Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/146

 to which p does not belong. Then a certain set of point-tracks belonging to π will intersect p; name it πp. Let P be any event-particle occupying some member of πp; then the point-track occupied by P and parallel to p will intersect every member of πp.

This theorem, the theorems of 43.2 and the corresponding theorem for two families of parallel rects are examples of the repetition property of parallelism. It is evident that, given any three event-particles not on one rect or one point-track, a parallelogram can be completed of which the three event-particles are three corners, any one of the event-particles being at the junction of the adjacent sides through the three corners. In such a parallelogram opposite sides are always of the same denomination, namely both rects or both point-tracks; but adjacent sides may be of opposite denominations.

43.4 The event-particles occupying a point p in the time-less space of a time-system α appear at the successive moments of α as successively occupying the same point p. If β be any other time-system, then the point p of the space of α intersects a series of points of the space of β in event-particles which lie on the successive moments of β. These event-particles of p thus occupy a succession of points of β at a succession of moments of β; and we shall find that this locus of points is what is meant by a straight line in the space of β. Thus the point p in the space of α correlates the successive points on a straight line of β with the successive moments of β. Thus in the space of β the point p of the space of α appears as exemplifying the kinematical conception of a moving material particle traversing a straight line. It will appear later