SATURN IN TRANSIT
BOUNDARIES OF MIND BODY AND SOUL
Erin Sullivan
1990
Page 10
unique to his or her personal ,background and the ordeal unlike anyone else's. With the same transit that brings forth the monsters comes the solution to the problem and it, too, is specific to what each individual seems to need to face at that time of life. As we shall see, the duality of Saturn often poses the problem and the solution in the same shape.
Finally, notwithstanding the conflicting reports of the battles of the Giants and the Titans, and the problematic interpretations of the wars'; Kronos' conclusive fate was to retire, rather gracefully considering the magnitude of his crimes and the circumstances of his banishment, to the Islands of the Blessed w~ere, on the Elysian Fields, he ruled benignly over returned heroes and other chthonian inhabitants. Saturn, born of earth, returned to earth.
The Mythic Kronos aild the Roman Saturn
The greatest difficulty we have in understanding myth today is our tendency to think of sequence or of duration in chronological time, which is what Mircea Eliade called 'profane time' because it exists in the realm of everyday life. The origins of myth lie in oral tradition (muthos), songs of adventures which took place in 'sacred time', in an atemporal realm, not in a literate society which measures time in a linear fashion.lo Because of the mythopoeic mind, the voice of the collective as heard through the poets of this ancient oral tradition, it is difficult for us to differentiate between what happened, when and to whom; for in sacred time there is no chronology. Eliade demonstrates that all human action is representative of an ancient ritual from a distant past, primarily actions of gods, and that we re-enact these prototype gestures in imitation of the gods in all subsequent behaviour. This view adds a dimension not only to astrology and planetary dynamics, but to our
THE EVOLUTION OF AN ARCHETYPE
11
modern lives also, insofar as it points to a sacred precedent. The horoscope is peopled by anCient gods, each with their individual nature. The relationship that each of these gods had with the others is enacted in the horoscope in a most astounding way. The actions ofthose gods seem uncannily like our own, even if not in detail - usually ours are not quite so dramatic; but they ,are identical in essence - they are archetypal experiences. Specifically, the anCient Greek Titan god, Kronos, has been one of the most durable of all the mythological images and symbols, sllrviving several cultural transitions and becoming assimilated into subsequent religions, philosophies and occult traditions as we shall see.
To begin with the marriage of heaven and earth: ~the union of Gaia and Ouranos - Earth and Sky - is paralleled by the marriage of their children Kronos and Rhea, and subsequently by Zeus and Hera. The archetypal mother and father' and the politics of such polarities in our own male-female relationships are analogous to the hieros gamos, the original, or sacred, marriage. Intrafamilial struggles become clearer when we see what gods did before us. Kronos' castration of his father is the sacred precedent for all father and son rites of maturation and separation, and the subsequent rejection of Kronos by his mother in preference for Rhea establishes an archetype for the necessary defection of the son from maternal rule. Saturn transits can sever one from the past as brutally as the infinite realm of Gaia and Ouranos was terminated. The seemingly paradoxical but entwined combination of fear of authority, the desire to overcome authority and the will to power, i.e. to become an authority, is embodied in the mythology and the archetype of Saturn.
The ambivalence of Saturn is better understood when we realize that he occupied concurrently two positions in the minds of the anCients: that of the benign, compassionate and liberal ruler of the Golden Age as well as the ruthless son who castrated his father and later, as a father himself, devoured his own children. A realized Saturn brings a peace of mind that cannot be paralleled, but an unacknowledged, unassimilated or projected Saturn can afflict an individual with chronic problems and power struggles. The covetous, callous ruler contrasts with the gentle monarch of
THE WISE WOUND
169
powerful .hypnotic' image indeed to which a woman by choice or training, may link her menstrual cycle. In addition modem work shows that the moon's light, or any other indirect night-time lighting at mid-cycle, may actually stimulate ovulation physiologically.
With these linked ideaS in mind, what evidence is there that these capacities have actually been used in the past, as they may be used in the future?
The name .Hera' means .Womb'. If, as has sometimes been said, .Hera' means 'Mistress' this is because she gives the laws, and the women's laws are the ways of the womb. .Thesmophoria', the great Greek women's fertility festival, means .Iaw-bearing'. These laws include las reglas, or the .way of all women', the menstrual rhythm. A reasonable derivation from Hesiod and Homer of the name 'Hera' is 'womb', and this interpretation is backed,by the fact that she is called panton genethla, 'origin of all things', which is the 'womb'. The great Goddess's name in most cultures in derivation means .womb' or .vulva': the Goddess is Genetrix. The womb gives birth, and it also menstruates.
'Astarte' or 'Ashtaroth' means 'womb', or 'that which issues from the womb'. 'Pallas Athena' means literally .Vulva-vulva'. The name of the Greek Goddess of childbirth is Eileithyia, and her name means .fluid of generation' which in this context is 'menses', which was thought to enter into the composition of the child and the milk.
The original home, or dwelling-place, was the womb of the woman, so cities may be called by the name of a goddess. Thus, the capital city of the ancient land of Og was called. Ashtaroth-Qamaim' which means .Womb of the Two Horns', since the human (and divine) womb is two-homed with its Fallopian tubes. It is the emblem of fertility and containment, and the sacred bucranium or"ox-head decorating Greek and Roman temples, was a womb-emblem. A locality is where you lie in childbed (Gk. lokhos) and produce the child and Jhe magical lochial blood of childbirth: the blood of the person's first place of arrival. The Queen is cwen or wife with the quim, which is a combe or cwm, the gune (woman) is a goddess when she#is gana and ;ani (woman) with a yoni, or cunt. Gens is wife, as in generation, or .great tribe'. This is all natural, as all human beings are born from a womb, and without this first .magic' there would be no consciousness and therefore no human religion, or anything else. So Hera's womb is literally panton genethla.1
Page 170
170
The Wise Wound
The Heraion, the temple of Hera, the cunt-place, was for centUries in Ancient Greece 'the sanctuary of the whole country, originally in the same way as the temple of Jerusalem, for instance, was a unique temple of Israel', says Carl Kerenyi. When you have an altar, you have to have a facing-partner: that is, an emblem of the diety whose rites are celebrated at this place. On the Christian altar this is a crucifix. A Poseidon altar would have the sea 'in its incalculable mobility'. An ahar of Helios might have the sun's ecliptic. The terrace of the Argive Heraioll was an immer.se cult stage for viewing the moon, and Hera herself was a single goddess in 'three phases'. Her myth in JakT times, as in Homer, is assGdated with Hera's 'sulks' but in the true cult sense this was a descent of Hera into the underworld, the low point (or, as we might say, PMT) being associated with the new moon. Hera is sometimes shown with a pomegranate, the red fruit full of seeds which associates her with the Queen of the Underworld, Persephone. Pausanias says the story of the pomegranate is 'rather secret'. These goddesses guided through the underworld. Prosymne is the new moon, and Prosymnos was Dionysos' guide to the underworld, as, according to Graves, Eurydice was the guide of Orpheus. Prosymne was Demeter's epithet, the earth-mother in her underworld aspect. The Goddess Prosymna was summoned in the name of the new moon when it lingered in the darkness. In Athens, Pallas Athena ('Vulvavulva') was reborn with the new moon, just as Hera was reborn from a bath in which she had her virginity restored, in the same manner as the womb is restored after the menstruation for a fresh cycle, with a fresh womb-lining. There was a ceremony of the washing of Athena's 'laundry'. In the Hera cult there would similarly be a pwcession of 'freed' women after the purification of a wooden figure of the goddess, shortly after the new moon had appeared, following 'the low point of Hera's periodic being'.
It is possible that the original temple on the Acropolis at Athens belonged to such a cult. Hera was worshipped in her 'great part-secret, part-public cult. .. in her transformations according to the Moon's phases' and so was Athena. That is to say, that the Acropolis was dedicated to Vulva. This is as shocking as supposing that where St Paul's great dome now stands in London was originally the site of moon-worship, and that the name 'London' was originally 'Laundon' in Celtic, or Moon-Town. There is however evidence for this also.2
Page 1 6 2
MYTHS TO LIVE BY
through which she is happily strolling is not of Heaven but H~ll. She is indignant. "I tell you, I know I am not in Hell," she insistS, "because I feel no pain." Well, if she likes (she is told), she can easily stroll on over the hill into Heaven. However, the strain of remaining there has been found intolerable (she is warned) .for those who are happy in Hell. There are a few-and they are mostly English-who nevertheless -remain, not because they are happy, but because they think they owe it to their position to be in Heaven. "An Englishman," states her informer, "thinks he is moral when he is only uncomfortable." And with that telling
Shavian quip, I am carried to my final reflections on this chapter's
theme. . .
For it was in the legend of the Holy Grail that the healing work was symbolized through which the world torn between honor and love, as represented in the Tristan legend, was to be cured of its irresolution. The intolerable spiritual disorder of the period was represented in this highly symbolic tale in the figure of a "waste land" -the same that T. S. Eliot in his poem of that name, published in 1922, adopted to characterize the condition of our own troubled time. Every natural impulse in that period of ecclesiastical despotism was branded as corrupt, with the only recognized means of "redemption" vested in sacraments administered by authorities who were themselves indeed corrupt. People were forced to profess and live by beliefs they did not always actu~lly hold. The imposed moral order held precedence over the claims of both truth and love. The pains of Hell were illustrated on earth in the torture of adulteresses, heretics, and other villains, torn apart or set afire in publk squares. And all hope of anything better was pitched high aloft to that celestial estate of which Gottfried spoke with such scorn, where those who could bear neither grief nor desire were to be bathed in a bliss everlasting.
In the legend of the Grail, as rendered in the Parzival of Gottfried's very great contemporary and leading literary rival, Wolfram von Eschenbach, this devastation of Christendom is symbolically attributed to the awesome wounding of the young
Page 184 I 8 f
MYTH S TO L I "y E :B Y
as ChunKuo, "of the Warring States," the first ruler of a united empire, Shih Huang Ti "(22 1-"207 B.C.), gov~rried, according to his claim, by the mandate of Heaven, under Heaven's law. ,.
