The present investigation -- which commenced some forty-five years ago -- led down many paths and embraced a variety of disciplines along the way. It also necessitated a degree of numerical analysis only made possible by the availability of powerful personal computers within the last two decades or so. More recently a rapid influx of information from the Internet supplied further impetus, but in chasing down this matter it was also necessary to range widely among the writings of the ancients, the Greek School especially (Homer, Hesiod, Ovid, Plato, Aristotle, Archimedes, Plutarch, Porphyry, Pythagoras, etc.,) and further include the perceptive contributions of Proclus [410-485 A.D.], certain later scholars (Averroes, Bradwardine, Campanus, Oresme) and Thomas Taylor [1785-1837] in particular.
   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).

   This said, it should also be emphasized that the Golden Ratio was not a pre-determined goal in the present series of essays. As the first three sections demonstrate, the investigation stems entirely from the rejection of Bode's "law" and the resulting need to develop a more workable approach to the structure of the Solar System. Simply stated, a mathematical problem concerning mean planetary periods results in the determination of a constant of linearity that somewhat surprisingly turns out ( via the quadratic equation: k 2 - k - 1 = 0 ) to be the Golden Ratio: Phi = 1.6180339887949. Moreover, and also as a direct consequence of the applied methodology, the resulting exponential planetary framework and associated spiral is necessary based on the larger constant Phi 2 = 2.6180339887949.

   Thus ends the initial phase of the inquiry, but on examining the historical ramifications of the matter it becomes increasingly apparent that it has long possessed its own inherent set of filters; readers and reckoners may reject the implications or pursue them as they please. But at this juncture it is necessary to re-state that the present work owed its origins to a relatively limited inquiry that started out as a criticism of Bode's Law and only later moved back in time to embrace such esoteric topics as the "Ouroborus" and "Alchemy." Neither were on the initial agenda, nor were they expected to be of any relevance, but they became inextricably interwoven with the general direction of the research nevertheless. Where possible it was preferable to deal with these topics in mathematical terms -- a treatment that some might wish to question, but on the other hand -- as in the case of Alchemy, for example, the resulting mathematical framework works far too well to be easily dismissed -- as indeed will be seen in later Sections.

    What then was Alchemy? Here one can only suggest a partial answer to a most complex question, but remaining with the astronomical and mathematical side of things, it is possible to suggest that the Alchemical transformation of "base metals" into "gold" was predominantly a mathematical one, and that the "gold" in question was in reality the Golden Section, i.e., Phi itself. Moreover, it also seems possible to suggest that the parameters in question were essentially those of the phi-based planetary framework derived in the first three sections of the present work, the complexity of the matter notwithstanding. This is undoubtedly a contentious issue, but it is also a testable one. Why then all the secrecy and "laboratory" diversions in Alchemy? Here again one can only theorize, but both the equiangular spiral and the exponential planetary framework are unequivocally heliocentric, and there can be little doubt that for many centuries it was indeed far more dangerous to promote such heresy than to be suspected of successfully converting base metals into gold. Those unfamiliar with "Alchemy" should perhaps pay a visit to the Alchemy Web Site; but those that do, be fore-warned -- do not to expect to comprehend the material collected there (more than 100 megabytes of texts, complex symbolic representations and scholarly articles, etc.) in a single afternoon, or even a Month of Sundays. Moreover, doubters and newcomers alike should also pause to dwell on what might best be termed "The Newton Corollary", for if the likes of Sir Isaac Newton, an indisputably erudite, highly competent scientist and mathematician became absorbed with Alchemy, what then does that really suggest about the subject itself? Surely that Alchemy was worthy of his attention and interest, as opposed to vague and unsubstantiated theorizings concerning his mental health, etc. As for the mathematical treatment of the astronomical component of Alchemy presented in the following Sections, objectors and nay-sayers are simply asked if the "Ouroborus" and "Alchemy" are not as suggested here, then what are the alternatives?

    In the same vein, it also appears necessary to ask why both Galileo and Kepler made so many references to Plato in their own works, whether Kepler actually discovered the Harmonic Law of Planetary motion, or whether he realized, as Galileo appears to have, that it was provided by Plato in the Republic and also implicit in the Timæus and the Epinomis. Those who object --Temporal Racists especially -- are invited to examine the question in more detail by considering the numbers of the Tyrant and the Bride. But if Plato does not mean the Harmonic Law of Planetary motion (The Square of the Mean period of Revolution equals the cube of the Mean Heliocentric Distance) when he invokes "distance," "squaring," and "cubing" in the Republic, then again, what does he mean? Notwithstanding the crass conceits, limited awareness and negative assertions of temporal racists, the heliocentric concept was undoubtedly known by the time of Aristarchus (ca. 300 BCE) and also almost certainly applied in Babylonian astronomy during the Seleucid Era (for technical details see: Babylonian Planetary Theory and the Heliocentric Concept).

    Lastly, the results of the present research are presented in condensed form in twelve sections with the main numerical and astronomical aspects in the first three. The sections that follow deal largely with the major historical ramifications of the matter, though they also unavoidably contain additional mathematical and astronomical details. Even so, there are a number of avenues left unstated or unexplored -- the topic is simply too vast and too complex for anything else at the present time. Yet the more the inquiry proceeds the more material seems to be applicable, in fact a substantial percentage of the writings of Plato, Aristotle, and many other ancient commentators seem to be of relevance in one way or another. An unlikely thesis at first acquaintance perhaps, but if Proclus was indeed correct in his assessment of the matter, that it specifically "pertains to the theory of the universe, discussing this from the beginning to the end" this would necessarily require an enormous amount of material and instruction. Moreover, although the spiral in question is undoubtedly complex (even in in modern terms) this still does not decisively eliminate the possibility of it being known in earlier times. Another contentious issue to be sure, yet on examination it becomes apparent that there is more to Archimedes [ca. 250 BCE] and his work on spirals than meets the eye; enough in fact to suggest that the spiral was indeed carried forward over the centuries by a perceptive few who understood its meaning and significance in what is essentially the present context.



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.(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

The 1853 Address of Benjamin Peirce

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.

III The Exponential Order
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.



IVd2b Spira Solaris and the 3-Fold Number
The Spiral of Pheidias; Pheidian/Golden Spirals Defined.
Pheidian Spirals and the Chemical Elements.

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).

Whirling Rectangles and The Golden Section (Animation I)

Whirling Rectangles and The Golden Section (Animation II)

Appendix: The Matter of Lost Light.
The works of Canon Mosely and Sir D'Arcy Wentworth Thompson.


Velocity Expansions of the Laws of Planetary Motion.

Kepler's Third Law of planetary motion: T2 = R3 (T = period in years, R = mean distance in astronomical units) may be extended to include the inverse of the mean speed Vi (in units of the inverse of the Earth's mean orbital speed) such that: R = Vi2 and T2 = R3 = Vi6. The first relation - found in Galileo's last major work, the Dialogues Concerning Two New Sciences (1638) - may also be restated and expanded to include relative speed Vr (in units of Earth's mean orbital speed k) and absolute speed Va = kVr. This paper explains the context of Galileo's velocity expansions of the laws of planetary motion and applies these relationships to the parameters of the Solar System. A related "percussive origins" theory of planetary formation is also discussed.

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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