 Section V. OF THE CONCEPT OF NATURE by Alfred North-Whitehead. 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 roots. By a root 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 the 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 makeup of a given event particle, those which are small enough, one of three cases must occur. Either one, all of these small events are entirely separate from the given event E, or two, all of these small events are parts of the event E, or three, 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 the 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 same 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 a 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. In 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. This 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. 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 root. We will not delay over any general consideration of these momental roots nor is it important for us to proceed to the still wider investigation of vagrant roots in general. There are, however, two simple sets of roots which are of vital importance. One is a set of momental roots and the other of vagrant roots. Both sets can be classed together as straight roots. We proceed to define them without any reference to the definitions of volumes and surfaces. The two types of straight roots will be called rectilinear roots and stations. Rectilinear roots are momental roots and stations are vagrant roots. Rectilinear roots are roots which in a sense lie in recs. 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 roots. 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 root 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 the respective moments of the event particles covered by it. Recs 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 conservationism 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 uncriticized 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. 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, that 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. When reviewing the factors of nature as immediately disclosed in sense-awareness we should note the fundamental character of the precept 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 locust stand-eye for mind involved in the sense-awareness. This locust stand-eye 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 precipiant event. This event is not the mind, that is to say, not the precipiant. It is in that 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 precipiant event which marks the where of awareness and the how of awareness. This precipiant 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 precipiant 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 on 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 envision 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-term 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 recipient mind and the green leaf and the relation of the 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 recipient 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. Even the short present duration of the here of the recipient event has a definite meaning of some sort. This meaning of here is the content of the special relation of the recipient event to its associated duration. I will call this relation co-gradients. Accordingly I ask for a description of the character of the relation of co-gradients. The presence snaps into a past and a present when the here of co-gradients 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 here's of sense-awareness within neighboring 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 presence, 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 reflection can seize it. We live in snippets 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 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 recipient 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 self-awareness. This peculiar relation is the relation of co-gredients between the recipient event and the duration. Co-gredients 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 which comprises change in the quality of its peculiar relation to the contained recipient 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 recipient event. It is only in the past that you can have been there with a standpoint distinct from your present here. Quarters 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 determinant station in the duration which belongs to an event which is here in one determinant sense of here is the same kind of quality of station which belongs to an event which is there in one determinant sense of there. Thus co-gredience 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 connection of a recipient event with the recipients of mind, but it has nothing to do with the relation of the recipient event to the duration which is the present whole of nature, posited as the disclosure of the recipients. Given the requisite biological character, the event in its character of a recipient event selects that duration with which the operative past of the event is practically co-gredient within the limits of the exactitude of observation, namely admit the alternative time systems which nature offers there will be one with a duration giving the best average of co-gredience for all the subordinate parts of the recipient event. This duration will be the whole of nature which is the terminus posited by sense awareness. Thus the character of the recipient event determines the time system immediately evident in nature. As the character of the recipient event changes with the passage of nature, or in other words as the recipient mind in its passage correlates itself with the passage of the recipient event into another recipient event, the time system correlated with the recipients of that mind may change. When the bulk of the events perceived our co-gredient in a duration other than that of the recipient event, the recipients