Spira Solaris logo


Jump to Spira Solaris Archytas-Mirabilis
Jump to Spira Solaris Sophia et Sapiens
To Spira Solaris Lux et Tenebris
To Spira Solaris Time and Tide
Oct 1st is Too Late
Madness of Mankind
Alchemical gold
As Above,





SPIRA SOLARIS: Form and Phyllotaxis


Spira Solaris, Top Right Spira Solaris, Top Left

SPIRA SOLARIS: Form and Phyllotaxis Assigned to the main heading SPIRA SOLARIS this January 2007 essay owes its origins to a number of events and issues, but the subtitle itself (Form and Phyllotaxis) arises from the neglected Solar System researches of American mathematician Benjamin Peirce (1809-1880). His investigations and conclusions resurfaced in September 2006 after the elevation of the asteroid Ceres and simultaneous demotion of the planet Pluto to the status of "Dwarf Planets." Essentially,with Pluto thus demoted Neptune once more becomes the outermost planet, as Peirce stipulated on theoretical grounds over 150 years ago. First published in 1850 and later recorded by his friend Louis Agassiz (1807-1873) in the latter's famousEssay on Classification (1857:131) Peirce theorised that "there can be no planet exterior to Neptune, but there may be one interior to Mercury."Revisiting my own attempts to come to terms with the fundamental structure of the Solar System in the first three sections of Spira Solaris Archytas-Mirabilis similarities between the two approaches now suggested that while I had unknowingly followed Peirce by omitting Pluto and adding an inter-Mercurial object, I had missed the point completely when it came to the significance of Neptune. Not so Benjamin Peirce. Nevertheless, although Agassiz provided additional details in his Essay on Classificationconcerning the latter's Solar System research including "the ratios of the laws of phyllotaxis" in this same astronomical context, Peirce's major conclusion was cast aside despite its potential significance. Although not stated in these precise terms, it was nothing less than the perception that

The Solar System is Pheidian in Form1and Phyllotactic2 in Nature
Just how right (or wrong)was Benjamin Peirce concerning the phyllotactic aspect of the Solar System? He was certainly one of the leading scholars of his day, and there is little doubt that he was also an influential scientist in his own right, as the following excerpts from his Scientific Biblography attest:

PEIRCE, BENJAMIN (b. Salem, Mass., 1809; d. Cambridge, Mass., 1880), mathematics, astronomy.
Graduated from Harvard (1829; M.A.,1833), where he was a tutor (1831-33) and professor (1832-80). In mathematics, he amended N. Bowditch's translation of Laplace's Mechanique celeste (1829-39); proved (1832) that there is no odd perfect number with fewer than four prime factors; published popular elementary textbooks; discussed possible systems of multiple algebras in Linear Associative Algebra ... and set forth, in A System of Analytic Mechanics (1855), the principles and methods of that science as a branch of mathematical theory, developed from the idea of the "potential." In astronomy, he studied comets; worked on revision of planetary theory and was the first to compute the perturbing influence of other planets on Neptune; and worked on the mathematics of the rings of Saturn, deducing that they were fluid. From 1852 he worked with the U.S. Coast Survey on longitude determination, ... became head of the survey (1867-74) (and) superintended measurement of the arc of the thirty-ninth parallel in order to join the Atlantic and Pacific systems of triangulation. Influential in founding the Smithsonian Institution and the National Academy of Sciences. (Concise Dictionary of Scientific Bibliography, Charles Scribner, New York 1981:540)

For my own part, I intend to show in this title essay that Benjamin Peirce's phyllotactic approach to the structure of the Solar system was indeed correct, all ramifications and consequences notwithstanding.  In short, the long-delayed legacy of Benjamin Peirce and the policies laid out in his closing remarks to the members of the American Association for the Advancement of Science concerning the founding of the Smithsonian Institution, and not least of all, the honest, unhindered pursuit of knowledge:

"Let us profit by the example of Smithson, and, instructed by the wisdom of this high-minded son of England, learn to confide in our own rulers. Let us be aroused to an earnest and harmonious effort to accomplish the plan proposed by our President at Albany, for the building up of an "institution for science, supplementary to existing institutions, to guide public action in reference to scientific matters." With the details of the plan and the arguments in its favor you are familiar. You know how useful it would be as a protection from the wasteful expenditure upon abortive attempts to reverse the laws of nature. You know how much it is required to sustain the purity and independence of science, even within its own proper domain. You know that in no age or country was there ever a more urgent call for a scientific society, in which scientific influence should predominate, where it should not be smothered by excess of patronage, and whence it should not be liable to banishment through any spirit or form of ostracism. If American genius is not fettered by the chains of necessity, and helplessly exposed to the assaults of envious mediocrity, but is generously nourished in the bosom of liberty, it will joyfully expand its free wings, and soar with the eagle to the conquest of the skies."

Concluding remarks from "The Address of Professor Benjamin Peirce, President of the American Association for the Year 1853, on retiring from the duties of President"
"Printed by Order of the Association" (1853:16-17)

SPIRA SOLARIS: Form and Phyllotaxis

1. Form: Pheidian growth per revolution, etc.,
2. phyllotaxes, phyllotaxies
    The arrangement of leaves on a plant stem.*
Derivative: phyllotacticadj . (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 (PDF: 80kb; also in HTML)

PART I   The Pierce  Planetary Framework (1850) Revisited. (1.272 Mb).
The inclusion of mean synodic periods between adjacent planets completes a stalled 19th Century model of the Solar System developed by American Benjamin Pierce (1809-1880). The model suggests that the Solar System may have been subject to disruptions in the past including the Mars-Jupiter gap, and also that at present Earth occupies an intermediate, synodic location between Venus and Mars, albeit not precisely. (Part I, Figure 1).  The resulting framework also suggests that complex elements of the Fibonacci series, Lucas series and Phi-series may underlie the dynamic structure of the Solar System. As a result, with twice the number of planetary periods now available, the incorporation of these three series and the synodic cycles permits the development of a working planetary framework for general testing.  A search for further enlightenment leads to Babylonian astronomy and a wider ranging series of of inquiries with positive and negative results sufficient enough to merit the inclusion of an optional excursus with Part I if required.
OPTION   Part I plus Historical /mathematical Excursus (1.313 Mb).

PART IIThe Pierce Planetary framework and External Systems. (313 Kb)
The revised framework was applied to a variety of external planetary systems with similarities encountered (e.g., in the structure of HR 8799) explained in part by synodic relations common to the Solar System inherent in the Phi-series. Other similarities between HR 8799 and the SolarSystem include the possible demise of the fifth planet and possible outward shifts by both HR 8799d and HR 8799e to intermediate, synodic locations. A theoretical inward extension for this system results in a period of 0.2405942 years (HR 8799_9) versus 0.24084445 years for the Solar System’s Mercury (planet #8). (Part II, Figures 1 & 2).

