《皇極經世》Vol. 22

Heaven's numbers are five, and Earth's numbers are also five; together they make the complete sum of ten. Heaven takes "one" as its root and transforms into four, and Earth likewise takes "one" as its root and transforms into four. Four has form, while "one" has none; this is the ultimate of what is called "being" (you) and "non-being" (wu). Heaven's substance-number is four, of which three function and one does not; Earth's substance-number is also four, of which three function and one does not. Therefore the formless "one" is used to model nature, the non-functioning "one" is used to model Dao (the Way), and the three functioning numbers are used to model Heaven, Earth, and Humanity.

Substance-numbers have eight transformations, and function-numbers have six. Thus, among the images of the Bagua (Eight Trigrams), four are fixed and two can be inverted; through the transformations of six lines, the eight trigrams are formed. Among the images of the doubled hexagrams, eight are fixed and twenty-eight can be inverted; through thirty-six transformations, the sixty-four hexagrams are formed. Hence the changes of the lines are limited to six, the changes of the trigrams end at eight, the changes of the stalk-count numbers are exhausted at thirty-six, and the doubled hexagrams reach their limit at sixty-four. Hexagrams are composed of the trigrams and, when doubled, become sixty-four; lines are composed of six, the stalk-count changes reach thirty-six, and the total number of lines in the doubled hexagrams is three hundred and eighty-four.

Heaven has four seasons, and each season has four months of forty days each. Four times forty makes one hundred and sixty; subtracting a ten from each, each season becomes three months, and each month thirty days. Four is the substance-number and three is the function-number. Although the substance-number is complete with four, one of them is often unused, so the numbers that actually function stop at three, and at their extreme reach nine. Substance-numbers are generally even, hence there are four and twelve; function-numbers are generally odd, hence there are three and nine.

Why do large numbers often appear insufficient while small numbers often appear full? Because large things are difficult to see, while small things can be seen. Therefore a year has four seasons, yet in reckoning months each season is counted as three, and each day is complete at ten double-hours (shichen). Thus the human body has four limbs, yet the hands have ten fingers.

Heaven appears in the south and is hidden at the North Pole; its motion is limited to six, and what exceeds it is counted as seven. Therefore people understand what lies before them but are unclear about what lies behind, and they often pay insufficient attention to what is on their left and right.

Heaven's substance-number is four and its function-number is three; Earth's substance-number is also four and its function-number is three. Heaven restrains Earth, and Earth restrains Heaven; yet Earth's restraint remains upon Earth, just as the remainder of daylight passes into night…

Heaven has the three luminaries (sanchen), and earth has the four phases (sixing). The "three luminaries" refer to the sun, moon, and stars, and the "four phases" refer to wood, fire, earth, and metal. Yet earth's movements are only partly visible, and the parts that do not appear may be what is called the surplus (yufen).

Heaven has two principal positions (zhengwei), and earth also has two. Together with two changes (bianhua), they form the Eight Trigrams (Bagua). Heaven has four principal positions, and earth also has four. Together with twenty-eight changes, they form the sixty-four hexagrams. Therefore, in the minor accomplished hexagrams (xiaocheng gua), there are four principal positions and two changes, six hexagrams in all. In the great accomplished hexagrams (dacheng gua), there are eight principal positions and twenty-eight changes, thirty-six hexagrams in all. Qian, Kun, Li, and Kan are the roots of the thirty-six hexagrams, while Dui, Zhen, Xun, and Gen are the roots of the other twenty-eight.

Qian has seven sons, Dui has six, Li has five, Zhen has four, Xun has three, Kan has two, and Gen has one. Kun is entirely yin, so it has no sons. Arranged again by yin-yang and firm-yielding (gangrou), Qian has seven sons, Kun has six, Dui has five, Gen has four, Li has three, Kan has two, and Zhen has one. Xun is firm, so it has no sons.

Qian and Kun each have seven changes, so the ultimate number of daily motion does not exceed seven fen. Dui and Gen each have six changes, and the moon's motion stops at six, for a total of twelve. Li and Kan each have five changes, and the sun's motion stops at five, for a total of ten. Zhen and Xun each have four changes, and bodily motion stops at four, for a total of eight.

The principal positions and changes of the hexagrams total thirty-six. Each hexagram has six lines, so the thirty-six hexagrams have 216 lines, the number used for calculation. Subtract four from thirty-six to get thirty-two, subtract four again to get twenty-eight, and subtract four once more to get twenty-four. Thus the number of the hexagrams is thirty-two, reached by subtracting four from the total. The number of heaven is twenty-eight, reached by subtracting eight. The number of earth is twenty-four, reached by subtracting twelve. Here the "four" refers to Qian, Kun, Li, and Kan. The "eight" also includes the four hexagrams Yi (Nourishment), Fu (Return), Daguo (Great Exceeding), and Xiaoguo (Small Exceeding). The "twelve" refers to Dui, Zhen, Tai (Peace), and Jiji (After Completion).

The sun has eight positions, but in practice the count goes only to seven, calculated by removing Qian. The moon has eight positions, but in practice the count goes only to six, calculated by removing Dui. The stars have eight positions, but in practice the count goes only to five, calculated by removing Li. The chen (the luminaries) have eight positions, but in practice the count goes only to four, calculated by removing Zhen.

The sun has eight positions, yet the count stops at seven, calculated by removing the Tai hexagram.

The months begin from the Dui position because the moon's cycle cannot keep pace with the sun's count, so the twelve months of the year usually leave twelve days over.

Yang numbers have no "ten," so they fall short at the end; Yin numbers have no "one," so they fall short at the beginning.

The Qian hexagram is Yang within Yang, and its number cannot change, so a year lists only twelve months. The Zhen hexagram is Yang within Yin, and its number likewise cannot change, so the twelve double-hours of the day cannot be made to appear. The Dui hexagram is Yin within Yang, and the Li hexagram is Yang within Yin. Both of their numbers can change, so the counts of days and months can be calculated separately.

Yin numbers begin with twelve and Yang numbers begin with thirty, and the two always keep the relation of "two and six," that is, Yin at two and Yang at six. By listing the years, one can see the months; by listing the months, one can see the dates; by listing the dates, one can see the double-hours. This is because Yang governs Yin.

Heaven has four kinds of change, and it also contains the four kinds of change of Earth. The changes of the Sun contain the changes of the Moon and the stars. Therefore, one hexagram contains four hexagrams.

The Sun, Moon, stars, and constellations each occupy one position: the Sun has four positions, the Moon has four, the stars have four, and the constellations also have four. Four times four makes sixteen positions. After this one change, the number of the movements of the Sun and Moon has been fully derived.

Heaven has four kinds of change, and Earth also has four kinds of change. Change includes growth and also diminution. After sixteen changes, the numbers of Heaven and Earth have been fully derived.

The Sun begins at "one," the Moon begins at "two," the stars begin at "three," and the double-hours begin at "four." From this derivation and expansion, Yang numbers are generally six and Yin numbers are generally two, and the numbers of the great and small cycles of movement are thereby carried to their limit.

Three hundred and sixty changes become twelve wan nine thousand six hundred; twelve wan nine thousand six hundred changes become one hundred and sixty-seven yi nine thousand six hundred; one hundred and sixty-seven yi nine thousand six hundred changes become twenty-eight thousand two hundred and eleven zhao, nine hundred and ninety wan, seven thousand four hundred and fifty-six yi.

Three hundred and sixty corresponds to one "shi" (a double-hour), and twelve wan nine thousand six hundred corresponds to one day. Dividing one hundred and sixty-seven yi nine thousand six hundred wan into thirty parts to match the number of months in a year, there appear advances and retreats of six days. Three hundred and sixty corresponds to one shi, and following the advances and retreats of the minor cycles, it corresponds to the hours of day and night.

Taking the number of the sixteen changes and removing the parts that interlace and coincide, and keeping the number that actually operates, one obtains twenty-eight thousand two hundred and eleven zhao, nine hundred and ninety-one wan, seven thousand four hundred and fifty-six yi.

The one hundred and fifty-six yi (15,600,000,000) is divided into twelve stages, the first six of increase and the last six of decrease. Each stage is 3,099,680,000. Of these, 16,796,160,000 is the figure for which each portion, when unfolded, advances the count by sixty days. Six stages unfolded into six portions advance three hundred and sixty days. Adding the one remaining portion and unfolding seven portions advances three hundred and sixty-six days. Reckoning backward follows the same principle.

From the total of 129,600, three portions are removed and called the "intersecting number" (jiaoshu). Seven of the portions are taken and called the "use number" (yongshu). Three portions of the use number combine into six, and adding the one remaining portion makes seven. The seven portions together total 90,720 years. Halving this gives 45,360 years, which is used to calculate an advance of six days.

