REPRESENTATION OF THREE-SPACE BY NAMES, AND IN A PLANE.
We may now ask ourselves the best way of passing on to a clear
comprehension of the facts of higher space. Something can be effected by
looking at these models; but it is improbable that more than a slight
sense of analogy will be obtained thus. Indeed, we have been trusting
hitherto to a method which has something vicious about it--we have been
trusting to our sense of what _must_ be. The plan adopted, as the
serious effort towards the comprehension of this subject, is to learn a
small portion of higher space. If any reader feel a difficulty in the
foregoing chapters, or if the subject is to be taught to young minds, it
is far better to abandon all attempt to see what higher space _must_ be,
and to learn what it _is_ from the following chapters.
NAMING A PIECE OF SPACE.
The diagram (Fig. 6) represents a block of 27 cubes, which form Set 1 of
the 81 cubes. The cubes are coloured, and it will be seen that the
colours are arranged after the pattern of Model 1 of previous chapters,
which will serve as a key to the block. In the diagram, G. denotes Gold,
O. Orange, F. Fawn, Br. Brown, and so on. We will give names to the
cubes of this block. They should not be learnt, but kept for reference.
We will write these names in three sets, the lowest consisting of the
cubes which touch the table, the next of those immediately above them,
and the third of those at the top. Thus the Gold cube is called Corvus,
the Orange, Cuspis, the Fawn, Nugæ, and the central one below, Syce. The
corresponding colours of the following set can easily be traced.
Olus Semita Lama
Via Mel Iter
Ilex Callis Sors
Bucina Murex Daps
Alvus Mala Proes
Arctos Mœna Far
Cista Cadus Crus
Dos Syce Bolus
Corvus Cuspis Nugæ
Thus the central or Light-buff cube is called Mala; the middle one of
the lower face is Syce; of the upper face Mel; of the right face, Proes;
of the left, Alvus; of the front, Mœna (the Dark-blue square of Model
1); and of the back, Murex (the Light-yellow square).
Now, if Model 1 be taken, and considered as representing a block of 64
cubes, the Gold corner as one cube, the Orange line as two cubes, the
Fawn point as one cube, the Dark-blue square as four cubes, the
Light-buff interior as eight cubes, and so on, it will correspond to the
diagram (Fig. 7). This block differs from the last in the number of
cubes, but the arrangement of the colours is the same. The following
table gives the names which we will use for these cubes. There are no
new names; they are only applied more than once to all cubes of the same
colour.
{Olus Semita Semita Lama
Fourth{Via Mel Mel Iter
Floor.{Via Mel Mel Iter
{Ilex Callis Callis Sors
{Bucina Murex Murex Daps
Third {Alvus Mala Mala Proes
Floor.{Alvus Mala Mala Proes
{Arctos Mœna Mœna Far
{Bucina Murex Murex Daps
Second{Alvus Mala Mala Proes
Floor.{Alvus Mala Mala Proes
{Arctos Mœna Mœna Far
{Cista Cadus Cadus Crus
First {Dos Syce Syce Bolus
Floor.{Dos Syce Syce Bolus
{Corvus Cuspis Cuspis Nugæ
[Illustration: Fig. 6.]
[Illustration: Fig. 7.]
[Illustration: Fig. 8.]
If we now consider Model 1 to represent a block, five cubes each way,
built up of inch cubes, and colour it in the same way, that is, with
similar colours for the corner-cubes, edge-cubes, face-cubes, and
interior-cubes, we obtain what is represented in the diagram (Fig. 8).
Here we have nine Dark-blue cubes called Mœna; that is, Mœna denotes the
nine Dark-blue cubes, forming a layer on the front of the cube, and
filling up the whole front except the edges and points. Cuspis denotes
three Orange, Dos three Blue, and Arctos three Brown cubes.
Now, the block of cubes can be similarly increased to any size we
please. The corners will always consist of single cubes; that is, Corvus
will remain a single cubic inch, even though the block be a hundred
inches each way. Cuspis, in that case, will be 98 inches long, and
consist of a row of 98 cubes; Arctos, also, will be a long thin line of
cubes standing up. Mœna will be a thin layer of cubes almost covering
the whole front of the block; the number of them will be 98 times 98.
