nature at an instant. In perception when we
see things moving in an approximation to an instantaneous space, the
future lines of motion as immediately perceived are rects which are
never traversed. These approximate rects are composed of small events,
namely approximate routes and event-particles, which are passed away
before the moving objects reach them. Assuming that our forecasts of
rectilinear motion are correct, these rects occupy the straight lines in
timeless space which are traversed. Thus the rects are symbols in
immediate sense-awareness of a future which can only be expressed in
terms of timeless space.

We are now in a position to explore the fundamental character of
perpendicularity. Consider the two time-systems {alpha} and {beta}, each
with its own timeless space and its own family of instantaneous moments
with their instantaneous spaces. Let M and N be respectively a moment of
{alpha} and a moment of {beta}. In M there is the direction of {beta}
and in N there is the direction of {alpha}. But M and N, being moments
of different time-systems, intersect in a level. Call this level
{lambda}. Then {lambda} is an instantaneous plane in the instantaneous
space of M and also in the instantaneous space of N. It is the locus of
all the event-particles which lie both in M and in N.

In the instantaneous space of M the level {lambda} is perpendicular to
the direction of {beta} in M, and in the instantaneous space of N the
level {lambda} is perpendicular to the direction of {alpha} in N. This
is the fundamental property which forms the definition of
perpendicularity. 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 ha