Asmodeus

tiling the planes




水者地之血氣如筋脉之通流者也

connecting the dots • more images

The Eightfold Way: The Beauty of Klein's Quartic Curve
On the Order-Seven Transformation of Elliptic Functions (PDF)
Über die Transformation siebenter Ordnung der elliptischen Funktionen
[PDF: Pages 90 to 136; Main Image: p126; Note 31: p133]

exact heptagon
Mandala • Proof

The Delian Problem and Transcendence in Theorem XX
of John Dee’s Hieroglyphic Monad

John Dee and Edward Kelley's Great Table, Part II
In Operibus Sigillo Dei Aemeth, III
image

I also draw the reader's attention to Brother Holden's original image of John Dee's Monas Hieroglyphica that adorns the cover page of each of the five books that make up Enochian Vision Magick. This image is in itself a most profound and historic element in the entire book for it represents a solution of a geometric puzzle going back to ancient Greek mathematical texts — that is, the production of a regular heptagon in a circle using only a compass and ruler. Please pay special attention to Clay's foreword to discover more. —Lon Milo DuQuette, Enochian Vision Magick

Monas Hieroglyphica: Theorem XXIII
Pythagorean quaternary • Artificial quaternary

see images in Klein, section 11. Replacement of the Riemann
Surface of Section 2 by a Regularly Tiled Cover
, especially Figures 6 & 7

Numogramming the Yi Jing

The Cosmos in the I Ching - The Yi-globe

Isomorphism between the I Ching table,
the 3×3×3 array of cubes and the Klein Configuration

Encoding of planetary distances and superstring structural parameters in the I Ching table
I Ching and the eight-fold way
Algebraic, arithmetic & geometric aspects of the I Ching table

Correlative Cosmology, linking Enochian and Pakua

Stargrams, Elements and Meridians

Figure 2: see Hua Shuo's Shi Si Jing Fa Hui (An Elucidation of the Fourteen Channels), providing two additional Zodiac points : 14th Harmonic [360/7 = 25.714286° ~ Semiseptile (aka, quattuordecimal)] · if you can't get to grips with a seventh harmonic chart, look at its expression in the fourteenth harmonic. This is the astrological representation of the manifestation of 7 (Harmonic Charts) · The acupoints are found only on the shi si jing (fourteen channels): the twelve jing mai (primary channels) and two of the qi jing mai, the ren mai and du mai. All other channels intersect acupoints on these fourteen channels and have no acupoints on their own (Chasing the Dragon's Tail) · The meridians, up and down, right and left, front and back, are twenty-eight in number (Chinese Acupuncture) • Figure 4: compare with Klein's Figure 10.

Celestial Lancets: A History and Rationale of Acupuncture and Moxa
Lu Gwei-Djen and Joseph Needham

{24} (i) The circulation of chhi and blood

So far we have been speaking mostly of the ching-lo [經絡] system, that of the acupuncture tracts and their branches, only too commonly thought of in the West as the be-all and end-all of the medical theory lying behind acupuncture. But in the ancient, medieval and traditional texts we meet frequently with another expression, much wider in significance, ching mo [經脈]. This may be translated as the tract-and-channel network system. Here we write it just as it appears in all the texts of the classical tradition; but there is another way of writing mo [脉], and this it is which has been adopted in contemporary Chinese printing, always avid of simplifications, for ching mo [經脉]. Since we have two orthographic forms at our disposal we propose to adopt a different and more useful convention. The word mo was used in a variety of senses. The first meaning that comes to the mind of an ordinary translator is 'pulse', and this is right enough, but the word also meant blood-vessels in general, the vessels which carried the hsüeh, and could even by extension refer to the heart. The characters mo [脈] and mo [脉] are normally completely interchangeable, but it will be convenient to use the first form mainly for statements about the pulse, restricting the second form mainly to talk about the channels of blood (with their contained hsüeh); and we shall try consistently to do this. Thus we write the typical phrases for the pulse of the blood in the arteries as forms of mo hsiang, but when we say that 'the blood moves in vessels' we write hsüeh hsing mo nei. Thus we shall write ching mo [經脈] to mean 'the tracts and the pulse', and ching mo [經脉] to mean 'the tracts and blood-channels' (more or less equivalent to the modern anatomically precise idea of blood-vessels and capillaries). But the widest meaning of ching mo in general remains 'the tract-and-channel network system' and this we shall often call it. The moment is now ripe for developing both these two conceptions a little, but first there is something more to be said about the relation of ching mo and ching lo.

