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Motion that is not cyclical is free and it has infinite range. This is represented by infinite speed as it is all over that range. On the other hand, cyclical motion means that the motion is repeating itself. The faster it repeats itself, the higher is its frequency.

This repetition puts a limitation on the freedom of motion. With increasing repetition the freedom becomes increasingly limited. This is represented by the speed of cyclical motion decreasing with increasing frequency. 

An example of cyclical motion is the oscillatory motion of a pendulum. When this pendulum moves while oscillating, the motion acquires the appearance of a wave that has a wavelength. The product of the frequency and wavelength gives a measure of its speed. The mathematical formulas for wave motion apply to the cyclical motion.

The cyclical motion means a certain fixedness because the same motion is repeating itself. As the frequency of this repetition increases it means that the motion is becoming more fixed. This fixedness appears as a consistency, which resists change.

This resistance to change is called inertia. The resistance (inertia) appears as force. This force can be felt. This is the basic nature of substance.

Underlying any substance there is force, and underlying that force is cyclical motion.

As the frequency of cyclical motion increases, it not only acquires increasing consistency, but it also acquires a curvature. This is because the range of free motion within a cycle is becoming smaller. We may say that the least cyclical motion represents the “surface” of the universe, which obviously has a curvature because it contains the universe. As one goes deeper into the universe, the motion becomes more consistent and substance-like. The overall picture of cyclical motion may appear somewhat like a “whirlpool.” This is what we see in the spiral shape of the galaxies. This we also see in the structure of atoms.

At the center of a whirlpool we have extremely dense and spinning motion. This anticipates black holes at the center of galaxies and nuclei at the center of the atoms.

The cyclical motion of a very large range, ultimately, condenses into a spinning motion of a very small range.

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Summary

Here we have the whole spectrum of substance created out of cyclical motion. As this cyclical motion increases in frequency, it gains consistency, inertia and substance. It acquires a curvature because of its narrowing range. We thus have a shrinking circumference. This gives it a look very similar to that of a whirlpool.

From the periphery of the atom to is nucleus, we have cyclical motion that is gradually increasing in consistency and shrinking in its circumference, ending up in a dense spinning nucleus at the center.

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Reference: Fundamentals of Physics

In classical mechanics, substance is recognized by its mass. But since Einstein’s 1905 paper on “light quanta,” substance has taken a wider significance. Light has force and momentum, therefore, it is substantial. Substance is anything that is substantial enough to be sensed.

Since term “mass” is recognized only in the context of matter, we need a broader term equivalent to it in the context of substance. Such a term is “consistency.” For example, light does not have mass, but it has consistency.

MASS represents “atomic substance” only. CONSISTENCY represents both atomic and non-atomic substance. When consistency is very high, as in the nucleus of an atom, it is recognized as mass.

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Consistency (Thickness)

The consistency of substance is defined as follows:

Consistency is the degree of substantiality of substance. It is recognized as density, firmness, or viscosity of the substance. For example, “Honey has higher consistency than water.” For radiant energy, consistency is measured per quanta, where quanta is determined by frequency. For matter, consistency is measured by the mass of its elementary particle, such as, proton or neutron, where the elementary particle is determined by its smallest discrete inertia. The quanta and the mass of elementary particle may be measured by the same unit. Thus, substance can have a gradient of consistencies.

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Measure of Consistency

The consistency may be measured in terms of doubling of frequency as follows:

Consistency (C) = log f / log 2

We may calculate the consistency of electromagnetic substance from its frequency. This may be listed as follows:

We may assume the consistency of matter to be very close to 77.6.

The planets and stars may also have consistencies close to 77.6. But we may calculate their “relative consistency” based on their momentum as follows

De Broglie Equation,       λ = h/p,

where h is Planck’s constant, and p is momentum of the object, which is made up of many particles.

Frequency:                      f = c/λ = (c/h) p = 4.528 x 10⁴¹ p

Total Consistency:         C_tot = (log f) / (log 2) = 138.4 + 3.322 log p

Thus, knowing the mass and velocity of Earth, we may calculate its total consistency as follows,

M_E = 5.972 x 10²⁴ kg, V_E = 3 x 10⁴ m/s, and  p = M_E V_E = 1.79 x 10²⁹

C_tot (Earth) = 138.4 + 3.322 log (1.79 x 10²⁹) = 235.6

Similarly, we may calculate, C_tot (Sun) = 256.6

In terms of relative consistencies, we may say,

C_rel (Earth) = 1

C_rel (Sun) = 1.089

This means that the Sun is a bit more fixed in space than the Earth.

The Black holes may have much higher consistencies. Their absolute consistency (consistency per particle) may be higher than the consistency of the neutron. As a result they are much more fixed in space. We may, therefore, expect to find the Black holes at the center of the galaxies.

