# Chapter XVI

# Quantitative Relations

It is quite apparent that the quantitative aspects of the physical universe are determined by specific laws and principles in just as definite a manner as the qualitative aspects that have been our principal concern in the preceding pages. No school of scientific thought contends that the magnitudes of the many physical “constants” of one kind or another are the result of pure chance, nor that the numerical values determined for such properties of matter as specific heat, viscosity, or refractive index are accidental. But, with few exceptions, these values cannot be derived from purely theoretical sources in present-day practice; they depend in one way or another on some measured quantity.

Much progress has been made in the development of mathematical expressions to represent the *variability of* physical properties under different conditions. For example, Van der Waals’ equation of state for gases, and the more complicated recent developments of the same kind, enable us to calculate the volume of a gas—methane, for example—under any specified conditions within a very wide range of temperatures and pressures. But in order to perform this calculation, using Van der Waals’ equation, we must start with two purely empirical constants, which for methane are a = 2.253 and b = 0.04278. Just why these particular values apply to this particular compound is completely unknown. In fact, it is quite likely that they are merely determinants of an approximate mathematical relation and have no real physical significance.

This is typical of the general situation as it now exists. Present-day physical science can give us mathematical expressions of an empirical or semi-theoretical nature which represent the behavior of the properties of matter under various conditions, but only rarely can it give us a numerical value on which to base our calculations. Almost always we have to start with measured values of the properties in question or with empirical constants of one kind or another. Returning to the example previously cited, Van der Waals’ equation is (P + a/V^{2}) (V - b) = RT. This is mathematical, but as it stands it is not quantitative, and until it is made quantitative its value is quite limited. The major purpose of such an expression is to enable us to calculate one of the three quantities P, V and T if we know the other two, but this cannot be done until we establish values for the constants a and b, something that the theory underlying the equation is unable to do. In order to make this mathematical expression quantitative we have to call upon empirical sources for help. It is correct to say, therefore, that present-day theory, of which this is a fair sample, is * mathematical,* but it is not truly *quantitative.*

One of the important features of the Reciprocal System is that it is actually quantitative in the sense in which this term is used in the foregoing paragraph. The theoretical development deals with numerical values from the very beginning, and a quantitative treatment goes hand in hand with the qualitative treatment as the system expands into more and more detail. At the present stage of the development the quantitative results are not always as specific as those of a qualitative nature are. In many cases the theoretical calculations lead to a set of possible values rather than to one unique result, and the choice from among these several possibilities is determined by probability considerations which have not yet been given adequate study. This situation is further complicated by the fact that the numerical values of physical quantities are not always determined by present conditions; past history may also enter into the picture. For example, where two or more crystal forms of a substance may coexist through a range of temperatures and pressures, it is not possible, as matters now stand, to determine the crystal form of a particular specimen from theory alone, and many of the properties of the substance are affected by this crystal form. Presumably future studies will bring the probability factors and the problems connected with past history within the scope of the theory, but in the meantime the Reciprocal System carries us a long way into the quantitative field, even though it is not yet fully developed from this standpoint.

This present chapter is not intended as an actual quantitative development, which would take us far beyond the limits of the current volume, but as a brief description of how the quantitative phase of the theoretical development is carried out. One of the first things that is required for this purpose is to break down the various physical quantities into space-time terms so that their relations to the general physical picture are clarified. We begin with one-dimensional space s and one-dimensional time t. Extension into additional dimensions then produces area s^{2} and volume s^{3}, together with the corresponding time quantities t^{2} and ts, which have not been named. Dividing space by time, we obtain velocity s/t, and an additional division of the same kind results in acceleration s/t^{2}. Velocity in two dimensions s^{2}/t^{2} and in three dimensions s^{3}/t^{3} are not identified by special names.

Next we will want to look at the inverse quantities. The inverse of motion in our material universe is resistance to motion. Resistance effective in three dimensions is inertia or mass t^{3}/s^{3}, the inverse of three-dimensional velocity. Multiplying mass by velocity (mv) we obtain momentum t^{2}/s^{2}, and another similar multiplication (mv^{2}) gives us energy t/s. Energy is thus the reciprocal of velocity. Inasmuch as energy (or the equivalent quantity, work) is the product of force and distance, we have the relation F = E/s, hence force is t/s^{2}. Force is thus the analog of acceleration, with the space and time terms interchanged.

