《天原发微(鲍云龙)》Vol. 6

I. Basic Units of the Calendrical Cycle

Taking nineteen years as one zhang (chapter), twenty-seven zhang make a total of 513 years. One full circuit of sun-moon conjunction is one hui (conjunction). Eighty-one zhang, or 1,539 years, are called one tong (union). Counting from the hour of Zi through Chen and then on to Shen, a total of 4,617 years is called one yuan (origin). One yuan contains three tong; each tong contains three hui; each hui contains twenty-seven zhang. Nine hui, or 243 zhang, bring the cycle to its end.

II. The End of the Cycle and the Way of the Xuan

By this method, when the total number of one yuan is divided by the numbers of zhang, hui, and tong, the cycles divide exactly. Specifically, within one zhang the intercalary remainder is exhausted; within one hui the lunar eclipse cycle is completed; within one tong the new-moon remainder is exhausted; and within one yuan the sixty jiazi cycle is also completed. This is the principle of the Xuan.

III. Calendrical Precession and Intercalation

The Xuan takes the dawn new moon of the Heavenly Origin jiazi as its starting point; the winter solstice begins at the starting point of the Qianniu lunar mansion, and the sun and moon move without cease. The circuit of heaven is calculated as 365 and one-quarter degrees; one month has thirty days; one year has twelve months. Intercalation is used to adjust the four seasons, so that the calendar accords with celestial phenomena.

A year is actually 365 and one-quarter days, yet there remains a surplus of less than a whole day. In the Zhuanxu Calendar the surplus is three-quarters of a day; in the Taichu Calendar it is half a day. The reason there is a surplus is that the yang number is already full while the yin number is still vacant. Therefore two supplementary values, qi and ying, are established to make the “Xuan numbers” complete and bring them into accord with the celestial measure. This is comparable to inserting an intercalary month in a year, so that the calendar’s day count matches celestial phenomena; only then can the calendar operate correctly.

IV. The Author’s Commentary

The author holds that although Shao Kangjie’s (Shao Yong’s) Jingshi Calendar and the calculation method in this book differ, both are deduced from similar principles and both attain a subtle and refined level.

V. Supplementary Notes

“Shaozi’s Pre-Heaven gua qi” refers to Shao Yong’s doctrine of Pre-Heaven gua qi, that is, the theory that matches the Bagua (Eight Trigrams) and sixty-four hexagrams of the Zhouyi with solar terms and seasonal order. “Qian, Kun, Kan, and Li” are four of the Bagua; this sentence is incomplete in the original, so only this brief explanation can be given here.

The four hexagrams Qian, Kun, Kan, and Li are respectively assigned to the four seasons, each governing a number of solar terms. When Kan moves to the midpoint of Zi and intersects with Li, its first line corresponds to the Winter Solstice and its top line to Awakening of Insects (Jingzhe), one of the twenty-four solar terms, around early March. When Li moves to the midpoint of Mao and intersects with Qian, its first line corresponds to the Spring Equinox and its top line to Grain in Ear (Mangzhong), a summer solar term. When Qian moves to the midpoint of Wu and intersects with Kan, its first line corresponds to the Summer Solstice and its top line to White Dew. When Kan moves to the midpoint of You and intersects with Kun, its first line corresponds to the Autumn Equinox and its top line to Major Snow. Here Zi, Wu, Mao, and You are the four cardinal positions among the twelve Earthly Branches, while “midpoint” means the center of each position.

Spring, summer, autumn, and winter each use the solar terms represented by the sixty-four hexagrams, all calculated from the midpoints of Zi, Wu, Mao, and You. These four midpoints correspond respectively to the Winter Solstice, Summer Solstice, Spring Equinox, and Autumn Equinox, and are called the “four midpoints, the two solstices and two equinoxes.” The four positions Yin, Shen, Si, and Hai correspond to the Beginnings of Spring, Autumn, Summer, and Winter, called the “four meng and four beginnings” (“meng” refers to the first month of each season).

According to the Jingshi Calendar (the calendar used in Shao Yong’s Huangji Jingshi of the Northern Song), the Winter Solstice is taken as the starting point of Heaven and Earth, namely the “yuan of yuan.” Therefore the four cardinal hexagrams are omitted, and only the three hundred sixty yao positions are used for calculation. The Gua Qi Diagram takes the Spring Equinox as the starting point of humans and the myriad things, which also belongs to the “yuan of yuan”; hence it uses the sixty-four hexagrams multiplied by four, yielding two hundred fifty-six positions. The Gua Qi Diagram further takes the midpoint of Zi at the Winter Solstice as the starting point of “shi,” the midpoint of Mao at the Spring Equinox as the starting point of “yuan,” the midpoint of Wu at the Summer Solstice as the starting point of “hui,” and the midpoint of You at the Autumn Equinox as the starting point of “yun.” (Shao Yong used yuan, hui, yun, and shi as units of time: one yuan contains twelve hui, one hui contains thirty yun, one yun contains twelve shi, and one shi is thirty years.) Each set of sixty-four hexagrams rotates in turn with the movement of the solar terms.

The governing sequence begins at the midpoint of Yin, initiating the myriad things and starting with Awakening of Insects, the solar term of early the second month; it closes and stores the myriad things at the midpoint of Xu, ending with the Beginning of Spring, the solar term of early the tenth month. Why is this so? Although the midpoints of Yin and Xu are spoken of in terms of the meeting of months, their function is actually based on solar terms. The Earth grows the myriad things, taking qi as the trigger; the qi of Heaven arrives first, and the things on Earth respond accordingly. The arrival of qi is usually half a month earlier than the lunar phase. Qi moves slowly and is often full; the moon moves quickly and is often contracted. Therefore Guan Lang (courtesy name Ziming) said: in the numbers of the current period, what exceeds is called “qi surplus,” and what is insufficient is called “new moon void.” Solar terms and the new and full moons have surplus and deficiency and are uneven; after long accumulation, the mid-qi sometimes falls at the end of a month, and at that point intercalation is used to adjust it, after which it returns to the normal order.

The Jingshi takes the study of the Pre-Heaven as its foundation, and so mid-qi and the new and full moons start hexagrams simultaneously. Based on the basic numerical values of the Pre-Heaven, it uses mid-qi to determine the months; yuan, hui, yun, and shi are all calculated from the midpoints. This is what is meant by “correcting the calendar by mid-qi.” The Gua Diagram says: the calculation of the major cycle should follow the Jingshi, beginning with Jiazi; the calculation of the minor cycle should follow the Gua Qi Diagram, beginning with Jiayin. Heaven governs through substance; the substance of the solar terms arises from the four midpoints. Therefore the major cycle begins at the Winter Solstice of a Jiazi year, and the starting points of the twenty-four solar terms all fall on the midpoints of Zi, Wu, Mao, and You. The function of the solar terms operates at the four beginnings, and therefore the minor cycle begins at the Beginning of Spring of a Jiayin year, and the starting points of the twenty-four solar terms all fall on the four meng positions of Yin, Shen, Si, and Hai.

Zhu Xi said: On the left side of the Xiantian Tu (Primordial Diagram), starting from the Zhen trigram corresponds to the winter solstice; Li and Dui occupy the center and correspond to the spring equinox, until at the end Qian connects with the summer solstice. On the right side, starting from the Xun trigram corresponds to the summer solstice; Kan and Gen occupy the center and correspond to the autumn equinox, until at the end Kun connects with the winter solstice.

(The Xiantian Tu is the Eight Trigrams diagram transmitted by Shao Yong of the Northern Song; Qian 1, Dui 2, Li 3, Zhen 4, Xun 5, Kan 6, Gen 7, Kun 8 are its numbering.)

The left-and-right ordering of the hexagrams in the diagram and their forward and reverse circulation have been discussed in great detail by Confucian scholars of earlier generations. The four trigrams Qian, Dui, Li, and Zhen (numbers one through four) are the “already generated” trigrams, ordered from south to north; viewed through gua qi (the theory matching the sixty-four hexagrams to the solar terms), the movement runs from north to south. The one-yang qi begins to arise from the Zhen trigram, so Shao Yong takes the middle of the zi hour at the winter solstice (zi ban) as the Fu hexagram.

In the eleventh month, at the beginning of the twelfth month’s chou period is Lesser Cold, corresponding to the Kan and Tun hexagrams; half a month later is Greater Cold, corresponding to Zhen and Shi He. At the beginning of the first month’s yin period is Start of Spring, corresponding to Wu Wang and Ming Yi; half a month later is Rain Water, corresponding to Bi and Ji Ji and Jia Ren. At the beginning of the second month’s mao period is Awakening of Insects, corresponding to Feng, Li, and Ge; half a month later is the Spring Equinox, corresponding to Tong Ren and Lin. At the beginning of the third month’s chen period is Pure Brightness, corresponding to Sun, Jie, and Fu; half a month later is Grain Rain, corresponding to Kui and Dui. At the beginning of the fourth month’s si period is Start of Summer, corresponding to Lü and Tai; half a month later is Lesser Fullness, corresponding to Da Chu, Xu, and Xiao Chu. At the beginning of the fifth month’s wu period is Grain in Ear, corresponding to Da Zhuang, Da You, and Guai. Where Qian at the end meets the summer solstice is exactly the middle of the wu hour (wu ban). These thirty-two hexagrams are yang, paired with spring and summer.

The four trigrams Xun, Kan, Gen, and Kun (numbers five through eight) are the “not yet generated” trigrams; on the diagram they are arranged from west toward north. Following gua qi circulation, the one-yin qi begins from the Xun trigram, so the middle of the wu hour at the summer solstice is the starting point. In the fifth month, at the beginning of the sixth month’s wei period is Lesser Heat, corresponding to Da Guo, Ding, and Heng; half a month later is Greater Heat, corresponding to Xun, Jing, and Gu. At the beginning of the seventh month’s shen period is Start of Autumn, corresponding to Sheng and Song; half a month later is End of Heat, corresponding to Kun, Wei Ji, and Jie. At the beginning of the eighth month’s you period is White Dew, corresponding to Guai, Kan, and Meng; half a month later is the Autumn Equinox, corresponding to Shi and Dun. At the beginning of the ninth month’s xu period is Cold Dew, corresponding to Xian, Lü, and Xiao Guo; half a month later is Frost’s Descent, corresponding to Jian, Jian, and Gen. At the beginning of the tenth month’s hai period is Start of Winter, corresponding to Qian and Pi; half a month later is Lesser Snow, corresponding to Cui, Jin, and Yu. At the beginning of the eleventh month’s zi period is Greater Snow, corresponding to Guan, Bi, and Bo. Where Kun at the end meets the winter solstice is exactly the middle of the zi hour. These thirty-two hexagrams are yin, paired with autumn and winter.

From zi through si belong to the six yang months of Qian, Dui, Li, and Zhen; their solar nodes are four: winter solstice, start of spring, spring equinox, and start of summer. From wu through hai belong to the six yin months of Xun, Kan, Gen, and Kun; their solar nodes are also four: summer solstice, start of autumn, autumn equinox, and start of winter. In one year there are eight nodes in all. Taking two hexagrams as one group gives sixteen hexagrams, with sixteen additional “qi” subdivisions; taking three hexagrams as one group gives forty-eight hexagrams—the two together make sixty-four hexagrams, used to correspond to the climate of one full cycle. Thus the seasons can be fixed, the year formed, ghosts and spirits set in motion, and transformation brought about.

Zhu Xi took the two solstices and two equinoxes as the framework; I follow gua qi to work out the detailed items within it.

This passage is the end of juan 11 of Tianyuan Fawen, together with the title of juan 12 and the opening subheading.

The collation note at the end of juan 11: the editors use the Siku Quanshu (Complete Library of the Four Treasuries) edition as the base text and note item by item the variant readings between the original text and other editions. For example, zhu (“host/master”) was originally wang (“king”); ni (“to look askance”) appears in other editions as yi (“to discuss”); shi (“to show”) appears in other editions as rong (“glory”); and wang (“prosperous”) was originally wang (“king”). They also supply omitted characters, such as the two characters sanbai (“three hundred”) and the character ke.

The title of juan 12: it attributes the compilation to Bao Yunlong of Luzhai (courtesy name Jingxiang), with corrections by Fang Hui of Xugu and Wanli.

