex may readily be
|
|
imagined. They anticipated with delight the confusion that would ensue. At
|
|
home they might hear political and ecclesiastical secrets intended not for
|
|
them but for their husbands and brothers, and might even issue some commands
|
|
in the name of a priestly Circle; out of doors the striking combination of red
|
|
and green without addition of any other col\-ours, would be sure to lead the
|
|
common people into endless mistakes, and the Woman would gain whatever the
|
|
Circles lost, in the deference of the passers by. As for the scandal that
|
|
would befall the Circular Class if the frivolous and unseemly conduct of the
|
|
Women were imputed to them, and as to the consequent subversion of the
|
|
Constitution, the Female Sex could not be expected to give a thought to these
|
|
considerations. Even in the households of the Circles, the Women were all in
|
|
favour of the Universal Colour Bill.
|
|
|
|
The second object aimed at by the Bill was the gradual demoralization of the
|
|
Circles themselves. In the general intellectual decay they still preserved
|
|
their pristine clearness and strength of understanding. From their earliest
|
|
childhood, familiarized in their Circular households with the total absence of
|
|
Colour, the Nobles alone preserved the Sacred Art of Sight Recognition, with
|
|
all the advantages that result from that admirable training of the intellect.
|
|
Hence, up to the date of the introduction of the Universal Colour Bill, the
|
|
Circles had not only held their own, but even increased their lead of the
|
|
other classes by abstinence from the popular fashion.
|
|
|
|
Now therefore the artful Irregular whom I described above as the real author
|
|
of this diabolical Bill, determined at one blow to lower the status of the
|
|
Hierarchy by forcing them to submit to the pollution of Colour, and at the
|
|
same time to destroy their domestic opportunities of training in the Art of
|
|
Sight Recognition, so as to enfeeble their intellects by depriving them of
|
|
their pure and colourless homes. Once subjected to the chromatic taint, every
|
|
parental and every childish Circle would demoralize each other. Only in
|
|
discerning between the Father and the Mother would the Circular infant find
|
|
problems for the exercise of his understanding --- problems too often likely to
|
|
be corrupted by maternal impostures with the result of shaking the child's
|
|
faith in all logical conclusions. Thus by degrees the intellectual lustre of
|
|
the Priestly Order would wane, and the road would then lie open for a total
|
|
destruction of all Aristocratic Legislature and for the subversion of our
|
|
Privileged Classes.
|
|
|
|
|
|
\chapter{Of the Suppression of the Chromatic Sedition}
|
|
The agitation for the Universal Colour Bill continued for three years; and up
|
|
to the last moment of that period it seemed as though Anarchy were destined to
|
|
triumph.
|
|
|
|
A whole army of Polygons, who turned out to fight as private soldiers, was
|
|
utterly annihilated by a superior force of Isosceles Triangles --- the Squares
|
|
and Pentagons meanwhile remaining neutral. Worse than all, some of the ablest
|
|
Circles fell a prey to conjugal fury. Infuriated by political animosity, the
|
|
wives in many a noble household wearied their lords with prayers to give up
|
|
their opposition to the Colour Bill; and some, finding their entreaties
|
|
fruitless, fell on and slaughtered their innocent children and husband,
|
|
perishing themselves in the act of carnage. It is recorded that during that
|
|
triennial agitation no less than twenty-three Circles perished in domestic
|
|
discord.
|
|
|
|
Great indeed was the peril. It seemed as though the Priests had no choice
|
|
between submission and extermination; when suddenly the course of events was
|
|
completely changed by one of those picturesque incidents which Statesmen ought
|
|
never to neglect, often to anticipate, and sometimes perhaps to originate,
|
|
because of the absurdly disproportionate power with which they appeal to the
|
|
sympathies of the populace.
|
|
|
|
It happened that an Isosceles of a low type, with a brain little if at all
|
|
above four degrees --- accidentally dabbling in the colours of some Tradesman
|
|
whose shop he had plundered --- painted himself, or caused himself to be painted
|
|
(for the story varies) with the twelve colours of a Dodecagon. Going into the
|
|
Market Place he accosted in a feigned voice a maiden, the orphan daughter of a
|
|
noble Polygon, whose affection in former days he had sought in vain; and by a
|
|
series of deceptions --- aided, on the one side, by a string of lucky accidents
|
|
too long to relate, and, on the other, by an almost inconceivable fatuity and
|
|
neglect of ordinary precautions on the part of the relations of the bride --- he
|
|
succeeded in consummating the marriage. The unhappy girl committed suicide on
|
|
discovering the fraud to which she had been subjected.
|
|
|
|
When the news of this catastrophe spread from State to State the minds of the
|
|
Women were violently agitated. Sympathy with the miserable victim and
|
|
anticipations of similar deceptions for themselves, their sisters, and their
|
|
daughters, made them now regard the Colour Bill in an entirely new aspect. Not
|
|
a few openly avowed themselves converted to antagonism; the rest needed only a
|
|
slight stimulus to make a similar avowal. Seizing this favourable opportunity,
|
|
the Circles hastily convened an extraordinary Assembly of the States; and
|
|
besides the usual guard of Convicts, they secured the attendance of a large
|
|
number of reactionary Women.
|
|
|
|
Amidst an unprecedented concourse, the Chief Circle of those days --- by name
|
|
Pantocyclus --- arose to find himself hissed and hooted by a hundred and twenty
|
|
thousand Isosceles. But he secured silence by declaring that henceforth the
|
|
Circles would enter on a policy of Concession; yielding to the wishes of the
|
|
majority, they would accept the Colour Bill. The uproar being at once
|
|
converted to applause, he invited Chromatistes, the leader of the Sedition,
|
|
into the centre of the hall, to receive in the name of his followers the
|
|
submission of the Hierarchy. Then followed a speech, a masterpiece of
|
|
rhetoric, which occupied nearly a day in the delivery, and to which no summary
|
|
can do justice.
|
|
|
|
With a grave appearance of impartiality he declared that as they were now
|
|
finally committing themselves to Reform or Innovation, it was desirable that
|
|
they should take one last view of the perimeter of the whole subject, its
|
|
defects as well as its advantages. Gradually introduction the mention of the
|
|
dangers to the Tradesmen, the Professional Classes and the Gentlemen, he
|
|
silenced the rising murmurs of the Isosceles by reminding them that, in spite
|
|
of all these defects, he was willing to accept the Bill if it was approved by
|
|
the majority. But it was manifest that all, except the Isosceles, were moved
|
|
by his words and were either neutral or averse to the Bill.
|
|
|
|
Turning now to the Workmen he asserted that their interests must not be
|
|
neglected, and that, if they intended to accept the Colour Bill, they ought at
|
|
least to do so with full view of the consequences. Many of them, he said, were
|
|
on the point of being admitted to the class of the Regular Triangles; others
|
|
anticipated for their children a distinction they could not hope for
|
|
themselves. That honourable ambition would now have to be sacrificed. With the
|
|
universal adoption of Colour, all distinctions would cease; Regularity would
|
|
be confused with Irregularity; development would give place to retrogression;
|
|
the Workman would in a few generations be degraded to the level of the
|
|
Military, or even the Convict Class; political power would be in the hands of
|
|
the greatest number, that is to say the Criminal Classes, who were already
|
|
more numerous than the Workmen, and would soon out-number all the other
|
|
Classes put together when the usual Compensative Laws of Nature were violated.
|
|
|
|
A subdued murmur of assent ran through the ranks of the Artisans, and
|
|
Chromatistes, in alarm, attempted to step forward and address them. But he
|
|
found himself encompassed with guards and forced to remain silent while the
|
|
Chief Circle in a few impassioned words made a final appeal to the Women,
|
|
exclaiming that, if the Colour Bill passed, no marriage would henceforth be
|
|
safe, no woman's honour secure; fraud, deception, hypocrisy would pervade
|
|
every household; domestic bliss would share the fate of the Constitution and
|
|
pass to speedy perdition. ``Sooner than this'', he cried, ``Come death''.
|
|
|
|
At these words, which were the preconcerted signal for action, the Isosceles
|
|
Convicts fell on and transfixed the wretched Chromatistes; the Regular
|
|
Classes, opening their ranks, made way for a band of Women who, under
|
|
direction of the Circles, moved back foremost, invisibly and unerringly upon
|
|
the unconscious soldiers; the Artisans, imitating the example of their
|
|
betters, also opened their ranks. Meantime bands of Convicts occupied every
|
|
entrance with an impenetrable phalanx.
|
|
|
|
The battle, or rather carnage, was of short duration. Under the skillful
|
|
generalship of the Circles almost every Woman's charge was fatal and very many
|
|
extracted their sting uninjured, ready for a second slaughter. But no second
|
|
blow was needed; the rabble of the Isosceles did the rest of the business for
|
|
themselves. Surprised, leader-less, attacked in front by invisible foes, and
|
|
finding egress cut off by the Convicts behind them, they at once --- after their
|
|
manner --- lost all presence of mind, and raised the cry of ``treachery''. This
|
|
sealed their fate. Every Isosceles now saw and felt a foe in every other. In
|
|
half an hour not one of that vast multitude was living; and the fragments of
|
|
seven score thousand of the Criminal Class slain by one another's angles
|
|
attested the triumph of Order.
|
|
|
|
The Circles delayed not to push their victory to the uttermost. The Working
|
|
Men they spared but decimated. The Militia of the Equilaterals was at once
|
|
called out, and every Triangle suspected of Irregularity on reasonable
|
|
grounds, was destroyed by Court Martial, without the formality of exact
|
|
measurement by the Social Board. The homes of the Military and Artisan classes
|
|
were inspected in a course of visitation extending through upwards of a year;
|
|
and during that period every town, village, and hamlet was systematically
|
|
purged of that excess of the lower orders which had been brought about by the
|
|
neglect to pay the tribute of Criminals to the Schools and University, and by
|
|
the violation of other natural Laws of the Constitution of Flatland. Thus the
|
|
balance of classes was again restored.
|
|
|
|
Needless to say that henceforth the use of Colour was abolished, and its
|
|
possession prohibited. Even the utterance of any word denoting Colour, except
|
|
by the Circles or by qualified scientific teachers, was punished by a severe
|
|
penalty. Only at our University in some of the very highest and most esoteric
|
|
classes --- which I myself have never been privileged to attend --- it is
|
|
understood that the sparing use of Colour is still sanctioned for the purpose
|
|
of illustrating some of the deeper problems of mathematics. But of this I can
|
|
only speak from hearsay.
|
|
|
|
Elsewhere in Flatland, Colour is now non-existent. The art of making it is
|
|
known to only one living person, the Chief Circle for the time being; and by
|
|
him it is handed down on his death-bed to none but his Successor. One
|
|
manufactory alone produces it; and, lest the secret should be betrayed, the
|
|
Workmen are annually consumed, and fresh ones introduced. So great is the
|
|
terror with which even now our Aristocracy looks back to the far-distant days
|
|
of the agitation for the Universal Colour Bill.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{Concerning our Priests}
|
|
It is high time that I should pass from these brief and discursive notes about
|
|
things in Flatland to the central event of this book, my initiation into the
|
|
mysteries of Space. That is my subject; all that has gone before is merely
|
|
preface.
|
|
|
|
For this reason I must omit many matters of which the explanation would not, I
|
|
flatter myself, be without interest for my Readers: as for example, our method
|
|
of propelling and stopping ourselves, although destitute of feet; the means by
|
|
which we give fixity to structures of wood, stone, or brick, although of
|
|
course we have no hands, nor can we lay foundations as you can, nor avail
|
|
ourselves of the lateral pressure of the earth; the manner in which the rain
|
|
originates in the intervals between our various zones, so that the northern
|
|
regions do not intercept the moisture falling on the southern; the nature of
|
|
our hills and mines, our trees and vegetables, our seasons and harvests; our
|
|
Alphabet and method of writing, adapted to our linear tablets; these and a
|
|
hundred other details of our physical existence I must pass over, nor do I
|
|
mention them now except to indicate to my readers that their omission proceeds
|
|
not from forgetfulness on the part of the author, but from his regard for the
|
|
time of the Reader.
|
|
|
|
Yet before I proceed to my legitimate subject some few final remarks will no
|
|
doubt be expected by my Readers upon these pillars and mainstays of the
|
|
Constitution of Flatland, the controllers of our conduct and shapers of our
|
|
destiny, the objects of universal homage and almost of adoration: need I say
|
|
that I mean our Circles or Priests?
|
|
|
|
When I call them Priests, let me not be understood as meaning no more than the
|
|
term denotes with you. With us, our Priests are Administrators of all
|
|
Business, Art, and Science; Directors of Trade, Commerce, Generalship,
|
|
Architecture, Engineering, Education, Statesmanship, Legislature, Morality,
|
|
Theology; doing nothing themselves, they are the Causes of everything worth
|
|
doing, that is done by others.
