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Updated: May 11, 2025
The doctrine of this paragraph is nothing else than another way of expressing the unresolvable multiple relation of an object to events. A complete time-system is formed by any one family of parallel durations. The excluded case is that of two durations overlapping so as to include in common an aggregate of finite events but including in common no other complete duration.
Such a set of event-particles will form a point in the timeless space of that time-system. Thus a point is really an absolute position in the timeless space of a given time-system. But there are alternative time-systems, and each time-system has its own peculiar group of points that is to say, its own peculiar definition of absolute position. This is exactly the theory which I will elaborate.
Accordingly the group of dynamical axes required for Newton's Laws of Motion is the outcome of the necessity of referring motion to a body at rest in the space of some one time-system in order to obtain a coherent account of physical properties.
Each such instantaneous space represents the ideal of nature at an instant and is also a moment of time. Each time-system thus possesses an aggregate of moments belonging to it alone. Each event-particle lies in one and only one moment of a given time-system. Cf. pp.
Namely, there is a definite whole of nature, simultaneously now present, whatever may be the character of its remote events. This consideration reinforces the previous conclusion. This conclusion leads to the assertion of the essential uniformity of the momentary spaces of the various time-systems, and thence to the uniformity of the timeless spaces of which there is one to each time-system.
Now another time-system, which I will name σ, can be found which is such that rest in its space is represented by the same magnitudes of velocities along and perpendicular to the α-direction in β as those velocities in α, along and perpendicular to the β-direction, which represent rest in π.
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.
Also a family of parallel moments is the family of moments of some one time-system. Thus we can enlarge our concept of a family of parallel levels so as to include levels in different moments of one time-system. With this enlarged concept we say that a complete family of parallel levels in a time-system α is the complete family of levels in which the moments of α intersect the moments of β.
This is completely explained by the fact that, the space-system and the time-system which we are using are in certain minute ways different from the space and the time relatively to the sun or relatively to any other body with respect to which it is moving.
The topic for this lecture is the continuation of the task of explaining the construction of spaces as abstracts from the facts of nature. It was noted at the close of the previous lecture that the question of congruence had not been considered, nor had the construction of a timeless space which should correlate the successive momentary spaces of a given time-system.
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