“Yingsuo”: this is the subheading of juan 12. In ancient calendrical astronomy, yingsuo refers to the variations in the speed of the sun and moon and the advance and retreat of their positions. This section contains only the heading; the content is to be seen in the following text.

Yang Surplus, Yin Contraction, and Intercalary Months

The surplus produced each year by the sun’s motion is called “yang surplus” (yangying), also called “solar qi surplus,” namely the five and a quarter days by which a year exceeds [the lunar count]. The moon’s new-moon/full-moon cycle is undercounted by six days, called “yin contraction” (yinsuo), also called “new-moon deficit” (shuoxu); the new moon is the first day of the lunar month, and shuoxu refers to the total number of days left vacant over the course of a year. When the remainder accumulated each year reaches twelve days, an intercalary month is established to adjust the calendar.

Calendar specialists hold two views on the direction of the sun, moon, and five planets. The Prior-Heaven theory (xiantian), which respects yang qi and takes heaven as primary, holds that the sun, moon, and five planets all revolve leftward with heaven. In this way the sun’s motion is slower than heaven and the moon’s motion slower than the sun, so that the numerical principles are coherent; Confucians regard this view as correct. The Posterior-Heaven theory (houtian), based on “renwei”—that is, following and complying with the movement of the celestial bodies—holds that the sun, moon, and five planets all move opposite to heaven, revolving rightward. In this way the sun moves slowly while the moon instead moves faster; calendar specialists advocate this view and hold that the calculations in the Yi are based on the Posterior Heaven. Confucians do not adopt the latter view, but for the time being both theories are presented side by side for the reader’s reference.

A passage in the “Canon of Yao” (Yaodian) says: “A year has three hundred and sixty-six days; by means of intercalary months the four seasons are determined, and the full order of the year is formed.”

A passage in the Shuzhuan (Commentary on the Documents) says: the original text is incomplete, and one can only tell that it speaks of the period from this year’s winter solstice to next year’s winter solstice, that is, one tropical year.

The Shuzhuan says: From this year’s winter solstice to next year’s winter solstice is called one ji (that is, a complete tropical year). Within one ji, the number of days counted by the calendar is 366 days, or a little more than 365 days. Heaven and the sun meet above, and so form one year; the whole year contains twenty-four solar terms, and this is so every two years, even when a leap month occurs.

The ancients used the “day divisor” (dividing one day into 940 parts) to measure the circuit of heaven (one revolution of the celestial sphere, measured in “degrees”). The excess beyond the circuit of heaven is one quarter of a day. One day is 940 parts; divided into four equal parts, each is 235 parts, exactly one degree.

Zhu Xi says: Calculated by solar terms, a year is 360 days; calculated by new and full moons (months based on lunar phases, with the first day as the new moon), a year is 354 days. Taking the middle number between the two, one therefore speaks of 360 days. The remaining increase or decrease does not exceed six fen. The “Yao Dian” (a chapter of the Shangshu) speaks in whole numbers, and hence says “three hundred and sixty-six days.”

Hu Shen says: By the four seasons, each season is a little more than ninety days; by the six qi (wind, cold, summerheat, dampness, dryness, fire), each qi is a little more than sixty days; by the Five Phases (metal, wood, water, fire, earth), each phase is a little more than seventy-two days, and all three calculations total 360 days. The stalk numbers of the Qian and Kun hexagrams (the counts used in Zhouyi divination) are also like this: Qian has 216 stalks and Kun has 144, totaling 360, which is the normal value of the number of Heaven. Exceeding this number is called “surplus” (ying); falling short of it is called “deficit” (suo).

The Yang principle has a surplus, so the sun’s one revolution makes a year, and the year regularly has more than five extra days; the Yin principle is deficient, so the moon’s one revolution makes a month, which regularly falls short of thirty days, and therefore there are small months. Twelve months accumulate into a year, which is about eleven days less than the solar year—the so-called “weak eleven days.” With these accumulated remainders, one leap month is inserted every three years, another every five years, and seven leap months in nineteen years, and then the Dao of Heaven is complete. The movements of the sun and moon inevitably have surplus and deficit, and those who make calendars cannot avoid error; therefore leap months are established so that the seasons and the order of the year coincide.

The Huangji Jingshi (written by Shao Yong of the Northern Song, famous for deducing cosmic time) also takes 360 as its basis: one yuan contains 360 yun, one hui contains 360 shi, one yun contains 360 years, one shi contains 360 months, one year contains 360 days, and one month contains 360 chen. The Yang number reaches 360 because six is taken as the base; six times sixty is 360.

Master Zhang of Hengqu (Zhang Zai, the Northern Song Neo-Confucian scholar) held that both the sun and the moon revolve leftward.

His explanation was this: Heaven moves extremely rapidly, completing one circuit in a day and night, approximately 365¼ degrees. The sun’s motion follows closely behind Heaven and is slightly faster than Heaven. For every degree Heaven moves, the sun also moves exactly one degree; relative to Heaven, therefore, it can only be counted as having fallen back one degree. When Heaven moves two degrees, the sun likewise falls back two degrees. Accumulating in this way to 365¼ days, the number of degrees Heaven has traveled exactly completes one circuit, and the number of degrees by which the sun has fallen back also exactly completes one full circuit; the two then come together again, forming one year. The moon’s motion, by contrast, is slower: in a day and night, relative to Heaven, it falls behind Heaven by more than thirteen degrees.

Zhu Xi (Master Zhu) objected: If one holds that Heaven makes one circuit in a day while the sun each day advances less than one degree, then why do the star patterns of the four seasons differ? If this were so, every day’s situation would be the same; what then would be used to determine the solar terms? Carried to its conclusion, this is as absurd and disordered as striking the third-watch drum at noon. He further said that in the subcommentary to the “Monthly Ordinances” in the Book of Rites there are two places where this is explained quite clearly, and the other calendrical works are all the same. If Heaven is regarded as a sphere enclosing [all things] within, then it can move only 365¼ degrees; if Heaven is regarded as being on the outside, then it becomes a matter of passing one degree in a day. In discussing the sun and moon, one should take their being within Heaven as the standard. If one observes from the Great Void (Taixu, the vast emptiness), Heaven naturally moves every day and no longer remains in its original position.

Zhu Xi also said: Both the sun and moon are calculated starting from the Horn Mansion. The sun each day completes one circuit and still returns to the Horn Mansion; Heaven, after completing one circuit, must pass a little beyond the Horn Mansion. Accumulating day by day, by the end of a year it comes together with the sun.

The author remarks: From the accounts of these two masters, it can be known that Heaven’s movement exceeds the proper number of degrees, the moon’s movement falls short, and only the sun’s movement is most regular. Therefore, in a year, 360 days are taken as the standard.

Shao Yong says: the yang qi still has surplus beyond the 360 days.

Within one year, when the sun completes a revolution and returns to the place where it meets Heaven, it exceeds 360 days by five days and 235 fen (the fen was a small unit in ancient time-reckoning; 940 fen make one day). This is because the motion of Heaven is fast and the motion of the sun is slow: Heaven turns leftward one circuit each day and travels one degree farther, while the sun moves rightward one degree each day, and thus cannot return to its original position at the same time as Heaven. Only after accumulating to more than 365 days can they meet. Therefore a year takes 360 days as its whole number, and the surplus is what is called “qi surplus.”

Zhu Xi says: the celestial body is perfectly round, with a circumference of 365 degrees and one quarter. Heaven revolves leftward around the earth, completing one circuit each day, while the sun falls behind by one degree each day. Thus the sun also circles the earth once a day, but travels one degree less than Heaven. Only after accumulating 365 days and 235/940 of a day does it meet Heaven; this is the number of degrees of the sun’s motion in one year.

Kong Yingda’s subcommentary says: Heaven is the formless Great Void; what is called Heaven refers only to the motion of the stars. Heaven encloses the earth from outside, like an eggshell enclosing the yolk. The stars move from east to west, and after 365 days and one quarter day they return to their original positions; the sun moves from west to east, and likewise only after 365 days and one quarter day does it return to its original lunar mansion. The twenty-eight lunar mansions are distributed in the four directions and constitute the celestial body. Heaven is measured by degrees, the year by days, and both have this number. It is only that the qi-number has excess and deficiency, and therefore there is waxing and waning.

Bao Yunlong comments: If the motion of Heaven had no excess but were the same as the sun, and the motion of the moon had no deficiency but were also the same as the sun, then each day the sun would return to the same position, and each year would be exactly 360 days with no remainder. In that case cold and heat would not be uneven, and there would be no changes in the size of the sun and moon or in drought and flood. If Heaven were like a rigid mold, it would not be the manner of creation and transformation; how then could there still be the mechanism by which ghosts and spirits reward good and punish evil?

It is also said: yin Qi contracts beyond 360 degrees.

A synodic month (from one new moon to the next) exceeds a whole number by about 441 fen; this is called shuoxu (the surplus fraction in the lunar-phase cycle). From the zi hour to the chou hour, the moon travels a total of 5,292 fen. The total shuoxu comes to about five days and 592 fen; rounded to a whole number, one says six days. This is because the moon moves more slowly than the sun.

Zhu Zi (Zhu Xi) says: the moon moves attached to the celestial body but is slower than Heaven, lagging each day by about 13 7/19 degrees. When 29 days and 499/940 of a day have accumulated, it meets the sun (this is called heshuo, conjunction at the new moon). After twelve such meetings, the whole days amount to 348 plus a remainder; the remainder accumulates to 5,988 fen. Dividing by the “day divisor” (the calculation unit in which 940 fen make one day), one obtains six days, with a remainder of 348 fen, giving a total of 354 days and 348/940 of a day. This is the number of the moon’s motion in one year.

Bao Yunlong remarks: the 7/19 degree is obtained by dividing 940 fen into nineteen parts, each about 49 fen, 4 li, 2 hao, 1 si, 5 hu, and 7 miao; it is used to compare the remainder by which the moon’s daily motion falls short of the constant degrees of Heaven and the sun. Calculated this way, the moon each day travels about 12 degrees and 346 fen less than the sun. When this accumulates each month to 29 days and 499 fen, its lag behind the sun is 365 degrees and 235 fen. Thereupon the degrees traveled by the sun exactly complete one circuit and obtain their original number; the degrees by which the moon lags also recede completely to their original number. The two exactly meet to form one month; the twelve months amount to twenty-nine days each. Counting the whole days as 348 and adding twelve times 499 fen, the total accumulates to 5,988 fen. Dividing by the day divisor of 940 fen gives six days and more than 300 fen.

Ancient calendrical theory held that the sun is Yang and the moon is Yin; Yang is often full, while Yin is often deficient. The sun’s motion constantly surpasses the moon’s, and the moon’s motion constantly falls behind the sun’s. Within a single year, because “qi surplus” (the excess of the tropical year over twelve synodic months, that is, the accumulated surplus of the solar terms) and “new-moon deficiency” (the amount by which twelve synodic months fall short of the tropical year, that is, the deficit in the lunar phase cycle) offset each other, the total comes to about 10 days and 827 parts; taking the whole number gives 12 days. This was used to determine the timing of intercalary months. In three years, three such values accumulate to about 32 days and 601 parts; taking the whole number gives 36 days, which is one intercalation every three years. In five years there is another intercalation; accumulating five such values gives about 54 days and 275 parts, rounded to 60 days. Broadly speaking, an intercalary month is established about every 32 days; although this is not exactly the same, the difference is not far off.

Therefore the sages established the calendar and used “guiyu” (merging the surplus remainders into the count) to make up for the discrepancies by which the moon’s motion falls short of the sun and the sun’s motion falls short of heaven. When the three are harmonized into one, after nineteen years (the original text says twenty-nine years), the solar terms and the new and full moons all accord with celestial phenomena without the slightest discrepancy, and human affairs can also be carried out on time.

Someone asks: whether a month is long or short is itself a calendar; it only requires people to calculate it. If the new crescent is seen on the second day, the preceding month was a long month (thirty days); if it is first seen on the third day, the preceding month was a short month (twenty-nine days). Just as swallows come in the second month and leave in the eighth, so it is with all things.

