CHAPTER 12

Scalar Motion

It was recognized from the beginning of the development of the theory of the universe of motion that the basic motions are necessarily scalar. This was stated specifically in the first published description of the theory, the original (1959) edition of The Structure of the Physical Universe. It was further recognized, and emphasized in that 1959 publication, that the rotational motion of the atoms of matter is one of these basic scalar motions, and therefore has an inward translational effect, which we can identify as gravitation. Throughout the early stages of the theoretical development, however, there was some question as to the exact status of rotation in a system of scalar motions, inasmuch as rotation, as ordinarily conceived, is directional, whereas scalar quantities, by definition, have no directions. At first this issue was not critical, but as the development of the theory was extended into additional physical areas, more types of motion of a rotational character were encountered, and it became necessary to clarify the nature of scalar rotation. A full scale investigation of the subject was therefore undertaken, the results of which were reported in The Neglected Facts of Science, published in 1982.

The existence of scalar motion is not recognized by present-day physics. Indeed, motion is usually defined in such a way that scalar motion is specifically excluded. This type of motion enters into observable physical phenomena in a rather unobtrusive manner, and it is not particularly surprising that its existence remained unrecognized for a long time. However, a quarter of a century has elapsed since that existence was brought to the attention of the scientific community in the first published description of the universe of motion, and it is hard to understand why so many individuals still seem unable to recognize that there are several observable types of motion that cannot be other than scalar.

For instance, the astronomers tell us that the distant galaxies are all moving radially outward away from each other. The full significance of this galactic motion is not apparent on casual consideration, as we see each of the distant galaxies moving outward from our own location, and we are able to locate each of the observed motions in our spatial reference system in the same manner as the familiar motions of our everyday experience. But the true character of this motion becomes apparent when we examine the relation of our Milky Way galaxy to this system of motions. Unless we take the stand that our galaxy is the only stationary object in the universe, an assumption that few scientists care to defend in this modern era, we must recognize that our galaxy is moving away from all of the others; that is, it is moving in all directions. And since it is conceded that our galaxy is not unique, it follows that all of the widely separated galaxies are moving outward in all directions. Such a motion, which takes place uniformly in all directions has no specific direction. It is completely defined by a magnitude (positive or negative), and is therefore scalar.

A close examination of gravitation shows that the gravitational motion is likewise scalar, differing from the motion of the galaxies only in that it is negative (inward) rather than positive (outward). The resemblance to the motion of the galaxies can easily be seen if we consider a system of gravitating objects isolated in space—perhaps a group of galaxies relatively close to each other. From our knowledge of the gravitational effects we can deduce that each of these objects will move inward toward all of the others. Here again the motion is scalar. Each object is moving inward in all directions.

A small-scale example of the same kind of motion can be seen in the motion of spots on the surface of an expanding balloon, often used as an analogy by those who undertake to explain the nature of the motions of the distant galaxies. Here, too, each individual is moving outward from all others. If the expansion is terminated, and succeeded by a contraction, the motions are reversed, and each spot then moves inward toward all others, as in the gravitational motion.

In the case of the expanding balloon there is a known physical mechanism that is causing the expansion, and our understanding of this mechanism makes it evident that all locations on the balloon surface are moving. The spots on this surface have no motion of their own. They are merely being carried along by the movement of the locations that they occupy. According to the astronomers’ current view, the recession of the distant galaxies is the same kind of a process. As Paul Davies explains:

Many people (including some scientists) think of the recession of the galaxies as due to the explosion of a lump of matter into a pre-existing void, with the galaxies as fragments rushing through space. This is quite wrong… The expanding universe is not the motion of the galaxies through space away from some centre, but is the steady expansion of space.²²

Here, again, it is the locations that are moving, carrying the galaxies along with them. But in this case there is no known physical mechanism to account for the movement. Like the expansion of the balloon, the “steady expansion of space” is merely a description, not an explanation, of the movement. All that the observations tell us is that an outward scalar motion of physical locations is taking place, carrying the galaxies with it.

The postulates of the Reciprocal System of theory, the theory of the universe of motion, generalize this type of motion. They define a universe in which scalar motion of physical locations is the basic form of motion from which all physical entities and phenomena are derived. The manner in which this type of motion manifests itself to observation therefore has an important bearing on the nature of fundamental physical phenomena.

