Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/150



The loci which bound the regions separating point-tracks from rects will be called ‘null-tracks.’ Their special properties will be considered later when congruence has been introduced. In any matrix there are two families of parallel null-tracks; and there is one member of each family passing through each event-particle on the rectilinear track. The order of event-particles on a null-track is derived from its intersection with systems of parallel rects [not co-momental] or of parallel point-tracks or from the orders on routes lying on it.

46. Straight Lines. 46.1 There is evidently an important theory of parallelism for families of matrices analogous to the theory of parallels for families of levels. The detailed properties need not be elaborated here.

Two matrices may either (i) be parallel, or (ii) intersect in one event-particle only, or (iii) intersect in a rect, or (iv) intersect in a point-track, or (v) intersect in a null-track. For the intersection of two levels only cases (i), (ii) and (iii) can occur; for the intersection of a level and a matrix only cases (ii) and (iii) can occur.

46.2 Each matrix contains various sets of parallel point-tracks. Any one such set is a locus of points in the space of some time-system. Such a locus of points is called a ‘straight line’ in the space of the time-system.

A matrix which contains the points of a straight line in the space of any time-system α will be called ‘an associated matrix for α,’ and it is called ‘the matrix including’ that straight line.

A matrix is an associated matrix for many time-systems, but it is the matrix including only one straight