may include a double consciousness of co-gredience, namely the consciousness of the whole within 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 co-gredience. This assertion of co-gredience is peculiarly evident when the duration to which the perceived event is co-gredient is the same as the duration which is the present whole of nature. In other words, when the event and the recipient event are both co-gredient to the same duration. We are now prepared to consider the meaning of stations in a duration where stations are a peculiar kind of roots 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, one, that the event is part of the duration and two, 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 co-gredient with the duration extends throughout that duration. It is not true that all the parts of an event co-gredient with the duration are also co-gredient with the duration. The relation of co-gredients 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 co-gredient with another duration which is part of the given duration though it is not co-gredient with the given duration itself. Such a part would be co-gredient 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 determinant root 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 co-gredient with a certain duration. A train traveling in it is part of that tunnel but is not itself at rest. If an event E be co-gredient with a duration D and D slash be any duration which is part of D then D slash belongs to the same time system as D. Also D slash intersects E in an event E slash which is part of E and is co-gredient with D slash. 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 co-gredient 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 one each of its events is co-gredient with the duration D and two 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 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 utilize the overlappings of the duration of one family that is of one time system, two prolonged 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 systems 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 then 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 or giving 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. He 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 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 the line in space alpha, which is in the direction of beta in space alpha. I have 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 as 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 REX. 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 allying 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 transversed by the being. These event particles are the instantaneous facts which pass with the instantaneous moment. What is really transversed are other event particles which at succeeding instance 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 transversed because the intrinsic characters of the latter events are in general so similar to those of the instantaneous road that we do not trouble to discriminate. But suppose a landmine 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 wrecks which are never traversed. These approximate wrecks are composed of small events namely approximate roots and event particles which are passed away before the moving objects reach them. Assuming that our forecasts of rectilinear motion are correct these wrecks occupy the straight lines in timeless space which are traversed. Thus the wrecks 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 time the 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 in the space of alpha which intersects m in event particles on rho is the straight line r of space alpha and the locus of those points of space alpha which intersects m in event particles on lambda is the plane one of space alpha. Then the plane one 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 apologize 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. CHAPTER VI. OF THE CONCEPT OF NATURE. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Recording by M.B. The concept of nature by Alfred North Whitehead. CHAPTER VI. CONGRUENCE. The aim of this lecture is to establish a theory of congruence. You must understand at once that congruence is a controversial question. It is the theory of measurement in space and in time. The question seems simple. In fact it is simple enough for a standard procedure to have been settled by active parliament. And devotion to metaphysical subtleties is almost the only crime which has never been imputed to any English parliament. But the procedure is one thing and its meaning is another. First let us fix attention on the purely mathematical question. When the segment between two points A and B is congruent to that between the two points C and D the quantitative measurements of the two segments are equal. The equality of the numerical measures and the congruence of the two segments are not always clearly discriminated and are lumped together under the term equality. But the procedure of measurement presupposes congruence. For example a yard measure is applied successively to measure two distances between two pairs of points on the floor of a room. It is of essence of the procedure of measurement that the yard measure remains unaltered as it is transferred from one position to another. Some objects can palpably alter as they move. For example an elastic thread. But a yard measure does not alter if made of the proper material. What is this but a judgment of congruence applied to the train of successive positions of the yard measure? We know that it does not alter because we judge it to be congruent to itself in various positions. In the case of the thread we can observe the loss of self congruence. Thus immediate judgments of congruence are presupposed in measurement and the process of measurement is merely a procedure to extend the recognition of congruence to cases where these immediate judgments are not available. Thus we cannot define