|
|
|
|
Although popularly everyone called a Circle is deemed a Circle, yet among the
|
|
better educated Classes it is known that no Circle is really a Circle, but
|
|
only a Polygon with a very large number of very small sides. As the number of
|
|
the sides increases, a Polygon approximates to a Circle; and, when the number
|
|
is very great indeed, say for example three or four hundred, it is extremely
|
|
difficult for the most delicate touch to feel any polygonal angles. Let me say
|
|
rather it would be difficult: for, as I have shown above, Recognition by
|
|
Feeling is unknown among the highest society, and to feel a Circle would be
|
|
considered a most audacious insult. This habit of abstention from Feeling in
|
|
the best society enables a Circle the more easily to sustain the veil of
|
|
mystery in which, from his earliest years, he is wont to enwrap the exact
|
|
nature of his Perimeter or Circumference. Three feet being the average
|
|
Perimeter it follows that, in a Polygon of three hundred sides each side will
|
|
be no more than the hundredth part of a foot in length, or little more than
|
|
the tenth part of an inch; and in a Polygon of six or seven hundred sides the
|
|
sides are little larger than the diameter of a Spaceland pin-head. It is
|
|
always assumed, by courtesy, that the Chief Circle for the time being has ten
|
|
thousand sides.
|
|
|
|
The ascent of the posterity of the Circles in the social scale is not
|
|
restricted, as it is among the lower Regular classes, by the Law of Nature
|
|
which limits the increase of sides to one in each generation. If it were so,
|
|
the number of sides in the Circle would be a mere question of pedigree and
|
|
arithmetic, and the four hundred and ninety-seventh descendant of an
|
|
Equilateral Triangle would necessarily be a polygon With five hundred sides.
|
|
But this is not the case. Nature's Law prescribes two antagonistic decrees
|
|
affecting Circular propagation; first, that as the race climbs higher in the
|
|
scale of development, so development shall proceed at an accelerated pace;
|
|
second, that in the same proportion, the race shall become less fertile.
|
|
Consequently in the home of a Polygon of four or five hundred sides it is rare
|
|
to find a son; more than one is never seen. On the other hand the son of a
|
|
five-hundred-sided Polygon has been known to possess five hundred and fifty,
|
|
or even six hundred sides.
|
|
|
|
Art also steps in to help the process of higher Evolution. Our physicians have
|
|
discovered that the small and tender sides of an infant Polygon of the higher
|
|
class can be fractured, and his whole frame re-set, with such exactness that a
|
|
Polygon of two or three hundred sides sometimes --- by no means always, for the
|
|
process is attended with serious risk --- but sometimes overleaps two or three
|
|
hundred generations, and as it were double at a stroke, the number of his
|
|
progenitors and the nobility of his descent.
|
|
|
|
Many a promising child is sacrificed in this way. Scarcely one out of ten
|
|
survives. Yet so strong is the parental ambition among those Polygons who are,
|
|
as it were, on the fringe of the Circular class, that it is very rare to find
|
|
the Nobleman of that position in society, who has neglected to place his
|
|
first-born in the Circular Neo-Therapeutic Gymnasium before he has attained
|
|
the age of a month.
|
|
|
|
One year determines success or failure. At the end of that time the child has,
|
|
in all probability, added one more to the tombstones that crowd the
|
|
Neo-Therapeutic Cemetery; but on rare occasional a glad procession bares back
|
|
the little one to his exultant parents, no longer a Polygon, but a Circle, at
|
|
least by courtesy: and a single instance of so blessed a result induces
|
|
multitudes of Polygonal parents to submit to similar domestic sacrifice, which
|
|
have a dissimilar issue.
|
|
|
|
|
|
\chapter{Of the Doctrine of our Priests}
|
|
As to the doctrine of the Circles it may briefly be summed up in a single
|
|
maxim, ``Attend to your Configuration''. Whether political, ecclesiastical, or
|
|
moral, all their teaching has for its object the improvement of individual and
|
|
collective Configuration --- with special reference of course to the
|
|
Configuration of the Circles, to which all other objects are subordinated.
|
|
|
|
It is the merit of the Circles that they have effectually suppressed those
|
|
ancient heresies which led men to waste energy and sympathy in the vain belief
|
|
that conduct depends upon will, effort, training, encouragement, praise, or
|
|
anything else but Configuration. It was Pantocyclus --- the illustrious Circle
|
|
mentioned above, as the queller of the Colour Revolt --- who first convinced
|
|
mankind that Configuration makes the man; that if, for example, you are born
|
|
an Isosceles with two uneven sides, you will assuredly go wrong unless you
|
|
have them made even --- for which purpose you must go to the Isosceles Hospital;
|
|
similarly, if you are a Triangle, or Square, or even a Polygon, born with any
|
|
Irregularity, you must be taken to one of the Regular Hospitals to have your
|
|
disease cured; otherwise you will end your days in the State Prison or by the
|
|
angle of the State Executioner.
|
|
|
|
All faults or defects, from the slightest misconduct to the most flagitious
|
|
crime, Pantocyclus attributed to some deviation from perfect Regularity in the
|
|
bodily figure, caused perhaps (if not congenital by some collision in a crowd;
|
|
by neglect to take exercise, or by taking too much of it; or even by a sudden
|
|
change of temperature, resulting in a shrinkage or expansion in some too
|
|
susceptible part of the frame. Therefore, concluded that illustrious
|
|
Philosopher, neither good conduct nor bad conduct is a fit subject, in any
|
|
sober estimation, for either praise or blame. For why should you praise, for
|
|
example, the integrity of a Square who faithfully defends the interests of his
|
|
client, when you ought in reality rather to admire the exact precision of his
|
|
right angles? Or again, why blame a lying, thievish Isosceles, when you ought
|
|
rather to deplore the incurable inequality of his sides?
|
|
|
|
Theoretically, this doctrine is unquestionable; but it has practical
|
|
drawbacks. In dealing with an Isosceles, if a rascal pleads that he cannot
|
|
help stealing because of his unevenness, you reply that for that very reason,
|
|
because he cannot help being a nuisance to his neighbours, you, the
|
|
Magistrate, cannot help sentencing him to be consumed --- and there's an end of
|
|
the matter. But in little domestic difficulties, when the penalty of
|
|
consumption, or death, is out of the question, this theory of Configuration
|
|
sometimes comes in awkwardly; and I must confess that occasionally when one of
|
|
my own Hexagonal Grandsons pleads as an excuse for his disobedience that a
|
|
sudden change of temperature has been too much for his Perimeter, and that I
|
|
ought to lay the blame not on him but on his Configuration, which can only be
|
|
strengthened by abundance of the choicest sweetmeats, I neither see my way
|
|
logically to reject, nor practically to accept, his conclusions.
|
|
|
|
For my own part, I find it best to assume that a good sound scolding or
|
|
castigation has some latent and strengthening influence on my Grandson's
|
|
Configuration; though I own that I have no grounds for thinking so. At all
|
|
events I am not alone in my way of extricating myself from this dilemma; for I
|
|
find that many of the highest Circles, sitting as Judges in law courts, use
|
|
praise and blame towards Regular and Irregular Figures; and in their homes I
|
|
know by experience that, when scolding their children, they speak about
|
|
``right'' and ``wrong'' as vehemently and passionately as if they believe that
|
|
these names represented real existence, and that a human Figure is really
|
|
capable of choosing between them.
|
|
|
|
Constantly carrying out their policy of making Configuration the leading idea
|
|
in every mind, the Circles reverse the nature of that Commandment which in
|
|
Spaceland regulates the relations between parents and children. With you,
|
|
children are taught to honour their parents; with us --- next to the Circles,
|
|
who are the chief object of universal homage --- a man is taught to honour his
|
|
Grandson, if he has one; or, if not, his Son. By ``honour'', however, is by no
|
|
means mean ``indulgence'', but a reverent regard for their highest interests:
|
|
and the Circles teach that the duty of fathers is to subordinate their own
|
|
interests to those of posterity, thereby advancing the welfare of the whole
|
|
State as well as that of their own immediate descendants.
|
|
|
|
The weak point in the system of the Circles --- if a humble Square may venture
|
|
to speak of anything Circular as containing any element of weakness --- appears
|
|
to me to be found in their relations with Women.
|
|
|
|
As it is of the utmost importance for Society that Irregular births should be
|
|
discouraged, it follows that no Woman who has any Irregularities in her
|
|
ancestry is a fit partner for one who desires that his posterity should rise
|
|
by regular degrees in the social scale.
|
|
|
|
Now the Irregularity of a Male is a matter of measurement; but as all Women
|
|
are straight, and therefore visibly Regular so to speak, one has to devise
|
|
some other means of ascertaining what I may call their invisible Irregularity,
|
|
that is to say their potential Irregularities as regards possible offspring.
|
|
This is effected by carefully-kept pedigrees, which are preserved and
|
|
supervised by the State; and without a certified pedigree no Woman is allowed
|
|
to marry.
|
|
|
|
Now it might have been supposed the a Circle --- proud of his ancestry and
|
|
regardful for a posterity which might possibly issue hereafter in a Chief
|
|
Circle --- would be more careful than any other to choose a wife who had no blot
|
|
on her escutcheon. But it is not so. The care in choosing a Regular wife
|
|
appears to diminish as one rises in the social scale. Nothing would induce an
|
|
aspiring Isosceles, who has hopes of generating an Equilateral Son, to take a
|
|
wife who reckoned a single Irregularity among her Ancestors; a Square or
|
|
Pentagon, who is confident that his family is steadily on the rise, does not
|
|
inquire above the five-hundredth generation; a Hexagon or Dodecagon is even
|
|
more careless of the wife's pedigree; but a Circle has been known deliberately
|
|
to take a wife who has had an Irregular Great-Grandfather, and all because of
|
|
some slight superiority of lustre, or because of the charms of a low voice ---
|
|
which, with us, even more than with you, is thought ``an excellent thing in a
|
|
Woman''.
|
|
|
|
Such ill-judged marriages are, as might be expected, barren, if they do not
|
|
result in positive Irregularity or in diminution of sides; but none of these
|
|
evils have hitherto provided sufficiently deterrent. The loss of a few sides
|
|
in a highly-developed Polygon is not easily noticed, and is sometimes
|
|
compensated by a successful operation in the Neo-Therapeutic Gymnasium, as I
|
|
have described above; and the Circles are too much disposed to acquiesce in
|
|
infecundity as a law of the superior development. Yet, if this evil be not
|
|
arrested, the gradual diminution of the Circular class may soon become more
|
|
rapid, and the time may not be far distant when, the race being no longer able
|
|
to produce a Chief Circle, the Constitution of Flatland must fall.
|
|
|
|
One other word of warning suggest itself to me, though I cannot so easily
|
|
mention a remedy; and this also refers to our relations with Women. About
|
|
three hundred years ago, it was decreed by the Chief Circle that, since women
|
|
are deficient in Reason but abundant in Emotion, they ought no longer to be
|
|
treated as rational, nor receive any mental education. The consequence was
|
|
that they were no longer taught to read, nor even to master Arithmetic enough
|
|
to enable them to count the angles of their husband or children; and hence
|
|
they sensibly declined during each generation in intellectual power. And this
|
|
system of female non-education or quietism still prevails.
|
|
|
|
My fear is that, with the best intentions, this policy has been carried so far
|
|
as to react injuriously on the Male Sex.
|
|
|
|
For the consequence is that, as things now are, we Males have to lead a kind
|
|
of bi-lingual, and I may almost say bimental, existence. With Women, we speak
|
|
of ``love'', ``duty'', ``right'', ``wrong'', ``pity'', ``hope'', and other irrational and
|
|
emotional conceptions, which have no existence, and the fiction of which has
|
|
no object except to control feminine exuberances; but among ourselves, and in
|
|
our books, we have an entirely different vocabulary and I may also say, idiom.
|
|
``Love'' them becomes ``the anticipation of benefits''; ``duty'' becomes ``necessity''
|
|
or ``fitness''; and other words are correspondingly transmuted. Moreover, among
|
|
Women, we use language implying the utmost deference for their Sex; and they
|
|
fully believe that the Chief Circle Himself is not more devoutly adored by us
|
|
than they are: but behind their backs they are both regarded and spoken of ---
|
|
by all but the very young --- as being little better than ``mindless organisms''.
|
|
|
|
Our Theology also in the Women's chambers is entirely different from our
|
|
Theology elsewhere.