The day has daylight and darkness, and the calendrical count also has tiaonü (the difference between the waxing and waning of the moon's motion and the calendar's calculation). Together they form twelve days. One day advances for every 3,600 years, so 43,200 years advance twelve days in total. The remaining 2,160 years are used to advance six parts of the surplus. Adding the 2,160 years from the intersecting number advances twelve parts, which serve as the intercalary number (runshu). Thus, each time the minor cycle has changed sixty times, it forms three hundred and sixty-six days.

Qian (the hexagram of pure yang in the Yijing [Classic of Changes]) is one. When the fifth line of Qian is separated, it becomes Dayou (Great Possession), which corresponds to the number 360. When the fourth line is separated, it becomes Xiaoxu (Small Accumulation), which corresponds to 129,600. When the third line is separated, it becomes Lü (Treading), which corresponds to 16,796,160,000. When the second line is separated, it becomes Tongren (Fellowship with Others), which corresponds to 28,211 trillion, 9,907,456 hundred million. When the sixth line of Qian is separated, it becomes Gou (Coming to Meet), and the number it corresponds to is... This is what is called the "division number" (fenshu): the greater is divided into the smaller, always in six layers from top to bottom, and therefore it is matched with yang numbers. For example, one divided becomes twelve, and eleven divided becomes three hundred and sixty.

Heaven governs the overall form of things, so after eight transformations it reaches sixteen. Earth governs the practical functions of things, so after six transformations it reaches twelve. Heaven, from its beginning, finally...

The earth number begins at twelve and finally reaches two hundred and four zi, six thousand nine hundred and eighty wan (ten thousand), seven thousand three hundred and eighty-one gai, five thousand four hundred and ninety-one wan, eight thousand four hundred and ninety-nine zhao (trillion), and seven million two hundred thousand yi (hundred million).

Splitting one into two yields the hexagram Guai (Breakthrough), which corresponds to the number twelve. Splitting two into four yields Da Zhuang (The Great Invigorating), corresponding to four thousand three hundred and twenty. Splitting four into eight yields Tai (Peace), corresponding to five hundred and fifty-nine million eight hundred and seventy-two thousand. Splitting eight into sixteen yields Lin (Approach), corresponding to nine hundred and forty-four zhao, three thousand six hundred and ninety-nine wan, six thousand nine hundred and fifteen yi, and two thousand wan. Splitting sixteen into thirty-two yields Fu (Return), whose corresponding number is missing from the original text. Splitting thirty-two into sixty-four yields Kun (The Receptive), which corresponds to the number of Wuji (Ultimateless). This is called the "long numbers." They grow from small to large, all derived from below upward, and so they are matched with the yin numbers.

Once there is earth, there is "two," and once there is "two," there is day and night. Yin and yang alternate and interact, forming the myriad things. Therefore the Yijing (Book of Changes) uses "two" to generate the numbers and "twelve" to unfold transformations. "One" is not itself a number, yet it serves as the foundation through which numbers come to be formed. The heavenly bodies move without pause and have never truly had day and night. Humans live on the earth, and so they take day and night as actual time. For this reason, the numbers humans use are based on the numbers of the earth.

Heaven's numbers counted upward from Lin and earth's counted upward from Shi belong to the "operational numbers" (yunshu). Heaven's numbers counted downward from Tongren and earth's counted downward from Dun belong to the "year numbers" (nianshu). The operational numbers belong to heaven, and the year numbers belong to earth. Heaven's numbers counted upward from Bi and earth's counted upward from Gen belong to the "use numbers" (yongshu). Heaven's numbers counted downward from Mingyi and earth's counted downward from Pi belong to the "exchange numbers" (jiaoshu). Heaven's numbers counted upward from Zhen and earth's counted upward from Jin belong to the "existing numbers" (youshu). Heaven's numbers counted downward from Yi and earth's counted downward from Yu belong to the "non-existing numbers" (wushu).

Heaven's numbers begin with Qian and end with Zhen, and the rest are classed as "non-existence," because some of the stars cannot be seen. Earth's numbers subtract "one" and begin at twelve, because the fire of the earth often lies hidden and does not show itself. Therefore heaven takes form as its foundation yet often conceals that foundation, and earth takes function as its root yet often conceals its function. A period is limited to three months, and a month to thirty days, and both remove the count of the stars. Thus eight multiplied by eight yields the sixty-four hexagrams, and this number of hexagrams does not change again.

There are eight fixed numbers and seven variable ones; seven multiplied by eight equals fifty-six, and the principles associated with this derive from it.

Yang lines (yang yao, the lines that represent yang in the Yi) correspond to the numbers of daytime, and yin lines correspond to the numbers of night. Heaven and earth join one another and yin and yang alternate in turn, so day and night are separated and firmness and yielding interlace. Spring and summer belong to yang, so the daytime numbers are greater and the nighttime numbers fewer. Autumn and winter belong to yin, so the daytime numbers are fewer and the nighttime numbers greater.

The body number (ti shu) of the calculation numbers (ce shu, the numbers used in computation) is three hundred and eighty-four. Subtracting the ce numbers of the four hexagrams Qian, Kun, Li, and Kan leaves three hundred and sixty, which is called the function number (yong shu). The function of the body number is two hundred and seventy. Subtracting the ce numbers of Qian, Li, and Kan leaves two hundred and fifty-two, which is called the function of the function number.

Of the two hundred and seventy that is the function of the body number, one hundred and fifty-six belong to yang and one hundred and fourteen to yin. After the ce number of Li is subtracted, the yang number is one hundred and fifty-two and the yin number is one hundred and twelve; this is the number actually employed. Broadly speaking, the yang number sets aside Li and uses Qian, and the yin number sets aside Kun and uses Kan. Thus the yang ce number of Heaven is one hundred and twelve, from which the yin numbers are removed. The yin ce number of Earth is also one hundred and twelve, and it has a further forty yang ce numbers, which are used to subtract the yang numbers of the south-north axis. In the far south lies the Great Heat (da shu), and in the far north the Great Cold (da han), where the myriad things cannot grow, so these numbers are subtracted. Are these forty numbers what remains of Heaven? Yang qi intrudes upon yin qi, and daytime intrudes upon night, so this portion falls to the earth. Together, the yang and yin numbers total yang one hundred and fifty-two and yin one hundred and twelve.

Subtracting the ce number of Qian from the yang numbers and the ce number of Kan from the yin numbers yields yang one hundred and forty-four and yin one hundred and eight; this is the function of the function number. The yang numbers are grouped in units of thirty-six, three groups making one hundred and eight. The yin numbers are likewise grouped in units of thirty-six, three groups again totaling one hundred and eight. This represents three portions of yang and three of yin, so that yin and yang each occupy half. The yang numbers also keep one remainder of thirty-six, which together with the rest gives yang one hundred and forty-four and yin one hundred and eight. Therefore, the function of the body number is two hundred and seventy, the number actually employed is two hundred and sixty-four, and the function of the function number is two hundred and fifty-two.

There are sixty-four hexagrams, but those actually employed are limited to thirty-six. There are three hundred and eighty-four lines, but those actually employed are limited to two hundred and sixteen. The sixty-four hexagrams, each divided into four parts, make two hundred and fifty-six; so when the first and top lines are removed from each hexagram, the total is again two hundred and fifty-six. This is the number by which the myriad things are generated. Therefore Li and Kan are the two hexagrams associated with the generation of the myriad things.

The lord of all things takes the four yang lines of Li (Fire, the sun) and the four yin lines of Kan (Water, the moon), so the number by which the myriad things are generated is based on four. The yang lines total one hundred and twelve, and the yin lines likewise total one hundred and twelve. After subtracting the lines of Li and Kan, the sum is two hundred and sixteen. Yin and yang are forty each, making two hundred and fifty-six in all. Thus each of the Eight Trigrams uses six lines and is governed by Qian and Kun, and among those six lines, four are put to use and are governed by Li and Kan.

Therefore, the heavenly seasons at dusk and at dawn do not give birth to the myriad things, but at midday they do. In terms of geography, the northern and southern extremes do not give birth to the myriad things, while the central lands do.

What is meant by "body number" (tishu)? The body number is the number by which the myriad things are generated. What is meant by "function number" (yongshu)? The function number is the number of operation and transformation. Operation and transformation belong to Heaven, and generation belongs to Earth. Heaven operates on its own, so the function numbers multiply one another, and their application determines the seasons of birth. Earth generates the myriad things through the union of Yin-Yang, so the yang number within the body numbers is multiplied by the yin number to yield the number of generation.

Heaven's numbers take three as their principle, so six multiplied by six and again by six gives the Qian counting-number (the counting-number of the Qian hexagram), two hundred and sixteen. Earth's numbers take two as their principle, so twelve multiplied by twelve gives the Kun counting-number, one hundred and forty-four. Qian uses nine: three times eight is twenty-four, and multiplied again by nine it is also two hundred and sixteen; two times eight is sixteen, and multiplied again by nine it is also one hundred and forty-four. Kun uses six: three times twelve is thirty-six, and multiplied again by six it is also two hundred and sixteen; two times twelve is twenty-four, and multiplied again by six it is also one hundred and forty-four. Taking twelve as the base for Kun and drawing numbers in the proportion of two, six, four, six, and one and a half as what remains for Qian, Qian comes to two hundred and fifty-two and Kun to one hundred and eight.

