CHAPTER XVMATERIAL OBJECTS56. Material Objects.56.1A material object is essentially a material object of a certain definite sort; namely, we define sorts of material objects, which are sets of objects with certain definite peculiarities, and a material object is such because it is a member of one of these sorts. For example a piece of wood is a material object because it belongs to the class of wooden objects and because this class possesses the requisite peculiarities. Similarly a charge of electricity is a material object for an analogous reason.The objects which compose a set () form a sort of 'material' objects when (i) the objects of the setare all uniform, (ii) not more than one member ofcan be located in any volume, (iii) no member ofcan be located in two volumes of the same moment, (iv) ifandbe two members ofrespectively located in non-overlapping volumes in the same moment, then any pair of situations ofandrespectively are separated events, (v) ifbe a member ofsituated in an event, and located in the volumewhich is a section of, andbe any volume which is a portion of, then there is a member ofwhich is located inand is a concurrent component of.56.2Ifbe a material object of a certain sort andbe a volume in whichis located andbe a portion of, then the material object of the same sort aswhich is located inis called an 'extensive component' of.56.3It is by means of the properties of material objects that the atomic properties of objects are combined in mathematical calculations with the extensive continuity of events. Apart from material objects mathematical physics as at present developed would be impossible. For example where the physicist sees the electron as an atomic whole, the mathematician sees a distribution of electricity continuous in time and in space and capable of division into component objects which are also analogous distributions.57. Stationary Events.57.1In order to understand the theory of the motion of material objects, it is first necessary to define the concept of a 'stationary' event. Consider some given time-system, and letdenote a volume lying in a certain momentof this time-system. Letbe a duration of bounded by momentsand, and inhered in by; so that,,are three parallel moments of the time-system, andlies betweenand. The volumeis the locus of a set of event-particles and each of these event-particles lies in one and only one station of the duration. Also each station ofeither does not intersector intersects it in one event-particle only. The assemblage of event-particles lying on stations of d which intersect[namely, each event-particle lying on one of these stations] is the complete set of event-particles analysing[8]an event. Such an event is called stationary in the time-systemand stretches throughout the duration. It can also be called 'stationary in,' sincedefines the time-system. Every event-particle within the event lies on a station of; and a station ofeither has all its event-particles lying within the event or none of them. The volumeis the section of the event by the moment. Furthermore if′ be any other moment of the time-systemit lying betweenand, it intersects the event in a volume′ which is a geometrical replica of the volume. The momentsandwhich bound the durationare the terminal moments of any event which is stationary in. The stations oflying in the event intersectandin terminal volumesandwhich are geometrical replicas ofand′. A volume, such as', in which a moment ofintersects an event stationary inis called a 'normal cross-section' of the event. A moment of another time-systemwhich intersects the stationary event in a volume, but does not intersect either of the terminal volumes, is said to intersect it in an 'oblique cross-section.' All the oblique cross-sections of a stationary event which are made by moments of the same time-system are geometrical replicas of each other.57.2Consider an eventstationary in the time-system, and letbe another time-system. Letbe the measure of the normal cross-sections ofand letbe the measure of the oblique cross-sections made by moments of. We require the ratio ofto. Take (as usual) mutual axes forand, and let the event-particle which is the origin lie inwhich is the antecedent terminal moment of. Thenis at the-time zero, and let(the subsequent terminal moment) be at the-time. Then if () be the-coordinates of the event-particle in which a station(of the set composing the event) intersects, the-coordinates of the other end (the subsequent end) ofinare ().Furthermore let () be the-coordinates of the antecedent end of, and let () be the-coordinates of the subsequent end of. Then by the usual formulae [cf. subarticle52.2]But, by analogous reasoning to that for the elementary case of geometrical parallelograms, the absolute extent of the eventcan be expressed asand as). Hence.57.3The stations ofduration oftime-systemare portions of points of the time-less space of[the-space].Thus