@ -169,11 +189,10 @@ He has, therefore, requested me to reply in his behalf to two special
objections, one of an intellectual, the other of a moral nature.
The first objection is, that a Flatlander, seeing a Line, sees something that
must be thick to the eye as well as long to the eye (otherwise it would not be
must be \textit{thick} to the eye as well as \textit{long} to the eye (otherwise it would not be
visible, if it had not some thickness); and consequently he ought (it is
argued) to acknowledge that his countrymen are not only long and broad, but
also (though doubtless to a very slight degree) thick or high. This objection
is plausible, and, to Spacelanders, almost irresistible, so that, I confess,
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,
when I first heard it, I knew not what to reply. But my poor old friend's
answer appears to me completely to meet it.
@ -189,16 +208,16 @@ even I cannot now comprehend it, nor realize it by the sense of sight or by
any process of reason; I can but apprehend it by faith.''
``The reason is obvious. Dimension implied direction, implies measurement,
implies the more and the less. Now, all our lines are equally and
infinitesimally thick (or high, whichever you like); consequently, there is
implies the more and the less. Now, all our lines are \textit{equally} and
\textit{infinitesimally} thick (or high, whichever you like); consequently, there is
nothing in them to lead our minds to the conception of that Dimension. No
`delicate micrometer' --- as has been suggested by one too hasty Spaceland
critic --- would in the least avail us; for we should not know what to measure,
nor in what direction. When we see a Line, we see something that is long and
bright; brightness, as well as length, is necessary to the existence of a
critic --- would in the least avail us; for we should not know \textit{what to measure,
nor in what direction}. When we see a Line, we see something that is long and
\textit{bright}; \textit{brightness}, as well as length, is necessary to the existence of a
Line; if the brightness vanishes, the Line is extinguished. Hence, all my
Flatland friends --- when I talk to them about the unrecognized Dimension which
is somehow visible in a Line --- say, `Ah, you mean brightness': and when I
is somehow visible in a Line --- say, `Ah, you mean \textit{brightness}': and when I
reply, `No, I mean a real Dimension,' they at once retort, `Then measure it,
or tell us in what direction it extends'; and this silences me, for I can do
neither. Only yesterday, when the Chief Circle (in other words our High
@ -211,8 +230,8 @@ challenge? I was crushed; and he left the room triumphant.''
``Does this still seem strange to you? Then put yourself in a similar position.
Suppose a person of the Fourth Dimension, condescending to visit you, were to
say, `Whenever you open your eyes, you see a Plane (which is of Two
Dimensions) and you infer a Solid (which is of Three); but in reality you also
say, `Whenever you open your eyes, you \textit{see} a Plane (which is of Two
Dimensions) and you \textit{infer} a Solid (which is of Three); but in reality you also
see (though you do not recognize) a Fourth Dimension, which is not colour nor
brightness nor anything of the kind, but a true Dimension, although I cannot
point out to you its direction, nor can you possibly measure it.' What would
@ -271,7 +290,15 @@ needs be precisely thus, and we know all about it.''
%b\mainmatter
\part{This World}
\part{This World}[
\vspace{2cm}
\begin{center}
\begin{normalsize}
\textit{'Be patient, for the world is broad and wide.'}
\end{normalsize}
\end{center}
]
\chapter{Of the Nature of Flatland} I call our world Flatland, not because we
call it so, but to make its nature clearer to you, my happy readers, who are
@ -395,7 +422,7 @@ Light from the world of three Dimensions --- as if I were the maddest of the
mad! But a truce to these painful digressions: let me return to our homes.
The most common form for the construction of a house is five-sided or
pentagonal, as in the annexed figure. The two Northern sides RO, OF,
pentagonal, as in the annexed figure. The two Northern sides \textit{RO}, \textit{OF},
constitute the roof, and for the most part have no doors; on the East is a
small door for the Women; on the West a much larger one for the Men; the South
side or floor is usually doorless.
@ -709,7 +736,7 @@ time immemorial --- and now has become a kind of instinct among the women of our
higher classes --- that the mothers and daughters should constantly keep their
eyes and mouths towards their husband and his male friends; and for a lady in
a family of distinction to turn her back upon her husband would be regarded as
a kind of portent, involving loss of status. But, as I shall soon shew, this
a kind of portent, involving loss of \textit{status}. But, as I shall soon shew, this
custom, though it has the advantage of safety, is not without disadvantages.