PART III Real-time Motions in the Solar System and the Golden Ratio. (3.284 Mb).
Further tests lead to real-time planetary motion in the Solar System with results which confirm pheidian similarities already encountered with the mean periods in Part I including historical aspects explored at length in the optional excursus.  Further concerns regarding possible 795-year cycles for the four major superior planets gives rise to the suspicion that such matters may have significant implications; phyllotaxis in such contexts is also considered. (Part III, Figs.1- 6e).

PART IVThe Fibonacci series, the Lucas series and Platonic Triangles. (477 Kb).
Searching for enlightenment Part IV deals with the two triangles in Plato’s Timaeus, the isosceles and the equilateral triangles. It is shown the former pertains to the Fibonacci series, the latter to the Lucas series and both with respect to the “Rotation of the Elements.” Again the results are pheidian and again apply to similar planetary frameworks. (Part IV, Tables 1 & 2).

PART V Time and Tide: The Spiral Form in Time and Place. (15.756 Mb).
The spiral form in time and place remains pheidian in form and interest while initially concentrating on ammonites, where David Raup’s neglected researches are re-examined and re-applied to ammonites after developing a series of double-precision pheidian test spirals. Later the study widens to include radiolarians and diodoms, but ends with concerns wihich are not so much about the Solar System and exoplanetary systems per se - but ourselves, our past, our present, and our increasingly uncertain future. With this in mind, a gentle, non-destructive option is suggested for those who might wish to embrace it.

( jnh, 21 December 2023 ).

Spira solaris: Archytas-mirabilis


Bode's Flaw Bode's "Law" - more correctly the Titius-Bode relationship - was an ad hoc scheme for approximating mean planetary distances originated by Johann Titius in 1866 and later 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.   Previous version: http://www.spirasolaris.ca/sbb4b.html ]

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.
[ Previous version: http://www.spirasolaris.ca/sbb4c.html ]


IV Spira Solaris Archytas-Mirabilis Most suitably represented in terms of exponential growth and a complex equiangular spiral, the Phi-series based planetary model appears to be new in one sense and yet quite ancient in another. Title graphic (91 kb, best viewed at 1280 x 1024 resolution).
Historical Digressions on the Golden Section

IVd2 Spira Solaris and The Middle Ages Ostensively the translation of Aristotle's De Caelo from medieval latin to French, Nicole Oresme's Le Livre du ciel et du monde [ ca.1375 CE] was more than a translation and a commentary. The numerous references in this work to the insights of the Arab scholar Ibn Rushd [Averroes,1128-1198 CE] lead back to Plato's Republic, Archimedes, Pythagoras, and the Golden Section in early Alchemical contexts.


IVd2b Spira Solaris and the 3-Fold Number "The three-fold number is present in all things whatsoever; nor did we ourselves invent this number, but rather nature teaches it to us." Haliotis Parva and the Golden Rectangle

IVd2c Spira Solaris and the Pheidian Planorbidae Applied to Nautiloid spirals, Ammonites, Snails and Seashells. Appendix: The Matter of Lost Light


IVc Spira Solaris, the Fourth Planet and Fifth Element PROCLUS [ Commentary on the Timaeus of Plato ]: "For through analogy the universe is completely rendered one, this having the power of making things that are divided to be one, of congregating things that are multiplied, and connecting things that are dissipated. Hence theologists surveying the cause of these things in the Gods, enclose Venus with Mars, and surround them with Vulcanian bonds; the difference which is in the world being connected through harmony and friendship. All this complication and connection likewise has Vulcan for its cause, who through demiurgic bonds connects sameness with difference, harmony with discord, and communion with contrariety."

V Spira Solaris.The Chaldean Oracles. Proclus, and Johannes Kepler Examines the role played by the Chaldean Oracles in influencing the Neoplatonist Proclus [410 - 485 CE] and the connection between the latter, The Harmonies of the World, and the Harmonic Law "conceived on the eighth day of the third month in the year 1618" by Johannes Kepler [1571-1630 CE].

VI Spira Solaris and the Universal Ouroboros Explores the historical antecedents and the relationship between the Equiangular Period Spiral, the Ouroboros, Alchemy, and the Sun.

VII Spira Solaris and the Three Parts of the Wisdom of the World The spiral configuration is found on rock art in regions as distant as Nicaragua, Italy, Indonesia, and the United States. It is also found on the Plains of Nazca in Peru, on Malta, in France, in Britain, and especially in Ireland, where a complex triple spiral is illuminated on the winter solstice inside the Neolithic Site at Newgrange. What is the significance of this complex motif, what message was it intended to convey, and where does the inquiry lead?


VII-II Spira Solaris: Where Virtue, Wisdom and Equity are Assembled [ Additional text: Tenebrosity; also in PDF.  Related Graphics: Out of the East (77kb; in full, single-page PDF). Out of the Dark (54 kb)]


Spira Solaris: Time and Tide


Did the Greenland Vikings simply fade away, or was there more to their story and more to the Viking Sagas in addition? It would seem that there was far more. In fact sufficient evidence exists to suggest that the last Vikings triumphed over the Northwest Passage and that the legendary lands of the Viking Sagas - Helluland, Markland and Vinland - are located on the West Coast of North America, not the East. Helluland extending from Etolin Island in Alaska south past the Bella Coola region of British Columbia; the Queen Charlotte Islands (Haida Gwai since 2010) more than meeting the basic requirements for Markland with British Columbia's Cowichan Valley at Duncan in the south-east corner of Vancouver Island providing  the most logical technical fit for Vinland itself.. But this is only part of this complex matter.

Part   1. Viking Press and Viking Ships

Part   2. West by Northwest
Part   3. Three Steps Back 
Part   4. The Nova Groenlandiae Map
Part   5. The Mysterious Akilinik of the Greenlanders
Part   6. Symbols, Markers and Indicators
Part   7. Reflections in the King's Mirror
Part   8. South by Southeast
Part   9. The Copper Canoe
Part 10. The Warp and the Weave
Part 11. Helluland, Markland and Vinland
Part 12. The Golden Apples of the Sun

Maps: Partial Map listing for The Last Viking

Other: Easter Island Stone Structures
Other: Waterstone of the Wild (HTML; also in PDF: 97 kb)

Postscript 1: A Fir Tree of the Mind (PDF: 37 kb)
Postscript 2. RongoRongo and the Raven's Tail
Postscript 3: The MacKenzie Sound Rock Carving Single-page in HTML (or PDF: 167 kb)
Postscript 4: A Dog's Life in the Carolinas Single-page in HTML (or PDF: 80 kb)


Curachmen  "For nearly an hour the men stood waiting to launch the curach. It was blowing fresh from the south-west, with hard squalls of wind and rain; and the breakers came charging in, rank after rank, in unending succession. From time to time the men would retreat before the biggest sea, afterwards shoving their curach well out into the water again. It looked as if the long-awaited deibhil, or lull, would never come..." [ G. J. Marcus. The Conquest of the North Atlantic, Oxford University Press, New York 1981:3-4.] (PDF: 26 kb; also in HTML)


MacKenzie Sound  "The rock carving in this photograph was found at a remote place on the north end of Vancouver Island about 50 miles from Alert Bay by Stephen Lablosky of 12320 Old Yale Road, Surrey. The figures are about 8 inches high and 1/4" deep on the face of a large granite boulder. What interests both he and I about this carving is that it can't be Indian, nor is it feasible that a logger would have the skill to do such work. Because of the religious symbolism and because of the skill necessary to carve in granite, we tend to believe that early Hudson's Bay explorers left it ... but for what reason?" [ Charles Lillard, "Notes and Queries," The Raincoast Chronicles FIRST FIVE, Numbers 2 and 4, Edited by Howard White, 1976, 1994:91,193 ]. (Single-page PDF: 167 kb; also in HTML)


Waterstone of the Wild. "All through the Peruvian and Bolivian Montana is to be found a small bird like a kingfisher, which makes its nest in neat round holes in the rocky escarpments above the river. These holes can plainly be seen, but are not usually accessible, and strangely enough they are found only where the birds are present. I once expressed surprise that they were lucky enough to find nesting-holes conveniently placed for them, and so neatly hollowed out as though with a drill.

