93361115 Egyptian Numerology the Pythagorean Triangle and Its Esoteric Meaning
May 6, 2019 | Author: Giuseppe Marsico | Category: N/A
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Esoteric meanings of the triangle in the pythagorean and egyptian tradition...
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Egyptian Numerology: the Pythagorean Triangle and Its Esoteric Meaning Some Historic Notes and Brief Comments on Sacred Geometry Ant Anton onie iett ttaa Franc rancin ini, i, S.R. S.R.C., C., M.D. M.D.,, wit with h Benefactor Benefactor Taciturnus , F.R. F.R.C. C.
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t is not always easy to fully grasp the wonder and the depth of Pythagoras’s insights and accomplishments with the scarce information that we actually have about his school in the south of Italy and his esoteric teachings. We can, however, explore the inspirations of one of his most famous theorems which still bears his name today, concerning the triangle and its many ramifications.
Most Western mathematical development and architecture derived rom Pythagoras’s capacity or a deep understanding o the wisdom o even more ancient cultures. Pythagoras, who traveled extensively and dedicated his lie to learning the arts and sciences o ancient traditions, was able to synthesize all o this learning into practical aspects o harmony, mathematics, and the art o living. Acco Accord rdin ingg to his his disci discipl plee and oll ollow ower er,, Plato, the circle and the interaction o two circles, where the center o each circle lies on the circumerence o the other (the so-called Vesica Piscis ), ), became the core o all solids. Following this train o thought, Geometry
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Vesica Piscis, which literally means “the bladder o a ish” in Latin.
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becomes Music, and Music becomes Cosmic Harmony and the Music o the Spheres. As part part o this this esoter esoteric ic teach teaching ing,, the the ancient Egyptian right triangle, having a primarily mystical meaning, became the oundation o mathematical calculations and construction. Here we will consider the history and transormation o the amous Pythagorean riangle and the mathematical and symbolic importance o the relationships between the numbers 3, 4, and 5, rom the viewpoint o modern scholarship, which is deeply imbued with with the the pri primor mordi dial al trad tradit itio ion. n. Egyptian Numerology he Egyptians believed in the importance o numbers.1 Perhaps the oremost proponent o this reality was the leading student o symbolist Egypt, Réné Schwaller de Lubicz (1887-1961) rom Alsac Alsacee-Lor Lorrai raine, ne, France rance:: “Schwaller de Lubicz’s second thesis is mathematical. Both the deliberate use o harmonic proportions in art and architecture and the numerical basis underlying Egyptian myth compelled him to a detailed reconsideration o Pythagoreanism, and to the construction o a system o thought consonant with the masterpieces o Egypt— with with the act o an empir empiree that that last lasted ed our our thousand years. “‘Number is All’, declared the Pythagoreans. What is today called Pythagorean
number mysticism is Egyptian in origin (i not older still) and corresponds to the underlying philosophy behind all the arts and sciences o Egypt. In eect, what Pythagoras did was to un-dramatize myth—a strategy that had the advantage o talking directly to those capable o thinking along these lines. “he work o Schwaller de Lubicz and the independent but complementary work o a ew other contemporary thinkers (J.G. Bennett, or example) has made it possible to re-express Pythagorean theory in a way acceptable to our thinking. When we reapply this to Egyptian myth it becomes clear that these curious tales are based upon an understanding o number and the interplay o number, not upon animism, tribal superstitions, priestly euds, the raw material o history or dreams.”2 he divine signiicance o numbers is personiied by the goddess Seshat, as the “Enumerator” or “he Lady o Builders.”3 he Hymn ound in the Leiden Papyrus I350 (Old Kingdom 2686-2134 BCE) demonstrates that number symbolism has been practiced since ancient times: “he Leiden Papyrus consists o an extended composition describing the principal aspects o the ancient creation narratives. he system o numeration in the Papyrus identiies the principle/aspect o creation and matches each one with its symbolic number.”4 Ancient Egyptians understood that everything in the universe is animated by lie orces. hereore, each particle is in constant movement and has interactions due to the eect o these lie orces. hese principles were called neteru (gods/goddesses). Numbers designated these energetic aspects o nature. Consequently, the entire universe is animated and vibrant, and each acet is considered either “male” or “emale.” We can deduce the sense o living interactions by the expressions used in the Rhind Mathematical Papyrus (ca. 1650 BCE):