In the electrical field, we have found that the unit quantity of electricity, the electron, is simply a rotating unit of space. Electrical quantity is therefore s. Current, as already explained, is quantity per unit of time, and is therefore s/t, equivalent to a velocity. Electrical energy is equivalent to and interchangeable with energy in other forms t/s. Electromotive force, or electric potential, is likewise merely one form of force in general t/s^{2}. Energy per unit time is power 1/s. The product of current and potential (IR) is resistance t^{2}/s^{3}. This quantity may also be defined in another way as mass per unit time.

Electric charge is not differentiated from electrical quantity in current practice because of the lack of recognition of the uncharged electron as the unit of current electricity, but these two quantities are quite distinct physically. The charge, being a unit of one-dimensional rotation, is the one-dimensional analog of mass and is therefore a quantity of energy t/s. Magnetic charge is similarly the two-dimensional analog of mass t^{2}/s^{2}. Magnetic potential, like electric potential, is charge divided by distance t^{2}/s^{3}. The potential gradient, or field intensity, for both electric and magnetic phenomena, is the potential divided by the distance. The electric field intensity is

t/s

^{2}× 1/s = t/s^{3}

t

^{2}/s^{3}× 1/s = t^{2}/s^{4}

Flux is defined in terms, which make it equal to charge, usually the product of the area and the field intensity:

t

^{2}/s^{4}× s^{2}= t^{2}/s^{2}

Another basic magnetic quantity is inductance (L), which is defined by the equation

F = -L dI/dt

In terms of space and time, the inductance is then

L = t/s

^{2}× t × t/s = t^{3}/s^{3}

Inductance is thus equivalent to mass. It is simply magnetic inertia. Because of the dimensional confusion now existing in magnetic relations it is usually regarded as equivalent to length, and is frequently expressed in centimeters. The true nature of this quantity can be seen if we compare the inductive force equation with the general force equation:

F = ma = m dv/dt = m d

^{2}s/dt^{2}F = L dI/dt = L d

^{2}Q/dt^{2}

The equations are identical. As we have previously found, I is a velocity and Q is space. It follows that m and L are equivalent. We may now express the inductive force equation in space-time terms as

F = L dI/dt = t

^{3}/s^{3}× s/t × 1/t = t/s^{2}

The consistency of the relationships defined in the foregoing paragraphs can be demonstrated by similarly breaking down other force equations into their space-time equivalents:

EMF of current (Rate of change of flux)

F = df/dt = t

^{2}/s^{2}× 1/t = t/s^{2}

Magnetic force of current (Ampere’s Law)

F = I dl M/r

^{2}= s/t × s × t^{2}/s^{2}× 1/s^{2}= t/s^{2}

Force on moving charge

F = H Q v = t

^{2}/s^{4}× s × s/t = t/s^{2}

Magnetic force on conductor

F = H I l = t

^{2}/s^{4}× s/t × s = t/s^{2}

The question naturally arises as to how it has been possible to express inductance in centimeters, or quantity of current in units of charge, or to utilize other such erroneous units, without getting into mathematical difficulties. An explanation that was used in a previous publication compares this with a situation in which we are undertaking to set up a system of units in which to express the properties of water. If it so happens that we fail to recognize that there is a difference between the properties of weight and volume, we will use the same unit—a cubic centimeter, let us say—for both. Since the specific gravity of water is fixed, this is equivalent to using a weight unit of one gram, and as long as we deal separately with weight and volume, each in its own context, the fact that the expression “cubic centimeter” has two entirely different meanings in our system will not cause any difficulty. But sooner or later we will want to use weight and volume at the same time, perhaps in order to utilize the concept of density, and then we will have trouble. Similarly, the conceptual errors in the electric and magnetic systems are of no consequence in ordinary practice, but they interpose some serious obstacles to a correct theoretical understanding of the phenomena that are involved.

