CHAPTER XIPOINTS AND STRAIGHT LINES41. Stations.41.1The fact that an event is 'cogredient' with a duration is a fundamental fact not to be explained purely in terms of extension. It has been pointed out inPart IIthat the exact concept of cogredience is 'Here throughout the duration' or 'There throughout the duration.' Let this fundamental relation of finite events to durations be denoted by ',' and let '' mean 'is a finite event which is cogredient with the duration.'41.2A 'stationary prime' within a durationis a prime whose formative condition () is that of being a simple abstractive class, such that each of its members extends over events which (i) are inhered in by some assigned event-particleinherent inand (ii) have the relationto. This formative condition is regular for primes. A 'station' within a durationis the abstractive element deduced from a stationary prime within.41.3Each event-particle in a duration is covered by one and only one station in that duration; and any event-particle covered by a station can be taken as the 'assigned event-particle' of the formative condition, inherent in every event which is a member of the station. Every station is a route; and also every station in a duration intersects every moment of that duration [i.e. inherent in it] in one and only one event-particle, and intersects no other moments of that time-system. It will be noted that a station is associated with a definite time-system, namely the time-system corresponding to its duration.41.4A station of one time-system either does not intersect a station of another time-system or intersects it in one event-particle only. Thus stations belong to the type of routes which have been denominated 'kinematic routes.' Each station exhibits an unchanging meaning of 'here' throughout the duration in which it is a station; namely, every event-particle in a station is 'here' in the duration in the same sense of 'here' as for every other event-particle in that station.42. Point-Tracks and Points.42.1Consider all the durations belonging to one time-system. Of these durations some intersect each other, and some are parts of others. Thus any event-particleis covered by many durations of this time-system, and lies in stations corresponding to these durations. We have now to consider the relations to each other of these various stations, each containing. The fundamental theorem is as follows: Ifand′ be durations of the same time-system, andextends over′, and ifbe an event-particle inhering in′ andand′ be the stations ofinand′ respectively, thencovers′. In other words used in less technical senses, If′ be part of, then′ is part of.42.2Any given stationin a durationcan thus be indefinitely prolonged throughout the time-system to whichbelongs. For letbe any other duration of the same time-system which intersectsin the duration′ and also extends beyond. Then the part ofwhich is included in′, namely′ (say), is a station in′. Also there is one and only one station in,(say), which covers′; and no other station incovers any event-particle of′. In this way the stationis prolonged in the time-system by the addition of the station, and so on indefinitely. The complete locus of event-particles thus defined by the indefinite prolongation of a station throughout its associated time-system is called a 'point-track.'A point-track intersects any moment of any time-system in one and only one event-particle.42.3Each point-track has a unique association with the time-system in which the routes lying on it are stations. A point-track is called a 'point' in the 'space of its associated time-system.' This space of a time-system is called 'time-less' because its points have no special relation to any one moment of its associated time-system.Each event-particle is contained in one and only one point of each time-system, and will be said to 'occupy' such a point. Two points of the same time-system never intersect; two point-tracks which are respectively points in the spaces of different time-systems either do not intersect or intersect in one event-particle only.Since each point-track intersects any moment in one and only one event-particle, two co-momental event-particles cannot lie on the same point-track. A pair of sequent event-particles lie in one and only one point-track, apart from exceptional cases when they lie in 'null-tracks.' Null-tracks are introduced later inarticle 45.42.4In the four-dimensional geometry of event-particles it has already been pointed out that rects have the character of straight lines, but that since sequent event-particles do not lie on the same rect there is a missing set of straight lines required to complete the geometry. Point-tracks [together with the exceptional set of loci termed 'null-tracks'] form this missing set of straight lines for this geometry of event-particles.The event-particles occupying a point-track have an order derived from the covering moments of any time-system. Those on a null-track have an order derived from routes which it is not necessary to discuss.43. Parallelism.43.1A theory of parallelism holds for point-tracks and