TERTIUM ORGANUM

THE THIRD CANON OF THOUGHT

A KEY TO THE ENIGMAS OF THE WORLD

CHAPTER 5

Four-dimensional space. 'Time-body' -Linga Sharira. Form of the human body from birth to death. Incommensurability of a three-dimensional and a four-dimensional body. Newton's fluents. Unreality of constant magnitudes in our world. Right and left hand in three-dimensional and a four-dimensional space. Differences between threedimensional and four-dimensional space. Not two different spaces, but two different modes of perception of one and the same world.
Four-dimensional space, if we attempt to represent it to ourselves, will be the infinite repetition of our space -of our infinite three-dimensional sphere -just as a line is the infinite repetition of a point.
A great deal of what has been said earlier will become much clearer for us if we take as our standpoint the view that the 'fourth dimension' should be looked for in time.
It will then become clear what is meant by saying that a four-dimensional body may be regarded as the trace of the movement in space of a threedimensional body in a direction not contained in it. The direction, not contained in three-dimensional space, in which every three-dimensional body moves, is the direction of time. By existing, every three-dimensional body moves in time, as it were, and leaves the trace of its motion in the form of a time-body, or a four-dimensional body. Because of the properties of our perceiving apparatus, we never see or sense this body; we only see its section, and this we call a three-dimensional body. Therefore, we are greatly mistaken in thinking that a three-dimensional body is something real. It is merely the projection of a four-dimensional body - its drawing, its image on our plane.
A four-dimensional body is an infinite number of three-dimensional bodies. In other words, a four-dimensional body is an infinite number of moments of existence of a three-dimensional body - of its states and positions. The threedimensional body which we see is only a figure on a cinema film, so to speak, one of a series of snapshots,
Four-dimensional space -time -is actually the distance between the forms, states and positions of one and the same body (and of different bodies, i.e. bodies which appear different to us). It separates those forms, states and positions from one another, and it also binds each one into some whole incomprehensible for us. This incomprehensible whole may be formed in time out of one physical body, or it may be formed out of different bodies.
It is easier for us to imagine such a time-'whole' if it refers to one physical body.
If we think of the physical body of a man, we shall find that, besides 'matter', there is something which, though altering, unquestionably remains the same from birth to death.
This something is the Linga Sharira of Indian philosophy, i.e. the form in which our physical body is moulded (The Secret Doctrine, H. P. Blavatsky). Eastern philosophy regards the physical body as something inconstant, something which is in a perpetual state of interchange with its surroundings. Particles come and go. The next second the body is no longer absolutely the same as it was a second earlier; today it is already quite different from what it was yesterday. After seven years it is an entirely different body. But, in spite of this, something always remains from birth to death; its aspect may change, but it remains the same. This is Linga Sharira.
Linga Sharira is the form, the image; it changes, but it remains the same. Any image of a man that we may portray to ourselves is not Linga Sharira. But if we try to form a mental picture of a man -stretched out in time, as it were - from birth to death, with all the details and features of childhood, maturity and old age, this will be Linga Sharira.
All things have form. We say that each separate thing consists of matter and form. As was already said by 'matter' we mean the causes of a long series of mixed sensations; but matter without form is not perceived by us; we cannot even think of matter without form. But we can visualize and think of form without matter.
A thing, i.e. a combination of form and matter, is never constant, it always changes in the course of time. This idea enabled Newton to evolve his theory affluents and fluxions.
Newton came to the conclusion that there are no constant magnitudes in nature. Only variable, flowing magnitudes exist -fluents. Newton named the rates of change of individual fluents, fluxions.
From the point of view of this theory all the things we know -people, plants, animals, planets - are fluents, and only differ from each other by the magnitude of their fluxions. But, while constantly changing in time, sometimes very radically and quickly, as for instance, a living body, a thing still remains the same. A man's body in youth, a man's body in old age - it is still the same body, although we know that in the old body not an atom of the young body is left.
Matter changes, but something remains the same notwithstanding all the changes. This something is Linga Sharira. Newton's theory is true for a threedimensional world existing in time. In this world nothing is constant. Everything is variable, because every moment a thing is no longer what it was before. We never see the body of Linga Sharira, we always see only its parts, and they appear to us variable. But if we look more closely, we shall see that this is an illusion. It is three-dimensional things that are unreal and variable. And they cannot be real, because, in actual fact, they do not exist, just as imaginary sections of a solid do not exist. Only four-dimensional bodies are real.
In one of his lectures collected in the book, A Pluralistic Universe, Professor James draws attention to an observation by Professor Bergson that science always studies only the t of the universe, i.e. not the universe as a whole, but only the moment, the 'time-section' of the universe.
The properties of four-dimensional space will become clearer for us if we make a detailed comparison of three-dimensional space with a surface and find out the differences that exist between them.
In his book, A New Era of Thought, Hinton examines these differences carefully. He imagines two equal right-angle triangles cut out of paper and placed on a plane surface with the right angles pointing in different directions. These triangles are exactly equal but, for some reason, they are quite different. One has its right angle pointing to the right, the other points to the left. If anyone wishes to make these triangles absolutely identical, it can only be done with the help of three-dimensional space. This means that one of the triangles must be picked up, turned over and replaced on the plane. Then they will be two equal and absolutely identical triangles. But to do this, it is necessary to lift one triangle from the plane into three-dimensional spaceand turn it over in that space. If this triangle is left on the plane, it can never be made identical with the other if, at the same time, the relation between the angles of the two triangles is to be kept. If the triangle is merely turned round on the plane, this relation will not be maintained. In our world there are figures completely analogous to these two triangles.
We know certain shapes which are equal the one to the other, which are exactly similar, and yet which we cannot make fit into the same portion of space, either practically or by imagination.
If we look at our hands we see quite clearly that our two hands are a very complicated case of non-symmetrical likeness. They are at the same time alike and quite different. One is right, the other is left. We can imagine only one way in which the two hands may be brought into complete likeness.
If we take the right-hand glove and the left-hand glove, they will not fit any more than the right hand will coincide with the left hand. But if we turn one glove inside out, then it will fit. Now, to suppose the same thing done with the solid hand as is done with the glove when it is turned inside out, we must suppose it, so to speak pulled through itself. ... If such an operation were possible, the right hand would be turned into an exact model of the left hand.*
But such an operation would be possible only in higher-dimensional space, just as the turning over of the triangle is possible only in a space higher than the plane. It is possible that, even granting the existence of four-dimensional space, a hand cannot be turned inside out and pulled through itself for reasons not dependent on geometrical conditions. But the example still holds good. Theoretically, things in the nature of the turning of a hand inside out should be possible in four-dimensional space, for in that space different, even very far removed points of our space and time should come into contact or be able to come into contact. All the points of a sheet of paper spread out on a table are separated from one another. But, if we lift the sheet off the table, we can fold it so as to bring any points we like into contact. If on one corner we write 'St Petersburg' and on another 'Madras', this will not prevent us from folding these corners together. Or, if on one corner the year 1812 is written, and on another the year 1912, these corners can also be made to touch. If the year on one corner is written in red ink and the ink is not yet dry, the figures may get imprinted on another corner. Then, if the sheet is once more opened out and placed on the table, to a man who does not know that it can be lifted off the table and folded in many different ways, it will appear quite incomprehensible how a figure on one corner could become imprinted on another corner. The possibility of any contact between distant points of the sheet will be incomprehensible for him and will remain incomprehensible for him so long as he thinks of the sheet in two-dimensional space only. As soon as he imagines the sheet in three-dimensional space, this possibility will become real and obvious for him.
* C. H. Hinton, A New Era of Thought, London, George Allen & Unwin, 1910, p. 44.

