《易图通变(雷思齐)》

The Daodejing, though only a few thousand words long, is a classic held in special reverence by our Daoist tradition. Since ancient times, no one has been able to set it aside. Yet commentaries have multiplied through the generations, with arguments so profound and tangled that readers struggle to grasp the essential meaning and see only a welter of competing views. Master Lei Siqi was deeply learned and able to discern what mattered. He studied the Daodejing with care, sought out its fundamental meaning, and wrote a commentary on it. He brought together the points where Confucianism and Daoism converge, while also examining their differences one by one, making the text’s lines of thought clear and coherent from beginning to end. It is like entering Laozi’s school, sitting beside him, and exchanging teachings.

Those who study this book, if they fully understand Master Lei’s interpretation, recognize “the profound within the profound” (the most deeply mysterious principles), keep their bearings in the search, and pass these teachings on to later generations, will greatly benefit our Daoist tradition.

In the bingxu year of the Zhiyuan era (1286), Jianzhai Zhang Zongyan, hereditary Celestial Master (a Zhang family Celestial Master who inherited the lineage of the Celestial Master tradition), wrote this preface.

Mr. Lei, whose personal name was Siqi and courtesy name was Qixian, was a renowned recluse of Linchuan. After the Song dynasty fell, he lived alone in the mountains. He wrote several dozen volumes, including The Meaning of Changes in the Diagrams and Divination of the Book of Changes, The Original Meaning of Laozi, and The Essential Meaning of Zhuangzi, as well as three volumes of poems in the manner of Tao Qian. He set aside the robes of a Confucian scholar and became a Daoist, devoting himself to religious practice. He was close to Zeng Ziliang, formerly magistrate of Chun’an County, and Wu Cheng, then a Hanlin Academician.

Famous scholars and officials from all parts of the country admired his character and often wrote to him or sent formal letters seeking his friendship. Those who heard him lecture were invariably struck with sudden insight, realizing how limited their previous understanding had been.

The late Yuan Jue, Hanlin Academician and Lecturer, was a gentleman of broad learning and refined character. He praised Mr. Lei’s writings, saying that their citations were apt and thorough, and that they inspired readers to strive forward. They explored the principles of change to the point of exhausting its workings, then used this understanding to illuminate the Great Dao. Yuan Jue also said, “Unless you understand Qixian himself, you cannot understand his aspirations.” He added that among the Confucian masters recognized throughout the realm today, none surpassed Master Wu, and among those renowned for writing books, none surpassed him either. Yuan Jue said that he and Mr. Lei had been entirely in accord when discussing the Laozi. He also praised Mr. Lei’s poetry as profound and finely wrought, vigorous and unusual, with the character of Du Fu and Han Yu. Everyone agreed that Yuan Jue’s assessment was well judged.

Alas, the ancients said that Yu Qing would not have devoted himself to writing had hardship and sorrow not beset him; Sima Qian, whose family had for generations held the office of Grand Astrologer, might not have completed the Records of the Grand Historian had he not suffered misfortune. To grasp Mr. Lei’s aspirations, one need only read his writings. Why explain them to people of the world? Why explain them to people of the world?

I have read Mr. Lei’s writings and imagined what kind of person he was for nearly fifty years. Now I have also obtained twenty volumes of his poetry and prose, brought to the capital by his disciples’ disciples, Fu Xingzhen and Zhou Weihe. I have therefore written this account to preserve the record.

In the third month of the third year of Zhishun, written by Jie Xisi.

In the past, after Emperor Shizu pacified the region south of the Yangtze, he first summoned the thirty-sixth Celestial Master—the thirty-sixth successor in the Zhang Celestial Master lineage—to court. Before long, the Celestial Master was ordered to oversee Daoist affairs. On returning to the mountains, he respectfully invited the Master to serve as a lecturer on Daoist learning and teach the younger generation. I was still young, but eager to learn, and so I became the Master’s student.

The Master once taught me: “Writing is merely one skill for approaching the Great Dao. People study in order to understand this Dao. Without understanding it, one cannot claim to have learned the way of the sages.” From then on, I often examined myself and applied myself ever more diligently to study. When I was about twenty, I entered court service under my former teacher. Later, when the Duke of Kaifu took office at court, I was unable to continue studying with the Master and complete my education.

The Master wrote commentaries on such works as The Meaning of the Changes in Divination through the Changes Diagram, The Original Meaning of the Laozi, and The Essentials of the Zhuangzi, amounting to several dozen volumes, as well as twenty volumes of poetry and prose. In the autumn of the second year of the Zhishun era, the Master’s disciple Fu Xingzhen sent his own student, Zhou Weihe, to the capital with these commentaries to show me. He said, “Although the Master’s writings already have prefaces by Mr. Huang Gongzhen, Mr. Zeng Ziliang, and Mr. Wu Cheng, they have yet to receive a few words from you explaining the Master’s learning. I venture to ask you to write a preface.”

I replied, “Alas, nearly thirty years have passed since the Master’s death. I can no longer see him, so it is enough to be able to see his writings. Yet could the Master’s learning be fully contained in these works? Later readers who study them will see for themselves that it went beyond them. I too am now old, and have long remained at court. I have often thought of gathering the Master’s writings and writing a preface for them, but have never managed to do so. Now that these writings have been compiled, how could I remain silent and fail to write a few words explaining the Master’s learning?” I have therefore written this preface.

The Master’s name was Siqi and his courtesy name was Qixian. Scholars respectfully called him Master Kongshan. He was from Linchuan, Fuzhou. Written by Wu Quanjie on the first day of the sixth month, ren-shen year, third year of the Zhishun era.

Lei Siqi (style name Qixian) said: “The Yellow River Chart is the Eight Trigrams. After the Yellow River Chart appeared, the sages took it as their model. Fuxi looked up to observe the signs in the heavens and down to examine the features of the earth. He took models from himself at close hand and from the myriad things at a distance. In this way, he understood the nature of the spirits and discerned the conditions of the myriad things. He first drew the Eight Trigrams on this basis and composed the Yi. Confucius said that the sages took the Yellow River Chart as their model. Could he have deceived us?

“The numbers of the Yellow River Chart take shape through their arrangement in the Eight Trigrams, interweaving and changing with one another. The principles called ‘taking three from Heaven and two from Earth’—using three and two as the basis for selecting numbers—and ‘interweaving three and five to bring about change’—using three and five in combination to derive transformations—arise naturally. Later generations failed to follow these numerical principles. In fact, the original chart contains only forty numbers, but they forced fifteen numbers into its center and falsely claimed that the numbers of Heaven and Earth add up to fifty-five. They also added numbers beyond the original forty according to their own ideas, trying to make the numbers fit the chart by force. The more they explained, the more confused the matter became, and the debate has continued to this day.

