reordering hexagrams mckenna mair kua music

7/28/2019 Reordering Hexagrams McKenna Mair Kua Music

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Stephen E. McKenna and Victor H . Mair A reordering of the hexa-grams of the I Ching

INTRODUCTIONThe importance of the I Chinga, or Book of Changes, in Chinese philosophycannot be overemphasized. It represents a unique attempt to create a totallyabstract representation of the real world. T he sixty-four hexagram s containedtherein cover all possible situation s which ca n arise , ' an d they, along with theinterpretations attache d to them, must be thought of as a system.

Most studies of the I Ching have focused attention on the interpretationsattached to the individual hexagrams, but some consideration has been givento the overall structure, and the relationships am ong the hexagrams. Since thesignificance of each hexagram should become clearer in the light of thathexagram's position in the overall structure, this is obviously not an un-important field of study. Blofeld sensed this when he stated that, "If, indealing with any hexagram, we glance at those immediately before and afterit, we shall be able to eva luate the total situatio n more effectively."' But thispresupposes that the hexagrams be arranged in a logically meaningfulsequence.

The purpose of this article is to discuss briefly orderings of the hexagramswhich have been previously put forward, and to present one of our ownmaking, hopefully showing its validity as a useful tool to understand betterthe hexagrams. This new ordering was devised under our belief that anordering ou ght t o exist which follows a logical system of development, both ingeometrical appearance a nd in the interpretations attached to the hexagrams.PREVIOUS ORDERINGS OF THE H E X A G R A M ST o discuss the factors which contributed to the making of this ordering, it isfirst necessary to review certain previously suggested ordering systems.

One factor which has commonly appeared is the idea of arranging thehexagrams in pairs. Two main methods of doing so were used. One was thech ' i e n - kuab in which each hexagram was paired with the one into which itwould chang e if turne d up side dow n (tha t is, "inversion"). Th e other was thep hng - t ' ung ' in which two hexagrams are paired if they are opposites linefor line, so that each position occupied by a strong (undivided) line inone hexagram is occupied by a weak (divided) line in the other (that is,"opposition"). T hu s the hexagram Kou, Coming to Meet, is paired with

Stephen E. McKenna was studying Applied Mathematics at Harvard University. Victor H. Mair isAssistant Professor of Chinese Literature at th e University of Pennsyvania.

A U T H O R S . NOTE: In this essay, M r . M cKen na has been responsible .for the conception, thecomputat ion, and the com ments on the hexagra ms, and M r. Ma irk role has been to provide amo dicu m of sinological suppor t. W e wish to expres s our profound gratitude t o Professors JosephNeedham and Hellmut Wilhelm who have read the article in manuscript and have ma de a number ofvaluable suggestions. The criticism of other scholars who have read parts of this essay is deeplyappreciated also. Any errors and mis c~ nce ptio nswhich remain are solely our responsibility.Phr iu~uphy a~r nd Wrsr 29 , no. 4 (October, 1979) C by T h e Univers i ty Press o f Hawaii. All rights reserved.


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422 McKenna and Mair

Kuai, Breakthrough, by ch'ien-kua, and with Fu, Return, by p h n g - t ' ~ n g . ~Ch ien-kua P 'ang-t ung

Kou t%i Kuai R Fu @The ordering which presently appears in the Book of Changes is the so-

called Chou ordering, traditionally attr ibuted to the founder of the Chou,King Wen. If he was in fact the autho r, that w ould d ate the ordering to abou t1150 B.C. K u o Mo-jod, however, has presented evidence to da te the text of theBook of Changes to the middle of the Spring and A utum n period, aroun d 600B . c . ~The "Ten Wings," o f course, da te from an even later p e r i ~ d . ~n anycase, it is clear that a different ordering was in use prior to it. Judging fromthe available evidence, Arthur Waley's opinion that "The text was probablycut up and shuffled a good many times before it reached its present order"seems well grounded. The Li chi records that Confucius found a volumedating from Shang (1766-1 112 B.c.) times called the K'un-ch'ienx.

Confucius said: "I wished to observe the way of the Hsia dy nasty an d, fortha t reason, went to Ch'i.' Th ou gh it did not fulfill my expectation^,^ I didobtain the Hsia Calendar. I wished to observe the way of the Shang dynastya n d, for t ha t reason, went to S ~ n g . ~hou gh it d id not fulfill my expectations,I did obtain the K'un-ch'ien.1 Thus d id I observe1' the way of the Hsia andthe way of the Shang by means of the intervals of the Hsia Calendar and themeanings of the K'un-ch'ien." l 2This title, K'un-ch'ien, is composed of the first two hexagrams in the Chouordering, but in the reverse order.13 In fact it is not surprising that this shouldbe the case, since early Chinese thought gave precedence to the feminine( yin"") force, and the hexagram K'un is made up entirely of divided lines,representing th e yin.14 The Chou dynasty was the most strongly patriarchalsociety to appea r in Ch ina up to th at time, so the reversal of this fundam entalprecedence would be expected if in fact King Wen was responsible for thereordering. If, as Kuo Mo-jo asserts, the present version was established afterConfucius, it is equally understandable that it has an even stronger patriar-chal impress.

Aside from this reversal of the precedence of the two basic hexagrams,there is only one apparen t logical factor in the ordering, th at of pairing. Mostof the hexagrams are paired according to the ch'ien-kua method. There are,however, eight hexagrams which remain unchanged when turned upsidedown; these are arranged into pairs by p'ang-t'ung. There appears to be nological reason be hind the progression from o ne pair to the next.