It is then liardly to be wondered if the,enthusiastic Hebrew <!u"tho(of IsaiaI140--"55, who was a contemporary of Cyrus,ihe-Great and living witness of the Persian restoration-to Jerusalem of its people, gives evidence in his prophecies of the influence of Zoroastrian ideas; for example, in the famous passages of Chapter 45: "Thus says the Lord to .his anointedlc tq,Cyru~ . . . 'I form light and create darkness, I make weal and create woe, -I am the Lord, who do all these things.' " It is in these chapter-s of the socalled Second or Deutero Isaiah that we find the earliest celeJ:>rations of Yahweh not simply as the greatest and most powerful god among.,gods, but as the one God of the universe, in whom not only Jews but also the gentiles a,re to find salvation: "Turn to me and be saved, all the ends ~of the earth!" we read, for instance. "For I am God; and there is nb other" (Isaiah 45:22). Moreover, whereas the earlier idea of the Messiah of the pre-exilic prophets had been simply of an ideal king on- David's throne, "to uphold it," as in Isaiah 9:6-7;; "with justice and with righteousness from this time forth and for evermore"; in the post-exilic period, and pa;ticularly In the very late, apocalyptic writings of the Alexandrian age-as, f(jr instance, in the Book of Daniel ~7: I3-27-there is the notion qf one who, at the end>pf historic time, should be given, over "all peoples, nations, and l~hguages," "an everlasting dominion, which shall not pass a\Xay." And at that time, furtherm.ore, "Many of those who sleep in the dust of the earth shall awake1; some to everlasting life, and some to shame and everlasting co!ltempt" (Daniel
)2:2)." - ,-'ii
There can be no doubt of the influence of Zoroastrian eschatol. ogy on such ideas as these of the.end of the world and resurrec
tion of the dead. Moreover, 'in the Essene Dead Sea Scrolls oLthe
- .
last century B.c.,,,,-the influel1c~ of Persian thought is apparent-""at
every turn. ~heir period itself, in fact, w~ one of such te~rible tu.. omIt that' the end~f the world and comirtg of the savior Saoshyant
Joseph Campbell
[ I 54
M Y T H S T 0 L I V E ,8 Y
God. And that Christ may not have actually suffered in that loving act we may take from a saying of the mystic Meister Eckhart: "To him who suffers but not for love, to suffer is suffering and hard to bear. But one who suffers for love suffers not, and his suffering is fruitful in God's sight."
Indeed, the very idea of a descent of God, into the world in love to invoke, in return, man's love to God, seems to me to imply exactly the contrary to the statement 1 have just quoted of Saint Paul. Implied, rather, it seems to me, is the idea that as mankind yearns for the grace of God, so God for the homage of mankind, the two yearnings being reciprocal. And the image,of the crucified as both true God and true man would then seem to bring to focus the matched terms of a mutual sacrifice-in the way not of atonement .in the penal sense, but of at-one-ment in the marital. And further: when extended to symbolize not only the one historic moment of Christ's crucifixion on Calvary, but the mystery through all time and space of God's presence and participation in the agony of all living things, the sign of the cross would then have to be looked, upon as the sign of an eternal affirmation of all that is, ever was, or shall ever be. One thinks of Christ's words reported
in the Gnostic Gospel According to Thomas: "Cleave a piece of wood, I am there; lift up the stone, you will find me there." Also, those of Plato in the Timaeus, where he states that time is "the moving image of Eternity." Or again, those of William Blake: "Eternity is in love with the productions of time." And there is a memorable passage in the writings of Thomas Mann, where he celebrates ,man as "a noble meeting [eine hohe Begegnung] of Spirit and Nature in their yearning way to each other."
, We can safely say, therefore, that whereas some moralists may find it possible to make a distinction between two spheres and reigns-one of flesh, the other of the spirit, one of time, the other of eternity-wherever love arises such definitions vanish, and a sense of life awakens in which all such oppositions are at one.
The most widely revered Oriental personification of such a world-affirming attitude, transcending opposites, is that figure of
Page 156
Page 157
The Mythology of Love e
I 5'7 ]
yourself!-.and even better, beyond that, in the words that I take to be the highest, the noblest and boldest, of the Christian teaching: "Love your enemies and pray for those who persecute y<>,u, so that you may be sons of your Father who is in heaven; for he makes his sun to rise on the evil and on the good, and sends rain on the just and on the unjust. . . ."
In all the great traditional representations of love as compassion, charity, .or agape, the operation of the virtue is described as general and impersonal, transcending differences and even loyalties..,And against this higher, spirityal order of love there is set
,generally in opposition the lower, of lust, or, as it is so often called, "animal passion," which is equally general and impersonal, transcending qifferences and even loyalties. Indeed, one could describe the latter most accurately, perhaps, simply as the zeal of the organs, male and female, for each other, and designate the writings of Sigmund Freud as the definitive modern text on the subject of such love. However, in the European twelfth and early thirteenth centurie~, in the poetry first of the troubadours of Provence, and then, with a new accent, of the Minnesingers, a way of experiencing love came to expression that was altogether different from either of those two as traditionally opposed. And since I regard this typically and exclusively European chapter of our subject as one., of the most important mutations not only of human feeling,
"
but also of the spiritual consciousness of our human race, I am
going to dwell on it a little, before proceeding to the final passages of this chapter.
To begin with, then: Marriage in the Middle Ages was almost exclusively asocial, family concern-as it has been forever, of
course, in Asia, and is to this day for many in the West. One was married according to family arrangements. Particularly in' aristocratic circles, young women hardly out of girlhood were married off as political pawns. And the Church, meanwhile, was sacramentalizing such unions with its inappropriately mystical language about the two that were now to be of one ft~sh, united thrQugh love and by God: and let no man put asunder what God. hath
Page 158
Page 159
The Mythology of Love
I 59 ]
that is to say, neither of Heaven nor of Hell, but of earth; grounded in the psyche of a particular individual and, specifically, the predilection of his eyes: their perception of another specific individual and communication of her image to his heart-which is to be (as we are told in other documents of the time) a "noble" or "gentle" heart, capable of the emotion of love, amor, not simply lust.
And what, then, would be the nat~re of a love so born?
In the various contexts of Oriental erotic mysticism, whether of
the Near East or of India, the woman is J!lystic~ly interpreted as an occasion for the lover to experience depths beyond depths of transcendent illumination-much in the way of Dante's appreciation of Beatrice. Not so among the troubadours. The beloved to them was a woman, not the manifestation of some divine principle; and specifically, that woman. The love was for her. And the celebrated experience was an agony of earthly love: an effect of the fact that the union of love can never be absolutely realized on this earth. Love's joy is in its savor of eternity; love's pain, the passage of time; so that (as in Gottfried's words) "bitter sweetness and dear grief" are of its essence. And for those "who cannot bear grief, and desire but to bathe in bliss," the ambrosial potion of this greatest gift of life is a drink too strong. Gottfried even deified Love as a goddess, and brought his bewildered couple to her hidden wilderness-chapel, known as "The Grotto for People in Love," where stood, in the place of an altar, the noble crystalline bed of love.
Moreover-and this, to me, is the most profoundly moving passage in Gottfried's version of the legend-when, on the ship sailing from Ireland (with which scene Wagner's opera commences), the young couple unwittingly drank the potion and became gradually aware of the love that for some time had been quietly growing in their hearts, Brangaene, the faithful servant who by chance had left the fateful ftask unattended, said to them in dire warning, "That ftask and what it contained will be the death of you both!" To which Tristan answered, "So then, God's will be-done, whether
SATURN IN TRANSIT
BOUNDARIES OF MIND BODY AND SOUL
Erin Sullivan 1991
Page 163"
All heroes enter a wasteland, and endure times of challenge and testing. There are distinct time periods that are critical for action which are followed by experimentation and then maintenance. If the maintenance period extends beyond its usefulness then another critical period of change-on-demand occurs, and- so on. The archetypal hero's journey with all of its phases and turning-points parallels the Saturn transit as it moves over the angles and travels through the quadrants of the horoscope."
SATURN IN TRANSIT
BOUNDARIES OF MIND BODY AND SOUL
Erin Sullivan 1991
THE PERSONAL HEROIC JOURNEY
THE ATONEMENT (IC)
"When he arrives at. the nadir of the mythological round, he undergoes a supreme ordeal and gains his reward. The triumph may be represented as the hero's sexual union with the goddess-mother of the world (sacred marriage), his recognition by the father-creator (father atonement), his own divinization (apotheosis), or again - if the powers have remained unfriendly to him - his theft of the boon he came to gain (bridge-theft, fire-theft); intrinsically it is an expansion of consciousness and therewith of being (illumination, transfiguration, freedom). Joseph Campbell 9
“I am He that lives, and was dead; and, behold, I am alive forevermore.” Revelation 1:18
No curved lines-A.E.F.H.I.K.L.M.N.T.V.W.X.Y.Z.No straight lines-C.O.S.No enclosed areas-C.E.F.G.H.I.J.K.L.M.N.S.T.U.V.W.X.Y.Z.Horizontal symmetry-B.C.D.E.H.I.K.O.X.Vertical symmetry-A.H.I.M.O.T.U.V.W.X.Y.Roman numerals-C.D.I.L.M.V.X.Just dots in Morse code-E.H.I.S.Just dashes in Morse code-M.O.T.Horizontal and vertical symmetry-H.I.O.X.Look the same upside down-H.I.N.O.S.X.Z.Can be drawn in one stroke-B.C.D.G.I.J.L.M.N.O.P.R.S.U.V.W.Z.
Capitals which look like lowercase-C.O.P.S.U.V.W.X.Z.
crystalinks.com/egyptreligion.
The ancient Greek word for interpreter/translator is Hermꮥus, directly related to the name of the god Hermes.
MYTHOLOGY
Hermes, the herald of the Olympian gods, is son of Zeus and the nymph Maia, daughter of Atlas and one of the Pleiades. Hermes is the god of shepherds, land travel, merchants, weights and measures, oratory, literature, athletics and thieves, and known for his cunning and shrewdness. Most importantly, he is the messenger of the gods. Besides that he was also a minor patron of poetry. He was worshipped throughout Greece - especially in Arcadia - and festivals in his honor were called Hermoea.