|
|
|
|
Now my humble fear is that this double training, in language as well as in
|
|
thought, imposes somewhat too heavy a burden upon the young, especially when,
|
|
at the age of three years old, they are taken from the maternal care and
|
|
taught to unlearn the old language --- except for the purpose of repeating it in
|
|
the presence of the Mothers and Nurses --- and to learn the vocabulary and idiom
|
|
of science. Already methinks I discern a weakness in the grasp of mathematical
|
|
truth at the present time as compared with the more robust intellect of our
|
|
ancestors three hundred years ago. I say nothing of the possible danger if a
|
|
Woman should ever surreptitiously learn to read and convey to her Sex the
|
|
result of her perusal of a single popular volume; nor of the possibility that
|
|
the indiscretion or disobedience of some infant Male might reveal to a Mother
|
|
the secrets of the logical dialect. On the simple ground of the enfeebling of
|
|
the male intellect, I rest this humble appeal to the highest Authorities to
|
|
reconsider the regulations of Female education.
|
|
|
|
|
|
|
|
|
|
\part{Other Worlds}
|
|
|
|
|
|
|
|
\chapter{How I had a Vision of Lineland}
|
|
It was the last day but one of the 1999th year of our era, and the first day
|
|
of the Long Vacation. Having amused myself till a late hour with my favourite
|
|
recreation of Geometry, I had retired to rest with an unsolved problem in my
|
|
mind. In the night I had a dream.
|
|
|
|
I saw before me a vast multitude of small Straight Lines (which I naturally
|
|
assumed to be Women) interspersed with other Beings still smaller and of the
|
|
nature of lustrous points --- all moving to and fro in one and the same Straight
|
|
Line, and, as nearly as I could judge, with the same velocity.
|
|
|
|
A noise of confused, multitudinous chirping or twittering issued from them at
|
|
intervals as long as they were moving; but sometimes they ceased from motion,
|
|
and then all was silence.
|
|
|
|
Approaching one of the largest of what I thought to be Women, I accosted her,
|
|
but received no answer. A second and third appeal on my part were equally
|
|
ineffectual. Losing patience at what appeared to me intolerable rudeness, I
|
|
brought my mouth to a position full in front of her mouth so as to intercept
|
|
her motion, and loudly repeated my question, ``Woman, what signifies this
|
|
concourse, and this strange and confused chirping, and this monotonous motion
|
|
to and fro in one and the same Straight Line?''
|
|
|
|
``I am no Woman'', replied the small Line: ``I am the Monarch of the world. But
|
|
thou, whence intrudest thou into my realm of Lineland?'' Receiving this abrupt
|
|
reply, I begged pardon if I had in any way startled or molested his Royal
|
|
Highness; and describing myself as a stranger I besought the King to give me
|
|
some account of his dominions. But I had the greatest possible difficulty in
|
|
obtaining any information on points that really interested me; for the Monarch
|
|
could not refrain from constantly assuming that whatever was familiar to him
|
|
must also be known to me and that I was simulating ignorance in jest. However,
|
|
by preserving questions I elicited the following facts:
|
|
|
|
\vspace*{5mm}
|
|
\includegraphics[scale=0.5]{fig6}
|
|
\vspace*{5mm}
|
|
|
|
|
|
It seemed that this poor ignorant Monarch --- as he called himself --- was
|
|
persuaded that the Straight Line which he called his Kingdom, and in which he
|
|
passed his existence, constituted the whole of the world, and indeed the whole
|
|
of Space. Not being able either to move or to see, save in his Straight Line,
|
|
he had no conception of anything out of it. Though he had heard my voice when
|
|
I first addressed him, the sounds had come to him in a manner so contrary to
|
|
his experience that he had made no answer, ``seeing no man,'' as he expressed
|
|
it, ``and hearing a voice as it were from my own intestines.'' Until the moment
|
|
when I placed my mouth in his World, he had neither seen me, nor heard
|
|
anything except confused sounds beating against, what I called his side, but
|
|
what he called his inside or stomach; nor had he even now the least conception
|
|
of the region from which I had come. Outside his World, or Line, all was a
|
|
blank to him; nay, not even a blank, for a blank implies Space; say, rather,
|
|
all was non-existent.
|
|
|
|
His subjects --- of whom the small Lines were men and the Points Women --- were
|
|
all alike confined in motion and eyesight to that single Straight Line, which
|
|
was their World. It need scarcely be added that the whole of their horizon was
|
|
limited to a Point; nor could any one ever see anything but a Point. Man,
|
|
woman, child, thing --- each as a Point to the eye of a Linelander. Only by the
|
|
sound of the voice could sex or age be distinguished. Moreover, as each
|
|
individual occupied the whole of the narrow path, so to speak, which
|
|
constituted his Universe, and no one could move to the right or left to make
|
|
way for passers by, it followed that no Linelander could ever pass another.
|
|
Once neighbours, always neighbours. Neighbourhood with them was like marriage
|
|
with us. Neighbours remained neighbours till death did them part.
|
|
|
|
Such a life, with all vision limited to a Point, and all motion to a Straight
|
|
Line, seemed to me inexpressibly dreary; and I was surprised to note that
|
|
vivacity and cheerfulness of the King. Wondering whether it was possible, amid
|
|
circumstances so unfavourable to domestic relations, to enjoy the pleasures of
|
|
conjugal union, I hesitated for some time to question his Royal Highness on so
|
|
delicate a subject; but at last I plunged into it by abruptly inquiring as to
|
|
the health of his family. ``My wives and children,'' he replied, ``are well and
|
|
happy.''
|
|
|
|
Staggered at this answer --- for in the immediate proximity of the Monarch (as I
|
|
had noted in my dream before I entered Lineland) there were none but Men --- I
|
|
ventured to reply, ``Pardon me, but I cannot imagine how your Royal Highness
|
|
can at any time either See or approach their Majesties, when there at least
|
|
half a dozen intervening individuals, whom you can neither see through, nor
|
|
pass by? Is it possible that in Lineland proximity is not necessary for
|
|
marriage and for the generation of children?''
|
|
|
|
``How can you ask so absurd a question?'' replied the Monarch. ``If it were
|
|
indeed as you suggest, the Universe would soon be depopulated. No, no;
|
|
neighbourhood is needless for the union of hearts; and the birth of children
|
|
is too important a matter to have been allowed to depend upon such an accident
|
|
as proximity. You cannot be ignorant of this. Yet since you are pleased to
|
|
affect ignorance, I will instruct you as if you were the veriest baby in
|
|
Lineland. Know, then, that marriages are consummated by means of the faculty
|
|
of sound and the sense of hearing.
|
|
|
|
``You are of course aware that every Man has two mouths or voices --- as well as
|
|
two eyes --- a bass at one and a tenor at the other of his extremities. I should
|
|
not mention this, but that I have been unable to distinguish your tenor in the
|
|
course of our conversation.'' I replied that I had but one voice, and that I
|
|
had not been aware that his Royal Highness had two. ``That confirms by
|
|
impression,'' said the King, ``that you are not a Man, but a feminine
|
|
Monstrosity with a bass voice, and an utterly uneducated ear. But to continue.
|
|
|
|
``Nature having herself ordained that every Man should wed two wives ---'' ``Why
|
|
two?'' asked I. ``You carry your affected simplicity too far,'' he cried. ``How
|
|
can there be a completely harmonious union without the combination of the Four
|
|
in One, viz. the Bass and Tenor of the Man and the Soprano and Contralto of
|
|
the two Women?'' ``But supposing,'' said I, ``that a man should prefer one wife or
|
|
three?'' ``It is impossible,'' he said; ``it is as inconceivable as that two and
|
|
one should make five, or that the human eye should see a Straight Line.'' I
|
|
would have interrupted him; but he proceeded as follows:
|
|
|
|
``Once in the middle of each week a Law of Nature compels us to move to and fro
|
|
with a rhythmic motion of more than usual violence, which continues for the
|
|
time you would take to count a hundred and one. In the midst of this choral
|
|
dance, at the fifty-first pulsation, the inhabitants of the Universe pause in
|
|
full career, and each individual sends forth his richest, fullest, sweetest
|
|
strain. It is in this decisive moment that all our marriages are made. So
|
|
exquisite is the adaptation of Bass and Treble, of Tenor to Contralto, that
|
|
oftentimes the Loved Ones, though twenty thousand leagues away, recognize at
|
|
once the responsive note of their destined Lover; and, penetrating the paltry
|
|
obstacles of distance, Love unites the three. The marriage in that instance
|
|
consummated results in a threefold Male and Female offspring which takes its
|
|
place in Lineland.''
|
|
|
|
``What! Always threefold?'' said I. ``Must one wife then always have twins?''
|
|
|
|
``Bass-voice Monstrosity! yes,'' replied the King. ``How else could the balance
|
|
of the Sexes be maintained, if two girls were not born for every boy? Would
|
|
you ignore the very Alphabet of Nature?'' He ceased, speechless for fury; and
|
|
some time elapsed before I could induce him to resume his narrative.
|
|
|
|
``You will not, of course, suppose that every bachelor among us finds his mates
|
|
at the first wooing in this universal Marriage Chorus. On the contrary, the
|
|
process is by most of us many times repeated. Few are the hearts whose happy
|
|
lot is at once to recognize in each other's voice the partner intended for
|
|
them by Providence, and to fly into a reciprocal and perfectly harmonious
|
|
embrace. With most of us the courtship is of long duration. The Wooer's voices
|
|
may perhaps accord with one of the future wives, but not with both; or not, at
|
|
first, with either; or the Soprano and Contralto may not quite harmonize. In
|
|
such cases Nature has provided that every weekly Chorus shall bring the three
|
|
Lovers into closer harmony. Each trial of voice, each fresh discovery of
|
|
discord, almost imperceptibly induces the less perfect to modify his or her
|
|
vocal utterance so as to approximate to the more perfect. And after many
|
|
trials and many approximations, the result is at last achieved. There comes a
|
|
day at last when, while the wonted Marriage Chorus goes forth from universal
|
|
Lineland, the three far-off Lovers suddenly find themselves in exact harmony,
|
|
and, before they are aware, the wedded Triplet is rapt vocally into a
|
|
duplicate embrace; and Nature rejoices over one more marriage and over three
|
|
more births.''
|
|
|
|
|
|
|
|
|
|
\chapter{How I vainly tried to explain the nature of Flatland}
|
|
Thinking that it was time to bring down the Monarch from his raptures to the
|
|
level of common sense, I determined to endeavour to open up to him some
|
|
glimpses of the truth, that is to say of the nature of things in Flatland. So
|
|
I began thus: ``How does your Royal Highness distinguish the shapes and
|
|
positions of his subjects? I for my part noticed by the sense of sight, before
|
|
I entered your Kingdom, that some of your people are lines and others Points;
|
|
and that some of the lines are larger ---'' ``You speak of an impossibility,''
|
|
interrupted the King; ``you must have seen a vision; for to detect the
|
|
difference between a Line and a Point by the sense of sight is, as every one
|
|
knows, in the nature of things, impossible; but it can be detected by the
|
|
snese of hearing, and by the same means my shape can be exactly ascertained.
|
|
Behold me --- I am a Line, the longest in Lineland, over six inches of Space ---''
|
|
``Of Length,'' I ventured to suggest. ``Fool,'' said he, ``Space is Length.
|
|
Interrupt me again, and I have done.''
|
|
|
|
I apologized; but he continued scornfully, ``Since you are impervious to
|
|
argument, you shall hear with your ears how by means of my two voices I reveal
|
|
my shape to my Wives, who are at this moment six thousand miles seventy yards
|
|
two feet eight inches away, the one to the North, the other to the South.
|
|
Listen, I call to them.''
|
|
|
|
He chirruped, and then complacently continued: ``My wives at this moment
|
|
receiving the sound of one of my voice, closely followed by the other, and
|
|
perceiving that the latter reaches them after an interval in which sound can
|
|
traverse 6.457 inches, infer that one of my mouths is 6.457 inches further
|
|
from them than the other, and accordingly know my shape to be 6.457 inches.
|
|
But you will of course understand that my wives do not make this calculation
|
|
every time they hear my two voices. They made it, once for all, before we were
|
|
married. But they could make it at any time. And in the same way I can
|
|
estimate the shape of any of my Male subjects by the sense of sound.''
|
|
|
|
``But how,'' said I, ``if a Man feigns a Woman's voice with one of his two
|
|
voices, or so disguises his Southern voice that it cannot be recognized as the
|
|
echo of the Northern? May not such deceptions cause great inconvenience? And
|
|
have you no means of checking frauds of this kind by commanding your
|
|
neighbouring subjects to feel one another?'' This of course was a very stupid
|
|
question, for feeling could not have answered the purpose; but I asked with
|
|
the view of irritating the Monarch, and I succeeded perfectly.
