-system {beta} is the
locus of those points of {beta} which all intersect some one point-track
which is a point in the space of some other time-system. Thus each point
in the space of a time-system {alpha} is associated with one and only
one straight line of the space of any other time-system {beta}.
Furthermore the set of straight lines in space {beta} which are thus
associated with points in space {alpha} form a complete family of
parallel straight lines in space {beta}. 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. 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. Let the
time-system be named {alpha}, and let the moment of time-system {alpha}
to which our quick perception of nature approximates be called M. Any
straight line r in space {alpha} is a locus of points and each point is
a point-track which is a locus of event-particles. Thus in the
four-dimensional geometry of all event-particles there is a
two-dimensional locus which is the locus of all event-particles on
points lying on the straight line r. I will call this locus of
event-