In terms of present-day thinking, motion is regarded as a process-taking place in space: a change in the spatial relation between objects. But if space and time are symmetrical, as the reciprocal postulate requires, then it is equally possible for the temporal relations between objects to undergo similar changes, and this process of temporal change constitutes motion in time.
The factors involved in the concepts of location in space and location in time have been examined in detail in previous publications, but since they are essential to an understanding of motion in time, we will review them briefly here. Let us first consider a distant galaxy, which at time t0 occupies location A in space, and is therefore receding from our Milky Way galaxy M, in the direction MA. During an interval t, the recession carries the galaxy outward in this direction to a new location B. But we know from observation of the nearer galaxies that these units also have random motions of their own, so that during time t the galaxy under consideration will have moved an additional distance to some location C which will not, unless by pure chance, lie on a prolongation of the line MAB. The actual displacement of the galaxy during time t is therefore the vector resultant of the distance AB and the distance BC.
We have already identified the recession of the galaxies as a manifestation of the progression of space, hence in view of the symmetry between space and time, we may deduce that exactly the same kind of a situation exists in time. While time is progressing from time location a to time location b, carrying a galaxy-our Milky Way galaxy, let us say-with it, the random motion of the galaxy in time takes it to some other time location c, and the total time displacement of the galaxy during this interval is not the time of the progression alone, but the vector resultant of this time ab and the random time displacement bc.
The progression of time which carries our galaxy from time location a to time location b is the quantity which we measure by means of a clock, and it is commonly called clock time. Since the progression of space that carries the distant galaxy from location A to location B is the space equivalent of clock time we may utilize the same terminology and call it clock space. The random motion of the distant galaxy from location B to location C takes place in the ordinary three-dimensional space of our everyday experience, and inasmuch as we usually represent this by some sort of a coordinate system, we may call it coordinate space. The displacement of our galaxy from time location b to time location c is the time equivalent of the space displacement BC, and it takes place in three-dimensional time. Again we may use the same terminology that we apply to space and refer to this as coordinate time.
Because the gravitational motion cancels the motion of the space-time progression in our local environment we do not detect the recession of our own galaxy and the only space that we recognize locally is coordinate space. When we observe, for instance, that our near neighbor among the stars, Alpha Centauri, is moving away from us with a radial velocity of approximately 20 km/see, this is a motion in coordinate space, the same kind of a motion that we observe in our everyday life. When we also observe that a galaxy in Ursa Major is moving away from us with a radial velocity of 15000 km/sec, it would appear on first consideration that this is exactly the same thing, aside from the substantial difference in the speed. But in reality there is a very significant difference between the two motions. The motion of Alpha Centauri that takes it away from us carries it toward any star located still farther away in this outward direction. Unless one is familiar with recent astronomical discoveries, he is quite likely to take the stand that this must necessarily be true. The fact is, however, that it is not true in the case of the motion of the distant galaxies. When the Ursa Major galaxy moves away from us, it is also moving away from all other galaxies, including those located diametrically opposite from us.
Obviously these are motions of a totally different nature. The explanation is that the galactic recession is not a motion in coordinate space, the kind of a motion with which we are familiar. Aside from a relatively slow motion of a random character, comparable to the motions which we observe in the galaxies of our local system, the Ursa Major galaxy remains stationary in coordinate space but it moves outward in clock space. Our own Milky Way galaxy is doing exactly the same thing but, of course, we do not recognize our own outward motion from direct observation. Because of the reciprocal relation between space and time the galaxies which move outward in clock space are moving inward in clock time, but here again the motion is outside the range of existing observational facilities.
In the local environment, where the gravitational motion exceeds that of the progression, the directions of motion are just the reverse. While the atoms of matter are moving inward in space under the influence of gravitational forces and thus gathering into aggregates that are localized in space, they are coincidentally moving outward in time in random directions. Hence the material aggregates are not localized in time; that is, the atoms of a star, for example, are clustered together in a stable configuration in a relatively small amount of space, but they are widely dispersed in time with no stable relationships between units. In our observation of time, therefore, we have no landmarks by which we can recognize positions in coordinate time, and the existence of this kind of time therefore remained undetected until it was discovered theoretically in this present investigation. The temporal relations between atoms occupying widely separated locations in time are comparable to the spatial relations between distant galaxies, and the aspect of time that is recognizable in our local environment is the clock time.