According to legend, Hermes was born in a cave on Mount Cyllene in Arcadia. Zeus had impregnated Maia at the dead of night while all other gods slept. When dawn broke amazingly he was born. Maia wrapped him in swaddling bands, then resting herself, fell fast asleep. Hermes, however, squirmed free and ran off to Thessaly. This is where Apollo, his brother, grazed his cattle. Hermes stole a number of the herd and drove them back to Greece. He hid them in a small grotto near to the city of Pylos and covered their tracks.
Before returning to the cave he caught a tortoise, killed it and removed its entrails. Using the intestines from a cow stolen from Apollo and the hollow tortoise shell, he made the first lyre. When he reached the cave he wrapped himself back into the swaddling bands. When Apollo realized he had been robbed he protested to Maia that it had been Hermes who had taken his cattle. Maia looked to Hermes and said it could not be, as he is still wrapped in swaddling bands. Zeus the all powerful intervened saying he had been watching and Hermes should return the cattle to Apollo. As the argument went on, Hermes began to play his lyre. The sweet music enchanted Apollo, and he offered Hermes to keep the cattle in exchange for the lyre. Apollo later became the grand master of the instrument, and it also became one of his symbols. Later while Hermes watched over his herd he invented the pipes known as a syrinx (pan-pipes), which he made from reeds. Hermes was also credited with inventing the flute. Apollo, also desired this instrument, so Hermes bartered with Apollo and received his golden wand which Hermes later used as his heralds staff. (In other versions Zeus gave Hermes his heralds staff).
Being the herald (messenger of the gods), it was his duty to guide the souls of the dead down to the underworld, which is known as a psychopomp. He was also closely connected with bringing dreams to mortals.
It was Hermes who liberated Io, the lover of Zeus, from the hundred-eyed giant Argus, who had been ordered by Hera, the jealous wife of Zeus, to watch over her. Hermes charmed the giant with his flute, and while Argos slept Hermes cut off his head and released Io. Hera, as a gesture of thanks to her loyal servant, scattered the hundred eyes of Argos over the tail of a peacock (Heras' sacred bird).
Known for his swiftness and athleticism, Hermes was given credit for inventing foot-racing and boxing. At Olympia a statue of him stood at the entrance to the stadium and his statues where in every gymnasium throughout Greece. Apart from herms, Hermes was a popular subject for artists. Both painted pottery and statuary show him in various forms, but the most fashionable depicted him as a good-looking young man, with an athletic body, and winged sandals and his heralds staff.
METAPHYSICS
Hermes was the Master of all arts and sciences, perfect in all crafts, Ruler of the Three Worlds, Scribe of the Gods, and Keeper of the Books of Life, Thoth Hermes Trismegistus - the Three Times, the "First Intelligencer" - was regarded by the ancient Egyptians as the embodiment of the Universal Mind. While in all probability there actually existed a great sage and educator by the name of Hermes, it is impossible to extricate the historical man from the mass of legendary accounts which attempt to identify him with the Cosmic Principle of Thought.
The Emerald Tablets of Thoth Hermes Trismegistus
Hermes was Thoth the Egyptian Scribe Who wrote the story of our reality
Hermes was Mercury - the Messenger of the Gods. Greek Gods of Mount Olympus - Mercury inherited Hermes' attributes
Hermes was the Trickster Representing the Duality of our Reality - Good vs. Evil. In the role of the Trickster he was also the god of thieves and deceit. His lessons are taught by tricking you.
Hermes was Merlin the Magician - linked to his role as the Trickster. His lessons are taught by Magic to help you see through the illusion of Time and Emotion.
Hermes - Ancient Mystery School Teachings - the Hermetic Orders - which is linked with Alchemy of Consciousness - - Moving to a higher frequency of thought - Releasing the Hermetic seals at the End of Time - Or the Illusion of Time - The release of the soul from the physical form.
An end time happens when the grid program of a reality - triggers a release into the next level of consciousness and all souls return to higher light.
HEINZ PAGELS
Page 90
Ateacller'" of2mathematics in 'pol'trevolutiopary"' Iran began his lec,ture onoprobabilify theory by holdfug- up a )lie which he was goingto~ use in a.-demo!}stration. 'Bewre-o he. could begin, an IslamiC fundamentalist student cried oUt, ""A satanic artifactr'~referring, of"-cours{;;' to. the die. Toe teach"et lost
5isry:ob 'and1alfuost his life.;The hotion olprobabilityis anti
tij:«&cal~o~th(jse interpr~tafions of"l~lam which maintain that Gq"<llknows eve~ing~there is 110 placefof,chance (or many
religi()U~fifndamentalists~ . ~~ -- ,-:' - -',
'" Had the teacher been pe!Illitled to give~ h~ lecture, we
'<;ilI1 im~giIfe <what'he 'would" have told the stu<ten.t~ He IDay Ilaye emphasized the application of probability theory to"lhe
reaI"Wodd and begun with :the operational d~finition of ptob--.,
ability. This kirid""of def!nition,is 'required because we do not hav~:: aii'iritrillsic' definition of randomness-something we learlledJrom the last chapter. We cannot determine whether Q)- not iiP actual process is truly .random. All we can do is~ to check and see if the process passes lots of testsJthat random
,processes are-supposed to pass.-so that it is "sufficiently random/' III practice this works -fine, but there is al'f,ays a problem in pHnciple-we ~ever'know.Jf someone will Gevise / Page 91 / ,,~cleveJ' new test that I:eveals thaL what we thoughf was '
~randouFreally is not. ~ - - iC
-g In spite' of th~ matllematical difficultY of deJlning~ ran-,""
domness, we. can take a'f-praginatic attitude, as dId. Richard
von MisJ!s. H~ said thatthe practical ~defInition oh randOlrt~ proces's'is:tl1at it is unbeatable. The pJ"actical defi!lition~orks~~ like this. Suppose that a gambling m:ichine'is built thaLwfu~~
iHt geQerates random numbers. Then over tl)e long run yoU::~ cannot beat it withimy strategy, and we could say that the~ numbers are really random ror.;; practical purposes. H there
~was it flaw in the machine and the numbers were not reallyrandom"and sO!lle"specmc number came up more'often, then we could use - this knowledge~ to .b~at the machine~ ReaJ rkdomness is unbeatable. This practical definition of ran-"'" domness is good for the real world. .GambJinghouses and= insurance companie,s altuse it. And because~rand9mness is unbeatable and they base their business on that fact; they ~lwayswin. ~.
. I( we look in nature for randomness we nnd the besf place to, look for chaos "is right in the atom-there is no" 0,
randomness like qU~tum randomn~s. H we test processes :.
like nuclearnfdioactive' decays w.to particles, they pass all the
Aests for randomnesS: ,When and where an atom decays is..
truly random. Wl:!!le we can lmagine a flaw in.;a g~bling'7 machine, physicisfs~find no. such flaw in the quantum' world. _. Quantum randomness cannot be beat; the God that plays dice is an honest gambler. But how do we study such randomness?
~; - Laplace and other mathematicians had the great Illsight that although individuaL random events ~ere mel!ningless; the distribution of those events was not and could be the~
subjectcof an exact science-probability theory. The~ntral
idea of ~probabj)ity theory is. the notion of a probability
'distribution-the assignment or probabilities to a set of re""" lated events. A simple example is coin flipping. The probabil
ity of heads is one-half and tails is one~half, and if you aHd upthe probabilities, they add up to one. A probability of one is the same as certainty: If you flip a coin it must come up=
either heads or tails. .
. Q!lce you have assigned the elementary probabilities, it is possible to go much further. It 'is possible. to comp!,1te probabilities for complex events. Take a single die., The probability of a single number from 1 to 6 coming up js assigned 1/6. Now suppose we throw two dice" as~is common
Page 92
THE COSMIC CODE
"In most dice games. The sum of the numbers on each die can run from 2 to 12. But they are not all equally probable. For example, the only way to throw a total of 2 is for each die to come up with a 1. Since the probability for each die to come up 1 is 1/6 and each event is independent, the joint probability is given by the product 1/6 X 1/6 = 1/36. On the average, throwing two dice, you will throw a 2 only once every 36 rolls.
.. You can throw a 3 in two different ways. The first die comes up 1 and the second die 2. This event also has a probability of 1/6 x 1/6 = 1/36. However, one could also achieve a total of 3 by the first die coming up 2 and the second die 1. This also has a probability 00/36. So the total probability of rolling a 3 is 1/36 + 1/36 = 1/18. Proceeding in this way of counting options, one can make a table of how the probability is distributed over the various throws.
The probability of different throws from a pair of dice. It seems as if
there is, a "force" or an "invisible hand" which makes the 7 come up
most often, but actually this is just a consequence of mathematical
probabilities
If you had to bet on a single number's coming up, the number to bet on is 7; it has the highest probability. If you had to bet on five out of the eleven numbers coming up, then bet on 5, 6, 7, 8, and 9 because they come up two out of three times; the remaining six numbers 2, 3, 4, 10, 11, and 12, . come up only one out of three times. What tells these numbers to turn up with these frequencies? We see that this probability distribution is just a consequence of mathematical combinatorics-adding up the different combinations by which a specific throw can be achieved. But it seems as if there is an "invisible hand" that pushes for the 7 more often than the other numbers. The remarkable feature of probability distri/ Page 94 /
mtions}ror re:[events is4hat the:o:distnbutio'h isn't tnaterlal
y~f it it manifested as a~kind 6f invisible force~ 011 material
mings"fil{e dice. ,-" "" s ~ "".