Earlier people said that the sun and moon meet between the dark of the moon and the new moon. It is best observed on the evening of the first day: when the sun has just set in the west, the faint moon sets with it. By the second day, the two have already separated slightly. After the first light appears on the third day, the two gradually move farther apart. By the fifteenth, the moon and sun are exactly opposite (full moon); at this time the sun has advanced about half the celestial circuit, and the moon has fallen behind by about half the celestial circuit. From the sixteenth to the end of the month, the sun completes the entire celestial circuit, and the moon’s lag also completes the entire celestial circuit. When the sun’s advance and the moon’s backward movement have both completed their original values, the two meet again.

Shao Yong says: the intercalation of a year is formed by the mutual coordination of six Yin and six Yang. Over three years the accumulated surplus is thirty-six days, so one intercalation is set every three years; over five years it accumulates to sixty days, so two intercalations are required in five years. After nineteen years, the courses of the sun and moon come together again.

The calendar therefore has no surplus fraction, taking nineteen years as one zhang (a complete cycle). Within these nineteen years there are seven intercalary months, totaling 228 months.

The principle of setting intercalations arises from the unequal pace of the sun and moon’s movements. The sun advances one degree each day along the ecliptic (the path of the sun on the celestial sphere), while the moon advances 13 and 7/19 degrees each day. Thirteen degrees is precisely the measure by which the twelve months of a year complete one circuit of heaven; the extra seven nineteenths must be supplied by intercalation. Thus the intercalation method pairs “seven” with “nineteen”: twelve times seven gives eighty-four, the number of months in seven years; twelve times nineteen gives 228, the number of months in nineteen years.

Calculated this way, each year leaves a surplus of about eleven days plus a fraction; over nineteen years the accumulated surplus is 209 days and 228 parts, which exactly forms a whole number, so after seven intercalations in nineteen years no remainder is left.

By this method one can calculate the following. To find an intercalary year, multiply the number of years by seven; when it reaches nineteen parts, that is an intercalary year. To find an intercalary month, multiply the number of months by eighty-four; when it reaches 228, that is the month in which an intercalation should be placed. To find the intercalary day, take seven parts of the intercalation; when they fill nineteen parts, one shi is obtained (in antiquity a day was divided into twelve shi, each shi being two hours), and this is the day of the intercalary new moon (the first day). To find the time of conjunction (the moment when sun and moon join and the first day begins), multiply the twelve shi by the number of a day, obtaining eighty-four parts; when it fills 228 parts, one shi is obtained.

The pairing of “seven and nineteen” is the rough value in the intercalation method; the pairing of “eighty-four and 228” is the precise value. Intercalation is originally an odd number of Heaven; it accumulates at seven and is completed at nineteen, so seven and nineteen, whether multiplied together or divided, both yield 133. The numbers of months and of shi are both first multiplied by twelve and then divided by twelve, thus yielding 228.

The Xici says, “Set aside the remainders between the fingers to image intercalation” (assigning the remainders to the odd numbers to symbolize intercalation). In the Xiantian [Former Heaven] system, the number of days uses 133 and the number of stars uses 105; both follow this principle. Intercalation is originally the odd number of Heaven, yet it must be sought through the moon; therefore it is said that “yang takes yin as its regulator,” and Yin and Yang mutually serve as substance and function. Multiply 228 by ten and then by two, obtaining 45,600,000; this is the total number of one yuan in the Sifen Calendar (the calendar in general use since the Eastern Han).

It is further said: the sun’s movement appears slow, and this is counted as “advance”; the moon’s movement appears rapid, and this is counted as “retreat.” At each conjunction of the sun and moon, half a day is added to the surplus and half a day subtracted from the deficit; the accumulated number is the “intercalary remainder” (the remainder used to determine intercalation). The number of days in one great cycle of the sun is 5,600,000, precisely the number of “one yuan” as stated in the Sifen Calendar (an ancient calendar).

It is further said: the sun’s apparently slow motion is “advance,” and the moon’s apparently rapid motion is “retreat.” At conjunction, half a day is added or subtracted, and from this the intercalary remainder is obtained. In one great cycle the sun advances six days and the moon retreats six days; this is the “intercalary difference.”

Heaven rotates to the left, while the sun and moon move slowly to the right; the discussion of their degrees has appeared above. Each month the moon makes one circuit of heaven, but it always moves on its own rather than rotating with heaven (“traveling alone”). When the moon catches up with the sun, this always occurs in the last day and a half, and it generally remains behind the sun. Thus the sun, though inherently slow, appears to “advance”; the moon, though inherently fast, appears to “retreat.” The sun and moon conjoin about once every thirty days, but the actual interval is only twenty-nine and a half days. Because the moon’s period is originally twenty-nine and a half days and the sun’s period is originally thirty and a half days, yet both are reckoned as thirty days, at each conjunction the sun has half a day in excess and the moon is short by half a day.

In one year the sun is originally six days more than twelve months, and another six days are added; the moon in one year is originally six days less than the sun, and now another six days are subtracted. Accumulating the days added and subtracted each time gives the intercalary remainder.

In one great cycle of the sun and moon, advance and retreat together make twelve days. Therefore there is one intercalary month in three years, two in five years; three in eight years, four in ten years, five in thirteen years, six in sixteen years, and seven in nineteen years. Calculated in this way, the coming and going are endless, and this is called “tong” (continuity). Ziyun (Yang Xiong) said, “The sui may be at peace or ailing, but the nian is diseased.” “Sui being at peace” means the number of days is sufficient and the harvest is secure; “sui being ailing” means the harvest year is sick and must be compensated by an intercalary month.

Zhu Xi said: intercalary months are determined by the zhongqi (mid-qi; the “mid-qi” among the twenty-four solar terms, such as the winter solstice and Rain Water).

Zhongqi and Intercalary Months

Each month contains one “zhongqi” (mid-season qi, a category among the twenty-four solar terms, such as Rain Water, Spring Equinox, and Summer Solstice; each month has one). If the zhongqi falls within that month, the month is a normal month. If the zhongqi is pushed beyond the end of the month, an intercalary month should be established for that month in order to adjust the discrepancy between the calendar and the seasons. To determine where the zhongqi lies, one observes the direction pointed to by the handle of the Northern Dipper: the direction pointed to by the handle is called “jian,” and when it points between two “chen” (names of twelve celestial directions), this is precisely where the sun and moon meet. If no zhongqi can be found, an intercalary month is set. The Dipper handle moves to the left (counterclockwise), while the sun and moon move relatively to the right, so “chen” and “jian” often coincide.

Boundaries of the Celestial Circuit and Fractions

The celestial circuit (zhoutian), one complete revolution of the celestial sphere, consists of 360 degrees. The ancients distributed the degrees of the celestial circuit among the twelve palaces (twelve regions of the sky), and each time the sun and moon passed through one palace, the fractions involved were calculated. In antiquity, one hundred seconds made one fen, and one hundred fen made one degree.

The Song scholar Wang Yan explained: The celestial circuit is divided into twelve palaces, each with its own fraction. A year is about 365 days, plus a remainder of several double-hours (one edition says “five shi”); at the onset of spring, it exactly matches the degrees of the celestial circuit. A synodic month is about thirty days, plus several double-hours; at the solar-term transition, it exactly matches the degrees of one palace. A degree is like a boundary; the celestial wheel has delimiting boundaries. The sun moves only one degree per day, so it takes 365 days and twenty-five ke (in antiquity one hundred ke made a day; twenty-five ke is about a quarter of a day) to traverse all the boundaries of the sky. The moon moves more than twelve degrees per month; astrologers take the round number and say thirteen degrees. The moon makes one circuit of the sky each month and traverses all the celestial boundaries, but a month is not exactly thirty days. Months are large and small; dividing fifty-nine days into two months exactly makes two circuits of the sky. Roughly speaking, a little more than twenty-nine and a half days makes one circuit of the sky, and a little more than fifty-nine days makes two circuits.

Wang Yan also said: In my view, ancient and modern calendrical specialists could only calculate the boundaries of Yin-Yang. That is why Confucius did not discuss calendrical details but only said, “Observe the seasons of Xia” (adopt the Xia calendar).

The Theory of Heaven Turning Left

Wang Yan also said: The theory that the celestial body turns left is like a large wheel outside and small wheels carrying the sun and moon inside. The large wheel turns quickly, while the small wheels turn slowly. Although all turn left, there are fast and slow movements, so it appears that the sun and moon turn right.

The movement of celestial bodies—the degrees of the sun and the degrees of the moon—should be viewed in four layers to be easily understood. Now, the great wheel represents the movement of heaven; it lies in the outermost layer, the case of what is called “heaven enclosing the earth.” The great wheel turns steadily and unfolds so as to display the degrees of heaven, amounting to about one degree and a quarter or more per day.

The small wheel carries the sun and moon and lies in an inner layer. The small wheel is smaller than the great wheel, so the inner layer contracts as it revolves. The moon’s movement also lags behind the sun, slower by about a quarter of a degree per day, or slightly less than that.

The outer layer encloses the inner layer; the outer is broad, the inner contracted. Their speeds and the layers of their bodily forms are inherently thus. Observing heaven divided into four layers makes it clear at a glance. In earlier years I saw a diagram drawn by my friend Wu Jiuxia, and it too accorded with this principle.

The outer first ring records the degrees of the full circuit of heaven, serving as the scale of the celestial bodies. The second ring takes heaven as reckoned from the Horn Mansion (Jiao), the first of the Twenty-Eight Mansions; each day it makes one revolution and again passes the Horn Mansion by one degree. On the third ring, the sun is also reckoned from the Horn Mansion and follows heaven in order; in one day it makes one circuit without exceeding the degree count, returning exactly to the Horn Mansion, and so it travels one degree less than heaven. Relative to heaven, the sun retrogrades one degree per day; after it has retrograded through its full number, heaven advances one degree per day and returns to the original place, matching the celestial degrees—this is one cycle.

On the fourth ring, the moon moves more slowly, traveling twelve degrees and a half part in 346 less than the sun; accumulating over twenty-nine and a half days, it completes one synodic month. The degrees by which the moon travels less than the sun accumulate to 365 degrees and 235 parts; the moon then again retrogrades through its full number and so meets the sun. Thus there are twelve conjunctions in a year (the original text says “one month, twelve conjunctions,” which should refer to a year). Calculating the amount by which the moon is less than the sun, it is about five days and 592 parts; combining with the new-moon deficiency and solar-qi surplus, the year has a total surplus of ten days and 827 parts, and intercalation is based on this. This will not be described in detail here.

Someone asked: What are the “hidden intercalation” and “manifest intercalation” spoken of by Shao Kangjie (Shao Yong, a Northern Song numerologist)? The answer: The details are not easy to explain clearly; let us speak only of what lies before us. A year has 360 days and six days remaining; these remainders are hidden within the sixty-day sexagenary cycle, one day hidden in every sixty days (about two months), so within 360 days a total of six days are hidden.

The so-called “manifest intercalation” is from the Opening of Things to the Closing of Things (Shao Yong’s terms for the time cycle), taking seven out of ten and removing three parts of the intersection number. The 360 days originally use 240 days, giving 252 days. The intercalary number twelve appears beyond the 240 days.

Zhu Xi said: The “Treatise on the Calendar” in the Records of the Grand Historian discusses the Sifen Calendar used by Zhuanxu; Liu Xin revised it as the Santong Calendar; Yixing’s Dayan Calendar is the most detailed and complete.

The Sitian kao of the Five Dynasties is also relatively concise and rigorous. In it, the calculations that added and subtracted 720 all showed errors after only two or three years. Among calendars ancient and modern, only Shao Yong’s (Kangjie) calendar uses 129,600 parts as the unit of one day, and so it is the most precise.

Below I will explain the calendars of former generations one by one.

The Sifen Calendar was established by Zhuanxu on the basis of the Lianshan Yi, which begins with the Gen trigram. It derives the numbers of the Hetu: from ten it extends to a hundred, divides the hundred into four parts, each part twenty-five, corresponding to one quarter of the number of Heaven. The Qin dynasty adopted this calendar and took the hai month as the beginning of the year. In Emperor Wu of Han’s time it was changed to the Taichu Calendar.

The Taichu Calendar was compiled after Emperor Wu ascended the throne by Tang Du, Luoxia Hong, and others. It took the yin month as the beginning of the year and changed the seventh year of Yuanfeng, a dingchou year, to the first year of Taichu; the year name was “Yanfeng Shetige” (that is, a jiayin year), and the month name was “Biju.” It took the new moon and winter solstice at midnight on a jiazi day as the starting point of the calendar. This was when Emperor Wu changed the reign title; it matched above with the original primordial epoch when Zhuanxu began the calendar in a jiayin year, in order to correspond to the auspicious omen of the conjunction of the sun, moon, and five planets.