“For many years I have studied this subject with dedication, examining a wide range of interpretations. I have grasped for myself the intent of the ancient sages, and so have written Tongbian (Comprehending Change), to pass on to scholars of the Yi in all regions and in later generations. I hope we will investigate together the truly correct principles it contains. The methods of divination that have been handed down also go against the intent of the ancients, so I have appended the relevant material to the following section. Those who share an interest in ancient principles should understand that I have not deliberately distorted the old teachings.”

In the ninth month of the Gengzi year of the Dade era of the Great Yuan (1300), the Daoist Lei Siqi, Qixian, wrote this preface.

The Yellow River Chart

The original numbers of the Yellow River Chart comprise the four cardinal directions and the four intermediate directions, making forty in all. They are arranged in a circle to form its shape. The number five of Heaven and the number ten of Earth are not actually set out in the chart; they are posited as a basis for working with the forty numbers. Thus, the numbers of Heaven and Earth total fifty-five.

Diagram Showing the Errors in the Claim That the Yellow River Chart Has Forty Numbers

The old diagram was altered: the Yellow River Chart was drawn as a square, and the number five of Heaven was placed in the center, making it impossible to derive and use the numbers. Here I first present the circular diagram to show the errors in the old account. Readers who examine the diagram that follows carefully will naturally see the point.

Diagram of the Yellow River Chart’s Numbers Arranged According to the Principle of Taking Three from Heaven and Two from Earth

The Yellow River Diagram was used as the basis for creating the Classic of Changes (Yijing). How did its numbers arise? Their source is simply the principle of one Yin and one Yang. The way of Yin and Yang in the Yijing likewise uses odd and even numbers to divide and combine with one another, bringing things into being and gradually forming them.

Yang numbers are odd, and Yin numbers are even. They change and interpenetrate, intertwining in a way that remains orderly. At root, these relationships are what the Commentary on the Changes describes as Heaven One and Earth Two, through to Heaven Five and Earth Ten.

The Yijing has the Taiji (Supreme Ultimate). Taiji is the original source at the center; it is also “one.” This central “one” differentiates into the Two Modes. “Mode” refers to things paired in correspondence, so the Two Modes are “two.” The Two Modes in turn give rise to the Four Images: one divides into two, then divides again into four. These four numbers are one, three, two, and four, and are called the “four generating numbers.” The Four Images then give rise to the Eight Trigrams. Together, the numbers six, seven, eight, and nine are called the “four completing numbers.”

The four odd numbers belong to Yang. Though they differ in how they are generated, each occupies a cardinal position in one of the four directions. The four even numbers belong to Yin; they too differ in how they are generated, but are placed at the four corners, in intermediate positions. Cardinal positions give rise to intermediate ones, and intermediate positions in turn help bring about cardinal ones. The odd numbers one and three are Yang’s generating numbers and depend on the even numbers six and eight for completion. The even numbers two and four are Yin’s generating numbers and likewise depend on the odd numbers seven and nine for completion.

From north to east, the numbers advance from one to three, showing the sequence in which Yang Qi arises and grows forward. From eight down to six, as one moves back from east to north, they show the sequence in which Yin Qi takes shape and recedes. This pattern is already contained within the arrangement. From west to south, the numbers advance from two to four, showing Yin Qi arising and growing forward. From nine down to seven, as one moves back from south to west, they show Yang Qi taking shape and receding. This pattern is likewise contained within the arrangement. At the four cardinal directions, Yang numbers (odd numbers) occupy the central positions; at the four intermediate directions, Yin numbers (even numbers) are placed to the sides.

The numbers of Heaven are five, and the numbers of Earth are ten. Both follow the principles of Yin and Yang and odd and even numbers, and both are called “generative numbers”—numbers that express generation and transformation. Why, then, are they not visible at the four directions? Because the Four Images (Young Yang, Great Yang, Young Yin, and Great Yin) have no fifth position, and the Eight Trigrams have no tenth position. Yet the reason the Classic of Changes can encompass all things and bring all affairs to completion is precisely that it uses the two notional numbers, five of Heaven and ten of Earth, to bring actual generative effects into play. When using the Four Images and Eight Trigrams to construct the Yellow River Diagram, one should not blindly follow conventional practice by drawing five empty dots at the center of the diagram and adding ten more, then mistakenly calling them “fifty,” without understanding how these two numbers actually function.

The Kan trigram begins the sequence in the due north. One combines with five to make six, the number of the Qian trigram in the northwest. The Kun trigram is placed in the southwest with two; two combines with five to make seven, the number of the Dui trigram in the due west. The Zhen trigram appears in the east with three; three combines with five to make eight, the number of the Gen trigram in the northeast. The Xun trigram is joined to the southeast with four; four combines with five to make nine, the number of the Li trigram in the due south.

Thus, adding five to an odd number turns it into an even number, while adding five to an even number turns it into an odd number. Adding five to a Yin number makes it a Yang number, and adding five to a Yang number makes it a Yin number. This shows how the numbers of generation produce the numbers of completion, and how the numbers of completion in turn generate new numbers. The numbers of generation are less than five, so they must rely on five to be adjusted and brought into balance. The Yang numbers one and three therefore form centered positions of their own beyond the numbers of completion represented by Qian six and Gen eight. Yin numbers are numerous while Yang numbers stand alone, which is why Kan is called “the Abyss.” “The Abyss” implies stagnation, and so calls for movement; hence the saying, “The Emperor emerges from Zhen”—the vitality of all things is set in motion from the east represented by Zhen.

The numbers of completion are greater than five and cannot attain balance by themselves. The Yin numbers two and four must therefore draw on five from within the Yang numbers of completion, Li nine and Dui seven, to complete themselves. Yang numbers are upright, while Yin numbers are off to the side; hence Li is called “Clinging,” meaning attachment. Movement serves to bring things together, but coming together requires dispersal. Li’s “Clinging” signifies compliance and mutual delight; hence the saying, “Joy is expressed in Dui.” Joy leads to differentiation, while differentiation arises from an earlier coming together.

The progression of Yang numbers begins with Kan, the first of the numbers of generation. It proceeds clockwise from the north, moving eastward to meet Xun’s four. The progression of Yin numbers begins with Kun, the second of the numbers of generation. It proceeds counterclockwise from the southwest, moving eastward to join Zhen’s three. Thus, the threes and fours among the Yin and Yang numbers of generation both advance and eventually meet.

In terms of the seasons, the fourth month corresponds to the Qian month, when all six lines of the Qian trigram are Yang; the Kun month corresponds to a time when all six lines are Yin. The relationship between Xun and Qian is therefore also mentioned here, along with the saying, “The Emperor emerges from Zhen and reaches order in Xun”: all things sprout in the east represented by Zhen, and become orderly in the southeast represented by Xun.

When the yang numbers move in retreat, they begin with the Li trigram. Li alone receives the middle position among the completed numbers. Moving in sequence from the south, it ultimately reaches Dui seven in the west. When the yin numbers move in retreat, they begin with the Gen trigram. In fact, they move in reverse from the northeast, pass through the north, then turn toward the west, finally reaching Qian six. Thus, the completed yin and yang numbers, six and seven, each separate only after moving through their courses.

