If to each vertical and to each horizontal of the mesh-system we assign consecutive whole numbers from zero on indefinitely, we see that the point of the plane which happens to coincide with the intersection of some particular horizontal and some particular vertical is defined by the numbers which represent the two lines, respectively. In this way [Pg 84] every point of intersection is defined by two numbers, and these numbers are called the Cartesian co-ordinates of the point. Of course, by this method we are unable to define the positions of points which do not happen to coincide with the corners of our squares. But there is nothing to prevent us from assuming that between the verticals and horizontals we have mentioned there lie an indefinite number of other similar lines, to which intermediary fractional numbers will be assigned. Henceforth every point of the plane can be regarded as defined by the intersection of some particular horizontal and some particular vertical. To what extent is it permissible to say that points on the plane have been defined by this method? If we disregard the existence of the co-ordinate system, nothing has been defined, but if we consider the co-ordinate system as given, then every point of the plane can be considered as defined unambiguously. In short, the points are defined not in the abstract, but in relation to the co-ordinate system. There is nothing mysterious about this method of defining the positions of points. Thus, in everyday life, when we agree to meet a friend at the corner of Fifth Avenue and Forty-second Street, we are inadvertently locating our point of meeting in terms of the Cartesian co-ordinate system defined by the avenues and streets. In the present case the co-ordinate system is not strictly Cartesian, since the streets and avenues may not enclose perfectly equal Euclideanly square blocks, but the general principle involved is the same. Needless to say, the definition of our point of meeting would convey no significance were the avenues and streets non-existent. Hence once again we see that it is only relative to the co-ordinate system that points can be defined.