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TERTIUM ORGANUM

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Crimson Glory
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« Reply #75 on: November 16, 2009, 12:32:31 am »

For the being living on a plane, everything above and below (if the plane be horizontal), and on the right or left (if the plane be vertical) will be existing in time, in the past and in the future: that which in reality is located outside of the plane will be regarded as non-existent, either as that which is already past, i.e., as something which has disappeared, ceased to be, will never return; or as in the future, i.e., as not existent, not manifested, as a thing in potentiality.

______

Let us imagine that a wheel with the spokes painted different colors is rotating through the plane upon which the plane being lives. To such a being all the motion of the wheel will appear as a variation of the color of the line of intersection of the wheel and the plane. The plane being will call this variation of the color of the line a phenomenon, and observing these phenomena he will notice in them a certain succession. He will know that the black line is followed by the white one, the white by the blue, the blue by the red, and so on. If simultaneously with the appearance of the white line some other phenomenon occurs—say the ringing of a

p. 66
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« Reply #76 on: November 16, 2009, 12:32:57 am »

bell—the two-dimensional being will say that the white line is the cause of that ringing. The change of the color of the lines, in the opinion of the two-dimensional being, will depend on causes lying right in his plane. Any pre-supposition of the possibility of the existence of causes lying outside of the plane he will characterize as fantastic and entirely unscientific. It will seem so to him because he will never be in a position to represent the wheel to himself, i.e., the parts of the wheel on both sides of the plane. After a rough study of the color of the lines, and knowing the order of their sequence, the plane being, perceiving one of them, say the blue one, will think that the black and the white ones have already passed, i.e., disappeared, ceased to exist, gone into the past; and that those lines which have not as yet appeared—the yellow, the green, and so on, and the new white and black ones still to come—do not yet exist, but lie in the future.

Therefore, though not conceiving the form of his universe, and regarding it as infinite in all directions, the plane being will nevertheless involuntarily think of the past as situated somewhere at one side of all, and of the future as somewhere at the other side of this totality. In such manner will the plane being conceive of the idea of time. We see that this idea arises because the two-dimensional being senses only two out of three dimensions of space; the third dimension he senses only after its effects become manifest upon the plane, and therefore he regards it as something different from the first two dimensions of space, calling it time.

______

Now let us imagine that through the plane upon which the two-dimensional being lives, two wheels with multi-colored spokes are rotating and are rotating in opposite directions. The spokes of one wheel come from above and go below; the spokes of the other come from below and go above.
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Crimson Glory
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« Reply #77 on: November 16, 2009, 12:33:33 am »

The plane being will never notice it.

He will never notice that where for one line (which he sees) there lies the past, for another line there lies the future. This thought will never even come into his head, because he will conceive of the past and the future very confusedly, regarding them as concepts, not as actual facts. But at the same time he will be firmly convinced that the past goes in one direction, and the future in another. Therefore

p. 67

it will seem to him a wild absurdity that on one side something past and something future can lie together, and on another side—and also beside these two—something future and something past. To the plane being the idea that some phenomena come whence others go, and vice versa, will seem equally absurd. He will tenaciously think that the future is that wherefrom everything comes, and the past is that whereto everything goes and wherefrom nothing returns. He will be totally unable to understand that events may arise from the past just as they do from the future.

Thus we see that the plane being will regard the changes of color of the lines lying on the plane very naively. The appearance of different spokes he will regard as the change of color of one and the same line, and the repeated appearance of the same colored spoke he will regard every time as a new appearance of a given color.

But nevertheless, having noticed periodicity in the change of the color of the lines upon the surface, having remembered the order of their appearance, and having learned to define the "time" of the appearance of certain spokes in relation to some other more constant phenomenon, the plane being will be in a position to foretell the change of the line from one color to another. Thereupon he will say that he has studied this phenomenon, that he can apply to it "the mathematical method"—can "calculate" it.

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Crimson Glory
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« Reply #78 on: November 16, 2009, 12:37:52 am »

If we ourselves enter the world of plane beings, then its inhabit-ants will sense the lines limiting the sections of our bodies. These sections will be for them living beings; they will not know from whence they appear, why they alter, or whither they disappear in such a miraculous manner. So also, the sections of all our inanimate but moving objects will seem independent living beings.

If the consciousness of a plane being should suspect our existence, and should come into some sort of communion with our consciousness, then to him we would appear as higher, omniscient, possibly omnipotent, but above all incomprehensible beings of a quite inconceivable category.

We could see his world just as it is, and not as it seems to him. We could see the past and the future; could foretell, direct, and even create events.

p. 68

We could know the very substance of things—could know what "matter" (the straight line) is, what "motion" (the broken line, the curve, the angle) is. We could see an angle, and we could see a centre. All this would give us an enormous advantage over the two-dimensional being.