congruence by measurement. In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes. In fact a complete theory of nonmetrical geometry. We then inquire about congruence and lay down the set of conditions or axioms as they are called which this relation satisfies. It has then been proved that there are alternative relations which satisfy these conditions equally well and that there is nothing intrinsic in the theory of space to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt. In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned. Poincaré, the great French mathematician, held that our actual choice among these geometries is guided purely by convention and that the effect of a change of choice would be simply to alter our expression of the physical laws of nature. By convention I understand Poincaré to mean that there is nothing inherent in nature itself giving any particular role to one of these congruence relations and that the choice of one particular relation is guided by the volitions of the mind at the other end of the sense awareness. The principle of guidance is intellectual convenience and not natural fact. The position has been misunderstood by many of Poincaré's expositors. They have muddled it up with another question namely that owing to the inexactitude of observation it is impossible to make an exact statement in the comparison of measures. It follows that a certain subset of closely allied congruence relations can be assigned of which each member equally well agrees with that statement of observed congruence when the statement is properly qualified within its limits of error. This is an entirely different question and it presupposes a rejection of Poincaré's position. The absolute indetermination of nature in respect of all the relations of congruence is replaced by the indetermination of observation with respect to a small subgroup of these relations. Poincaré's position is a strong one. He in effect challenges anyone to point out any factor in nature which gives a preeminent status to the congruence relation which mankind has actually adopted. But undeniably the position is very paradoxical. Bertrand Russell had controversy with him on this question and pointed out that on Poincaré's principles there was nothing in nature to determine whether the earth is larger or smaller than some assigned billiard ball. Poincaré replied that the attempt to find relations in nature for the selection of a definitive congruence relation in space is like trying to determine the position of a ship in the ocean by counting the crew and observing the color of the captain's eyes. In my opinion both disputants were right assuming the grounds on which the discussion was based. Russell in effect pointed out that apart from minor inexactitudes a determinate congruence relation is among the factors in nature which our sense awareness posits for us. Poincaré asks for information as to the factor in nature which might lead any particular congruence relation to play a preeminent role among the factors posited in sense awareness. I cannot see the answer to either of these contentions provided that you admit the materialistic theory of nature. With this theory nature at an instant in space is an independent fact. Thus we have to look for our preeminent congruence relation amid nature in instantaneous space. And Poincaré is undoubtedly right in saying that nature on this hypothesis gives us no help in finding it. On the other hand Russell is in an equally strong position when he asserts that as a fact of observation we do find it and what is more agree in finding the same congruence relation. On this basis it is one of the most extraordinary facts of human experience that all mankind without any assignable reason should agree in fixing attention on just one congruence relation amid the indefinite number of indistinguishable competitors for notice. One would have expected disagreement on this fundamental choice to have divided nations and to have rent families. But the difficulty was not even discovered till the close of the 19th century by a few mathematical philosophers and philosophic mathematicians. The case is not like that of our agreement on some fundamental fact of nature such as the three dimensions of space. If space has only three dimensions we should expect all mankind to be aware of the fact as they are aware of it. But in the case of congruence mankind agree in an arbitrary interpretation of sense awareness when there is nothing in nature to guide it. I look on it as no slight recommendation of the theory of nature which I am expounding to you that it gives a solution of this difficulty by pointing out the factor in nature which issues in the preeminence of one congruence relation over the indefinite herd of other such relations. The reason for this result is that nature is no longer confined within space at an instant. Space and time are now interconnected and this peculiar factor of time which is so immediately distinguished among the deliverances of our sense awareness relates itself to one particular congruence relation in space. Congruence is a particular example of the fundamental fact of recognition. In perception we recognize this recognition does not merely concern the comparison of a factor of nature posited by memory with a factor posited by immediate sense awareness. Recognition takes place within the present without any intervention of pure memory. For the present fact is a duration with its antecedent and consequent durations which are parts of itself. The discrimination in sense awareness of a finite event with its quality of passage is also accompanied by the discrimination of other factors of nature which do not share in the passage of events. Whatever passes is an event but we find entities in nature which do not pass namely we recognize sameness in nature. Recognition is not primarily an intellectual act of comparison. It is in its essence merely sense awareness in its capacity of positing before us factors in nature which