Vesica Piscis with igures within
“I go three times into the hekat (a bushel, unit o volume), a seventh o me is added to me and I return ully satisied.” 5 he same Papyrus also recommends that there are “Rules or enquiring into nature and or knowing all that exists, every mystery, every secret.”6
Ancient Egyptians understood that everything in the universe is animated by lie orces. Tereore, each particle is in constant movement and has interactions due to the eect o these lie orces. he concept o the relationships between 3, 4, and 5 is as ancient as Egypt, and as old as the Great Pyramid o Giza. his powerul, magniicent structure conceals deeply mystical traditions that only recently have begun to be recognized by archeological research. We know today exactly why the relationships between the number 3, 4, and 5 are so important, but it was a well-kept secret in ancient times. “Since the Pythagoreans considered the irst ten numbers to be seed patterns or all the principles o the cosmos, a geometer needs only create their shapes to model all the universal rhythms. he irst three shapes to emerge rom the vesica piscis , the triangle, square and pentagon (3, 4, 5) orm the only relationship, or ratios, required to generate all the rest (except or sevenness). hese relationships called the square roots … are expressible not as whole numbers but Page xli
as never-ending decimals...these ongoing relationships hold the structural pattern or all numbers and shapes that ollow.”7 he classical numbers 3, 4, 5 are represented in the very structure o the pyramid. he Great Pyramid o Giza was built in approximately 2560 BCE, at the time o the Old Kingdom: • “ThenumberOneisthewhole structure itsel. • The triangular faces represent the number 3. • Thesquarebaseisthenumber4 and inally • The four corners plus its apex complete the number 5.”8
“he upright, thereore, may be likened to the male, the base to the emale, and the hypotenuse to the child o both, and so Ausar [Osiris] may be regarded as the origin, Auset [Isis] as the recipient, and Heru [Horus] as perected result.”9 he short side o the right angle triangle is named “Ausar,” which corresponds to Osiris, the Father. he longer side is named “Auset,” corresponding to Isis, the Mother. And inally, the hypotenuse is called “Heru,” or Horus, the Son.10
The Egyptian Triangle From this image the constant relationships between the One, as the whole structure and 3-4-5 as its indivisible components are clearly shown. hese numbers had a proound mystical symbolism that becomes explicit in the explanations related to the Pythagorean triangle. he Egyptian 3-4-5 triangle is irst described by Plutarch in Moralia Vol. V :
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Pythagorean riangle with Egyptian Attributions. Figures ©2007 by Je Dahl/Wikimedia Commons.
hese names were assigned according to speciic criteria: the upright side (3) is likened to the “male,” the Father Osiris. he horizontal base (4) is related to the “emale,” the Mother Isis, and the hypotenuse (5) corresponds to the Son, Horus. he irst is the “origin,” the second is the “recipient,” and the Son is the “result.”11 As Plutarch describes: “hree is the irst perect odd number: our is a square whose side is the even number two; but ive is in some ways like to its ather, and in some ways like to its mother, being made up o three and two. And panta , (all), is a derivative o pente (ive), and they speak o counting as ‘numbering by ives.’ Five makes a square o itsel.”12 Pythagoras’s Contribution
H. Spencer Lewis and the AMORC Egyptian Expedition on the Giza Plateau, 1929. Photo rom the Rosicrucian Archives ©1929.
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Pythagoras’s contribution (sixth century BCE) consisted in translating this concept
he Pythagorean heorem: a 2 + b2 = c2
into the mathematical statement (as we would write it today) a 2 + b2 = c2, in which the square o the hypotenuse o a right triangle is equal to the sum o the squares on the other two sides. Pythagoras spread this notion by means o his teachings at the school o Crotona in southern Italy. he geometrical and mathematical demonstration o the Pythagorean theorem is explained by sacred geometrician Robert Lawlor in relation to the gnomonic expansion.13 “he contemplation o this igure leads to an understanding o one o nature’s most common orms o growth, growth by accretion or accumulative increase, in which the old orm is contained within the new. his is the way the more permanent tissues o the animal body, such as bones, teeth, horns and shells develop, in contrast to the sot tissue which is discarded and replaced. “his method o iguring the gnomon shows its relationships to the Pythagorean ormula. Shown here is the gnomonic increase rom the square surace area o 4 to the
Gnomonic Expansion related to the Pythagorean heorem.
