I must amplify this by saying that, "Two directions in space are to be considered as independent when they are so related that no movement, however great, along one of them will result in the slightest movement along, or parallel to, the other. That is to say, at right angles, or perpendicular to one another."
Fig. 1
Thus in Fig. 1 AOA´ and BOB´ are independent directions. One might move for ever along OA or OA´ and yet one would not have moved in the very least in the direction of OB or of OB´.
Now on a flat surface, such as a sheet of paper, it is not possible to draw more than two such directions. Any other line that can be drawn, XOX´ for instance, is in a compound direction, so to speak. That is to say it is partly in the direction AOA´ and partly in the direction BOB´ and it is possible to reach any point in it, Y for example, by moving along OA´ to a and then moving in the direction of OB´ a distance equal to Ob, or vice versa or by doing the two simultaneously.
For the benefit of those who are absolutely ignorant of the rudiments of Geometrical knowledge, I would point out that Parallel lines are said to point, in fact do point, in the same direction.
Fig. 2
Thus, in Fig. 2, the direction of the line ZZ´ is the same as that of AOA´ and the direction of the line PP´ is the same as that of XOX´.
Thus we see that in a flat surface we find only two dimensions and consequently we can refer to a flat surface as "Space of two dimensions" or "Two-dimensional space."