The four yang trigrams together contain twelve lines, of which eight are yang and four are yin. Multiplying the yang lines by thirty-six and the yin lines by twenty-four gives a total of three hundred and eighty-four.

Each hexagram and its reverse (zheng fu) are opposites, each consisting of six yang and six yin lines. In the Yi, there are twelve such pairs of six-yang and six-yin hexagrams. After the four trigram-hexagrams are removed, the remaining hexagrams with eight yang and four yin lines, and those with eight yin and four yang lines, form six pairs each. Those with ten yang and two yin lines, and those with ten yin and two yang lines, form three pairs each.

The body number of the hexagrams is three hundred and eighty-four, yet the function number is only three hundred and sixty. This is because Qian, Kun, Li, and Kan are not counted in the function number. As to why these four hexagrams are excluded, the original text has not yet finished explaining.

Qian, Kun, Li, and Kan (the four hexagrams of the Yi [Classic of Changes]) do not take part in a certain calculation, precisely so that the use of the whole number three hundred and sixty may be accomplished. Therefore, all things change, and only these four remain unchanged. Because they are unchanging, they can be used to explain change.

The numbers in use stop at three hundred and sixty, so why is it said that there are three hundred and sixty-six? This is because numbers have a remainder. What use is the remainder? It indicates the full operation of the Qian hexagram. If Qian and Kun do not take part in the operation, then Li and Kan each bear half of it. Why can Qian operate in full? Because Yang governs fullness and growth. Why do Qian and Kun not take part in the operation? Because there is only yang qi, and the myriad things cannot be born; yin qi is too scant, and the myriad things cannot be completed either. Why do Li and Kan each operate by half? Because Li lies in the east and Kan in the west, occupying the middle position where Yin-Yang alternate. They are respectively the gates of spring and autumn, and of day and night.

Sometimes Qian is used, and sometimes Li and Kan. Why? Speaking from the standpoint of yang qi in fullness, one uses Qian. If yang fullness is considered in divided terms, it becomes yang qi invading yin qi and daylight invading darkness, and so Li and Kan are used. Qian represents yang fullness and therefore operates in full, while Kun represents yin emptiness and therefore does not operate at all. Yang invades yin, and yin also invades yang, so Li and Kan each operate by half.

Hence, when the celestial bodies reach the south, they are all visible; when they reach the north, they are wholly invisible; in the east and the west, each is half visible. Li and Kan represent the boundary of Yin-Yang, so Li corresponds to the earthly branch Yin and Kan corresponds to the branch Shen. Yet the actual count often exceeds these two limits, because the qi of Yin and Yang overflows. Nevertheless, the number of operation does not exceed the position of Yin, and the number of intersection does not exceed the position of Shen. (Another view holds that Li corresponds to Mao and Kan to You.)

Qian has forty-eight numbers, of which one quarter is overcome by yin. Kun also has forty-eight numbers, of which one quarter is used to restrain yang. Therefore Qian obtains thirty-six and Kun obtains twelve. Yang governs advance, so the number of advance is three hundred and sixty days. Yin governs decline, so among the twelve months of the year, the yin number removes twelve days.

The forward counting runs: Qian one, Dui two, Li three, Zhen four, Xun five, Kan six, Gen seven, Kun eight. The reverse counting runs: Zhen one, Li and Dui two, Qian three, Xun, Kan and Gen four, Kun six. The numbers assigned to each hexagram are: Qian forty-eight, Dui thirty, Li twenty-four, Zhen ten, Xun forty, Kan thirty-six, Gen twenty, and Kun twelve. Another set of numbers gives Qian thirty-six and Kun twelve, with Li, Dui, and Xun each twenty-eight, and Kan, Gen, and Zhen each twenty. (The original text carries a collation note here: the order and the numbers may contain errors.)

Circular numbers take one as their base, and square numbers take two as their base. This embodies the distinction between odd and even numbers: six may be reduced to one, and twelve may be reduced to two.

Heaven is round and earth is square. The numbers of the round accumulate from one to six, and the numbers of the square also begin from one and accumulate to eight. After transformation, the numbers of the round begin from four and accumulate to twelve. The number six generally changes in the order of six, the number eight in the order of eight, and the number twelve also in the order of eight. This is the natural Dao (the Way).

Eight is the number of the forms of heaven and earth, six is the number of heaven's function, and twelve is the number of earth's function. Heaven turns the square into the round yet always retains the one within it. Earth divides one into four yet always keeps its square form. Heaven changes form without changing function, and earth changes function without changing form.

Six plus one becomes seven, and twelve plus four becomes sixteen. Yang governs advance, so heaven adds one to make seven. Yin governs retreat, so earth subtracts four and stops at twelve. This means that yang always retains the one, while yin constantly conceals it. Therefore the number of heaven and earth's forms ends at eight, the number of heaven's function reaches seven, and the number of earth's function stops at twelve.

The round works by paring away the corners of the square. One becomes four, and four minus one is three; three becomes nine, and nine minus three is six. The square takes the round into itself as its form. One becomes three, and three joined with one is four; four becomes twelve, and twelve joined with four is sixteen. Therefore the number of function is formed from three and its ultimate number is six, while the number of form is formed from four and its ultimate number is sixteen. Thus the diameter of the round is one and its circumference is three, and its numbers begin at one and accumulate to six. The square divides one into four, and then divides four into sixteen. These are all the natural Dao.

One applies two to produce three, and three minus one is two. Three produces nine, and nine minus one is eight, and minus three is six. Therefore one applies three, and three in turn applies two. Three applies nine, and nine in turn resolves into eight and six. From this, two produces four, eight produces sixteen, and six produces twelve. Three plus one is four, nine plus three is twelve, and twelve plus four is sixteen. Thus four takes one as its root and three as its function, twelve takes three as its root and nine as its function, and sixteen takes four as its root and twelve as its function. (This point may be considered further.)

Yang is honored, and spirit (shen) is also honored. Spirit can command the myriad things and can also conceal and gather its functions. Therefore the Dao generates heaven……

Heaven, earth, and the myriad things are born by relying on the Dao, yet they cannot manifest the Dao itself. The myriad things also all take the Dao as their model.

Yang is the function of the Dao, and Yin is its substance. The function of Yang takes Yin as its root, and the function of Yin in turn relies on Yang. If Yang is taken as the function, then Yin is honored; if Yin is taken as the function, then Yang is honored. Yin is close to the Dao, so Yin is used to figure the Dao.

Six transformations can be extended to thirty-six kinds; eight transformations form sixty-four; twelve transformations form three hundred eighty-four. Changing by sixes (six times six) extends to three hundred eighty-four kinds; changing by eights (eight times eight) yields sixty-four; changing by sevens (seven times seven, forty-nine) also extends to three hundred eighty-four.

The circle takes six as the basis of its transformations. Advancing by sixes gives sixty variations, which correspond to the three hundred and sixty. The square takes eight as its basis, so eight times eight gives sixty-four. Yang governs advance, so the advancing number is carried up to sixty.

The circle symbolizes the stars of heaven, and the methods of calendar calculation may have arisen from it. The square symbolizes the earth, and the methods of dividing the regions and laying out the fields may likewise have been modeled on it. The circle corresponds to the numbers of the Hetu (River Chart), and the square corresponds to the writings of the Luoshu (Luo River Writing). Therefore Fuxi and King Wen composed the Yi (Book of Changes) on this basis, and Great Yu and Jizi arranged the Hongfan (Great Plan) accordingly.

In yarrow divination (shi), the count uses seven rather than six. This is because the remainder is also counted in. Removing the remainder leaves six, so the yarrow-stalk number is thirty-six. Fifty stalks correspond to the sixty-four hexagrams and also symbolize the counting of intercalary years. Using forty-nine stalks corresponds to the sixty-four hexagrams and symbolizes the counting of a single year. In divination, the remainders are merged, and the resulting hexagram is like the intercalary month within a year. Why is the number of hexagrams reduced by four? Because heaven changes and earth follows it. So the yarrow number is reduced by one, and the hexagram number is reduced by four.

Here the circumference and diameter of circles and squares are used to explain the numbers. A circle with a diameter of one has a circumference of about three. Doubling this relationship of the circle gives six. A square with a diameter of one has a perimeter of four. Doubling it gives eight.

Cutting a square and transforming it into a circle symbolizes the motion of the celestial bodies. Dividing the large into smaller parts symbolizes the earth giving birth to the myriad things. Therefore heaven uses the transformations of six, and earth uses the transformations of four.