Syce will be a similar square layer of cubes on the ground, so also
Mel, Alvus, Proes, and Murex in their respective places. Mala, the
interior of the cube, will consist of 98 times 98 times 98 inch cubes.
[Illustration: Fig. 9]
Now, if we continued in this manner till we had a very large block of
thousands of cubes in each side Corvus would, in comparison to the whole
block, be a minute point of a cubic shape, and Cuspis would be a mere
line of minute cubes, which would have length, but very small depth or
height. Next, if we suppose this much sub-divided block to be reduced in
size till it becomes one measuring an inch each way, the cubes of which
it consists must each of them become extremely minute, and the corner
cubes and line cubes would be scarcely discernible. But the cubes on the
faces would be just as visible as before. For instance, the cubes
composing Mœna would stretch out on the face of the cube so as to fill
it up. They would form a layer of extreme thinness, but would cover the
face of the cube (all of it except the minute lines and points). Thus we
may use the words Corvus and Nugæ, etc., to denote the corner-points of
the cube, the words Mœna, Syce, Mel, Alvus, Proes, Murex, to denote the
faces. It must be remembered that these faces have a thickness, but it
is extremely minute compared with the cube. Mala would denote all the
cubes of the interior except those, which compose the faces, edges, and
points. Thus, Mala would practically mean the whole cube except the
colouring on it. And it is in this sense that these words will be used.
In the models, the Gold point is intended to be a Corvus, only it is
made large to be visible; so too the Orange line is meant for Cuspis,
but magnified for the same reason. Finally, the 27 names of cubes, with
which we began, come to be the names of the points, lines, and faces of
a cube, as shown in the diagram (Fig. 9). With these names it is easy
to express what a plane-being would see of any cube. Let us suppose that
Mœna is only of the thickness of his matter. We suppose his matter to be
composed of particles, which slip about on his plane, and are so thin
that he cannot by any means discern any thickness in them. So he has no
idea of thickness. But we know that his matter must have some thickness,
and we suppose Mœna to be of that degree of thickness. If the cube be
placed so that Mœna is in his plane, Corvus, Cuspis, Nugæ, Far, Sors,
Callis, Ilex and Arctos will just come into his apprehension; they will
be like bits of his matter, while all that is beyond them in the
direction he does not know, will be hidden from him. Thus a plane-being
can only perceive the Mœna or Syce or some one other face of a cube;
that is, he would take the Mœna of a cube to be a solid in his
plane-space, and he would see the lines Cuspis, Far, Callis, Arctos. To
him they would bound it. The points Corvus, Nugæ, Sors, and Ilex, he
would not see, for they are only as long as the thickness of his matter,
and that is so slight as to be indiscernible to him.
We must now go with great care through the exact processes by which a
plane-being would study a cube. For this purpose we use square slabs
which have a certain thickness, but are supposed to be as thin as a
plane-being’s matter. Now, let us take the first set of 81 cubes again,
and build them from 1 to 27. We must realize clearly that two kinds of
blocks can be built. It may be built of 27 cubes, each similar to Model
1, in which case each cube has its regions coloured, but all the cubes
are alike. Or it may be built of 27 differently coloured cubes like Set
1, in which case each cube is coloured wholly with one colour in all its
regions. If the latter set be used, we can still use the names Mœna,
Alvus, etc. to denote the front, side, etc., of any one of the cubes,
whatever be its colour. When they are built up, place a piece of card
against the front to represent the plane on which the plane-being lives.
The front of each of the cubes in the front of the block touches the
plane. In previous chapters we have supposed Mœna to be a Blue square.
But we can apply the name to the front of a cube of any colour. Let us
say the Mœna of each front cube is in the plane; the Mœna of the Gold
cube is Gold, and so on. To represent this, take nine slabs of the same
colours as the cubes. Place a stiff piece of cardboard (or a book-cover)
slanting from you, and put the slabs on it. They can be supported on the
incline so as to prevent their slipping down away from you by a thin
book, or another sheet of cardboard, which stands for the surface of the
plane-being’s earth.
