CHAPTER IIITHE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE
Having now obtained the conception of a four-dimensional space, and having formed the analogy which, without any further geometrical difficulties, enables us to enquire into its properties, I will refer the reader, whose interest is principally in the mechanical aspect, to Chapters VI. and VII. In the present chapter I will deal with the general significance of the enquiry, and in the next with the historical origin of the idea.
First, with regard to the question of whether there is any evidence that we are really in four-dimensional space, I will go back to the analogy of the plane world.
A being in a plane world could not have any experience of three-dimensional shapes, but he could have an experience of three-dimensional movements.
We have seen that his matter must be supposed to have an extension, though a very small one, in the third dimension. And thus, in the small particles of his matter, three-dimensional movements may well be conceived to take place. Of these movements he would only perceive the resultants. Since all movements of an observable size in the plane world are two-dimensional, he would only perceive the resultants in two dimensions of the small three-dimensional movements. Thus, there would be phenomena which he could not explain by histheory of mechanics—motions would take place which he could not explain by his theory of motion. Hence, to determine if we are in a four-dimensional world, we must examine the phenomena of motion in our space. If movements occur which are not explicable on the suppositions of our three-dimensional mechanics, we should have an indication of a possible four-dimensional motion, and if, moreover, it could be shown that such movements would be a consequence of a four-dimensional motion in the minute particles of bodies or of the ether, we should have a strong presumption in favour of the reality of the fourth dimension.
By proceeding in the direction of finer and finer subdivision, we come to forms of matter possessing properties different to those of the larger masses. It is probable that at some stage in this process we should come to a form of matter of such minute subdivision that its particles possess a freedom of movement in four dimensions. This form of matter I speak of as four-dimensional ether, and attribute to it properties approximating to those of a perfect liquid.
Deferring the detailed discussion of this form of matter to Chapter VI., we will now examine the means by which a plane being would come to the conclusion that three-dimensional movements existed in his world, and point out the analogy by which we can conclude the existence of four-dimensional movements in our world. Since the dimensions of the matter in his world are small in the third direction, the phenomena in which he would detect the motion would be those of the small particles of matter.
Suppose that there is a ring in his plane. We can imagine currents flowing round the ring in either of two opposite directions. These would produce unlike effects, and give rise to two different fields of influence. If thering with a current in it in one direction be taken up and turned over, and put down again on the plane, it would be identical with the ring having a current in the opposite direction. An operation of this kind would be impossible to the plane being. Hence he would have in his space two irreconcilable objects, namely, the two fields of influence due to the two rings with currents in them in opposite directions. By irreconcilable objects in the plane I mean objects which cannot be thought of as transformed one into the other by any movement in the plane.
Instead of currents flowing in the rings we can imagine a different kind of current. Imagine a number of small rings strung on the original ring. A current round these secondary rings would give two varieties of effect, or two different fields of influence, according to its direction. These two varieties of current could be turned one into the other by taking one of the rings up, turning it over, and putting it down again in the plane. This operation is impossible to the plane being, hence in this case also there would be two irreconcilable fields in the plane. Now, if the plane being found two such irreconcilable fields and could prove that they could not be accounted for by currents in the rings, he would have to admit the existence of currents round the rings—that is, in rings strung on the primary ring. Thus he would come to admit the existence of a three-dimensional motion, for such a disposition of currents is in three dimensions.
Now in our space there are two fields of different properties, which can be produced by an electric current flowing in a closed circuit or ring. These two fields can be changed one into the other by reversing the currents, but they cannot be changed one into the other by any turning about of the rings in our space; for the disposition of the field with regard to the ring itself is different when weturn the ring, over and when we reverse the direction of the current in the ring.
As hypotheses to explain the differences of these two fields and their effects we can suppose the following kinds of space motions:—First, a current along the conductor; second, a current round the conductor—that is, of rings of currents strung on the conductor as an axis. Neither of these suppositions accounts for facts of observation.
Hence we have to make the supposition of a four-dimensional motion. We find that a four-dimensional rotation of the nature explained in a subsequent chapter, has the following characteristics:—First, it would give us two fields of influence, the one of which could be turned into the other by taking the circuit up into the fourth dimension, turning it over, and putting it down in our space again, precisely as the two kinds of fields in the plane could be turned one into the other by a reversal of the current in our space. Second, it involves a phenomenon precisely identical with that most remarkable and mysterious feature of an electric current, namely that it is a field of action, the rim of which necessarily abuts on a continuous boundary formed by a conductor. Hence, on the assumption of a four-dimensional movement in the region of the minute particles of matter, we should expect to find a motion analogous to electricity.
Now, a phenomenon of such universal occurrence as electricity cannot be due to matter and motion in any very complex relation, but ought to be seen as a simple and natural consequence of their properties. I infer that the difficulty in its theory is due to the attempt to explain a four-dimensional phenomenon by a three-dimensional geometry.
