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The other reason for failure arises from the four-dimensional extension of events so that there is no determinate route of transition of events in linear series. For example, the tunnel of a tube railway is an event at rest in a certain time-system, that is to say, it is cogredient with a certain duration. A train travelling in it is part of that tunnel, but is not itself at rest.
Thus the position of the event-particle in the temporal series α is defined by this moment, which we will call M. The position of the particle in the space of M is then fixed in the ordinary way by three levels which intersect in it and in it only. This procedure of fixing the position of an event-particle shows that the aggregate of event-particles forms a four-dimensional manifold.
Now the ordinary mind cannot grasp this idea in the least, though some few who have made a special study of the subject have gradually come to be able to realize one or two very simple four-dimensional figures.
The phenomena can be explained in terms of higher space. If Hinton's hypothesis be the true explanation, the universality of electro-magnetic action would again point to the conclusion that our three-dimensional world is superficial the surface, that is, of a four-dimensional universe.
This is the ultimate foundation in fact which enables us to speak with meaning of the mensuration, in Riemann's sense of the word, of the four-dimensional continuum of space-time.
Accordingly they express for us the demands of an ideal accuracy, and of an ideal simplicity in the exposition of relations. These event-particles are the ultimate elements of the four-dimensional space-time manifold which the theory of relativity presupposes. You will have observed that each event-particle is as much an instant of time as it is a point of space.
And because the more remote these are from the things of sense, from knowledge and experience, the projected figures of four-dimensional geometry would lend themselves to these uses with an especial grace. Color without form is as a soul without a body; yet the body of light must be without any taint of materiality.
We have got to find the way of expressing the field of activity of events in the neighbourhood of some definite event-particle E of the four-dimensional manifold. I bring in a fundamental physical idea which I call the 'impetus' to express this physical field. The event-particle E is related to any neighbouring event-particle P by an element of impetus.
In such a space it would of course be possible to establish four axial lines, all intersecting at a point, and all mutually at right angles with one another. Every hyper-solid of four-dimensional space has these four axes. He should learn to lure them forth along them axis lines. That is, let him build up his figures, space by space, developing them from lower spaces to higher.
The problem of ugliness and evil would seem at first thought to be totally unrelated to the subject of space hyperdimensionality, but there is at least a symbolical relation. This was suggested to the author by the endeavor of two friends whose interests were pre-eminently mathematical to discover what certain four-dimensional figures would look like in three-dimensional space.
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