can be connected with the analogous theory for rects. Point-tracks which are points in the space of the same time-system are called 'parallel.' Thus a complete family of parallel point-tracks is merely a complete family of points in the space of some time-system. The parallelism of point-tracks is evidently transitive, symmetrical and reflexive. The definition of the parallelism of stations is derived from that of point-tracks.43.2The parallelism of point-tracks and the parallelism of rects and moments are interconnected. Letbe any rect in a moment, and letbe any family of parallel point-tracks. Then a certain set of point-tracks belonging towill intersect, and this set will intersect any moment parallel toin a rect parallel to. Again letbe any point-track and letbe any complete family of parallel rects. Then a certain set of rects belonging towill intersect; name it. Letbe any event-particle on some member of; then the point-track containingand parallel towill intersect every member of.43.3A theorem analogous to those of43.2also holds for two families of point-tracks. Letbe any point-track and letbe any family of parallel point-tracks to whichdoes not belong. Then a certain set of point-tracks belonging towill intersect; name it. Let P be any event-particle occupying some member of; then the point-track occupied byand parallel towill intersect every member of.This theorem, the theorems of43.2and the corresponding theorem for two families of parallel rects are examples of the repetition property of parallelism. It is evident that, given any three event-particles not on one rect or one point-track, a parallelogram can be completed of which the three event-particles are three corners, any one of the event-particles being at the junction of the adjacent sides through the three corners. In such a parallelogram opposite sides are always of the same denomination, namely both rects or both point-tracks; but adjacent sides may be of opposite denominations.43.4The event-particles occupying a pointin the time-less space of a time-system a appear at the successive moments ofas successively occupying the same point. Ifbe any other time-system, then the pointof the space ofintersects a series of points of the space ofin event-particles which lie on the successive moments of. These event-particles ofthus occupy a succession of points ofat a succession of moments of; and we shall find that this locus of points is what is meant by a straight line in the space of. Thus the pointin the space ofcorrelates the successive points on a straight line ofwith the successive moments of. Thus in the space ofthe pointof the space ofappears as exemplifying the kinematical conception of a moving material particle traversing a straight line. It will appear later that, owing to the 'repetition property' of parallelism, the motion is uniform.44. Matrices.44.1A level is obtained by taking a rectand an event-particleco-momental with, and by forming the locus of event-particles on rects throughand intersecting, including also particles on the rect throughand parallel to.The same level would be obtained by taking the particles on the rects intersectingand parallel to some one rect throughwhich intersects.44.2Analogously to levels, a locus of event-particles called a 'matrix' is obtained by taking a rectand an event-particlewhich is not co-momental with, and by forming the locus of event-particles on rects or point-tracks throughand intersecting, including also the event-particles on the rect throughand parallel to.A 'matrix' is a two-dimensional plane in the four-dimensional geometry of event-particles. Levels and matrices together make up the complete set of such two-dimensional planes, and have the usual properties of such planes which need not be detailed here.44.3Matrices are also obtained by taking an event-particleand a point-track, and by forming the locus of event-particles on rects or point-tracks throughand intersecting, including also event-particles on the point-track throughand parallel to. Any matrix can be generated in either of the two ways. Furthermore matrices can be generated by the use of parallels in the same way as levels are generated as explained in44.1and as assumed in43.4.45. Null-Tracks.45.1The relations between rects and point-tracks are best understood by taking a rectand a particlewhich is not co-level with. In this way a matrix is obtained as explained in44.2.fig09Fig. 9.Then in respect tothe rectis divided into three (logical) parts by two event-particlesand. The segment betweenandhas the property that any event-particle on it is joined toby a point-track [e.g.in the figure]; and either of the two infinite segments, namely that beyondand that beyond, is such that any event-particle on it is joined toby a rect [e.g.