This situation is a good example of the way in which important information is often overlooked because no one spends the time and effort that are required in order to make a thorough study of a seemingly unimportant observation. It has long been recognized that the motion of spots on the surface of an expanding balloon is, in some way, different from the ordinary motions of our everyday experience. The mere fact that this balloon motion is so widely used as an analogy in explaining the recession of the distant galaxies is clear evidence of this general recognition. But the galaxies seem to be a special case, and expanding balloons do not play any significant part in normal physical activity. Consequently, no one has been much interested in the physics of these objects, and this admittedly unique kind of motion was never subjected to a critical examination prior to the investigation of scalar motion that was undertaken in the course of the theoretical development reported in the several volumes of this work. The finding that the fundamental motion of the universe is scalar revolutionizes this situation. The motions of the galaxies, gravitating objects, and spots on the surface of an expanding balloon are obviously the kind of motions—scalar motions—that our theory identifies as fundamental.

Scientists are usually, with ample justification, reluctant to accept a hypothesis that postulates the existence of phenomena that are unknown to observation. It should therefore be emphasized that scalar motion is not an unobserved phenomenon; it is an observed phenomenon that has not heretofore been recognized in its true character. Once the motions identified in the foregoing paragraphs have been critically examined, and their scalar character has been recognized, the existence of scalar motion is no longer a hypothesis; it is a demonstrated physical fact. The existence of other scalar motions, as required by the theory of the universe of motion is then a natural and logical corollary, and those observed phenomena that have the theoretical properties of scalar motion can legitimately be identified as scalar motions.

A one-dimensional scalar motion of a physical location is defined by a magnitude, and can therefore be represented one-dimensionally as a point, or an assemblage of points, moving along a straight line. Introduction of a reference point—that is, coupling the motion to the reference system at a specific point in that system—enables distinguishing between positive motion, outward from the reference point, and negative motion, inward toward the reference point. The direction imputed to the motion may be a constant direction, as in the case of the translational motion of the photon, the direction of which is determined by chance at the time of emission, unless external factors intervene. The key point disclosed by our investigation is that the direction is not necessarily constant. A discontinuous, or non-uniform, change of direction could be maintained only by a repeated application of an external force, but it has been known from the time of Galileo that a continuous and uniform change of position or direction is just as permanent and just as self-sustaining as a condition of rest. Our finding merely extends this principle to the assignment of direction to scalar motion.

As an illustration, let us consider the motion of point X, originating at point A, and initially proceeding in the direction AB in three-dimensional space. Then let us assume that line AB is rotated around an axis perpendicular to it, and passing through point A. This does not change the inherent nature or magnitude of the motion of point X, which is still moving radially outward from point A at the same speed as before. What has been changed is the direction of the movement, which is not a property of the motion itself, but a feature of the relation between the motion and three-dimensional space. Instead of continuing to move outward from A in the direction AB, point X now moves outward in all directions in the plane of rotation. If that plane is then rotated around another perpendicular axis, the outward motion of point X is distributed over all directions in space. It is then a rotationally distributed scalar motion.

The results of such a distributed scalar motion are totally different from those produced by a combination of vectorial motions in different directions. The combined effects of the magnitudes and directions of vectorial motions can be expressed as vectors. The results of addition of these vectors are highly sensitive to the effects of direction. For example, a vectorial motion AB added to a vectorial motion AB’ of equal magnitude, but diametrically opposite direction, produces a zero resultant. Similarly, vectorial motions of equal magnitude in all directions from a given point add up to zero. But a scalar motion retains the same positive (outward) or negative (inward) magnitude regardless of the manner in which it is directionally distributed.

None of the types of scalar motion that have been identified can be represented in a fixed spatial reference system in its true character. Such a reference system cannot represent simultaneous motion in all directions. Indeed, it cannot represent motion in more than one direction. In order to represent a system of two or more scalar motions in a spatial reference system it is necessary to define a reference point for the system as a whole; that is, the scalar system must be coupled to the reference system in such a way that one of the moving locations in the scalar system is arbitrarily defined as motionless (from the scalar standpoint) relative to the reference system. The direction imputed to the motion of each of the other objects, or physical locations, in the scalar system is then its direction relative to the reference point.

