AOH :: 4THDEM.TXT|
Another World; or The Fourth Dimension. A PD Etext.
ANOTHER WORLD, by A.T. SCHOFIELD, M.D.
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THE FOURTH DIMENSION.
A.T. SCHOFIELD, M.D.
LONDON: GEORGE ALLEN & UNWIN LTD.
RUSKIN HOUSE 40 MUSEUM STREET, W.C.
FIRST EDITION JUNE, 1888.
PROFESSOR J.H. GLADSTONE,
PH.D., F.R.S., ETC., ETC.,
IN GRATEFUL RECOGNITION
OF VALUED HELP,
THIS BOOK IS DEDICATED
THE FOURTH DIMENSION.
IT is undoubtedly the cherished belief of the vast majority of
mankind, whether they be Christian, Mohammedan, Hindoo, or
Heathen, whether they be savage or civilized, in every: quarter
of the globe, that there is another world besides the material
universe in which we live.
All unite in considering that world to be a higher sphere than
ours, and its inhabitants to be more or less spiritual beings.
It is also generally believed that the beings of that spirit
world can and do visit ours, manifesting themselves in a human
or animal shape.
When we come, however, to further details of this higher world,
we have every diversity of belief and superstition.
The only account and description of it to which we, as
Christians, attach any credence, is found in the Bible, a book
which we regard as a revelation of its rulers, inhabitants, and
laws, given to man by the supreme Ruler, not only of the
spiritual, but of our material world, God.
In our own persons we get confirmation of the existence of a
higher sphere, in being able consciously to distinguish between
our spiritual, intellectual, and moral selves and our bodies
and brains, through which we act and by which we live..
Materialists will, we know, have none of this. To them, if true
to their creed, there is, and can be, nothing beyond the
material. Mind, morals, feelings, passions, are to them only
protoplasmic changes of. ganglion nerve cells, producing
carbonic acid gas and water.
To them the almost universal consensus of opinion in favour of
a spirit world goes for nothing, unless such a world can be
demonstrated, handled, and weighed.
We therefore propose, in the following pages to discuss from a
somewhat new point of view the question of the existence of
such a world, what are its powers, its laws, and its
relationship, with this universe, and in doing so, will observe
how far these powers and laws, deduced by analogy from
mathematics, correspond to the spiritual claims of the
I would here take the opportunity of acknowledging my deep
indebtedness to the anonymous author of a small book, called
"Flatland," which I have used extensively throughout, and
without which I am quite sure the public would never have been
troubled with these remarks; my object being to carry on the
line of argument there brought forward, to what seems to me
its true and necessary conclusion.
Finally, let me ask the indulgence of my more advanced
mathematical readers for the many fallacies and "non-sequiturs"
that doubtless abound, in spite of my true endeavours
simply and impartially to draw none but legitimate and logical
conclusions from the arguments and facts I have advanced.
PREFACE TO THIRD EDITION.
MANY speculations concerning the fourth dimension have been
made since this book was first issued, notably that by Mr.
Wells that it is "Time." But no theory carries conviction, and
indeed the whole is a speculation, the interest however of
which remains untouched in the close parallel afforded between
what would be true of a fourth dimension and all that is
written or known concerning the spirit world. A few additions
have been made in this edition.
A. T. SCHOFIELD.
THE LAND OF NO DIMENSION
WE are all so habituated to take visible realities around us as
a matter of course, and so accustomed to every variety of solid
or material form, that why all the universe should be limited
to solidity, or three dimensions, is only asked at rare
intervals by a few of the more thoughtful among us.
To make this plain. Even those unaccustomed to algebra will
understand that if x represents three inches, or a line of this
length, x^2 (x square) represents 3 x 3, or nine square inches
on a flat surface, three inches each way; x^3 (x cube), again,
represents 3 x 3 x 3, or twenty-seven cubic or solid inches, or
a solid body measuring three inches every way. Hence we
consider x as representing lines, x^2 squares, x^3 solids, and
then comes the question, What does x^4 represent? for
mathematics passes as easily from x^3 to x^4 as from x^2 to
x^3, and yet while x, x^2, x^3, refer to objects known to all
of us, the wisest can form no possible conception of what x^4,
or a world of four dimensions, is like.
Perhaps, however, before disturbing our minds, and entering
seriously upon the question as to whether there can be and is
any object or world represented by x^4, and whether or no we
can comprehend it, my reader will not be offended if, for the
benefit of those less learned than himself, I labour in the
simplest language further to explain these various dimensions.
To begin: _No dimension, or size in no direction_, is
represented mathematically by a point, which is an object
described as having no parts or magnitude, thus:-- . . . . .
_One dimension_ (x), or _size in one direction_, is
represented mathematically by a straight line, which is
described as having length without breadth,
_Two dimensions_ (x^2), or _size in two directions_, is
represented mathematically by a superficies or surface, which
is described as having length and breadth without thickness,
FIGURE OF DIAMOND RECTANGLE AND A CIRCLE
_Three dimensions_ (x^3), or _size in three directions_, is
represented mathematically by a _solid body_, which is described
as having length, breadth, and thickness, thus:--
FIGURE OF BOX, SPHERE AND A 3-D CROSS
_Four dimensions_ (x^4), or _size in four directions_, we cannot
represent mathematically, nor can we describe in what direction
its fourth dimension lies, nor can we draw, or even imagine it;
the fact being that the whole material world which we can see,
and of which we can speak, is a world of three dimensions (or
x^3) and no more, nor is it possible for the mind of man to
indicate or imagine any other direction than three--length,
breadth, and height (or depth or thickness, etc.).
On this account it is that so many have denied the possibility
of there being anything higher than a solid. To show the
fallacy of this argument, then, we will consider the imaginary
case of an inhabitant of a country where nothing but perfectly
flat objects exist, when an endeavour is made to explain to him
our own world of solids; and by putting ourselves in his place,
and carefully observing the difficulty he, accustomed only to
x^2 or flatness, would have in grasping x^3, or solidity, which
nevertheless exists, we may understand better that the
difficulty we in x^3, or solidity, have in our turn of grasping
x^4, or the fourth dimension, is no argument whatever against
the existence of such a world.
First of all, however, we will consider the still lower
conditions of _no dimensions_ and of _one dimension_.
Imagine, then, a world or universe consisting entirely and
absolutely of a single POINT, a country which therefore
possesses neither length, breadth, depth, nor height. Imagine
(if you can) the sole being in such a world, and observe what
his experience would be, as described in "Flatland."
"He is himself his own world, his own universe; of any other
than himself he can form no conception; he knows not length, or
breadth, or height, for he has no experience of them; he has no
cognizance even of the number two; for he is himself one and
all, being really nothing. Yet mark his perfect
self-complacency, and hence learn this lesson, that to be
self-contented is to be vile and ignorant, and that to aspire
is better than to be blindly and impotently happy. Now listen!
There arose from the little buzzing creature a tiny, low,
monotonous tinkling, from which I caught these words. `IT fills
all space, and what IT fills IT is, what IT thinks that IT
utters; and what IT utters, that IT hears, and IT itself is
thinker, utterer, hearer. IT is the one, and yet the all in all.'"
This then gives us an idea of what a world would be that
consisted only of one being, and that being having no parts or size.
Having duly performed this excruciating effort of imagination,
and succeeded in realizing what nothing, or "Pointland,"
really is, the exhausted reader had better pause for five
minutes before taking the next step higher into the more
interesting world of one dimension, or "Lineland."
THE LAND OF ONE DIMENSION
LET now my faithful reader, somewhat recruited from the study
of Chapter I., proceed to picture a world of one dimension--a
universe that consists only of innumerable straight lines, long
and short, all arranged in one and the same interminable
straight line--nothing else at all, no deviation to right or
left, no right or left even existing to this linear world,
still less any height or depth. To duly appreciate and grasp
the phenomena of such a world, it would greatly assist the mind
if my reader were to arrange a number of pencils or matches in
one long line, end to end, and follow the fragments with his
eye. Let not any think that these preliminary studies are
needless, for every link in the chain of analogy must be
carefully followed, if we are to reach the important
conclusions we are here aiming at.