The Santong Calendar was made by Liu Xin revising the Taichu Calendar. Three threes make nine; nine nines make eighty-one; this is the “mother of the systems” (tongmu). It takes Huangzhong as the heavenly system, with jiazi as the first year; Linzhong as the earthly system, with jiachen as the first year; and Taicu as the human system, with jiashen as the first year. Its epoch is the same as that of the Taichu Calendar, both beginning from the Huangzhong pitch (see the annotation to the Xuantu above for details).

The Taixuan Calendar was made by Yang Xiong of the Han dynasty and corresponds mutually with the Santong Calendar and the Zhuanxu Calendar. The Taichu Calendar derives its numbers by nine-times-nine and takes one quarter of the Luoshu; this is the method of summing up actual numbers. The Zhuanxu Calendar derives by ten-times-ten and takes one quarter of the Hetu; this is the method of empty containment in the center. The Taichu Calendar takes eighty-one as the day divisor, namely the number of nine times nine.

The Taixuan uses seventy-two as the “day divisor” (the base for calculation), and seventy-two comes from eight times nine. The Taichu Calendar uses thirty-two as the “denominator” (fractional divisor), and thirty-two comes from eight times four. The Taixuan also uses thirty-six as a “denominator,” and thirty-six comes from nine times four. Comparing the Taixuan with the Taichu Calendar, one subtracts one miao from the “nine,” the other adds one miao to the “nine,” and in the end both obtain 2,592 miao. Their starting methods differ, but the result is the same; see the Gua Qi (the theory of matching hexagrams with solar terms) for details.

The Dayan Calendar (written “Taiyan Calendar” in the original) was made by Yixing, a monk and astronomer of the Tang dynasty. From the Han through the Sui, calendars all followed the numerology of the pitch pipes, but Yixing relied specifically on the Dayan and took gua qi as its foundation. The Dayan takes the numbers from the empty center of the Hetu: it pairs the numbers of the four directions—one and six, two and seven, three and eight, four and nine—to symbolize the four seasons; it takes the central five and ten combined as the “derivative mother” (base number); and it further takes nine, six, seven, and eight to represent old yang, old yin, young yang, and young yin respectively for calculation. The numbers of the lines correspond to the sixty stalks, and the calculations operate within four hundred twenty. This is the “Dayan,” the pivot of the movements of heaven and earth. Later authors could not alter it.

The Tongtian Calendar of the Xiande era was made by Wang Pu of the Five Dynasties. Ouyang Xiu included it in the Sitian kao, where it formed a system of its own. The yang number is thirty-six stalks and the yin number twenty-four stalks; when the yin and yang numbers combine and transform, together they yield seventy-two. Multiplying again yields 720, again yields 720,000, and again yields 72,000,000. Thus the “yuanji” (the initial epoch for calendrical reckoning) gives rise to the yuan, and years, months, days, and hours are all recorded by the jiazi (sexagenary cycle). The sun, moon, and five planets gather at the zi position, exactly within the sequence of “waxing and waning” (the variations in the speed of the sun and moon), and the seven regulators (the sun, moon, and five planets) are all set in order.

Bao Yunlong (the author, referring to himself as “the ignorant one”) holds that among calendars ancient and modern, none in earlier ages surpassed the Han Taichu Calendar and none in later ages surpassed the Tang Dayan Calendar. Both derive from the Hetu; the various calendars in between merely added to, subtracted from, and patched their foundations.

The ancients established the “six pitch standards” (liulü) of music according to the Luoshu (Luo Writing), with counting beginning from odd numbers; they arranged the Bagua (Eight Trigrams) according to the Hetu (Yellow River Chart), with counting beginning from even numbers. Although the methods adopted in these two cases differed, both broadly belonged to “Later Heaven Yi” (Houtian Yi), the system based on King Wen’s Bagua. At that time the “Former Heaven Diagram” (Xiantian Tu), the Bagua sequence attributed to Fuxi, was still concealed and had not yet appeared.

The so-called “Former Heaven calendar” (Xiantian calendar) is the calendrical system on which Shao Yong (posthumous title Kangjie, commonly called Shaozi) based his Huangji Jingshi. Its method combines the Heavenly Stems—the ten stems jia, yi, bing, ding, and so on—with the Earthly Branches—the twelve branches zi, chou, yin, mao, and so on—for calculation: jia is paired with day, zi with month, jia with stars, and chen with zi; these are called the “Four Images” (sixiang). In large numbers it is divided into yuan, hui, yun, and shi, the largest units of time; in small numbers it is divided into sui, yue, ri, and chen, the smaller units of time.

The specific calculation is as follows: for days, jia serves as the governing stem, and one governing unit contains twelve months; for stars, jia serves as the governing stem, and one governing unit contains 360 cycles; for chen counted from zi, one governing unit contains 4,320. The thirty-six jia are evenly distributed among the twelve chen, each chen receiving 360. The Former Heaven and Later Heaven calendars use different methods of calculation; when intercalary months are included, one yuan comprises twelve hui, 360 yun, and 4,320 shi, totaling 129,600 years.

This is precisely why Zhu Xi praised Shao Kangjie’s calendar so strongly. The author Bao Yunlong arranged the various calendrical systems and concluded with this. Cheng Yi also said: “If in regulating the calendar one can comprehend the principles involved, then what one understands will be extensive.” Only Shao Yaofu—that is, Shao Yong—established the “method of differences” (chafa); at the moments of yin-yang interaction, its refinement surpasses both ancient and modern times. As for calculating the movements of the sun and moon according to the principle of “yang fullness and yin deficit,” no deviation will occur; it is only in these subtle details that errors easily arise.

Fascicle Twelve ends here. The following are collation notes explaining modifications to the text of the base edition: the first is the character zhu ("vermilion"), which the base edition originally had as wei; the second is the character li, originally jing; the third is the character miao, originally chao; the fourth is the character fen, originally ri; the fifth is the character kan, originally zhe. All the above characters have been revised according to the Siku Quanshu edition.

The title of Fascicle Thirteen is Tianyuan Fawei, fascicle thirteen. The work is attributed to Bao Yunlong, courtesy name Jingxiang, style name Luzhai, a Yuan dynasty scholar. In the attribution, Fang Hui (courtesy name Wanli, style name Xugu) was responsible for collation.

This fascicle is titled "Xiangshu" (images and numbers). "Xiang" refers to hexagram images and the forms of all things in heaven and earth; "shu" refers to the numerical principles derived from astronomy and calendrical calculation. "Xiangshu" is an important orientation in Yijing studies, and this fascicle will develop its discussion around this theme.

The pattern that emerged floating in the Yellow River—the Hetu (Yellow River Diagram)—made it possible to depict the Bagua (Eight Trigrams), and Yin-Yang first acquired concrete form; the writing that appeared in the Luo River—the Luoshu (Luo River Writing)—made it possible to set forth the nine categories of governing the realm, and the Five Phases first received their numerical determinations.

If images are not paired (even), they cannot be established; if numbers are not odd, they cannot operate. The two may depend on each other, but neither can be lacking.

Zhu Zi (Zhu Xi) said: Heaven and Earth do not speak, and so a sage must come forth to explain the principles. If Heaven and Earth could speak, they would surely speak even better. The Hetu and Luoshu are the Yi (Changes) that Heaven and Earth themselves drew. Fuxi looked up to observe the celestial patterns and looked down to examine the earth’s order, seeing that the principles of the Yi were everywhere between Heaven and Earth. When the Hetu appeared, it exactly matched his mind, and so he drew the Bagua according to it. That the study of images and numbers became unclear in later ages has its reasons.

Zhu Zhen of Hanshang said: Shang Qu (a disciple of Confucius) learned the Yi from Confucius; after it was transmitted to Ding Kuan, one branch split into Meng Xi and Jing Fang. By the Tang dynasty its source and development could still be traced. Yixing (a Tang Buddhist monk skilled in astronomy and calendrics) gathered the learning of the Meng and Jing schools and discussed doctrines such as gua qi (hexagram qi), najia (stem-embracing), and the Five Phases; all of these came from the Great Commentary (Dazhuan) of the Zhouyi—that is, the Yi Commentary, including the Xici, Shuogua, and other sections. Afterward Ma Rong, Zheng Xuan, Xun Shuang, and Yu Fan each established their own traditions, but they were still not far from the source of image-number learning. In the Wei dynasty, Wang Bi and Zhong Hui studied together; they completely discarded the old explanations and mixed in the words of Laozi and Zhuangzi, devoting themselves to wording and meaning-principle and no longer investigating the roots. From this point the source of the Great Commentary became unclear, and the Way of Heaven and the Way of Humanity split apart.

The Great Commentary of the Yi says: The Yellow River brought forth the Diagram, and the Luo River brought forth the Writing; the sage took them as his model.

Zhu Xi said: Kong Anguo of the Han dynasty held that when Fuxi ruled the world, a dragon-horse emerged from the Yellow River, and Fuxi drew the Bagua according to the markings on its back; this was the Hetu (River Chart). When Yu the Great controlled the waters, a divine turtle carrying a diagram and writing emerged from the Luo River, and the patterns on its shell contained numbers from one through nine. Yu accordingly arranged their order and derived nine categories of governing methods—the Nine Categories of the Hongfan—and this was the Luoshu (Luo Writing).

I believe that at the beginning of heaven and earth, when chaos was undivided, the images and numbers of the myriad things had not yet appeared; the Hetu and Luoshu were what Heaven used to reveal the sages. The Analects says, “The River does not bring forth the Chart”; the Zhouyi says, “The River and Luo bring forth the charts and writings”; the Shangshu speaks of “the Heavenly Sphere and the Hetu.” From this we know that the Hetu and Luoshu are textual images naturally formed by heaven and earth, regarded since antiquity as auspicious omens, and not something human power could fabricate. As for the doubts raised by Ma Duanlin, Ouyang Xiu, and others, what need is there to argue further?

Still others recklessly fabricated bizarre accounts: the Huainanzi calls it “cinnabar writing and a green chart,” while the Chunqiu wei among the apocryphal texts says it consists of nine chapters or six chapters. Gu Yewang held that the Luoshu was eighteen characters such as “Nong yong” and “Jing yong,” written on the back of the divine turtle. Ban Gu held that the main text of the Luoshu, from “Chu yi” to “Liu ji,” totals sixty-five characters. Kong Yingda of the Tang said that Heaven’s words are concise and essential, and would not be so repeatedly admonishing and so convolutedly elaborate. Only Kong Anguo’s account is most appropriate, and so Zhu Xi adopted it.

Zhu Xi also said: Heaven one, Earth two, Heaven three, Earth four, Heaven five, Earth six, Heaven seven, Earth eight, Heaven nine, Earth ten. The Heaven numbers are five, and the Earth numbers are also five. The numbers of heaven and earth pair with one another, and each has its correspondence. The Heaven numbers sum to twenty-five, and the Earth numbers sum to thirty; this is the root that brings about the transformation of the myriad things and drives the wondrous operation of Yin-Yang creation.

Zhu Xi says: This is the principle by which Confucius explained the numbers of the Hetu (Yellow River Diagram). In the Hetu, one and six occupy the north (below), two and seven the south (above), three and eight the east (left), and four and nine the west (right); the five in the center is the mother substance from which the myriad numbers are derived, and ten follows it as the child it generates. Next, one, two, three, and four are the positions of the Four Images (Sixiang), while six, seven, eight, and nine are the numbers of the Four Images. The two old yin (laoyin) are located in the northwest, and the two young yin (shaoyin) in the southwest. Their numbers, each according to category, intersect and combine on the outside; the four characters “mutually obtaining and forming a union” cover the entire numerical principle of the Hetu.

(Note: The Four Images refer to Taiyin (greater yin), Shaoyin (lesser yin), Taiyang (greater yang), and Shaoyang (lesser yang), the four basic states formed by stacking two Yin-Yang lines. The Hetu is the diagram said to have emerged on the back of a dragon-horse in the age of Fuxi, and it is a foundation of Yijing studies and Daoist numerology.)