One times eight is nine, which trimmed becomes seven. Eight trimmed becomes six. Sixteen trimmed becomes twelve. Twenty-four trimmed becomes eighteen. Thirty-two trimmed becomes twenty-four. Forty trimmed becomes thirty. Forty-eight trimmed becomes thirty-six. Fifty-six trimmed becomes forty-two. Sixty-four trimmed becomes forty-eight. The final sentence, "bā yī fēn wéi sì" (eight-one divided into four), is corrupted in its wording, and its meaning cannot be determined.

The number of hexagrams can be carried further: thirty-two plus sixteen makes forty-eight, thirty-two plus thirty-two makes sixty-four, and adding another thirty-two gives ninety-six. Taken together, the hexagrams may also be reckoned as 384 lines (yao).

Beginning from One (yi), one generates six, six generates twelve, twelve generates eighteen, eighteen generates twenty-four, twenty-four generates thirty, and thirty generates thirty-six. Extending this process through six further transformations yields 360. This is the number of the movement and transformation of the myriad things.

A second sequence of generation runs as follows: four generates twelve, twelve generates twenty, twenty generates twenty-eight, and twenty-eight generates thirty-six. This is the number of the generation of the myriad things. Accordingly, the yang counting-number (yang ce) of Qian is thirty-six; those of Dui, Li, and Xun are twenty-eight; those of Zhen, Kan, and Gen are twenty; and that of Kun is twelve.

Each change in the circular numbers adds six. If the number of changes is instead subtracted, the sequence is five, ten, fifteen, twenty, twenty-five, and thirty. In other words, when the numbers are retained they are added in sixes, and when they are subtracted they are added in fives. Five minus one leaves four, four minus one leaves three, and two minus one leaves one. Thus one plus one makes two and two minus one leaves one; two plus one makes three and three minus one leaves two; three plus one makes four and four minus one leaves three; four plus one makes five and five minus one leaves four. Therefore two takes one as its root, three takes two, four takes three, five takes four, and six takes five. (The reasoning here requires further consideration.)

Each change in the square numbers becomes four. Four generates eight, and adding four and eight gives twelve. Eight generates twelve, and adding eight and twelve gives twenty. Twelve generates sixteen, and adding twelve and sixteen gives twenty-eight. Sixteen generates twenty, and adding sixteen and twenty gives thirty-six. Put another way, one plus three makes four, twelve plus twenty makes thirty-two, and twenty-eight plus thirty-six makes sixty-four. (The reasoning here requires further consideration.)

How were the numbers of the "Great Expansion" (dayan) in the Yijing (Classic of Changes) obtained? This was the method the sages used to derive the number of hexagrams. Heaven's number is twenty-five, which when summed gives fifty; earth's number is thirty, which when summed gives sixty. Therefore it is said...

The five positions work together, each with its own corresponding relationships. Fifty is the total number of yarrow stalks used in divination, while sixty is the number of hexagrams. Five is the lesser derivation of the yarrow number, and fifty is the greater derivation; eight is the lesser completion of the hexagrams, and sixty-four is the greater completion.

The yarrow number takes the image of the round to model the numbers of Heaven, so seven times seven gives forty-nine. When fifty is mentioned, the one stalk set aside is included in the count. The hexagram number takes the image of the square to model the numbers of Earth, so eight times eight gives sixty-four. When sixty is mentioned, four are subtracted from sixty-four. The yarrow stalks are the number of application, while the hexagrams are the number of form. Since the number of application is grounded in the number of form, one is left aside; since the number of form takes the number of application as its basis, four are subtracted. The circle has one as its basis, and the square has four; hence one is left aside from the yarrow number, and four are subtracted from the hexagram number.

The number used in yarrow divination is seven. Even when there is a remainder, it carries the meaning of leaving one aside. “Setting one aside”—removing one stalk from the yarrow bundle to symbolize the Three Powers of Heaven, Earth, and Humanity—also carries the meaning of subtracting one. In yarrow divination, one stalk is set aside to symbolize the Three Powers; the remaining forty-eight stalks, the units used for counting yarrow, make up the stalk count of one hexagram: four times twelve is forty-eight.

Subtract three from twelve, leaving nine. Four times three gives twelve stalks set aside; four times nine gives thirty-six stalks used, corresponding to the thirty-six yang lines of the Qian hexagram. Subtract five from twelve, leaving seven. Four times five gives twenty stalks set aside; four times seven gives twenty-eight stalks used, corresponding to the twenty-eight yang lines of the Dui and Li hexagrams. Subtract six from twelve, leaving six. Four times six gives twenty-four stalks set aside; four times six also gives twenty-four stalks used, corresponding to half of the Kun hexagram—that is, its twenty-four yin lines. Subtract four from twelve, leaving eight. Four times four gives sixteen stalks set aside; four times eight gives thirty-two stalks used, corresponding to the twenty-four lines of the Gen and Kan hexagrams, plus the eight yin lines of the upper trigram, for a total of thirty-four lines.

Thus, seven and nine belong to Yang, while six and eight belong to Yin. Nine is the ultimate number of Yang, and six is the ultimate number of Yin. When a number reaches its limit, it changes; these are therefore the numbers through which hexagrams change. Zhen and Xun have no stalk counts because they correspond to numbers that are not used. Heaven takes strength and vigor as its De (Virtue/Power), so softness does not manifest; Earth takes softness and compliance as its form, so hardness does not arise. Thus, Zhen and Xun are not used. The Qian hexagram uses nine, so its stalk count is nine. Multiplying by four makes it correspond to the four seasons; each season…

A period lasts ninety days. The Kun hexagram uses “six,” so the number of stalks corresponding to it is also taken as six.

There are four odd numbers: one, two, three, and four. There are also four stalk numbers: six, seven, eight, and nine. Together, the two groups make eight numbers, corresponding to the eight variations of the square numbers. The numbers obtained by combining the remainder with the hexagram fall into six patterns: five combined with four, and four combined with four; nine combined with eight; five combined with four and eight; nine combined with four and eight; five combined with eight and eight; and nine combined with four and four. These correspond to the six variations of the circular numbers.

The range of odd numbers ends at four; five is not used. The range of stalk numbers ends at nine; ten is not used. Five corresponds to one, and ten to two. Thus five and ten are set aside, and only forty-nine is used. Five is excluded among the odd numbers, and ten among the stalk numbers, because they represent the extreme conditions of “being” and “nonbeing,” serving as analogies for the operation of natural numerical principles.

There are sixty-four hexagrams, but only sixty are actually used. Why? The sixty hexagrams have three hundred and sixty lines, corresponding exactly to the three hundred and sixty days of the year. The sexagenary cycle used to mark the days also ends at sixty. After six cycles of the sixty stems and branches, the cycle of the Way of Heaven begins again. The numbers of stalks therefore correspond in the same way: thirty-six plus twenty-four equals sixty, and thirty-two plus twenty-eight also equals sixty.

The Qian hexagram has forty-eight, Kun has twelve, Zhen has twenty, Xun has forty, Li and Dui each have thirty-two, and Kan and Gen each have twenty-eight. Together they total sixty. The number of yarrow stalks is complete, so it belongs to the Yang stalks. Thirty-six plus twenty-eight equals sixty-four. Since four numbers are removed from the hexagram count, among the Yin stalks twenty-four plus thirty-two equals fifty-six.

By derivation, nine yields thirty-six, all Yang numbers, and so is Yang within Yang. Seven yields twenty-eight, first Yang and then Yin, and so is Yin within Yang. Six yields twenty-four, all Yin numbers, and so is Yin within Yin. Eight yields thirty-two, first Yin and then Yang, and so is Yang within Yin.

The stalk numbers advance by fours and can be derived to one hundred; the hexagram numbers also advance by fours and can be derived to one hundred and twenty. One hundred is counted in tens, and one hundred and twenty in twelves.

For the numbers obtained by combining the remainder with the hexagram: if the combination is five with four and four, the stalk number is forty-nine; if it is nine with eight, the stalk number is forty-six; if it is five with eight and eight, or nine with four and eight, the stalk number is forty-seven; if it is nine with four and four, or five…

When paired with four and eight, the stalk numbers are all forty-eight. Taking nine is one change, used to correspond to Qian; taking six is also one change, used to correspond to Kun; taking seven is two changes, used to correspond to Dui and Li; taking eight is also two changes, used to correspond to Gen and Kan.

Five paired with four and four, after subtracting one for the hexagram, gives four eights, thirty-two; nine paired with eight and eight, after subtracting one for the hexagram, gives four sixes, twenty-four; five paired with eight and eight, and nine paired with four and eight, after subtracting one for the hexagram, give four fives, twenty; nine paired with four and four, and five paired with four and eight, after subtracting one for the hexagram, give four fours, sixteen. Therefore, removing the numbers three, four, five, and six yields the stalk numbers nine, eight, seven, and six.