Examining the relation of the fourth dimension to the three dimensions known to us, we must admit that our geometry is obviously inadequate for the investigation of higher space.
It was pointed out earlier that a four-dimensional body is incommensurable with a three-dimensional one, just as a year is incommensurable with St Petersburg.
It is quite clear why this is so. A four-dimensional body consists of an infinitely great number of three-dimensional bodies; therefore, they can have no common measure. In comparison with a four-dimensional body, a threedimensional body is analogous to a point as compared with a line.
And, as a point is incommensurable with a line, as a line is incommensurable with a surface, as a surface is incommensurable with a solid - so a three-dimensional body is incommensurable with a four-dimensional one.
It is also clear why the geometry of three dimensions is not sufficient to define the position of the domain of the fourth dimension in relation to threedimensional space. Just as in one-dimensional geometry, i.e. on a line, it is impossible to define the position of the surface of which the given line is a side; just as on the surface - two-dimensional geometry -it is impossible to define the position of the solid of which the given surface is a side, so in threedimensional geometry, in three-dimensional space, it is impossible to define four-dimensional space. Putting it briefly, as planimetry is inadequate for the study of questions of stereometry, so stereometry is inadequate for the studyof four-dimensional space.
As a deduction from everything that has been said, it may be repeated that each point of our space is a cross-section of a line of a higher space, or as Riemann put it: the material atom is the entry of the fourth dimension into three-dimensional space.
In order to come nearer to this problem of higher dimensions and higher space it is first of all necessary to understand the essence of the domain of higher dimensions and its properties as compared with the domain of three dimensions. Only then will it be possible to investigate this domain more precisely and find out the laws which operate in it.
What is it that we have to understand?
It seems to me that, before anything else, it is necessary to understand that here it is not a question of two spatially different domains — or of two domains, one of which (again spatially, 'geometrically') constitutes a part of the other -but of two modes of perception of the same one world of one space.
Further, it is necessary to understand that all the objects known to us exist not only in the categories in which we perceive them, but in an infinite number of others in which we do not know, or are unable to know, how to sense them. So first of all we must learn to think of things in other categories, then represent them to ourselves as far as we can in these other categories. Then and then only we may develop the capacity for perceiving things in higher space, and of sensing 'higher space' itself.
Or, perhaps, the first thing required is a direct perception of everything in the surrounding world that is not included within the framework of three dimensions, that exists outside the category of time and space -everything, therefore, that we are accustomed to regard as non-existent. It variability is a sign of the three-dimensional world, we must seek for that which is constant, and in this way we may come closer to an understanding of the fourdimensional world. Moreover, we are accustomed to regard as really existingonly that which can be measured in length, breadth and height. But, as has been pointed out already, it is necessary to widen the boundaries of the reallyexisting. Mensurability is too crude a criterion of existence, because mensurability itself is too conditioned a concept. So we may say that any approach to an exact investigation of the domain of higher dimensions probably requires the conviction, derived from direct sensation, that manythings that cannot be measured have a real existence, more real indeed than many things that can be measured.



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