bond which connects the motion of β in space α with the motions of α in space β. Motion is essentially a relation between some object of nature and the one timeless space of a time-system. An instantaneous space is static, being related to the static 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 α and β, 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 α and a moment of β. In M there is the direction of β and in N there is the direction of α. But M and N, being moments of different time-systems, intersect in a level. Call this level λ. Then λ 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 λ is perpendicular to the direction of β in M, and in the instantaneous space of N the level λ is perpendicular to the direction of α 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 α follows immediately from this theory of perpendicularity in each of its instantaneous spaces. Let ρ be any rect in the moment M of α and let λ be a level in M which is perpendicular to ρ. The locus of those points of the space of α which intersect M in event-particles on ρ is the straight line r of space α, and the locus of those points of the space of α which intersect M in event-particles on λ is the plane l of space α. 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. We shall find that this discovery of definite unique properties defining perpendicularity is of critical importance in the theory of congruence which is the topic for the next lecture.