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HORUS VOL II. Issue 2

Planetary Motions, Egyptian Unit Fractions and the Fibonacci Series
(c) 1985 George R. Douglas, Jr.

It was well known among the ancients there were a few bodies that moved about the stars and these were called 'planets' or 'wanderers'.

According to the Soviet Encyclopedia, records of astronomical and meteorological observations stimulated the development of astrology, not until the first millennium B.C., and astronomy, between the fourth and first millennia B.C. The planets were identified and, unlike the fixed stars which were compared to calmly grazing sheep, planets were called 'bibba,' or 'goats.' Each planet was given its own special name, except Mercury which was called bibbu, that is, "planet". Venus was called Dilbat; Jupiter, Mulubabbar ("star-sun"); Mars, Zalbatanu; and Saturn, Kaimanu. It was at that time that the movements of the planets began to be observed. Specifically, texts devoted to the movements of Venus are extant. [1]

An actual period of a planet's revolution around the Sun is called the sidereal period. As seen from the Sun, a planet will again be in the same position relative to the stars after one sidereal period. The time between successive conjunctions of a planet with the Sun, as seen from the Earth, is called a synodic period. Thus,

1/P - 1/E - 1/S = 0, Eq.(1)

where P is the sidereal period of the planet, E is the sidereal period of the Earth and S is the planets synodic period.[2] Equation (1) may be referred to as a synodic equation and it is an equation comprised of unit fractions.

Egyptians were extensively manipulators of unit fractions, fractions which solely possessed a unit numerator, or if not, possessed methods or tables that reduced nonunitary fractions to sets of unit fractions.[3]

Babylonians used fractions which did not necessarily possess a unit numerator and they also had a reciprocal table.[4] Josephus has it that Abraham during his travels to Egypt as is recorded in Genesis brought to Egypt the mathematics of Babylon.[5]

Fibonacci numbers

Additive series sometimes known as Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, 34,... , in which each term is the sum of the preceding two terms, may clearly be said to be known and used in Egypt's Old Kingdom times. Dividing any number in the series by the preceding one gives an approximate value for [phi], 1.6180... . Leonardo Fibonacci is said to have learned this relation of numbers during his travels in Egypt, and he then published them in Italy in 1202. This ratio [phi] is found inherent in the proportions and measurements of Egyptian architecture, such as the Temple of Luxor and the Great Pyramid of Cheops.[6]

Other additive series ought not be ignored, whether in mathematics or the astronomy of the ancients; and William J. Douglas admits the additive series 1, 5, 6, 11, 17, 28, 45, 73,...,[7] and we can infer the additive series 1, 6, 7, 13 .... 225, 364, 589... .[8] There must be more; these can be shown in Table 1.

Fascinatingly, a conclusion reached here is congruous to that of Douglas, that while Babylonians perceived integral fractions essentially throughout, it is tempting to speculate whether the Babylonians also understood the synodic equation and whether they were calculating as well as observing orbital periods and synodic periods. Babylonian mathematics used tables of reciprocals and concerned itself with various types of the mean, including the harmonic mean - all of which appear to bear upon the synodic equation.[9] Babylonians may have become more concerned with operational data than with observational data, particularly if the operational data consisted of integral fractions and the observational data did not: Babylonians should not be faulted for failing to derive wholly accurate observational data.[10]

[*!* Image: Table 1 is a columnar display of various additive series including Fibonacci and Lucas, and in which a row K shows values of |xz-y2| of Equation (5). Inset 1a is an extension of rows 1, 2 and 7 of Table 1.]

