A World of Points
To deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, as P, by measuring its distances, X and Y, from these “coordinate axes.” The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distances
and
between the points P and Q in these directions, instead of measuring the single distance PQ. We do the same thing when we say that the railroad station is five blocks north and two east.
The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distances
