SPACE AND MOTION
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. Furthermore it was also noted that there were many
spatial abstractive elements which we had not yet defined. We will first
consider the definition of some of these abstractive elements, namely
the definitions of solids, of areas, and of routes. By a 'route' I mean
a linear segment, whether straight or curved. The exposition of these
definitions and the preliminary explanations necessary will, I hope,
serve as a general explanation of the function of event-particles in the
analysis of nature.
We note that event-particles have 'position' in respect to each other.
In the last lecture I explained that 'position' was quality gained by a
spatial element in virtue of the intersecting moments which covered it.
Thus each event-particle has position in this sense. The simplest mode
of expressing the position in nature of an event-particle is by first
fixing on any definite time-system. Call it {alpha}. There will be one
moment of the temporal series of {alpha} which covers the given
event-particle. Thus the position of the event-particle in the temporal
series {alpha} is defined by this moment, which we will call M. The
position of the particle in the space of M is then fixed in the ordinary
way by three levels which intersect in it and in it only. This procedure
of fixing the position of an event-particle shows that the aggregate of
event-particles forms a four-dimensional manifold. A finite event
occupies a limited chunk of this manifold in a sense which I now proceed
to explain.
Let e be any given event. The manifold of event-particles falls into
three sets in reference to e. Each event-particle is a group of equal
abstractive sets and each abstractive set towards its small-end is
composed of smaller and smaller finite events. When we select from these
finite events which enter into the make-up of a given event-particle
those which are small enough, one of three cases must occur. Either (i)
all of these small events are entirely separate from the given event
e, or (ii) all of these small events are parts of the event e, or
(iii) all of these small events overlap the event e but are not parts
of it. In the first case the event-particle will be said to 'lie
outside' the event e, in the second case the event-particle will be
said to 'lie inside' the event e, and in the third case the
event-particle will be said to be a 'boundary-particle' of the event
e. Thus there are three sets of particles, namely the set of those
which lie outside the event e, the set of those which lie inside the
event e, and the boundary of the event e which is the set of
boundary-particles of e. Since an event is four-dimensional, the
boundary of an event is a three-dimensional manifold. For a finite event
there is a continuity of boundary; for a duration the boundary consists
of those event-particles which are covered by either of the two bounding
moments. Thus the boundary of a duration consists of two momentary
three-dimensional spaces. An event will be said to 'occupy' the
aggregate of event-particles which lie within it.
Two events which have 'junction' in the sense in which junction was
described in my last lecture, and yet are separated so that neither
event either overlaps or is part of the other event, are said to be
'adjoined.'
This relation of adjunction issues in a peculiar relation between the
boundaries of the two events. The two boundaries must have a common
portion which is in fact a continuous three-dimensional locus of
event-particles in the four-dimensional manifold.
A three-dimensional locus of event-particles which is the common portion
of the boundary of two adjoined events will be called a 'solid.' A solid
may or may not lie completely in one moment. A solid which does not lie
in one moment will be called 'vagrant.' A solid which does lie in one
moment will be called a volume. A volume may be defined as the locus of
the event-particles in which a moment intersects an event, provided that
the two do intersect. The intersection of a moment and an event will
evidently consist of those event-particles which are covered by the
moment and lie in the event. The identity of the two definitions of a
volume is evident when we remember that an intersecting moment divides
the event into two adjoined events.
A solid as thus defined, whether it be vagrant or be a volume, is a mere
aggregate of event-particles illustrating a certain quality of position.
We can also define a solid as an abstractive element. In order to do so
we recur to the theory of primes explained in the preceding lecture. Let
the condition named {sigma} stand for the fact that each of the events
of any abstractive set satisfying it has all the event-particles of some
particular solid lying in it. Then the group of all the {sigma}-primes
is the abstractive element which is associated with the given solid. I
will call this abstractive element the solid as an abstractive element,
and I will call the aggregate of event-particles the solid as a locus.