The expression ching mo was often used almost synonymously with ching lo, partly because the ancient Chinese physicians had a great deal more at the back of their minds than the modern student of Chinese medical history, and partly because they did not always make the same kind of sharp distinctions that we find in modern science, even modern biology and physiology. It will be remembered that the word mo also occurred in inter-connections between the tracts (ching) as lo mo, though their dendritic capillary terminations resumed the name of lo as sun lo. As we shall see in a moment, the Yang circulated through the chhi tracts, ching, while the Yin was carried round in the blood through the vessels. So just as a mo appeared on the Yang side, a lo appeared on the Yin one, for the blood-containing capillaries in the epidermis were termed hsüeh lo. Thus there was a certain terminological overlap, and one can only say that on the whole, and broadly speaking, the ching and lo were channels for Yang and chhi, while the mo were channels for blood and Yin. Undoubtedly the typical abbreviation ching lo can be confusing to those uninitiated in classical Chinese medical literature, since mo is left out of it and hence the basic theoretical parallelism is obscured. Of course practical acupuncture itself was done mainly at points along the tracts, but it was based on a much wider body of theory which involved channels other than tracts. It also involved sphygmological diagnosis (mo hsiang), and the study of the inter-relations between the internal organs (tsang hsiang) in accordance with the principles of patefact and subdite (piao li) phenomena.

The expression ching mo [經脈] is an important one, too much overlooked in modern attempts to introduce therapeutic acupuncture into the Western world. 'Tracts and pulse' had a most intimate relation, because it was only the very delicate observation of the 28 different types of pulse which could give a clue as to which tracts were needing acupuncture. After all, these were the waves stemming from the heart beat, and that would be expected to be responsive to diseased conditions in various other parts of the body, especially the viscera. In a page or two we shall learn from original texts how intimately connected the Yang and Yin were thought to be — by scrutinising the latter in blood, therefore, it ought to be possible to know what was going wrong with the former in the tracts. This was the link, then, which the physicians were arguing about in the Warring States and Early Han periods. Only the blood side could tell you what to do with the needles on the tract side (Fig. 8).

AQ 345 = TILING OF THE PLANES = ROD OF APPOINTMENT (HEB-168 מטד מוסדה).

AQ 605 = GOD IS AT THE CENTER OF THIS FULCRUM = HAY DALET SAMEKH VAU MEM DALET TET MEM.

AQ 1187 = THE EIGHTFOLD WAY: THE BEAUTY OF KLEIN'S QUARTIC CURVE (SILVIO LEVY, ED.) = CARMARA (HO) IS ALSO BALIGON WHO IS THE FIRST OF THE 49 SPIRITS (SEE TABLE AT 41A).

AQ 2786 = TWO ANGELS - THE KING CARMARA AND HIS PRINCE HAGONEL - ARE SPECIAL IN THE HEPTARCHY. THEIR NAMES DO NOT BEGIN WITH THE NOTORIOUS "B" AS IT IS THE CASE WITH ALL OTHER ANGELIC NAMES = LOVE IS LIKE A WILLFUL BIRD, DO YOU WANT IT? IT FLIES AWAY! YET, WHEN YOU LEAST EXPECT ITS BLISS, IT TURNS AROUND AND IT'S HERE TO STAY! —DR. BRONNER'S PEPPERMINT SOAP.

AQ 2745 = COME THEREFORE, O THOU CLOUD, AND WRETCHED DARKNESS, COME FORTH I SAY OUT OF THIS TABLE: FOR THE LORD AGAIN HATH OPENED THE EARTH: AND SHE SHALL BECOME KNOWN TO THE WORTHY = THE KLEIN QUARTIC CAN BE VIEWED AS AN ALGEBRAIC CURVE OVER THE COMPLEX NUMBERS C, DEFINED BY THE FOLLOWING QUARTIC EQUATION IN HOMOGENEOUS COORDINATES: X^3Y + Y^3Z + Z^3X = 0.

