In connection with interval, it is worth while to compare its formal characteristics with those of similarity. We saw that, in the generalized geometry with which Eddington ends, we want a relation of four neighbouring points, expressing the fact that they form a parallelogram. But we met with certain difficulties owing to the fact that this is only supposed to be possible for an infinitesimal quadrilateral, which is a figment of the mathematical imagination, and that it was not wholly easy to see how to substitute a procedure by means of limits. We were led to the suggestion that, instead of saying "[Pg 118] is a parallelogram," we should have to say " is more nearly a parallelogram than ." Perhaps this could be somewhat simplified. Suppose we say: " is more nearly a parallelogram than ." And perhaps this could be still further simplified so as to take the form: " is more like than is." We here suppose that between any two points there is a relation, which we will not call distance, but (say) "separation," and that this relation, like a shade of colour, is capable of a greater or less resemblance to another of the same kind. In a Euclidean space, two finite separations finitely separated may be exactly similar in the relevant respects; we then have a finite parallelogram. But in the generalized geometry that we are considering, we shall say that no two separations are exactly alike, though they are capable of indefinite approximation to exact likeness. Let us see how far this will take us. In the case of similarity, we have a relation which is capable of degrees, and may be called "quasi-transitive"—i.e. if is very like , and is very like , then must be rather like . This is just the sort of thing required for Weyl's geometry. Consider four points, , , , , and suppose that is rather like . Take a series of points forming a continuous route from to , without loops; this can be done by purely ordinal methods to be explained later. Suppose that among these points there are some, such as which make more like than is. We may suppose that these points have a limit or last term, which we will call . We can then similarly proceed along to a point which gives more like than for any other point on . We have then done nearly as well as possible, if not quite, with the three points , , as starting-points. By means of suitable postulates, we could[Pg 119] insure that a construction of the above sort, carried out repeatedly without changing the points , , , should at last end with a definite point such that is more like than any other distance from is. We may call the figure a "quasi-parallelogram." Now let , , ... , ... be a series of points on a route from to . Then proceed to take points , , ... between and on some route, and form the quasi-parallelograms having one corner at , one corner at and one at , the fourth being called .