Let my reader, then, now retire into his inner consciousness,
and proceed to imagine a kingdom or world, as we have said,
consisting of an infinite number of inhabitants, each one
being a shorter or longer straight line, and all arranged in
one and the same straight line, thus:
------ ---- --- -- -- -- -- -- --------
If one end of these creatures or lines be furnished with an
eye, it is obvious they will each see the end of the line next
in front of them, which will be a simple point.
None, therefore, in this line (or world) can ever see anything
beyond a point. To see a line one must obviously be out of the
line (or the Land of One Dimension) altogether.
If this is not clear, place your eye at the end of any straight
line (a needle or knitting needle), and you will only see a
Let the mind now proceed to picture a being of two dimensions,
such as a square (illustrating it at the same time by a piece
of cardboard), furnished with an eye at one of its angles,
approaching this world of Lineland (Slide the cardboard square
along the table towards the long line of pencils or matches,
etc.); and then listen to the following remarks from our
unknown author. The square speaks.
"I saw before me a vast multitude of small straight lines,
-- -- -- -- -- -- all moving to and fro in one and the same
straight line. Approaching the largest, I accosted it (Here
bring the square close to a match), but received no answer.
Losing patience at what appeared to me intolerable rudeness, I
brought my mouth into a position full in front of it (Here
slide a corner of the square into the line in front of the
match), and repeated my question. `What signifies this
monotonous motion to and fro in one and the same straight line?'
"`I am the Monarch of the World,' replied the small line. `But
thou, whence intrudest thou into my realm of Lineland?'
"Receiving this abrupt reply, I begged pardon; and by
persevering questions extracted the following facts:--
"It seems that this poor ignorant monarch, as he calls himself,
was fully persuaded that the straight line which he called his
kingdom, and in which he passed his existence, constituted the
whole of the world, and indeed the whole of space. Not being
able either to move or see, save in his straight line, he had
no conception of anything out of it.
"Though he had heard my voice when I first addressed him,
the sound had come to him in a manner so contrary to his
own experience, that he had made no answer, `seeing no man,'
as he expressed it, and `hearing a voice, as it were, from
his own inside.'
"Until the moment when I placed my mouth in his world, he had
not seen me; nor had he now the least conception of the region
from which I had come. Outside his world or line all was blank
to him; nay, not even a blank, for a blank implies space; say
rather, all was non-existent. Such a life, with all vision
limited to a point, and all motion to a straight line, seemed
to me inexpressibly dreary, and I was surprised to note the
vivacity and cheerfulness of the king."
Such were the observations of our supposed living square, and
such would be the only life possible, were the world but one
straight line. Our square, however, rejoicing in his own two
dimensions of breadth as well as length, tries to enlighten the
king of Lineland, and proceeds:--
"Thinking that it was time to bring down the monarch from his
raptures to the level of common sense, I determined to
endeavour to open up to him some glimpses of the truth; that is
to say, of the nature of flat things, or two dimensions.
"So I began thus: `Before I entered your kingdom, I noticed
that some of the lines were larger----'
"`You speak of an impossibility,' interrupted the king; `you
must have seen a vision, for to detect the difference (even)
between a line and a point by the sense of sight is, as every
one knows, in the nature of things, impossible. How could you
see a line, that is to say, the inside of any man?'
"`_I_ can discern a line from a point, and let me prove it.
Before I came into your kingdom I saw you dancing backwards and
forwards, with seven lines and a dot in front of you, and eight
lines and a dot behind you.'
"He then proceeds to tell the king that there is another motion
possible, besides backwards and forwards; namely, from left to right.
"`Let me ask,' said the king, `what you mean by these words
"left" and "right." I suppose it is your way of saying
northward (forwards) and southward (backwards).'
"`Not so' replied I; `besides your motion of forwards and
backwards' there is another motion; which I call from left to
"`Exhibit to me, if you please, this motion from left to right.'
"'Nay, that I cannot do, unless you could step out of your line
"`Out of my line? Do you mean out of the world? Out of space?'
"`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? 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?'
"`Never! And what do you mean? How can a man move in the
direction of his inside?'
"`Well then, I will try deeds; I will gradually move out of
Lineland in the direction which I desire to indicate to you.'
"At this word I began to move my body out of Lineland. As long
as any part of me remained in his dominion, and in his view,
the king kept exclaiming, `I see you.' But when I had at last
moved myself out of his line, he cried, `He is dead.' (Move card
slowly out of the straight line.)
"`I am not dead,' replied I; `I am simply out of Lineland, that
is to say, out of the straight line which you call space, and
at this moment I can see your line, or side, or inside, as you
are pleased to call it'
"But the monarch replied" If you were a man of sense, you would
listen to reason. You ask me to believe that there is another
line beside that which my senses indicate, and another motion
beside that of which I am daily conscious. I, in return, ask
you to describe in words, or to indicate by motion, that other
line of which you speak. Instead of moving, you merely exercise
some magic art of vanishing and returning to sight. Acknowledge
your folly, or depart from my dominion.'
"Furious at his perversity, I retorted, `Besotted being, you
think yourself the perfection of existence, while you are in
reality the most imperfect and imbecile. You profess to see,
whereas you can see nothing but a point'"
We have given these extracts at length, in order that the
reader may fully grasp what would be the general conditions of
life, prospects, and intelligence in a world of one dimension,
and also the necessary impossibility of one in such a world
being able to understand the existence of another by argument
or illustration. Various other suggestive analogies present
themselves here, but we will defer their discussion until we
have the other dimensions before us, and then consider them all
together. In the next chapter, therefore, we will move a step
higher, and attentively view life in a world of two dimensions.
THE LAND OF TWO DIMENSIONS
WE must now again tax the imagination of our readers, and ask
them to picture a country of two dimensions, where only length
and breadth are known. This country our author aptly calls
"Flatland," and in order to present it vividly before our
readers, we must again quote extensively. Our old friend, the
animated square, speaks.
"Imagine a vast sheet of paper, on which straight lines,
triangles, squares, and circles, instead of remaining fixed in
their places, move freely about on the surface--very much like
shadows--and you will have a pretty correct notion of my
country and countrymen. In such a country you will perceive at
once that it is impossible that there should be anything of
what you call a `solid' kind, but I daresay you will suppose
that we could at least see the triangles, squares, and other
figures moving, about as I have described them. On the
contrary, nothing was visible, nor could be visible to us,
except straight lines."
Our readers will see the strict analogy here: that just as
those in one dimension could only see points, not lines,--so
those in two dimensions can only see lines, not squares, etc.
if the eye is placed on a level (that is, in the same world)
with the edge of one of the cardboard figures, whatever its
shape, only a straight line will be seen; for it is only as we
rise above or go below it--that is, enter the third
dimension--that we see the shape of the figure.
The houses in "Flatland," according to our author and to
reason, consist of spaces enclosed by lines, openings being
left for doors. Of course the idea of a roof to such houses is
necessarily absurd, there being no space except in length and
breadth in that world; hence the houses are to our ideas open.
There is a north, south, east and west; the first two being
equivalent to length and the latter two to breadth.
Of course, any being in such a house, when the door was shut,
though inaccessible and invisible to any inhabitant of
Flatland, could be as easily touched and seen by us if outside
the house. (One of the pieces of cardboard placed inside a
circle of thread will illustrate this.)
Such a world, then, being imagined, thickly peopled with flat
figures gliding incessantly to and fro on the surface, or in
and out these spaces surrounded by lines, which they call
houses, we will now try and understand the extraordinary
experiences of our animated square in Flatland, when, after
having tried and failed to enlighten the king of one dimension,
he is, in his turn, instructed by a being from our world of
solids, or three dimensions. The incident occurred thus:--
"It was the last day of the year 1999 of our era. My four sons
and two orphan grandchildren had retired to their several
apartments, and my wife alone remained with me to see the old
millennium out and the new one in.