do not pass. For example green is perceived as situated in a certain finite event within the present duration. This green preserves its self-identity throughout whereas the event passes and thereby obtains the property of breaking into parts. The green patch has parts but in talking of the green patch we are speaking of the event in its sole capacity of being for us the situation of green. The green itself is numerically one self-identical entity without parts because it is without passage. Factors in nature which are without passage will be called objects. There are radically different kinds of objects which will be considered in this succeeding lecture. Recognition is reflected into the intellect as comparison. The recognized objects of one event are compared with the recognized objects of another event. The comparison may be between two events in the present or it may be between two events of which one is posited by memory awareness and the other by immediate sense awareness. But it is not the events which are compared. For each event is essentially unique and incomparable. What are compared are the objects and relations of objects situated in events. The event considered as a relation between objects has lost its passage and in this aspect is itself an object. This object is not the event but only an intellectual abstraction. The same object can be situated in many events and in this sense even the whole event viewed as an object can recur though not the very event itself with its passage and its relations to other events. Objects which are not posited by sense awareness may be known to the intellect. For example, relations between objects and relations between relations may be factors in nature not disclosed in sense awareness but known by logical inference as necessarily in being. Thus objects for our knowledge may be merely logical abstractions. For example, a complete event is never disclosed in sense awareness and thus the object which is the sum total of objects situated in an event as thus interrelated is a mere abstract concept. Again, a right angle is a perceived object which can be situated in many events but though rectangularity is posited by sense awareness the majority of geometrical relations are not so posited. Also, rectangularity is in fact often not perceived when it can be proved to have been there for perception. Thus an object is often known merely as an abstract relation not directly posited in sense awareness although it is there in nature. The identity of quality between congruent segments is generally of this character. In certain special cases this identity of quality can be directly perceived but in general it is inferred by a process of measurement depending on our distinct sense awareness of selected cases and a logical inference from the transitive character of congruence. Congruence depends on motion and thereby is generated the connection between spatial congruence and temporal congruence. Motion along a straight line has a symmetry round that line. This symmetry is expressed by the symmetrical geometrical relations of the line to the family of planes normal to it. Also, another symmetry in the theory of motion arises from the fact that rest in the points of beta corresponds to uniform motion along a definite family of parallel straight lines in the space of alpha. We must note the three characteristics. One, of the uniformity of the motion corresponding to any point of beta along its correlated straight line in alpha and two of the equality in magnitude of the velocities along the various lines of alpha correlated to rest in the various points of beta and three of the parallelism of the lines of this family. We are now in possession of a theory of parallels and a theory of perpendicular and a theory of motion and from these theories the theory of congruence can be constructed. It will be remembered that a family of parallel levels in any moment is the family of levels in which that moment is intersected by the family of moments of some other time system. Also, a family of parallel moments is the family of moments of some one time system. Thus we can enlarge our concept of a family of parallel levels so as to include levels in different moments of one time system. With this enlarged concept we say that a complete family of parallel levels in a time system alpha is the complete family of levels in which the moments of alpha intersect the moments of beta. This complete family of parallel levels is also evidently a family lying in the moments of the time system beta. By introducing a third system gamma parallel rects are obtained. Also all the points of any one time system form a family of parallel point tracks. Thus there are three types of parallelograms in the four dimensional manifold of event particles. In parallelograms of the first type the two pairs of parallel sides are both of them pairs of rects. In parallelograms of the second type one pair of parallel sides is a pair of rects and the other pair is a pair of point tracks. In parallelograms of the third type the two pairs of parallel sides are both of them pairs of point tracks. The first axiom of congruence is that the opposite sides of any parallelogram are congruent. This axiom enables us to compare the lengths of any two segments either respectively on parallel rects or on the same rect. Also it enables us to compare the lengths of any two segments either respectively on parallel point tracks or on the same point track. It follows from this axiom that two objects at rest in any two points of a time system beta are moving with equal velocities in any other time system alpha along parallel lines. Thus we can speak of the velocity in alpha due to the time system beta without specifying any particular point in beta. The axiom also enables us to measure time in any time system but does not enable us to compare times in different time systems. The second axiom of congruence concerns parallelograms on congruent bases and between the same parallels which have