But even though the local situation is such that we normally recognize only clock time and coordinate space, clock space and coordinate time have some very significant effects under what we consider extreme conditions. At extreme distances the space progression, the motion in clock space, manifests itself as the recession of the galaxies. At extreme velocities the motion in coordinate time manifests itself by the deviations from Newton’s Laws of Motion that were revealed by the Michelson-Morley experiment. Under conditions of extremely small space or time separations there is a replacement of motion in space by motion in time and vice versa which has some important effects that will be discussed later in this chapter.
The expression “motion in time,” as it will be used in this work, refers to motion in coordinate time: a change of location in three dimensional time analogous in all respects to the change of location in three-dimensional coordinate space which constitutes motion in space. The total time corresponding to any specific clock time is the vector resultant of this coordinate time and the corresponding clock time.
The quantitative measure of motion in space is velocity, and the mathematical definition of velocity is v = s/t. Here the term s is a vector quantity representing the displacement in space. The term t is scalar, since time has no direction in space, and the term v is then a vector quantity representing the space displacement per unit of time. The analogous quantitative expression for motion in time is D = t/s or delta is equal to t divided by s. Here the term t is a vector quantity representing the displacement in time. The term s is scalar, since space has no direction in time. The term D is then a vector quantity representing the time displacement per unit of space, this being the time-oriented quantity corresponding to velocity in space-oriented phenomena.
From the foregoing, the reason for the scalar nature of the space-time progression, which is motion in both space and time, is evident. Since time has no direction in space and is therefore a scalar quantity so far as motion in space is concerned, and for similar reasons, space is a scalar quantity so far as motion in time is concerned, it follows that motion in both space and time cannot have a direction in either space or time. Hence motion in space-time is scalar.
We may summarize the conclusions with respect to the directional characteristics of the various kinds of motion as follows:
Some readers of the previous volumes of this series have found it difficult to accept the idea that time can be three-dimensional because this makes any time interval a vector quantity and presumably leads to situations in which we are called upon to divide one vector quantity by another. As indicated in the foregoing discussion, however, such situations are non-existent. If we are dealing with spatial relations, time is scalar because time has no direction in space. If we are dealing with temporal relations, space is scalar because space has no direction in time. Either space or time can be vectorial, but there is no physical situation in which both are vectorial.
Since the property which we are calling “direction in time” is something quite distinct from “direction” as we ordinarily use the term in the sense of “direction in space,” there might be some good arguments in favor of coining a new name and not using the word “direction” for this purpose. This would do doubt contribute toward clarifying such issues as the reason why time is always scalar in the equations of space motion irrespective of the number of dimensions which time itself may have. On the other hand, it should not be difficult to get away from the habit of interpreting “direction” as meaning “direction in space.” As a matter of fact, we already recognize that the spatial connotation that we give to the word “direction” is in many cases fictitious. A scalar quantity is specifically defined as one, which has no direction, yet we habitually use the word “direction” and directional terms of one kind or another in speaking of scalar quantities, or even in connection with items, which cannot be expressed in physical terms at all. We speak of wages and prices as moving in the same direction, temperature as going up or down, a change in the direction of our thinking, and so on. Here we recognize, consciously or unconsciously, that in our definition of a scalar quantity we are using the term “direction” in the sense of “direction in space,” whereas when we are talking about the direction of price change or something similar we are using the word “direction” without any spatial significance. It should not require any great mental effort to apply the same viewpoint to temporal quantities and to recognize that here also the term “direction” has no spatial significance.
On the other side of the ledger, there are some very definite advantages to be gained by utilizing the term “direction” in reference to time as well as in reference to space. Because of the symmetry of space and time, the property of time that corresponds to the familiar property of space that we call “direction” has exactly the same characteristics as the latter, and by using the term “direction in time” as a name for this property we convey an immediate understanding of its nature and characteristics that would otherwise take a great deal of discussion and explanation. All that is then necessary is to keep in mind that although direction in time is like direction in space, it is not direction in space.
