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
perinea palconcern 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 presentday 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 Æ 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 = .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.
Presentday physical science can give us mathematical expressions of an
empirical or semitheoretical 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/V²) (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 presentday 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 spacetime terms so that their relations to the general physical
picture are clarified. We begin with onedimensional space s and onedimensional
time t. Extension into additional dimensions then produces area s² and
volume s³, together with the corresponding time quantities t² 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/t². Velocity in two dimensions s²/t² and in three dimensions s³/t³ 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 t³/s³, the
inverse of threedimensional velocity. Multiplying mass by velocity (mv)
we obtain momentum t²/s², and another similar multiplication (mv²) 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/s². 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/s². Energy per unit time is power
1/s. The product of current and potential (IR) is resistance t²/s³. 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 onedimensional
rotation, is the onedimensional analog of mass and is therefore a quantity
of energy t/s. Magnetic charge is similarly the twodimensional analog
of mass t²/s². Magnetic potential, like electric potential, is charge
divided by distance t²/s³. Does the distance divide the potential the
potential gradient, or field intensity, for both electric and magnetic
phenomena. The electric field intensity is
t/s² × 1/s = t/s³
Magnetic field intensity is
t²/s³ × 1/s = t²/s^{4}
Flux is defined in terms, which make it equal to charge,
usually the product of the area and the field intensity:
t²/s^{4} × s² = t²/s²
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/s² × t × t/s = t³/s³
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 d²s/dt²
F = L dI/dt = L d²Q/dt²
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 spacetime terms as
F = L dI/dt = t³/s³ × s/t × 1/t = t/s²
The consistency of the relationships defined in the foregoing
paragraphs can be demonstrated by similarly breaking down other force
equations into their spacetime equivalents:
EMF of current (Rate of change of flux)
F = df/dt = t²/s² × 1/t = t/s²
Magnetic force of current (Ampere’s Law)
F = I dl M/r² = s/t × s × t²/s² × 1/s² = t/s²
Force on moving charge
F = HQv = t²/s^{4} × s × s/t = t/s²
Magnetic force on conductor
F = HIl = t²/s^{4} × s/t × s = t/s²
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 undertakeing
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 core rect theoretical understanding
of the phenomena that are involved.
An analogous situation exists with respect to Planck’s
constant h. This constant is expressed in ergseconds, the product of
energy and time, or action, as it is called. In spacetime terms
action is t²/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 spacetime 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 x 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 socalled
“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 spacetime terms
h = t/s × t/s = t²/s²
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 = 1/2mv² = (t³/s³ × s/t) × s/t = t/s
E = hv = t²/s² × 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 multidimensional 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 d² units of distance
to 1² 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 spacetime
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 = k × m × a, 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 threedimensional 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
predetermined 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 halfintegral
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 (R³+). 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 threedimensional 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 twodimensional figure, a
disk. Rotation in another dimension generates a threedimensional figure,
and in order to make threedimensional 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 twodimensional.
Once a twodimensional 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 twodimensional
unit around the third of the three perpendicular axes. This oppositely
directed onedimensional rotation is not necessary; that is, the
twodimensional 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
twodimensional units and units with both onedimensional and twodimensional
rotation.
Another important point is that two separate twodimensional
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 onedimensional
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 twodimensional rotation can
and do exist. But, as a general principle, symmetrical combinations are
more probable than asymmetrical combinations. A 11 combination, for example,
is inherently more probable than a 20 combination, and if a second twodimensional
displacement unit is added to a 10 combination–a unit with a single
twodimensional rotation–it is highly probable that the new unit
will generate a rotation around the second axis, bringing the combination
to the 11 status, rather than raising it to 20 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 10 to 11
is a simple absorption of the entire rotating unit, whereas a change from
10 to 20 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 twodimensional 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 subatomic 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
onedimensional rotation will be designated as electric and the
twodimensional 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 223 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
100 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 110, which we identify as the neutron, the neutral magnetic
subatomic particle.
Still another magnetic displacement unit results in 210.
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 210 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

100

Rotational base

110

Neutron

210

Helium

220

Neon

320

Argon

330

Krypton

430

Xenon

440

Radon

540

Unstable

The electric equivalent of a magnetic displacement n
is n² 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 spacetime come portents progresses,
the other remaining fixed at the unit value. The progression of 1/n, for
instance, is 1/1, 1/2, 1/3, and so on. The progression of the reciprocals
of 1/2 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 2n², this does not refer to the total from zero to n; it is the equivalent
of the nth 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 2n²
units, whereupon the relative probabilities would result in the conversion
of these 2n² 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 n² 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 21(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 100 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

211

Lithium

3

212

Beryllium

4

213

Boron

5

214

Carbon

6

22(4)


22(3)

Nitrogen

7

22(2)

Oxygen

8

22(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
shortlived 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

211

Sodium

11

224

Silicon

14

32(1)

Chlorine

17

321

Potassium

19

329

Cobalt

27

33(1)

Bromine

35

331

Rubidium

37

339

Rhodium

45

43(1)

Iodine

53

431

Cesium

55

4316

Ytterbium

70

44(1)

Astatine

85

441

Francium

87

4416

Nobelium

102

54(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 subatomic particles in the material system as follows:
Displacement

Designation

10(1)

Electron

100

Rotational base

101

Positron





11(1)

Neutrino

110

Neutron

111

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 interatomic 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 interatomic 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.