Prooilbili!y= distributions"'of real, events like tho~eAor 0: thrOwing- dice are part of the invisible world. The cdis-tribu-
C'~tiR~s are invisible not because like atoms theyare-:mat~rially
small but because they art?"11ot material at all. What isvil'ible~ is~1ndividual material events like the throw o[dice. Pro1Ja:bil-~
ity;distril:mtions, are~like invisible hands thafdo not touch. A
c g<fo<I exanJple';iS" the slow, invisib1e processoof biological- evo--;
: lution. =-Tlfis process'becomes real only-when.we go beyond
th~, ~eemingly, random events and, examin~ the ,probability ~
di~tributJonstliat gi,[5\-objectivesignmcanceto environmental : pressure on a species to'-evolve into another species more likely to survive in - that environI:IJPnt. Distributions of ev~nts ~ seem to .have an obje6tivity nqCpossessedby anAndividual random event. -In the microscopic world of atoms wethave already'seen tfiat it is the distribution of events that is speci- os fled by the quantum-'- theory; not the~ individual event. The quantum pro~bility distributiofis, the--invisible hands af tl)e atomic level, are actUally responsible for the chemical forces.
that _bind'atoms tog~ther., ~
- We might imagine that"probability distributions, beFause they haye some kind of objectivity, have an exisfenceiQ.dep~fi.denfof toe individual event. This errpr can result In _'f-= thinking thatth~ .distributlon""cau~es"the events to fall inter-a :is; spe~mccpatteI1l" Thisis secular fatalism-the belief that Ilrob~ ~ ability distribri.!ions influence toe outcome of single events,,- ;Bur this is "backwar-d" reasoning, because it is the sin~e events 'which establish the distiibution, not the other~way aroung. By introducing a nonrandom element, an,..elemeIit of ~ organizatiim on the lev_el of individual events, one changes
the probability distribution. If you load dice they~will fall - : differently. -:0
While the invisibility and objectivity of distributions is amazing, another remarkable feature of probability distributionsis their stability, whether they are distriqutions of atomic. motions in matter, chemical reactions, or biological and social events. A stable distribution, one which does not change/With
time, is called", ail equilibrium distribution. ;The probability distributions of dice-throws we do not expect to- change in' tiIpe, because the dice are not subject to temporal forces., But what abQ!lt the probability of breaking a leg in a ski accident
Joseph Campbell
[ I 54
M Y T H S T 0 L I V E ,8 Y
God. And that Christ may not have actually suffered in that loving act we may take from a saying of the mystic Meister Eckhart: "To him who suffers but not for love, to suffer is suffering and hard to bear. But one who suffers for love suffers not, and his suffering is fruitful in God's sight."
Indeed, the very idea of a descent of God, into the world in love to invoke, in return, man's love to God, seems to me to imply exactly the contrary to the statement 1 have just quoted of Saint Paul. Implied, rather, it seems to me, is the idea that as mankind yearns for the grace of God, so God for the homage of mankind, the two yearnings being reciprocal. And the image,of the crucified as both true God and true man would then seem to bring to focus the matched terms of a mutual sacrifice-in the way not of atonement .in the penal sense, but of at-one-ment in the marital. And further: when extended to symbolize not only the one historic moment of Christ's crucifixion on Calvary, but the mystery through all time and space of God's presence and participation in the agony of all living things, the sign of the cross would then have to be looked, upon as the sign of an eternal affirmation of all that is, ever was, or shall ever be. One thinks of Christ's words reported
in the Gnostic Gospel According to Thomas: "Cleave a piece of wood, I am there; lift up the stone, you will find me there." Also, those of Plato in the Timaeus, where he states that time is "the moving image of Eternity." Or again, those of William Blake: "Eternity is in love with the productions of time." And there is a memorable passage in the writings of Thomas Mann, where he celebrates ,man as "a noble meeting [eine hohe Begegnung] of Spirit and Nature in their yearning way to each other."
, We can safely say, therefore, that whereas some moralists may find it possible to make a distinction between two spheres and reigns-one of flesh, the other of the spirit, one of time, the other of eternity-wherever love arises such definitions vanish, and a sense of life awakens in which all such oppositions are at one.
The most widely revered Oriental personification of such a world-affirming attitude, transcending opposites, is that figure of
Page 156
Page 157
The Mythology of Love e
I 5'7 ]
yourself!-.and even better, beyond that, in the words that I take to be the highest, the noblest and boldest, of the Christian teaching: "Love your enemies and pray for those who persecute y<>,u, so that you may be sons of your Father who is in heaven; for he makes his sun to rise on the evil and on the good, and sends rain on the just and on the unjust. . . ."
In all the great traditional representations of love as compassion, charity, .or agape, the operation of the virtue is described as general and impersonal, transcending differences and even loyalties..,And against this higher, spirityal order of love there is set
,generally in opposition the lower, of lust, or, as it is so often called, "animal passion," which is equally general and impersonal, transcending qifferences and even loyalties. Indeed, one could describe the latter most accurately, perhaps, simply as the zeal of the organs, male and female, for each other, and designate the writings of Sigmund Freud as the definitive modern text on the subject of such love. However, in the European twelfth and early thirteenth centurie~, in the poetry first of the troubadours of Provence, and then, with a new accent, of the Minnesingers, a way of experiencing love came to expression that was altogether different from either of those two as traditionally opposed. And since I regard this typically and exclusively European chapter of our subject as one., of the most important mutations not only of human feeling,
"
but also of the spiritual consciousness of our human race, I am
going to dwell on it a little, before proceeding to the final passages of this chapter.
To begin with, then: Marriage in the Middle Ages was almost exclusively asocial, family concern-as it has been forever, of
course, in Asia, and is to this day for many in the West. One was married according to family arrangements. Particularly in' aristocratic circles, young women hardly out of girlhood were married off as political pawns. And the Church, meanwhile, was sacramentalizing such unions with its inappropriately mystical language about the two that were now to be of one ft~sh, united thrQugh love and by God: and let no man put asunder what God. hath
Page 158
Page 159
The Mythology of Love
I 59 ]
that is to say, neither of Heaven nor of Hell, but of earth; grounded in the psyche of a particular individual and, specifically, the predilection of his eyes: their perception of another specific individual and communication of her image to his heart-which is to be (as we are told in other documents of the time) a "noble" or "gentle" heart, capable of the emotion of love, amor, not simply lust.
And what, then, would be the nat~re of a love so born?
In the various contexts of Oriental erotic mysticism, whether of
the Near East or of India, the woman is J!lystic~ly interpreted as an occasion for the lover to experience depths beyond depths of transcendent illumination-much in the way of Dante's appreciation of Beatrice. Not so among the troubadours. The beloved to them was a woman, not the manifestation of some divine principle; and specifically, that woman. The love was for her. And the celebrated experience was an agony of earthly love: an effect of the fact that the union of love can never be absolutely realized on this earth. Love's joy is in its savor of eternity; love's pain, the passage of time; so that (as in Gottfried's words) "bitter sweetness and dear grief" are of its essence. And for those "who cannot bear grief, and desire but to bathe in bliss," the ambrosial potion of this greatest gift of life is a drink too strong. Gottfried even deified Love as a goddess, and brought his bewildered couple to her hidden wilderness-chapel, known as "The Grotto for People in Love," where stood, in the place of an altar, the noble crystalline bed of love.
Moreover-and this, to me, is the most profoundly moving passage in Gottfried's version of the legend-when, on the ship sailing from Ireland (with which scene Wagner's opera commences), the young couple unwittingly drank the potion and became gradually aware of the love that for some time had been quietly growing in their hearts, Brangaene, the faithful servant who by chance had left the fateful ftask unattended, said to them in dire warning, "That ftask and what it contained will be the death of you both!" To which Tristan answered, "So then, God's will be-done, whether
SECRET CHAMBER
THE QUEST FOR THE HALL OF RECORDS
Robert Bauval
1
999
The Hall of Records
Page 167
"Gerald Massey published between 1883 and 1907. Massey, who is described as 'a poet, Shakespearean scholar, mythographer and radical Egyptologist', wrote this of the Sphinx of Giza:
The great Sphinx as keeper of these secrets was couched in mountainous repose upon the horizon in the eastern equinox, when the gate of 'fair exit' was in the lion sign and the gates of 'fair entrancc' was in Aquarius, the water sign, that is figured over the abyss of source on the celestial globe. The Sphinx thcn is a figure
of the double horizon and the duality of Har-Makhu when the place of conjunction was at the point of precession in thc lion sign. And if, as is the Egyptian way, the fact was registered forthwith, we may date the Sphinx as a monument which was reared by thcse great builders and thinkers, who lived so largely out of themselves, some thirteen thousand years ago. 58
Other ideas in Cayce's readings, such as the 'Great White Brotherhood' and the initiatory function of the Great Pyramid of the 'last' or higher 'degrees', can be found in the popular literature of Helena Blavatsky and the Theosophical movement that was also widely circulating at the times Edgar Cayce gave his first readings on the pyramids in 1931. In two particular works by Blavatsky, The Secret Doctrine and Isis Unveiled, both published in the late 1870s, there is much talk of the 'root races' of the world, the White Brotherhood and the Masonic style by degree initiations in the Great Pyramid of Giza. 59 Blavatsky also claimed that the empty coffer in the King's Chamber of the Great Pyramid was 'a baptismal font upon emerging from which the neophyte was born again and became an adept', meaning that the neophyte experienced death and learnt of its mystery.60 Blavatsky herself had apparently spent a whole night alone in the King's Chamber in 1886 where 'she had some remarkable experiences'.61 This idea, too, is expressed in the Cayce readings; for when asked about the meaning of the empty sarcophagus in the King's Chamber and what could be learnt from it, the following answer was given:
That there will be no more death. Don't misunderstand or misintcrpret! But the interpretation of dcath will be made plain.62"
NAMES OF GOD
TETRAGRAMMATON
The most important and most often written name of God in Judaism is the Tetragrammaton, the four-letter name of God. "Tetragrammaton" derives from the Greek prefix tetra- ("four") and gramma
http://en.wikipedia.org/wiki/Names of God in Judaism
A
MAZE
IN
ZAZAZA ENTER AZAZAZ
AZAZAZAZAZAZAZZAZAZAZAZAZAZA
ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ
THE
MAGICALALPHABET
ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA
12345678910111213141516171819202122232425262625242322212019181716151413121110987654321
WORK DAYS OF GOD
Herbert W Morris D.D.circa 1883
Page 22
"As all the words in the English language are composed out of the twenty-six letters of the alphabet,.."
LIGHT AND LIFE
Lars Olof Bjorn 1976
Page 197
"By writing the 26 letters of the alphabet in a certain order one may put down almost any message (this book 'is written with the same letters' as the Encyclopaedia Britannica and Winnie the Pooh, only the order of the letters differs). In the same way Nature is able to convey with her language how a cell and a whole organism is to be constructed and how it is to function. Nature has succeeded better than we humans; for the genetic code there is only one universal language which is the same in a man, a bean plant and a bacterium."
