Let us observe, to begin with, that events may be divided into zones with respect to a given event. There are first those that are compresent with a given event, then those not compresent with it, but compresent with an event compresent with it, and so on. The th zone will consist of events that can be reached in steps, but not in , "step" being taken as the passage from an event to another which is compresent with it. We will call two points "connected" when there is an event which is a member of both. The passage from event to event by the relation of compresence may be replaced by the passage from point to point by the relation of connection. Thus points also can be collected into zones. If there is a minimum to the size of events, we may assume that it is always possible to pass from one event to another by a finite number of "steps." If so, there must be a smallest number of steps in which the passage can be made; thus every event will belong to some definite zone with respect to a given event. This is useful in the introduction of order, because we can agree that the th zone is to be nearer the origin than the th if so that it only remains to introduce order among the members of a given zone. And even here we only want such order as is involved in analysis situs, not such more rigid order as is involved, e.g., in projective geometry. When an event can be reached from another in steps but not in , we may regard the intermediate events as forming a sort of quantized geodesic route between the two events. In virtue of the above division into zones, which can be effected with respect to any point as origin, we can define a rather small region of space-time by means of four integers, representing the number of steps in which any point in the region can be reached from four given points. It is only within a small region of this sort, therefore, that we need the[Pg 305] more delicate methods of microscopic order, to which we shall now proceed. Given two points and , let us denote by "" their logical product, i.e. the events which are members of both, or, in geometrical language, the events which contain both. It is obvious that, taking the view of events explained at the beginning of the preceding chapter, will be null unless and are fairly near together. As already stated, we say that and are "connected" when is not null. Microscopic order is confined to connected points, at any rate to begin with.