PART II.
CHAPTER I.
THREE-SPACE. GENESIS OF A CUBE. APPEARANCES OF A CUBE TO A PLANE-BEING.
The models consist of a set of eight and a set of four cubes. They are marked with different colours, so as to show the properties of the figure in Higher Space, to which they belong.
The simplest figure in one-dimensional space, that is, in a straight line, is a straight line bounded at the two extremities. The figure in this case consists of a length bounded by two points.
Looking at Cube 1, and placing it so that the figure 1 is uppermost, we notice a straight line in contact with the table, which is coloured Orange. It begins in a Gold point and ends in a Fawn point. The Orange extends to some distance on two faces of the Cube; but for our present purpose we suppose it to be simply a thin line.
This line we conceive to be generated in the following way. Let a point move and trace out a line. Let the point be the Gold point, and let it, moving, trace out the Orange line and terminate in the Fawn point. Thus the figure consists of the point at which it begins, the point at which it ends, and the portion between. We may suppose the point to start as a Gold point, to change its colour to Orange during the motion, and when it stops to become Fawn. The motion we suppose from left to right, and its direction we call X.
If, now, this Orange line move away from us at right angles, it will trace out a square. Let this be the Black square, which is seen underneath Model 1. The points, which bound the line, will during this motion trace out lines, and to these lines there will be terminal points. Also, the Square will be terminated by a line on the opposite side. Let the Gold point in moving away trace out a Blue line and end in a Buff point; the Fawn point a Crimson line ending in a Terracotta point. The Orange line, having traced a Black square, ends in a Green-grey line. This direction, away from the observer, we call Y.
Now, let the whole Black square traced out by the Orange line move upwards at right angles. It will trace out a new figure, a Cube. And the edges of the square, while moving upwards, will trace out squares. Bounding the cube, and opposite to the Black square, will be another square. Let the Orange line moving upwards trace a Dark Blue square and end in a Reddish line. The Gold point traces a Brown line; the Fawn point traces a French-grey line, and these lines end in a Light-blue and a Dull-purple point. Let the Blue line trace a Vermilion square and end in a Deep-yellow line. Let the Buff point trace a Green line, and end in a Red point. The Green-grey line traces a Light-yellow square and ends in a Leaden line; the Terracotta point traces a Dark-slate line and ends in a Deep-blue point. The Crimson line traces a Blue-green square and ends in a Bright-blue line.
Finally, the Black square traces a Cube, the colour of which is invisible, and ends in a white square. We suppose the colour of the cube to be a Light-buff. The upward direction we call Z. Thus we say: The Gold point moved Z, traces a Brown line, and ends in a Light-blue point.
We can now clearly realize and refer to each region of the cube by a colour.
At the Gold point, lines from three directions meet, the X line Orange, the Y line Blue, the Z line Brown.
Thus we began with a figure of one dimension, a line, we passed on to a figure of two dimensions, a square, and ended with a figure of three dimensions, a cube.
The square represents a figure in two dimensions; but if we want to realize what it is to a being in two dimensions, we must not look down on it. Such a view could not be taken by a plane-being.
Let us suppose a being moving on the surface of the table and unable to rise from it. Let it not know that it is upon anything, but let it believe that the two directions and compounds of those two directions are all possible directions. Moreover, let it not ask the question: “On what am I supported?” Let it see no reason for any such question, but simply call the smooth surface, along which it moves, Space.
Such a being could not tell the colour of the square traced by the Orange line. The square would be bounded by the lines which surround it, and only by breaking through one of those lines could the plane-being discover the colour of the square.
In trying to realize the experience of a plane-being it is best to suppose that its two dimensions are upwards and sideways, i.e., Z and X, because, if there be any matter in the plane-world, it will, like matter in the solid world, exert attractions and repulsions. The matter, like the beings, must be supposed very thin, that is, of so slight thickness that it is quite unnoticed by the being. Now, if there be a very large mass of such matter lying on the table, and a plane-being be free to move about it, he will be attracted to it in every direction. “Towards this huge mass” would be “Down,” and “Away from it” would be “Up,” just as “Towards the earth” is to solid beings “Down,” and “Away from it” is “Up,” at whatever part of the globe they may be. Hence, if we want to realize a plane-being’s feelings, we must keep the sense of up and down. Therefore we must use the Z direction, and it is more convenient to take Z and X than Z and Y.
Any direction lying between these is said to be compounded of the two; for, if we move slantwise for some distance, the point reached might have been also reached by going a certain distance X, and then a certain distance Z, or vice versâ.
Let us suppose the Orange line has moved Z, and traced the Dark-blue square ending in the Reddish line. If we now place a piece of stiff paper against the Dark-blue square, and suppose the plane-beings to move to and fro on that surface of the paper, which touches the square, we shall have means of representing their experience.
To obtain a more consistent view of their existence, let us suppose the piece of paper extended, so that it cuts through our earth and comes out at the antipodes, thus cutting the earth in two. Then suppose all the earth removed away, both hemispheres vanishing, and only a very thin layer of matter left upon the paper on that side which touches the Dark-blue square. This represents what the world would be to a plane-being.
It is of some importance to get the notion of the directions in a plane-world, as great difficulty arises from our notions of up and down. We miss the right analogy if we conceive of a plane-world without the conception of up and down.
A good plan is, to use a slanting surface, a stiff card or book cover, so placed that it slopes upwards to the eye. Then gravity acts as two forces. It acts (1) as a force pressing all particles upon the slanting surface into it, and (2) as a force of gravity along the plane, making particles tend to slip down its incline. We may suppose that in a plane-world there are two such forces, one keeping the beings thereon to the plane, the other acting between bodies in it, and of such a nature that by virtue of it any large mass of plane-matter produces on small particles around it the same effects as the large mass of solid matter called our earth produces on small objects like our bodies situated around it. In both cases the larger draws the smaller to itself, and creates the sensations of up and down.
If we hold the cube so that its Dark-blue side touches a sheet of paper held upwards to the eye, and if we then look straight down along the paper, confining our view to that which is in actual contact with the paper, we see the same view of the cube as a plane-being would get. We see a Light-blue point, a Reddish line, and a Dull-purple point. The plane-being only sees a line, just as we only see a square of the cube.
The line where the paper rests on the table may be taken as representative of the surface of the plane-being’s earth. It would be merely a line to him, but it would have the same property in relation to the plane-world, as a square has in relation to a solid world; in neither case can the notion of what in the latter is termed solidity be quite excluded. If the plane-being broke through the line bounding his earth, he would find more matter beyond it.
Let us now leave out of consideration the question of “up and down” in a plane-world. Let us no longer consider it in the vertical, or ZX, position, but simply take the surface (XY) of the table as that which supports a plane-world. Let us represent its inhabitants by thin pieces of paper, which are free to move over the surface of the table, but cannot rise from it. Also, let the thickness (i.e., height above the surface) of these beings be so small that they cannot discern it. Lastly let us premise there is no attraction in their world, so that they have not any up and down.
Placing Cube 1 in front of us, let us now ask how a plane-being could apprehend such a cube. The Black face he could easily study. He would find it bounded by Gold point, Orange line, Fawn point, Crimson line, and so on. And he would discover it was Black by cutting through any of these lines and entering it. (This operation would be equivalent to the mining of a solid being).
But of what came above the Black square he would be completely ignorant. Let us now suppose a square hole to be made in the table, so that the cube could pass through, and let the cube fit the opening so exactly that no trace of the cutting of the table be visible to the plane-being. If the cube began to pass through, it would seem to him simply to change, for of its motion he could not be aware, as he would not know the direction in which it moved. Let it pass down till the White square be just on a level with the surface of the table. The plane-being would then perceive a Light-blue point, a Reddish line, a Dull-purple point, a Bright-blue line, and so on. These would surround a White square, which belonged to the same body as that to which the Black square belonged. But in this body there would be a dimension, which was not in the square. Our upward direction would not be apprehended by him directly. Motion from above downwards would only be apprehended as a change in the figure before him. He would not say that he had before him different sections of a cube, but only a changing square. If he wanted to look at the upper square, he could only do so when the Black square had gone an inch below his plane. To study the upper square simultaneously with the lower, he would have to make a model of it, and then he could place it beside the lower one.
Looking at the cube, we see that the Reddish line corresponds precisely to the Orange line, and the Deep-yellow to the Blue line. But if the plane-being had a model of the upper square, and placed it on the right-hand side of the Black square, the Deep-yellow line would come next to the Crimson line of the Black square. There would be a discontinuity about it. All that he could do would be to observe which part in the one square corresponded to which part in the other. Obviously too there lies something between the Black square and the White.
The plane-being would notice that when a line moves in a direction not its own, it traces out a square. When the Orange line is moved away, it traces out the Black square. The conception of a new direction thus obtained, he would understand that the Orange line moving so would trace out a square, and the Blue line moving so would do the same. To us these squares are visible as wholes, the Dark-blue, and the Vermilion. To him they would be matters of verbal definition rather than ascertained facts. However, given that he had the experience of a cube being pushed through his plane, he would know there was some figure, whereof his square was part, which was bounded by his square on one side, and by a White square on another side. We have supposed him to make models of these boundaries, a Black square and a White square. The Black square, which is his solid matter, is only one boundary of a figure in Higher Space.
But we can suppose the cube to be presented to him otherwise than by passing through his plane. It can be turned round the Orange line, in which case the Blue line goes out, and, after a time, the Brown line comes in. It must be noticed that the Brown line comes into a direction opposite to that in which the Blue line ran. These two lines are at right angles to each other, and, if one be moved upwards till it is at right angles to the surface of the table, the other comes on to the surface, but runs in a direction opposite to that in which the first ran. Thus, by turning the cube about the Orange line and the Blue line, different sides of it can be shown to a plane-being. By combining the two processes of turning and pushing through the plane, all the sides can be shown to the plane-being. For instance, if the cube be turned so that the Dark-blue square be on the plane, and it be then passed through, the Light-yellow square will come in.
Now, if the plane-being made a set of models of these different appearances and studied them, he could form some rational idea of the Higher Solid which produced them. He would become able to give some consistent account of the properties of this new kind of existence; he could say what came into his plane space, if the other space penetrated the plane edge-wise or corner-wise, and could describe all that would come in as it turned about in any way.
He would have six models. Let us consider two of them—the Black and the White squares. We can observe them on the cube. Every colour on the one is different from every colour on the other. If we now ask what lies between the Orange line and the Reddish line, we know it is a square, for the Orange line moving in any direction gives a square. And, if the six models were before the plane-being, he could easily select that which showed what he wanted. For that which lies between Orange line and Reddish line must be bounded by Orange and Reddish lines. He would search among the six models lying beside each other on his plane, till he found the Dark-blue square. It is evident that only one other square differs in all its colours from the Black square, viz., the White square. For it is entirely separate. The others meet it in one of their lines. This total difference exists in all the pairs of opposite surfaces on the cube.
Now, suppose the plane-being asked himself what would appear if the cube turned round the Blue line. The cube would begin to pass through his space. The Crimson line would disappear beneath the plane and the Blue-green square would cut it, so that opposite to the Blue line in the plane there would be a Blue-green line. The French-grey line and the Dark-slate line would be cut in points, and from the Gold point to the French-grey point would be a Dark-blue line; and opposite to it would be a Light-yellow line, from the Buff point to the Dark-slate point. Thus the figure in the plane world would be an oblong instead of a square, and the interior of it would be of the same Light-buff colour as the interior of the cube. It is assumed that the plane closes up round the passing cube, as the surface of a liquid does round any object immersed.
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
But, in order to apprehend what would take place when this twisting round the Blue line began, the plane-being would have to set to work by parts. He has no conception of what a solid would do in twisting, but he knows what a plane does. Let him, then, instead of thinking of the whole Black square, think only of the Orange line. The Dark-blue square stands on it. As far as this square is concerned, twisting round the Blue line is the same as twisting round the Gold point. Let him imagine himself in that plane at right angles to his plane-world, which contains the Dark-blue square. Let him keep his attention fixed on the line where the two planes meet, viz., that which is at first marked by the Orange line. We will call this line the line of his plane, for all that he knows of his own plane is this line. Now, let the Dark-blue square turn round the Gold point. The Orange line at once dips below the line of his plane, and the Dark-blue square passes through it. Therefore, in his plane he will see a Dark-blue line in place of the Orange one. And in place of the Fawn point, only further off from the Gold point, will be a French-grey point. The Diagrams ([1]), ([2]) show how the cube appears as it is before and after the turning. G is the Gold, F the Fawn point. In ([2]) G is unmoved, and the plane is cut by the French-grey line, Gr.
Instead of imagining a direction he did not know, the plane-being could think of the Dark-blue square as lying in his plane. But in this case the Black square would be out off his plane, and only the Orange line would remain in it. Diagram ([3]) shows the Dark-blue square lying in his plane, and Diagram ([4]) shows it turning round the Gold point. Here, instead of thinking about his plane and also that at right angles to it, he has only to think how the square turning round the Gold point will cut the line, which runs left to right from G, viz., the dotted line. The French-grey line is cut by the dotted line in a point. To find out what would come in at other parts, he need only treat a number of the plane sections of the cube perpendicular to the Black square in the same manner as he had treated the Dark-blue square. Every such section would turn round a point, as the whole cube turned round the Blue line. Thus he would treat the cube as a number of squares by taking parallel sections from the Dark-blue to the Light-yellow square, and he would turn each of these round a corner of the same colour as the Blue line. Combining these series of appearances, he would discover what came into his plane as the cube turned round the Blue line. Thus, the problem of the turning of the cube could be settled by the consideration of the turnings of a number of squares.
As the cube turned, a number of different appearances would be presented to the plane-being. The Black square would change into a Light-buff oblong, with Dark-blue, Blue-green, Light-yellow, and Blue sides, and would gradually elongate itself until it became as long as the diagonal of the square side of the cube; and then the bounding line opposite to the Blue line would change from Blue-green to Bright-blue, the other lines remaining the same colour. If the cube then turned still further, the Bright-blue line would become White, and the oblong would diminish in length. It would in time become a Vermilion square, with a Deep-yellow line opposite to the Blue line. It would then pass wholly below the plane, and only the Blue line would remain.