    ' They make the holes themselves.'

The words were spoken by a man who had spent a quarter of a century in the forests. ' I've seen how they do it, many a time. I've watched, I have, and seen the birds come to the cliff with leaves of some sort in their beaks, and cling to the rock like woodpeckers to a tree while they rubbed the leaves in a circular motion over the surface. Then they would fly off, and come back with more leaves, and carry on with the rubbing process. After three or four repetitions they dropped the leaves and started pecking at the place with their sharp beaks, and here's the marvelous part, they would soon open out a round hole in the stone. Then off they'd go again, and go through the rubbing process with leaves several times before continuing to peck. It took several days, but finally they had opened out holes deep enough to contain their nests. I've climbed up and taken a look at them, and, believe me, a man couldn't drill a neater hole !' "  [ Brian Fawcett, Exploration Fawcett. The Companion Book Club, London, 1954:105-106 ].  HTML; also in  PDF (88 kb).  If such a bird does exist, what might it be? A hole-making Bolivian Ovenbird perhaps? A "Common miner" (Geositta cunicularia) or a "Puna miner"(geositta punensis)?

From a wider, earlier and more distant perspective there is something else to be considered. Namely, whether the repeated pouring of a "softening" liquid of this sort while drilling could provide an explanation for
"Petrie's Core #7," a slightly conical, cyclindrical residual core drilled from a solid block of stone block by ancient Egyptians that requires modern power tools to emulate.  


For the Record "A spendid cast of the deeply carved face of Wi te Manewha was taken by Lindauer and Sir Walter Buller. His moko was perfect, the lines cut exceptionally deeply, even to the eyelids. It was said that the tattooing was done twice to make deeper markings. The cast was made during the lifetime of the chief. It is of considerable ethnological value because it is the only known instance of an old-time Maori permitting such a thing, since the head was considered sacred." [ An extract from MAORI PAINTINGS: Pictures from the Partridge Collection of Paintings by Gottfried Lindauer. Edited by J. C. Graham, A. H & A.W. Reed, Wellington, 1956]. (HTML; also in PDF: 440 kb)


Should you wander down an increasingly urban pathway along the western bank of British Columbia's Capilano River (the natural boundary separating North and West Vancouver), you may come across a small inscription on a rock by the wayside. Carved with care, it is clearly modern, inscription and text alike.  It says sadly and simply:

 Strength in your weeping,
 Tears that come seeping,
Down the old canyons,
Back to the sea.
From The River by Jean Gowland

Passing under a bridge the path continues along the pavement of a road that leads around a large Mall, where (now separated from the river by a wire and split-board fence) the path curves south again into a small wooded area.  Finally, one eventually reaches the rivermouth and following the shoreline to the west, a small bay bounded by a pair of rocky outcroppings. Augmented and strengthened, these too are modern, and often visited for they extend outwards from the beach area of West Vancouver's Ambleside Park. At the end of the largest outcropping, however, there is another recent inscription, this time on a bronze bi-lingual plaque. But here the text is accompanied by something else, namely a large wooden figure looking out over the Inlet towards the southwest with both arms outstretched. This is the Coast Salish Welcome Figure erected by the Squamish First Nation to commemorate the first "Gathering of Ocean Canoes" in the summer of 2001. 
For this is their land; this is their will, and this is their generous welcome.



A Witnessing "Yet it was in one way the most dramatic dance of the whole night, for the drummers came down out of the bleachers to join those on the floor and make an avenue of sound through which the old dancer progressed, with hundreds of voices shouting out her song, and her attendants scattering handfuls of coins among the drummers and the singers. It was obviously a farewell, for we felt no doubt that this was the old woman's last dance, and that she and everyone else knew it. But it was also the kind of assertion of continuity, for here was a person who had been a child in the last flourishing of the old native culture, and by supporting her in her dance the rest of the people were not only proclaiming their continuity with the past but also celebrating the revival of the old ways." [A Witnessing: The concluding chapter of George Woodcock's PEOPLES OF THE COAST : The Indians of the Pacific Northwest, Hurtig Publishers, Edmonton, 1977:209-214.] (HTML; also in PDF: 56kb)


Quantum Entanglement: "I hoisted the anchor and arrived at Barney's to find him collapsed and hemorrhaging. I took him in my boat to the nearest settlement, leaving the two starfish in a bucket of sea water in the shack. Barney later died in the Prince Rupert hospital.  When I got back to Kettle Inlet, the starfish were dead. The water should have been changed every twenty-four hours or oftener. The years have passed and I've never related this story for fear that I might become a candidate for the funny house. Quite recently I read an article similar in regard to plants which gave me the courage to contribute my story at this time." (C.H. Doane [ "The Starfish Telegraph," The Raincoast Chronicles SIX/TEN, Number 10. Edited by Howard White, 1976, 1994:237 ]. Single-page PDF: 29 kb; also in HTML)


The Times and Tides of John Davis: Selections from The Voyages and Works of John Davis the Navigator (1880). [ PDF: One page Index (as below) ]
The original 1880 publication of The Voyages and Works of John Davis the Navigator, more often than not restricted to Special Collections and Rare Book Departments, etc., is currently available on the Internet as a large, graphics-based GoogleDigital PDF file. Because of its sheer size, however, the various topics in this 515-page volume may still tend to overwhelm the casual reader, especially in this particular format. For this reason, selected segments from the above are provided here in smaller, searchable PDF files that retain the original antiquated English texts, typesetting and pagination. The choice of material, however, does not center on the voyages of John Davis per se, but on his use of the heliocentric concept and the impact of this understanding on the development of Elizabethan navigation in general. Thus, in addition to The Worlde's Hydrographical Discription (1595) and Seaman's Secrets (1594,1607) by John Davis himself the selection includes the  "Note on the New Map" of A. D. 1600 by Charles Henry Coote and an extensive Bibliographical List of Works on Navigation during the Reign of Elizabeth prepared by Albert Hastings Markham. Lastly, an additional supplement: Introduction, Symbols and Abbreviations, and a Short Bibliography to Copernicus and Kepler (1952) by Charles Glenn Wallis has also been included to provide a necessary background for the opposing geocentric and heliocentric planetary theories of the time[ Sources: John Davis, Albert.Hastings Markham and Charles Henry Coote in The Voyages and Works of John Davis, the Navigator, The Hakluyt Society, No. LIX, London, 1880.  Charles Glenn Wallis in Great Books of the Western World 16, Editor-in-Chief Robert Maynard Hutchison, William Benton, Chicago 1952:481-495. ]