The Heaven numbers are 1, 3, 5, 7, and 9; the Earth numbers are 2, 4, 6, 8, and 10. Interweaving these numbers and calculating by reference to the changes of the Heaven and Earth numbers is like Heaven and Earth joining and day and night alternating.

One is the beginning of number, yet it cannot yet be counted as a concrete number. Thus two times two gives four, three times three gives nine, four times four gives sixteen, five times five gives twenty-five, six times six gives thirty-six, seven times seven gives forty-nine, eight times eight gives sixty-four, and nine times nine gives eighty-one. One cannot change further; a hundred returns to ten, and ten again returns to one, and neither can change further. Therefore, in calculation one is discarded; the range of change reaches at most nine, and what is used are all numbers of change.

Five times five gives twenty-five, which is a Heaven number; six times six gives thirty-six, which is the stalk number of the Qian hexagram; seven times seven gives forty-nine, which is the operative number of the Great Expansion; eight times eight gives sixty-four, which is the number of hexagrams; nine times nine gives eighty-one, which is the number of the Xuanfan.

The number of the Great Expansion is probably the root of calculation. Therefore, the starting points of calculation are none other than the forms of square, circle, curve, and straight.

Among Yin numbers there is no one; among Yang numbers there is no ten.

Multiplication increases numbers, while division decreases numbers. Although there are many methods of calculation, none go beyond these two.

Yang receives Yin and is generated; Yin receives Yang and takes form. Therefore, yarrow-stalk numbers take four as a unit and form nine; hexagram numbers take four as a unit and form ten. This is like the interweaving of the Heavenly Stems and Earthly Branches: the Heavenly Stems end a cycle at six, and the Earthly Branches end a cycle at five.

Three times four gives twelve, and two times six also gives twelve; twelve times two gives twenty-four, three times eight gives twenty-four, and four times six also gives twenty-four.

Twelve times thirty-six equals four hundred thirty-two; nine times thirty-six equals three hundred twenty-four; and six times sixty equals three hundred sixty. Four times twelve equals forty-eight; three times sixteen also equals forty-eight; and six times eight equals forty-eight. Five times twelve equals sixty; three times twenty also equals sixty; and six times ten equals sixty. These numbers correspond with one another.

Four times nine is thirty-six, and six times six is also thirty-six. Six is a Yang number, and when it is combined with half of the Yin number six, the result is nine. As for six, Yin and Yang each have three. In terms of the three lines of a trigram, Heaven, Earth, and humanity also each have three. Yin and Yang each contain Heaven, Earth, and humanity, and Heaven, Earth, and humanity each contain Yin and Yang. Thus, the number of Heaven is taken as three and the number of Earth as two, and numbers are derived from these.

After the Supreme Ultimate (Taiji) differentiates, the two forms of Yin and Yang are established. Yang Qi descends and joins with Yin Qi, while Yin Qi rises and joins with Yang Qi; thus the Four Images come into being. The union of Yang and Yin, and of firmness and gentleness, respectively produce the four images of Heaven and the four images of Earth, from which the Eight Trigrams (Bagua) are formed. Only when the Eight Trigrams interweave do the myriad things arise.

Thus, one divides into two, two into four, four into eight, eight into sixteen, sixteen into thirty-two, and thirty-two into sixty-four. This is what is meant by distinguishing Yin and Yang and alternating the use of gentleness and firmness: the positions of the six lines change, forming the ordered patterns of the hexagrams. Numbers continue to unfold: ten becomes a hundred, a hundred becomes a thousand, and a thousand becomes ten thousand. This is like a tree growing a trunk from its root, branches from its trunk, and leaves from its branches: the larger the divisions, the fewer the branches; the finer the divisions, the more numerous the branches. Gathered together, all is one; unfolded, it becomes the myriad things. Thus, Qian (Heaven, creative activity) differentiates things, while Kun (Earth, receptive support) brings them together. Zhen makes things grow, and Xun makes them decline. Growth leads to differentiation, differentiation leads to decline, and after decline comes gathering.

Qian and Kun establish the directions; Zhen and Xun undergo one union; Dui, Li, Kan, and Gen undergo two unions. Thus, in the Zhen trigram, Yang is scarce while Yin remains predominant; in the Xun trigram, Yin is scarce while Yang remains predominant. The Yang in the Dui and Li trigrams gradually increases, while the Yin in the Kan and Gen trigrams gradually increases. Hence Chen and fire do not appear.

When the single Qi differentiates, Yin and Yang are thereby distinguished. Where Yang Qi predominates, Heaven is formed; where Yin Qi predominates, Earth is formed. Thus, when Yin and Yang are balanced, form and substance become complete; when one outweighs the other, dispositions differ. Once form and substance have each differentiated, the side with more Yang Qi becomes firm...

Yin represents softness, from which nature and emotion also differentiate. The hexagram with the greatest abundance of Yang Qi is one in which Yin Qi is stronger within Yang; the one with the greatest abundance of Yin Qi is one in which Yang Qi is stronger within Yin. In the hexagrams Dui, Li, and Xun, Yang lines predominate; in Gen, Kan, and Zhen, Yin lines predominate. These hexagrams therefore carry out the workings of Heaven and Earth. Qian is pure Yang, and Kun pure Yin, so neither directly takes part in these workings.

Qian gives one of its four parts to Kun, and Kun likewise offers one of its four parts to Qian. When Qian and Kun unite, they generate the Six Children—the six hexagrams formed through the transformations of Qian and Kun. The three male hexagrams belong to Yang, and the three female hexagrams to Yin. Dui gives one Yang line to Gen, and Kan gives one Yin line to Li; Zhen and Xun exchange two lines. Taken together, Yin and Yang each make up half. Thus water and fire can both generate and constrain one another, and all things take shape.

The names and positions of Qian and Kun cannot change. The names of Kan and Li may be exchanged, but their positions cannot; the positions of Zhen and Xun may be exchanged, but their names cannot; both the names and positions of Dui and Gen may be exchanged. The image of Li resembles Qian, and Kan resembles Kun; Zhongfu resembles Qian, Yi resembles Li, Xiaoguo resembles Kun, and Daguo resembles Kan. Therefore, the hexagrams Qian, Kun, Li, Kan, Zhongfu, Yi, Daguo, and Xiaoguo cannot change.

In the celestial image, Li corresponds to night, so Yin is contained within Yang. In the terrestrial image, Kan corresponds to daytime, so Yang is contained within Yin. Zhen represents Yin Qi beginning to enter into union, from which Yang Qi emerges; Xun represents Yang Qi beginning to wane, from which Yin Qi emerges. Dui represents the growth of Yang Qi, and Gen the growth of Yin Qi. Zhen and Dui are Yin within Heaven, while Xun and Gen are Yang within Earth. Thus, the images of Zhen and Dui have Yin above and Yang below, whereas Xun and Gen have Yang above and Yin below.

When speaking of Heaven from the perspective of the beginning of all things, Yin is above and Yang below, symbolizing the union of Yin and Yang and their free, harmonious flow. When speaking of Earth from the perspective of things already formed, Yang is above and Yin below, symbolizing high and low each occupying its proper place.

Qian and Kun establish the positions above and below; Li and Kan stand at the gates on the left and right, symbolizing Heaven and Earth opening and closing, and the rising and setting of the sun and moon. From this come spring, summer, autumn, and winter; the dark and new moons, the first and last quarters, and the full moon; the changing lengths of day and night; and the sun and moon’s differing speeds, fullness, and decline.

Before distinct forms appear, Yang is contained within Yin; once forms have emerged, Yang Qi differentiates into Yin. Yin is the mother of Yang, and Yang the father of Yin. The mother conceives the eldest son, forming the Fu hexagram; the father begets the eldest daughter, forming the Gou hexagram. Thus, Yang begins to arise from Fu, and Yin begins to arise from Gou.

Without form, nature cannot become substance; without nature, substance cannot generate life. Yang takes Yin as its substance, while Yin takes Yang as its nature. The active aspect is nature; the still aspect is substance. In Heaven, Yang moves and Yin is still; on Earth, Yang is still and Yin moves. Nature depends on substance to become settled and still, while substance moves in accordance with nature. Thus Yang unfolds slowly, while Yin transforms quickly. (This point will be explained in greater detail below.)

Yang cannot stand on its own and must be supported by Yin, so Yang takes Yin as its foundation. Yin cannot manifest on its own and must await Yang to set it in motion, so Yin takes Yang as its guide. Yang initiates things and enjoys the fruits of their completion; Yin acts according to Yang’s laws and bears the final labor.

Yang can know, but Yin cannot; Yang can manifest, but Yin cannot. Whatever can know and manifest belongs to “being,” so Yang’s nature is being, while Yin’s nature is nonbeing. Yang reaches only a limited number of places, whereas Yin is everywhere. Yang departs, while Yin remains. What is everywhere and endures is what is real; therefore, Yang is empty in substance, while Yin is substantial.