Consider the seasonal cycle: the tenth lunar month now belongs to the Kun month, when the hexagram of six yin lines reaches its end; the Qian month marks the beginning of the hexagram of six yang lines. Hence the saying, “Qian begins with Xun; its words are spoken at Dui; it contends at Qian.” Yang Qi begins with the Qian trigram, then passes through Xun and Dui, arriving at Qian’s position, where it contends with yin Qi.

The heavenly numbers begin with one. One represents Taiji, the undivided whole of the cosmos, and is also yang’s proper position. When one divides into two, this represents Taiji’s differentiation; two represents yin’s secondary position. One and two add up to three: yang Qi comes first, and three is odd. One and three add up to four: yin Qi follows, and four is even. One and four first respond to each other and combine to make five; two and three then correspond and likewise combine to make five. The two fives joined together make ten. When the five among the heavenly numbers is also included, this is called “combining threes and fives.” In ancient writing, the character for five resembles forms crossing and joining; in both ancient and modern writing, the character for ten is written as ten. These forms express the underlying numerical principles they symbolize. Once the transformations of these principles are complete, the generating numbers become the completed numbers, and the completed numbers in turn give rise to the generating numbers.

Kan one and Xun four add up to five, so Qian six and Li nine add up to fifteen, making twenty altogether. Kun two and Zhen three also add up to five, so Dui seven and Gen eight add up to fifteen, likewise making twenty. Thus, the four generating numbers—one, two, three, and four—add up to ten, and generate the four completed numbers—six, seven, eight, and nine—for a total of thirty. The numbers in the Hetu, the diagram that represents how Heaven and Earth generate things, therefore extend to forty. When the heavenly five and earthly ten are then set aside, the result is the fifty-five numbers of Heaven and Earth.

Two and five among the generating numbers combine to make ten. Although one is Yin and the other Yang, both use a Yang number from the generating numbers to carry a Yin number from the completed numbers. In this way, the North and East are sustained, and the Center is established. Adding two fives together makes thirty, and Yin and Yang then interweave. The Yin numbers among the completed numbers also support the Yang numbers among the generating numbers, sustaining the South and West and setting the Center in motion. “Establishing” means placing actual numbers within empty numbers so that substance can take shape; “moving” means spreading actual numbers and using empty numbers to set them in motion.

One, two, three, and four interweave to form the Four Images—the four forms of old and young Yin and Yang. The four directions interweave through the empty five within them, making ten and forming the basis of the generating numbers. Six, seven, eight, and nine together contain the Eight Trigrams. The actual five within them is then divided into four parts, making twenty and serving as the application of the completed numbers. Two fives combine to make ten; interwoven across different directions and positions, they form the components of twenty. Dividing and recombining the five within twenty then encompasses the basis of the generating numbers. Six, seven, eight, and nine together contain the Eight Trigrams; the actual five within them is divided into four parts, making twenty and serving as the application of the completed numbers. Two fives combine to make ten, forming the components of twenty. Separating and reuniting the five within twenty then encompasses the full operation of Heaven’s Five—the numerical pattern with five as its center. Ten interweaves to form the shape of five, while five extends throughout and gives effect to ten.

Kan one and Li nine add up to ten; Kun two and Gen eight add up to ten; Zhen three and Dui seven add up to ten; Xun four and Qian six add up to ten. Each pair consists of one generating number and one completed number. One must pair with nine, and three with seven: the four odd numbers of the cardinal directions combine to make ten and belong to Yang. Two must pair with eight, and four with six: the four even numbers of the intermediate directions combine to make ten and belong to Yin. Thus each of the four pairs contains a five, making twenty altogether; now each pair independently adds up to ten. Pairing five with ten separates the number of the Center from the actual numbers among the completed numbers; pairing ten with five gathers the empty numbers among the generating numbers back into the Center. In this way, five and ten combine, and ten is in turn divided into fives. Interwoven and set in relation across vertical, horizontal, diagonal, and straight arrangements, the numerical pattern remains fifteen whether the figures are set upright, reversed, or inverted.

When fives are paired to make ten, each pair totals ten. One and nine combine to make ten, corresponding respectively to Kan and Li. Kan one lies in the Yin direction of the North, where Yang energy first begins to arise; it is Yang within Yin. This shows that when Yin energy reaches its extreme, Yang begins to emerge. The Yang number one occupies the initial position; even when the arrangement is reversed or inverted, its numerical pattern remains fifteen.

When two numbers are added to make ten, each pair is complementary. One and nine correspond respectively to the Kan trigram (Water) and the Li trigram (Fire). The number of Kan is one. It occupies the northern position, associated with Yin, yet symbolizes Yang Qi beginning to arise. This means that when Yin Qi reaches its extreme, Yang Qi begins to emerge. The newly arisen Yang Qi dwells alone in the central position. The number of Li is nine. It occupies the southern position, associated with Yang, and symbolizes Yang Qi developing to its fullest. Yet when Yang Qi reaches its extreme, Yin Qi also begins to emerge, lodged within the Yin numbers represented by the Xun and Kun trigrams. Yin Qi does not appear on its own: when the Yang Qi symbolized by nine is at its peak, Yin Qi joins with it and moderates it. The “Shuogua” chapter of the Classic of Changes, which explains the symbolic meanings of the trigrams, says, “Water and fire approach one another without clashing.” This refers to water and fire drawing near and coming into harmony without conflict. It corresponds to the winter and summer solstices: at the winter solstice, Yang Qi within the Yin of Kan begins to arise; at the summer solstice, Yang Qi reaches its height, and Yin Qi begins to emerge within it.

Two and eight add up to ten and correspond to the Kun and Gen trigrams; three and seven add up to ten and correspond to the Zhen and Dui trigrams. Yang Qi rises from below, moving from the north through the east and finally reaching the south. Yin Qi descends from above, moving from the south through the west and finally reaching the north. Yin and Yang advance and retreat in alternation in this way, bringing about harmony. They cross one another along the east–west axis, forming a path of recurring opening and closing. This also corresponds to the alternating periods of blockage and passage, and of the Pi and Tai hexagrams. The Zhen trigram symbolizes spring’s rising vitality, while the Kun trigram symbolizes Yin Qi gradually taking shape. Joy, associated with the Dui trigram, corresponds to speech; the mountain trigram, Gen, and the lake trigram, Dui, communicate with one another. Together, these correspond to the turning points between spring and autumn. Through this cycle of change, the workings of Yin and Yang unfold in winding courses and extend in all directions.

As for four and six, their sum of ten corresponds to the Xun and Qian trigrams and symbolizes the beginning and end of Yin and Yang. Ending and beginning are pivotal to the transformations of Heaven and Earth; both change and passage arise from them. Why is this so? Yang Qi starts from the Qian and Kan trigrams, passes through Gen and Zhen, and moves from back to front, finally joining with Xun in the southeast. Yin Qi starts from Xun and Li, passes through Kun and Dui, and moves from front to back, finally spreading into Qian in the northwest.