For example, if we denote our galaxy as A, the direction of the motion of distant galaxy X, as we see it, is AX. But observers in galaxy B, if there are any, see galaxy X as moving in the different direction BX, those in galaxy C see the direction as CX, and so on. The significance of this dependence of the direction on the reference point can be appreciated when it is contrasted with the corresponding aspect of vectorial motion. If an object X is moving vectorially in the direction AX when viewed from location A, it is also moving in this same direction AX when viewed from any other location in the reference system.

It should be understood that the immobilization of the reference point in the reference system applies only to the representation of the scalar motion. There is nothing to prevent an object located at the reference point, the reference object we may call it, from acquiring an additional motion of a vectorial character. For example, the expanding balloon may be resting on the floor of a moving vehicle, in which case the reference point is in motion vectorially. Where an additional motion of this nature exists, it is subject to the same considerations as any other vectorial motion.

The coupling of a system of scalar motions to a fixed reference system at a reference point does not alter the rate of separation of the members of the scalar system. The arbitrary designation of the reference point as motionless (from the scalar standpoint) therefore makes it necessary to attribute the motion of the reference point, or object, to the other points or objects in the system.

This conclusion that the observed change of position of an object B is due, in part, to the motion of some different object A may be hard for those who are thinking in terms of the conventional view of the nature of motion to accept, but it can easily be verified by consideration of a specific example. Any two spots on the surface of an expanding balloon, for instance, are moving away from each other; that is, they are both moving. While spot X moves away from spot Y, spot Y is coincidentally moving away from spot X. Placing the balloon in a reference system does not alter these motions. The balloon continues expanding in exactly the same way as before. The distance XY continues to increase at the same rate, but if X is the reference point, it is motionless in the reference system (so far as the scalar motion is concerned), and the entire increase in the distance XY, including that due to the motion of X, has to be attributed to the motion of Y.

The same is true of the motions of the distant galaxies. The recession that is measured is merely the increase in distance between our galaxy and the one that is receding from us. It follows that a part of the increase in separation that we attribute to the recession of the other galaxy is actually due to motion of our own galaxy. This is not difficult to understand when, as in the case of the galaxies, the reason why objects appear to move faster than they actually do is obviously the arbitrary assumption that our location is stationary. What is now needed is a recognition that this is a general proposition. The same result follows whenever a moving object is arbitrarily taken as stationary for reference purposes. The motion of any reference point of a scalar motion is seen, by the reference system, in the same way in which we view our motion in the galactic system; that is, the motion that is frozen by the reference system is seen as motion of the distant objects.

This transfer of motion from one object to another by reason of the manner in which scalar motion is represented in the reference system has no significant consequences in the galactic situation, as it makes no particular difference to us whether galaxy X is receding from us, or we are receding from it, or both. But the questions as to which objects are actually moving, and how much they are moving, have an important bearing on other scalar motions, such as gravitation. With the benefit of the information now available, it is evident that the rotation of the atoms of matter described in Volume I is a rotationally distributed negative (inward) scalar motion. By virtue of that motion, each atom, irrespective of how it may be moving, or not moving, vectorially, is moving inward toward all other atoms of matter. This inward motion can obviously be identified as gravitation. Here, then, we have the answer to the question as to the origin of gravitation. The same thing that makes an atom an atom—the scalar rotation—causes it to gravitate.

Although Newton specifically disclaimed making any inference as to the mechanism of gravitation, the fact that there is no time term in his equation implies that the gravitational effect is instantaneous. This, in turn, leads to the conclusion that gravitation is “action at a distance,” a process in which one mass acts upon another distant mass without an intervening connection. There is no experimental or observational evidence contradicting the instantaneous action. As noted in Volume I, even in astronomy, where it might be presumed that any inaccuracy would be serious, in view of the great magnitudes involved, “Newtonian theory is still employed almost exclusively to calculate the motions of celestial bodies.”²³

However, instantaneous action at a distance is philosophically unacceptable to most physicists, and they are willing to go to almost any lengths to avoid conceding its existence. The hypothesis of transmission through a “luminiferous ether” served this purpose when it was first proposed, but as further studies were made, it became obvious that no physical substance could have the contradictory properties that were required of this hypothetical medium.