This is a general situation that applies all through the space-time relations. The impact of time on our consciousness is vague and elusive, and it is therefore quite difficult to visualize any concrete physical situations involving three-dimensional time. The most effective way of grasping the essentials of the participation of time in such situations is to visualize the corresponding space phenomenon and then recognize that the time phenomenon is exactly the same except that space and time are reversed. In order to do this easily and efficiently it is necessary to utilize the same terms for the corresponding space and time entities, and this policy has been followed throughout the present series of publications.
As brought out in Chapter V, the developments of this work, revolutionary as they may be in some of their aspects, have very little effect on the physical processes and relations of everyday life. The reason is now apparent. In our everyday experience we are dealing with motion in space, and the laws and principles governing motion in space are already firmly established. Any correct theoretical development must necessarily lead to these same laws and principles. The discovery of the possibility of motion in time and the formulation of the analogous laws and principles governing this type of motion has no effect on any situation where all motion is in space.
But physical science is now penetrating into regions where motion in time plays an important, often controlling, part in physical phenomena, and much of the difficulty that the present-day physicist is encountering in his attempts to systematize knowledge in these regions is due to his attempt to treat the phenomena in these regions by means of the relations applicable to motion in space, while the motion with which he is dealing is actually, in whole or in part, motion in time.
The problem, which Einstein faced in setting up his Special Theory of Relativity, is typical. In the light of the information that has been developed in this work we may compare it to the problem that would confront anyone who knew nothing of the concept of direction and who attempted to devise a scalar equation to relate speed and velocity. The latter would, of course, be nothing but another scalar quantity to anyone who is unaware of the existence of direction, just as the time entering into a high velocity was to Einstein no different from the time entering into a low velocity, aside from the difference in magnitude. Such an investigator would find it entirely possible to devise an accurate mathematical relation that would apply to some special speed-velocity situation, in the same way that the Lorentz equations apply to uniform translational motion, but it is obvious that there can be no general relation of a scalar character connecting speed and velocity. If a “Special Theory” of the speed-velocity relation is devised for some special case, sinusoidal motion, let us say, and the success of the theory in this particular area leads its originators and supporters to believe that it can be extended to motion in general, the outcome will inevitably be either that the attempts are ultimately given up as hopeless, or that successive ad hoc modifications of the original theory finally result in a “General Theory” of the speed-velocity relation which is so vague and confused conceptually and so complex mathematically that no one can pin it down closely enough to reveal its true character.
A valid general solution of the hypothetical speed-velocity problem cannot be obtained until someone discovers the three-dimensionality of space and introduces the concept of direction. Similarly no general solution for Einstein’s problem was possible until this present investigation discovered the three-dimensionality of time and introduced the concept of motion in time. But in both cases the answer is simple and obvious as soon as the necessary conceptual foundation has been laid.
Hesse sums up the position in which physics stands without the concept of motion in time as follows:
The fundamental logical proposition to which all such theories must conform is that one cannot assert both that the velocity of light is invariant for all possible reference frames, and that the geometry of light rays is Euclidean.54
The general and uncritical acceptance of this so-called “logical proposition” is the factor that has forced physical theory into the uncomfortable and untenable position that it occupies today. The truth is that this is not logical at all. It could be logical only if coupled with a proviso that the existing concepts of the nature of space, time, and motion must be maintained unchanged, and there is clearly no physical justification for such a proviso. There has never been any assurance that these concepts are physically valid; on the contrary, they are pure assumptions, and the development of the Reciprocal System has now demonstrated that they are erroneous assumptions. If they are replaced by concepts that are physically valid, then it is possible to formulate a logical and self-consistent alternative to the Special Theory which does just exactly what Hesse claims is impossible; that is, it reconciles Euclidean geometry with the constant velocity of light.