Moving with ease and smoothness in uttering words; of rapid speech;
@ -742,11 +769,9 @@ their existence and the basis of the constitution of Flatland.
\chapter{Of our Methods of Recognizing one another}
\chapter{Of our Methods of Recognizing one another} You, who are blessed with
You, who are blessed with
shade as well as light, you, who are gifted with two eyes, endowed with a
knowledge of perspective, and charmed with the enjoyment of various colours,
you, who can actually see an angle, and contemplate the complete circumference
@ -755,7 +780,7 @@ clear to you the extreme difficulty which we in Flatland experience in
recognizing one another's configuration?
Recall what I told you above. All beings in Flatland, animate and inanimate,
no matter what their form, present to our view the same, or nearly the same,
no matter what their form, present \textit{to our view} the same, or nearly the same,
appearance, viz. that of a straight Line. How then can one be distinguished
from another, where all appear the same?
@ -774,7 +799,7 @@ degree more than correspondent with those of hearing, so that an Isosceles can
easily feign the voice of a Polygon, and, with some training, that of a Circle
himself. A second method is therefore more commonly resorted to.
Feeling is, among our Women and lower classes --- about our upper classes I
\textit{Feeling} is, among our Women and lower classes --- about our upper classes I
shall speak presently --- the principal test of recognition, at all events
between strangers, and when the question is, not as to the individual, but as
to the class. What therefore ``introduction'' is among the higher classes in
@ -830,15 +855,15 @@ out of seven votes from the Sanitary and Social Board for passing him into the
class of the Equal-sided --- often deplored, with a tear in his venerable eye, a
miscarriage of this kind, which had occurred to his
great-great-great-Grandfather, a respectable Working Man with an angle or
brain of 59 degrees 30 minutes. According to his account, my unfortunate
brain of 59$^{\circ}$ 30$'$. According to his account, my unfortunate
Ancestor, being afflicted with rheumatism, and in the act of being felt by a
Polygon, by one sudden start accidentally transfixed the Great Man through the
diagonal and thereby, partly in consequence of his long imprisonment and
degradation, and partly because of the moral shock which pervaded the whole of
my Ancestor's relations, threw back our family a degree and a half in their
ascent towards better things. The result was that in the next generation the
family brain was registered at only 58 degrees, and not till the lapse of five
generations was the lost ground recovered, the full 60 degrees attained, and
family brain was registered at only 58$^{\circ}$, and not till the lapse of five
generations was the lost ground recovered, the full 60$^{\circ}$ attained, and
the Ascent from the Isosceles finally achieved. And all this series of
calamities from one little accident in the process of Feeling.
@ -850,23 +875,23 @@ a time, or at all events only a number of bits of straight lines all in one
straight line, --- how can you ever discern any angle, and much less register
angles of different sizes?''.
I answer that though we cannot see angles, we can infer them, and this with
I answer that though we cannot \textit{see} angles, we can \textit{infer} them, and this with
great precision. Our sense of touch, stimulated by necessity, and developed by
long training, enables us to distinguish angles far more accurately than your
sense of sight, when unaided by a rule or measure of angles. Nor must I omit
to explain that we have great natural helps. It is with us a Law of Nature
that the brain of the Isosceles class shall begin at half a degree, or thirty
minutes, and shall increase (if it increases at all) by half a degree in every
generation until the goal of 60 degrees is reached, when the condition of
generation until the goal of 60$^{\circ}$ is reached, when the condition of
serfdom is quitted, and the freeman enters the class of Regulars.
Consequently, Nature herself supplies us with an ascending scale or Alphabet
of angles for half a degree up to 60 degrees, Specimens of which are placed in
of angles for half a degree up to 60$^{\circ}$, Specimens of which are placed in
every Elementary School throughout the land. Owing to occasional
retrogressions, to still more frequent moral and intellectual stagnation, and
to the extraordinary fecundity of the Criminal and Vagabond classes, there is
always a vast superfluity of individuals of the half degree and single degree
class, and a fair abundance of Specimens up to 10 degrees. These are
class, and a fair abundance of Specimens up to 10$^{\circ}$. These are
absolutely destitute of civic rights; and a great number of them, not having
even intelligence enough for the purposes of warfare, are devoted by the
States to the service of education. Fettered immovably so as to remove all
@ -949,21 +974,18 @@ Equilateral Triangle and a Pentagon; how am I to distinguish them?