Einstein’s solution was to abandon the concept of the ether as a “substance”—something physical—and to introduce the idea of a quasi-physical entity, a phantom medium that is assumed to have the capabilities of a physical medium without those limitations that are imposed by physical existence. He identifies this phantom medium with space, but concedes that the difference between his space and the ether is mainly semantic. He explains, “We shall say our space has the physical property of transmitting waves, and so omit the use of a word (ether) we have decided to avoid.”²⁴ Since this space (or ether) must exert physical effects without being physical, Einstein has difficulty defining its relation to physical reality. At one time he asserts that “according to the general theory of relativity space is endowed with physical qualities,”²⁵ while in another connection he says that “The ether of the general theory of relativity [which he identifies as space] is a medium which is itself devoid of all mechanical and kinematical qualities.”²⁶ Elsewhere, in a more candid statement, he concedes, in effect, that his explanations are not persuasive, and advises us just to “take for granted the fact that space has the physical property of transmitting electromagnetic waves, and not to bother too much about the meaning of this statement.”²⁷

Einstein’s successors have added another dimension to the confusion of ideas by retaining this concept of space as quasi-physical, something that can be “curved” or otherwise manipulated by physical influences, but transferring the “ether-like” functions of Einstein’s “space” to “fields.” According to this more recent view, matter exerts a gravitational effect that creates a gravitational field, this field transmits the effect at the speed of light, and finally the field acts upon the distant object. Various other fields—electric, magnetic, etc.—are presumed to coexist with the gravitational field, and to act in a similar manner.

The present-day “field” is just as intangible as Einstein’s “space.” There is no physical evidence of its existence. All that we know is that if a test object of an appropriate type is placed within a certain region, it experiences a force whose magnitude can be correlated with the distance to the location of the originating object. What existed before the test object was introduced is wholly speculative. Faraday’s hypothesis was that the field is a condition of stress in the ether. Present-day physicists have transferred the stress to space in order to be able to discard the ether, a change that has little identifiable meaning. As R. H. Dicke puts it, “One suspects that, with empty space having so many properties, all that had been accomplished in destroying the ether was a semantic trick. The ether had been renamed the vacuum.”²⁸ P. W. Bridgman, who reviewed this situation in considerable detail, arrived at a similar conclusion. The results of analysis, he says, “suggest that the role played by the field concept is that of an intellectual dummy, which cancels out of the final result.”²⁹

The theory of the universe of motion gives us a totally different view of this situation. In this universe the reality is motion. Space and time have a real existence only where, and to the extent that, they actually exist as components of motion. On this basis, extension space, the space that is represented by the conventional reference system, is no more than a frame of reference for the spatial magnitudes and directions of the entity, motion, that actually exists. It follows that extension space cannot have any physical properties. It cannot be “curved” or modified in any other way by physical means. Of course, the reference system, being nothing but a human contrivance, could be altered conceptually, but such a change would have no physical significance.

The status of extension space as a purely mental concept devised for reference purposes, rather than a physical entity, likewise means that this space is not a container, or background, for the physical activity of the universe, as assumed by conventional science. In that conventional view, everything physically real is contained within the space and time of the spatio-temporal reference system. When it becomes necessary to postulate something outside these limits in order to meet the demands of theory construction, it is assumed that such phenomena are, in some way, unreal. As Werner Heisenberg puts it, they do not “exist objectively in the same sense as stones or trees exist.”³⁰

The development of the theory of the universe of motion now shows that the conventional spatio-temporal system of reference does not contain everything that is physically real. On the contrary, it is an incomplete system that is not capable of representing the full range of motions which exist in the physical universe. It cannot represent motion in more than one scalar dimension; it cannot represent a scalar system in which all elements are moving; nor can it correctly represent the position of an individual object that is moving in all directions simultaneously (that is, an object whose motion is scalar, and therefore has no specific direction). Many of the other shortcomings of this reference system will not become apparent until we examine the effects of very high speed motion in Volume III, but those that have been mentioned have a significant impact on the phenomena that we are now examining.

The inability to represent more than one dimension of scalar motion is a particularly serious deficiency, inasmuch as the postulated three-dimensionality of the universe of motion necessarily permits the existence of three dimensions of motion. Only one dimension of vectorial motion is possible, because all three dimensions of space are required in order to represent the directions of this one-dimensional motion, but scalar motion has magnitude only, and a three-dimensional universe can accommodate scalar motion in all three of its dimensions.