by prolonging the stations which constitute the stationary eventwe obtain the assemblage of-points which is the complete assemblage of-points intersecting the cross-sections of, each event-particle in each cross-section lying on one and only one such-point and each of these-points intersecting each cross-section in one event-particle. The assemblage of these-points is a volume of the-space, and the successive instantaneous volumes which are the normal cross-sections of[stationary in] each occupy this same volume in the-space. Thus the stationary eventduring the lapse of-time throughout which it endures is happening at the same place in the-space.But the successive oblique cross-sections offormed by moments of another time-systemare instantaneous volumes which successively occupy different volumes in the-space. These instantaneous volumes travel in the-space, sweeping over it with the uniform velocity, namely the velocity due to the time-systemin the-space.57.4A 'normal slice' of a stationary event is the slice of it cut off between any two normal cross-sections. An 'oblique slice' of a stationary event is the slice of it cut off between any two parallel oblique cross-sections. A normal slice of a stationary event is itself a stationary event in the same time-system.58. Motion of Objects.58.1A material object is 'motionless' within a duration when throughout that duration the material object and its extensive components are all situated in stationary events.In the case of a motionless material object, Law I for uniform objects can be made more precise, as follows:Ifbe a material object motionless in the durationandbe the stationary event extending throughoutin which it is situated, thenis situated in any oblique slice of.The accompanying figures illustrate (i) the kind of slice which is included in this law and (ii) the kind of slice which is excluded.fig19Fig. 19.It immediately follows that—with the nomenclature of the enunciation of the law—is located in every oblique cross-section of.Ifbe the time-system of the duration in whichis motionless and, in some other time-system,′ be the duration of maximum extent which intersectsin an oblique slice, then throughout′ in the time-less space of a the material objecthas a uniform motion of translation with the velocity ofin.58.2This property, possessed by a material object which is motionless in a time-system, of being situated in every oblique slice of its stationary situation is a fundamental physical law of nature. Namely, percipients cogredient with different time-systems can 'recognise' the same material objects. In other words, the character of a material object is not altered by its motion.58.3The motion of a material objectis 'regular' when ifbe any volume in which it is located andbe any event-particle in, and′ be any variable volume which containsand is a portion of, and′ be the extensive component ofwhich is located in′, then, as′ is progressively diminished without limit, a time-systemcan be found such that the errors of calculations, respecting magnitudes exhibited by′ which assume that′ is motionless in, tend to the limit zero, provided that the time-lapse of the durations inwithin which′ is motionless is also correspondingly diminished without limit.The above definition of regular motion is a description of the assumptions in the ordinary mathematical treatment of the motion of a material object (not necessarily rigid) which is not moving with a uniform motion of translation. Ifbe the standard time-system to which motions are referred, then the velocity ofinis the velocity at the event-particle[i.e. at the-space pointat the-time] of the material object.59. Extensive Magnitude.59.1A theory of extensive magnitude is required to complete the theory of material objects.Letand′ be two objects (material or otherwise), then the statement thatand′ possess quantities of a certain kind and that the ratio of the quantityto the quantity′ has a certain definite numerical value is a reference to some determinate method of comparison ofto′ which is the defining characteristic of that kind of quantity[9].The quantity of a certain kind possessed by a material objectis called 'extensive' when it is a determinate function of the quantities of the same kind possessed by any two of its extensive components which (i) are exhaustive ofand (ii) are non-overlapping [i.e. have no extensive component in common].If the determinate function be that of simple addition [so that,being the quantities possessed respectively byand its two extensive components,], then the kind of quantity will be called 'absolutely' extensive. When an extensive quantity is not absolutely extensive, it will be called 'semi-extensive.'59.2It is usual in philosophical discussions to con- fine the term 'extensive quantity' to what is here defined as 'absolutely extensive quantity,' and to ignore entirely the occurrence of semi-extensive