It will be obvious, to every child in Spaceland who has touched the threshold
of Geometrical Studies, that, if I can bring my eye so that its glance may
bisect an angle (A) of the approaching stranger, my view will lie as it were
evenly between his two sides that are next to me (viz. CA and AB), so that I
bisect an angle (\textsc{a}) of the approaching stranger, my view will lie as it were
evenly between his two sides that are next to me (viz. \textsc{ca} and \textsc{ab}), so that I
shall contemplate the two impartially, and both will appear of the same size.
Now in the case of (1) the Merchant, what shall I see? I shall see a straight
line DAE, in which the middle point (A) will be very bright because it is
nearest to me; but on either side the line will shade away rapidly to dimness,
because the sides AC and AB recede rapidly into the fog and what appear to me
as the Merchant's extremities, viz. D and E, will be very dim indeed.
line \textsc{dae}, in which the middle point (\textsc{a}) will be very bright because it is
nearest to me; but on either side the line will shade away \textit{rapidly to dimness},
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}.
On the other hand in the case of (2) the Physician, though I shall here also
see a line (D'A'E') with a bright centre (A'), yet it will shade away less
rapidly to dimness, because the sides (A'C', A'B') recede less rapidly into
the fog: and what appear to me the Physician's extremities, viz. D' and E',
will not be not so dim as the extremities of the Merchant.
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
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.
The Reader will probably understand from these
two instances how --- after a very long training supplemented by constant
@ -984,11 +1006,12 @@ present his side to me instead of his angle, then, until I have asked him to
rotate, or until I have edged my eye around him, I am for the moment doubtful
whether he may not be a Straight Line, or, in other words, a Woman. Again,
when I am in the company of one of my two hexagonal Grandsons, contemplating
one of his sides (AB) full front, it will be evident from the accompanying
diagram that I shall see one whole line (AB) in comparative brightness
(shading off hardly at all at the ends) and two smaller lines (CA and BD) dim
throughout and shading away into greater dimness towards the extremities C and
D.
one of his sides (\textsc{ab}) full front, it will be evident from the accompanying
diagram that I shall see one whole line (\textsc{ab}) in comparative brightness
(shading off hardly at all at the ends) and two smaller lines (\textsc{ca} and \textsc{bd}) dim
throughout and shading away into greater dimness towards the extremities \textsc{c} and
@ -1004,12 +1027,10 @@ directions, as for example in a ball-room or conversazione --- must be of a
nature to task the angularity of the most intellectual, and amply justify the
rich endowments of the Learned Professors of Geometry, both Static and
Kinetic, in the illustrious University of Wentbridge, where the Science and
Art of Sight Recognition are regularly taught to large classes of the elite of
Art of Sight Recognition are regularly taught to large classes of the \textit{élite} of
the States.
It is only a few of the scions of our
noblest and wealthiest houses, who are able to give the time and money
necessary for the thorough prosecution of this noble and valuable Art. Even to
It is only a few of the scions of our noblest and wealthiest houses, who are able to give the time and money necessary for the thorough prosecution of this noble and valuable Art. Even to
me, a Mathematician of no mean standing, and the Grandfather of two most
hopeful and perfectly regular Hexagons, to find myself in the midst of a crowd
of rotating Polygons of the higher classes, is occasionally very perplexing.
@ -1184,7 +1205,7 @@ minutes: and the loss of their precious lives would have been an irreparable
injury to the State. The art of healing also has achieved some of its most
glorious triumphs in the compressions, extensions, trepannings, colligations,
and other surgical or diaetetic operations by which Irregularity has been
partly or wholly cured. Advocating therefore a Via Media, I would lay down no
partly or wholly cured. Advocating therefore a \textit{Via Media}, I would lay down no
fixed or absolute line of demarcation; but at the period when the frame is
just beginning to set, and when the Medical Board has reported that recovery
is improbably, I would suggest that the Irregular offspring be painlessly and
@ -1244,7 +1265,7 @@ by Nature to imply distinction of colours'' --- such was the sophism which in
those days flew from mouth to mouth, converting whole towns at a time to a new
culture. But manifestly to our Priests and Women this adage did not apply. The
latter had only one side, and therefore --- plurally and pedantically speaking ---
no sides. The former --- if at least they would assert their claim to be readily
\textit{no sides}. The former --- if at least they would assert their claim to be readily
and truly Circles, and not mere high-class Polygons, with an infinitely large
number of infinitesimally small sides --- were in the habit of boasting (what
Women confessed and deplored) that they also had no sides, being blessed with
@ -1345,19 +1366,19 @@ Priests and Women, under a new Legislation, may not be recognized; if so, a
word or two will make it obvious.