Among the Ten Wings is an appendix, the Hsii kuaaY "The Sequence ofthe Hexagrams") which att em pts to explain the progression in ideas from on e


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In his philosophy, Leibniz identified 0 with the nothingness of unformedChaos and 1 with G od . He felt tha t Sha o Yung's ordering system showed thedevelopment from the one to the other, and in fact recommended using thesystem for purposes of evangelization. At that time there was a commontendency for missionaries in China to try to present the Christian God assomething which was not foreign t o Chinese thought, and attempts to identifyHim with T ao were common.

The Sh ao Yung-Leibniz orderin g system is mathem atically logical in theprogression from one hexagram to the next; it is not logical, however, inits progression from one interpretation to the next. When the situationsdescribed by the hexagrams are viewed in this order, the development fromone to the next seems to follow no pat ter n at all.THE DEVELOPMENT OF THE PROPOSED OR DER ING

In seeking a logical orde ring system, we began by looking at th e way in whichone hexagram becomes another by "moving lines." A moving line is a linewhich is so strong that it changes to its opposite. In a very strong hexagram,all the lines would be moving lines, an d in ma ny cases it is easy to see how thesituatio n represented by a hexagram could change into that represented by itsp hng-t 'ung opposite. We arra nged the hexagram s in pairs in this way.

The phrase 'b 'ang-t 'ung" appears in the W en-yenb f commentary on thehexagram Ch'ien, where it reads: "The six individual lines open up andunfold the thoug ht, so that the cha racte r of the whole is explained through itsdifferent sides." P'ang-t 'ung , which is tran sla ted here as "different sides,"literally means "lateral interchange." This phrase has been interpreted tomean that there is a correspondence between Ch'ien and the six hexagramswhich can be formed by a change of only one line. However, the great C h'ingscholar of the Book of Changes, Chia o H su nb g (1763-1820), theorized t ha tphng-t 'ung means changing all the lines at once, so that each hexagram ispaired with its oppo site.22 Certainly this is not an unreasona ble suppositionconsidering tha t K'un an d Ch'ien were paired in both the Chou and Shangorderings. Although Chiao Hsun proceeded t o develop the concept of p'ang-t 'ung in such a manner that it could be applied to all of the hexagrams, hefollowed traditio n in keeping Ch'ien in the first place.13

We have chosen the feminine hexagram, K'un, to sta rt our sequence out ofa desire to return to what we believe were the original ideas involved in theshaping of the I. The early Chinese philosophies emphasized the motherimage as the origin of all things, and the existence of a version of the I Chingduring the Shang period shows that the origin of the I predates the patriar-chal society of Cho u.

The next step in constructing a logical ordering system was to find amethod of development from one pair of hexagrams to the next. It seemedmost logical to have a minimal change between the hexagrams, that is, only


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Figure 1. Th e Binary G ray C ode Sequence

one line changed. What was needed was a system by which all possibilitiescould be covered thro ugh an orderly progression while changing only one lineat each step. We chose, like Leibniz, to view the hexagrams as analogous tobinary numbers. There does exist a type of binary number system whichprogresses while changing only one digit in a perfectly orderly manner. It isknown as the G ra y Code, for Frank Gray, a theoret ical mathematician w hodeveloped it .24When the hexagrams were arranged in p'ang-t'ung pairs, andthe progression from each pair to the next was according to a Gray Codeprogression, a logical sequence of development became apparent in theinterpretations of the hexagrams. Figure 1 shows the sequence of develop-ment for the initial sixty-four binary numbers in the Gray Code; Figure 2shows how this combination of phng- t ' ung pairing and Gray Code pro-gression translates in to an arrangem ent of the hexagrams.

In his correlation between the binary number system and the hexagrams,Leibniz identified the rightmost digit in a number with the top line of thehexagram; he did this because tha t is the way Sha o Yung ha d developed hisprogression of the hexagrams. Obviously the correspon dence could have gonethe oppo site way, with the rightmost digit analogo us to the bo ttom line of thehexagram. We chose to view it that way for the followirlg reasons: In anynumbering system the rightmost digit changes most frequently while theleftmost changes least frequently. It seemed logical that the most importantline of the hexagram, the fifth, should be the last to change. Note thatnowhere in the sequence does the sixth line alone change.

There is some reason to speculate that the proposed ordering may be arediscovery of an older order. The strongest argument in favor of thisassertion is the logical progression of the in terpre tation s, which we shall detailpresently. However, it is first necessary to establish that the ancient Chinese


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Figure 2. Ordering of the Hexagrams


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son), L i (second daughter), Kgn (third son), Tu i (third daughter). Thissequence ca n easily be set up in a table such as Figu re 2:

This shares several obvious features with the proposed ordering of thehexagrams, differing mainly in its choice of Ch'ien, not K'un, to begin. Itdevelops, as the proposed ordering does, by a p'ang-t'ung switch followed by thechange of a single line. Although it is not exactly a Gray Code, it does showsimilarities: it matches perfectly the succession of the lower trigrams ofhexagra ms #21-28 in Figure 2. Because of this, we believe we are welljustified in stating that the concept of developing through p'ang-t'ung fol-lowed by the change of a single line would not be foreign to C hinese thoug ht.T H E S E Q U E N C E O F T H E I N T E R P RE T A T I O NWe consider one of the strongest arguments in favor of the proposedreordering t o be the logical way in which the interpre tations attached to thehexagrams develop. Each hexagram describes a situation; not only is there aclear progression from one s itua tion to th e next (tha t is, it is easy to see howone situation could develop into the next), but the starting a nd ending pointsare also logical.

