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In parallelograms of the first type the two pairs of parallel sides are both of them pairs of rects. In parallelograms of the second type one pair of parallel sides is a pair of rects and the other pair is a pair of point-tracks. In parallelograms of the third type the two pairs of parallel sides are both of them pairs of point-tracks.
Now in the third assumption this sharpness of distinction is adequately preserved. There is a fundamental distinction between the metrical properties of point-tracks and rects. But in the fourth assumption this fundamental distinction vanishes.
Rectilinear routes are momental routes and stations are vagrant routes. Rectilinear routes are routes which in a sense lie in rects. Any two event-particles on a rect define the set of event-particles which lie between them on that rect.
Meanwhile we will ignore them. Also I will always speak of 'event-particles' in preference to 'puncts, the latter being an artificial word for which I have no great affection. Parallelism among rects and levels is now explicable. Consider the instantaneous space belonging to a moment A, and let A belong to the temporal series of moments which I will call α.
This complete family of parallel levels is also evidently a family lying in the moments of the time-system β. By introducing a third time-system γ, parallel rects are obtained. Also all the points of any one time-system form a family of parallel point-tracks. Thus there are three types of parallelograms in the four-dimensional manifold of event-particles.
A volume however has not yet been defined. This definition will be given in the next lecture. Evidently levels, rects, and puncts in their capacity as infinite aggregates cannot be the termini of sense-awareness, nor can they be limits which are approximated to in sense-awareness.
None of these levels can intersect, and they form a family of parallel instantaneous planes in the instantaneous space of moment A. Thus the parallelism of moments in a temporal series begets the parallelism of levels in an instantaneous space, and thence as it is easy to see the parallelism of rects. Accordingly the Euclidean property of space arises from the parabolic property of time.
Accordingly we can think of rects and levels as merely loci of event-particles. In so doing we are also cutting out those abstractive elements which cover sets of event-particles, without these elements being event-particles themselves. There are classes of these abstractive elements which are of great importance. I will consider them later on in this and in other lectures.
The first axiom of congruence is that the opposite sides of any parallelogram are congruent. This axiom enables us to compare the lengths of any two segments either respectively on parallel rects or on the same rect. Also it enables us to compare the lengths of any two segments either respectively on parallel point-tracks or on the same point-track.
The first of these axioms, which is the third axiom of congruence, is that if ABC is a triangle of rects in any moment and D is the middle event-particle of the base BC, then the level through D perpendicular to BC contains A when and only when AB is congruent to AC. This axiom evidently expresses the symmetry of perpendicularity, and is the essence of the famous pons asinorum expressed as an axiom.
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