Imagine a woman duly decorated, according to the new Code; with the front half
(i.e., the half containing the eye and mouth) red, and with the hinder half
green. Look at her from one side. Obviously you will see a straight line, half
red, half green.
(\textit{i.e.}, the half containing the eye and mouth) red, and with the hinder half
green. Look at her from one side. Obviously you will see a straight line, \textit{half
Now imagine a Priest, whose mouth is at M, and whose front semicircle (AMB) is
Now imagine a Priest, whose mouth is at M, and whose front semicircle (\textsc{amb}) is
consequently coloured red, while his hinder semicircle is green; so that the
diameter AB divides the green from the red. If you contemplate the Great Man
so as to have your eye in the same straight line as his dividing diameter
(AB), what you will see will be a straight line (CBD), of which one half (CB)
will be red, and the other (BD) green. The whole line (CD) will be rather
(\textsc{ab}), what you will see will be a straight line (\textsc{cbd}), of which \textit{one half} (\textsc{cb})
\textit{will be red}, \textit{and the other} (\textsc{bd}) \textit{green}. The whole line (\textsc{cd}) will be rather
shorter perhaps than that of a full-sized Woman, and will shade off more
rapidly towards its extremities; but the identity of the colours would give
you an immediate impression of identity in Class, making you neglectful of
@ -1554,7 +1575,7 @@ of the agitation for the Universal Colour Bill.
\chapter{Concerning our Priests}
It is high time that I should pass from these brief and discursive notes about
things in Flatland to the central event of this book, my initiation into the
mysteries of Space. That is my subject; all that has gone before is merely
mysteries of Space. \textit{That} is my subject; all that has gone before is merely
preface.
For this reason I must omit many matters of which the explanation would not, I
@ -1591,8 +1612,8 @@ only a Polygon with a very large number of very small sides. As the number of
the sides increases, a Polygon approximates to a Circle; and, when the number
is very great indeed, say for example three or four hundred, it is extremely
difficult for the most delicate touch to feel any polygonal angles. Let me say
rather it would be difficult: for, as I have shown above, Recognition by
Feeling is unknown among the highest society, and to feel a Circle would be
rather it \textit{would} be difficult: for, as I have shown above, Recognition by
Feeling is unknown among the highest society, and to \textit{feel} a Circle would be
considered a most audacious insult. This habit of abstention from Feeling in
the best society enables a Circle the more easily to sustain the veil of
mystery in which, from his earliest years, he is wont to enwrap the exact
@ -1802,7 +1823,16 @@ reconsider the regulations of Female education.
\part{Other Worlds}
\part[Other Worlds]{Other Worlds}[
\vspace{2cm}
\begin{flushright}
\begin{normalsize}
\textit{'O brave new worlds, \\ That have such people in them!'}
\end{normalsize}
\end{flushright}
]
\chapter{How I had a Vision of Lineland}
It was the last day but one of the 1999th year of our era, and the first day
@ -1995,7 +2025,7 @@ question, for feeling could not have answered the purpose; but I asked with
the view of irritating the Monarch, and I succeeded perfectly.
``What!'' cried he in horror, ``explain your meaning.'' ``Feel, touch, come into
contact,'' I replied.. ``If you mean by feeling,`` said the King, ``approaching so
contact,'' I replied. ``If you mean by \textit{feeling},`` said the King, ``approaching so
close as to leave no space between two individuals, know, Stranger, that this
offence is punishable in my dominions by death. And the reason is obvious. The
frail form of a Woman, being liable to be shattered by such an approximation,
@ -2012,7 +2042,7 @@ is non-existent: for the Voice, being the essence of one's Being, cannot be
thus changed at will. But come, suppose that I had the power of passing
through solid things, so that I could penetrate my subjects, one after
another, even to the number of a billion, verifying the size and distance of
each by the sense of feeling: How much time and energy would be wasted in this
each by the sense of \textit{feeling:} How much time and energy would be wasted in this
clumsy and inaccurate method! Whereas now, in one moment of audition, I take
as it were the census and statistics, local, corporeal, mental and spiritual,
of every living being in Lineland. Hark, only hark!''