An analogous situation exists with respect to Planck’s constant h. This constant is expressed in erg-seconds, the product of energy and time, or *action,* as it is called. In space-time terms action is t^{2}/s. This is, in itself, an oddity, as none of the other basic quantities that have been discussed in the previous pages has a higher power in the numerator of the space-time relation than in the denominator, and it strongly suggests that action does not have the same kind of physical significance as these more familiar quantities. Furthermore, the concept of action does not help us to *understand* the relation between energy and frequency; all that we have here is * mathematical* knowledge, the knowledge that there is a definite proportionality between the two quantities. Giving the constant of proportionality the name “action” and the dimensions energy × time does not contribute any *conceptual* information about the relationship.

In order to gain a genuine understanding of the situation it is necessary to recognize the true physical nature of the so-called “frequency.” From the facts previously developed it is apparent that this is actually a *velocity,* a relation of space to time, not a function of time alone. The true statement of Planck’s radiation energy equation is therefore E = hv. The constant h is then equal to E/v, or in space-time terms

h = t/s × t/s = t

^{2}/s^{2}

These are simply the reversing dimensions required to convert velocity s/t into its reciprocal, energy t/s. The constant h in the radiant energy equation has the same dimensions and serves the same purpose as the momentum mv in the kinetic energy equation:

E = ½mv

^{2}= (t^{3}/s^{3}× s/t) × s/t = t/sE = hv = t

^{2}/s^{2}× s/t = t/s

Another source of dimensional confusion is the lack of recognition of the dimensionless character of some of the terms in the equations for electric, magnetic, and gravitational forces. The dimensions of electric potential, for example, are given in the handbooks as e^{-½} × m^{½} × l^{½} × t^{-1} on the assumption that the mass and charge terms in Coulomb’s Law and its multi-dimensional counterparts all have mass and charge dimensions. This is not correct, as a close examination of the gravitational equation clearly shows. If such an assumption is applied to this equation, the gravitational force is proportional to the square of the mass, whereas the general force equation F = m × a says that *any* force is directly proportional to the mass involved. Obviously there is something wrong.

An analysis shows that the dimensional situation has not been properly assessed in arriving at the currently accepted conclusions. A mass m exerts m units of force on unit mass at unit distance, and it exerts m’ times as many units of force on m’ units of mass at the same distance, not m’ grams times as many units of force but merely m’ times as many. In other words, one of the mass terms in the equation, m’ in this case, is simply a dimensionless ratio, the ratio of m’ units of mass to one unit of mass. The distance term in the equation is likewise a ratio, the ratio of d^{2} units of distance to 1^{2} units of distance, as can easily be seen on consideration of the derivation of the inverse square relation.

If we eliminate the dimensionless terms from the gravitational equation; that is, set up an equation for the force exerted by a mass m on unit mass at unit distance, we have F = k × m. A previously published study of this relation indicates that acceleration also enters into this situation, but that it is the acceleration due to the force of the space-time progression, and it always has unit value, hence has no numerical effect, and for this reason has not hitherto been recognized. The acceleration term should be inserted to complete the gravitational equation from a dimensional standpoint, and we then have F = kma, which is identical with the general force equation as, of course, it should be.

The key to the quantitative aspect of the new theoretical system is the postulate that space and time exist only in discrete units. From this basic principle it follows that the displacements of space and time which are responsible for the existence of all physical phenomena consist of a specific number of units of space (or time) associated with a specific number of units of time (or space), and the successive additions of different kinds of motion which build up the compound unit as shown on Chart E are actually additions of *units* of motion (displacement). Matter, for instance, does not exist in an infinite gradation of quantities; it exists only in a series in which the first member has one net unit of positive three-dimensional rotation and each succeeding member adds one more unit of the same kind. The properties of the individual members of this series, the series of chemical elements, are determined by the way in which the rotation around each of the three axes of the atom is built up by successive addition of separate units of motion, with the sequence of assignment of units to the different axes following a specific pre-determined pattern.

The same kind of an orderly addition of units exists all through the theoretical development, and the specific number of units of each kind of motion present in an atom or other physical entity is the factor that determines the quantitative properties of that entity. All subsequent numerical values result from mathematical relations between the basic entities and are derived entirely from combinations and other systematic modifications of the original displacement values. Because of this manner in which the quantitative relations are developed, no arbitrary or measured numerical values are introduced at any point. Aside from the conversion constants which are required if the results are to be expressed in some conventional system of units rather than in the natural units in which they originally appear, all numerical constants which enter into the theoretical relations are structural constants: integral or half-integral values which represent the actual numbers of the various types of physical units entering into the particular phenomenon under consideration.

