Representations of Motion
The “Frame of Reference” from which observations are made, and therefore, the “Frame” in which mathematical representations of all of our observations must be made, involves the three dimensional framework called space and the scalar quantity called time. Observations require classification of the movements of things as being vectorially in linear translation, linear oscillation, unidirectional rotation, rotational oscillation, or some combination of these four classifications of movement. To mathematically define movement in any mode requires specification of a reference point or line and the designation of three mutually perpendicular dimensions having origin at the specified reference point or orientation to the line. Any motion in such a system is represented by a vector which has both magnitude and direction, and always in one specific direction which requires all three Euclidean dimensions to adequately set boundaries by which to define relations.
The mathematical limitations are a direct result of limitations on existence in this three dimensional spatial reference system. Anything which is observed to have motion in space is observed to have movement in only one direction of one dimension and/or rotation in a specific direction around an axis that is oriented in a single direction of one dimension only. Any attempt to give an object any compound motion (movement in two or three dimensions concurrently) results in merely changing the direction in space of the one dimensional linear movement of the atoms of the object. As a direct result of this limitation of having any spatial motion reduced to one dimensional movement, the ability to observe effects of scalar motion is severely limited, even to the point of failing to recognize scalar motions as such or that they really exit as something different from vectorial motions.
Space and Time
Space as it is known and used in equations of physical processes is threedimensional. The sector of the universe in which observation is made definitely presents a three dimensional aspect for those observations—no more, no less. Space as it is found from experimentation, not as it may be interpreted to be from any theoretical viewpoint, is homogeneous. So far as can be determined each part of space is exactly like every other part of space. It is isotropic; its behavior is the same in all directions. Space is threedimensional, homogeneous, and isotropic in the local environment that is accessible to direct experimental observation. Mathematical continuity does not require infinite divisibility, but even if it did, such divisibility would be beyond observational range, both practically and theoretically. Certainly, infinite space cannot be verified either.
Considering the little that is known about space, direct knowledge of time is still more limited. The most conspicuous feature of time (as we observe it regardless of how we define it) is that it progresses. It is only as a progression that time is known at all. Whatever properties are recognized for time are simply characteristics of the progression; so far as can be determined, the progression is uniform. The most obvious mathematical property observed is that in the context of the familiar phenomena of everyday life, time is scalar. In the velocity equation v = s/t the term t is a scalar quantity. As observed, time appears to flow steadily onward in the same scalar direction. The formulation of the Second Law of Thermodynamics gives expression to this empirical observation. In spite of the fact of contrary assertions contained in various verbal descriptive statements of the Second Law^{1}, the term t is mathematically reversible. At present that interpretation in the equations representing various physical phenomena is not allowed. In spite of the constant direction of “Time’s Arrow” in the everyday phenomena of this local environment, it would be presumptuous to be other than cautious about extrapolating the same constancy of direction to all regions of the universe.
In the equation v = s/t any change of location in space s is a vector quantity because it has direction in space. It follows that the velocity v also has a direction in space, and thus the relation is a space velocity equation. In this equation the term t is necessarily scalar because time has no direction in space. Time is scalar in this space velocity equation irrespective of its own dimensions, because no matter how many dimensions it may have, one or many, time has no direction in space. Time is definitely directional in its flow characteristic in that we experience only the “now” of time subsequent to the “past” and prior to the “future”. That which is called the direction of the flow of time is not identifiable with any direction in space, and therefore, time has no direction in space. Time is a scalar quantity relative to the dimensionality of space.
If time is multidimensional, then that property which corresponds to the spatial property that is called “dimension” becomes a temporal property. Whatever the temporal properties are called, whether they are referred to as “dimensions of time” and “magnitudes of time” or given altogether different names, those properties are temporal properties, not spatial properties. The names by which the properties of time are described does not give magnitudes or directions of time any direction or magnitude in space. Regardless of how many dimensions time may have, time cannot be a vector quantity in any equation in which the property of having direction in space is that which qualifies the quantity as vectorial.
Although these items constitute all that is actually known about space and time individually from direct observation of each alone, there is one more source of direct information. There, is some observational knowledge of the RELATION between space and time. The first item of this nature is that the relation between space and time in this sector of the universe is motion. The second item is that in any motion illustrating this relation, the space and time thereof are reciprocally related from a scalar standpoint. This means that moving a greater distance in the same time has exactly the same effect on the speed, the scalar measure of the motion, as moving the same distance in less time.
SUMMARIZING these observations from the local environment:^{2}

Space is three dimensional, homogeneous, and isotropic.