If the turning were continued till half a revolution had been accomplished, the Black square would come in again. But now it would come up into the plane from underneath. It would appear as a Black square exactly similar to the first; but the Orange line, instead of running left to right from Gold point, would run right to left. The square would be the same, only differently disposed with regard to the Blue line. It would be the looking-glass image of the first square. There would be a difference in respect of the lie of the particles of which it was composed. If the plane-being could examine its thickness, he would find that particles which, in the first case, lay above others, now lay below them. But, if he were really a plane-being, he would have no idea of thickness in his squares, and he would find them both quite identical. Only the one would be to the other as if it had been pulled through itself. In this phenomenon of symmetry he would apprehend the difference of the lie of the line, which went in the, to him, unknown direction of up-and-down.
CHAPTER II.
FURTHER APPEARANCES OF A CUBE TO A PLANE-BEING.
Before leaving the observation of the cube, it is well to look at it for a moment as it would appear to a plane-being, in whose world there was such a fact as attraction. To do this, let the cube rest on the table, so that its Dark-blue face is perpendicular in front of us. Now, let a sheet of paper be placed in contact with the Dark-blue square. Let up and sideways be the two dimensions of the plane-being, and away the unknown direction. Let the line where the paper meets the table, represent the surface of his earth. Then, there is to him, as all that he can apprehend of the cube, a Dark-blue square standing upright; and, when we look over the edge of the paper, and regard merely the part in contact with the paper, we see what the plane-being would see.
If the cube be turned round the up line, the Brown line, the Orange line will pass to the near side of the paper, and the section made by the cube in the paper will be an oblong. Such an oblong can be cut out; and when the cube is fitted into it, it can be seen that it is bounded by a Brown line and a Blue-green line opposite thereto, while the other boundaries are Black and White lines. Next, if we take a section half-way between the Black and White squares, we shall have a square cutting the plane of the aforesaid paper in a single line. With regard to this section, all we have to inquire is, What will take the place of this line as the cube turns? Obviously, the line will elongate. From a Dark-blue line it will change to a Light-buff line, the colour of the inside of the section, and will terminate in a Blue-green point instead of a French-grey. Again, it is obvious that, if the cube turns round the Orange line, it will give rise to a series of oblongs, stretching upwards. This turning can be continued till the cube is wholly on the near side of the paper, and only the Orange line remains. And, when the cube has made half a revolution, the Dark-blue square will return into the plane; but it will run downwards instead of upwards as at first. Thereafter, if the cube turn further, a series of oblongs will appear, all running downwards from the Orange line. Hence, if all the appearances produced by the revolution of the cube have to be shown, it must be supposed to be raised some distance above the plane-being’s earth, so that those appearances may be shown which occur when it is turned round the Orange line downwards, as well as when it is turned upwards. The unknown direction comes into the plane either upwards or downwards, but there is no special connection between it and either of these directions. If it come in upwards, the Brown line goes nearwards or -Y; if it come in downwards, or -Z, the Brown line goes away, or Y.
Let us consider more closely the directions which the plane-being would have. Firstly, he would have up-and-down, that is, away from his earth and towards it on the plane of the paper, the surface of his earth being the line where the paper meets the table. Then, if he moved along the surface of his earth, there would only be a line for him to move in, the line running right and left. But, being the direction of his movement, he would say it ran forwards and backwards. Thus he would simply have the words up and down, forwards and backwards, and the expressions right and left would have no meaning for him. If he were to frame a notion of a world in higher dimensions, he must invent new words for distinctions not within his experience.
To repeat the observations already made, let the cube be held in front of the observer, and suppose the Dark-blue square extended on every side so as to form a plane. Then let this plane be considered as independent of the Dark-blue square. Now, holding the Brown line between finger and thumb, and touching its extremities, the Gold and Light-blue points, turn the cube round the Brown line. The Dark-blue square will leave the plane, the Orange line will tend towards the -Y direction, and the Blue line will finally come into the plane pointing in the +X direction. If we move the cube so that the line which leaves the plane runs +Y, then the line which before ran +Y will come into the plane in the direction opposite to that of the line which has left the plane. The Blue line, which runs in the unknown direction can come into either of the two known directions of the plane. It can take the place of the Orange line by turning the cube round the Brown line, or the place of the Brown line by turning it round the Orange line. If the plane-being made models to represent these two appearances of the cube, he would have identically the same line, the Blue line, running in one of his known directions in the first model, and in the other of his known directions in the second. In studying the cube he would find it best to turn it so that the line of unknown direction ran in that direction in the positive sense. In that case, it would come into the plane in the negative sense of the known directions.
Starting with the cube in front of the observer, there are two ways in which the Vermilion square can be brought into the imaginary plane, that is the extension of the Dark-blue square. If the cube turn round the Brown line so that the Orange line goes away, (i.e. +Y), the Vermilion square comes in on the left of the Brown line. If it turn in the opposite direction, the Vermilion square comes in on the right of the Brown line. Thus, if we identify the plane-being with the Brown line, the Vermilion square would appear either behind or before him. These two appearances of the Vermilion square would seem identical, but they could not be made to coincide by any movement in the plane. The diagram ([Fig. 5.]) shows the difference in them. It is obvious that no turn in the plane could put one in the place of the other, part for part. Thus the plane-being apprehends the reversal of the unknown direction by the disposition of his figures. If a figure, which lay on one side of a line, changed into an identical figure on the other side of it, he could be sure that a line of the figure, which at first ran in the positive unknown direction, now ran in the negative unknown direction.
We have dwelt at great length on the appearances, which a cube would present to a plane-being, and it will be found that all the points which would be likely to cause difficulty hereafter, have been explained in this obvious case.
There is, however, one other way, open to a plane-being of studying a cube, to which we must attend. This is, by steady motion. Let the cube come into the imaginary plane, which is the extension of the Dark-blue square, i.e. let it touch the piece of paper which is standing vertical on the table. Then let it travel through this plane at right angles to it at the rate of an inch a minute. The plane-being would first perceive a Dark-blue square, that is, he would see the coloured lines bounding that square, and enclosed therein would be what he would call a Dark-blue solid. In the movement of the cube, however, this Dark-blue square would not last for more than a flash of time. (The edges and points on the models are made very large; in reality they must be supposed very minute.) This Dark-blue square would be succeeded by one of the colour of the cube’s interior, i.e. by a Light-buff square. But this colour of the interior would not be visible to the plane-being. He would go round the square on his plane, and would see the bounding lines, viz. Vermilion, White, Blue-green, Black. And at the corners he would see Deep-yellow, Bright-blue, Crimson, and Blue points. These lines and points would really be those parts of the faces and lines of the cube, which were on the point of passing through his plane. Now, there would be one difference between the Dark-blue square and the Light-buff with their respective boundaries. The first only lasted for a flash; the second would last for a minute or all but a minute. Consider the Vermilion square. It appears to the plane-being as a line. The Brown line also appears to him as a line. But there is a difference between them. The Brown line only lasts for a flash, whereas the Vermilion line lasts for a minute. Hence, in this mode of presentation, we may say that for a plane-being a lasting line is the mode of apprehending a plane, and a lasting plane (which is a plane-being’s solid) is the mode of apprehending our solids. In the same way, the Blue line, as it passes through his plane, gives rise to a point. This point lasts for a minute, whereas the Gold point only lasted for a flash.
CHAPTER III.
FOUR-SPACE. GENESIS OF A TESSARACT. ITS REPRESENTATION IN THREE-SPACE.
Hitherto we have only looked at Model 1. This, with the next seven, represent what we can observe of the simplest body in Higher Space. A few words will explain their construction. A point by its motion traces a line. A line by its motion traces either a longer line or an area; if it moves at right angles to its own direction, it traces a rectangle. For the sake of simplicity, we will suppose all movements to be an inch in length and at right angles to each other. Thus, a point moving traces a line an inch long; a line moving traces a square inch; a square moving traces a cubic inch. In these cases each of these movements produces something intrinsically different from what we had before. A square is not a longer line, nor a cube a larger square. When the cube moves, we are unable to see any new direction in which it can move, and are compelled to make it move in a direction which has previously been used. Let us suppose there is an unknown direction at right angles to all our known directions, just as a third direction would be unknown to a being confined to the surface of the table. And let the cube move in this unknown direction for an inch. We call the figure it traces a Tessaract. The models are representations of the appearances a Tessaract would present to us if shown in various ways. Consider for a moment what happens to a square when moved to form a cube. Each of its lines, moved in the new direction, traces a square; the square itself traces a new figure, a cube, which ends in another square. Now, our cube, moved in a new direction, will trace a tessaract, whereof the cube itself is the beginning, and another cube the end. These two cubes are to the tessaract as the Black square and White square are to the cube. A plane-being could not see both those squares at once, but he could make models of them and let them both rest in his plane at once. So also we can make models of the beginning and end of the tessaract. Model 1 is the cube, which is its beginning; Model 2 is the cube which is its end. It will be noticed that there are no two colours alike in the two models. The Silver point corresponds to the Gold point, that is, the Silver point is the termination of the line traced by the Gold point moving in the new direction. The sides correspond in the following manner:—
Sides.
| Model 1. | Model 2. | ||
|---|---|---|---|
| Black | corresponds | to | Bright-green |
| White | „ | „ | Light-grey |
| Vermilion | „ | „ | Indian-red |
| Blue-green | „ | „ | Yellow-ochre |
| Dark-blue | „ | „ | Burnt-sienna |
| Light-yellow | „ | „ | Dun |
The two cubes should be looked at and compared long enough to ensure that the corresponding sides can be found quickly. Then there are the following correspondencies in points and lines.
Points.
| Model 1. | Model 2. | ||
|---|---|---|---|
| Gold | corresponds | to | Silver |
| Fawn | „ | „ | Turquoise |
| Terra-cotta | „ | „ | Earthen |
| Buff | „ | „ | Blue tint |
| Light-blue | „ | „ | Quaker-green |
| Dull-purple | „ | „ | Peacock-blue |
| Deep-blue | „ | „ | Orange-vermilion |
| Red | „ | „ | Purple |
Lines
| Model 1. | Model 2. | ||
|---|---|---|---|
| Orange | corresponds | to | Leaf-green |
| Crimson | „ | „ | Dull-green |
| Green-grey | „ | „ | Dark-purple |
| Blue | „ | „ | Purple-brown |
| Brown | „ | „ | Dull-blue |
| French-grey | „ | „ | Dark-pink |
| Dark-slate | „ | „ | Pale-pink |
| Green | „ | „ | Indigo |
| Reddish | „ | „ | Brown-green |
| Bright-blue | „ | „ | Dark-green |
| Leaden | „ | „ | Pale-yellow |
| Deep-yellow | „ | „ | Dark |
The colour of the cube itself is invisible, as it is covered by its boundaries. We suppose it to be Sage-green.
These two cubes are just as disconnected when looked at by us as the black and white squares would be to a plane-being if placed side by side on his plane. He cannot see the squares in their right position with regard to each other, nor can we see the cubes in theirs.
Let us now consider the vermilion side of Model 1. If it move in the X direction, it traces the cube of Model 1. Its Gold point travels along the Orange line, and itself, after tracing the cube, ends in the Blue-green square. But if it moves in the new direction, it will also trace a cube, for the new direction is at right angles to the up and away directions, in which the Brown and Blue lines run. Let this square, then, move in the unknown direction, and trace a cube. This cube we cannot see, because the unknown direction runs out of our space at once, just as the up direction runs out of the plane of the table. But a plane-being could see the square, which the Blue line traces when moved upwards, by the cube being turned round the Blue line, the Orange line going upwards; then the Brown line comes into the plane of the table in the -X direction. So also with our cube. As treated above, it runs from the Vermilion square out of our space. But if the tessaract were turned so that the line which runs from the Gold point in the unknown direction lay in our space, and the Orange line lay in the unknown direction, we could then see the cube which is formed by the movement of the Vermilion square in the new direction.
Take Model 5. There is on it a Vermilion square. Place this so that it touches the Vermilion square on Model 1. All the marks of the two squares are identical. This Cube 5, is the one traced by the Vermilion square moving in the unknown direction. In Model 5, the whole figure, the tessaract, produced by the movement of the cube in the unknown direction, is supposed to be so turned that the Orange line passes into the unknown direction, and that the line which went in the unknown direction, runs opposite to the old direction of the Orange line. Looking at this new cube, we see that there is a Stone line running to the left from the Gold point. This line is that which the Gold point traces when moving in the unknown direction.
It is obvious that, if the Tessaract turns so as to show us the side, of which Cube 5 is a model, then Cube 1 will no longer be visible. The Orange line will run in the unknown or fourth direction, and be out of our sight, together with the whole cube which the Vermilion square generates, when the Gold point moves along the Orange line. Hence, if we consider these models as real portions of the tessaract, we must not have more than one before us at once. When we look at one, the others must necessarily be beyond our sight and touch. But we may consider them simply as models, and, as such, we may let them lie alongside of each other. In this case, we must remember that their real relationships are not those in which we see them.
We now enumerate the sides of the new Cube 5, so that, when we refer to it, any colour may be recognised by name.
The square Vermilion traces a Pale-green cube, and ends in an Indian-red square.
(The colour Pale-green of this cube is not seen, as it is entirely surrounded by squares and lines of colour.)