These interpretations, which rely heavily on the Wilhelm-Baynes trans-lation, are based on the Judgments of the hexagrams as a whole. TheJudgments on the lines deal with individual components of the generalsituations and, for simplicity's sake, are not discussed here. A completeexposition of the hexagram s in light of the proposed reordering would requirefar more extensive treatment than we are able to afford them here. Thefollowing sequence of interpretations is, therefore, intended to serve as asuggestive outline only and should not be considered as a full and finalstatem ent of the meaning of the reordered hexagrams.1. K'unbq, The Receptive (No. 2 in Chou order). This is the logical startingpoint for anyone beginning a new endeavor. The situation described is one ofweakness, when one is not yet ready t o begin active work. T her e is no point intrying to accomplish som ething if one doe s not have t he strength necessary to


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complete the task. The text here recommends that a person in this situationshould find a strong person und er wh om he can serve. Althoug h the text do esnot explicitly say, this is also an ideal position for developing one's strength,by learning from the example of the person followed; such a position isanalogous to apprenticeship, wherein one gradually reaches mastery of anability in one's ow n right.2. Ch'ienb', The Creative ( 1 ) . Even after achieving strength, he continues tofortify himself; he is now re ady to b egin wo rk.3. Koub; Coming to Meet (44). After reaching a position of authority, hebegins to exert his influence on those in lower positions. Here he encountershis first challenge. He m ust deal w ith less worthy, but powerful, m en w ho seekto gain ascendancy. H e must work vigorously to prevent any gain in powe r bysuch people. Since he has the power to accomplish this, there is no reason forany sort of compromise.4. Fub', Ret urn (24). Wh en the problem h as been dea lt with, he should rest t oprepare himself for futu re endeavors.5. Linbu, Approach (19). Again there is a time for vigorous action, andsuccess is assured; however, the situation is not stable, and the time for actionwill come to an end when obstacles arise. He should be watchful, so that hecan sense the change in fortune and be prepared to cease his activity. If hecontinues to act, mindless of the change in situation, it could be dang erous.6. Tunbv ,Retreat (33). When the obstacles begin to appear, their nature willbe clearly delineated before they become a d irect th rea t. Since he is not s trongenough to face them directly he retreats to strengthen himself, keeping thethreatening forces at a distance un til he can deal with them .7. T'ung JCnbw, Fellowship with Men (13). The most obvious way to gainstrength is to gather o ther people arou nd him t o supp ort his defense.8. Shihbx,Th e Army (7). Th e people m ust be well organized t o be a n effectiveforce; a mob is a useless, and in fact dangerous, weapon. For this reason hemust ta ke care that he is strong enough t o control the people he has gathered.9. Shengby,Pushing Upward (46). When the proper preparations have beenmade, action is easily successful, and he gains a position wherein he isrespected even by tho se in higher positions.10. W u W a ng bz , nnocence (25). When outside thre ats have been eliminated,he can turn his attention to within his own society, working to benefit hispeople.11. P'i'", Standstill (12). It is well th at his atte ntion has been turn ed to w ithinhis own society, for it is from there that the next problem arises. Inferior mengain power, and he is unable to prevent it or do anything to remedy thesituation. All he can do is to sit tight and wait for the situation to resolveitself.12 . T'a iCb,Peace (11). Waiting eventually ends, and he is once again incontrol. The situation here strongly resembles that of # 10; once again he


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devotes himself to the w elfare of his people. T he difference is th at here there isa very strong warning tha t his character and his actions mu st be proper.13. Ming I", Darkening of the Light (36). Again the situation changes as hisauthority in society is no longer respected. His only course of action is to tryto continue to work, but to do so quietly, without drawing attention tohimself; otherwise he m ay be considered a thre at by those in power.14. Sungcd , Conflict (6). The situation worsens; this is why the previoushexagram recommended continuing to act in whatever manner possible,unlike the nonaction suggested in No 11. In that hexagram the situationwould resolve itself; here just the opposite happens: the evil forces becomestronger, and it takes strong a ction even to achieve a comprom ise.15. Liice,Treading (10). U nd er th e comprom ise, it is possible t o gain pow er;however, to do so he must show himself totally worthy of authority byabsolute blamelessness and humility. With any wrong move there will be evilmen ready t o grab at any excuse to throw him dow n. Th e idea expressed hereis similar to our modern aphorism that a politician must be "like Caesar'swife." It was in anticipation of this that the warning was attached to No. 12tha t he should carefully watch his actions an d character. Pow er can be gainedthis way.16. Ch'iencf,Modesty (15). Having gained power through his own humility,he now seeks to e stablish this ord er a m ong th e rest of society, giving pow er tothe humble, and taking power aw ay from those who are too boastful.17. Hsiao Kuocg,Preponderance of the Small (62). His power continues toincrease, but he is still hampered by the evil men in strong positions,therefore, it is a good time to concentrate his attention on less importantmatters. As always, he must pay close attention to his appe arance.18. Chung FuCh, nner Truth (61). The power of the evil men is nearing anend, but for the moment he must continue to try to work against them insmall, unnoticed ways. The Image reads, "Thus the superior man discussescriminal cases in order to delay executions." Richard Wilhelm interprets thisas an indication of the care which should be taken in the administration ofjustice, but in the light of the sequence it seems more likely that the superiorman here is simply stalling to prevent an injustice within the system beingperpetrated by those in power.19. Huanci , Dispersion (59). Finally the evil men have lost power and he isonce again able to exercise his authority freely. The period under poorleadership ha s led t o disunity amo ng the people; his job now is t o bring th emtogether again through a common purpose, such as participation in thereligious rites.20. F2ngcJ,Abundance (55). Having reached again a position of high au-thority, he should exercise it strongly, standing as a n inspiration t o his peoplean d being thoroug h in his adm inistratio n of justice.21. Ta Chuangck,The Power of the Great (34). He must take care, however,