A brief summary of the detailed explanations of this situation that have been published elsewhere can be given with the aid of the diagram, Figure 1. Let us assume that a ray of light from a distant source S passes from A to B and from A’ to B’ in two parallel systems. Then let us assume that the systems AB and A’B’ are in motion in opposite directions as shown, and are in coincidence as the light ray passes A and A’. Because of the motions of the respective systems, point B will have moved to some point C closer to A by the time the light reaches it, whereas B’ will have moved to some more distant point C’. Yet if the results of the Michelson-Morley experiment are to be believed, the velocity of the incoming ray at C is identical with the velocity of the incoming ray at C’; that is, the velocity of light is independent of the reference system.
The interpretation that has been placed on the results of this experiment by the physicists is that the time required for the light ray to pass from A to C is the same as the time required to pass from A’ to C’. From a common sense viewpoint this conclusion is absurd, and the feeling of discomfort which most laymen, and many scientists, experience on contact with Relativity theory is basically due to the fact that Einstein made this contradiction of common sense the cornerstone of his theory. With the benefit of the discussion earlier in this chapter, it is now evident that the physicists’ interpretation of the Michelson-Morley experiment is wrong, and that the conflict with common sense was wholly unnecessary. The time AC is not the same as the time A’C’. It is only the clock time that is the same in both systems, and the clock time is only one component of the total time.
In our observations of the distant galaxies we can ignore the random motion of these objects—the motion in coordinate space—because it is so small that its effect is negligible compared to the effect of the motion of the recession—the motion in clock space. But if this random motion were taking place at a velocity in the neighborhood of that of light, the situation would be quite different. This motion in coordinate space would then have a very appreciable effect on the total displacement of the galaxy during any interval of observation and we could no longer ignore it.
Similarly, the change in temporal location—the displacement in coordinate time—at the relatively low velocities of our ordinary experience is negligible in comparison with the time of the progression—the clock time—and we can disregard it. But here again, this ceases to be true at the velocity of light. At high velocities the coordinate time has a finite magnitude, and the total time, the time that actually enters into physical relations, is the vector sum of the clock time and the coordinate time. The total time required for the light ray to pass from A to C in the moving system is the clock time ab minus the coordinate time cb, or ac, and the velocity is the space displacement AC divided by the net total time act The total time for the ray to pass from A’ to C’ is the sum of the clock time a’b’ and the coordinate time b’c’, or a’c’, and the velocity is the space displacement A’C’ divided by the total time a’c’. The velocity of light is thus AB/ab for a system at rest, AC/ac in the system moving toward the light source, and A’C’/a’c’ in the system moving away from the light source.
Inasmuch as one unit of time is equivalent to one unit of space, according to the postulates of the Reciprocal System, this means that the velocity of light is unity—one unit of space per unit of time—in all three cases. And the theory further tells us that this velocity cannot be other than unity under any circumstances, because a light photon has no motion of its own. The photon stays permanently in the same space-time unit in which it originates and is carried along by the progression of space-time itself. The progression moves one unit of space per unit of time simply because one unit of space is one unit of time and the equivalence of unit space and unit time is the progression. Even in passing through matter, where the measured light velocity is less than unity, the true velocity still remains one unit of space per unit of time. The factors, which cause the measured velocities to diverge from unity under these conditions, will be considered in a subsequent chapter.
As can be seen from the foregoing explanation, the concept of motion in time, which is one of the necessary consequences of the postulates that were derived by extrapolating the observed properties of space and time, solves Einstein’s problem, the problem posed by the results of the Michelson-Morley experiment, in an easy and natural way without any distortion of established physical principles. Notwithstanding Hesse’s statement that the constant velocity of light is logically incompatible with Euclidean geometry, this solution of the problem, number six in our list of Outstanding Achievements; is completely in harmony with both. In spite of the general belief that the Michelson-Morley results cannot be explained without giving up the idea of absolute space and time, Einstein’s positive assertion, on the same grounds, that “moving rods must change their length, moving clocks must change their rhythm,”88 and all of the chorus of similar statements throughout scientific literature, the findings of the Reciprocal System are entirely consistent with both the constant velocity of light and the existence of absolute magnitudes of space and time.