also their other pairs of sides parallel. The axiom asserts that the rect joining the two event particles of intersection of the diagonals is parallel to the rect on which the bases lie. By the aid of this axiom it easily follows that the diagonals of a parallelogram bisect each other. Congruence is extended in any space beyond parallel rects to all rects by two axioms depending on perpendicularity. The first of these axioms which is the third axiom of congruence is that if ABC is a triangle of rects in any moment and D is the middle event particle of the base BC then the level through D perpendicular to BC contains A when and only when AB is congruent to AC. This axiom evidently expresses the symmetry of perpendicularity and is the essence of the famous Pawn's Asynorum expressed as an axiom. The second axiom depending on perpendicularity and the fourth axiom of congruence is that if R and A be a rect and an event particle in the same moment and AB and AC be a pair of rectangular rects intersecting R in B and C and AD and AE be another pair of rectangular rects intersecting R in D and E then either D or E lies in the segment BC and the other one of the two does not lie in this segment. Also as a particular case of this axiom if AB be perpendicular to R and in consequence AC be parallel to R then D and E lie on opposite sides of B respectively. By the aid of these two axioms the theory of congruence can be extended so as to compare lengths of segments on any two rects. Accordingly Euclidean metrical geometry in space is completely established and lengths in the spaces of different time systems are comparable as the result of definite properties of nature which indicate just that particular method of comparison. The comparison of time measurements in diverse time systems requires two other axioms. The first of these axioms forming the fifth axiom of congruence will be called the axiom of kinetic symmetry. It expresses the symmetry of the quantitative relations between two time systems when the times and lengths in the two systems are measured in congruent units. The axiom can be explained as follows. Let alpha and beta be the names of two time systems. The directions of motion in the space of alpha due to rest in a point of beta is called the beta direction in alpha and the direction of motion in the space of beta due to rest in a point of alpha is called the alpha direction in beta. Consider a motion in the space of alpha consisting of a certain velocity in the beta direction of alpha and a certain velocity at right angles to it. This motion represents rest in the space of another time system. Call it pi. Rest in pi will also be represented in the space of beta by a certain velocity in the alpha direction in beta and a certain velocity at right angles to this alpha direction. Thus a certain motion in the space of alpha is correlated to a certain motion in the space of beta as both representing the same fact which can also be represented by rest in pi. Now another time system which I will call sigma can be found which is such that rest in its space is represented by the same magnitudes of velocities along and perpendicular to the alpha direction in beta as those velocities in alpha along and perpendicular to the beta direction which represent rest in pi. The required axiom of kinetic symmetry is that rest in sigma will be represented in alpha by the same velocities along and perpendicular to the beta direction in alpha as those velocities in beta along and perpendicular to the alpha direction which represent rest in pi. A particular case of this axiom is that relative velocities are equal and opposite. Namely rest in alpha is represented in beta by a velocity along the alpha direction which is equal to the velocity along the beta direction in alpha which represents rest in beta. Finally the sixth axiom of congruence is that the relation of congruence is transitive. So far as this axiom applies to space it is superfluous for the property follows from our previous axioms. It is however necessary for time as a supplement to the axiom of kinetic symmetry. The meaning of the axiom is that if the time unit of system alpha is congruent to the time unit of system beta and the time unit of system beta is congruent to the time unit of system gamma then the time units of alpha and gamma are also congruent. By means of these axioms formulae for the transformation of measurements made in one time system to measurements of the same facts of nature made in another time system can be deduced. These formulae will be found to involve one arbitrary constant which I will call kappa. It is of the dimensions of the square of a velocity. Accordingly four cases arise. In the first case kappa is zero. This case produces nonsensical results in opposition to the elementary deliverances of experience. We can put this case aside. In the second case kappa is infinite. This case yields the ordinary formulae for transformation in relative motion. Namely those formulae which are to be found in every elementary book on dynamics. In the third case kappa is negative. Let us call it negative c squared where c will be the dimensions of a velocity. This case yields the formulae of transformation which Lormore discovered for the transformation of Maxwell's equations of the electromagnetic field. These formulae were extended by H. A. Lawrence and used by Einstein and Minkowski as the basis of their novel theory of relativity. I am not now speaking of Einstein's more recent theory of general relativity by which he deduces his modification of the law of gravitation. If this be the case which applies to nature, then c must be a close approximation to the velocity of light in vacuo. Perhaps it is this actual velocity. In this connection in vacuo must not mean an absence of events, namely the absence of the all-pervading ether of events. It must mean the absence of certain types of objects. In the fourth case kappa is positive. Let us call it H2 where H will be of the dimensions of a velocity. This