The Worlde's Hydrographical Discription  (PDF: 256 kb)
Seaman's Secrets : Book I . (Degrees, Times, and Tides, etc. PDF: 1.33 Mb)
Seaman's Secrets : Book II. (The Globe and Other Instruments. PDF: 1.49 Mb)

John Davis the Navigator [Introduction plus Note and 3rd Voyage(1587). PDF: 664 kb]
Letters of Marque and Other Matters
(PDF: 61 kb)
Bibliographical List of Works on Navigation
(PDF: 215 kb)

"Note on the New Map" of A. D. 1600 (PDF: 113 kb)

"Introduction to Copernicus and Kepler, etc." (PDF: 147 kb)

The World Encompassed by Sir Francis Drake "Collected out of the Notes of Master Francis Fletcher, Preacher in this employment, compared with divers others notes that went in the same VOYAGE.  Printed at London for Nicholas Bourne, dwelling at the south entrance of the Royal Exchange, 1652." (PDF: 856 kb)


" Now that there is no longer a western frontier, we are slowly recognizing the American "melting pot" and the Canadian "mosaic" for what they are, stewpan and collage. Also, we are learning our ancestors rode roughshod over the land with little consideration for the future. They took the land, built their towns and villages, and then, knowing they had civilized a new world, died. What we are no longer sure of is our own place in this new world of theirs. The question asked by Margaret Atwood in Survival belongs to all of us: "what do you do for a past if you are white, relatively new to a continent, and rootless?" Personally, I doubt anyone will answer Atwood's question satisfactorily. The wording is too exquisitely subjective, every answer leads to a new question. A maze this large and confusing suggests we no longer believe in the Noble Pioneer, this creature has gone the way of the Noble Democrat and the Noble Savage - older myths created by earlier North American historians. If this is so, is it not possible that the history we have learned from our text books, this history we call ours, may be ambiguous? [ Charles Lillard: Introduction to WARRIORS OF THE NORTH PACIIFIC: Missionary Accounts of the Northwest Coast, the Skeena and Stikine Rivers and the Klondike, 1829–1900, Edited and Annotated by Charles Lillard, Sono Nis Press, Victoria, B.C. 1984:10. PDF: 124 Kb; also in HTML ] 



Spira Solaris: Sophia et Sapiens


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; also in PDF.


Time Series Graphics Examples of chaotic and resonant planetary relationships in the Solar System and a possible link with Solar Activity.


The Tyrant and the Bride The Number of the Tyrant is 9. What is the Number of the Bride? Plato's Republic is an enduring and much admired work, but the problems presented here have still confuted hundreds, if not thousands of inquiring minds since Plato's time [427-347 BC]. So be forewarned, this is not a simple matter; nor is it a matter of simple arithmetic, either. Clues abound everywhere, but watch for phantoms and misdirections nevertheless. (HTML). March 2019 expanded version in PDF: 561 kb)


Pythagorean Fragments Selections from the Doxographers, Fragments of Philolaus, Fragments of Archytas, Metaphysical and Political Fragments. (HTML; also in PDF: 204 kb)


Time and Eternity II (HTML; also in PDF: 48 kb)

"Time is the moving image of Eternity,
Plato remarked among the Stars.
Eternity is the sudden wholeness of Time,
Apollo answers amid the Flowers."

The first and concluding chapters from The Origins of Alchemy in Graeco-Roman Egypt by Jack Lindsay, Ebenezer Baylis and Son, Trinity Press, London 1970 ( PDF: 230 kb).


The Star of Bethlehem and Babylon From a technical viewpoint the detection of the faint but unquestionably visible planet URANUS by Babylonian astronomers in 9 BC provides one of the more logical mechanistic explanations for the phenomenon. A non-denominational discourse for open minds.


Babylonian Mathematics and Sexagesimal Notation Base-60 has its advantages, e.g., the Maya relationship of 81 synodic months = 2392 days results in a decimal value of 29.530864... days for the mean synodic month. In Base-60 it is exactly 29;31,51,6,40 days (the modern estimate is 29;31,50,7,30 days) as opposed to the rounded Babylonian estimate of 29;31,50,8,20 days. The latter, allied with the Babylonian mean sidereal month of 27;19,18 days produces an excellent value for the sidereal year (essentially the heliocentric motion of Earth in this context) of 365.25647 days (the modern estimate is 365.25636 days). For more concerning this calculation and additional lunisolar cycles see the limited introduction to "Babylonian Planetary Theory and the Heliocentric Concept" below.

At present a number of oddities remain in the mathematical cuneiform texts of the Old Babylonian Period (1900 BCE-1650 BCE). Included here is an identical yet apparently redundant operation in procedures dealing with the determinaton of the cube roots of the integer square numbers: 27, 64, 125 and 216 (Kazuo Muroi, "Extraction of Cube Roots in Babylonian Mathematics," Centaurus, Vol. 31,1989:181-188; see also Abraham Sachs, "Babylonian Mathematical Texts II-III," Journal of Cuneiform Studies, Vol. 6, No, 4, 1952:151-155; VAT 8547 and YBC 6295). Either way there is something to be said for repetition, especially concerning squares and cubes, with not only tables of both attested for this earlier period, but also tables of fractional exponents and (apparently), even logarithms (MLC 2078, Neugebauer and Sachs, Mathematical Cuneiform Texts, New Haven, 1945:33-34). At the other end of the scale is
"Plimpton 322," an exceedingly complex Old Babylonian text that "tabulates the answers to a problem containing Pythagorean numbers (or Pythagorean triangles)" considered to be "the oldest preserved document in number theory." (Neugebauer and Sachs, Mathematical Cuneiform Texts,1945:38-41).

JORAN FRIBERG (2005, 2006, 2007)
In contrast to the limited selection mentioned above the following wide-ranging collations and commentaries
on Babylonian mathematics provide not only additional material, but also a major expansion of the historical, cultural and scientific complexities that attend this matter.

Friberg, Joran. Amazing Traces of a Babylonian Origin in Greek Mathematics (World Scientific Publishing, Singapore, 2007).
Friberg, Joran. A Remarkable Collection of Babylonian Mathematical texts (Springer, New York. 2006).
Friberg, Joran.
Unexpected Links between Egyptian and Babylonian Mathematics (World Scientific Publishing, Singapore, 2005).