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developing for a long time suddenly becomes apparent, bu t he has the supp ortof the people in working t o remedy the situation .36. Suicz, Following (17). His talents as a leader bring about success in hisendeavors, but he realizes that he cann ot lead at all t imes, and takes time outto rest.37 . Tuid", The Joyous (58). His leisure time includes gathering with hisfriends, but even in such social gathering he includes time for practicing, inorder to be constantly prepared .38. K2ndb, Keeping Still (52). He remains constantly calm, and does notworry about problems which are outside his abili ty to correct, but concernshimself with the im med iate situa tion only.39. Pidc ,Grace (22). He decides minor matters on the basis of appearances,acting according to what is considered proper. Thus he conserves his energyso that he may devote as much attention as is necessary to the importantproblems without being distracted by the minor ma tters.40. K'undd,Oppression (47). When a time of adversity arises, he is prepared,and can deal with it without being discouraged by the helpless appearance ofthe s i tuat ion.41. Ta Kuode ,Preponderance of the Great (28). Even when he stands alone,he has th e inner strength necessary to persevere an d a tta in success.42. Id' , Th e Corners of the M ou th (27). T o remain prepared, he is careful tonourish himself, bu t exercises temperance .43. Pod g ,Splitting Apart (23). Unw orthy men gain power within the society.He does not have the power openly to dismiss them, but they are weak-willedmen w hom he can prevent from causin g trouble by simple payoffs.44 . Kuaidh,Break through (43). The situ ation improv es slightly so tha t he canbegin to tak e quiet action, but he does not yet have strength enough t o startan open conflict. As he begins this quiet action, he continues the payoffs, sotha t the evil men will not realize th at he is working against the m.45. Kod i ,Revolution (49). When the time for action comes, he is successfulbecause he has been making p reparation s, and the positions of au thority arepurged of unworthy men.46. MBngd', Youthful Folly (4). Having brought society back to its properstate, he once again turn s his attention to his own development and becomesmore mature.47. Sundk,Decrease (41). W hen faced with a time of scarcity, he rema ins calman d uses this position of adversity to develop his character.48. Hsiend',Influence (31). It is now an auspiciou s time for gath ering togeth erwith o ther people, tha t he m ay learn from them.49. Ciziendm,Obstruction (39). When faced with troubles, however, he doesnot look to others fo r answers, but seeks the source of the problems which arewithin himself.50 . K'ueid", Opposition (38). In this way, while remaining among other


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people, he retains his individuality. Wh en he finds himself in the com pan y ofinferior people, he keeps his own superior character, even if the settingprevents him from behaving in a proper man ner.51. Wei Chid0 ,Before Completion (64). When abo ut to take a m ajor step, hefirst seeks to understand all the factors in the situation, so that he can dealwith them properly. In seeing all of the factors, he can then understand thesitua tion as a whole.52. Chi Chid", After Completion (63). Having attained success, he realizesthat this success is only temporary, as some new problem will arise, so hebegins to prepa re himself for whateve r may co nfro nt him next.53. Hsiidq, Waiting (5). Meanwhile he strengthens himself through nourish-men t a nd waits cheerfully.54. Chindr',Progress (35). His position is strong enough that, unlike thesituati on in No. 15, he can let his greatness be appar ent.55. Shih Hod'; Biting Through (21). He lets his strength show in the adminis-tration of justice. This is no t a time fo r a show of mercy.56 . Chingd' , The Well (48). He organizes the people in their work, for thebenefit of all.57. K h n d u ,Th e Abysm al (29). He also devotes himself t o the educa tion of thepeople, since this helps to bring a bo ut harm ony in society.58. Lid " ,Th e Clinging (30). On e way of teaching is by exam ple, an d so he letshis greatness be not only app aren t, as suggested in No. 54, but in fact shiningforth like a light. The ide a here is very much like that of Christ 's "You are thelight of the world." (M att . 5: 14- 16)59. Li id w ,Th e W andere r (56). Jus t as the previous hexagram carries forwardthe idea of N o. 54, this one develops the idea of No . 55. There the main pointwas tha t the adm inistration of justice should be strict; here the point is m adetha t it should not be overly severe, an d shou ld be take n care of quickly.60. Chiehdx,Limitation (60). In all these matters, he should determine theproper limits of conduct, taking care that they are firm but reasonable, thesam e qualities necessary to goo d justice.61. Chundy,Difficulty at the Beginning (3). It is not now a time for action, butrather for planning. After determining what the proper limits of conductshould be, he now considers what the proper places of people within societyshould be. This is analogous to the situation in No. 16. There he decidedwhat places people should occupy on the basis of a simple factor: humility.Here he is deciding on the basis of the proper limits of conduct, which he hasworked out on his own, because the more complex a situation gets, the lessapplicable are simple cut-and-dried rules; now he must go by rules which areparticularly approp riate t o the society which has been developed.62. Tingdz ,Th e Ca ldro n (50). Before he can pu t others in their proper places,he must first mak e sure his own position is correct.


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Surely in any situation one would want to know if one has the necessaryqualities, and there is no reason t o single Pi ou t in this respect.

