The symmetry of perpendicularity is a particular instance of the symmetry of the mutual relations between two time-systems.

We shall find in the next lecture that it is from this symmetry that the theory of congruence is deduced.
The theory of perpendicularity in the timeless space of any time-system {alpha} follows immediately from this theory of perpendicularity in each of its instantaneous spaces.

Let {rho} be any rect in the moment M of {alpha} and let {lambda} be a level in M which is perpendicular to {rho}.

The locus of those points of the space of {alpha} which intersect M in event-particles on {rho} is the straight line r of space {alpha}, and the locus of those points of the space of {alpha} which intersect M in event-particles on {lambda} is the plane l of space {alpha}.

Then the plane l is perpendicular to the line r.
In this way we have pointed out unique and definite properties in nature which correspond to perpendicularity.