Each Line traces a Square and ends in a Line.
| The | Blue | line | | - | traces a | - | | Light-brown | square | | - | and ends in a | - | | Purple-brown | line |
| „ | Brown | „ | Yellow | „ | Dull-blue | „ | ||||||||||
| „ | Deep-yellow | „ | Light-red | „ | Dark | „ | ||||||||||
| „ | Green | „ | Deep-crimson | „ | Indigo | „. |
Each Point traces a Line and ends in a Point.
| The | Gold | point | | - | traces a | - | | Stone | line | | - | and ends in a | - | | Silver | point |
| „ | Buff | „ | Light-green | „ | Blue-tint | „ | ||||||||||
| „ | Light-blue | „ | Rich-red | „ | Quaker-green | „ | ||||||||||
| „ | Red | „ | Emerald | „ | Purple | „. |
It will be noticed that besides the Vermilion square of this cube another square of it has been seen before. A moment’s comparison with the experience of a plane-being will make this more clear. If a plane-being has before him models of the Black and White squares of the Cube, he sees that all the colours of the one are different from all the colours of the other. Next, if he looks at a model of the Vermilion square, he sees that it starts from the Blue line and ends in a line of the White square, the Deep-yellow line. In this square he has two lines which he had before, the Blue line with Gold and Buff points, the Deep-yellow line with Light-blue and Red points. To him the Black and White squares are his Models 1 and 2, and the Vermilion square is to him as our Model 5 is to us. The left-hand square of Model 5 is Indian-red, and is identical with that of the same colour on the left-hand side of Model 2. In fact, Model 5 shows us what lies between the Vermilion face of 1, and the Indian-red face of 2.
From the Gold point we suppose four perfectly independent lines to spring forth, each of them at right angles to all the others. In our space there is only room for three lines mutually at right angles. It will be found, if we try to introduce a fourth at right angles to each of three, that we fail; hence, of these four lines one must go out of the space we know. The colours of these four lines are Brown, Orange, Blue, Stone. In Model 1 are shown the Brown, Orange, and Blue. In Model 5 are shown the Brown, Blue, and Stone. These lines might have had any directions at first, but we chose to begin with the Brown line going up, or Z, the Orange going X, the Blue going Y, and the Stone line going in the unknown direction, which we will call W.
Consider for a moment the Stone and the Orange lines. They can be seen together on Model 7 by looking at the lower face of it. They are at right angles to each other, and if the Orange line be turned to take the place of the Stone line, the latter will run into the negative part of the direction previously occupied by the former. This is the reason that the Models 3, 5, and 7 are made with the Stone line always running in the reverse direction of that line of Model 1, which is wanting in each respectively. It will now be easy to find out Models 3 and 7. All that has to be done is, to discover what faces they have in common with 1 and 2, and these faces will show from which planes of 1 they are generated by motion in the unknown direction.
Take Model 7. On one side of it there is a Dark-blue square, which is identical with the Dark-blue square of Model 1. Placing it so that it coincides with 1 by this square line for line, we see that the square nearest to us is Burnt-sienna, the same as the near square on Model 2. Hence this cube is a model of what the Dark-blue square traces on moving in the unknown direction. Here the unknown direction coincides with the negative away direction. In fact, to see this cube, we have been obliged to suppose the Blue line turned into the unknown direction, for we cannot look at more than three of these rectangular lines at once in our space, and in this Model 7 we have the Brown, Orange, and Stone lines. The faces, lines, and points of Cube 7 can be identified by the following list.
The Dark-blue square traces a Dark-stone cube (whose interior is rendered invisible by the bounding squares), and ends in a Burnt-sienna square.
Each Line traces a Square and ends in a Line.
| The | Orange | line | | - | traces an | - | | Azure | square | | - | and ends in a | - | | Leaf-green | line |
| „ | Brown | „ | Yellow | „ | Dull-blue | „ | ||||||||||
| „ | French-grey | „ | Yellow-green | „ | Dark-pink | „ | ||||||||||
| „ | Reddish | „ | Ochre | „ | Brown-green | „. |
Each Point traces a Line and ends in a Point.
| The | Gold | point | | - | traces a | - | | Stone | line | | - | and ends in a | - | | Silver | point |
| „ | Fawn | „ | Smoke | „ | Turquoise | „ | ||||||||||
| „ | Light-blue | „ | Rich-red | „ | Quaker-green | „ | ||||||||||
| „ | Dull-purple | „ | Green-blue | „ | Peacock-blue | „. |
If we now take Model 3, we see that it has a Black square uppermost, and has Blue and Orange lines. Hence, it evidently proceeds from the Black square in Model 1; and it has in it Blue and Orange lines, which proceed from the Gold point. But besides these, it has running downwards a Stone line. The line wanting is the Brown line, and, as in the other cases, when one of the three lines of Model 1 turns out into the unknown direction, the Stone line turns into the direction opposite to that from which the line has turned. Take this Model 3 and place it underneath Model 1, raising the latter so that the Black squares on the two coincide line for line. Then we see what would come into our view if the Brown line were to turn into the unknown direction, and the Stone line come into our space downwards. Looking at this cube, we see that the following parts of the tessaract have been generated.
The Black square traces a Brick-red cube (invisible because covered by its own sides and edges), and ends in a Bright-green square.
Each Line traces a Square and ends in a Line.
| The | Orange | line | | - | traces an | - | | Azure | square | | - | and ends in a | - | | Leaf-green | line |
| „ | Crimson | „ | Rose | „ | Dull-green | „ | ||||||||||
| „ | Green-grey | „ | Sea-blue | „ | Dark-purple | „ | ||||||||||
| „ | Blue | „ | Light-brown | „ | Purple-brown | „. |
Each Point traces a Line and ends in a Point.
| The | Gold | point | | - | traces a | - | | Stone | line | | - | and ends in a | - | | Silver | point |
| „ | Fawn | „ | Smoke | „ | Turquoise | „ | ||||||||||
| „ | Terra-cotta | „ | Magenta | „ | Earthen | „ | ||||||||||
| „ | Buff | „ | Light-green | „ | Blue-tint | „. |
This completes the enumeration of the regions of Cube 3. It may seem a little unnatural that it should come in downwards; but it must be remembered that the new fourth direction has no more relation to up-and-down than to right-and-left or to near-and-far.
And if, instead of thinking of a plane-being as living on the surface of a table, we suppose his world to be the surface of the sheet of paper touching the Dark-blue square of Cube 1, then we see that a turn round the Orange line, which makes the Brown line go into the plane-being’s unknown direction, brings the Blue line into his downwards direction.
There still remain to be described Models 4, 6, and 8. It will be shown that Model 4 is to Model 3 what Model 2 is to Model 1. That is, if, when 3 is in our space, it be moved so as to trace a tessaract, 4 will be the opposite cube in which the tessaract ends. There is no colour common to 3 and 4. Similarly, 6 is the opposite boundary of the tessaract generated by 5, and 8 of that by 7.
A little closer consideration will reveal several points. Looking at Cube 5, we see proceeding from the Gold point a Brown, a Blue, and a Stone line. The Orange line is wanting; therefore, it goes in the unknown direction. If we want to discover what exists in the unknown direction from Cube 5, we can get help from Cube 1. For, since the Orange line lies in the unknown direction from Cube 5, the Gold point will, if moved along the Orange line, pass in the unknown direction. So also, the Vermilion square, if moved along in the direction of the Orange line, will proceed in the unknown direction. Looking at Cube 1 we see that the Vermilion square thus moved ends in a Blue-green square. Then, looking at Model 6, on it, corresponding to the Vermilion square on Cube 5, is a Blue-green square.
Cube 6 thus shows what exists an inch beyond 5 in the unknown direction. Between the right-hand face on 5 and the right-hand face on 6 lies a cube, the one which is shown in Model 1. Model 1 is traced by the Vermilion square moving an inch along the direction of the Orange line. In Model 5, the Orange line goes in the unknown direction; and looking at Model 6 we see what we should get at the end of a movement of one inch in that direction. We have still to enumerate the colours of Cubes 4, 6, and 8, and we do so in the following list. In the first column is designated the part of the cube; in the columns under 4, 6, 8, come the colours which 4, 6, 8, respectively have in the parts designated in the corresponding line in the first column.
Cube itself:—
| 4 | 6 | 8 | |
|---|---|---|---|
| Chocolate | Oak-yellow | Salmon |
Squares:—
| Lower face | Light-grey | Rose | Sea-blue |
| Upper | White | Deep-brown | Deep-green |
| Left-hand | Light-red | Yellow-ochre | Deep-crimson |
| Right-hand | Deep-brown | Blue-green | Dark-grey |
| Near | Ochre | Yellow-green | Dun |
| Far | Deep-green | Dark-grey | Light-yellow |
Lines:—
On ground, going round the square from left to right:—
| 4 | 6 | 8 | |
|---|---|---|---|
| 1. | Brown-green | Smoke | Dark-purple |
| 2. | Dark-green | Crimson | Magenta |
| 3. | Pale-yellow | Magenta | Green-grey |
| 4. | Dark | Dull-green | Light-green |
Vertical, going round the sides from left to right:—
| 1. | Rich-red | Dark-pink | Indigo |
| 2. | Green-blue | French-grey | Pale-pink |
| 3. | Sea-green | Dark-slate | Dark-slate |
| 4. | Emerald | Pale-pink | Green |
Round upper face in same order:—
| 1. | Reddish | Green-blue | Pale-yellow |
| 2. | Bright-blue | Bright-blue | Sea-green |
| 3. | Leaden | Sea-green | Leaden |
| 4. | Deep-yellow | Dark-green | Emerald |
Points:—
On lower face, going from left to right:—
| 1. | Quaker-green | Turquoise | Blue-tint |
| 2. | Peacock-blue | Fawn | Earthen |
| 3. | Orange-vermilion | Terra-cotta | Terra-cotta |
| 4. | Purple | Earthen | Buff |
On upper face:—
| 1. | Light-blue | Peacock-blue | Purple |
| 2. | Dull-purple | Dull-purple | Orange-vermilion |
| 3. | Deep-blue | Deep-blue | Deep-blue |
| 4. | Red | Orange-vermilion | Red |
If any one of these cubes be taken at random, it is easy enough to find out to what part of the Tessaract it belongs. In all of them, except 2, there will be one face, which is a copy of a face on 1; this face is, in fact, identical with the face on 1 which it resembles. And the model shows what lies in the unknown direction from that face. This unknown direction is turned into our space, so that we can see and touch the result of moving a square in it. And we have sacrificed one of the three original directions in order to do this. It will be found that the line, which in 1 goes in the 4th direction, in the other models always runs in a negative direction.
Let us take Model 8, for instance. Searching it for a face we know, we come to a Light-yellow face away from us. We place this face parallel with the Light-yellow face on Cube 1, and we see that it has a Green line going up, and a Green-grey line going to the right from the Buff point. In these respects it is identical with the Light-yellow face on Cube 1. But instead of a Blue line coming towards us from the Buff point, there is a Light-green line. This Light-green line, then, is that which proceeds in the unknown direction from the Buff point. The line is turned towards us in this Model 8 in the negative Y direction; and looking at the model, we see exactly what is formed when in the motion of the whole cube in the unknown direction, the Light-yellow face is moved an inch in that direction. It traces out a Salmon cube (v. Table on [p. 127]), and it has Sea-blue and Deep-green sides below and above, and Deep-crimson and Dark-grey sides left and right, and Dun and Light-yellow sides near and far. If we want to verify the correctness of any of these details, we must turn to Models 1 and 2. What lies an inch from the Light-yellow square in the unknown direction? Model 2 tells us, a Dun square. Now, looking at 8, we see that towards us lies a Dun square. This is what lies an inch in the unknown direction from the Light-yellow square. It is here turned to face us, and we can see what lies between it and the Light-yellow square.
CHAPTER IV.
TESSARACT MOVING THROUGH THREE-SPACE. MODELS OF THE SECTIONS.
In order to obtain a clear conception of the higher solid, a certain amount of familiarity with the facts shown in these models is necessary. But the best way of obtaining a systematic knowledge is shown hereafter. What these models enable us to do, is to take a general review of the subject. In all of them we see simply the boundaries of the tessaract in our space; we can no more see or touch the tessaract’s solidity than a plane-being can touch the cube’s solidity.
There remain the four models 9, 10, 11, 12. Model 9 represents what lies between 1 and 2. If 1 be moved an inch in the unknown direction, it traces out the tessaract and ends in 2. But, obviously, between 1 and 2 there must be an infinite number of exactly similar solid sections; these are all like Model 9.
Take the case of a plane-being on the table. He sees the Black square,—that is, he sees the lines round it,—and he knows that, if it moves an inch in some mysterious direction, it traces a new kind of figure, the opposite boundary whereof is the White square. If, then, he has models of the White and Black squares, he has before him the end and beginning of our cube. But between these squares are any number of others, the plane sections of the cube. We can see what they are. The interior of each is a Light-buff (the colour of the substance of the cube), the sides are of the colours of the vertical faces of the cube, and the points of the colours of the vertical lines of the cube, viz., Dark-blue, Blue-green, Light-yellow, Vermilion lines, and Brown, French-grey, Dark-slate, Green points. Thus, the square, in moving in the unknown direction, traces out a succession of squares, the assemblage of which makes the cube in layers. So also the cube, moving in the unknown direction, will at any point of its motion, still be a cube; and the assemblage of cubes thus placed constitutes the tessaract in layers. We suppose the cube to change its colour directly it begins to move. Its colour between 1 and 2 we can easily determine by finding what colours its different parts assume, as they move in the unknown direction. The Gold point immediately begins to trace a Stone-line. We will look at Cube 5 to see what the Vermilion face becomes; we know the interior of that cube is Pale-green (v. Table, [p. 122]). Hence, as it moves in the unknown direction, the Vermilion square forms in its course a series of Pale-green squares. The Brown line gives rise to a Yellow square; hence, at every point of its course in the fourth direction, it is a Yellow line, until, on taking its final position, it becomes a Dull-blue line. Looking at Cube 5, we see that the Deep yellow line becomes a Light-red line, the Green line a Deep Crimson one, the Gold point a Stone one, the Light-blue point a Rich-red one, the Red point an Emerald one, and the Buff point a Light-green one. Now, take the Model 9. Looking at the left side of it, we see exactly that into which the Vermilion square is transformed, as it moves in the unknown direction. The left side is an exact copy of a section of Cube 5, parallel to the Vermilion face.