"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER
ONE MAY PUT DOWN ALMOST ANY MESSAGE"
A |
B |
C |
D |
E |
F |
G |
H |
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
1+0 |
1+1 |
1+2 |
1+3 |
1+4 |
1+5 |
1+6 |
1+7 |
1+8 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
S |
T |
U |
V |
W |
X |
Y |
Z |
I |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
9 |
1+9 |
2+0 |
2+1 |
2+2 |
2+3 |
2+4 |
2+5 |
2+6 |
ME |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
= |
I |
ME |
I |
ME |
I |
ME |
I |
ME |
I |
9 |
18 |
9 |
18 |
9 |
18 |
9 |
18 |
9 |
= |
1+8 |
= |
1+8 |
= |
1+8 |
= |
1+8 |
= |
= |
9 |
= |
9 |
= |
9 |
= |
9 |
= |
I |
ME |
I |
ME |
I |
ME |
I |
ME |
1 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
9 |
I |
ME |
I |
ME |
I |
ME |
I |
ME |
1 |
"BY WRITING THE 26 LETTERS OF THE
ALPHABET IN A CERTAIN ORDER
ONE MAY PUT DOWN ALMOST ANY MESSAGE"
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
- |
- |
- |
- |
- |
- |
- |
- |
- |
1+0 |
1+1 |
1+2 |
1+3 |
1+4 |
1+5 |
1+6 |
1+7 |
1+8 |
1+9 |
2+0 |
2+1 |
2+2 |
2+3 |
2+4 |
2+5 |
2+6 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
A |
B |
C |
D |
E |
F |
G |
H |
|
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
A
HISTORY OF GOD
Karen Armstrong 1993
The God of the Mystics
Page 250
"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically;
the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by
combining the letters with the sacred numbers, rearranging them in
endless configurations, the mystic weaned his mind away from the normal connotations of words."
THIS IS THE SCENE OF THE SCENE UNSEEN
THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE
3 |
THE |
33 |
15 |
6 |
4 |
MIND |
40 |
22 |
4 |
2 |
OF |
21 |
12 |
3 |
9 |
HUMANKIND |
95 |
41 |
5 |
18 |
First Total |
|
|
|
1+8 |
Add to Reduce |
1+8+9 |
9+0 |
1+8 |
9 |
Second Total |
|
|
|
|
Reduce to Deduce |
1+8 |
- |
- |
|
Essence of Number |
|
|
|
THE
FAR YONDER SCRIBE
AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE
THE
ZED ALIZ ZED
IN
SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS
NUMBER
9
THE SEARCH FOR THE SIGMA CODE
Cecil Balmond 1998
Cycles and Patterns
Page 165
Patterns
"The essence of mathematics is to look for patterns.
Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.
These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!
Searching out patterns is a pure delight.
Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.
And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."
FINGERPRINTS OF THE GODS
A QUEST FOR THE BEGINNING AND THE END
Graham Hancock 1995
Chapter 32
Speaking to the Unborn
Page 285
"It is understandable that a huge range of myths from all over the ancient world should describe geological catastrophes in graphic detail. Mankind survived the horror of the last Ice Age, and the most plausible source for our enduring traditions of flooding and freezing, massive volcanism and devastating earthquakes is in the tumultuous upheavals unleashed during the great meltdown of 15,000 to 8000 BC. The final retreat of the ice sheets, and the consequent 300-400 foot rise in global sea levels, took place only a few thousand years before the beginning of the historical period. It is therefore not surprising that all our early civilizations should have retained vivid memories of the vast cataclysms that had terrified their forefathers.
Much harder to explain is the peculiar but distinctive way the myths of cataclysm seem to bear the intelligent imprint of a guiding hand.l Indeed the degree of convergence between such ancient stories is frequently remarkable enough to raise the suspicion that they must all have been 'written' by the same 'author'.
Could that author have had anything to do with the wondrous deity, or superhuman, spoken of in so many of the myths we have reviewed, who appears immediately after the world has been shattered by a horrifying geological catastrophe and brings comfort and the gifts of civilization to the shocked and demoralized survivors?
White and bearded, Osiris is the Egyptian manifestation of this / Page 286 /
universal figure, and it may not be an accident that one of the first acts he is remembered for in myth is the abolition of cannibalism among the primitive inhabitants of the Nile Valley.2 Viracocha, in South America, was said to have begun his civilizing mission immediately after a great flood; Quetzalcoatl, the discoverer of maize, brought the benefits of crops, mathematics, astronomy and a refined culture to Mexico after the Fourth Sun had been overwhelmed by a destroying deluge.
Could these strange myths contain a record of encounters between scattered palaeolithic tribes which survived the last Ice Age and an as yet unidentified high civilization which passed through the same epoch?
And could the myths be attempts to communicate?
A message in the bottle of time"
'Of all the other stupendous inventions,' Galileo once remarked,
what sublimity of mind must have been his who conceived how to communicate his most secret thoughts to any other person, though very distant either in time or place, speaking with those who are in the Indies, speaking to those who are not yet born, nor shall be this thousand or ten thousand years? And with no greater difficulty than the various arrangements of two dozen little signs on paper? Let this be the seal of all the admirable inventions of men.3
If the 'precessional message' identified by scholars like Santillana, von Dechend and Jane Sellers is indeed a deliberate attempt at communication by some lost civilization of antiquity, how come it wasn't just written down and left for us to find? Wouldn't that have been easier than encoding it in myths? Perhaps.
Nevertheless, suppose that whatever the message was written on got destroyed or worn away after many thousands of years? Or suppose that the language in which it was inscribed was later forgotten utterly (like the enigmatic Indus Valley script, which has been studied closely for more than half a century but has so far resisted all attempts at decoding)? It must be obvious that in such circumstances a written / Page 287 / legacy to the future would be of no value at all, because nobody would be able to make sense of it.
What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them - and the city of Teotihuacan may be the calling-card of a lost civilization written in the eternal language of mathematics.
Geodetic data, related to the exact positioning of fixed geographical points and to the shape and size of the earth, would also remain valid and recognizable for tens of thousands of years, and might be most conveniently expressed by means of cartography (or in the construction of giant geodetic monuments like the Great Pyramid of Egypt, as
we shall see).
Another 'constant' in our solar system is the language of time: the great but regular intervals of time calibrated by the inch-worm creep of precessional motion. Now, or ten thousand years in the future, a message that prints out numbers like 72 or 2160 or 4320 or 25,920 should be instantly intelligible to any civilization that has evolved a modest talent for mathematics and the ability to detect and measure the almost imperceptible reverse wobble that the sun appears to make along the ecliptic against the background of the fixed stars..."
"What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them"
"WRITTEN IN THE ETERNAL LANGUAGE OF MATHEMATICS"
THE LIGHT IS RISING RISING IS THE LIGHT
26 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8 |
9 |
|
|
|
|
5 |
6 |
|
|
|
1 |
|
|
|
|
6 |
|
8 |
+ |
= |
|
4+3 |
= |
|
|
|
|
|
|
|
|
|
|
|
|
|
8 |
9 |
|
|
|
|
14 |
15 |
|
|
|
19 |
|
|
|
|
24 |
|
26 |
+ |
= |
|
1+1+5 |
= |
|
|
|
|
|
26 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
|
1 |
2 |
3 |
4 |
|
|
7 |
8 |
9 |
|
2 |
3 |
4 |
5 |
|
7 |
|
+ |
= |
|
8+3 |
= |
|
1+1 |
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
|
10 |
11 |
12 |
13 |
|
|
16 |
17 |
18 |
|
20 |
21 |
22 |
23 |
|
25 |
|
+ |
= |
|
2+3+6 |
= |
|
1+1 |
|
|
|
26 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
+ |
= |
|
3+5+1 |
= |
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
+ |
= |
|
1+2+6 |
= |
|
|
|
|
|
26 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
+ |
= |
|
occurs |
x |
3 |
= |
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
+ |
= |
|
occurs |
x |
3 |
= |
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
+ |
= |
|
occurs |
x |
3 |
= |
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
4 |
|
|
|
|
+ |
= |
|
occurs |
x |
3 |
= |
|
1+2 |
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
5 |
|
|
|
+ |
= |
|
occurs |
x |
3 |
= |
|
1+5 |
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
6 |
|
|
+ |
= |
|
occurs |
x |
3 |
= |
|
1+8 |
|
|
|
|
|
|
|
|
7 |
|
|
|
|
|
|
|
|
7 |
|
|
|
|
|
|
|
|
7 |
|
+ |
= |
|
occurs |
x |
3 |
= |
|
2+1 |
|
|
|
|
|
|
|
|
|
8 |
|
|
|
|
|
|
|
|
8 |
|
|
|
|
|
|
|
|
8 |
+ |
= |
|
occurs |
x |
3 |
= |
|
2+4 |
|
|
|
|
|
|
|
|
|
|
9 |
|
|
|
|
|
|
|
|
9 |
- |
- |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
2 |
= |
|
1+8 |
|
26 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4+5 |
|
|
2+6 |
|
1+2+6 |
|
5+4 |
26 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
|
|
|
|
|
|
|
|
|
26 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
9 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8 |
9 |
+ |
= |
|
1+7 |
= |
|
|
|
|
|
|
|
|
|
|
|
8 |
9 |
+ |
= |
|
1+7 |
= |
|
|
|
9 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
|
+ |
= |
|
2+8 |
= |
|
1+0 |
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
|
+ |
= |
|
2+8 |
= |
|
1+0 |
|
9 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
+ |
= |
|
4+5 |
= |
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
+ |
= |
|
4+5 |
= |
|
|
|
9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
