Hinton - The Fourth Dimension.pdf
Hinton - The Fourth Dimension.pdf Hinton - The Fourth Dimension.pdf
28 THE FOURTH DIMENSION Transferring our conceptions to those of an existence in a higher dimensionality traversed by a space of conciousness, we have an illustration of a thought which has found frequent and varied expression. When, however, we ask ourselves what degree of truth there lies in it, we must admit that, at far as we can see, it is merely symbolical. The true path in the investigation of a higher dimensionality lies in another direction. The significance of the Parmenidean doctrine lies in this: that here, as again and again, we find that those conceptions which man introduces of himself, which he does not derive from the mere record of his outward experience, have a striking and significant correspondence to the conception of a physical existence in a world of a higher space. How close we come to Parmenides’ thought by this manner of representation it is impossible to say. What I want to point out is the adequateness of the illustration, not only to give a static model of his doctrine, but one capable as it were, of a plastic modification into a correspondence into kindred forms of thought. Either one of two things must be true—that four-dimensional conceptions give a wonderful power of representing the thought of the East, or that the thinkers of the East must have been looking at and regarding four-dimensional existence. Coming now to the main stream of thought we must dwell in some detail on Pythagoras, not because of his direct relation to the subject, but because of his relation to the investigators who come later. Pythagoras invented the two-way counting. Let us represent the single-way counting by the points aa, ab, ac, ad, using these pairs of letters instead of the numbers 1, 2, 3, 4. I put an a in each case first for a reason which will immediately appear. We have a sequence and order. There is no conception of distance necessarily involved. The difference
THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 29 between the posits is one of order not of distance-- only when identified with a number of equal material things in juxtaposition does the notion of distance arise. Now, besides the simple series I can have, starting from aa, ba, ca, da, from ab, bb, cb, db, and so on, and forming a scheme: da db dc dd ca cb cc cd ba bb bc bd aa ab ac ad This complex or manifold gives a two-way order. I can represent it by a set of points, if I am on my guard against assuming any relation of distance. Pythagoras studied this two-fold way of counting on reference to material bodies, and discovered that most remarkable property of the combination of number and matter that Fig. 15. bears his name. The Pythagorean property of an extended material system can be exhibited in a manner which will be of use to us afterwards, and which therefore I will employ now instead of using the kind of figure which he himself employed. Consider a two-fold field of points arranged in regular rows. Such a field will be presupposed in the following argument. It is evident that in fig. 16 four of the points determine a square, which square we may take as the 1 2 Fig. 16. unit of measurement for areas. But we can also measure areas in another way. Fig. 16 (1) shows four points determining a square. But four squares also meet in a point, fig. 16 (2). Hence a point at the corner of a square belongs equally to four squares.
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THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 29<br />
between the posits is one of order not of distance--<br />
only when identified with a number of equal material<br />
things in juxtaposition does the notion of distance arise.<br />
Now, besides the simple series I can have, starting from<br />
aa, ba, ca, da, from ab, bb, cb, db, and so on, and forming<br />
a scheme:<br />
da db dc dd<br />
ca cb cc cd<br />
ba bb bc bd<br />
aa ab ac ad<br />
This complex or manifold gives a two-way order. I can<br />
represent it by a set of points, if I am on my guard<br />
against assuming any relation of distance.<br />
Pythagoras studied this two-fold way of<br />
counting on reference to material bodies, and<br />
discovered that most remarkable property of<br />
the combination of number and matter that<br />
Fig. 15.<br />
bears his name.<br />
<strong>The</strong> Pythagorean property of an extended material<br />
system can be exhibited in a manner which will be of<br />
use to us afterwards, and which therefore I will employ<br />
now instead of using the kind of figure which he himself<br />
employed.<br />
Consider a two-fold field of points arranged in regular<br />
rows. Such a field will be presupposed in the following<br />
argument.<br />
It is evident that in fig. 16 four<br />
of the points determine a square,<br />
which square we may take as the<br />
1 2<br />
Fig. 16.<br />
unit of measurement for areas.<br />
But we can also measure areas<br />
in another way.<br />
Fig. 16 (1) shows four points determining a square.<br />
But four squares also meet in a point, fig. 16 (2).<br />
Hence a point at the corner of a square belongs equally<br />
to four squares.
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