AQ 3078 = ICH WÜNSCHE NUN NOCH ZUM SCHLUSSE ZU ZEIGEN, WIE WEIT DIESE LAGE[N]VERHÄLTNISSE HERVORTRETEN, WENN MAN DIE REELLEN PUNKTE DER KURVE VIERTER ORDNUNG Λ³Μ + Μ³Ν + Ν³ = 0 INS AUGE FAßT = NOW COMETH OUT OF THE TABLE A DARK SMOKE, AND THERE REMAINETH ON THE TABLE A GOLDISH SLIME: AND THE THINGS WHICH HOVERED IN THE AIRE DO NOW COME, AND LIGHT DOWN ON THAT SLIME, AND SO MOUNT UP AGAIN.

AQ 6804 = BY CONSIDERING THE ACTION OF THE MODULAR GROUP ON THE COMPLEX PLANE, KLEIN SHOWED THAT THE FUNDAMENTAL REGION IS MOVED AROUND TO TESSELLATE THE PLANE. IN 1879 HE LOOKED AT THE ACTION OF PSL(2,7), THOUGHT OF AS AN IMAGE OF THE MODULAR GROUP, AND OBTAINED AN EXPLICIT REPRESENTATION OF A RIEMANN SURFACE. HE SHOWED IT HAD EQUATION X3Y + Y3Z + Z3X = 0 AS A CURVE IN PROJECTIVE SPACE AND ITS GROUP OF SYMMETRIES WAS PSL(2,7) OF ORDER 168 = …THE INITIAL TIME-CIRCUIT NUMMOGRAM THING… THE ENOCHIAN SYSTEM AND THE PANDEMONIUM MATRIX/NUMMOGRAM TO BE THE TWO GREATEST OCCULT MATHEMATICAL MYSTERIES OF OUR TIME, SO ANY CROSS RELATING OF THE TWO IS SURE TO BE A HYPERSTITIONALLY POTENT MIXTURE… LOOKING UP MORE ON ALL THE MATH… TO SOLVE THIS OR MAKE ANY MAJOR CONTRIBUTIONS… HAVE A HIGHER DEGREE OF MATHEMATICAL AND PHILOSOPHICAL COMPETENCE.

AQ 1568 = QUEL PIVOT, J'ENTENDS, DANS CES CONTRASTES, À L'INTELLIGIBILITÉ? IL FAUT UNE GARANTIE—LA SYNTAXE (Le Mystère, dans les Lettres :: What fulcrum is there, I mean, for intelligibility in these contrasts? A guarantee is necessary—Syntax • What pivot is there, I mean within these contrasts, for intelligibility? A guarantee is needed—Syntax) = WATER OF WATER :: TDIM·TAAD·MAGL·NLRX • YTPA·RZLA·TNBR·XGSD • PHRA·BOZA·OCNC·ASMT • ANAA·DOPA·PSAC·ZIZA (64 Letters of the Kerubic Squares).

AQ 304 = DEE IS THE FULCRUM = 45-LETTER NAME OF GOD = CELESTIAL LANCETS = FRIEDRICH SCHLEGEL = GALACTIC ALIGNMENT = HECATE TRIFORMIS = HEXAGRAM XIAO GUO = MAYA DECIPHERMENT = PESTIS SOLIDUS = REDEEMING THE TIME = THE BELL CURVEBALL = THE BOOK OF THE KOM = THE PLATINUM NET = THE WAYS OF BALANCE = TREASURE TROVE = TRIANGULAR TRADE.


Understanding the I Ching: The Wilhelm Lectures on the Book of Changes
Hellmut Wilhelm, Richard Wilhelm

{103} Contributions came from the Taoists, who thus carried forward not only the work of their great master Lao-tse, whose view of the world was very close to that of the I Ching, but also the work of their later philosophers. Contributions were also made by a school of natural philosophers whose symbols and systems up to that time had moved along courses diverging in part from the I Ching.

In this way, for instance, a different, more rigid system of the world, the system of the "five stages of change," wu hsing, gained entrance to the I Ching. This system of "five stages of change" or activating forces, that is, water, fire, wood, metal, and earth, is also very ancient, going back to the Chou era at least, and perhaps even further. The earliest combination of the two systems is to be found in the two tablets [i, 332-33] , the Ho T'u, the Yellow River Map, and the Lo Shu, the Writing from the Lo River, both of which were held to have originated in mythical times and under magical circumstances, though in all probability their origin does not antedate the last part of the Chou era.

The five stages of change — with which had been co-ordinated the above-mentioned elements, and in addition the colors, tones, points of the compass, virtues, parts of the body, and everything else imaginable — were made to conform with the numerical symbols from the commentaries of the I Ching in these two tablets or diagrams.