|
|
|
|
``What!'' cried he in horror, ``explain your meaning.'' ``Feel, touch, come into
|
|
contact,'' I replied.. ``If you mean by feeling,`` said the King, ``approaching so
|
|
close as to leave no space between two individuals, know, Stranger, that this
|
|
offence is punishable in my dominions by death. And the reason is obvious. The
|
|
frail form of a Woman, being liable to be shattered by such an approximation,
|
|
must be preserved by the State; but since Women cannot be distinguished by the
|
|
sense of sight from Men, the Law ordains universally that neither Man nor
|
|
Woman shall be approached so closely as to destroy the interval between the
|
|
approximator and the approximated.
|
|
|
|
``And indeed what possible purpose would be served by this illegal and
|
|
unnatural excess of approximation which you call touching, when all the ends
|
|
of so brutal and course a process are attained at once more easily and more
|
|
exactly by the sense of hearing? As to your suggested danger of deception, it
|
|
is non-existent: for the Voice, being the essence of one's Being, cannot be
|
|
thus changed at will. But come, suppose that I had the power of passing
|
|
through solid things, so that I could penetrate my subjects, one after
|
|
another, even to the number of a billion, verifying the size and distance of
|
|
each by the sense of feeling: How much time and energy would be wasted in this
|
|
clumsy and inaccurate method! Whereas now, in one moment of audition, I take
|
|
as it were the census and statistics, local, corporeal, mental and spiritual,
|
|
of every living being in Lineland. Hark, only hark!''
|
|
|
|
So saying he paused and listened, as if in an ecstasy, to a sound which seemed
|
|
to me no better than a tiny chirping from an innumerable multitude of
|
|
lilliputian grasshoppers.
|
|
|
|
``Truly,'' replied I, ``your sense of hearing serves you in good stead, and fills
|
|
up many of your deficiencies. But permit me to point out that your life in
|
|
Lineland must be deplorably dull. To see nothing but a Point! Not even to be
|
|
able to contemplate a Straight Line! Nay, not even to know what a Straight
|
|
Line is! To see, yet to be cut off from those Linear prospects which are
|
|
vouchsafed to us in Flatland! Better surely to have no sense of sight at all
|
|
than to see so little! I grant you I have not your discriminative faculty of
|
|
hearing; for the concert of all Lineland which gives you such intense
|
|
pleasure, is to me no better than a multitudinous twittering or chirping. But
|
|
at least I can discern, by sight, a Line from a Point. And let me prove it.
|
|
Just before I came into your kingdom, I saw you dancing from left to right,
|
|
and then from right to left, with Seven Men and a Woman in your immediate
|
|
proximity on the left, and eight Men and two Women on your right. Is not this
|
|
correct?''
|
|
|
|
``It is correct,'' said the King, ``so far as the numbers and sexes are
|
|
cocnerned, though I know not what you mean by `right' and `left.' But I deny
|
|
that you saw these things. For how could you see the Line, that is to say the
|
|
inside, of any Man? But you must have heard these things, and then dreamed
|
|
that you saw them. And let me ask what you mean by those words `left' and
|
|
`right'. I suppose it is your way of saying Northward and Southward.''
|
|
|
|
``Not so,'' replied I; ``besides your motion of Northward and Southward, there is
|
|
another motion which I call from right to left.''
|
|
|
|
King. Exhibit to me, if you please, this motion from left to right.
|
|
|
|
I. Nay, that I cannot do, unless you could setp out of your Line altogether.
|
|
|
|
King. Out of my Line? Do you mean out of the world? Out of Space?
|
|
|
|
I. Well, yes. Out of your world. Out of your Space. For your Space is not the
|
|
true Space. True Space is a Plane; but your Space is only a Line.
|
|
|
|
King. If you cannot indicate this motion from left to right by yourself moving
|
|
in it, then I beg you to describe it to me in words.
|
|
|
|
I. If you cannot tell your right side from your left, I fear that no words of
|
|
mine can make my meaning clearer to you. But surely you cannot be ignorant of
|
|
so simple a distinction.
|
|
|
|
King. I do not in the least understand you.
|
|
|
|
I. Alas! How shall I make it clear? When you move straight on, does it not
|
|
sometimes occur to you that you could move in some other way, turning your eye
|
|
round so as to look in the direction towards which your side is now fronting?
|
|
In other words, instead of always moving in the direction of one of your
|
|
extremities, do you never feel a desire to move in the direction, so to speak,
|
|
of your side?
|
|
|
|
King. Never. And what do you mean? How can a man's inside ``front'' in any
|
|
direction? Or how can a man move in the direction of his inside?
|
|
|
|
I. Well then, since words cannot explain the matter, I will try deeds, and
|
|
will move gradually out of Lineland in the direction which I desire to
|
|
indicate to you.
|
|
|
|
At the word I began to move my body out of Lineland. As long as any part of me
|
|
remained in his dominion and in his view, the King kept exclaiming, ``I see
|
|
you, I see you still; you are not moving.'' But when I had at last moved myself
|
|
out of his Line, he cried in his shrillest voice, ``She is vanished; she is
|
|
dead.'' ``I am not dead,'' replied I; ``I am simply out of Lineland, that is to
|
|
say, out of the Straight Line which you call Space, and in the true Space,
|
|
where I can see things as they are. And at this moment I can see your Line, or
|
|
side --- or inside as you are pleased to call it; and I can see also the Men and
|
|
Women on the North and South of you, whom I will now enumerate, describing
|
|
their order, their size, and the interval between each.''
|
|
|
|
\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig7}
|
|
|
|
When I had done this at great length, I cried triumphantly, ``Does that at last
|
|
convince you?'' And, with that, I once more entered Lineland, taking up the
|
|
same position as before.
|
|
|
|
But the Monarch replied, ``If you were a Man of sense --- though, as you appear
|
|
to have only one voice I have little doubt you are not a Man but a Woman ---
|
|
but, if you had a particle of sense, you would listen to reason. You ask me to
|
|
believe that there is another Line besides that which my senses indicate, and
|
|
another motion besides that of which I am daily conscious. I, in return, ask
|
|
you to describe in words or indicate by motion that other Line of which you
|
|
speak. Instead of moving, you merely exercise some magic art of vanishing and
|
|
returning to sight; and instead of any lucid description of your new World,
|
|
you simply tell me the numbers and sizes of some forty of my retinue, facts
|
|
known to any child in my capital. Can anything be more irrational or
|
|
audacious? Acknowledge your folly or depart from my dominions.''
|
|
|
|
Furious at his perversity, and especially indignant that he professed to be
|
|
ignorant of my sex, I retorted in no measured terms, ``Besotted Being! You
|
|
think yourself the perfection of existence, while you are in reality the most
|
|
imperfect and imbecile. You profess to see, whereas you see nothing but a
|
|
Point! You plume yourself on inferring the existence of a Straight Line; but I
|
|
can see Straight Lines, and infer the existence of Angles, Triangles, Squares,
|
|
Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I
|
|
am the completion of your incomplete self. You are a Line, but I am a Line of
|
|
Lines called in my country a Square: and even I, infinitely superior though I
|
|
am to you, am of little account among the great nobles of Flatland, whence I
|
|
have come to visit you, in the hope of enightening your ignorance.''
|
|
|
|
Hearing these words the King advanced towards me with a menacing cry as if to
|
|
pierce me through the diagonal; and in that same movement there arose from
|
|
myriads of his subjects a multitudinous war-cry, increasing in vehemence till
|
|
at last methought it rivalled the roar of an army of a hundred thousand
|
|
Isosceles, and the artillery of a thousand Pentagons. Spell-bound and
|
|
motionless, I could neither speak nor move to avert the impending destruction;
|
|
and still the noise grew louder, and the King came closer, when I awoke to
|
|
find the breakfast-bell recalling me to the realities of Flatland.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{Concerning a Stranger from Spaceland}
|
|
From dreams I proceed to facts.
|
|
|
|
It was the last day of our 1999th year of our era. The patterning of the rain
|
|
had long ago announced nightfall; and I was sitting \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 feet, we can no more ``sit''
|
|
nor ``stand'' (in your sense of the word) than one of your soles or flounders.
|
|
|
|
Nevertheless, we perfectly well recognize the different mental states of
|
|
volition implied by ``lying,'' ``sitting,'' and ``standing,'' which are to some
|
|
extent indicated to a beholder by a slight increase of lustre corresponding to
|
|
the increase of volition.
|
|
|
|
But on this, and a thousand other kindred subjects, time forbids me to dwell.}
|
|
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.
|
|
|
|
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 the new one in.
|
|
|
|
I was rapt 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 Sight Recognition, turning ourselves upon
|
|
our centres, 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 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 the number of inches in the side: ``and thus,'' said I, ``we know
|
|
that $ 3^2 $, or 9, represents the number of square inches in a
|
|
Square whose side is 3 inches long.''
|
|
|
|
The little Hexagon meditated on this a while and then said to me; ``But you
|
|
have been teaching me to raise numbers to the third power: I suppose
|
|
$ 3^3 $ 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 shew the boy how a Point by moving
|
|
through a length of three inches makes a Line of three inches, which may be
|
|
represented by 3; 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 $ 3^2 $.
|
|
|
|
Upon this, my Grandson, 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 3; and if a straight Line of
|
|
three inches, moving parallel to itself, makes a Square of three inches every
|
|
way, represented by $ 3^2 $; 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 $ 3^3 $.''
|
|
|
|
``Go to bed,'' said I, a little ruffled by this 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,
|
|
endeavouring to form a retrospect of the year 1999 and of the possibilities of
|
|
the year 2000; but not quite able to shake of 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 glass Northward
|
|
for the last time in the old Millennium; and in the act, I exclaimed aloud,
|
|
``The boy is 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 dishonouring your own
|
|
Grandson.'' But I took no notice of her. Looking around in every direction I
|
|
could see nothing; yet still I \textit{felt} a Presence, and shivered as the cold
|
|
whisper came again. I started up. ``What is the matter?'' said my Wife, ``there
|
|
is no draught; what are you looking for? There is nothing.'' There was nothing;
|
|
and I resumed my seat, again exclaiming, ``The boy is a fool, I say;
|
|
$ 3^3 $ can have no meaning in Geometry.'' At once there came a
|
|
distinctly audible reply, ``The boy is not a fool; and $ 3^3 $ has
|
|
an obvious Geometrical meaning.''
|
|
|
|
My Wife as well as myself heard the words, although she did not 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 shewed me
|
|
that the extremities passed into dimness 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 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 they any,'' 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, 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 and be felt by ---'' then, suddenly
|
|
recoiling, ``Oh! it is not a Woman, and there are no angles either, not a trace
|
|
of one. Can it be that I have so misbehaved to a perfect Circle?''
|
|
|
|
``I am indeed, in a certain sense a Circle,'' replied the Voice, ``and a more
|
|
perfect Circle than any in Flatland; but to speak more accurately, I am many
|
|
Circles in one.'' Then 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 of her own retirement had long passed, with many
|
|
reiterated apologies for her recent indiscretion, she at last retreated to her
|
|
apartment.
|
|
|
|
I glanced at the half-hour glass. The last sands had fallen. The third
|
|
Millennium had begun.