Babylonian Mathematics from The Open University (2011)


Babylonian Planetary Theory
Over the past century or so the parameters and methods of Babylonian planetary theory have been gradually assembled from the analysis of astronomical cuneiform texts from the Seleucid Era (310 BCE-75 CE). Unfortunately, such texts do not belong to a carefully preserved, well-organized library replete with a detailed description of the underlying theory. Nevertheless the available cuneiform texts - often damaged and scattered - have yielded perhaps an end product in the form of ephemerides for the Sun, Moon and the Planets and a collection of "procedure" texts that describe how the mean parameters and varying motions were determined and applied. Although not fully explained in modern terms the latter have been described in considerable detail, notably in Otto Neugebauer's largely unreadable Astronomical Cuneiform Texts (Lund Humphreys, London, 1955); in related sections of the latter's A History of Ancient Mathematical Astronomy (1975); in Science Awakening II (Bartel van der Wearden, 1978); and more recently in The Babylonian Theory of the Planets by Noel Swerdlow (1998). Still difficult to comprehend, further information has been published in various Journals, especially The Journal of Cuneiform Studies (JCS) with additional contributions by Otto Neugebauer, Abraham J. Sachs, Asger Aaboe and others. 
Beyond such specialised attention the subject of Babylonian astronomy seems to have been of little interest in this modern age, largely (one might suggest) because researchers have consistantly treated the subject from a non-fictive, non-model viewpoint. Given the limitations posed by partial recovery and lack of historical guidance this is perhaps understandable, but whether this negative viewpoint should have persisted so long is another matter. Against this complaint a number of reasons can be offered to explain the apparent stagnation, including an unduly cautious approach influenced in part by the terminology and the unfamiliar methodology adopted by the Babylonians themselves. And also, perhaps, a general unwillingness to apply "Occam's Razor" to the impressive simplicity of the Babylonian methodology reinforced (consciously or unconsciously)  by "Temporal Racism," i.e., premature dismissal based on the belief that "They (whomever) could not know that, or do this (whatever) in those days (whenever)." 
But this said, it also seems likely that at least part of the current stagnation arose from insufficient "prior analytics," including failure to consider the various complexities and attendant sidereal components inherent in the Babylonian treatment of mean and varying synodic motion. Moreover, the fundamental parameters of time and motion applied in Babylonian planetary theory were also (it would seem) not fully defined in either modern or ancient contexts.
Why was this material not taken more seriously? Perhaps the unusual methodology, unfamiliar terminology, and - not least of all - the sexagesimal system applied well beyond modern usage. Thus not simply the "minute" and the "second," but sequentially on down through the "thirds", "fourths", "fifths" and "sixths," etc., and not only for units of time and degrees of motion. The general application of base-60 is not particularly difficult, just unfamiliar territory at first acquaintance, but in the pre-computer era of the twentieth century it was likely still somewhat difficult to handle in this specific context. On the other hand, Babylonian use of 360 degrees for revolutions, rotations and subdivisions of the same causes no difficulty since all have been retained to the present day, albeit occasionally defaulting to decimal values at the lower end (e.g., time-keeping, latitudes and longitudes, etc.)

Babylonian concerns with varying motion seem to have followed from the determination of accurate mean values for all four types of month. Thus 29;31,50,8,20 days for the Mean synodic month, 27;33,16,20 days for the Anomalistic month; 27;19,18 days for the Mean sidereal month and 27;12,43,56 days for the Draconic month. It is not thought that the Babylonians possessed a value for the Tropical month, but nevertheless additional values inherent in the Babylonian material can be suggested.
More generally, a year of 365.256469 days (essentially the heliocentric motion of Earth) is readily obtained from
the application of the general synodic formula (the product of two periods divided by their difference) to the Babylonian mean synodic month (MSM) and mean sidereal month (MSID), i.e, the relation (MSM*MSID)/(MSM-MSID). In identical manner the Babylonian Mean synodic month and Draconic month yields an eclipse cycle of 346.616572 days (modern estimate: 346.62005 days). Finally, in the same way the Babylonian Anomalistic month produces a Babylonian anomalistic cycle of 411.78054 days (modern: 411.78387 days). Although far more can be said concerning Babylonian methodology and parameters in this context the details are at present best left to discussions concerning the Babylonian Lunar texts. In short, however, in addition to the possible availability of the above set as well as both mean and varying values, Babylonian interest in the various complex motions of the Moon and the 18.6 year "Saros" cycle is generally accepted, but perhaps still not fully understood.

There seems little doubt that fundamental period relations played a pivotal role in Babylonian planetary theory, and also, that these periods could have been determined in perhaps a century or less of continued observation. It could also have taken considereably longer, but nevertheless ample time exists from the beginning of the Old Babylonian Period (1900 BCE-1650 BCE) down to the much later Seleucid Era (310 BCE-75 CE). Moreover, although no comparable astronomical texts from the earlier period have been recovered, there is nothing of a mathematical nature lacking in the Old Babylonian Period. As it is now - perhaps more a function of partial recovery than anything else - it is the material in the astronomical cuneiform texts from the latest era that contain the methodology and the details applied so successfully in Babylonian astronomy.
For the superior planets Mars, Jupiter and Saturn It is known that Babylonian astronomers employed disarmingly simple "Long" integer period relations derived from pairs of near-integer periods with corrections of opposite sign to obtain an integer period in years to which corresponded an integer number of revolutions (sidereal periods) and corresponding numbers of synodic periods. Thus, as stated in Section 1 of No. 813, a Babylonian procedure text for Jupiter (Neugebauer, Astronomical Cuneiform Texts, 1955:402, translation by A. Sachs) with decimal values substituted for clarity the following instructions and results for this planet:

    "Compute(?) for the whole zodiac (or for each sign) the according to the day and the velocity.
    in 12 years you add 4;10, in 71 (years) you subtract 5, in 427 (years) longitude (returns) to its (original) longitude."

provide the following fundamental relationships for the determination of the mean synodic arcs from the division of the total sidereal motion by the total numbers of synodic events (thus both the mean synodic arc and also the mean synodic period):

JUPITER:  427 Years, 36 Sidereal  Revolutions(36 x 360 degrees) and 391 Mean  Synodic Arcs .
SATURN:  265 Years, 9 Sidereal Revolutions (9 x 360 degrees) and 256 Mean Synodic Arcs.
MARS:  284 Years, 151 Sidereal  Revolutions (151 x 360 degrees) and 133 Mean Synodic Arcs.