Movement from below to above is called ascent; movement from above to below is called descent. Ascent means growth, and descent means decline. Thus Yang arises from below, while Yin arises from above. The generation of all things always follows a pattern in which opposites give rise to one another: Yin gives rise to Yang, and Yang in turn gives rise to Yin; Yin then gives rise to Yang, and Yang to Yin again, in an unending cycle.

The foundation of Heaven and Earth probably begins at the center. Thus, Qian and Kun, the trigrams symbolizing Heaven and Earth, undergo repeated changes yet never depart from the center. Human beings dwell between Heaven and Earth, and the heart occupies the center of the human being. The sun is at its fullest at noon, and the moon is at its fullest at midmonth; therefore, the noble person values the center.

At its root there is only one Qi: as it flourishes, it becomes Yang; as it recedes, it becomes Yin. Yin and Yang, then, are ultimately one. The six lines of a hexagram can be reduced to three, and the Eight Trigrams to four. Thus, when speaking of Heaven, one does not speak of Earth; when speaking of the ruler, one does not speak of the minister; when speaking of the father, one does not speak of the son; and when speaking of the husband, one does not speak of the wife. Yet with Earth, Heaven can bring forth the growth of all things; with ministers, the ruler can put the myriad transformations into effect; and with sons and wives, fathers and husbands can establish the order of the family. Thus, where there is one, there is two; where there is two, there is four; where there are three, there are six; and where there are four, there are eight.

When Yin and Yang divide, they form the Two Modes, Heaven and Earth, the two basic forms of Yin and Yang. When the Two Modes interact, they form the Four Images: Lesser Yang, Greater Yang, Lesser Yin, and Greater Yin. The Four Images interact to form the Eight Trigrams; when the Eight Trigrams combine with one another, they give rise to all things. Thus, the Two Modes generate Heaven and Earth.

Things are classified according to the “Four Images”—Yin, Yang, and their changing forms. The Four Images establish the forms of Heaven and Earth, and in turn give rise to the Eight Trigrams, which use Yin-Yang symbols to represent the patterns of change among all things. The Eight Trigrams establish the forms of the sun and moon and give rise to the classes of all things. By overlapping the trigrams to form the sixty-four hexagrams, we can explain the forms of all things.

“Class” refers to the order in which things come into being; “form” refers to the shape produced when different images combine. To investigate the class of a thing, one must begin with how it came into being. To observe its form, one must look to the image it displays. Tracing the process of generation allows us to infer what has not yet occurred; observing an image that has already taken shape reveals its patterns through what has already come to pass. Thus, the sun and moon belong to the same class: both arise from the same source, yet occupy different positions. Though their positions differ, they display the same image. If we extend this reasoning, what change among all things could escape this pattern?

When the seasons of Heaven change, the myriad things on Earth respond. As the seasons turn, Yin shifts into Yang; as things change, Yang shifts into Yin. The sequence of the seasons can therefore be traced backward to make predictions, while the myriad things must take shape gradually, following their processes. Thus Yang Qi moves first, and Yin Qi follows; Yin Qi moves in reverse, while Yang Qi moves forward. In terms of form, Heaven differentiates into Earth, and Earth differentiates into the myriad things, but the Dao cannot be divided. At the end, the myriad things return to Earth, Earth returns to Heaven, and Heaven returns to the Dao. This is why the noble person values the Dao.

Whenever there is change, there must be a corresponding response. A change within elicits a reaction outside; a change outside elicits a reaction within. Change below brings a response above, and change above brings a response below. When Heaven changes, the sun responds. Thus, the source of change lies in Heaven, while the sun provides the measure of response. The sun marks the movements of the stars, and the moon joins them according to their sequence. Water arises from earth, and fire is concealed in stone; birds roost in trees, and beasts depend on plants; the heart is connected to the lungs, and the liver is linked to the gallbladder. These all illustrate how things resonate with and respond to one another.

Things that originate in Heaven are drawn toward what is above; those that originate in Earth are drawn toward what is below. Thus, change and response generally proceed in opposite directions.

When Yang and Yin unite, they produce animals with hooves or horns; when hardness and softness unite, they produce the roots of plants and trees. When Yin and Yang unite, they produce animals with feathers or wings; when softness and hardness unite, they produce the stems and branches of plants and trees. Heaven and Earth unite, and Earth also unites with Heaven. Thus, some animals have feathers yet walk, while others have feet yet leap through the air; among grasses there is wood, and within wood there is grass. By tracing things according to their classes, we find that living things fall within set numerical patterns. Beasts are suited to move on the ground, and birds are suited to move through the air: each follows the characteristics of its own class.

Forms corresponding to creatures on land can also be found in water, just as a shadow corresponds to the object that casts it. Creatures on land are often adept at walking, while creatures in water are often adept at darting about; this results from the interaction between different environments. Thus, creatures that are large on land are often small in water, while creatures that are large in water are often small on land.

The hair of tigers and leopards, like grasses and plants, belongs to the category of “grass.” The feathers of birds such as eagles and hawks, like trees, belong to “wood.”

Trees are a subcategory of the stars, so fruit is used to symbolize them.

Leaves belong to Yin, while flowers and fruit belong to Yang. Branches and leaves are soft; roots and trunks are hard.

Human bones are large, and the body has many parts. Trees have thick trunks and abundant branches and leaves. All these correspond to the numerical patterns of Heaven and Earth.

Animals that move have bodies that extend horizontally; plants that grow upright extend vertically. Humans are naturally suited to moving horizontally, yet walk upright instead.

Animals that fly have wings, while those that run have toes. A person’s hands resemble wings, and their feet resemble toes.

Flying animals eat trees, while running animals eat grass. Humans combine both ways of life and also eat birds and beasts, making them the most honored of all things.

The parts of the body must join in pairs to come into being: Yang joining with firmness produces the heart and lungs; Yang joining with softness produces the liver and gallbladder; softness joining with Yin produces the kidneys and bladder; firmness joining with Yin produces the spleen and stomach. The heart gives rise to the eyes, the gallbladder to the ears, the spleen to the nose, the kidneys to the mouth, the lungs to the bones, the liver to the flesh, the stomach to the marrow, and the bladder to the blood. Thus, the Qian trigram corresponds to the heart, Dui to the spleen, Li to the gallbladder, Zhen to the kidneys, Kun to the blood, Gen to the flesh, Kan to the marrow, and Xun to the bones. The Tai hexagram corresponds to the eyes, Zhongfu to the nose, Jiji to the ears, Yi to the mouth, Dajia to the lungs, Weiji to the stomach, Xiaoguo to the liver, and Pi to the bladder.

Heaven and Earth have eight kinds of images, while humans have sixteen. Why? Humans are formed from the union of Heaven and Earth and born from the joining of their parents, so they have sixteen kinds of images.

Why are the heart above the lungs and the gallbladder above the liver? When discussing human nature, one must trace it back to Heaven; when discussing the human body, one must relate it to Earth. Heaven is concealed within Earth, just as fire is concealed within stone. The heart and gallbladder symbolize this relationship. Why do the heart and gallbladder seem to hang upside down? Grasses and trees follow the Earth’s natural form, while humans grow in the opposite direction, so they appear inverted. Why do the mouth and eyes grow horizontally, while the nose grows vertically? The parts of the body must intersect: animals that move should naturally grow vertically, yet extend horizontally; plants that grow upright should naturally extend horizontally, yet grow vertically. These are all expressions of intersection.

Heaven has four seasons, Earth has four directions, and humans have four limbs. The patterns of Heaven can therefore be observed in the joints of the fingers, and the patterns of Earth examined in the lines of the palm. The principles of Heaven and Earth are all reflected in the fingers and palm—how could they not be regarded as important?

Shen is governed by the heart, Qi by the kidneys, and the body by the head. When form and Qi join, Shen dwells at the center and directs them. This is the Dao of the Three Powers: Heaven, Earth, and humanity.

The human limbs each have their own channels. Each channel is divided into three sections, and each section into three pulses, corresponding to the numerical principles of the Dao of Heaven.

The heart stores Shen (spirit), the kidneys store Jing (essence), the spleen stores Hun (ethereal soul), and the gallbladder stores Po (corporeal soul). The stomach receives and digests food, sending Qi (vital energy) to the lungs and blood to the liver, then conveying water and grain—that is, food and drink—to the bladder and intestines.

Heaven is round and Earth is square. Heaven is high in the south and low in the north, so from a distance it looks like a sloping canopy spread overhead. The land is low in the southeast and high in the northwest; therefore the southeast has abundant water, while the northwest has many mountains. Heaven covers Earth, and Earth supports Heaven. They contain one another, so Earth exists in Heaven and Heaven exists on Earth.

Heaven revolves in a vast, all-encompassing motion that is difficult to measure. People therefore observe the movements of the Northern Dipper to infer celestial phenomena. The direction in which the Dipper points marks the movement of the heavenly bodies: the Dipper’s bowl points toward the zi position, and its handle toward the yin position. Celestial observations take the yin hour as the start of daytime. The Northern Dipper has seven stars, so the divisions of daytime number no more than seven. (This will be explained in greater detail below.)

