§ 7. (_a_) _The Puzzle of Continuity._ Continuity is, strictly speaking, a property of certain series, and may be defined for purposes of reference much as follows. A series is continuous when any term divides the whole series unambiguously into two mutually exclusive parts which between them comprise all the terms of the series, and when every term which so divides the series is itself a term of the series. From this second condition it obviously follows that a number of intermediate terms can always be inserted between any two terms whatever of a continuous series; no term of the series has a _next_ term. This is the peculiarity of the continuous with which we shall be specially concerned. Thus the series of points on a straight line is continuous because (1) any point P on the line divides it into two collections of points in such a way that every point of the one is to the left of every point of the other, and every point of the second to the right of every point of the former; and (2) every point which divides the line in this way is a point on the line. Again, the whole series of real numbers is continuous for the same reason. Every member of the number-series divides it into two classes, so that every number of one is less than every number of the other, and every number which thus divides the series is itself a term of the number-series. But the series of _rational_ real numbers is not continuous, because it can be divided into mutually exclusive classes by terms which are not themselves members of the series. (_E.g._ √2 is not a member of the series of rational numbers, but we can exhaustively divide all rational numbers into the two mutually exclusive classes, rational numbers _less_ than √2 and rational numbers _not less_ than √2.)[105] From the continuity of the series of real numbers it follows that any other series which corresponds point for point with the terms of the number-series will be continuous. Now one such series is that of the successive parts of time. Every moment of time divides the whole series of moments into two mutually exclusive classes, the moments _before_ itself and the moments which are _not before_ itself. And whatever thus divides the time-series is itself a moment in that series. Hence from the continuity of the time-series it follows that any puzzles created by this property of continuousness will apply to the case of Causation. In what follows I shall not discuss the general problem of the continuous, a problem which requires special mathematical equipment for its efficient handling, but shall confine myself to the difficulties introduced by continuity into the scientific concept of causal relation.