We will now give names to the cubes of Block 1 of the 81 Set. We call
each one Mala, to denote that it is a cube. They are written in the
following list in floors or layers, and are supposed to run backwards or
away from the reader. Thus, in the first layer, Frenum Mala is behind or
farther away than Urna Mala; in the second layer, Ostrum is in front,
Uncus behind it, and Ala behind Uncus.
Third, or {Mars Mala Merces Mala Tyro Mala
Top {Spicula Mala Mora Mala Oliva Mala
Floor. {Comes Mala Tibicen Mala Vestis Mala
Second, or{Ala Mala Cortis Mala Aer Mala
Middle {Uncus Mala Pallor Mala Tergum Mala
Floor. {Ostrum Mala Bidens Mala Scena Mala
First, or {Sector Mala Hama Mala Remus Mala
Bottom {Frenum Mala Plebs Mala Sypho Mala
Floor. {Urna Mala Moles Mala Saltus Mala
These names should be learnt so that the different cubes in the block
can be referred to quite easily and immediately by name. They must be
learnt in every order, that is, in each of the three directions
backwards and forwards, _e.g._ Urna to Saltus, Urna to Sector, Urna to
Comes; and the same reversed, viz., Comes to Urna, Sector to Urna, etc.
Only by so learning them can the mind identify any one individually
without even a momentary reference to the others around it. It is well
to make it a rule not to proceed from one cube to a distant one without
naming the intermediate cubes. For, in Space we cannot pass from one
part to another without going through the intermediate portions. And, in
thinking of Space, it is well to accustom our minds to the same
limitations.
Urna Mala is supposed to be solid Gold an inch each way; so too all the
cubes are supposed to be entirely of the colour which they show on their
faces. Thus any section of Moles Mala will be Orange, of Plebs Mala
Black, and so on.
[Illustration: Fig. 10.]
Let us now draw a pair of lines on a piece of paper or cardboard like
those in the diagram (Fig. 10). In this diagram the top of the page is
supposed to rest on the table, and the bottom of the page to be raised
and brought near the eye, so that the plane of the diagram slopes
upwards to the reader. Let Z denote the upward direction, and X the
direction from left to right. Let us turn the Block of cubes with its
front upon this slope _i.e._ so that Urna fits upon the square marked
Urna. Moles will be to the right and Ostrum above Urna, _i.e._ nearer
the eye. We might leave the block as it stands and put the piece of
cardboard against it; in this case our plane-world would be vertical. It
is difficult to fix the cubes in this position on the plane, and
therefore more convenient if the cardboard be so inclined that they will
not slip off. But the upward direction must be identified with Z. Now,
taking the slabs, let us compose what a plane-being would see of the
Block. He would perceive just the front faces of the cubes of the Block,
as it comes into his plane; these front faces we may call the Moenas of
the cubes. Let each of the slabs represent the Moena of its
corresponding cube, the Gold slab of the Gold cube and so on. They are
thicker than they should be; but we must overlook this and suppose we
simply see the thickness as a line. We thus build a square of nine slabs
to represent the appearance to a plane-being of the front face of the
Block. The middle one, Bidens Moena, would be completely hidden from him
by the others on all its sides, and he would see the edges of the eight
outer squares. If the Block now begin to move through the plane, that
is, to cut through the piece of paper at right angles to it, it will not
for some time appear any different. For the sections of Urna are all
Gold like the front face Moena, so that the appearance of Urna at any
point in its passage will be a Gold square exactly like Urna Moena, seen
by the plane-being as a line. Thus, if the speed of the Block’s passage
be one inch a minute, the plane-being will see no change for a minute.
In other words, this set of slabs lasting one minute will represent what
he sees.
When the Block has passed one inch, a different set of cubes appears.