Here again, as in so many other situations that have been discussed in the several volumes of this series, previous investigators have simply failed to examine possible alternatives. It is true that in this case the correct alternative was more effectively hidden below many layers of long-standing habits of thought than in most other instances, and overlooking it was more a matter of inadvertence than a deliberate ignoring of unpopular alternatives, as in some other instances, but failure to be on the alert for alternatives is a serious defect in scientific practice. The consequences of error are always the same. Nature rewards only the correct answers; it gives no credit for effort, nor does it make allowances for extenuating circumstances. The plain truth is that the physicists should have recognized the possibility of a more satisfactory alternative to the Special Theory, irrespective of whether or not they were able to formulate anything acceptable, and a substantial part of the immense amount of effort that has been wasted in a futile attempt to patch up the weak spots in that theory should have been applied to searching for such an alternative. Failure to find the correct answer is, of course, excusable—even the most diligent efforts by the most competent individuals do not always reach their goals—but failure to look for the correct answer, because of unwillingness to admit that currently accepted ideas could be wrong, is a fair target for criticism.
The revolutionary change in outlook that accompanies the application of the concept of motion in time to the Relativity problem is typical of the way in which this concept clears up confusion and contradictions in all of the far-out regions: the realm of the very small (atomic physics and quantum theory), the realm of the very fast (gravitation and Relativity Theory), and the realm of the very large (astronomy and cosmology). All through these areas hitherto complex and difficult problems are reduced to simple and understandable terms when they are viewed in the context of the new conceptual structure.
Phenomena at the atomic level have been particularly baffling to the physicist. From the very first, those who sought to formulate theories and mathematical formulae to represent atomic processes have found it necessary to abandon established physical principles and to base their theoretical developments on ad hoc assumptions and specially devised principles of impotence. In spite of the unparalleled freedom of thought thus attained (or perhaps, more accurately, because of it) the theories which the physicists have developed have encountered serious difficulties at every turn, and have never been adequate to meet the constantly growing demands upon them. The experimental branch of physics is continually discovering new phenomena and new characteristics of previously known phenomena, which the theoretical branch did not anticipate and cannot reconcile with their existing theories. Each experimental advance thus requires a wholesale modification of these theories and a new batch of ad hoc assumptions.
The difficulties in which current atomic theory is now enmeshed were discussed in detail in The Case Against the Nuclear Atom, but that work was purely a critique of existing thought in the atomic field and it did not offer any new explanations of the matters at issue. The preceding discussion has now laid the foundation for a new theoretical understanding of motion in the atomic region.
Let us consider an atom A in motion toward another atom B through free space (Figure 2). According to accepted ideas, atom A will continue to move in the direction AB until the atoms, or the force fields surrounding them, if such fields exist, are in contact. The postulates of the Reciprocal System specify, however, that space exists only in units, hence when atom A reaches point x, one unit of space distant from B. it cannot move any closer to B in space. It is, however, free to change its position in time relative to the time location occupied by atom B. The reciprocal relation between space and time makes an increase in time separation equivalent to a decrease in space separation, and while atom A cannot move any closer to atom B in space, it can move to the equivalent of a spatial position that is closer to B by moving outward in coordinate time. When the time separation between the two atoms has increased to n units, space remaining unchanged, the equivalent space separation, the quantity that will be determined by the usual methods of measurement, is then 1/n units. In this way the measured distance, area, or volume may be a fraction of a natural unit, even though the actual one, two, or three-dimensional space cannot be less than one unit in any case.
So far as the inter-atomic distance itself is concerned, it is not very material whether this is an actual spatial separation or merely the equivalent of such a separation, but the fact that the movement of the atoms changes from a motion in space to a motion in time at the unit level has some important consequences from other standpoints. For instance, the spatial direction AB in which atom A was originally moving no longer has any significance inside unit distance, as the motion in time which replaces the previous motion in space has no spatial direction. It does have what we choose to call a direction in time, but this temporal direction has no relation at all to the spatial direction of the previous motion. No matter what the spatial direction of the motion of the atom may have been before unit distance was reached, the temporal direction of the motion after it makes the transition to motion in time is determined purely by chance.
Any kind of action originating in the region where all motion is in time is also subject to significant modifications if it reaches the unit boundary and enters the region of space motion. For example, the connection between space motion and time motion is scalar” and consequently only one dimension of a two-dimensional or three-dimensional motion can be transmitted across the boundary. This point will have an important bearing on one of the phenomena that will be discussed later.
