![[]](/sites/all/modules/ubercart/uc_cart/images/bullet-arrow-up.gif "Expand cart block."){kind=icon}
There are no products in your shopping cart.
Circle and Square: Explorations in Symbology (Part II)
Links Across Asia
In the Old Turkic religion, the Earth is seen as square, covered by a circumscribed sky dome, the four corners of the earth lying outside the shelter of the sky. The Mongol yurt dwelling (a circular tent) with its pillar an axis mundi, or world axis, is seen as a microcosm of the universe. Some questions concerning the origin of the circle-and-square cosmology—whether, for example, the Turks and Mongols got their cosmological notions from the Indo-Iranians or from the Chinese—may be difficult to answer. One link which seems easy to identify is the influence of Indo-Iranian or Indo-European concepts on Buddhist cosmology and architecture. The plan of the Buddhist stupa was based on the square (Earth) and the circle (Heaven). The use of the circle and the square in complex ways is obvious in the ground plans of the Rawak Vihara at Khotan, in Xinjiang, of the Kanishka stupa at Peshawar, in Pakistan, and of the vast Tope-e-Rustam stupa at Balkh, in Afghanistan. The stupa at the center of the vihara (monastery) at Rawak is based on a large square platform with staircases projecting at the quarters, creating a cruciform shape. The ruins of Miran, south of Lop Nor in Xinjiang, excavated by Aurel Stein show the principal building was a massive square. On a circular base inside a round room stood a stupa not visible from the outside. Wall paintings there date from 300 A.D. And finally, the word ‘mandala’ means ‘circle.’ The mandalas central to so many forms of Buddhism are either a circle within a square or a square within a circle within a square. We will return later to the questions of historical links between similar cosmological concepts in cultures distant from each other, and will suggest that this evidence shows the religious unity of the ancient world.
Clipeus, Tholos, Pomerium and More—Classical Parallels
We turn now to the Classical world of the Mediterranean lands for more circle-and-square cosmology and symbols. H.P L’Orange, Norwegian archaeologist and art historian, wrote that the ancient world conceived the clipeus, the round shield, as an image of the cosmos, citing Roman poet Ovid on Achilles’ shield as an imago mundi. The ancient East saw the world as a circle or clipeus and placed the cosmocrator, god and king, at its center. Seals of Achaemenian Persia show Ahura Mazda in the middle of the ‘world ring,’ and the same forms appear in reliefs at Persepolis. This is the imago clipeata of the Byzantine and other classical peoples.
The Greek tholoi, which had conical roofs, may have been heroa (a center of a hero cult). The Doric tholos at Marmaria (ca.390 B.C.), the most ornate sacred building since the Parthenon, stimulated the creation of great circular buildings at Epidaurus, Olympia and Samothrace. The tholos of Epidaurus consisted of three concentric circular foundations supporting a ‘triple crown’ of columns, a cella wall, and an innermost circle of 14 columns. Tholos is also the word for the beehive-shaped tombs of the Mycenean Greeks; the mounds were a re-creation of the archetypal cosmic mountain, the source of life, center of the world and the link between the worlds of the gods, the living, and the dead. The round hat with a high crown worn in ancient Greece was called a tholia, after this tholos.
In Plutarch’s account, Romulus slew Remus after the marking (with a plow drawn by a bull and a cow) of the pomerium, the sacred circle which delineated Roma Quadrata, the square Rome, because Remus violated the pomerium. For Plutarch, Rome was both a circle and a square (urbs quadrata), a mandala.
According to Plutarch, Romulus learned from Etruscan builders “as in the mysteries,” or by a secret rite, how to create the city. According to Varro, the pomerium was a circle marked by a wall of stones, which marked a Roman sacred space, the realm of the imperium domi, the jurisdiction of civil power.
The infraction for which Remus was caused to forfeit his life may have involved a violation of an oath. In the Indo-European cultural sphere, oath-taking was highly ritualized. We know that among Germanic peoples, trials were conducted by the thing, or legislative assembly. “An oath circle would be inscribed on the ground and the person swearing required to stand inside it. Under no conditions could he leave the circle or even place a foot outside its perimeter until the ritual was ended.”
More examples from the Classical world:
*in the Timaeus and the Critias, Plato describes Atlantis as having a palace enclosed by circular walls, on a rectangular plain divided into square plots.
*One Roman signa (flag) showed a wolf mounted on a cube (Earth) and on a sphere (Heaven).
*On the other hand, the sanctuary of the Roman goddess Vesta was round, a fact which some Roman writers attempted to explain by comparing the goddess with the earth, seemingly a reversal of symbolism.
*The rogus consecrationis of the Roman emperors after Augustus was an immense square four-tiered pyre, around which priests and horsemen would move in a circle (the decursio).
Tons Brunes and the ‘Sacred Cut’
Danish engineer Tons Brunes, in The Secrets of Ancient Geometry and Its Use (Copenhagen: 1967) describes how a square can be divided into nine equal square parts using a set square and pair of compasses. He calls the middle square of this nine-part grid a sacred cut and claimed that the geometry of the sacred cut is part of the design of many ancient buildings, including the Parthenon and the Roman Pantheon. Brunes traced the application of the ‘sacred cut’ from the ancient Egyptians through the Greeks and Romans to medieval Europe and believed that the idea was transmitted from Egypt to Greece by Pythagoras. For Brunes the ‘cut’ is ‘sacred’ because it employs both circle and square, uniting the earthly with the divine; it squares the circle; and it yields an octagon (the traditional shape of Christian baptistries).
The nine-part square was also foundational in China, where the center sector is also sacred, and in India, where it became the swastika. We shall see below that the combining of the nine-part square with numerology to create the ‘magic square’ lies at the heart of circle-and-square cosmology.
Pan-psychism or Pan-diffusionism?
From China across Eurasia to Rome, the ancient world was unified by the cosmology of circle and square, heaven and earth. What can account for this phenomenon? Carl Jung believed that the circle-square pattern and antithesis is archetypal in a universal human sub-consciousness. And some phenomena seem to support Jung’s theory of archetypes. We find, for example, the circle and square cosmology in the Americas, examples of which would include:
*the tukipa temple of the Huichol of the Sierra Madre Occidental of Mexico, which is circular.
*The circular White Path of the xingwikaon (‘Big House’) of the Delaware, upon which singers and dancers walk;
*the sacred camp circle, which for the Plains Indians is an image of the cosmos;
*the sacred circle within the central, circular sacred teepee which was the center of Sioux ritual performances; and
*the circle of dancers of the Ghost Dance at Wounded Knee Creek in 1890 —all these parallels
seem remote enough from Eurasia to recommend a Jungian explanation.
