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  1. \title{Flatland} \author{Edwin A. Abbott}
  2. \date{\today}
  3. \documentclass[10pt, twoside]{extreport}
  4. \usepackage[english]{babel}
  5. \usepackage{graphicx}
  6. \usepackage[T1]{fontenc}
  7. \usepackage[hidelinks]{hyperref} % Hide boxes in TOC
  8. \usepackage[fulloldstylenums]{kpfonts}
  9. % Adjust line spacing
  10. \usepackage{setspace}
  11. % \onehalfspacing
  12. \setstretch{1.1}
  13. % Remove space after paragraphs; adjust indent
  14. \usepackage[skip=0pt, indent=7mm]{parskip}
  15. \usepackage{geometry}
  16. \geometry{
  17. paperheight=8.5in,
  18. paperwidth=5.5in,
  19. heightrounded,
  20. margin=0.6in,
  21. headsep=0.15in
  22. }
  23. % Adjust the top and bottom margins
  24. %
  25. \addtolength{\topmargin}{0.3in}
  26. \addtolength{\textheight}{-0.2in}
  27. % Fancy headers
  28. \usepackage{fancyhdr}
  29. \pagestyle{fancy}
  30. \fancyhf{} % remove everything
  31. \renewcommand{\headrulewidth}{0.5pt}
  32. \fancyhead[EL, OR]{\thepage}
  33. \fancyhead[RE]{\textit{Flatland:}}
  34. \fancyhead[LO]{\textit{A Romance of Many Dimensions}}
  35. % Include page numbers on chapter pages
  36. \usepackage[]{titlesec}
  37. \assignpagestyle{\chapter}{fancy}
  38. % Insert blank pages
  39. % See
  40. \usepackage{afterpage}
  41. \newcommand\blankpage{%
  42. \null
  43. \thispagestyle{empty}%
  44. \newpage}
  45. % Remove page numbers from "part" pages
  46. \usepackage{nonumonpart}
  47. \urlstyle{same}
  48. % Add additional text to Part pages
  49. %
  50. \makeatletter
  51. \let\old@endpart\@endpart
  52. \renewcommand\@endpart[1][]{%
  53. \begin{quote}#1\end{quote}%
  54. \old@endpart}
  55. \makeatother
  56. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  57. % Begin the document %
  58. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  59. \begin{document} % \maketitle
  60. \thispagestyle{empty}
  61. \vspace*{2cm}
  62. \includegraphics[scale=0.45]{flatland_cover}
  63. \vspace*{2cm}
  64. \begin{center}
  65. \textit{``Fie, fie, how franticly I square my talk!''}
  66. \end{center}
  67. \clearpage
  68. \thispagestyle{empty}
  69. \noindent{} This work is in the public domain in the United States because it was
  70. published before January 1, 1923.
  71. The author died in 1926, so this work is also in the public domain in
  72. countries and areas where the copyright term is the author's life plus 80
  73. years or less. This work may also be in the public domain in countries and
  74. areas with longer native copyright terms that apply the rule of the shorter
  75. term to foreign works.
  76. \vspace{5mm}
  77. \noindent\emph{Written by} Edwin A. Abbott \\
  78. \emph{Illustrated by} Edwin A. Abbott \\
  79. \emph{Typeset by} Ives van der Flaas\\
  80. \emph{Additional typesetting by} Kenneth John Odle\\ \\
  81. Fifth edition
  82. \chapter*{A Note From The Typesetter}
  83. \thispagestyle{empty}
  84. Although Edwin A. Abbott's essay ``Flatland'' is readily available on the
  85. internet, I failed to find a nicely typeset version. At this time,
  86. the rendering quality of internet browsers doesn't come anywhere near the
  87. quality of a nice book.
  88. For these reasons, as well as a healthy portion of boredom, I made the version
  89. of ``Flatland'' you are currently reading, in \LaTeX. The \LaTeX\ source can be
  90. found on \url{} (feel free to make issues
  91. or send me pull requests when you find bugs or have improvements). It should
  92. also be fairly easy --- a matter of minutes --- to produce a book-sized
  93. version, ready to send to a printer.
  94. \vspace{1cm}
  95. \hfill Ives van der Flaas <>, 2011-09-20
  96. \vspace{1.5cm}
  97. \noindent{}\textbf{An additional note:}\\
  98. \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.
  99. 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.
  100. The source can be found on \texttt{\url{}}.
  101. \begin{flushright}
  102. \textit{Kenneth John Odle}
  103. \end{flushright}
  104. %\newpage \ \newpage
  105. %\thispagestyle{empty}
  106. \afterpage{\blankpage}
  107. \addtocontents{toc}{\protect\thispagestyle{empty}}
  108. \tableofcontents{}
  109. \thispagestyle{empty}
  110. % \addcontentsline{toc}{chapter}{Dedication}
  111. \chapter*{Dedication}
  112. \thispagestyle{empty}
  113. \begin{center}
  114. To\\
  115. The Inhabitants of \textsc{Space in General}\\
  116. And \textsc{H.C. in Particular}\\
  117. This Work is Dedicated\\
  118. By a Humble Native of Flatland\\
  119. In the Hope that\\
  120. Even as he was Initiated into the Mysteries\\
  121. Of \textsc{Three Dimensions}\\
  122. Having been previously conversant\\
  123. With \textsc{Only Tow}\\
  124. So the Citizens of that Celestial Region\\
  125. May aspire yet higher and higher\\
  126. To the Secrets of \textsc{Four Five or Even Six} Dimensions\\
  127. Thereby contributing\\
  128. To the Enlargement of \textsc{the Imagination}\\
  129. And the possible Development\\
  130. Of that most rare and excellent Gift of \textsc{Modesty}\\
  131. Among the Superior Races\\
  132. Of \textsc{Solid Humanity}\\
  133. \end{center}
  134. \afterpage{\blankpage}
  135. % \addcontentsline{toc}{chapter}{Preface}
  136. \chapter*{Preface}
  137. \thispagestyle{empty}
  138. If my poor Flatland friend retained the vigour of mind which he enjoyed when
  139. he began to compose these Memoirs, I should not now need to represent him in
  140. this preface, in which he desires, fully, to return his thanks to his readers
  141. and critics in Spaceland, whose appreciation has, with unexpected celerity,
  142. required a second edition of this work; secondly, to apologize for certain
  143. errors and misprints (for which, however, he is not entirely responsible);
  144. and, thirdly, to explain on or two misconceptions. But he is not the Square he
  145. once was. Years of imprisonment, and the still heavier burden of general
  146. incredulity and mockery, have combined with the thoughts and notions, and much
  147. also of the terminology, which he acquired during his short stay in spaceland.
  148. He has, therefore, requested me to reply in his behalf to two special
  149. objections, one of an intellectual, the other of a moral nature.
  150. The first objection is, that a Flatlander, seeing a Line, sees something that
  151. must be \textit{thick} to the eye as well as \textit{long} to the eye (otherwise it would not be
  152. visible, if it had not some thickness); and consequently he ought (it is
  153. argued) to acknowledge that his countrymen are not only long and broad, but
  154. also (though doubtless to a very slight degree) \textit{thick} or \textit{high}. This objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess,
  155. when I first heard it, I knew not what to reply. But my poor old friend's
  156. answer appears to me completely to meet it.
  157. ``I admit,'' said he --- when I mentioned to him this objection --- ``I admit the
  158. truth of your critic's facts, but I deny his conclusions. It is true that we
  159. have really in Flatland a Third unrecognized Dimension called `height,' just
  160. as it also is true that you have really in Spaceland a Fourth unrecognized
  161. Dimension, called by no name at present, but which I will call `extra-height.'
  162. But we can no more take cognizance of our `height' than you can of your
  163. `extra-height.' Even I --- who have been in Spaceland, and have had the
  164. privilege of understanding for twenty-four hours the meaning of `height' ---
  165. even I cannot now comprehend it, nor realize it by the sense of sight or by
  166. any process of reason; I can but apprehend it by faith.''
  167. ``The reason is obvious. Dimension implied direction, implies measurement,
  168. implies the more and the less. Now, all our lines are \textit{equally} and
  169. \textit{infinitesimally} thick (or high, whichever you like); consequently, there is
  170. nothing in them to lead our minds to the conception of that Dimension. No
  171. `delicate micrometer' --- as has been suggested by one too hasty Spaceland
  172. critic --- would in the least avail us; for we should not know \textit{what to measure,
  173. nor in what direction}. When we see a Line, we see something that is long and
  174. \textit{bright}; \textit{brightness}, as well as length, is necessary to the existence of a
  175. Line; if the brightness vanishes, the Line is extinguished. Hence, all my
  176. Flatland friends --- when I talk to them about the unrecognized Dimension which
  177. is somehow visible in a Line --- say, `Ah, you mean \textit{brightness}': and when I
  178. reply, `No, I mean a real Dimension,' they at once retort, `Then measure it,
  179. or tell us in what direction it extends'; and this silences me, for I can do
  180. neither. Only yesterday, when the Chief Circle (in other words our High
  181. Priest) came to inspect the State Prison and paid me his seventh annual visit,
  182. and when for the seventh time he put me the question, `Was I any better?' I
  183. tried to prove to him that he was `high', as well as long and broad, although
  184. he did not know it. But what was his reply? `You say I am ``high''; measure my
  185. ``high-ness'' and I will believe you.'. What could I do? How could I meet his
  186. challenge? I was crushed; and he left the room triumphant.''
  187. ``Does this still seem strange to you? Then put yourself in a similar position.
  188. Suppose a person of the Fourth Dimension, condescending to visit you, were to
  189. say, `Whenever you open your eyes, you \textit{see} a Plane (which is of Two
  190. Dimensions) and you \textit{infer} a Solid (which is of Three); but in reality you also
  191. see (though you do not recognize) a Fourth Dimension, which is not colour nor
  192. brightness nor anything of the kind, but a true Dimension, although I cannot
  193. point out to you its direction, nor can you possibly measure it.' What would
  194. you say to such a visitor? Would not you have him locked up? Well, that is my
  195. fate: and it is as natural for us Flatlanders to lock up a Square for
  196. preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube
  197. for preaching the Fourth. Alas, how strong a family likeness runs through
  198. blind and persecuting humanity in all Dimensions! Points, Lines, Squares,
  199. Cubes, Extra-Cubes --- we are all liable to the same errors, all alike the
  200. Slavers of our respective Dimensional prejudices, as one of our Spaceland
  201. poets has said --- `One touch of Nature makes all worlds akin.' '' \footnote{The
  202. Author desires me to add, that the misconceptions of some of his critics on
  203. this matter has induced him to insert in his dialogue with
  204. the Sphere, certain remarks which have a bearing on the point in question and
  205. which he had previously omitted as being tedious and unnecessary.}
  206. On this point the defence of the Square seems to me to be impregnable. I wish
  207. I could say that his answer to the second (or moral) objection was equally
  208. clear and cogent. It has been objected that he is a woman-hater; and as this
  209. objection has been vehemently urged by those whom Nature's decree has
  210. constituted the somewhat larger half of the Spaceland race, I should like to
  211. remove it, so far as I can honestly do so. But the Square is so unaccustomed
  212. to the use of the moral terminology of Spaceland that I should be doing him an
  213. injustice if I were literally to transcribe his defence against this charge.
  214. Acting, therefore, as his interpreter and summarizer, I gather that in the
  215. course of an imprisonment of seven years he has himself modified his own
  216. personal views, both as regards Women and as regards the Isosceles or Lower
  217. Classes. Personally, he now inclines to the opinion of the Sphere that the
  218. Straight Lines are in many important respects superior to the
  219. Circles. But, writing as a Historian, he has identified himself (perhaps too
  220. closely) with the views generally adopted by Flatland, and (as he has been
  221. informed) even by Spaceland, Historians; in whose pages (until very recent
  222. times) the destinies of Women and of the masses of mankind have seldom been
  223. deemed worthy of mention and never of careful consideration.
  224. In a still more obscure passage he now desires to disavow the Circular or
  225. aristocratic tendencies with which some critics have naturally credited him.
  226. While doing justice to the intellectual power with which a few Circles have
  227. for many generations maintained their supremacy over immense multitudes of
  228. their countrymen, he believes that the facts of Flatland, speaking for
  229. themselves without comment on his part, declare that Revolutions cannot always
  230. be suppressed by slaughter, and that Nature, in sentencing the Circles to
  231. infecundity, has condemned them to ultimate failure --- ``and herein,'' he says,
  232. ``I see a fulfillment of the great Law of all worlds, that while the wisdom of
  233. Man thinks it is working one thing, the wisdom of Nature constrains it to work
  234. another, and quite a different and far better thing.'' For the rest, he begs
  235. his readers not to suppose that every minute detail in the daily life of
  236. Flatland must needs correspond to some other detail in Spaceland; and yet he
  237. hopes that, taken as a whole, his work may prove suggestive as well as
  238. amusing, to those Spacelanders of moderate and modest minds who --- speaking of
  239. that which is of the highest importance, but lies beyond experience --- decline
  240. to say on the one hand, ``This can never be,'' and on the other hand, ``It must
  241. needs be precisely thus, and we know all about it.''
  242. %b\mainmatter
  243. \part{This World}[
  244. \vspace{2cm}
  245. \begin{center}
  246. \begin{normalsize}
  247. \textit{'Be patient, for the world is broad and wide.'}
  248. \end{normalsize}
  249. \end{center}
  250. ]
  251. \chapter{Of the Nature of Flatland} I call our world Flatland, not because we
  252. call it so, but to make its nature clearer to you, my happy readers, who are
  253. privileged to live in Space.
  254. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares,
  255. Pentagons, Hexagons, and other figures, instead of remaining fixed in their
  256. places, move freely about, on or in the surface, but without the power of
  257. rising above or sinking below it, very much like shadows --- only hard with
  258. luminous edges --- and you will then have a pretty correct notion of my country
  259. and countrymen. Alas, a few years ago, I should have said ``my universe'': but
  260. now my mind has been opened to higher views of things.
  261. In such a country, you will perceive at once that it is impossible that there
  262. should be anything of what you call a ``solid'' kind; but I dare say you will
  263. suppose that we could at least distinguish by sight the Triangles, Squares,
  264. and other figures, moving about as I have described them. On the contrary, we
  265. could see nothing of the kind, not at least so as to distinguish one figure
  266. from another. Nothing was visible, nor could be visible, to us, except
  267. Straight Lines; and the necessity of this I will speedily demonstrate.
  268. Place a penny on the middle of one of your tables in Space; and leaning over
  269. it, look down upon it. It will appear a circle.
  270. But now, drawling back to the edge of the table, gradually lower your eye
  271. (thus bringing yourself more and more into the condition of the inhabitants of
  272. Flatland), and you will find the penny becoming more and more oval to your
  273. view, and at last when you have placed your eye exactly on the edge of the
  274. table (so that you are, as it were, actually a Flatlander) the penny will then
  275. have ceased to appear oval at all, and will have become, so far as you can
  276. see, a straight line.
  277. The same thing would happen if you were to treat in the same way a Triangle,
  278. or a Square, or any other figure cut out of pasteboard. As soon as you look at
  279. it with your eye on the edge of the table, you will find that it ceases to
  280. appear to you as a figure, and that it becomes in appearance a straight line.
  281. Take for example an equilateral Triangle --- who represents with us a Tradesman
  282. of the respectable class. Figure 1 represents the Tradesman as you would see
  283. him while you were bending over him from above; figures 2 and 3 represent the
  284. Tradesman, as you would see him if your eye were close to the level, or all
  285. but on the level of the table; and if your eye were quite on the level of the
  286. table (and that is how we see him in Flatland) you would see nothing but a
  287. straight line.
  288. \includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig1}
  289. When I was in Spaceland I heard that your sailors have very similar
  290. experiences while they traverse your seas and discern some distant island or
  291. coast lying on the horizon. The far-off land may have bays, forelands, angles
  292. in and out to any number and extent; yet at a distance you see none of these
  293. (unless indeed your sun shines bright upon them revealing the projections and
  294. retirements by means of light and shade), nothing but a grey unbroken line
  295. upon the water.
  296. Well, that is just what we see when one of our triangular or other
  297. acquaintances comes towards us in Flatland. As there is neither sun with us,
  298. nor any light of such a kind as to make shadows, we have none of the helps to
  299. the sight that you have in Spaceland. If our friend comes closer to us we see
  300. his line becomes larger; if he leaves us it becomes smaller; but still he
  301. looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon,
  302. Circle, what you will --- a straight Line he looks and nothing else.
  303. You may perhaps ask how under these disadvantageous circumstances we are able
  304. to distinguish our friends from one another: but the answer to this very
  305. natural question will be more fitly and easily given when I come to describe
  306. the inhabitants of Flatland. For the present let me defer this subject, and
  307. say a word or two about the climate and houses in our country.
  308. \chapter{Of the Climate and Houses in Flatland}
  309. As with you, so also with us, there are four points of the compass North,
  310. South, East, and West.
  311. There being no sun nor other heavenly bodies,
  312. it is impossible for us to determine the North in the usual way; but we have a
  313. method of our own. By a Law of Nature with us, there is a constant attraction
  314. to the South; and, although in temperate climates this is very slight --- so
  315. that even a Woman in reasonable health can journey several furlongs northward
  316. without much difficulty --- yet the hampering effect of the southward attraction
  317. is quite sufficient to serve as a compass in most parts of our earth.
  318. Moreover, the rain (which falls at stated intervals) coming always from the
  319. North, is an additional assistance; and in the towns we have the guidance of
  320. the houses, which of course have their side-walls running for the most part
  321. North and South, so that the roofs may keep off the rain from the North. In
  322. the country, where there are no houses, the trunks of the trees serve as some
  323. sort of guide. Altogether, we have not so much difficulty as might be expected
  324. in determining our bearings.
  325. Yet in our more temperate regions, in which the southward attraction is hardly
  326. felt, walking sometimes in a perfectly desolate plain where there have been no
  327. houses nor trees to guide me, I have been occasionally compelled to remain
  328. stationary for hours together, waiting till the rain came before continuing my
  329. journey. On the weak and aged, and especially on delicate Females, the force
  330. of attraction tells much more heavily than on the robust of the Male Sex, so
  331. that it is a point of breeding, if you meet a Lady in the street, always to
  332. give her the North side of the way --- by no means an easy thing to do always at
  333. short notice when you are in rude health and in a climate where it is
  334. difficult to tell your North from your South.
  335. Windows there are none in our houses: for the light comes to us alike in our
  336. homes and out of them, by day and by night, equally at all times and in all
  337. places, whence we know not. It was in old days, with our learned men, an
  338. interesting and oft-investigate question, ``What is the origin of light?'' and
  339. the solution of it has been repeatedly attempted, with no other result than to
  340. crowd our lunatic asylums with the would-be solvers. Hence, after fruitless
  341. attempts to suppress such investigations indirectly by making them liable to a
  342. heavy tax, the Legislature, in comparatively recent times, absolutely
  343. prohibited them. I --- alas, I alone in Flatland --- know now only too well the
  344. true solution of this mysterious problem; but my knowledge cannot be made
  345. intelligible to a single one of my countrymen; and I am mocked at --- I, the
  346. sole possessor of the truths of Space and of the theory of the introduction of
  347. Light from the world of three Dimensions --- as if I were the maddest of the
  348. mad! But a truce to these painful digressions: let me return to our homes.
  349. The most common form for the construction of a house is five-sided or
  350. pentagonal, as in the annexed figure. The two Northern sides \textit{RO}, \textit{OF},
  351. constitute the roof, and for the most part have no doors; on the East is a
  352. small door for the Women; on the West a much larger one for the Men; the South
  353. side or floor is usually doorless.
  354. \begin{center}
  355. \includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig2}
  356. \end{center}
  357. Square and triangular houses are not allowed, and for this reason. The angles
  358. of a Square (and still more those of an equilateral Triangle,) being much more
  359. pointed than those of a Pentagon, and the lines of inanimate objects (such as
  360. houses) being dimmer than the lines of Men and Women, it follows that there is
  361. no little danger lest the points of a square of triangular house residence
  362. might do serious injury to an inconsiderate or perhaps absentminded traveller
  363. suddenly running against them: and therefore, as early as the eleventh century
  364. of our era, triangular houses were universally forbidden by Law, the only
  365. exceptions being fortifications, powder-magazines, barracks, and other state
  366. buildings, which is not desirable that the general public should approach
  367. without circumspection.
  368. At this period, square houses were still everywhere permitted, though
  369. discouraged by a special tax. But, about three centuries afterwards, the Law
  370. decided that in all towns containing a population above ten thousand, the
  371. angle of a Pentagon was the smallest house-angle that could be allowed
  372. consistently with the public safety. The good sense of the community has
  373. seconded the efforts of the Legislature; and now, even in the country, the
  374. pentagonal construction has superseded every other. It is only now and then in
  375. some very remote and backward agricultural district that an antiquarian may
  376. still discover a square house.
  377. \chapter{Concerning the Inhabitants of Flatland}
  378. The greatest length or breadth of a full grown inhabitant of Flatland may be
  379. estimated at about eleven of your inches. Twelve inches may be regarded as a
  380. maximum.
  381. Our Women are Straight Lines.
  382. Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides,
  383. each about eleven inches long, and a base or third side so short (often not
  384. exceeding half an inch) that they form at their vertices a very sharp and
  385. formidable angle. Indeed when their bases are of the most degraded type (not
  386. more than the eighth part of an inch in size), they can hardly be
  387. distinguished from Straight lines or Women; so extremely pointed are their
  388. vertices. With us, as with you, these Triangles are distinguished from others
  389. by being called Isosceles; and by this name I shall refer to them in the
  390. following pages.
  391. Our Middle Class consists of Equilateral or Equal-Sided Triangles.
  392. Our Professional Men and Gentlemen are Squares (to which class I myself
  393. belong) and Five-Sided Figures or Pentagons.
  394. Next above these come the Nobility, of whom there are several degrees,
  395. beginning at Six-Sided Figures, or Hexagons, and from thence rising in the
  396. number of their sides till they receive the honourable title of Polygonal, or
  397. many-Sided. Finally when the number of the sides becomes so numerous, and the
  398. sides themselves so small, that the figure cannot be distinguished from a
  399. circle, he is included in the Circular or Priestly order; and this is the
  400. highest class of all.
  401. It is a Law of Nature with us that a male child shall have one more side than
  402. his father, so that each generation shall rise (as a rule) one step in the
  403. scale of development and nobility. Thus the son of a Square is a Pentagon; the
  404. son of a Pentagon, a Hexagon; and so on.
  405. But this rule applies not always to the Tradesman, and still less often to the
  406. Soldiers, and to the Workmen; who indeed can hardly be said to deserve the
  407. name of human Figures, since they have not all their sides equal. With them
  408. therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a
  409. Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope
  410. is not shut out, even from the Isosceles, that his posterity may ultimately
  411. rise above his degraded condition. For, after a long series of military
  412. successes, or diligent and skillful labours, it is generally found that the
  413. more intelligent among the Artisan and Soldier classes manifest a slight
  414. increase of their third side or base, and a shrinkage of the two other sides.
  415. Intermarriages (arranged by the Priests) between the sons and daughters of
  416. these more intellectual members of the lower classes generally result in an
  417. offspring approximating still more to the type of the Equal-Sided Triangle.