"I was wrapt in thought, pondering in my mind some words that
had casually issued from the mouth of my youngest grandson.
Taking nine squares, each an inch every way, I had put them
together so as to form one large square with a side of three
inches, and I had proved to my grandson that, though it was
impossible for us to see the inside of this, or indeed of any
square, yet we might ascertain the number of square inches in
a square, simply by squaring the number of inches in the side;
`and thus,' said I, `we know that 3^2, or 9, represents the number
of square inches in a square whose side is three inches long.'
"The little Hexagon (my grandson) meditated on this awhile, and
then said to me: `But you have been teaching me also to raise
numbers to the third power; I suppose 3^3 must mean something in
geometry. What does it mean?'
"`Nothing at all,' replied I, `not at least in geometry; for
geometry has only two dimensions.' And then I began to show the
boy how a point, by moving through a length of three inches,
makes a line of three inches, which may be represented by 3;
and how a line of three inches, moving parallel to itself
through a length of three inches, makes a square of three
inches every way, which may be represented by 3^2.
"Upon this my grandson, again returning to his former
suggestion, took me up rather suddenly, and exclaimed,--
"`Well, then, if a point by moving three inches makes a line of
three inches, represented by 3, and if a straight line of three
inches moving parallel to itself makes a square of three inches
every way, represented by 3^2,--it must be that a square of
three inches every way moving somehow parallel to itself (but
I don't see how) must make a something else (but I don't see
what) of three inches every way,--and this must be
represented by 3^3.'"
Let the reader observe here how the Hexagon, by reasoning
strictly by analogy, thus discovers and describes a cube or
"`Go to bed,' said I, a little ruffled by his interruption. `If
you would talk less nonsense, you would remember more sense.'
"So my grandson had disappeared in disgrace. Rousing myself
from my reverie, I exclaimed, `The boy is a fool!'
"Straightway I became conscious of a presence in the room, and
a chilly breath thrilled through my very being. Looking round
in every direction, I could see nothing. I resumed my seat
again, exclaiming, `The boy is a fool, I say; 3^3 can have no
distinct meaning in geometry.'
"At once there came a distinctly audible reply, `The boy is not
a fool, and 3^3 has an obvious geometrical meaning.'
"My wife, as well as myself, heard the words, although she did
not understand their meaning; and both of us sprang forward in
the direction of the sound. What was our horror when we beheld
before us a figure!
"My wife retreated to her apartment. I began to approach the
stranger with the intention of taking a nearer view. He
remained motionless while I walked round him, beginning from
his eye, and returning to it again. Circular he was throughout;
there could not be a doubt of it. Then followed a dialogue.
(The reader will be much helped if he will illustrate this by
first placing a cardboard square inside a large circle of
thread, and then place a ball inside the circle; or, better
still, let the surface of a basin of water represent Flatland,
and a floating circle of thread and piece of cardboard the
house and its inhabitant, and then a ball, half immersed, the
visitor--capable of sinking through or rising out of Flatland
"_I_. `Before your lordship enters into further communication,
would he deign to satisfy the curiosity of one who would gladly
know whence his visitor came?'
"Stranger. `From space, sir; whence else?'
"_I_. `Pardon me, my lord; but is not your lordship already in
space--even at this moment?'
"Stranger. `Pooh! What do you know of space? Define space.'
"_I_. `Space, my lord, is length and breadth, indefinitely
"Stranger. `Exactly. You see you do not even know what space
is. You think it is of two dimensions, only; but I have come to
announce to you a third--height, as well as breadth and
"_I_. `Your lordship is pleased to be merry. We also speak of
length and height (which are the same), or breadth and
thickness (which are the same), thus denoting two dimensions by
"Stranger. `But I mean not only three names, but three
"_I_. `Would your lordship indicate or explain to me in what
direction is the third dimension?
"Stranger. `I came from it. It is up above and down below.'
"_I_. `My lord means, seemingly, that it is northward and
"Stranger. `I mean nothing of the kind. I mean a direction in
which you cannot look, because you have no eye in your side.'"
(if the reader makes a dot for an eye on the cardboard square,
he will see that such an eye in the "side" of the square
would look upwards. Observe also that the borders of the square
form its outside, and all of it that can be seen by any one on
the same level, and that the surface of the square is its
inside, enclosed by the four borders.)
"_I_. `Pardon me, my lord; a moments inspection will convince
your lordship that I have a perfect luminary at the junction of
two of my sides.'"
The reader will see the square calls his borders sides, whereas
the stranger refers to the surface of the square. Both may be
called sides; thus a cardboard square has four sides, or two
"Stranger. `Yes; but in order to see into space you ought to
have an eye, not in your border, but in your side that is, what
you would probably call your inside; but we in Spaceland call
it your side.'
"_I_. `An eye in my inside! An eye in my stomach!! Your
"Stranger. `I am in no jesting humour. I tell you I came from
space. From that position of advantage I discerned your houses,
yea, even your insides, all lying open to my view.'
"_I_. `Such assertions are easily made, my lord.'
"Stranger. `How shall I convince him? Surely a plain statement
of facts, followed by ocular demonstration, ought to suffice.
Now, sir, listen to me. You are living in a plane. I am not a
plane (or flat) figure, but a solid. You call me a circle, but
I am a sphere. Your country of two dimensions is not spacious
enough to represent me,--a being of three; but can only exhibit
a slice or section of me, which is what you call a circle. See,
now I will rise, and the effect on your eye will be that my
circle will become smaller and smaller, till it dwindles to a
point, and finally vanishes.'
"There was no `rising' that I could see; but he diminished, and
finally vanished, and then, after a while, reappeared and
regained his original size. He heaved a deep sigh, for he
perceived I had altogether failed to comprehend him. Indeed, I
was now inclining to the belief that he was an extremely clever
"After a long pause he continued our dialogue.
"`How many sides has a square, and how many angles?'
"_I_. `Four sides and four angles.'
"_Sphere_. `Now stretch your imagination a little, and conceive
a square in Flatland (you are a square) with its side, or what
you call its inside, moving parallel to itself, upwards.'"
(The reader performs this by just gradually raising the
cardboard square from the table and parallel with it.)
"_I_. `What! northward?'
"_Sphere_. `No; not northward; upward--out of Flatland altogether.'
"Restraining my impatience, I replied: `And what may be the
nature of the figure which I am to shape out by this motion
which you are pleased to denote by the word "upward"?'
"_Sphere_. `A cube, with eight terminal points (or angles).'
"_I_. `And how many sides will pertain to this being whom I am
to generate by the motion of my "inside" in an "upward" direction?'
"_Sphere_. `The cube which you will generate will be bounded by
six sides--that is to say, six of your insides. You see it all
"`Monster!' I shrieked, `be thou juggler, enchanter, dream or
devil, no more will I endure thy mockeries. Either thou or I
"And saying these words, I precipitated myself upon him. It was
in vain. I could feel him slowly slipping from my contact--not
edging to the right or left, but moving somehow out of the
world, and vanishing to nothing. But I still heard the
"_Sphere_. `Why will you refuse to listen to reason? I had hoped
to find in you a fit apostle for the gospel of three
dimensions. Listen, my friend. I have told you I can see from
my position in space the inside of all things that you consider
closed. For example, I see in yonder cupboard, near which you
are standing, several of what you call boxes (but like
everything else in Flatland, they have no tops or bottoms) full
of money. I see also two tablets of accounts. I am about to
descend into that cupboard, and to bring you one of those
tablets. I saw you lock the cupboard half an hour ago, and I
know you have the key in your possession. But I descend from
space; the door, you see, remains unmoved. Now I am in the
cupboard, and am taking the tablet. Now I have it. Now I ascend
"I rushed to the closet, and dashed the door open. One of the
tablets was gone. At the same time it appeared on the floor of
the room." All this, however, failed to convince our square,
who at last threw himself in impotent rage upon the apparent
The sphere then, unwilling to leave him in his ignorance, as
a last resource lifted our poor square right up out of
Flatland--out of the land of two dimensions altogether--into
our world of space of three dimensions. Here we will follow him
in the next chapter.