- |
- |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
1 |
= |
|
|
- |
2 |
- |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
2 |
= |
|
|
- |
- |
3 |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
3 |
= |
|
|
- |
- |
- |
4 |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
4 |
= |
|
|
- |
- |
- |
- |
5 |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
5 |
= |
|
|
- |
- |
- |
- |
- |
6 |
- |
- |
- |
+ |
= |
|
occurs |
x |
6 |
= |
|
|
- |
- |
- |
- |
- |
- |
7 |
- |
- |
+ |
= |
|
occurs |
x |
7 |
= |
|
|
- |
- |
- |
- |
- |
- |
- |
8 |
- |
+ |
= |
|
occurs |
x |
8 |
= |
|
|
- |
- |
- |
- |
- |
- |
- |
- |
9 |
+ |
= |
|
occurs |
x |
9 |
= |
|
9 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4+5 |
|
|
4+5 |
|
4+5 |
9 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
|
|
|
|
|
|
9 |
|
|
|
|
|
|
|
|
I |
|
|
|
|
|
|
|
|
9 |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
6 |
|
|
|
+ |
= |
|
1+1 |
= |
|
|
|
|
|
|
|
|
14 |
15 |
|
|
|
+ |
= |
|
1+7 |
= |
|
|
|
9 |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
1 |
2 |
3 |
4 |
|
|
7 |
8 |
9 |
+ |
= |
|
3+4 |
= |
|
|
|
|
10 |
11 |
12 |
13 |
|
|
16 |
17 |
18 |
+ |
= |
|
9+7 |
= |
|
1+6 |
|
9 |
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
+ |
= |
|
1+2+6 |
= |
|
|
|
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
+ |
= |
|
4+5 |
= |
|
|
|
|
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
1 |
- |
- |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
1 |
= |
|
|
- |
2 |
- |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
2 |
= |
|
|
- |
- |
3 |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
3 |
= |
|
|
- |
- |
- |
4 |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
4 |
= |
|
|
- |
- |
- |
- |
5 |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
5 |
= |
|
|
- |
- |
- |
- |
- |
6 |
- |
- |
- |
+ |
= |
|
occurs |
x |
6 |
= |
|
|
- |
- |
- |
- |
- |
- |
7 |
- |
- |
+ |
= |
|
occurs |
x |
7 |
= |
|
|
- |
- |
- |
- |
- |
- |
- |
8 |
- |
+ |
= |
|
occurs |
x |
8 |
= |
|
9 |
- |
- |
- |
- |
- |
- |
- |
- |
9 |
+ |
= |
|
occurs |
x |
9 |
= |
|
|
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
9 |
|
|
|
|
|
|
|
|
|
|
|
4+5 |
|
|
4+5 |
|
4+5 |
|
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
9 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
- |
8 |
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
- |
|
1 |
|
|
|
- |
6 |
|
8 |
+ |
= |
|
1+5 |
= |
|
|
|
- |
|
19 |
|
|
|
- |
24 |
|
26 |
+ |
= |
|
6+9 |
= |
|
|
|
- |
8 |
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
- |
|
- |
2 |
3 |
4 |
5 |
|
7 |
- |
+ |
= |
|
2+1 |
= |
|
|
|
- |
|
- |
20 |
21 |
22 |
23 |
|
25 |
- |
+ |
= |
|
1+1+1 |
= |
|
- |
|
- |
8 |
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
- |
|
19 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
+ |
= |
|
1+8+0 |
= |
|
|
|
- |
- |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
+ |
= |
|
3+6 |
= |
|
|
|
- |
8 |
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
- |
|
1 |
- |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
1 |
= |
|
- |
|
- |
2 |
- |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
2 |
= |
|
- |
|
- |
- |
3 |
- |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
3 |
= |
|
- |
|
- |
- |
- |
4 |
- |
- |
- |
- |
+ |
= |
|
occurs |
x |
4 |
= |
|
- |
|
- |
- |
- |
- |
5 |
- |
- |
- |
+ |
= |
|
occurs |
x |
5 |
= |
|
- |
|
- |
- |
- |
- |
- |
6 |
- |
- |
+ |
= |
|
occurs |
x |
6 |
= |
|
- |
|
- |
- |
- |
- |
- |
- |
7 |
- |
+ |
= |
|
occurs |
x |
7 |
= |
|
- |
|
- |
- |
- |
- |
- |
- |
- |
8 |
+ |
= |
|
occurs |
x |
8 |
= |
|
9 |
|
|
|
|
|
|
|
|
|
+ |
= |
|
occurs |
x |
9 |
= |
|
9 |
8 |
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
- |
- |
|
|
|
|
|
|
|
|
|
|
4+5 |
|
|
4+5 |
|
4+5 |
9 |
|
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
|
- |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
|
|
|
|
|
|
|
9 |
8 |
S |
T |
U |
V |
W |
X |
Y |
Z |
|
|
|
|
|
|
|
|
THE DEATH OF GODS IN ANCIENT EGYPT
Jane B. Sellars 1992
Page 204
"The overwhelming awe that accompanies the realization, of the measurable orderliness of the universe strikes modern man as well. Admiral Weiland E. Byrd, alone In the Antarctic for five months of polar darkness, wrote these phrases of intense feeling:
Here were the imponderable processes and forces of the cosmos, harmonious and soundless. Harmony, that was it! I could feel no doubt of oneness with the universe. The conviction came that the rhythm was too orderly. too harmonious, too perfect to be a product of blind chance - that, therefore there must be purpose in the whole and that man was part of that whole and not an accidental offshoot. It was a feeling that transcended reason; that went to the heart of man's despair and found it groundless. The universe was a cosmos, not a chaos; man was as rightfully a part of that cosmos as were the day and night.10
Returning to the account of the story of Osiris, son of Cronos god of' Measurable Time, Plutarch takes, pains to remind the reader of the original Egyptian year consisting of 360 days.
Phrases are used that prompt simple mental. calculations and an attention to numbers, for example, the 360-day year is described as being '12 months of 30 days each'. Then we are told that, Osiris leaves on a long journey, during which Seth, his evil brother, plots with 72 companions to slay Osiris: He also secretly obtained the measure of Osiris and made ready a chest in which to entrap him.
The, interesting thing about this part of the-account is that nowhere in the original texts of the Egyptians are we told that Seth, has 72 companions. We have already been encouraged to equate Osiris with the concept of measured time; his father being Cronos. It is also an observable fact that Cronos-Saturn has the longest sidereal period of the known planets at that time, an orbit. of 30 years. Saturn is absent from a specific constellation for that length of time.
A simple mathematical fact has been revealed to any that are even remotely sensitive to numbers: if you multiply 72 by 30, the years of Saturn's absence (and the mention of Osiris's absence prompts one to recall this other), the resulting product is 2,160: the number of years required, for one 30° shift, or a shift: through one complete sign of the zodiac. This number multplied by the /Page205 / 12 signs also gives 25,920. (And Plutarch has reminded us of 12)
If you multiply the unusual number 72 by 360, a number that Plutarch mentions several times, the product will be 25,920, again the number of years symbolizing the ultimate rebirth.
This 'Eternal Return' is the return of, say, Taurus to the position of marking the vernal equinox by 'riding in the solar bark with. Re' after having relinquished this honoured position to Aries, and subsequently to the to other zodiacal constellations.
Such a return after 25,920 years is indeed a revisit to a Golden Age, golden not only because of a remarkable symmetry In the heavens, but golden because it existed before the Egyptians experienced heaven's changeability.
But now to inform the reader of a fact he or she may already know. Hipparaus did: not really have the exact figures: he was a
trifle off in his observations and calculations. In his published work, On the Displacement of the Solstitial and Equinoctial Signs, he
gave figures of 45" to 46" a year, while the truer precessional
lag along the ecliptic is about 50 seconds. The exact measurement for the lag, based on the correct annual lag of 50'274" is 1° in 71.6 years, or 360° in 25,776 years, only 144 years less
than the figure of 25,920.
With Hipparchus's incorrect figures a 'Great Year' takes from 28,173.9 to 28,800 years, Incorrect by a difference of from 2,397.9 years to 3,024.
Since Nicholas Copernicus (AD 1473-1543) has always been credited with giving the correct numbers (although Arabic astronomer Nasir al-Din Tusi,11 born AD 1201, is known to have fixed the Precession at 50°), we may correctly ask, and with justifiable astonishment 'Just whose information was Plutarch transmitting'
AN IMPORTANT POSTSCRIPT
Of course, using our own notational system, all the important numbers have digits that reduce to that amazing number 9 a number that has always delighted budding mathematician.
Page 206
Somewhere along the way, according to Robert Graves, 9 became the number of lunar wisdom.12
This number is found often in the mythologies of the world. the Viking god Odin hung for nine days and nights on the World Tree in order to acquire the secret of the runes, those magic symbols out of which writing and numbers grew. Only a terrible sacrifice would give away this secret, which conveyed upon its owner power and dominion over all, so Odin hung from his neck those long 9 days and nights over the 'bottomless abyss'. In the tree were 9 worlds, and another god was said to have been born of 9 mothers.
Robert Graves, in his White Goddess, Is intrigued by the seemingly recurring quality of the number 72 in early myth and ritual. Graves tells his reader that 72 is always connected with the number 5, which reflects, among other things, the five Celtic dialects that he was investigating. Of course, 5 x 72= 360, 360 x 72= 25,920. Five is also the number of the planets known to the ancient world, that is, Saturn, Jupiter, Mars, Venus Mercury.
Graves suggests a religious mystery bound up with two ancient Celtic 'Tree Alphabets' or cipher alphabets, which as genuine articles of Druidism were orally preserved and transmitted for centuries. He argues convincingly that the ancient poetry of Europe
was ultimately based on what its composers believed to be magical principles, the rudiments of which formed a close religious secret for centuries. In time these were-garbled, discredited and forgotten.