Table 1 shows PS-E2 of Equation (1) as K values which suggest various cycles or periods; K values are shown in Table I as 1, 11, 19, 29,.-701,....,[11]

From the Fibonacci series, Douglas interestingly observes that during eight years required for five synodic revolutions, Venus makes thirteen revolutions around the Sun and these numbers are successive terms of one of the additive series shown above. Douglas then introduces these terms into a synodic equation, and if this concept is applied to Equation (1), the generalized equation ought to be

1/P - 1/E - 1/S + (PS-E2)/PES = 0 Eq.(2)

where P,E,S are successive terms of any of the additive series shown in the columns of Table 1. Wall calls for the Metonic cycle of 19 years and two hours, - the period of a cycle in which the phases of the Moon repeat on the same day of the month, discovered by the Greek astronomer Meton in 433 B.C.[12] Then, the synodic equation of Douglas would be modified computing a 19-year period by Equation (2).

1/28 - 1/45 - 1/73 + 19/91,980 = 0 Eq.(3)

and for the Venus/Earth orbital ratio 73/45, Douglas can be modified further in the use of the synodic equation to compute a corresponding 29-year period in view of Table 1 and Equation (2), so this is

1/225 - 1/364 - 1/509 + 29/48,239,100 = 0 Eq.(4)

This study has not extended its analyses to all periods of uniformitarian or catastrophic events, but McCanney[13] cites a Nature article that lists the following ages of worldwide catastrophic events corresponding to mass extinctions, the ages being given in periods of millions of years; namely 1, 13, 25, 35, 58, 63, 135, 181, 230, 280, 345, and 405. McCanney's interpretation is that such events are governed strictly by chance and a random distribution is suggested.

Here, however, the conclusion is proposed that essentially all these periods are on Table 1 where K is 109, 131 or above and where N ranges from 0 to but not in excess of 7, a few of the terms being approximate, which this writer assumes the events listed by McCanney to be.[13] That a million years should be taken as a unit value of K in Table 1 is suggested but is not determined.[14]

Generally, as pointed out by Dwardu Cardona, it is conceded that no fully theoretical explanation of commensurabilities, or harmonic orbits, or orbited resonance phenomena, or any workable Titius-Bode "Law" seems to exist in celestial mechanics,15 and his essay reflects the fervent need in celestial mechanics for an expression of a more simplistic relation.

What seems to be needed is the concept that a harmonic relation be presupposed as follows:

1 1 1 Cyclic function, K

  -  
  -  
  +  

  = 0 (Eq. 5)
[Q Planet's Synodic
Period, x]
[Reference Planet
in years, y]
[Revolution of
Q Planet around
the Sun, z]
[Product], xyz

where x, y and z are ascending and sequential Fibonacci numbers or some other additive series in Table 1.

Three of the above fractions in Equations (5) are unit fractions, the only kind believed to have been used in Egyptian antiquity, and the other is an integral unit fraction, a kind of allowable fraction in the Reciprocity Table of Fractions known to the Babylonians.

Observing that a ratio of two sequential numbers of any of the above series approach a limit of 0, 0.6180 .... or if reciprocated, then 1 + [phi], or 1.6180 .... and responds to the nugget that Plato in his Timaeus considered 0, the ratio 0.6180... to be the key to the physics of the universe.[16]

It is an unanswered question whether Babylonians and Egyptians understood a completed synodic equation, or whether we do today, or whether it was part of a revealed knowledge transmitted from the Chinese or other source. Babylonians used reciprocal fractions and they were aware of binomial sums, and, of course, they were concerned with various types of means and ratios, including the harmonic mean; and they may be seen to bear a likelihood of resemblance to the synodic equation.[17]

To point out inadequacies in the accuracy of derived numerical values determined by modern calculations of [pi], [phi] and other constants and values,[18] over calculations of Babylonian, Egyptian and Greek antiquity may posit a lack of understanding, but if there appears to be a presence of good or superb operational theory of the synodic equation in their application of unit fractions, credit ought to be extended to these ancient mathematicians.

Finally, there is the matter of the peculiar, apparent resonance in Venus' axial rotation with the Earth,[19] or other possible planets[20] and it is an unanswered conclusion why this particularly seems to apply to Venus and none other of the planetary motions.