gives a perfectly possible type of transformation formulae but not one which explains any facts of experience. It has also another disadvantage. With the assumption of this fourth case the distinction between space and time becomes unduly blurred. The whole object of these lectures has been to enforce the doctrine that space and time spring from a common root and that the ultimate fact of experience is a space-time fact. But after all, mankind does distinguish very sharply between space and time and it is owing to this sharpness of distinction that the doctrine of these letters is somewhat of a paradox. Now in the third assumption this sharpness of distinction is adequately preserved. There is a fundamental distinction between the metrical properties of point tracks and rects but in the fourth assumption this fundamental distinction vanishes. Neither the third nor the fourth assumption can agree with experience unless we assume that the velocity c of the third assumption and the velocity h of the fourth assumption are extremely large compared to the velocities of ordinary experience. It must be the case the formulae of both assumptions will obviously reduce to a close approximation to the formulae of the second assumption which are the ordinary formulae of dynamical textbooks. For the sake of a name I will call these textbook formulae the orthodox formulae. There can be no question as to the general correctness of the orthodox formulae. It would be merely silly to raised doubts on this point but the determination of the status of these formulae is by no means settled by this admission. The independence of time and space is an unquestioned presupposition of the orthodox thought which has produced the orthodox formulae. With this presupposition and given the absolute points of one absolute space the orthodox formulae are immediate deductions. Accordingly these formulae are presented to our imaginations as facts which cannot be otherwise time and space being what they are. The orthodox formulae have therefore attained to the status of necessities which cannot be questioned in science. Any attempt to replace these formulae by others was to abandon the role of physical explanation and to have recourse to mere mathematical formulae. But even in physical science difficulties have accumulated around the orthodox formulae. In the first place Maxwell's equations of the electromagnetic field are not invariant for the transformations of the orthodox formulae whereas they are invariant for the transformations of the formulae arising from the third of the four cases mentioned above provided that a velocity C is identified with a famous electromagnetic constant quantity. Again the known results of the delicate experiments to detect the earth's variations of motion through the ether in its orbital path are explained immediately by the formulae of the third case. But if we assume the orthodox formulae we have to make a special and arbitrary assumption as to the contraction of matter during motion. I mean the Fitzgerald Lorentz assumption. Lastly Fresnel's coefficient of drag which represents the variation of the velocity of light in a moving medium is explained by the formulae of the third case and requires another arbitrary assumption if we use the orthodox formulae. It appears therefore that on the mere basis of physical explanation there are advantages in the formulae of the third case as compared with the orthodox formulae. But the way is blocked by the ingrained belief that these latter formulae possess a character of necessity. It is therefore an urgent requisite for physical science and for philosophy to examine critically the grounds for this supposed necessity. The only satisfactory method of scrutiny is to recur to the first principles of our knowledge of nature. This is exactly what I am attempting to do in these lectures. I ask what it is that we are aware of in our sense perception of nature. I then proceed to examine those factors in nature which lead us to conceive nature as occupying space and persisting through time. This procedure has led to an investigation of the characters of space and time. It results from these investigations that the formulae of the third case and the orthodox formulae are on a level as possible formulae resulting from the basic character of our knowledge of nature. The orthodox formulae have thus lost any advantage as necessity which they enjoyed over the serial group. The way is thus open to adopt whichever of the two groups best accords with observation. I take this opportunity of pausing for a moment in the course of my argument and of reflecting on the general character which my doctrine ascribes to some familiar concepts of science. I have no doubt that some of you have felt that in certain aspects this character is very paradoxical. This vein of paradox is partly due to the fact that educated language has been made to conform to the prevalent orthodox theory. We are thus in expounding an alternative doctrine driven to the use of either strange terms or of familiar words with unusual meanings. This victory of the orthodox theory over language is very natural. Events are named after the prominent objects situated in them and thus both in language and in thought the event sinks behind the object and becomes the mere play of its relations. The theory of space is then converted into a theory of relations of objects instead of a theory of the relations of events. But objects have not the passage of events. Accordingly space as a relation between objects is devoid of any connection with time. It is space at an instant without any determinant relations between the spaces at successive instance. It cannot be one timeless space because the relations between objects change. A few minutes ago in speaking of the deduction of the orthodox formulae for relative motion I said that they followed as an immediate deduction from the assumption of absolute points in absolute space. This reference to absolute space was not an oversight. I know