  418. Rarely --- in proportion to the vast numbers of Isosceles births --- is a genuine
  419. and certifiable Equal-Sided Triangle produced from Isosceles parents
  420. \footnote{ ``What need of a certificate?'' a Spaceland critic may ask: ``Is not
  421. the procreation of a Square Son a certificate from Nature herself, proving the
  422. Equal-sidedness of the Father?'' I reply that no Lady of any position will
  423. marry an uncertified Triangle. Square offspring has sometimes resulted from a
  424. slightly Irregular Triangle; but in almost every such case the Irregularity of
  425. the first generation is visited on the third; which either fails to attain the
  426. Pentagonal rank, or relapses to the Triangular.}. Such a birth requires, as
  427. its antecedents, not only a series of carefully arranged intermarriages, but
  428. also a long-continued exercise of frugality and self-control on the part of
  429. the would-be ancestors of the coming Equilateral, and a patient, systematic,
  430. and continuous development of the Isosceles intellect through many
  431. generations.
  432. The birth of a True Equilateral Triangle from Isosceles parents is the subject
  433. of rejoicing in our country for many furlongs around. After a strict
  434. examination conducted by the Sanitary and Social Board, the infant, if
  435. certified as Regular, is with solemn ceremonial admitted into the class of
  436. Equilaterals. He is then immediately taken from his proud yet sorrowing
  437. parents and adopted by some childless Equilateral, who is bound by oath never
  438. to permit the child henceforth to enter his former home or so much as to look
  439. upon his relations again, for fear lest the freshly developed organism may, by
  440. force of unconscious imitation, fall back again into his hereditary level.
  441. The occasional emergence of an Equilateral from the ranks of his serf-born
  442. ancestors is welcomed, not only by the poor serfs themselves, as a gleam of
  443. light and hope shed upon the monotonous squalor of their existence, but also
  444. by the Aristocracy at large; for all the higher classes are well aware that
  445. these rare phenomena, while they do little or nothing to vulgarize their own
  446. privileges, serve as almost useful barrier against revolution from below.
  447. Had the acute-angled rabble been all, without exception, absolutely destitute
  448. of hope and of ambition, they might have found leaders in some of their many
  449. seditious outbreaks, so able as to render their superior numbers and strength
  450. too much even for the wisdom of the Circles. But a wise ordinance of Nature
  451. has decreed that, in proportion as the working-classes increase in
  452. intelligence, knowledge, and all virtue, in that same proportion their acute
  453. angle (which makes them physically terrible) shall increase also and
  454. approximate to their comparatively harmless angle of the Equilateral Triangle.
  455. Thus, in the most brutal and formidable of the soldier class --- creatures
  456. almost on a level with women in their lack of intelligence --- it is found that,
  457. as they wax in the mental ability necessary to employ their tremendous
  458. penetrating power to advantage, so do they wane in the power of penetration
  459. itself.
  460. How admirable is this Law of Compensation! And how perfect a proof of the
  461. natural fitness and, I may almost say, the divine origin of the aristocratic
  462. constitution of the States of Flatland! By a judicious use of this Law of
  463. Nature, the Polygons and Circles are almost always able to stifle sedition in
  464. its very cradle, taking advantage of the irrepressible and boundless
  465. hopefulness of the human mind. Art also comes to the aid of Law and Order. It
  466. is generally found possible --- by a little artificial compression or expansion
  467. on the part of the State physicians --- to make some of the more intelligent
  468. leaders of a rebellion perfectly Regular, and to admit them at once into the
  469. privileged classes; a much larger number, who are still below the standard,
  470. allured by the prospect of being ultimately ennobled, are induced to enter the
  471. State Hospitals, where they are kept in honourable confinement for life; one
  472. or two alone of the most obstinate, foolish, and hopelessly irregular are led
  473. to execution.
  474. Then the wretched rabble of the Isosceles, planless and leaderless, are ether
  475. transfixed without resistance by the small body of their brethren whom the
  476. Chief Circle keeps in pay for emergencies of this kind; or else more often, by
  477. means of jealousies and suspicious skillfully fomented among them by the
  478. Circular party, they are stirred to mutual warfare, and perish by one
  479. another's angles. No less than one hundred and twenty rebellions are recorded
  480. in our annals, besides minor outbreaks numbered at two hundred and
  481. thirty-five; and they have all ended thus.
  482. \chapter{Concerning the Women}
  483. If our highly pointed Triangles of the Soldier class are formidable, it may be
  484. readily inferred that far more formidable are our Women. For, if a Soldier is
  485. a wedge, a Woman is a needle; being, so to speak, all point, at least at the
  486. two extremities. Add to this the power of making herself practically invisible
  487. at will, and you will perceive that a Female, in Flatland, is a creature by no
  488. means to be trifled with.
  489. But here, perhaps, some of my younger Readers may ask how a woman in Flatland
  490. can make herself invisible. This ought, I think, to be apparent without any
  491. explanation. However, a few words will make it clear to the most unreflecting.
  492. Place a needle on the table. Then, with your eye on the level of the table,
  493. look at it side-ways, and you see the whole length of it; but look at it
  494. end-ways, and you see nothing but a point, it has become practically
  495. invisible. Just so is it with one of our Women. When her side is turned
  496. towards us, we see her as a straight line; when the end containing her eye or
  497. mouth --- for with us these two organs are identical --- is the part that meets
  498. our eye, then we see nothing but a highly lustrous point; but when the back is
  499. presented to our view, then --- being only sub-lustrous, and, indeed, almost as
  500. dim as an inanimate object --- her hinder extremity serves her as a kind of
  501. Invisible Cap.
  502. The dangers to which we are exposed from our Women must now be manifest to the
  503. meanest capacity in Spaceland. If even the angle of a respectable Triangle in
  504. the middle class is not without its dangers; if to run against a Working Man
  505. involves a gash; if collision with an Officer of the military class
  506. necessitates a serious wound; if a mere touch from the vertex of a Private
  507. Soldier brings with it danger of death; --- what can it be to run against a
  508. woman, except absolute and immediate destruction? And when a Woman is
  509. invisible, or visible only as a dim sub-lustrous point, how difficult must it
  510. be, even for the most cautious, always to avoid collision!
  511. Many are the enactments made at different times in the different States of
  512. Flatland, in order to minimize this peril; and in the Southern and less
  513. temperate climates, where the force of gravitation is greater, and human
  514. beings more liable to casual and involuntary motions, the Laws concerning
  515. Women are naturally much more stringent. But a general view of the Code may be
  516. obtained from the following summary: ---
  517. Every house shall have one entrance on the Eastern side, for the use of
  518. Females only; by which all females shall enter ``in a becoming and respectful
  519. manner'' \footnote{ When I was in Spaceland I understood that some of your
  520. Priestly circles have in the same way a separate entrance for Villagers,
  521. Farmers and Teachers of Board Schools (Spectator, Sept. 1884, P. 1255) that
  522. they may ``approach in a becoming and respectful manner.''} and not by the Men's
  523. or Western door. No Female shall walk in any public place without continually
  524. keeping up her Peace-cry, under penalty of death. Any Female, duly certified
  525. to be suffering from St. Vitus's Dance, fits, chronic cold accompanied by
  526. violent sneezing, or any disease necessitating involuntary motions, shall be
  527. instantly destroyed. In some of the States there is an additional Law
  528. forbidding Females, under penalty of death, from walking or standing in any
  529. public place without moving their backs constantly from right to left so as to
  530. indicate their presence to those behind them; others oblige a Woman, when
  531. travelling, to be followed by one of her sons, or servants, or by her husband;
  532. others confine Women altogether to their houses except during the religious
  533. festivals. But it has been found by the wisest of our Circles or Statesmen
  534. that the multiplication of restrictions on Females tends not only to the
  535. debilitation and diminution of the race, but also to the increase of domestic
  536. murders to such an extent that a State loses more than it gains by a too
  537. prohibitive Code.
  538. For whenever the temper of the Women is thus exasperated by confinement at
  539. home or hampering regulations abroad, they are apt to vent their spleen upon
  540. their husbands and children; and in the less temperate climates the whole male
  541. population of a village has been sometimes destroyed in one or two hours of
  542. simultaneous female outbreak. Hence the Three Laws, mentioned above, suffice
  543. for the better regulated States, and may be accepted as a rough
  544. exemplification of our Female Code.
  545. After all, our principal safeguard is found, not in Legislature, but in the
  546. interests of the Women themselves. For, although they can inflict
  547. instantaneous death by a retrograde movement, yet unless they can at once
  548. disengage their stinging extremity from the struggling body of their victim,
  549. their own frail bodies are liable to be shattered.
  550. The power of Fashion is also on our side. I pointed
  551. out that in some less civilized States no female is suffered to stand in any
  552. public place without swaying her back from right to left. This practice has
  553. been universal among ladies of any pretensions to breeding in all
  554. well-governed States, as far back as the memory of Figures can reach. It is
  555. considered a disgrace to any state that legislation should have to enforce
  556. what ought to be, and is in every respectable female, a natural instinct. The
  557. rhythmical and, if I may so say, well-modulated undulation of the back in our
  558. ladies of Circular rank is envied and imitated by the wife of a common
  559. Equilateral, who can achieve nothing beyond a mere monotonous swing, like the
  560. ticking of a pendulum; and the regular tick of the Equilateral is no less
  561. admired and copied by the wife of the progressive and aspiring Isosceles, in
  562. the females of whose family no ``back-motion'' of any kind has become as yet a
  563. necessity of life. Hence, in every family of position and consideration, ``back
  564. motion'' is as prevalent as time itself; and the husbands and sons in these
  565. households enjoy immunity at least from invisible attacks.
  566. Not that it must be for a moment supposed that our Women are destitute of
  567. affection. But unfortunately the passion of the moment predominates, in the
  568. Frail Sex, over every other consideration. This is, of course, a necessity
  569. arising from their unfortunate conformation. For as they have no pretensions
  570. to an angle, being inferior in this respect to the very lowest of the
  571. Isosceles, they are consequently wholly devoid of brainpower, and have neither
  572. reflection, judgment nor forethought, and hardly any memory. Hence, in their
  573. fits of fury, they remember no claims and recognize no distinctions. I have
  574. actually known a case where a Woman has exterminated her whole household, and
  575. half an hour afterwards, when her rage was over and the fragments swept away,
  576. has asked what has become of her husband and her children.
  577. Obviously then a Woman is not to be irritated as long as she is in a position
  578. where she can turn round. When you have them in their apartments --- which are
  579. constructed with a view to denying them that power --- you can say and do what
  580. you like; for they are then wholly impotent for mischief, and will not
  581. remember a few minutes hence the incident for which they may be at this moment
  582. threatening you with death, nor the promises which you may have found it
  583. necessary to make in order to pacify their fury.
  584. On the whole we got on pretty smoothly in our domestic
  585. relations, except in the lower strata of the Military Classes. There the want
  586. of tact and discretion on the part of the husbands produces at times
  587. indescribable disasters. Relying too much on the offensive weapons of their
  588. acute angles instead of the defensive organs of good sense and seasonable
  589. simulations, these reckless creatures too often neglect the prescribed
  590. construction of the women's apartments, or irritate their wives by ill-advised
  591. expressions out of doors, which they refuse immediately to retract. Moreover a
  592. blunt and stolid regard for literal truth indisposes them to make those lavish
  593. promises by which the more judicious Circle can in a moment pacify his
  594. consort. The result is massacre; not, however, without its advantages, as it
  595. eliminates the more brutal and troublesome of the Isosceles; and by many of
  596. our Circles the destructiveness of the Thinner Sex is regarded as one among
  597. many providential arrangements for suppressing redundant population, and
  598. nipping Revolution in the bud.
  599. Yet even in our best regulated and most approximately Circular families I
  600. cannot say that the ideal of family life is so high as with you in Spaceland.
  601. There is peace, in so far as the absence of slaughter may be called by that
  602. name, but there is necessarily little harmony of tastes or pursuits; and the
  603. cautious wisdom of the Circles has ensured safety at the cost of domestic
  604. comfort. In every Circular or Polygonal household it has been a habit from
  605. time immemorial --- and now has become a kind of instinct among the women of our
  606. higher classes --- that the mothers and daughters should constantly keep their
  607. eyes and mouths towards their husband and his male friends; and for a lady in
  608. a family of distinction to turn her back upon her husband would be regarded as
  609. a kind of portent, involving loss of \textit{status}. But, as I shall soon shew, this
  610. custom, though it has the advantage of safety, is not without disadvantages.
  611. Moving with ease and smoothness in uttering words; of rapid speech;
  612. nimble in speaking; glib; as, a flippant, voluble, tongue. In the house of
  613. the Working Man or respectable Tradesman --- where the wife is allowed to turn
  614. her back upon her husband, while pursuing her household avocations --- there are
  615. at least intervals of quiet, when the wife is neither seen nor heard, except
  616. for the humming sound of the continuous Peace-cry; but in the homes of the
  617. upper classes there is too often no peace. There the voluble mouth and bright
  618. penetrating eye are ever directed toward the Master of the household; and
  619. light itself is not more persistent than the stream of Feminine discourse. The
  620. tact and skill which suffice to avert a Woman's sting are unequal to the task
  621. of stopping a Woman's mouth; and as the wife has absolutely nothing to say,
  622. and absolutely no constraint of wit, sense, or conscience to prevent her from
  623. saying it, not a few cynics have been found to aver that they prefer the
  624. danger of the death-dealing but inaudible sting to the safe sonorousness of a
  625. Woman's other end.
  626. To my readers in Spaceland the condition of our Women may seen truly
  627. deplorable, and so indeed it is. A Male of the lowest type of the Isosceles
  628. may look forward to some improvement of his angle, and to the ultimate
  629. elevation of the whole of his degraded caste; but no Woman can entertain such
  630. hopes for her sex. ``Once a Woman, always a Woman'' is a Decree of Nature; and
  631. the very Laws of Evolution seem suspended in her disfavour. Yet at least we
  632. can admire the wise Prearrangement which has ordained that, as they have no
  633. hopes, so they shall have no memory to recall, and no forethought to
  634. anticipate, the miseries and humiliations which are at once a necessity of
  635. their existence and the basis of the constitution of Flatland.
  636. \chapter{Of our Methods of Recognizing one another}
  637. You, who are blessed with
  638. shade as well as light, you, who are gifted with two eyes, endowed with a
  639. knowledge of perspective, and charmed with the enjoyment of various colours,
  640. you, who can actually see an angle, and contemplate the complete circumference
  641. of a Circle in the happy region of the Three Dimensions --- how shall I make it
  642. clear to you the extreme difficulty which we in Flatland experience in
  643. recognizing one another's configuration?
  644. Recall what I told you above. All beings in Flatland, animate and inanimate,
  645. no matter what their form, present \textit{to our view} the same, or nearly the same,
  646. appearance, viz. that of a straight Line. How then can one be distinguished
  647. from another, where all appear the same?
  648. The answer is threefold. The first means of recognition is the sense of
  649. hearing; which with us is far more highly developed than with you, and which
  650. enables us not only to distinguish by the voice our personal friends, but even
  651. to discriminate between different classes, at least so far as concerns the
  652. three lowest orders, the Equilateral, the Square, and the Pentagon --- for the
  653. Isosceles I take no account. But as we ascend the social scale, the process of
  654. discriminating and being discriminated by hearing increases in difficulty,
  655. partly because voices are assimilated, partly because the faculty of
  656. voice-discrimination is a plebeian virtue not much developed among the
  657. Aristocracy. And wherever there is any danger of imposture we cannot trust to
  658. this method. Amongst our lowest orders, the vocal organs are developed to a
  659. degree more than correspondent with those of hearing, so that an Isosceles can
  660. easily feign the voice of a Polygon, and, with some training, that of a Circle
  661. himself. A second method is therefore more commonly resorted to.
  662. \textit{Feeling} is, among our Women and lower classes --- about our upper classes I
  663. shall speak presently --- the principal test of recognition, at all events
  664. between strangers, and when the question is, not as to the individual, but as
  665. to the class. What therefore ``introduction'' is among the higher classes in
  666. Spaceland, that the process of ``feeling'' is with us. ``Permit me to ask you to
  667. feel and be felt by my friend Mr. So-and-so'' --- is still, among the more
  668. old-fashioned of our country gentlemen in districts remote from towns, the
  669. customary formula for a Flatland introduction. But in the towns, and among men
  670. of business, the words ``be felt by'' are omitted and the sentence is
  671. abbreviated to, ``Let me ask you to feel Mr. So-and-so''; although it is
  672. assumed, of course, that the ``feeling'' is to be reciprocal. Among our still
  673. more modern and dashing young gentlemen --- who are extremely averse to
  674. superfluous effort and supremely indifferent to the purity of their native
  675. language --- the formula is still further curtailed by the use of ``to feel'' in a
  676. technical sense, meaning, ``to
  677. recommend-for-the-purposes-of-feeling-and-being-felt''; and at this moment the
  678. ``slang'' of polite or fast society in the upper classes sanctions such a
  679. barbarism as ``Mr. Smith, permit me to feel Mr. Jones.''.
  680. Let not my Reader however suppose that ``feeling'' is with us the tedious
  681. process that it would be with you, or that we find it necessary to feel right
  682. round all the sides of every individual before we determine the class to which
  683. he belongs. Long practice and training, begun in the schools and continued in
  684. the experience of daily life, enable us to discriminate at once by the sense
  685. of touch, between the angles of an equal-sided Triangle, Square, and Pentagon;
  686. and I need not say that the brainless vertex of an acute-angled Isosceles is
  687. obvious to the dullest touch. It is therefore not necessary, as a rule, to do
  688. more than feel a single angle of an individual; and this, once ascertained,
  689. tells us the class of the person whom we are addressing, unless indeed he
  690. belongs to the higher sections of the nobility. There the difficulty is much
  691. greater. Even a Master of Arts in our University of Wentbridge has been known
  692. to confuse a ten-sided with a twelve-sided Polygon; and there is hardly a
  693. Doctor of Science in or out of that famous University who could pretend to
  694. decide promptly and unhesitatingly between a twenty-sided and a twenty-four
  695. sided member of the Aristocracy.
  696. Those of my readers who recall the extracts I gave above from the Legislative
  697. code concerning Women, will readily perceive that the process of introduction
  698. by contact requires some care and discretion. Otherwise the angles might
  699. inflict on the unwary Feeler irreparable injury. It is essential for the
  700. safety of the Feeler that the Felt should stand perfectly still. A start, a
  701. fidgety shifting of the position, yes, even a violent sneeze, has been known
  702. before now to prove fatal to the incautious, and to nip in the bud many a
  703. promising friendship. Especially is this true among the lower classes of the
  704. Triangles. With them, the eye is situated so far from their vertex that they
  705. can scarcely take cognizance of what goes on at that extremity of their frame.
  706. They are, moreover, of a rough coarse nature, not sensitive to the delicate
  707. touch of the highly organized Polygon. What wonder then if an involuntary toss
  708. of the head has ere now deprived the State of a valuable life!
  709. I have heard that my excellent Grandfather --- one of the least irregular of his
  710. unhappy Isosceles class, who indeed obtained, shortly before his decease, four
  711. out of seven votes from the Sanitary and Social Board for passing him into the
  712. class of the Equal-sided --- often deplored, with a tear in his venerable eye, a
  713. miscarriage of this kind, which had occurred to his
  714. great-great-great-Grandfather, a respectable Working Man with an angle or
  715. brain of 59$^{\circ}$ 30$'$. According to his account, my unfortunate
  716. Ancestor, being afflicted with rheumatism, and in the act of being felt by a
  717. Polygon, by one sudden start accidentally transfixed the Great Man through the
  718. diagonal and thereby, partly in consequence of his long imprisonment and
  719. degradation, and partly because of the moral shock which pervaded the whole of
  720. my Ancestor's relations, threw back our family a degree and a half in their
  721. ascent towards better things. The result was that in the next generation the
  722. family brain was registered at only 58$^{\circ}$, and not till the lapse of five
  723. generations was the lost ground recovered, the full 60$^{\circ}$ attained, and
  724. the Ascent from the Isosceles finally achieved. And all this series of
  725. calamities from one little accident in the process of Feeling.
  726. At this point I think I hear some of my better educated readers exclaim, ``How
  727. could you in Flatland know anything about angles and degrees, or minutes? We
  728. see an angle, because we, in the region of Space, can see two straight lines
  729. inclined to one another; but you, who can see nothing but on straight line at
  730. a time, or at all events only a number of bits of straight lines all in one
  731. straight line, --- how can you ever discern any angle, and much less register
  732. angles of different sizes?''.
  733. I answer that though we cannot \textit{see} angles, we can \textit{infer} them, and this with
  734. great precision. Our sense of touch, stimulated by necessity, and developed by
  735. long training, enables us to distinguish angles far more accurately than your
  736. sense of sight, when unaided by a rule or measure of angles. Nor must I omit
  737. to explain that we have great natural helps. It is with us a Law of Nature
  738. that the brain of the Isosceles class shall begin at half a degree, or thirty
  739. minutes, and shall increase (if it increases at all) by half a degree in every
  740. generation until the goal of 60$^{\circ}$ is reached, when the condition of
  741. serfdom is quitted, and the freeman enters the class of Regulars.
  742. Consequently, Nature herself supplies us with an ascending scale or Alphabet
  743. of angles for half a degree up to 60$^{\circ}$, Specimens of which are placed in
  744. every Elementary School throughout the land. Owing to occasional
  745. retrogressions, to still more frequent moral and intellectual stagnation, and
  746. to the extraordinary fecundity of the Criminal and Vagabond classes, there is
  747. always a vast superfluity of individuals of the half degree and single degree
  748. class, and a fair abundance of Specimens up to 10$^{\circ}$. These are
  749. absolutely destitute of civic rights; and a great number of them, not having
  750. even intelligence enough for the purposes of warfare, are devoted by the
  751. States to the service of education. Fettered immovably so as to remove all
  752. possibility of danger, they are placed in the classrooms of our Infant
  753. Schools, and there they are utilized by the Board of Education for the purpose
  754. of imparting to the offspring of the Middle Classes that tact and intelligence
  755. of which these wretched creatures themselves are utterly devoid.
  756. In some States the Specimens are occasionally fed and suffered to exist for
  757. several years; but in the more temperate and better regulated regions, it is
  758. found in the long run more advantageous for the educational interests of the
  759. young, to dispense with food, and to renew the Specimens every month --- which
  760. is about the average duration of the foodless existence of the Criminal class.
  761. In the cheaper schools, what is gained by the longer existence of the Specimen
  762. is lost, partly in the expenditure for food, and partly in the diminished
  763. accuracy of the angles, which are impaired after a few weeks of constant
  764. ``feeling''. Nor must we forget to add, in enumerating the advantages of the
  765. more expensive system, that it tends, though slightly yet perceptibly, to the
  766. diminution of the redundant Isosceles population --- an object which every
  767. statesman in Flatland constantly keeps in view. On the whole therefore ---
  768. although I am not ignorant that, in many popularly elected School Boards,
  769. there is a reaction in favour of ``the cheap system'' as it is called --- I am
  770. myself disposed to think that this is one of the many cases in which expense
  771. is the truest economy.
  772. But I must not allow questions of School Board politics to divert me from my
  773. subject. Enough has been said, I trust, to shew that Recognition by feeling is
  774. not so tedious or indecisive a process as might have been supposed; and it is
  775. obviously more trustworthy than Recognition by hearing. Still there remains,
  776. as has been pointed out above, the objection that this method is not without
  777. danger. For this reason many in the Middle and Lower classes, and all without
  778. exception in the Polygonal and Circular orders, prefer a third method, the
  779. description of which shall be reserved for the next section.
  780. \chapter{Of Recognition by Sight}
  781. I am about to appear very inconsistent. In previous sections I have said that
  782. all figures in Flatland present the appearance of a straight line; and it was
  783. added or implied, that it is consequently impossible to distinguish by the
  784. visual organ between individuals of different classes: yet now I am about to
  785. explain to my Spaceland critics how we are able to recognize one another by
  786. the sense of sight.