square o 5, where the gnomon o the larger square 5 is equal to 1/4 o the initial square o 4. “Kepler, the amous astronomer and ormulator o the laws o planetary motion, stated: ‘Geometry has two great treasures: one is the theorem o Pythagoras, the other is the division o a line into mean and extreme ratios, that is phi , the Golden Mean. he irst may be compared to a measure o gold, the second to a precious jewel.’”14 The Egyptian Alchemical Triangle Egyptian concepts o number symbolism were popularized in the West and later became classic ormulas among medieval cathedral builders.
Egyptian Alchemical riangle
he alchemists also elaborated their theories using the same triangle. hey assumed that the whole igure represented the all-pervasive First Matter. he vertical side named by Osiris represented the three vital principles: Salt, Sulur, and Mercury. he long horizontal side named by Isis was the expression o the our basic elements: Fire, Water, Air, and Earth. Finally, the hypotenuse named by Horus showed ive divisions, representing the ive stages o the development o Lie: Minerals, Plants, Animals, Humans, and Enlightened Ones.15 he essential eature or the alchemist was that the diagram relected the vibrations o Spirit maniesting as Matter. From the First Matter, related to Osiris, Lie maniested by means o the our elements, into the physical Page xliii
world o minerals, vegetation, animals, completes the Great Work by becoming one with the Father, even as the line o Horus in and humans. he ive sections demonstrated the this triangle, completes itsel by arriving at the 16 theoretical distance separating the dierent point whence the line o Osiris descends.” he line o Osiris shows three alchemical realms. One ith is the evolution rom mineral to plant, one ith rom plant to divisions: animal, one ith rom animal to human, but • The first division of the Osiris the true ulillment o the ull humanity needs line corresponds to Mercury, since the last ith o the scale to be completed. this represents the higher consciousness—pure knowing—which is the irst emanation rom the one source Te concept o the relationships o being.17 between 3, 4, and 5 is as ancient • TheseconddivisionoftheOsiris as Egypt, and as old as the Great line shows the symbol o Sulur. Pyramid o Giza. Tis powerul, Sulur is related to sel-consciousness. magnifcent structure conceals deeply With respect to this symbol, Bernard revisan reminds us: “Gold is nothing mystical traditions that only recently else but quicksilver congealed by have begun to be recognized by its Sulur.”18 archeological research. • Thethirddivisionofthevertical line is represented by the symbol o Salt. he basis o the Great Work is he distance separating the evolution balance. By its symmetry, the symbol o plant to human is the same as the one o Salt represents the ancient adage, needed by an average human being to attain “As Above, So Below.”19 the high level o an adept. Similarly, the he line o Isis shows our alchemical neophyte, aspiring to become an adept, has a diicult road to climb beore coming close divisions. hey are the well known symbolical to the vertex, or highest point that represents elements that are the building blocks o all maniestation: Fire, Air, Water, and Earth. the desired union with the Divine. he alchemist considered this diagram as a teaching tool or explaining the rules o discipline, meditation, renunciation, and silence that were imposed on the aspirants beore they were considered ready or the most sacred and proound teachings. his same diagram also indicated the need to be humble and patient, since the irst steps on the path did not justiy a superiority o the neophyte over an ordinary human being. Only a long and serious preparation would permit true ulillment o the desired union with the Divine: “Above the natural human being is the he Christ in a Vesica Piscis o Light: The Transfiguration Rosicrucian one who has made himsel [or hersel] the of Christ . From an iconostasis in the Constantinople Digest style, mid-twelth century, Saint Catherine’s Monastery, subject o the Great Work, who is adept, No. 1 Sinai (Egypt). master and Magus. he Human being who 2009 Page xliv
Te shape o the human eye itsel is a Vesica Piscis. Te spiritual signifcance o “seeing eye to eye” to the “mirror o the soul” was highly regarded by numerous Renaissance artists who used this orm extensively in art and architecture.
