We shall say that an event is "at" an instant when it is a member of the group by which the instant is constituted; and we shall say that one instant is before another if the group which is the one instant contains an event which is earlier than, but not simultaneous with, some event in the group which is the other instant. When one event is earlier than, but not simultaneous with another, we shall say that it "wholly precedes" the other. Now we know that of two events which are not simultaneous, there must be one which wholly precedes the other, and in that case the other cannot also wholly precede the one; we also know that, if one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third. From these facts it is easy to deduce that the instants as we have defined them form a series. We have next to show that every event is "at" at least one instant, _i.e._ that, given any event, there is at least one class, such as we used in defining instants, of which it is a member. For this purpose, consider all the events which are simultaneous with a given event, and do not begin later, _i.e._ are not wholly after anything simultaneous with it. We will call these the "initial contemporaries" of the given event. It will be found that this class of events is the first instant at which the given event exists, provided every event wholly after some contemporary of the given event is wholly after some _initial_ contemporary of it. Finally, the series of instants will be compact if, given any two events of which one wholly precedes the other, there are events wholly after the one and simultaneous with something wholly before the other. Whether this is the case or not, is an empirical question; but if it is not, there is no reason to expect the time-series to be compact.[17] [17] The assumptions made concerning time-relations in the above are as follows:-- I. In order to secure that instants form a series, we assume: (a) No event wholly precedes itself. (An "event" is defined as whatever is simultaneous with something or other.) (b) If one event wholly precedes another, and the other wholly precedes a third, then the first wholly precedes the third. (c) If one event wholly precedes another, it is not simultaneous with it. (d) Of two events which are not simultaneous, one must wholly precede the other. II. In order to secure that the initial contemporaries of a given event should form an instant, we assume: (e) An event wholly after some contemporary of a given event is wholly after some _initial_ contemporary of the given event. III. In order to secure that the series of instants shall be compact, we assume: (f) If one event wholly precedes another, there is an event wholly after the one and simultaneous with something wholly before the other.