IN STARRY SKIES

Evening Post, Volume CXXI, Issue 20, 24 January 1936, Page 16

IN STARRY SKIES

SIGNIFICANCE OF "FOUR DIMENSIONAL GEOMETRY"

(By "Omega Centauri.")

Fifteen or twenty years ago we were all eagerly reading the so-called popular books on relativity. Those of us whose ideas became more or less fixed in early Victorian days did not regard some aspects of the new presentation of fundamental facts in an altogether kindly spirit. The disrespectful references to the work of Newton and, in fact, of all who thought or wrote before the dawn of the twentieth century, aroused some feelings of resentment. I have re-read some of these books today, and the old question marks and exclamation signs in the margin still hold good. All was plain sailing up to a certain point and then the trouble began with startling suddenness. We had long realised that everything is in motion, that we do not know of any fixed point or fixed direction in space, that we can study the motion of a particular.body only in relation to that of others. The fact that light takes a finite time to travel any particular distance was, of course, well known, and as a consequence it was clear that we could not observe distant objects as they are at the moment of observation. Up to a certain point the books dealt with what seemed to be self-evident facts. Owing to the finite velocity of light, since radiation is our only method of receiv-

ing information from the distant parts of the universe, it is inevitable that we must get our measurements of time and space to a certain extent involved with one another. If we agree with certain assumptions the formulae which enable us to allow for relative velocities in parallel lines follow readily.

But at this point, suddenly, without •warning, the blow fell. • We found we ■were expected to believe, not that space and time become mixed in our measurements, but that neither space nor time independently exists, that there is only a space-time continuum that is finite though unbounded, and strangely curved. No proof whatever was given. A few quotations from a book published in 1920 will illustrate the position. "Light then does not travel in straight lines, as was formerly suppose^, but in curves, being much knocked about by various planetary systems whose spheres of ' influence it invades. Despite Euclid and geometry, two parallel lines probably meet long before infinity is reached." "Newton was not wholly in the wrong; he was only approximately right." "Minkowski and Einstein picture time as the fourth dimension. To them time occupies no more important position than length, breadth, or thickness, and is as intimately related to these three as the three are to one another.'.' "This force (gravitation) brings about a distortion .or strain' in- world • lines;1 or,' what amounts to the same thing, a distortion or strain of time and space." "At this point Newton's conceptions fail," for his views and his laws do not include, strains, in space."

Now, in the November number of "Scientia" there is an extremely ■: in-: terestjng. article on the "Signifiance of Four-dimensional Geometry " by H. V. Metcalf.^of College Hill, Clinton, New York. ;If this had been written fifteen years ago it would have saved a great deal of misconception. Mr. Metcalf writes as one who.fully appreciates, the value of-the new extensions of knowledge, but he ' shows clearly that writers on four-dimen-sional geometry use common words with quite unusual meanings, and he concludes that "Four-dimensional space is a purely • abstract mathematical creation, having no physical existence and . expressing no possible concrete concept." He starts by asking: "Are the unquestionably valid and useful concepts'of four-dimensional geometry inconsistent with our intuition of three-dimensional space as including every position that can exist in the universe." 2e approaches the subject by considering the equation XY/Z equals: R where X, V, and Z are variables, and R a constant. A certain geometrical figure, occupying three dimensions, affords one concrete illustration of the relations expressedby this equation; X, V, and Z being the three rectangular co-ordinates of every point in the figure. But another entirely distinct case in which the equation holds good is afforded by the behaviour of an ideal gas, X, representing its volume, V its pressure, and Z its absolute temperature. Neither of these concrete illustrations has any more fundamental relation

to the law expressed by the equation than the other. Similarly various concrete illustrations might be found in which four variables obey the law expressed by an equation. "But no illustration of this law can be drawn from geometry, since geometrical space involves only three dimensions, and therefore furnishes only three variables for our equation." But an equation involving four or more variables may bear an analogy in form to an equation which involves only three, and in such cases words, which strictly apply only to the geometrical ideas, are used in relation to the former also. In four dimensional geometry the words space, dimension, perpendicular, figure, etc., are used in senses quite distinct from the ordinary geometrical meaning. After explaining the position very fully and carefully Mr. Metcalf remarks: "It seems unfortunate that mathematicians have done so much redefining the words in accordance with suggested analogies, instead of coining new words for important new concepts. He makes it I perfectly clear that a feur-dimensional figure is a pure abstraction, artificially defined, which corresponds to no possible concrete geometrical figure. "Geometry and four-dimensional geometry are quite distinct in their nature. Geometry is a concrete branch of mathematics dealing with concrete quantities only. Four-dimensional geometry is purely analytical in its nature, founded on definitions that are pure mathematical abstractions to which no concrete geometrical concepts can possibly be attached. "We hear occasionally of a man try-

Ing to picture in his imagination the configuration of a four-dimensional figure, and thinking that he has almost succeeded.' This seems to be essentially absurd. It seems to indicate a failure to grasp the real significance of four-dimensional geometry. There is no form to be imagined, only definitions to be understood."

Talking of space that is curved and finite though boundless, and discussing whether space is positively or negotively 'curved, when applying these statements to external physical space, seems to be using words without expressing any possible meaning. The unfortunate thing is that people are apt to forget that the origin and true meaning of the terms are such that they "do not apply to space in the geometrical and tuitional sense of the word, that is to pure extension, the space in which all physical measurements are made, and-in which all physical phenomena occur." With regard to time being used as a fourth dimension Metcalf says: "When rightly interpreted it has 'a real meaning. In polydimensional geometry the fourth dimension means nothing more or less than a fourth variable entering into our equations. Now in certain equations of motion, time and the three dimensions of space enter symmetrically as four independent variables. This makes it.cd* yenient to handle the sub j ect by the methods: of four-dimensional geometry, dealing with time as one of the dimensions. But to argue from this that time and space are alike in their ultimate nature is a curious lapse of clear thought." The author concludes his article by pointing out that there is no conflict between the mathematical concept of the fourth dimension of space and our intuition of space as having only three, possible dimensions. They are simply dealing with two different things. Each is sound in its own field and not at all in conflict with the other in its field:

; Of course the arguments in this paper apply equally to polydimensional geometry. A full page of quotations is addea to show that many writers make it clear that the space they,treat of is a mathematical abstraction. From these^quotations we take the following sentences: — "We should regard the n-dimensional space as a mere convenience, we should not give it a physical significance. .'. . When we speak of surfaces and waves in n-dimensions we speak by analogy." Flint, Wave Mechanics. . .' "In using these terms we do not propose even to raise the question whether in any geometrical sense there is.such a thing as space of more than three dimensions."—Bocher, Higher Algebra. "In the synthetic study of. ' fourdimensional geometry we'are forced to give up intuition and rely entirely on logic."—Manning,.Geometry of Four i Dimensions.

"It is convenient to speak of such regions by analogy with space of two or three dimensions; but the foundation for the definition of such a region must be sought in an analytic formulation."— Osgood, Advanced Calculus.