On the other hand , when the hexagrams are arranged so that Pi com es lastin the sequence, there is a clear reason why such an instruction would appear:here is the only case among the sixty-four hexagrams in which there is nological subsequent step. A second consultation of the oracle is necessary toanswer the question, "Whither next?"T H E PHILOSOPHY B E H I N D T H E PROPOSED ORDE RI NGW e believe som e justification of this orde ring reg arding th e philosophybehind it is appropriate. In the first place, it fits in remarkably well with thedominant theme in Chinese thought of Order and Pattern, referred to byJoseph Needham in the second volume of Science and Civilisation in China asOrganism. It would also appear to be compatible with a dialectic principlesimilar to that put forth by Hegel and used frequently by Marx and Engels.Hegel's theory on the development of ideas begins with an initial idea calledthe thesis.30 This thesis prod uce s its opposite, the antithesis. Th e thesis an dthe antithesis interact t o produ ce th e synthesis, a com bination o r compromiseof the tw o. Th is synthesis can then be conside red as a thesis, which gives riseto a new antithesis, and then a new synthesis. The process continues untilsome final synthesis is reached. The correspondence between this dialecticmeth od an d the proposed ordering is apparen t. Th e sequence begins with themost im mediately obvious thesis a nd antithesis, the tw o primary hexagram s,K'un and Ch'ien. The first synthesis is Kou, which represents th e first mix tureof strong a nd weak lines. This then p roduce s its own antithesis, Fu, and so the"dialogue" continues. Eac h progression from on e pair of hexagrams to thenext involves a mixture of the two hexagrams in the first pair, for when oneline changes in the hexagram , it is equivalent to having acquired one line fromthe other hexagram in the pair. Thus in Kou, the five strong lines come fromCh'ien, while the bottom weak line comes from K'un.

In this dialectical development, the last synthesis is Ta Yu , which thenproduces its antithesis Pi. However, the philosophy of Hegelian dialecticsdoes not preclude the possibility of the final synthesis being the antithesis ofthe most recent thesis o r synthesis. Th e hexagram Pi does describe the idealcombination of strong and weak lines, as has already been shown, theultimate synthesis. It is interesting to note that an etymologically literaldefinition of the word "synthesis" (from the Gr eek obv and Sqotc,) is veryclose to "holding together," the translation of Pi given by Wilhelm. It means,literally, "together-placing." 'CONCLUSIONWe have presented here an ordering system which we believe to be useful for


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better understanding the I Ching. It is also hoped that, with this essentiallypolar arrangement of the hexagrams, some new insights into the dominantmode of thought in China around the t ime of the formation of the earlyversions of the I Ching might be gained. On the question of whether thisordering may have existed previously, we can only speculate. But, regardlessof its historicity, we d o hope that the validity of this ordering as a way ofviewing the system of the sixty-four hexagrams has, at least, been de-monstrated and that i t may be of some use to individuals who wish to domo re detailed analyses of the hexagrams.

NOTES1. "In general it can be said that the hexagrams arranged in their proper order symbolize the

entire sequence of changes through which everything in the universe, at all levels from themicrocosmic to th e macrocosmic, passes in continu ous cycles." Joh n B lofeld, I Ching (T he Booko f c h a n g e ) (New Y ork: E . P. D utt on and Co. , 1968), p. 48 ( i tal ics ours) .

2. Ibid ., p. 52.3. Hellmut Wilhelm, Chan ge: Eight Lec tures on the I Ching, trans. Cary F . Baynes, Bollingen

Series 62 (Prince ton, New Jers ey: Prince ton U niversity Press, 1960), pp. 84-85. H . Wilhelm,Heaven, Earth, and M an in the Book o f Changes (Seattle, Washington: University of WashingtonPress, 1977), pp. 108-1 14.

4 . K uo Mo- jod , Chou-i te kou-ch'eng shih-tai' (D e L'Ppoque a Laquelle fit PlaborP Le TcheouY i ) (Cha ngsh a: Com mer cial Press, 1940), pp. 20-28 (Fren ch and Chinese). Joseph Nee dham ,Science and Civilisation in China (Cam bridg e: At the U niversity Press, 1965), 2: 306-309, providesa very useful survey of scholarly opinion on the dates and origins of the various parts of the IChing. For an interpretation of the ultimate origin of divination in China, see Taguti HukusirCif[Tagu chi FukushirCi], Shlueki no kigeng [The origin o f th e Ch ou-i] (Tokyo: Meiji shoin, 1960). It isimpossible for us here to got into such important but difficult questions as the relationshipbetween oracle bone and milfoil stalk divination, the difference between the Fu-hsih (Izsien-t'ierz',"prior t o heaven") and King W eni (hou-t ' ienk,"subsequent to heaven") arrang eme nts of thetrigrams, and so forth, which are peripheral to ou r enterprise.

5. Confer Hellmut Wilhelm, "I-ching Oracles in the Tso-chuan and the Kuo-yii," JAOS, 79, 4(October-Decemb er, 1959): 275-280 and Li Chin g-ch'ih ' , "I ch uan t 'an-yiianm" ["A n inquiryinto the origins of the appendices t o the Book ofChange.r"], originally published in the 1930 Shih-hsiieh nien-pao" [Ye arb ook ofhistoriccrl studies] of Yenching University and reprinted in Ku C hieh-kango, ed., Ku-s hihpie nD Discussions ofanc ient history] (Peipin g: P'u she, 1932; 2d ed.), 3: 95-132.6. "The Book of Changes," B M F E A , 5 (1933): 141. It should be noted that Waley, followingthe lead of iconoclastic C hinese scholars of the twenties an d thirties, regarded the earliest layersof the texts of the I Ching as folk proverbs dealing with prescience. A similar view is held byNa tha n Sivin, "Review of Blofeld, The Book of Change," HJ AS , 26 (1966): 290-298: "Th ere is bynow a consensus that the I ching is a jumble of straightfor ward divin ation judg men ts ( 'Profitableif to the southwest, unprofitable if to the northeast ') and rhymed b ut often truncated proverbs orsayings ( 'When the wild goose skims [? or advances] over the land, the husband will go to waran d not r etu rn; the wife will be gravid b ut will not deliver')" (p.293). Such a view neithersubsta ntiates nor invalidates our atte mp t to find a mor e logical ordering for the hexa gram s whichcould h ave existed independently of an y of the texts. Th e da te at which the texts becameattach ed to the hexagrams, while highly significant, is not crucial insofar as the basic question ofarrangement is concerned. Indeed, an examination of the oracles mentioned in the Tso-chuanseems to indicate the existence of a different scheme, the nature of which it is impossible todetermine precisely. Confer Tanigawa Ryiizanq (1774-1831), Sa k ok u ek i ikkagen' [A personal