THE LAND OF THREE DIMENSIONS
THE first object that met the bewildered gaze of our square,
when thus finally translated from the world of two dimensions
into that of three, was the perfect figure of the sphere beside
him, still appearing as a curiously shaded flat circle, this
being the first surface he had ever gazed upon; all flat
objects, when in his own country, appearing, as we have seen,
as straight lines.
He then turned his wondering eyes downwards, and beheld to his
amazement Flatland as it really was, with its flat inhabitants
of different shapes all snugly ensconced in their different
rooms of their roofless houses, all of which were of course now
perfectly open to his view. He could gaze down upon his own
house and the room he had just quitted, and could see his wife
and children. He, in his turn, now could look into his own
locked cabinet, and discern the very tablets already spoken of
But as he was carried higher he saw more. His whole native
city, hitherto known to him only as lines, lay revealed, with
the shape of every inhabitant equally plainly to be seen,
whether in the street or within doors.
Naturally he thought at first he had become a god, in thus
seeing all that he had only surmised before.
With the sphere as his guide, he then travelled on through
space, till beneath him he saw the interior of the great
judgment hall of Flatland, with all its wise men assembled. He
then heard the following decree, to his dismay, read out before
"Whereas the States had been troubled by divers ill-intentioned
persons pretending to have received revelations from another
world, it has been for this cause unanimously resolved by the
Grand Council to make strict search for such misguided persons,
to scourge and imprison any triangle, and to arrest any one of
higher rank, to be examined and judged by the council."
"You hear your fate," the sphere remarked; "death or
imprisonment awaits the apostle of the gospel of three
"Not so," replied our square; "the matter is now so clear to
me, the nature of real space so palpable, that methinks I could
make a child understand it. Permit me but to descend at this
moment and enlighten them."
"Not yet," said the sphere, who then taking our friend with him
further into space, proceeded to introduce him to solid
figures, beginning with a cube.
Taking a number of square cards (the reader can do this if he
have enough), each the shape of his friend, he placed them one
on another till they were as high as they were broad, and thus
he built up a cube.
To the uneducated eye of the square, however, accustomed only
to see lines and points, and to whom the sight of even a flat
surface was a new revelation, this solid form (like the sphere)
appeared to be an irregular six-sided flat figure thus, (1)--not
a solid like this, (2)--
Fig. 1. Fig. 2.
The reader can verify this by closing one eye, and drawing the
outlines of a cube seen sideways, on paper.
It was not until some time after, when he had by the direction
of his friend carefully felt its six sides and its eight
angles, and walked round and round and under and over it, and
had many views of it in different lights, that the stupendous
fact began to dawn upon him, that this new world which he had
entered, not only enabled him to see all objects in his own
familiar Flatland in a new and truer light, but contained
bodies of a fresh and glorious order, utterly transcending all
his powers of imagination or description, and of a form so
novel, so unexpected, as to be incredible, were it not that his
senses convinced him of their existence. It took, indeed, a
long time for him to understand that the surfaces he saw of the
sphere and cube, thus,--
were not their interiors, thus:--
SHADED CIRCLE SHADED SQUARE
Once our friend the square had, however, fairly grasped, as far
as he could, the fact that he now beheld in actual fact the
realization of the mathematical formula of x^3, and of that
problem of his grandson he had scouted as being alike
unreasonable and impossible, he was not content to stop here.
See now in the words of our author to what he aspired.
"I thirsted," says he, "for yet deeper knowledge than he (the
sphere) was offering to me.
"Pardon me," said I, "O thou whom I must no longer address as
the perfection of beauty; let me beg of thee to vouchsafe thy
servant a sight of thine interior."
"_Sphere_. `My what?'
"_I_. `Thine interior, thy stomach!'
"_Sphere_. `Whence this ill-timed, impertinent request? And what
mean you by saying that I am no longer the perfection of all beauty?'
"_I_. `My lord, your own wisdom has taught me to aspire to one
even more great, more beautiful, than yourself. As you
yourself, superior to all Flatland forms, combine many circles
in one, so doubtless there is one above you, who combines many
spheres in one supreme existence, surpassing even the solids of
Spaceland. And even as we who are now in space look down on
Flatland, and see the insides of all things, so of a certainty
there is yet above us some higher, purer region, whither thou
dost surely purpose to lead me, from the vantage ground of which
we shall look down upon the revealed insides of all solid things.'
"_Sphere_. `Pooh! Stuff! Enough of this trifling.'
"_I_. `Nay--deny me not what I know it is in thy power to
perform. Grant me but one glimpse of thine interior.'
"_Sphere_. `Well then, to content and silence you, let me say at
once, I cannot. Would you have me turn my stomach inside out to
"_I_. `But my lord has shown me the insides of all my
countrymen in the land of two dimensions by taking me into the
land of three. What therefore more easy than to take his
servant a second journey into the blessed region of the fourth
dimension, where I shall look down with him once more upon this
land of three dimensions, and see the inside of every
three-dimensioned house, and the inside of every solid living
"_Sphere_. `But where is this land of four dimensions?'
"_I_. `I know not; but doubtless my teacher knows.'
"_Sphere_. `Not I. There is no such land. The very idea of it is
"_I_. `Trifle not with me, my lord. I crave and thirst for more
knowledge. Doubtless we cannot _see_ that other higher Spaceland
now, because we have no eye in our stomachs. But, just as there
_was_ the realm of Flatland, though that poor puny Lineland
monarch could turn neither to left nor right, and just as there
_was_ close at hand, touching my frame, the land of three
dimensions, though I, blind and senseless wretch, had no power
to touch it, no eye in my interior to discern it; so, of a
surety, there is a fourth dimension, which my lord perceives
with the inner eye of thought.
"`In _one_ dimension did not a moving point produce a line with
two terminal points?
"`In _two_ dimensions did not a moving line produce a square
with four terminal points?
"`In _three_ dimensions did not a moving square produce a cube
with eight terminal points?
"`And in _four_ dimensions shall not a cube alas for analogy,
and alas for the progress of truth, if it be not so--result in
a still more divine organization with sixteen terminal points?
Behold the infallible confirmation of the series, 2, 4, 8, 16.
Is not this a geometrical progression strictly according to
analogy? I ask, therefore, is it, or is it not, a fact that ere
now _your_ countrymen also have witnessed the descent of beings
of a higher order than their own, entering closed rooms, even
as your lordship entered mine, without the opening of doors or
windows, and appearing and vanishing at will? On the reply to
this question I am ready to stake everything.'
"_Sphere_. `It is reported so. But men are divided in opinion as
to the facts. Therefore pray have done with this trifling, and
let us return to business.'
"_I_. `I was certain of it.'
"_Sphere_. `But most people say these visions arose from the brain.'
"_I_. `Say they so? Oh! believe them not; or if indeed it be
so, that this other space is really Thoughtland, then take me
to that blessed region where----'
"My words were cut short by a crash outside, which impelled me
through space--down--down--down to Flatland. Then a darkness,
and when I came to myself, I was once more a common creeping
square, in my study at home.
"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. I would begin with my wife.
"Just as I had decided, I heard a herald's proclamation.
Listening attentively, I recognized the words of the resolution
of the council, enjoining the arrest or imprisonment of any who
should pervert the minds of the people by delusions, and by
professing to have received revelations from another world. I
reflected the danger was not to be trifled with. Why not
therefore make my first experiment with my little grandson,
with whom I should be in perfect safety, for he would know
nothing of the proclamation of the council?
"I therefore immediately sent for my grandson, and taught him
once more how a point by motion in one dimension produces a
line; and how a straight line in two dimensions produces a
square. After this, forcing a laugh, I said "And now, you
scamp, you wanted to make me believe that a square may in the
same way, by motion "upward, not northward," produce another
figure, a sort of extra square in three dimensions.'
