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 ZXY. 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 positionZaXōYdWc(the minus sign over theomeaningthat Ops runs in the negative direction), and its Vespers lie in the followingorder:—
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.
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 MoenaWall 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 XY, letting Arctos run Z, Cuspis X, and OpsY. 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, orY, 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):
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 theZor 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 nowcome into it an inch downwards, orZ, 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.
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 (Model10), 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.
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 byZaXcYdWoor more briefly bya c d o, we can form any other arrangement.
To confirm the meaning of the symbola c d ofor 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 writea 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 ZXY 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 Malacubes 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 positiona 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 bytaking 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.
Let us denote the original position of the cube, that wherein Arctos goes Z, Cuspis X, and Dos Y, by the expression,
(1)
If the cube be turned round Cuspis, Dos goesZ, Cuspis remains unchanged, and Arctos goes Y, and we have the position,
whereZdmeans that Dos runs in the negative direction of the Z axis from the point where the axes intersect. We might writeZdbut it is preferable to writeZd. If we next turn the cube round the line, which runs Y, that is, round Arctos, we obtain the position,
(2)
and by means of this double turn we have putcanddin the places ofaandc. Moreover, we have no negative directions. This position we call simplyc d a. If from it we turn the cube rounda, which runs Y, we getZdXcYa, and if, then, we turn it round Dos we getZdXaYcor simplyd a c. This last is another position inwhich all the lines are positive, and the projections, instead of lying in different quadrants, will be contained in one.
The arrangement of cubes ina c dwe know. That inc d ais:
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.
Thed a cposition gives the following arrangement:
The sides, which touch the vertical plane in the first position, are respectively, ina c dMoena, inc d aSyce, ind a cAlvus.
Take the shape Urna, Ostrum, Moles, Saltus, Scena, Sypho, Remus, Aer, Tyro. This gives ina c dthe 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 positionc d ais, of course, expressed in the same words, but it has a different appearance. The front face consists of the Syces of
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 positiond a c, the cubes in the front face have their Alvus sides against the plane, and are:
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 thea c dprojection be the Moenas of Urna, Ostrum, Bidens, Scena, Vestis. Let thec d abe the Syces of Urna, Frenum, Plebs, Sypho, and thed a cbe the Alvus of Urna, Frenum, Uncus, Spicula. Now, froma c dwehave Urna, Frenum, Sector, Ostrum, Uncus, Ala, Bidens, Pallor, Cortis, Scena, Tergum, Aer, Vestis, Oliva, Tyro. Fromc d awe 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 thea c dsolution as if it were a set of cubes in thec d aarrangement. 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 thea c dsolution 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 wholec d aface is filled up in the projection, as far as this solution is concerned. But in thec d asolution 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 thed a cprojection, we have Urna, Ostrum, Frenum, Uncus and Pallor and Tergum, Oliva. And of these Ostrum Alvus is wanting in thed a cprojection 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.
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 atonce 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
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.
TABLE B.Vesper presentation of 81 Tessaracts.
TABLE C.Pluvium presentation of 81 Tessaracts.
TABLE D.Lar presentation of 81 Tessaracts.
The relation between the four different arrangements shown in thetables A,B,C, andD, will be understood from what has been said inchapter 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—
In the Pluvium presentation:
The tessaracts—
In the Lar presentation:
The tessaracts—
This relation was already treated inchapter 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 Malain 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 inTable 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 projectionof 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 inTable 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.