The Concept of Nature — the Tarner Lectures, 1920

This property of the unique intersection of a moment and a point-track is not confined to the case when the moment and the point-track belong to the same timesystem. Any two event-particles on a point-track are sequential, so that they cannot lie in the same moment. Accordingly no moment can intersect a point-track more than once, and every moment intersects a pointtrack in one event-particle.

Anyone who at the successive moments of a should be at the event-particles where those moments intersect a given point of a will be at rest in the timeless space of time-system a. But in any other timeless space belonging to another time-system he will be at a different point at each succeeding moment of that timesystem. In other words he will be moving. He will be moving in a straight line with uniform velocity. We might take this as the definition of a straight line. Namely, a straight line in the space of time-system j8 is the locus of those points of jS which all intersect some one point-track which is a point in the space of some

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other time-system. Thus each point in the space of a time-system a is associated with one and only one straight Hne of the space of any other time-system j8. Furthermore the set of straight Hnes in space /8 which are thus associated with points in space a form a complete family of parallel straight lines in space j8. Thus there is a one-to-one correlation of points in space a with the straight lines of a certain definite family of parallel straight lines in space /S. Conversely there is an analogous one-to-one correlation of the points in space ^ with the straight lines of a certain family of parallel straight lines in space a. These families will be called respectively the family of parallels in /8 associated with a , and the family of parallels in a associated with ^S , The direction in the space of ^ indicated by the family of parallels in j8 will be called the direction of a in space /3, and the family of parallels in a is the direction of ^ in space «. Thus a being at rest at a point of space a will be moving uniformly along a line in space ^ which is in the direction of a in space j3, and a being at rest at a point of space ^ will be moving uniformly along a line in space a which is in the direction of ^ in space a . I have been speaking of the timeless spaces which are associated with time-systems. These are the spaces of physical science and of any concept of space as eternal and unchanging. But what we actually perceive is an approximation to the instantaneous space indicated by event-particles which lie within some moment of the time-system associated with our awareness. The points of such an instantaneous space are event-particles and the straight lines are rects. Let the time-system be named a, and let the moment of time-system a to which our quick perception of nature approximates be

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called M. Any straight line r in space a is a locus of points and each point is a point-track which is a locus of event-particles. Thus in the four- dimensional geometry of all event-particles there is a two-dimensional locus which is the locus of all event-particles on points lying on the straight line r. I will call this locus of event-particles the matrix of the straight line r. A matrix intersects any moment in a rect. Thus the matrix of r intersects the moment M in a rect p . Thus p is the instantaneous rect in M which occupies at the moment M the straight line r in the space of a. Accordingly when one sees instantaneously a moving being and its path ahead of it, what one really sees is the being at some event-particle A lying in the rect p which is the apparent path on the assumption of uniform motion. But the actual rect p which is a locus of event-particles is never traversed by the being. These event-particles are the instantaneous facts which pass with the instantaneous moment. What is really traversed are other event-particles which at succeeding instants occupy the same points of space a as those occupied by the event-particles of the rect /». For example, we see a stretch of road and a lorry moving along it. The instantaneously seen road is a portion of the rect p — of course only an approximation to it. The lorry is the moving object. But the road as seen is never traversed. It is thought of as being traversed because the intrinsic characters of the later events are in general so similar to those of the instantaneous road that we do not trouble to discriminate. But suppose a land mine under the road has been exploded before the lorry gets there. Then it is fairly obvious that the lorry does not traverse what we saw at first. Suppose the lorry is at rest inv] SPACE AND MOTION 117 space ^. Then the straight line r of space a is in the direction of ^ in space a, and the rect p is the representative in the moment M of the Une r of space a. The direction of p in the instantaneous space of the moment M is the direction of /3 in M, where M is a moment of time-system a. Again the matrix of the line r of space a will also be the matrix of some line ^ of space j3 which will be in the direction of a in space p. Thus if the lorry halts at some point P of space a which lies on the line r, it is now moving along the line s of space /S. This is the theory of relative motion; the common matrix is the bond which connects the motion of y8 in space a with the motions of a in space jS.