Remove the front layer of cubes. There will now be in contact with the
paper nine new cubes, whose names we write in the order in which we
should see them through a piece of glass standing upright in front of
the Block:
Spicula Mala Mora Mala Oliva Mala
Uncus Mala Pallor Mala Tergum Mala
Frenum Mala Plebs Mala Sypho Mala
We pick out nine slabs to represent the Moenas of these cubes, and
placed in order they show what the plane-being sees of the second set
of cubes as they pass through. Similarly the third wall of the Block
will come into the plane, and looking at them similarly, as it were
through an upright piece of glass, we write their names:
Mars Mala Merces Mala Tyro Mala
Ala Mala Cortis Mala Aer Mala
Sector Mala Hama Mala Remus Mala
Now, it is evident that these slabs stand at different times for
different parts of the cubes. We can imagine them to stand for the Moena
of each cube as it passes through. In that case, the first set of slabs,
which we put up, represents the Moenas of the front wall of cubes; the
next set, the Moenas of the second wall. Thus, if all the three sets of
slabs be together on the table, we have a representation of the sections
of the cube. For some purposes it would be better to have four sets of
slabs, the fourth set representing the Murex of the third wall; for the
three sets only show the front faces of the cubes, and therefore would
not indicate anything about the back faces of the Block. For instance,
if a line passed through the Block diagonally from the point Corvus
(Gold) to the point Lama (Deep-blue), it would be represented on the
slabs by a point at the bottom left-hand corner of the Gold slab, a
second point at the same corner of the Light-buff slab, and a third at
the same corner of the Deep-blue slab. Thus, we should have the points
mapped at which the line entered the fronts of the walls of cubes, but
not the point in Lama at which it would leave the Block.
Let the Diagrams 1, 2, 3 (Fig. 11), be the three sets of slabs. To see
the diagrams properly, the reader must set the top of the page on the
table, and look along the page from the bottom of it. The line in
question, which runs from the bottom left-hand near corner to the top
right-hand far corner of the Block will be represented in the three sets
of slabs by the points A, B, C. To complete the diagram of its course,
we need a fourth set of slabs for the Murex of the third wall; the same
object might be attained, if we had another Block of 27 cubes behind the
first Block and represented its front or Moenas by a set of slabs. For
the point, at which the line leaves the first Block is identical with
that at which it enters the second Block.
[Illustration: Fig. 11.]
If we suppose a sheet of glass to be the plane-world, the Diagrams 1, 2,
3 (Fig. 11), may be drawn more naturally to us as Diagrams α, β, γ (Fig.
12). Here α represents the Moenas of the first wall, β those of the
second, γ those of the third. But to get the plane-being’s view we must
look over the edge of the glass down the Z axis.
[Illustration: Fig. 12.]
Set 2 of slabs represent the Moenas of Wall 2. These Moenas are in
contact with the Murex of Wall 1. Thus Set 2 will show where the line
issues from Wall 1 as well as where it enters Wall 2.
The plane-being, therefore, could get an idea of the Block of cubes by
learning these slabs. He ought not to call the Gold slab Urna Mala, but
Urna Moena, and so on, because all that he learns are Moenas, merely the
thin faces of the cubes. By introducing the course of time, he can
represent the Block more nearly. For, if he supposes it to be passing an
inch a minute, he may give the name Urna Mala to the Gold slab enduring
for a minute.
But, when he has learnt the slabs in this position and sequence, he has
only a very partial view of the Block. Let the Block turn round the Z
axis, as Model 1 turns round the Brown line. A different set of cubes
comes into his plane, and now they come in on the Alvus faces.
(Alvus is here used to denote the left-hand faces of the cubes, and is
not supposed to be Vermilion; it is simply the thinnest slice on the
left hand of the cube and of the same colour as the cube.) To represent
this, the plane-being should employ a fresh set of slabs, for there is
nothing common to the Moena and Alvus faces except an edge. But, since
each cube is of the same colour throughout, the same slab may be used
for its different faces. Thus the Alvus of Urna Mala can be represented
by a Gold slab. Only it must never be forgotten that it is meant to be a
new slab, and is not identical with the same slab used for Moena.
[Illustration: Fig. 13.]
[Illustration: Fig. 14.]
Now, when the Block of cubes has turned round the Brown line into the
plane, it is clear that they will be on the side of the Z axis opposite
to that on which were the Moena slabs. The line, which ran Y, now runs
-X. Thus the slabs will occupy the second quadrant marked by the axes,
as shown in the diagram (Fig. 13). Each of these slabs we will name
Alvus. In this view, as before, the book is supposed to be tilted up
towards the reader, so that the Z axis runs from O to his eye. Then, if
the Block be passed at right angles through the plane, there will come
into view the two sets of slabs represented in the Diagrams (Fig. 13).