Support for Jungian archetypes can certainly be found in Black Elk’s dictum: “the Power of the World always works in circles, and everything tries to be round.” Black Elk disliked the white man’s houses, saying: “It is a bad way to live, for there can be no power in a square.” And his statement that “Everything the Power of the World does is done in a circle” can be taken either as the empirical observation of a spiritual adept, or as evidence of world-wide diffusion of cosmological concepts, or as evidence for the existence of Jungian archetypes, depending on one’s point of view.
Psychological insight as well as historical fact is implied in Marshall McLuhan’s generalization that “Men lived in round houses until they became sedentary and specialized in their work organization... The square room or house speaks the language of the sedentary specialist, while the round hut...tells of the integral nomadic ways of food-gathering peoples.” Maekawa Kaname analyzed residential settlements of medieval Japan and found “a symbolic aspect”—“Squares [such as castles] meant domination and circles [farming villages] community.” This anthropological evidence supports Jung’s claim for the archetypal status of these symbols.
On the other hand, pan-diffusionists such as American anthropologist Terence Greider find evidence that even the Americas were included in three waves of world-wide diffusion of culture. The Third Wave culture focused on astronomy, astrology, and calendrics—the systematic observation and recording of the phenomena of the celestial realm. Its characteristic signs were the symbols circle and square. Greider points out that for millennia the circle had represented the earth in prehistoric art, but in Third Wave art the circle changed its reference from the earth to the heavens, and the square replaced it as the earth symbol, and the quincunx pattern (center and four corners) became the ideal pattern for human occupation of the earth. Greider notes this pattern in Shang China, Inca Peru, and in the Navajo and Mayan concepts of the land. Circle and square evidently represent Heaven and Earth even in the Big Horn Medicine Wheel in Wyoming.
Newton Was Right
To the medieval alchemists, quadratura circuli (‘squaring the circle’) was the ultimate goal—the key to overcoming the opposition of matter and spirit, to the transmutation of base metals to gold. “From a man and a woman make a circle, then a square, then a triangle, finally a circle, and you will obtain the Philosopher’s Stone.” American historian B.J.T. Dobbs wrote that Sir Isaac Newton probed alchemy “as it has never been probed before or since.” Newton believed that in the beginning of the world, God had imparted the secrets of ‘natural philosophy’ (science and mathematics) and of the true religion to a select few mortals. The knowledge was subsequently lost, but then partially recovered later and incorporated into myths and mysteries, where it would remain hidden but recoverable to one who possessed the key of knowledge.
For the ancient Egyptians, one key to knowledge was expressed in the architecture of the pyramids. Apparently, the first true pyramid with straight rather than stepped sides was attempted at Meidum by the pharaoh Sneferu (r. 2613 to 2589 B.C.). The builders tried to construct the sides at an angle of almost 52 degrees—a pyramid with this angle will have a height equal to the radius of a circle of circumference equal to the perimeter of its base. In other words, the Egyptian pyramid is an architectural squaring of the circle. The sides of the Meidum pyramid collapsed during the latter stages of construction—the angle seems to have been too steep for the construction techniques of the time, and it was not until the construction of the Great Pyramid at Giza, still the largest structure ever built by humankind, at the time of Cheops (Khufu, r. 2589 to 2566 B.C.), that the Egyptians were sufficiently confident to try again to build pyramids with the ‘magic angle’ of 52 degrees.
Isaac Newton’s interest in the Ancient Mysteries followed that of the Renaissance Christian Hermeticists Marsilio Ficino, Pico della Mirandola, Giordano Bruno, Campanella, Paracelsus and others who rejected Scholasticism in favor of looking back to the Pythagoreans, the Kabbalah, ‘Hermes Trismegistus,’ and the Mystery Religions of the Greco-Roman world to “rediscover the primordial revelations of Egypt and Asia and demonstrate their common ground and single source.” They believed in a ‘Prisca Theologia,’ that a single, true, theology exists, vestiges of which can be found all religious traditions, and which was given by God to humankind in antiquity.
The Magic Square
Independent scholar Robert Dickter has identified the ‘Magic Square’ known to the ancient Chinese as the Luo-shu (洛 書 ) as the basis of both Chinese and Pythagorean numerology. An explication of Dickter’s argument is beyond the scope of this essay, but if Dickter and his sources are correct, the mathematics implicit in this pattern underlies the circle-and-square cosmology and reveals the meaning and significance of many of the topics of this essay—the Chinese TLV mirrors, jade bi discs, and Ming Tang temple; the gnomon or carpenter’s square, the mandala, the Indo-Iranian vara, the Buddhist stupa, the ‘sacred cut’ and Classical architecture, medieval alchemy, and more. If so, then the mathematics of the Magic Square, as the ancestor of modern mathematics and science, is the Primal Revelation sought by Sir Isaac Newton.
Conclusion
In Chinese legend, at the time of the Emperor Yu the Great, founder (ca.2070 B.C.) of the Xia dynasty, a magical turtle with the Luo-shu pattern inscribed on its shell emerged from the flooding Luo River. Dickter attributes to the Luo-shu the eventual emergence of Pythagorean mathematics, over a millennium later. But since Pharaoh Sneferu, whose pyramid was an attempt to square the circle, evidently preceded Emperor Yu by some centuries, a more likely scenario would push the origin of Magic Square mathematics, and circle-and-square cosmology, deeper into pre-history and attribute its appearance in Egypt, and then in China, to diffusion from a single source, since it is highly unlikely that anything as complex as Magic Square mathematics could be archetypal in the Jungian sense.
REFERENCES
1. See Alonzo L. Gaskill, The Lost Language of Symbolism (Salt Lake City: Deseret Book, 2003) for a more comprehensive introduction to symbology. An earlier version of this essay was read by the author at the August 1990 Sunstone Symposium in Salt Lake City.
2. Henri Maspero Taoism and Chinese Religion, trans. Frank Kiernan Jr. (Amherst: U. Mass., 1981), p.435; the two feet side by side form a square. See also Mircea Eliade, ed., Encyclopedia of Religion (NY: Collier Macmillan, 1987) 16 vols., XIV:290, 299.