The instantaneous volumes in instantaneous space which are the ideals of
our sense-perception are volumes as abstractive elements. What we really
perceive with all our efforts after exactness are small events far
enough down some abstractive set belonging to the volume as an
abstractive element.
It is difficult to know how far we approximate to any perception of
vagrant solids. We certainly do not think that we make any such
approximation. But then our thoughts--in the case of people who do think
about such topics--are so much under the control of the materialistic
theory of nature that they hardly count for evidence. If Einstein's
theory of gravitation has any truth in it, vagrant solids are of great
importance in science. The whole boundary of a finite event may be
looked on as a particular example of a vagrant solid as a locus. Its
particular property of being closed prevents it from being definable as
an abstractive element.
When a moment intersects an event, it also intersects the boundary of
that event. This locus, which is the portion of the boundary contained
in the moment, is the bounding surface of the corresponding volume of
that event contained in the moment. It is a two-dimensional locus.
The fact that every volume has a bounding surface is the origin of the
Dedekindian continuity of space.
Another event may be cut by the same moment in another volume and this
volume will also have its boundary. These two volumes in the
instantaneous space of one moment may mutually overlap in the familiar
way which I need not describe in detail and thus cut off portions from
each other's surfaces. These portions of surfaces are 'momental areas.'
It is unnecessary at this stage to enter into the complexity of a
definition of vagrant areas. Their definition is simple enough when the
four-dimensional manifold of event-particles has been more fully
explored as to its properties.
Momental areas can evidently be defined as abstractive elements by
exactly the same method as applied to solids. We have merely to
substitute 'area' for a 'solid' in the words of the definition already
given. Also, exactly as in the analogous case of a solid, what we
perceive as an approximation to our ideal of an area is a small event
far enough down towards the small end of one of the equal abstractive
sets which belongs to the area as an abstractive element.
Two momental areas lying in the same moment can cut each other in a
momental segment which is not necessarily rectilinear. Such a segment
can also be defined as an abstractive element. It is then called a
'momental route.' We will not delay over any general consideration of
these momental routes, nor is it important for us to proceed to the
still wider investigation of vagrant routes in general. There are
however two simple sets of routes which are of vital importance. One is
a set of momental routes and the other of vagrant routes. Both sets can
be classed together as straight routes. We proceed to define them
without any reference to the definitions of volumes and surfaces.
The two types of straight routes will be called rectilinear routes and
stations. Rectilinear routes are momental routes and stations are
vagrant routes. Rectilinear routes are routes which in a sense lie in
rects. Any two event-particles on a rect define the set of
event-particles which lie between them on that rect. Let the
satisfaction of the condition {sigma} by an abstractive set mean that
the two given event-particles and the event-particles lying between them
on the rect all lie in every event belonging to the abstractive set. The
group of {sigma}-primes, where {sigma} has this meaning, form an
abstractive element. Such abstractive elements are rectilinear routes.
They are the segments of instantaneous straight lines which are the
ideals of exact perception. Our actual perception, however exact, will
be the perception of a small event sufficiently far down one of the
abstractive sets of the abstractive element.
A station is a vagrant route and no moment can intersect any station in
more than one event-particle. Thus a station carries with it a
comparison of the positions in their respective moments of the
event-particles covered by it. Rects arise from the intersection of
moments. But as yet no properties of events have been mentioned by which
any analogous vagrant loci can be found out.
The general problem for our investigation is to determine a method of
comparison of position in one instantaneous space with positions in
other instantaneous spaces. We may limit ourselves to the spaces of the
parallel moments of one time-system. How are positions in these various
spaces to be compared? In other words, What do we mean by motion? It is
the fundamental question to be asked of any theory of relative space,
and like many other fundamental questions it is apt to be left
unanswered. It is not an answer to reply, that we all know what we mean
by motion. Of course we do, so far as sense-awareness is concerned. I am
asking that your theory of space should provide nature with something to
be observed. You have not settled the question by bringing forward a
theory according to which there is nothing to be observed, and by then
reiterating that nevertheless we do observe this non-existent fact.