Another important fact is that the effective direction of the space-time progression reverses at the unit level. Outside unit space the progression carries all objects outward in space away from each other. Inside unit space only time progresses and since an increase in time, with space remaining constant, is equivalent to a decrease in space, time progression in this region, the time region, as we will call it, moves all objects to locations which, in effect, are closer together. The gravitational motion always opposes the progression and hence the direction of this motion also reverses at the unit boundary. As it is ordinarily observed in the time-space region, the region outside unit distance, gravitation is an inward motion, moving objects closer together. In the time region it acts in the outward direction, moving objects farther apart.
On first consideration it may seem illogical for the same force to act in opposite directions in different regions, but from the natural viewpoint these are not different directions. As brought out in Chapter VI, the natural datum is unity, not zero, and the progression therefore always acts in the same natural direction: away from unity. In the time-space region away from unity is also away from zero, but in the time region away from unity is toward zero. Gravitation likewise has the same natural direction in both regions, in this case toward unity.
It is this reversal of direction at the unit level, which enables the atoms to take up equilibrium positions and form solid and liquid aggregates. No such equilibrium can be established where the space-time progression is outward, because in this case the effect of any change in the distance between the atoms resulting from an unbalance of forces is to accentuate that unbalance. If the inward-directed gravitational force exceeds the outward-directed progression, an inward motion takes place, making the gravitational force still stronger. Conversely, if the gravitational force is the smaller, the resulting motion is outward, which further weakens the already inadequate gravitational force. Under these conditions there can be no equilibrium.
In the time region, however, the effect of a change in relative position opposes the unbalanced force, which caused the change. If the gravitational force (outward in this region) is the greater, an outward motion takes place, weakening this gravitational force and ultimately reducing it to equality with the constant inward-directed force of the progression. Similarly, if the force of the progression is the greater, the movement is inward, and this increases the gravitational force until equilibrium is reached.
This is a most important finding: one that is unquestionably entitled to be designated as Outstanding Achievement Number Seven. The equilibrium that must necessarily be established between the atoms of matter inside unit distance in the RS universe obviously corresponds to the observed inter-atomic equilibrium that prevails in solids and, with certain modifications, in liquids. Here, then, is the explanation of solid and liquid cohesion that we derive from the Reciprocal System: the first comprehensive and completely self-consistent theory of this phenomenon that has ever been formulated. The mere fact that it is far superior in all respects to the currently accepted electrical theory of matter is not, in itself, very significant, inasmuch as the electrical hypothesis is definitely one of the less successful segments of present-day physical theory, but a comparison of the two theories should nevertheless be of interest from the standpoint of demonstrating how great an advance the new theoretical system actually accomplishes in this particular field.
The first item to be noted is that we are not replacing the electrical theory of matter with another “theory of matter.” The Reciprocal System is a complete general theory of the universe; it contains no hypotheses other than those relating to the nature of space and time and it produces an explanation of solid cohesion in the same way that it derives logical and comprehensive explanations of other physical phenomena, simply by development of the consequences of the space-time postulates. We therefore do not have to call upon any additional force of a hypothetical nature to account for the cohesion; the same thing that makes the atom an atom—the rotation in three dimensions—accounts for both the inward force of gravitation in the region outside unit distance and the existence of the inter-atomic equilibrium inside this distance.
Another significant point is that the new theory identifies both of the forces that enter into the inter-atomic equilibrium. One of the major defects of the electrical theory is that it only provides one force, the hypothetical electrical force of attraction, whereas two opposed forces are required to explain the observed situation. Originally it was assumed that the atoms were impenetrable and that the electrical forces simply held them in contact, but present-day experimental knowledge of compressibility and other properties of solids has demolished this hypothesis and it is now evident that there must be what Karl Darrow calls an “antagonist” to counter the attractive force, whatever it may be, and produce an equilibrium. Physicists have been unable to find any such force and, as Darrow says, they take refuge in evasion, and “manage to avoid the question, by using “words not conveying directly the notion of force.”89
A particularly important advantage of the Reciprocal System is that it utilizes the same mechanism for the cohesion of all substances. The basic hypothesis of the electrical theory—that of a force of attraction between a hypothetical positively charged constituent of the molecule and a hypothetical negatively charged constituent—is applicable only to a restricted class of solid substances and it is necessary to call upon weird ideas such as “shared electrons” to replace the basic hypothesis where the latter is clearly inapplicable. Even with this extraordinary kind of latitude, the theory is still in serious trouble. It is freely admitted, for instance, that there is still no plausible explanation for the cohesion of metallic aggregates. The superiority of an all-inclusive theory is obvious.
