CHAPTER VI

HIGHER SPACE AND PHYSICAL SCIENCE.

In an earlier chapter I defined a valid hypothesis as one which explained at least _some_ of the observed facts and did not contradict any of them.

Since then I have been trying to show that the Higher Space ideas do throw a certain amount of light on quite a number of difficulties and enable us to clear up certain anomalies and dilemmas which seem to be insoluble without its aid.

We must now consider rather more definitely than we have hitherto done whether there is any thing in the hypothesis to conflict with those established conclusions of scientists which are the nearest approach we have to absolute certainties. I think we shall find not only that there is no such conflict but that there are here and there distinct indications that the higher space ideas may some day find applications in the exegesis of even the most strictly physical sciences.

These indications are admittedly very nebulous at present, it may be that they are all illusory and as will appear later they cannot _all_ lead to anything, for some are mutually exclusive.

I do not propose to express any very definite opinions on their comparative values but shall simply state them and leave it to my readers to decide what they are worth.

It must be remembered throughout that we cannot expect to find any very definite indications of the existence of higher space as a reality for the simple reason that physical science is concerned solely with those phenomena of matter and force which are "_ex hypothesi_" essentially three-dimensional.

It is worth noting at the outset that physical scientists have evinced no especial hostility to the concept of the fourth dimension, as such, however much they may have opposed to the more definitely Psychic researches which I, personally, believe to be closely associated with it.

Lord Kelvin, for instance, saw in it nothing repugnant to scientific thought and professed himself quite willing to adopt it should such a course seem to be indicated by the evidence. Another distinguished physicist has gone so far as to evolve a theory of "ether squirts" from the direction of the fourth dimension in connection with the ultimate constitution of matter.

Again M. Poincaré the distinguished French Physicist has said "The characteristic property of space, that of having three dimensions is only ... a property residing, so to speak, in human intelligence."

Mathematical physicists also find that certain experimental anomalies are resolved if they refer phenomena to four interchangeable axes involving homogeneous co-ordinates instead of to three space axes and one time axis. If this is not dealing in four-dimensional space it is first cousin to it.

M. Poincaré also pointed out that the postulates of Euclid are not experimentally verifiable facts and as a matter of fact much work has been done in the elaboration of non-Euclidean geometries. This is too mathematical a subject to be dealt with in detail here, but I can indicate the general drift of it, so far as it is relevant to the present discussion by means of the time honoured analogy of the two-dimensional world.

Most of my readers will know what are meant by the terms "latitude" and "longitude" and that the lines of longitude are "great circles" which pass through the poles and cut the earth's equator at right angles. It is also a matter of common knowledge that if on a plane surface two lines are drawn each of which cuts another line at right angles these two lines will be parallel--that is to say they will never meet however far they may be produced. This holds good provided that the surface in which they are drawn is truly plane--_i.e._, flat. But it breaks down, as we see in the case of the "great circles" of longitude, if the lines are drawn on a sphere. Now imagine two-dimensional beings, having no conception of the existence of a third dimension, living on the surface of a very large sphere. They might discover this principle about parallel lines and all would go well until they began making measurements over very large distances. Then their Geometry would begin to go wrong. They would find that lines drawn in their surface which ought not to meet however far produced would begin to show a tendency to do so. This would be an indication to them that there was such a thing as a third dimension of space and that their two-dimensional world was curved in this third dimension.

Now if a two-dimensional space can be curved in three dimensions there is no sort of reason why three-dimensional space should not be curved in four and in a precisely similar way three-dimensional geometry would, if such were the case, begin to "go wrong" where very large measurements were involved. Now, the largest measurements we ever make are astronomical measurements and as a matter of fact, according to Mr. Bragdon, there does seem to be a tendency for Geometry to go wrong in certain cases. He says that the number of negative parallaxes of stars is larger than would be expected having regard to the probable experimental errors. The parallax of an object is the angle which it subtends at two different points of observation, and so long as it is at a finite distance from these two points--which in the case of a star are the two opposite ends of the earth's orbit--this angle must be positive. That is to say the lines drawn in the observed direction of the star from the two points must converge.