  787. If however the Reader will take the trouble to refer to the passage in which
  788. Recognition by Feeling is stated to be universal, he will find this
  789. qualification --- ``among the lower classes''. It is only among the higher classes
  790. and in our more temperate climates that Sight Recognition is practised.
  791. That this power exists in any regions and for any classes is the result of
  792. Fog; which prevails during the greater part of the year in all parts save the
  793. torrid zones. That which is with you in Spaceland an unmixed evil, blotting
  794. out the landscape, depressing the spirits, and enfeebling the health, is by us
  795. recognized as a blessing scarcely inferior to air itself, and as the Nurse of
  796. arts and Parent of sciences. But let me explain my meaning, without further
  797. eulogies on this beneficent Element.
  798. If Fog were non-existent, all lines would appear equally and indistinguishably
  799. clear; and this is actually the case in those unhappy countries in which the
  800. atmosphere is perfectly dry and transparent. But wherever there is a rich
  801. supply of Fog, objects that are at a distance, say of three feet, are
  802. appreciably dimmer than those at the distance of two feet eleven inches; and
  803. the result is that by careful and constant experimental observation of
  804. comparative dimness and clearness, we are enabled to infer with great
  805. exactness the configuration of the object observed.
  806. An instance will do more than a volume of generalities to make my meaning
  807. clear.
  808. Suppose I see two individuals approaching whose rank I wish to ascertain. They
  809. are, we will suppose, a Merchant and a Physician, or in other words, an
  810. Equilateral Triangle and a Pentagon; how am I to distinguish them?
  811. \includegraphics[trim=20mm 0mm 0mm 0mm, width=\linewidth]{fig3}
  812. It will be obvious, to every child in Spaceland who has touched the threshold
  813. of Geometrical Studies, that, if I can bring my eye so that its glance may
  814. bisect an angle (\textsc{a}) of the approaching stranger, my view will lie as it were
  815. evenly between his two sides that are next to me (viz. \textsc{ca} and \textsc{ab}), so that I
  816. shall contemplate the two impartially, and both will appear of the same size.
  817. Now in the case of (1) the Merchant, what shall I see? I shall see a straight
  818. line \textsc{dae}, in which the middle point (\textsc{a}) will be very bright because it is
  819. nearest to me; but on either side the line will shade away \textit{rapidly to dimness},
  820. because the sides \textsc{ac} and \textsc{ab} \textit{recede rapidly into the fog} and what appear to me as the Merchant's extremities, viz. \textsc{d} and \textsc{e}, will be \textit{very dim indeed}.
  821. On the other hand in the case of (2) the Physician, though I shall here also
  822. see a line (\textsc{d}$'$\textsc{a}$'$\textsc{e}$'$) with a bright centre (\textsc{a}$'$), yet it will shade away \textit{less rapidly} to dimness, because the sides (\textsc{a}$'$\textsc{c}$'$, \textsc{a}$'$\textsc{b}$'$) \textit{recede less rapidly into
  823. the fog}; and what appear to me the Physician's extremities, viz. \textsc{d}$'$ and \textsc{e}$'$, will not be \textit{not so dim} as the extremities of the Merchant.
  824. The Reader will probably understand from these
  825. two instances how --- after a very long training supplemented by constant
  826. experience --- it is possible for the well-educated classes among us to
  827. discriminate with fair accuracy between the middle and lowest orders, by the
  828. sense of sight. If my Spaceland Patrons have grasped this general conception,
  829. so far as to conceive the possibility of it and not to reject my account as
  830. altogether incredible --- I shall have attained all I can reasonably expect.
  831. Were I to attempt further details I should only perplex. Yet for the sake of
  832. the young and inexperienced, who may perchance infer --- from the two simple
  833. instances I have given above, of the manner in which I should recognize my
  834. Father and my Sons --- that Recognition by sight is an easy affair, it may be
  835. needful to point out that in actual life most of the problems of Sight
  836. Recognition are far more subtle and complex.
  837. If for example, when my Father, the Triangle, approaches me, he happens to
  838. present his side to me instead of his angle, then, until I have asked him to
  839. rotate, or until I have edged my eye around him, I am for the moment doubtful
  840. whether he may not be a Straight Line, or, in other words, a Woman. Again,
  841. when I am in the company of one of my two hexagonal Grandsons, contemplating
  842. one of his sides (\textsc{ab}) full front, it will be evident from the accompanying
  843. diagram that I shall see one whole line (\textsc{ab}) in comparative brightness
  844. (shading off hardly at all at the ends) and two smaller lines (\textsc{ca} and \textsc{bd}) dim
  845. throughout and shading away into greater dimness towards the extremities \textsc{c} and
  846. \textsc{d}.
  847. \begin{center}
  848. \includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig4}
  849. \end{center}
  850. But I must not give way to the temptation of enlarging on these
  851. topics. The meanest mathematician in Spaceland will readily believe me when I
  852. assert that the problems of life, which present themselves to the
  853. well-educated --- when they are themselves in motion, rotating, advancing or
  854. retreating, and at the same time attempting to discriminate by the sense of
  855. sight between a number of Polygons of high rank moving in different
  856. directions, as for example in a ball-room or conversazione --- must be of a
  857. nature to task the angularity of the most intellectual, and amply justify the
  858. rich endowments of the Learned Professors of Geometry, both Static and
  859. Kinetic, in the illustrious University of Wentbridge, where the Science and
  860. Art of Sight Recognition are regularly taught to large classes of the \textit{élite} of
  861. the States.
  862. 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
  863. me, a Mathematician of no mean standing, and the Grandfather of two most
  864. hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd
  865. of rotating Polygons of the higher classes, is occasionally very perplexing.
  866. And of course to a common Tradesman, or Serf, such a sight is almost as
  867. unintelligible as it would be to you, my Reader, were you suddenly transported
  868. to our country.
  869. In such a crowd you could see on all sides of you nothing but a Line,
  870. apparently straight, but of which the parts would vary irregularly and
  871. perpetually in brightness or dimness. Even if you had completed your third
  872. year in the Pentagonal and Hexagonal classes in the University, and were
  873. perfect in the theory of the subject, you would still find there was need of
  874. many years of experience, before you could move in a fashionable crowd without
  875. jostling against your betters, whom it is against etiquette to ask to ``feel'',
  876. and who, by their superior culture and breeding, know all about your
  877. movements, while you know very little or nothing about theirs. In a word, to
  878. comport oneself with perfect propriety in Polygonal society, one ought to be a
  879. Polygon oneself. Such at least is the painful teaching of my experience.
  880. It is astonishing how much the Art --- or I may almost
  881. call it instinct --- of Sight Recognition is developed by the habitual practice
  882. of it and by the avoidance of the custom of ``Feeling''. Just as, with you, the
  883. deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet,
  884. will never acquire the more difficult but far more valuable art of lip-speech
  885. and lip-reading, so it is with us as regards ``Seeing'' and ``Feeling''. None who
  886. in early life resort to ``Feeling'' will ever learn ``Seeing'' in perfection.
  887. For this reason, among our Higher Classes, ``Feeling'' is discouraged or
  888. absolutely forbidden. From the cradle their children, instead of going to the
  889. Public Elementary schools (where the art of Feeling is taught,) are sent to
  890. higher Seminaries of an exclusive character; and at our illustrious
  891. University, to ``feel'' is regarded as a most serious fault, involving
  892. Rustication for the first offence, and Expulsion for the second.
  893. But among the lower classes the art of Sight Recognition is regarded as an
  894. unattainable luxury. A common Tradesman cannot afford to let his son spend a
  895. third of his life in abstract studies. The children of the poor are therefore
  896. allowed to ``feel'' from their earliest years, and they gain thereby a precocity
  897. and an early vivacity which contrast at first most favourably with the inert,
  898. undeveloped, and listless behaviour of the half-instructed youths of the
  899. Polygonal class; but when the latter have at last completed their University
  900. course, and are prepared to put their theory into practice, the change that
  901. comes over them may almost be described as a new birth, and in every art,
  902. science, and social pursuit they rapidly overtake and distance their
  903. Triangular competitors.
  904. Only a few of the Polygonal Class fail to pass the Final Test or Leaving
  905. Examination at the University. The condition of the unsuccessful minority is
  906. truly pitiable. Rejected from the higher class, they are also despised by the
  907. lower. They have neither the matured and systematically trained powers of the
  908. Polygonal Bachelors and Masters of Arts, nor yet the native precocity and
  909. mercurial versatility of the youthful Tradesman. The professions, the public
  910. services, are closed against them, and though in most States they are not
  911. actually debarred from marriage, yet they have the greatest difficulty in
  912. forming suitable alliances, as experience shews that the offspring of such
  913. unfortunate and ill-endowed parents is generally itself unfortunate, if not
  914. positively Irregular.
  915. It is from these specimens of the refuse of our Nobility that the great
  916. Tumults and Seditions of past ages have generally derived their leaders; and
  917. so great is the mischief thence arising that an increasing minority of our
  918. more progressive Statesmen are of opinion that true mercy would dictate their
  919. entire suppression, by enacting that all who fail to pass the Final
  920. Examination of the University should be either imprisoned for life, or
  921. extinguished by a painless death.
  922. But I find myself digressing into the subject of Irregularities, a matter of
  923. such vital interest that it demands a separate section.
  924. \chapter{Concerning Irregular Figures}
  925. Throughout the previous pages I have been assuming --- what perhaps should have
  926. been laid down at the beginning as a distinct and fundamental proposition ---
  927. that every human being in Flatland is a Regular Figure, that is to say of
  928. regular construction. By this I mean that a Woman must not only be a line, but
  929. a straight line; that an Artisan or Soldier must have two of his sides equal;
  930. that Tradesmen must have three sides equal; Lawyers (of which class I am a
  931. humble member), four sides equal, and, generally, that in every Polygon, all
  932. the sides must be equal.
  933. The sizes of the sides would of course depend upon the age of the individual.
  934. A Female at birth would be about an inch long, while a tall adult Woman might
  935. extend to a foot. As to the Males of every class, it may be roughly said that
  936. the length of an adult's size, when added together, is two feet or a little
  937. more. But the size of our sides is not under consideration. I am speaking of
  938. the equality of sides, and it does not need much reflection to see that the
  939. whole of the social life in Flatland rests upon the fundamental fact that
  940. Nature wills all Figures to have their sides equal.
  941. If our sides were unequal our angles might be unequal. Instead of its being
  942. sufficient to feel, or estimate by sight, a single angle in order to determine
  943. the form of an individual, it would be necessary to ascertain each angle by
  944. the experiment of Feeling. But life would be too short for such a tedious
  945. groping. The whole science and art of Sight Recognition would at once perish;
  946. Feeling, so far as it is an art, would not long survive; intercourse would
  947. become perilous or impossible; there would be an end to all confidence, all
  948. forethought; no one would be safe in making the most simple social
  949. arrangements; in a word, civilization might relapse into barbarism.
  950. Am I going too fast to carry my Readers with me to these obvious conclusions?
  951. Surely a moment's reflection, and a single instance from common life, must
  952. convince every one that our social system is based upon Regularity, or
  953. Equality of Angles. You meet, for example, two or three Tradesmen in the
  954. street, whom your recognize at once to be Tradesman by a glance at their
  955. angles and rapidly bedimmed sides, and you ask them to step into your house to
  956. lunch. This you do at present with perfect confidence, because everyone knows
  957. to an inch or two the area occupied by an adult Triangle: but imagine that
  958. your Tradesman drags behind his regular and respectable vertex, a
  959. parallelogram of twelve or thirteen inches in diagonal: --- what are you to do
  960. with such a monster sticking fast in your house door?
  961. But I am insulting the intelligence of my Readers by accumulating details
  962. which must be patent to everyone who enjoys the advantages of a Residence in
  963. Spaceland. Obviously the measurements of a single angle would no longer be
  964. sufficient under such portentous circumstances; one's whole life would be
  965. taken up in feeling or surveying the perimeter of one's acquaintances. Already
  966. the difficulties of avoiding a collision in a crowd are enough to tax the
  967. sagacity of even a well-educated Square; but if no one could calculate the
  968. Regularity of a single figure in the company, all would be chaos and
  969. confusion, and the slightest panic would cause serious injuries, or --- if there
  970. happened to be any Women or Soldiers present --- perhaps considerable loss of
  971. life.
  972. Expediency therefore concurs with Nature in stamping the seal of its approval
  973. upon Regularity of conformation: nor has the Law been backward in seconding
  974. their efforts. ``Irregularity of Figure'' means with us the same as, or more
  975. than, a combination of moral obliquity and criminality with you, and is
  976. treated accordingly. There are not wanting, it is true, some promulgators of
  977. paradoxes who maintain that there is no necessary connection between
  978. geometrical and moral Irregularity. ``The Irregular,'' they say, ``is from his
  979. birth scouted by his own parents, derided by his brothers and sisters,
  980. neglected by the domestics, scorned and suspected by society, and excluded
  981. from all posts of responsibility, trust, and useful activity. His every
  982. movement is jealously watched by the police till he comes of age and presents
  983. himself for inspection; then he is either destroyed, if he is found to exceed
  984. the fixed margin of deviation, at an uninteresting occupation for a miserable
  985. stipend; obliged to live and board at the office, and to take even his
  986. vacation under close supervision; what wonder that human nature, even in the
  987. best and purest, is embittered and perverted by such surroundings!''
  988. All this very plausible reasoning does not convince me, as it has not
  989. convinced the wisest of our Statesmen, that our ancestors erred in laying it
  990. down as an axiom of policy that the toleration of Irregularity is incompatible
  991. with the safety of the State. Doubtless, the life of an Irregular is hard; but
  992. the interests of the Greater Number require that it shall be hard. If a man
  993. with a triangular front and a polygonal back were allowed to exist and to
  994. propagate a still more Irregular posterity, what would become of the arts of
  995. life? Are the houses and doors and churches in Flatland to be altered in order
  996. to accommodate such monsters? Are our ticket-collectors to be required to
  997. measure every man's perimeter before they allow him to enter a theatre, or to
  998. take his place in a lecture room? Is an Irregular to be exempted from the
  999. militia? And if not, how is he to be prevented from carrying desolation into
  1000. the ranks of his comrades? Again, what irresistible temptations to fraudulent
  1001. impostures must needs beset such a creature! How easy for him to enter a shop
  1002. with his polygonal front foremost, and to order goods to any extent from a
  1003. confiding tradesman! Let the advocates of a falsely called Philanthropy plead
  1004. as they may for the abrogation of the Irregular Penal Laws, I for my part have
  1005. never known an Irregular who was not also what Nature evidently intended him
  1006. to be --- a hypocrite, a misanthropist, and, up to the limits of his power, a
  1007. perpetrator of all manner of mischief.
  1008. Not that I should be disposed to recommend (at present) the extreme measures
  1009. adopted by some States, where an infant whose angle deviates by half a degree
  1010. from the correct angularity is summarily destroyed at birth. Some of our
  1011. highest and ablest men, men of real genius, have during their earliest days
  1012. laboured under deviations as great as, or even greater than forty-five
  1013. minutes: and the loss of their precious lives would have been an irreparable
  1014. injury to the State. The art of healing also has achieved some of its most
  1015. glorious triumphs in the compressions, extensions, trepannings, colligations,
  1016. and other surgical or diaetetic operations by which Irregularity has been
  1017. partly or wholly cured. Advocating therefore a \textit{Via Media}, I would lay down no
  1018. fixed or absolute line of demarcation; but at the period when the frame is
  1019. just beginning to set, and when the Medical Board has reported that recovery
  1020. is improbably, I would suggest that the Irregular offspring be painlessly and
  1021. mercifully consumed.
  1022. \chapter{Of the Ancient Practice of Painting}
  1023. If my Readers have followed me with any attention up to this point, they will
  1024. not be surprised to hear that life is somewhat dull in Flatland. I do not, of
  1025. course, mean that there are not battles, conspiracies, tumults, factions, and
  1026. all those other phenomena which are supposed to make History interesting; nor
  1027. would I deny that the strange mixture of the problems of life and the problems
  1028. of Mathematics, continually inducing conjecture and giving an opportunity of
  1029. immediate verification, imparts to our existence a zest which you in Spaceland
  1030. can hardly comprehend. I speak now from the aesthetic and artistic point of
  1031. view when I say that life with us is dull; aesthetically and artistically,
  1032. very dull indeed.
  1033. How can it be otherwise, when all one's prospect, all one's landscapes,
  1034. historical pieces, portraits, flowers, still life, are nothing but a single
  1035. line, with no varieties except degrees of brightness and obscurity?
  1036. It was not always thus. Colour, if Tradition speaks the truth, once for the
  1037. space of half a dozen centuries or more, threw a transient splendour over the
  1038. lives of our ancestors in the remotest ages. Some private individual --- a
  1039. Pentagon whose name is variously reported --- having casually discovered the
  1040. constituents of the simpler colours and a rudimentary method of painting, is
  1041. said to have begun by decorating first his house, then his slaves, then his
  1042. Father, his Sons, and Grandsons, lastly himself. The convenience as well as
  1043. the beauty of the results commended themselves to all. Wherever Chromatistes,
  1044. --- for by that name the most trustworthy authorities concur in calling him, ---
  1045. turned his variegated frame, there he at once excited attention, and attracted
  1046. respect. No one now needed to ``feel'' him; no one mistook his front for his
  1047. back; all his movements were readily ascertained by his neighbours without the
  1048. slightest strain on their powers of calculation; no one jostled him, or failed
  1049. to make way for him; his voice was saved the labour of that exhausting
  1050. utterance by which we colourless Squares and Pentagons are often forced to
  1051. proclaim our individuality when we move amid a crowd of ignorant Isosceles.
  1052. The fashion spread like wildfire. Before a week was over, every Square and
  1053. Triangle in the district had copied the example of Chromatistes, and only a
  1054. few of the more conservative Pentagons still held out. A month or two found
  1055. even the Dodecagons infected with the innovation. A year had not elapsed
  1056. before the habit had spread to all but the very highest of the Nobility.
  1057. Needless to say, the custom soon made its way from the district of
  1058. Chromatistes to surrounding regions; and within two generations no one in all
  1059. Flatland was colourless except the Women and the Priests.
  1060. Here Nature herself appeared to erect a barrier, and to plead against
  1061. extending the innovations to these two classes. Many-sidedness was almost
  1062. essential as a pretext for the Innovators. ``Distinction of sides is intended
  1063. by Nature to imply distinction of colours'' --- such was the sophism which in
  1064. those days flew from mouth to mouth, converting whole towns at a time to a new
  1065. culture. But manifestly to our Priests and Women this adage did not apply. The
  1066. latter had only one side, and therefore --- plurally and pedantically speaking ---
  1067. \textit{no sides}. The former --- if at least they would assert their claim to be readily
  1068. and truly Circles, and not mere high-class Polygons, with an infinitely large
  1069. number of infinitesimally small sides --- were in the habit of boasting (what
  1070. Women confessed and deplored) that they also had no sides, being blessed with
  1071. a perimeter of only one line, or, in other words, a Circumference. Hence it
  1072. came to pass that these two Classes could see no force in the so-called axiom
  1073. about ``Distinction of Sides implying Distinction of Colour;'' and when all
  1074. others had succumbed to the fascinations of corporal decoration, the Priests
  1075. and the Women alone still remained pure from the pollution of paint.
  1076. Immoral, licentious, anarchical, unscientific --- call them by what names you
  1077. will --- yet, from an aesthetic point of view, those ancient days of the Colour
  1078. Revolt were the glorious childhood of Art in Flatland --- a childhood, alas,
  1079. that never ripened into manhood, nor even reached the blossom of youth. To
  1080. live then in itself a delight, because living implied seeing. Even at a small
  1081. party, the company was a pleasure to behold; the richly varied hues of the
  1082. assembly in a church or theatre are said to have more than once proved too
  1083. distracting from our greatest teachers and actors; but most ravishing of all
  1084. is said to have been the unspeakable magnificence of a military review.
  1085. The sight of a line of battle of twenty thousand
  1086. Isosceles suddenly facing about, and exchanging the sombre black of their
  1087. bases for the orange of the two sides including their acute angle; the militia
  1088. of the Equilateral Triangles tricoloured in red, white, and blue; the mauve,
  1089. ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly
  1090. rotating near their vermillion guns; the dashing and flashing of the
  1091. five-coloured and six-coloured Pentagons and Hexagons careering across the
  1092. field in their offices of surgeons, geometricians and aides-de-camp --- all
  1093. these may well have been sufficient to render credible the famous story how an
  1094. illustrious Circle, overcome by the artistic beauty of the forces under his
  1095. command, threw aside his marshal's baton and his royal crown, exclaiming that
  1096. he henceforth exchanged them for the artist's pencil. How great and glorious
  1097. the sensuous development of these days must have been is in part indicated by
  1098. the very language and vocabulary of the period. The commonest utterances of
  1099. the commonest citizens in the time of the Colour Revolt seem to have been
  1100. suffused with a richer tinge of word or thought; and to that era we are even
  1101. now indebted for our finest poetry and for whatever rhythm still remains in
  1102. the more scientific utterance of those modern days.
  1103. \chapter{Of the Universal Colour Bill}
  1104. But meanwhile the intellectual Arts were fast decaying.
  1105. The Art of Sight Recognition, being no longer needed, was no longer practised;
  1106. and the studies of Geometry, Statics, Kinetics, and other kindred subjects,
  1107. came soon to be considered superfluous, and fell into disrespect and neglect
  1108. even at our University. The inferior Art of Feeling speedily experienced the
  1109. same fate at our Elementary Schools. Then the Isosceles classes, asserting
  1110. that the Specimens were no longer used nor needed, and refusing to pay the
  1111. customary tribute from the Criminal classes to the service of Education, waxed
  1112. daily more numerous and more insolent on the strength of their immunity from
  1113. the old burden which had formerly exercised the twofold wholesome effect of at
  1114. once taming their brutal nature and thinning their excessive numbers.
  1115. Year by year the Soldiers and Artisans began more vehemently to assert --- and
  1116. with increasing truth --- that there was no great difference between them and
  1117. the very highest class of Polygons, now that they were raised to an equality
  1118. with the latter, and enabled to grapple with all the difficulties and solve
  1119. all the problems of life, whether Statical or Kinetical, by the simple process
  1120. of Colour Recognition. Not content with the natural neglect into which Sight
  1121. Recognition was falling, they began boldly to demand the legal prohibition of
  1122. all ``monopolizing and aristocratic Arts'' and the consequent abolition of all
  1123. endowments for the studies of Sight Recognition, Mathematics, and Feeling.
  1124. Soon, they began to insist that inasmuch as Colour, which was a second Nature,
  1125. had destroyed the need of aristocratic distinctions, the Law should follow in
  1126. the same path, and that henceforth all individuals and all classes should be
  1127. recognized as absolutely equal and entitled to equal rights.
  1128. Finding the higher Orders wavering and undecided, the leaders of the
  1129. Revolution advanced still further in their requirements, and at last demanded
  1130. that all classes alike, the Priests and the Women not excepted, should do
  1131. homage to Colour by submitting to be painted. When it was objected that
  1132. Priests and Women had no sides, they retorted that Nature and Expediency
  1133. concurred in dictating that the front half of every human being (that is to
  1134. say, the half containing his eye and mouth) should be distinguishable from his
  1135. hinder half. They therefore brought before a general and extraordinary
  1136. Assembly of all the States of Flatland a Bill proposing that in every Woman
  1137. the half containing the eye and mouth should be coloured red, and the other
  1138. half green. The Priests were to be painted in the same way, red being applied
  1139. to that semicircle in which the eye and mouth formed the middle point; while
  1140. the other or hinder semicircle was to be coloured green.