Among the many signs of the transmission of special numbers he points out that the aggregate number of letter strokes for the complete 22-letter Ogham alphabet that he is studying is 72 and that this number is the multiple of 9, 'the number of lunar wisdom'. . . . he then mentions something about 'the seventy day season during which Venus moves successively from. maximum eastern elongation 'to inferior conjunction and maximum western elongation'.13
Page 207
"...Feniusa Farsa, Graves equates this hero with Dionysus Farsa has 72 assistants who helped him master the 72 languages created at the confusion of Babel, the tower of which is said to be built of 9 different materials
We are also reminded of the miraculous translation into Greek of the Five Books of Moses that was done by 72 scholars working for 72 days, Although the symbol for the Septuagint is LXX, legend, according to the fictional letter of Aristeas, records 72. The translation was done for Ptolemy Philadelphus (c.250 BC), by Hellenistic Jews, possibly from Alexandra.14
Graves did not know why this number was necessary, but he points
out that he understands Frazer's Golden Bough to be a a book hinting
that 'the secret involves the truth that the Christian
dogma, and rituals, are the refinement of a great body of
primitive beliefs, and that the only original element in Christianity- is the personality of Christ.15
Frances A. Yates, historian of Renaissance hermetisma tells, us
the cabala had 72 angels through which the sephiroth (the powers
of God) are believed to be approached, and further, she supplies the information that although the Cabala supplied a set of 48 conclusions purporting to confirm the Christian religion from the foundation of ancient wisdom, Pico Della Mirandola, a Renaissance magus, introduced instead 72, which were his 'own opinion' of the correct number. Yates writes, 'It is no accident
there are seventy-two of Pico's Cabalist conclusions, for the
conclusion shows that he knew something of the mystery of the Name of God with seventy-two letters.'16
In Hamlet's Mill de Santillarta adds the facts that 432,000 is the number of syllables in the Rig-Veda, which when multiplied by the soss
(60) gives 25,920" (The reader is forgiven for a bit of laughter at this point)
Thee Bible has not escaped his pursuit. A prominent Assyriologist of the last century insisted that the total of the years recounted
mounted in Genesis for the lifetimes of patriarchs from
the Flood also contained the needed secret numbers. (He showed that in the 1,656.years recounted in the Bible there are 86,400 7 day weeks, and dividing this number yields / Page 208 / 43,200.)
In Indian yogic schools it is held that all living beings exhale and inhale 21,600 times a day, .multiply this by 2 and again we have.the necessary 432 digits.
Joseph Campbell discerns the secret in the date set for the coming of Patrick to Ireland. Myth-gives this date-as.- the interest-
ing number of AD.432.18
Whatever one may think-of some of these number coincidences, it becomes. difficult to escape the suspicion that many signs (number and otherwise) -indicate that early man observed the results.. of the movement of Precession . and that the-.transmission of this information was .considered of prime importance.
'With the awareness of the phenomenon, observers would certainly have tried for its measure, and such an endeavour would
have constituted the construction-of a 'Unified Field Theory' for nothing .less than Creation itself. Once determined, it would have been information worthy of secrecy and worthy of the passing on to future adepts.
But one last word about mankind's romance with number coincidences.The antagonist in John Updike's novel, Roger's Version, is a computer hacker, who, convinced.,that scientific evidence of God's existence is accumulating, endeavours to prove it by feeding -all the available scientific information. into a comuter. In his search for God 'breaking, through', he has become fascinated by certain numbers that have continually been cropping up. He explains them excitedly as 'the terms of Creation':
"...after a while I noticed that all over the sheet there seemed to hit these twenty-fours Jumping out at me. Two four; two,four.Planck time, for instance, divided by the radiation constant yields a figure near eight times ten again to the negative twenty-fourth, and the permittivity of free space, or electric constant, into the Bohr radiusekla almost exactly six times ten to the negative twenty-fourth. On positive side, the electromagnetic line-structure constant times Hubble radius - that is, the size of the universe as we now perceive it gives us something quite close to ten to the twenty-fourth, and the
strong-force constant times the charge on the proton produces two point four times ten to the negative eighteenth, for another I began to circle twenty-four wherever it appeared on the Printout here' - he held it up. his piece of striped and striped wallpaper, decorated / Page 209 /
with a number of scarlet circles - 'you can see it's more than random.'19
This inhabitant of the twentieth century is convinced that the striking occurrences of 2 and 4 reveal the sacred numbers by which God is speaking to us.
So much for any scorn directed to ancient man's fascination with number coincidences. That fascination is alive and well, Just a bit more incomprehensible"
OF TIME AND STARS
Arthur C. Clarke 1972
FOREWORD
"'Into the Comet' and 'The Nine Billion Names of God' both involve computers and the troubles they may cause us. While writing this preface, I had occasion to call upon my own HP 9100A computer, Hal Junior, to answer an interesting question. Looking at my records, I find that I have now written just about one hundred short stories. This volume contains eighteen of them: therefore, how many possible 18-story collections will I be able to put together? The answer as I am sure will be instantly obvious to you - is 100 x 99. . . x 84 x 83 divided by 18 x 17 x 16 ... x .2 x 1. This is an impressive number - Hal Junior tells me that it is approximately 20,772,733,124,605,000,000.
Page 15
The Nine Billion Names of God
'This is a slightly unusual request,' said Dr Wagner, with what he hoped was commendable restraint. 'As far as I know, it's the first time anyone's been asked to supply a Tibetan monastery with an Automatic Sequence Computer. I don't wish to be inquisitive, but I should hardly have thought that your - ah - establishment had much use for such a machine. Could you explain just what you intend to do with it?'
'Gladly,' replied the lama, readjusting his silk robes and carefully putting away the slide rule he had been using far currency conversions. 'Your Mark V Computer can carry out any routine mathematical operation involving up to ten digits. However, for our work we are interested in letters, not numbers. As we wish you to modify the output circuits, the machine will be printing words, not columns of figures.'
'I don't quite understand. . .'
'This is a project on which we have been working for the last three centuries - since the lamasery was founded, in fact. It is somewhat alien to your way of thought, so I hope you will listen with an open mind while I explain it.'
'Naturally.'
'It is really quite simple. We have been compiling a list which shall contain all the possible names of God.'
'I beg your pardon?'
Page16
'We have reason to believe,' continued the lama imperturbably, 'that all such names can be written with not more than nine letters in an alphabet we have devised.'
'And you have been doing this for three centuries?'
'Yes: we expected it would take us about fifteen thousand years to complete the task.'
'Oh,' Dr Wagner looked a little dazed. 'Now I see why you wanted to hire one of our machines. But what exactly is the purpose of this project?'
The lama hesitated for a fraction of a second, and Wagner wondered if he had offended him. If so, there was no trace of annoyance in the reply.
'Call it ritual, if you like, but it's a fundamental part of our belief. All the many names of the Supreme Being - God Jehova, Allah, and so on - they are only man-made labels. There is a philosophical problem of some difficulty here, which I do not propose to discuss, but somewhere among all the possible combinations of letters that can occur are what one may call the real names of God. By systematic permutation of letters, we have been trying to list them all.'
'I see. You've been starting at AAAAAAA . . . and working up to ZZZZZZZZ . . .'
'Exactly - though we use a special alphabet of our own. Modifying the electromatic typewriters to deal with this is, of course, trivial. A rather more interesting problem is that of devising suitable circuits to eliminate ridiculous combinations. For example, no letter must occur more than three times in succession.'
,'Three? Surely you mean two.'
'Three is correct: I am afraid it would take too long to explain why, even if you understood our language.' "
Page 68
Into the Comet
"Pickett's fingers danced over the beads, sliding them up and down the wires with lightning speed. There were twelve wires in all, so that the abacus could handle numbers up to 999,999,999,999 - or could be divided into separate sections where several independent calculations could be carried out simultaneously.
'374072,' said Pickett, after an incredibly brief interval of time. 'Now see how long you take to do it, with pencil and paper.'
There was a much longer delay before Martens, who like most mathematicians was poor at arithmetic, called out '375072'. A hasty check soon confirmed that Martens had taken at least three times as long as Pickett to arrive at the wrong answer.
The atronomer's face was a study in mingled chagrin, astonishment, and curiosity.
'Where did you learn that trick?' he asked. 'I thought those things could only add and subtract.'
'Well - multiplication's only repeated addition, isn't it? All I did was to add 856 seven times in the unit column, three times in the tens column, and four times in the hundreds column. You do the same thing when you use pencil and paper. Of course, there are some short cuts, but if you think I'm fast, you should have seen my granduncle. He used to work in a Yokohama bank, and you couldn't see his fingers / Page 69 / when he was going at speed"
All about the planets in our Solar System. The nine planets that orbit the sun are (in order from the sun): Mercury,Venus, Earth, Mars, Jupiter, Saturn, ... www.enchantedlearning.com/subjects/astronomy/planets
Our s
ola
r syst
em consists of the
sun, e
ight p
lanets, m
oons, dwa
rf p
lanets, an
astero
id belt,
comets,
meteo
rs, a
nd othe
rs. The
sun is the cente
r of
our s
ola
r syst
em; the p
lanets, the
ir m
oons, the
astero
ids,
comets, a
nd othe
r rocks a
nd gas all o
rb
it the
sun.
The nine planets that orbit the sun are (in order from the sun): Mercury,Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto (a dwarf planet). A belt of asteroids (minor planets made of rock and metal) lies between Mars and Jupiter. These objects all orbit the sun in roughly circular orbits that lie in the same plane, the ecliptic (Pluto is an exception; it has an elliptical orbit tilted over 17° from the ecliptic).