"`Dear grandpapa,' he said, `that was only my fun, and of
course I meant nothing at all by it; and I don't think I said
anything about the third dimension; and I am sure I did not say
one word about "upward, not northward," for that would be
such nonsense, you know. How could a thing move upward, not
northward? Even if I were a baby, I could not be so absurd as
that. How silly it is! Ha! ha! ha!'
"`Not at all silly,' said I, losing my temper; `here, for
example, I take this square,'--and at the word I gasped a
movable square which was lying at hand,--`and move it, you see,
not northward but, yes, I move it upward--that is to say, not
northward, but I move it somewhere--not exactly like this--but
"Here I brought my sentence to an inane conclusion, shaking the
square about in a purposeless manner, much to the amusement of
my grandson, who burst out laughing louder than ever, and
declaring I was joking with him, ran away. Thus ended my first
attempt to convert a pupil to the gospel of three dimensions."
Our poor square then shut himself up and tried to write a book
on the subject, but was greatly hampered for want of
illustrations, which he found impossible to draw, or words to
convey his meaning, which he found he could not coin.
Meanwhile, his life was under a cloud. He could not help
comparing what he saw in two dimensions with the reality of
Flatland as seen from three. One day he tried to see a cube
with his eyes shut, but was not quite certain he had realized
the original. This urged him to take some further steps to make
the revelation known, but how to begin he knew not.
At times he could not restrain dangerous utterances, dropping
such expressions as the "eye that discerns the interior of
things," "the all-seeing one," and "the third and fourth
dimensions"; and at last he was drawn at a debating society,
one day, to give a full account of his glorious journey into
Space and of all he had seen and learned there.
He was at once arrested, and taken before the great council, to
whom he retold all his story. At the close of a long
examination he was finally asked two questions:--
1. Whether he could indicate the direction which he meant when
he used the words, "upward, not northward"?
2. Whether, by any diagrams or descriptions (other than the
enumeration of imaginary sides and angles), he could indicate
the figure he called a cube?
As it was obviously impossible for him to comply with either of
these apparently reasonable demands, our unfortunate square was
finally sentenced to perpetual imprisonment.
Here, for many years, he ceaselessly tried to teach the gospel
of three dimensions to his fellow-prisoners, but alas! without
the slightest effect, being universally regarded as a harmless
Here, then, we bid our square friend a final adieu, and leave
the little book in which his story is enshrined, to consider
further the laws of a fourth dimension.
THE LAND OF FOUR DIMENSIONS
IN now summing up the result of all that has been said, and
trying to carry the facts that have been observed in the
relation of the first to the second, and the second to the
third dimension into the relations of the third to the fourth,
we will first of all consider this higher and unknown dimension
as a mathematical figure, and secondly enumerate some of the
probable laws of a world of such dimensions and its
inhabitants, as deduced by analogy, and their possible
relations with our world and its inhabitants.
Then we may further consider the actual facts around us bearing
on the question, and compare these deduced laws of the fourth
dimension with some of the claims of Christianity as stated in
Let us then, first of all, consider the mathematical or
geometrical side of the question, and inquire what would be the
character of regular figures in the fourth dimension, arguing
And in so doing, we must warn the reader that the subject is
necessarily somewhat involved and intricate; but that
nevertheless the conclusions arrived at are so fascinating and
novel, that if he will only traverse the preliminary Sahara in
patience, he will probably feel rewarded by the subsequent
oasis he reaches in the summing up and application of the whole
Let us therefore proceed to set forth the facts in order.
IN ONE DIMENSION we get--
(1) Straight lines,
(2) Varying only in one direction--length;
(3) Having two terminal points (or sides or outsides, the line
between these being the inside); and
(4) Seen only (by a single eye in line with them) as points.
IN TWO DiMENSIONS we get--
(1) Surface or flat figures,
(2) Varying in two directions--length and breadth, also in
number of sides and angles (we also get irregular figures of
one dimension, but lying in two, as curved or crooked lines;)
(3) Having not less than three terminal points or angles,
and not less than three borders or boundary lines, or sides or
outsides (the surface of the figure being the inside); and
(4) Seen only (by a single eye on a level with them) as lines.
IN THREE DIMENSiONS we get--
(2) Varying only in three directions--length, breadth, depth,
also in number and regularity of sides and angles (we also get
irregular figures of two dimensions, but lying in three, as
curved or crooked surfaces);
(3) Having not less than four[57a] terminal points or angles,
and not less than four borders, surfaces, or sides or outsides
(the contents being the insides); and
(4) Seen only (by a single eye[57b]) as surfaces.
IN FOUR DIMENSIONS we get (by analogy)--
(1) Unnamed bodies,
(2) Varying only in four directions, length, breath, depth,
and----, also in number and regularity of size and angles (we
also get irregular bodies of three dimensions, but lying in
(3) Having not less than five terminal points or angles, and
not less than five borders, solids, or sides or outsides; and
(4) Seen only (by a double eye) as solids.
Turning now to consider some of the probable laws deducible
by analogy from these data and the foregoing chapters, we may
suggest the following, the general truth of which the reader
will probably be now prepared to admit.
SOME OF THE RELATIONS OF A BEING IN ONE DIMENSION, WITH THE
DIMENSION BELOW HIM AND THE BEINGS IN IT, _e.g._, A BEING IN A
FOURTH DIMENSION WITH THE THIRD (OUR WORLD) AND THOSE IN IT, ARE:--
1. He can enter or leave the world below him, that is, appear and
disappear at will, and that without changing his form (pp. 14, 33).
2. However near to the world below him, he remains invisible
till actually in it.
3. He can be in closest proximity with the beings in the world
below, and yet outside that world altogether, and therefore
4. From his dimension he can see and enter at will the inside
of every living being and thing in the world below him.
5. When he enters the world below, he can never be wholly seen,
and that part of him that is seen is always in the form of the
world below him which he enters.
6. His voice, while still in his own dimension, would be heard
(if hearing were possible) by a being of the world below as an
internal voice, or a voice from his own inside (p. 16).
7. His appearance and disappearance in the world below are not
caused by any change of form or substance, but by his entering
or leaving that world.
8. A world and beings of any dimension include all the shapes
and characters of those below them, adding to them that further
shape and character peculiar to the added dimension.
THE RELATIONS OF A BEING IN ONE DIMENSION WITH THAT ABOVE HIM
AND ITS INHABITANTS, _e.g._, ONE IN THE THIRD DIMENSION (OUR
WORLD) WITH THE FOURTH.
1. All conception of a higher dimension is impossible, though
capable of mathematical demonstration.
2. However vast and populous the dimension, to him it is
absolutely and necessarily non-existent.
3. If he could hear such beings, the sound would appear to come
from his inner consciousness, and not from his own world without.
4. If such beings enter his world, he can only see and
comprehend that part of them that enters it. Such beings may
directly enter his own inside.
5. And to him such part _always appears in the likeness of an
inhabitant of his world_ (the inhabitants of one world being
always a partial likeness, or the likeness of a part, of those
in the world above them).
6. He can never, by his own power, leave his own dimension or world.
7. While in his world, he can never see the true appearance or
shape of any being in it, but only its exterior.
8. If raised into the dimension above, he at once perceives the
true dimension and shape of every being in his own world.
9. The beings of the dimension into which he is raised, at
first present the same appearance as the beings (now first
truly seen) in his own dimension.
10. By close inspection and careful comparison the real
difference can be discerned.
11. Even if the dimension above be visited and understood, it
is impossible to describe it in the language, or to draw it in
the figures, of his own dimension.
12. All such attempts are necessarily unintelligible, and sound
foolish and irrational.
13. All attempts to understand or grasp the dimension above,
without having entered it, are futile.
14. An eye in one's inside would, according to analogy, look in
the direction of the dimension above.
15. Each dimension adds one new direction of size, space,
capacity, and form to the one below.
16. The visibility of a being _does not depend on physical
properties_, but on its position inside or outside of the world
ANALOGICAL TABLE OF DIMENSIONS.