  1141. There was no little cunning in this proposal, which indeed emanated not from
  1142. any Isosceles --- for no being so degraded would have angularity enough to
  1143. appreciate, much less to devise, such a model of state-craft --- but from an
  1144. Irregular Circle who, instead of being destroyed in his childhood, was
  1145. reserved by a foolish indulgence to bring desolation on his country and
  1146. destruction on myriads of followers.
  1147. On the one hand the proposition was calculated to bring the Women in all
  1148. classes over to the side of the Chromatic Innovation. For by assigning to the
  1149. Women the same two colours as were assigned to the Priests, the Revolutionists
  1150. thereby ensured that, in certain positions, every Woman would appear as a
  1151. Priest, and be treated with corresponding respect and deference --- a prospect
  1152. that could not fail to attract the Female Sex in a mass.
  1153. But by some of my Readers the possibility of the identical appearance of
  1154. Priests and Women, under a new Legislation, may not be recognized; if so, a
  1155. word or two will make it obvious.
  1156. Imagine a woman duly decorated, according to the new Code; with the front half
  1157. (\textit{i.e.}, the half containing the eye and mouth) red, and with the hinder half
  1158. green. Look at her from one side. Obviously you will see a straight line, \textit{half
  1159. red}, \textit{half green}.
  1160. \begin{center}
  1161. \includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig5}
  1162. \end{center}
  1163. Now imagine a Priest, whose mouth is at M, and whose front semicircle (\textsc{amb}) is
  1164. consequently coloured red, while his hinder semicircle is green; so that the
  1165. diameter AB divides the green from the red. If you contemplate the Great Man
  1166. so as to have your eye in the same straight line as his dividing diameter
  1167. (\textsc{ab}), what you will see will be a straight line (\textsc{cbd}), of which \textit{one half} (\textsc{cb})
  1168. \textit{will be red}, \textit{and the other} (\textsc{bd}) \textit{green}. The whole line (\textsc{cd}) will be rather
  1169. shorter perhaps than that of a full-sized Woman, and will shade off more
  1170. rapidly towards its extremities; but the identity of the colours would give
  1171. you an immediate impression of identity in Class, making you neglectful of
  1172. other details. Bear in mind the decay of Sight Recognition which threatened
  1173. society at the time of the Colour revolt; add too the certainty that Woman
  1174. would speedily learn to shade off their extremities so as to imitate the
  1175. Circles; it must then be surely obvious to you, my dear Reader, that the
  1176. Colour Bill placed us under a great danger of confounding a Priest with a
  1177. young Woman.
  1178. How attractive this prospect must have been to the Frail Sex may readily be
  1179. imagined. They anticipated with delight the confusion that would ensue. At
  1180. home they might hear political and ecclesiastical secrets intended not for
  1181. them but for their husbands and brothers, and might even issue some commands
  1182. in the name of a priestly Circle; out of doors the striking combination of red
  1183. and green without addition of any other col\-ours, would be sure to lead the
  1184. common people into endless mistakes, and the Woman would gain whatever the
  1185. Circles lost, in the deference of the passers by. As for the scandal that
  1186. would befall the Circular Class if the frivolous and unseemly conduct of the
  1187. Women were imputed to them, and as to the consequent subversion of the
  1188. Constitution, the Female Sex could not be expected to give a thought to these
  1189. considerations. Even in the households of the Circles, the Women were all in
  1190. favour of the Universal Colour Bill.
  1191. The second object aimed at by the Bill was the gradual demoralization of the
  1192. Circles themselves. In the general intellectual decay they still preserved
  1193. their pristine clearness and strength of understanding. From their earliest
  1194. childhood, familiarized in their Circular households with the total absence of
  1195. Colour, the Nobles alone preserved the Sacred Art of Sight Recognition, with
  1196. all the advantages that result from that admirable training of the intellect.
  1197. Hence, up to the date of the introduction of the Universal Colour Bill, the
  1198. Circles had not only held their own, but even increased their lead of the
  1199. other classes by abstinence from the popular fashion.
  1200. Now therefore the artful Irregular whom I described above as the real author
  1201. of this diabolical Bill, determined at one blow to lower the status of the
  1202. Hierarchy by forcing them to submit to the pollution of Colour, and at the
  1203. same time to destroy their domestic opportunities of training in the Art of
  1204. Sight Recognition, so as to enfeeble their intellects by depriving them of
  1205. their pure and colourless homes. Once subjected to the chromatic taint, every
  1206. parental and every childish Circle would demoralize each other. Only in
  1207. discerning between the Father and the Mother would the Circular infant find
  1208. problems for the exercise of his understanding --- problems too often likely to
  1209. be corrupted by maternal impostures with the result of shaking the child's
  1210. faith in all logical conclusions. Thus by degrees the intellectual lustre of
  1211. the Priestly Order would wane, and the road would then lie open for a total
  1212. destruction of all Aristocratic Legislature and for the subversion of our
  1213. Privileged Classes.
  1214. \chapter{Of the Suppression of the Chromatic Sedition}
  1215. The agitation for the Universal Colour Bill continued for three years; and up
  1216. to the last moment of that period it seemed as though Anarchy were destined to
  1217. triumph.
  1218. A whole army of Polygons, who turned out to fight as private soldiers, was
  1219. utterly annihilated by a superior force of Isosceles Triangles --- the Squares
  1220. and Pentagons meanwhile remaining neutral. Worse than all, some of the ablest
  1221. Circles fell a prey to conjugal fury. Infuriated by political animosity, the
  1222. wives in many a noble household wearied their lords with prayers to give up
  1223. their opposition to the Colour Bill; and some, finding their entreaties
  1224. fruitless, fell on and slaughtered their innocent children and husband,
  1225. perishing themselves in the act of carnage. It is recorded that during that
  1226. triennial agitation no less than twenty-three Circles perished in domestic
  1227. discord.
  1228. Great indeed was the peril. It seemed as though the Priests had no choice
  1229. between submission and extermination; when suddenly the course of events was
  1230. completely changed by one of those picturesque incidents which Statesmen ought
  1231. never to neglect, often to anticipate, and sometimes perhaps to originate,
  1232. because of the absurdly disproportionate power with which they appeal to the
  1233. sympathies of the populace.
  1234. It happened that an Isosceles of a low type, with a brain little if at all
  1235. above four degrees --- accidentally dabbling in the colours of some Tradesman
  1236. whose shop he had plundered --- painted himself, or caused himself to be painted
  1237. (for the story varies) with the twelve colours of a Dodecagon. Going into the
  1238. Market Place he accosted in a feigned voice a maiden, the orphan daughter of a
  1239. noble Polygon, whose affection in former days he had sought in vain; and by a
  1240. series of deceptions --- aided, on the one side, by a string of lucky accidents
  1241. too long to relate, and, on the other, by an almost inconceivable fatuity and
  1242. neglect of ordinary precautions on the part of the relations of the bride --- he
  1243. succeeded in consummating the marriage. The unhappy girl committed suicide on
  1244. discovering the fraud to which she had been subjected.
  1245. When the news of this catastrophe spread from State to State the minds of the
  1246. Women were violently agitated. Sympathy with the miserable victim and
  1247. anticipations of similar deceptions for themselves, their sisters, and their
  1248. daughters, made them now regard the Colour Bill in an entirely new aspect. Not
  1249. a few openly avowed themselves converted to antagonism; the rest needed only a
  1250. slight stimulus to make a similar avowal. Seizing this favourable opportunity,
  1251. the Circles hastily convened an extraordinary Assembly of the States; and
  1252. besides the usual guard of Convicts, they secured the attendance of a large
  1253. number of reactionary Women.
  1254. Amidst an unprecedented concourse, the Chief Circle of those days --- by name
  1255. Pantocyclus --- arose to find himself hissed and hooted by a hundred and twenty
  1256. thousand Isosceles. But he secured silence by declaring that henceforth the
  1257. Circles would enter on a policy of Concession; yielding to the wishes of the
  1258. majority, they would accept the Colour Bill. The uproar being at once
  1259. converted to applause, he invited Chromatistes, the leader of the Sedition,
  1260. into the centre of the hall, to receive in the name of his followers the
  1261. submission of the Hierarchy. Then followed a speech, a masterpiece of
  1262. rhetoric, which occupied nearly a day in the delivery, and to which no summary
  1263. can do justice.
  1264. With a grave appearance of impartiality he declared that as they were now
  1265. finally committing themselves to Reform or Innovation, it was desirable that
  1266. they should take one last view of the perimeter of the whole subject, its
  1267. defects as well as its advantages. Gradually introduction the mention of the
  1268. dangers to the Tradesmen, the Professional Classes and the Gentlemen, he
  1269. silenced the rising murmurs of the Isosceles by reminding them that, in spite
  1270. of all these defects, he was willing to accept the Bill if it was approved by
  1271. the majority. But it was manifest that all, except the Isosceles, were moved
  1272. by his words and were either neutral or averse to the Bill.
  1273. Turning now to the Workmen he asserted that their interests must not be
  1274. neglected, and that, if they intended to accept the Colour Bill, they ought at
  1275. least to do so with full view of the consequences. Many of them, he said, were
  1276. on the point of being admitted to the class of the Regular Triangles; others
  1277. anticipated for their children a distinction they could not hope for
  1278. themselves. That honourable ambition would now have to be sacrificed. With the
  1279. universal adoption of Colour, all distinctions would cease; Regularity would
  1280. be confused with Irregularity; development would give place to retrogression;
  1281. the Workman would in a few generations be degraded to the level of the
  1282. Military, or even the Convict Class; political power would be in the hands of
  1283. the greatest number, that is to say the Criminal Classes, who were already
  1284. more numerous than the Workmen, and would soon out-number all the other
  1285. Classes put together when the usual Compensative Laws of Nature were violated.
  1286. A subdued murmur of assent ran through the ranks of the Artisans, and
  1287. Chromatistes, in alarm, attempted to step forward and address them. But he
  1288. found himself encompassed with guards and forced to remain silent while the
  1289. Chief Circle in a few impassioned words made a final appeal to the Women,
  1290. exclaiming that, if the Colour Bill passed, no marriage would henceforth be
  1291. safe, no woman's honour secure; fraud, deception, hypocrisy would pervade
  1292. every household; domestic bliss would share the fate of the Constitution and
  1293. pass to speedy perdition. ``Sooner than this'', he cried, ``Come death''.
  1294. At these words, which were the preconcerted signal for action, the Isosceles
  1295. Convicts fell on and transfixed the wretched Chromatistes; the Regular
  1296. Classes, opening their ranks, made way for a band of Women who, under
  1297. direction of the Circles, moved back foremost, invisibly and unerringly upon
  1298. the unconscious soldiers; the Artisans, imitating the example of their
  1299. betters, also opened their ranks. Meantime bands of Convicts occupied every
  1300. entrance with an impenetrable phalanx.
  1301. The battle, or rather carnage, was of short duration. Under the skillful
  1302. generalship of the Circles almost every Woman's charge was fatal and very many
  1303. extracted their sting uninjured, ready for a second slaughter. But no second
  1304. blow was needed; the rabble of the Isosceles did the rest of the business for
  1305. themselves. Surprised, leader-less, attacked in front by invisible foes, and
  1306. finding egress cut off by the Convicts behind them, they at once --- after their
  1307. manner --- lost all presence of mind, and raised the cry of ``treachery''. This
  1308. sealed their fate. Every Isosceles now saw and felt a foe in every other. In
  1309. half an hour not one of that vast multitude was living; and the fragments of
  1310. seven score thousand of the Criminal Class slain by one another's angles
  1311. attested the triumph of Order.
  1312. The Circles delayed not to push their victory to the uttermost. The Working
  1313. Men they spared but decimated. The Militia of the Equilaterals was at once
  1314. called out, and every Triangle suspected of Irregularity on reasonable
  1315. grounds, was destroyed by Court Martial, without the formality of exact
  1316. measurement by the Social Board. The homes of the Military and Artisan classes
  1317. were inspected in a course of visitation extending through upwards of a year;
  1318. and during that period every town, village, and hamlet was systematically
  1319. purged of that excess of the lower orders which had been brought about by the
  1320. neglect to pay the tribute of Criminals to the Schools and University, and by
  1321. the violation of other natural Laws of the Constitution of Flatland. Thus the
  1322. balance of classes was again restored.
  1323. Needless to say that henceforth the use of Colour was abolished, and its
  1324. possession prohibited. Even the utterance of any word denoting Colour, except
  1325. by the Circles or by qualified scientific teachers, was punished by a severe
  1326. penalty. Only at our University in some of the very highest and most esoteric
  1327. classes --- which I myself have never been privileged to attend --- it is
  1328. understood that the sparing use of Colour is still sanctioned for the purpose
  1329. of illustrating some of the deeper problems of mathematics. But of this I can
  1330. only speak from hearsay.
  1331. Elsewhere in Flatland, Colour is now non-existent. The art of making it is
  1332. known to only one living person, the Chief Circle for the time being; and by
  1333. him it is handed down on his death-bed to none but his Successor. One
  1334. manufactory alone produces it; and, lest the secret should be betrayed, the
  1335. Workmen are annually consumed, and fresh ones introduced. So great is the
  1336. terror with which even now our Aristocracy looks back to the far-distant days
  1337. of the agitation for the Universal Colour Bill.
  1338. \chapter{Concerning our Priests}
  1339. It is high time that I should pass from these brief and discursive notes about
  1340. things in Flatland to the central event of this book, my initiation into the
  1341. mysteries of Space. \textit{That} is my subject; all that has gone before is merely
  1342. preface.
  1343. For this reason I must omit many matters of which the explanation would not, I
  1344. flatter myself, be without interest for my Readers: as for example, our method
  1345. of propelling and stopping ourselves, although destitute of feet; the means by
  1346. which we give fixity to structures of wood, stone, or brick, although of
  1347. course we have no hands, nor can we lay foundations as you can, nor avail
  1348. ourselves of the lateral pressure of the earth; the manner in which the rain
  1349. originates in the intervals between our various zones, so that the northern
  1350. regions do not intercept the moisture falling on the southern; the nature of
  1351. our hills and mines, our trees and vegetables, our seasons and harvests; our
  1352. Alphabet and method of writing, adapted to our linear tablets; these and a
  1353. hundred other details of our physical existence I must pass over, nor do I
  1354. mention them now except to indicate to my readers that their omission proceeds
  1355. not from forgetfulness on the part of the author, but from his regard for the
  1356. time of the Reader.
  1357. Yet before I proceed to my legitimate subject some few final remarks will no
  1358. doubt be expected by my Readers upon these pillars and mainstays of the
  1359. Constitution of Flatland, the controllers of our conduct and shapers of our
  1360. destiny, the objects of universal homage and almost of adoration: need I say
  1361. that I mean our Circles or Priests?
  1362. When I call them Priests, let me not be understood as meaning no more than the
  1363. term denotes with you. With us, our Priests are Administrators of all
  1364. Business, Art, and Science; Directors of Trade, Commerce, Generalship,
  1365. Architecture, Engineering, Education, Statesmanship, Legislature, Morality,
  1366. Theology; doing nothing themselves, they are the Causes of everything worth
  1367. doing, that is done by others.
  1368. Although popularly everyone called a Circle is deemed a Circle, yet among the
  1369. better educated Classes it is known that no Circle is really a Circle, but
  1370. only a Polygon with a very large number of very small sides. As the number of
  1371. the sides increases, a Polygon approximates to a Circle; and, when the number
  1372. is very great indeed, say for example three or four hundred, it is extremely
  1373. difficult for the most delicate touch to feel any polygonal angles. Let me say
  1374. rather it \textit{would} be difficult: for, as I have shown above, Recognition by
  1375. Feeling is unknown among the highest society, and to \textit{feel} a Circle would be
  1376. considered a most audacious insult. This habit of abstention from Feeling in
  1377. the best society enables a Circle the more easily to sustain the veil of
  1378. mystery in which, from his earliest years, he is wont to enwrap the exact
  1379. nature of his Perimeter or Circumference. Three feet being the average
  1380. Perimeter it follows that, in a Polygon of three hundred sides each side will
  1381. be no more than the hundredth part of a foot in length, or little more than
  1382. the tenth part of an inch; and in a Polygon of six or seven hundred sides the
  1383. sides are little larger than the diameter of a Spaceland pin-head. It is
  1384. always assumed, by courtesy, that the Chief Circle for the time being has ten
  1385. thousand sides.
  1386. The ascent of the posterity of the Circles in the social scale is not
  1387. restricted, as it is among the lower Regular classes, by the Law of Nature
  1388. which limits the increase of sides to one in each generation. If it were so,
  1389. the number of sides in the Circle would be a mere question of pedigree and
  1390. arithmetic, and the four hundred and ninety-seventh descendant of an
  1391. Equilateral Triangle would necessarily be a polygon With five hundred sides.
  1392. But this is not the case. Nature's Law prescribes two antagonistic decrees
  1393. affecting Circular propagation; first, that as the race climbs higher in the
  1394. scale of development, so development shall proceed at an accelerated pace;
  1395. second, that in the same proportion, the race shall become less fertile.
  1396. Consequently in the home of a Polygon of four or five hundred sides it is rare
  1397. to find a son; more than one is never seen. On the other hand the son of a
  1398. five-hundred-sided Polygon has been known to possess five hundred and fifty,
  1399. or even six hundred sides.
  1400. Art also steps in to help the process of higher Evolution. Our physicians have
  1401. discovered that the small and tender sides of an infant Polygon of the higher
  1402. class can be fractured, and his whole frame re-set, with such exactness that a
  1403. Polygon of two or three hundred sides sometimes --- by no means always, for the
  1404. process is attended with serious risk --- but sometimes overleaps two or three
  1405. hundred generations, and as it were double at a stroke, the number of his
  1406. progenitors and the nobility of his descent.
  1407. Many a promising child is sacrificed in this way. Scarcely one out of ten
  1408. survives. Yet so strong is the parental ambition among those Polygons who are,
  1409. as it were, on the fringe of the Circular class, that it is very rare to find
  1410. the Nobleman of that position in society, who has neglected to place his
  1411. first-born in the Circular Neo-Therapeutic Gymnasium before he has attained
  1412. the age of a month.
  1413. One year determines success or failure. At the end of that time the child has,
  1414. in all probability, added one more to the tombstones that crowd the
  1415. Neo-Therapeutic Cemetery; but on rare occasional a glad procession bares back
  1416. the little one to his exultant parents, no longer a Polygon, but a Circle, at
  1417. least by courtesy: and a single instance of so blessed a result induces
  1418. multitudes of Polygonal parents to submit to similar domestic sacrifice, which
  1419. have a dissimilar issue.
  1420. \chapter{Of the Doctrine of our Priests}
  1421. As to the doctrine of the Circles it may briefly be summed up in a single
  1422. maxim, ``Attend to your Configuration''. Whether political, ecclesiastical, or
  1423. moral, all their teaching has for its object the improvement of individual and
  1424. collective Configuration --- with special reference of course to the
  1425. Configuration of the Circles, to which all other objects are subordinated.
  1426. It is the merit of the Circles that they have effectually suppressed those
  1427. ancient heresies which led men to waste energy and sympathy in the vain belief
  1428. that conduct depends upon will, effort, training, encouragement, praise, or
  1429. anything else but Configuration. It was Pantocyclus --- the illustrious Circle
  1430. mentioned above, as the queller of the Colour Revolt --- who first convinced
  1431. mankind that Configuration makes the man; that if, for example, you are born
  1432. an Isosceles with two uneven sides, you will assuredly go wrong unless you
  1433. have them made even --- for which purpose you must go to the Isosceles Hospital;
  1434. similarly, if you are a Triangle, or Square, or even a Polygon, born with any
  1435. Irregularity, you must be taken to one of the Regular Hospitals to have your
  1436. disease cured; otherwise you will end your days in the State Prison or by the
  1437. angle of the State Executioner.
  1438. All faults or defects, from the slightest misconduct to the most flagitious
  1439. crime, Pantocyclus attributed to some deviation from perfect Regularity in the
  1440. bodily figure, caused perhaps (if not congenital by some collision in a crowd;
  1441. by neglect to take exercise, or by taking too much of it; or even by a sudden
  1442. change of temperature, resulting in a shrinkage or expansion in some too
  1443. susceptible part of the frame. Therefore, concluded that illustrious
  1444. Philosopher, neither good conduct nor bad conduct is a fit subject, in any
  1445. sober estimation, for either praise or blame. For why should you praise, for
  1446. example, the integrity of a Square who faithfully defends the interests of his
  1447. client, when you ought in reality rather to admire the exact precision of his
  1448. right angles? Or again, why blame a lying, thievish Isosceles, when you ought
  1449. rather to deplore the incurable inequality of his sides?
  1450. Theoretically, this doctrine is unquestionable; but it has practical
  1451. drawbacks. In dealing with an Isosceles, if a rascal pleads that he cannot
  1452. help stealing because of his unevenness, you reply that for that very reason,
  1453. because he cannot help being a nuisance to his neighbours, you, the
  1454. Magistrate, cannot help sentencing him to be consumed --- and there's an end of
  1455. the matter. But in little domestic difficulties, when the penalty of
  1456. consumption, or death, is out of the question, this theory of Configuration
  1457. sometimes comes in awkwardly; and I must confess that occasionally when one of
  1458. my own Hexagonal Grandsons pleads as an excuse for his disobedience that a
  1459. sudden change of temperature has been too much for his Perimeter, and that I
  1460. ought to lay the blame not on him but on his Configuration, which can only be
  1461. strengthened by abundance of the choicest sweetmeats, I neither see my way
  1462. logically to reject, nor practically to accept, his conclusions.
  1463. For my own part, I find it best to assume that a good sound scolding or
  1464. castigation has some latent and strengthening influence on my Grandson's
  1465. Configuration; though I own that I have no grounds for thinking so. At all
  1466. events I am not alone in my way of extricating myself from this dilemma; for I
  1467. find that many of the highest Circles, sitting as Judges in law courts, use
  1468. praise and blame towards Regular and Irregular Figures; and in their homes I
  1469. know by experience that, when scolding their children, they speak about
  1470. ``right'' and ``wrong'' as vehemently and passionately as if they believe that
  1471. these names represented real existence, and that a human Figure is really
  1472. capable of choosing between them.
  1473. Constantly carrying out their policy of making Configuration the leading idea
  1474. in every mind, the Circles reverse the nature of that Commandment which in
  1475. Spaceland regulates the relations between parents and children. With you,
  1476. children are taught to honour their parents; with us --- next to the Circles,
  1477. who are the chief object of universal homage --- a man is taught to honour his
  1478. Grandson, if he has one; or, if not, his Son. By ``honour'', however, is by no
  1479. means mean ``indulgence'', but a reverent regard for their highest interests:
  1480. and the Circles teach that the duty of fathers is to subordinate their own
  1481. interests to those of posterity, thereby advancing the welfare of the whole
  1482. State as well as that of their own immediate descendants.
  1483. The weak point in the system of the Circles --- if a humble Square may venture
  1484. to speak of anything Circular as containing any element of weakness --- appears
  1485. to me to be found in their relations with Women.
  1486. As it is of the utmost importance for Society that Irregular births should be
  1487. discouraged, it follows that no Woman who has any Irregularities in her
  1488. ancestry is a fit partner for one who desires that his posterity should rise
  1489. by regular degrees in the social scale.
  1490. Now the Irregularity of a Male is a matter of measurement; but as all Women
  1491. are straight, and therefore visibly Regular so to speak, one has to devise
  1492. some other means of ascertaining what I may call their invisible Irregularity,
  1493. that is to say their potential Irregularities as regards possible offspring.
  1494. This is effected by carefully-kept pedigrees, which are preserved and
  1495. supervised by the State; and without a certified pedigree no Woman is allowed
  1496. to marry.