Time progresses uniformly and (perhaps only in the local region) unidirectionally.

The scalar relation between space and time is reciprocal, and this relation constitutes motion.
GENERALIZING these findings and expressing them as hypotheses applicable to the entire universe, we have:^{3}

Space is threedimensional, homogeneous, and isotropic throughout the universe.

Time progresses uniformly throughout the universe.

Throughout the universe, the relation between space and time is reciprocal, and this relation constitutes motion.
Deductions from the Hypotheses
The first consequence deduced from the extrapolated hypotheses is that a general reciprocal relation exists between space and time and that there must be complete symmetry of representation for these two entities in order to have a scalar reciprocity be completely generalized. Symmetry of representation implies that all properties which are possessed by either space or time individually are properties of both space and time. The interpretation of this is that both space and time are threedimensional, homogeneous and isotropic, and both progress at a uniform rate relative to the other.
A GENERAL reciprocity has to include all characteristics observed concerning both aspects involved in the relation called motion. The idea of scalar reciprocity is not at all difficult to grasp, but the idea of reciprocity of dimensionality is, for most people, virtually impossible. Reciprocal dimensionality of the aspects of motion leads to the observed inability to represent the dimensionality of the other while representing the dimensionality of the one; thus, the three dimensionality of time is exactly like the three dimensionality of space; it just isn’t observable. Reciprocity of progression is also not particularly difficult to grasp because all that is specified is a progression, not a progression in a particular direction. Therefore, the progression required is scalar, not vectorial, although it must be represented in a dimensional system. The difference between the usual scalar reciprocity and reciprocity of progression will become clearer as the discussion continues.
A conclusion that all properties of either space or time are properties of both space and time would be demolished immediately if any of the properties extrapolated from one to the other could be shown to be inconsistent with established facts. In view of the great differences which appear to exist between space and time as we ordinarily envision them, it would seem that discrepancies of this kind should be easy to locate.
It is true that the concept of threedimensional time is in conflict with prevailing ideas, but it is only conflicts with established facts that are fatal to any conceptual hypothesis. The historical record of human ideas as to the dimensions of time does not make any idea factual for any theoretical interpretation of actuality.
A dimension of time is not a dimension in space, it is not anything in space; it is a property of time itself. As previously pointed out, the scalar nature of the time term in the equations of motion is not a result of time being one dimensional. The scalar nature of time results from the fact that time has no direction in space, regardless of how many dimensions or directions it may have of its own. There is nothing at all in any of our observations that precludes time from being three dimensional.^{4}
To those who are accustomed to thinking along different lines, the idea of a progression of space similar to the observed progression of time may seem even more outrageous than does the concept of threedimensional time. The fact is that there is actual observational evidence of a spatial progression.
Since a spatial location can be of any size, any identifiable portion of a reference system may be called a location in that system. Because of the hypothesized three dimensional homogeneous isotropy of space, a scalar progression of space from one location to another would require all locations to progress outward away from all other similarly defined locations at a uniform rate. Jumping ahead for a moment, a location might contain observable “things” which have a simultaneous inward scalar progression. If such is the case, the combination of a uniform outward scalar progression of similarly defined locations which also contain objects having an inward scalar progression brings about an equality between the two opposing scalar progressions at some distance from each “thing” or object. For objects at distances greater than that at which an equality of the two opposing progressions exists, this theoretical development requires that the objects involved should be moving steadily radially outward away from each other at a rate proportional to the distance of their separation. That rate is the difference between the effects of the two progressions at the specified distance of separation. Of course, locations in space cannot be seen, but objects which occupy locations in space can be seen.