The generating numbers of Yang are “one and three,” which add up to four; the Xun trigram also takes this number. The generating numbers of Yin are “two and four,” which add up to six; the Qian trigram also takes this number. Doubling and distributing four yields the eight seasonal divisions. Their arrangement resembles the sequence of the eight basic trigrams, each made up of three lines: it extends and develops the original sequence. Doubling and distributing six yields six Yin and six Yang, corresponding to the twelve months of the year. This follows the arrangement of doubled hexagrams, each formed by stacking two basic trigrams, and likewise develops the original number.

The six of the Qian trigram, though formed by combining two and four, can also be formed by combining one and five. Five, positioned outside, joins with one to make six: five still belongs to the generating numbers, while six is already a completed number. This reveals another principle of generation and completion. Six also combines with the “one” of the Kan trigram, placing it exactly at the center. Thus, in the Qian trigram, old Yin changes into nine.

If the Xun trigram takes only the two generating numbers one and three, with four as the limit, the number of Yang Qi is insufficient to reach the center by itself. It must therefore draw on the five among the nine numbers of the Li trigram to reach the center and stand opposite the Kan trigram. Four and five combine to make nine, reaching the utmost limit of Yang numbers. Once it reaches this extreme, it turns back. Thus Xun and Li take their numbers together, and nine serves as old Yang, which changes into six.

Here, the opposition between Qian and Xun is used to explain changes in number, with the transformations of nine and six understood to arise from it. But the ways Qian and Xun divide and combine, and serve in turn as beginning and end, are not limited to this one account.

When the number of Zhen, three, combines with the number of Xun, four, thunder and wind press and stir against each other. This phenomenon appears in the Hexagram Heng. Xun corresponds to the fourth month, as stated earlier. If the numbers of Dui, seven, and Qian, six, are assigned to the northwest, and Qian corresponds to the tenth month, then the opposition of Qian and Xun, whose numbers add to ten, cannot account for every kind of change.

Qian belongs to Yang, yet the number assigned to it is the Yin number six, and its position is before Kan. Dui belongs to Yin, yet the number assigned to it is the Yang number seven, and its position is after Kun. Thus, in the numbers six and seven of Qian and Dui, five is used in each case to work out the transformations. As Qian’s six undergoes change, one Qian appears before the process of combining and comparing numbers by fives. It lies before Kan, signifying that Qian’s one comes forth from the outside and becomes Kan’s one. As Dui’s seven undergoes change, two Dui appear after that process. They lie after Kun, signifying that Kun’s two enters from within and becomes Dui’s two.

Although Qian and Dui divide from one another through the completed numbers six and seven, Kan and Kun stir one another within them through the generating numbers one and two. Yet Dui is not called “two,” while Kun is; Kan is called “one,” while Qian is not. Perhaps Qian does not call itself “one” because its contained function is not limited by number and encompasses the generative work of five and ten among the Eight Trigrams, forming the complete Taiji (Supreme Ultimate).

In terms of number, this is “Heaven is one, Earth is two,” through which Heaven and Earth establish their respective positions. In terms of Qi, it is the division of Yin and Yang, which interact and stir one another. The movement of the Changes begins from this: the number that advances as Yang Qi arises becomes the number that retreats as Yin Qi takes form; likewise, the number that advances as Yin Qi arises becomes the number that retreats as Yang Qi takes form. Yin and Yang advance and retreat in turn, generating and bringing things to completion. This ceaseless transformation is the Dao (the Way) of one Yin and one Yang, without end. Is it not subtle, wondrous in its changes, and numinous?

The Yi takes three lines as a group. Through division and combination, these form the Eight Trigrams (Bagua); then two groups of three lines are interwoven to form the sixty-four hexagrams. Once drawn, each hexagram has only six lines. Why, then, are the three lines of the Qian hexagram singled out and called “nine”—even described as “nine”?

When the hexagram lines are arranged in a diagram, five is placed at the center. Nine numbers are needed for the fifth number to fall at the center. With five at the center, six becomes the topmost line; this can be understood as the remainder left after the Qi of the hexagram has run its course. The generating numbers end here, and the completing numbers begin. The meanings of nine and six as empty and substantial are also lodged in this arrangement. Kun begins with the completing number six and belongs to the substantial; Qian reaches its limit at the completing number nine and belongs to the empty. Six is where Kun’s number first reaches completion, yet in the diagram it also serves as a vessel for Qian’s nine. Nine is where Qian’s number reaches completion, yet in the hexagram it corresponds to Kun’s six. Six stands at the end of the descending numbers, yet moves forward into the Kan hexagram; nine stands at the end of the ascending numbers, yet is attached backward to the Li hexagram. Thus, six has its own place in the hexagrams, while nine has no independent place and is lodged in six. Nine and six alternate in use: nine belongs to the empty, six to the substantial, and together they encompass both the form and function of the hexagrams.

Qian is old Yang and Kun is old Yin; the two are like parents. Young Yin and young Yang are like their six children, the male and female represented by the six hexagrams. The numbers of young Yin and young Yang are seven and eight. Qian’s fundamental function takes nine, and Kun’s takes six. Considered across all sixty-four hexagrams, seven and eight are present, yet only nine and six are used. This is because the male and female also arise through transformation from the parents, and so they change accordingly. The generation of all things depends on the mother’s body, but the virtue embodied by the father is difficult to point to directly. Thus, Kun’s form can be seen in its six lines; although Qian functions through nine, it is difficult to trace the origin of its form. This shows that Qian’s source is wondrous and formless, and that the transformations presented by the Yi have no fixed form.

The heavenly numbers one and five combine to make six, while the earthly numbers two and four also combine to make six. Five and six occupy the middle, where the numbers of Heaven and Earth’s generation meet. In the method of Na Jia (assigning the Heavenly Stems to the trigrams), Wu and Ji are regarded as the central numbers: Wu is assigned to the middle line of Kan, and Ji to the middle line of Li. Wu and Ji correspond to five and six. Placing them in the middle of the Kan and Li trigrams reveals how five and six occupy the center in relation to each other.

The earthly numbers two and four combine to make six, signifying the beginning of generation in the realm of Earth. In the Kun hexagram, this appears as “dragons fight in the wild”: Yin Qi battles Yang, the force of Qian. Yin seems like Yang, yet cannot become pure Yang. This shows that Yin’s operation depends on Yang, and that when the Yin Way reaches its limit, it is exhausted. The Kun hexagram speaks of “using six” and uses “favorable to enduring constancy” to show that the Yin Way should remain upright and firm, without exceeding its proper bounds.