that the doctrine of the relativity of space at present holds the field both in science and philosophy. But I do not think that its inevitable consequences are understood when we face them the paradox of the presentation of the character of space which I have elaborated is greatly mitigated. If there is no absolute position a point must cease to be a simple entity. What is a point to one man in a balloon with his eyes fixed on an instrument is a track of points to an observer on the earth who is watching the balloon through a telescope and is another track of points to an observer in the sun who is watching the balloon through some instrument suited to such a being. Accordingly if I am reproached with the paradox of my theory of points as classes of event particles and of my theory of event particles as groups of abstractive sets I ask my critic to explain exactly what he means by a point. While you explain your meaning about anything however simple it is always apt to look subtle and fine spun. I have at least explained exactly what I do mean by a point what relations it involves and what entities are the relata. If you admit the relativity of space you also must admit that points are complex entities logical constructs involving other entities and their relation. Produce your theory not in a few vague phrases of indefinite meaning but explain it step by step in definite terms referring to assigned relations and assigned relata. Also show that your theory of points issues in a theory of space. Furthermore note that the example of the man in the balloon and the observer on earth and the observer in the sun shows that every assumption of relative rest requires a timeless space with radically different points from those which issue from every other such assumption. The theory of the relativity of space is inconsistent with any doctrine of one unique set of points of one timeless space. The fact is that there is no paradox in my doctrine of the nature of space which is not in essence inherent in the theory of the relativity of space. But this doctrine has never really been accepted in science whatever people say. What appears in our dynamical treatises is Newton's doctrine of relative motion based on the doctrine of differential motion in absolute space. When you once admit that the points are radically different entities for differing assumptions of rest then the orthodox formulae lose all their obviousness. They were only obvious because you were really thinking of something else. When discussing this topic you can only avoid paradox by taking refuge from the flood of criticism in the comfortable arc of no meaning. The new theory provides a definition of the congruence of periods of time. The prevalent view provides no such definition. Its position is that if we take such time measurements so that familiar velocities will seem to us to be uniform or uniform then the laws of motion are true. Now in the first place no change could appear either as uniform or non-uniform without involving a definite determination of the congruence for time periods. So in appealing to familiar phenomena it allows that there is some factor in nature which we can intellectually construct as a congruence theory. It does not however say anything about it except that the laws of motion are then true. Suppose that with some expositors we cut out the reference to familiar velocities such as the rate of rotation of the earth. We are then driven to admit that there is no meaning in temporal congruence except that certain assumptions make the laws of motion true. Such a statement is historically false. King Alfred the Great was ignorant of the laws of motion but knew very well what he meant by the measurement of time and achieved his purpose by means of burning candles. Also no one in past ages justified the use of sand in our glasses by saying that some centuries later interesting laws of motion would be discovered which would give a meaning to the statement that sand was emptied from the bulbs in equal times. Uniformity in change is directly perceived and it follows that mankind perceives in nature factors from which a theory of temporal congruence can be formed. The prevalent theory entirely fails to produce such factors. The mention of the laws of motion raises another point where the prevalent theory has nothing to say and the new theory gives a complete explanation. It is well known that the laws of motion are not valid for any axes of reference which you may choose to take fixed in any rigid body. You must choose a body which is not rotating and has no acceleration. For example they do not really apply to axes fixed in the earth because of the diurnal rotation of that body. The law which fails when you assume the wrong axes as at rest is the third law that action and reaction are equal and opposite. With the wrong axes uncompensated centrifugal forces and uncompensated composite centrifugal forces appear due to rotation. The influence of these forces can be demonstrated by many facts on the earth's surface. Foucault's pendulum, the shape of the earth, the fixed directions of the rotations of cyclones and anti-cyclones. It is difficult to take seriously the suggestion that these domestic phenomena on the earth are due to the influence of the fixed stars. I cannot persuade myself to believe that a little star in its twinkling turned round Foucault's pendulum in the Paris exhibition of 1861. Of course anything is believable when a definite physical connection has been demonstrated. For example the influence of sunspots. Here all demonstration is lacking in the form of any coherent theory. According to the theory of these lectures the axes to which motion is to be referred are axes at rest in the space of some time system. For example consider the space of a time system alpha. There are sets of axes at rest in the space of alpha. These are suitable dynamical axes. Also a set of axes in this space which is moving with uniform velocity without rotation is another suitable set. All the