NONE ONE TWO THREE FOUR
No dim. may One Two Three Four
be dimension dimensions dimensions dimensions
represented may be may be may be may be
by a point represented represented represented represented
of no size by a finite by a square by a cube of by a . . . .
which has 0 straight line of 2 linear two linear of 2 linear
sides and of 2 linear inches which inches which inches which
one point inches which has 4 sides has 6 sides has 8 sides
and 0 has 2 sides or lines and or surfaces or solids
borders, and or points 4 terminal and 8 and 16
is and 2 points or terminal terminal
represented terminal angles and 4 points or points or
by 0, and points and 2 borders, and angles and angles and
may be borders, and is 12 borders, 48 borders,
formed by a is represented and is and is
fixed represented by a 2^2 or represented represented
luminous by 2, or 2 2X2, or 4 by 2^3 or by 2^4 or
point linear square 2X2X2, or 2X2X2X2, or
inches, may inches, and 8 cubic 16 Fourth
be formed by may be inches, and dim. inches,
a luminous formed by a may be and may be
point moving luminous formed by a formed by a
through 2 line moving luminous luminous
linear through 2 square cube moving
inches. linear moving through 2
inches. through 2 linear
OBSERVE THE FOUR ASCENDING SERIES.
Sides 0 -- -- 2 -- -- 4 -- -- 6 -- -- 8 --
Angles 1 -- -- 2 -- -- 4 -- -- 8 -- -- 16 --
Borders 0 -- -- 2 -- -- 4 -- -- 12 -- -- 48 --
Contents 0 -- -- 2 -- -- 4 -- -- 8 -- -- 16 --
THE LAND OF FOUR DIMENSIONS IN RELATION
TO OURS OF THREE.
TURNING now from analogies and theories to facts, we find in
the first place an almost universal consensus of opinion
amongst all nations, throughout all ages (with few and curious
exceptions), that there does exist a higher world than ours,
invisible to mortal eyes.
Those among civilized nations who have doubted or denied its
existence have done so in spite of their own feelings, and in
virtue of a reasoning that denies anything that cannot be
apprehended by the senses, in short, anything that is not
"matter." The narrowness of such reasoning gives it all its
exactness, and the materialist finds a satisfaction in denying
all he cannot account for, or where the clear but limited light
of his understanding fails to penetrate. Some minds, I suppose,
prefer the well-trimmed order of a London square within its
iron railing, or a well-stocked kitchen garden with its four
high brick walls, to the boundless prairie or the rolling
moorland. The known can at any rate be made to yield a tribute
to the complacent human wisdom which can classify, analyse, and
otherwise ticket and name it; while the unknown is denied by
our little philosophers, partly because the human mind cannot
fully grasp it, and finds it easier to ignore it, and partly
because the unknown refuses to be measured, weighed, and
arranged, and thus furnish another trophy to the greatness of
It must not be supposed, however, that our patient reader has
been asked to wade through all these pages merely to prove to
our materialists that there is a world that finds no place in
their philosophy; for the reader himself doubtless already
accepts the fact of this world in a general way, and the number
of absolute materialists is too small, and their convictions
too strong, to be much shaken by the humble methods adopted
here. We seek to do far more than this; we hope to show by
analogy how the powers of this higher world, in many an
unlooked-for particular, correspond with those that may justly
be supposed to belong to x^4.
Let us now proceed to consider some of the phenomena of this
unseen world, as current in tradition, as experienced by
individuals, and as recorded in books--mainly in the Bible,
this being the authoritative history accepted by all Christians
of the spiritual kingdom.
All believe that this world is a higher one than ours; higher
in the sense of being greater, wiser, more powerful; that it,
like ours, contains inhabitants good and bad, and regions fair,
and dark, and terrible. But we all feel that the goodness of
some of its inhabitants on the one hand, and the evil of the
rest on the other, alike transcend in every way all standards
of good and evil here; and that, in the same way, both the
fairness and the foulness and horror of its different regions
transcend all ever seen by mortal eye, or that can be pictured
by the human mind.
Most believe this unseen world to be densely peopled, and that
in some way it rules over our own with a sway in every way
greater, again stronger, and more comprehensive than that of
any known earthly government.
Another curiously universal, instinctive belief and one by no
means confined to Christianity, is, that when a man dies, part
of him (his soul, or spirit) leaves this world altogether, to
enter the higher one. And here we may turn aside to remark that
the general belief that man has a spiritual nature--something
beyond and above the highest ganglion cell in his brain,
something that leaves the body at death, but abides in it
through life--may be well illustrated by algebra.
Let, for example, the body, material and solid, be represented
fairly enough by x^3, and the spirit, higher and possessing an
unknown power, by x^4. Then (x^3+x^4) represents the man in
life, while (x^3 +x^4)-x^4 represents the departure of the
spirit (x^4) at death, which returns to its own dimension,
while the body (x^3), which is left, returns to the earth to
which it belongs.
If this, then, be true, as is surely believed amongst all
Christians, that man _is_ at any rate a complex being, having as
definite a relation with the unseen world above him as with the
visible world around him, a relation which is realized by all
after death, then is explained the instinctive craving of all
the human race, even apart from Bible revelation, after a
higher world; hence, also, the capability to receive and
understand its mysteries, and the possibility of communion
with it even now.
Turning from tradition to experience, we have not only
unnumbered instances of communion between our spirits and the
inhabitants of the higher world, but equally numerous instances
of the entrance of these higher beings, and their consequent
appearance in our world.
Speaking of communion, and turning to the Bible and to the
lives of the saints and of all good men in ancient and modern
days, and, on the other hand, to certain events in the lives of
bad men, especially in connection with great crimes, no student
of the subject can doubt that the expressions, "We see Jesus,"
"David sat before the Lord," "God spake to Moses," "Satan
tempted him," "Daniel cried unto the Lord," "I sought the
Lord, and He heard me," and hundreds of similar utterances in
biographies and from the lips of living men, represent the
_fact_ of communion and intercourse between the two worlds, just
as faith, the evidence of things not seen, prayer,
contemplation and abstraction represent the _means_.
Then, again, as to appearances. The Psychical Research Society
may be unable to discover a single authentic ghost, but
nevertheless innumerable appearances from the spirit world are
everywhere believed in, and, we think, credibly attested.
The testimony of the Bible alone (if believed) is of course
overwhelming on the point. Angels come and go at will, God
Himself is seen in Old Testament times in human form, and in
New Testament times, when our Lord takes a spiritual body, He
appears or disappears in this world of ours at will. A hand
wrote on Belshazzar's wall. The form of the Son of God was seen in
the fiery furnace. Since then appearances have been seen and voices
heard that cannot be explained by anything in three dimensions.
Passing on to consider the history of this higher world, more
especially as recorded in the Bible, we find its superiority as
to its inhabitants, its regions, and its powers, all amply
confirmed. Whether we consider the attributes of God, or of an
angel, or of a devil, whether we read about heaven or hell, we
are made conscious throughout, that all, from the omnipotent
Ruler of this higher world down to its meanest servants,
transcend our ideas in every way. We find omniscience and
omnividence claimed, "all things being naked and open to Him
with whom we have to do" (Heb. iv. 13). We find the angels
described as unseen messengers of good and evil, surrounding
our path on every side, and carrying out the will of their
Master for weal or woe. We find indications that this unseen
world itself surrounds us on every side. We are positively told
that our soul definitely enters it at death, when it is "absent
from the body."
We have more. In the twelfth chapter of the second epistle to
the Corinthians we find a detailed account given us by an
educated man, well read in the philosophy of his day--Paul, of
the fact of his being caught up into the higher world
(supposed, by referring to the date which he gives, to have
been when stoned and dragged out of a city and left for dead),
and the curious statement made that although he saw and heard
much, he found it impossible to describe or relate anything in
human language, on his return to this world.
