For the plane being such a point, moving in a circle in its plane, would probably constitute a cosmical phenomenon, something like the motion of a planet in its orbit.

Suppose now the spiral to be still and the film to move vertically upward, the whole spiral will be represented in the film in the consecutive positions of the point of intersection.

If instead of one spiral we take a complicated construction consisting of spirals, inclined and straight lines, broken and curved lines, and if the film move vertically upward we shall have an entire universe of moving points the movements of which will appear to the plane being as original. The plane being will explain these movements as depending one upon another, and indeed he will never happen to think that these movements are fictitious and are dependent upon the spirals and other lines lying outside his space.*

Returning to the plane being and his perception of the world, and analyzing his relations to the three-dimensional world, we see that for the two-dimensional or plane being it will be very difficult to understand all the complexity of the phenomena of our world, as it appears to us. He (the plane being) is accustomed to perceive the world as being too simple.

Taking into consideration the sections of figures instead of the figures themselves, the plane being will compare them in relation to their length and their greater or lesser curvature, i. e., their for him more or less rapid motion.

The differences between the objects of our world, as they exist for us he would not understand. The functions of the objects of our world would be completely mysterious to his mind—incomprehensible, "supernatural."

* C. H. Hinton, "The Fourth Dimension," pp. 23, 24 and 25.

THE GULF BETWEEN

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Let us imagine that a coin, and a candle the diameter of which is equal to that of the coin, are on the plane upon which the twodimensional being lives. To the plane being they will seem two equal circles, i. e., two moving, and absolutely identical lines; he will never discover any difference between them. The functions of the coin and of the candle in our world—these are for him absolutely a terra incognita. If we try to imagine what an enormous evolution the plane being must pass through in order to understand the function of the coin and of the candle and the difference between these functions, we shall understand the nature of the division between the plane world and the world of three dimensions, and the complete impossibility of even imagining, on the plane, anything at all like the three-dimensional world, with its manifoldness of function.

The properties of the phenomena of the plane world will be extremely monotonous; they will differ by the order of their appearance, their duration, and their periodicity. Solids, and the things of this world will be flat and uniform, like shadows, i. e., like the shadows of quite different solids, which seem to us uniform. Even if the plane being could come in contact with our consciousness, he would never be in a position to understand all the manifoldness and richness the phenomena of our world and the variety of function of the things of that world.

Plane beings would not be in a position to master our most ordinary concepts.

It would be extremely difficult for them to understand that phenomena, identical for them, are in reality different; and on the other hand, that phenomena quite separate for them are in reality parts of one great phenomenon, and even of one object or one being.

This last will be one of the most difficult things for the plane being to understand. If we imagine our plane being to be inhabiting a horizontal plane, intersecting the top of a tree, and parallel to the surface of the earth, then for such a being each of the various sections of the branches will appear as a quite separate phenomenon or object. The idea of the tree and its branches will never occur to him.

Generally speaking, the understanding of the most fundamental and simple things of our world will be infinitely long and difficult

to the plane being. He would have to entirely reconstruct his concepts of space and time. This would be the first step. Unless it is taken, nothing is accomplished. Until the plane being shall imagine all our universe as existing in time, i. e., until he refers to time everything lying on both sides of his plane, he will never understand anything. In order to begin to understand "the third dimension" the inhabitant of the plane must conceive of his time concepts spatially, that is, translate his time into space.

To achieve even the spark of a true understanding of our world he will have to reconstruct completely all his ideas—to revaluate all values, to revise all concepts, to dissever the uniting concepts, to unite those which are dissevered; and, what is most important, to create an infinite number of new ones.

If we put down the five fingers of one hand on the plane of the two-dimensional being they will be for him five separate phe

nomena.

Let us try to imagine what an enormous mental evolution he would have to undergo in order to understand that these five separate phenomena on his plane are the finger-tips of the hand of a large, active and intelligent being—man.