3. Mircea Eliade, A History of Religious Ideas, Vol. II, trans. William R. Trask (U. Chicago: 1982), pp.11-17
4. Encyclopedia of Religion, V:511; hereafter cited as ER
5. William Watson, Art of Dynastic China (London: Thames and Hudson, 1981), pp.510-11; the tri-level Temple of Annual Prayers (Tian-tan) is commonly said to have been constructed without the use of even a single nail. For more description of the three (odd number) terraces of the Altar of Heaven and the two (even number) quadrangular terraces of the Altar for the Adoration of the Earth, see Encyclopedia of World Art (NY: McGraw-Hill, 1959-1968), s.v. ‘China,’ III:426-7.
6. Ann Paludan, Chronicles of the Chinese Emperors (London: Thames & Hudson, 1998), p.168. The blue roof tiles of the Temple of Annual Prayers also symbolize Heaven.
7. Watson (1981), p.32
8. The two essential vessels of all early Chinese ritual—funerals, sacrifices and feasts—were the square ‘fu’ vessels and round ‘gui’ vessels. For references, see my essay “Compasses, Square, Level and Line,” note 16.
9. Michael Loewe, Ways to Paradise: The Chinese Quest for Immortality (London: Allen & Unwin, 1979), p.40. For more references, see my “Compasses...” notes 17 and 18.
10. Maxwell K. Hearn, Splendors of Imperial China (NY: Metropolitan Museum of Art, 1996), pp.9, 12
11. ER, VII:504
12. Gina L. Barnes, China, Korea and Japan: The Rise of Civilization in East Asia (London: Thames and Hudson, 1993) pp.114-5, illlustrated; for a color photo of a Liangzhu cong, see http://www.sinoarts.net/home.php?cat=259
13. William E. Soothill, The Hall of Light: A Study of Early Chinese Kingship (London, 1951), pp.201-2; the information comes from the Zhou-li. For a recent account and interpretation of the cong and bi, see Jessica Rawson, Mysteries of Ancient China (London: British Museum, 1996), pp.52-4.
14. Yu, Weichao, ed., A Journey into China’s Antiquity (Beijing: Morning Glory, 1997) 3 vols., I:80, 87, 94, 100
15. Encyclopedia of World Art (NY: McGraw-Hill, 1959-1968), II:80 and Plate 31; the proof of this is said to be found in the large bi at the Brussels Royal Museum of Art and History. For more on cong and bi jades, see EWA XII:894, and II:131, VI:plate 49, XII:832.
16. Kubo, Noritada, Doukyou-shi (Tokyo: Yamakawa, 1977), p.92; the symbol of Tai-yi is said to be the flaming pearl, a symbol found also in Japan, where it is the symbol of the Inari shrines (Jean Herbert, Shinto [1967], p.506) and in the Holy Land, as a Samaritan symbol (Merrill C. Tenney. ed., Zondervan Pictorial Bible Dictionary [Grand Rapids, 1967], p.762). For an introduction to Tai-yi, see Rawson (1996), pp.149-50; also: http://www.patheos.com/Library/Taoism/Ritual-Worship-Devotion-Symbolism/... .
17. in The Archaeology of Music in Ancient China (NY: Paragon House, 1990), p.104
18. Ibid., p.82
19. C.A.S.Williams, Outlines of Chinese Symbolism and Art Motives, 3rd rev. ed. (Shanghai: Kelly and Walsh, 1941; multiple reprints), pp.284, 288, 376
20. Ibid., p.82; for more on this topic, see my essay “Compasses, Square, Level and Line.” For more on compasses, plumb line, and the circle and square see Williams, op.cit., pp.33, 176, 211. The use of compasses and square as didactic symbols has been handed down among Chinese and Japanese carpenters’ guilds in much the same way as these symbols have been transmitted in Western Freemasonry. The use of these symbols among Chinese secret societies such as the Tiandi-hui, or ‘Triads,’ parallels the ancient Confucian texts, modern Chinese and Japanese carpenters’ guilds, and Western Freemasonry. Williams generalizes (op.cit., p.349), “There is said to be a considerable resemblance to Freemasonry in the ritual of many of the Chinese secret societies.”
21. The first published notice of the similarities between the rituals and symbols of Western Freemasonry and Chinese secret societies seems to have been “Some Account of a Secret Association in China, Entitled the Triad Society,” by Dr. William Milne (d. 1822); published posthumously by Robert Morrison in Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. 1, No. 2 (1826), pp. 240-250. Gustave Schlegel wrote, in Thian Ti Hwui: the Hung League, or Heaven-Earth League, a Secret Society with the Chinese in China and India (Batavia: Lange, 1866; repr. Scotland: Tynron Press, 1991), p.ix, “Every person who has read anything of the secret societies in China, must have been struck with the resemblance between them and the society of Freemasons.” Schlegel’s book was the first full-length study on the topic.
22. W. G. Stirling and J.S.M. Ward, The Hung Society or the Society of Heaven and Earth (London: Baskerville Press, 1925-26), 3 vols., further promoted the idea that the two traditions have descended from a common ancestor, “the Ancient Mysteries;” see, e.g., p.232. Worthy of note also is William Stanton, The Triad Society or Heaven and Earth Association (Hong Kong: Kelly & Walsh, 1900).
23. For a more recent account, see The Origins of the Tiandihui: The Chinese Triads in Legend and History, by Dian H. Murray in collaboration with Qin Biaoqi (Stanford U., 1994)
24. Watson (1981), p.87, and Color Plate 3
25. http://en.wikipedia.org/wiki/TLV_mirror , illustrated
26. http://we.magma.jp/~ark/sinkyou_utyuu/doukyou.html , in Japanese
27. http://akatonbo-jo.cocolog-nifty.com/jo/2007/10/post_0b6c.html , citing scholar 来 村 多 加 史 (Kimura Takashi), in Japanese; for the 圭 表 , see
http://baike.baidu.com/view/41639.htm , in Chinese (simplified characters). For authoritative interpretation of the TLV mirror symbols see Loewe (1979), p.40.
28. Barnes (1993), pp.199, 202; the illustration at Loewe (1979), p.85 is especially useful
29. Rawson (1996), pp.159-161, illustrated; see also http://en.wikipedia.org/wiki/Liubo
30. Loewe (1979), pp.75ff; the illustration on p.80 is particularly useful.
31. Ibid., pp.80ff.
32. Encyclopedia of World Art, XII:705
33. Encyclopedia of World Art, I:39
34. H.P. L’Orange, “Expressions of Cosmic Kingship in the Ancient World,” in VIIIth Inter- national Congress for the History of Religions (Rome, April 1955), The Sacral Kingship (Leiden: Brill, 1959), pp.481-3. Hatra, a Parthian caravan city destroyed by the Sassanians in 241 A.D., was circular with a rectangular sanctuary in its center (Nara Museum of Art, Silk Road, Steppe Routes[1988], p.186). A relief from Nineveh in the British Museum shows a circular fortified encampment divided into quarters (Helen Rosenau, The Ideal City: Its Architectural Evolution [NY: Harper & Row, 1972], p.20).