quantities. But in physical science semi-extensive quantities are well known. For example, consider a sphere of radiusuniformly charged with electricity throughout its volume. Divide the sphere into two parts, namely a concentric nucleus of radiusand a shell of thickness. Then the electromagnetic mass of the whole sphere is not the sum of the electromagnetic masses of these two parts, but is to be calculated by a quadratic law from the charges.A material object expresses the spatial distribution of a quantity of 'material,' when the quantity is absolutely extensive.59.3The volume-density, at a timein the-space of a time-system, of the distribution of any absolutely extensive quantity possessed by a material objectis calculated by the ordinary mathematical formula. Consider any event-particleoccupying the-pointat the-time. Letbe the measure of a volume in the-space which contains; and let′ be the extensive component oflocated in, if there be such an extensive component. Letbe the measure of the quantity possessed by′. Then the limit of the ratio ofto, asis indefinitely diminished, is the density atat the timeof the material [i.e. of the absolutely extensive quantity].59.4The above definitions contemplate quantities immediately possessed by the extensive objects as such, for example, charges of electricity and intensities of sense-objects. But there are also quantities which are only mediately possessed by the objects, but are immediately possessed by the events which are their situations. Such quantities may vary with the variation in the situation of the object mediately possessing them.A mediately possessed quantity may for a certain type of material objects satisfy the characteristic condition for an extensive quantity. In that case it is an extensive quantity mediately possessed by that type of material objects. All variable extensive quantities are of this mediate character. A quantity mediately possessed by a material objectat a moment[i.e. at a time] of a time-systemis the limit of the quantity possessed by the successive converging situations ofin the successive durations of an abstractive class (of durations in the time-systemwhich converges to.The volume-density, at a timein the-space of a time-system, of the distribution of any absolutely extensive quantity mediately possessed by a material objectis calculated according to the preceding definition for the case of immediately possessed quantities, except that the 'quantity mediately possessed by(or by an extensive component of)at the time' must be substituted everywhere for the 'quantity possessed by(or by an extensive component of).'59.5We can compare the volume-densitiesandof an absolutely extensive quantity for two time-systemsandrespectively at a given event-particle, assuming, as we may assume, that the motion of the material object possessing (mediately or immediately) the quantity is regular.Letbe the time-system in which the objectis stationary at, and letbe the volume-density atfor the time-system. Letbe the moments in,, andrespectively which contain. Letbe the measure of a small volume inwhich contains[and therefore the measure of the volume in the timeless-space which this instantaneous volume occupies]. Consider the event () stationary inof which this small volumeis a normal cross-section, and bounded by terminal moments′ and″ on either side ofand both near. Then, by the theory of regular motion, we can take this stationary event () as the situation of an extensive component of, whenis small enough and the duration bounded by′ and″ is short enough. Letandbe the measures of the volumes which are the oblique cross-sections ofmade byand. Then ultimately,, andare expressions for the measure of the quantity possessed by.But by equation (1) of57.2of this chapter,Now take the mutual axes forand, and let () and () be the coordinates ofinandrespectively, and let () and () be the velocities due to it inandrespectively. Then by equation (1) of52.6,59.6Now letdenote differentiation following the motion () at () and letdenote differentiation at the point ().Then it is easily proved thatHence from equation (2) of59.5aboveAgain by using the formulae ofarticle 52, we can prove thatFrom these results we immediately deduceNow the condition that the total extensive quantity which is the 'charge' of any extensive component never varies when conceived as distributed through the-space isThis is the well-known equation of continuity. Now equation (4) shows that if this equation holds for the space of any time-system, it holds for the spaces of all time-systems.When the equation of continuity holds, the 'charge' of any extensive component of the material object under consideration never varies. Hence it is a mere matter of words and definition whether the charge is said to be mediately possessed by the object or immediately possessed.[8]Cf.subarticle 37.3, Chapter X, Part III.[9]Cf.Principia Mathematica.