But we have only accounted for one side of our Model 9. There are five other sides. Take the near side corresponding to the Dark-blue square on Cube 1. When the Dark-blue square moves, it traces a Dark-stone cube, of which we have a copy in Cube 7. Looking at 7 (v. Table, [p. 124]), we see that, as soon as the Dark-blue square begins to move, it becomes of a Dark-stone colour, and has Yellow, Ochre, Yellow-green, and Azure sides, and Stone, Rich-red, Green-blue, Smoke lines running in the unknown direction from it. Now, the side of Model 9, which faces us, has these colours the squares being seen as lines, and the lines as points. Hence Model 9 is a copy of what the cube becomes, so far as the Vermilion and Dark-blue sides are concerned, when, moving in the unknown direction, it traces the tessaract.
We will now look at the lower square of our model. It is a Brick-red square, with Azure, Rose, Sea-blue, and Light-brown lines, and with Stone, Smoke, Magenta, and Light-green points. This, then, is what the Black square should change into, as it moves in the unknown direction. Let us look at Model 3. Here the Stone line, which is the line in the unknown direction, runs downwards. It is turned into the downwards direction, so that the cube traced by the Black square may be in our space. The colour of this cube is Brick-red; the Orange line has traced an Azure, the Blue line a Light-brown, the Crimson line a Rose, and the Green-grey line a Sea-blue square. Hence, the lower square of Model 9 shows what the Black square becomes, as it traces the tessaract; or, in other words, the section of Model 3 between the Black and Bright-green squares exactly corresponds to the lower face of Model 9.
Therefore, it appears that Model 9 is a model of a section of the tessaract, that it is to the tessaract what a square between the Black and White squares is to the cube.
To prove the other sides correct, we have to see what the White, Blue-green, and Light-yellow squares of Cube 1 become, as the cube moves in the unknown direction. This can be effected by means of the Models 4, 6, 8. Each cube can be used as an index for showing the changes through which any side of the first model passes, as it moves in the unknown direction till it becomes Cube 2. Thus, what becomes of the White square? Look at Cube 4. From the Light-blue corner of its White square runs downwards the Rich-red line in the unknown direction. If we take a parallel section below the White square, we have a square bounded by Ochre, Deep-brown, Deep-green, and Light-red lines; and by Rich-red, Green-blue, Sea-green, and Emerald points. The colour of the cube is Chocolate, and therefore its section is Chocolate. This description is exactly true of the upper surface of Model 9.
There still remain two sides, those corresponding to the Light-yellow and Blue-green of Cube 1. What the Blue-green square becomes midway between Cubes 1 and 2 can be seen on Model 6. The colour of the last-named is Oak-yellow, and a section parallel to its Blue-green side is surrounded by Yellow-green, Deep-brown, Dark-grey and Rose lines and by Green-blue, Smoke, Magenta, and Sea-green points. This is exactly similar to the right side of Model 9. Lastly, that which becomes of the Light-yellow side can be seen on Model 8. The section of the cube is a Salmon square bounded by Deep-crimson, Deep-green, Dark-grey and Sea-blue lines and by Emerald, Sea-green, Magenta, and Light-green points.
Thus the models can be used to answer any question about sections. For we have simply to take, instead of the whole cube, a plane, and the relation of the whole tessaract to that plane can be told by looking at the model, which, starting with that plane, stretches from it in the unknown direction.
We have not as yet settled the colour of the interior of Model 9. It is that part of the tessaract which is traced out by the interior of Cube 1. The unknown direction starts equally and simultaneously from every point of every part of Cube 1, just as the up direction starts equally and simultaneously from every point of a square. Let us suppose that the cube, which is Light-buff, changes to a Wood-colour directly it begins to trace the tessaract. Then the internal part of the section between 1 and 2 will be a Wood-colour. The sides of the Model 9 are of the greatest importance. They are the colour of the six cubes, 3, 4, 5, 6, 7, and 8. The colours of 1 and 2 are wanting, viz. Light-buff and Sage-green. Thus the section between 1 and 2 can be found by its wanting the colours of the Cubes 1 and 2.
Looking at Models 10, 11, and 12 in a similar manner, the reader will find they represent the sections between Cubes 3 and 4, Cubes 5 and 6, and Cubes 7 and 8 respectively.
CHAPTER V.
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.
| Fourth Floor. | - | | Olus | Semita | Semita | Lama |
| Via | Mel | Mel | Iter | |||
| Via | Mel | Mel | Iter | |||
| Ilex | Callis | Callis | Sors | |||
| Third Floor. | - | | Bucina | Murex | Murex | Daps |
| Alvus | Mala | Mala | Proes | |||
| Alvus | Mala | Mala | Proes | |||
| Arctos | Mœna | Mœna | Far | |||
| Second Floor. | - | | Bucina | Murex | Murex | Daps |
| Alvus | Mala | Mala | Proes | |||
| Alvus | Mala | Mala | Proes | |||
| Arctos | Mœna | Mœna | Far | |||
| First Floor. | - | | Cista | Cadus | Cadus | Crus |
| Dos | Syce | Syce | Bolus | |||
| Dos | Syce | Syce | Bolus | |||
| Corvus | Cuspis | Cuspis | Nugæ | |||
Fig. 6.
Fig. 7.
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.
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 Top Floor. | - | | Mars Mala | Merces Mala | Tyro Mala | |
| Spicula Mala | Mora Mala | Oliva Mala | ||||
| Comes Mala | Tibicen Mala | Vestis Mala | ||||
| Second, or Middle Floor. | - | | Ala Mala | Cortis Mala | Aer Mala | |
| Uncus Mala | Pallor Mala | Tergum Mala | ||||
| Ostrum Mala | Bidens Mala | Scena Mala | ||||
| First, or Bottom Floor. | - | | Sector Mala | Hama Mala | Remus Mala | |
| Frenum Mala | Plebs Mala | Sypho Mala | ||||
| 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.
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.
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.
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.
Fig. 13.
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.
Fig. 14.
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.
Fig. 15.
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.
CHAPTER VI.
THE MEANS BY WHICH A PLANE-BEING WOULD ACQUIRE A CONCEPTION OF OUR FIGURES.
Take the Block of twenty-seven Mala cubes, and build up the following shape ([Fig. 18]):—
Urna Mala, Moles Mala, Plebs Mala, Pallor Mala, Mora Mala.
If this shape, passed through the vertical plane, the plane-being would perceive:—
(1) The squares Urna Moena and Moles Moena.
(2) The three squares Plebs Moena, Pallor Moena, Mora Moena,
and then the whole figure would have passed through his plane.
If the whole Block were turned round the Z axis till the Alvus sides entered, and the figure built up as it would be in that disposition of the cubes, the plane-being would perceive during its passage through the plane:—
(1) Urna Alvus;
(2) Moles Alvus, Plebs Alvus, Pallor Alvus, Mora Alvus, which would all enter on the left side of the Z axis.
Again, if the Block were turned round the X axis, the Syce side would enter, and the cubes appear in the following order:—
(1) Urna Syce, Moles Syce, Plebs Syce;
(2) Pallor Syce;
(3) Mora Syce.
Fig. 17.
Fig. 18.
A comparison of these three sets of appearances would give the plane-being a full account of the figure. It is that which can produce these various appearances.
Let us now suppose a glass plate placed in front of the Block in its first position. On this plate let the axes X and Z be drawn. They divide the surface into four parts, to which we give the following names ([Fig. 17]):—
Z X = that quarter defined by the positive Z and positive X axis.
Z X = that quarter defined by the positive Z and negative X axis (which is called “Z negative X”).
Z X = that quarter defined by the negative Z and negative X axis.
Z X = that quarter defined by the negative Z and positive X axis.
The Block appears in these different quarters or quadrants, as it is turned round the different axes. In Z X the Moenas appear, in Z X the Alvus faces, in Z X the Syces. In each quadrant are drawn nine squares, to receive the faces of the cubes when they enter. For instance, in Z X we have the Moenas of:—
| Z | |||||
| Comes | Tibicen | Vestis | |||
| Ostrum | Bidens | Scena | |||
| Urna | Moles | Saltus | |||
| X | |||||
And in Z X we have the Alvus of:—
| Z | |||||
| Mars | Spicula | Comes | |||
| Ala | Uncus | Ostrum | |||
| Sector | Frenum | Urna | |||
| -X | |||||
And in the Z X we have the Syces of:—
| X | |||||
| Urna | Moles | Saltus | |||
| Frenum | Plebs | Sypho | |||
| Sector | Hama | Remus | |||
| -Z | |||||
Now, if the shape taken at the beginning of this chapter be looked at through the glass, and the distance of the second and third walls of the shape behind the glass be considered of no account—that is, if they be treated as close up to the glass—we get a plane outline, which occupies the squares Urna Moena, Moles Moena, Bidens Moena, Tibicen Moena. This outline is called a projection of the figure. To see it like a plane-being, we should have to look down on it along the Z axis.
It is obvious that one projection does not give the shape. For instance, the square Bidens Moena might be filled by either Pallor or Cortis. All that a square in the room of Bidens Moena denotes, is that there is a cube somewhere in the Y, or unknown, direction from Bidens Moena. This view, just taken, we should call the front view in our space; we are then looking at it along the negative Y axis.
When the same shape is turned round on the Z axis, so as to be projected on the Z X quadrant, we have the squares—Urna Alvus, Frenum Alvus, Uncus Alvus, Spicula Alvus. When it is turned round the X axis, and projected on Z X, we have the squares, Urna Syce, Moles Syce, Plebs Syce, and no more. This is what is ordinarily called the ground plan; but we have set it in a different position from that in which it is usually drawn.
Fig. 19.
Now, the best method for a plane-being of familiarizing himself with shapes in our space, would be to practise the realization of them from their different projections in his plane. Thus, given the three projections just mentioned, he should be able to construct the figure from which they are derived. The projections ([Fig. 19]) are drawn below the perspective pictures of the shape ([Fig. 18]). From the front, or Moena view, he would conclude that the shape was Urna Mala, Moles Mala, Bidens Mala, Tibicen Mala; or instead of these, or also in addition to them, any of the cubes running in the Y direction from the plane. That is, from the Moena projection he might infer the presence of all the following cubes (the word Mala is omitted for brevity): Urna, Frenum, Sector, Moles, Plebs, Hama, Bidens, Pallor, Cortis, Tibicen, Mora, Merces.
Next, the Alvus view or projection might be given by the cubes (the word Mala being again omitted): Urna, Moles, Saltus, Frenum, Plebs, Sypho, Uncus, Pallor, Tergum, Spicula, Mora, Oliva. Lastly, looking at the ground plan or Syce view, he would infer the possible presence of Urna, Ostrum, Comes, Moles, Bidens, Tibicen, Plebs, Pallor, Mora.
Now, the shape in higher space, which is usually there, is that which is common to all these three appearances. It can be determined, therefore, by rejecting those cubes which are not present in all three lists of cubes possible from the projections. And by this process the plane-being could arrive at the enumeration of the cubes which belong to the shape of which he had the projections. After a time, when he had experience of the cubes (which, though invisible to him as wholes, he could see part by part in turn entering his space), the projections would have more meaning to him, and he might comprehend the shape they expressed fragmentarily in his space. To practise the realization from projections, we should proceed in this way. First, we should think of the possibilities involved in the Moena view, and build them up in cubes before us. Secondly, we should build up the cubes possible from the Alvus view. Again, taking the shape at the beginning of the chapter, we should find that the shape of the Alvus possibilities intersected that of the Moena possibilities in Urna, Moles, Frenum, Plebs, Pallor, Mora; or, in other words, these cubes are common to both. Thirdly, we should build up the Syce possibilities, and, comparing their shape with those of the Moena and Alvus projections, we should find Urna, Moles, Plebs, Pallor, Mora, of the Syce view coinciding with the same cubes of the other views, the only cube present in the intersection of the Moena and Alvus possibilities, and not present in the Syce view, being Frenum.
The determination of the figure denoted by the three projections, may be more easily effected by treating each projection as an indication of what cubes are to be cut away. Taking the same shape as before, we have in the Moena projection Urna, Moles, Bidens, Tibicen; and the possibilities from them are Urna, Frenum, Sector, Moles, Plebs, Hama, Bidens, Pallor, Cortis, Tibicen, Mora, Merces. This may aptly be called the Moena solution. Now, from the Syce projection, we learn at once that those cubes, which in it would produce Frenum, Sector, Hama, Remus, Sypho, Saltus, are not in the shape. This absence of Frenum and Sector in the Syce view proves that their presence in the Moena solution is superfluous. The absence of Hama removes the possibility of Hama, Cortis, Merces. The absence of Remus, Sypho, Saltus, makes no difference, as neither they nor any of their Syce possibilities are present in the Moena solution. Hence, the result of comparison of the Moena and Syce projections and possibilities is the shape: Urna, Moles, Plebs, Bidens, Pallor, Tibicen, Mora. This may be aptly called the Moena-Syce solution. Now, in the Alvus projection we see that Ostrum, Comes, Sector, Ala, and Mars are absent. The absence of Sector, Ala, and Mars has no effect on our Moena-Syce solution; as it does not contain any of their Alvus possibilities. But the absence of Ostrum and Comes proves that in the Moena-Syce solution Bidens and Tibicen are superfluous, since their presence in the original shape would give Ostrum and Comes in the Alvus projection. Thus we arrive at the Moena-Alvus-Syce solution, which gives us the shape: Urna, Moles, Plebs, Pallor, Mora.
It will be obvious on trial that a shape can be instantly recognised from its three projections, if the Block be thoroughly well known in all three positions. Any difficulty in the realization of the shapes comes from the arbitrary habit of associating the cubes with some one direction in which they happen to go with regard to us. If we remember Ostrum as above Urna, we are not remembering the Block, but only one particular relation of the Block to us. That position of Ostrum is a fact as much related to ourselves as to the Block. There is, of course, some information about the Block implied in that position; but it is so mixed with information about ourselves as to be ineffectual knowledge of the Block. It is of the highest importance to enter minutely into all the details of solution written above. For, corresponding to every operation necessary to a plane-being for the comprehension of our world, there is an operation, with which we have to become familiar, if in our turn we would enter into some comprehension of a world higher than our own. Every cube of the Block ought to be thoroughly known in all its relations. And the Block must be regarded, not as a formless mass out of which shapes can be made, but as the sum of all possible shapes, from which any one we may choose is a selection. In fact, to be familiar with the Block, we ought to know every shape that could be made by any selection of its cubes; or, in other words, we ought to make an exhaustive study of it. In the Appendix is given a set of exercises in the use of these names (which form a language of shape), and in various kinds of projections. The projections studied in this chapter are not the only, nor the most natural, projections by which a plane-being would study higher space. But they suffice as an illustration of our present purpose. If the reader will go through the exercises in the Appendix, and form others for himself, he will find every bit of manipulation done will be of service to him in the comprehension of higher space.