he iveold symmetry o the hypotenuse symbolizes the development o lie in its evolution toward the Light. It is related to the basic aspects o the ourold symmetry o elemental nature represented by the line o Isis. he process described by sacred geometer Robert Lawlor can be summarized succinctly: “at the origin and end all dierentiation merges toward unity.” An object a and its human perceiver b are contained within an all-pervasive unity, that can be symbolized by the number 1. his can be expressed by the ormula: a:b = b:1, in which the object is related to the human, as the human is related to the whole.20 Lawlor suggests that this is a proportion upon which the “experience” o knowledge can be ormulated and that this is “the goal o dynamic meditation.”21
he Vesica Piscis is ormed by the intersection o two circles or spheres whose centers intersect in such a way that each center’s circle lies on the circumerence o the other circle. It was known in the early civilizations o Mesopotamia, Arica, and Asia, and requently used in Christian iconography. It is revealed in the construction o ancient doors and portals.23 “It has been the subject o mystical speculation at several periods o history, perhaps irst among the Pythagoreans. he mathematical ratio o its width (measured to the endpoints o the ‘body,’ not including the ‘tail’) to its height was reportedly believed by them to be 265:153 (namely 1.73203...). he geometric ratio o these dimensions is actually the square root o 3, or 1.73205... (since i straight lines are drawn connecting the centers o the two circles with each other, with the two points where the circles intersect, two equilateral triangles join along an edge). he ratio 265:153 is an approximation to the square root o 3, with the property that no better approximation can be obtained with smaller whole numbers.”24
The Profound Secret of the Pythagorean School he proound secrets o the Pythagorean School can now be discussed in simple diagrams. We can consider this as a graphic demonstration o the Divine Light embedded into the human body. he explanation must begin by showing the square root o 3, or the Vesica Piscis (Latin: “bladder o a ish”).22
Hindu Goddess Kali and God Bhairava in Union in a Vesica Piscis , 18th century. Watercolor rom Nepal, git o Dr. and Mrs. Robert S. Coles to Los Angeles County Museum o Art
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Platonic Solids and other images contained in Metatron’s Cube, created by Sloth Monkey/Wikimedia Commons
Modern researchers have commented on the signiicance o this deeply symbolic orm: “his symbolic intersection represents the ‘common ground,’ ‘shared vision,’ or ‘mutual understanding’ between equal individuals. he shape o the human eye itsel is a Vesica Piscis. he spiritual signiicance o ‘seeing eye to eye’ to the ‘mirror o the soul’ was highly regarded by numerous Renaissance artists who used this orm extensively in art and architecture. he ratio o the axes o the orm is the square root o 3, which alludes to the deepest nature o the triune which cannot be adequately expressed by rational language alone.”25 “his almond space is the crucible o the creating process.”26 “he irst three shapes to emerge rom the vesica piscis, the triangle, square and pentagon (3,4,5) orm the only relationship, or ratios, required to generate all the rest
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Metatron’s Cube
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(except or sevenness)...these ongoing relationships hold the structural pattern or all numbers and shapes that ollow.”27 The Metatron Cube Understanding the ratios o the triangle, square, and pentagon permits us to grasp the importance o a more complex igure, such as the Metatron’s Cube, which contains twodimensional images o the Platonic Solids 28 and many other primal orms.29 he two large triangles orm a hexagram that is symbolic o the harmony o the opposites. One triangle points downwards and one points upward. his is popularly called the Star of David and is symbolic o ininite celestial light. We can imagine that this light is all pervasive and circumscribes everything and every orm inside it. he diagram shows that this ininite light is also maniested light, as it is symbolically included in a hexagon, a six sided igure. In Qabala, six is the number o the sephira ipereth, and thereore this hexagon represents the beauty and spiritual love maniested in the natural universe. At irst glance, we can perceive that the whole universe is permeated by this well balanced divine light symbolizing peace and harmony. Inside, at the very core o the construction we can detect another pentagon in which a pentagram is inscribed. he pentagram is the symbol o humanity maniested, as many traditions relate ive to the human person. In the Metetron’s Cube, the central circle is white and empty, but we can easily mentally draw another hexagram inside it using the points where the surrounding
Heinrich Cornelius Agrippa, Image o a human body in a pentagram rom Libri Tres de Occulta Philosophia , 153133. Symbols o the sun and moon are in center, while the other ive classical “planets” are around the edge.
