SPACE RELATIONS

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We learn what cannot be in four-dimensional space, and this permits us to set forth what can be there.

In his book, The Fourth Dimension, Hinton makes an interesting statement concerning the method by which we may approach the problem of higher dimensions. He says:

Our space itself bears within it relations through which we can establish relations to other (higher) spaces.

For within space are given the conception of point and line, line and plane, which really involve the relation of space to a higher space.

Let us consider these relations within our space, and see what conclusions we can derive from their investigation.

We know that our geometry regards the line as a tracing of the movement of a point; the surface as a tracing of the movement of a line; and the solid as a tracing of the movement of a surface. On these premises we put to ourselves this question: Is it not possible to regard the "four-dimensional body" as a tracing of the movement of a three-dimensional body?

But what is this movement, and in what direction?

The point, moving in space, and leaving the tracing of its movement, a line, moves in a direction not contained in it, because in a point there is no direction whatsoever.

The line, moving in space, and leaving the tracing of its movement, the surface, moves in a direction not contained in it because, moving in a direction contained in it, a line will continue to be a line.

The surface, moving in space, and leaving a tracing of its movement, the solid, moves also in a direction not contained in it. If it should move otherwise, it would remain always the surface. In order to leave a tracing of itself as a "solid," or three-dimensional figure, it must set off from itself, move in a direction which in itself it has not.

In analogy with all this, the solid, in order to leave as the tracing of its movement, the four-dimensional figure (hypersolid) shall move in a direction not confined in it; or in other words it shall come out of itself, set off from itself, move in a direction which is not present in it. Later on it will be shown in what manner we shall understand this.

But for the present we can say that the direction of the movement in the fourth dimension lies out of all those directions which are possible in a three-dimensional figure.

We consider the line as an infinite number of points; the surface as an infinite number of lines; the solid as an infinite number of surfaces.

In analogy with this it is possible to consider that it is necessary to regard a four-dimensional body as an infinite number of threedimensional bodies, and four-dimensional space as an infinite number of three-dimensional spaces.

Moreover, we know that the line is limited by points, that the surface is limited by lines, that the solid is limited by surfaces. It is possible that a four-dimensional body is limited by threedimensional bodies.

Or it is possible to say that the line is the distance between two points; the surface the distance between two lines; the solid—between two surfaces.

Or again, that the line separates two points or several points from one another (for a straight line is the shortest distance between two points); that the surface separates two or several lines from each other; that the solid separates several surfaces one from another; as the cube separates six flat surfaces one from another—its faces.

The line binds several separate points into a certain whole (the straight, the curved, the broken line); the surface binds several lines into a certain whole (the quadrilateral, the triangle); the solid binds several surfaces into a certain whole (the cube, the pyramid).

It is possible that four-dimensional space is the distance between a group of solids, separating these solids, yet at the same time binding them into some to us inconceivable whole, even though they seem to be separate from one another.

Moreover, we regard the point as a section of a line; the line as a section of a surface; the surface as a section of a solid.

By analogy, it is possible to regard the solid (the cube, sphere, pyramid) as a section of a four-dimensional body, and our entire three-dimensional space as a section of a four-dimensional space.

If every three-dimensional body is the section of a four-dimensional one, then every point of a three-dimensional body is the

PLANE PROJECTIONS OF SOLIDS 37 section of a four-dimensional line. It is possible to regard an "atom" of a physical body, not as something material, but as an intersection of a four-dimensional line by the plane of our consciousness.

The view of a three-dimensional body as the section of a fourdimensional one leads to the thought that many (for us) separate bodies may be the sections of parts of one four-dimensional body.

A simple example will clarify this thought. If we imagine a horizontal plane, intersecting the top of a tree, and parallel to the surface of the earth, then upon this plane the sections of branches will seem separate, and not bound to one another. Yet in our space, from our standpoint, these are sections of branches of one tree, comprising together one top, nourished from one root, casting one shadow.

