The present investigation -- which commenced some
years ago -- led down many paths and embraced a variety of disciplines
the way. It also necessitated a degree of numerical analysis only made
by the availability of powerful personal computers within the last two
or so. More recently a rapid influx of information from the Internet
further impetus, but in chasing down this matter it was also necessary
range widely among the writings of the ancients, the Greek School
(Homer, Hesiod, Ovid, Plato, Aristotle, Archimedes,
Porphyry, Pythagoras, etc.,) and further include the perceptive
of Proclus [410-485 A.D.], certain later scholars (Averroes,
Bradwardine, Campanus, Oresme) and Thomas Taylor
Moreover, in considering the astronomical elements in ancients works it was also deemed necessary to examine the origins of later scientific advances associated in one way or another with these earlier sources. Further, as an important corollary, it was also felt that if one wished to understand the astronomical elements in Plato's Dialogues and related materials, then one should address the works of those who appear to have benefited most from them -- Johannes Kepler [1571-1530 CE] and Galilei Galileo [1564-1642 CE] especially in light of their acknowledged appreciation and utilization of material from Plato's Timæus.
As far as links back to Plato and the Pythagoreans are concerned, readers may judge for themselves whether there was indeed such a thing as "The Doctrine of the Timaeus" and to what degree the neoplatonist Proclus was justified in stating:
That the design of the Platonic Timaeus embraces the whole of physiology, and that it pertains to the theory of the universe, discussing this from the beginning to the end, appears to me to be clearly evident to those who are not entirely illiterate.And also, perhaps, whether Thomas Taylor himself was one more dedicated link in the "golden chain of philosophers" to which he refers in the following:
This sublime theology, though it was scientifically disseminated by Plato, yet conformably to the custom of the most ancient philosophers, was delivered by him synoptically, and in such a way as to be inaccessible to the vulgar; but when, in consequence of the commencement of a degraded and barren period, this theology became corrupted through the negligence and confusion of its votaries, then such of his disciples as happened to live when it was thus degraded and deformed found it necessary to unfold it more fully, in order to prevent its becoming utterly extinct. The men by whom this arduous task was accomplished were the last of the disciples of Plato; men who, though they lived in a base age, possessed a divine genius, and who having happily fathomed the depth of their great master's works, luminously and copiously developed their recondite meaning, and benevolently communicated it in their writings for the general good. From this golden chain of philosophers, as they have been justly called, my elucidations of the present mystic hymns are principally derived: for I know of no other genuine sources, if it be admitted (and it must by every intelligent reader), that the theology of Orpheus is the same as that of Pythagoras and Plato.Although important as well as extensive, the Pythagorean material was still only a single facet of a much wider investigation. Another concerned the precise technical details long-buried in the Babylonian astronomical cuneiform texts of the Seleucid Era [310 BCE - 75 CE] -- information that only surfaced during the latter part of the Nineteenth Century and has yet to see the full light of day even now. Irrespective of how little this neglected corpus of knowledge was regarded, there nonetheless remained the leading question why Babylonian astronomy was so obviously concerned with synodic motion and varying orbital velocity. The first question in fact resulted in the application of the general synodic formula that provided the key to the present discourse (see Section II).
AND QUALIFICATION I
THE GOLDEN SECTION AND THE PHYLLOTACTIC STRUCTURE OF THE SOLAR SYSTEM
It is well known that the arrangement of the leaves in plants may be expressed by very simple series of fractions, all of which are gradual approximations to, or the natural means between 1/2 or 1/3, which two fractions are themselves the maximum and the minimum divergence between two single successive leaves. The normal series of fractions which expresses the various combinations most frequently observed among the leaves of plants is as follows: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55, etc. Now upon comparing this arrangement of the leaves in plants with the revolutions of the members of our solar system, Peirce has discovered the most perfect identity between the fundamental laws which regulate both.1 (Louis Agassiz, ESSAY ON CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127; emphases supplied)
SPIRA SOLARIS: Form and Phyllotaxis
1. phyllotaxes, phyllotaxies
The arrangement of leaves on a plant stem.*Derivative: phyllotactic adj . (Source: ALLWords.com)
* Phyllotactic fractions were applied by Benjamin Peirce to the mean periods of revolution alone; the present dynamic treatment is more complex.
Associated graphics: Solar System Phyllotactic Resonant Triples, Neptune to Mercury
I Bode's Flaw
Bode's "Law" - more correctly the Titius-Bode relationship - was an ad hoc scheme for approximating mean planetary distances that was originated by Johann Titius in 1866 and popularized by Johann Bode in 1871. The " law " later failed in the cases of the outermost planets Neptune and Pluto, but it was flawed from the outset with respect to distances of both MERCURY and EARTH, as Titius was perhaps aware.
II The Alternative
Describes an alternative approach to the structure of the Solar System that employs logarithmic data, orbital velocity, synodic motion, and mean planetary periods in contrast to ad hoc methodology and the use of mean heliocentric distances alone.
The constant of linearity for the resulting planetary framework is the ubiquitous constant Phi known since antiquity. Major departures from the theoretical norm are the ASTEROID BELT, NEPTUNE, and EARTH in a resonant synodic position between VENUS and MARS. Fibonacci/Golden Section Resonances in the Solar System.
Spira Solaris and
the plan-view of the Milky Way.
AND QUALIFICATION II
GOLDEN SECTION SPIRALS IN NATURE, TIME AND PLACE
Notes on the
Logarithmic Spiral (Jay Hambidge; R. C. Archibald)
R.C. Archibald's Golden Bibliography.
The Whirlpool Galaxy (M51) (BW: 100kb)
The Whirlpool Galaxy (M51)(Colour: 200kb)
The Phyllotaxic approach to the structure of the Solar System of Benjamin Pierce (1750)
IVd2c Spira Solaris and
the Pheidian Planorbidae.
Applied to Nautiloid spirals, Ammonites, Snails and Seashells.
The Phedian Planorbidae in Astronomical context; Orbital velocity, Mass and Angular Momentum.
Ammonites and Seashells (Beginning excerpt).
and The Golden Section (Animation I)
and The Golden Section (Animation II)
Appendix: The Matter of Lost
The works of Canon Mosely and Sir D'Arcy Wentworth Thompson.
Laws of Planetary Motion.
Note: This paper (which deals with the resurrection of the Fourth Law of Planetary Motion, i.e., the velocity component) was written north of the 70th parallel during the Summer of 1988.
It was subsequently published in the Journal of the Royal Astronomical Society of Canada (JRASC) the following year. It is reproduced here with the permission of the Editor of the Journal.
Times Series Analysis.
The advent of modern computers permits the investigation of planetary motion on an unprecedented scale.
It is now feasible to treat single events sequentially and apply detailed time-series analyses to the results.
Time Series Graphics
Examples of chaotic and resonant planetary relationships in the Solar System and a possible link with Solar Activity.
Copyright © 1997. John N. Harris, M.A. (CMNS). Revised April 2, 2009.
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