Comment: The Heaven numbers are the sum of five odd numbers, and the Earth numbers are the sum of five even numbers. One and two are paired to form a couple, three and four are paired, five and six are paired, seven and eight are paired, and nine and ten are paired. Each follows in odd-even succession, paired two by two like brothers; this follows their order and cannot be disordered.

One and six combine to form Water, two and seven to form Fire, three and eight to form Wood, four and nine to form Metal, and five and ten to form Earth. This likewise follows order and cannot be disordered.

Each is paired according to Yin-Yang, two by two like husband and wife; this accords with their nature and admits no doubt. The Heavenly Stems are likewise so: Jia is yang, and obtaining its counterpart in yin Yi, it forms Wood; Jia combines with Ji and further transforms into Earth; Bing is yang, and with yin Ding it forms Fire; Bing combines with Xin and further transforms into Water; Wu is yang, and with yin Ji it forms Earth; Wu combines with Gui and further transforms into Fire; Geng is yang, and with yin Xin it forms Metal; Yi comes to combine with Geng and further transforms into Metal; Ren is yang, and with yin Gui it forms Water; Ding comes to combine with Ren and further transforms into Wood. All this is pairing the Five Phases two by two and returning them to the One.

Pairing necessarily entails union, and within union there are also mutual generation, mutual overcoming, sameness, difference, and uneven situations. This is precisely the principle by which the five relationships (wulun) of ruler and minister, father and son, elder and younger brother, husband and wife, and senior and junior are joined with the Five Phases and cannot be separated. Heaven and Earth are the greatest manifestation of the union of Yin and Yang: when one, three, five, seven, and nine combine as the Heaven numbers, none is separate from five; when two, four, six, eight, and ten combine as the Earth numbers, none is separate from five either. The five numbers of Yin-Yang combine wondrously and generate human beings and the myriad things; therefore the total of the Heaven and Earth numbers is fifty-five. Five is the center; being centered, it can change; changing, it can transform; its Dao is inexhaustible. One, once changed, becomes six; two, once transformed, becomes seven; three changes and becomes eight; four changes and becomes nine; five changes and becomes ten. Change leads to transformation, and transformation leads to change; odd and even generate and complete each other; contraction and extension, coming and going, all do not go beyond the scope of the Hetu. How could the principle of the five numbers of Heaven and Earth be compared with the later talk of ghosts and spirits and discussions of change that lapse into absurd and unfounded claims?

(Note: The sentence “Five is the center” in the text means that five occupies the center and serves as the pivot of change; “fifty-five” is the sum of the Heaven and Earth numbers from one to ten, that is, the sum of the odd and even numbers in 1+2+…+10: 25+30=55.)

Zhu Xi said that the Hetu uses the five generative numbers to govern the five completive numbers, while the Luoshu uses the five odd numbers to govern the four even numbers. He held that the generative numbers are all within, since generation is the fundamental heart of Heaven and Earth; the completive numbers are all without, since completion is the function of Heaven and Earth. This statement is extremely apt.

Bao Yunlong comments: The Hetu distinguishes Yin and Yang by generative and completive numbers. One, two, three, four, and five are generative numbers, belonging to Yang and dwelling within; six, seven, eight, nine, and ten are completive numbers, belonging to Yin and dwelling without. To use the generative to govern the completive, the inner to join the outer, with the center as host and the outer as guest—this is what is meant by “the body has fixed norms, and each occupies its proper place.” The Luoshu distinguishes Yin and Yang by odd and even numbers. One, three, seven, and nine occupy the four cardinal positions and are paired with the four Yang trigrams; two, four, six, and eight occupy the four corners and are paired with the four Yin trigrams. To use the odd to govern the even, the noble to preside over the lowly, with the upright as ruler and the lateral as minister—this means that numbers undergo change yet each attains its place.

However, the constant and the changing in the two diagrams cannot be rigidly confined to one aspect. In the case of the Hetu, Lesser Yin and Old Yang remain unmoved in their positions, while Lesser Yang occupies the south and corresponds to the Qian position, and Old Yang occupies the west and corresponds to the Kan position, which does not agree with the order of trigram generation in the Former Heaven arrangement. (Note: Old Yang, Lesser Yang, and Lesser Yin are the numbers nine, seven, and eight in the Zhouyi.) In the case of the Luoshu, Old Yang occupies the south and corresponds to Qian, and Lesser Yang occupies the west and corresponds to Kan, which instead agrees with the trigram-generation order. This is precisely what is meant by “there is change within the constant, and constancy within change”; the subtle excellence of this is difficult to perceive.

Shaozi (Shao Yong) said: What is round is the stars; the numerical principles of calendrical reckoning probably begin here.

The calendrical treatise recorded in the Tang Zhi (the calendrical section of the Tang histories) says: The numbers of Heaven begin with one, and the numbers of Earth begin with two; by combining the two beginnings one can determine the firm and the yielding. The numbers of Heaven center on five, and the numbers of Earth center on six; by combining the two centers one can penetrate the pitch pipes and the calendar. Heaven is the stars; the numerical recording of the calendar probably begins here as well.

The Tang Zhi (referring to the “Li zhi” [Treatise on Calendrical Astronomy] in the New Book of Tang, which contains discussions of calendrical methods) says: The numbers of Heaven begin with one; the numbers of Earth begin with two. When the two are combined, the firm and the yielding (yang and yin) are determined. The numbers of Heaven center on five; the numbers of Earth center on six. When two and five combine, calendrical astronomy and the pitch pipes are brought into coherence. Heaven has five tones, by which it governs the days; Earth has six pitch pipes, by which it governs chen (the times represented by the twelve earthly branches). Zhu Xi has already explained this in detail, so it will not be repeated here.

Shao Yong said: The Li trigram images stars, and the Zhen trigram images chen. The number of stars corresponds to the ten of days, and the ten of days is five; chen are the twelve earthly branches, which, paired two by two, yield six. Five and six occupy the center of yin and yang, and so the Huangji Jingshi uses them to calculate the calendar. Nine and ten draw on each other, and from this the calculation method derives the intervals of time. This has been so ever since calendrical and astronomical images first existed.

The circular image symbolizes stars. The Hetu (Yellow River Diagram) has no four corners; following the shape of the stars, it naturally forms a circle, and the numbers in the diagram correspond to the number of stars.

Mr. Zhang said: The number of celestial bodies is seven. The Northern Dipper has seven stars; the sun, moon, and five planets together make seven; the twenty-eight lunar mansions are four sevens; adding the six stars of the Southern Dipper gives a total of fifty-five. From this one knows that the numbers of Heaven, Earth, humanity, and their patterns and principles all correspond to the Hetu.

He also said: The square symbolizes earth. The ancient well-field method of dividing land probably arose from this.

Zhu Xi said: A district has nine wells and nine hundred mu; this corresponds to the numbers of the Hetu and is the basis for dividing well-fields.

I believe: The square originally belongs to the content of the Luoshu (Luo River Chart). The earth has four directions; the Shangshu places the yang numbers in the four cardinal positions, symbolizing the shape of a square-li well-field. The Luoshu takes nine as its master and also symbolizes the nine hundred mu of the well-field. Heaven bestowed the nine categories of the Hongfan (the nine great laws of governing the state); Yu the Great obtained this book because of his flood control, and with it opened up territory,divided the nine provinces, collected tribute according to the land’s produce, and devoted effort to canals and waterworks. Yu achieved the nine meritorious works and brought benefit to the world for all generations; from this one can know the importance of the nine-chapter book of the Hongfan.

It is also said: Roundness is the numerical principle of the Hetu, while squareness is the textual structure of the Luoshu. Therefore Fuxi and King Wen of Zhou created the Yi according to the Hetu and Luoshu, while Yu of Xia and Jizi wrote the Hongfan accordingly. (The Hetu and Luoshu are traditionally said to have emerged from the Yellow River and the Luo River respectively, and what they record is the numerical principle of Heaven and Earth. The Hongfan is a chapter in the Book of Documents, recording the nine categories of governing principles—the “Nine Categories”—that Jizi set forth to King Wu of Zhou.)

Numbers begin with one yin and one yang. The image of yang is round, so the body of the Hetu is round, yet its numbers are arranged by directional positions; the image of yin is square, so the body of the Luoshu is square, yet its numbers are arranged in a circular rotation. This is what is meant by “three matching Heaven and two matching Earth” (matching three with Heaven and two with Earth, combining the numbers of Heaven and Earth).

When numbers reach ten and are arranged by directional positions, that is the Hetu; therefore the Great Commentary on the Yi (i.e., the Yizhuan) has the statement “the numbers of Heaven and Earth total fifty-five.” When numbers reach nine and are arranged in circular rotation, that is the Luoshu; therefore the Hongfan has the entry “Heaven bestowed the Nine Categories.”

In the distant primeval chaos, Heaven and Earth, yin and yang, originally should each have had their images, but at first there were no fixed numbers. Only when the Hetu appeared did there come to be numbers such as “fifty-five,” and the generation and transformation of odd and even numbers became clear and visible. This is precisely the principle that the sage, by his unique wisdom, alone could expound profoundly.

Zhu Xi said: Counting from one to ten gives the Hetu; leaving a space in the middle, one derives the Yi from it.

An Interpretation of Tianyuan Fawei (Bao Yunlong)

The Hetu (Yellow River Diagram) is circular in shape and symbolizes Heaven; its numbers are grouped by tens and arranged in a square, symbolizing Earth. In the Hetu, the odd numbers (yang numbers) sum to twenty-five and the even numbers (yin numbers) sum to thirty, making fifty-five in all. The total of the even numbers is somewhat greater, while the odd numbers are relatively deficient. If the central ten is subtracted, one obtains the Luoshu’s (Luo River Chart’s) forty-five; if the central five is subtracted, one obtains the Dayan (Great Expansion) number fifty. Five added to ten gives the sum of each vertical and horizontal row in the Luoshu. Multiplying five by ten and ten by five also both yield the Dayan number.

The Hetu is based on the “generating numbers” (one through five); starting from the north and proceeding in a leftward rotation, they generate one another in sequence: water generates wood, wood generates fire, fire generates earth, earth generates metal, and metal again generates water. Viewed from the opposing directions, one and six water in the north overcome Bing and Ding fire in the south; four and nine metal in the west overcome Jia and Yi wood in the east. Thus “mutual overcoming” is already contained within “mutual generation.” The Yizhuan’s saying “The ceaseless generation of life is called Yi” (the continuous growth and transformation of the myriad things is Yi) expresses precisely this principle. If the center is hollowed out, it is Taiji (the Supreme Ultimate, the origin of the universe) generating the two modes (Liangyi; Yin-Yang), the Four Images (Sixiang; four basic phenomena), and the Bagua (Eight Trigrams); hence it is said, “one generates two, two generates four, four generates eight.” The Hetu also contains the essential purport of the Luoshu.

The text poses the question: If Heaven and Earth only generated and did not overcome, by what would the myriad things take form and grow? If there were only overcoming and no generation, humankind would perish. This shows that the way of creation and transformation must have generation and overcoming coexisting; therefore the Hetu and Luoshu are used to manifest their meaning.

Another account says: the arrangement of one through nine is the Luoshu; if the center is filled in solid, it becomes “Fan” (norm, pattern).

The Luoshu is square in shape and symbolizes Earth; its number is nine, and it is arranged in a circle, symbolizing Heaven. In the Luoshu, the odd numbers sum to twenty-five and the even numbers sum to twenty, totaling forty-five, so the odd numbers are in excess and the even numbers deficient. When the center is empty, it likewise symbolizes Taiji; the odd numbers around the periphery sum to twenty, corresponding to the two modes. One, two, three, and four, together with six, seven, eight, and nine, are distributed on the two sides; the sums of the vertical and horizontal rows are all fifteen, also corresponding to the Four Images. The five at the center, combined with the remaining numbers, again yields the Dayan number. Five plus ten makes fifteen, which is interconnected with the Hetu.

This section discusses the numerical relationship between the Hetu (River Chart) and the Luoshu (Luo River Writing); the two complement each other.

The numbers of the Luoshu are arranged according to the “completion numbers” (chengshu), rotating to the right from the north. The order in which the Five Phases (wuxing) overcome one another is: water overcomes fire, fire overcomes metal, metal overcomes wood, wood overcomes earth, earth overcomes water, and water again overcomes fire, joining end to beginning in an unending cycle. If viewed in terms of opposing directions, the four and nine of the southeast belong to metal and generate the one and six of the northwest, which belong to water; the three and eight of the northeast (written “shubei” in the original) belong to wood and generate the two and seven of the southwest, which belong to fire. It can be seen that the generating relationship is already contained within the overcoming relationship; overcoming is precisely what enables the myriad things to be completed and to fulfill one another.