moving points fixed in these moving axes are really tracing out parallel lines with one uniform velocity. In other words they are reflections in the space of alpha of a set of fixed axes in the space of some other time system beta. Accordingly the group of dynamical axes required for Newton's laws of motion is the outcome of the necessity of referring motion to a body at rest in the space of some one time system in order to obtain a coherent account of physical properties. If we do not do so the meaning of the motion of one portion of our physical configuration is different from the meaning of the motion of another portion of the same configuration. Thus the meaning of motion being what it is in order to describe the motion of any system of objects without changing the meaning of your terms as you proceed with your description you are bound to take one of these sets of axes as axes of reference. Though you may choose their reflections into the space of any time system which you wish to adopt. A definite physical reason is thereby assigned for the peculiar property of the dynamical group of axes. On the orthodox theory the position of the equations of motion is most ambiguous. The space to which they refer is completely undetermined and so is the measurement of the lapse of time. Science is simply setting out on a fishing expedition to see whether it cannot find some procedure which it can call the measurement of space and some procedure which it can call the measurement of time and something which it can call a system of forces and something which it can call masses so that these formulae may be satisfied. The only reason on this theory why anyone should want to satisfy these formulae is a sentimental regard for Galileo, Newton, Euler and Lagrange. The theory so far from founding science on a sound observational basis forces everything to conform to a mere mathematical preference for a certain simple formulae. I do not for a moment believe that this is a true account of the real status of the laws of motion. These equations want some slight adjustment for the new formulae of relativity. But with these adjustments imperceptible in ordinary use the laws deal with fundamental physical quantities which we know very well and wish to correlate. The measurement of time was known to all civilized nations long before the laws were thought of. It is this time, as thus measured, that the laws are concerned with. Also they deal with the space of our daily life. When we approach to an accuracy of measurement beyond that of observation adjustment is allowable. But within the limits of observation we know what we mean when we speak of measurements of space and measurements of time and uniformity of change. It is for science to give an intellectual account of what is so evident in sense awareness. It is to me, thoroughly incredible that the ultimate fact beyond which there is no deeper explanation is that mankind has really been swayed by an unconscious desire to satisfy the mathematical formulae which we call the laws of motion. Formulae completely unknown till the 17th century of our epoch. The correlation of the facts of sense experience affected by the alternative account of nature extends beyond the physical properties of motion and the properties of congruence. It gives an account of the meaning of the geometrical entities such as points, straight lines and volumes and connects the kindred ideas of extension in time and extension in space. The theory satisfies the true purpose of an intellectual explanation in the sphere of natural philosophy. This purpose is to exhibit the interconnections of nature and to show that one set of ingredients in nature requires for the exhibition of its character the presence of the other set of ingredients. The false idea which we have to get rid of is that of nature as a mere aggregate of independent entities each capable of isolation. According to this conception these entities whose characters are capable of isolated definition come together and by their accidental relations form the system of nature. This system is thus thoroughly accidental and even if it be subject to a mechanical fate it is only accidentally so subject. With this theory space might be without time and time might be without space. The theory admittedly breaks down when we come to the relations of matter and space. The relational theory of space is an admission that we cannot know space without matter or matter without space but the seclusion of both from time is still jealously guarded. The relations between portions of matter in space are accidental facts owing to the absence of any coherent account of how space springs from matter or how matter springs from space. Also what we really observe in nature its colors and its sounds and its touches are secondary qualities. In other words they are not in nature at all but are accidental products of the relations between nature and mind. The explanation of nature which I urge as an alternative ideal to this accidental view of nature is that nothing in nature could be what it is except as an ingredient in nature as it is. The whole which is present for discrimination is posited in sense awareness as necessary for the discriminated parts. An isolated event is not an event because every event is a factor in a larger whole and is significant of that whole. There can be no time apart from space and no space apart from time and no space and no time apart from the passage of the events of nature. The isolation of an entity in thought when we think of it as a bare it has no counterpart in any corresponding isolation such isolation is merely part of the procedure of intellectual knowledge. The laws of nature are the outcome of the characters of the entities which we find in nature. The entities being what they are the laws must be what they are and conversely the entities follow from the laws. We are a long way from the attainment of such an ideal but it remains as the abiding goal of theoretical science. End of chapter 6