One of the principal difficulties that is encountered in explaining the Reciprocal System of theory, or portions thereof, is a general tendency on the part of readers or listeners to assume that the author or speaker, whoever he may be, does not actually mean what he says. No previous major theory is purely theoretical; every one takes certain empirical information as a given element in the premises of the theory. The conventional theory of matter, for example, takes the existence of matter as given. It then assumes that this matter is composed of “elementary particles,” which it attempts to identify with observed material particles. On the basis of this assumption, together with the empirical information introduced into the theory, it then attempts to explain the observed range of structural characteristics. Inasmuch as all previous theories of major scope have been constructed on this pattern, there is a general impression that physical theories *must* be so constructed, and it is therefore assumed that when reference is made to the fact that the Reciprocal System utilizes *no* empirical data of any kind, this statement must have some meaning other than its literal significance.

The theoretical development in the preceding chapters should dispose of this misapprehension so far as the qualitative aspect of the universe is concerned. While the task is still only in the early stages, enough of the basic features of the physical universe—radiation, matter, gravitation, etc,.—have been derived by deduction from the postulates, without the aid of further assumptions, or of empirical information, to demonstrate that a purely theoretical qualitative development is, in fact, feasible. But a complete account of a theoretical universe must necessarily include the quantitative aspects of physical phenomena as well as the qualitative aspects.

Here is another place where the way in which the development of theory * has* taken place is mistakenly regarded as the way in which this development *must* take place. The theoretical products of the Newtonian era, the so-called “classical” physics, were capable of being expressed in simple mathematical terms. But some deviations from the classical laws have been encountered in the far-out regions that have been reached by observation and experiment in recent years, and the physicists have not been able to account for these deviations without employing extremely complex mathematical processes, together with conceptual artifices of a rather dubious character, such as Einstein’s “rubber yardstick” , or fudge factor. In the light of the points brought out in the preceding chapter it is now evident that the difficulties are due to a misunderstanding of the basic nature of the far-out phenomena, but since the modern theorists have not realized this, they have concluded that the true relationships of the universe are extremely complex, and that they cannot be expressed by anything other than very complex mathematics.

The general acceptance of this view of the situation has led a large segment of the scientific community, particularly the theoretical physicists, to the further conclusion that any treatment of the subject matter by means of simple mathematics is necessarily wrong, and can safely be dismissed without examination. Indeed, many of these individuals go a step farther, and characterize such a treatment as “non-mathematical.” This attitude is, of course, preposterous, and it cannot be defended, but it is nevertheless so widespread that it constitutes a serious obstacle in the way of a full appreciation of the merits of any *simple* mathematical treatment.

In beginning the quantitative development of the Reciprocal System of theory it is therefore necessary to emphasize that simplicity is a virtue, not a defect. It is so recognized, in principle, by scientists in general, including those who are now contending that the universe is fundamentally complex, or even, as expressed by P. W. Bridgman, that it “is not intrinsically reasonable or understandable.”^{56} In its entirety, the universe is indeed complex, extremely so, but as the initial steps in the development of the Reciprocal System in the preceding pages have already begun to demonstrate from a qualitative standpoint, it is actually a complex aggregate of interrelated simple elements.

The principal advantage of mathematical treatment of physical subject matter is the precision with which knowledge of a mathematical character can be developed and expressed. This is offset to a considerable degree, however, by the fact that mathematical knowledge of physical phenomena is incomplete, and from the physical standpoint, ambiguous. No mathematical statement of a physical relation is complete in itself. As Bridgman frequently pointed out, it must be accompanied by a “text” that tells us what the mathematics mean, and how they are to be applied. There is no definite and fixed relation between this text and the mathematics; that is, every mathematical statement of a physical relation is capable of different *interpretations.*

The importance of this point in the present connection lies in the fact that the Reciprocal System makes relatively few changes in the mathematical aspects of current physical theory. The changes that it calls for are primarily *conceptual*. They require different interpretations of the mathematics, changes in the text, as Bridgman would say. Such changes, modifications of our ideas as to what the mathematics *mean,* obviously cannot be represented by alterations in the mathematical expressions. These expressions will have to stand as they are. Many readers of the first edition have asked that the new ideas be “put in mathematical form.” But what these individuals really mean is that they want the theory put into some *different* mathematical form. They are, in effect, demanding that we change the mathematics and leave the concepts alone. This, we cannot do. The errors in current physical thought are primarily conceptual, not mathematical, and the corrections have to be made where the errors are, not somewhere else.