The heavenly numbers one, five, and nine signify beginning, middle, and end in sequence; nine is the endpoint of generation among the heavenly numbers. The Qian hexagram uses “an overreaching dragon will have cause for regret” to show that an excess of Yang Qi cannot last, and that it should not place itself at the head of all things. The Qian hexagram speaks of “using nine,” and uses “the world is well governed” to show that Yin and Yang each occupy their proper place, thereby bringing stability to the world. This reflects the vigorous operation of the Way of Heaven, which accomplishes its work without relying on Yin Qi.

One and nine combine to make Qian, totaling ten. Ten is implicit beyond the heavenly number nine and cannot directly manifest its function. When the Way of Earth has not yet reached completion, the Way of Heaven must bring it to an end. There is a deeper principle here as well. From this, one can infer the general meaning of nine and six combining to form the great ten.

The numerical pattern set out in the Hetu originally served only to explain the basic trigrams. Here, the discussion extends to the substance and function of the doubled hexagrams, formed by stacking two basic trigrams into a six-line figure. Neither the basic trigrams nor the doubled hexagrams can do without the interplay of one Yin and one Yang. Through the division and combination of odd and even numbers, Yin and Yang give rise to change from beginning to end.

One, two, three, four, and five are all generative numbers: one through four are actual numbers, while five is an assumed number that serves a function. Six, seven, eight, nine, and ten are all completion numbers: six through nine are actual numbers, while ten is an assumed number that serves a function. From this perspective, the Four Images—the four basic forms produced by the transformations of Yin and Yang—have five as their root, while the Eight Trigrams, formed from Yin and Yang lines, have ten as their root. The meaning is clear.

Each of the four directions contains ten numbers, making forty altogether; each of the four corners contains five, making twenty altogether. Twenty consists of four fives, half of which are numbers used in a virtual sense. Forty consists of eight fives, and its actual numbers together make up the full number. Thus the Four Images and the Eight Trigrams have forty numbers altogether. Five establishes the substance of the center and is the root from which the generative numbers arise; ten carries out the function of the center and brings the completion numbers to fulfillment.

Five is distributed among the completion numbers six, seven, eight, and nine, in which Yin numbers are fewer and Yang numbers more numerous. Yin occupies a secondary position, while Yang holds the primary one. Five does not occupy the center, but is lodged in the off-center positions at the four corners. Ten combines with the generative numbers one, two, three, and four, in which Yang numbers are fewer and Yin numbers more numerous. Here too, Yin occupies a secondary position, while Yang holds the primary one. Ten establishes the central position in each case, occupying the primary positions in the four directions. This shows that the primary position is the central position, and the central position is also the primary one.

At first, five establishes the position of the virtual center. Ten, in turn, stands at the center beyond the realm of actual function. Together, they embody the middle course of one Yin and one Yang. Yin and Yang each contain Yin and Yang within themselves; therefore, within substance and function there are further distinctions of substance and function, and of virtuality and actuality. Virtuality and actuality transform into each other in an unbroken cycle, ending and beginning again. Transformation and generation respond to one another, forming a process that never ceases.

Moreover, five is the middle course among the five Yang numbers. Yin and Yang each contain Yin and Yang within themselves; therefore, within substance and function there are further distinctions of substance and function, and of virtuality and actuality. Actuality and virtuality, beginning and ending, transform into each other, ending and beginning again. Transformation and generation work together in an unceasing cycle.

Even multiplying five by five to obtain twenty-five still gives a number of Heaven. Multiplying six by five, five by six, three by ten, or ten by three gives thirty, all numbers of Earth. Add one and nine, two and eight, three and seven, and four and six respectively, and each pair totals ten. Then add the ten formed by combining the actual number with the void number. Pairing these numbers according to five, and then multiplying ten by five, yields the Great Expansion number of fifty.

Yet the “one” that begins divination, and the “five” lodged within it, are both hidden in the Heavenly Way at the source of all things. They lie beyond the scope of fifty and fifteen, and so are not among the numbers actually used in the Great Expansion. Who would have thought that it is precisely this “use of what is not used” that makes the Great Expansion number work? From this we can see that the changing interplay of Yin and Yang is the Dao. By combining and interweaving the numbers of Heaven and Earth, totaling fifty-five, one can bring about change and set the subtle workings of spirits and ghosts in motion. In this way, one can discern the forms and transformations of things in the world and understand the source of their wondrous workings. This is why Fuxi, the legendary sage king of antiquity, took the Yellow River Diagram as his model and composed the Yi.

Confucius praised the Yi, saying that it is profound and endlessly transformative, ultimately reaching the wondrous realm of the spirit. Ah, this is precisely what makes the Yi so wondrous.

End of Volume One of The Comprehensive Changes of the Yi Diagrams.

Textual notes:

1. The original reading “zhe zhong” has been corrected to “zhong zhe” on the basis of the Siku edition.

2. The Siku edition reads “Zhiyuan bingchen” where the text has “Zhiyuan bingxu”; both readings seem to be erroneous. According to the Handbook of Chinese Reign Periods, there was no bingxu or bingchen year during the Zhiyuan era of Emperor Shundi of the Yuan dynasty, though there was a bingxu year during the Zhizheng era.

3. The Siku edition reads “jingqie” where the text has “qie zhi”; the reading may instead be “jingqie zhi.”

4. The Siku edition reads “si wen zong zhu” where the text has “si wen zong.”

5. The Siku edition reads “te” where the text has “dan.”

The second volume of Yitu Tongbian, written by Lei Siqi, a Daoist priest from Linchuan.

The textual note explains: The Siku edition has “te” where the text has “dan.” The original edition also has “qi” where the text has “qi”; the reading here has been corrected according to the Siku edition.

The River Diagram (Hetu), a pattern said to embody the principles governing Heaven and Earth, contains the principles of how they operate. Throughout it, numbers are in flux. At times there is an excess, at times a deficiency; precisely because of these excesses and deficiencies, balance must be restored through moderation. Thus, although the odd and even numbers of Yin-Yang divide and combine, are empty and full, and advance and retreat, they naturally generate one another and attain the Way of harmony.

“Alternating Yin and Yang is called the Dao” means that Yin and Yang separate to form Heaven and Earth, then join above and below to form Qian and Kun, names for Heaven and Earth. Heaven and Earth are arrayed above and below across the four directions, while the center connects those directions. When the Yin and Yang of Qian and Kun combine, they form the four seasons; within the seasons are further divisions into solar terms. These changes take different forms, yet follow the same principle.

Viewed in terms of physical form, Heaven and Earth extend along the north-south axis, with Heaven above and Earth below. Things with material form—Water, Fire, Wood, and Metal—can each be represented in the diagram. Nothing surpasses Heaven and Earth as a model to follow, and their vastness is matched by the diagram.

Viewed in terms of the movement of Qi, Yin and Yang extend along the east-west axis, with Qian belonging to Yang and Kun to Yin. The Qi of the four seasons—winter, summer, spring, and autumn—can likewise be harmonized according to principle. The pattern by which change flows freely is most clearly seen in the four seasons, which correspond to this flow of change.