To illustrate how the quantitative aspects of the various physical phenomena originate, the development of the series of chemical elements will serve as a good example. As brought out in Chapter VI, the atom of matter is a linear oscillation (LV^{1+}) rotating in three dimensions (R^{3+}). When we examine the details of this rotation, however, we find that, although it is rotation in three dimensions, it is not, strictly speaking, a three-dimensional rotation. A line cannot rotate around itself as an axis, as such a rotation would be indistinguishable from no rotation at all. The linear oscillation therefore rotates around its midpoint. One such rotation generates a two-dimensional figure, a disk. Rotation in another dimension generates a three-dimensional figure, and in order to make three-dimensional rotation possible it would be necessary to have a fourth dimension available. Since there is no fourth dimension in the physical universe, the basic rotation of the atom is two-dimensional.

Once a two-dimensional rotating unit is in existence, however, it is possible to add oppositely directed motions of various kinds in the same way that the compound motions of Chart E are built up. One of the possibilities that is open is the rotation of the two-dimensional unit around the third of the three perpendicular axes. This oppositely directed one-dimensional rotation is not *necessary;* that is, the two-dimensional unit may exist without any effective rotational displacement in this third dimension. (A rotation at unit velocity is the rotational zero, just as unit linear velocity is the physical zero for translational motion.) The possible rotational combinations therefore include both purely two-dimensional units and units with both one-dimensional and two-dimensional rotation.

Another important point is that two separate two-dimensional rotations may be combined in one physical unit. The nature of this combination can be clearly illustrated by two cardboard disks interpenetrated along a common diameter C. The diameter A perpendicular to C in disk A represents one linear oscillation, and the disk A is the figure generated by a one-dimensional rotation of this oscillation around an axis B perpendicular to both A and C. Rotation of a second linear oscillation, represented by the diameter B. around axis A generates the disk B. It is then obvious that the primary rotation represented by disk A may be given a secondary rotation around axis A and the primary rotation represented by disk B may be similarly rotated around axis B without interference at any point, as long as the rotational velocities are equal.

Here, again, the second rotation is not necessary for stability. Units in which there is only one two-dimensional rotation can and do exist. But, as a general principle, symmetrical combinations are more probable than asymmetrical combinations. A 1-1 combination, for example, is inherently more probable than a 2-0 combination, and if a second two-dimensional displacement unit is added to a 1-0 combination—a unit with a single two-dimensional rotation—it is highly probable that the new unit will generate a rotation around the second axis, bringing the combination to the 1-1 status, rather than raising it to 2-0 by adding to the existing rotation. This probability is further heightened by the fact that the rotation that is to be absorbed will itself be a combination of a linear oscillation and an added rotation so that the increase from 1-0 to 1-1 is a simple absorption of the entire rotating unit, whereas a change from 1-0 to 2-0 involves some kind of a readjustment. The addition of these two factors creates such a strong bias in favor of the symmetrical distribution that the alternative distribution is, in effect, barred. The combinations with only one two-dimensional rotation are therefore confined to those, which do not possess more than one unit of rotational displacement.

With the benefit of the foregoing information, we are now ready to start identifying the possible rotational combinations; that is, the possible material atoms and sub-atomic particles. As a preliminary step, however, it will be desirable to define some convenient terms and symbols which will facilitate the discussion. For reasons, which should be apparent from the points brought out in the preceding chapters, the one-dimensional rotation will be designated as * electric* and the two-dimensional rotation as *magnetic.* In order to avoid interference it is necessary that the two rotating systems of the atom have the same velocities. Each added unit of magnetic displacement therefore increases the rotation of both systems in one magnetic dimension rather than one system in both dimensions. In those cases where the displacements in the two dimensions are unequal the rotation is distributed in the form of a spheroid, and where this is true the rotation which is effective in two dimensions of the spheroid will be called the *principal* magnetic rotation and the other the * subordinate* magnetic rotation.