There is nothing extraordinary about the close correlation between the mathematical aspects of the Reciprocal System and those of current theory. The conventional mathematical relations were, for the most part, derived empirically, and any *correct* theory of a more general nature must necessarily arrive at these same mathematics. But there is no guarantee that the prevailing interpretation of these mathematical results is correct. On the contrary, as Jeans pointed out in the statement previously quoted, the physical interpretations of correct mathematical formulae have often been “badly wrong.”

Correction of the errors that have been made in the interpretation of the mathematical expressions often has very significant consequences, not so much in the particular area to which such an expression is directly applicable, but in collateral areas. The interpretation is usually tailored to fit the immediate physical situation reasonably well, but if it is not correct it becomes an impediment to progress in related areas. If it does not actually lead to erroneous conclusions such as the limitation on speed that Einstein derived from a wrong interpretation of the mathematics of acceleration at high speeds, it at least misses all of the significant collateral implications of the true explanation.

For example, the mathematical statement of the recession of the distant galaxies merely tells us that these galaxies are receding at speeds directly proportional to their distances. The currently popular interpretation of this mathematical relation *assumes* that the recession is an ordinary vectorial motion. The problem in accounting for it then becomes a matter of identifying (or inventing) a force of sufficient magnitude to produce the extremely high speeds of the most distant objects. The accepted hypothesis is that they were produced by a gigantic explosion of the entire contents of the universe at some unique stage of its history. The Reciprocal System is in agreement with the mathematical aspects of current theory. It arrives theoretically at the conclusion that the distant galaxies must recede at speeds proportional to their respective distances: the same conclusion that present-day astronomy derives empirically. But the new theoretical system says that this recession is *not* a vectorial motion imparted to the galaxies by some powerful force. It is a scalar outward motion that results from viewing the galaxies in the context of a stationary spatial frame of reference rather than in the natural moving system of reference to which all physical objects actually conform.

So far as the recession phenomenon itself is concerned, it makes little difference, aside from the implications for cosmology, which interpretation of the mathematical relation between speed and distance is accepted, but on the basis of the currently popular hypothesis, this relation has no further significance, whereas on the basis of the explanation derived from the postulates of the Reciprocal System, the same forces that apply to the distant galaxies are applicable to *all* atoms and aggregates of matter, producing effects which vary with the relative magnitudes of the different forces involved. On the basis of this new information, the mathematical relation, which applies to the galaxies, is one of far-reaching importance.

This present chapter will initiate a demonstration that the very complex mathematical relations that are encountered in many physical areas are the result of permutations and combinations of simple basic elements, rather than a reflection of a complex fundamental reality. The process whereby the compound unit of motion that we call an atom is produced by applying a rotational motion to a previously existing vibrational motion, the photon, is typical of the manner in which the complex phenomena of the universe are built up from simple foundations. We start with a uniform linear, or translational, motion at unit speed. Then by directional reversals we produce a simple harmonic motion, or vibration. Next the vibrating unit is caused to rotate. The addition of this motion of a different type alters the behavior of the unit—gives it different properties, as we say—and puts it into a new physical category. All of the more complex physical entities with which we will deal in the subsequent pages are similarly built up by compounding the simpler motions.

The first phase of this mathematical development is a striking example of the way in which a few very simple mathematical premises quickly proliferate into a large number and variety of mathematical consequences. The development will begin with nothing more than the series of cardinal numbers and the geometry of three dimensions. By subjecting these to simple mathematical processes, the applicability of which to the physical universe of motion is specified in the fundamental postulates, the combinations of rotational motions that can exist in the theoretical universe will be ascertained. It will then be shown that these rotational combinations that theoretically can exist can be individually identified with the atoms of the chemical elements and the sub-atomic particles that are observed to exist in the physical universe.