These two accounts differ in how they are expressed, but share the same underlying principle. The four directions contain a center, which establishes the foundation of the whole; from this, the Four Images and Eight Trigrams are formed. The four seasons of the year, in turn, are arranged according to the number five. Five occupies the center and has its fixed place, and the seasons and eight seasonal divisions proceed accordingly.

When Confucius compiled the “Appended Statements” of the Book of Changes (Yijing), often called the “Great Treatise,” he did not preserve the River Diagram, but specifically retained the numbers of Heaven and Earth. His purpose was to show that once one understands the principles of number, the contents of the River Diagram are already encompassed.

In the Hetu (River Chart), an ancient diagram of number and form, the numbers arise with one from Heaven as their beginning and six from Earth completing them; two from Earth begin and seven from Heaven complete; three from Heaven begin and eight from Earth complete; four from Earth begin and nine from Heaven complete. Then how should we determine the origins of five from Heaven and ten from Earth? To search for them only on the surface of the diagram is like looking for a horse by following a picture: the meaning is hard to grasp. If five is placed in the empty center of the Bagua (Eight Trigrams), it can still accommodate that number; place ten there as well, and there is no room for it. From this we can see that to know only that numbers arise, without understanding why they arise as they do, is to see merely their outward appearance. It is still not to understand how the sages of antiquity established foundations for Heaven and Earth and set standards for the people. For thousands of years, this has been a source of deep regret; the principles faced by the ancients and by people today are no different.

Odd numbers belong to Heaven, and even numbers belong to Earth. What begins to arise from the numbers of Heaven must be completed by the numbers of Earth; what begins to arise from the numbers of Earth must be completed by the numbers of Heaven. To belong to Heaven or Earth means simply the coming and going, advance and retreat of Yin-Yang, and the relation between substance and function, separation and union, among odd and even numbers. The “Shuogua” (“Discussion of the Trigrams”), a section of the Zhouyi that explains the Eight Trigrams, says: “Take three from the numbers of Heaven and two from the numbers of Earth to derive numerical principles; observe the changes of Yin and Yang to establish the trigrams, and bring forth the changes of firmness and yielding to produce the lines.” The “Xici zhuan” (“Commentary on the Appended Statements”), a section of the Zhouyi that explains the principles of change, also says: “By mutually testing and verifying, the transformations unfold; their numbers are applied in interwoven and combined ways. When these changes are brought into connection, the patterns of Heaven and Earth take shape; when their numerical principles are investigated, the symbols of all things under Heaven are determined.” If these were not the most profound changes under Heaven, who could take part in them? The method of the Hetu is what so fully unfolds these subtle workings.

Number begins with one, takes five as its midpoint, and ends with ten. To “take five as the midpoint” means to gather the four off-center positions into the middle; to “take ten as the endpoint” means to bring the eight cardinal positions together and complete them.

One is the starting point of number. Yet as a whole, it is neither in excess nor deficient, and cannot be applied directly. It therefore becomes manifest only when it reaches the middle position. Thus, one is lodged in five, and then lodged in ten, thereby completing the endpoint of number. Accordingly, the Four Images—the four basic forms produced by the transformations of Yin and Yang—are numbered by five, while the Eight Trigrams are numbered by ten. Though each has its own number, neither has a fixed arrangement of positions: five gives form to the off-center positions, and ten completes the cardinal positions. In this way, the original one is difficult to find directly; it can be discerned only through the central position expressed by five and ten.

I searched many times for the principle behind this, but never found an answer. Later, I happened to remember how, when I was a child tending livestock, Jian asked me whether I had seen a game called Bagexi. Its board was nothing more than a cross drawn inside a square. At the time, I looked down on the game, thinking it crude and lowly, and never seriously considered its underlying principle. I never imagined that I would understand its meaning only after turning seventy. The ancients’ saying that “people use it every day without knowing its principle” truly did not deceive me.

What we now call “two” was written in ancient times as a circle; what we now call “five” was represented by a certain symbol in ancient script; and what we now call “ten” is still the ancient character for ten, simply written in the form we use today. From this, we can glimpse the method of “fifteen.” Add another horizontal stroke beneath the form for ten, and one can see the ancient form for “earth,” which represented “center.” From this, the expression “central earth” becomes clear.

Add a vertical stroke inside the form for “center,” and one can see north and south as the meridian, with Heaven above and Earth below, each in its proper place. Add a horizontal stroke, and east and west appear as the parallel; Yin and Yang are then divided and arranged. Following these directions in turn, the positions at the four corners come into balance, and the four quarters are established. Qi (vital energy) advances and retreats in relation to one another, generating through interaction; all of this follows the numbers “five” and “ten” to establish the central Way.

Thus, the “fifteen” in the Hetu, the traditional cosmic diagram, and the “fifty” in the great derivation number used in Yijing divination both employ “ten” and “five” as hypothetical numbers. Their actual basis is an invisible system of spatial orientations. To understand why “ten” and “five” can form the center, begin with “one”: it starts from one, is centered in five, and ends in ten. Ultimately, ten returns to one. By using ten to seek one, one can also inwardly comprehend the existence of that one.

Among those who have examined how people in successive generations devised their own interpretations of the Study of the Yi, Sima’s Qianxu seems to come closest to explaining “completing ten” as occupying the center and belonging to Earth. Yet Qianxu takes “nine” as its foundation and “ten” as its tomb, then divides the numbers into fifty-five entries and uses them as names. In this way, the number “center” is given a real place; it is not merely posited. This cannot accord with the principle of the Hetu, which places the “empty five” in “ten.”

The numbers of the Hetu total fifty-five, but only forty occupy actual positions. The remaining fifteen are used emptily within those forty numbers to connect their transformations; they have no actual positions of their own.

The Jing speaks of “joining three and two” and “joining three with five”: these connect numerical transformations and put them to use. “Joining three” arises through a transformation of one and two: one is divided, with a portion set apart as two, and this is applied among the forty actual numbers. This is simply a way to connect numerical principles; it does not establish another actual position.

When one is divided from itself to form two, and an opposing counterpart is established, the two becomes visible while the one that is its root remains unseen. This one, in turn, is brought together from within the two, and so becomes three. By the method of “joining three,” this forms a small completion—that is, the initial formation of a three-line trigram. Humans participate between Heaven and Earth, and all things arise from this. When three such small completions accumulate and three is joined in turn, the result accords with the Book of Changes’ principle of “the Qian origin employing nine.” This is how the method of “joining three” was first established, and why Heaven is represented by the odd yang numbers.

“Joining two” means that one gives rise to two, which stand opposite one another and are established together, symbolizing the Two Fundamental Forms (Yin and Yang). Each of the two is then divided in two, symbolizing the Four Images (young yang, greater yang, young yin, and greater yin). All things ultimately spring from the one; this, too, is why Heaven is represented by the odd yang numbers.