The heavenly one united with the earthly six and begat water, which stands in the north. The earthly two and the heavenly seven produced fire, which stands in the south. From the three of heaven and the eight of earth came wood in the east; from the {104} four of the earth and the nine of heaven came metal in the west; while the heavenly five and the earthly ten produced the material element of earth (arable soil), which is in the center. The Lo Shu then takes the numbers, separates them, and, omitting the number ten assigns one to each of the eight trigrams. In this procedure, only the central five (as the most complex of the numbers, the one that unites the two basic elements, heaven and earth: three and two) is immune from this kind of allocation by characteristics.

Originally there seem to have been texts accompanying these tablets, but except for insignificant fragments they have been lost. We have still another group of books from this period, however, reflecting a similar spirit, which were accepted as secondary commentaries on the Book of Changes. These are the so-called eight apocrypha, in Chinese wei, which means the woof in weaving, as against the warp, ching, the classic book itself. The most important of these apocrypha is the I-wei ch'ien tso-tu, a discourse inspired by the hexagram Ch'ien, The Creative. The following apocrypha will give an idea of the line of thought followed:

The holy men of ancient times followed the yin and the yang and from them determined decay and growth; they established Ch'ien and K'un and thereby grasped heaven and earth. Thus formed things originate from the formless. But from what do Ch'ien and K'un originate? It is said (by Lieh-Tsǔ): There is a primal change, a primal beginning, a primal origin, and a primal creation. The primal change is the state in which energy does not yet express itself. The primal beginning is the {105} state in which energy originates; the primal origin is the state in which form originates; the primal creation is the state in which matter originates. When energy, form and matter are present, but not yet separate, we call this chaos. Chaos means the state in which all things still exist mixed together and have not yet separated one from another. If one looks at it there is nothing to see; if one listens to it there is nothing to hear; if one follows it, one obtains nothing. [Cf. Tao Tê Ching, ch. 14.]

Change has no limiting form. Change transforms itself and becomes one; one transforms itself and becomes seven; seven transforms itself and becomes nine. Nine is the end point of this transformation. Change then transforms itself again, however, and becomes two; two transforms itself and becomes six; six transforms itself and becomes eight. One is the origin of the transforming form. What is pure and light rises up and becomes heaven; what is turbid and heavy sinks down and becomes earth. Things have an origin, a time of completeness, and an end. Therefore three lines form the trigram Ch'ien. When Ch'ien and K'un come together all life originates, for things are composed of yin and yang. Therefore it is doubled and six lines form a hexagram.

Yang in movement advances forward; yin in movement draws back. Thus yang has seven and yin eight for a symbol. A yin and a yang together make fifteen. This is called tao. Now yang transforms seven to nine and yin transforms eight to six and together they also make fifteen. Thus the sum of the symbol and that of the transformed symbol is the same. The five tones, the six sound pipes, and the seven planets are made on this pattern. The number of the great expansion is fifty. The {106} changes take place according to it, and the spirits and genii act in accord with it. The day has ten stems, these are the five tones. There are twelve zodiacal places, there are the six tone-pipes. There are twenty-eight lunar mansions: these are the seven planets. Therefore fifty is the great gate through which things come forth. Confucius said: "Yang is three, yin is four. That is the correct place."

Fifty was the number of yarrow stalks used in consulting the oracle. In the struggle to unite the existing systems and groups at all costs, the numbers gradually degenerated from a symbol pregnant with meaning to a means for speculative, idle play. Considerable acuteness was required to formulate all these combinations and groups of combinations in a convincing way, and here and there, together with much rubbish of logical errors and banalities, we come on shining crystals whose many facets refract a ray of real meaning. Twenty-one different commentaries on the Book of Changes and a series of theoretical works, all of which are based on the apocrypha, have come down to us from the Han era [206 B.C. – A.D. 221].