There is one point of view in the study of the Block, by means of slabs, which is of some interest. The cubes of the Block, and therefore also the representative slabs of their faces, can be regarded as forming rows and columns. There are three sets of them. If we take the Moena view, they represent the views of the three walls of the Block, as they pass through the plane. To form the Alvus view, we only have to rearrange the slabs, and form new sets. The first new set is formed by taking the first, or left-hand, column of each of the Moena sets. The second Alvus set is formed by taking the second or middle columns of the three Moena sets. The third will consist of the remaining or right-hand columns of the Moenas.
Similarly, the three Syce sets may be formed from the three horizontal rows or floors of the Moena sets.
Hence, it appears that the plane-being would study our space by taking all the possible combinations of the corresponding rows and columns. He would break up the first three sets into other sets, and the study of the Block would practically become to him the study of these various arrangements.
CHAPTER VII.
FOUR-SPACE: ITS REPRESENTATION IN THREE-SPACE.
We now come to the essential difficulty of our task. All that has gone before is preliminary. We have now to frame the method by which we shall introduce through our space-figures the figures of a higher space. When a plane-being studies our shapes of cubes, he has to use squares. He is limited at the outset. A cube appears to him as a square. On Model 1 we see the various squares as which the cube can appear to him. We suppose the plane-being to look from the extremity of the Z axis down a vertical plane. First, there is the Moena square. Then there is the square given by a section parallel to Moena, which he recognises by the variation of the bounding lines as soon as the cube begins to pass through his plane. Then comes the Murex square. Next, if the cube be turned round the Z axis and passed through, he sees the Alvus and Proes squares and the intermediate section. So too with the Syce and Mel squares and the section between them.
Now, dealing with figures in higher space, we are in an analogous position. We cannot grasp the element of which they are composed. We can conceive a cube; but that which corresponds to a cube in higher space is beyond our grasp. But the plane-being was obliged to use two-dimensional figures, squares, in arriving at a notion of a three-dimensional figure; so also must we use three-dimensional figures to arrive at the notion of a four-dimensional. Let us call the figure which corresponds to a square in a plane and a cube in our space, a tessaract. Model 1 is a cube. Let us assume a tessaract generated from it. Let us call the tessaract Urna. The generating cube may then be aptly called Urna Mala. We may use cubes to represent parts of four-space, but we must always remember that they are to us, in our study, only what squares are to a plane-being with respect to a cube.
Let us again examine the mode in which a plane-being represents a Block of cubes with slabs. Take Block 1 of the 81 Set of cubes. The plane-being represents this by nine slabs, which represent the Moena face of the block. Then, omitting the solidity of these first nine cubes, he takes another set of nine slabs to represent the next wall of cubes. Lastly, he represents the third wall by a third set, omitting the solidity of both second and third walls. In this manner, he evidently represents the extension of the Block upwards and sideways, in the Z and X directions; but in the Y direction he is powerless, and is compelled to represent extension in that direction by setting the three sets of slabs alongside in his plane. The second and third sets denote the height and breadth of the respective walls, but not their depth or thickness. Now, note that the Block extends three inches in each of the three directions. The plane-being can represent two of these dimensions on his plane; but the unknown direction he has to represent by a repetition of his plane figures. The cube extends three inches in the Y direction. He has to use 3 sets of slabs.
The Block is built up arbitrarily in this manner: Starting from Urna Mala and going up, we come to a Brown cube, and then to a Light-blue cube. Starting from Urna Mala and going right, we come to an Orange and a Fawn cube. Starting from Urna Mala and going away from us, we come to a Blue and a Buff cube. Now, the plane-being represents the Brown and Orange cubes by squares lying next to the square which represents Urna Mala. The Blue cube is as close as the Brown cube to Urna Mala, but he can find no place in the plane where he can place a Blue square so as to show this co-equal proximity of both cubes to the first. So he is forced to put a Blue square anywhere in his plane and say of it: “This Blue square represents what I should arrive at, if I started from Urna Mala and moved away, that is in the Y or unknown direction.” Now, just as there are three cubes going up, so there are three going away. Hence, besides the Blue square placed anywhere on the plane, he must also place a Buff square beyond it, to show that the Block extends as far away as it does upwards and sideways. (Each cube being a different colour, there will be as many different colours of squares as of cubes.) It will easily be seen that not only the Gold square, but also the Orange and every other square in the first set of slabs must have two other squares set somewhere beyond it on the plane to represent the extension of the Block away, or in the unknown Y direction.
Coming now to the representation of a four-dimensional block, we see that we can show only three dimensions by cubic blocks, and that the fourth can only be represented by repetitions of such blocks. There must be a certain amount of arbitrary naming and colouring. The colours have been chosen as now stated. Take the first Block of the 81 Set. We are familiar with its colours, and they can be found at any time by reference to Model 1. Now, suppose the Gold cube to represent what we can see in our space of a Gold tessaract; the other cubes of Block 1 give the colours of the tessaracts which lie in the three directions X, Y, and Z from the Gold one. But what is the colour of the tessaract which lies next to the Gold in the unknown direction, W? Let us suppose it to be Stone in colour. Taking out Block 2 of the 81 Set and arranging it on the pattern of Model 9, we find in it a Stone cube. But, just as there are three tessaracts in the X, Y, and Z directions, as shown by the cubes in Block 1, so also must there be three tessaracts in the unknown direction, W. Take Block 3 of the 81 Set. This Block can be arranged on the pattern of Model 2. In it there is a Silver cube where the Gold cube lies in Block 1. Hence, we may say, the tessaract which comes next to the Stone one in the unknown direction from the Gold, is of a Silver colour. Now, a cube in all these cases represents a tessaract. Between the Gold and Stone cubes there is an inch in the unknown direction. The Gold tessaract is supposed to be Gold throughout in all four directions, and so also is the Stone. We may imagine it in this way. Suppose the set of three tessaracts, the Gold, the Stone, and the Silver to move through our space at the rate of an inch a minute. We should first see the Gold cube which would last a minute, then the Stone cube for a minute, and lastly the Silver cube a minute. (This is precisely analogous to the appearance of passing cubes to the plane-being as successive squares lasting a minute.) After that, nothing would be visible.
Now, just as we must suppose that there are three tessaracts proceeding from the Gold cube in the unknown direction, so there must be three tessaracts extending in the unknown direction from every one of the 27 cubes of the first Block. The Block of 27 cubes is not a Block of 27 tessaracts, but it represents as much of them as we can see at once in our space; and they form the first portion or layer (like the first wall of cubes to the plane-being) of a set of eighty-one tessaracts, extending to equal distances in all four directions. Thus, to represent the whole Block of tessaracts there are 81 cubes, or three Blocks of 27 each.
Now, it is obvious that, just as a cube has various plane boundaries, so a tessaract has various cube boundaries. The cubes of the tessaract, which we have been regarding, have been those containing the X, Y, and Z directions, just as the plane-being regarded the Moena faces containing the X and Z directions. And, as long as the tessaract is unchanged in its position with regard to our space, we can never see any portion of it which is in the unknown direction. Similarly, we saw that a plane-being could not see the parts of a cube which went in the third direction, until the cube was turned round one of its edges. In order to make it quite clear what parts of a cube came into the plane, we gave distinct names to them. Thus, the squares containing the Z and X directions were called Moena and Murex; those containing the Z and Y, Alvus and Proes; and those the X and Y, Syce and Mel. Now, similarly with our four axes, any three will determine a cube. Let the tessaract in its normal position have the cube Mala determined by the axes Z, X, Y. Let the cube Lar be that which is determined by X, Y, W, that is, the cube which, starting from the X Y plane, stretches one inch in the unknown or W direction. Let Vesper be the cube determined by Z, Y, W, and Pluvium by Z, X, W. And let these cubes have opposite cubes of the following names:
| Mala | has | Margo |
| Lar | „ | Velum |
| Vesper | „ | Idus |
| Pluvium | „ | Tela |
Another way of looking at the matter is this. When a cube is generated from a square, each of the lines bounding the square becomes a square, and the square itself becomes a cube, giving two squares in its initial and final positions. When a cube moves in the new and unknown direction, each of its planes traces a cube and it generates a tessaract, giving in its initial and final positions two cubes. Thus there are eight cubes bounding the tessaract, six of them from the six plane sides and two from the cube itself. These latter two are Mala and Margo. The cubes from the six sides are: Lar from Syce, Velum from Mel, Vesper from Alvus, Idus from Proes, Pluvium from Moena, Tela from Murex. And just as a plane-being can only see the squares of a cube, so we can only see the cubes of a tessaract. It may be said that the cube can be pushed partly through the plane, so that the plane-being sees a section between Moena and Murex. Similarly, the tessaract can be pushed through our space so that we can see a section between Mala and Margo.
There is a method of approaching the matter, which settles all difficulties, and provides us with a nomenclature for every part of the tessaract. We have seen how by writing down the names of the cubes of a block, and then supposing that their number increases, certain sets of the names come to denote points, lines, planes, and solid. Similarly, if we write down a set of names of tessaracts in a block, it will be found that, when their number is increased, certain sets of the names come to denote the various parts of a tessaract.
For this purpose, let us take the 81 Set, and use the cubes to represent tessaracts. The whole of the 81 cubes make one single tessaractic set extending three inches in each of the four directions. The names must be remembered to denote tessaracts. Thus, Corvus is a tessaract which has the tessaracts Cuspis and Nugæ to the right, Arctos and Ilex above it, Dos and Cista away from it, and Ops and Spira in the fourth or unknown direction from it. It will be evident at once, that to write these names in any representative order we must adopt an arbitrary system. We require them running in four directions; we have only two on paper. The X direction (from left to right) and the Y (from the bottom towards the top of the page) will be assumed to be truly represented. The Z direction will be symbolized by writing the names in floors, the upper floors always preceding the lower. Lastly, the fourth, or W, direction (which has to be symbolized in three-dimensional space by setting the solids in an arbitrary position) will be signified by writing the names in blocks, the name which stands in any one place in any block being next in the W direction to that which occupies the same position in the block before or after it. Thus, Ops is written in the same place in the Second Block, Spira in the Third Block, as Corvus occupies in the First Block.
Since there are an equal number of tessaracts in each of the four directions, there will be three floors Z when there are three X and Y. Also, there will be three Blocks W. If there be four tessaracts in each direction, there will be four floors Z, and four blocks W. Thus, when the number in each direction is enlarged, the number of blocks W is equal to the number of tessaracts in each known direction.
On [pp. 136], [137] were given the names as used for a cubic block of 27 or 64. Using the same and more names for a tessaractic Set, and remembering that each name now represents, not a cube, but a tessaract, we obtain the following nomenclature, the order in which the names are written being that stated above:
| Third Block. | |||||
| Upper Floor. | - | Solia | Livor | Talus | |
| Lensa | Lares | Calor | |||
| Felis | Tholus | Passer | |||
| Middle Floor. | - | Lixa | Portica | Vena | |
| Crux | Margo | Sal | |||
| Pagus | Silex | Onager | |||
| Lower Floor. | - | Panax | Mensura | Mugil | |
| Opex | Lappa | Mappa | |||
| Spira | Luca | Ancilla | |||
| Second Block. | |||||
| Upper Floor. | - | Orsa | Mango | Libera | |
| Creta | Velum | Meatus | |||
| Lucta | Limbus | Pator | |||
| Middle Floor. | - | Camoena | Tela | Orca | |
| Vesper | Tessaract | Idus | |||
| Pagina | Pluvium | Pactum | |||
| Lower Floor. | - | Lis | Lorica | Offex | |
| Lua | Lar | Olla | |||
| Ops | Lotus | Limus | |||
| First Block. | |||||
| Upper Floor. | - | Olus | Semita | Lama | |
| Via | Mel | Iter | |||
| Ilex | Callis | Sors | |||
| Middle Floor. | - | Bucina | Murex | Daps | |
| Alvus | Mala | Proes | |||
| Arctos | Moena | Far | |||
| Lower Floor. | - | Cista | Cadus | Crus | |
| Dos | Syce | Bolus | |||
| Corvus | Cuspis | Nugæ | |||
It is evident that this set of tessaracts could be increased to the number of four in each direction, the names being used as before for the cubic blocks on pp. 136, 137, and in that case the Second Block would be duplicated to make the four blocks required in the unknown direction. Comparing such an 81 Set and 256 Set, we should find that Cuspis, which was a single tessaract in the 81 Set became two tessaracts in the 256 Set. And, if we introduced a larger number, it would simply become longer, and not increase in any other dimension. Thus, Cuspis would become the name of an edge. Similarly, Dos would become the name of an edge, and also Arctos. Ops, which is found in the Middle Block of the 81 Set, occurs both in the Second and Third Blocks of the 256 Set; that is, it also tends to elongate and not extend in any other direction, and may therefore be used as the name of an edge of a tessaract.
Looking at the cubes which represent the Syce tessaracts, we find that, though they increase in number, they increase only in two directions; therefore, Syce may be taken to signify a square. But, looking at what comes from Syce in the W direction, we find in the Middle Block of the 81 Set one Lar, and in the Second and Third Blocks of the 256 Set four Lars each. Hence, Lar extends in three directions, X, Y, W, and becomes a cube. Similarly, Moena is a plane; but Pluvium, which proceeds from it, extends not only sideways and upwards like Moena, but in the unknown direction also. It occurs in both Middle Blocks of the 256 Set. Hence, it also is a cube. We have now considered such parts of the Sets as contain one, two, and three dimensions. But there is one part which contains four. It is that named Tessaract. In the 256 Set there are eight such cubes in the Second, and eight in the Third Block; that is, they extend Z, X, Y, and also W. They may, therefore, be considered to represent that part of a tessaract or tessaractic Set, which is analogous to the interior of a cube.