We have also the account of Elijah and Enoch and Christ
suddenly leaving this world for the higher one, while
In some parts of the Bible, notably in the Revelation, a
definite endeavour is made to describe some of these higher
glories in human language, and all that can be done is to
picture them by the commonest earthly symbols--gold, glass,
precious stones, pearls, thrones, palms, lamps, trumpets, white
linen, swords, suppers, and so forth. No words existing to
pourtray the glories of the spiritual world.
This world is described at length as passing away altogether,
and yet the foundations of that world are not even shaken, it
being described as a "kingdom that cannot be moved."
The descent, as we have seen, of beings from it to our earth is
constantly recorded, their appearance and disappearance spoken
of, the former always in human form, though this latter is
never spoken of as being assumed for the occasion. When another
form, as that of a dove, is assumed, the fact on the contrary
is always expressly mentioned.
Not only are mysterious appearances and disappearances
constantly recorded, but very definitely in the case of our
Lord, as entering a room in a body "with flesh and bones,"
though all entrance to it was barred. Also, at another time,
when, sitting at supper, He vanished out of their sight, though
in a body and capable of eating and drinking.
The Bible speaks also of our relationship with that world. it
tells us that the apprehension of its glories are not by means
of the seeing eye or the hearing ear, but by revelation of the
Spirit of God. It reiterates the fact that the natural (or
finite) mind, though linked with the fourth dimension, cannot
of itself grasp spiritual realities, but that they must be
revealed to us by spiritual means, and that those alone to whom
this is vouchsafed can discern, judge, and weigh all earthly
things in their true and real light.
The third chapter of John tells us more. It lays down the fact
that by no education can any man mentally or morally enter this
higher sphere. He must, in the language of our Saviour, be
positively born "again," or, as better rendered, "from above,"
that is, introduced as truly into the higher world by birth, as
he was first introduced into this world in the same manner.
All spiritual perception of this world is by internal eyesight,
the "eyes of our understanding."[76a]
The language and descriptions of those who profess to have been
thus introduced, mentally and morally (though not yet
physically) into this other world, are mostly unintelligible,
and foolishness to the inhabitants of this.[76b]
Attempts, always unsuccessful, to penetrate its mysteries
beyond, or apart from what is directly revealed, only help to
fill our asylums and lists of suicides, while, on the other
hand, we are everywhere surrounded with large bodies of sane
people who claim to have been introduced into it, to obey its
laws, and to enjoy its privileges, and some of its powers.
With regard to our future relationship with this higher world,
the Bible is equally clear. It not only plainly shows that the
soul in affinity with it in life, definitely enters it at death,
but points to a mysterious time, in the hope of which thousands
have closed their eyes on this world, of a resurrection, when
the spirit shall be clothed again with a body, but differing
from the present one in its origin and in its properties and
powers, being called a spiritual body, and fitted to enter
physically, for the first time, those higher regions already
familiar to the departed soul.
These few detached remarks may serve to point out some leading
features of the world "to come," in relation to our own; we
will now consider them in connection with the mutual relations
of the various dimensions.
THE LAND OF FOUR DIMENSIONS.
FACTS AND ANALOGIES.
THOSE who have closely followed the allegory of the different
dimensions in the earlier chapters of this treatise, must
have been struck, in the first place, by the absolute
impossibility of any inhabitant belonging purely and simply to
any one dimension even conceiving the existence of a higher
world than his own; which to him, be it point, or line, or
surface, or solid space, is all that there is, or can be.
In the second place, a moment's reflection will have shown
them that in the very nature of things, it obviously must be so.
To one living entirely on a surface and in a world where there
is nothing but length and breadth indefinitely prolonged, the
idea of height and depth are absurd and impossible, and there
exists no mode of demonstrating them, unless they are
Having these facts before us, and applying them to ourselves,
we find, to our surprise, they are _not_ fully borne out in our
We _can_ to some extent understand the existence of another
world, even apart from all revelation or entrance into it, and
this is because _we are something more than mere forms of
three dimensions_. Were we such only, no such traditions as we
have alluded to would be common to the human race; the idea of
a higher world would be as impossible to us as to the brute
But there is a part of us that has been made in the likeness of
God, a part breathed into us by the Divine breath, through
which we instinctively perceive the higher sphere, and by means
of which we are partly able to apprehend its teachings.
Still, to a great extent, the analogies hold good. We, like our
friend the square, in Flatland, can see spiritual beings when
they enter our world, and like him, explain their appearing and
vanishing by magic or miracle, rather than by the simple fact
of their entering or leaving our dimension.
Again, analogy has shown us how near, how very near, the new
dimension that characterizes the world above us may be, with
its inhabitants, and yet be outside ours altogether.
On the other hand, analogy shows us that just as a point is
comprehended in a line, a line in a square, a square in a cube,
so is our world of three dimensions completely included and
swallowed up in the universe of four.
Analogy points out how onmnividence is an almost necessary
property of a higher world.
The careful comparison of the analogies of the third and fourth
dimensions with the revealed relation of our world to the
spirit world, shows such a likeness between the two, that it is
not too much to say that if we call our world a world of three
dimensions, we may fairly consider the spirit world in many
respects a world of four.
We conclude, therefore, that a higher world than ours is not
only conceivably possible, but probable; secondly, that such a
world may be considered as a world of four dimensions; and,
thirdly, that the spiritual world agrees largely in its
mysterious laws, in its language which is foolishness to us, in
its miraculous appearances and interpositions, in its high and
lofty claims of omniscience, omnividence, etc., and in other
particulars, with what by analogy would be the laws, language,
and claims of a fourth dimension.
Once these conclusions are admitted, and our eventual
destination, body and soul, seen to be in this higher world,
the transcendent importance of understanding all about it, the
intense and real interest of all connected with it, becomes
If it be true that we are everywhere surrounded by another
world, which is our final goal, how foolish to stop our ears to
its history, to shut our eyes to its facts, as recorded in what
is believed by us to be an authoritative statement of them!
The honest materialist has some excuse for the total neglect of
a Bible he disbelieves; but what shall we say of those who,
professing to accept these stupendous realities, are utterly
indifferent to them and the Book that reveals them?
Surely the study of what we will term this fourth dimension far
transcends the highest earthly subjects, and dwarfs to their
proper level all objects of human ambition, for we see at once
that the lowest inhabitant of the fourth dimension is
necessarily of a different and a higher order than the greatest
monarch in the third. The lowliest plant is of a higher order
than, and different beauty from, the most precious mineral,
possessing as it does one sort of life; the feeblest animal,
again, is of a higher order than, and different beauty from,
the oak or the cedar, possessing as it does another sort of
life; and in the same way the humblest subject of God's
spiritual kingdom is of a higher order than, and different
beauty from the highest animal, possessing as he does yet
another sort of life.
Another great advantage these considerations give is that, if
admitted, they at once rescue Christianity from being degraded
to a code of ethics, whereby men can better adorn this third
dimension, and present it in its true and proper character of
a new world and kingdom, with its invisible inhabitants, laws,
houses, and rulers; in a word, it becomes objective instead of
subjective. Light is also thrown on the mysterious connection
of soul and body in us personally, and on the entrance of the
former into another world the moment it is released from the
body by death.
Conversion, the new birth, salvation, or whatever the entrance
of the light of Christianity into the heart of man is called,
is now seen _not_ to be a process of education in morality, in
order to produce better members of society, and of this world,
but something infinitely higher--a positive resurrection into
a higher and purer world, where Christ now is, an instruction
in the heavenly principles, and a glimpse into its transcendent
glories, coupled with a view into the hearts of men and of the
real nature of all earthly things, that reveal their true
value, the result being undoubtedly to elevate the tone, life,
and manners of the individual, but the object being to fit
him to be shortly removed altogether to that higher sphere to
which he now belongs, there to be clothed with a spiritual (or
fourth dimension) body.