•    •    •    •

{126} …the idea that access to this wisdom is to be obtained through the manipulation of yarrow stalks. The connection between the two is difficult for us to understand. That a number or group of numbers arrived at by counting off stalks should form the gateway to such insights is something we do not readily accept. Wang Fu-chih, the greatest I Ching scholar of the Ch'ing era, tried to explain it as follows:

Between heaven and earth there exists nothing but law and energy. The energy carries the law and the law regulates the energy. Law does not manifest itself (has no form); it is only through energy that the image is formed, and the image yields the number. (Image here equals idea, number is the intelligible aspect of law as embodied in the idea.) If this law becomes blurred the image is not right and the number is not clear. This reveals itself in great things and expresses itself in small things. Thus only a man of the highest integrity can understand this law; basing himself on its revelation he can grasp the symbols, and observing its small expressions, he can understand the auguries. In this {127} way the art of the image and number (that is, consulting the oracle) comes about by itself. [Chou I Nei-chuan]

To Wang Fu-chih, therefore, number is the phenomenal form of the law, that one of its expressions in which it is intelligible. Indeed, to seek the law through numbers and to base oneself on it is a principle by no means alien to us either, although in our case it is a different kind of law we are intent on understanding and mastering. From here it is no great step to the idea that even spheres of life to which Western science has not yet applied such methods are governed by laws, indeed that the totality of life is based on law. This step was never taken in the West, though many men have intuitively striven in that direction. Leibniz is a case in point.

•    •    •    •

{8} If nonetheless we undertake to devote some hours to this book, it is because we have important reasons. A book that has stood in such high repute among the men who have determined the fate of China, and that beginning with Leibniz (we shall have more to say on this point later), has had so much influence on the leading minds of Europe, will have something to offer us also. If we can obtain from it some insight into the minds of our hosts in this country whose guests we are, that in itself will be rewarding. And if, moreover, we should succeed in deriving from it not only understanding, but also genuine illumination, our undertaking will have been entirely justified. We shall do well, though, to keep our doubts in mind as we proceed, rather than suppress them. So may we avoid overenthusiasm and its resulting undertow.

{114} The second of the great masters of Sung Confucianism, Shao Yung, was a speculative genius, and his philosophical deductions from the fundamental concepts of the I Ching are so concise and charged with meaning that it is virtually a sensuous pleasure to follow them; at the same time, they are so wide-ranging as to reach into every corner of the world. Shao Yung's mathematical exactitude led him to work out a different I Ching table, in which he arranges the hexagrams in a natural system.

He starts with the two primary lines, the light and the dark, then adds to each again a light and a dark line, thus obtaining four two-line complexes:

Above each of these a light line and again, alternately a dark line is added, so that the eight trigrams stand in the following arrangement:

Continuing in the same way, he obtains first complexes of four, then of five, and finally of six lines, that is, the {115} hexagram, as shown in Figure 1. This so-called natural order can be arranged in a sequence, one hexagram after the other, or in a square of eight times eight hexagrams, in which counting begins at the lower right-hand corner and continues through to the lower left, then starts again at the right on the second line from below and continues to the left, and so on. Finally, the arrangement can also be a circle; one half was separated and inverted in order to make it harmonize with an earlier arrangement of the eight trigrams known as the Sequence of Earlier Heaven. [I, 285] (Cf. Figure 2.) Earlier Heaven here does not signify something antecedent in time but the transcendental world of ideas — in other words, an a priori arrangement.

Shao Yung's schema has led to one of the most extraordinary episodes in the history of the human mind, and to this day it has never been satisfactorily cleared up. More than six hundred years after its origin, Shao's diagram fell into the hands of Leibniz through the agency of Jesuit missionaries, and he recognized in it a system that had previously sprung from his own mathematical genius. To facilitate the solution of certain mathematical problems, Leibniz had thought out the so-called binary, or dyadic, numeral system, which makes use of two numbers only, instead of ten, but otherwise follows the same principle as the decimal system. The two figures are 0 and 1. The numerical sequence of the binary system would looks as follows:

      1, 10, 11, 100, 110, 111, 1000, etc.

In the Sequence of Earlier Heaven Leibniz now rediscovered his own dyadic system, though he had to begin with zero for the correspondence to emerge. He took the broken line for a zero, and the unbroken for a 1. Thus the hexagram Po was 1, if zeros preceding 1 are disregarded, and stood in the first place in his system;

{116–117} Figure 1. Shao Yung's Sequence [Binary/Decimal Values : last line, reading right-left corresponds to square in Figure 2. The Circular Sequence (page 118) starting from top-left: 2, 23, 8, 20, 16, 35, 45, 12, 15, 52, 39, 53, 62, 56, 31, 33, 7, 4, 29, 59, 40, 64, 47, 6, 46, 18, 48, 57, 32, 50, 28, 44, 24, 27, 3, 42, 51, 21, 17, 25, 36, 22, 63, 37, 55, 30, 49, 13, 19, 41, 60, 61, 54, 38, 58, 10, 11, 26, 5, 9, 34, 14, 43, 1 ~ (Yijing hexagram sequences)].