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland}
|
|
As soon as the sound of the Peace-cry of my departing Wife had died away, I
|
|
began to approach the Stranger with the intention of taking a nearer view and
|
|
of bidding him be seated: but his appearance struck me dumb and motionless
|
|
with astonishment. Without the slightest symptoms of angularity he
|
|
nevertheless varied every instant with graduations 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 me a burglar or cut-throat,
|
|
some monstrous Irregular Isosceles, 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), 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 slightest
|
|
roughness or inequality: never in my life had I met with a more perfect
|
|
Circle. He remained motionless while I walked around him, beginning from his
|
|
eye and returning to it again. Circular he was throughout, a perfectly
|
|
satisfactory Circle; there could not be a doubt of it. Then followed a
|
|
dialogue, which I will endeavour 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 the Stranger with some impatience at the
|
|
lengthiness of my introductory process.
|
|
|
|
\textit{Stranger}. Have you felt me enough by this time? Are you not introduced to me
|
|
yet?
|
|
|
|
\textit{I}. Most illustrious Sir, excuse my awkwardness, which arises not from
|
|
ignorance of the usages of polite society, but from a little surprise and
|
|
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 curiosity of one who would gladly know whence his visitor came?
|
|
|
|
\textit{Stranger}. From Space, from Space, Sir: whence else?
|
|
|
|
\textit{I}. Pardon me, my Lord, but is not your Lordship already in Space, your
|
|
Lordship and his humble servant, even at this moment?
|
|
|
|
\textit{Stranger}. Pooh! what do you know of Space? Define Space.
|
|
|
|
\textit{I}. Space, my Lord, is height and breadth indefinitely prolonged.
|
|
|
|
\textit{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, and length.
|
|
|
|
\textit{I}. Your Lordship is pleased to be merry. We also speak of length and height,
|
|
or breadth and thickness, thus denoting Two Dimensions by four names.
|
|
|
|
\textit{Stranger}. But I mean not only three names, but Three Dimensions.
|
|
|
|
\textit{I}. Would your Lordship indicate or explain to me in what direction is the
|
|
Third Dimension, unknown to me?
|
|
|
|
\textit{Stranger}. I came from it. It is up above and down below.
|
|
|
|
\textit{I}. My Lord means seemingly that it is Northward and Southward.
|
|
|
|
\textit{Stranger}. I mean nothing of the kind. I mean a direction in which you cannot
|
|
look, because you have no eye in your side.
|
|
|
|
\textit{I}. Pardon me, my Lord, a moment's inspection will convince your Lordship that
|
|
I have a perfectly luminary at the juncture of my two sides.
|
|
|
|
\textit{Stranger}: Yes: 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.
|
|
|
|
\textit{I}. An eye in my inside! An eye in my stomach! Your Lordship jests.
|
|
|
|
\textit{Stranger}. I am in no jesting humour. 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 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.
|
|
|
|
\textit{I}. Such assertions are easily made, my Lord.
|
|
|
|
\textit{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?
|
|
|
|
\textit{I}. Through the roof, I suppose.
|
|
|
|
\textit{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 and household?
|
|
|
|
\textit{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.
|
|
|
|
\textit{Stranger}. (To himself.) What must I do? Stay; one more argument suggests
|
|
itself to me. When you see a Straight Line --- your wife, for example --- how many
|
|
Dimensions do you attribute to her?
|
|
|
|
\textit{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).
|
|
|
|
\textit{Stranger}. But the very fact that a Line is visible implies that it possesses
|
|
yet another Dimension.
|
|
|
|
\textit{I}. My Lord, I have just acknowledge 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.
|
|
|
|
\textit{Stranger}. You do not understand me. I mean that when you see a Woman, you
|
|
ought --- besides inferring her breadth --- to 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?
|
|
|
|
\textit{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''?
|
|
|
|
\textit{Stranger}. No, indeed. By ``height'' I mean a Dimension like your length: only,
|
|
with you, ``height'' is not so easily perceptible, being extremely small.
|
|
|
|
\textit{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 understanding must hold me excused.
|
|
|
|
\textit{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
|
|
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 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 manifest himself as a Circle.
|
|
|
|
Do you not remember --- for 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 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 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 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 sections 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.
|
|
|
|
\includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig8}
|
|
|
|
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 voice --- close
|
|
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 my section 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 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.
|
|
|
|
\textit{Sphere}. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a
|
|
luminous wake, what name would you give to the wake?
|
|
|
|
\textit{I}. A straight Line.
|
|
|
|
\textit{Sphere}. And a straight Line has how many extremities?
|
|
|
|
\textit{I}. Two.
|
|
|
|
\textit{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?
|
|
|
|
\textit{I}. A square.
|
|
|
|
\textit{Sphere}. And how many sides has a Square? How many angles?
|
|
|
|
\textit{I}. Four sides and four angles.
|
|
|
|
\textit{Sphere}. Now stretch your imagination a little, and conceive a Square in
|
|
Flatland, moving parallel to itself upward.
|
|
|
|
\textit{I}. What? Northward?
|
|
|
|
\textit{Sphere}. No, not Northward; upward; out of Flatland altogether.
|
|
|
|
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 you --- for you are a Square and will serve 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 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 Flatland,
|
|
anywhere, so that I could get rid of him --- I 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.''
|
|
|
|
\textit{Sphere}. Oh, certainly. It is all plain and simple, and in strict accordance
|
|
with Analogy --- only, 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 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?
|
|
|
|
\textit{I}. Eight.
|
|
|
|
\textit{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?
|
|
|
|
\textit{I}. And has this Creature sides, as well as Angles or what you call ``terminal
|
|
Points''?
|
|
|
|
\textit{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 solids.
|
|
|
|
\textit{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?
|
|
|
|
\textit{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?
|
|
|
|
\textit{I}. Arithmetical.
|
|
|
|
\textit{Sphere}. And what is the next number?
|
|
|
|
\textit{I}. Six.
|
|
|
|
\textit{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.
|
|
|
|
|
|
|
|
\chapter{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.
|
|
|
|
\textit{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 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 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 have no tops or bottom) full of money; I see also two 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. Now I have
|
|
it. Now I ascent 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. 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 sense; but the
|
|
Stranger continued: ``Surely you must now see that my explanation, and no
|
|
other, suits the phenomena. What you call Solid things are 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 the Plane yourself, if you could but summon up the necessary
|
|
volition. A slight 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 neighbour the Hexagon and his family in their several
|
|
apartments; now I see the inside of the Theatre, 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 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 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 have not hurt you much, have I? If you are not convinced now, 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 against
|
|
the wall till help came!
|
|
|
|
Once more I dashed my 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
|
|
any case he remained motionless, while I, hearing, as I thought, the sound of
|
|
some help approaching, pressed against him with redoubled vigor, and continued
|
|
to shout for assistance.
|
|
|
|
A convulsive shudder ran through the Sphere. ``This must not be,'' I thought I
|
|
heard him say: ``either he must listen to reason, 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 thus must the
|
|
fruits of one thousand years of waiting be thrown away. I hear her coming.
|
|
Back! back! Away from me, or you must go with me --- wither you know not --- into
|
|
the Land of Three Dimensions!''
|
|
|
|
``Fool! Madman! Irregular!'' I exclaimed; ``never will 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!''
|
|
|
|
|
|
|
|
\chapter{How I came to Spaceland, and what I saw there}
|
|
An unspeakable horror seized me. There was a darkness; then a dizzy, sickening
|
|
sensation of sight that was not like seeing; I saw a Line that was no Line;
|
|
Space that was not Space: I was myself, and not myself. When I could find
|
|
voice, I shrieked loud in agony, ``Either this is madness or it is Hell.'' ``It
|
|
is neither, calmly replied the voice of the Sphere, ``it is Knowledge; it is
|
|
Three Dimensions: open your eye once again and try to look steadily.''
|
|
|
|
I looked, and, behold, a new world! There stood before me, visibly
|
|
incorporate, all that I had before inferred, conjectured, dreamed, of perfect
|
|
Circular beauty. What seemed the centre of the Stranger's form lay open to my
|
|
view: yet I could see no heart, lungs, nor arteries, only a beautiful
|
|
harmonious Something --- for which I had no words; but you, my Readers in
|
|
Spaceland, would call it the surface of the Sphere.
|
|
|
|
Prostrating myself mentally before my Guide, I cried, ``How it is, O divine
|
|
ideal of consummate loveliness and wisdom that I see thy inside, and yet
|
|
cannot discern thy heart, thy lungs, thy arteries, thy liver?'' ``What you think
|
|
you see, you see not,'' he replied; ``it is not giving to you, nor to any other
|
|
Being, to behold my internal parts. I am of a different order of Beings from
|
|
those in Flatland. Where I a Circle, you could discern my intestines, but I am
|
|
a Being, composed as I told you before, of many Circles, the Many in the One,
|
|
called in this country a Sphere. And, just as the outside of a Cube is a
|
|
Square, so the outside of a Sphere represents the appearance of a Circle.''
|
|
|
|
Bewildered though I was by my Teacher's enigmatic utterance, I no longer
|
|
chafed against it, but worshipped him in silent adoration. He continued, with
|
|
more mildness in his voice. ``Distress not yourself if you cannot at first
|
|
understand the deeper mysteries of Spaceland. By degrees they will dawn upon
|
|
you. Let us begin by casting back a glance at the region whence you came.
|
|
Return with me a while to the plains of Flatland and I will shew you that
|
|
which you have often reasoned and thought about, but never seen with the sense
|
|
of sight --- a visible angle.'' ``Impossible!'' I cried; but, the Sphere leading
|
|
the way, I followed as if in a dream, till once more his voice arrested me:
|
|
``Look yonder, and behold your own Pentagonal house, and all its inmates.''
|
|
\begin{center}
|
|
\includegraphics[trim=20mm 0mm 20mm 0mm,scale=0.7]{fig9}
|
|
\end{center}
|
|
|
|
|
|
I looked below, and saw with my physical eye all that domestic individuality
|
|
which I had hitherto merely inferred with the understanding. And how poor and
|
|
shadowy was the inferred conjecture in comparison with the reality which I now
|
|
behold! My four Sons calmly asleep in the North-Western rooms, my two orphan
|
|
Grandsons to the South; the Servants, the Butler, my Daughter, all in their
|
|
several apartments. Only my affectionate Wife, alarmed by my continued
|
|
absence, had quitted her room and was roving up and down in the Hall,
|
|
anxiously awaiting my return. Also the Page, aroused by my cries, had left his
|
|
room, and under pretext of ascertaining whether I had fallen somewhere in a
|
|
faint, was prying into the cabinet in my study. All this I could now see, not
|
|
merely infer; and as we came nearer and nearer, I could discern even the
|
|
contents of my cabinet, and the two chests of gold, and the tablets of which
|
|
the Sphere had made mention.
|
|
|
|
Touched by my Wife's distress, I would have sprung downward to reassure her,
|
|
but I found myself incapable of motion. ``Trouble not yourself about your
|
|
Wife,'' said my Guide: ``she will not be long left in anxiety; meantime, let us
|
|
take a survey of Flatland.''
|
|
|
|
Once more I felt myself rising through space. It was even as the Sphere had
|
|
said. The further we receded from the object we beheld, the larger became the
|
|
field of vision. My native city, with the interior of every house and every
|
|
creature therein, lay open to my view in miniature. We mounted higher, and lo,
|
|
the secrets of the earth, the depths of the mines and inmost caverns of the
|
|
hills, were bared before me.
|
|
|
|
Awestruck at the sight of the mysteries of the earth, thus unveiled before my
|
|
unworthy eye, I said to my Companion, ``Behold, I am become as a God. For the
|
|
wise men in our country say that to see all things, or as they express it,
|
|
omnividence, is the attribute of God alone.'' There was something of scorn in
|
|
the voice of my Teacher as he made answer: ``it is so indeed? Then the very
|
|
pick-pockets and cut-throats of my country are to be worshipped by your wise
|
|
men as being Gods: for there is not one of them that does not see as much as
|
|
you see now. But trust me, your wise men are wrong.''
|
|
|
|
\textit{I}. Then is omnividence the attribute of
|
|
others besides Gods?
|
|
|
|
\textit{Sphere}. I do not know. But, if a pick-pocket or a cut-throat of our country
|
|
can see everything that is in your country, surely that is no reason why the
|
|
pick-pocket or cut-throat should be accepted by you as a God. This
|
|
omnividence, as you call it --- it is not a common word in Spaceland --- does it
|
|
make you more just, more merciful, less selfish, more loving? Not in the
|
|
least. Then how does it make you more divine?
|
|
|
|
\textit{I}. ``More merciful, more loving!'' But these are the qualities of women! And we
|
|
know that a Circle is a higher Being than a Straight Line, in so far as
|
|
knowledge and wisdom are more to be esteemed than mere affection.
|
|
|
|
\textit{Sphere}. It is not for me to classify human faculties according to merit. Yet
|
|
many of the best and wisest in Spaceland think more of the affections than of
|
|
the understanding, more of your despised Straight Lines than of your belauded
|
|
Circles. But enough of this. Look yonder. Do you know that building?
|
|
|
|
I looked, and afar off I saw an immense Polygonal structure, in which I
|
|
recognized the General Assembly Hall of the States of Flatland, surrounded by
|
|
dense lines of Pentagonal buildings at right angles to each other, which I
|
|
knew to be streets; and I perceived that I was approaching the great
|
|
Metropolis.
|
|
|
|
``Here we descend,'' said my Guide. It was now morning, the first hour of the
|
|
first day of the two thousandth year of our era. Acting, as was their wont, in
|
|
strict accordance with precedent, the highest Circles of the realm were
|
|
meeting in solemn conclave, as they had met on the first hour of the first day
|
|
of the year 1000, and also on the first hour of the first day of the year 0.