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view of the references to the B ook of Changes in the Tso-chua n and in the Kuo-yii] (Kyoto:Yamashiroya, 1818) and Mao Ch'i-lings (1623-1716), Ch' un-c h'iu chan-shih shu' [Oracles in theSpring and Autumn Annals], in Ts'ung-shu chi-ch'eng ch'u-pienu [Assemblage of collectanea-Firstseries] (Shanghai: Commercial Press, 1935-40), ts'e 707. Hu Tzu-fengv, Hsien-ch'in chu-tzu i-shuot'ung-k h o w [ A thorough exarninarion of the e.uplanations of pre-Ch'in philosophers regarding theBook of Changes] (Taipei: Wen shih che ch'u-pan she, 1974) is a helpful collection of earlyreferences to th e I Ching in various Cho u texts.7. The descendants of the Hsia were supposed to have settled in Ch'i during the Choudynasty.

8. Following K'ung Ying-taY 574-648) who , in the subcommentary, glosses cheng' as cheng-yen"" (21.9a-see note 12). Confu cius was unable to veri f :~wha t he ha d gone to investigate.

9. Th e descen dants of the Shang house settled in Sun g during the Chou .10. Cheng Hsuan's commentary (21.8a) states that Confucius "obtained the book on ),in-yang

of the Shang dynasty. Th is book survives as the Kuei-ts hnga b[Re turn to the hidden]." The Kuei-ts'ang is mentioned in the Chou-liac[Ritua l of' the Cho u], in Thirteen Classics, 24.1 1b: "(GrandDiviner). He is in charge of the three methods for determining changes. The first is called Lien-shunad [Connected Mou ntains], the second is called Kuei- tshng, and the third Chou-i"' [ChouChunges]." Confer ~ d o u a r d i ot, tr ans. , Le Tcheou-li ou Rites des Tcheou (Paris: ImprimerieNa tiona le, 185 I), 2 :70. The com men tary (24.11b) of Cheng H suana' (127-200) explains the titleKuei-tshng as meaning t hat "all things return and hide in its center." By "it," Cheng probab lyintended that which encompasses change or is beyond change. The subcommentary of ChiaKung-yenag (fl . 650-655) confirms the meaning of the title given by Cheng H su an bu t specifiesthat it is the earth to which all things revert. The earth being k 'un (female principle), it makeseminent sense for the "Kuei-ts'ang Changes to give the foremost position to pure k 'unah" .One isreminded of the frequent references to "the Mo ther" in the Tao -te ching (1, 20, 25, 52, and 59) asthe origin of the phenomenal world. The two fragmentary Kuei-ts'ang in Yii-han shan-fang chi i-shu"' [L ost boo ks collected at the ,/ade case retrea t],compiled by M a Kuo-hana', (Changsha: Lang-huan kuan, 1883), ts'e 1 and in Hun-wei i-shu ch'aoak[Copies of' lost bo ok sfr om the Hun and Weidynasties], compiled by Wang Mo"', (Chin-hsi: Wang shih, 1798), ts'e 1 are forged and are of novalue in discussing the pre-Chou ordering of the hexagrams. See Hsii-hsiu ssu-k'u ch'iian-shu t'i-yao am [ A continuation of 't he synopsis of the catalogue t o the library in four branches of'lirerature](Taipei: Tai wa n Com mercial Press, 1972), 1: 1-2 and , for a very full account of the subject,Chang Hsin-ch'engan, Wei-shu t 'ung-k hoao [A rhorough examination offorged books] (Shanghai:Com mercia l Press, 1957; originally pu blished 1939), 2: 39-45.

11. Cheng Hsiian's com men tary (21.8a) implies that he observed the way of the Hsia an d theway of the Sh ang through th e two books.12 . Li chiaP[Records of ritual], in Shih-sun ching chu-shuaq [T he thirteen classics with com -mmtaries and sub-commentaries] (Kian gsi: Nan-ch'ang hsueh-t 'ang, 1815 [actually H un an : Pao-ch'ing wu-pen shu-chu, 1896 recutting]; (Taipei: I-wen, 1976, facsimile reprint), 21.8a. ConferJames Legge, trans., The L i K i (Oxfo rd: Clarend on Press, 1885), 1 : 368. A similar passage occursin Analects, III . ix but no mention is made of the Hsia Calendar and the K'un-ch'ien. See Legge,The Chinese Classics, 5 (Hong Kong: Hong Kong University Press, 1960), p. 158. See als oAnalect.~,III.xiv (Legge, p. 160) for a related passage. The appearance of the specific titles in theLi chi and their absence in the Analecrs makes them suspect. However, in the Doctrine of theMean , XXVI1I.v (Legge, Classics, 5:4 24 ), we read: "The M aster said , 'I may describe theceremonies of the Hsid dynasty, but Chi cannot sufficiently attest my words. I have learned theceremonies of the Yin dynasty, and in Sung they still continue. I have learned the ceremonies ofC h iu , which are now used, an d I follow Chdu"' (italics ours). While the evidence is notconclusive, the references (see note 11) in the classics to a Sha ng man ual of change indicate eitherthat the compilers of the classics knew firsthand of such a work or that there was a strongtradition for the earlier existence of such a work tha t persisted into the late Chou and Han. EvenCh'ing commentators did not call the Li chi reference to a K'un-ch'ien into question. See, forexam ple, Liu Pao -na nar (1791-1855), Lun- Yii cheng-yia"[Th e correct interpretation of th e Analects](Sh ang hai: Comm ercial Press, 1930; 2d ed., 1934), 1:62-63.


