observation. Namely, amid the alternative time-systems which nature offers there will be one with a duration giving the best average of cogredience for all the sub- ordinate parts of the percipient event. This duration will be the whole of nature which is the terminus posited by sense-awareness. Thus the character of the percipient event determines the time-system immediately evident in nature. 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. When the bulk of the events perceived are cogredient in a duration other than that of the percipient event, the percipience may include a double consciousness of cogredience, namely the consciousness of the whole within which the observer in the train is 'here,' and the consciousness of the whole within which the trees and bridges and telegraph posts are definitely 'there.' Thus in per- ceptions under certain circumstances the events dis- criminated assert their own relations of cogredience. This assertion of cogredience is peculiarly evident when the duration to which the perceived event is cogredient is the same as the duration which is the present whole of nature — in other words, when the event and the per- cipient event are both cogredient to the same duration.
We are now prepared to consider the meaning of
stations in a duration, where stations are a peculiar
kind of routes, which define absolute position in the
associated timeless space.
There are however some preliminary explanations. A finite event will be said to extend throughout a
112 THE CONCEPT OF NATUKt, l^«.
duration when it is part of the duration and is inter-
sected by any moment which Hes in the duration. Such
an event begins with the duration and ends with it.
Furthermore every event which begins with the dura-
tion and ends with it, extends throughout the duration.
This is an axiom based on the continuity of events. By
beginning with a duration and ending with it, I mean
(i) that the event is part of the duration, and (ii) that
both the initial and final boundary moments of the
duration cover some event-particles on the boundary of
the event.
Every event which is cogredient with a duration extends throughout that duration.
It is not true that all the parts of an event cogredient with a duration are also cogredient with the duration. The relation of cogredience may fail in either of two ways. One reason for failure may be that the part does not extend throughout the duration. In this case the part may be cogredient with another duration which is part of the given duration, though it is not cogredient with the given duration itself. Such a part would be cogredient if its existence were sufficiently prolonged in that 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 travel- ling in it is part of that tunnel, but is not itself at rest.
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' .
v] SPACE AND MOTION 113
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. Each of these events occupies its own aggregate of event-particles. These aggregates will have a common portion, namely the class of event-particle lying in all of them. This class of event-particles is what I call the 'station' of the event-particle P in the duration d. This is the station in the character of a locus. A station can also be defined in the character of an abstractive element. Let the pro- perty a be the name of the property which an abstractive set possesses when (i) each of its events is cogredient with the duration d and (ii) the event-particle P lies in each of its events. Then the group of a -primes, where a has this meaning, is an abstractive element and is the station oi P in d as an abstractive element. The locus of event-particles covered by the station oi P in d as an abstractive element is the station of P in ^ as a locus. A station has accordingly the usual three characters, namely, its character of position, its ex- trinsic character as an abstractive element, and its intrinsic character.
It follows from the peculiar properties of rest that two stations belonging to the same duration cannot intersect. Accordingly every event-particle on a station of a duration has that station as its station in the duration. Also every duration which is part of a given duration intersects the stations of the given duration in loci which are its own stations. By means of these properties we can utilise the overlappings of the durations of one family — that is, of one time-system — ^to prolong stations in- definitely backwards and forwards. Such a prolonged station will be called a point-track. A point-track is a
W.N. 8
114 THE CONCEPT OF JNAiUKt i^ctt.
locus of event-particles. It is defined by reference to
one particular time-system, a say. Corresponding to
any other time-system these will be a different group
of point-tracks. Every event-particle will lie on one
and only one point-track of the group belonging to any
one time-system. The group of point-tracks of the time-
system a is the group of points of the timeless space of a .
Each such point indicates a certain quality of absolute
position in reference to the durations of the family
associated with a, and thence in reference to the suc-
cessive instantaneous spaces lying in the successive
moments of a. Each moment of a will intersect a
point-track in one and only one event-particle.
This property of the unique intersection of a moment
and a point-track is not confined to the case when the
moment and the point-track belong to the same time-
system. Any two event-particles on a point-track are
sequential, so that they cannot lie in the same moment.