  1497. Now it might have been supposed the a Circle --- proud of his ancestry and
  1498. regardful for a posterity which might possibly issue hereafter in a Chief
  1499. Circle --- would be more careful than any other to choose a wife who had no blot
  1500. on her escutcheon. But it is not so. The care in choosing a Regular wife
  1501. appears to diminish as one rises in the social scale. Nothing would induce an
  1502. aspiring Isosceles, who has hopes of generating an Equilateral Son, to take a
  1503. wife who reckoned a single Irregularity among her Ancestors; a Square or
  1504. Pentagon, who is confident that his family is steadily on the rise, does not
  1505. inquire above the five-hundredth generation; a Hexagon or Dodecagon is even
  1506. more careless of the wife's pedigree; but a Circle has been known deliberately
  1507. to take a wife who has had an Irregular Great-Grandfather, and all because of
  1508. some slight superiority of lustre, or because of the charms of a low voice ---
  1509. which, with us, even more than with you, is thought ``an excellent thing in a
  1510. Woman''.
  1511. Such ill-judged marriages are, as might be expected, barren, if they do not
  1512. result in positive Irregularity or in diminution of sides; but none of these
  1513. evils have hitherto provided sufficiently deterrent. The loss of a few sides
  1514. in a highly-developed Polygon is not easily noticed, and is sometimes
  1515. compensated by a successful operation in the Neo-Therapeutic Gymnasium, as I
  1516. have described above; and the Circles are too much disposed to acquiesce in
  1517. infecundity as a law of the superior development. Yet, if this evil be not
  1518. arrested, the gradual diminution of the Circular class may soon become more
  1519. rapid, and the time may not be far distant when, the race being no longer able
  1520. to produce a Chief Circle, the Constitution of Flatland must fall.
  1521. One other word of warning suggest itself to me, though I cannot so easily
  1522. mention a remedy; and this also refers to our relations with Women. About
  1523. three hundred years ago, it was decreed by the Chief Circle that, since women
  1524. are deficient in Reason but abundant in Emotion, they ought no longer to be
  1525. treated as rational, nor receive any mental education. The consequence was
  1526. that they were no longer taught to read, nor even to master Arithmetic enough
  1527. to enable them to count the angles of their husband or children; and hence
  1528. they sensibly declined during each generation in intellectual power. And this
  1529. system of female non-education or quietism still prevails.
  1530. My fear is that, with the best intentions, this policy has been carried so far
  1531. as to react injuriously on the Male Sex.
  1532. For the consequence is that, as things now are, we Males have to lead a kind
  1533. of bi-lingual, and I may almost say bimental, existence. With Women, we speak
  1534. of ``love'', ``duty'', ``right'', ``wrong'', ``pity'', ``hope'', and other irrational and
  1535. emotional conceptions, which have no existence, and the fiction of which has
  1536. no object except to control feminine exuberances; but among ourselves, and in
  1537. our books, we have an entirely different vocabulary and I may also say, idiom.
  1538. ``Love'' them becomes ``the anticipation of benefits''; ``duty'' becomes ``necessity''
  1539. or ``fitness''; and other words are correspondingly transmuted. Moreover, among
  1540. Women, we use language implying the utmost deference for their Sex; and they
  1541. fully believe that the Chief Circle Himself is not more devoutly adored by us
  1542. than they are: but behind their backs they are both regarded and spoken of ---
  1543. by all but the very young --- as being little better than ``mindless organisms''.
  1544. Our Theology also in the Women's chambers is entirely different from our
  1545. Theology elsewhere.
  1546. Now my humble fear is that this double training, in language as well as in
  1547. thought, imposes somewhat too heavy a burden upon the young, especially when,
  1548. at the age of three years old, they are taken from the maternal care and
  1549. taught to unlearn the old language --- except for the purpose of repeating it in
  1550. the presence of the Mothers and Nurses --- and to learn the vocabulary and idiom
  1551. of science. Already methinks I discern a weakness in the grasp of mathematical
  1552. truth at the present time as compared with the more robust intellect of our
  1553. ancestors three hundred years ago. I say nothing of the possible danger if a
  1554. Woman should ever surreptitiously learn to read and convey to her Sex the
  1555. result of her perusal of a single popular volume; nor of the possibility that
  1556. the indiscretion or disobedience of some infant Male might reveal to a Mother
  1557. the secrets of the logical dialect. On the simple ground of the enfeebling of
  1558. the male intellect, I rest this humble appeal to the highest Authorities to
  1559. reconsider the regulations of Female education.
  1560. \part[Other Worlds]{Other Worlds}[
  1561. \vspace{2cm}
  1562. \begin{flushright}
  1563. \begin{normalsize}
  1564. \textit{'O brave new worlds, \\ That have such people in them!'}
  1565. \end{normalsize}
  1566. \end{flushright}
  1567. ]
  1568. \chapter{How I had a Vision of Lineland}
  1569. It was the last day but one of the 1999th year of our era, and the first day
  1570. of the Long Vacation. Having amused myself till a late hour with my favourite
  1571. recreation of Geometry, I had retired to rest with an unsolved problem in my
  1572. mind. In the night I had a dream.
  1573. I saw before me a vast multitude of small Straight Lines (which I naturally
  1574. assumed to be Women) interspersed with other Beings still smaller and of the
  1575. nature of lustrous points --- all moving to and fro in one and the same Straight
  1576. Line, and, as nearly as I could judge, with the same velocity.
  1577. A noise of confused, multitudinous chirping or twittering issued from them at
  1578. intervals as long as they were moving; but sometimes they ceased from motion,
  1579. and then all was silence.
  1580. Approaching one of the largest of what I thought to be Women, I accosted her,
  1581. but received no answer. A second and third appeal on my part were equally
  1582. ineffectual. Losing patience at what appeared to me intolerable rudeness, I
  1583. brought my mouth to a position full in front of her mouth so as to intercept
  1584. her motion, and loudly repeated my question, ``Woman, what signifies this
  1585. concourse, and this strange and confused chirping, and this monotonous motion
  1586. to and fro in one and the same Straight Line?''
  1587. ``I am no Woman'', replied the small Line: ``I am the Monarch of the world. But
  1588. thou, whence intrudest thou into my realm of Lineland?'' Receiving this abrupt
  1589. reply, I begged pardon if I had in any way startled or molested his Royal
  1590. Highness; and describing myself as a stranger I besought the King to give me
  1591. some account of his dominions. But I had the greatest possible difficulty in
  1592. obtaining any information on points that really interested me; for the Monarch
  1593. could not refrain from constantly assuming that whatever was familiar to him
  1594. must also be known to me and that I was simulating ignorance in jest. However,
  1595. by preserving questions I elicited the following facts:
  1596. \vspace*{5mm}
  1597. \includegraphics[scale=0.5]{fig6}
  1598. \vspace*{5mm}
  1599. It seemed that this poor ignorant Monarch --- as he called himself --- was
  1600. persuaded that the Straight Line which he called his Kingdom, and in which he
  1601. passed his existence, constituted the whole of the world, and indeed the whole
  1602. of Space. Not being able either to move or to see, save in his Straight Line,
  1603. he had no conception of anything out of it. Though he had heard my voice when
  1604. I first addressed him, the sounds had come to him in a manner so contrary to
  1605. his experience that he had made no answer, ``seeing no man,'' as he expressed
  1606. it, ``and hearing a voice as it were from my own intestines.'' Until the moment
  1607. when I placed my mouth in his World, he had neither seen me, nor heard
  1608. anything except confused sounds beating against, what I called his side, but
  1609. what he called his inside or stomach; nor had he even now the least conception
  1610. of the region from which I had come. Outside his World, or Line, all was a
  1611. blank to him; nay, not even a blank, for a blank implies Space; say, rather,
  1612. all was non-existent.
  1613. His subjects --- of whom the small Lines were men and the Points Women --- were
  1614. all alike confined in motion and eyesight to that single Straight Line, which
  1615. was their World. It need scarcely be added that the whole of their horizon was
  1616. limited to a Point; nor could any one ever see anything but a Point. Man,
  1617. woman, child, thing --- each as a Point to the eye of a Linelander. Only by the
  1618. sound of the voice could sex or age be distinguished. Moreover, as each
  1619. individual occupied the whole of the narrow path, so to speak, which
  1620. constituted his Universe, and no one could move to the right or left to make
  1621. way for passers by, it followed that no Linelander could ever pass another.
  1622. Once neighbours, always neighbours. Neighbourhood with them was like marriage
  1623. with us. Neighbours remained neighbours till death did them part.
  1624. Such a life, with all vision limited to a Point, and all motion to a Straight
  1625. Line, seemed to me inexpressibly dreary; and I was surprised to note that
  1626. vivacity and cheerfulness of the King. Wondering whether it was possible, amid
  1627. circumstances so unfavourable to domestic relations, to enjoy the pleasures of
  1628. conjugal union, I hesitated for some time to question his Royal Highness on so
  1629. delicate a subject; but at last I plunged into it by abruptly inquiring as to
  1630. the health of his family. ``My wives and children,'' he replied, ``are well and
  1631. happy.''
  1632. Staggered at this answer --- for in the immediate proximity of the Monarch (as I
  1633. had noted in my dream before I entered Lineland) there were none but Men --- I
  1634. ventured to reply, ``Pardon me, but I cannot imagine how your Royal Highness
  1635. can at any time either See or approach their Majesties, when there at least
  1636. half a dozen intervening individuals, whom you can neither see through, nor
  1637. pass by? Is it possible that in Lineland proximity is not necessary for
  1638. marriage and for the generation of children?''
  1639. ``How can you ask so absurd a question?'' replied the Monarch. ``If it were
  1640. indeed as you suggest, the Universe would soon be depopulated. No, no;
  1641. neighbourhood is needless for the union of hearts; and the birth of children
  1642. is too important a matter to have been allowed to depend upon such an accident
  1643. as proximity. You cannot be ignorant of this. Yet since you are pleased to
  1644. affect ignorance, I will instruct you as if you were the veriest baby in
  1645. Lineland. Know, then, that marriages are consummated by means of the faculty
  1646. of sound and the sense of hearing.
  1647. ``You are of course aware that every Man has two mouths or voices --- as well as
  1648. two eyes --- a bass at one and a tenor at the other of his extremities. I should
  1649. not mention this, but that I have been unable to distinguish your tenor in the
  1650. course of our conversation.'' I replied that I had but one voice, and that I
  1651. had not been aware that his Royal Highness had two. ``That confirms by
  1652. impression,'' said the King, ``that you are not a Man, but a feminine
  1653. Monstrosity with a bass voice, and an utterly uneducated ear. But to continue.
  1654. ``Nature having herself ordained that every Man should wed two wives ---'' ``Why
  1655. two?'' asked I. ``You carry your affected simplicity too far,'' he cried. ``How
  1656. can there be a completely harmonious union without the combination of the Four
  1657. in One, viz. the Bass and Tenor of the Man and the Soprano and Contralto of
  1658. the two Women?'' ``But supposing,'' said I, ``that a man should prefer one wife or
  1659. three?'' ``It is impossible,'' he said; ``it is as inconceivable as that two and
  1660. one should make five, or that the human eye should see a Straight Line.'' I
  1661. would have interrupted him; but he proceeded as follows:
  1662. ``Once in the middle of each week a Law of Nature compels us to move to and fro
  1663. with a rhythmic motion of more than usual violence, which continues for the
  1664. time you would take to count a hundred and one. In the midst of this choral
  1665. dance, at the fifty-first pulsation, the inhabitants of the Universe pause in
  1666. full career, and each individual sends forth his richest, fullest, sweetest
  1667. strain. It is in this decisive moment that all our marriages are made. So
  1668. exquisite is the adaptation of Bass and Treble, of Tenor to Contralto, that
  1669. oftentimes the Loved Ones, though twenty thousand leagues away, recognize at
  1670. once the responsive note of their destined Lover; and, penetrating the paltry
  1671. obstacles of distance, Love unites the three. The marriage in that instance
  1672. consummated results in a threefold Male and Female offspring which takes its
  1673. place in Lineland.''
  1674. ``What! Always threefold?'' said I. ``Must one wife then always have twins?''
  1675. ``Bass-voice Monstrosity! yes,'' replied the King. ``How else could the balance
  1676. of the Sexes be maintained, if two girls were not born for every boy? Would
  1677. you ignore the very Alphabet of Nature?'' He ceased, speechless for fury; and
  1678. some time elapsed before I could induce him to resume his narrative.
  1679. ``You will not, of course, suppose that every bachelor among us finds his mates
  1680. at the first wooing in this universal Marriage Chorus. On the contrary, the
  1681. process is by most of us many times repeated. Few are the hearts whose happy
  1682. lot is at once to recognize in each other's voice the partner intended for
  1683. them by Providence, and to fly into a reciprocal and perfectly harmonious
  1684. embrace. With most of us the courtship is of long duration. The Wooer's voices
  1685. may perhaps accord with one of the future wives, but not with both; or not, at
  1686. first, with either; or the Soprano and Contralto may not quite harmonize. In
  1687. such cases Nature has provided that every weekly Chorus shall bring the three
  1688. Lovers into closer harmony. Each trial of voice, each fresh discovery of
  1689. discord, almost imperceptibly induces the less perfect to modify his or her
  1690. vocal utterance so as to approximate to the more perfect. And after many
  1691. trials and many approximations, the result is at last achieved. There comes a
  1692. day at last when, while the wonted Marriage Chorus goes forth from universal
  1693. Lineland, the three far-off Lovers suddenly find themselves in exact harmony,
  1694. and, before they are aware, the wedded Triplet is rapt vocally into a
  1695. duplicate embrace; and Nature rejoices over one more marriage and over three
  1696. more births.''
  1697. \chapter{How I vainly tried to explain the nature of Flatland}
  1698. Thinking that it was time to bring down the Monarch from his raptures to the
  1699. level of common sense, I determined to endeavour to open up to him some
  1700. glimpses of the truth, that is to say of the nature of things in Flatland. So
  1701. I began thus: ``How does your Royal Highness distinguish the shapes and
  1702. positions of his subjects? I for my part noticed by the sense of sight, before
  1703. I entered your Kingdom, that some of your people are lines and others Points;
  1704. and that some of the lines are larger ---'' ``You speak of an impossibility,''
  1705. interrupted the King; ``you must have seen a vision; for to detect the
  1706. difference between a Line and a Point by the sense of sight is, as every one
  1707. knows, in the nature of things, impossible; but it can be detected by the
  1708. snese of hearing, and by the same means my shape can be exactly ascertained.
  1709. Behold me --- I am a Line, the longest in Lineland, over six inches of Space ---''
  1710. ``Of Length,'' I ventured to suggest. ``Fool,'' said he, ``Space is Length.
  1711. Interrupt me again, and I have done.''
  1712. I apologized; but he continued scornfully, ``Since you are impervious to
  1713. argument, you shall hear with your ears how by means of my two voices I reveal
  1714. my shape to my Wives, who are at this moment six thousand miles seventy yards
  1715. two feet eight inches away, the one to the North, the other to the South.
  1716. Listen, I call to them.''
  1717. He chirruped, and then complacently continued: ``My wives at this moment
  1718. receiving the sound of one of my voice, closely followed by the other, and
  1719. perceiving that the latter reaches them after an interval in which sound can
  1720. traverse 6.457 inches, infer that one of my mouths is 6.457 inches further
  1721. from them than the other, and accordingly know my shape to be 6.457 inches.
  1722. But you will of course understand that my wives do not make this calculation
  1723. every time they hear my two voices. They made it, once for all, before we were
  1724. married. But they could make it at any time. And in the same way I can
  1725. estimate the shape of any of my Male subjects by the sense of sound.''
  1726. ``But how,'' said I, ``if a Man feigns a Woman's voice with one of his two
  1727. voices, or so disguises his Southern voice that it cannot be recognized as the
  1728. echo of the Northern? May not such deceptions cause great inconvenience? And
  1729. have you no means of checking frauds of this kind by commanding your
  1730. neighbouring subjects to feel one another?'' This of course was a very stupid
  1731. question, for feeling could not have answered the purpose; but I asked with
  1732. the view of irritating the Monarch, and I succeeded perfectly.
  1733. ``What!'' cried he in horror, ``explain your meaning.'' ``Feel, touch, come into
  1734. contact,'' I replied. ``If you mean by \textit{feeling},`` said the King, ``approaching so
  1735. close as to leave no space between two individuals, know, Stranger, that this
  1736. offence is punishable in my dominions by death. And the reason is obvious. The
  1737. frail form of a Woman, being liable to be shattered by such an approximation,
  1738. must be preserved by the State; but since Women cannot be distinguished by the
  1739. sense of sight from Men, the Law ordains universally that neither Man nor
  1740. Woman shall be approached so closely as to destroy the interval between the
  1741. approximator and the approximated.
  1742. ``And indeed what possible purpose would be served by this illegal and
  1743. unnatural excess of approximation which you call touching, when all the ends
  1744. of so brutal and course a process are attained at once more easily and more
  1745. exactly by the sense of hearing? As to your suggested danger of deception, it
  1746. is non-existent: for the Voice, being the essence of one's Being, cannot be
  1747. thus changed at will. But come, suppose that I had the power of passing
  1748. through solid things, so that I could penetrate my subjects, one after
  1749. another, even to the number of a billion, verifying the size and distance of
  1750. each by the sense of \textit{feeling:} How much time and energy would be wasted in this
  1751. clumsy and inaccurate method! Whereas now, in one moment of audition, I take
  1752. as it were the census and statistics, local, corporeal, mental and spiritual,
  1753. of every living being in Lineland. Hark, only hark!''
  1754. So saying he paused and listened, as if in an ecstasy, to a sound which seemed
  1755. to me no better than a tiny chirping from an innumerable multitude of
  1756. lilliputian grasshoppers.
  1757. ``Truly,'' replied I, ``your sense of hearing serves you in good stead, and fills
  1758. up many of your deficiencies. But permit me to point out that your life in
  1759. Lineland must be deplorably dull. To see nothing but a Point! Not even to be
  1760. able to contemplate a Straight Line! Nay, not even to know what a Straight
  1761. Line is! To see, yet to be cut off from those Linear prospects which are
  1762. vouchsafed to us in Flatland! Better surely to have no sense of sight at all
  1763. than to see so little! I grant you I have not your discriminative faculty of
  1764. hearing; for the concert of all Lineland which gives you such intense
  1765. pleasure, is to me no better than a multitudinous twittering or chirping. But
  1766. at least I can discern, by sight, a Line from a Point. And let me prove it.
  1767. Just before I came into your kingdom, I saw you dancing from left to right,
  1768. and then from right to left, with Seven Men and a Woman in your immediate
  1769. proximity on the left, and eight Men and two Women on your right. Is not this
  1770. correct?''
  1771. ``It is correct,'' said the King, ``so far as the numbers and sexes are
  1772. cocnerned, though I know not what you mean by `right' and `left.' But I deny
  1773. that you saw these things. For how could you see the Line, that is to say the
  1774. inside, of any Man? But you must have heard these things, and then dreamed
  1775. that you saw them. And let me ask what you mean by those words `left' and
  1776. `right'. I suppose it is your way of saying Northward and Southward.''
  1777. ``Not so,'' replied I; ``besides your motion of Northward and Southward, there is
  1778. another motion which I call from right to left.''
  1779. \textit{King}. Exhibit to me, if you please, this motion from left to right.
  1780. \textit{I}. Nay, that I cannot do, unless you could step out of your Line altogether.
  1781. \textit{King}. Out of my Line? Do you mean out of the world? Out of Space?
  1782. \textit{I}. Well, yes. Out of \textit{your} world. Out of \textit{your} Space. For your Space is not the true Space. True Space is a Plane; but your Space is only a Line.
  1783. \textit{King}. If you cannot indicate this motion from left to right by yourself moving
  1784. in it, then I beg you to describe it to me in words.
  1785. \textit{I}. If you cannot tell your right side from your left, I fear that no words of
  1786. mine can make my meaning clearer to you. But surely you cannot be ignorant of
  1787. so simple a distinction.
  1788. \textit{King}. I do not in the least understand you.
  1789. \textit{I}. Alas! How shall I make it clear? When you move straight on, does it not
  1790. sometimes occur to you that you \textit{could} move in some other way, turning your eye
  1791. round so as to look in the direction towards which your side is now fronting?
  1792. In other words, instead of always moving in the direction of one of your
  1793. extremities, do you never feel a desire to move in the direction, so to speak,
  1794. of your side?
  1795. \textit{King}. Never. And what do you mean? How can a man's inside ``front'' in any
  1796. direction? Or how can a man move in the direction of his inside?
  1797. \textit{I}. Well then, since words cannot explain the matter, I will try deeds, and
  1798. will move gradually out of Lineland in the direction which I desire to
  1799. indicate to you.
  1800. At the word I began to move my body out of Lineland. As long as any part of me
  1801. remained in his dominion and in his view, the King kept exclaiming, ``I see
  1802. you, I see you still; you are not moving.'' But when I had at last moved myself
  1803. out of his Line, he cried in his shrillest voice, ``She is vanished; she is
  1804. dead.'' ``I am not dead,'' replied I; ``I am simply out of Lineland, that is to
  1805. say, out of the Straight Line which you call Space, and in the true Space,
  1806. where I can see things as they are. And at this moment I can see your Line, or
  1807. side --- or inside as you are pleased to call it; and I can see also the Men and
  1808. Women on the North and South of you, whom I will now enumerate, describing
  1809. their order, their size, and the interval between each.''
  1810. \includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig7}
  1811. When I had done this at great length, I cried triumphantly, ``Does that at last
  1812. convince you?'' And, with that, I once more entered Lineland, taking up the
  1813. same position as before.
  1814. But the Monarch replied, ``If you were a Man of sense --- though, as you appear
  1815. to have only one voice I have little doubt you are not a Man but a Woman ---
  1816. but, if you had a particle of sense, you would listen to reason. You ask me to
  1817. believe that there is another Line besides that which my senses indicate, and
  1818. another motion besides that of which I am daily conscious. I, in return, ask
  1819. you to describe in words or indicate by motion that other Line of which you
  1820. speak. Instead of moving, you merely exercise some magic art of vanishing and
  1821. returning to sight; and instead of any lucid description of your new World,
  1822. you simply tell me the numbers and sizes of some forty of my retinue, facts
  1823. known to any child in my capital. Can anything be more irrational or
  1824. audacious? Acknowledge your folly or depart from my dominions.''
  1825. Furious at his perversity, and especially indignant that he professed to be
  1826. ignorant of my sex, I retorted in no measured terms, ``Besotted Being! You
  1827. think yourself the perfection of existence, while you are in reality the most
  1828. imperfect and imbecile. You profess to see, whereas you see nothing but a
  1829. Point! You plume yourself on inferring the existence of a Straight Line; but I
  1830. can see Straight Lines, and infer the existence of Angles, Triangles, Squares,
  1831. Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I
  1832. am the completion of your incomplete self. You are a Line, but I am a Line of
  1833. Lines called in my country a Square: and even I, infinitely superior though I
  1834. am to you, am of little account among the great nobles of Flatland, whence I
  1835. have come to visit you, in the hope of enightening your ignorance.''
  1836. Hearing these words the King advanced towards me with a menacing cry as if to
  1837. pierce me through the diagonal; and in that same movement there arose from
  1838. myriads of his subjects a multitudinous war-cry, increasing in vehemence till
  1839. at last methought it rivalled the roar of an army of a hundred thousand
  1840. Isosceles, and the artillery of a thousand Pentagons. Spell-bound and
  1841. motionless, I could neither speak nor move to avert the impending destruction;
  1842. and still the noise grew louder, and the King came closer, when I awoke to
  1843. find the breakfast-bell recalling me to the realities of Flatland.