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13. The subco mm entary (21.9a) , ci t ing a M r. Hsiung" ', says that "The Shang dynasty Changesgave precedence to the female principle. Therefore they put k 'u n first and ch'ien after it."

14. By no means d o we assert tha t there was an established conception of ),in-jang cosmologybefore the "Ten Wings" and the Tao-t e ching came into being. On this subject, we are in completeagreement with Li-Han-sana', Hsien-ch'in liang-Han chih yin-yang ~' u- hs in gsiieh-shuo"" [Theorieson yin-yang and the five phases during the pre-Ch'in period and the We ster n and Eastern Handynasties] (Taipei: Chung-ting wen-hua ch'u-pan kung-ssu, 1968). On the precedence of thefemale in early Chinese philosophy, see Ellen Marie Chen, "Tao as the Great Mother and thelnRuence of Motherly Love in the Sh aping of Chinese Philosophy," History of Religions 14, no. 1(Au gust-J anua ry, 1974): 51-64. Th e matriarch al n ature of early Chinese society is well estab-lished. See, for example, H o Ping-ti, The Cradle qf the Eu.rt (Chicago, Illinois: The University ofChicago Press, 1975), pp . 274-281; K u o Mo-jo , Chu ng-ku o shih kao"" [D ra) history of China](Peking: Jen-min c h'u-pa n she, 1962), 1 : 15-47; Marcel Granet, Chinese Civilization (London:Routledge an d Ke gan Paul, 1930), p. 185; Gr ane t, La polygynie sororale et le sororat dans la Chinefbodale. ~ t u d eur le sfo rm es anciennes de la poljgamie'chinoise (Paris: Lerou x, 1920), reproducedin ~ t u d r sociologiques .rur la Chine, Bibliotheque de Sociologie Contemporaine (Paris: PressesUniversitaires de Fran ce, 1953). pp. 28-30; an d Gra net , The Religion of the Chinese People, trans.and ed. Maurice F reed man (Ne w York : Harper and Row, 1975; original ly published 1922), pp.39, 51, 85, 87, 114, an d 52, where we rea d: "Since wom en conceived in their nata l hom es,reincarnation must be in the uterine line: a newborn child was none other than an ancestor who,after a s tay in M othe r Ear th, the stuffcomm on to maternal forbears, took on individual l ife againand reappeared within the living section of the family. At the same time as the idea of MotherEarth was elaborated-basis of kinship in a family attached t o a plot of land of its own andorganized according to the system of descent through women-the belief was formed, in thefamily groups fixed to domestic Soil and confident in their perennity, that the family substancewas a s eternal in the sam e way as was their Soil an d like it ever unchanging."

15. In Lucius Chapin Porter, comp., Aids to the Stu dy o f Chinese Philosophy (Peking: YenchingUniv ersity, 1934), p. 72, note 4 (italics our s).

16 . The I Ching, or Book o f Changes, The Richard Wilhelm translat ion rendered into Englishby C ary F . Baynes, 3d ed ., Bollingen Series 19 (Princ eton, New Jersey: Princeton U niversity Press1967), p. 260; hereafter cited as W ilhelm-Baynes, The I Ching.

17. 1 Ching: Book of Changes, t rans. James Legge, eds. Ch'u Chai and Winberg Chai (NewYork: University Books), p. 54.18. Chou-i (SP TK ed.), 9.6b.19. There have, of course, been numerous other systematic arrangements of the hexagrams

which a re useful for sorting them, such a s Ching Fang'sbb (77-37 B.c.) grouping "by houses," butnone of which we are aw are involve a logical concept of dev elopment t ha t can g enera te all sixty-four of the hexagrams.20. O n the subject of Leibniz an d Sh ao Yun g, see Joseph N eedh am, Science and Civilisation inChina, 2 : 340-345; Hellmut W ilhelm, Change, pp. 89-91; H. Wilhelm, Heaven, Earth, and Ma n,pp. 6-10. Tw o recent studies of the relationship between Leibniz and Chinese philosophy areDavid E. Mungello's Leibniz and Confucianism: The Search ,for Accord (Honolulu , Hawai i : Th eUniversity Press of H awa ii, 1977) an d G ottfried Wilhelm Leibniz' , Dircourse on the NaturalTheology of the Chine.re, t ranslated and with an introduction and commentaries by HenryRosem ont, Jr . , and Daniel J . Co ok, Monog raph No . 4 of the Society for Asian a nd Com parat ivePhilosophy (Ho nolulu , Haw aii: Th e University Press of Haw aii, 1977). Both of these booksdiscuss the correspondences between Leibniz' binary notation and the I Ching. Addit ionalreferences may be found in H. Wilhelm, "The Book of Changes in the Western Tradition: ASelective B ibliography," Parerga 2 (Seatt le: Inst i tute for Comparat ive an d Foreign A rea Studies,University of Wa shingto n, 1975), pp. 13-14. Also see the first note in R. F. Merkel, "Leibniz undChina," Leibniz zu seinem 300. Geburtstag, 1646-1946 (Berlin: Walter de Gr uy ter & Co., 1952).No. 8. Since Leibniz, numerous individuals have attempted to elucidate the mathematics of the IChing. Among the more recent at tempts are Rene Barde, "Recherches sur les OriginesArithmetiques du Yi-King," Archives Internationales d'Histoire des Sciences (continuation of
http:///reader/full/Chine.rehttp:///reader/full/Chine.re