In copying this arrangement with the slabs, the cardboard on which they
are arranged must slant upwards to the eye, _i.e._, OZ must run up to
the eye, and the sides of the slabs seen are in Diagram 2 (Fig. 13), the
upper edges of Tibicen, Mora, Merces; in Diagram 3, the upper edges of
Vestis, Oliva, Tyro.
There is another view of the Block possible to a plane-being. If the
Block be turned round the X axis, the lower face comes into the vertical
plane. This corresponds to turning Model 1 round the Orange line. On
referring to the diagram (Fig. 14), we now see that the name of the
faces of the cubes coming into the plane is Syce. Here the plane-being
looks from the extremity of the Z axis and the squares, which he sees
run from him in the -Z direction. (As this turn of the Block brings its
Syce into the vertical plane so that it extends three inches below the
base line of its Moena, it is evident that the turn is only possible if
the Moena be originally at a height of at least three inches above the
plane-being’s earth line in the vertical plane.) Next, if the Block be
passed through the plane, the sections shown in the Diagrams 2 and 3
(Fig. 14) are brought into view.
Thus, there are three distinct ways of regarding the cubic Block, each
of them equally primary; and if the plane-being is to have a correct
idea of the Block, he must be equally familiar with each view. By means
of the slabs each aspect can be represented; but we must remember in
each of the three cases, that the slabs represent different parts of the
cube.
When we look at the cube Pallor Mala in space, we see that it touches
six other cubes by its six faces. But the plane-being could only arrive
at this fact by comparing different views. Taking the three Moena
sections of the Block, he can see that Pallor Mala Moena touches Plebs
Moena, Mora Moena, Uncus Moena, and Tergum Moena by lines. And it takes
the place of Bidens Moena, and is itself displaced by Cortis Moena as
the Block passes through the plane. Next, this same Pallor Mala can
appear to him as an Alvus. In this case, it touches Plebs Alvus, Mora
Alvus, Bidens Alvus, and Cortis Alvus by lines, takes the place of Uncus
Alvus, and is itself displaced by Tergum Alvus as the Block moves.
Similarly he can observe the relations, if the Syce of the Block be in
his plane.
Hence, this unknown body Pallor Mala appears to him now as one
plane-figure now as another, and comes before him in different
connections. Pallor Mala is that which satisfies all these relations.
Each of them he can in turn present to sense; but the total conception
of Pallor Mala itself can only, if at all, grow up in his mind. The way
for him to form this mental conception, is to go through all the
practical possibilities which Pallor Mala would afford him by its
various movements and turns. In our world these various relations are
found by the most simple observations; but a plane-being could only
acquire them by considerable labour. And if he determined to obtain a
knowledge of the physical existence of a higher world than his own, he
must pass through such discipline.
* * * * *
[Illustration: Fig. 15.]
[Illustration: Fig. 16.]
We will see what change could be introduced into the shapes he builds by
the movements, which he does not know in his world. Let us build up this
shape with the cubes of the Block: Urna Mala, Moles Mala, Bidens Mala,
Tibicen Mala. To the plane-being this shape would be the slabs, Urna
Moena, Moles Moena, Bidens Moena, Tibicen Moena (Fig. 15). Now let the
Block be turned round the Z axis, so that it goes past the position, in
which the Alvus sides enter the vertical plane. Let it move until,
passing through the plane, the same Moena sides come in again. The mass
of the Block will now have cut through the plane and be on the near side
of it towards us; but the Moena faces only will be on the plane-being’s
side of it. The diagram (Fig. 16) shows what he will see, and it will
seem to him similar to the first shape (Fig. 15) in every respect except
its disposition with regard to the Z axis. It lies in the direction -X,
opposite to that of the first figure. However much he turn these two
figures about in the plane, he cannot make one occupy the place of the
other, part for part. Hence it appears that, if we turn the
plane-being’s figure about a line, it undergoes an operation which is to
him quite mysterious. He cannot by any turn in his plane produce the
change in the figure produced by us. A little observation will show that
a plane-being can only turn round a point. Turning round a line is a
process unknown to him. Therefore one of the elements in a plane-being’s
knowledge of a space higher than his own, will be the conception of a
kind of turning which will change his solid bodies into their own
images.