{119} the next, Pi, was 10, that is our 2; and so on. Leibniz placed the zero (=K'un) at the beginning of the sequence, and so Shao Yung's system corresponded point for point with the binary system, right up to the last hexagram, Ch'ien, which for Leibniz was 111111, or 63. The only difference is that this correspondence is not a direct but an inverted one, that is, in order to obtain it, one must begin at the end of the series, which serves to emphasize once more the fact that parallel cultural phenomena in East and West are as mirror images to each other! Nonetheless, the correspondence arrived at by these two great minds independently, each having started from a completely different basis, is truly an astonishing phenomenon. To Leibniz, the key to the problems before him was number; to Shao Yung, it was the hexagram. And the intellectual means by which these two kindred spirits tackled their problems took on the same form in both. For a long time Leibniz had been trying to validate spiritual truths in mathematical terms, thus making them, as he thought, irrefutable. It is easy to imagine the enthusiasm aroused in him by the discovery of this correspondence. [For a more detailed treatment, see Hellmut Wilhelm, "The Concept of Time in the Book of Changes," Man and Time (Papers from the Eranos Yearbooks 3; New York and London, 1957), 215ff.]

another translation…

American Civilization Portrayed in Ancient Confucianism
Wei-Bin Zhang

{22} Leibniz first published his binary system in a paper entitled De Progressione Dyadica in 1679. In 1698, one of the Jesuit missionaries in China brought to Leibniz's notice the hexagrams of the I Ching as offering a parallel to his binary system.¹ When Leibniz read copies of Shao Yung's arrangements of the Sixty-four Hexagrams, he immediately recognized its similarity to his own binary arithmetic. In 1713, in a book called Two Letters on the Binary Number and the Chinese Philosophy, Leibniz claimed that the I Ching was parallel to his own binary code; indeed, he called his invention a "rediscovery" of the principles of the I Ching. But there is more. He conceived this system not only as a purely mathematical scheme but also as a universal calculus. It was supposed to be a language that could help to reconcile the various religious factions in the Europe as well as the nations of Asia and Europe.

__________________
¹ Substituting a zero for each broken line and a 1 for each unbroken line. He placed the zero for the Kun at the beginning of his sequence. The hexagram Po was 1 (as zeros preceding 1 are disregarded), and stood in the first place in his system. The next, Pi, was 10, that is, our usual 2. The procedure may be continued until all the Sixty-four Hexagrams are numbered in his binary system. In his sequence, the last one is Chien, transcribed as 111111, or 63.

•    •    •    •

{128} It was natural that yarrow stalks in particular were chosen, because they grew wild in the common which in ancient times was set apart for sacred rituals. The plant was ready to hand, it grew in a hallowed spot and no other justification was required. Since the common has ceased to play this role, it has been customary to gather the yarrow stalks from some other hallowed place, for instance, from the grave of Confucius or of Mencius.

Fifty yarrow stalks make up the set used for consulting the oracle. However, of these fifty only forty-nine are used, one is set aside right at the beginning and plays no further part. I shall not go into the speculations on number symbolism that were attached to this and to the remaining procedures; they can perfectly well be omitted from a description of the method as such.

{14} …it is important to point out that the oracle of the Book of Changes uses as its key the forces of vegetable rather than animal life. It was a plant growing in sacred places, the yarrow, whose stalks gave access to the oracle when manipulated in a certain way.

•    •    •    •

How to Form a Hexagram and Consult the I Ching
Shih-Chuan Chen

Let us examine Conspectus A, Chapter IX, for procedures to be followed when consulting the I Ching: 1. The number of the Great Elucidation (ta yen) is fifty. Of these, forty-nine are used. They are divided into two portions, to represent heaven and earth. Hereupon one is set apart, to represent man. They are counted off by fours, to represent the intercalary month. There are two intercalations in five years and, therefore, there are two such operations; afterwards the whole process is repeated. 2. Heaven is one, earth is two; heaven is three, earth is four; heaven is five, earth is six; heaven is seven, earth is eight; heaven is nine, earth is ten. 3. There are five heavenly numbers. There are also five earthly numbers. One series of the five numbers corresponds to the other series of the five numbers and each number of one series is paired with a number of another series. The sum of the heavenly numbers is twenty-five, that of the earthly numbers is thirty. It is within this number (fifty-five) that the changes and transformations are effected and the spirit-like agencies function. 4. The numbers that yield the creative total 216, while those which yield the receptive total 144, making in all 360. They correspond to the days of a year. 5. The numbers of the stalks used for the sixty-four hexagrams amount to 11,520, which corresponds to the ten thousand things. 6. Therefore, by means of the four constituents (6, 7, 8, 9) the changes are made possible; eighteen operations yield a hexagram.