The arrangement of colours corresponding to these names is seen on Model 1 corresponding to Mala, Model 2 to Margo, and Model 9 to the intermediate block.
When we take the view of the tessaract with which we commenced, and in which Arctos goes Z, Cuspis X, Dos Y, and Ops W, we see Mala in our space. But when the tessaract is turned so that the Ops line goes -X, while Cuspis is turned W, the other two remaining as they were, then we do not see Mala, but that cube which, in the original position of the tessaract, contains the Z, Y, W, directions, that is, the Vesper cube.
A plane-being may begin to study a block of cubes by their Syce squares; but if the block be turned round Dos, he will have Alvus squares in his space, and he must then use them to represent the cubic Block. So, when the tessaractic Set is turned round, Mala cubes leave our space, and Vespers enter.
There are two ways which can be followed in studying the Set of tessaracts.
I. Each tessaract of one inch every way can be supposed to be of the same colour throughout, so that, whichever way it be turned, whichever of its edges coincide with our known axes, it appears to us as a cube of one uniform colour. Thus, if Urna be the tessaract, Urna Mala would be a Gold cube, Urna Vesper a Gold cube, and so on. This method is, for the most part, adopted in the following pages. In this case, a whole Set of 4 × 4 × 4 × 4 tessaracts would in colours resemble a set composed of four cubes like Models 1, 9, 9, and 2. But, when any question about a particular tessaract has to be settled, it is advantageous, for the sake of distinctness, to suppose it coloured in its different regions as the whole set is coloured.
II. The other plan is, to start with the cubic sides of the inch tessaract, each coloured according to the scheme of the Models 1 to 8. In this case, the lines, if shown at all, should be very thin. For, in fact, only the faces would be seen, as the width of the lines would only be equal to the thickness of our matter in the fourth dimension, which is indistinguishable to the senses. If such completely coloured cubes be used, less error is likely to creep in; but it is a disadvantage that each cube in the several blocks is exactly like the others in that block. If the reader make such a set to work with for a time, he will gain greatly, for the real way of acquiring a sense of higher space is to obtain those experiences of the senses exactly, which the observation of a four-dimensional body would give. These Models 1-8 are called sides of the tessaract.
To make the matter perfectly clear, it is best to suppose that any tessaract or set of tessaracts which we examine, has a duplicate exactly similar in shape and arrangement of parts, but different in their colouring. In the first, let each tessaract have one colour throughout, so that all its cubes, apprehended in turn in our space, will be of one and the same colour. In the duplicate, let each tessaract be so coloured as to show its different cubic sides by their different colours. Then, when we have it turned to us in different aspects, we shall see different cubes, and when we try to trace the contacts of the tessaracts with each other, we shall be helped by realizing each part of every tessaract in its own colour.
CHAPTER VIII.
REPRESENTATION OF FOUR-SPACE BY NAME. STUDY OF TESSARACTS.
We have now surveyed all the preliminary ground, and can study the masses of tessaracts without obscurity.
We require a scaffold or framework for this purpose, which in three dimensions will consist of eight cubic spaces or octants assembled round one point, as in two dimensions it consisted of four squares or quadrants round a point.
These eight octants lie between the three axes Z, X, Y, which intersect at the given point, and can be named according to their positions between the positive and negative directions of those axes. Thus the octant Z, X, Y, is that which is contained by the positive portions of all three axes; the octant Z, X, Y, that which is to the left of Z, X, Y, and between the positive parts of Z and Y and the negative of X. To illustrate this quite clearly, let us take the eight cubes—Urna, Moles, Plebs, Frenum, Uncus, Pallor, Bidens, Ostrum—and place them in the eight octants. Let them be placed round the point of intersection of the axes; Pallor Corvus, Plebs Ilex, etc., will be at that point. Their positions will then be:—
| Urna | in the | Octant | Z | X | Y |
| Moles | „ | „ | Z | X | Y |
| Plebs | „ | „ | Z | X | Y |
| Frenum | „ | „ | Z | X | Y |
| Uncus | „ | „ | Z | X | Y |
| Pallor | „ | „ | Z | X | Y |
| Bidens | „ | „ | Z | X | Y |
| Ostrum | „ | „ | Z | X | Y |
The names used for the cubes, as they are before us, are as follows:—
| Third Block. | |||||
| Third Floor. | - | Arcus Mala | Ovis Mala | Portio Mala | |
| Laurus Mala | Tigris Mala | Segmen Mala | |||
| Axis Mala | Troja Mala | Aries Mala | |||
| Second Floor. | - | Postis Mala | Clipeus Mala | Tabula Mala | |
| Orcus Mala | Lacerta Mala | Testudo Mala | |||
| Verbum Mala | Luctus Mala | Anguis Mala | |||
| First Floor. | - | Telum Mala | Nepos Mala | Angusta Mala | |
| Polus Mala | Penates Mala | Vulcan Mala | |||
| Cervix Mala | Securis Mala | Vinculum Mala | |||
| Second Block. | |||||
| Third Floor. | - | Ara Mala | Vomer Mala | Pluma Mala | |
| Praeda Mala | Sacerdos Mala | Hydra Mala | |||
| Cortex Mala | Mica Mala | Flagellum Mala | |||
| Second Floor. | - | Pilum Mala | Glans Mala | Colus Mala | |
| Ocrea Mala | Tessera Mala | Domitor Mala | |||
| Cardo Mala | Cudo Mala | Malleus Mala | |||
| First Floor. | - | Agmen Mala | Lacus Mala | Arvus Mala | |
| Crates Mala | Cura Mala | Limen Mala | |||
| Thyrsus Mala | Vitta Mala | Sceptrum Mala | |||
| First Block. | |||||
| Third Floor. | - | Mars Mala | Merces Mala | Tyro Mala | |
| Spicula Mala | Mora Mala | Oliva Mala | |||
| Comes Mala | Tibicen Mala | Vestis Mala | |||
| Second Floor. | - | Ala Mala | Cortis Mala | Aer Mala | |
| Uncus Mala | Pallor Mala | Tergum Mala | |||
| Ostrum Mala | Bidens Mala | Scena Mala | |||
| First Floor. | - | Sector Mala | Hama Mala | Remus Mala | |
| Frenum Mala | Plebs Mala | Sypho Mala | |||
| Urna Mala | Moles Mala | Saltus Mala | |||
Their colours can be found by reference to the Models 1, 9, 2, which correspond respectively to the First, Second, and Third Blocks. Thus, Urna Mala is Gold; Moles, Orange; Saltus, Fawn; Thyrsus, Stone; Cervix, Silver. The cubes whose colours are not shown in the Models, are Pallor Mala, Tessera Mala, and Lacerta Mala, which are equivalent to the interiors of the Model cubes, and are respectively Light-buff, Wooden, and Sage-green. These 81 cubes are the cubic sides and sections of the tessaracts of an 81 tessaractic Set, which measures three inches in every direction. We suppose it to pass through our space. Let us call the positive unknown direction Ana (i.e., +W) and the negative unknown direction Kata (-W). Then, as the whole tessaract moves Kata at the rate of an inch a minute, we see first the First Block of 27 cubes for one minute, then the Second, and lastly the Third, each lasting one minute.
Now, when the First Block stands in the normal position, the edges of the tessaract that run from the Corvus corner of Urna Mala, are: Arctos in Z, Cuspis in X, Dos in Y, Ops in W. Hence, we denote this position by the following symbol:—
| Z | X | Y | W |
| a | c | d | o |
where a stands for Arctos, c for Cuspis, d for Dos, and o for Ops, and the other letters for the four axes in space. a, c, d, o are the axes of the tessaract, and can take up different directions in space with regard to us.
Let us now take a smaller four-dimensional set. Of the 81 Set let us take the following:—
| Z | X | Y | W |
| a | c | d | o |
| Second Block. | ||||
| Second Floor. | - | Ocrea Mala | Tessera Mala | |
| Cardo Mala | Cudo Mala | |||
| First Floor. | - | Crates Mala | Cura Mala | |
| Thyrsus Mala | Vitta Mala | |||
| First Block. | ||||
| Second Floor. | - | Uncus Mala | Pallor Mala | |
| Ostrum Mala | Bidens Mala | |||
| First Floor. | - | Frenum Mala | Plebs Mala | |
| Urna Mala | Moles Mala | |||
Let the First Block be put up before us in Z X Y, (Urna Corvus is at the junction of our axes Z X Y). The Second Block is now one inch distant in the unknown direction; and, if we suppose the tessaractic Set to move through our space at the rate of one inch a minute, the Second will enter in one minute, and replace the first. But, instead of this, let us suppose the tessaracts to turn so that Ops, which now goes W, shall go -X. Then we can see in our space that cubic side of each tessaract which is contained by the lines Arctos, Dos, and Ops, the cube Vesper; and we shall no longer have the Mala sides but the Vesper sides of the tessaractic Set in our space. We will now build it up in its Vesper view (as we built up the cubic Block in its Alvus view). Take the Gold cube, which now means Urna Vesper, and place it on the left hand of its former position as Urna Mala, that is, in the octant Z X Y. Thyrsus Vesper, which previously lay just beyond Urna Vesper in the unknown direction, will now lie just beyond it in the -X direction, that is, to the left of it. The tessaractic Set is now in the position Za Xō Yd Wc (the minus sign over the o meaning that Ops runs in the negative direction), and its Vespers lie in the following order:—
| Second Block. | ||||
| Second Floor. | - | Tessara | Pallor | |
| Cudo | Bidens | |||
| First Floor. | - | Cura | Plebs | |
| Vitta | Moles | |||
| First Block. | ||||
| Second Floor. | - | Ocrea | Uncus | |
| Cardo | Ostrum | |||
| First Floor. | - | Crates | Frenum | |
| Thyrsus | Urna | |||
The name Vesper is left out in the above list for the sake of brevity, but should be used in studying the positions.
Fig. 20.
On comparing the two lists of the Mala view and Vesper view, it will be seen that the cubes presented in the Vesper view are new sides of the tessaract, and that the arrangement of them is different from that in the Mala view. (This is analogous to the changes in the slabs from the Moena to Alvus view of the cubic Block.) Of course, the Vespers of all these tessaracts are not visible at once in our space, any more than are the Moenas of all three walls of a cubic Block to a plane-being. But if the tessaractic Set be supposed to move through space in the unknown direction at the rate of an inch a minute, the Second Block will present its Vespers after the First Block has lasted a minute. The relative position of the Mala Block and the Vesper Block may be represented in our space as in the diagram, [Fig. 20]. But it must be distinctly remembered that this arrangement is quite conventional, no more real than a plane-being’s symbolization of the Moena Wall and the Alvus Wall of the cubic Block by the arrangement of their Moena and Alvus faces, with the solidity omitted, along one of his known directions.
The Vespers of the First and Second Blocks cannot be in our space simultaneously, any more than the Moenas of all three walls in plane space. To render their simultaneous presence possible, the cubic or tessaractic Block or Set must be broken up, and its parts no longer retain their relations. This fact is of supreme importance in considering higher space. Endless fallacies creep in as soon as it is forgotten that the cubes are merely representative as the slabs were, and the positions in our space merely conventional and symbolical, like those of the slabs along the plane. And these fallacies are so much fostered by again symbolizing the cubic symbols and their symbolical positions in perspective drawings or diagrams, that the reader should surrender all hope of learning space from this book or the drawings alone, and work every thought out with the cubes themselves.
If we want to see what each individual cube of the tessaractic faces presented to us in the last example is like, we have only to consider each of the Malas similar in its parts to Model 1, and each of the Vespers to Model 5. And it must always be remembered that the cubes, though used to represent both Mala and Vesper faces of the tessaract, mean as great a difference as the slabs used for the Moena and Alvus faces of the cube.
If the tessaractic Set move Kata through our space, when the Vesper faces are presented to us, we see the following parts of the tessaract Urna (and, therefore, also the same parts of the other tessaracts):
(1) Urna Vesper, which is Model 5.
(2) A parallel section between Urna Vesper and Urna Idus, which is Model 11.
(3) Urna Idus, which is Model 6.
When Urna Idus has passed Kata our space, Moles Vesper enters it; then a section between Moles Vesper and Moles Idus, and then Moles Idus. Here we have evidently observed the tessaract more minutely; as it passes Kata through our space, starting on its Vesper side, we have seen the parts which would be generated by Vesper moving along Cuspis—that is Ana.
Two other arrangements of the tessaracts have to be learnt besides those from the Mala and Vesper aspect. One of them is the Pluvium aspect. Build up the Set in Z X Y, letting Arctos run Z, Cuspis X, and Ops Y. In the common plane Moena, Urna Pluvium coincides with Urna Mala, though they cannot be in our space together; so too Moles Pluvium with Moles Mala, Ostrum Pluvium with Ostrum Mala. And lying towards us, or Y, is now that tessaract which before lay in the W direction from Urna, viz., Thyrsus. The order will therefore be the following (a star denotes the cube whose corner is at point of intersection of the axes, and the name Pluvium must be understood to follow each of the names):
| Z | X | Y | W |
| a | c | ō | d |
| Second Block. | ||||||
| Second Floor. | - | Uncus | Pallor | |||
| Ocrea | Tessera | |||||
| First Floor. | - | Frenum | Plebs | |||
| Crates | Cura | |||||
| First Block. | ||||||
| Second Floor. | - | Ostrum | Bidens | |||
| Cardo | Cudo | |||||
| First Floor. | - | * | Urna | Moles | ||
| Thyrsus | Vitta | |||||
Thus the wall of cubes in contact with that wall of the Mala position which contains the Urna, Moles, Ostrum, and Bidens Malas, is a wall composed of the Pluviums of Urna, Moles, Ostrum, and Bidens. The wall next to this, and nearer to us, is of Thyrsus, Vitta, Cardo, Cudo, Pluviums. The Second Block is one inch out of our Space, and only enters it if the Block moves Kata. Model 7 shows the Pluvium cube; and each of the cubes of the tessaracts seen in the Pluvium position is a Pluvium. If the tessaractic Set moved Kata, we would see the Section between Pluvium and Tela for all but a minute; and then Tela would enter our space, and the Tela of each tessaract would be seen. Model 12 shows the Section from Pluvium to Tela. Model 8 is Tela. Tela only lasts for a flash, as it has only the minutest magnitude in the unknown or Ana direction. Then, Frenum Pluvium takes the place of Urna Tela; and, when it passes through, we see a similar section between Frenum Pluvium and Frenum Tela, and lastly Frenum Tela. Then the tessaractic Set passes out, or Kata, our space. A similar process takes place with every other tessaract, when the Set of tessaracts moves through our space.