Unless motion is something as a fact in nature, kinetic energy and
momentum and all that depends on these physical concepts evaporate from
our list of physical realities. Even in this revolutionary age my
conservatism resolutely opposes the identification of momentum and
moonshine.
Accordingly I assume it as an axiom, that motion is a physical fact. It
is something that we perceive as in nature. Motion presupposes rest.
Until theory arose to vitiate immediate intuition, that is to say to
vitiate the uncriticised judgments which immediately arise from
sense-awareness, no one doubted that in motion you leave behind that
which is at rest. Abraham in his wanderings left his birthplace where it
had ever been. A theory of motion and a theory of rest are the same
thing viewed from different aspects with altered emphasis.
Now you cannot have a theory of rest without in some sense admitting a
theory of absolute position. It is usually assumed that relative space
implies that there is no absolute position. This is, according to my
creed, a mistake. The assumption arises from the failure to make another
distinction; namely, that there may be alternative definitions of
absolute position. This possibility enters with the admission of
alternative time-systems. Thus the series of spaces in the parallel
moments of one temporal series may have their own definition of absolute
position correlating sets of event-particles in these successive spaces,
so that each set consists of event-particles, one from each space, all
with the property of possessing the same absolute position in that
series of spaces. 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.
In looking to nature for evidence of absolute position it is of no use
to recur to the four-dimensional manifold of event-particles. This
manifold has been obtained by the extension of thought beyond the
immediacy of observation. We shall find nothing in it except what we
have put there to represent the ideas in thought which arise from our
direct sense-awareness of nature. To find evidence of the properties
which are to be found in the manifold of event-particles we must always
recur to the observation of relations between events. Our problem is to
determine those relations between events which issue in the property of
absolute position in a timeless space. This is in fact the problem of
the determination of the very meaning of the timeless spaces of physical
science.
In reviewing the factors of nature as immediately disclosed in
sense-awareness, we should note the fundamental character of the percept
of 'being here.' We discern an event merely as a factor in a determinate
complex in which each factor has its own peculiar share.
There are two factors which are always ingredient in this complex, one
is the duration which is represented in thought by the concept of all
nature that is present now, and the other is the peculiar _locus standi_
for mind involved in the sense-awareness. This _locus standi_ in nature
is what is represented in thought by the concept of 'here,' namely of an
'event here.'
This is the concept of a definite factor in nature. This factor is an
event in nature which is the focus in nature for that act of awareness,
and the other events are perceived as referred to it. This event is part
of the associated duration. I call it the 'percipient event.' This event
is not the mind, that is to say, not the percipient. It is that in
nature from which the mind perceives. The complete foothold of the mind
in nature is represented by the pair of events, namely, the present
duration which marks the 'when' of awareness and the percipient event
which marks the 'where' of awareness and the 'how' of awareness. This
percipient event is roughly speaking the bodily life of the incarnate
mind. But this identification is only a rough one. For the functions of
the body shade off into those of other events in nature; so that for
some purposes the percipient event is to be reckoned as merely part of
the bodily life and for other purposes it may even be reckoned as more
than the bodily life. In many respects the demarcation is purely
arbitrary, depending upon where in a sliding scale you choose to draw
the line.
I have already in my previous lecture on Time discussed the association
of mind with nature. The difficulty of the discussion lies in the
liability of constant factors to be overlooked. We never note them by
contrast with their absences. The purpose of a discussion of such
factors may be described as being to make obvious things look odd. We
cannot envisage them unless we manage to invest them with some of the
freshness which is due to strangeness.