If, as in certain cases seems to happen, they _diverge_, then one of three things must be the case; either the observations are wrong or else light does not, as is commonly believed, travel in straight lines (for after all what we call a straight line in astronomy is only the path of a ray of light) or else our geometry is breaking down and we must suppose that our space is curved, which would necessitate the acceptance of the existence of a fourth dimension.

It must be admitted that the explanation of negative parallaxes is more likely to be found in one or both of the two first alternatives than in the third.

Mr. Hinton has a good deal to say in his books about various four-dimensional theories of electricity involving four-dimensional vortices. These are highly ingenious but there does not seem to be any considerable reason for supposing them to be anything more and I shall therefore not describe them here. Two of his ideas however are so striking, although for different reasons, that I think a brief outline will not be out of place.

In his book "A new Era of Thought" he points out the remarkable analogy which exists between the properties of ether as postulated by physicists and those which a perfectly smooth solid sheet would present to the intelligence of two-dimensional beings living on it.

The hypothesis of the ether was introduced to account for the transmission of light, heat, electricity, and so forth, and has proved of the utmost service to physicists. Most of my readers are probably acquainted with the general idea and I need not therefore discuss it in detail.

It will be sufficient here to say that it is supposed to be a weightless, homogeneous medium extending throughout all space and permeating all bodies. Indeed Matter itself is supposed to be no more than the result of more or less complex disturbances in it.

But although it accounts for the phenomena in connection with which it was called into being it is necessary to ascribe to it very contradictory properties. On the one hand it has been calculated that in order for it to transmit the forces which we know that it does transmit, for instance the force of gravitation, it must possess a rigidity some 3,000 times greater than that of the strongest known steel. On the other hand we must suppose it to be of a tenuity far in excess of the most perfect vacuum which we can obtain, for otherwise the earth and other planets which are moving at immense speed through this medium would be slowed down; which is not in practice the case.

Now Hinton points out that to a two-dimensional being, a perfectly smooth solid sheet on the surface of which he lived would possess many of these properties. Being perfectly smooth it would be imperceptible to him and would offer no opposition to the passage of bodies over it. Yet it could, being solid, transmit vibration just as we know the ether does for us. Also it could be as rigid as you please without losing any of its imperceptibility. It could not be weighed and it could not be eliminated from any vessel no matter what care was taken to do so.

The analogy is striking but it does not appeal to me and I do not think that even Mr. Hinton means it to be taken strictly, for in other passages he gives quite different suggestions as to the ether.

One of the latter is derived from a consideration of the phenomena of rotation in four-dimensional space and is of some intrinsic interest.

In two space rotation takes place about a point, in three space about a line and we should therefore expect that in four space it would do so about a plane. This is easily shown to be the case although I do not propose to go into the proof here. The only important point is that whereas it is impossible to conceive a mass of three-dimensional spheres in a state of continuous rotation,--because they would be trying to drive each other in different directions and so would prevent the rotation,--in four dimensions this is not the case and a mass of "hyper-spheres" could be "self-driving," that is to say the rotation of each could be such as to assist and not to retard that of its neighbours. This fact is of interest because Lord Kelvin showed that the contradictory properties of the ether referred to above could only be reconciled by supposing it to be animated throughout by a motion of a vortical character.

This "self-driving" effect of rotating hyper-spheres is worth glancing at a little more closely. It arises from the fact that there are two distinct sorts of rotation which such a sphere may possess. In three-dimensional rotation the motion may take place about any axis we please and the other two axes which can be drawn will change one into the other, so to speak, as the rotation takes place. But in four-dimensional space we have four axes and while the X and Y axes change place, say, there is nothing to prevent the W and Z axes doing so too. Thus we might have the X axis changing into the Y and the W into the Z. To reverse both of these motions so as to have the Y axis changing into the X and the Z into the W does not give us a new kind of motion any more than reversing the direction of an ordinary three-dimensional rotation does--it is only equivalent to looking at it from a different point of view. But if in the case of the four-dimensional rotation we reverse one only of the two rotational components we do get a new kind of motion, and this is of interest in view of the fact that electricity like other forces is regarded as a mode of etheric motion, and if this be so there would seem to be a certain need for two distinct kinds of it in order to correspond to positive and negative electricity respectively.