References:

1. Great Soviet Encyclopedia, 4.9d (3rd ed. 1974).

2. H. Spencer Jones, General Astronomy, 200 (2nd ed., 1934).

3. L. N. H. Bunt, P. S. Jones, and J. D. Bedient, The Historical Roots of Elementary Mathematics, (1976), pp. 1530.

4. Bunt, 45-47, 49, 239.

5. Whiston, in his index under Abraham, lists that Abram. instructs the Egyptians in the mathematical sciences; William Whiston, "The Complete Works of Flavius Josephus," Antiquities of the Jews, Book I, Chapter viii, Section 2, 39, 857 (1858, 1860) and Josephus' Complete Works, 33, 747, (Kregel 1960). Warmington in his translation of Josephus' works (Josephus IV Jewish Antiquities) refers to a note citing Artapanus' (ca. 2nd Cent. B.C.) statement that Abraham migrated with his household to Egypt and taught astrology. Really, the translators may have missed the point in saying Abraham "communicated to them arithmetic, and delivered to them science of astronomy" (Whiston), or "he introduced them to arithmetic and transmitted to them the laws of astronomy" (Warmington). A contextual reading implies that Abraham provided the Egyptians with uniform knowledge and reasonings on arithmetic and astronomy. Respecting Artapanus, the above reference note #1 gives a date later than Abraham for astrology, while an earlier date is provided for astronomy.

6. William James Douglas, "The Pentagram of Venus," HORUS I:1, 1985) pp. 15-21,18.

7. Ibid., 20.

8. Ibid., 18,20; the Babylonian Ninsianna tablets imply that Venus made 313 revolutions around the Sun for 193 revolutions made by the Earth. With a minor adjustment, Douglas simplifies the 313/193 ratio to 73/45, which are numbers of the additive series starting with 1 and 5. Similarly, the 73/45 ratio is converted to 365/225 and the synodic equation is used to compute the corresponding synodic period; however the ratio is more nearly seen derived simply from the numbers of the additive series starting with 1 and 6 and shown in Table 1.

9. Ibid., 20.

10. Ibid., 21.

11.Velikovsky has noted a long-term psychological and behavioral cycle of around 700 years (coinciding with a proposed interval between two great catastrophes) reaching even into medieval times; according to Victor Clube, it certainly does seem to provide a plausible background to widespread messianic predictions around the year 0 of the common era, and the general atmosphere of terror (sic) generated by comets at this time. Victor Clube, "Cometary Catastrophes and the Ideas of Immanuel Velikovsky," Journal of the Society for Interdisciplinary Studies, V:4, (1984), pp. 106-111. Clube, while urging that numerology is dangerous stuff to get too excited about, interestingly notes that Halley's comet and Mars have such an association; further, Clube demonstrates there are several long-term periodicities in the Babylonian records which modern astronomers seek to explain in terms of repeated planetary conjunctions.

12. Alban Wall, "The Book of the Secrets of Enoch," HORUS I:1, (1985),pp. 7-10. Bunt, [see note #3] mentions the relation 1/n - 1/(n+1) - 1/n(n+l) .

13. See Nature, (Nov. 1979), cited in J. M. McCanney, "The Nature and Origin of Comets and the Evolution of Celestial Bodies, Part ll," KRONOS IX:3, (1984). In criticism to reference #7 of McCanney, C. Leroy Ellenberger says that series of twelve numbers cited are not the "ages of world events," but merely the generally accepted horizons for genealogical ages. C. Leroy Ellenberger, "The New Solar System: Selected Criticisms," KRONOS IX:3, (1984).

14. Ibid.

15. Dwardu Cardona, Ejections, Resonances and Inversions, KRONOS X:2, (1985), particularly pp. 6162.

16. Douglas, p. 18

17. Ibid., p. 20.

18. Ibid., pp. 20, 21.

19. Utter of Kirk L. Thompson, Some Comments on "Still Facing Many Problems", KRONOS X:3, (Summer 1985).

20. Letter of Robert A. Bureau, "Planet X and the Sunspot Cycle, KRONOS X:1, (1984).

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