@ -2046,35 +2076,34 @@ that you saw them. And let me ask what you mean by those words `left' and
``Not so,'' replied I; ``besides your motion of Northward and Southward, there is
another motion which I call from right to left.''
King. Exhibit to me, if you please, this motion from left to right.
\textit{King}. Exhibit to me, if you please, this motion from left to right.
I. Nay, that I cannot do, unless you could setp out of your Line altogether.
\textit{I}. Nay, that I cannot do, unless you could step out of your Line altogether.
King. Out of my Line? Do you mean out of the world? Out of Space?
\textit{King}. Out of my Line? Do you mean out of the world? Out of Space?
I. Well, yes. Out of your world. Out of your Space. For your Space is not the
true Space. True Space is a Plane; but your Space is only a Line.
\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.
King. If you cannot indicate this motion from left to right by yourself moving
\textit{King}. If you cannot indicate this motion from left to right by yourself moving
in it, then I beg you to describe it to me in words.
I. If you cannot tell your right side from your left, I fear that no words of
\textit{I}. If you cannot tell your right side from your left, I fear that no words of
mine can make my meaning clearer to you. But surely you cannot be ignorant of
so simple a distinction.
King. I do not in the least understand you.
\textit{King}. I do not in the least understand you.
I. Alas! How shall I make it clear? When you move straight on, does it not
sometimes occur to you that you could move in some other way, turning your eye
\textit{I}. Alas! How shall I make it clear? When you move straight on, does it not
sometimes occur to you that you \textit{could} move in some other way, turning your eye
round so as to look in the direction towards which your side is now fronting?
In other words, instead of always moving in the direction of one of your
extremities, do you never feel a desire to move in the direction, so to speak,
of your side?
King. Never. And what do you mean? How can a man's inside ``front'' in any
\textit{King}. Never. And what do you mean? How can a man's inside ``front'' in any
direction? Or how can a man move in the direction of his inside?
I. Well then, since words cannot explain the matter, I will try deeds, and
\textit{I}. Well then, since words cannot explain the matter, I will try deeds, and
will move gradually out of Lineland in the direction which I desire to
indicate to you.
@ -2166,7 +2195,7 @@ reward him by giving him a few hints on Arithmetic, as applied to Geometry.
Taking nine Squares, each an inch every way, I had put them together so as to
make one large Square, with a side of three inches, and I had hence proved to
my little Grandson that --- though it was impossible for us to see the inside of
my little Grandson that --- though it was impossible for us to \textit{see} the inside of
the Square --- yet we might ascertain the number of square inches in a Square by
simply squaring the number of inches in the side: ``and thus,'' said I, ``we know
that $3^2$, or 9, represents the number of square inches in a
@ -2335,7 +2364,7 @@ your inside; but we in Spaceland should call it your side.
since you will not understand what Space means, from the Land of Three
Dimensions whence I but lately looked down upon your Plane which you call
Space forsooth. From that position of advantage I discerned all that you speak
of as solid (by which you mean ``enclosed on four sides''), your houses, your
of as \textit{solid} (by which you mean ``enclosed on four sides''), your houses, your
churches, your very chests and safes, yes even your insides and stomachs, all
lying open and exposed to my view.
@ -2381,13 +2410,13 @@ see her length, we infer her breadth; which, though very slight, is capable of
measurement.
\textit{Stranger}. You do not understand me. I mean that when you see a Woman, you
ought --- besides inferring her breadth --- to see her length, and to see what we
call her height; although the last Dimension is infinitesimal in your country.
ought --- besides inferring her breadth --- to see her length, and to \textit{see} what we
call her \textit{height}; although the last Dimension is infinitesimal in your country.
If a Line were mere length without ``height,'' it would cease to occupy Space
and would become invisible. Surely you must recognize this?
\textit{I}. I must indeed confess that I do not in the least understand your Lordship.
When we in Flatland see a Line, we see length and brightness. If the
When we in Flatland see a Line, we see length and \textit{brightness}. If the
brightness disappears, the Line is extinguished, and, as you say, ceases to
occupy Space. But am I to suppose that your Lordship gives the brightness the
title of a Dimension, and that what we call ``bright'' you call ``high''?
@ -2530,14 +2559,13 @@ evidently in Geometrical Progression. What is the next number?
\textit{I}. Eight.
\textit{Sphere}. Exactly. The one Square produces a
with \textit{eight} terminal Points. Now are you convinced?
\textit{I}. And has this Creature sides, as well as Angles or what you call ``terminal
Points''?