In view of this piece of evidence we cannot disregard that afforded by the existence of symmetry. In this connection I will allude to the simple way of producingthe images of insects, sometimes practised by children. They put a few blots of ink in a straight line on a piece of paper, fold the paper along the blots, and on opening it the lifelike presentment of an insect is obtained. If we were to find a multitude of these figures, we should conclude that they had originated from a process of folding over; the chances against this kind of reduplication of parts is too great to admit of the assumption that they had been formed in any other way.
The production of the symmetrical forms of organised beings, though not of course due to a turning over of bodies of any appreciable size in four-dimensional space, can well be imagined as due to a disposition in that manner of the smallest living particles from which they are built up. Thus, not only electricity, but life, and the processes by which we think and feel, must be attributed to that region of magnitude in which four-dimensional movements take place.
I do not mean, however, that life can be explained as a four-dimensional movement. It seems to me that the whole bias of thought, which tends to explain the phenomena of life and volition, as due to matter and motion in some peculiar relation, is adopted rather in the interests of the explicability of things than with any regard to probability.
Of course, if we could show that life were a phenomenon of motion, we should be able to explain a great deal that is at present obscure. But there are two great difficulties in the way. It would be necessary to show that in a germ capable of developing into a living being, there were modifications of structure capable of determining in the developed germ all the characteristics of its form, and not only this, but of determining those of all the descendants of such a form in an infinite series. Such a complexity of mechanical relations, undeniable though it be, cannotsurely be the best way of grouping the phenomena and giving a practical account of them. And another difficulty is this, that no amount of mechanical adaptation would give that element of consciousness which we possess, and which is shared in to a modified degree by the animal world.
In those complex structures which men build up and direct, such as a ship or a railway train (and which, if seen by an observer of such a size that the men guiding them were invisible, would seem to present some of the phenomena of life) the appearance of animation is not due to any diffusion of life in the material parts of the structure, but to the presence of a living being.
The old hypothesis of a soul, a living organism within the visible one, appears to me much more rational than the attempt to explain life as a form of motion. And when we consider the region of extreme minuteness characterised by four-dimensional motion the difficulty of conceiving such an organism alongside the bodily one disappears. Lord Kelvin supposes that matter is formed from the ether. We may very well suppose that the living organisms directing the material ones are co-ordinate with them, not composed of matter, but consisting of etherial bodies, and as such capable of motion through the ether, and able to originate material living bodies throughout the mineral.
Hypotheses such as these find no immediate ground for proof or disproof in the physical world. Let us, therefore, turn to a different field, and, assuming that the human soul is a four-dimensional being, capable in itself of four dimensional movements, but in its experiences through the senses limited to three dimensions, ask if the history of thought, of these productivities which characterise man, correspond to our assumption. Let us pass in review those steps by which man, presumably a four-dimensionalbeing, despite his bodily environment, has come to recognise the fact of four-dimensional existence.
Deferring this enquiry to another chapter, I will here recapitulate the argument in order to show that our purpose is entirely practical and independent of any philosophical or metaphysical considerations.
If two shots are fired at a target, and the second bullet hits it at a different place to the first, we suppose that there was some difference in the conditions under which the second shot was fired from those affecting the first shot. The force of the powder, the direction of aim, the strength of the wind, or some condition must have been different in the second case, if the course of the bullet was not exactly the same as in the first case. Corresponding to every difference in a result there must be some difference in the antecedent material conditions. By tracing out this chain of relations we explain nature.
But there is also another mode of explanation which we apply. If we ask what was the cause that a certain ship was built, or that a certain structure was erected, we might proceed to investigate the changes in the brain cells of the men who designed the works. Every variation in one ship or building from another ship or building is accompanied by a variation in the processes that go on in the brain matter of the designers. But practically this would be a very long task.
A more effective mode of explaining the production of the ship or building would be to enquire into the motives, plans, and aims of the men who constructed them. We obtain a cumulative and consistent body of knowledge much more easily and effectively in the latter way.
Sometimes we apply the one, sometimes the other mode of explanation.
But it must be observed that the method of explanation founded on aim, purpose, volition, always presupposesa mechanical system on which the volition and aim works. The conception of man as willing and acting from motives involves that of a number of uniform processes of nature which he can modify, and of which he can make application. In the mechanical conditions of the three-dimensional world, the only volitional agency which we can demonstrate is the human agency. But when we consider the four-dimensional world the conclusion remains perfectly open.
The method of explanation founded on purpose and aim does not, surely, suddenly begin with man and end with him. There is as much behind the exhibition of will and motive which we see in man as there is behind the phenomena of movement; they are co-ordinate, neither to be resolved into the other. And the commencement of the investigation of that will and motive which lies behind the will and motive manifested in the three-dimensional mechanical field is in the conception of a soul—a four-dimensional organism, which expresses its higher physical being in the symmetry of the body, and gives the aims and motives of human existence.
Our primary task is to form a systematic knowledge of the phenomena of a four-dimensional world and find those points in which this knowledge must be called in to complete our mechanical explanation of the universe. But a subsidiary contribution towards the verification of the hypothesis may be made by passing in review the history of human thought, and enquiring if it presents such features as would be naturally expected on this assumption.