Thus there is a one-to-one correlation of points in space {alpha} with the straight lines of a certain definite family of parallel straight lines in space {beta}.
Conversely there is an analogous one-to-one correlation of the points in space {beta} with the straight lines of a certain family of parallel straight lines in space {alpha}.

These families will be called respectively the family of parallels in {beta} associated with {alpha}, and the family of parallels in {alpha} associated with {beta}.

The direction in the space of {beta} indicated by the family of parallels in {beta} will be called the direction of {alpha} in space {beta}, and the family of parallels in {alpha} is the direction of {beta} in space {alpha}.

Thus a being at rest at a point of space {alpha} will be moving uniformly along a line in space {beta} which is in the direction of {alpha} in space {beta}, and a being at rest at a point of space {beta} will be moving uniformly along a line in space {alpha} which is in the direction of {beta} in space {alpha}.
I have been speaking of the timeless spaces which are associated with time-systems.

These are the spaces of physical science and of any concept of space as eternal and unchanging.