It is just possible that there is some connection, as Mr. Hinton suggests, between this need and the two kinds of four-dimensional rotation referred to above.

* * * * *

Most writers on the subject of higher space make great play with the phenomena of symmetry and adduce its occurrence in nature as evidence of the existence of a fourth dimension. This view is not warranted by the facts and I shall therefore touch on it only very briefly.

The point arises in the following way. Consider the two triangles ABC and DEF in Fig. 9. If these were cut out and laid on a smooth surface exactly as shown, no amount of sliding about would enable us to fit one exactly over the other. In order to do this it would be necessary to pick one up out of the plane of the paper and turn it over. In a precisely similar manner two asymmetrical three-dimensional objects such as a right and left hand, each of which is the mirror image of the other, could not be made to coincide unless one of them were to be turned over in four-dimensional space. The point made by Mr. Hinton and other writers who attach importance to the phenomena of symmetry, is that there seems to be a general tendency in nature towards a right and left handed symmetry in which the whole organism is symmetrical about a central plane, each half being the mirror image of the other and that this symmetry is unlikely to have arisen through equal increments on either side of the central plane. They suppose as an alternative that "the ultimate elements of living matter" are not right and left handed _ab initio_, but become so by virtue of some of them being "folded over" in four-dimensional space.

This view seems to me to lack foundation especially in view of the fact that the work of Le Bel and Van't Hoff fully cleared up the analogous phenomena in the case of crystals without introducing the concept of higher space at all. In general therefore I agree with Schubert who says:--

" ... the only inference we can here make is that the idea of a four-dimensioned space is competent, from a mathematical point of view, to throw some light on the phenomena of symmetry."

(Mathematical Essays, p. 91.)

None the less Bragdon is right in his contention that "Could it be shown that the two-dimensional symmetry in nature is the result of a three dimensional movement, the right and left-handed symmetry of solids would by analogy be the result of a four-dimensional movement."

I need hardly say that if we could experimentally obtain the changing of an asymmetrical right-handed object into the corresponding left-handed one it would be of the very first importance as a proof of the reality of higher space.

Far more important than any of the foregoing, however, are the considerations arising from what is known as the Principle of Relativity. This subject, which has received much attention at the hands of mathematical physicists in recent years, is far too abstruse to be dealt with in detail here and a partial and popularised account would almost certainly fail to satisfy those who are not wholly ignorant of mathematical physics and would weary those who are. I propose, therefore, to dismiss it in very few words in spite of its great importance and relevance.

"The Principle of Relativity is the hypothesis that it is impossible by means of physical experiments to determine the absolute velocity of a body through space." (Cunningham "Relativity and the Electron Theory," p. 2).

We cannot, for example, determine the velocity of the earth relative to the ether.

This is of importance when we are dealing with the idea of "simultaneity"--an idea which, as we saw in Chapter IV. is closely associated with our notion of Time. For our criterion of simultaneity has in practice been based on optical communication. (Cp. Ibid, pp. 5 and 28). But it is easy to show that "the setting up of a standard of simultaneity by means of light signals is not possible until a definite velocity is assigned to the observer. Thus the hypothesis of relativity requires a reconsideration of the way in which we measure time." (Ibid, pp. 5, 28, 29).

"This again reacts on the measurement of the length of a material body, the 'distance between two points' being the distance between simultaneous positions of those points. Thus it becomes necessary also to examine the way in which we measure space. It becomes impossible to consider space and time separately; the two measures are interrelated to such an extent that Minkowski felt himself constrained to say that 'from henceforth time by itself and space by itself are mere shadows, that they are only two aspects of a single and indivisible manner of co-ordinating the facts of the physical world.'" (Ibid, pp. 5 and 6.)

When it is remembered that the Principle of Relativity is firmly established in scientific thought it will be realised that this conclusion arrived at as a result of purely physical considerations is of the very utmost importance as an independent confirmation of the general line of thought developed in the preceding pages.

I therefore feel it legitimate to claim that in so far as physical science throws any light on the subject at all its testimony is distinctly favourable.