  1844. \chapter{Concerning a Stranger from Spaceland}
  1845. From dreams I proceed to facts.
  1846. It was the last day of our 1999th year of our era. The patterning of the rain
  1847. had long ago announced nightfall; and I was sitting \footnote{ When I say
  1848. ``sitting,'' of course I do not mean any change of attitude such as you in
  1849. Spaceland signify by that word; for as we have no feet, we can no more ``sit''
  1850. nor ``stand'' (in your sense of the word) than one of your soles or flounders.
  1851. Nevertheless, we perfectly well recognize the different mental states of
  1852. volition implied by ``lying,'' ``sitting,'' and ``standing,'' which are to some
  1853. extent indicated to a beholder by a slight increase of lustre corresponding to
  1854. the increase of volition.
  1855. But on this, and a thousand other kindred subjects, time forbids me to dwell.}
  1856. in the company of my wife, musing on the events of the past and the prospects
  1857. of the coming year, the coming century, the coming Millennium.
  1858. My four Sons and two orphan Grandchildren had retired to their several
  1859. apartments; and my wife alone remained with me to see the old Millennium out
  1860. and the new one in.
  1861. I was rapt in thought, pondering in my mind some words that had casually
  1862. issued from the mouth of my youngest Grandson, a most promising young Hexagon
  1863. of unusual brilliancy and perfect angularity. His uncles and I had been giving
  1864. him his usual practical lesson in Sight Recognition, turning ourselves upon
  1865. our centres, now rapidly, now more slowly, and questioning him as to our
  1866. positions; and his answers had been so satisfactory that I had been induced to
  1867. reward him by giving him a few hints on Arithmetic, as applied to Geometry.
  1868. Taking nine Squares, each an inch every way, I had put them together so as to
  1869. make one large Square, with a side of three inches, and I had hence proved to
  1870. my little Grandson that --- though it was impossible for us to \textit{see} the inside of
  1871. the Square --- yet we might ascertain the number of square inches in a Square by
  1872. simply squaring the number of inches in the side: ``and thus,'' said I, ``we know
  1873. that $ 3^2 $, or 9, represents the number of square inches in a
  1874. Square whose side is 3 inches long.''
  1875. The little Hexagon meditated on this a while and then said to me; ``But you
  1876. have been teaching me to raise numbers to the third power: I suppose
  1877. $ 3^3 $ must mean something in Geometry; what does it mean?''
  1878. ``Nothing at all,'' replied I, ``not at least in Geometry; for Geometry has only
  1879. Two Dimensions.'' And then I began to shew the boy how a Point by moving
  1880. through a length of three inches makes a Line of three inches, which may be
  1881. represented by 3; and how a Line of three inches, moving parallel to
  1882. itself through a length of three inches, makes a Square of three inches every
  1883. way, which may be represented by $ 3^2 $.
  1884. Upon this, my Grandson, again returning to his former suggestion, took me up
  1885. rather suddenly and exclaimed, ``Well, then, if a Point by moving three inches,
  1886. makes a Line of three inches represented by 3; and if a straight Line of
  1887. three inches, moving parallel to itself, makes a Square of three inches every
  1888. way, represented by $ 3^2 $; it must be that a Square of three
  1889. inches every way, moving somehow parallel to itself (but I don't see how) must
  1890. make Something else (but I don't see what) of three inches every way --- and
  1891. this must be represented by $ 3^3 $.''
  1892. ``Go to bed,'' said I, a little ruffled by this interruption: ``if you would talk
  1893. less nonsense, you would remember more sense.''
  1894. So my Grandson had disappeared in disgrace; and there I sat by my Wife's side,
  1895. endeavouring to form a retrospect of the year 1999 and of the possibilities of
  1896. the year 2000; but not quite able to shake of the thoughts suggested by the
  1897. prattle of my bright little Hexagon. Only a few sands now remained in the
  1898. half-hour glass. Rousing myself from my reverie I turned the glass Northward
  1899. for the last time in the old Millennium; and in the act, I exclaimed aloud,
  1900. ``The boy is a fool.''
  1901. Straightway I became conscious of a Presence in the room, and a chilling
  1902. breath thrilled through my very being. ``He is no such thing,'' cried my Wife,
  1903. ``and you are breaking the Commandments in thus dishonouring your own
  1904. Grandson.'' But I took no notice of her. Looking around in every direction I
  1905. could see nothing; yet still I \textit{felt} a Presence, and shivered as the cold
  1906. whisper came again. I started up. ``What is the matter?'' said my Wife, ``there
  1907. is no draught; what are you looking for? There is nothing.'' There was nothing;
  1908. and I resumed my seat, again exclaiming, ``The boy is a fool, I say;
  1909. $ 3^3 $ can have no meaning in Geometry.'' At once there came a
  1910. distinctly audible reply, ``The boy is not a fool; and $ 3^3 $ has
  1911. an obvious Geometrical meaning.''
  1912. My Wife as well as myself heard the words, although she did not understand
  1913. their meaning, and both of us sprang forward in the direction of the sound.
  1914. What was our horror when we saw before us a Figure! At the first glance it
  1915. appeared to be a Woman, seen sideways; but a moment's observation shewed me
  1916. that the extremities passed into dimness too rapidly to represent one of the
  1917. Female Sex; and I should have thought it a Circle, only that it seemed to
  1918. change its size in a manner impossible for a Circle or for any regular Figure
  1919. of which I had had experience.
  1920. But my Wife had not my experience, nor the coolness necessary to note these
  1921. characteristics. With the usual hastiness and unreasoning jealousy of her Sex,
  1922. she flew at once to the conclusion that a Woman had entered the house through
  1923. some small aperture. ``How comes this person here?'' she exclaimed, ``you
  1924. promised me, my dear, that there should be no ventilators in our new house.''
  1925. ``Nor are they any,'' said I; ``but what makes you think that the stranger is a
  1926. Woman? I see by my power of Sight Recognition ---'' ``Oh, I have no patience with
  1927. your Sight Recognition,'' replied she, ```Feeling is believing' and `A Straight
  1928. Line to the touch is worth a Circle to the sight''' --- two Proverbs, very common
  1929. with the Frailer Sex in Flatland.
  1930. ``Well,'' said I, for I was afraid of irritating her, ``if it must be so, demand
  1931. an introduction.'' Assuming her most gracious manner, my Wife advanced towards
  1932. the Stranger, ``Permit me, Madam to feel and be felt by ---'' then, suddenly
  1933. recoiling, ``Oh! it is not a Woman, and there are no angles either, not a trace
  1934. of one. Can it be that I have so misbehaved to a perfect Circle?''
  1935. ``I am indeed, in a certain sense a Circle,'' replied the Voice, ``and a more
  1936. perfect Circle than any in Flatland; but to speak more accurately, I am many
  1937. Circles in one.'' Then he added more mildly, ``I have a message, dear Madam, to
  1938. your husband, which I must not deliver in your presence; and, if you would
  1939. suffer us to retire for a few minutes ---'' But my wife would not listen to the
  1940. proposal that our august Visitor should so incommode himself, and assuring the
  1941. Circle that the hour of her own retirement had long passed, with many
  1942. reiterated apologies for her recent indiscretion, she at last retreated to her
  1943. apartment.
  1944. I glanced at the half-hour glass. The last sands had fallen. The third
  1945. Millennium had begun.
  1946. \chapter{How the Stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland}
  1947. As soon as the sound of the Peace-cry of my departing Wife had died away, I
  1948. began to approach the Stranger with the intention of taking a nearer view and
  1949. of bidding him be seated: but his appearance struck me dumb and motionless
  1950. with astonishment. Without the slightest symptoms of angularity he
  1951. nevertheless varied every instant with graduations of size and brightness
  1952. scarcely possible for any Figure within the scope of my experience. The
  1953. thought flashed across me that I might have before me a burglar or cut-throat,
  1954. some monstrous Irregular Isosceles, who, by feigning the voice of a Circle,
  1955. had obtained admission somehow into the house, and was now preparing to stab
  1956. me with his acute angle.
  1957. In a sitting-room, the absence of Fog (and the season happened to be
  1958. remarkably dry), made it difficult for me to trust to Sight Recognition,
  1959. especially at the short distance at which I was standing. Desperate with fear,
  1960. I rushed forward with an unceremonious, ``You must permit me, Sir ---'' and felt
  1961. him. My Wife was right. There was not the trace of an angle, not the slightest
  1962. roughness or inequality: never in my life had I met with a more perfect
  1963. Circle. He remained motionless while I walked around him, beginning from his
  1964. eye and returning to it again. Circular he was throughout, a perfectly
  1965. satisfactory Circle; there could not be a doubt of it. Then followed a
  1966. dialogue, which I will endeavour to set down as near as I can recollect it,
  1967. omitting only some of my profuse apologies --- for I was covered with shame and
  1968. humiliation that I, a Square, should have been guilty of the impertinence of
  1969. feeling a Circle. It was commenced by the Stranger with some impatience at the
  1970. lengthiness of my introductory process.
  1971. \textit{Stranger}. Have you felt me enough by this time? Are you not introduced to me
  1972. yet?
  1973. \textit{I}. Most illustrious Sir, excuse my awkwardness, which arises not from
  1974. ignorance of the usages of polite society, but from a little surprise and
  1975. nervousness, consequent on this somewhat unexpected visit. And I beseech you
  1976. to reveal my indiscretion to no one, and especially not to my Wife. But before
  1977. your Lordship enters into further communications, would he deign to satisfy
  1978. the curiosity of one who would gladly know whence his visitor came?
  1979. \textit{Stranger}. From Space, from Space, Sir: whence else?
  1980. \textit{I}. Pardon me, my Lord, but is not your Lordship already in Space, your
  1981. Lordship and his humble servant, even at this moment?
  1982. \textit{Stranger}. Pooh! what do you know of Space? Define Space.
  1983. \textit{I}. Space, my Lord, is height and breadth indefinitely prolonged.
  1984. \textit{Stranger}. Exactly: you see you do not even know what Space is. You think it is
  1985. of Two Dimensions only; but I have come to announce to you a Third --- height,
  1986. breadth, and length.
  1987. \textit{I}. Your Lordship is pleased to be merry. We also speak of length and height,
  1988. or breadth and thickness, thus denoting Two Dimensions by four names.
  1989. \textit{Stranger}. But I mean not only three names, but Three Dimensions.
  1990. \textit{I}. Would your Lordship indicate or explain to me in what direction is the
  1991. Third Dimension, unknown to me?
  1992. \textit{Stranger}. I came from it. It is up above and down below.
  1993. \textit{I}. My Lord means seemingly that it is Northward and Southward.
  1994. \textit{Stranger}. I mean nothing of the kind. I mean a direction in which you cannot
  1995. look, because you have no eye in your side.
  1996. \textit{I}. Pardon me, my Lord, a moment's inspection will convince your Lordship that
  1997. I have a perfectly luminary at the juncture of my two sides.
  1998. \textit{Stranger}: Yes: but in order to see into Space you ought to have an eye, not on
  1999. your Perimeter, but on your side, that is, on what you would probably call
  2000. your inside; but we in Spaceland should call it your side.
  2001. \textit{I}. An eye in my inside! An eye in my stomach! Your Lordship jests.
  2002. \textit{Stranger}. I am in no jesting humour. I tell you that I come from Space, or,
  2003. since you will not understand what Space means, from the Land of Three
  2004. Dimensions whence I but lately looked down upon your Plane which you call
  2005. Space forsooth. From that position of advantage I discerned all that you speak
  2006. of as \textit{solid} (by which you mean ``enclosed on four sides''), your houses, your
  2007. churches, your very chests and safes, yes even your insides and stomachs, all
  2008. lying open and exposed to my view.
  2009. \textit{I}. Such assertions are easily made, my Lord.
  2010. \textit{Stranger}. But not easily proved, you mean. But I mean to prove mine.
  2011. When I descended here, I saw your four Sons, the Pentagons, each in his
  2012. apartment, and your two Grandsons the Hexagons; I saw your youngest Hexagon
  2013. remain a while with you and then retire to his room, leaving you and your Wife
  2014. alone. I saw your Isosceles servants, three in number, in the kitchen at
  2015. supper, and the little Page in the scullery. Then I came here, and how do you
  2016. think I came?
  2017. \textit{I}. Through the roof, I suppose.
  2018. \textit{Stranger}. Not so. Your roof, as you know very well, has been recently
  2019. repaired, and has no aperture by which even a Woman could penetrate. I tell
  2020. you I come from Space. Are you not convinced by what I have told you of your
  2021. children and household?
  2022. \textit{I}. Your Lordship must be aware that such facts touching the belongings of his
  2023. humble servant might be easily ascertained by any one of the neighbourhood
  2024. possessing your Lordship's ample means of information.
  2025. \textit{Stranger}. (\textit{To himself}.) What must I do? Stay; one more argument suggests
  2026. itself to me. When you see a Straight Line --- your wife, for example --- how many
  2027. Dimensions do you attribute to her?
  2028. \textit{I}. Your Lordship would treat me as if I were one of the vulgar who, being
  2029. ignorant of Mathematics, suppose that a Woman is really a Straight Line, and
  2030. only of One Dimension. No, no, my Lord; we Squares are better advised, and are
  2031. as well aware of your Lordship that a Woman, though popularly called a
  2032. Straight Line, is, really and scientifically, a very thin Parallelogram,
  2033. possessing Two Dimensions, like the rest of us, viz., length and breadth (or
  2034. thickness).
  2035. \textit{Stranger}. But the very fact that a Line is visible implies that it possesses
  2036. yet another Dimension.
  2037. \textit{I}. My Lord, I have just acknowledged that a Woman is broad as well as long. We
  2038. see her length, we infer her breadth; which, though very slight, is capable of
  2039. measurement.
  2040. \textit{Stranger}. You do not understand me. I mean that when you see a Woman, you
  2041. ought --- besides inferring her breadth --- to see her length, and to \textit{see} what we
  2042. call her \textit{height}; although the last Dimension is infinitesimal in your country.
  2043. If a Line were mere length without ``height,'' it would cease to occupy Space
  2044. and would become invisible. Surely you must recognize this?
  2045. \textit{I}. I must indeed confess that I do not in the least understand your Lordship.
  2046. When we in Flatland see a Line, we see length and \textit{brightness}. If the
  2047. brightness disappears, the Line is extinguished, and, as you say, ceases to
  2048. occupy Space. But am I to suppose that your Lordship gives the brightness the
  2049. title of a Dimension, and that what we call ``bright'' you call ``high''?
  2050. \textit{Stranger}. No, indeed. By ``height'' I mean a Dimension like your length: only,
  2051. with you, ``height'' is not so easily perceptible, being extremely small.
  2052. \textit{I}. My Lord, your assertion is easily put to the test. You say I have a Third
  2053. Dimension, which you call ``height.'' Now, Dimension implies direction and
  2054. measurement. Do but measure my ``height,'' or merely indicate to me the
  2055. direction in which my ``height'' extends, and I will become your convert.
  2056. Otherwise, your Lordship's own understanding must hold me excused.
  2057. \textit{Stranger}. (\textit{To himself}.) I can do neither. How shall I convince him? Surely a
  2058. plain statement of facts followed by ocular demonstration ought to suffice. ---
  2059. Now, Sir; listen to me.
  2060. You are living on a Plane. What you style Flatland is the vast level surface
  2061. of what I may call a fluid, or in, the top of which you and your countrymen
  2062. move about, without rising above or falling below it.
  2063. I am not a plane Figure, but a Solid. You call me a Circle; but in reality I
  2064. am not a Circle, but an infinite number of Circles, of size varying from a
  2065. Point to a Circle of thirteen inches in diameter, one placed on the top of the
  2066. other. When I cut through your plane as I am now doing, I make in your plane a
  2067. section which you, very rightly, call a Circle. For even a Sphere --- which is
  2068. my proper name in my own country --- if he manifest himself at all to an
  2069. inhabitant of Flatland --- must needs manifest himself as a Circle.
  2070. Do you not remember --- for I, who see all things, discerned last night the
  2071. phantasmal vision of Lineland written upon your brain --- do you not remember, I
  2072. say, how when you entered the realm of Lineland, you were compelled to
  2073. manifest yourself to the King, not as a Square, but as a Line, because that
  2074. Linear Realm had not Dimensions enough to represent the whole of you, but only
  2075. a slice or section of you? In precisely the same way, your country of Two
  2076. Dimensions is not spacious enough to represent me, a being of Three, but can
  2077. only exhibit a slice or section of me, which is what you call a Circle.
  2078. The diminished brightness of your eye indicates incredulity. But now prepare
  2079. to receive proof positive of the truth of my assertions. You cannot indeed see
  2080. more than one of my sections, or Circles, at a time; for you have no power to
  2081. raise your eye out of the plane of Flatland; but you can at least see that, as
  2082. I rise in Space, so my sections become smaller. See now, I will rise; and the
  2083. effect upon your eye will be that my Circle will become smaller and smaller
  2084. till it dwindles to a point and finally vanishes.
  2085. \includegraphics[trim=20mm 0mm 0mm 0mm,width=\linewidth]{fig8}
  2086. There was no ``rising'' that I could see; but he diminished and finally
  2087. vanished. I winked once or twice to make sure that I was not dreaming. But it
  2088. was no dream. For from the depths of nowhere came forth a hollow voice --- close
  2089. to my heart it seemed --- ``Am I quite gone? Are you convinced now? Well, now I
  2090. will gradually return to Flatland and you shall see my section become larger
  2091. and larger.''
  2092. Every reader in Spaceland will easily understand that my mysterious Guest was
  2093. speaking the language of truth and even of simplicity. But to me, proficient
  2094. though I was in Flatland Mathematics, it was by no means a simple matter. The
  2095. rough diagram given above will make it clear to any Spaceland child that the
  2096. Sphere, ascending in the three positions indicated there, must needs have
  2097. manifested himself to me, or to any Flatlander, as a Circle, at first of full
  2098. size, then small, and at last very small indeed, approaching to a Point. But
  2099. to me, although I saw the facts before me, the causes were as dark as ever.
  2100. All that I could comprehend was, that the Circle had made himself smaller and
  2101. vanished, and that he had now re-appeared and was rapidly making himself
  2102. larger.
  2103. When he regained his original size, he heaved a deep sigh; for he perceived by
  2104. my silence that I had altogether failed to comprehend him. And indeed I was
  2105. now inclining to the belief that he must be no Circle at all, but some
  2106. extremely clever juggler; or else that the old wives' tales were true, and
  2107. that after all there were such people as Enchanters and Magicians.
  2108. After a long pause he muttered to himself, ``One resource alone remains, if I
  2109. am not to resort to action. I must try the method of Analogy.'' Then followed a
  2110. still longer silence, after which he continued our dialogue.
  2111. \textit{Sphere}. Tell me, Mr. Mathematician; if a Point moves Northward, and leaves a
  2112. luminous wake, what name would you give to the wake?
  2113. \textit{I}. A straight Line.
  2114. \textit{Sphere}. And a straight Line has how many extremities?
  2115. \textit{I}. Two.
  2116. \textit{Sphere}. Now conceive the Northward straight Line moving parallel to itself,
  2117. East and West, so that every point in it leaves behind it the wake of a
  2118. straight Line. What name will you give to the Figure thereby formed? We will
  2119. suppose that it moves through a distance equal to the original straight line.
  2120. --- What name, I say?
  2121. \textit{I}. A square.
  2122. \textit{Sphere}. And how many sides has a Square? How many angles?
  2123. \textit{I}. Four sides and four angles.
  2124. \textit{Sphere}. Now stretch your imagination a little, and conceive a Square in
  2125. Flatland, moving parallel to itself upward.
  2126. \textit{I}. What? Northward?
  2127. \textit{Sphere}. No, not Northward; upward; out of Flatland altogether.
  2128. If it moved Northward, the Southern points in the Square would have to move
  2129. through the positions previously occupied by the Northern points. But that is
  2130. not my meaning.
  2131. I mean that every Point in you --- for you are a Square and will serve the
  2132. purpose of my illustration --- every Point in you, that is to say in what you
  2133. call your inside, is to pass upwards through Space in such a way that no Point
  2134. shall pass through the position previously occupied by any other Point; but
  2135. each Point shall describe a straight Line of its own. This is all in
  2136. accordance with Analogy; surely it must be clear to you.
  2137. Restraining my impatience --- for I was now under a strong temptation to rush
  2138. blindly at my Visitor and to precipitate him into Space, or out of Flatland,
  2139. anywhere, so that I could get rid of him --- I replied: ---
  2140. ``And what may be the nature of the Figure which I am to shape out by this
  2141. motion which you are pleased to denote by the word `upward'? I presume it is
  2142. describable in the language of Flatland.''
  2143. \textit{Sphere}. Oh, certainly. It is all plain and simple, and in strict accordance
  2144. with Analogy --- only, by the way, you must not speak of the result as being a
  2145. Figure, but as a Solid. But I will describe it to you. Or rather not I, but
  2146. Analogy.
  2147. We began with a single Point, which of course --- being itself a Point --- has
  2148. only one terminal Point.
  2149. One Point produces a Line with two terminal Points.
  2150. One Line produces a Square with four terminal Points.
  2151. Now you can give yourself the answer to your own question: 1, 2, 4, are
  2152. evidently in Geometrical Progression. What is the next number?
  2153. \textit{I}. Eight.
  2154. \textit{Sphere}. Exactly. The one Square produces a
  2155. \textit{Something-which-you-do-not-as-yet-know-a-name-for-but-which-we-call-a-Cube}
  2156. with \textit{eight} terminal Points. Now are you convinced?
  2157. \textit{I}. And has this Creature sides, as well as Angles or what you call ``terminal
  2158. Points''?
  2159. \textit{Sphere}. Of course; and all according to Analogy. But, by the way, not what \textit{you} call sides, but what \textit{we} call sides. You would call them solids.
  2160. \textit{I}. And how many solids or sides will appertain to this Being whom I am to
  2161. generate by the motion of my inside in an ``upward'' direction, and whom you
  2162. call a Cube?
  2163. \textit{Sphere}. How can you ask? And you a mathematician! The side of anything is
  2164. always, if I may so say, one Dimension behind the thing. Consequently, as
  2165. there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may so
  2166. say, has 2 sides (for the points of a Line may be called by courtesy, its
  2167. sides); a Square has 4 sides; 0, 2, 4; what Progression do you call that?
  2168. \textit{I}. Arithmetical.
  2169. \textit{Sphere}. And what is the next number?
  2170. \textit{I}. Six.
  2171. \textit{Sphere}. Exactly. Then you see you have answered your own question. The Cube
  2172. which you will generate will be bounded by six sides, that is to say, six of
  2173. your insides. You see it all now, eh?
  2174. ``Monster,'' I shrieked, ``be thou juggler, enchanter, dream, or devil, no more
  2175. will I endure thy mockeries. Either thou or I must perish.'' And saying these
  2176. words I precipitated myself upon him.
  2177. \chapter{How the Sphere, having in vain tried words, resorted to deeds}
  2178. It was in vain. I brought my hardest right angle into violent collision with
  2179. the Stranger, pressing on him with a force sufficient to have destroyed any
  2180. ordinary Circle: but I could feel him slowly and unarrestably slipping from my
  2181. contact; not edging to the right nor to the left, but moving somehow out of
  2182. the world, and vanishing into nothing. Soon there was a blank. But still I
  2183. heard the Intruder's voice.
  2184. \textit{Sphere}. Why will you refuse to listen to reason? I had hoped to find in you ---
  2185. as being a man of sense and an accomplished mathematician --- a fit apostle for
  2186. the Gospel of the Three Dimensions, which I am allowed to preach once only in
  2187. a thousand years: but now I know not how to convince you. Stay, I have it.