“Two” is generated from “one”: beginning with one, it divides into two, which stand in opposition to each other and can therefore symbolize the two powers, Yin and Yang. Divide each of the two again, and the Four Images take shape. What was originally one divided into two can also be examined by treating one side as “three.” Moreover, “two” is already the opposite of “one”—so why could we not divide each of the “three” into two and set the resulting parts in opposition to each other? Thus, when “three” is divided and each part is given two sides, the number six is formed. This accords with what the Kun hexagram says: “The primal power of Kun makes use of six.” The principle of dividing Yin and Yang is thereby established, while also incorporating the Three Powers of Heaven, Earth, and humanity: each of the three has Yin and Yang, making six. The number six then becomes the six lines of a hexagram, arranged into a pattern. The idea that Earth belongs to Yin and is represented by even numbers also comes from this.

On this basis, divide each of two groups of “two” again and combine them in alternating ways to obtain “five”; double each of the two groups of “two” to obtain “ten.” Ten appears beyond “three” and “nine” as the number in which “one” is lodged and through which it operates. Why say this? Dividing numbers into three parts and into two, and thereby forming five and ten, are all arrangements of numerical principles made by human beings. Yet they also accord with nature, Heaven, and Earth: the pattern of division into three resembles a blood vessel, while the pattern of division into two resembles slanting rain. From these arrangements of five and ten, we can see how the ancients conceived of placing humanity at the center and relating it to what surrounds it. We can also grasp their purpose in establishing principles of thought and measures of the cosmos.

Only by dividing each “three” into three, and each “two” into two, is the number of “three and two” initially completed. One must then divide “three” into three once more, calling this “taking three as Heaven,” and divide “two” into two once more, calling this “taking two as Earth.” This is where the ancients’ purpose in establishing numerical principles lies.

Dividing each “three” into three and each “two” into two is the initial completion of the number of “three and two.” Then divide “three” into three, called “taking three as Heaven,” and divide “two” into two, called “taking two as Earth.” In this way, the numbers of Heaven and Earth are first separated and then joined. The “one,” which at first is not used directly in calculation, is lodged within “ten” and runs through it, allowing numbers both to separate and to come together, thus forming a complete numerical structure. If we look for this “one” among the numbers in the diagram, it is like the starting point of a circle: we cannot tell exactly where it is, and can see only “ten.” Dividing each “three” into three and each “two” into two together makes up the greater number of “three and two.” From this, we can discern the fundamental principle by which Heaven and Earth arise as an undivided whole and cycle back and forth. The so-called “interweaving and crisscrossing of three and five”—a numerical method of combining elements and varying them both vertically and horizontally—is not further explained here.

“Combining in threes and fives and arranging in an interwoven pattern” here means that numbers below five are smaller than five and therefore do not yet have a “five” paired with them; numbers above five are greater than five, so five must be brought into relation with them. Five is placed among the numbers, which are then interwoven and examined both vertically and horizontally. In this way, one applies the rule of taking the middle as the model.

Yet the Commentary on the Changes says, “Combine in threes and fives to bring about change; interweave and arrange the numbers.” It also says that only by understanding change can one reveal the patterns of Heaven and Earth, and only by probing the principles of number can one determine the forms of all things under Heaven. Clearly, the reasoning involved is not easy to grasp. Since the Commentary on the Changes teaches people to combine in threes and fives and arrange numbers in an interwoven pattern, how could one fail to investigate its underlying principles in depth?

The author examines the transmission of Yi learning and observes that nearly two thousand years have passed, yet few scholars have investigated this level of the subject. Even in the later Dragon Diagram—a scheme developed further from the numbers of the River Diagram—no one had explained the principle. Deeply astonished by this, the author devoted himself to seeking it out and probing its finer details, and in doing so came to understand it. There was no other cause.

The Commentary on the Changes says, “Heaven’s numbers are five, and Earth’s numbers are five; the five positions correspond to one another, each paired with its own counterpart.” This refers to the numbers in the diagram. Heaven’s number one begins at the position of Kan (north, Water), while Earth’s number six is attached to the position of Kun (southwest, Earth); one and six thus pair together. When calculated vertically and horizontally, they total seven. Heaven’s number five is then brought into relation with them, set in the middle, and examined alongside them. One number lies at each end. Thus the number one at Kan belongs to Yang and serves as the root of the northern starting point; the number six at Qian belongs to Yin and is attached to the northwestern corner.

Heaven’s number three emerges from the position of Zhen (east, Thunder), while Earth’s number eight is attached to the position of Gen (northeast, Mountain); three and eight thus pair together. When calculated vertically and horizontally, they total eleven. Heaven’s number five is then brought into relation with them, set in the middle, and examined alongside them. Three numbers lie at each end. Thus the number three at Zhen belongs to Yang and occupies the due east; the number eight at Gen belongs to Yin and is attached to the northeastern corner. Both pairs place Yang before Yin.

The earthly number “two” lies in the southwest, corresponding to the Kun trigram; the heavenly number “seven” is placed in the west, corresponding to the Dui trigram. Two and seven add up to nine. When they are arranged in alternation and examined together, each end has a two. The number two associated with Kun belongs to Yin, so it is placed first in the southwest corner; the number seven associated with Dui belongs to Yang, so it is set in the due west.

The earthly number “four” lies somewhat apart in the southeast, at the Xun position; the heavenly number “nine” lies due south, at the Li position. Four and nine add up to thirteen. When the heavenly number “five” is placed alternately in the middle and the arrangement is examined side by side, each end has a four. The number four associated with Xun belongs to Yin, so it withdraws to the southeast corner; the number nine associated with Li belongs to Yang, so it faces south and stands in the most important position. In both pairs, Yin comes first and Yang follows.

From this we can see why Qian, Dui, Gen, and Li take the numbers six, seven, eight, and nine respectively: each is formed by adding five to the number of Kan, Kun, Zhen, and Xun respectively. Adding five to a Yang number makes it a Yin number; adding five to a Yin number makes it a Yang number. Adding five to an even number makes it odd; adding five to an odd number makes it even. Qian’s six is one plus five; Dui’s seven is two plus five; Gen’s eight is three plus five; Li’s nine is four plus five.

This also helps explain “alternating and combining to produce change” (canwu yi bian): pair Qian with Kan, and Gen with Zhen, to trace the pattern from the north toward the east; pair Dui with Kun, and Li with Xun, to trace it from the west toward the south. Arranging them separately is called “can”; arranging them together is called “wu.” This is what is meant by “alternating and combining to produce change.” By separating and alternating them, we can see the relationships of “can”; by combining them in turn, we can see the positions of the numbers. This is called “interweaving and ordering the numbers” (cuozong qi shu). “Interweaving and ordering” means “alternating and combining.” When the numbers are dispersed and arranged in alternation, we can see one, two, three, and four layered one upon another.