Needless to say, if required, the mean sidereal arc could just as easily be obtained from the division of the total sidereal motion by the corresponding numbers of revolutions in all three cases.
The simplicity of these period relations belies their accuracy. For example, from the 284-year period of Mars the period of revolution is simply 284/151 = 1.880794 years (modern estimate: 1.880744 years), an impressive result given the speed and proximity of Mars to Earth and the eccentricity of the former in particular. Moreover, this favourable comparison was obtained without defining the length of the Babylonian year. This central parameter is, unfortunately, a further source of complexity since it involves the motion of the moon and consequently a year expressed in mean synodic months applied in planetary contexts. There are better estimates for the year utilizing this parameter and there also exist superior alternative derivations. Nevertheless, although a little on the high side a year of
12;22,8 mean synodic months = 365.2606377 days was chosen for its mathematical convenience. In passing it may be noted that from the above period relation for Jupiter the mean synodic period obtained from the division of the long period of 427 years and the total number of synodic arcs (391) yields a value of 427/391 = 1.0920716 years. With a year of 12;22,8 months the Babylonian rounded value for this period is given as 13;30,27,46 months in Section 1 of Jupiter procedure text No. 812 (Astronomical Cuneiform Texts, 1955:392-393), and though not fully completed in the text, 13;30,27,46,16,40 months in the second section to provide an alternative approach for both the mean and varying motions of this planet. Similar procedures were adopted for Saturn and Mars using the same length of year, again simplicity in a practical sense, yet on a more complex level, perhaps also resonance-related concerns embracing both the Moon and the annual/diurnal motions of Earth itself.

How were the planets tracked over "the whole zodiac ... according to the day and the velocity"? As far as the above derivation may be concerned, although unusual at first acquaintance, Babylonian astronomers employed five "characteristic (synodic) phenomena" to account for the motions of the superior planets Mars, Jupiter and Saturn as observed from Earth, thus the apparent synodic motion of the planets in the night sky. Of the five Babylonian phenomena only "opposition" remains with us today to any extent, whereas the other four phenomena are additional defining intervals for synodic movements that necessarily incorporate the lap-cycles of swifter-moving Earth. All of which in practice become further complicated by the elliptical orbits and varying orbital velocities of not only the planets in question, but also those of Earth itself. Nevertheless, this too was fully addressed, again with impressive simplicity.
Precisely how, when, and where the Babylonian approach to planetary motion first originated remains unknown, but on examination it can seen that their inquiry was both logical and supremely sensible, commencing (perhaps) with the various motions of the two major luminaries (the Sun and the Moon), the development of units of time and measure in this context, and further (likely concurrent) investigation of the five visible planets known in Antiquity.Thus over the years, after the determination of the mean motions and discernment of variations in position and velocity, the establishment of the extremal synodic arcs (the maxima and minima of Babylonian Systems A and B along with their periodic times).
In short, the establishment of a comprehensive framework for further refinement of the synodic cycles with orbital variations factored in for both Earth and the superior planets. Thus the eventual ability to account for variations in repetitive synodic phenomena throughout the entire 360-degree zodiacal round
apparently observed against reference frames provided by the local horizon and some 33 "Normal" stars distributed around the plane of the ecliptic. Historically correct or otherwise, on examination there can be little doubt that the Babylonian characteristic phenomena so used are best understood in terms of a fictive, heliocentric planetary model that incorporates the dual west-to-east movements of Earth. Thus (1): the west-to-east diurnal rotation of Earth about its own axis, and (2) the likewise orbital revolution of Earth and the planets around the center of the Solar System with time variously defined (as done today) by seconds, minutes, hours, days, months and years.

Just how successful was the Babylonian approach to the motions of the Planets, Sun and Moon? This has yet to be fully established. However, although not immediately obvious, the precise determination of mean orbital velocities followed by accurate extremal limits and lines of apsides have sidereal equivalents that provide room for both speculation and further expansion.
In any event, there would seem to have been ample time available for the development of an underlying planetary theory by the Seleucid Era (310 BCE-75 CE), based perhaps on data obtained from an extensive Babylonian observation program known to have extended from approximately 700 BCE into the 1st Century of the present era. For details see the links immediately below.

INTRODUCTION to ASTRONOMICAL DIARIES AND RELATED TEXTS FROM BABYLONIA (by Abraham J. Sachs and Herman Hunger, 1984) with added notes. Updated on  31 August 2021.(PDF:1.08 Mb kb).
Babylonian "Normal" Stars and the Zodiac  plus standard zodiacal signs and dual names (graphic). Single-page reference in PDF: 129  kb.).

INTRODUCTION plus VAT4956 The Introduction plus a short astronomical diary for the year 567 BCE. (PDF: 285 kb)

Asger Aaboe, Journal of Cuneiform Studies, Volune 18, 1964:31-34. 
(Single-page PDF:  575 kb)

The unexplained trapezoid in two Babylonian astronomical cuneiform texts for JUPITER from the Seleucid Era (310 BCE-75 CE) Partial analysis by Otto Neugebauer (Astronomical Cuneiform Texts, 1955:405,430-31; single-page PDF, 34 kb)

A modern, far-reaching analysis of the trapezoid in ACT 817, ACT 813 and new related Babylonian texts by Mathieu Ossendrijver (2016).
"Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph."
This is a complex treatment of a complex matter, and also, since it is a fictive approach to Jupiter's motion, a major step forward, especially since the data used here has an almost certain heliocentric component.


BY J. N. HARRIS.  Cape Parry, N. W. T. (1989).

Journal of the Royal Astronomical Society of Canada,
Volume 83, No. 3, pp. 207-218, June 1989.

"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

The alerting material for the above paper was a critical dialogue between "Sagredo" and "Salviati" in Galileo's Dialogues Concerning Two New Sciences (1638):

"Sagredo. Allow me, please, to interrupt in order that I may point out the beautiful agreement between this thought of the Author [Galileo] and the views of Plato concerning the origin of the various uniform speeds with which the heavenly bodies revolve. The latter chanced upon the idea that a body could not pass from rest to any given speed and maintain it uniformly except by passing through all the degrees of speed intermediate between the given speed and rest. Plato thought that God, after having created the heavenly bodies, assigned them the proper and uniform speeds with which they were forever to revolve; and that He made them start from rest and move over definite distances under a natural and rectilinear acceleration such as governs the motion of terrestrial bodies. He added that once these bodies had gained their proper and permanent speed, their rectilinear motion was converted into a circular one, the only motion capable of maintaining uniformity, a motion in which the body revolves without either receding from or approaching its desired goal. This conception is truly worthy of Plato; and it is to be all the more highly prized since its underlying principles remained hidden until discovered by our Author who removed from them the mask and poetical dress and set forth the idea in correct historical perspective. In view of the fact that astronomical science furnishes us such complete information concerning the size of the planetary orbits, the distances of these bodies from their centers of revolution, and their velocities, I cannot help thinking that our Author (to whom this idea of Plato was not unknown) had some curiosity to discover whether or not a definite "sublimity" might be assigned to each planet, such that, if it were to start from rest at this particular height and to fall with naturally accelerated motion along a straight line, and were later to change the speed thus acquired into uniform motion, the size of its orbit and its period of revolution would be those actually observed.

Salviati. I think I remember his having told me that he once made the computation and found a satisfactory correspondence with observation. But he did not wish to speak of it, lest in view of the odium which his many new discoveries had already brought upon him, this might be adding fuel to the fire. But if any one desires such information he can obtain it for himself from the theory set forth in the present treatment."