35. Ibid., pp.483-5
36. H.P. L’Orange, Studies on the Iconography of Cosmic Kingship in the Ancient World (Oslo and Cambridge MA: 1953), p.23
37. Encyclopedia of World Art, XI:109; hereafter, EWA
38. Jane Turner, ed., Grove Dictionary of Art (1996), I:190; Georgina Herrmann, Monuments of Merv (London: Soc. of Antiquaries, 1999)
39. B.G. Gafurov, ed., Kushan Studies in the USSR (Calcutta, 1970), pp.84-5
40. Gregoire Frumkin, Archaeology of Soviet Central Asia (Leiden: Brill, 1970), pp.10, 94-5, and plates XXIX, XXX
41. Leonid T. Yablonsky, in Jeannine Davis-Kimball, et al., eds., Nomads of the Eurasian Steppe in the Early Iron Age (Berkeley: Zinat Press, 1995), p.249
42. Hence the comparison to Noah’s Ark; see the on-line Jewish Encyclopedia, s.v. ‘Zoroastrianism,’ or ‘Persia.’ See also Elena E. Kuz’mina, The Origin of the Indo-Europeans, ed. J.P. Mallory and trans. S. Pitina and P. Prudovsky (Leiden: Brill, 2007), vol. 3, pp.34-5
43. Kuz’mina (2007), vol. 3, pp.34-5
44. ER, XV:89
45. ER, VII:49
46. Benjamin Rowland, The Art of Central Asia (1970), pp.122-4
47. A photo of the ruins of the Miran stupa may be accessed at: www.theorientalcaravan.com/images/Xinjiang04/miran_stupa.jpg .
48. ER, IX:139
49. H.P. L’Orange, Iconography of Cosmic Kingship in the Ancient World (Oslo, 1953), pp.90, 9
50. H.P. L’Orange (1959), pp.488-90
51. Grove Dictionary of Art, 13:380, 390 and 8:694; for the Thymele at Epidaurus and the round temples of Samothrace and Athens, see James Hastings, ed., Encyclopedia of Religion and Ethics, I:730b, 736a, 739a-740b
52. EWA XI:435, 437
53. ER, XIV:553
54. EWA, IV:24; the Hittite king wore a cone-shaped helmet, as did his soldiers (ibid., p.21); and the Israelite high priest wore a high, white conical hat (ibid., p.22). One is tempted to speculate here about possible links to the steeple hennin and the witches’ hat of Europe in the Late Middle Ages.
55. ER, XV:105; Aniela Jaffe, “Symbolism in the Visual Arts,” in Carl Jung, ed., Man and His Symbols (NY: Dell, 1964), p.242; Plutarch, Lives, trans. John Dryden (NY: Modern Library), pp.30-1
56. ER, XII:449-50
57. ER, XV:304
58. Cited by Helen Rosenau, The Ideal City: Its Architectural Evolution (NY: Harper & Row, 1972), pp.20-1; John’s Holy City, the New Jerusalem, was cubical (Revelation 21:16).
59. ER, XV:431
60. ER, XV:250-1
61. ER, I:361
62. Donald J. and Carol Martin Watts, “A Roman Apartment Complex,” Scientific American, Vol. 255, No. 6 (December 1986), pp.132-139
63. Paul Calter, “Ad Quadratum, the Sacred Cut & Roman Architecture,” in Geometry in Art and Architecture, Dartmouth College, on-line at: http://www.dartmouth.edu/~matc/math5.geometry/unit7/unit7.html . Calter explains how the ‘sacred cut’ squares the circle; his work also includes units on the Golden Ratio, the Pythagoreans, Squaring the Circle in the Great Pyramid, the Roman architect Vitruvius—all important aspects of the subject of this essay.
64. Paul Wheatley, The Pivot of the Four Quarters (1971)
65. Terence Greider, Origins of Pre-Columbian Art (Austin: U. Texas, 1982), pp.108-9
66. ER, VI:493
67. ER, VII:17-8
68. ER, VII:17
69. John G. Niehardt, Black Elk Speaks (1932), pp.165, 186, 189, 206
70. Ibid., p.237
71. The quotations from Black Elk come from Niehardt (1932), p.194. Aristotle’s claim that “What is eternal is circular and what is circular is eternal” (cited in Gorgio de Santillana and Herta von Dechend, Hamlet’s Mill [Boston, 1969], pp.48-9) seems more like a statement of a cosmological premise.
72. Understanding Media (1964), pp.118-9
73. The orthogonal land allotment system imposed by the government differed greatly from the traditional settlement and field layouts. Maekawa Kaname, “Moated Sites in Medieval Japan: Notions of Square and Circle,” Abstracts of Papers Presented at the Second Worldwide SEAA Conference, www.eastasianarchaeology.oreg/news/seaaconfsbstracts00.htm , accessed by the author on 10/03/00.
74. Terence Greider, Origins of Pre-Columbian Art (Austin: U. Texas, 1982); Eliade found similar patterns in villages in Bali and New Guinea and in the sacred lodges of the Algonquin (The Sacred and the Profane, trans. Willard R. Trask [Harcourt, Brace & World, many repr.], p.45).
75. Mircea Eliade, Forge and Crucible: Origins and Structure of Alchemy (U. Chicago, 1979), p.123
76. B.J.T. Dobbs, The Foundations of Newton’s Alchemy (Cambridge, 1975), cited in Mircea Eliade, A History of Religious Ideas, Vol. III (U. Chicago, 1988), p.260
77. Joseph F. Baugher, On Civilized Stars: The Search for Intelligence in Outer Space (Prentice-Hall, 1985), p.170
78. The quotation is from Mircea Eliade (1988, pp.250-1); for a link to standard introductions to eight Greco-Roman Mystery traditions, see http://en.wikipedia.org/wiki/Category:Mystery_religions . In 1990 (see footnote i above) the author pointed out that Joseph Smith had in common with Isaac Newton a commitment to the significance of the Primal Revelation. Eliade (1979, p.149) says of the Mystery religions: “The meaning and end of the Mysteries were the transmutation of man. By experiencing initiatory death and resurrection, the initiate changed his mode of being (he became immortal).”
79. Dickter’s work, entitled Number, Time and Archetype, is available as an on-line e-book at www.luo-shu.com/book/contents . On the mathematics of magic squares, see http://the-magic-square.blogspot.com/ . For a basic introduction to the Luo-shu, see http://en.wikipedia.org/wiki/Lo_Shu_Square .