The core of these relationships is the number five, the root of the Huangji (Royal Ultimate)—the centrally situated governing great norm among the “Nine Categories” (jiuchou) in the Hongfan chapter of the Shangshu. Kong Yingda of the Tang dynasty explained that the Huangji itself is not separately assigned a number because it encompasses the Nine Categories, using the one at the center to govern the eight directions and occupying the true central position between Heaven and Earth. The Five Phases and the Five Affairs each have five items; the Eight Administrations have eight; the Five Records have five; but the Huangji alone occupies the central position. The Three Virtues have three items, the examination of doubts has seven, the various government affairs have ten, and blessings and extremes together with what they generate are all subdivisions of the Nine Categories and are likewise contained within the numerical principles of the Hetu.

It is also said that both the Hetu and the Luoshu place five at the center. The reason five is the root is that it accords with the statement “Heaven is three, Earth is two, and on this the numbers depend” (from the Xici of the Yi; three is the number of Heaven, two is the number of Earth, and their combination forms five), and it is the foundation of number.

The numerology of five embodies the principle of “taking three as Heaven and two as Earth and relying on numbers.” (The phrase “taking three as Heaven and two as Earth” comes from the Xici of the Zhouyi and refers to using the heavenly number three and the earthly number two as the foundation for deducing change.)

Zhu Xi said: The Hetu takes the five generating numbers (one through five) as primary, and so its five central dots also contain the emblems of the five generating numbers. The lower dot represents Heavenly One, and one joins with four; the left dot represents Heavenly Three, and three joins with eight; the right dot represents Earthly Four, and four joins with nine; the central dot represents Heavenly Five, and five joins with ten. Five occupies the center; the generating numbers are on the outer ring, and the completing numbers lie outside the generating numbers. One plus five gives six, two plus five gives seven, three plus five gives eight, and four plus five gives nine. One through four arise through five, and so are called generating numbers; six through nine are completed through five, and so are called completing numbers. Although the numbers are paired opposite one another, they also interlace. From this it can be known that the fifty-five numbers of the Hetu are all contained within the five central dots.

The Luoshu takes odd numbers as primary, and so its central five also contains the emblems of the five odd numbers. The lower dot is likewise Heavenly One, and one is paired with the even number two; the left dot is Heavenly Three, and three is paired with the even number four; the central dot is Heavenly Five, and five must be paired with the even number ten; the right dot is Heavenly Seven, and seven is paired with the even number six; the upper dot is Heavenly Nine, and nine must be paired with the even number eight. Five occupies the center; the odd numbers occupy the four cardinal positions, while the even numbers are arranged in the four corner positions. One plus five gives six, located at the rear right; two plus five gives seven, located on the right; three plus five gives eight, located at the rear left; four plus five gives nine, located at the front. One, three, seven, and nine are the odd numbers generated by five; two, four, six, and eight are the even numbers paired with five. The numbers interlace vertically and horizontally, yet each has its own order and none is disordered. From this it can be known that the forty-five numbers of the Luoshu are also all contained within the five central dots.

Zhu Xi said: “The old and young of Yin-Yang each conceal the dwelling place of the other.” He also said: “The numbers seven, eight, nine, and six are each different.”

In the Hetu (River Chart), one, two, three, and four are taken as positions, while six, seven, eight, and nine are taken as image-numbers (symbolic numbers). In the Hetu, one and six combine, two and seven combine, three and eight combine, and four and nine combine; the central five with five, and ten with fifty-five—these numbers contain one another. Dong Panjian’s explanation of the principle that “Yin and Yang hide each other” is the most detailed. One is the position of old yang, and six is the number of old yin; yet “one” contains “nine,” so the number of old yang is already hidden within it. Four is the position of old yin, and nine is the number of old yang; yet “four” contains “six,” so the number of old yin is already hidden within it. The cases of lesser yin and lesser yang are likewise so. If Yin and Yang could not depend on each other and hide within each other, then the operation of Heaven and Earth’s creative transformation would cease.

Furthermore, seven and nine belong to Yang. Yang governs advance: from lesser yang (seven) it advances to above eight, and when advance reaches its extreme it becomes old yang (nine), while seven has no place left to go. When Yang reaches its extreme, Yin is born; therefore six and eight belong to Yin. Yin governs retreat: from lesser yin (eight) it retreats to below seven, and when retreat reaches its extreme it becomes old yin (six), while eight has no place left to go. Thus when Yin reaches its extreme, Yang is again generated from within it. Lesser yang and lesser yin transform into old yang and old yin; the old ones change, and the young ones continue to grow. This too is the principle of the ceaseless generation and regeneration of creative transformation.

However, when Yang advances there is surplus, so old yang yields the position of eight, and lesser yang yields the position of six; the Yang numbers are constantly full. When Yin retreats there is insufficiency, so old yin lacks seven, and lesser yin lacks nine; the Yin numbers are constantly reduced. The Zhouyi says: “The fullness and depletion of Heaven and Earth wax and wane with the seasons.” One who understands this principle is probably one who understands the great Dao.

He also said: “In the Luoshu (Luo Writing), the numbers in each vertical and horizontal row all add up to fifteen, and the numbers alternately wane and wax.”

Vernacular Explanation of Tianyuan Fawei (Bao Yunlong)

The Book (Shangshu), by means of one, two, three, and four, encompasses seven, eight, nine, and six; these are the Yin-Yang of the Four Images. The old and young [Yin-Yang lines] and the Bagua (Eight Trigrams) arise from this. Their alternating waxing and waning never goes beyond these four numbers: seven, eight, nine, and six.

Shao Yong, quoting the Yizhuan (Commentary on the Changes), says: The Zhen hexagram causes the myriad things to grow; the Qian hexagram causes them to differentiate; the Xun hexagram causes them to disperse; the Kun hexagram causes them to gather. This passage gives the general meaning of the waxing and waning of Yin-Yang.

The Zhen hexagram begins within the yin lines and gives rise to yang Qi (vital energy); when yang Qi increases to the nine of the Qian hexagram, yang Qi reaches its utmost flourishing and then ages. The Xun hexagram begins to diminish yang Qi and gives rise to yin Qi; when yin Qi increases to the six of the Kun hexagram, yin Qi reaches its utmost flourishing and then ages.

Mr. Zhang says: Seven represents yang Qi from youth to maturity; as it grows from six to nine, eight then represents the process by which yin Qi diminishes the nine. When it reaches six, yin Qi contracts; when it reaches nine, yang Qi ages. He also says: The growth of yang Qi proceeds from seven through eight to nine; the diminution of yin Qi proceeds from eight through seven to six. This is said in general terms, referring to the waxing and waning of one Qi. If Zhen and Xun are correlated with seven and eight, this refers to their general positional relation among the hexagrams.

If one calculates according to the hexagram positions of the Luoshu, the Zhen hexagram instead corresponds to eight, while Xun occupies the position of nine. Their alternating waxing and waning is as follows: one plus five makes six, which alternately waxes and wanes with the nine of the south; four plus five makes nine, which alternately waxes and wanes with the six of the north. When six advances and becomes nine, nine grows and six diminishes; when nine retreats and becomes six, nine instead diminishes and six again grows. Three plus five makes eight, which alternately waxes and wanes with the seven of the west; two plus five makes seven, which alternately waxes and wanes with the eight of the north. When seven advances and becomes eight, eight grows and seven diminishes; when eight retreats and becomes seven, eight instead diminishes and seven again grows.

Broadly speaking, when numbers advance upward they grow, and when they retreat downward they diminish. A growing number, after retreating, will again diminish; a diminishing number, after advancing, will again grow. The numbers of the four corners and the four cardinal positions, added vertically and horizontally, all make fifteen; remove the central five, and the vertical and horizontal sums all make ten. Measured against ten, nine is the remainder after subtracting one, eight is the remainder after subtracting two, seven is the remainder after subtracting three, and six is the remainder after subtracting four (as Zhu Xi says).

Interwoven and interlaced, no matter how they change, they all meet with seven, eight, nine, and six converging here.

Notes: The Book refers to the Shangshu; what is quoted here is the numerology related to the Luoshu. The Four Images in the Zhouyi refer to the four states of Greater Yang (Taiyang), Lesser Yin (Shaoyin), Lesser Yang (Shaoyang), and Greater Yin (Taiyin); here they are represented by old yang (nine), lesser yin (eight), lesser yang (seven), and old yin (six). Master Shao is Shao Yong, a Northern Song Neo-Confucian scholar. The Luoshu is one of the legendary Hetu and Luoshu; it uses a nine-palace square array to represent numerology and is a foundational diagram of ancient divination and numerology.

Vernacular Interpretation

Someone asked Zhu Xi: The diagram of the Hetu (Yellow River Chart) is not similar to the line-figures of the Bagua (Eight Trigrams); on what basis did Fuxi draw the trigrams?

I (Bao Yunlong) hold that the Great Commentary to the Changes says: “The Changes has Taiji (the Supreme Ultimate, the primordial undifferentiated chaos at the beginning of the cosmos), which produces the two modes (the two basic states of Yin-Yang). The two modes produce the Four Images (the four basic forms: Greater Yang, Lesser Yin, Lesser Yang, and Greater Yin). The Four Images produce the Bagua.” This is precisely Confucius’s explanation of the principle by which Fuxi drew the Bagua on the basis of the Hetu. The phrase “The Changes has Taiji” refers to the five and ten at the center of the Hetu; concealed within the diagram, they symbolize the position of the central palace.

The phrase “produces the two modes” means that in the Hetu, one, three, seven, and nine are the Yang mode, while two, four, six, and eight are the Yin mode. The phrase “produces the Four Images” means the images formed by one, two, three, and four as the generating numbers (the numbers first produced), together with the images formed by six, seven, eight, and nine as the completing numbers (numbers derived from the generating numbers). The phrase “produces the Bagua” means that seven, six, eight, and nine correspond respectively to the four trigrams Qian, Kun, Li, and Kan, occupying the four cardinal positions of east, south, west, and north; then one, two, three, and four correspond to Gen, Dui, Zhen, and Xun, filling the gaps at the four corners. It was precisely on the basis of this single diagram that Fuxi brought to completion all the line-figures of Yin-Yang, the Four Images, and the Bagua.

“One” is Taiji. Only because there is Taiji can the myriad things be generated: one produces two, two produces four, four produces eight, and eight produces the sixty-four hexagrams; the endless generation of life all arises from this center.

I also heard my teacher say: These few sentences are originally the foundation of the making of the Changes. Then how should these numbers be understood specifically? The teacher said: Do not explain them in a vague, general way; one should trace back to the beginning of Heaven and Earth, and then the subtle essentials of the sage’s making of the Changes will all be found therein. “One” is the beginning of Yang; therefore the sage took one odd number (that is, the yang line “—”) and drew it as the first line of the Qian trigram. Whatever is Yang among the lines of the trigrams is all assigned to Qian.

This passage discusses the order in which the Bagua (Eight Trigrams) of the Zhouyi (Book of Changes) are generated, and how the numerology of the Hetu (Yellow River Chart) serves as their fundamental basis.

First consider the origins of the yang and yin trigrams. Zhen, Kan, and Gen all take “one” as their main element, and in each of their numerical combinations an odd number (yang) is present as the dominant factor. When yin and yang first divided, one odd number was taken to draw the Qian trigram (☰, three yang lines); therefore all trigrams belonging to yang may be seen as transformations of Qian. By the same principle, the yin trigrams—in contrast to Zhen, Kan, and Gen—take “two” as their main element; one even number was taken to draw the Kun trigram (☷, three yin lines), and all trigrams belonging to yin may be seen as transformations of Kun. Xun, Li, and Dui likewise take “two” as dominant, and the line figures of the Bagua are thereby determined.