The movement of the heavenly bodies produces day and night; the Sun’s movement brings cold and heat. In summer, the Sun’s course lies closer to the Earth; in winter, it lies farther away. This is a change brought about by the meeting of Heaven and Earth. Heaven revolves to the left, while the Sun and Moon move to the right: their paths cross the movement of Heaven.

The Sun is in the east in the morning and in the west in the evening, following the movement of the heavenly bodies. In summer it lies farther north, and in winter farther south, shifting with the changing meeting of Heaven and Earth. With each complete revolution of the heavenly bodies, the Sun’s position passes one lunar lodge; this corresponds to the Sun’s movement. At the Spring Equinox, the Sun meets Heaven and Earth at the you hour; at the Summer Solstice, at the wu hour; at the Autumn Equinox, at the mao hour; and at the Winter Solstice, at the zi hour.

The Sun moves more slowly, so its position appears to advance; the Moon moves more quickly, so its position appears to fall behind. Each time the Sun and Moon meet, the elapsed time gains or loses half a day, giving rise to the intercalary remainder. After the Sun completes a great cycle, it advances by six days; after the Moon completes one, it falls back by six days. This gives rise to the intercalary discrepancy.

When the Sun moves through the yang degrees, its shadow lengthens; when it moves through the yin degrees, its shadow shortens. This is the principle by which host and guest relate. As the Moon moves away from the Sun, its light gradually brightens and its movement slows; as it approaches the Sun, the Moon begins to appear and moves faster. This is the proper relationship between ruler and minister.

As Yang Qi wanes, Yin Qi arises; therefore, when the Sun is below, the Moon rises in the west. When Yin Qi grows strong enough to contend with Yang Qi, the Moon rises in the east at the full moon. Heaven is the father and the Sun the son, so Heaven revolves to the left while the Sun moves to the right. The Sun is the husband and the Moon the wife, so the Sun rises in the east while the Moon appears in the west.

Solar and lunar eclipses result from the paths of the Sun and Moon crossing. When the Sun is directly opposite the Moon, the Moon is obscured and a lunar eclipse occurs; when the Moon obscures the Sun, a solar eclipse occurs. This is like Wood and Fire restraining one another. Thus, the noble person acts through wisdom, while the petty person acts through brute force.

The sun follows the course of the heavens, the moon follows the sun, and the stars appear with the moon. Thus the stars take the moon as their model, the moon takes the sun as its model, and the sun takes the heavens as its model. The heavens are half bright and half dark; the sun is half full and half waning; the moon is half full and half incomplete; and the stars are half in motion and half at rest. These are all manifestations of Yin-Yang change.

Day and night alternate in the sky: the sun appears by day and the moon by night, but each remains unseen for half the time; half the stars appear at night. This reflects differences in rank and status. Since the moon can also be seen by day, it is Yin within Yang; since the stars can be seen at night, they are Yang within Yin.

The heavens are associated with odd numbers, and the earth with even numbers. Therefore, to observe the heavens, one need only look at the stars; to examine the earth, one need only look at its mountains and waters. The stars reveal the form of the celestial body, while mountains and waters reveal the form of the earth. The heavens contain all things, and the earth bears all things, so the forms of both the heavens and the earth closely approach the Dao (the Way).

In the far south, heat reaches its extreme; in the far north, cold reaches its extreme. Thus the south, where things melt, and the north, where things congeal, are both places where life struggles to survive. In summer, the sun moves northward in accordance with the position of the Northern Dipper; in winter, it moves southward in accordance with the Dipper. Only when Heaven and Earth meet can cold and heat be brought into balance; only when cold and heat are balanced can all things grow.

Heaven takes strength and vigor as its De (Virtue/Power), so things that are soft and weak cannot become manifest. Earth takes yielding softness as its nature, so things that are hard and rigid cannot grow there. Thus Zhen and Xun, names of trigrams in the Book of Changes associated with Heaven’s Yang Qi, belong to Heaven’s Yang, while Earth belongs to Yin. When Yang Qi appears, Yin Qi responds. Chen, a designation for times and directions among the Twelve Earthly Branches, is utmost Yin; the sun is utmost Yang. Both are in the heavens, while on Earth there are only water and fire. Things on the earth’s surface all have form and substance: when Yin Qi is concealed within Yang Qi, physical form arises; when Yang Qi is concealed within Yin Qi, innate nature arises.

Thus Yang can give rise to Yin, and Yin can also give rise to Yang; Yang overcomes Yin, and Yin also overcomes Yang. What cannot be concealed within Yang does not appear on Earth; what cannot be subdued within Yin does not appear in the heavens. When little Yang Qi is concealed within something, its form is soft and weak; it therefore fears Yang Qi and is governed by it. When much Yang Qi is concealed within something, its form is strong and firm; it can therefore resist Yang Qi and instead be governed by Yin Qi. Water and fire are thus active and follow Yang, while earth and stone are still and follow Yin.

Yang gives rise to Yin, so water takes shape first; Yin gives rise to Yang, so fire takes shape later. Yin and Yang generate one another, and physical form and innate nature depend on each other. Thus, when Yang Qi departs, Yin Qi withers; when Yin Qi is exhausted, Yang Qi also perishes.

When metal comes into contact with fire, it melts into liquid; when wood meets fire, it burns. This is the response of like to like. Water freezes when it encounters cold and dries up when it encounters fire: this is the influence of what overcomes it.

When Yin is present within Yang, rain forms; when Yang is present within Yin, wind forms. When firmness contains softness, clouds form; when softness contains firmness, thunder forms. Without Yin, rain cannot form; without Yang, wind and thunder cannot arise. Softness belongs to Yin, but Yin cannot stand on its own, so it waits for Yang Qi to arrive before it can rise. Thunder is vigorous and firm, belonging to physical form; physical form cannot act by itself, but must wait for Yang Qi to arrive before it can erupt.

Only when an idea arises in the mind can words be spoken. Once words are spoken, images can appear; once images appear, their underlying patterns and numbers can be sought. When the patterns and numbers are established, the images become clear; when the images are clear, words become useful; and when words serve their purpose, the idea can be expressed. Images and patterns are like the fish basket and rabbit snare; words and ideas are like the fish and rabbit caught in them. Once you have the fish and rabbit, you can forget the basket and snare—that is the right approach. If you rely only on the basket and snare to seek fish and rabbits, you may not catch any.

When the times of Heaven change, people model themselves on them. Thus Origination, Flourishing, Advantage, and Perseverance—the four stages of process and qualities described in the Book of Changes—embody the Changes’ transformations. When people’s actions change, Heaven responds. Thus good fortune, misfortune, regret, and difficulty embody the Changes’ responses. If Origination and Flourishing are viewed as transformations, then Advantage and Perseverance are responses. If good fortune and misfortune are viewed as responses, then regret and difficulty are transformations. Origination brings good fortune, and good fortune brings advantage: this is a response. If Flourishing turns toward misfortune, misfortune is the response. Persevering in what is right allows the possibility of regret and correction, while stubborn attachment leads toward danger. Thus transformations contain responses, and responses contain transformations.

Responses within transformation belong to the Way of Heaven: Origination is transformation, and Flourishing responds to it; Advantage is transformation, and Perseverance responds to it. Transformations within response belong to human affairs: if a transformation leads to danger, a fitting response can turn it toward good fortune; if it leads to difficulty, a response can bring regret and correction. Regret and correction are signs of good fortune, while difficulty is the root of misfortune. Therefore, the noble person follows the Way of Heaven and does not simply comply with human wishes.

Origination corresponds to spring and benevolence. Spring begins the four seasons, and benevolence is foremost among the virtues. At this time, seasonal Qi has not yet grown vigorous, but benevolence is sufficient to nourish all people. For this reason, benevolence is named, rather than the season. Flourishing corresponds to summer and ritual propriety. Summer is the season at its height, and ritual propriety is the refinement of virtue. What reaches its height must decline; refinement cannot prevent this decline. For this reason, the season is named, rather than virtue. Hence the saying, “How great is Qian’s Origination,” followed by an emphasis on “regret.”

Advantage corresponds to autumn and righteousness. Autumn is when all things mature, and righteousness provides the standards for virtue. All things are maturing and receiving their harvest; if righteousness does not extend to what is beneficial, it is difficult to accomplish anything. For this reason, the season is named, rather than virtue. Perseverance corresponds to winter and wisdom. Winter marks the year’s end, and wisdom is the gathering in of virtue. Persevering in what is right brings good fortune; failing to do so brings misfortune. For this reason, virtue is named, rather than the season. Thus Advantage and Perseverance concern the nature and circumstances of things.