In the Old Turkic religion, the Earth is seen as square, covered by a circumscribed sky dome, the four corners of the earth lying outside the shelter of the sky. The Mongol yurt dwelling (a circular tent) with its pillar an axis mundi, or world axis, is seen as a microcosm of the universe. Some questions concerning the origin of the circle-and-square cosmology—whether, for example, the Turks and Mongols got their cosmological notions from the Indo-Iranians or from the Chinese—may be difficult to answer. One link which seems easy to identify is the influence of Indo-Iranian or Indo-European concepts on Buddhist cosmology and architecture. The plan of the Buddhist stupa was based on the square (Earth) and the circle (Heaven). The use of the circle and the square in complex ways is obvious in the ground plans of the Rawak Vihara at Khotan, in Xinjiang, of the Kanishka stupa at Peshawar, in Pakistan, and of the vast Tope-e-Rustam stupa at Balkh, in Afghanistan. The stupa at the center of the vihara (monastery) at Rawak is based on a large square platform with staircases projecting at the quarters, creating a cruciform shape. The ruins of Miran, south of Lop Nor in Xinjiang, excavated by Aurel Stein show the principal building was a massive square. On a circular base inside a round room stood a stupa not visible from the outside. Wall paintings there date from 300 A.D. And finally, the word ‘mandala’ means ‘circle.’ The mandalas central to so many forms of Buddhism are either a circle within a square or a square within a circle within a square. We will return later to the questions of historical links between similar cosmological concepts in cultures distant from each other, and will suggest that this evidence shows the religious unity of the ancient world.
Clipeus, Tholos, Pomerium and More—Classical Parallels
We turn now to the Classical world of the Mediterranean lands for more circle-and-square cosmology and symbols. H.P L’Orange, Norwegian archaeologist and art historian, wrote that the ancient world conceived the clipeus, the round shield, as an image of the cosmos, citing Roman poet Ovid on Achilles’ shield as an imago mundi. The ancient East saw the world as a circle or clipeus and placed the cosmocrator, god and king, at its center. Seals of Achaemenian Persia show Ahura Mazda in the middle of the ‘world ring,’ and the same forms appear in reliefs at Persepolis. This is the imago clipeata of the Byzantine and other classical peoples.
The Greek tholoi, which had conical roofs, may have been heroa (a center of a hero cult). The Doric tholos at Marmaria (ca.390 B.C.), the most ornate sacred building since the Parthenon, stimulated the creation of great circular buildings at Epidaurus, Olympia and Samothrace. The tholos of Epidaurus consisted of three concentric circular foundations supporting a ‘triple crown’ of columns, a cella wall, and an innermost circle of 14 columns. Tholos is also the word for the beehive-shaped tombs of the Mycenean Greeks; the mounds were a re-creation of the archetypal cosmic mountain, the source of life, center of the world and the link between the worlds of the gods, the living, and the dead. The round hat with a high crown worn in ancient Greece was called a tholia, after this tholos.
In Plutarch’s account, Romulus slew Remus after the marking (with a plow drawn by a bull and a cow) of the pomerium, the sacred circle which delineated Roma Quadrata, the square Rome, because Remus violated the pomerium. For Plutarch, Rome was both a circle and a square (urbs quadrata), a mandala.
According to Plutarch, Romulus learned from Etruscan builders “as in the mysteries,” or by a secret rite, how to create the city. According to Varro, the pomerium was a circle marked by a wall of stones, which marked a Roman sacred space, the realm of the imperium domi, the jurisdiction of civil power.
The infraction for which Remus was caused to forfeit his life may have involved a violation of an oath. In the Indo-European cultural sphere, oath-taking was highly ritualized. We know that among Germanic peoples, trials were conducted by the thing, or legislative assembly. “An oath circle would be inscribed on the ground and the person swearing required to stand inside it. Under no conditions could he leave the circle or even place a foot outside its perimeter until the ritual was ended.”
More examples from the Classical world:
*in the Timaeus and the Critias, Plato describes Atlantis as having a palace enclosed by circular walls, on a rectangular plain divided into square plots.
*One Roman signa (flag) showed a wolf mounted on a cube (Earth) and on a sphere (Heaven).
*On the other hand, the sanctuary of the Roman goddess Vesta was round, a fact which some Roman writers attempted to explain by comparing the goddess with the earth, seemingly a reversal of symbolism.
*The rogus consecrationis of the Roman emperors after Augustus was an immense square four-tiered pyre, around which priests and horsemen would move in a circle (the decursio).
Tons Brunes and the ‘Sacred Cut’
Danish engineer Tons Brunes, in The Secrets of Ancient Geometry and Its Use (Copenhagen: 1967) describes how a square can be divided into nine equal square parts using a set square and pair of compasses. He calls the middle square of this nine-part grid a sacred cut and claimed that the geometry of the sacred cut is part of the design of many ancient buildings, including the Parthenon and the Roman Pantheon. Brunes traced the application of the ‘sacred cut’ from the ancient Egyptians through the Greeks and Romans to medieval Europe and believed that the idea was transmitted from Egypt to Greece by Pythagoras. For Brunes the ‘cut’ is ‘sacred’ because it employs both circle and square, uniting the earthly with the divine; it squares the circle; and it yields an octagon (the traditional shape of Christian baptistries).
The nine-part square was also foundational in China, where the center sector is also sacred, and in India, where it became the swastika. We shall see below that the combining of the nine-part square with numerology to create the ‘magic square’ lies at the heart of circle-and-square cosmology.
Pan-psychism or Pan-diffusionism?
From China across Eurasia to Rome, the ancient world was unified by the cosmology of circle and square, heaven and earth. What can account for this phenomenon? Carl Jung believed that the circle-square pattern and antithesis is archetypal in a universal human sub-consciousness. And some phenomena seem to support Jung’s theory of archetypes. We find, for example, the circle and square cosmology in the Americas, examples of which would include:
*the tukipa temple of the Huichol of the Sierra Madre Occidental of Mexico, which is circular.
*The circular White Path of the xingwikaon (‘Big House’) of the Delaware, upon which singers and dancers walk;
*the sacred camp circle, which for the Plains Indians is an image of the cosmos;
*the sacred circle within the central, circular sacred teepee which was the center of Sioux ritual performances; and
*the circle of dancers of the Ghost Dance at Wounded Knee Creek in 1890 —all these parallels
seem remote enough from Eurasia to recommend a Jungian explanation.