′ and″ in the figure]. The above diagram and succeeding diagrams have the defect of representing matrices by levels, and thus of giving the conceptions an undeserved air of paradox.Again we may take an event-particleand a point-tracknot containing. In this way a matrix is obtained as explained in44.3.fig10Fig. 10.Then in respect tothe point-trackis divided by two event-particlesandinto three (logical) parts. The segment betweenandhas the property that any event-particle on it is joined toby a rect [e.g.in the figure]; and either of the two infinite segments, respectively beyondand beyond, is such that any event-particle on it is joined toby a point-track [e.g.′ and″ in the figure].45.2It is evident therefore that a matrix in respect to an event-particle P lying on it is separated into four regions by two lociandwhich may equally well be termed rects or point-tracks.fig11Fig. 11.The event-particles in the vertically opposed regionsandare joined toby rects; and the event-particles in the vertically opposed regionsandare joined toby point-tracks.The loci which bound the regions separating point-tracks from rects will be called 'null-tracks.' Their special properties will be considered later when congruence has been introduced. In any matrix there are two families of parallel null-tracks; and there is one member of each family passing through each event-particle on the rectilinear track. The order of event-particles on a null-track is derived from its intersection with systems of parallel rects [not co-momental] or of parallel point-tracks or from the orders on routes lying on it.46. Straight Lines.46.1There is evidently an important theory of parallelism for families of matrices analogous to the theory of parallels for families of levels. The detailed properties need not be elaborated here.Two matrices may either (i) be parallel, or (ii) intersect in one event-particle only, or (iii) intersect in a rect, or (iv) intersect in a point-track, or (v) intersect in a null-track. For the intersection of two levels only cases (i), (ii) and (iii) can occur; for the intersection of a level and a matrix only cases (ii) and (iii) can occur.46.2Each matrix contains various sets of parallel point-tracks. Any one such set is a locus of points in the space of some time-system. Such a locus of points is called a 'straight line' in the space of the time-system.A matrix which contains the points of a straight line in the space of any time-systemwill be called 'an associated matrix for,' and it is called 'the matrix including' that straight line.A matrix is an associated matrix for many time-systems, but it is the matrix including only one straight line in each corresponding space. The family of time-systems for which a given matrix is an associated matrix is called a 'collinear' family. A whole family of parallel matrices are associated matrices for the same collinear family of time-systems, if any one matrix of the family is thus associated. In the space of any one time-system the straight lines included by a family of parallel associated matrices are said to be parallel.46.3A matrix intersects a moment in a rect. If the moment belong to a time-system with which the matrix is associated, this rect in the moment corresponds to the straight line included by the matrix in the sense that it has one particle occupying each of its points. A rect thus associated with a straight line will be said to 'occupy' it.Thus the event-particles on a matrixassociated with a time-systemcan be exhaustively grouped into mutually exclusive subsets in two distinct ways: (i)They can be grouped into the points ofwhich lie on; this locus of points is the included straight line in the space of, which we will name: (ii) The event-particles oncan be grouped into the sets of parallel rects which are the intersections ofwith the moments of, and thus each of these rects occupies.46.4There are three different types of meaning which can be given to the idea of 'space' in connection with external nature, (i) There is the four-dimensional space of which event-particles are the points and the rects and point-tracks and null-tracks are the straight lines. In the geometry of this space there is a lack of uniformity between the congruence theories for rects and for point-tracks, and no such theory for null-tracks, (ii) There are the three-dimensional momentary (instantaneous) spaces in the moments of any time-system, of which event-particles are the points and rects are the straight lines. The observed space of ordinary perception is an approximation to this exact concept, (iii) There is the time-less three-dimensional space of the time-system, of which point-tracks are the points and matrices include the straight lines. This is the space of physical science.There is an exact correlation between the time-less space of a time-system and any momentary space of the same time-system. For any point of the momentary space is an event-particle which occupies one and only one point of the time-less space; and any straight line of the momentary space is a rect which lies in one associated matrix including one straight line of the time-less space, or (in other words) each straight line of the momentary space occupies a straight line of the time-less space.A time-system corresponds to a consentient set of the Newtonian group, and the time-less space of the time-system is the space of the corresponding consentient group.
