Given any three points on a straight line in ordinary space, there must be one of them which is between the other two. This will not be the case with the points on a circle or any other closed curve, because, given any three points on a circle, we can travel from any one to any other without passing through the third. In fact, the notion "between" is characteristic of open series—or series in the strict sense—as opposed to what may be called [Pg 38] "cyclic" series, where, as with people at the dinner-table, a sufficient journey brings us back to our starting-point. This notion of "between" may be chosen as the fundamental notion of ordinary geometry; but for the present we will only consider its application to a single straight line and to the ordering of the points on a straight line.[11] Taking any two points , , the line consists of three parts (besides and themselves): [11]Cf. Rivista di Matematica, IV. pp. 55 ff.; Principles of Mathematics, p. 394 ( 375). (1) Points between and . (2) Points such that is between and . (3) Points such that is between and . Thus the line can be defined in terms of the relation "between." In order that this relation "between" may arrange the points of the line in an order from left to right, we need certain assumptions, namely, the following:— (1) If anything is between and , and are not identical. (2) Anything between and is also between and . (3) Anything between and is not identical with (nor, consequently, with , in virtue of (2)). (4) If is between and , anything between and is also between and . (5) If is between and , and is between and , then is between and . (6) If and are between and , then either and are identical, or is between and , or is between and . (7) If is between and and also between and , then either and are identical, or is between and , or is between and . These seven properties are obviously verified in the case of points on a straight line in ordinary space. Any three-term relation which verifies them gives rise to series, as may be seen from the following definitions. For the sake of definiteness, let us assume [Pg 39] that is to the left of . Then the points of the line are (1) those between which and , lies—these we will call to the left of ; (2) itself; (3) those between and ; (4) itself; (5) those between which and lies —these we will call to the right of . We may now define generally that of two points , , on the line , we shall say that is "to the left of" in any of the following cases:— (1) When and are both to the left of , and is between and ; (2) When is to the left of , and is or or between and or to the right of ; (3) When is , and is between and or is or is to the right of ; (4) When and are both between and , and is between and ; (5) When is between and , and is or to the right of ; (6) When is and is to the right of ;