In all the phenomena of the world of the two-dimensional being we could see considerably more than he sees—or could see quite other things than he.

And we could tell him very much that was new, amazing, and unexpected about the phenomena of his world, provided indeed that be could hear us and understand us.
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« Reply #79 on: November 16, 2009, 12:38:41 am »

First of all we could tell him that what he regards as phenomena—angles and curves, for instance—are properties of higher figures; that other "phenomena" of his world are not phenomena, but only "parts" or "sections" of phenomena; that what he calls "solids" are only sections of solids—and many things besides.

We would be able to tell him that on both sides of his plane (i.e., of his space or ether) lies infinite space (which the plane being calls time); and that in this space lie the causes of all his phenomena, and the phenomena themselves, the past as well as the future ones; moreover, we might add that "phenomena" themselves are not something happening and then ceasing to be, but combinations of properties of higher solids.

But we should experience considerable difficulty in explaining anything to the plane being; and it would be very difficult for him to understand us. First of all it would be difficult because he would not have the concepts corresponding to our concepts. He would lack "necessary words."

For instance, "section"—this would be for him a quite new and inconceivable word; then "angle"—again an inconceivable word; "centre"—still more inconceivable; the third perpendicular—something incomprehensible, lying outside of his geometry.
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« Reply #80 on: November 16, 2009, 12:38:58 am »

The fallacy of his conception of time would be the most difficult thing for the plane being to understand. He could never understand that that which has passed and that which is to be are existing simultaneously on the lines perpendicular to his plane. And he could never conceive the idea that the past is identical with the future, because phenomena come from both sides and go in both directions.

p. 69

But the most difficult thing for the plane being would be to conceive the idea that "time" includes in itself two ideas: the idea of space, and the idea of motion upon this space.

We have shown that what the two-dimensional being living on the plane calls motion has for us a quite different aspect.

In his book The Fourth Dimension, under the heading "The First Chapter in the History of Four-space," Hinton writes:

Parmenides, and the Asiatic thinkers with whom he is in close affinity, propound a theory of existence which is in close accord with a conception of a possible relation between a higher and lower dimensional space. . . . It is one which in all ages has had a strong attraction for pure intellect, and is the natural mode of thought for those who refrain from projecting their own volition into nature under the guise of causality.
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« Reply #81 on: November 16, 2009, 12:39:22 am »

According to Parmenides of the school of Elea the all is one, unmoving and unchanging. The permanent amid the transient—that foothold for thought, that solid ground for feeling, on the discovery of which depends all our life—is no phantom; it is the image amidst deception of true being, the eternal, the unmoved, the one. Thus says Parmenides.

But how is it possible to explain the shifting scene, these mutations of things?

"Illusion," answers Parmenides. Distinguishing between truth and error, he tells of the true doctrine of the one—the false opinion of a changing world. He is no less memorable for the manner of his advocacy than for the cause he advocates.

Can the mind conceive a more delightful intellectual picture than that of Parmenides pointing to the one, the true, the unchanging, and yet on the other hand ready to discuss all manner of false opinion! . . .

In support of the true opinion he proceeded by the negative way of showing the self-contradictions in the ideas of change and motion. . . . To express his doctrine in the ponderous modern way we must make the statement that motion is phenomenal, not real.

Let us represent his doctrine.

Imagine a sheet of still water into which a slanting stick is being lowered with a motion vertically downward. Let 1, 2, 3 (Fig. 1) be three consecutive positions of the stick. A, B, C will be three connective positions of the meeting of the stick with the surface of the water. As the stick passes down, the meeting will move from A on to B and C.

Suppose now all the water to be removed except a film. At the meeting of the film and the stick there will be an interruption of the film. If we suppose the film to have a property, like that of a soap bubble, of closing up round any penetrating object, then as the stick goes vertically down-ward the interruption in the film will move on. If we pass a spiral through the film the intersection will give a point moving in a circle (shown by the dotted lines in Fig. 2).

p. 70

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.
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« Reply #82 on: November 16, 2009, 12:39:53 am »

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« Reply #83 on: November 16, 2009, 12:43:08 am »

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« Reply #84 on: November 16, 2009, 12:43:28 am »

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. 1

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.
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« Reply #85 on: November 16, 2009, 12:43:54 am »

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."

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« Reply #86 on: November 16, 2009, 12:44:25 am »

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 two-dimensional 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.
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« Reply #87 on: November 16, 2009, 12:45:13 am »

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

p. 72

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.
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« Reply #88 on: November 16, 2009, 12:45:37 am »

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 phenomena.

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.
Footnotes

70:1 C. H. Hinton, "The Fourth Dimension," pp. 23, 24 and 25.

http://www.sacred-texts.com/eso/to/to09.htm
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