When it came to King Wen’s doubling of the trigrams (stacking the Bagua two by two into the sixty-four hexagrams), it was no more than an adjustment of the upper and lower positions of the line figures. Taking one line of Qian and exchanging it with Kun, Kun becomes Kan and occupies the north; taking the second line of Kun and exchanging it with Qian, Qian becomes Li and occupies the south. Zhen occupies the east, containing one yang and two yin; Dui occupies the west, containing two yang and one yin. The northwest is Qian (three yang), the southwest is Kun (three yin), the northeast is Gen (two yin and one yang), and the southeast is Xun (one yin and two yang). The arrangements of the four cardinal directions (east, west, south, north) and the four diagonals (the four corner positions) are all progressively deduced according to the sequence of one and two.

Fuxi drew the Prior Heaven Bagua on the basis of the Hetu, and King Wen, emulating Fuxi, derived the Later Heaven Yi. Yu the Great received the Hongfan Nine Categories (the nine great laws of governing the state) as recorded in the Luoshu (Luo River Writing), and began with controlling the waters, causing the land to be leveled; Jizi expounded the Hongfan, first speaking of “one is water, two is fire,” so that human relations were brought into harmony. These all derive from the numerology of “Heaven one, Earth two.” If we set aside the Hetu, what else can we rely on?

The Commentary says: The Hetu and the Luoshu, one vertical and one horizontal, serve as warp and weft for each other; the Bagua and the Nine Chapters, one outer and one inner, serve as surface and core for each other.

Zhu Xi said: Fifty-five (the sum of the numbers in the Hetu) is the root, and the changes of forty-five (the sum of the numbers in the Luoshu) can be derived from it. Taken as a whole, the Hetu (the diagram said to have emerged from the Yellow River in the time of Fuxi) has the image of the “Nine Categories” (the nine great norms of governance recorded in the Hongfan chapter of the Book of Documents), while the Luoshu (the diagram said to have emerged from the Luo River in the time of Yu the Great) has the image of the Five Phases. The Hetu consists of constant numbers; the Luoshu consists of changing numbers. The generating numbers (one through five) govern the completing numbers (six through ten), and these are the backbone of the constant numbers; odd numbers govern even numbers, and these are the function of the changing numbers. The Hetu takes generating numbers as primary; the Luoshu takes odd numbers as primary. On the surface, the Hetu can be used to draw the trigrams, while inwardly it can set forth the Nine Categories; on the surface, the Luoshu can set forth the Nine Categories, while inwardly it can be used to draw the trigrams. The Hetu pairs generating and completing numbers with Yin-Yang; within that pairing there remains distinction, and so the inner and outer patterns are divided into host and guest. The Luoshu distinguishes Yin-Yang by odd and even numbers; within that distinction there remains pairing, and so within opposition lies the subtle wonder of fluid change.

The so-called jing and wei do not mean taking what is above as jing and what is left and right as wei; rather, jing refers to the normative and wei to the changing. Likewise, the so-called surface and inner do not mean that one is inner and the other outer; rather, within the Hetu there is the Luoshu, and within the Luoshu there is the Hetu. The Hetu and Luoshu are mutually normative and changing, and which is normative and which is changing depends on which is taken as primary: if the Hetu is taken as primary, then the Hetu is normative and the Luoshu is changing; if the Luoshu is taken as primary, then the Luoshu is normative and the Hetu is changing. The same applies to the idea of surface and inner.

Zhu Xi also said: As for the numbers and positions of the Hetu and Luoshu, three places are the same and two places differ.

The Hetu (Yellow River Diagram) assigns One and Six to Water in the north, Two and Seven to Fire in the south, Three and Eight to Wood in the east, Four and Nine to Metal in the west, and Five and Ten to Earth in the center. Each of these five occupies the original position of the direction to which it corresponds; this is the substance of number. Within this there is also change. The Luoshu (Luo River Writing) is precisely what is used to explain this change: in the Luoshu, the three positions of Earth, Water, and Wood are the same as in the Hetu and do not change. This means that Heavenly Five resides in the center and generates Earth; this is Taiji (the Supreme Ultimate). Once Taiji moves, it generates Heavenly One Water in the north; emerging again in the east, it is Heavenly Three generating Wood. This is Yang Qi: it lies hidden in winter and arises and grows in spring. Its substance cannot change; by this it nourishes roots and sprouts. This is said with regard to the internal generative numbers. As for Two and Seven, which originally belong to the southern position of Fire, when summer arrives, Qian Metal (Heavenly One) then generates the Yang of Si and takes its place in high summer, so as to nurture the myriad things and extend the work of generation. It is like a ruler personally attending the Mingtang (Bright Hall) and holding the great governance. Therefore the generative numbers Two and Seven of Lesser Yang (Zhen and Kan) cannot but defer to the honored position of Qian, the Father, and withdraw to the west. When autumn arrives, Qian and Dui return again to the west, and Xun and Kan return again to the south, each following its own substance and keeping to its constant way. Broadly speaking, supporting Yang and restraining Yin is precisely the original mind ofHeaven and Earth, which delights in giving life. The Hetu takes Ten as its master: Yang is in surplus, so odd numbers are in excess and even numbers are deficient. Heaven and Earth pair to form a body; only after the body is established can its functions be carried out. Therefore the Luoshu takes Nine as its master: odd numbers are in excess and even numbers are deficient; this is Heaven encompassing Earth and using it. Thus Yin numbers are restrained and made to withdraw to twenty (with the completion numbers of Yang numbers assisting from outside); seven and nine are changing, while six and eight are unchanging. Heaven and Earth borrow these numbers and lodge them in the Hetu and Luoshu; the sage follows them in order to elucidate the subtleties within. The discussions of former Confucians are already quite detailed, so I will not repeat them here; I will merely state my own view. See in the Yanyi (Expanded Meaning) the statements “the five Earths are lodged by means of these numbers” and “Water one, Wood three, Earth five.”

Annotations: The Hetu and Luoshu are two numerical diagrams said to have emerged from the Yellow River in the time of Fuxi and from the Luo River in the time of Yu the Great; they are the foundation of ancient Yijing studies and image-number studies. “One-Six Water, Two-Seven Fire…” means that the Hetu pairs one through ten with the five directions (north, south, east, west, center) and the Five Phases (Wood, Fire, Earth, Metal, Water), and the numbers of Heaven and Earth each have their assigned category. Generative numbers and completion numbers: one through five are generative numbers, and six through ten are completion numbers; generative numbers are the root, while completion numbers are the increase of the generative numbers. Qian, the Father, and Lesser Yang: this borrows hexagrams from the Yijing to explain the movement of Yin-Yang and the Five Phases. Qian is Heaven; Lesser Yang refers to the initial arising of Yang Qi (vital energy) in Yang hexagrams such as Zhen and Kan. Taiji: here it refers to Heavenly Five residing in the center and generating Earth, the root source of all transformation.

Master Cai of Jiezhai (the quoted author) says: The Yi takes “images” as primary, while the Hongfan takes “numbers” as primary. The numbers of the Hetu (Yellow River Diagram) are even; even numbers represent mutual opposition, and so the Yi takes the two Qi of Yin and Yang as its root. The numbers of the Luoshu (Luo River Chart) are odd; odd numbers represent circulation and operation, and so the Hongfan takes the Five Phases as its root.

Even numbers belong to stillness; stillness takes movement as its function. Therefore, in the Hetu the sums of each paired numbers are all odd: one and six, two and seven, three and eight, four and nine, five and ten. Odd numbers belong to movement; movement takes stillness as its function. Therefore, in the Luoshu the sums of each paired numbers are all even: one and nine, two and eight, three and seven, four and six. From this we know that good and bad fortune in the Yi arises from movement, while the Hetu actively grasps change and can perceive thesubtle omens of things in advance; thus it first attains good fortune, and is called “Prior Heaven.” Good and bad fortune in the Hongfan appears in stillness; stillness must await movement before it can take form, and so the Luoshu follows the Heavenly Dao and can only clearly discern good and bad fortune; thus it is called “Posterior Heaven.”

Qi has two categories, Yin and Yang, and the Five Phases are distributed within them. One, three, five, seven, and nine are the operation of Yang numbers. In the Xici, Confucius summarizes the five Yang numbers of the heavenly numbers and arrives at “the number of Heaven, twenty-five.” Two, four, six, eight, and ten are the operation of Yin numbers; summing the five Yin numbers of the earthly numbers yields “the number of Earth, thirty.” If Yin lacks Yang’s coordination, it cannot transform; if Yang lacks Yin’s coordination, it cannot act on its own. This shows that only when Yin and Yang coordinate can transformation arise. What can drive ghosts and spirits and govern transformation lies in “five”—that is, the Five Phases.

The fixed substance in which Heaven and Earth and Yin and Yang stand in mutual opposition is the order of Yin and Yang’s circulation from one to ten. Distinguished by odd and even, one, three, five, seven, and nine are Yang; two, four, six, eight, and ten are Yin. Ordered by sequence, one through five are the generating numbers (Yang), and six through ten are the completing numbers (Yin). Numbers operate within Yin and Yang before form is determined; images appear within Yin and Yang after form has been completed. The Taiji (Supreme Ultimate, the cosmic source) in the Yi possesses the principle of Yin and Yang but is not yet formed. Images belong to Yin and to stillness; numbers belong to Yang and to movement. The Taiji is neither still nor moving, but takes stillness as primary. Before the Taiji, images and numbers have no priority of before and after. When movement arises and generates Yang, numbers begin from this, and images also begin with it; when stillness arises and generates Yin, images then take form, and numbers are completed here.

The division and union of Yin and Yang, and their stillness and movement, are not the same. In terms of “establishing” (establishing the substance), stillness is the time when Yin and Yang stand in mutual opposition, while movement is the time when the one Qi circulates. In terms of “operating,” stillness is the time when one Yin and one Yang unite in virtue, while movement is Yang first and Yin after, the two meeting and operating.

The author’s commentary says: This passage is exquisite; readers should ponder it deeply.

Interpretation of Tianyuan Fawei (Bao Yunlong)

I. The Xiantian Horizontal Diagram

Fuxi drew the Xiantian (Prior Heaven) horizontal diagram on the basis of the Hetu (Yellow River Diagram). The horizontal diagram takes one, two, three, four, five, six, seven, and eight as its sequence; the hexagram order is arranged from right to left, exactly opposite to the forward and reverse order of the circular diagram.

The Four Images—Old Yang, Young Yin, Young Yang, and Old Yin—each occupy one position: Old Yang occupies the first position and divides into Qian and Dui; Young Yin occupies the second position and divides into Li and Zhen; Young Yang occupies the third position and divides into Xun and Kan; Old Yin occupies the fourth position and divides into Gen and Kun. This is the arrangement viewed from top to bottom.

If viewed from bottom to top and from left to right, however, it is a relation of generation: Gen and Kun are generated by the one and six of Old Yin; Xun and Kan are generated by the two and seven of Young Yang; Li and Zhen are generated by the three and eight of Young Yin; Qian and Dui are generated by the four and nine of Old Yang.

Specifically, above Taiyang (a single yang line, namely “—”) one odd and one even are generated, yielding Qian and Dui; above Young Yin (“--”) one odd and one even are generated, yielding Li and Zhen; above Young Yang one odd and one even are generated, yielding Xun and Kan; above Taiyin (that is, Old Yin) one odd and one even are generated, yielding Gen and Kun. This is the Xiantian horizontal diagram. From one to eight, it was originally only the order in which the hexagrams are generated.

II. The Xiantian Circular Diagram

Fuxi drew the Xiantian circular diagram on the basis of the Hetu.

Zhu Xi said: in the horizontal diagram of the sixty-four hexagrams, the four hexagrams Zhen, Xun, Fu, and Gou happen to be located exactly in the middle. Taking the middle as the turning point, one therefore folds the horizontal diagram into a circular diagram: first one proceeds backward starting from Zhen and Fu until Qian; then one proceeds forward starting from Xun and Gou. In this way the Xiantian circular diagram is formed.

Notes: Hetu—a diagram said to have appeared in the Yellow River in the time of Fuxi, using black and white dots to represent the numbers one through ten; later Yixue (Yijing scholarship) regarded it as the source of image-number thought. Xiantian—refers to the order that hexagram images already possessed before the generation of heaven and earth and the myriad things, as opposed to “Houtian” (Posterior Heaven, the order of what has already been generated as represented by King Wen’s Bagua). Four Images—the four line-images formed by pairing yin and yang in old and young forms. Old Yang is nine, Young Yang is seven, Young Yin is eight, and Old Yin is six.