Accordingly no moment can intersect a point-track
more than once, and every moment intersects a point-
track in one event-particle.
Anyone who at the successive moments of a should
be at the event-particles where those moments intersect
a given point of a will be at rest in the timeless space
of time-system a. But in any other timeless space
belonging to another time-system he will be at a
different point at each succeeding moment of that time-
system. In other words he will be moving. He will be
moving in a straight line with uniform velocity. We
might take this as the definition of a straight line.
Namely, a straight line in the space of time-system j8 is
the locus of those points of jS which all intersect some
one point-track which is a point in the space of some
V] Space and Motion 115
ii6 THE CONCEPT OF NATURE ii;«.
called M. Any straight line r in space a is a locus of points and each point is a point-track which is a locus of event-particles. Thus in the four- dimensional geo- metry 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-particles the matrix of the straight line r. A matrix intersects any moment in a rect. Thus the matrix of r intersects the moment M in a rect p . Thus p is the instantaneous rect in M which occupies at the moment M the straight line r in the space of a. Accordingly when one sees instantaneously a moving being and its path ahead of it, what one really sees is the being at some event-particle A lying in the rect p which is the apparent path on the assumption of uniform motion. But the actual rect p which is a locus of event-particles is never traversed by the being. These event-parti- cles are the instantaneous facts which pass with the instantaneous moment. What is really traversed are other event-particles which at succeeding instants occupy the same points of space a as those occupied by the event-particles of the rect /». For example, we see a stretch of road and a lorry moving along it. The in- stantaneously seen road is a portion of the rect p — of course only an approximation to it. The lorry is the moving object. But the road as seen is never traversed. It is thought of as being traversed because the intrinsic characters of the later events are in general so similar to those of the instantaneous road that we do not trouble to discriminate. But suppose a land mine under the road has been exploded before the lorry gets there. Then it is fairly obvious that the lorry does not traverse what we saw at first. Suppose the lorry is at rest in
v] SPACE AND MOTION 117
space ^. Then the straight line r of space a is in the
direction of ^ in space a, and the rect p is the repre-
sentative in the moment M of the Une r of space a.
The direction of p in the instantaneous space of the
moment M is the direction of /3 in M, where M is a
moment of time-system a. Again the matrix of the
line r of space a will also be the matrix of some line ^
of space j3 which will be in the direction of a in space p.
Thus if the lorry halts at some point P of space a which
lies on the line r, it is now moving along the line s of
space /S. This is the theory of relative motion; the
common matrix is the bond which connects the motion
of y8 in space a with the motions of a in space jS.
Motion is essentially a relation between some object
of nature and the one timeless space of a time-system.
An instantaneous space is static, being related to the
static nature at an instant. In perception when we see
things moving in an approximation to an instantaneous
space, the future lines of motion as immediately per-
ceived are rects which are never traversed. These
approximate rects are composed of small events, namely
approximate routes and event-particles, which are
passed away before the moving objects reach them.
Assuming that our forecasts of rectilinear motion are
correct, these rects occupy the straight lines in timeless
space which are traversed. Thus the rects are symbols
in immediate sense-awareness of a future which can
only be expressed in terms of timeless space.
We are now in a position to explore the fundamental
character of perpendicularity. Consider the two time-
systems a and jS, each with its own timeless space and
its own family of instantaneous moments with their
instantaneous spaces. Let M and N be respectively a
ii8 THE CONCEPT OF NATURE [ch.
moment of a and a moment of /3. In M there is the
direction of ^ and in A^" there is the direction of a.
But M and N, being moments of different time-systems,
intersect in a level. Call this level A. Then A is an
instantaneous plane in the instantaneous space of M
and also in the instantaneous space of N. It is the locus
of all the event-particles which lie both in M and in N.
In the instantaneous space of M the level A is per-
pendicular to the direction of /S in M, and in the
instantaneous space of N the level A is perpendicular
to the direction of a in N. This is the fundamental
property which forms the definition of perpendicularity.
The symmetry of perpendicularity is a particular in-
stance of the symmetry of the mutual relations between
two time-systems. We shall find in the next lecture
that it is from this symmetry that the theory of con-
gruence is deduced.