In Chapter IX in addition to the constants of the heavenly numbers (1,3,5,7,9) and the earthly numbers (2, 4, 6, 8, 10) there are two more constants, 50 and 55. The number of the Great Elucidation is 50, which deals with the operations in making a hexagram. The total sum of the heavenly numbers and the earthly numbers is 55, which provides the calculation by which its "related hexagram" is determined. They were assigned two different functions by the Chou Chinese.
. . . . . .
Following what is given in Conspectus A, Chapter IX, we find a more natural, consistent, and simpler way (in contrast with the arbitrary way of Chu Hsi), to form a hexagram and to determine its "related hexagram." We believe that it was the way by which the Chou Chinese consulted The I Ching.

Let the diviner take out the fifty yarrow stalks from his container and keep one stalk aside as the unused one. There are 49 stalks. This follows Conspectus A, Chapter IX where it states, "The number of the Great Elucidation is fifty. Of these, forty-nine are used."
. . . . . .
About how the "related hexagram" is found or, in other words, how a hexagram is transformed into another no rule is given in Conspectus A & B or in the other part of the "Ten Wings." However, there are a few scattered hints. It is said in Conspectus A, Chapter IX, 3, that "The total sum of heavenly numbers and earthly numbers is fifty-five. It is within this number that the changes and transformations are effected and the spirit-like agencies function." These are very significant statements as far as divinatory procedures are concerned.
. . . . . .
Table 1 may help us to find out which monogram is called upon to change in order to acquire the "related hexagram." The calculations in the table show all the possibilities, but fewer possibilities than those listed have been verified by the cases recorded in the Tso Chuan and Kuo Yü.

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In terms of searching for "meaningful coincidences" in a world governed by chance, the idea of "synchronicity" may be more fruitful than the concept of a linear causal linkage (causality). However, behind "synchronicity" there is a deep conviction of the Chinese that no lines of demarcation exist between man and heaven and earth. The Principle of The Three Participants (san ts'ai chih tao) is demonstrated in a trigram or a hexagram and is repeatedly referred to in the "Ten Wings." Because of this principle the ancient Chinese avoided becoming victims of the fallacy of the bifurcation of nature. The derivatives from the bifurcation, namely: the separation of the subjective and the objective, the distinction of the primary and secondary qualities, and the confrontation of the ego and non-ego, have not tortured the Chinese mind. According to the Principle of The Three Participants, the ancient Chinese viewed the world neither as phenomena nor as noumena but as reality. The cognitive, the perceptive, the conceptive or the appreciative is not regarded as an act of projection from the subject onto the object but as one of the manifestations in the universe. From the macrocosmic point of view the universe is ageless, well balanced and, itself, a Grand Equilibrium. From the microcosmic point of view, changes and transformations resulting from the mutual responses of the three participants take place everywhere and at all times. Basing themselves upon their deep-rooted conviction, the ancient Chinese searched for a device (operations on a set of constants) to predict the possible consequences of their actions in this world governed by chance. Scapulimancy, the Chou I and other oracle books were devices which met their demand.

In a world governed by chance the ancient Chinese sought not only the "meaningful coincidences (most of them were the historical events since the immemorial past)" but the "significant symbols" of either the natural phenomena, or qualities of things or images to strengthen his psyche at the moment when a final decision was announced. We must realize that the mental process of decision-making is complex. No matter how much information and data are piped into the headquarters of the army, the Chief Commander has to make hasty decisions as situations develop. In our lifetime situations force us to make vital decisions in the nick of time. We can hardly exercise our "free choice" by assorting and evaluating the information and data. Rather, we make decisions based upon our convictions. "Synchronicity" provided a "chance hit" device for the ancient Chinese; but the conviction that whatever a man does will affect heaven and earth, and that how ever the heaven and earth behave will affect man induced them to consult the I Ching.

see also:

Stick Dice for the I Ching
An Interpretation of the Divinatory Inscriptions on Early Chou Bronzes (PDF)