56. Material Objects.56.1A material object is essentially a material object of a certain definite sort; namely, we define sorts of material objects, which are sets of objects with certain definite peculiarities, and a material object is such because it is a member of one of these sorts. For example a piece of wood is a material object because it belongs to the class of wooden objects and because this class possesses the requisite peculiarities. Similarly a charge of electricity is a material object for an analogous reason.
The objects which compose a set () form a sort of 'material' objects when (i) the objects of the setare all uniform, (ii) not more than one member ofcan be located in any volume, (iii) no member ofcan be located in two volumes of the same moment, (iv) ifandbe two members ofrespectively located in non-overlapping volumes in the same moment, then any pair of situations ofandrespectively are separated events, (v) ifbe a member ofsituated in an event, and located in the volumewhich is a section of, andbe any volume which is a portion of, then there is a member ofwhich is located inand is a concurrent component of.
56.2Ifbe a material object of a certain sort andbe a volume in whichis located andbe a portion of, then the material object of the same sort aswhich is located inis called an 'extensive component' of.
56.3It is by means of the properties of material objects that the atomic properties of objects are combined in mathematical calculations with the extensive continuity of events. Apart from material objects mathematical physics as at present developed would be impossible. For example where the physicist sees the electron as an atomic whole, the mathematician sees a distribution of electricity continuous in time and in space and capable of division into component objects which are also analogous distributions.
57. Stationary Events.57.1In order to understand the theory of the motion of material objects, it is first necessary to define the concept of a 'stationary' event. Consider some given time-system, and letdenote a volume lying in a certain momentof this time-system. Letbe a duration of bounded by momentsand, and inhered in by; so that,,are three parallel moments of the time-system, andlies betweenand. The volumeis the locus of a set of event-particles and each of these event-particles lies in one and only one station of the duration. Also each station ofeither does not intersector intersects it in one event-particle only. The assemblage of event-particles lying on stations of d which intersect[namely, each event-particle lying on one of these stations] is the complete set of event-particles analysing[8]an event. Such an event is called stationary in the time-systemand stretches throughout the duration. It can also be called 'stationary in,' sincedefines the time-system. Every event-particle within the event lies on a station of; and a station ofeither has all its event-particles lying within the event or none of them. The volumeis the section of the event by the moment. Furthermore if′ be any other moment of the time-systemit lying betweenand, it intersects the event in a volume′ which is a geometrical replica of the volume. The momentsandwhich bound the durationare the terminal moments of any event which is stationary in. The stations oflying in the event intersectandin terminal volumesandwhich are geometrical replicas ofand′. A volume, such as', in which a moment ofintersects an event stationary inis called a 'normal cross-section' of the event. A moment of another time-systemwhich intersects the stationary event in a volume, but does not intersect either of the terminal volumes, is said to intersect it in an 'oblique cross-section.' All the oblique cross-sections of a stationary event which are made by moments of the same time-system are geometrical replicas of each other.
57.2Consider an eventstationary in the time-system, and letbe another time-system. Letbe the measure of the normal cross-sections ofand letbe the measure of the oblique cross-sections made by moments of. We require the ratio ofto. Take (as usual) mutual axes forand, and let the event-particle which is the origin lie inwhich is the antecedent terminal moment of. Thenis at the-time zero, and let(the subsequent terminal moment) be at the-time. Then if () be the-coordinates of the event-particle in which a station(of the set composing the event) intersects, the-coordinates of the other end (the subsequent end) ofinare ().
Furthermore let () be the-coordinates of the antecedent end of, and let () be the-coordinates of the subsequent end of. Then by the usual formulae [cf. subarticle52.2]
But, by analogous reasoning to that for the elementary case of geometrical parallelograms, the absolute extent of the eventcan be expressed asand as). Hence.
57.3The stations ofduration oftime-systemare portions of points of the time-less space of[the-space].