(Dialogues Concerning Two New Sciences published in 1638 (translation by Henry Crew and Alfonso de Salvio, Fourth Day,1914:261-262; emphases supplied)

As indeed one can, once alerted to the primary source, the materials and the methodology provided by Galileo in the following condensed paper published in 1989 in the
Journal of the Royal Astronomical Society of Canada reproduced here wth the permisson of the Editor
Velocity Expansions of the Laws of Planetary Motion [HTML] or PDF (475 kb).

Galilei Galileo's Dialogues Concerning Two New Sciences (1638) translated by Henry Crew and Alfonso de Salvio (1914).
INTRODUCTION Title Page, Contents, Transcriber's Notes (added) Translator's Preface, Introduction, Galileo's Dedication, Publisher's Remarks and Index. PDF (929 kb).
FIRST DAY "Treating of the resistence which solid bodies offer to fracture."  PDF (938 kb).
SECOND DAY "Concerning the cause of cohesion."  PDF (944 kb) .
THIRD DAY "Second new science, treating of motion (movemeti locali), Uniform motion, Naturally accelerated motion."  PDF (1.2 mb).                                                  
FOURTH DAY concerning:"Violent motions. Projectiles." (Includes the reference to PLATO in the above astronomical context). PDF (805 kb).
ADDED ( or "FIFTH" ? ) DAY
from the Stillman Drake translation of the Two New Sciences (1974). PDF (175 kb).
APPENDIX to the FOURTH DAY also included by Stillman Drake (1974). PDF (385 kb).

May 20, 2011 Update: With slight modification Galileo's "percussive" theory might also be applied to the origins of recently discovered "rogue planets" thought at present to be roaming the universe in surprising large numbers.


Spira Solaris: Lux et Tenebris


Alchemy is a subject of which we have all heard of, and which is yet a mystery. In some ways it belongs to the world of mystery stories from the past. We remember perhaps the shade and wonderful light of Rembrandt's etching of the Alchemist, or the stories we read in Chaucer or Ben Jonson. Was the alchemist a philosopher, deep in a mysterious study where he discovered the secrets of transmutation? Was he just a charlatan involved in a particularly fantastic kind of mumbo-jumbo? Was he simply a forerunner of the scientists of today? Or had he an occult knowledge which we cannot hope to acquire in our materialistic environment? [ Introduction to The Arts of the Alchemists, C.A. Burland, 1986-1; for more on the scope and complexity of this subject see: Timeline of Alchemical Books and Various Alchemical Texts from the extensive collection of material available at the The Alchemy Web Site ]


Chapter 19 " You may have heard me called an atheist, but that's not quite true. Atheism is unprovable, so uninteresting... My field of interest is the psychopathology known as Religion .... Lucretius hit it on the nail when he said that religion was the by-product of fear - a reaction to a mysterious and often hostile universe. For much of human prehistory, it may have been a necessary evil - but why was it so much more evil than necessary - and why did it survive when it was no longer necessary? "[ condensed from Chapter 19: "The Madness of Mankind," 3001: The Final Odyssey by Sir Arthur C. Clarke, 1997:136-142; PDF: 27 kb ].


Grave e Mesto " The time for departure came. We all agreed that delay would be bad. I took one last look around. There was the electronic box, the thing I had come to think of as a piano ... I had a strong urge to play on it for one last time. I told the others, saying I would prefer to be alone, that I would follow in a few minutes. Melea answered: ' Don't be too long. There isn't much time.' I began to play. I realized that only in music could I find the answer I was seeking to the questions of the previous evening. Argument I could follow, it weighed with me, yet I could decide nothing from it. I did not know exactly what the music was, it was an improvisation not so much on a musical theme as on the agony of the destiny of man. I continued to play on and on, aware at last that I had made my commitment. I was playing the Schubert Andantino when Melea returned." [ The end of Chapter 14, October the First is Too Late by Sir Fred Hoyle, 1968:158-172; PDF: 57 kb ]


Man and the Sun " In following the solar cycle of this book, I have honoured those who worshipped the Sun God in his many forms. Yet I have also honoured those scientists whose probing minds have dispelled the simple divinity of the star. The members of the Holy Office were right to be fearful of the ideas of Copernicus to see that they would lead to the destruction of many of the old religious forms. They were wrong as well as ridiculous trying to turn back the tide of science, of man's efforts to comprehend the physical universe, for that pursuit is a part of what is divine in humanity. We have to honour both the King of Heaven and Prometheus. The present peril and despair of humanity show that we cannot live without religious meaning although we may do without religious institutions. (The time may come when even those few who still follow them turn against priests who in gem-encrusted copes and mitres, serve Him who taught poverty and humility, who betray Him who taught love of one enemy by raising no murmur against a holocaust of hate.) If we cannot find god in the world, we lose Him in ourselves and become contemptible in our own eyes. We become mere statistics. For this is the greatest evil coming from the unbalanced Apollonian mind. Science has won power over the universe of matter by breaking down and down, by numbering and measuring. So at last everything that cannot be broken down, numbered and measured must be deemed not to exist. Science is uniting man with the sun in a totality of energy and matter. That is communion at the lowest level of being. But we have always been right to seek it also at the highest." [ Quotation from the concluding chapter of Man and the Sun by Jacquetta Hawkes, 1962:239-241. Also in PDF ]



The Hinges of History (1995). "As we, the people of the First World, the Romans of the twentieth century, look out across our Earth, we see some signs for hope, many more for despair. Technology proceeds apace, delivering the marvels that knit our world together-- the conquering of diseases that plagued every age but ours and the consequent lowering of mortality rates, revolutions in crop yields that continue to feed expanding populations, the contemplated "information highway" that will soon enable all of us to retrieve information and communicate with one another in ways so instant and complete that they would dazzle those who built the Roman roads, the first great information system." [ Rationale and closing paragraphs from HOW THE IRISH SAVED CIVILIZATION: The Untold Story of Ireland’s Heroic Role from the Fall of Rome to the Rise of Medieval Europe by Thomas Cahill, 1995: 216-218PDF:  3 pages, 75 kb .