41. Stations.41.1The fact that an event is 'cogredient' with a duration is a fundamental fact not to be explained purely in terms of extension. It has been pointed out inPart IIthat the exact concept of cogredience is 'Here throughout the duration' or 'There throughout the duration.' Let this fundamental relation of finite events to durations be denoted by ',' and let '' mean 'is a finite event which is cogredient with the duration.'
41.2A 'stationary prime' within a durationis a prime whose formative condition () is that of being a simple abstractive class, such that each of its members extends over events which (i) are inhered in by some assigned event-particleinherent inand (ii) have the relationto. This formative condition is regular for primes. A 'station' within a durationis the abstractive element deduced from a stationary prime within.
41.3Each event-particle in a duration is covered by one and only one station in that duration; and any event-particle covered by a station can be taken as the 'assigned event-particle' of the formative condition, inherent in every event which is a member of the station. Every station is a route; and also every station in a duration intersects every moment of that duration [i.e. inherent in it] in one and only one event-particle, and intersects no other moments of that time-system. It will be noted that a station is associated with a definite time-system, namely the time-system corresponding to its duration.
41.4A station of one time-system either does not intersect a station of another time-system or intersects it in one event-particle only. Thus stations belong to the type of routes which have been denominated 'kinematic routes.' Each station exhibits an unchanging meaning of 'here' throughout the duration in which it is a station; namely, every event-particle in a station is 'here' in the duration in the same sense of 'here' as for every other event-particle in that station.
42. Point-Tracks and Points.42.1Consider all the durations belonging to one time-system. Of these durations some intersect each other, and some are parts of others. Thus any event-particleis covered by many durations of this time-system, and lies in stations corresponding to these durations. We have now to consider the relations to each other of these various stations, each containing. The fundamental theorem is as follows: Ifand′ be durations of the same time-system, andextends over′, and ifbe an event-particle inhering in′ andand′ be the stations ofinand′ respectively, thencovers′. In other words used in less technical senses, If′ be part of, then′ is part of.
42.2Any given stationin a durationcan thus be indefinitely prolonged throughout the time-system to whichbelongs. For letbe any other duration of the same time-system which intersectsin the duration′ and also extends beyond. Then the part ofwhich is included in′, namely′ (say), is a station in′. Also there is one and only one station in,(say), which covers′; and no other station incovers any event-particle of′. In this way the stationis prolonged in the time-system by the addition of the station, and so on indefinitely. The complete locus of event-particles thus defined by the indefinite prolongation of a station throughout its associated time-system is called a 'point-track.'
A point-track intersects any moment of any time-system in one and only one event-particle.
42.3Each point-track has a unique association with the time-system in which the routes lying on it are stations. A point-track is called a 'point' in the 'space of its associated time-system.' This space of a time-system is called 'time-less' because its points have no special relation to any one moment of its associated time-system.
Each event-particle is contained in one and only one point of each time-system, and will be said to 'occupy' such a point. Two points of the same time-system never intersect; two point-tracks which are respectively points in the spaces of different time-systems either do not intersect or intersect in one event-particle only.
Since each point-track intersects any moment in one and only one event-particle, two co-momental event-particles cannot lie on the same point-track. A pair of sequent event-particles lie in one and only one point-track, apart from exceptional cases when they lie in 'null-tracks.' Null-tracks are introduced later inarticle 45.
42.4In the four-dimensional geometry of event-particles it has already been pointed out that rects have the character of straight lines, but that since sequent event-particles do not lie on the same rect there is a missing set of straight lines required to complete the geometry. Point-tracks [together with the exceptional set of loci termed 'null-tracks'] form this missing set of straight lines for this geometry of event-particles.
The event-particles occupying a point-track have an order derived from the covering moments of any time-system. Those on a null-track have an order derived from routes which it is not necessary to discuss.
43. Parallelism.43.1A theory of parallelism holds for point-tracks and can be connected with the analogous theory for rects. Point-tracks which are points in the space of the same time-system are called 'parallel.' Thus a complete family of parallel point-tracks is merely a complete family of points in the space of some time-system. The parallelism of point-tracks is evidently transitive, symmetrical and reflexive. The definition of the parallelism of stations is derived from that of point-tracks.