circles touch each other. he symbolism o this inner balance o opposites maniests the potentiality o the human beings o bringing to shine inside themselves the same Light o Peace and Harmony that surrounds them: “Horus is above me, at my right side, at my let side, Horus is in ront o me, behind me, below me Horus is everywhere around me, Horus is inside me.”30
systematic maniestations o natural laws. his proves that the early thinkers did not look on lie as being illed with accidental occurrences and uncontrolled; rather they are a Divine or perect scheme o supreme conception.”31 he images we have described in this brie introduction, and many others derived rom these mystical proportions are always available or our meditation and inspiration. heir power is undiminished over the centuries, and concentration upon them as mandalas can yield deep insights. As the spirit o Pythagoras is very much alive in the Rosicrucian Order, we can ind in these “images as mandalas” a path or discovery o the supreme conception o the cosmic whole, which is a perect scheme in which everything is in order, and is beautiul, both above and below.
A Divine Scheme of Supreme Conception H. Spencer Lewis, ormer AMORC Imperator, wrote: “ime has proven that many o these ideas were correctly ormed.... I we agree with the early philosophers that the whole universe can be explained according to geometrical laws and principles, then, we should not be surprised to ind the numbers to have a peculiar signiicance, as Pythagoras taught. “It is quite evident that the earliest thinkers were determined to ind law and order and a mathematical basis or the
H. Spencer Lewis and his wie Martha on the AMORC Grand our o Egypt, 1937. Photo rom the Rosicrucian Archives ©1937.
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EndnotEs 1See works o René Schwaller de Lubicz, especially
A Study of Numbers: a Guide to the Constant Creation of the Universe (Rochester V: Inner raditions, 1986); The Temple of Man: Apet of the South at Luxor , 2 vols (Rochester V: Inner raditions, 1998). 2 John Anthony West, Serpent in the Sky (New York: he Julian Press, 1987), 42, 44. See also Carmen Blacker and Michael Loewe, eds.; with contributions by J.M. Plumley et al., Ancient Cosmologies (London: Allen & Unwin, 1975), 24. 3Moustaa Gadalla, The Egyptian Sacred Numerology , published online at: www.egypttehuti.org/articles/sacred-numerology.html. 4Ibid. 5Ibid. 6Ibid. 7Michael S. Schneider, A Beginner’s Guide to Constructing the Universe (New York: HarperCollins 1994), 35. 8Paul Foster Case, The Esoteric Keys of Alchemy (Vancouver: Ishtar Publishing, 1932), 54. 9Plutarch, “On Isis and Osiris,” 56 Moralia Book 5. ranslated by Moustaa Gadalla, The Egyptian Sacred Numerology . See endnote 3. 10riangle and discussion ater description in Moustaa Gadalla, The Egyptian Sacred Numerology . See endnote 3. 11Ibid. 12Plutarch, “On Isis and Osiris,” 56 Moralia Book 5. ranslated by Moustaa Gadalla, The Egyptian Sacred Numerology . See endnote 3.
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13Robert
Lawlor, Sacred Geometry, Philosophy and Practice (London: hames and Hudson, 1982), 64. Image based on description by Lawlor. 14Ibid., 53, 64. 15riangle image and discussion rom a description in Case, Esoteric Keys , 55. 16Ibid. 17Ibid., 79. 18Bernard revisan, quoted in Case, Esoteric Keys , 85. 19Case, Esoteric Keys , 93 20See Lawlor, Sacred Geometry , 42, 46. 21Ibid., 46. 22Bruce Rawles, Sacred Geometry home page, www.geometrycode.com/sg/index.shtml. 23Ibid. 24“Vesica Pisces” at Wikipedia , http://en.wikipedia. org/wiki/Vesica_piscis. 25Rawles, Sacred Geometry home page. 26Schneider, Beginner’s Guide , 32. 27Ibid., 35. 28See “Platonic Solids,” at Wikipedia , http:// en.wikipedia.org/wiki/Platonic_solids. 29Rawles, Sacred Geometry home page. 30 Julie Scott, Grand Master o the English Grand Lodge o AMORC, during meditation in Egypt, December 2008. 31H. Spencer Lewis, “Mystical Numbers” in Essays of a Modern Mystic , 6th ed. (San Jose: Rosicrucian Order, AMORC, 1978), 171.
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