There is still one more arrangement to be learnt. If the line of the tessaract, which in the Mala position goes Ana, or W, be changed into the Z or downwards direction, the tessaract will then appear in our space below the Mala position; and the side presented to us will not be Mala, but that which contains the lines Dos, Cuspis, and Ops. This side is Model 3, and is called Lar. Underneath the place which was occupied by Urna Mala, will come Urna Lar; under the place of Moles Mala, Moles Lar; under the place of Frenum Mala, Frenum Lar. The tessaract, which in the Mala position was an inch out of our space Ana, or W, from Urna Mala, will now come into it an inch downwards, or Z, below Urna Mala, with its Lar presented to us; that is, Thyrsus Lar will be below Urna Lar. In the whole arrangement of them written below, the highest floors are written first, for now they stretch downwards instead of upwards. The name Lar is understood after each.
| Z | X | Y | W |
| ō | c | d | a |
| Second Block. | ||||||
| Second Floor. | - | Uncus | Pallor | |||
| Ostrum | Bidens | |||||
| First Floor. | - | Ocrea | Tessera | |||
| Cardo | Cudo | |||||
| First Block. | ||||||
| Second Floor. | - | Frenum | Plebs | |||
| * | Urna | Moles | ||||
| First Floor. | - | Crates | Cura | |||
| Thyrsus | Vitta | |||||
Here it is evident that what was the lower floor of Malas, Urna, Moles, Plebs, Frenum, now appears as if carried downwards instead of upwards, Lars being presented in our space instead of Malas. This Block of Lars is what we see of the tessaract Set when the Arctos line, which in the Mala position goes up, is turned into the Ana, or W, direction, and the Ops line comes in downwards.
The rest of the tessaracts, which consists of the cubes opposite to the four treated above, and of the tessaractic space between them, is all Ana in our space. If the tessaract be moved through our space—for instance, when the Lars are present in it—we see, taking Urna alone, first the section between Urna Lar and Urna Velum (Model 10), and then Urna Velum (Model 4), and similarly the sections and Velums of each tessaract in the Set. When the First Block has passed Kata our space, Ostrum Lar enters; and the Lars of the Second Block of tessaracts occupy the places just vacated by the Velums of the First Block. Then, as the tessaractic Set moves on Kata, the sections between Velums and Lars of the Second Block of tessaracts enter our space, and finally their Velums. Then the whole tessaractic Set disappears from our space.
When we have learnt all these aspects and passages, we have experienced some of the principal features of this small Set of tessaracts.
CHAPTER IX.
FURTHER STUDY OF TESSARACTS.
When the arrangement of a small set has been mastered, the different views of the whole 81 Set should be learnt. It is now clear to us that, in the list of the names of the eighty-one tessaracts given above, those which lie in the W direction appear in different blocks, while those that lie in the Z, X, or Y directions can be found in the same block. Therefore, from the arrangement given, which is denoted by Za Xc Yd Wo or more briefly by a c d o, we can form any other arrangement.
To confirm the meaning of the symbol a c d o for position, let us remember that the order of the axes known in our space will invariably be Z X Y, and the unknown direction will be stated last, thus: Z X Y W. Hence, if we write a ō d c, we know that the position or aspect intended is that in which Arctos (a) goes Z, Ops (ō) negative X, Dos (d) Y, and Cuspis (c) W. And such an arrangement can be made by shifting the nine cubes on the left side of the First Block of the eighty-one tessaracts, and putting them into the Z X Y octant, so that they just touch their former position. Next to them, to their left, we set the nine of the left side of the Second Block of the 81 Set; and next to these again, on their left, the nine of the left side of the Third Block. This Block of twenty-seven now represents Vesper Cubes, which have only one square identical with the Mala cubes of the previous blocks, from which they were taken.
Similarly the Block which is one inch Ana, can be made by taking the nine cubes which come vertically in the middle of each of the Blocks in the first position, and arranging them in a similar manner. Lastly, the Block which lies two inches Ana, can be made by taking the right sides of nine cubes each from each of the three original Blocks, and arranging them so that those in the Second original Block go to the left of those in the First, and those in the Third to their left. In this manner we should obtain three new Blocks, which represent what we can see of the tessaracts, when the direction in which Urna, Moles, Saltus lie in the original Set, is turned W.
The Pluvium Block we can make by taking the front wall of each original Block, and setting each an inch nearer to us, that is -Y. The far sides of these cubes are Moenas of Pluviums. By continuing this treatment of the other walls of the three original Blocks parallel to the front wall, we obtain two other Blocks of tessaracts. The three together are the tessaractic position a c ō d, for in all of them Ops lies in the -Y direction, and Dos has been turned W.
The Lar position is more difficult to construct. To put the Lars of the Blocks in their natural position in our space, we must start with the original Mala Blocks, at least three inches above the table. The First Lar Block is made by taking the lowest floors of the three Mala Blocks, and placing them so that that of the Second is below that of the First, and that of the Third below that of the Second. The floor of cubes whose diagonal runs from Urna Lar to Remus Lar, will be at the top of the Block of Lars; and that whose diagonal goes from Cervix Lar to Angusta Lar, will be at the bottom. The next Block of Lars will be made by taking the middle horizontal floors of the three original Blocks, and placing them in a similar succession—the floor from Ostrum Lar to Aer Lar being at the top, that from Cardo Lar to Colus Lar in the middle, and Verbum Lar to Tabula Lar at the bottom. The Third Lar Block is composed of the top floor of the First Block on the top—that is, of Comes Lar to Tyro Lar, of Cortex Lar to Pluma Lar in the middle, and Axis Lar to Portio Lar at the bottom.
CHAPTER X.
CYCLICAL PROJECTIONS.
Let us denote the original position of the cube, that wherein Arctos goes Z, Cuspis X, and Dos Y, by the expression,
| Z | X | Y |
| a | c | d |
(1)
If the cube be turned round Cuspis, Dos goes Z, Cuspis remains unchanged, and Arctos goes Y, and we have the position,
| Z | X | Y |
| d | c | a |
where Zd means that Dos runs in the negative direction of the Z axis from the point where the axes intersect. We might write Zd but it is preferable to write Zd. If we next turn the cube round the line, which runs Y, that is, round Arctos, we obtain the position,
| Z | X | Y |
| c | d | a |
(2)
and by means of this double turn we have put c and d in the places of a and c. Moreover, we have no negative directions. This position we call simply c d a. If from it we turn the cube round a, which runs Y, we get Zd Xc Ya, and if, then, we turn it round Dos we get Zd Xa Yc or simply d a c. This last is another position in which all the lines are positive, and the projections, instead of lying in different quadrants, will be contained in one.
The arrangement of cubes in a c d we know. That in c d a is:
| Third Floor. | - | Vestis | Oliva | Tyro | |
| Scena | Tergum | Aer | |||
| Saltus | Sypho | Remus | |||
| Second Floor. | - | Tibicen | Mora | Merces | |
| Bidens | Pallor | Cortis | |||
| Moles | Plebs | Hama | |||
| First Floor. | - | Comes | Spicula | Mars | |
| Ostrum | Uncus | Ala | |||
| Urna | Frenum | Sector | |||
It will be found that learning the cubes in this position gives a great advantage, for thereby the axes of the cube become dissociated with particular directions in space.
The d a c position gives the following arrangement:
| Remus | Aer | Tyro |
| Hama | Cortis | Merces |
| Sector | Ala | Mars |
| Sypho | Tergum | Oliva |
| Plebs | Pallor | Mora |
| Frenum | Uncus | Spicula |
| Saltus | Scena | Vestis |
| Moles | Bidens | Tibicen |
| Urna | Ostrum | Comes |
The sides, which touch the vertical plane in the first position, are respectively, in a c d Moena, in c d a Syce, in d a c Alvus.
Take the shape Urna, Ostrum, Moles, Saltus, Scena, Sypho, Remus, Aer, Tyro. This gives in a c d the projection: Urna Moena, Ostrum Moena, Moles Moena, Saltus Moena, Scena Moena, Vestis Moena. (If the different positions of the cube are not well known, it is best to have a list of the names before one, but in every case the block should also be built, as well as the names used.) The same shape in the position c d a is, of course, expressed in the same words, but it has a different appearance. The front face consists of the Syces of
| Saltus | Sypho | Remus |
| Moles | Plebs | Hama |
| Urna | Frenum | Sector |
And taking the shape we find we have Urna, and we know that Ostrum lies behind Urna, and does not come in; next we have Moles, Saltus, and we know that Scena lies behind Saltus and does not come in; lastly, we have Sypho and Remus, and we know that Aer and Tyro are in the Y direction from Remus, and so do not come in. Hence, altogether the projection will consist only of the Syces of Urna, Moles, Saltus, Sypho, and Remus.
Next, taking the position d a c, the cubes in the front face have their Alvus sides against the plane, and are:
| Sector | Ala | Mars |
| Frenum | Uncus | Spicula |
| Urna | Ostrum | Comes |
And, taking the shape, we find Urna, Ostrum; Moles and Saltus are hidden by Urna, Scena is behind Ostrum, Sypho gives Frenum, Remus gives Sector, Aer gives Ala, and Tyro gives Mars. All these are Alvus sides.
Let us now take the reverse problem, and, given the three cyclical projections, determine the shape. Let the a c d projection be the Moenas of Urna, Ostrum, Bidens, Scena, Vestis. Let the c d a be the Syces of Urna, Frenum, Plebs, Sypho, and the d a c be the Alvus of Urna, Frenum, Uncus, Spicula. Now, from a c d we have Urna, Frenum, Sector, Ostrum, Uncus, Ala, Bidens, Pallor, Cortis, Scena, Tergum, Aer, Vestis, Oliva, Tyro. From c d a we have Urna, Ostrum, Comes, Frenum, Uncus, Spicula, Plebs, Pallor, Mora, Sypho, Tergum, Oliva. In order to see how these will modify each other, let us consider the a c d solution as if it were a set of cubes in the c d a arrangement. Here, those that go in the Arctos direction, go away from the plane of projection, and must be represented by the Syce of the cube in contact with the plane. Looking at the a c d solution we write down (keeping those together which go away from the plane of projection): Urna and Ostrum, Frenum and Uncus, Sector and Ala, Bidens, Pallor, Cortis, Scena and Vestis, Tergum and Oliva, Aer and Tyro. Here we see that the whole c d a face is filled up in the projection, as far as this solution is concerned. But in the c d a solution we have only Syces of Urna, Frenum, Plebs, Sypho. These Syces only indicate the presence of a certain number of the cubes stated above as possible from the Moena projection, and those are Urna, Ostrum, Frenum, Uncus, Pallor, Tergum, Oliva. This is the result of a comparison of the Moena projection with the Syce projection. Now, writing these last named as they come in the d a c projection, we have Urna, Ostrum, Frenum, Uncus and Pallor and Tergum, Oliva. And of these Ostrum Alvus is wanting in the d a c projection as given above. Hence Ostrum will be wanting in the final shape, and we have as the final solution: Urna, Frenum, Uncus, Pallor, Tergum, Oliva.
CHAPTER XI.
A TESSARACTIC FIGURE AND ITS PROJECTIONS.
We will now consider a fourth-dimensional shape composed of tessaracts, and the manner in which we can obtain a conception of it. The operation is precisely analogous to that described in chapter VI., by which a plane being could obtain a conception of solid shapes. It is only a little more difficult in that we have to deal with one dimension or direction more, and can only do so symbolically.
We will assume the shape to consist of a certain number of the 81 tessaracts, whose names we have given on p. 168. Let it consist of the thirteen tessaracts: Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo, Vitta, Cura, Penates, Polus, Orcus, Lacerta.
Firstly, we will consider what appearances or projections these tessaracts will present to us according as the tessaractic set touches our space with its (a) Mala cubes, (b) Vesper cubes, (c) Pluvium cubes, or (d) Lar cubes. Secondly, we will treat the converse question, how the shape can be determined when the projections in each of those views are given.
Let us build up in cubes the four different arrangements of the tessaracts according as they enter our space on their Mala, Vesper, Pluvium or Lar sides. They can only be printed by symbolizing two of the directions. In the following tabulations the directions Y, X will at once be understood. The direction Z (expressed by the wavy line) indicates that the floors of nine, each printed nearer the top of the page, lie above those printed nearer the bottom of it. The direction W is indicated by the dotted line, which shows that the floors of nine lying to the left or right are in the W direction (Ana) or the -W direction (Kata) from those which lie to the right or left. For instance, in the arrangement of the tessaracts, as Malas (Table A) the tessaract Tessara, which is exactly in the middle of the eighty-one tessaracts has
| Domitor on its right | side | or in the | X | direction. |
| Ocrea on its left | „ | „ | -X | „ |
| Glans away from us | „ | „ | Y | „ |
| Cudo nearer to us | „ | „ | -Y | „ |
| Sacerdos above it | „ | „ | Z | „ |
| Cura below it | „ | „ | -Z | „ |
| Lacerta in the Ana or | W | „ | ||
| Pallor in the Kata or | -W | „ | ||
Similarly Cervix lies in the Ana or W direction from Urna, with Thyrsus between them. And to take one more instance, a journey from Saltus to Arcus would be made by travelling Y to Remus, thence -X to Sector, thence Z to Mars, and finally W to Arcus. A line from Saltus to Arcus is therefore a diagonal of the set of 81 tessaracts, because the full length of its side has been traversed in each of the four directions to reach one from the other, i.e. Saltus to Remus, Remus to Sector, Sector to Mars, Mars to Arcus.