A unique group of numbers representing the different rotational components will be derived for each of these combinations. The set of numbers applying to each element or type of particle theoretically determines the properties of that substance, inasmuch as these properties, like all other quantitative features of a universe of motion, are functions of the magnitudes of the motions that constitute the material substances. It will be shown in this and the following chapter that this theoretical assertion is valid for some of the simpler properties, including those, which depend upon the position of the element in the periodic table. The application of these numerical factors to other properties will be discussed from time to time as consideration of these other properties is undertaken later in the development.

One preliminary step that will have to be taken is to revise present measurement procedures and units in order to accommodate them to the natural moving system of reference. Because of the status of unity as the natural reference datum, a deviation of n-l units downward from unity to a speed 1/n has the same *natural* magnitude as a deviation of n-l units upward to a speed n/1, even though, when measured from zero speed in the conventional manner, the changes are wholly disproportionate. When n is 4, for example, the upward change is from 1 to 4, an increase of 3 units, whereas the downward change is from 1 to ¼, a decrease of only ¾ unit.

In order to reflect the fact that these deviations are actually * equal* in magnitude from the natural standpoint, the basis on which the fundamental processes of the universe take place, it is necessary to set up a new system of speed measurement, in which we express the magnitude of the speed in terms of the deviation, upward or downward, from unit speed, instead of measuring from some zero in the conventional manner. Inasmuch as the units in which speeds are measured on this basis are not commensurable with those of speed as measured from zero, it would lead to complete confusion if the units of the new system were called units of speed. For this reason, when reference is made to speed in *terms of its natural magnitude in* any of the publications dealing with the Reciprocal System of theory, it is not called speed. Instead, the term “speed displacement” is used, the units of this displacement being natural units of deviation from unity.

In practice, the term “speed displacement” is usually shortened to “displacement,” and this has led to some criticism of the terminology on the ground that “displacement” already has other scientific meanings. But it is highly desirable, as an aid to understanding, that the idea of a deviation from a norm should be clearly indicated in the language that is used, and there are not many English words that meet the requirements. Under the circumstances, “displacement,” appears to be the best choice. The sense in which this term is used will almost always be indicated by the context in which it appears, and in the few cases where there might be some question, the possibility of confusion can be avoided by employing the full name, “speed displacement.”

Another reason for the use of a distinctive term in designating natural speed magnitudes is that this is necessary in order to make the addition of speeds meaningful. Conventional physics claims that it recognizes speed as a scalar quantity, but in actual practice gives it no more than a quasi-scalar status. True scalar quantities are additive. If we have five gallons of gasoline in one container and ten gallons in another, the total, the quantity in which we are most interested, is fifteen gallons. The corresponding sum of two speeds of the same object—rotational and translational, for example—has no meaning at all in current physical thought. In the universe of motion described by the Reciprocal System of theory, however, the scalar total of all of the speeds of an object is one of the most important properties of that object. Thus, even though speed has the same basic significance in the Reciprocal System as in conventional theory—that is, it is a measure of the magnitude of motion—the manner in which speed enters into physical phenomena is so different in the two systems that it would be inappropriate to express it in the same units of measurement in both cases, even if this were not ruled out for other reasons.

It would, of course, be somewhat simpler if we could say “speed” whenever we mean speed, and not have to use two different terms for the same thing. But the meaning of whatever is said should be clear in all cases if it is kept in mind that whenever reference is made to “displacement,” this means “speed,” but not speed as ordinarily measured. It is speed measured in different quantities, and from a different reference datum.

A decrease in speed from 1/1 to 1/n involves a *positive displacement* of n-1 units; that is, an addition of n-1 units of motion in which time is unidirectional while the space direction alternates, thus, in effect, adding n-1 units of time to the original speed 1/1. Similarly, an increase in speed from 1/1 to n/1 involves a n*egative displacement,* an addition of n-1 units of motion in which space is unidirectional while the time direction alternates; thus, in effect, adding n-1 units of space to the original speed 1/1.