Support for Jungian archetypes can certainly be found in Black Elk’s dictum: “the Power of the World always works in circles, and everything tries to be round.” Black Elk disliked the white man’s houses, saying: “It is a bad way to live, for there can be no power in a square.” And his statement that “Everything the Power of the World does is done in a circle” can be taken either as the empirical observation of a spiritual adept, or as evidence of world-wide diffusion of cosmological concepts, or as evidence for the existence of Jungian archetypes, depending on one’s point of view.
Psychological insight as well as historical fact is implied in Marshall McLuhan’s generalization that “Men lived in round houses until they became sedentary and specialized in their work organization... The square room or house speaks the language of the sedentary specialist, while the round hut...tells of the integral nomadic ways of food-gathering peoples.” Maekawa Kaname analyzed residential settlements of medieval Japan and found “a symbolic aspect”—“Squares [such as castles] meant domination and circles [farming villages] community.” This anthropological evidence supports Jung’s claim for the archetypal status of these symbols.
On the other hand, pan-diffusionists such as American anthropologist Terence Greider find evidence that even the Americas were included in three waves of world-wide diffusion of culture. The Third Wave culture focused on astronomy, astrology, and calendrics—the systematic observation and recording of the phenomena of the celestial realm. Its characteristic signs were the symbols circle and square. Greider points out that for millennia the circle had represented the earth in prehistoric art, but in Third Wave art the circle changed its reference from the earth to the heavens, and the square replaced it as the earth symbol, and the quincunx pattern (center and four corners) became the ideal pattern for human occupation of the earth. Greider notes this pattern in Shang China, Inca Peru, and in the Navajo and Mayan concepts of the land. Circle and square evidently represent Heaven and Earth even in the Big Horn Medicine Wheel in Wyoming.
Newton Was Right
To the medieval alchemists, quadratura circuli (‘squaring the circle’) was the ultimate goal—the key to overcoming the opposition of matter and spirit, to the transmutation of base metals to gold. “From a man and a woman make a circle, then a square, then a triangle, finally a circle, and you will obtain the Philosopher’s Stone.” American historian B.J.T. Dobbs wrote that Sir Isaac Newton probed alchemy “as it has never been probed before or since.” Newton believed that in the beginning of the world, God had imparted the secrets of ‘natural philosophy’ (science and mathematics) and of the true religion to a select few mortals. The knowledge was subsequently lost, but then partially recovered later and incorporated into myths and mysteries, where it would remain hidden but recoverable to one who possessed the key of knowledge.
For the ancient Egyptians, one key to knowledge was expressed in the architecture of the pyramids. Apparently, the first true pyramid with straight rather than stepped sides was attempted at Meidum by the pharaoh Sneferu (r. 2613 to 2589 B.C.). The builders tried to construct the sides at an angle of almost 52 degrees—a pyramid with this angle will have a height equal to the radius of a circle of circumference equal to the perimeter of its base. In other words, the Egyptian pyramid is an architectural squaring of the circle. The sides of the Meidum pyramid collapsed during the latter stages of construction—the angle seems to have been too steep for the construction techniques of the time, and it was not until the construction of the Great Pyramid at Giza, still the largest structure ever built by humankind, at the time of Cheops (Khufu, r. 2589 to 2566 B.C.), that the Egyptians were sufficiently confident to try again to build pyramids with the ‘magic angle’ of 52 degrees.
Isaac Newton’s interest in the Ancient Mysteries followed that of the Renaissance Christian Hermeticists Marsilio Ficino, Pico della Mirandola, Giordano Bruno, Campanella, Paracelsus and others who rejected Scholasticism in favor of looking back to the Pythagoreans, the Kabbalah, ‘Hermes Trismegistus,’ and the Mystery Religions of the Greco-Roman world to “rediscover the primordial revelations of Egypt and Asia and demonstrate their common ground and single source.” They believed in a ‘Prisca Theologia,’ that a single, true, theology exists, vestiges of which can be found all religious traditions, and which was given by God to humankind in antiquity.
The Magic Square
Independent scholar Robert Dickter has identified the ‘Magic Square’ known to the ancient Chinese as the Luo-shu (洛 書 ) as the basis of both Chinese and Pythagorean numerology. An explication of Dickter’s argument is beyond the scope of this essay, but if Dickter and his sources are correct, the mathematics implicit in this pattern underlies the circle-and-square cosmology and reveals the meaning and significance of many of the topics of this essay—the Chinese TLV mirrors, jade bi discs, and Ming Tang temple; the gnomon or carpenter’s square, the mandala, the Indo-Iranian vara, the Buddhist stupa, the ‘sacred cut’ and Classical architecture, medieval alchemy, and more. If so, then the mathematics of the Magic Square, as the ancestor of modern mathematics and science, is the Primal Revelation sought by Sir Isaac Newton.
Conclusion
In Chinese legend, at the time of the Emperor Yu the Great, founder (ca.2070 B.C.) of the Xia dynasty, a magical turtle with the Luo-shu pattern inscribed on its shell emerged from the flooding Luo River. Dickter attributes to the Luo-shu the eventual emergence of Pythagorean mathematics, over a millennium later. But since Pharaoh Sneferu, whose pyramid was an attempt to square the circle, evidently preceded Emperor Yu by some centuries, a more likely scenario would push the origin of Magic Square mathematics, and circle-and-square cosmology, deeper into pre-history and attribute its appearance in Egypt, and then in China, to diffusion from a single source, since it is highly unlikely that anything as complex as Magic Square mathematics could be archetypal in the Jungian sense.
REFERENCES
1. See Alonzo L. Gaskill, The Lost Language of Symbolism (Salt Lake City: Deseret Book, 2003) for a more comprehensive introduction to symbology. An earlier version of this essay was read by the author at the August 1990 Sunstone Symposium in Salt Lake City.
2. Henri Maspero Taoism and Chinese Religion, trans. Frank Kiernan Jr. (Amherst: U. Mass., 1981), p.435; the two feet side by side form a square. See also Mircea Eliade, ed., Encyclopedia of Religion (NY: Collier Macmillan, 1987) 16 vols., XIV:290, 299.
3. Mircea Eliade, A History of Religious Ideas, Vol. II, trans. William R. Trask (U. Chicago: 1982), pp.11-17
4. Encyclopedia of Religion, V:511; hereafter cited as ER
5. William Watson, Art of Dynastic China (London: Thames and Hudson, 1981), pp.510-11; the tri-level Temple of Annual Prayers (Tian-tan) is commonly said to have been constructed without the use of even a single nail. For more description of the three (odd number) terraces of the Altar of Heaven and the two (even number) quadrangular terraces of the Altar for the Adoration of the Earth, see Encyclopedia of World Art (NY: McGraw-Hill, 1959-1968), s.v. ‘China,’ III:426-7.
