49. Analytic Geometry. 49.1 Consider any time-system : we will term the space of this time-system '-space' and its moments '-moments'; also the points and straight lines of -space will be termed '-points' and '-lines,' and rects and levels which lie in -moments will be termed '-rects' and '-levels.' If be any event-particle, then will denote the -moment which covers . If be any other time-system, there are no -moments which are also -moments, and no -points which are also -points; but there are -levels which are also -levels and -rects which are also -rects. For the two moments and intersect in a common level which will be called . Then rects lying in are both -rects and -rects. In particular through in the level pairs of mutually normal rects exist, and every rect through and is a member of one such pair. 49.2 Let be any arbitrarily chosen event-particle, which we will term the origin; and let be the -point occupied by ; and let , , be any triad of mutually rectangular -rects in the moment , each containing . In this notation , , etc., do not denote any particular entities, but the symbols such as and are each to be taken as one whole. Let denote the matrix containing and , with analogous meanings for and ; and let , and denote respectively the levels containing and , and , and . Let be any other event-particle occupying the point , and let be the -rects through respectively parallel to , , . Fig. 16. In the diagram the third dimension of the moments and namely the -dimension, is suppressed, so that these moments are diagrammatically represented as two-dimensional. Point-tracks (in this case -points) are represented by dotted lines. The diagram has the defect of representing matrices, such as , by levels, and is thus liable to lead to unfounded assumptions. 49.3 Lengths on all rects, whether or no they be -rects, are measurable in terms of one unit length. But time-lapses between -moments—or, what is the same thing, time-lapses along -points—must be measured in a time-unit peculiar to the time-system , since as yet no means of obtaining congruent time-units in different time-systems has been disclosed. We will suppose at present that in each time-system there is a given arbitrarily chosen unit for time-measurement. 49.4 Let the momentary space of be referred to the three rectangular -rects , , as axes of coordinates; and let the momentary space of be referred to the three rectangular -rects , , as axes of coordinates; and let the time-less space of a [the -space] be referred to the three rectangular -lines respectively included in the matrices , , as axes of coordinates; and let the four-dimensional space of all particles be referred to the four axes consisting of the three -rects , , , and of the -point as axes of co-ordinates.