In the first edition of this work the displacements here designated positive and negative were called “time displacement” and “space displacement” respectively, to emphasize the fact that the positive displacement represents an increased amount of time in association with one unit of space, while the reverse is true in negative displacement. Experience has shown, however, that the original terminology tends to be confusing, particularly in that it is frequently interpreted as indicating addition of independent quantities of time or space to the phenomena under consideration, whereas, in fact, it is the *speed* that is being increased or decreased. As pointed out in Chapter 2, in a universe of motion there is no such thing as physical space or time independent of motion. We can *abstract* the space aspect of motion mentally, and *imagine* it existing independently, as a reference system (extension space) or otherwise, but we cannot add or subtract space or time in actual practice except by superimposing a new motion on the motion we wish to alter.

If we were dealing with speed measured from the mathematical zero it would be logical to apply the term positive to an addition to the speed, but where we measure from unity the values increase in both directions, and there is no reason why one increase should be considered any more “positive” than the other. The choice has therefore been made on a convenience basis, and the “positive” designation has been applied to the displacements on the low speed side of the unit speed datum because these are the displacements of the material system of phenomena. We will find, as we proceed, that the displacements toward higher speeds, where they occur at all in the material sector, do so mainly as negative modifications of the predominantly low speed motion combinations.

Inasmuch as the units of positive displacement and of negative displacement are simply units of deviation from the natural speed datum they are additive algebraically. Thus, if there exists a motion in time with a negative speed displacement of n-1 units (equivalent to n units of speed in conventional terms) we can reduce the speed to zero, relative to the natural datum, by adding a motion with a positive speed displacement of n-1 units. Addition of further positive displacement will result in a net speed below unity; that is, a motion in space. But there is no way by which we can alter either the time aspect or the space aspect of the motion independently. The variable in a universe of motion is *speed,* and the variation occurs only in displacement units. The change in terminology has been made in the hope that it will contribute toward a full realization that what we are dealing with are units of speed, even though, for technical reasons, we cannot call it speed.

In the case of radiation, there is no inherent upper limit to the speed displacement (conventionally measured as frequency), but in actual practice a limit is imposed by the capabilities of the processes that produce the radiation, examination of which will be deferred until after further groundwork has been laid. The range of radiation frequencies is so wide that, except near 1/1, where the steps from n to n + 1 are relatively large, the frequency spectrum is practically continuous.

The rotational situation is very different. In contrast to the almost unlimited number of possible vibrational frequencies, the maximum number of units of rotational displacement that can participate in any one combination of rotations is relatively small, for reasons which will appear in the course of the discussion. Furthermore, probability considerations dictate the distribution of the total number of rotational displacement units among the different rotations in each individual case, so that in general there is only one stable combination among the various mathematically possible ways of distributing a given total rotational displacement. This limits the possible rotational combinations that we identify as material atoms and particles to a relatively small series, the successive members of which differ initially by one displacement unit, and at a later stage by two of the single displacement units.

With this understanding of the fundamentals, let us now proceed to an examination of the general characteristics of the combinations of rotational motions. The existence of different rotational patterns is clear from the start, as the rotation can not only take place at different speeds (displacements), but, in a three-dimensional universe, can also take place independently in the different dimensions. As we will see in our investigation, however, some restrictions are imposed by geometry.

The photon cannot rotate around the line of vibration as an axis. Such a rotation would be indistinguishable from no rotation at all. But it can rotate around either or both of the two axes perpendicular to the line of vibration and to each other. One such rotation of the one-dimensional photon generates a two-dimensional figure: a disk. Rotation of the disk around the second available axis then generates a three-dimensional figure: a sphere. This exhausts the available dimensions, and no further rotation of the same nature can take place. The basic rotation of the atom or particle is therefore two-dimensional, and, as brought out in Chapter 5, it is in the inward scalar direction. But after the two-dimensional rotation is in existence it is possible to give the entire combination of vibrational and rotational motions a rotation around the third axis, which is also inward from the scalar standpoint, but is opposed to the two-dimensional rotation vectorially. This reverse rotation is optional, as the basic rotation is distributed over all three dimensions, and nothing further is required for stability. A rotating system therefore consists of a photon rotating two-dimensionally, with or without a reverse rotation in the third dimension.