|
|
|
|
The minutes of the previous meetings were now read by one whom I at once
|
|
recognized as my brother, a perfectly Symmetrical Square, and the Chief Clerk
|
|
of the High Council. It was found recorded on each occasion that: ``Whereas the
|
|
States had been troubled by divers ill-intentioned persons pretending to have
|
|
received revelations from another World, and professing to produce
|
|
demonstrations whereby they had instigated to frenzy both themselves and
|
|
others, it had been for this cause unanimously resolved by the Grand Council
|
|
that on the first day of each millenary, special injunctions be sent to the
|
|
Prefects in the several districts of Flatland, to make strict search for such
|
|
misguided persons, and without formality of mathematical examination, to
|
|
destroy all such as were Isosceles of any degree, to scourge and imprison any
|
|
regular Triangle, to cause any Square or Pentagon to be sent to the district
|
|
Asylum, and to arrest any one of higher rank, sending him straightway to the
|
|
Capital to be examined and judged by the Council.''
|
|
|
|
``You hear your fate,'' said the Sphere to me, while the Council was passing for
|
|
the third time the formal resolution. ``Death or imprisonment awaits the
|
|
Apostle of the Gospel of Three Dimensions.'' ``Not so,'' replied I, ``the matter
|
|
is now so clear to me, the nature of real space so palpable, that methinks I
|
|
could make a child understand it. Permit me but to descend at this moment and
|
|
enlighten them.'' ``Not yet,'' said my Guide, ``the time will come for that.
|
|
Meantime I must perform my mission. Stay thou there in thy place.'' Saying
|
|
these words, he leaped with great dexterity into the sea (if I may so call it)
|
|
of Flatland, right in the midst of the ring of Counsellors. ``I come,'' said he,
|
|
``to proclaim that there is a land of Three Dimensions.''
|
|
|
|
I could see many of the younger Counsellors start back in manifest horror, as
|
|
the Sphere's circular section widened before them. But on a sign from the
|
|
presiding Circle --- who shewed not the slightest alarm or surprise --- six
|
|
Isosceles of a low type from six different quarters rushed upon the Sphere.
|
|
``We have him,'' they cried; ``No; yes; we have him still! he's going! he's
|
|
gone!''
|
|
|
|
``My Lords,'' said the President to the Junior Circles of the Council, ``there is
|
|
not the slightest need for surprise; the secret archives, to which I alone
|
|
have access, tell me that a similar occurrence happened on the last two
|
|
millennial commencements. You will, of course, say nothing of these trifles
|
|
outside the Cabinet.''
|
|
|
|
Raising his voice, he now summoned the guards. ``Arrest the policemen; gag
|
|
them. You know your duty.'' After he had consigned to their fate the wretched
|
|
policemen --- ill-fated and unwilling witnesses of a State-secret which they
|
|
were not to be permitted to reveal --- he again addressed the Counsellors. ``My
|
|
Lords, the business of the Council being concluded, I have only to wish you a
|
|
happy New Year.'' Before departing, he expressed, at some length, to the Clerk,
|
|
my excellent but most unfortunate brother, his sincere regret that, in
|
|
accordance with precedent and for the sake of secrecy, he must condemn him to
|
|
perpetual imprisonment, but added his satisfaction that, unless some mention
|
|
were made by him of that day's incident, his life would be spared.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{How, though the Sphere
|
|
showed me other mysteries of Spaceland, I still desire more; and what came of
|
|
it}
|
|
When I saw my poor brother led away to imprisonment, I attempted to leap down
|
|
into the Council Chamber, desiring to intercede on his behalf, or at least bid
|
|
him farewell. But I found that I had no motion of my own. I absolutely
|
|
depended on the volition of my Guide, who said in gloomy tones, ``Heed not thy
|
|
brother; haply thou shalt have ample time hereafter to condole with him.
|
|
Follow me.''
|
|
\begin{center}
|
|
\includegraphics[trim=20mm 0mm 20mm 0mm,scale=0.6]{fig10}
|
|
\end{center}
|
|
|
|
Once more we ascended into space. ``Hitherto,'' said the Sphere, ``I have shewn
|
|
you naught save Plane Figures and their interiors. Now I must introduce you to
|
|
Solids, and reveal to you the plan upon which they are constructed. Behold
|
|
this multitude of moveable square cards. See, I put one on another, not, as
|
|
you supposed, Northward of the other, but on the other. Now a second, now a
|
|
third. See, I am building up a Solid by a multitude of Squares parallel to one
|
|
another. Now the Solid is complete, being as high as it is long and broad, and
|
|
we call it a Cube.''
|
|
|
|
``Pardon me, my Lord,'' replied I; ``but to my eye the appearance is as of an
|
|
Irregular Figure whose inside is laid open to view; in other words, methinks I
|
|
see no Solid, but a Plane such as we infer in Flatland; only of an
|
|
Irregularity which betokens some monstrous criminal, so that the very sight of
|
|
it is painful to my eyes.''
|
|
|
|
``True,'' said the Sphere; ``it appears to you a Plane, because you are not
|
|
accustomed to light and shade and perspective; just as in Flatland a Hexagon
|
|
would appear a Straight Line to one who has not the Art of Sight Recognition.
|
|
But in reality it is a Solid, as you shall learn by the sense of Feeling.''
|
|
|
|
He then introduced me to the Cube, and I found that this marvellous Being was
|
|
indeed no Plane, but a Solid; and that he was endowed with six plane sides and
|
|
eight terminal points called solid angles; and I remembered the saying of the
|
|
Sphere that just such a Creature as this would be formed by the Square moving,
|
|
in Space, parallel to himself: and I rejoiced to think that so insignificant a
|
|
Creature as I could in some sense be called the Progenitor of so illustrious
|
|
an offspring.
|
|
|
|
But still I could not fully understand the meaning of what my Teacher had told
|
|
me concerning ``light'' and ``shade'' and ``perspective''; and I did not hesitate to
|
|
put my difficulties before him.
|
|
|
|
Were I to give the Sphere's explanation of these matters, succinct and clear
|
|
though it was, it would be tedious to an inhabitant of Space, who knows these
|
|
things already. Suffice it, that by his lucid statements, and by changing the
|
|
position of objects and lights, and by allowing me to feel the several objects
|
|
and even his own sacred Person, he at last made all things clear to me, so
|
|
that I could now readily distinguish between a Circle and a Sphere, a Plane
|
|
Figure and a Solid.
|
|
|
|
This was the Climax, the Paradise, of my strange eventful History. Henceforth
|
|
I have to relate the story of my miserable Fall: --- most miserable, yet surely
|
|
most undeserved! For why should the thirst for knowledge be aroused, only to
|
|
be disappointed and punished? My volition shrinks from the painful task of
|
|
recalling my humiliation; yet, like a second Prometheus, I will endure this
|
|
and worse, if by any means I may arouse in the interiors of Plane and Solid
|
|
Humanity a spirit of rebellion against the Conceit which would limit our
|
|
Dimensions to Two or Three or any number short of Infinity. Away then with all
|
|
personal considerations! Let me continue to the end, as I began, without
|
|
further digressions or anticipations, pursuing the plain path of dispassionate
|
|
History. The exact facts, the exact words, --- and they are burnt in upon my
|
|
brain, --- shall be set down without alteration of an iota; and let my Readers
|
|
judge between me and Destiny.
|
|
|
|
The Sphere would willingly have continued his lessons by indoctrinating me in
|
|
the conformation of all regular Solids, Cylinders, Cones, Pyramids,
|
|
Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to
|
|
interrupt him. Not that I was wearied of knowledge. On the contrary, I
|
|
thirsted for yet deeper and fuller draughts than he was offering to me.
|
|
|
|
``Pardon me,'' said I, ``O Thou Whom I must no longer address as the Perfection
|
|
of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine
|
|
interior.''
|
|
|
|
\textit{Sphere}. My what?
|
|
|
|
\textit{I}. Thine interior: thy stomach, thy intestines.
|
|
|
|
\textit{Sphere}. Whence this ill-timed impertinent request? And what mean you by saying
|
|
that I am no longer the Perfection of all Beauty?
|
|
|
|
\textit{I}. My Lord, your own wisdom has taught me to aspire to One even more great,
|
|
more beautiful, and more closely approximate to Perfection than yourself. As
|
|
you yourself, superior to all Flatland forms, combine many Circles in One, so
|
|
doubtless there is One above you who combines many Spheres in One Supreme
|
|
Existence, surpassing even the Solids of Spaceland. And even as we, who are
|
|
now in Space, look down on Flatland and see the insides of all things, so of a
|
|
certainty there is yet above us some higher, purer region, whither thou dost
|
|
surely purpose to lead me --- O Thou Whom I shall always call, everywhere and in
|
|
all Dimensions, my Priest, Philosopher, and Friend --- some yet more spacious
|
|
Space, some more dimensionable Dimensionality, from the vantage-ground of
|
|
which we shall look down together upon the revealed insides of Solid things,
|
|
and where thine own intestines, and those of thy kindred Spheres, will lie
|
|
exposed to the view of the poor wandering exile from Flatland, to whom so much
|
|
has already been vouchsafed.
|
|
|
|
\textit{Sphere}. Pooh! Stuff! Enough of this trifling! The time is short, and much
|
|
remains to be done before you are fit to proclaim the Gospel of Three
|
|
Dimensions to your blind benighted countrymen in Flatland.
|
|
|
|
\textit{I}. Nay, gracious Teacher, deny me not what I know it is in thy power to
|
|
perform. Grant me but one glimpse of thine interior, and I am satisfied for
|
|
ever, remaining henceforth thy docile pupil, thy unemacipable slave, ready to
|
|
receive all thy teachings and to feed upon the words that fall from thy lips.
|
|
|
|
\textit{Sphere}. Well, then, to content and silence you, let me say at once, I would
|
|
shew you what you wish if I could; but I cannot. Would you have me turn my
|
|
stomach inside out to oblige you?
|
|
|
|
\textit{I}. But my Lord has shewn me the intestines of all my countrymen in the Land of
|
|
Two Dimensions by taking me with him into the Land of Three. What therefore
|
|
more easy than now to take his servant on a second journey into the blessed
|
|
region of the Fourth Dimension, where I shall look down with him once more
|
|
upon this land of Three Dimensions, and see the inside of every
|
|
three-dimensioned house, the secrets of the solid earth, the treasures of the
|
|
mines of Spaceland, and the intestines of every solid living creature, even
|
|
the noble and adorable Spheres.
|
|
|
|
\textit{Sphere}. But where is this land of Four Dimensions?
|
|
|
|
\textit{I}. I know not: but doubtless my Teacher knows.
|
|
|
|
\textit{Sphere}. Not I. There is no such land. The very idea of it is utterly
|
|
inconceivable.
|
|
|
|
\textit{I}. Not inconceivable, my Lord, to me, and therefore still less inconceivable
|
|
to my Master. Nay, I despair not that, even here, in this region of Three
|
|
Dimensions, your Lordship's art may make the Fourth Dimension visible to me;
|
|
just as in the Land of Two Dimensions my Teacher's skill would fain have
|
|
opened the eyes of his blind servant to the invisible presence of a Third
|
|
Dimension, though I saw it not.
|
|
|
|
Let me recall the past. Was I not taught below that when I saw a Line and
|
|
inferred a Plane, I in reality saw a Third unrecognized Dimension, not the
|
|
same as brightness, called ``height''? And does it not now follow that, in this
|
|
region, when I see a Plane and infer a Solid, I really see a Fourth
|
|
unrecognized Dimension, not the same as colour, but existent, though
|
|
infinitesimal and incapable of measurement?
|
|
|
|
And besides this, there is the Argument from Analogy of Figures.
|
|
|
|
\textit{Sphere}. Analogy! Nonsense: what analogy?
|
|
|
|
\textit{I}. Your Lordship tempts his servant to see whether he remembers the
|
|
revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst,
|
|
for more knowledge. Doubtless we cannot see that other higher Spaceland now,
|
|
because we have no eye in our stomachs. But, just as there was the realm of
|
|
Flatland, though that poor puny Lineland Monarch could neither turn to left
|
|
nor right to discern it, and just as there was close at hand, and touching my
|
|
frame, the land of Three Dimensions, though I, blind senseless wretch, had no
|
|
power to touch it, no eye in my interior to discern it, so of a surety there
|
|
is a Fourth Dimension, which my Lord perceives with the inner eye of thought.
|
|
And that it must exist my Lord himself has taught me. Or can he have forgotten
|
|
what he himself imparted to his servant?
|
|
|
|
In One Dimension, did not a moving Point produce a Line with two terminal
|
|
points?
|
|
|
|
In Two Dimensions, did not a moving Line produce a Square with four terminal
|
|
points?
|
|
|
|
In Three Dimensions, did not a moving Square produce --- did not this eye of
|
|
mine behold it --- that blessed Being, a Cube, with eight terminal points?