It is because of this habit of letting constant factors slip from
consciousness that we constantly fall into the error of thinking of the
sense-awareness of a particular factor in nature as being a two-termed
relation between the mind and the factor. For example, I perceive a
green leaf. Language in this statement suppresses all reference to any
factors other than the percipient mind and the green leaf and the
relation of sense-awareness. It discards the obvious inevitable factors
which are essential elements in the perception. I am here, the leaf is
there; and the event here and the event which is the life of the leaf
there are both embedded in a totality of nature which is now, and within
this totality there are other discriminated factors which it is
irrelevant to mention. Thus language habitually sets before the mind a
misleading abstract of the indefinite complexity of the fact of
sense-awareness.
What I now want to discuss is the special relation of the percipient
event which is 'here' to the duration which is 'now.' This relation is a
fact in nature, namely the mind is aware of nature as being with these
two factors in this relation.
Within the short present duration the 'here' of the percipient event has
a definite meaning of some sort. This meaning of 'here' is the content
of the special relation of the percipient event to its associated
duration. I will call this relation 'cogredience.' Accordingly I ask for
a description of the character of the relation of cogredience. The
present snaps into a past and a present when the 'here' of cogredience
loses its single determinate meaning. There has been a passage of nature
from the 'here' of perception within the past duration to the different
'here' of perception within the present duration. But the two 'heres' of
sense-awareness within neighbouring durations may be indistinguishable.
In this case there has been a passage from the past to the present, but
a more retentive perceptive force might have retained the passing nature
as one complete present instead of letting the earlier duration slip
into the past. Namely, the sense of rest helps the integration of
durations into a prolonged present, and the sense of motion
differentiates nature into a succession of shortened durations. As we
look out of a railway carriage in an express train, the present is past
before reflexion can seize it. We live in snippits too quick for
thought. On the other hand the immediate present is prolonged according
as nature presents itself to us in an aspect of unbroken rest. Any
change in nature provides ground for a differentiation among durations
so as to shorten the present. But there is a great distinction between
self-change in nature and change in external nature. Self-change in
nature is change in the quality of the standpoint of the percipient
event. It is the break up of the 'here' which necessitates the break up
of the present duration. Change in external nature is compatible with a
prolongation of the present of contemplation rooted in a given
standpoint. What I want to bring out is that the preservation of a
peculiar relation to a duration is a necessary condition for the
function of that duration as a present duration for sense-awareness.
This peculiar relation is the relation of cogredience between the
percipient event and the duration. Cogredience is the preservation of
unbroken quality of standpoint within the duration. It is the
continuance of identity of station within the whole of nature which is
the terminus of sense-awareness. The duration may comprise change within
itself, but cannot--so far as it is one present duration--comprise
change in the quality of its peculiar relation to the contained
percipient event.
In other words, perception is always 'here,' and a duration can only be
posited as present for sense-awareness on condition that it affords one
unbroken meaning of 'here' in its relation to the percipient event. It
is only in the past that you can have been 'there' with a standpoint
distinct from your present 'here.'
Events there and events here are facts of nature, and the qualities of
being 'there' and 'here' are not merely qualities of awareness as a
relation between nature and mind. The quality of determinate station in
the duration which belongs to an event which is 'here' in one
determinate sense of 'here' is the same kind of quality of station which
belongs to an event which is 'there' in one determinate sense of
'there.' Thus cogredience has nothing to do with any biological
character of the event which is related by it to the associated
duration. This biological character is apparently a further condition
for the peculiar connexion of a percipient event with the percipience of
mind; but it has nothing to do with the relation of the percipient event
to the duration which is the present whole of nature posited as the
disclosure of the percipience.
Given the requisite biological character, the event in its character of
a percipient event selects that duration with which the operative past
of the event is practically cogredient within the limits of the
exactitude of 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 subordinate 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 perceptions
under certain circumstances the events discriminated 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 percipient 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 duration when it is part of the duration
and is intersected by any moment which lies in the duration. Such an
event begins with the duration and ends with it. Furthermore every event
which begins with the duration 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 travelling 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'.