43.2The parallelism of point-tracks and the parallelism of rects and moments are interconnected. Letbe any rect in a moment, and letbe any family of parallel point-tracks. Then a certain set of point-tracks belonging towill intersect, and this set will intersect any moment parallel toin a rect parallel to. Again letbe any point-track and letbe any complete family of parallel rects. Then a certain set of rects belonging towill intersect; name it. Letbe any event-particle on some member of; then the point-track containingand parallel towill intersect every member of.
43.3A theorem analogous to those of43.2also holds for two families of point-tracks. Letbe any point-track and letbe any family of parallel point-tracks to whichdoes not belong. Then a certain set of point-tracks belonging towill intersect; name it. Let P be any event-particle occupying some member of; then the point-track occupied byand parallel towill intersect every member of.
This theorem, the theorems of43.2and the corresponding theorem for two families of parallel rects are examples of the repetition property of parallelism. It is evident that, given any three event-particles not on one rect or one point-track, a parallelogram can be completed of which the three event-particles are three corners, any one of the event-particles being at the junction of the adjacent sides through the three corners. In such a parallelogram opposite sides are always of the same denomination, namely both rects or both point-tracks; but adjacent sides may be of opposite denominations.
43.4The event-particles occupying a pointin the time-less space of a time-system a appear at the successive moments ofas successively occupying the same point. Ifbe any other time-system, then the pointof the space ofintersects a series of points of the space ofin event-particles which lie on the successive moments of. These event-particles ofthus occupy a succession of points ofat a succession of moments of; and we shall find that this locus of points is what is meant by a straight line in the space of. Thus the pointin the space ofcorrelates the successive points on a straight line ofwith the successive moments of. Thus in the space ofthe pointof the space ofappears as exemplifying the kinematical conception of a moving material particle traversing a straight line. It will appear later that, owing to the 'repetition property' of parallelism, the motion is uniform.
44. Matrices.44.1A level is obtained by taking a rectand an event-particleco-momental with, and by forming the locus of event-particles on rects throughand intersecting, including also particles on the rect throughand parallel to.
The same level would be obtained by taking the particles on the rects intersectingand parallel to some one rect throughwhich intersects.
44.2Analogously to levels, a locus of event-particles called a 'matrix' is obtained by taking a rectand an event-particlewhich is not co-momental with, and by forming the locus of event-particles on rects or point-tracks throughand intersecting, including also the event-particles on the rect throughand parallel to.
A 'matrix' is a two-dimensional plane in the four-dimensional geometry of event-particles. Levels and matrices together make up the complete set of such two-dimensional planes, and have the usual properties of such planes which need not be detailed here.
44.3Matrices are also obtained by taking an event-particleand a point-track, and by forming the locus of event-particles on rects or point-tracks throughand intersecting, including also event-particles on the point-track throughand parallel to. Any matrix can be generated in either of the two ways. Furthermore matrices can be generated by the use of parallels in the same way as levels are generated as explained in44.1and as assumed in43.4.
45. Null-Tracks.45.1The relations between rects and point-tracks are best understood by taking a rectand a particlewhich is not co-level with. In this way a matrix is obtained as explained in44.2.
fig09Fig. 9.
Fig. 9.
Fig. 9.
Then in respect tothe rectis divided into three (logical) parts by two event-particlesand. The segment betweenandhas the property that any event-particle on it is joined toby a point-track [e.g.in the figure]; and either of the two infinite segments, namely that beyondand that beyond, is such that any event-particle on it is joined toby a rect [e.g.′ and″ in the figure]. The above diagram and succeeding diagrams have the defect of representing matrices by levels, and thus of giving the conceptions an undeserved air of paradox.
Again we may take an event-particleand a point-tracknot containing. In this way a matrix is obtained as explained in44.3.
fig10Fig. 10.
Fig. 10.
Fig. 10.