The directional characteristics of the inter-atomic forces are also explained in an easy and natural way by the Reciprocal System, whereas the electrical theory has practically nothing to contribute in this respect. Inasmuch as the atomic forces, according to the new system, result from rotation in three dimensions, and each dimension has its own specific rotational velocity, it is evident that a change in the inter-atomic orientation will alter the force, which the atom exerts on its neighbor.
Although the foregoing items add up to a very impressive total, the greatest triumph of the new system in this particular area is mathematical. By means of methods outlined in Chapter XV it is possible to determine the magnitudes of the rotational motions in each of the three dimensions of the atoms of the different chemical elements and the magnitudes of the corresponding directional forces. Equating these forces to the constant force of the space-time progression and solving for the equilibrium distance then gives us the inter-atomic distance or distances for each element or compound. As shown in The Structure of the Physical Universe, where these calculations are carried out in detail, there are uncertainties in the structures of the more complex substances that cause the result to take the form of a set of possible values in each instance rather than a specific figure, but for the simpler substances the calculated distance is definite and unequivocal, and in all of these cases it agrees with the experimental results within the margin of uncertainty of the latter. Additional studies, as yet unpublished, show that this agreement between the theoretical and experimental values holds good not only at normal pressures but also throughout the entire experimental pressure range, up to 100,000 atmospheres static pressure and to several million atmospheres by the recently developed shock wave techniques.
These results, the quantitative as well as the qualitative, are derived solely from the postulates concerning the nature of space and time, without any additional assumptions or any reference to the data of observation. Development of the consequences of the postulates reveals that there must exist certain combinations of rotational and vibrational motion, that the possible combinations of this type form a series in which each successive member possesses one more net unit of motion than its immediate predecessor, that, for example, a unit of number 19 in this series will form an equilibrium structure with other units of the same kind, in which each individual is separated from its immediate neighbors by the equivalent of 1.541 natural units of distance, and that number 19 will also combine with number 35 in the series, and the resulting 19-35 combination will form an equilibrium structure in which each individual is the equivalent of 1.133 natural units distant from its nearest neighbor.
All of the foregoing is purely theoretical and entirely independent of anything that may exist in the actual physical universe. It is simply a description of conditions and mathematical relations, which are necessary consequences of the Fundamental Postulates of the Reciprocal System and which therefore, must exist in the theoretical RS universe. But having come this far, we can now compare these features of the RS universe with the corresponding features of the observed physical universe. When we do this we find an exact agreement all the way down the line. The units of compound motion are atoms of matter, the series of these combinations is the series of chemical elements, numbers 19 and 35 in this series are the elements potassium and bromine respectively, their combination is potassium bromide, and the equilibrium structures which are formed are solid aggregates of these substances. By making some one measurement, either of an actual inter-atomic distance or of an appropriate combination of basic physical constants, we can identify the ratio of conventional units (Angstrom units) to natural units of distance as 2.914. Applying this ratio to the inter-atomic distances calculated theoretically, we obtain 4.49 for potassium and 3.30 for potassium bromide. Representative values from experiment are 3.29 for potassium bromide and 4.50 to 4.54 for potassium.
This, then, is another demonstration that the theoretical RS universe derived from the postulates of the Reciprocal System is identical with the observed physical universe: a demonstration which is particularly significant in that it involves a long chain of theoretical deductions, which are in complete agreement with the observed facts at every step of the way, and which ultimately lead to specific numerical values which can be compared directly with the corresponding experimental results.