To make out, step by step, how the plane being would attain to an understanding of our world, lying in the region of the to him mysterious third dimension—i. e., partly in the past, partly in the future would be interesting in the highest degree. First of all, in order to understand the world of three dimensions, he must cease to be two-dimensional—he must become three-dimensional himself, or in other words he must feel an interest in the life of three-dimensional space. After having felt the interest of this life, he will by so doing transcend his plane, and will never be in a position thereafter to return to it. Entering more and more within the circle of ideas and concepts which were entirely incomprehensible to him before, he will have already become, not two-dimensional, but three-dimensional. But all along the plane being will have been essentially three-dimensional, that is, he will have had the third dimension, without his being conscious of it himself. To become three-dimensional he must be three-dimensional. Then as the end of ends he can address himself to the self-liberation from the illusion of the two-dimensionality of himself and the world, and to the apprehension of the three-dimensional world.

CHAPTER VII

The impossibility of the mathematical definition of dimensions. Why does not mathematics sense dimensions? The entire conditionality of the representation of dimensions by powers. The possibility of representing all powers on a line. Kant and Lobachevsky. The difference between non-Euclidian geometry and metageometry. Where shall we find the explanation of the three-dimensionality of the world, if Kant's ideas are true? Are not the conditions of the three-dimensionality of the world confined to our receptive apparatus, to our psyche?

NR

OW that we have studied those "relations which our space itself bears within it" we shall return to the questions: But what in reality do the dimensions of space represent —and why are there three of them?

The fact that it is impossible to define three-dimensionality mathematically must appear most strange.

We are little conscious of this, and it seems to us a paradox, because we speak of the dimensions of space, but it remains a fact that mathematics does not sense the dimensions of space.

The question arises, how can such a fine instrument of analysis as mathematics not feel dimensions, if they represent some real properties of space?

Speaking of mathematics, it is necessary to recognize first of all, as a fundamental premise, that correspondent to each mathematical expression is always the relation of some realities.

If there is no such a thing, if it be not true then there is no mathematics. This is its principal substance, its principal contents. To express the correlations of magnitudes is the problem of mathematics. But these correlations must be between something. Instead of algebraical a, b and c it must be possible to substitute some reality. This is the ABC of all mathematics; a, b and c are credit bills; they can be good ones only if behind them there is a real something, and they can be counterfeited if behind them there is no reality whatever.

"Dimensions" play here a very strange role. If we designate them by the algebraic symbols a, b and c, they have the character of counterfeit credit bills. For this a, b and c it is impossible to substitute any real magnitudes which are capable of expressing the correlations of dimensions.

Usually dimensions are represented by powers: the first, the second, the third; that is, if a line is called a, then a square, the sides of which are equal to this line, is called a2, and a cube, the face of which is equal to this square, is called a3.

This among other things gave Hinton the foundation on which he constructed his theory of tesseracts, four-dimensional solids—a*. But this is pure fantasy. First of all, because the representation of "dimensions" by powers is entirely conditional. It is possible to represent all powers on a line. For example, take the segment of a line equal to five millimetres; then a segment equal to twentyfive millimetres will be the square of it, i. e., a2 and a segment of one hundred and twenty-five millimetres will be the cube—a3.

How shall we understand that mathematics does not feel dimensions that it is impossible to express mathematically the difference between dimensions?

It is possible to understand and explain it by one thing only— namely, that this difference does not exist.

We really know that all three dimensions are in substance identical, that it is possible to regard each of the three dimensions either as following the sequence, the first, the second, the third, or the other way about. This alone proves that dimensions are not mathematical magnitudes. All the real properties of a thing can be expressed mathematically as quantities, i. e., numbers, showing the relation of these properties to other properties.

But in the matter of dimensions it is as if mathematics sees more than we do, or farther than we do, through some boundaries which arrest us but not it—and sees that no realities whatever correspond to our concepts of dimensions.

If the three dimensions really corresponded to three powers, then we should have the right to say that only these three powers refer to geometry, and that all the other higher powers, beginning with the fourth, lie beyond geometry.