  2188. Deeds, and not words, shall proclaim the truth. Listen, my friend.
  2189. I have told you I can see from my position in Space the inside of all things
  2190. that you consider closed. For example, I see in yonder cupboard near which you
  2191. are standing, several of what you call boxes (but like everything else in
  2192. Flatland, they have no tops or bottom) full of money; I see also two tablets
  2193. of accounts. I am about to descend into that cupboard and to bring you one of
  2194. those tablets. I saw you lock the cupboard half an hour ago, and I know you
  2195. have the key in your possession. But I descend from Space; the doors, you see,
  2196. remain unmoved. Now I am in the cupboard and am taking the tablet. Now I have
  2197. it. Now I ascend with it.
  2198. I rushed to the closet and dashed the door open. One of the tablets was gone.
  2199. With a mocking laugh, the Stranger appeared in the other corner of the room,
  2200. and at the same time the tablet appeared upon the floor. I took it up. There
  2201. could be no doubt --- it was the missing tablet.
  2202. I groaned with horror, doubting whether I was not out of my sense; but the
  2203. Stranger continued: ``Surely you must now see that my explanation, and no
  2204. other, suits the phenomena. What you call Solid things are really superficial;
  2205. what you call Space is really nothing but a great Plane. I am in Space, and
  2206. look down upon the insides of the things of which you only see the outsides.
  2207. You could leave the Plane yourself, if you could but summon up the necessary
  2208. volition. A slight upward or downward motion would enable you to see all that
  2209. I can see.
  2210. ``The higher I mount, and the further I go from your Plane, the more I can see,
  2211. though of course I see it on a smaller scale. For example, I am ascending; now
  2212. I can see your neighbour the Hexagon and his family in their several
  2213. apartments; now I see the inside of the Theatre, ten doors off, from which the
  2214. audience is only just departing; and on the other side a Circle in his study,
  2215. sitting at his books. Now I shall come back to you. And, as a crowning proof,
  2216. what do you say to my giving you a touch, just the least touch, in your
  2217. stomach? It will not seriously injure you, and the slight pain you may suffer
  2218. cannot be compared with the mental benefit you will receive.''
  2219. Before I could utter a word of remonstrance, I felt a shooting pain in my
  2220. inside, and a demoniacal laugh seemed to issue from within me. A moment
  2221. afterwards the sharp agony had ceased, leaving nothing but a dull ache behind,
  2222. and the Stranger began to reappear, saying, as he gradually increased in size,
  2223. ``There, I have not hurt you much, have I? If you are not convinced now, I
  2224. don't know what will convince you. What say you?''
  2225. My resolution was taken. It seemed intolerable that I should endure existence
  2226. subject to the arbitrary visitations of a Magician who could thus play tricks
  2227. with one's very stomach. If only I could in any way manage to pin him against
  2228. the wall till help came!
  2229. Once more I dashed my hardest angle against him, at the same time alarming the
  2230. whole household by my cries for aid. I believe, at the moment of my onset, the
  2231. Stranger had sunk below our Plane, and really found difficulty in rising. In
  2232. any case he remained motionless, while I, hearing, as I thought, the sound of
  2233. some help approaching, pressed against him with redoubled vigor, and continued
  2234. to shout for assistance.
  2235. A convulsive shudder ran through the Sphere. ``This must not be,'' I thought I
  2236. heard him say: ``either he must listen to reason, or I must have recourse to
  2237. the last resource of civilization.'' Then, addressing me in a louder tone, he
  2238. hurriedly exclaimed, ``Listen: no stranger must witness what you have
  2239. witnessed. Send your Wife back at once, before she enters the apartment. The
  2240. Gospel of Three Dimensions must not be thus frustrated. Not thus must the
  2241. fruits of one thousand years of waiting be thrown away. I hear her coming.
  2242. Back! back! Away from me, or you must go with me --- wither you know not --- into
  2243. the Land of Three Dimensions!''
  2244. ``Fool! Madman! Irregular!'' I exclaimed; ``never will I release thee; thou shalt
  2245. pay the penalty of thine impostures.''
  2246. ``Ha! Is it come to this?'' thundered the Stranger: ``then meet your fate: out of
  2247. your Plane you go. Once, twice, thrice! 'Tis done!''
  2248. \chapter{How I came to Spaceland, and what I saw there}
  2249. An unspeakable horror seized me. There was a darkness; then a dizzy, sickening
  2250. sensation of sight that was not like seeing; I saw a Line that was no Line;
  2251. Space that was not Space: I was myself, and not myself. When I could find
  2252. voice, I shrieked loud in agony, ``Either this is madness or it is Hell.'' ``It
  2253. is neither, calmly replied the voice of the Sphere, ``it is Knowledge; it is
  2254. Three Dimensions: open your eye once again and try to look steadily.''
  2255. I looked, and, behold, a new world! There stood before me, visibly
  2256. incorporate, all that I had before inferred, conjectured, dreamed, of perfect
  2257. Circular beauty. What seemed the centre of the Stranger's form lay open to my
  2258. view: yet I could see no heart, lungs, nor arteries, only a beautiful
  2259. harmonious Something --- for which I had no words; but you, my Readers in
  2260. Spaceland, would call it the surface of the Sphere.
  2261. Prostrating myself mentally before my Guide, I cried, ``How it is, O divine
  2262. ideal of consummate loveliness and wisdom that I see thy inside, and yet
  2263. cannot discern thy heart, thy lungs, thy arteries, thy liver?'' ``What you think
  2264. you see, you see not,'' he replied; ``it is not giving to you, nor to any other
  2265. Being, to behold my internal parts. I am of a different order of Beings from
  2266. those in Flatland. Were I a Circle, you could discern my intestines, but I am
  2267. a Being, composed as I told you before, of many Circles, the Many in the One,
  2268. called in this country a Sphere. And, just as the outside of a Cube is a
  2269. Square, so the outside of a Sphere represents the appearance of a Circle.''
  2270. Bewildered though I was by my Teacher's enigmatic utterance, I no longer
  2271. chafed against it, but worshipped him in silent adoration. He continued, with
  2272. more mildness in his voice. ``Distress not yourself if you cannot at first
  2273. understand the deeper mysteries of Spaceland. By degrees they will dawn upon
  2274. you. Let us begin by casting back a glance at the region whence you came.
  2275. Return with me a while to the plains of Flatland and I will shew you that
  2276. which you have often reasoned and thought about, but never seen with the sense
  2277. of sight --- a visible angle.'' ``Impossible!'' I cried; but, the Sphere leading
  2278. the way, I followed as if in a dream, till once more his voice arrested me:
  2279. ``Look yonder, and behold your own Pentagonal house, and all its inmates.''
  2280. \begin{center}
  2281. \includegraphics[trim=20mm 0mm 20mm 0mm,scale=0.7]{fig9}
  2282. \end{center}
  2283. I looked below, and saw with my physical eye all that domestic individuality
  2284. which I had hitherto merely inferred with the understanding. And how poor and
  2285. shadowy was the inferred conjecture in comparison with the reality which I now
  2286. behold! My four Sons calmly asleep in the North-Western rooms, my two orphan
  2287. Grandsons to the South; the Servants, the Butler, my Daughter, all in their
  2288. several apartments. Only my affectionate Wife, alarmed by my continued
  2289. absence, had quitted her room and was roving up and down in the Hall,
  2290. anxiously awaiting my return. Also the Page, aroused by my cries, had left his
  2291. room, and under pretext of ascertaining whether I had fallen somewhere in a
  2292. faint, was prying into the cabinet in my study. All this I could now \textit{see}, not
  2293. merely infer; and as we came nearer and nearer, I could discern even the
  2294. contents of my cabinet, and the two chests of gold, and the tablets of which
  2295. the Sphere had made mention.
  2296. Touched by my Wife's distress, I would have sprung downward to reassure her,
  2297. but I found myself incapable of motion. ``Trouble not yourself about your
  2298. Wife,'' said my Guide: ``she will not be long left in anxiety; meantime, let us
  2299. take a survey of Flatland.''
  2300. Once more I felt myself rising through space. It was even as the Sphere had
  2301. said. The further we receded from the object we beheld, the larger became the
  2302. field of vision. My native city, with the interior of every house and every
  2303. creature therein, lay open to my view in miniature. We mounted higher, and lo,
  2304. the secrets of the earth, the depths of the mines and inmost caverns of the
  2305. hills, were bared before me.
  2306. Awestruck at the sight of the mysteries of the earth, thus unveiled before my
  2307. unworthy eye, I said to my Companion, ``Behold, I am become as a God. For the
  2308. wise men in our country say that to see all things, or as they express it,
  2309. \textit{omnividence}, is the attribute of God alone.'' There was something of scorn in
  2310. the voice of my Teacher as he made answer: ``Is it so indeed? Then the very
  2311. pick-pockets and cut-throats of my country are to be worshipped by your wise
  2312. men as being Gods: for there is not one of them that does not see as much as
  2313. you see now. But trust me, your wise men are wrong.''
  2314. \textit{I}. Then is omnividence the attribute of
  2315. others besides Gods?
  2316. \textit{Sphere}. I do not know. But, if a pick-pocket or a cut-throat of our country
  2317. can see everything that is in your country, surely that is no reason why the
  2318. pick-pocket or cut-throat should be accepted by you as a God. This
  2319. omnividence, as you call it --- it is not a common word in Spaceland --- does it
  2320. make you more just, more merciful, less selfish, more loving? Not in the
  2321. least. Then how does it make you more divine?
  2322. \textit{I}. ``More merciful, more loving!'' But these are the qualities of women! And we
  2323. know that a Circle is a higher Being than a Straight Line, in so far as
  2324. knowledge and wisdom are more to be esteemed than mere affection.
  2325. \textit{Sphere}. It is not for me to classify human faculties according to merit. Yet
  2326. many of the best and wisest in Spaceland think more of the affections than of
  2327. the understanding, more of your despised Straight Lines than of your belauded
  2328. Circles. But enough of this. Look yonder. Do you know that building?
  2329. I looked, and afar off I saw an immense Polygonal structure, in which I
  2330. recognized the General Assembly Hall of the States of Flatland, surrounded by
  2331. dense lines of Pentagonal buildings at right angles to each other, which I
  2332. knew to be streets; and I perceived that I was approaching the great
  2333. Metropolis.
  2334. ``Here we descend,'' said my Guide. It was now morning, the first hour of the
  2335. first day of the two thousandth year of our era. Acting, as was their wont, in
  2336. strict accordance with precedent, the highest Circles of the realm were
  2337. meeting in solemn conclave, as they had met on the first hour of the first day
  2338. of the year 1000, and also on the first hour of the first day of the year 0.
  2339. The minutes of the previous meetings were now read by one whom I at once
  2340. recognized as my brother, a perfectly Symmetrical Square, and the Chief Clerk
  2341. of the High Council. It was found recorded on each occasion that: ``Whereas the
  2342. States had been troubled by divers ill-intentioned persons pretending to have
  2343. received revelations from another World, and professing to produce
  2344. demonstrations whereby they had instigated to frenzy both themselves and
  2345. others, it had been for this cause unanimously resolved by the Grand Council
  2346. that on the first day of each millenary, special injunctions be sent to the
  2347. Prefects in the several districts of Flatland, to make strict search for such
  2348. misguided persons, and without formality of mathematical examination, to
  2349. destroy all such as were Isosceles of any degree, to scourge and imprison any
  2350. regular Triangle, to cause any Square or Pentagon to be sent to the district
  2351. Asylum, and to arrest any one of higher rank, sending him straightway to the
  2352. Capital to be examined and judged by the Council.''
  2353. ``You hear your fate,'' said the Sphere to me, while the Council was passing for
  2354. the third time the formal resolution. ``Death or imprisonment awaits the
  2355. Apostle of the Gospel of Three Dimensions.'' ``Not so,'' replied I, ``the matter
  2356. is now so clear to me, the nature of real space so palpable, that methinks I
  2357. could make a child understand it. Permit me but to descend at this moment and
  2358. enlighten them.'' ``Not yet,'' said my Guide, ``the time will come for that.
  2359. Meantime I must perform my mission. Stay thou there in thy place.'' Saying
  2360. these words, he leaped with great dexterity into the sea (if I may so call it)
  2361. of Flatland, right in the midst of the ring of Counsellors. ``I come,'' said he,
  2362. ``to proclaim that there is a land of Three Dimensions.''
  2363. I could see many of the younger Counsellors start back in manifest horror, as
  2364. the Sphere's circular section widened before them. But on a sign from the
  2365. presiding Circle --- who shewed not the slightest alarm or surprise --- six
  2366. Isosceles of a low type from six different quarters rushed upon the Sphere.
  2367. ``We have him,'' they cried; ``No; yes; we have him still! he's going! he's
  2368. gone!''
  2369. ``My Lords,'' said the President to the Junior Circles of the Council, ``there is
  2370. not the slightest need for surprise; the secret archives, to which I alone
  2371. have access, tell me that a similar occurrence happened on the last two
  2372. millennial commencements. You will, of course, say nothing of these trifles
  2373. outside the Cabinet.''
  2374. Raising his voice, he now summoned the guards. ``Arrest the policemen; gag
  2375. them. You know your duty.'' After he had consigned to their fate the wretched
  2376. policemen --- ill-fated and unwilling witnesses of a State-secret which they
  2377. were not to be permitted to reveal --- he again addressed the Counsellors. ``My
  2378. Lords, the business of the Council being concluded, I have only to wish you a
  2379. happy New Year.'' Before departing, he expressed, at some length, to the Clerk,
  2380. my excellent but most unfortunate brother, his sincere regret that, in
  2381. accordance with precedent and for the sake of secrecy, he must condemn him to
  2382. perpetual imprisonment, but added his satisfaction that, unless some mention
  2383. were made by him of that day's incident, his life would be spared.
  2384. \chapter{How, though the Sphere showed me other mysteries of Spaceland, I still desire more; and what came of it}
  2385. When I saw my poor brother led away to imprisonment, I attempted to leap down
  2386. into the Council Chamber, desiring to intercede on his behalf, or at least bid
  2387. him farewell. But I found that I had no motion of my own. I absolutely
  2388. depended on the volition of my Guide, who said in gloomy tones, ``Heed not thy
  2389. brother; haply thou shalt have ample time hereafter to condole with him.
  2390. Follow me.''
  2391. \begin{center}
  2392. \includegraphics[trim=20mm 0mm 20mm 0mm,scale=0.6]{fig10}
  2393. \end{center}
  2394. Once more we ascended into space. ``Hitherto,'' said the Sphere, ``I have shewn
  2395. you naught save Plane Figures and their interiors. Now I must introduce you to
  2396. Solids, and reveal to you the plan upon which they are constructed. Behold
  2397. this multitude of moveable square cards. See, I put one on another, not, as
  2398. you supposed, Northward of the other, but \textit{on} the other. Now a second, now a
  2399. third. See, I am building up a Solid by a multitude of Squares parallel to one
  2400. another. Now the Solid is complete, being as high as it is long and broad, and
  2401. we call it a Cube.''
  2402. ``Pardon me, my Lord,'' replied I; ``but to my eye the appearance is as of an
  2403. Irregular Figure whose inside is laid open to view; in other words, methinks I
  2404. see no Solid, but a Plane such as we infer in Flatland; only of an
  2405. Irregularity which betokens some monstrous criminal, so that the very sight of
  2406. it is painful to my eyes.''
  2407. ``True,'' said the Sphere; ``it appears to you a Plane, because you are not
  2408. accustomed to light and shade and perspective; just as in Flatland a Hexagon
  2409. would appear a Straight Line to one who has not the Art of Sight Recognition.
  2410. But in reality it is a Solid, as you shall learn by the sense of Feeling.''
  2411. He then introduced me to the Cube, and I found that this marvellous Being was
  2412. indeed no Plane, but a Solid; and that he was endowed with six plane sides and
  2413. eight terminal points called solid angles; and I remembered the saying of the
  2414. Sphere that just such a Creature as this would be formed by the Square moving,
  2415. in Space, parallel to himself: and I rejoiced to think that so insignificant a
  2416. Creature as I could in some sense be called the Progenitor of so illustrious
  2417. an offspring.
  2418. But still I could not fully understand the meaning of what my Teacher had told
  2419. me concerning ``light'' and ``shade'' and ``perspective''; and I did not hesitate to
  2420. put my difficulties before him.
  2421. Were I to give the Sphere's explanation of these matters, succinct and clear
  2422. though it was, it would be tedious to an inhabitant of Space, who knows these
  2423. things already. Suffice it, that by his lucid statements, and by changing the
  2424. position of objects and lights, and by allowing me to feel the several objects
  2425. and even his own sacred Person, he at last made all things clear to me, so
  2426. that I could now readily distinguish between a Circle and a Sphere, a Plane
  2427. Figure and a Solid.
  2428. This was the Climax, the Paradise, of my strange eventful History. Henceforth
  2429. I have to relate the story of my miserable Fall: --- most miserable, yet surely
  2430. most undeserved! For why should the thirst for knowledge be aroused, only to
  2431. be disappointed and punished? My volition shrinks from the painful task of
  2432. recalling my humiliation; yet, like a second Prometheus, I will endure this
  2433. and worse, if by any means I may arouse in the interiors of Plane and Solid
  2434. Humanity a spirit of rebellion against the Conceit which would limit our
  2435. Dimensions to Two or Three or any number short of Infinity. Away then with all
  2436. personal considerations! Let me continue to the end, as I began, without
  2437. further digressions or anticipations, pursuing the plain path of dispassionate
  2438. History. The exact facts, the exact words, --- and they are burnt in upon my
  2439. brain, --- shall be set down without alteration of an iota; and let my Readers
  2440. judge between me and Destiny.
  2441. The Sphere would willingly have continued his lessons by indoctrinating me in
  2442. the conformation of all regular Solids, Cylinders, Cones, Pyramids,
  2443. Pentahedrons, Hexahedrons, Dodecahedrons, and Spheres: but I ventured to
  2444. interrupt him. Not that I was wearied of knowledge. On the contrary, I
  2445. thirsted for yet deeper and fuller draughts than he was offering to me.
  2446. ``Pardon me,'' said I, ``O Thou Whom I must no longer address as the Perfection
  2447. of all Beauty; but let me beg thee to vouchsafe thy servant a sight of thine
  2448. interior.''
  2449. \textit{Sphere}. My what?
  2450. \textit{I}. Thine interior: thy stomach, thy intestines.
  2451. \textit{Sphere}. Whence this ill-timed impertinent request? And what mean you by saying
  2452. that I am no longer the Perfection of all Beauty?
  2453. \textit{I}. My Lord, your own wisdom has taught me to aspire to One even more great,
  2454. more beautiful, and more closely approximate to Perfection than yourself. As
  2455. you yourself, superior to all Flatland forms, combine many Circles in One, so
  2456. doubtless there is One above you who combines many Spheres in One Supreme
  2457. Existence, surpassing even the Solids of Spaceland. And even as we, who are
  2458. now in Space, look down on Flatland and see the insides of all things, so of a
  2459. certainty there is yet above us some higher, purer region, whither thou dost
  2460. surely purpose to lead me --- O Thou Whom I shall always call, everywhere and in
  2461. all Dimensions, my Priest, Philosopher, and Friend --- some yet more spacious
  2462. Space, some more dimensionable Dimensionality, from the vantage-ground of
  2463. which we shall look down together upon the revealed insides of Solid things,
  2464. and where thine own intestines, and those of thy kindred Spheres, will lie
  2465. exposed to the view of the poor wandering exile from Flatland, to whom so much
  2466. has already been vouchsafed.
  2467. \textit{Sphere}. Pooh! Stuff! Enough of this trifling! The time is short, and much
  2468. remains to be done before you are fit to proclaim the Gospel of Three
  2469. Dimensions to your blind benighted countrymen in Flatland.
  2470. \textit{I}. Nay, gracious Teacher, deny me not what I know it is in thy power to
  2471. perform. Grant me but one glimpse of thine interior, and I am satisfied for
  2472. ever, remaining henceforth thy docile pupil, thy unemacipable slave, ready to
  2473. receive all thy teachings and to feed upon the words that fall from thy lips.
  2474. \textit{Sphere}. Well, then, to content and silence you, let me say at once, I would
  2475. shew you what you wish if I could; but I cannot. Would you have me turn my
  2476. stomach inside out to oblige you?
  2477. \textit{I}. But my Lord has shewn me the intestines of all my countrymen in the Land of
  2478. Two Dimensions by taking me with him into the Land of Three. What therefore
  2479. more easy than now to take his servant on a second journey into the blessed
  2480. region of the Fourth Dimension, where I shall look down with him once more
  2481. upon this land of Three Dimensions, and see the inside of every
  2482. three-dimensioned house, the secrets of the solid earth, the treasures of the
  2483. mines of Spaceland, and the intestines of every solid living creature, even
  2484. the noble and adorable Spheres.
  2485. \textit{Sphere}. But where is this land of Four Dimensions?
  2486. \textit{I}. I know not: but doubtless my Teacher knows.
  2487. \textit{Sphere}. Not I. There is no such land. The very idea of it is utterly
  2488. inconceivable.
  2489. \textit{I}. Not inconceivable, my Lord, to me, and therefore still less inconceivable
  2490. to my Master. Nay, I despair not that, even here, in this region of Three
  2491. Dimensions, your Lordship's art may make the Fourth Dimension visible to me;
  2492. just as in the Land of Two Dimensions my Teacher's skill would fain have
  2493. opened the eyes of his blind servant to the invisible presence of a Third
  2494. Dimension, though I saw it not.
  2495. Let me recall the past. Was I not taught below that when I saw a Line and
  2496. inferred a Plane, I in reality saw a Third unrecognized Dimension, not the
  2497. same as brightness, called ``height''? And does it not now follow that, in this
  2498. region, when I see a Plane and infer a Solid, I really see a Fourth
  2499. unrecognized Dimension, not the same as colour, but existent, though
  2500. infinitesimal and incapable of measurement?
  2501. And besides this, there is the Argument from Analogy of Figures.
  2502. \textit{Sphere}. Analogy! Nonsense: what analogy?
  2503. \textit{I}. Your Lordship tempts his servant to see whether he remembers the
  2504. revelations imparted to him. Trifle not with me, my Lord; I crave, I thirst,
  2505. for more knowledge. Doubtless we cannot see that other higher Spaceland now,
  2506. because we have no eye in our stomachs. But, just as there was the realm of
  2507. Flatland, though that poor puny Lineland Monarch could neither turn to left
  2508. nor right to discern it, and just as there was close at hand, and touching my
  2509. frame, the land of Three Dimensions, though I, blind senseless wretch, had no
  2510. power to touch it, no eye in my interior to discern it, so of a surety there
  2511. is a Fourth Dimension, which my Lord perceives with the inner eye of thought.
  2512. And that it must exist my Lord himself has taught me. Or can he have forgotten
  2513. what he himself imparted to his servant?
  2514. In One Dimension, did not a moving Point produce a Line with two terminal
  2515. points?
  2516. In Two Dimensions, did not a moving Line produce a Square with four terminal
  2517. points?
  2518. In Three Dimensions, did not a moving Square produce --- did not this eye of
  2519. mine behold it --- that blessed Being, a Cube, with eight terminal points?
  2520. And in Four Dimensions shall not a moving Cube --- alas, for Analogy, and alas
  2521. for the Progress of Truth, if it be not so --- shall not, I say, the motion of a
  2522. divine Cube result in a still more divine Organization with \textit{sixteen} terminal
  2523. points?
  2524. Behold the infallible confirmation of the Series, 2, 4,
  2525. 8, 16: is not this a Geometrical Progression? Is not this --- if I might quote
  2526. my Lord's own words --- ``strictly according to Analogy''?