In referring to the various combinations of rotational displacement a notation in the form 2-2-3 will be utilized, the three figures representing the displacements in the principal magnetic, the subordinate magnetic, and the electric rotational dimensions respectively. Where the displacement is in space instead of in time, the appropriate figure will be enclosed in parentheses. It should be understood that the terms “electric” and “magnetic” refer only to rotation in one and two dimensions respectively, and neither term implies the existence of a charge. Where a charge is present, this will be so stated.

This may also be an appropriate place to insert another reminder of the nature of such presentations as the one that follows. This is not a *hypothesis* as to what exists in the *actual* world; it is a description of what *actually* exists in the * hypothetical* world. The general principles controlling the combinations of rotational motions, as set forth in the preceding paragraphs, are the principles, which *must* hold good in the theoretical RS universe. Of course, the general proof of the identity of the RS universe and the observed physical universe means that all of the conclusions also apply to the latter but it should be remembered that the description which follows is entirely theoretical; no part of it is derived from observation or from inferences based on observation. The exact agreement with the observed facts is therefore a reflection of the accuracy of the theory, not a testimony to the ingenuity of the originator.

If a linear oscillation is given a rotational motion with a single unit of magnetic displacement, the resulting combination 1-0-0 is the rotational base. In this combination the single rotational displacement merely neutralizes the vibrational displacement in space, and the net displacement is zero; that is, this unit is the rotational equivalent of nothing at all. In accordance with the general principles previously stated, the addition of another magnetic displacement unit produces 1-1-0, which we identify as the neutron, the neutral magnetic sub-atomic particle.

Still another magnetic displacement unit results in 2-1-0. Here, for the first time, we have an *effective* displacement in both magnetic dimensions, and this combination therefore has some properties, which are quite different from those of the neutron. These properties we identify as the characteristics of * matter* and we identify the combination 2-1-0 as the element helium. Further additions of magnetic displacement, going alternately to the two magnetic dimensions, produce a series of elements, which we identify as the *inert gases.* The complete series is as follows:

Displacement | Designation |
---|---|

1-0-0 | Rotational base |

1-1-0 | Neutron |

2-1-0 | Helium |

2-2-0 | Neon |

3-2-0 | Argon |

3-3-0 | Krypton |

4-3-0 | Xenon |

4-4-0 | Radon |

5-4-0 | Unstable |

The electric equivalent of a magnetic displacement n is n^{2} in each dimension. The symmetry principle therefore tells us that the magnetic rotation is more probable than electric rotation where the option exists. As a consequence, the role of electric rotation is confined to filling in the intervals between the members of the foregoing series.

Here there is a mathematical point that must be taken into consideration. In the undisplaced condition, all progression is by units. We have first one unit, then another similar unit, yet another, and so on, the total up to any specific point being n units. There is no individual term with the value n; this value appears only as the total. The progression of displacements follows a different mathematical pattern because in this case only one of the space-time come portents progresses, the other remaining fixed at the unit value. The progression of 1/n, for instance, is 1/1, ½, 1/3, and so on. The progression of the reciprocals of ½ is 1, 2, 3… n. Here the quantity n is the final term, not the total. For the total we must sum up all of the individual terms. Similarly, when we find that the electric equivalent of a magnetic displacement n is 2n^{2}, this does not refer to the total from zero to n; it is the equivalent of the n^{th} term alone. From the foregoing it is evident that if all rotational displacement were in time, the complete series of elements would start with the lowest possible magnetic combination, helium, and the electric displacement would increase step by step until it reached a total of 2n^{2} units, whereupon the relative probabilities would result in the conversion of these 2n^{2} units of electric displacement into one additional unit of magnetic displacement, and the building up of the electric displacement would then be resumed. This behavior is modified, however, by the fact that electric displacement in matter, unlike magnetic displacement, may be in space rather than in time.

The net rotational displacement of any material atom must be in time in order to give rise to those properties which are characteristic of matter. It necessarily follows that the magnetic displacement, which is the larger component of the total, must also be in time. But the smaller component, the electric displacement, may be in space without affecting the direction of the net total displacement. Which direction the electric displacement will actually take in any particular situation then becomes a matter of probability. Since the probability factors favor the lower number of units, we can deduce that successive additions to the net total time displacement from any inert gas base will take the form of electric displacement in time until n^{2} units have been added. At this point the probabilities are nearly equal and the alternate situation may exist. As the net displacement rises still farther, the alternate arrangement becomes more probable, and in the second half of each group the magnetic displacement is increased by one unit and an appropriate number of units of the oppositely directed displacement in space brings the net total down to the required figure. Successive units of this space displacement are then eliminated to move up the atomic series.