"For over forty years Alexander Thorn, Emeritus Professor of Engineering Science and Emeritus Fellow of Brasenose College, Oxford, has surveyed and planned stone circles and other prehistoric settings of standing stones in Great Britain and Brittany. He was assisted by his son, Dr. Archibald Stevenson Thorn not only in the surveying but also in the preparation of many of the subsequently published papers. His standards of planning have been far higher than those of the average archaeologist and certainly superior to most of the 19th century antiquarians whose plans of circles were often inaccurate and slipshod. It has been the misfortune of those interested in megalithic rings that often only such inexact plans have been available. For a good plan three criteria apply. The survey itself must have been precisely done, using adequate equipment, measuring to several points around each stone at ground level; an accurate scale must be provided on the drawn plan; and True North must be shown, preferably towards the top of the page. Alexander Thom's plans fulfil all these conditions and they must be regarded as the finest and largest collection of stone circle plans ever assembled by an individual. Yet many of them have never been published. Others have been so reduced that their value has been diminished. Consequently, all his plans of British circles have been gathered together here, nearly always one to a page with an accompanying text opposite giving information about the site. The only omissions, at Thom's request, are the Callanish rings, recently surveyed by the Geography Department of Glasgow University, and Stenness of which a plan has been published in the Proceedings of the Society of Antiquaries of Scotland 107, 1978. The resulting collection of plans should be invaluable to anyone wishing to study stone circles.  During his researches Professor Thorn concluded that the prehistoric builders of these rings used an almost universal unit of measurement, the Megalithic Yard of 2.72 ft or 0.83 metres; sometimes laid out non-circular rings in flattened circles, ellipses, egg-shapes and compound figures whose construction was based on the understanding and use of right-angled triangles with integral sides; and had an empirical knowledge of astronomical phenomena. The plans usually include the geometry Thorn has deduced for the sites, and the accompanying notes contain Thom's conclusions about the lengths of the diameters and sometimes the astronomy, if any, for each ring. These ideas remain contentious but this book is not concerned with the controversy. None of it detracts from the excellence of the plans, over 200 of them, which will be indispensable for anyone researching the British stone circles.[ Aubrey Burl, introduction to Megalithic Rings: Plans and Data for 229 monuments, by A. & A.S. Thom, collated, with archaeological notes by Aubrey Burl, BAR International Series 81, Oxford, 1980. ].

Alexander Thom's statistical methodology (including Broadbent's Criterion), background materials, analyses and far-reaching conclusions were published in Megalithic Sites in Britain (1967, 1971).
Details from the latter work are provided below for those who might wish to judge this complex matter for themselves:
PART 1: Introduction, Statistical, Mathematical and Astronomical Backgrounds, Megalithic Yard, Conclusions. (PDF, 4.10 Mb).
PART 2: Circles, Rings, Megalithic Astronomy. (PDF, 4.80 Mb).
PART 3: The Calendar, Indications of Lunar Declinations. (PDF, 2.93 Mb).
PART 4: The Outer Hebrides, A Variety of Sites. (PDF, 3.43 Mb ).
APPENDIX:  A Remarkable 4000 Year-Old Egyptian Ship with hinged, portable A-frame mast, dual-purpose sail and other sophisticated features.
Single page Graphic with text 
(PDF); also in HTML.


Jack Anawak, MP for Nunatsiaq (Hansard,1993): " The history of this land claim goes back many years. It spans several federal governments and numerous ministers of Indian and northern affairs. Very few people realise that prior to 1973 the Government of Canada did not have a policy to negotiate land claims.  It was the current leader of the opposition, under whom I am proud to serve, who, when he was Minister of Indian Affairs and Northen Development, brought forward the first policy to  negotiate and resolve land claims. In 1975 the Inuit of Nunavik achieved the James Bay and Northern Quebec Agreement. The following year, in 1976, the Inuit of the Northwest Territories, as represented by the Inuit Tapirisat, presented their land claim to the federal government for negotiation. Their submission included the proposal for the creation of the Nunavut Territory. The claim proposal was subsequently revised in 1977. In that same year, the Inuvialuit of the western Arctic filed their own land claim. In 1978 they signed an agreement in principle with the federal government and the final agreement was reached in 1984. Between 1976 and 1978 the Inuit of the central and eastern Arctic experienced difficulty with their negotations. There was an impasse over dealing with Nunavut at the land claims table. In 1980 a breakthrough was achieved. Agreement was reached to deal with the Inuit proposals on Nunavut through a political development process in the Northwest Territories separate from but parallel with the land claims negotiations. In 1982 the Tungavik Federation of Nunavut was formed for the specific purpose of negotiating the Inuit land claims. In April 1990 an agreement was reached. Article 4 of that agreement in principle affirmed federal, territorial and Inuit support for the creation of Nunavut as soon as possibler. In December 1991 negotiaitons were finalized on outstanding items in the land claims, including the creation of Nunavut. In November 1992 the Inuit of Nunavut voted to ratify their agreement.
It has been a long journey filled with many rough spots and roadblocks. I want to focus now on the actual land claim and some of the obstacles the Inuit encountered in their negotiations with the government. From the beginning the government set out all  kinds of preconditions and restrictions. In return the Inuit were more than generous. Some of the comments I am about to make I have made on other occasions over the past several years and many of the people listening or watching today will be familiar with them. However, I feel I must restate them for the record. First. I want to take issue with the term "land claim". It is highly inappropriate. I wish there were a better term to use but I will try to explain what I mean by inappropriate. When aboriginal peoples talk about their lands, we are talking about our homelands. We are talking about the territories and resources upon which our people have survived for thousands of years. We are talking first and foremost about our cultures and our way of life on these territories. The land, the waters, the wildlife, and we the people, are one and the same. We are not separate from our environment. We are part of it, and it is part of us. Yet non-aboriginal governments have looked upon land claim negotiations as real-estate transactions.This is not our view. It is difficult for us to understand the non-aboriginal  concept of individual land title and ownership. We see these negotiations primarily as the means to preserve our relationship with the land and ensure our survival as peoples in the larger society surrounding us. Therefore we are also talking about economic and political power. We require the economical and political means to control what happens on our lands. In claim negotiations Inuit peoples are not seeking someting that someone else already owns. We dispute that implication. We are not asking the government to give us something that does not belong to us. We are only seeking recognition of what is rightfully ours. We are trying to take back what was taken away from us by governments without consent in the past. We are reasonable peoples. We have always been willing to share our lands and our resources. We recognise that all peoples and all governments must work together for the benefit of all. This is why Inuit and other aboriginal peoples have entered into land claims negotiations  . . . ."
PDF, 4.71 Mb.  Note: Nunavut means "Our Land". 
Reduced English version  
(5-page pdf) .


Scholium: Newton's closing paragraph " And now we might add something concerning a certain most subtle spirit which pervades and lies hid in all gross bodies; by the force and action of which spirit the particles of bodies attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighboring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic spirit operates." (also in PDF)

Why did Sir Isaac Newton end his opus magnum in this way? A simple statement of faith? Or was it something more encompassing?





Inuit Wisdom: Only One Great Thing
Rumi and the Rising Sun  +PDF
Great Solitudes  +  PDF
Abide With Me  +PDF
Sisters of Softlight  + PDF
The Ten Rules of the Canoe  +PDF
Sandburg Dream Girl  Endless Night  Real Being  Stromata
The Theory of Everything  + PDF
The Statendam et La mer
Deep Contemplations
 Believe in your Heart+PDF
Subject Not Thy mind + PDF
Time, Eternal, Infinite  + PDF
Only Temperate Souls  + PDF
Visions (Ode to Scipio)  +  PDF
  E-MAIL *Kessinger Publishing   O Canada 2010
Quantum entanglement and the STARFISH TELEGRAPH

John N. Harris, M.A. Last updated  July 16, 2024. Best viewed at 1280 x 1024,10 point latin fonts, or whatever.

" Incredulity is given to the world as a punishment "
Source :
The Crowning of Nature

" The weary paddler resting is still ballast "
Source :
The Ten Rules of the Canoe

" Criticism Without Light Condemns Us All to Further Darkness "