Thus by prolonging the stations which constitute the stationary eventwe obtain the assemblage of-points which is the complete assemblage of-points intersecting the cross-sections of, each event-particle in each cross-section lying on one and only one such-point and each of these-points intersecting each cross-section in one event-particle. The assemblage of these-points is a volume of the-space, and the successive instantaneous volumes which are the normal cross-sections of[stationary in] each occupy this same volume in the-space. Thus the stationary eventduring the lapse of-time throughout which it endures is happening at the same place in the-space.
But the successive oblique cross-sections offormed by moments of another time-systemare instantaneous volumes which successively occupy different volumes in the-space. These instantaneous volumes travel in the-space, sweeping over it with the uniform velocity, namely the velocity due to the time-systemin the-space.
57.4A 'normal slice' of a stationary event is the slice of it cut off between any two normal cross-sections. An 'oblique slice' of a stationary event is the slice of it cut off between any two parallel oblique cross-sections. A normal slice of a stationary event is itself a stationary event in the same time-system.
58. Motion of Objects.58.1A material object is 'motionless' within a duration when throughout that duration the material object and its extensive components are all situated in stationary events.
In the case of a motionless material object, Law I for uniform objects can be made more precise, as follows:
Ifbe a material object motionless in the durationandbe the stationary event extending throughoutin which it is situated, thenis situated in any oblique slice of.
The accompanying figures illustrate (i) the kind of slice which is included in this law and (ii) the kind of slice which is excluded.
fig19Fig. 19.
Fig. 19.
Fig. 19.
It immediately follows that—with the nomenclature of the enunciation of the law—is located in every oblique cross-section of.
Ifbe the time-system of the duration in whichis motionless and, in some other time-system,′ be the duration of maximum extent which intersectsin an oblique slice, then throughout′ in the time-less space of a the material objecthas a uniform motion of translation with the velocity ofin.
58.2This property, possessed by a material object which is motionless in a time-system, of being situated in every oblique slice of its stationary situation is a fundamental physical law of nature. Namely, percipients cogredient with different time-systems can 'recognise' the same material objects. In other words, the character of a material object is not altered by its motion.
58.3The motion of a material objectis 'regular' when ifbe any volume in which it is located andbe any event-particle in, and′ be any variable volume which containsand is a portion of, and′ be the extensive component ofwhich is located in′, then, as′ is progressively diminished without limit, a time-systemcan be found such that the errors of calculations, respecting magnitudes exhibited by′ which assume that′ is motionless in, tend to the limit zero, provided that the time-lapse of the durations inwithin which′ is motionless is also correspondingly diminished without limit.
The above definition of regular motion is a description of the assumptions in the ordinary mathematical treatment of the motion of a material object (not necessarily rigid) which is not moving with a uniform motion of translation. Ifbe the standard time-system to which motions are referred, then the velocity ofinis the velocity at the event-particle[i.e. at the-space pointat the-time] of the material object.
59. Extensive Magnitude.59.1A theory of extensive magnitude is required to complete the theory of material objects.
Letand′ be two objects (material or otherwise), then the statement thatand′ possess quantities of a certain kind and that the ratio of the quantityto the quantity′ has a certain definite numerical value is a reference to some determinate method of comparison ofto′ which is the defining characteristic of that kind of quantity[9].
The quantity of a certain kind possessed by a material objectis called 'extensive' when it is a determinate function of the quantities of the same kind possessed by any two of its extensive components which (i) are exhaustive ofand (ii) are non-overlapping [i.e. have no extensive component in common].
If the determinate function be that of simple addition [so that,being the quantities possessed respectively byand its two extensive components,], then the kind of quantity will be called 'absolutely' extensive. When an extensive quantity is not absolutely extensive, it will be called 'semi-extensive.'
59.2It is usual in philosophical discussions to con- fine the term 'extensive quantity' to what is here defined as 'absolutely extensive quantity,' and to ignore entirely the occurrence of semi-extensive quantities. But in physical science semi-extensive quantities are well known. For example, consider a sphere of radiusuniformly charged with electricity throughout its volume. Divide the sphere into two parts, namely a concentric nucleus of radiusand a shell of thickness. Then the electromagnetic mass of the whole sphere is not the sum of the electromagnetic masses of these two parts, but is to be calculated by a quadratic law from the charges.