By reason of this availability of electric space displacement as a component of the atomic structure, an element with a net displacement less than that of helium becomes possible. This element 2-1-(1), which we identify as hydrogen, is the first member of the series of chemical elements. Each succeeding member of the series adds one unit of electric time displacement or the equivalent thereof. Helium is element number two. At this point the displacement is one unit above the initial level of 1-0-0 in each magnetic dimension and any further increase in the magnetic displacement requires the addition of a second unit in one of the dimensions. Where n = 2, the electric equivalent of the added magnetic unit is 8, and hence there are eight elements in the next group, as follows:

Displacement | Element | Atomic Number |
---|---|---|

2-1-1 | Lithium | 3 |

2-1-2 | Beryllium | 4 |

2-1-3 | Boron | 5 |

2-1-4 | Carbon | 6 |

2-2-(4) | ||

2-2-(3) | Nitrogen | 7 |

2-2-(2) | Oxygen | 8 |

2-2-(1) | Fluorine | 9 |

Another similar group with one additional unit of magnetic displacement follows, then two groups of 18 units each (n = 3) and two groups of 32 elements each (n = 4). As indicated in Chapter XIII, the atoms of the last of these groups are radioactive, and the instability increases rapidly as the atomic number approaches 100. The relatively few elements near and above 100 that have been identified are therefore known mainly through artificial production of extremely short-lived isotopes. A full listing of the elements of these upper groups does not appear necessary for present purposes, but the following tabulation shows the first and last members of each group and the element at the midpoint:

Displacement | Element | Atomic Number |
---|---|---|

2-1-1 | Sodium | 11 |

2-2-4 | Silicon | 14 |

3-2-(1) | Chlorine | 17 |

3-2-1 | Potassium | 19 |

3-2-9 | Cobalt | 27 |

3-3-(1) | Bromine | 35 |

3-3-1 | Rubidium | 37 |

3-3-9 | Rhodium | 45 |

4-3-(1) | Iodine | 53 |

4-3-1 | Cesium | 55 |

4-3-16 | Ytterbium | 70 |

4-4-(1) | Astatine | 85 |

4-4-1 | Francium | 87 |

4-4-16 | Nobelium | 102 |

5-4-(1) | Unknown | 117 |

By a similar process of addition of electric displacement in time and space to the rotational base and the neutron we may complete the list of sub-atomic particles in the material system as follows:

Displacement | Designation |
---|---|

1-0-(1) | Electron |

1-0-0 | Rotational base |

1-0-1 | Positron |

1-1-(1) | Neutrino |

1-1-0 | Neutron |

1-1-1 | Unnamed particle |

The development of this systematic quantitative explanation of the series of chemical elements is clearly entitled to be designated as Outstanding Achievement Number Fifteen, not only because it is important in itself, inasmuch as it provides definite answers to longstanding questions as to why such a series exists, why it includes these specific elements and not others, why some, but not all, of the properties of these elements are periodic, and what factors determine the magnitudes of those properties, but also because it provides a point of departure for a host of other quantitative developments.

Through the addition of successive units of rotational displacement in accordance with the sequence that has been described, the *only* way in which these units can be added in conformity with the principles of the Reciprocal System, each of the chemical elements acquires its own individual pattern of displacements in the different dimensions. The numerical values of these displacements are then the figures that enter into the quantitative expressions of the various physical properties, and since each element has its own set of figures each has a unique quantitative pattern.

The series of elements is first established by successive additions of units of displacement, beginning with one effective unit and ending with 117. This gives each element one unique numerical value which enters into a great variety of mathematical expressions of physical properties and makes the quantitative result for that element different from that of any other. But for other purposes the significant value is not the net total displacement but the displacement in some one or more of the rotational dimensions, and because of the definite and specific factors which determine the particular rotation to which each successive displacement addition goes, each element also has its own individual pattern of rotation values. The quantitative aspects of the elementary physical properties such as mass, volume, etc., are determined directly by one or more of these four numerical values that characterize the individual elements. More complex quantitative relations are then established by interaction between the elementary values in much the same manner as the proliferation of qualitative relations previously discussed.