6. Ann Paludan, Chronicles of the Chinese Emperors (London: Thames & Hudson, 1998), p.168. The blue roof tiles of the Temple of Annual Prayers also symbolize Heaven.
7. Watson (1981), p.32
8. The two essential vessels of all early Chinese ritual—funerals, sacrifices and feasts—were the square ‘fu’ vessels and round ‘gui’ vessels. For references, see my essay “Compasses, Square, Level and Line,” note 16.
9. Michael Loewe, Ways to Paradise: The Chinese Quest for Immortality (London: Allen & Unwin, 1979), p.40. For more references, see my “Compasses...” notes 17 and 18.
10. Maxwell K. Hearn, Splendors of Imperial China (NY: Metropolitan Museum of Art, 1996), pp.9, 12
11. ER, VII:504
12. Gina L. Barnes, China, Korea and Japan: The Rise of Civilization in East Asia (London: Thames and Hudson, 1993) pp.114-5, illlustrated; for a color photo of a Liangzhu cong, see http://www.sinoarts.net/home.php?cat=259
13. William E. Soothill, The Hall of Light: A Study of Early Chinese Kingship (London, 1951), pp.201-2; the information comes from the Zhou-li. For a recent account and interpretation of the cong and bi, see Jessica Rawson, Mysteries of Ancient China (London: British Museum, 1996), pp.52-4.
14. Yu, Weichao, ed., A Journey into China’s Antiquity (Beijing: Morning Glory, 1997) 3 vols., I:80, 87, 94, 100
15. Encyclopedia of World Art (NY: McGraw-Hill, 1959-1968), II:80 and Plate 31; the proof of this is said to be found in the large bi at the Brussels Royal Museum of Art and History. For more on cong and bi jades, see EWA XII:894, and II:131, VI:plate 49, XII:832.
16. Kubo, Noritada, Doukyou-shi (Tokyo: Yamakawa, 1977), p.92; the symbol of Tai-yi is said to be the flaming pearl, a symbol found also in Japan, where it is the symbol of the Inari shrines (Jean Herbert, Shinto [1967], p.506) and in the Holy Land, as a Samaritan symbol (Merrill C. Tenney. ed., Zondervan Pictorial Bible Dictionary [Grand Rapids, 1967], p.762). For an introduction to Tai-yi, see Rawson (1996), pp.149-50; also: http://www.patheos.com/Library/Taoism/Ritual-Worship-Devotion-Symbolism/... .
17. in The Archaeology of Music in Ancient China (NY: Paragon House, 1990), p.104
18. Ibid., p.82
19. C.A.S.Williams, Outlines of Chinese Symbolism and Art Motives, 3rd rev. ed. (Shanghai: Kelly and Walsh, 1941; multiple reprints), pp.284, 288, 376
20. Ibid., p.82; for more on this topic, see my essay “Compasses, Square, Level and Line.” For more on compasses, plumb line, and the circle and square see Williams, op.cit., pp.33, 176, 211. The use of compasses and square as didactic symbols has been handed down among Chinese and Japanese carpenters’ guilds in much the same way as these symbols have been transmitted in Western Freemasonry. The use of these symbols among Chinese secret societies such as the Tiandi-hui, or ‘Triads,’ parallels the ancient Confucian texts, modern Chinese and Japanese carpenters’ guilds, and Western Freemasonry. Williams generalizes (op.cit., p.349), “There is said to be a considerable resemblance to Freemasonry in the ritual of many of the Chinese secret societies.”
21. The first published notice of the similarities between the rituals and symbols of Western Freemasonry and Chinese secret societies seems to have been “Some Account of a Secret Association in China, Entitled the Triad Society,” by Dr. William Milne (d. 1822); published posthumously by Robert Morrison in Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. 1, No. 2 (1826), pp. 240-250. Gustave Schlegel wrote, in Thian Ti Hwui: the Hung League, or Heaven-Earth League, a Secret Society with the Chinese in China and India (Batavia: Lange, 1866; repr. Scotland: Tynron Press, 1991), p.ix, “Every person who has read anything of the secret societies in China, must have been struck with the resemblance between them and the society of Freemasons.” Schlegel’s book was the first full-length study on the topic.
22. W. G. Stirling and J.S.M. Ward, The Hung Society or the Society of Heaven and Earth (London: Baskerville Press, 1925-26), 3 vols., further promoted the idea that the two traditions have descended from a common ancestor, “the Ancient Mysteries;” see, e.g., p.232. Worthy of note also is William Stanton, The Triad Society or Heaven and Earth Association (Hong Kong: Kelly & Walsh, 1900).
23. For a more recent account, see The Origins of the Tiandihui: The Chinese Triads in Legend and History, by Dian H. Murray in collaboration with Qin Biaoqi (Stanford U., 1994)
24. Watson (1981), p.87, and Color Plate 3
25. http://en.wikipedia.org/wiki/TLV_mirror , illustrated
26. http://we.magma.jp/~ark/sinkyou_utyuu/doukyou.html , in Japanese
27. http://akatonbo-jo.cocolog-nifty.com/jo/2007/10/post_0b6c.html , citing scholar 来 村 多 加 史 (Kimura Takashi), in Japanese; for the 圭 表 , see
http://baike.baidu.com/view/41639.htm , in Chinese (simplified characters). For authoritative interpretation of the TLV mirror symbols see Loewe (1979), p.40.
28. Barnes (1993), pp.199, 202; the illustration at Loewe (1979), p.85 is especially useful
29. Rawson (1996), pp.159-161, illustrated; see also http://en.wikipedia.org/wiki/Liubo
30. Loewe (1979), pp.75ff; the illustration on p.80 is particularly useful.
31. Ibid., pp.80ff.
32. Encyclopedia of World Art, XII:705
33. Encyclopedia of World Art, I:39
34. H.P. L’Orange, “Expressions of Cosmic Kingship in the Ancient World,” in VIIIth Inter- national Congress for the History of Religions (Rome, April 1955), The Sacral Kingship (Leiden: Brill, 1959), pp.481-3. Hatra, a Parthian caravan city destroyed by the Sassanians in 241 A.D., was circular with a rectangular sanctuary in its center (Nara Museum of Art, Silk Road, Steppe Routes[1988], p.186). A relief from Nineveh in the British Museum shows a circular fortified encampment divided into quarters (Helen Rosenau, The Ideal City: Its Architectural Evolution [NY: Harper & Row, 1972], p.20).
