Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/147

 that, owing to the ‘repetition property’ of parallelism, the motion is uniform.

44. Matrices. 44.1 A level is obtained by taking a rect r and an event-particle P co-momental with r, and by forming the locus of event-particles on rects through P and intersecting r, including also particles on the rect through P and parallel to r.

The same level would be obtained by taking the particles on the rects intersecting r and parallel to some one rect through P which intersects r.

44.2 Analogously to levels, a locus of event-particles called a ‘matrix’ is obtained by taking a rect r and an event-particle P which is not co-momental with r, and by forming the locus of event-particles on rects or point-tracks through P and intersecting r, including also the event-particles on the rect through P and parallel to r.

A ‘matrix’ is a two-dimensional plane in the four-dimensional geometry of event-particles. Levels and matrices together make up the complete set of such two-dimensional planes, and have the usual properties of such planes which need not be detailed here.

44.3 Matrices are also obtained by taking an event-particle P and a point-track p, and by forming the locus of event-particles on rects or point-tracks through P and intersecting p, including also event-particles on the point-track through P and parallel to p. Any matrix can be generated in either of the two ways. Furthermore matrices can be generated by the use of parallels in the same way as levels are generated as explained in 44.1 and as assumed in 43.4.

45. Null-Tracks. 45.1 The relations between rects and point-tracks are best understood by taking a rect