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Precession of the Planets

or

What was Plato Writing About?

by

V.L.Pakhomov

 

PART TWO

 

 

 

The Precession Law of the Planets System

 

"The chief aim of all investigations of the external world
should be to discover the rational order and harmony
which has been imposed on it by God and
which He revealed to us in the language of mathematics."

— Johannes Kepler (1571-1630)

 

Now we shall proceed to the analysis of galactic latitude of the North Pole of the planets (see tab. 1).

As we are interested only in the slope angle of the rotation axis of the planets to the galactic plane, for Pluto it is possible to change the sign of the galactic latitude, that is, to use the South Pole latitude (all the more that Pluto has a retrograde rotation).

In the following table, planets are ordered on the increase of the slope angle of their rotation axis to the galactic plane.

 

Table 3

 

 

bg

1

Mars

3.27

2

Neptune

7.50

3

Uranus

14.55

4

Saturn

21.35

5 Pluto 21.65
6

Sun

22.77

7

Mercury

24.48

8

Earth

27.10

9

Venus

28.74

10

Jupiter

30.63

 

These data are shown on the following diagram.

 

Figure 8

 

To find the analytical formula for this dependence we shall present the same data on a logarithmic scale.

 

Figure 9

 

As you can see, two straight segments of this diagram [1, 3] and [4, 10] allow the approximation by the function b = a·pk.

 

If we shift the origin of the coordinates to the point (4, 0) then the initial diagram will look so.

 

Figure 10

 

Here two segments are easily visible. Please note: Plato also takes two sequences of numbers.

Using the method of least squares we get the values of parameters a and p.

 

bk = a·pk , k = 0, 1,... ,6

where:

a = a1 = 20.5839, p = p1 = 1.0673

 

for the group of seven planets (fig. 10).

 

For the group of four planets (k = -3, -2, -1, 0) there will be the same formula, but with the values of parameters

 

a = a2 = 24.006, p = p2 = 1.876

 

The parameter p2 is close to 2, as well as was specified by Plato when he wrote about the geometrical progression (2, 4, 8) with a factor 2.

 

Approximation on the first three planets (k = -3, -2, -1) gives the parameter p2' = 2.109.

 

On the following diagram, the obtained approximation is shown.

 

Figure 11

 

Here the source data are drawn in blue. The average relative error is equal to 2.43%.

 

Taking into account the data of ancient science explained by Plato, and that this dependence remain same today, it is possible to state that during the precession of the planets, the values of parameters p1 and p2 remain constant.

 

The Precession Law of the Planets System

During the precession of the planets of the solar system,
the parameters p1 and p2 remain constant.

 

Parameters p1 and p2 can be named the harmonic constants of the precession.

This law allows us to calculate the parameters of the precession of all planets and the Sun using the known data about the precession of one planet (for example, the Earth).

 

Further, I shall present information which leads to a deeper understanding of parameters p1 and p2.

 

 

Diatonic Solar System

 

"...the people of their island [Laputa]
had their ears adapted to hear the music of the spheres..."
— Jonathan Swift (1667–1745)

 

The theory that planets at their rotation around of the Earth make some sounds differing from each other depending on their size, velocity, and distances, was accepted by the ancient Greeks.

 

Pythagoras and his followers, and later also Kepler believed that the universe is completely organized on a system of the musical harmony. To them everything in the universe is vibrating in tune with the larger things that contain it and we are literally living inside this giant musical instrument, which is playing notes, chords, and scales.

In 1619, Johannes Kepler wrote "Harmonice Mundi" (Harmonies of the World). It was Kepler's attempt to relate a musical harmony to planetary motion and, more specifically, to find a "musically harmonious" relation between the distances of the planets from the sun.

The universe as a monochord
by Robert Fludd, 1627

 

 

The diatonic scale is of ancient origin, but the particular tuning incorporated into modern just intonation is due to Ptolemy (Claudius Ptolemaeus, Greek astronomer, ca. 100 - ca. 170). He gave it as one of a dozen or so possible tunings for the diatonic scale (calling it the "syntonic diatonic").

There are three intervals in the diatonic scale: 9/8 (1.125), 10/9 (1.111), and 16/15 (1.067). The first two are called whole steps and the third a half step (or semitone).

 

As you can see, the parameter p1 = 1.0673, obtained in the chapter "The Precession Law of the Planets System", really coincides with the coefficient (interval) of the diatonic musical scale (semitone)!

 

The intervals mentioned by Plato (which determines the structure of planetary spheres), in music are named: 3/2 - a quint, 4/3 - a quart, 9/8 - a second. And the fraction 256/243 = 1.0535... is the basis of construction of the modern musical scale — the equally tempered scale, created only after 2000 years [2].

Modern music uses an equal tempered scale with the coefficient 122 = 1.0595... The appropriate sound interval is named a semitone. Equal temperament makes the ratio between each semitone a constant. I wrote about it in the book [3].

In 1722, J. S. Bach, having learned of the equal-tempered 12 note per octave scale described in 1691 by the German scientist and musician Andreas Werckmeister (1645-1706), wrote book 1 of The Well Tempered Clavier. It was the first experience of application of the equal tempered scale in the history of music.

 

Thus, if we accept that Mars corresponds to the note 'do' of the 1-st octave, then on the modern musical scale, the planets can be arranged so:

 

Figure 12

 

All of these sound simultaneously as one chord. The accidental symbol — #  (sharp), means on a halftone above. For example, #do is read 'do sharp', etc.

Figure 12 is the musical approximation of the diagram of the galactic latitude of the North Pole of planets (see fig. 8). Still further, the approximations shown in figures 11 and 12 are completely equivalent in their numerical expression.

 

Taking into account the above-stated, it is possible to tell that all 10 "planets" take an interval in 4 octaves on a musical scale.

  • 1-st octave: do — Mars

  • 2-nd octave: re — Neptune

  • 3-rd octave: re — Uranus, #sol — Saturn, la — Pluto, #la — Sun, si — Mercury

  • 4-th octave: #do — Earth, re — Venus, #re — Jupiter

Thus, Pythagorean "harmony" and "the music of the spheres" express the real law of celestial mechanics.

 

What force caused the planets to precess according to the musical scale, and organized the solar system as a well-tuned instrument?

 

 

To be continued (see Part Three)

 

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