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Updated: May 11, 2025
All the moving points fixed in these moving axes are really tracing out parallel lines with one uniform velocity. In other words they are the reflections in the space of α of a set of fixed axes in the space of some other time-system β.
As the character of the percipient event changes with the passage of nature or, in other words, as the percipient mind in its passage correlates itself with the passage of the percipient event into another percipient event the time-system correlated with the percipience of that mind may change.
It will be convenient to reserve the ordinary spatial terms 'plane, 'straight line, 'point' for the elements of the timeless space of a time-system.
We are now in possession of a theory of parallels and a theory of perpendiculars and a theory of motion, and from these theories the theory of congruence can be constructed. It will be remembered that a family of parallel levels in any moment is the family of levels in which that moment is intersected by the family of moments of some other time-system.
The other reason for failure arises from the four-dimensional extension of events so that there is no determinate route of transition of events in linear series. For example, the tunnel of a tube railway is an event at rest in a certain time-system, that is to say, it is cogredient with a certain duration. A train travelling in it is part of that tunnel, but is not itself at rest.
These are the spaces of physical science and of any concept of space as eternal and unchanging. But what we actually perceive is an approximation to the instantaneous space indicated by event-particles which lie within some moment of the time-system associated with our awareness. The points of such an instantaneous space are event-particles and the straight lines are rects.
In this way each time-system possesses its own permanent timeless space peculiar to it alone, and each such space is composed of timeless points which belong to that time-system and to no other.
If an event e be cogredient with a duration d, and d′ be any duration which is part of d. Then d′ belongs to the same time-system as d. Also d′ intersects e in an event e′ which is part of e and is cogredient with d′. Let P be any event-particle lying in a given duration d. Consider the aggregate of events in which P lies and which are also cogredient with d.
It follows from this axiom that two objects at rest in any two points of a time-system β are moving with equal velocities in any other time-system α along parallel lines. Thus we can speak of the velocity in α due to the time-system β without specifying any particular point in β.
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