King Wen’s creation of the Classic of Changes was truly magnificent! Surely he understood the laws governing the workings of Heaven and Earth. When the Qian and Kun hexagrams join, they form Tai; when Kan and Li join, they form Ji Ji. Qian begins at the Zi hour, and Kun at the Wu hour; Kan ends at the Yin hour, and Li at the Shen hour. This corresponds to the times of Heaven.

Qian is placed in the northwest, while Kun is set back in the southwest. The eldest son takes charge, and the eldest daughter takes the mother’s place. Kan and Li each occupy their proper positions, while Dui and Gen are paired as consorts. This corresponds to the directions of the Earth. The principles on which a ruler governs the state are, in broad terms, all contained here.

Qian and Kun are the foundations of Heaven and Earth; Li and Kan are the workings through which Heaven and Earth operate. Thus the Classic of Changes begins with Qian and Kun, places Li and Kan in its middle, and ends with Ji Ji and Wei Ji. Tai and Pi stand midway through the Upper Classic, while Xian and Heng open the Lower Classic. All of these describe the workings of change in Heaven and Earth.

Kun governs the three female hexagrams in the southwest, while Qian governs the three male hexagrams in the northeast. The Upper Classic begins with “three,” and the Lower Classic ends with “four”; both carry the sense of Yin and Yang joining, communicating, and harmonizing. The Classic of Changes therefore speaks of function: Qian functions through nine, Kun through six, and the number of the Great Expansion is forty-nine. “The hidden dragon must not be used” speaks of the dragon lying concealed, when it must not be put to use. How vast is the principle of function! Through it, I have glimpsed the intent of the sages.

The Dao generates Heaven, and Heaven generates Earth. Once Heaven and Earth have completed their work, they withdraw. Thus a son succeeds his father, and a minister receives the ruler’s abdication, just as Qian steps back by one position.

The union of Qian and Kun forms Tai; after transformation, they become the Miscellaneous Hexagrams. Qian, Kun, Kan, and Li are the hexagrams whose functions are developed in the first section; Dui, Gen, Zhen, and Xun are those whose functions are developed in the second. The four hexagrams Yi, Zhong Fu, Da Guo, and Xiao Guo are the fitting and appropriate parts of the three sections.

The Classic of Changes describes Yin and Yang joining and changing. Zhen and Dui are the hexagrams in which Yin and Yang first join, so they occupy the positions of morning and evening. Li and Kan are the hexagrams in which their union reaches its height, so they occupy the positions of Zi and Wu. Although Xun and Gen do not join directly, Yin and Yang are still intermingled within them; therefore they occupy the secondary positions within the workings. Qian and Kun represent pure Yang and pure Yin, so they occupy positions where no function is exercised.

Qian and Kun are arranged vertically, while the six child hexagrams are arranged horizontally: this is the foundation of the Classic of Changes. Zhen and Dui are arranged horizontally, while the other six hexagrams are arranged vertically: this is its function.

Hexagram images arise from concrete forms, numbers from the nature of things, names from words, and meaning becomes apparent through function. All numbers in the world arise from the principles underlying things; to go against those principles is to fall into the tricks of numerology. People rely on numbers alone in pursuit of such tricks, and so lose sight of the underlying principles.

If one deals with the affairs of the world by following the Dao, then blessings and misfortunes, worries and joys cannot trouble one’s own person.

Heaven’s Yang Qi lies in the south and its Yin Qi in the north; Earth’s Yin Qi lies in the south and its Yang Qi in the north.

The Yang Qi of people in the north is above, and their Yin Qi is below. When the two meet, the Yang Qi descends and the Yin Qi rises.

The principles of Heaven and Earth can be fully understood, but their physical forms cannot be completely grasped through calculation. If we use the huntian method—a way of calculating the movements of celestial bodies by viewing the heavens as a sphere—and reason from their physical forms, can we fully account for the heavens?

Chen is a collective term for the twelve chen. The points where the Sun and Moon meet are called chen; chen also refers to the domain through which celestial bodies move. Among the celestial bodies, there are no concrete substances, only Qi.

By investigating principles deeply, one can enter the wondrous realm of spirit and put what one has learned into practice. Without such investigation, one cannot reach this realm; without reaching it, one cannot put it to practical use.

Methods of governance must be able to respond to change; they must not be confined to fixed practices. Just as the policies of winter cannot simply be applied in spring.

Yang numbers begin with one and extend to ten, as in the Ten Heavenly Stems: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, and Gui. Yin numbers begin with two and extend to twelve, as in the Twelve Earthly Branches—Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, and Hai—and the twelve months.

Yuan, heng, li, and zhen—the four virtues of the Book of Changes—each encompass circumstances of good fortune, misfortune, regret, and distress. Even when one acts in accordance with these virtues, going against the times can still bring danger.

The first and top lines of a hexagram share the same quality, yet the top line’s progress is still inferior to that of the first. The second and fifth lines share the same quality, yet the second line, in a Yin position, is less fittingly centered than the fifth, which occupies a Yang position. The third and fourth lines share the same quality, yet the third, at the top of the lower trigram, is less advantageous than the fourth, which is closer to the fifth.

The Yang Qi of Heaven lies in the south, so the Sun travels in the south. The firm, vigorous Qi of Earth lies in the north, so mountains are mostly found there. This gives rise to a configuration in which the land is higher in the northwest and the heavens are higher in the southeast.

Heavenly spirits reside in the Sun; the human Shen (spirit) resides in the eyes. When a person is awake, the spirit resides in the heart; when asleep, it resides in the kidneys. This serves as an analogy for the day-and-night movement of the primordial Qi in the celestial bodies.

The movement of clouds and the descent of rain, the flash of lightning and the rumble of thunder, all express how things of the same kind respond to and interact with one another.

Breathing out, spitting, sighing, and exhaling, along with wind, rain, clouds, and mist, should likewise be understood as mutually corresponding.

Each thing has its own sequence of Taiji, the Two Modes, the Four Images, and the Eight Trigrams, and each displays patterns of change across past and present.

Clouds contain the qualities of water, fire, earth, and stone; the categories of transformation in other things are similar.

The winter solstice and summer solstice are separated by 180 degrees east to west and 60 degrees north to south.

The Sun’s path during the month of the winter solstice resembles its path on the day of the summer solstice; the Sun’s path during the month of the summer solstice, in turn, resembles its path on the day of the winter solstice. Qian, Kun, Kan, and Li—the four cardinal trigrams—are called the “Four Cardinal Trigrams” because their images show no repeated changes.

The branches and trunks of trees are formed from the Qi of earth and stone, so they are not easily changed. Their leaves and flowers are formed from the Qi of water and fire, so they change and are readily replaced.

What happens naturally is the Way of Heaven; only a sage can investigate and emulate it. Acting or stopping in response to the times may be human conduct, but it also reflects the Way of Heaven.

Innate endowment belongs to Heaven; the form acquired later belongs to Earth.

When the sun sinks below the horizon, it presents the scene of the essences of Yin and Yang in Heaven and Earth joining together.

All things take “four” as their substance and “six” as the number of their transformations, containing both spirit and Qi. The transformations of Qi must follow a cycle of six, hence the number three hundred and sixty.

When anything reaches its utmost point, it can stop at ten-tenths. For example, on the summer solstice, the sun’s course reaches the position of sixty; with dawn and dusk apportioned as well, this amounts to roughly seven-tenths of the whole.

The east is red, the south white, the west yellow, and the north black. These are the colors of the four directions among the five colors. They can be observed at dawn, noon, dusk, and night.

Though the diagram has no words, I speak about it all day and never depart from it, for it contains the principles of Heaven, Earth, and all things.

The winter solstice corresponds to the middle of the Zi hour and is the extreme of Yin Qi; the spring equinox corresponds to the middle of the Mao hour and is the midpoint of Yang Qi; the summer solstice corresponds to the middle of the Wu hour and is the extreme of Yang Qi; the autumn equinox corresponds to the middle of the You hour and is the midpoint of Yin Qi. Of the year’s three hundred and sixty days, one hundred and eighty lie in each half. This is the number of days between the spring and autumn equinoxes.

The shifting boundary between light and shadow in a lamp’s glow symbolizes the phenomena of the sun and moon.

The moon is the sun’s shadow, and emotion is the shadow of innate nature. The heart embodies nature, and the gallbladder embodies emotion; nature belongs to Shen, while emotion belongs to ghosts. Water is the foundation on which fire depends, and fire is the Qi that water manifests. Black is the foundation of white, and cold is the foundation of heat.

The heart may be called Taiji (the Supreme Ultimate); one may also say that the Dao is Taiji.

The body can be divided, but Shen cannot.

Beasts that crouch in the grass have coats like the grass; birds that perch in the woods have feathers like the leaves. Their surroundings make them this way.

Yin’s influence is greater, because Yang is one and Yin is two.

After the winter solstice, Heaven and Earth begin to “exhale”; after the summer solstice, they begin to “inhale.” This is Heaven and Earth’s breathing over the course of a year.