Observations of distant galaxies indicate that they are massive objects which have a gravitational characteristic and that their observed locations are indeed being carried outward away from our location. They are so far away that any lateral motions which they possess are unobservable, and the effect of mutual gravitational movements are attenuated to the point that the gravitational movement due to either or both galaxies is no longer the controlling factor in their relative movements. Mutual gravitation is merely a modifier for the rate of outward movement in exact accord with this theory.
The scalar progression of space is derived by theoretical reasoning based on extrapolation of our observations of space and time in our everyday experience. A hypothesis such as the scalar progression of space that is corroborated by an entirely different phenomenon extremely remote from our daily experience is of an entirely different character than any of the many ad hoc hypotheses presently found in modern physics. This corroboration for the progression of space takes the hypotheses out of the realm of being strictly hypothesis and puts them into the realm of experimental observation. We are now in a position to assert that we have increased factual knowledge of the physical universe. From this we can look forward with confidence to additional applications of the progression hypothesis in other physical areas, which will not only represent further advances in those areas of scientific knowledge, but will reinforce the already strong position of the hypotheses as absolute knowledge for all areas of science.^{5}
Other Inferences
To answer the question of whether space and time are continuous or exist in discrete step type units, one can refer to fundamental language definitions of concepts. Motion must be continuous if it is to conform to the basic concept from which the word is derived (L. motionis) as well as be isotropic and homogeneous. For application to physical concepts ordinary mathematics requires the use of real numbers in its definitions. The concepts of motion must conform to the rules of ordinary commutative arithmetic if motion is to be a principal ingredient of physical phenomena. Since space and time are hypothesized to be reciprocally related in a general concept of motion, they must be represented by real number units. Identification of abstract points within units of motion does not imply step discreteness any more than identification of phase differences implies the necessity of different frequencies. The concept of abstract points is merely a tool for analysis of effects.
Considering the observed relation of space and time, the question is encountered as to whether we should consider space and time as separate (as did both Einstein and Newton) but related entities (as Einstein later recognized) or as two different aspects of the same basic entity by which neither can exist without the other. The question of relatedness has no bearing on the development of thought beyond the question of whether space can exist without time or vice versa.
Since we are hypothesizing that space and time are reciprocally related throughout the universe as well as being homogeneous and isotropic, it is appropriate and necessary for logical consistency to hypothesize that space and time are the two different aspects of the one thing defined by their relation, motion. By this proposition, neither exists without the other. In this way the questions of infinite or zero motion or effect of motion are avoided by it being impossible to have infinite space and zero time and vice versa. Although extremely high velocities, s/t ratios, and infinitesimal velocities are possibilities, values closer to one are more probable than large values or very small values, maximum probability exists at the one for one ratio of space to time.
It is observed that throughout the history of science there has been a steady growth in the recognition of discontinuity in the physical world. At the time the atomic nature of matter was first proposed, all primary physical phenomena were thought to be continuous and infinitely divisible. As knowledge has grown, more and more phenomena have been found to exist only in discrete units. The discrete unit nature of electric charge and of radiant energy are already well confirmed, and there is increasing evidence for the existence of basic units in other phenomena. However, since the subsequent theoretical development is not an outgrowth of experience and observation, but is deductively derived from the consequences of the postulates utilizing only the assumptions, hypotheses and logic that originally led to the postulates for the Reciprocal System of theory, experience and observation occupy only a corroborative role for those consequences.
The Postulates
The basic postulates for the development of a theoretical physical universe of motion are:^{6}