TABLE A.
Mala presentation of 81 Tessaracts.
| Z | W | -W | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Arcus | Ovis | Portio | Ara | Vomer | Pluma | Mars | Merces | Tyro | |||||||||||||||||
| Laurus | Tigris | Segmen | Praeda | Sacerdos | Hydra | Spicula | Mora | Oliva | |||||||||||||||||
| Axis | Troja | Aries | Cortex | Mica | Flagellum | Comes | Tibicen | Vestis | |||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Postis | Clipeus | Tabula | Pilum | Glans | Coins | Ala | Cortis | Aer | |||||||||||||||||
| Orcus | Lacerta | Testudo | Ocrea | Tessera | Domitor | Uncus‡ | Pallor‡ | Tergum | |||||||||||||||||
| Verbum | Luctus | Anguis | Cardo | Cudo | Malleus | Ostrum | Bidens‡ | Scena | |||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Telum | Nepos | Angusta | Agmen | Lacus | Arvus | Sector | Hama | Remus | |||||||||||||||||
| Polus | Penates | Vulcan | Crates | Cura | Limen | Frenum‡ | Plebs‡ | Sypho | |||||||||||||||||
| Cervix | Securis | Vinculum | Thyrsus | Vitta | Sceptrum | Urna‡ | Moles‡ | Saltus | |||||||||||||||||
| -Z | X | X | X | ||||||||||||||||||||||
TABLE B.
Vesper presentation of 81 Tessaracts.
| Z | W | -W | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Portio | Pluma | Tyro | Ovis | Vomer | Merces | Arcus | Ara | Mars | |||||||||||||||||
| Segmen | Hydra | Oliva | Tigris | Sacerdos | Mora | Laurus | Praeda | Spicula | |||||||||||||||||
| Aries | Flagellum | Vestis | Troja | Mica | Tibicen | Axis | Cortex | Comes | |||||||||||||||||
| -X | -X | -X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Tabula | Colus | Aer | Clipeus | Glans | Cortis | Postis | Pilum | Ala | |||||||||||||||||
| Testudo | Domitor | Tergum | Lacerta* | Tessera* | Pallor* | Orcus* | Ocrea* | Uncus* | |||||||||||||||||
| Anguis | Malleus | Scena | Luctus* | Cudo* | Bidens* | Verbum† | Cardo† | Ostrum† | |||||||||||||||||
| -X | -X | -X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Angusta | Arvus | Remus | Nepos | Lacus | Hama | Telum | Agmen | Sector | |||||||||||||||||
| Vulcan | Limen | Sypho | Penates* | Cura* | Plebs* | Polus* | Crates* | Frenum* | |||||||||||||||||
| Vinculum | Sceptrum | Saltus | Securis* | Vitta* | Moles* | Cervix* | Thyrsus* | Urna* | |||||||||||||||||
| -Z | -X | -X | -X | ||||||||||||||||||||||
TABLE C.
Pluvium presentation of 81 Tessaracts.
| Z | W | -W | |||||||||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Mars | Merces | Tyro | Spicula | Mora | Oliva | Comes | Tibicen | Vestis | |||||||||||||||||
| Ara | Vomer | Pluma | Praeda | Sacerdos | Hydra | Cortex | Mica | Flagellum | |||||||||||||||||
| Arcus | Ovis | Portio | Laurus | Tigris | Segmen | Axis | Troja | Aries | |||||||||||||||||
| -Y | -Y | -Y | |||||||||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Ala | Cortis | Aer | Uncus* | Pallor* | Tergum | Ostrum† | Bidens† | Scena | |||||||||||||||||
| Pilum | Glans | Colus | Ocrea* | Tessera* | Domitor | Cardo† | Cudo* | Malleus | |||||||||||||||||
| Postis | Clipeus | Tabula | Orcus* | Lacerta* | Testudo | Verbum† | Luctus† | Anguis | |||||||||||||||||
| -Y | -Y | -Y | |||||||||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Sector | Hama | Remus | Frenum* | Plebs* | Sypho | Urna* | Moles* | Saltus | |||||||||||||||||
| Agmen | Lacus | Arvus | Crates* | Cura* | Limen | Thyrsus* | Vitta* | Sceptrum | |||||||||||||||||
| Telum | Nepos | Angusta | Polus* | Penates* | Vulcan | Cervix† | Securis† | Vinculum | |||||||||||||||||
| -Z | -Y | -Y | -Y | ||||||||||||||||||||||
TABLE D.
Lar presentation of 81 Tessaracts.
| Z | W | -W | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Mars | Merces | Tyro | Ala | Cortis | Aer | Sector | Hama | Remus | |||||||||||||||||
| Spicula | Mora | Oliva | Uncus | Pallor* | Tergum | Frenum* | Plebs* | Sypho | |||||||||||||||||
| Comes | Tibicen | Vestis | Ostrum | Bidens | Scena | Urna* | Moles* | Saltus | |||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Ara | Vomer | Pluma | Pilum | Glans | Colus | Agmen | Laurus | Arvus | |||||||||||||||||
| Proeda | Sacerdos | Hydra | Ocrea | Tessera* | Domitor | Crates | Cura* | Limen | |||||||||||||||||
| Cortex | Mica | Flagellum | Cardo | Cudo* | Malleus | Thyrsus | Vitta* | Sceptrum | |||||||||||||||||
| X | X | X | |||||||||||||||||||||||
| Y | Y | Y | |||||||||||||||||||||||
| Arcus | Ovis | Portio | Postis | Clipeus | Tabula | Telum | Nepos | Angusta | |||||||||||||||||
| Laurus | Tigris | Segmen | Orcus* | Lacerta* | Testudo | Polus* | Penates* | Vulcan | |||||||||||||||||
| Axis | Troja | Aries | Verbum | Luctus | Anguis | Cervix | Securis | Vinculum | |||||||||||||||||
| -Z | X | X | X | ||||||||||||||||||||||
The relation between the four different arrangements shown in the [tables A], [B], [C], and [D], will be understood from what has been said in [chapter VIII.] about a small set of sixteen tessaracts. A glance at the lines, which indicate the directions in each, will show the changes effected by turning the tessaracts from the Mala presentation.
In the Vesper presentation:
The tessaracts—
| (e.g. Urna, Ostrum, Comes), | which ran | Z | still run | Z. |
| (e.g. Urna, Frenum, Sector), | „ | Y | „ | Y. |
| (e.g. Urna, Moles, Saltus), | „ | X | now run | W. |
| (e.g. Urna, Thyrsus, Cervix), | „ | W | „ | -X. |
In the Pluvium presentation:
The tessaracts—
| (e.g. Urna, Ostrum, Comes), | which ran | Z | still run | Z. |
| (e.g. Urna, Moles, Saltus), | „ | X | „ | X. |
| (e.g. Urna, Frenum, Sector), | „ | Y | now run | W. |
| (e.g. Urna, Thyrsus, Cervix), | „ | W | „ | -Y. |
In the Lar presentation:
The tessaracts—
| (e.g. Urna, Moles, Saltus), | which ran | X | still run | X. |
| (e.g. Urna, Frenum, Sector), | „ | Y | „ | Y. |
| (e.g. Urna, Ostrum, Comes), | „ | Z | now run | W. |
| (e.g. Urna, Thyrsus, Cervix), | „ | W | „ | -Z. |
This relation was already treated in [chapter IX.], but it is well to have it very clear for our present purpose. For it is the apparent change of the relative positions of the tessaracts in each presentation, which enables us to determine any body of them.
In considering the projections, we always suppose ourselves to be situated Ana or W towards the tessaracts, and any movement to be Kata or -W through our space. For instance, in the Mala presentation we have first in our space the Malas of that block of tessaracts, which is the last in the -W direction. Thus, the Mala projection of any given tessaract of the set is that Mala in the extreme -W block, whose place its (the given tessaract’s) Mala would occupy, if the tessaractic set moved Kata until the given tessaract reached our space. Or, in other words, if all the tessaracts were transparent except those which constitute the body under consideration, and if a light shone through Four-space from the Ana (W) side to the Kata (-W) side, there would be darkness in each of those Malas, which would be occupied by the Mala of any opaque tessaract, if the tessaractic set moved Kata.
Let us look at the set of 81 tessaracts we have built up in the Mala arrangements, and trace the projections in the extreme -W block of the thirteen of our shape. The latter are printed in italics in [Table A], and their projections are marked ‡.
Thus the cube Uncus Mala is the projection of the tessaract Orcus, Pallor Mala of Pallor and Tessera and Tacerta, Bidens Mala of Cudo, Frenum Mala of Frenum and Polus, Plebs Mala of Plebs and Cura and Penates, Moles Mala of Moles and Vitta, Urna Mala of Urna.
Similarly, we can trace the Vesper projections ([Table B]). Orcus Vesper is the projection of the tessaracts Orcus and Lacerta, Ocrea Vesper of Tessera, Uncus Vesper of Pallor, Cardo Vesper of Cudo, Polus Vesper of Polus and Penates, Crates Vesper of Cura, Frenum Vesper of Frenum and Plebs, Urna Vesper of Urna and Moles, Thyrsus Vesper of Vitta. Next in the Pluvium presentation ([Table C]) we find that Bidens Pluvium is the projection of the tessaract Pallor, Cudo Pluvium of Cudo and Tessera, Luctus Pluvium of Lacerta, Verbum Pluvium of Orcus, Urna Pluvium of Urna and Frenum, Moles Pluvium of Moles and Plebs, Vitta Pluvium of Vitta and Cura, Securis Pluvium of Penates, Cervix Pluvium of Polus. Lastly, in the Lar presentation ([Table D]) we observe that Frenum Lar is the projection of Frenum, Plebs Lar of Plebs and Pallor, Moles Lar of Moles, Urna Lar of Urna, Cura Lar of Cura and Tessara, Vitta Lar of Vitta and Cudo, Penates Lar of Penates and Lacerta, Polur Lar of Polus and Orcus.
Secondly, we will treat the converse problem, how to determine the shape when the projections in each presentation are given. Looking back at the list just given above, let us write down in each presentation the projections only.
Mala projections:
Uncus, Pallor, Bidens, Frenum, Plebs, Moles, Urna.
Vesper projections:
Orcus, Ocrea, Uncus, Cardo, Polus, Crates, Frenum, Urna, Thyrsus.
Pluvium projections:
Bidens, Cudo, Luctus, Verbum, Urna, Moles, Vitta, Securis, Cervix.
Lar projections:
Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates.
Now let us determine the shape indicated by these projections. In now using the same tables we must not notice the italics, as the shape is supposed to be unknown. It is assumed that the reader is building the problem in cubes. From the Mala projections we might infer the presence of all or any of the tessaracts written in the brackets in the following list of the Mala presentation.
(Uncus, Ocrea, Orcus); (Pallor, Tessera, Lacerta);
(Bidens, Cudo, Luctus); (Frenum, Crates, Polus);
(Plebs, Cura, Penates); (Moles, Vitta, Securis);
(Urna, Thyrsus, Cervix).
Let us suppose them all to be present in our shape, and observe what their appearance would be in the Vesper presentation. We mark them all with an asterisk in [Table B]. In addition to those already marked we must mark (†) Verbum, Cardo, Ostrum, and then we see all the Vesper projections, which would be formed by all the tessaracts possible from the Mala projections. Let us compare these Vesper projections, viz. Orcus, Ocrea, Uncus, Verbum, Cardo, Ostrum, Polus, Crates, Frenum, Cervix, Thyrsus, Urna, with the given Vesper projections. We see at once that Verbum, Ostrum, and Cervix are absent. Therefore, we may conclude that all the tessaracts, which would be implied as possible by their presence, are absent, and of the Mala possibilities may exclude the tessaracts Bidens, Luctus, Securis, and Cervix itself. Thus, of the 21 tessaracts possible in the Mala view, there remain only 17 possible, both in the Mala and Vesper views, viz. Uncus, Ocrea, Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Crates, Polus, Plebs, Cura, Penates, Moles, Vitta, Urna, Thyrsus. This we call the Mala-Vesper solution.
Next let us take the Pluvium presentation. We again mark with an asterisk in Table C the possibilities inferred from the Mala-Vesper solution, and take the projections those possibilities would produce. The additional projections are again marked (†). There are twelve Pluvium projections altogether, viz. Bidens, Ostrum, Cudo, Cardo, Luctus, Verbum, Urna, Moles, Vitta, Thyrsus, Securis, Cervix. Again we compare these with the given Pluvium projections, and find three are absent, viz. Ostrum, Cardo, Thyrsus. Hence the tessaracts implied by Ostrum and Cardo and Thyrsus cannot be in our shape, viz. Uncus, Ocrea, Crates, nor Thyrsus itself. Excluding these four from the seventeen possibilities of the Mala-Vesper solution we have left the thirteen tessaracts: Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Polus, Plebs, Cura, Penates, Moles, Vitta, Urna. This we call the Mala-Vesper-Pluvium solution.
Lastly, we have to consider whether these thirteen tessaracts are consistent with the given Lar projections. We mark them again on Table D with an asterisk, and we find that the projections are exactly those given, viz. Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates. Therefore, we have not to exclude any of the thirteen, and can infer that they constitute the shape, which produces the four different given views or projections.
In fine, any shape in space consists of the possibilities common to the projections of its parts upon the boundaries of that space, whatever be the number of its dimensions. Hence the simple rule for the determination of the shape would be to write down all the possibilities of the sets of projections, and then cancel all those possibilities which are not common to all. But the process adopted above is much preferable, as through it we may realize the gradual delimitation of the shape view by view. For once more we must remind ourselves that our great object is, not to arrive at results by symbolical operations, but to realize those results piece by piece through realized processes.