|
|
|
|
And in Four Dimensions shall not a moving Cube --- alas, for Analogy, and alas
|
|
for the Progress of Truth, if it be not so --- shall not, I say, the motion of a
|
|
divine Cube result in a still more divine Organization with sixteen terminal
|
|
points?
|
|
|
|
Behold the infallible confirmation of the Series, 2, 4,
|
|
8, 16: is not this a Geometrical Progression? Is not this --- if I might quote
|
|
my Lord's own words --- ``strictly according to Analogy''?
|
|
|
|
Again, was I not taught by my Lord that as in a Line there are two bounding
|
|
Points, and in a Square there are four bounding Lines, so in a Cube there must
|
|
be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is
|
|
not this an Arithmetical Progression? And consequently does it not of
|
|
necessity follow that the more divine offspring of the divine Cube in the Land
|
|
of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my
|
|
Lord has taught me to believe, ``strictly according to Analogy''?
|
|
|
|
O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not
|
|
knowing the facts; and I appeal to your Lordship to confirm or deny my logical
|
|
anticipations. If I am wrong, I yield, and will no longer demand a Fourth
|
|
Dimension; but, if I am right, my Lord will listen to reason.
|
|
|
|
I ask therefore, is it, or is it not, the fact, that ere now your countrymen
|
|
also have witnessed the descent of Beings of a higher order than their own,
|
|
entering closed rooms, even as your Lordship entered mine, without the opening
|
|
of doors or windows, and appearing and vanishing at will? On the reply to this
|
|
question I am ready to stake everything. Deny it, and I am henceforth silent.
|
|
Only vouchsafe an answer.
|
|
|
|
\textit{Sphere} (after a pause). It is reported so. But men are divided in opinion as
|
|
to the facts. And even granting the facts, they explain them in different
|
|
ways. And in any case, however great may be the number of different
|
|
explanations, no one has adopted or suggested the theory of a Fourth
|
|
Dimension. Therefore, pray have done with this trifling, and let us return to
|
|
business.
|
|
|
|
\textit{I}. I was certain of it. I was certain that my anticipations would be
|
|
fulfilled. And now have patience with me and answer me yet one more question,
|
|
best of Teachers! Those who have thus appeared --- no one knows whence --- and
|
|
have returned --- no one knows whither --- have they also contracted their
|
|
sections and vanished somehow into that more Spacious Space, whither I now
|
|
entreat you to conduct me?
|
|
|
|
\textit{Sphere} (moodily). They have vanished, certainly --- if they ever appeared. But
|
|
most people say that these visions arose from the thought --- you will not
|
|
understand me --- from the brain; from the perturbed angularity of the Seer.
|
|
|
|
\textit{I}. Say they so? Oh, believe them not. Or if it indeed be so, that this other
|
|
Space is really Thoughtland, then take me to that blessed Region where I in
|
|
Thought shall see the insides of all solid things. There, before my ravished
|
|
eye, a Cube moving in some altogether new direction, but strictly according to
|
|
Analogy, so as to make every particle of his interior pass through a new kind
|
|
of Space, with a wake of its own --- shall create a still more perfect
|
|
perfection than himself, with sixteen terminal Extra-solid angles, and Eight
|
|
solid Cubes for his Perimeter. And once there, shall we stay our upward
|
|
course? In that blessed region of Four Dimensions, shall we linger at the
|
|
threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve
|
|
that our ambition shall soar with our corporal ascent. Then, yielding to our
|
|
intellectual onset, the gates of the Six Dimension shall fly open; after that
|
|
a Seventh, and then an Eighth ---
|
|
|
|
How long I should have continued I know not. In vain did the Sphere, in his
|
|
voice of thunder, reiterate his command of silence, and threaten me with the
|
|
direst penalties if I persisted. Nothing could stem the flood of my ecstatic
|
|
aspirations. Perhaps I was to blame; but indeed I was intoxicated with the
|
|
recent draughts of Truth to which he himself had introduced me. However, the
|
|
end was not long in coming. My words were cut short by a crash outside, and a
|
|
simultaneous crash inside me, which impelled me through space with a velocity
|
|
that precluded speech. Down! down! down! I was rapidly descending; and I knew
|
|
that return to Flatland was my doom. One glimpse, one last and
|
|
never-to-be-forgotten glimpse I had of that dull level wilderness --- which was
|
|
now to become my Universe again --- spread out before my eye. Then a darkness.
|
|
Then a final, all-consummating thunder-peal; and, when I came to myself, I was
|
|
once more a common creeping Square, in my Study at home, listening to the
|
|
Peace-Cry of my approaching Wife.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\chapter{How the Sphere encouraged me in a Vision}
|
|
Although I had less than a minute for reflection, I felt, by a kind of
|
|
instinct, that I must conceal my experiences from my Wife. Not that I
|
|
apprehended, at the moment, any danger from her divulging my secret, but I
|
|
knew that to any Woman in Flatland the narrative of my adventures must needs
|
|
be unintelligible. So I endeavoured to reassure her by some story, invented
|
|
for the occasion, that I had accidentally fallen through the trap-door of the
|
|
cellar, and had there lain stunned.
|
|
|
|
The Southward attraction in our country is so slight that even to a Woman my
|
|
tale necessarily appeared extraordinary and well-nigh incredible; but my Wife,
|
|
whose good sense far exceeds that of the average of her Sex, and who perceived
|
|
that I was unusually excited, did not argue with me on the subject, but
|
|
insisted that I was ill and required repose. I was glad of an excuse for
|
|
retiring to my chamber to think quietly over what had happened. When I was at
|
|
last by myself, a drowsy sensation fell on me; but before my eyes closed I
|
|
endeavoured to reproduce the Third Dimension, and especially the process by
|
|
which a Cube is constructed through the motion of a Square. It was not so
|
|
clear as I could have wished; but I remembered that it must be ``Upward, and
|
|
yet not Northward,'' and I determined steadfastly to retain these words as the
|
|
clue which, if firmly grasped, could not fail to guide me to the solution. So
|
|
mechanically repeating, like a charm, the words, ``Upward, yet not Northward,''
|
|
I fell into a sound refreshing sleep.
|
|
|
|
During my slumber I had a dream. I thought I was once more by the side of the
|
|
Sphere, whose lustrous hue betokened that he had exchanged his wrath against
|
|
me for perfectly placability. We were moving together towards a bright but
|
|
infinitesimally small Point, to which my Master directed my attention. As we
|
|
approached, methought there issued from it a slight humming noise as from one
|
|
of your Spaceland bluebottles, only less resonant by far, so slight indeed
|
|
that even in the perfect stillness of the Vacuum through which we soared, the
|
|
sound reached not our ears till we checked our flight at a distant from it of
|
|
something under twenty human diagonals.
|
|
|
|
``Look yonder,'' said my Guide, ``in Flatland thou hast lived; of Lineland thou
|
|
hast received a vision; thou hast soared with me to the heights of Spaceland;
|
|
now, in order to complete the range of thy experience, I conduct thee downward
|
|
to the lowest depth of existence, even to the realm of Pointland, the Abyss of
|
|
No dimensions.
|
|
|
|
``Behold yon miserable creature. That Point is a Being like ourselves, but
|
|
confined to the non-dimensional Gulf. He is himself his own World, his own
|
|
Universe; of any other than himself he can form no conception; he knows not
|
|
Length, nor Breadth, nor Height, for he has had no experience of them; he has
|
|
no cognizance even of the number Two; nor has he a thought of Plurality; for
|
|
he is himself his One and All, being really Nothing. Yet mark his perfect
|
|
self-contentment, and hence learn his lesson, that to be self-contented is to
|
|
be vile and ignorant, and that to aspire is better than to be blindly and
|
|
impotently happy. Now listen.''
|
|
|
|
He ceased; and there arose from the little buzzing creature a tiny, low,
|
|
monotonous, but distinct tinkling, as from one of your Spaceland phonographs,
|
|
from which I caught these words, ``Infinite beatitude of existence! It is; and
|
|
there is nothing else beside It.''
|
|
|
|
``What,'' said I, ``does the puny creature mean by `it'?'' ``He means himself,''
|
|
said the Sphere: ``have you not noticed before now, that babies and babyish
|
|
people who cannot distinguish themselves from the world, speak of themselves
|
|
in the Third Person? But hush!''
|
|
|
|
``It fills all Space,'' continued the little soliloquizing Creature, ``and what
|
|
It fills, It is. What It thinks, that It utters; and what It utters, that It
|
|
hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it
|
|
is the One, and yet the All in All. Ah, the happiness, ah, the happiness of
|
|
Being!''
|
|
|
|
``Can you not startle the little thing out of its complacency?'' said I. ``Tell
|
|
it what it really is, as you told me; reveal to it the narrow limitations of
|
|
Pointland, and lead it up to something higher.'' ``That is no easy task,'' said
|
|
my Master; ``try you.''
|
|
|
|
Hereon, raising by voice to the uttermost, I addressed the Point as follows:
|
|
|
|
``Silence, silence, contemptible Creature. You call yourself the All in All,
|
|
but you are the Nothing: your so-called Universe is a mere speck in a Line,
|
|
and a Line is a mere shadow as compared with ---'' ``Hush, hush, you have said
|
|
enough,'' interrupted the Sphere, ``now listen, and mark the effect of your
|
|
harangue on the King of Pointland.''
|
|
|
|
The lustre of the Monarch, who beamed more brightly than ever upon hearing my
|
|
words, shewed clearly that he retained his complacency; and I had hardly
|
|
ceased when he took up his strain again. ``Ah, the joy, ah, the joy of Thought!
|
|
What can It not achieve by thinking! Its own Thought coming to Itself,
|
|
suggestive of its disparagement, thereby to enhance Its happiness! Sweet
|
|
rebellion stirred up to result in triumph! Ah, the divine creative power of
|
|
the All in One! Ah, the joy, the joy of Being!''
|
|
|
|
``You see,'' said my Teacher, ``how little your words have done. So far as the
|
|
Monarch understand them at all, he accepts them as his own --- for he cannot
|
|
conceive of any other except himself --- and plumes himself upon the variety of
|
|
`Its Thought' as an instance of creative Power. Let us leave this God of
|
|
Pointland to the ignorant fruition of his omnipresence and omniscience:
|
|
nothing that you or I can do can rescue him from his self-satisfaction.''
|
|
|
|
After this, as we floated gently back to Flatland, I could hear the mild voice
|
|
of my Companion pointing the moral of my vision, and stimulating me to aspire,
|
|
and to teach others to aspire. He had been angered at first --- he confessed ---
|
|
by my ambition to soar to Dimensions above the Third; but, since then, he had
|
|
received fresh insight, and he was not too proud to acknowledge his error to a
|
|
Pupil. Then he proceeded to initiate me into mysteries yet higher than those I
|
|
had witnessed, shewing me how to construct Extra-Solids by the motion of
|
|
Solids, and Double Extra-Solids by the motion of Extra-Solids, and all
|
|
``strictly according to Analogy,'' all by methods so simple, so easy, as to be
|
|
patent even to the Female Sex.?
|
|
|
|
|
|
|
|
\chapter{How I tried to teach the
|
|
Theory of Three Dimensions to my Grandson, and with what success}
|
|
I awoke rejoicing, and began to reflect on the glorious career before me. I
|
|
would go forth, methought, at once, and evangelize the whole of Flatland. Even
|
|
to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I
|
|
would begin with my Wife.
|
|
|
|
Just as I had decided on the plan of my operations, I heard the sound of many
|
|
voices in the street commanding silence. Then followed a louder voice. It was
|
|
a herald's proclamation. Listening attentively, I recognized the words of the
|
|
Resolution of the Council, enjoining the arrest, imprisonment, or execution of
|
|
any one who should pervert the minds of people by delusions, and by professing
|
|
to have received revelations from another World.
|
|
|
|
I reflected. This danger was not to be trifled with. It would be better to
|
|
avoid it by omitting all mention of my Revelation, and by proceeding on the
|
|
path of Demonstration --- which after all, seemed so simple and so conclusive
|
|
that nothing would be lost by discarding the former means. ``Upward, not
|
|
Northward'' --- was the clue to the whole proof. It had seemed to me fairly clear
|
|
before I fell asleep; and when I first awoke, fresh from my dream, it had
|
|
appeared as patent as Arithmetic; but somehow it did not seem to me quite so
|
|
obvious now. Though my Wife entered the room opportunely at just that moment,
|
|
I decided, after we had exchanged a few words of commonplace conversation, not
|
|
to begin with her.