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Archeion), 5, no. 21 (July-D ecem ber, 1952): 234-281, and Leo Reisinger, Dus I Cing: Einefi~rmalnissenschaftlicheUntersuchung des chinesischen Orakels, Acta Ethnologica et Linguistica,no. 25 (Vienna: Engelbert Stiglmayr, 1972), who, on page 96, states that "an irreducible ergodicMarkov-chain can be constructed" from the sixty-four symbols. Li K'ai-hsuanbd, I-shu ch'ien-.rhuobeAn Elementary E,xplanation o f t he M athemat ics of th e Book of Changes (Taipei: Ming-shanch'u-pa n she. 1975) con tain s several essays dealing with th e I ching and basic principles ofcom puter science. Howe ver, so long as the existing o rder of the hexagrams is not challenged, anyattempts to apply mathematical logic to the Book of C hunge . ~are doomed to end in failure. Atbest, they serve only to specify the probabilities involved in divining with the received text. AsMart in Gardner has said in "Mathematical Games," Scientific American (January, 1974):1 0 8 113, "From t ime to t ime a s tudent of the I Ching announces his discovery of a mathe matic alscheme underlying the arrangement of pairs, but on closer inspection it turns out that so manyarbitrary assumptions are made that in effect the order must be assumed before it emerges fromthe analysis. As far as anyone knows, the pairs of the King Wen sequence are in random order,and there is no know n basis for determining which memb er of a pair precedes the other" (p.108).21. Wilhelm-Baynes, The I Ching, p. 378.22. C onsult th e first dia gra m of "I-t'u luehbh " ["A synopsis of the diag ram s of the changes"],in I-h.riieh sun s hub i [T hr ee treatises on studies of the Book of Changes],preface 1813, from Chiaoshih ts'ung-.rhubJ [CoNe cted works o f Chiao H siin] (Tiao-ku lou ed. of 1876). ts'e 4-5, l . la-8a.Confer Wei Tat , An Exposition of the I Ching or Book of Changes (Taipei: Institute of CulturalStudies, 1970), pp. 302-303 an d Th om e H. Fa ng (Fang Tung -meibk), "I chih lo-chi wen-t'ib'"["The problem of logic in the Book o f Changes"], in Che-hsiieh sun huibm [Three types o f wisdom inphilosophy] (Taipei: San-m in shu-chu, 1971), pp. 1 0 91 4 3, where (p. 133)p'ang-i 'ung is defined aslooking at the hexag rams in units of two in which the yin and yang lines are paired against eachother. Before Chiao Hsun. whom Wei Tat and Th om e Fang rely on , Yu F an bn (164-233) hadearlier gained a partial understanding of the concept o f p h n g - t ' u n g . He applied it , however, toonly twenty-eight of the hexagrams and stopped short of developing it into a principle forexplaining one aspect of the entire set of hexagrams as did Chiao Hsun. For a detailedcomparison of the contr ibut ions of Chiao Hsun and Yu Fan in the matter of the p'ang-t 'ungpairing of hexagrams, see Ch'eng Ch'i-p'anb", Tia o-k u lou i-ibP [T hepurport of siu die .~on th e B ookof C hungesfrom the Tiao-ku studio of Chia o H.riin] (Cha ngsh a: Comm ercial Press, 1940).23. In general, the ordering worked ou t by Chiao H sun to illustrate the concept o f p ' ang - i ' ungresembles the reorder ing propo sed below in one aspect only-both consist of thirty-tw o pairs ofhexagrams which, line for line, are exact opposites.24. Fra nk G ray , U.S. Patent No. 2632058, listed in Oficia l Gazet te of ihe United Siates PatentOffice, vol. 668, no. 3 (Washington, D.C.: U.S. Government Printing Office, March 17, 1953),pp. 823-824. Th e Gr ay cod e is explained in Montgomery Phister. Jr. , Logical Design of DigitalCornpuiers (New York an d Lo nd on : Joh n Wiley and Sons, 1958), pp. 232-234 and Rob ert Baronand Albert Piccirilli, Digiial Logic and Computer Operaiions (New York: McGraw-Hil l, 1967), pp.115-116. Ma rtin G ard ner has written a fascinating essay in which he describes the applicationsof the G ra y Cod e to the solution of various ancient puzzles, among them the "Chinese rings" andthe Tower of Hanoi puzzle. See his "Mathematical Games," Scien t i jc American (August, 1972):10 61 09 , in which he gives other imp ortant references to the Gra y C ode and i ts applicat ions. Inthe same issue of Scient i jc American, pp. 76-83, there is an article by F. G . Heath entitled"Origins of the Binary Code" in which the Gra y C od e is discussed o n pp. 81-82.25. Wilhelm-Baynes, The I Ching, p. 274.26. Wilhelm-Baynes, The I Ching, p. 292.27. Ibid., p. 378.28. Wei Ta t, op. cit. , pp. 300-301.29. Wilhelm-Baynes, The I Ching, p. 36.30. T he terms "thesis," "antithesis," an d "synthesis" a re not Hegel's but were used by latercommentator to clarify his theories. See John McTaggart, Ellis McTaggart, Studies in HegelianDialectic (Cam bridg e: Cam bridg e University Press, 1922), 1.1, pp. 1-2, and G . W. F. H egel,Enq,clopedia of the Philosophical Sciences, trans. William Wallace as The Logic of Hegel, 2d ed.(L on do n: Oxfo rd University Press, 1892), $81, pp. 147-152).


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31. We are indebted to David Aronson, who first noted the correspondence with Hegeliandialectics.

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