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 property {sigma} 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 {sigma}-primes, where {sigma} has this meaning, is an
abstractive element and is the station of P in d as an abstractive
element. The locus of event-particles covered by the station of P in d
as an abstractive element is the station of P in d as a locus. A station
has accordingly the usual three characters, namely, its character of
position, its extrinsic 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 indefinitely backwards and forwards.
Such a prolonged station will be called a point-track. A point-track is
a locus of event-particles. It is defined by reference to one particular
time-system, {alpha} 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 {alpha} is the
group of points of the timeless space of {alpha}. Each such point
indicates a certain quality of absolute position in reference to the
durations of the family associated with {alpha}, and thence in reference
to the successive instantaneous spaces lying in the successive moments
of {alpha}. Each moment of {alpha} 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 {alpha} should be at the
event-particles where those moments intersect a given point of {alpha}
will be at rest in the timeless space of time-system {alpha}. 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 {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-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 {rho}. Thus {rho} is the instantaneous rect in M which occupies at
the moment M the straight line r in the space of {alpha}. 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 {rho} which is the apparent path on the assumption of uniform
motion. But the actual rect {rho} which is a locus of event-particles is
never traversed by the being. These event-particles 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 {alpha} as those occupied by the
event-particles of the rect {rho}. For example, we see a stretch of road
and a lorry moving along it. The instantaneously seen road is a portion
of the rect {rho}--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 space {beta}. Then the straight line r
of space {alpha} is in the direction of {beta} in space {alpha}, and the
rect {rho} is the representative in the moment M of the line r of space
{alpha}. The direction of {rho} in the instantaneous space of the moment
M is the direction of {beta} in M, where M is a moment of time-system
{alpha}. Again the matrix of the line r of space {alpha} will also be
the matrix of some line s of space {beta} which will be in the direction
of {alpha} in space {beta}. Thus if the lorry halts at some point P of
space {alpha} which lies on the line r, it is now moving along the line
s of space {beta}. This is the theory of relative motion; the common
matrix is the bond which connects the motion of {beta} in space {alpha}
with the motions of {alpha} in space {beta}.
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 perceived 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 {alpha} and {beta}, each
with its own timeless space and its own family of instantaneous moments
with their instantaneous spaces. Let M and N be respectively a moment of
{alpha} and a moment of {beta}. In M there is the direction of {beta}
and in N there is the direction of {alpha}. But M and N, being moments
of different time-systems, intersect in a level. Call this level
{lambda}. Then {lambda} 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 {lambda} is perpendicular to
the direction of {beta} in M, and in the instantaneous space of N the
level {lambda} is perpendicular to the direction of {alpha} in N. This
is the fundamental property which forms the definition of
perpendicularity. The symmetry of perpendicularity is a particular
instance 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 congruence is deduced.
The theory of perpendicularity in the timeless space of any time-system
{alpha} follows immediately from this theory of perpendicularity in each
of its instantaneous spaces. Let {rho} be any rect in the moment M of
{alpha} and let {lambda} be a level in M which is perpendicular to
{rho}. The locus of those points of the space of {alpha} which intersect
M in event-particles on {rho} is the straight line r of space {alpha},
and the locus of those points of the space of {alpha} which intersect M
in event-particles on {lambda} is the plane l of space {alpha}. Then the
plane l is perpendicular to the line r.
In this way we have pointed out unique and definite properties in nature
which correspond to perpendicularity. We shall find that this discovery
of definite unique properties defining perpendicularity is of critical
importance in the theory of congruence which is the topic for the next
lecture.
I regret that it has been necessary for me in this lecture to administer
such a large dose of four-dimensional geometry. I do not apologise,
because I am really not responsible for the fact that nature in its most
fundamental aspect is four-dimensional. Things are what they are; and it
is useless to disguise the fact that 'what things are' is often very
difficult for our intellects to follow. It is a mere evasion of the
ultimate problems to shirk such obstacles.