The mass of the normal atom, for example, is a function of the net total time displacement, and it increases continuously with the atomic number. The index of refraction and the diamagnetic susceptibility are functions of the magnetic displacement alone, and they also increase as the atomic number rises, but discontinuously and much more slowly. The inter-atomic distance in the solid state is likewise a direct function of the magnetic displacement, but it is also an inverse function of the electric displacement and this makes it a periodic property. The atoms are widely separated in the first element of each rotational group (sodium, etc.) and the inter-atomic distance decreases to a minimum at the midpoint of the group, increasing again thereafter, so that the separation in the last element (chlorine, etc.) approximates that in the first.

In the past the biggest obstacle to the development of a quantitative system which could reproduce the magnitudes of physical properties by means of relations based entirely on theoretical foundations has been the lack of any known numerical characteristics of the elements other than the atomic number. Many *ad hoc* constructions, such as “electron shells“, for example, have been devised in an attempt to provide these additional numerical values but, from a practical standpoint, it is virtually impossible to solve a * complicated* problem by *ad hoc* methods. The typical results of this method of approach are graphically, if somewhat unintentionally, portrayed by an author who summarizes the application of quantum mechanics to the “electron shell” and related atomic and molecular concepts with the enthusiastic statement that “Quantum mechanics… gives the solution, in principle, to almost every chemical problem,” and then almost in the same breath, admits that “Very unfortunately, however, there is an enormous gap between this solution in principle and the practical calculation of the properties of any specific molecule.”^{128}

Such solutions “in principle” are fraudulent. If a “solution” does not produce the right answers, then it is not a solution * in fact, *and calling it a solution in *principle is* a gross misuse of the word “solution.” The purpose of a physical theory is to produce the right answers; if it produces the wrong answers then the theory itself is wrong, both in principle and in fact. The only feasible route to success in solving a very complex problem such as that of the properties of matter is by way of a theory which is correct in all respects from the very beginning. This is what the Reciprocal System now provides. In this system the basic combinations of values applicable to each element are not derived from an *ad hoc* manipulation of empirical results, but are derived theoretically *from the properties of space and time.* Thus they have the firm theoretical basis that is essential in order to produce the correct results in such a vast and complex field.

Furthermore, the scope of the practical applicability of the new system is broadened to a very considerable degree by the extraordinary simplicity of the basic relations that have been established. This was a rather unexpected aspect of the theoretical development, since the phenomena under consideration are, as a rule, very complicated, and it is only natural to assume that this is the result of a complex underlying relationship. One of the reasons why we hear so much about solutions “in principle” is that the mathematical expressions which have been developed in an attempt to express these complex relations become so unwieldy in practical application that they cannot be handled by available mathematical techniques, and it is normally impossible to carry the calculations to completion so that it can be determined specifically whether the theories are correct or not. But the new development now shows that in most cases where properties of a basic nature are involved, the complex relation that we observe is actually a combination of two or more relatively simple relations.

For example, the volumetric pattern of water is so complicated that P. W. Bridgman, the foremost investigator in this field, exe pressed serious doubt that it would *ever* be possible to reproduce this pattern mathematically.^{129} A study based on the principles of the Reciprocal System reveals, however, that this is not *one complex pattern* as Bridgman assumed; it is a composite of *four simple patterns:* the characteristic linear relation of the liquid state, a probability distribution representing the proportion of solid molecules in the lower liquid range, another probability distribution representing the proportion of critical molecules in the upper liquid range, and a third probability distribution originating from the fact that there are two forms of the water molecule—a high temperature form and a low temperature form—within the liquid range. When the four factors are separated and each is treated according to its own simple rules of behavior, Bridgman’s “impossible” task can be, and has been, accomplished.

Unraveling such a tangle of relations is, of course, a major undertaking, but it is a relatively straightforward task that can hardly fail to reach its objectives if sufficient effort is applied. Since the new theoretical system accomplishes a similar simplification in a great many physical fields, it opens the door to almost unlimited progress along quantitative lines.