Although the two dimensions of the basic rotation have been treated separately for descriptive purposes, first generating a disk by one rotation, and then a sphere by the second, it should be understood that there are not two one-dimensional rotations; there is one two-dimensional rotation. This distinction has a significant bearing on the properties of the rotational combinations. The combined magnitude of two one-dimensional rotations of n displacement units each is 2n. The magnitude of a two-dimensional rotation in which the displacement is n in each dimension is n^{2}. It is not essential that all of the rotations be *effective in* the physical sense. Unless there is effective rotation in at least one dimension it is meaningless to speak of rotation, as such motion cannot be distinguished from translation. But if there is effective rotation—that is, rotation with a speed differing from unity—in at least one dimension, there can be rotation at unit speed (zero displacement) in the other dimension or dimensions.

The vibrational speed displacement of the basic photon may be either negative (greater than unity) or positive (less than unity). Let us consider the case of a photon with a negative displacement, to which we propose to add a unit of rotational displacement (rotate the photon). Inasmuch as the individual units of vibrational displacement are discrete (that is, they are not tied together in any way), the one applied unit of rotational motion results in rotation of only one of the vibrational units. Because of the lack of any connection between the vibrational units there is no force resisting separation. When the one unit starts moving inward by reason of the rotation it therefore moves away from the remainder of the photon, which continues to be carried outward by the progression of the natural reference system. Irrespective of the number of vibrational units in the photon to which the rotational displacement was added, the compound motion produced by this addition thus contains only the vibrational units that are being rotated. The remaining vibrational units of the original photon continue as a photon of lower displacement.

When a compound motion of this type, rotation of a vibration, is formed, the inward motion due to the rotation replaces the outward motion of the progression of the reference system. Thus the components of the compound motion are not subject to oppositely directed motions in the manner of the multi-unit rotating photons, and these components do not separate spontaneously. However, the vibrational displacement of the photon now under consideration is negative. If the rotational displacement applied to this photon is also negative, the displacement units, being units of the same scalar nature, are additive in the same manner as the vibrational units of a photon. Like the photon units, they are easily separated when even a relatively small force is applied, and the rotational displacement is therefore readily transferred from the original photon to some other object, under appropriate conditions. For this reason, combinations of negative vibrational and negative rotational displacements are inherently unstable. On the other hand, if the applied rotational displacement is positive, equal numbers of the positive and negative displacement units neutralize each other. In this case the combination has no net displacement. A motion that *does* have a net displacement cannot be extracted from such a combination without the intervention of some outside agency. It is simple enough to separate one negative unit from an aggregate of n negative units, but getting one negative unit out of nothing at all is not so easily accomplished. A combination of a negative vibration and a positive rotation (or vice versa) is therefore inherently stable.

All that has been said about additions to a photon with negative displacement applies with equal force, but in the inverse manner, to the addition of rotation to a photon with positive displacement. We therefore arrive at the conclusion that in order to produce stable combinations photons oscillating in time (negative displacement) must be rotated in space (positive displacement), whereas photons oscillating in space must be rotated in time. This alternation of positive and negative displacements is a general requirement for stability of compound motions, and it will play an important part in the theoretical development in the subsequent pages. It should be understood, however, that stability is dependent on the environment. Any combination will break up if the environmental conditions are sufficiently unfavorable. Conversely, there are situations, to be examined later, in which environmental influences create conditions that confer stability on combinations that are normally unstable.

The combinations in which the net rotation is in space (positive displacement) can be identified with the relatively stable atoms and particles of our local environment, and constitute what we will call the * material* system. For the present we will confine the discussion to the members of this material system, and will leave the inverse type of combination, the *cosmic* system, as we will call it, for later consideration.