|
|
|
|
My Pentagonal Sons were men of character and standing, and physicians of no
|
|
mean reputation, but not great in mathematics, and, in that respect, unfit for
|
|
my purpose. But it occurred to me that a young and docile Hexagon, with a
|
|
mathematical turn, would be a most suitable pupil. Why therefore not make my
|
|
first experiment with my little precocious Grandson, whose casual remarks on
|
|
the meaning of three-to-the-third had met with the approval of the Sphere?
|
|
Discussing the matter with him, a mere boy, I should be in perfect safety; for
|
|
he would know nothing of the Proclamation of the Council; whereas I could not
|
|
feel sure that my Sons --- so greatly did their patriotism and reverence for the
|
|
Circles predominate over mere blind affection --- might not feel compelled to
|
|
hand me over to the Prefect, if they found me seriously maintaining the
|
|
seditious heresy of the Third Dimension.
|
|
|
|
But the first thing to be done was to satisfy in some way the curiosity of my
|
|
Wife, who naturally wished to know something of the reasons for which the
|
|
Circle had desired that mysterious interview, and of the means by which he had
|
|
entered the house. Without entering into the details of the elaborate account
|
|
I gave her, --- an account, I fear, not quite so consistent with truth as my
|
|
Readers in Spaceland might desire, --- I must be content with saying that I
|
|
succeeded at last in persuading her to return quitely to her household duties
|
|
without eliciting from me any reference to the World of Three Dimensions. This
|
|
done, I immediately sent for my Grandson; for, to confess the truth, I felt
|
|
that all that I had seen and heard was in some strange way slipping away from
|
|
me, like the image of a half- grasped, tantalizing dream, and I longed to
|
|
essay my skill in making a first disciple.
|
|
|
|
When my Grandson entered the room I carefully secured the door. Then, sitting
|
|
down by his side and taking our mathematical tablets, --- or, as you would call
|
|
them, Lines --- I told him we would resume the lesson of yesterday. I taught him
|
|
once more how a Point by motion in One Dimension produces a Line, and how a
|
|
straight Line in Two Dimensions produces a Square. After this, forcing a
|
|
laugh, I said, ``And now, you scamp, you wanted to make believe that a Square
|
|
may in the same way by motion `Upward, not Northward' produce another figure,
|
|
a sort of extra square in Three Dimensions. Say that again, you young rascal.''
|
|
|
|
At this moment we heard once more the herald's ``O yes! O yes!'' outside in the
|
|
street proclaiming the Resolution of the Council. Young though he was, my
|
|
Grandson --- who was unusually intelligent for his age, and bred up in perfect
|
|
reverence for the authority of the Circles --- took in the situation with an
|
|
acuteness for which I was quite unprepared. He remained silent till the last
|
|
words of the Proclamation had died away, and then, bursting into tears, ``Dear
|
|
Grandpapa,'' he said, ``that was only my fun, and of course I meant nothing at
|
|
all by it; and we did not know anything then about the new Law; and I don't
|
|
think I said anything about the Third Dimension; and I am sure I did not say
|
|
one word about `Upward, not Northward,' for that would be such nonsense, you
|
|
know. How could a thing move Upward, and not Northward? Upward and not
|
|
Northward! Even if I were a baby, I could not be so absurd as that. How silly
|
|
it is! Ha! ha! ha!''
|
|
|
|
``Not at all silly,'' said I, losing my temper; ``here for example, I take this
|
|
Square,'' and, at the word, I grasped a moveable Square, which was lying at
|
|
hand --- ``and I move it, you see, not Northward but --- yes, I move it Upward ---
|
|
that is to say, Northward but I move it somewhere --- not exactly like this, but
|
|
somehow ---'' Here I brought my sentence to an inane conclusion, shaking the
|
|
Square about in a purposeless manner, much to the amusement of my Grandson,
|
|
who burst out laughing louder than ever, and declared that I was not teaching
|
|
him, but joking with him; and so saying he unlocked the door and ran out of
|
|
the room. Thus ended my first attempt to convert a pupil to the Gospel of
|
|
Three Dimensions.
|
|
|
|
|
|
|
|
|
|
\chapter{How I then tried to diffuse the Theory of Three Dimensions by other
|
|
means, and of the result}
|
|
My failure with my Grandson did not encourage me to communicate my secret to
|
|
others of my household; yet neither was I led by it to despair of success.
|
|
Only I saw that I must not wholly rely on the catch-phrase, ``Upward, not
|
|
Northward,'' but must rather endeavour to seek a demonstration by setting
|
|
before the public a clear view of the whole subject; and for this purpose it
|
|
seemed necessary to resort to writing.
|
|
|
|
So I devoted several months in privacy to the composition of a treatise on the
|
|
mysteries of Three Dimensions. Only, with the view of evading the Law, if
|
|
possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in
|
|
theory, a Figure could look down upon Flatland and see simultaneously the
|
|
insides of all things, and where it was possible that there might be supposed
|
|
to exist a Figure environed, as it were, with six Squares, and containing
|
|
eight terminal Points. But in writing this book I found myself sadly hampered
|
|
by the impossibility of drawing such diagrams as were necessary for my
|
|
purpose: for of course, in our country of Flatland, there are no tablets but
|
|
Lines, and no diagrams but Lines, all in one straight Line and only
|
|
distinguishable by difference of size and brightness; so that, when I had
|
|
finished my treatise (which I entitled, ``Through Flatland to Thoughtland'') I
|
|
could not feel certain that many would understand my meaning.
|
|
|
|
Meanwhile my life was under a cloud. All pleasures palled upon me; all sights
|
|
tantalized and tempted me to outspoken treason, because I could not but
|
|
compare what I saw in Two Dimensions with what it really was if seen in Three,
|
|
and could hardly refrain from making my comparisons aloud. I neglected my
|
|
clients and my own business to give myself to the contemplation of the
|
|
mysteries which I had once beheld, yet which I could impart to no one, and
|
|
found daily more difficult to reproduce even before my own mental vision.
|
|
|
|
One day, about eleven months after my return from Spaceland, I tried to see a
|
|
Cube with my eye closed, but failed; and though I succeeded afterwards, I was
|
|
not then quite certain (nor have I been ever afterwards) that I had exactly
|
|
realized the original. This made me more melancholy than before, and
|
|
determined me to take some step; yet what, I knew not. I felt that I would
|
|
have been willing to sacrifice my life for the Cause, if thereby I could have
|
|
produced conviction. But if I could not convince my Grandson, how could I
|
|
convince the highest and most developed Circles in the land?
|
|
|
|
And yet at times my spirit was too strong for me, and I gave vent to
|
|
dangerous utterances. Already I was considered heterodox if not treasonable,
|
|
and I was keenly alive to the danger of my position; nevertheless I could not
|
|
at times refrain from bursting out into suspicious or half-seditious
|
|
utterances, even among the highest Polygonal or Circular society. When, for
|
|
example, the question arose about the treatment of those lunatics who said
|
|
that they had received the power of seeing the insides of things, I would
|
|
quote the saying of an ancient Circle, who declared that prophets and inspired
|
|
people are always considered by the majority to be mad; and I could not help
|
|
occasionally dropping such expressions as ``the eye that discerns the interiors
|
|
of things,'' and ``the all-seeing land''; once or twice I even let fall the
|
|
forbidden terms ``the Third and Fourth Dimensions.'' At last, to complete a
|
|
series of minor indiscretions, at a meeting of our Local Speculative Society
|
|
held at the palace of the Prefect himself, --- some extremely silly person
|
|
having read an elaborate paper exhibiting the precise reasons why Providence
|
|
has limited the number of Dimensions to Two, and why the attribute of
|
|
omnividence is assigned to the Supreme alone --- I so far forgot myself as to
|
|
give an exact account of the whole of my voyage with the Sphere into Space,
|
|
and to the Assembly Hall in our Metropolis, and then to Space again, and of my
|
|
return home, and of everything that I had seen and heard in fact or vision. At
|
|
first, indeed, I pretended that I was describing the imaginary experiences of
|
|
a fictitious person; but my enthusiasm soon forced me to throw off all
|
|
disguise, and finally, in a fervent peroration, I exhorted all my hearers to
|
|
divest themselves of prejudice and to become believers in the Third Dimension.
|
|
|
|
Need I say that I was at once arrested and taken before the Council?
|
|
|
|
Next morning, standing in the very place where but a very few months ago the
|
|
Sphere had stood in my company, I was allowed to begin and to continue my
|
|
narration unquestioned and uninterrupted. But from the first I foresaw my
|
|
fate; for the President, noting that a guard of the better sort of Policemen
|
|
was in attendance, of angularity little, if at all, under 55 degrees, ordered
|
|
them to be relieved before I began my defence, by an inferior class of 2 or 3
|
|
degrees. I knew only too well what that meant. I was to be executed or
|
|
imprisoned, and my story was to be kept secret from the world by the
|
|
simultaneous destruction of the officials who had heard it; and, this being
|
|
the case, the President desired to substitute the cheaper for the more
|
|
expensive victims.
|
|
|
|
After I had concluded my defence, the President, perhaps perceiving that some
|
|
of the junior Circles had been moved by evident earnestness, asked me two
|
|
questions: ---
|
|
|
|
\begin{enumerate}
|
|
\item Whether I could indicate the direction which I meant when I used the
|
|
words ``Upward, not Northward''?
|
|
\item Whether I could by any diagrams or descriptions (other than the
|
|
enumeration of imaginary sides and angles) indicate the Figure I was
|
|
pleased to call a Cube? I declared that I could say nothing more, and
|
|
that I must commit myself to the Truth, whose cause would surely
|
|
prevail in the end.
|
|
\end{enumerate}
|
|
|
|
The President replied that he quite concurred in my sentiment, and that I
|
|
could not do better. I must be sentenced to perpetual imprisonment; but if the
|
|
Truth intended that I should emerge from prison and evangelize the world, the
|
|
Truth might be trusted to bring that result to pass. Meanwhile I should be
|
|
subjected to no discomfort that was not necessary to preclude escape, and,
|
|
unless I forfeited the privilege by misconduct, I should be occasionally
|
|
permitted to see my brother who had preceded me to my prison.
|
|
|
|
Seven years have elapsed and I am still a prisoner, and --- if I except the
|
|
occasional visits of my brother --- debarred from all companionship save that of
|
|
my jailers. My brother is one of the best of Squares, just sensible, cheerful,
|
|
and not without fraternal affection; yet I confess that my weekly interviews,
|
|
at least in one respect, cause me the bitterest pain. He was present when the
|
|
Sphere manifested himself in the Council Chamber; he saw the Sphere's changing
|
|
sections; he heard the explanation of the phenomena then give to the Circles.
|
|
Since that time, scarcely a week has passed during seven whole years, without
|
|
his hearing from me a repetition of the part I played in that manifestation,
|
|
together with ample descriptions of all the phenomena in Spaceland, and the
|
|
arguments for the existence of Solid things derivable from Analogy. Yet --- I
|
|
take shame to be forced to confess it --- my brother has not yet grasped the
|
|
nature of Three Dimensions, and frankly avows his disbelief in the existence
|
|
of a Sphere.
|
|
|
|
Hence I am absolutely destitute of converts, and, for
|
|
aught that I can see, the millennial Revelation has been made to me for
|
|
nothing. Prometheus up in Spaceland was bound for bringing down fire for
|
|
mortals, but I --- poor Flatland Prometheus --- lie here in prison for bringing
|
|
down nothing to my countrymen. Yet I exist in the hope that these memoirs, in
|
|
some manner, I know not how, may find their way to the minds of humanity in
|
|
Some Dimension, and may stir up a race of rebels who shall refuse to be
|
|
confined to limited Dimensionality.
|
|
|
|
That is the hope of my brighter moments. Alas, it is
|
|
not always so. Heavily weights on me at times the burdensome reflection that I
|
|
cannot honestly say I am confident as to the exact shape of the once-seen,
|
|
oft-regretted Cube; and in my nightly visions the mysterious precept, ``Upward,
|
|
not Northward,'' haunts me like a soul-devouring Sphinx. It is part of the
|
|
martyrdom which I endure for the cause of Truth that there are seasons of
|
|
mental weakness, when Cubes and Spheres flit away into the background of
|
|
scarce-possible existences; when the Land of Three Dimensions seems almost as
|
|
visionary as the Land of One or None; nay, when even this hard wall that bars
|
|
me from my freedom, these very tablets on which I am writing, and all the
|
|
substantial realities of Flatland itself, appear no better than the offspring
|
|
of a diseased imagination, or the baseless fabric of a dream.
|
|
|
|
\begin{center}
|
|
\includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig11}
|
|
\end{center}
|
|
|
|
\end{document}
|
|
|
|
|