35. Ibid., pp.483-5
36. H.P. L’Orange, Studies on the Iconography of Cosmic Kingship in the Ancient World (Oslo and Cambridge MA: 1953), p.23
37. Encyclopedia of World Art, XI:109; hereafter, EWA
38. Jane Turner, ed., Grove Dictionary of Art (1996), I:190; Georgina Herrmann, Monuments of Merv (London: Soc. of Antiquaries, 1999)
39. B.G. Gafurov, ed., Kushan Studies in the USSR (Calcutta, 1970), pp.84-5
40. Gregoire Frumkin, Archaeology of Soviet Central Asia (Leiden: Brill, 1970), pp.10, 94-5, and plates XXIX, XXX
41. Leonid T. Yablonsky, in Jeannine Davis-Kimball, et al., eds., Nomads of the Eurasian Steppe in the Early Iron Age (Berkeley: Zinat Press, 1995), p.249
42. Hence the comparison to Noah’s Ark; see the on-line Jewish Encyclopedia, s.v. ‘Zoroastrianism,’ or ‘Persia.’ See also Elena E. Kuz’mina, The Origin of the Indo-Europeans, ed. J.P. Mallory and trans. S. Pitina and P. Prudovsky (Leiden: Brill, 2007), vol. 3, pp.34-5
43. Kuz’mina (2007), vol. 3, pp.34-5
44. ER, XV:89
45. ER, VII:49
46. Benjamin Rowland, The Art of Central Asia (1970), pp.122-4
47. A photo of the ruins of the Miran stupa may be accessed at: www.theorientalcaravan.com/images/Xinjiang04/miran_stupa.jpg .
48. ER, IX:139
49. H.P. L’Orange, Iconography of Cosmic Kingship in the Ancient World (Oslo, 1953), pp.90, 9
50. H.P. L’Orange (1959), pp.488-90
51. Grove Dictionary of Art, 13:380, 390 and 8:694; for the Thymele at Epidaurus and the round temples of Samothrace and Athens, see James Hastings, ed., Encyclopedia of Religion and Ethics, I:730b, 736a, 739a-740b
52. EWA XI:435, 437
53. ER, XIV:553
54. EWA, IV:24; the Hittite king wore a cone-shaped helmet, as did his soldiers (ibid., p.21); and the Israelite high priest wore a high, white conical hat (ibid., p.22). One is tempted to speculate here about possible links to the steeple hennin and the witches’ hat of Europe in the Late Middle Ages.
55. ER, XV:105; Aniela Jaffe, “Symbolism in the Visual Arts,” in Carl Jung, ed., Man and His Symbols (NY: Dell, 1964), p.242; Plutarch, Lives, trans. John Dryden (NY: Modern Library), pp.30-1
56. ER, XII:449-50
57. ER, XV:304
58. Cited by Helen Rosenau, The Ideal City: Its Architectural Evolution (NY: Harper & Row, 1972), pp.20-1; John’s Holy City, the New Jerusalem, was cubical (Revelation 21:16).
59. ER, XV:431
60. ER, XV:250-1
61. ER, I:361
62. Donald J. and Carol Martin Watts, “A Roman Apartment Complex,” Scientific American, Vol. 255, No. 6 (December 1986), pp.132-139
63. Paul Calter, “Ad Quadratum, the Sacred Cut & Roman Architecture,” in Geometry in Art and Architecture, Dartmouth College, on-line at: http://www.dartmouth.edu/~matc/math5.geometry/unit7/unit7.html . Calter explains how the ‘sacred cut’ squares the circle; his work also includes units on the Golden Ratio, the Pythagoreans, Squaring the Circle in the Great Pyramid, the Roman architect Vitruvius—all important aspects of the subject of this essay.
64. Paul Wheatley, The Pivot of the Four Quarters (1971)
65. Terence Greider, Origins of Pre-Columbian Art (Austin: U. Texas, 1982), pp.108-9
66. ER, VI:493
67. ER, VII:17-8
68. ER, VII:17
69. John G. Niehardt, Black Elk Speaks (1932), pp.165, 186, 189, 206
70. Ibid., p.237
71. The quotations from Black Elk come from Niehardt (1932), p.194. Aristotle’s claim that “What is eternal is circular and what is circular is eternal” (cited in Gorgio de Santillana and Herta von Dechend, Hamlet’s Mill [Boston, 1969], pp.48-9) seems more like a statement of a cosmological premise.
72. Understanding Media (1964), pp.118-9
73. The orthogonal land allotment system imposed by the government differed greatly from the traditional settlement and field layouts. Maekawa Kaname, “Moated Sites in Medieval Japan: Notions of Square and Circle,” Abstracts of Papers Presented at the Second Worldwide SEAA Conference, www.eastasianarchaeology.oreg/news/seaaconfsbstracts00.htm , accessed by the author on 10/03/00.
74. Terence Greider, Origins of Pre-Columbian Art (Austin: U. Texas, 1982); Eliade found similar patterns in villages in Bali and New Guinea and in the sacred lodges of the Algonquin (The Sacred and the Profane, trans. Willard R. Trask [Harcourt, Brace & World, many repr.], p.45).
75. Mircea Eliade, Forge and Crucible: Origins and Structure of Alchemy (U. Chicago, 1979), p.123
76. B.J.T. Dobbs, The Foundations of Newton’s Alchemy (Cambridge, 1975), cited in Mircea Eliade, A History of Religious Ideas, Vol. III (U. Chicago, 1988), p.260
77. Joseph F. Baugher, On Civilized Stars: The Search for Intelligence in Outer Space (Prentice-Hall, 1985), p.170
78. The quotation is from Mircea Eliade (1988, pp.250-1); for a link to standard introductions to eight Greco-Roman Mystery traditions, see http://en.wikipedia.org/wiki/Category:Mystery_religions . In 1990 (see footnote i above) the author pointed out that Joseph Smith had in common with Isaac Newton a commitment to the significance of the Primal Revelation. Eliade (1979, p.149) says of the Mystery religions: “The meaning and end of the Mysteries were the transmutation of man. By experiencing initiatory death and resurrection, the initiate changed his mode of being (he became immortal).”
79. Dickter’s work, entitled Number, Time and Archetype, is available as an on-line e-book at www.luo-shu.com/book/contents . On the mathematics of magic squares, see http://the-magic-square.blogspot.com/ . For a basic introduction to the Luo-shu, see http://en.wikipedia.org/wiki/Lo_Shu_Square .