Inasmuch as the oscillating photon is being rotated in two dimensions (the basic positive rotation), one unit of two-dimensional positive displacement is required to neutralize the negative vibrational displacement of the photon, and reduce the net total displacement to zero. Because of its lack of any effective deviation from unit speed (the reference datum) this combination of motions has no observable physical properties, and for that reason it was somewhat facetiously called “the rotational equivalent of nothing” in the first edition. But this understates the significance of the combination. While it has no effective net total magnitude, its rotational component does have a direction. The idea of a motion that has direction but no magnitude sounds something like a physicist’s version of the Cheshire cat, but the zero effective magnitude is a property of the structure as a whole, while the rotational direction of the two-dimensional motion, which makes the addition of further positive rotational displacement possible, is a property of one component of the total structure. Thus, even though this combination of motions can do nothing itself, it does constitute a base from which something (a material particle) can be constructed that cannot be formed directly from a linear type of motion. We will therefore call it the *rotational base.*

There are actually two of the rotational bases. The one we have been discussing is the base of the material system. The structures of the cosmic system are constructed from a different base; one that is just the inverse of the material base. In this inverse combination the photon is oscillating in space (positive displacement) and rotating in time (negative displacement).

Successive additions of positive displacement to the rotational base produce the combinations of motions that we identify as the sub-atomic particles and the atoms of the chemical elements. The next two chapters will describe the structures of the individual combinations. Before beginning this description, however, it will be in order to make some general comments about the implications of the theoretical conclusion that the atoms and particles of matter are systems of rotational motions.

One of the most significant results of the new concept of the structure of atoms and particles that has been developed from the postulates of the Reciprocal System is that it is no longer necessary to invoke the aid of spirits or demons or their modern equivalents: mysterious hypothetical forces of a purely ad hoc nature—to explain how the parts of the atom hold together. There is nothing to explain, because the atom has no separate parts. It is one integral unit, and the special and distinctive characteristics of each kind of atom are not due to the way in which separate “parts” are put together, but are due to the nature and magnitude of the several distinct *motions* of which each atom is composed.

At the same time, this explanation of the structure of the atom tells us why such a unit can expel particles, or disintegrate into smaller units, even though it has no separate parts; how it can act, in some respects, as if it were an aggregate of sub-atomic units, even though it is actually a single integral entity. Such a structure can obviously part with some of its motion, or absorb additional units of motion, without in any way altering the fact that it is a single entity, not a collection of parts. When the pitcher throws a curve ball, it is still a single unit—it is a baseball—even though it now has both a translational motion and a rotational motion, which it did not have while still in his hand. We do not have to worry about what kind of a force holds the rotational “part,” the translational “part,” and the horsehide covered “nucleus” together.

There has been a general impression that if we can get particles *out* of an atom, then there must be particles in atoms; that is, the atom must be constructed of particles. This conclusion seems so natural and logical that it has survived what would ordinarily be a fatal blow: the discovery that the particles which *emanate from* the atom in the process of radioactivity and otherwise are not the *constituents of* the atom; that is, they do not have the properties which are required of the constituents. Furthermore, it is now clear that a great variety of particles that cannot be regarded as constituents of normal atoms can be produced from these atoms by appropriate processes. The whole situation is now in a state of confusion. As Heisenberg commented:

Wrong questions and wrong pictures creep automatically into particle physics and lead to developments that do not fit the real situation in nature.

^{27}

It is now apparent that all of this confusion has resulted from the wholly gratuitous, but rarely questioned, assumption that the sub-atomic particles have the characteristics of “parts” ; that is, they exist as particles in the structure of the atom, they require something that has the nature of a “force” to keep them in position, and so on. When we substitute *motions* for *parts,* in accordance with the findings of the Reciprocal System, the entire situation automatically clears up. Atoms are compound motions, sub-atomic particles are less complex motions of the same general nature, and photons are simple motions. An atom, even though it is a single unitary structure without separate parts, can eject some of its motion, or transfer it to some other structure. If the motion which separates from the atom is translational it reappears as translational motion of some other unit; if it is simple linear vibration it reappears as radiation; if it is a rotational motion of less than atomic complexity it reappears as a sub-atomic particle; if it is a complex rotational motion it reappears as a smaller atom. In any of these cases, the status of the original atom changes according to the nature and magnitude of the motion that is lost.

The explanation of the observed interconvertibility of the various physical entities is now obvious. All of these entities are forms of motion or combinations of different forms, hence any of them can be changed into some other form or combination of motion by appropriate means. *Motion is the common denominator of the physical universe.** *

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