Zhu Xi said: In the horizontal diagram of the sixty-four hexagrams, the four hexagrams Zhen, Xun, Fu, and Gou are exactly in the middle. When one folds or turns it from the middle, the circular diagram is drawn. First counting backward from Zhen and Fu to Qian, and then counting forward from Xun and Gou to Kun, forms the circular diagram. Spring, summer, autumn, and winter; the moon’s dark, new, quarter, and full phases; and the day’s night, day, dusk, and dawn can all be arranged in sequence within the diagram—this is the main rationale for making the diagram. He also said: In the horizontal diagram, all is counted in forward order; the directions of the square diagram and the circular diagram are counted in reverse order; in the circular diagram, only half is counted in reverse order. Details are seen in the “Left and Right Chapter.”

The Earlier Heaven Bagua (the trigram order traditionally attributed to Fuxi) corresponds with the Luoshu (the numerical diagram said to have emerged from the Luo River).

Old Yang, nine (Qian), is due south; four is Dui in the southeast; Lesser Yin, three, is Li due east; eight is Zhen in the northeast; Lesser Yang, two, is Xun in the southwest; seven is Kan due west; Old Yin, one, is Kun due north; six is Gen in the northwest. In the circular diagram, Qian, Dui, Li, and Zhen are the four yang trigrams and occupy the left; Xun, Kan, Gen, and Kun are the four yin trigrams and occupy the right. The Luoshu also places Qian, Kun, Kan, and Li in the four cardinal positions, and Zhen, Dui, Xun, and Gen in the four corners (the four diagonal positions). The trigram positions in the diagram and in the book both agree. There is only one difference: Old Yang and Lesser Yang are interchanged in the southwest position; this is the result of flexible adjustment.

The Later Heaven Bagua (the trigram order traditionally attributed to King Wen) corresponds with the numbers of the Hetu (the numerical diagram said to have emerged from the Yellow River).

The directions of the Later Heaven Bagua also correspond to the numbers of the Hetu (Yellow River Chart)—traditionally, during the time of Fuxi, a dragon-horse emerged from the Yellow River bearing a chart, a numerical array recorded on it. One and Six belong to Water and correspond to the Kan trigram, occupying the north; Two and Seven belong to Fire and correspond to Li, occupying the south; Three and Eight belong to Wood, with Zhen occupying due east and Xun the southeast; Four and Nine belong to Metal, with Dui occupying due west and Qian the northwest; Five and Ten belong to Earth, with Gen (Five) occupying the northeast and Kun (Ten) the southwest, while Five and Ten penetrate the central palace and transform into Earth. From this it can be seen that the Later Heaven numbers invariably correspond with the Hetu.

Some, seeing that Two and Seven occupy the south and are called Li, and that Four and Nine occupy the northwest and are called Qian and Dui, conclude that Ten alone is the Luoshu (Luo River Writing)—traditionally, when Yu the Great controlled the waters, a divine turtle emerged from the Luo River bearing numbers on its shell. Seeing also that Nine is borne above in the south and called Qian, and that Two and Seven occupy the southwest and are called Xun and Kan, they conclude that Nine alone is the Hetu. Some therefore say that King Wen modeled himself on this diagram and established the Later Heaven Yi; the two diagrammatic schemes thus became intermingled and confused, and the theories of the Northern Song Yijing scholar Liu Mu subsequently became current. Readers should distinguish them clearly.

The yin-yang numbers of the Hetu and Luoshu are distinguished by odd and even; odd and even numbers interlace with one another.

In the Hetu (River Chart), one, two, three, and four, with seven, eight, nine, and six contained within them, together form ten. In the Luoshu (Luo Writing), one, two, three, and four are set opposite seven, eight, nine, and six, together forming ten. Ten is “two-five” (two times five); both sets of numbers are generated through multiplication by five and completed by five. Five is the central earth position and represents the central palace.

Consider first the Hetu. Old Yang (the most flourishing of the yang numbers), one, occupies the north and contains nine within; its completing number—the number that pairs with it to make ten—is in the west. Young Yin, two, occupies the south and contains eight within; its completing number is in the east. Young Yang, three, occupies the east and contains seven within; its completing number is in the south. Old Yin, four, occupies the west and contains six within; its completing number is in the north. The numbers of the four directions interlace and mutually generate one another, and so they are said to “mutually generate.”

Now consider the Luoshu. Its arrangement is one of direct opposition: one and nine face each other north and south, while three and seven face each other east and west. Beyond the four cardinal positions, the four corners are also paired in opposition: the northeast corner opposes the southwest corner, namely two and eight; the southeast corner opposes the northwest corner, namely four and six. In this way they crisscross vertically and horizontally, forming warp and weft.

Both the Hetu and the Luoshu conceal ten at the center; this is to pair the Five Phases two by two so that they mutually complete one another. Master Dong, a scholar who annotated the Hetu and Luoshu, explained this solely through the positions and numbers of the Four Images (Sixiang), and his account is also quite good.

Zhu Xi said: “The Yi was probably first obtained by Fuxi from the Hetu; originally it had no need to rely on the Luoshu. The Hongfan, by contrast, was obtained by Yu the Great alone from the Luoshu; it was not necessarily something that followed and imitated the Hetu.”

Master Zheng said: When the sage drew the Bagua (Eight Trigrams) according to the Hetu (Yellow River Diagram) and set forth the Nine Categories (Jiuchou) according to the Luoshu (Luo River Writing)—the Nine Categories being the nine great norms of governance recorded in the “Hongfan” chapter of the Shangshu (Book of Documents)—could this mean that they were copying a ready-made pattern, like drawing a circle with a compass or a square with a carpenter’s square? The underlying principles are connected, but this is not how it happened. When Fuxi drew the trigrams according to the Hetu, he had not previously seen the Luoshu, yet the numbers of the trigrams naturally coincided with the Luoshu. When Yu the Great set forth the Nine Categories according to the Luoshu, he did not further investigate the numbers of the Hetu, yet the numbers naturally matched them. In my view, the directional arrangement of the Hetu corresponds to Former Heaven (Xiantian)—the original order before things have formed, by which they may be inferred—while the directional arrangement of the Luoshu corresponds to Later Heaven (Houtian)—the order after things have manifested, by which they may be verified. Although their arrangements differ as earlier and later, and their numbers differ as ten and nine, both take the Five Phases (Wood, Fire, Earth, Metal, Water) as their root and the central five as their pivot; in truth there is only one principle.

Transmitted by Master Chen Xiyi of Pushang.

Master Xiyi, whose name was Chen Tuan, transmitted the study of numbers to Mu Xiu (courtesy name Bochang). Mu Xiu transmitted it to Li Zhicai, and Li Zhicai to Shao Kangjie (Shao Yong), who thus wrote the Huangji Jingshi (Supreme Principles Governing the World). Bochang in turn transmitted the Taiji (Supreme Ultimate) Diagram to Zhou Lianxi (Zhou Dunyi). Zhu Hanshang (Zhu Zhen, whose sobriquet was Hanshang) said that Zhou Dunyi transmitted it to the two Cheng brothers (Cheng Hao and Cheng Yi). At that time, Zhang Zai’s teaching stood between that of the two Cheng brothers and Shao Yong. Thus Zhou Dunyi wrote the Tongshu, Cheng Yi wrote the Yizhuan (Commentary on the Changes), and Zhang Zai wrote “Taihe” (Great Harmony) and two or three other pieces. Chen Tuan also transmitted the study of images (xiangxue) to Shen Fang; Shen Fang transmitted it to Xu Jian of Lujiang, and this branch of the lineage thus reached the South.

Some also say that the Former Heaven square and circular diagrams originated in Master Ma’s inner realization. Master Zhu (Zhu Xi) clarified that this account is incorrect, holding that these diagrams were all made by Fuxi himself and contained only the drawn trigrams, with no words or written text. After the deaths of Confucius and Mencius, these diagrams were secretly concealed by Fangshi (esoteric adepts) and not transmitted, until Master Xiyi first brought them into circulation.

Cai Xishan (Cai Yuanding) said: The diagrams of the Hetu and Luoshu were identified in this way from Kong Anguo and Liu Xin of the Han dynasty, Guan Lang (courtesy name Ziming) of Wei in the Three Kingdoms, down to Shao Kangjie (Shao Yong) and Shao Yaofu of the Song. It was only Liu Mu who exchanged the names of the two, and later people followed him. Therefore we now restore the old account and follow the original order in all respects.

In records ancient and modern, Kong Anguo, Liu Xiang and his son, and Ban Gu all held that the Hetu was bestowed on Fuxi and the Luoshu on Yu the Great. Guan Ziming and Shao Kangjie also took the number ten as the Hetu and the number nine as the Luoshu. The numbers of the Nine Palaces are “nine on top, one below, three on the left, seven on the right, two and four as shoulders, six and eight as feet,” exactly the pattern on a turtle’s back. Zhu Zi (Zhu Xi) said: In reading the “Mingtang” chapter of the Da Dai Liji, there is the phrase “two, nine, four, seven, five, three, six, eight, one”; Zheng Xuan’s commentary says this imitates the markings on a turtle’s back. From this it is clear that Han dynasty people originally took nine as the Luoshu.

Liu Mu, however, took nine as the Hetu and ten as the Luoshu, completely reversing the account of earlier Confucians. He further claimed that this diagram came from Chen Tuan (Xiyi) and was not committed to writing, saying that Xu Jian transmitted it to Kuachang, and Kuachang in turn to Liu Mu. Many people in the world followed his account, and it spread throughout the realm. Today, if one looks at the Yizhuan by Hanshang (Zhu Zhen), the Qiyi by Zhang Xingcheng of Shu, and the works of the Zheng clan of Puyang and others, they all adopted Liu Mu’s account; all were written before the correction had been made. It was only when Zhu Zi appeared that, together with Cai Xishan, he examined and verified the matter, taking ten as the Hetu and nine as the Luoshu and restoring the old account; only then did people throughout the realm accept it.

Nevertheless, Liu Mu’s doctrine should not be left unknown; one or two passages are excerpted below. He cites the Great Commentary to the Changes and holds that both the Hetu and Luoshu emerged in Fuxi’s time, no longer saying that they were bestowed on Fuxi. This passage is a hypothetical argument, its gist being: even if later generations cannot see the relevant words in the Xici, this is no reason to conclude that the sixty-four hexagrams were first re-derived by King Wen.

In ancient documents, the record in the Hongfan (Great Plan) of “Heaven bestowing the Nine Categories” is no longer to be found. Some hold that the Nine Categories were composed by Jizi. But they have not considered that the principles of “quyu” and “quyang” were already present long before King Wen made the hexagrams, and that the offices of Situ and Sikong had already been established when Shun appointed the Nine Officials. From this it can be seen that both the Hetu (Yellow River Chart) and the Luoshu (Luo River Writing) existed in high antiquity; Fuxi merely drew the hexagrams in accordance with them, and Yu the Great followed them and set them forth.

I do not think this view is without reason. But if it is said that the Luoshu did not appear when Yu the Great was controlling the waters, then the sentence “Heaven bestowed on Yu the Great the Nine Categories of the Hongfan” cannot be made to make sense. In the Xinwei examination of recent years, Liu Mengjian adopted this view and thereby took first place in the Nangong (the Ministry of Rites metropolitan examination). At that time most scholars followed Zhu Xi’s (Wengong’s) doctrine concerning the Hetu and Luoshu, while those who advocated this view were instead ranked behind.

Moreover, the Yugong only says “guide the Luo” and does not say “obtained the book”; the Shundian only says “hui chou” and does not say “nine categories.” What is called “Heaven bestowed it on Yu” merely means that Heaven revealed it.

Liu Mu further says: the number of Heaven and Earth is fifty-five; the number of the Hetu is forty-five. The number ten is left out and not used because the four corners and the four cardinal positions, when added, each make ten; adding the central five, the horizontal and vertical lines each make fifteen, and the total is also fifty-five. The number five is left out and not used because Heaven One plus five makes six, Earth Two plus five makes seven, Heaven Three plus five makes eight, Earth Four plus five makes nine, and Heaven Five plus five makes ten; therefore it is said, “Earth Ten completes it.”

Again, the number of the Bagua is thirty-six. The number nine is left out and not used because the numbers of Qian and Kun are both nine, those of Zhen and Xun are both nine, and those of Kan and Li, Gen and Dui are also both nine.