Then in respect tothe point-trackis divided by two event-particlesandinto three (logical) parts. The segment betweenandhas the property that any event-particle on it is joined toby a rect [e.g.in the figure]; and either of the two infinite segments, respectively beyondand beyond, is such that any event-particle on it is joined toby a point-track [e.g.′ and″ in the figure].
45.2It is evident therefore that a matrix in respect to an event-particle P lying on it is separated into four regions by two lociandwhich may equally well be termed rects or point-tracks.
fig11Fig. 11.
Fig. 11.
Fig. 11.
The event-particles in the vertically opposed regionsandare joined toby rects; and the event-particles in the vertically opposed regionsandare joined toby point-tracks.
The loci which bound the regions separating point-tracks from rects will be called 'null-tracks.' Their special properties will be considered later when congruence has been introduced. In any matrix there are two families of parallel null-tracks; and there is one member of each family passing through each event-particle on the rectilinear track. The order of event-particles on a null-track is derived from its intersection with systems of parallel rects [not co-momental] or of parallel point-tracks or from the orders on routes lying on it.
46. Straight Lines.46.1There is evidently an important theory of parallelism for families of matrices analogous to the theory of parallels for families of levels. The detailed properties need not be elaborated here.
Two matrices may either (i) be parallel, or (ii) intersect in one event-particle only, or (iii) intersect in a rect, or (iv) intersect in a point-track, or (v) intersect in a null-track. For the intersection of two levels only cases (i), (ii) and (iii) can occur; for the intersection of a level and a matrix only cases (ii) and (iii) can occur.
46.2Each matrix contains various sets of parallel point-tracks. Any one such set is a locus of points in the space of some time-system. Such a locus of points is called a 'straight line' in the space of the time-system.
A matrix which contains the points of a straight line in the space of any time-systemwill be called 'an associated matrix for,' and it is called 'the matrix including' that straight line.
A matrix is an associated matrix for many time-systems, but it is the matrix including only one straight line in each corresponding space. The family of time-systems for which a given matrix is an associated matrix is called a 'collinear' family. A whole family of parallel matrices are associated matrices for the same collinear family of time-systems, if any one matrix of the family is thus associated. In the space of any one time-system the straight lines included by a family of parallel associated matrices are said to be parallel.
46.3A matrix intersects a moment in a rect. If the moment belong to a time-system with which the matrix is associated, this rect in the moment corresponds to the straight line included by the matrix in the sense that it has one particle occupying each of its points. A rect thus associated with a straight line will be said to 'occupy' it.
Thus the event-particles on a matrixassociated with a time-systemcan be exhaustively grouped into mutually exclusive subsets in two distinct ways: (i)They can be grouped into the points ofwhich lie on; this locus of points is the included straight line in the space of, which we will name: (ii) The event-particles oncan be grouped into the sets of parallel rects which are the intersections ofwith the moments of, and thus each of these rects occupies.
46.4There are three different types of meaning which can be given to the idea of 'space' in connection with external nature, (i) There is the four-dimensional space of which event-particles are the points and the rects and point-tracks and null-tracks are the straight lines. In the geometry of this space there is a lack of uniformity between the congruence theories for rects and for point-tracks, and no such theory for null-tracks, (ii) There are the three-dimensional momentary (instantaneous) spaces in the moments of any time-system, of which event-particles are the points and rects are the straight lines. The observed space of ordinary perception is an approximation to this exact concept, (iii) There is the time-less three-dimensional space of the time-system, of which point-tracks are the points and matrices include the straight lines. This is the space of physical science.
There is an exact correlation between the time-less space of a time-system and any momentary space of the same time-system. For any point of the momentary space is an event-particle which occupies one and only one point of the time-less space; and any straight line of the momentary space is a rect which lies in one associated matrix including one straight line of the time-less space, or (in other words) each straight line of the momentary space occupies a straight line of the time-less space.
A time-system corresponds to a consentient set of the Newtonian group, and the time-less space of the time-system is the space of the corresponding consentient group.