The physical universe is composed entirely of one component, MOTION, existing in three dimensions, in discrete units, and with two reciprocal aspects, space and time.
Motion is defined as the relation between two uniformly progressing reciprocal quantities, space and time.

The physical universe conforms to the relations of ordinary commutative mathematics, its primary magnitudes are absolute, and its geometry is Euclidean.
The net result of the basic postulates for the theoretical universe of motion, plus the limitations of the previously specified assumptions and observations is to assert the existence of any kind of motion that is not excluded by those assumptions whether it is observable or not. The effect of this interpretation of the postulates is to stipulate that in the theoretical universe of motion anything that can exist theoretically does exist.
Initial Consequences
The first postulate specifies “motion”. It does not specify vectorial motion only, nor does it specify scalar motion only, therefore, both kinds of motion must be observed either directly or as an effect. The first postulate states that motion exists in three dimensions; therefore, the representation of both scalar and vectorial motions must be in three dimensions. This statement is NOT equivalent to saying that there are three dimensions of scalar motion and three dimensions of vectorial motion.
The basic entities of which the theoretical universe of motion is constructed are units of motion, and the existence of different observable entities and phenomena is due to the fact that scalar motion or the effect of a scalar motion necessarily assumes a specific direction when it becomes manifest in the context of a dimensional frame of reference.
One of the original assumptions was that the generally accepted principles of mathematical analysis are valid. Even so, it has been found necessary to state specifically as a postulate that the theoretical universe of motion, in general, conforms to the relationships of ordinary commutative mathematics including probability relations. By making the original assumption part of the postulates the magnitude of the primary quantity is absolute, the geometry of motion is Euclidean, and the fundamental characteristics of the mathematical system cause the modes and sequence of representation for the relations among primary and displacement motions to have very specific values less than unit primary motion in the aspect of representation.
The determination of what entities, phenomena, and processes can exist in the theoretical universe reduces to a matter of determining what kinds of motions and combinations of motions can exist in such a universe, and what changes can take place in and among the three dimensional representations of these motions and their effects. The physical processes of the theoretical universe thus developed include a continuing series of interchanges among the representations and combinations of scalar motions and their effects in the three dimensionality of space. In all of these interchanges, causality is maintained; no motions of any type occur except as a result of previously existing motions and the consequences of the possibilities and probabilities for representation of the combined units of motion. The only sense in which determinism applies to the universe of motion is in the degree to which an added unit of scalar motion can affect the resulting vectorial representation required by a three dimensional frame of reference. In many situations, the directional representation of scalar motions or their effects in three dimensional space are continually being redetermined by chance processes, with apparently initial results being chaos.
A point of considerable significance is that the postulates imply the existence of independent motions, although they do not provide any mechanism for originating or terminating the existence of independent motions. Consequently, the number of effective units of such motion now existing can neither be increased nor decreased by any process within the physical system. The only thing that any physical or chemical process can do is to shift the associations of the already existing units of motion. This inability to alter the existing number of effective units of independent motion is the basis for what will be called the general conservation law, and the various subsidiary conservation laws applying to specific physical phenomena.
The question of how the universe came into being and its ultimate fate is not addressed by the Reciprocal System of theory. It is, therefore, completely neutral on the question of creation. The subsequent development in this discussion of the structures of subatomic particles and atoms of matter is not a description of how they were formed or came into being, but is merely a description of their structures. The discussion of chemical and physical phenomena is greatly simplified and is not intended to be a definitive description either of the theory or how motion becomes manifest.
Essential Considerations
One of the first essentials for an understanding of the system of motions that constitutes the theoretical universe of motion defined by the Reciprocal System of theory is to relate all motions to the natural reference system. Eliminating the confusion that has been introduced in all theoretical, as well as experimental, scientific inquiry by the use of the fixed reference system of everyday experience will probably be the most difficult task which the reader faces. This is not to say that limited application of a three dimensional framework has not been proper in its place; it is just that insistence upon continuing to use a limited concept of three dimensionality in areas in which those limitations may not apply is inappropriate and not truly scientific. Scalar motions are either positively or negatively directed. It is the three dimensionality of space and time individually that proliferates the appearances and lets us observe translations, rotations, and oscillations, as well as, numerous other effects.
To understand this system, the reader must accept the postulates and their validity as a working premise. This is the only appropriate approach for study of any theoretical construct based on any theory. There are several points about which the reader must withhold argument:

Motion is the relation of two reciprocally related quantities called space and time. The term “motion” has no other significance. It is not of something; motion is nothing other than the relationship between space and time. It is a concept; it is not a “thing”.

Both space and time have a three dimensional characteristic; a dimension of time is NOT a dimension in space, NOR is a dimension of space a dimension in time; related to the other, yes; but in the other, no!

Both space and time have a flow characteristic, but the flow of time is not a flow in space nor is the flow of space a flow in time, although each is like the other. All properties of space and time are reciprocally related to the corresponding property of the other.

Motion is unitary, it exists only in units. Space and time manifest only in units because motion is unitary. Units of motion are a progression of one unit of space for a progression of one unit of time.

The universe is three dimensional, not six dimensional, and not four dimensional (time is not dimensional or even quasidimensional in three dimensional space, it is scalar).

Motion is either purely scalar with absolutely no preferential direction or it is scalar plus vectorial due to effects of dimensionalizing scalar motion in a generalized three dimensional reference system of space or of time. The effect in the generalized dimensional system of space is determined by the representation in the individual three dimensional system.

Progressions of scalar values are either outward from a reference point or inward toward the reference point. Since no two geometric locations can be closer than zero separation in either aspect and both aspects must be present to have motion, and thus, at least one unit of each must be present, the outward direction from one unit is the normal or natural direction for progression of the natural reference system with respect to a dimensional system.
Because each of these points follows from the previous points, the apparent progression must be outward from unit value of motion and be represented by greater values of motion or by lesser values of motion. This is accomplished for effective values of displacement motion greater than unity by having more space represented than time, and vice versa for lesser values of motion. Representation of lesser values can be accomplished in two ways: by the quantity of space represented remaining effectively at unit value and time progressing toward larger values, or by the net value of displacement in all directions being less than the value of primary motion.