Or here is another interesting example expressing the same idea, given by Mr. Leadbeater, the theosophical writer, in one of his books. If we touch the surface of a table with our finger tips, then upon the surface will be just five circles, and from this plane presentment it is impossible to construe any idea of the hand, and of the man to whom this hand belongs. Upon the table's surface will be five separate circles. How from them is it possible to imagine a man, with all the richness of his physical and spiritual life? It is impossible. Our relation to the four-dimensional world will be analogous to the relation of that consciousness which sees five circles upon the table to a man. We see just "finger tips"—to us the fourth dimension is inconceivable.

We know that it is possible to represent a three-dimensional body upon a plane, that it is possible to draw a cube, a polyhedron or a sphere. This will not be a real cube or a real sphere, but the projection of a cube or of a sphere on a plane. We may conceive of the three-dimensional bodies of our space somewhat in the nature of images in our space of to us incomprehensible four-dimensional bodies.

CHAPTER IV

In what direction may the fourth dimension lie? What is motion? Two kinds of motion—motion in space and motion in time—which are contained in every movement. What is time? Two ideas contained in the conception of time. The new dimension of space, and motion upon that dimension. Time as the fourth dimension of space. Impossibility of understanding the fourth dimension without the idea of motion. The idea of motion and the "time-sense." The time-sense as a limit (surface) of the "space-sense." Hinton on the law of surfaces. The "ether" as a surface. Riemann's idea concerning the translation of time into space in the fourth dimension. Present, past, and future. Why we do not see the past and the future. Life as a feeling of one's way. Wundt on the subject of our sensuous knowledge.

W

E have established by a comparison of the relation of lower dimensional figures to higher dimensional ones that it is possible to regard a four-dimensional body as the tracing of the motion of a three-dimensional body upon the dimension not contained in it; i. e., that the direction of the motion upon the fourth dimension lies outside of all the directions which are possible in three-dimensional space.

But in what direction is it?

In order to answer this question it will be necessary to discover whether we do not know some motion not confined in three-dimensional space.

We know that every motion in space is accompanied by that which we call motion in time. Moveover, we know that everything existing, even if not moving in space, moves eternally in time.

And equally in all cases, whether speaking of motion or absence of motion, we have in mind an idea of what was before, what now becomes, and what will follow after. In other words, we have in mind the idea of time. The idea of motion of any kind, also the idea of absence of motion, is indissolubly bound up with the idea of time. Any motion or absence of motion proceeds in time and

WHAT IS TIME?

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cannot proceed out of time. Consequently, before speaking of what motion is, we must answer the question, what is time?

Time is the most formidable and difficult problem which confronts humanity.

Kant regards time as he does space: as a subjective form of our receptivity; i. e., he says that we create time ourselves, as a function of our receptive apparatus, for convenience in perceiving the outside world. Reality is continuous and constant, but in order to make possible the perception of it, we must dissever it into separate moments; imagine it as an infinite series of separate moments out of which there exists for us only one. In other words, we perceive reality as if through a narrow slit, and what we are seeing through this slit we call the present; what we did see and now do not see the past; and what we do not quite see but are expecting— the future.

Regarding each phenomenon as an effect of another, or others, and this in its turn as a cause of a third; that is, regarding all phenomena in functional interdependence one upon another, by this very act we are contemplating them in time, because we picture to ourselves quite clearly and precisely first a cause, then an effect; first an action, then its function; and cannot contemplate them otherwise. Thus we may say that the idea of time is bound up with the idea of causation and functional interdependence. Without time, causation cannot exist, just as without time, motion or the absence of motion cannot exist.

But our perception concerning our "being in time" is entangled and misty up to improbability.

First of all let us analyze our relation toward the past, present and future. Usually we think that the past already does not exist. It has passed, disappeared, altered, transformed itself into something else. The future also does not exist—it does not exist as yet. It has not arrived, has not formed. By the present we mean the moment of transition of the future into the past, i. e., the moment of transition of a phenomenon from one non-existence into another non-existence. For that moment only does the phenomenon exist for us in reality; before, it existed in potentiality, afterward it will exist in remembrance. But this short moment is after all only a fiction: it has no measurement. We have a full right to say that