We will now define a set of points as "collinear" if every pair of the set are connected, and every triad , , are such that either is contained in , or is contained in . We will define a set of points as a "line" if (1) it is collinear, (2) it is not contained in any larger collinear group with the same extremities. It will be seen that this definition is analogous to that of points. We may define a set of events[Pg 307] as "co-punctual" when every quintet of the set are co-punctual; and we can then define a set of events as a "point" when (1) it is co-punctual, (2) it is not contained in any larger co-punctual group. This way of stating our previous definition of "points" brings out the analogy. The "lines" that we are defining are not to be supposed "straight"; straightness is a notion wholly foreign to the geometry we are developing. Perhaps it might be better to call them "routes"; but there is no harm in calling them "lines" provided we remember that they are not supposed to be straight. For the present, we shall not be concerned with lines, but only with collinear groups of points. Let us define a set of points as "-collinear" if (1) every pair of the set is connected; (2) given any two, , , either is between or , or is between and . We shall want such axioms as will enable us to show that such a set of points is collinear, not merely -collinear, and that their order is independent of . It is obvious that, if we put before whenever is between and , we obtain a serial order of any set of points which is -collinear. But to insure that the order shall be independent of we require the following three axioms: (1) If , , , are points, and is contained in , and is contained in , and and are distinct, then is not contained in . (2) If is contained in , and is contained in , than is contained in the sum of and . (It follows at once that is contained in .) (3) If is contained in , and is contained in , then is contained in the sum of and . (It follows at once that is contained in .) The practical effects of these three axioms are: [Pg 308] (1) If and are between and , and is between and , then is not between and . (2) If is between and , and is between and , then and are between and . (3) If is between and , and is between and , then is between and .