Since the conventional reference system cannot represent all of the distributed scalar motions, and present-day science does not recognize the existence of any motions that cannot be represented in that system, it has been necessary for the theorists to make some arbitrary assumptions as a means of compensating for the distortion of the physical picture due to this deficiency of the reference system. One of the principal steps taken in this direction is the introduction of the concept of “fundamental forces,” autonomous entities that exist in their own right, and not as properties of something more basic. The present tendency is to regard these so-called fundamental forces as the sources of all physical activity, and the currently popular goal of the theoretical physicists, the formulation of a “grand unified theory,” is limited to finding a common denominator of these forces.

Gravitation is, in a way, an exception, as the currently popular hypothesis as to the nature of the gravitational force, Einstein’s general theory of relativity, does attempt an explanation of its origin. According to this theory, the gravitational force is due to a distortion of space resulting from the presence of matter. So far as can be determined from the scientific literature, no one has the slightest idea as to how such a distortion of space could be accomplished. Arthur Eddington expressed the casual attitude of the scientific community toward this issue in the following statement: “We do not ask how mass gets a grip on space-time and causes the curvature which our theory postulates.”³¹ But unless the question is asked, the answer is not forthcoming. In Newton’s theory the gravitational force originates from mass in a totally unexplained manner. In Einstein’s theory it is a result of a distortion, or “curvature,” of space that is produced by mass in a totally unexplained manner. Thus, whatever its other merits may be, the current theory (general relativity) accomplishes no more toward accounting for the origin of the gravitational force than its predecessor.

In order to arrive at such an explanation we need to recognize that force is not an autonomous entity; it is a property of motion. The motion of an individual mass unit is measured in terms of speed (or velocity). The total amount of motion in a material aggregate is then the product of the speed and the number of mass units, a quantity formerly called “quantity of motion,” but now known as momentum. The rate of change of the motion of the individual unit is acceleration; that of the total quantity of motion is force. The force is thus the total quantity of acceleration.

The significance of this, in the present connection, is that force not only produces an acceleration when applied to a mass (a fact that is currently recognized), it is an acceleration prior to that application (a fact that is currently overlooked or disregarded). In other words, the acceleration is simply transferred. For example, when a rocket is fired, the total “quantity of acceleration” available for application to the rocket (the force) is the sum of the quantities of acceleration of the individual particles of the gas produced from the propellant. The division of this total quantity among the mass units of the rocket determines the acceleration of each individual unit. and therefore of the rocket as a whole.

Since force is a property of a motion rather than an autonomous entity, it follows that wherever there is a force there must also be a motion of which the force is a property. This leads to the conclusion that a gravitational force field is a region of space in which gravitational motions exist. In the context of conventional physical thought this conclusion is unacceptable, since there are no moving entities in an unoccupied field.

The information about the nature of scalar motion developed in the earlier pages clarifies this situation. A material aggregate is moving gravitationally in all directions, but the conventional spatial reference system is unable to represent a system of motions of this nature in its true character. As previously indicated, where the scalar motion AB of object A (the massive object now under consideration) toward object B (the test mass) cannot be represented in the reference system because of the limitations of that system, this motion AB is shown as a motion BA; that is, a motion of the test mass B toward the massive object A, constituting an addition to the actual motion of that test mass. Because of the spherical distribution of the scalar motions of the atoms of mass A, the magnitude of the motion imputed to mass B depends on its distance from A, and is inversely proportional to the square of that distance. Thus each point in the region surrounding A corresponds to a specific fraction of the motion of A, representing the amount of motion that would be imputed to a unit mass, if that mass is actually placed at this particular point.

Here, then, is the explanation of the gravitational field (and, by extension, all other fields of the same nature). The field is not something physically real in the space; “for the modern physicist as real as the chair on which he sits,”³² as asserted by Einstein. Nor is it, as Faraday surmised, a stress in the ether. Neither is it some kind of a change in the properties of space, as envisioned by present-day theorists. It is simply the pattern of the magnitudes of the motions of one mass that have to be imputed to other masses because of the inability of the reference system to represent scalar motion as it actually exists.

No doubt this assertion that what appears to be a motion of one object is actually, in large part, a motion of a different object is somewhat confusing to those who are accustomed to conventional ideas about motion. But once it is realized that scalar motion exists, and because it has no inherent direction it may be distributed over all directions, it is evident that the reference system cannot represent this motion in its true character. In the preceding analysis we have determined just how the reference system does represent this motion that it cannot represent correctly.