  2527. Again, was I not taught by my Lord that as in a Line there are \textit{two} bounding
  2528. Points, and in a Square there are \textit{four} bounding Lines, so in a Cube there must
  2529. be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is
  2530. not this an Arithmetical Progression? And consequently does it not of
  2531. necessity follow that the more divine offspring of the divine Cube in the Land
  2532. of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my
  2533. Lord has taught me to believe, ``strictly according to Analogy''?
  2534. O, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not
  2535. knowing the facts; and I appeal to your Lordship to confirm or deny my logical
  2536. anticipations. If I am wrong, I yield, and will no longer demand a Fourth
  2537. Dimension; but, if I am right, my Lord will listen to reason.
  2538. I ask therefore, is it, or is it not, the fact, that ere now your countrymen
  2539. also have witnessed the descent of Beings of a higher order than their own,
  2540. entering closed rooms, even as your Lordship entered mine, without the opening
  2541. of doors or windows, and appearing and vanishing at will? On the reply to this
  2542. question I am ready to stake everything. Deny it, and I am henceforth silent.
  2543. Only vouchsafe an answer.
  2544. \textit{Sphere} (\textit{after a pause}). It is reported so. But men are divided in opinion as
  2545. to the facts. And even granting the facts, they explain them in different
  2546. ways. And in any case, however great may be the number of different
  2547. explanations, no one has adopted or suggested the theory of a Fourth
  2548. Dimension. Therefore, pray have done with this trifling, and let us return to
  2549. business.
  2550. \textit{I}. I was certain of it. I was certain that my anticipations would be
  2551. fulfilled. And now have patience with me and answer me yet one more question,
  2552. best of Teachers! Those who have thus appeared --- no one knows whence --- and
  2553. have returned --- no one knows whither --- have they also contracted their
  2554. sections and vanished somehow into that more Spacious Space, whither I now
  2555. entreat you to conduct me?
  2556. \textit{Sphere} (\textit{moodily}). They have vanished, certainly --- if they ever appeared. But
  2557. most people say that these visions arose from the thought --- you will not
  2558. understand me --- from the brain; from the perturbed angularity of the Seer.
  2559. \textit{I}. Say they so? Oh, believe them not. Or if it indeed be so, that this other
  2560. Space is really Thoughtland, then take me to that blessed Region where I in
  2561. Thought shall see the insides of all solid things. There, before my ravished
  2562. eye, a Cube moving in some altogether new direction, but strictly according to
  2563. Analogy, so as to make every particle of his interior pass through a new kind
  2564. of Space, with a wake of its own --- shall create a still more perfect
  2565. perfection than himself, with sixteen terminal Extra-solid angles, and Eight
  2566. solid Cubes for his Perimeter. And once there, shall we stay our upward
  2567. course? In that blessed region of Four Dimensions, shall we linger at the
  2568. threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve
  2569. that our ambition shall soar with our corporal ascent. Then, yielding to our
  2570. intellectual onset, the gates of the Six Dimension shall fly open; after that
  2571. a Seventh, and then an Eighth ---
  2572. How long I should have continued I know not. In vain did the Sphere, in his
  2573. voice of thunder, reiterate his command of silence, and threaten me with the
  2574. direst penalties if I persisted. Nothing could stem the flood of my ecstatic
  2575. aspirations. Perhaps I was to blame; but indeed I was intoxicated with the
  2576. recent draughts of Truth to which he himself had introduced me. However, the
  2577. end was not long in coming. My words were cut short by a crash outside, and a
  2578. simultaneous crash inside me, which impelled me through space with a velocity
  2579. that precluded speech. Down! down! down! I was rapidly descending; and I knew
  2580. that return to Flatland was my doom. One glimpse, one last and
  2581. never-to-be-forgotten glimpse I had of that dull level wilderness --- which was
  2582. now to become my Universe again --- spread out before my eye. Then a darkness.
  2583. Then a final, all-consummating thunder-peal; and, when I came to myself, I was
  2584. once more a common creeping Square, in my Study at home, listening to the
  2585. Peace-Cry of my approaching Wife.
  2586. \chapter{How the Sphere encouraged me in a Vision}
  2587. Although I had less than a minute for reflection, I felt, by a kind of
  2588. instinct, that I must conceal my experiences from my Wife. Not that I
  2589. apprehended, at the moment, any danger from her divulging my secret, but I
  2590. knew that to any Woman in Flatland the narrative of my adventures must needs
  2591. be unintelligible. So I endeavoured to reassure her by some story, invented
  2592. for the occasion, that I had accidentally fallen through the trap-door of the
  2593. cellar, and had there lain stunned.
  2594. The Southward attraction in our country is so slight that even to a Woman my
  2595. tale necessarily appeared extraordinary and well-nigh incredible; but my Wife,
  2596. whose good sense far exceeds that of the average of her Sex, and who perceived
  2597. that I was unusually excited, did not argue with me on the subject, but
  2598. insisted that I was ill and required repose. I was glad of an excuse for
  2599. retiring to my chamber to think quietly over what had happened. When I was at
  2600. last by myself, a drowsy sensation fell on me; but before my eyes closed I
  2601. endeavoured to reproduce the Third Dimension, and especially the process by
  2602. which a Cube is constructed through the motion of a Square. It was not so
  2603. clear as I could have wished; but I remembered that it must be ``Upward, and
  2604. yet not Northward,'' and I determined steadfastly to retain these words as the
  2605. clue which, if firmly grasped, could not fail to guide me to the solution. So
  2606. mechanically repeating, like a charm, the words, ``Upward, yet not Northward,''
  2607. I fell into a sound refreshing sleep.
  2608. During my slumber I had a dream. I thought I was once more by the side of the
  2609. Sphere, whose lustrous hue betokened that he had exchanged his wrath against
  2610. me for perfectly placability. We were moving together towards a bright but
  2611. infinitesimally small Point, to which my Master directed my attention. As we
  2612. approached, methought there issued from it a slight humming noise as from one
  2613. of your Spaceland bluebottles, only less resonant by far, so slight indeed
  2614. that even in the perfect stillness of the Vacuum through which we soared, the
  2615. sound reached not our ears till we checked our flight at a distant from it of
  2616. something under twenty human diagonals.
  2617. ``Look yonder,'' said my Guide, ``in Flatland thou hast lived; of Lineland thou
  2618. hast received a vision; thou hast soared with me to the heights of Spaceland;
  2619. now, in order to complete the range of thy experience, I conduct thee downward
  2620. to the lowest depth of existence, even to the realm of Pointland, the Abyss of
  2621. No dimensions.
  2622. ``Behold yon miserable creature. That Point is a Being like ourselves, but
  2623. confined to the non-dimensional Gulf. He is himself his own World, his own
  2624. Universe; of any other than himself he can form no conception; he knows not
  2625. Length, nor Breadth, nor Height, for he has had no experience of them; he has
  2626. no cognizance even of the number Two; nor has he a thought of Plurality; for
  2627. he is himself his One and All, being really Nothing. Yet mark his perfect
  2628. self-contentment, and hence learn his lesson, that to be self-contented is to
  2629. be vile and ignorant, and that to aspire is better than to be blindly and
  2630. impotently happy. Now listen.''
  2631. He ceased; and there arose from the little buzzing creature a tiny, low,
  2632. monotonous, but distinct tinkling, as from one of your Spaceland phonographs,
  2633. from which I caught these words, ``Infinite beatitude of existence! It is; and
  2634. there is nothing else beside It.''
  2635. ``What,'' said I, ``does the puny creature mean by `it'?'' ``He means himself,''
  2636. said the Sphere: ``have you not noticed before now, that babies and babyish
  2637. people who cannot distinguish themselves from the world, speak of themselves
  2638. in the Third Person? But hush!''
  2639. ``It fills all Space,'' continued the little soliloquizing Creature, ``and what
  2640. It fills, It is. What It thinks, that It utters; and what It utters, that It
  2641. hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it
  2642. is the One, and yet the All in All. Ah, the happiness, ah, the happiness of
  2643. Being!''
  2644. ``Can you not startle the little thing out of its complacency?'' said I. ``Tell
  2645. it what it really is, as you told me; reveal to it the narrow limitations of
  2646. Pointland, and lead it up to something higher.'' ``That is no easy task,'' said
  2647. my Master; ``try you.''
  2648. Hereon, raising by voice to the uttermost, I addressed the Point as follows:
  2649. ``Silence, silence, contemptible Creature. You call yourself the All in All,
  2650. but you are the Nothing: your so-called Universe is a mere speck in a Line,
  2651. and a Line is a mere shadow as compared with ---'' ``Hush, hush, you have said
  2652. enough,'' interrupted the Sphere, ``now listen, and mark the effect of your
  2653. harangue on the King of Pointland.''
  2654. The lustre of the Monarch, who beamed more brightly than ever upon hearing my
  2655. words, shewed clearly that he retained his complacency; and I had hardly
  2656. ceased when he took up his strain again. ``Ah, the joy, ah, the joy of Thought!
  2657. What can It not achieve by thinking! Its own Thought coming to Itself,
  2658. suggestive of its disparagement, thereby to enhance Its happiness! Sweet
  2659. rebellion stirred up to result in triumph! Ah, the divine creative power of
  2660. the All in One! Ah, the joy, the joy of Being!''
  2661. ``You see,'' said my Teacher, ``how little your words have done. So far as the
  2662. Monarch understand them at all, he accepts them as his own --- for he cannot
  2663. conceive of any other except himself --- and plumes himself upon the variety of
  2664. `Its Thought' as an instance of creative Power. Let us leave this God of
  2665. Pointland to the ignorant fruition of his omnipresence and omniscience:
  2666. nothing that you or I can do can rescue him from his self-satisfaction.''
  2667. After this, as we floated gently back to Flatland, I could hear the mild voice
  2668. of my Companion pointing the moral of my vision, and stimulating me to aspire,
  2669. and to teach others to aspire. He had been angered at first --- he confessed ---
  2670. by my ambition to soar to Dimensions above the Third; but, since then, he had
  2671. received fresh insight, and he was not too proud to acknowledge his error to a
  2672. Pupil. Then he proceeded to initiate me into mysteries yet higher than those I
  2673. had witnessed, shewing me how to construct Extra-Solids by the motion of
  2674. Solids, and Double Extra-Solids by the motion of Extra-Solids, and all
  2675. ``strictly according to Analogy,'' all by methods so simple, so easy, as to be
  2676. patent even to the Female Sex.
  2677. \chapter{How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success}
  2678. I awoke rejoicing, and began to reflect on the glorious career before me. I
  2679. would go forth, methought, at once, and evangelize the whole of Flatland. Even
  2680. to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I
  2681. would begin with my Wife.
  2682. Just as I had decided on the plan of my operations, I heard the sound of many
  2683. voices in the street commanding silence. Then followed a louder voice. It was
  2684. a herald's proclamation. Listening attentively, I recognized the words of the
  2685. Resolution of the Council, enjoining the arrest, imprisonment, or execution of
  2686. any one who should pervert the minds of people by delusions, and by professing
  2687. to have received revelations from another World.
  2688. I reflected. This danger was not to be trifled with. It would be better to
  2689. avoid it by omitting all mention of my Revelation, and by proceeding on the
  2690. path of Demonstration --- which after all, seemed so simple and so conclusive
  2691. that nothing would be lost by discarding the former means. ``Upward, not
  2692. Northward'' --- was the clue to the whole proof. It had seemed to me fairly clear
  2693. before I fell asleep; and when I first awoke, fresh from my dream, it had
  2694. appeared as patent as Arithmetic; but somehow it did not seem to me quite so
  2695. obvious now. Though my Wife entered the room opportunely at just that moment,
  2696. I decided, after we had exchanged a few words of commonplace conversation, not
  2697. to begin with her.
  2698. My Pentagonal Sons were men of character and standing, and physicians of no
  2699. mean reputation, but not great in mathematics, and, in that respect, unfit for
  2700. my purpose. But it occurred to me that a young and docile Hexagon, with a
  2701. mathematical turn, would be a most suitable pupil. Why therefore not make my
  2702. first experiment with my little precocious Grandson, whose casual remarks on
  2703. the meaning of $ 3^3 $ had met with the approval of the Sphere?
  2704. Discussing the matter with him, a mere boy, I should be in perfect safety; for
  2705. he would know nothing of the Proclamation of the Council; whereas I could not
  2706. feel sure that my Sons --- so greatly did their patriotism and reverence for the
  2707. Circles predominate over mere blind affection --- might not feel compelled to
  2708. hand me over to the Prefect, if they found me seriously maintaining the
  2709. seditious heresy of the Third Dimension.
  2710. But the first thing to be done was to satisfy in some way the curiosity of my
  2711. Wife, who naturally wished to know something of the reasons for which the
  2712. Circle had desired that mysterious interview, and of the means by which he had
  2713. entered the house. Without entering into the details of the elaborate account
  2714. I gave her, --- an account, I fear, not quite so consistent with truth as my
  2715. Readers in Spaceland might desire, --- I must be content with saying that I
  2716. succeeded at last in persuading her to return quitely to her household duties
  2717. without eliciting from me any reference to the World of Three Dimensions. This
  2718. done, I immediately sent for my Grandson; for, to confess the truth, I felt
  2719. that all that I had seen and heard was in some strange way slipping away from
  2720. me, like the image of a half- grasped, tantalizing dream, and I longed to
  2721. essay my skill in making a first disciple.
  2722. When my Grandson entered the room I carefully secured the door. Then, sitting
  2723. down by his side and taking our mathematical tablets, --- or, as you would call
  2724. them, Lines --- I told him we would resume the lesson of yesterday. I taught him
  2725. once more how a Point by motion in One Dimension produces a Line, and how a
  2726. straight Line in Two Dimensions produces a Square. After this, forcing a
  2727. laugh, I said, ``And now, you scamp, you wanted to make believe that a Square
  2728. may in the same way by motion `Upward, not Northward' produce another figure,
  2729. a sort of extra square in Three Dimensions. Say that again, you young rascal.''
  2730. At this moment we heard once more the herald's ``O yes! O yes!'' outside in the
  2731. street proclaiming the Resolution of the Council. Young though he was, my
  2732. Grandson --- who was unusually intelligent for his age, and bred up in perfect
  2733. reverence for the authority of the Circles --- took in the situation with an
  2734. acuteness for which I was quite unprepared. He remained silent till the last
  2735. words of the Proclamation had died away, and then, bursting into tears, ``Dear
  2736. Grandpapa,'' he said, ``that was only my fun, and of course I meant nothing at
  2737. all by it; and we did not know anything then about the new Law; and I don't
  2738. think I said anything about the Third Dimension; and I am sure I did not say
  2739. one word about `Upward, not Northward,' for that would be such nonsense, you
  2740. know. How could a thing move Upward, and not Northward? Upward and not
  2741. Northward! Even if I were a baby, I could not be so absurd as that. How silly
  2742. it is! Ha! ha! ha!''
  2743. ``Not at all silly,'' said I, losing my temper; ``here for example, I take this
  2744. Square,'' and, at the word, I grasped a moveable Square, which was lying at
  2745. hand --- ``and I move it, you see, not Northward but --- yes, I move it Upward ---
  2746. that is to say, Northward but I move it somewhere --- not exactly like this, but
  2747. somehow ---'' Here I brought my sentence to an inane conclusion, shaking the
  2748. Square about in a purposeless manner, much to the amusement of my Grandson,
  2749. who burst out laughing louder than ever, and declared that I was not teaching
  2750. him, but joking with him; and so saying he unlocked the door and ran out of
  2751. the room. Thus ended my first attempt to convert a pupil to the Gospel of
  2752. Three Dimensions.
  2753. \chapter{How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result}
  2754. My failure with my Grandson did not encourage me to communicate my secret to
  2755. others of my household; yet neither was I led by it to despair of success.
  2756. Only I saw that I must not wholly rely on the catch-phrase, ``Upward, not
  2757. Northward,'' but must rather endeavour to seek a demonstration by setting
  2758. before the public a clear view of the whole subject; and for this purpose it
  2759. seemed necessary to resort to writing.
  2760. So I devoted several months in privacy to the composition of a treatise on the
  2761. mysteries of Three Dimensions. Only, with the view of evading the Law, if
  2762. possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in
  2763. theory, a Figure could look down upon Flatland and see simultaneously the
  2764. insides of all things, and where it was possible that there might be supposed
  2765. to exist a Figure environed, as it were, with six Squares, and containing
  2766. eight terminal Points. But in writing this book I found myself sadly hampered
  2767. by the impossibility of drawing such diagrams as were necessary for my
  2768. purpose: for of course, in our country of Flatland, there are no tablets but
  2769. Lines, and no diagrams but Lines, all in one straight Line and only
  2770. distinguishable by difference of size and brightness; so that, when I had
  2771. finished my treatise (which I entitled, ``Through Flatland to Thoughtland'') I
  2772. could not feel certain that many would understand my meaning.
  2773. Meanwhile my life was under a cloud. All pleasures palled upon me; all sights
  2774. tantalized and tempted me to outspoken treason, because I could not but
  2775. compare what I saw in Two Dimensions with what it really was if seen in Three,
  2776. and could hardly refrain from making my comparisons aloud. I neglected my
  2777. clients and my own business to give myself to the contemplation of the
  2778. mysteries which I had once beheld, yet which I could impart to no one, and
  2779. found daily more difficult to reproduce even before my own mental vision.
  2780. One day, about eleven months after my return from Spaceland, I tried to see a
  2781. Cube with my eye closed, but failed; and though I succeeded afterwards, I was
  2782. not then quite certain (nor have I been ever afterwards) that I had exactly
  2783. realized the original. This made me more melancholy than before, and
  2784. determined me to take some step; yet what, I knew not. I felt that I would
  2785. have been willing to sacrifice my life for the Cause, if thereby I could have
  2786. produced conviction. But if I could not convince my Grandson, how could I
  2787. convince the highest and most developed Circles in the land?
  2788. And yet at times my spirit was too strong for me, and I gave vent to
  2789. dangerous utterances. Already I was considered heterodox if not treasonable,
  2790. and I was keenly alive to the danger of my position; nevertheless I could not
  2791. at times refrain from bursting out into suspicious or half-seditious
  2792. utterances, even among the highest Polygonal or Circular society. When, for
  2793. example, the question arose about the treatment of those lunatics who said
  2794. that they had received the power of seeing the insides of things, I would
  2795. quote the saying of an ancient Circle, who declared that prophets and inspired
  2796. people are always considered by the majority to be mad; and I could not help
  2797. occasionally dropping such expressions as ``the eye that discerns the interiors
  2798. of things,'' and ``the all-seeing land''; once or twice I even let fall the
  2799. forbidden terms ``the Third and Fourth Dimensions.'' At last, to complete a
  2800. series of minor indiscretions, at a meeting of our Local Speculative Society
  2801. held at the palace of the Prefect himself, --- some extremely silly person
  2802. having read an elaborate paper exhibiting the precise reasons why Providence
  2803. has limited the number of Dimensions to Two, and why the attribute of
  2804. omnividence is assigned to the Supreme alone --- I so far forgot myself as to
  2805. give an exact account of the whole of my voyage with the Sphere into Space,
  2806. and to the Assembly Hall in our Metropolis, and then to Space again, and of my
  2807. return home, and of everything that I had seen and heard in fact or vision. At
  2808. first, indeed, I pretended that I was describing the imaginary experiences of
  2809. a fictitious person; but my enthusiasm soon forced me to throw off all
  2810. disguise, and finally, in a fervent peroration, I exhorted all my hearers to
  2811. divest themselves of prejudice and to become believers in the Third Dimension.
  2812. Need I say that I was at once arrested and taken before the Council?
  2813. Next morning, standing in the very place where but a very few months ago the
  2814. Sphere had stood in my company, I was allowed to begin and to continue my
  2815. narration unquestioned and uninterrupted. But from the first I foresaw my
  2816. fate; for the President, noting that a guard of the better sort of Policemen
  2817. was in attendance, of angularity little, if at all, under 55 degrees, ordered
  2818. them to be relieved before I began my defence, by an inferior class of 2 or 3
  2819. degrees. I knew only too well what that meant. I was to be executed or
  2820. imprisoned, and my story was to be kept secret from the world by the
  2821. simultaneous destruction of the officials who had heard it; and, this being
  2822. the case, the President desired to substitute the cheaper for the more
  2823. expensive victims.
  2824. After I had concluded my defence, the President, perhaps perceiving that some
  2825. of the junior Circles had been moved by evident earnestness, asked me two
  2826. questions: ---
  2827. \begin{enumerate}
  2828. \item Whether I could indicate the direction which I meant when I used the
  2829. words ``Upward, not Northward''?
  2830. \item Whether I could by any diagrams or descriptions (other than the
  2831. enumeration of imaginary sides and angles) indicate the Figure I was
  2832. pleased to call a Cube? I declared that I could say nothing more, and
  2833. that I must commit myself to the Truth, whose cause would surely
  2834. prevail in the end.
  2835. \end{enumerate}
  2836. The President replied that he quite concurred in my sentiment, and that I
  2837. could not do better. I must be sentenced to perpetual imprisonment; but if the
  2838. Truth intended that I should emerge from prison and evangelize the world, the
  2839. Truth might be trusted to bring that result to pass. Meanwhile I should be
  2840. subjected to no discomfort that was not necessary to preclude escape, and,
  2841. unless I forfeited the privilege by misconduct, I should be occasionally
  2842. permitted to see my brother who had preceded me to my prison.
  2843. Seven years have elapsed and I am still a prisoner, and --- if I except the
  2844. occasional visits of my brother --- debarred from all companionship save that of
  2845. my jailers. My brother is one of the best of Squares, just sensible, cheerful,
  2846. and not without fraternal affection; yet I confess that my weekly interviews,
  2847. at least in one respect, cause me the bitterest pain. He was present when the
  2848. Sphere manifested himself in the Council Chamber; he saw the Sphere's changing
  2849. sections; he heard the explanation of the phenomena then give to the Circles.
  2850. Since that time, scarcely a week has passed during seven whole years, without
  2851. his hearing from me a repetition of the part I played in that manifestation,
  2852. together with ample descriptions of all the phenomena in Spaceland, and the
  2853. arguments for the existence of Solid things derivable from Analogy. Yet --- I
  2854. take shame to be forced to confess it --- my brother has not yet grasped the
  2855. nature of Three Dimensions, and frankly avows his disbelief in the existence
  2856. of a Sphere.
  2857. Hence I am absolutely destitute of converts, and, for
  2858. aught that I can see, the millennial Revelation has been made to me for
  2859. nothing. Prometheus up in Spaceland was bound for bringing down fire for
  2860. mortals, but I --- poor Flatland Prometheus --- lie here in prison for bringing
  2861. down nothing to my countrymen. Yet I exist in the hope that these memoirs, in
  2862. some manner, I know not how, may find their way to the minds of humanity in
  2863. Some Dimension, and may stir up a race of rebels who shall refuse to be
  2864. confined to limited Dimensionality.
  2865. That is the hope of my brighter moments. Alas, it is
  2866. not always so. Heavily weighs on me at times the burdensome reflection that I
  2867. cannot honestly say I am confident as to the exact shape of the once-seen,
  2868. oft-regretted Cube; and in my nightly visions the mysterious precept, ``Upward,
  2869. not Northward,'' haunts me like a soul-devouring Sphinx. It is part of the
  2870. martyrdom which I endure for the cause of Truth that there are seasons of
  2871. mental weakness, when Cubes and Spheres flit away into the background of
  2872. scarce-possible existences; when the Land of Three Dimensions seems almost as
  2873. visionary as the Land of One or None; nay, when even this hard wall that bars
  2874. me from my freedom, these very tablets on which I am writing, and all the
  2875. substantial realities of Flatland itself, appear no better than the offspring
  2876. of a diseased imagination, or the baseless fabric of a dream.
  2877. \begin{center}
  2878. \includegraphics[trim=20mm 0mm 0mm 0mm, scale=0.5]{fig11}
  2879. \end{center}
  2880. \end{document}