\textit{Sphere}. Of course; and all according to Analogy. But, by the way, not what you
call sides, but what we call sides. You would call them solids.
\textit{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.
\textit{I}. And how many solids or sides will appertain to this Being whom I am to
generate by the motion of my inside in an ``upward'' direction, and whom you
@ -2565,8 +2593,7 @@ words I precipitated myself upon him.
\chapter{How the Sphere, having in vain
tried words, resorted to deeds}
\chapter{How the Sphere, having in vain tried words, resorted to deeds}
It was in vain. I brought my hardest right angle into violent collision with
the Stranger, pressing on him with a force sufficient to have destroyed any
ordinary Circle: but I could feel him slowly and unarrestably slipping from my
@ -2700,7 +2727,7 @@ several apartments. Only my affectionate Wife, alarmed by my continued
absence, had quitted her room and was roving up and down in the Hall,
anxiously awaiting my return. Also the Page, aroused by my cries, had left his
room, and under pretext of ascertaining whether I had fallen somewhere in a
faint, was prying into the cabinet in my study. All this I could now see, not
faint, was prying into the cabinet in my study. All this I could now \textit{see}, not
merely infer; and as we came nearer and nearer, I could discern even the
contents of my cabinet, and the two chests of gold, and the tablets of which
the Sphere had made mention.
@ -2812,9 +2839,8 @@ were made by him of that day's incident, his life would be spared.
\chapter{How, though the Sphere
showed me other mysteries of Spaceland, I still desire more; and what came of
it}
\chapter{How, though the Sphere showed me other mysteries of Spaceland, I still desire more; and what came of it}
When I saw my poor brother led away to imprisonment, I attempted to leap down
into the Council Chamber, desiring to intercede on his behalf, or at least bid
him farewell. But I found that I had no motion of my own. I absolutely
@ -2829,7 +2855,7 @@ Once more we ascended into space. ``Hitherto,'' said the Sphere, ``I have shewn
you naught save Plane Figures and their interiors. Now I must introduce you to
Solids, and reveal to you the plan upon which they are constructed. Behold
this multitude of moveable square cards. See, I put one on another, not, as
you supposed, Northward of the other, but on the other. Now a second, now a
you supposed, Northward of the other, but \textit{on} the other. Now a second, now a
third. See, I am building up a Solid by a multitude of Squares parallel to one
another. Now the Solid is complete, being as high as it is long and broad, and
we call it a Cube.''
@ -2981,15 +3007,15 @@ mine behold it --- that blessed Being, a Cube, with eight terminal points?
And in Four Dimensions shall not a moving Cube --- alas, for Analogy, and alas
for the Progress of Truth, if it be not so --- shall not, I say, the motion of a
divine Cube result in a still more divine Organization with sixteen terminal
divine Cube result in a still more divine Organization with \textit{sixteen} terminal
points?
Behold the infallible confirmation of the Series, 2, 4,
8, 16: is not this a Geometrical Progression? Is not this --- if I might quote
my Lord's own words --- ``strictly according to Analogy''?
Again, was I not taught by my Lord that as in a Line there are two bounding
Points, and in a Square there are four bounding Lines, so in a Cube there must
Again, was I not taught by my Lord that as in a Line there are \textit{two} bounding
Points, and in a Square there are \textit{four} bounding Lines, so in a Cube there must
be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is
not this an Arithmetical Progression? And consequently does it not of
necessity follow that the more divine offspring of the divine Cube in the Land
@ -3168,8 +3194,8 @@ patent even to the Female Sex.
\chapter{How I tried to teach the
Theory of Three Dimensions to my Grandson, and with what success}
\chapter{How I tried to teach the Theory of Three Dimensions to my Grandson, and with what success}
I awoke rejoicing, and began to reflect on the glorious career before me. I
would go forth, methought, at once, and evangelize the whole of Flatland. Even
to Women and Soldiers should the Gospel of Three Dimensions be proclaimed. I
@ -3256,8 +3282,8 @@ Three Dimensions.
\chapter{How I then tried to diffuse the Theory of Three Dimensions by other
means, and of the result}
\chapter{How I then tried to diffuse the Theory of Three Dimensions by other means, and of the result}
My failure with my Grandson did not encourage me to communicate my secret to
others of my household; yet neither was I led by it to despair of success.
Only I saw that I must not wholly rely on the catch-phrase, ``Upward, not