Those who have followed, and who accept the preceding lines of
argument and thought, will undoubtedly thus see more clearly
the reason and cause of a great many distinctive features of
Christianity. They will understand its lofty claims, and know
why, when its laws are truly proclaimed, they are to men
foolishness, where, as, when adulterated with the wisdom of
this world (of the third dimension), they are more or less
intelligible. They will see why it is insisted on so strongly
that the natural man cannot receive its mysteries (being of the
third dimension), and that they are only spiritually discerned
(that is, by revelation), why we must be born again, or
introduced by the power of God into this new world.
They will now see how it is possible this kingdom can be within
us, and yet surrounding us; how angels may be by our very side,
and yet outside this world of space altogether.
They will see the impossibility of guessing the direction of
heaven or hell, seeing there is an unknown direction around us,
which we cannot conceive, and the puerility of assuming that it
must be "up above" or "down below."
They will see that though the glorious material universe
extends beyond the utmost limits of our vision, even
artificially aided by the most powerful telescopes, that does
not prevent the spiritual world and its beings, and heaven and
hell being by our very side.
They will see that, far from these spiritual regions occupying
some small corner of the material universe, as surely as the
greater includes the less, so surely is the material universe,
vast as it is, swallowed up in the spiritual.
The indications of the vast unknown extent of this spiritual
kingdom will be more clearly understood in such references as
Ephesians i. 21. They will now more clearly discern "the
powers of the world to come," whereof we speak, and understand
the mysterious appearances in the Bible of spiritual beings,
always in human form, necessarily so to be seen in the third
dimension. It will be no difficulty to them to believe all
thoughts and hearts are naked and open to the Ruler of this
world, still less that every closed object and the inside of
every solid thing is and must be clearly seen.
Believing as we do that the soul, or immortal part, of man is
connected with this fourth dimension, while the body belongs to
the third, the phenomena of death is clearly seen to be the
separation of these two dimensions, the body refining in this
world while the soul enters the other.
The simple and almost childish language of Revelation, already
alluded to, will no longer appear strange, when it is seen that
it is an inspired attempt to put the glories of the fourth
dimension into the language of the third; hence the necessary
use of such words as glass, gold, etc. Nor will the language of
Paul, in 2 Cor. xii., fail to be better understood as to what
he heard when caught up out of the third dimension into the
fourth being impossible to utter or render into human language.
The arrogance of man will receive a severe and salutary check
when it is seen how, in the very nature of things, it is
impossible he can understand even the new direction in which
this glorious world lies, while the Christian will quite see
why he is constantly misunderstood, and always so, indeed,
when he lives in the region of the fourth dimension; and hence
that saying is and must be true, that he "discerns all things,
yet he himself is discerned of no man."
It is hardly too much to say that when the possibility is
proved of there being another world in close proximity to ours,
but necessarily invisible to us, save as its beings enter or
leave ours, and when we discern a few of the leading laws,
that by strict analogy may be taken as found in such a world in
relation to ours; that nearly the whole of Christianity becomes
clearer to us, its language more intelligible, and some of its
most difficult statements almost axiomatic.
If we consider such scriptures, for instance, as Ephesians i.,
Colossians i., 2 Corinthians v., and 1 Corinthians xv., we
find, just as Adam is the principal being in the third
dimension, so is Christ in the fourth, and hence with
appropriateness is called "the Second Man."
The new creation is seen to be as literal and real an
introduction of beings into the fourth dimension as the old was
into the third, and such a verse as Colossians i. 16,
descriptive of the Creator's power in both dimensions, here
designated visible and invisible, is apprehended.
The power whereby Christians are lifted out of the third into
the fourth, mentally, at any rate, is graphically portrayed in
Colossians ii. 20-iii. 4. They are there spoken of as dead and
risen with Christ (into the fourth dimension), and are to be
occupied with the superior glories of their new sphere.
It is but little wonder, therefore, that those who have really
been made thus alive should speak somewhat slightingly of the
glories of this world, when they consider the higher glories of
their own, or that they should be enthusiastic in describing
it, or earnest in endeavouring to introduce others into it;
nor, on the other hand, that by those who are not thus alive,
they should be accounted fools and fanatics, and their language
extravagant and unintelligible. The wonder rather is, that
those who are thus alive should not be more enthusiastic than
they are, and appear more foolish than they do.
In conclusion, we would briefly emphasize these following points.
If we have to any degree succeeded in showing the probability
of that other world being of a higher dimension than our own,
and that we have a link with it naturally in the spiritual part
of our beings; we see most clearly established by analogy,
that by no development of our mental faculties, by no
advancement in science, by no cultivation of conduct or morals,
in short, by no education or improvement of the human race,
_per se_, can we understand, enter, or view this higher kingdom.
Any comprehension, in short, of it, is not by cultivation, or
strengthening even of that link we already have with it in our
souls, but by a distinct revelation from that world to these
powers within us, and a consequent elevation of these powers
into this higher dimension. In relation therefore with
Christianity (as we call this scheme of revelation), we see why
the most highly cultured in the learning of the third dimension
possess little if any advantage (nay, often the reverse) over
the wayfaring man, though a fool, inasmuch as it is to both of
them a distinct revelation, more easily received indeed in the
latter case, since there is here no force of intellect to set
aside, for the meaning of our Lord's saying is now clearly
apparent, that except we become as little children, we shall
_in no wise_ enter the kingdom of heaven.
If then these few remarks, and these mathematical analogies,
serve to show that the scriptural way of entering the Kingdom
of God is the only way possible; if they assist to rouse
enthusiasm in believers, to convert unbelievers, and to silence
materialists, the object of the writer will be fully gained.
 "Flatland." Seeleys.
 A line having no breadth, its outsides (so to speak) are its
two extremities, that which lies between being the inside of the
line; and this inside is naked and open to the eye of our square
in two dimensions, but can never be seen by being in one. This
will become clearer as we proceed.
 This chapter will be better understood if the reader provides
himself with a few squares, circles, triangles, etc., cut out of
cardboard, to represent the inhabitants,--the country being
represented by the top of the table on which they are laid; while
a house in flatland may be easily made by enclosing a space with
bits of cotton.
 By analogy these are of course "insides."
 This is exactly what the grandson suggested.
 Observe the inside of one dimension is always the outside
of the dimension higher.
 Observe the surface of a higher dimension appears to be
the interior to the dimension below.
 This diagram shows what is meant by "upward not northward;"
upward being the direction of the third dimension, a direction
impossible to be even conceived by an inhabitant of two
dimensions, familiar as it is to us.
 No flat figure can have less than three angles and three
borders, viz., a triangle; for two straight lines cannot enclose
a space. (Circles and curved lines are not considered, being
really an infinite number of straight lines.)
[57a] A solid body cannot have fewer than four angles and sides,
viz., a solid triangle. (Circular and curved bodies are not
considered, being composed of an infinite number of sides.)
[57b] We see bodies as solids, not surfaces, simply because
we have two eyes, and can see them from two points of view
at once. The stereoscope is founded on this fact.
 By higher is meant greater in qualities and powers. In
speaking of this world, though the whole of it is included,
it is mainly with that part of it that constitutes God's
spiritual kingdom that we are concerned.
 In taking x^4 here to represent spirits and hereafter
the spirit world, it must be remembered that we are absolutely
ignorant of what is really involved by this formula. As far as
we know, the "material" is strictly limited to three dimensions,
nothing in one or two being material, or having any substance
whatever. It must therefore be distinctly understood that we
firmly believe God is a spirit, and the other world a spiritual
one, and that we have no wish or intention of materializing it
in enforcing the truth of some of its laws by means of analogies
drawn from a supposed fourth dimension.
 Hebrews xi. 1.
 We would ask the reader most especially to note this in
connection with paragraph 1, page 59, and paragraph 16 page 62.
 I Corinthians i. 9-13.
[76a] Ephesians i. 18.
[76b] I Corinthians ii. 14.
 This shows also the folly of those who, reasoning on
"three dimension" lines, assert that the spiritual world
must be beyond the confines of the material, and hence millions
of miles away, and farther than the farthest star.
 I Corinthians ii., end of chapter.
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