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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."

But if we refuse to be restricted to a flat surface we find that it is possible to draw a third line through O which is quite "independent" of the directions of the two lines we have previously drawn. We can do this by drawing it vertically, that is to say, perpendicular to the plane of the paper. Call this line COC´.

Fig. 3

I have shown it in perspective in Fig. 3. This line fulfils the definition we gave of an independent direction in space for it is at right angles both to AOA´ and to BOB´. But we have now exhausted our resources. Try as we will we are unable to draw a fourth line which shall be at right angles to AOA´, BOB´, and COC´ simultaneously.