This may appear to be a return to the action at a distance that is so distasteful to most scientists, but, in fact, the apparent action on distant objects is an illusion created by the introduction of the concept of autonomous forces to compensate for the shortcomings of the reference system. If the reference system were capable of representing all of the scalar motions in their true character, there would be no problem. Each mass would then be seen to be pursuing its own course, moving inward in space independently of other objects.

In this case, accepted scientific theory has gone wrong because prejudice supported by abstract theory has been allowed to override the results of physical observations. The observers keep calling attention to the absence of evidence of the finite propagation time that current theory ascribes to the gravitational effect, as in this extract from a news report of a conference at which the subject was discussed:

When it [the distance] is astronomical, the difficulty arises that the intermediaries need a measurable time to cross, while the forces in fact seem to appear instantaneously.³³

But it is assumed that we must accept either a finite propagation time or action at a distance, which, as Bridgman once said, is “a concept to which many physicists have a violent allergy.”³⁴ Einstein’s theory, which supports the propagation hypothesis, has therefore been accorded a status superior to the observations. The following statement from a physicist brings this point out explicitly:

Nowadays we are also convinced that gravitation progresses with the speed of light. This conviction, however, does not stem from a new experiment or a new observation, it is a result solely of the theory of relativity.³⁵

This is another example of a practice that has been the subject of critical comment in several different connections in the preceding pages of this and the earlier volume. Overconfidence in the existing body of scientific knowledge has led the investigators to assume that all alternatives in a given situation have been considered. It is then concluded that an obviously flawed hypothesis must be accepted, in spite of its shortcomings, because “there is no other way.” Time and again in the earlier pages, development of the theory of the universe of motion has shown that there is another way, one that is free from the objectionable features. So it is in this case. It is not necessary either to contradict observation by assuming a finite speed of propagation or to accept action at a distance.

Some of the most significant consequences of the existence of scalar motion are related to its dimensions. This term is used in several different senses, two of which are utilized extensively in this work. When physical quantities are resolved into component quantities of a fundamental nature, these component quantities are called dimensions. Identification of the dimensions, in this sense, of the basic physical quantities has been an important feature of the development of theory in the preceding pages. In a different sense of the term, it is generally recognized that space is three-dimensional.

Conventional physics recognizes motion in three-dimensional space, and represents motions of this nature by lines in a three-dimensional spatial coordinate system. But these motions which exist in three dimensions of space are only one-dimensional motions. Each individual motion of this kind can be characterized by a vector, and the resultant of any number of these vectors is a one-dimensional motion defined by the vector sum. All three dimensions of the spatial reference system are required for the representation of one-dimensional motion, and there is no way by which the system can indicate a change of position in any other dimension. However, the postulate that the universe of motion is three-dimensional carries with it the existence of three dimensions of motion. Thus there are two dimensions of motion in the physical universe that cannot be represented in the conventional spatial reference system.

In common usage the word “dimensions” is taken to mean spatial dimensions, and reference to three dimensions is ordinarily interpreted geometrically. It should be realized, however, that the geometric pattern is merely a graphical representation of the relevant physical magnitudes and directions. From the mathematical standpoint an n-dimensional quantity is one that requires n independent magnitudes for a complete definition. Thus a scalar motion in three dimensions, the maximum in a three-dimensional universe, is defined in terms of three independent magnitudes One of these magnitudes—that is, the magnitude of one of the dimensions of scalar motion—can be further divided dimensionally by the introduction of directions relative to a spatial reference system. This expedient resolves the one-dimensional scalar magnitude into three orthogonally related sub-magnitudes, which together with the directions, constitute vectors. But no more than one of the three scalar magnitudes that define a three-dimensional scalar motion can be sub-divided vectorially in this manner.

Here is a place where recognition of the existence of scalar motion changes the physical picture radically. As long as motion is viewed entirely in vectorial terms—that is, as a change of position relative to the spatial reference system—there can be no motion other than that represented in that system. But since scalar motion has magnitude only, there can be motion of this character in all three of the existing dimensions of the physical universe. It should be emphasized that these dimensions of scalar motion are mathematical dimensions. They can be represented geometrically only in part, because of the limitations of geometrical representation. In order to distinguish these mathematical dimensions of motion from the geometric dimensions of space in which one dimension of the motion takes place, we are using the term “scalar dimension” in a manner analogous to the use of the term “scalar direction” in the earlier pages of this and the preceding volume.