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Reference: Fundamentals of Physics

Doctoral Thesis completed in April 1905 and revised in 1906.
Published in Annalen der Physik 19 (1906) 289-305
A New Determination of Molecular Dimensions
Highlights in bold and comments in color by Vinaire

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Summary

In this paper Einstein explores a new theoretical method to calculate the molecular dimensions through the observation of physical phenomenon in liquids. The main part of the thesis involves hydrodynamics and the relation between coefficients of viscosity, followed by a determination of diffusion-coefficient from fundamentals.

Einstein’s thesis serves to validate his theoretical approach and the fundamental expressions derived for the viscosities and the diffusion coefficient.

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Some of the mathematical symbols used by Einstein in his thesis are as follows:

k: coefficient of viscosity of the solvent
k*: coefficient of viscosity of the solution
x, y, z, t: the coordinates and time
x₀, y₀, z₀: an arbitrary point (center of the molecule; molecule has a contact layer)
u, v, w: velocity components of the solvent
u₀, v₀, w₀: dilation motion in the absence of the sphere—a particle of solute (at infinity)
u₁, v₁, w₁: dilation motion in the presence of the sphere—would have to vanish at infinity
p: hydrostatic pressure (Section 1)
G: small region around this arbitrary point x₀, y₀, z₀
P: radius of the rigid sphere
R: a sphere of radius R, where R is indefinitely large compared with P
X_n, Y_n, Z_n: the components of the pressure exerted on the surface of the sphere of radius R
n: the number of solute molecules per unit volume
W: mechanical work done on the liquid (per unit of time) in the liquid lying within the sphere R = the energy which is transformed into heat
ds: surface element
ϕ: the fraction of the volume occupied by the spheres.
s: specific volume of the sugar present in solution
p: the osmotic pressure of the dissolved substance (Section 4)
ρ: grams of solute in a unit volume (section 3)
m: molecular weight of the solute
N: the number of actual molecules’ in a gram-molecule (Avogadro’s Number)
P: the hydrodynamically-effective radius of the solute molecule (Section 3)

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Here is a summary of the steps taken by Einstein in his thesis:

1. When a very small sphere is suspended in a liquid, how does it effect the motion of that liquid? 

• Set up hydrodynamical equations for viscous fluids for random motions in the liquid.
• Solve it using the method given by Kirchhoff getting expressions for p (hydrostatic pressure) and for u, v, w (the velocity components of solvent). 
• These solutions are unique for the boundary conditions at the surface of the sphere (a particle of solute).
• Use these solutions to calculate the mechanical work done in the liquid (per unit of time).
• The energy expression is obtained in terms of the viscosity of the solvent when the sphere is present.
• Obtain the energy expression when the sphere is absent. There is a lot of mathematics and simplifying assumptions.

2. Calculation of the viscosity-coefficient of the liquid in which a large number of small spheres are suspended in irregular distribution.

• Use the results from above to calculate the energy for a large number of spheres suspended in the liquid.
• Express the energy in terms of the parameters for the solution in a manner similar to above.
• Compare the energy expressions for the solvent and the solution, to determine the ratio of their viscosities.
• This ratio of viscosities is obtained in terms of the total volume of the solute present in the solution per unit volume.
• Conclusion: If very small rigid spheres are suspended in a liquid, the coefficient of internal friction is thereby increased by a fraction which is equal to 2.5 times the total volume of the spheres suspended in a unit volume, provided that this total volume is very small.

3. On the volume of a dissolved substance of molecular volume large in comparison with that of the solvent.

• Einstein applies the above conclusion to the viscosity data for dilute solution of sugar in water.
• A gram of sugar dissolved in water has the same effect on the viscosity as small suspended rigid spheres of total volume 0.98 cc. 
• But a gram of solid sugar has a volume of 0.61 cc.
• Einsteins reasons that the sugar molecules present in solution limit the mobility of the water immediately adjacent, so that a quantity of water, whose volume is approximately one-half the volume of the sugar-molecule, is bound on to the sugar-molecule
• Using the molecular weight of sugar, Einstein calculates the volume of a sugar molecule in terms of the Avogadro’s number.

4. On the diffusion of an undissociated substance in solution in a liquid

• Use Kirchhoff’s equation to calculate the diffusion-coefficient of an undissociated solution.
• In this calculation osmotic pressure is treated as a force acting on the individual molecules.
• This expression of diffusion-coefficient includes “the product of the number N of actual molecules in a gram-molecule and of the hydrodynamically-effective radius P of the molecule. 

5. Determination of molecular dimensions with the help of relations already obtained

• The expressions of the ratio of viscosities and the expression of the diffusion-coefficient give us two equations in “N” and “P”.
• Einstein solves these equations for the aqueous sugar solution.
• The values of “N” and “P” are found to have the same order of magnitude as determined by other methods.

Einstein’s thesis serves to validate his theoretical approach and the fundamental expressions derived for the viscosities and for the diffusion coefficient.

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Other references to this Paper:

1. Something Interesting about Albert Einstein’s Ph.D. Thesis
2. On Einstein’s Doctoral Thesis
3. Einstein’s PhD thesis

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