A material object expresses the spatial distribution of a quantity of 'material,' when the quantity is absolutely extensive.
59.3The volume-density, at a timein the-space of a time-system, of the distribution of any absolutely extensive quantity possessed by a material objectis calculated by the ordinary mathematical formula. Consider any event-particleoccupying the-pointat the-time. Letbe the measure of a volume in the-space which contains; and let′ be the extensive component oflocated in, if there be such an extensive component. Letbe the measure of the quantity possessed by′. Then the limit of the ratio ofto, asis indefinitely diminished, is the density atat the timeof the material [i.e. of the absolutely extensive quantity].
59.4The above definitions contemplate quantities immediately possessed by the extensive objects as such, for example, charges of electricity and intensities of sense-objects. But there are also quantities which are only mediately possessed by the objects, but are immediately possessed by the events which are their situations. Such quantities may vary with the variation in the situation of the object mediately possessing them.
A mediately possessed quantity may for a certain type of material objects satisfy the characteristic condition for an extensive quantity. In that case it is an extensive quantity mediately possessed by that type of material objects. All variable extensive quantities are of this mediate character. A quantity mediately possessed by a material objectat a moment[i.e. at a time] of a time-systemis the limit of the quantity possessed by the successive converging situations ofin the successive durations of an abstractive class (of durations in the time-systemwhich converges to.
The volume-density, at a timein the-space of a time-system, of the distribution of any absolutely extensive quantity mediately possessed by a material objectis calculated according to the preceding definition for the case of immediately possessed quantities, except that the 'quantity mediately possessed by(or by an extensive component of)at the time' must be substituted everywhere for the 'quantity possessed by(or by an extensive component of).'
59.5We can compare the volume-densitiesandof an absolutely extensive quantity for two time-systemsandrespectively at a given event-particle, assuming, as we may assume, that the motion of the material object possessing (mediately or immediately) the quantity is regular.
Letbe the time-system in which the objectis stationary at, and letbe the volume-density atfor the time-system. Letbe the moments in,, andrespectively which contain. Letbe the measure of a small volume inwhich contains[and therefore the measure of the volume in the timeless-space which this instantaneous volume occupies]. Consider the event () stationary inof which this small volumeis a normal cross-section, and bounded by terminal moments′ and″ on either side ofand both near. Then, by the theory of regular motion, we can take this stationary event () as the situation of an extensive component of, whenis small enough and the duration bounded by′ and″ is short enough. Letandbe the measures of the volumes which are the oblique cross-sections ofmade byand. Then ultimately,, andare expressions for the measure of the quantity possessed by.
But by equation (1) of57.2of this chapter,
Now take the mutual axes forand, and let () and () be the coordinates ofinandrespectively, and let () and () be the velocities due to it inandrespectively. Then by equation (1) of52.6,
59.6Now letdenote differentiation following the motion () at () and letdenote differentiation at the point ().
Then it is easily proved that
Hence from equation (2) of59.5above
Again by using the formulae ofarticle 52, we can prove that
From these results we immediately deduce
Now the condition that the total extensive quantity which is the 'charge' of any extensive component never varies when conceived as distributed through the-space is
This is the well-known equation of continuity. Now equation (4) shows that if this equation holds for the space of any time-system, it holds for the spaces of all time-systems.
When the equation of continuity holds, the 'charge' of any extensive component of the material object under consideration never varies. Hence it is a mere matter of words and definition whether the charge is said to be mediately possessed by the object or immediately possessed.
[8]Cf.subarticle 37.3, Chapter X, Part III.
[8]Cf.subarticle 37.3, Chapter X, Part III.
[9]Cf.Principia Mathematica.
[9]Cf.Principia Mathematica.