- |
6 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
- |
- |
|
- |
- |
- |
9 |
- |
- |
6 |
- |
|
|
|
1+5 |
|
|
= |
|
|
|
- |
- |
- |
9 |
- |
- |
15 |
- |
|
|
|
2+4 |
|
|
= |
6 |
|
|
- |
6 |
M |
I |
R |
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
- |
4 |
- |
9 |
9 |
- |
9 |
|
|
|
3+1 |
|
|
= |
|
|
|
- |
- |
13 |
- |
18 |
18 |
- |
18 |
|
|
|
6+7 |
|
|
1+3 |
4 |
|
|
- |
6 |
M |
I |
R |
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
- |
13 |
9 |
18 |
18 |
15 |
18 |
|
|
|
9+1 |
|
|
1+0 |
|
|
|
- |
- |
4 |
9 |
9 |
9 |
6 |
9 |
|
|
|
4+6 |
|
|
1+0 |
|
|
|
- |
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
|
|
|
|
|
|
|
|
|
|
1 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
|
|
2 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
|
|
3 |
- |
- |
- |
- |
- |
- |
- |
- |
|
4 |
- |
- |
|
|
|
- |
|
|
occurs |
x |
|
= |
|
= |
|
|
|
|
|
|
|
|
|
|
|
5 |
- |
- |
- |
- |
- |
- |
- |
- |
|
- |
- |
- |
|
|
|
- |
|
|
occurs |
x |
|
= |
6 |
= |
|
|
|
|
- |
- |
|
|
|
- |
|
7 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
- |
- |
|
|
|
- |
|
8 |
- |
- |
- |
- |
- |
- |
- |
- |
|
- |
9 |
9 |
|
|
|
|
|
|
occurs |
x |
|
= |
36 |
3+6 |
|
26 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2+6 |
|
- |
9 |
9 |
|
|
|
|
|
1+9 |
|
|
- |
|
4+6 |
|
1+9 |
8 |
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
4 |
9 |
9 |
9 |
6 |
9 |
|
|
1+0 |
|
|
- |
|
1+0 |
|
1+0 |
8 |
6 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
- |
- |
|
- |
- |
9 |
- |
- |
6 |
- |
|
|
|
1+5 |
|
|
= |
|
|
|
- |
- |
9 |
- |
- |
15 |
- |
|
|
|
2+4 |
|
|
= |
6 |
|
|
6 |
M |
I |
R |
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
4 |
- |
9 |
9 |
- |
9 |
|
|
|
3+1 |
|
|
= |
|
|
|
- |
13 |
- |
18 |
18 |
- |
18 |
|
|
|
6+7 |
|
|
1+3 |
4 |
|
|
6 |
M |
I |
R |
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
13 |
9 |
18 |
18 |
15 |
18 |
|
|
|
9+1 |
|
|
1+0 |
|
|
|
- |
4 |
9 |
9 |
9 |
6 |
9 |
|
|
|
4+6 |
|
|
1+0 |
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
|
4 |
- |
- |
|
|
|
- |
|
|
occurs |
x |
|
= |
|
= |
|
|
- |
- |
- |
|
|
|
- |
|
|
occurs |
x |
|
= |
6 |
= |
|
|
- |
9 |
9 |
|
|
|
|
|
|
occurs |
x |
|
= |
36 |
3+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
9 |
9 |
|
|
|
|
|
1+9 |
|
|
- |
|
4+6 |
|
1+9 |
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
9 |
9 |
9 |
6 |
9 |
|
|
1+0 |
|
|
- |
|
1+0 |
|
1+0 |
6 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
11 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
- |
- |
- |
9 |
- |
- |
6 |
- |
|
9 |
|
|
|
|
|
|
|
2+4 |
|
|
= |
|
|
|
- |
- |
- |
9 |
- |
- |
15 |
- |
|
9 |
|
|
|
|
|
|
|
3+3 |
|
|
= |
6 |
|
|
- |
11 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
- |
4 |
- |
9 |
9 |
- |
9 |
|
|
4 |
1 |
7 |
5 |
|
|
|
4+8 |
|
|
1+2 |
3 |
|
|
- |
- |
13 |
- |
18 |
18 |
- |
18 |
|
|
13 |
1 |
7 |
5 |
|
|
|
9+3 |
|
|
1+2 |
3 |
|
|
- |
11 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
- |
13 |
9 |
18 |
18 |
15 |
18 |
|
9 |
13 |
1 |
7 |
5 |
|
|
|
1+2+6 |
|
|
|
|
|
|
- |
- |
4 |
9 |
9 |
9 |
6 |
9 |
|
9 |
4 |
1 |
7 |
5 |
|
|
|
7+2 |
|
|
|
|
|
|
- |
11 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
occurs |
x |
|
= |
1 |
= |
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
- |
- |
|
|
2 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
|
- |
- |
- |
- |
- |
|
|
3 |
- |
- |
- |
- |
- |
- |
- |
- |
|
4 |
- |
- |
|
|
|
- |
|
|
|
|
|
- |
|
|
occurs |
x |
|
= |
|
= |
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
occurs |
x |
|
= |
5 |
= |
|
- |
|
- |
- |
- |
|
|
|
- |
- |
- |
- |
- |
- |
- |
|
|
occurs |
x |
|
= |
6 |
= |
|
|
- |
|
- |
- |
|
|
|
- |
|
|
|
|
|
- |
|
7 |
occurs |
x |
|
= |
7 |
= |
|
|
|
|
- |
- |
|
|
|
- |
- |
- |
- |
- |
- |
- |
|
8 |
- |
- |
- |
- |
- |
- |
- |
- |
|
- |
9 |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
45 |
3+6 |
|
13 |
11 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
11 |
|
|
|
|
1+3 |
1+1 |
- |
9 |
9 |
|
|
|
|
|
|
|
|
|
|
|
3+2 |
|
|
1+1 |
|
7+2 |
|
3+6 |
4 |
2 |
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
- |
- |
4 |
9 |
9 |
9 |
6 |
9 |
|
9 |
4 |
1 |
7 |
5 |
|
|
|
|
|
- |
|
- |
|
- |
4 |
2 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
11 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
- |
- |
9 |
- |
- |
6 |
- |
|
9 |
|
|
|
|
|
|
|
2+4 |
|
|
= |
|
|
|
- |
- |
9 |
- |
- |
15 |
- |
|
9 |
|
|
|
|
|
|
|
3+3 |
|
|
= |
6 |
|
|
11 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
4 |
- |
9 |
9 |
- |
9 |
|
|
4 |
1 |
7 |
5 |
|
|
|
4+8 |
|
|
1+2 |
3 |
|
|
- |
13 |
- |
18 |
18 |
- |
18 |
|
|
13 |
1 |
7 |
5 |
|
|
|
9+3 |
|
|
1+2 |
3 |
|
|
11 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
- |
- |
- |
|
- |
13 |
9 |
18 |
18 |
15 |
18 |
|
9 |
13 |
1 |
7 |
5 |
|
|
|
1+2+6 |
|
|
|
|
|
|
- |
4 |
9 |
9 |
9 |
6 |
9 |
|
9 |
4 |
1 |
7 |
5 |
|
|
|
7+2 |
|
|
|
|
|
|
11 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
occurs |
x |
|
= |
1 |
= |
|
- |
4 |
- |
- |
|
|
|
- |
|
|
|
|
|
- |
|
|
occurs |
x |
|
= |
|
= |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
occurs |
x |
|
= |
5 |
= |
|
- |
- |
- |
- |
|
|
|
- |
- |
- |
- |
- |
- |
- |
|
|
occurs |
x |
|
= |
6 |
= |
|
- |
|
- |
- |
|
|
|
- |
|
|
|
|
|
- |
|
7 |
occurs |
x |
|
= |
7 |
= |
|
|
- |
9 |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
45 |
3+6 |
|
11 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
11 |
|
|
|
|
1+1 |
- |
9 |
9 |
|
|
|
|
|
|
|
|
|
|
|
3+2 |
|
|
1+1 |
|
7+2 |
|
3+6 |
2 |
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
- |
4 |
9 |
9 |
9 |
6 |
9 |
|
9 |
4 |
1 |
7 |
5 |
|
|
|
|
|
- |
|
- |
|
- |
2 |
M |
I |
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
9 |
|
|
|
|
|
|
|
1+0 |
|
|
= |
|
- |
- |
9 |
|
|
|
|
|
|
|
2+8 |
|
|
1+0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
|
4 |
1 |
7 |
5 |
|
|
|
1+7 |
|
|
|
|
- |
- |
|
13 |
1 |
7 |
5 |
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
9 |
13 |
1 |
7 |
5 |
|
|
|
3+5 |
|
|
|
|
- |
- |
9 |
4 |
1 |
7 |
5 |
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
|
|
|
- |
- |
- |
- |
- |
- |
- |
2 |
- |
- |
- |
- |
- |
|
|
- |
- |
- |
- |
- |
- |
- |
3 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
4 |
- |
- |
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
5 |
|
|
- |
- |
- |
- |
- |
- |
- |
6 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
7 |
8 |
|
- |
- |
- |
- |
- |
- |
- |
8 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1+9 |
|
|
|
|
|
|
|
|
2+6 |
|
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1+0 |
- |
9 |
4 |
1 |
7 |
5 |
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
9 |
|
|
|
|
|
|
|
1+0 |
|
|
= |
|
- |
9 |
|
|
|
|
|
|
|
2+8 |
|
|
1+0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
4 |
1 |
7 |
5 |
|
|
|
1+7 |
|
|
|
|
- |
|
13 |
1 |
7 |
5 |
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
9 |
13 |
1 |
7 |
5 |
|
|
|
3+5 |
|
|
|
|
- |
9 |
4 |
1 |
7 |
5 |
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
4 |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
5 |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
7 |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2+6 |
|
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
9 |
4 |
1 |
7 |
5 |
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
9 |
|
|
|
|
1 |
|
|
|
1+0 |
|
|
= |
|
- |
- |
9 |
|
|
|
|
19 |
|
|
|
2+8 |
|
|
1+0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
|
4 |
1 |
7 |
5 |
|
|
|
|
1+7 |
|
|
|
|
- |
- |
|
13 |
1 |
7 |
5 |
|
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
- |
9 |
13 |
1 |
7 |
5 |
19 |
|
|
|
5+4 |
|
|
|
|
- |
- |
9 |
4 |
1 |
7 |
5 |
1 |
|
|
|
2+7 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
|
|
|
- |
- |
- |
- |
- |
- |
- |
- |
2 |
- |
- |
- |
- |
- |
|
|
- |
- |
- |
- |
- |
- |
- |
- |
3 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
4 |
- |
- |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
5 |
|
|
- |
- |
- |
- |
- |
- |
- |
- |
6 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
7 |
8 |
|
- |
- |
- |
- |
- |
- |
- |
- |
8 |
- |
- |
- |
- |
- |
- |
- |
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1+9 |
|
|
|
|
|
|
|
|
|
2+6 |
|
|
|
|
2+7 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1+0 |
- |
9 |
4 |
1 |
7 |
5 |
1 |
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
9 |
|
|
|
|
1 |
|
|
|
1+0 |
|
|
= |
|
- |
9 |
|
|
|
|
19 |
|
|
|
2+8 |
|
|
1+0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
|
4 |
1 |
7 |
5 |
|
|
|
|
1+7 |
|
|
|
|
- |
|
13 |
1 |
7 |
5 |
|
|
|
|
2+6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
- |
9 |
13 |
1 |
7 |
5 |
19 |
|
|
|
5+4 |
|
|
|
|
- |
9 |
4 |
1 |
7 |
5 |
1 |
|
|
|
2+7 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
|
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
4 |
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
5 |
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
7 |
|
|
|
|
|
|
|
|
|
|
occurs |
x |
|
= |
9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2+6 |
|
|
|
|
2+7 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
- |
9 |
4 |
1 |
7 |
5 |
1 |
|
|
|
|
|
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