When the numbers are separated, gathered together, and then considered as a whole, the result is the interlocking combinations of six, seven, eight, and nine. This is what is meant by “the five positions correspond with one another, each having its own match” (the numbers of the five directions fit together, each with its corresponding number). How could it be as later interpreters claimed, that five empty dots were added to the center of the diagram and called “Heaven’s Five”? We can see from this that the numbers of the Yellow River Chart arise naturally, whether read vertically or horizontally, in forward or reverse order. They were not assembled by human hands or forced into arbitrary patterns.

The numbers increase from the north, pass through the east, and reach the south. They decrease from the south, pass through the west, and eventually return to the north. These movements are all the circulation of Yang numbers, with Yin numbers following them. When Yin numbers increase, they proceed from Kun Two to Xun Four; when they decrease, they proceed from Gen Eight to Qian Six. Yin numbers always follow Yang numbers. However varied the changes may be, they never go beyond the interplay of one Yin and one Yang.

Thus, when Kan One is paired to form Qian Six in the north, Kun Two is paired to form Dui Seven in the west, Zhen Three is paired to form Gen Eight in the east, and Xun Four is paired to form Li Nine in the south, we can see that the basic arrangement of the numbers arises from combining Three and Five.

Next, subtract the number after the Zhen trigram to make a descending sequence; increase Gen’s full number, Eight, to Nine, then multiply three by three. This is “sān,” the method of assigning numbers on the basis of three. Separate the two before the Zhen trigram and combine them with Xun Four to make Six, then divide Six into two parts. This is the initial method of “two.” The full number of the Li trigram is already Nine, and it lies between the beginning and the end; multiplying three by three expresses the “three” at the midpoint. Then combine Kun Two with the four before the Dui trigram to make Six. This is the method by which “two” continues. Finally, remove the three after the Dui trigram to make up Qian Six; multiplying three by three expresses the “three” at the end.

Here we can see the guiding principle of “three for Heaven, two for Earth” (assigning three to Heaven and two to Earth). Whether numbers are staggered, separated, combined, or brought together, the method of using three or two must always be grounded in five. This is the meaning of “interweaving three and five”: bringing change into a coherent pattern and investigating the principles of numbers. Confucius’s words, “Comprehend their changes, and the patterns of Heaven and Earth take form; exhaust their numbers, and the forms of all under Heaven are determined,” refer precisely to understanding change and probing the principles of numbers in order to determine the forms of all things in the world.

“If someone were not the most subtle, skillful at transformation, and wondrous person under Heaven, who could accomplish this? These words are indeed well said.”

Thus, taking Heaven as the model (“joining with Heaven”) and forming the trigrams refers to the Qian trigram. Confucius’s words, “Understand transformation and bring to completion the patterns between Heaven and Earth; exhaust the principles of number and establish the forms of all things under Heaven,” speak precisely of this.

Taking Heaven as the model, one joins with the yang number three and uses it to encompass the principle of “using the nine” of the Qian origin—the fundamental power by which the Qian trigram brings forth the myriad things. The number three contains the number two within it, so yin is also revealed within yang. Taking Earth as the model (“joining with Earth”), one layers the yin number two, following the Dao of Qian. One then combines and doubles the two to make six, bringing together the principle of “using the six” of the Kun origin—the fundamental power by which the Kun trigram bears the myriad things. The number two also makes use of the number three, so yang is revealed within yin. This shows that yin and yang transform into and interpenetrate one another, acting throughout Heaven and Earth. Thus the principles of number must include three and two, and must also include nine and six.

The Taiji (Supreme Ultimate), the fundamental state of the cosmos before differentiation, divides into the Two Modes (yin and yang), which give rise to the Four Images and the Eight Trigrams. The numbers underlying this process are simply the generating numbers one, two, three, and four, and the completed numbers six, seven, eight, and nine. As for five and ten, though they too belong to the sequence of numbers, they serve only as empty numbers and are not included among the substantive forms of the trigrams.

When one divides into two, two combines to form three, and three combines to form four, the Four Images are established. The Four Images take five as their root: the heavenly five occupies the center as an empty number, serving as the root. Yet one combined with four makes five, and two combined with three also makes five, forming two corresponding pairs of numbers. Taking the heavenly five as the root for further derivation is what is meant by coordinating the numbers in alternating combinations. The sum of one, two, three, and four is naturally ten; with the heavenly five at the center, alternating combinations yield fifteen.

The heavenly numbers one, three, and five sum to nine. The earthly numbers one, two, and three sum to six, as do one and five. One plus six makes seven, as does two plus five; one plus seven makes eight, as does three plus five; one plus eight makes nine, as does four plus five. In this way, the sequence of the Eight Trigrams is completed.

One plus nine makes ten, and the sum of one, two, three, and four is likewise ten. The principles of the number ten already contain the Eight Trigrams. Taken as a whole, ten uses one as its root in practice. When numbers such as one, three, and four are used in derivation, ten appears in the form of nine. Ten becomes nine so that it may ultimately return to one; therefore ten is also one.

In the Hetu (a diagram used in ancient times to represent the numerical relations between Heaven and Earth), the numbers in the four directions pair off to make ten: Kan one faces Li nine, Kun two faces Gen eight, Zhen three faces Dui seven, and Xun four faces Qian six. Each pair uses Heavenly Five—the number five symbolizing Heaven—as a notional number, so each can be counted as fifteen.

From this we can see that Heavenly Five and Earthly Ten—the number ten symbolizing Earth—are not included among the actual numbers assigned to the directions, yet they can serve as auxiliary numbers in calculation. Their underlying forms and positions are properly established; the numbers are simply posited notionally and matched with the actual numbers to form the pairing of ten and five. This is the principle of “interweaving five and three,” in which different numbers are combined in alternating patterns. This method also expresses the relationship of “three and two,” pairing three with two. Thus, the four directional sets actually shown in the Hetu total forty. When Heavenly Five and Earthly Ten are added as notional numbers used throughout the calculations, the numbers of Heaven and Earth total fifty-five.

Seven is young Yang and eight is young Yin. Though they add up to fifteen, their generative numbers are incomplete, so they cannot bring about change. Nine is old Yang and six is old Yin; they too add up to fifteen. Old Yang, nine, is formed by adding one, three, and five; old Yin, six, is formed by adding two and four. Thus six changes into nine, and nine changes into six. The line statements in the hexagrams of the Yijing (Classic of Changes), formed by stacking two trigrams, do not use seven or eight; their fundamental numbers are nine and six. Change through interweaving numbers, matching three with Heaven and two with Earth—all of these refer to the same principle.

The numbers in the four directions are each ten. Combining them with nine yields twenty-seven. Then add the two sixes to obtain twelve, making thirty-nine in all. Only one number remains unused. Where, then, should this “one” be used? It is therefore kept within the number ten, making forty altogether.

The Kan trigram, which represents the north, water, and related images, takes “one” as its point of orientation. It gathers the numbers of ten back into itself to establish the “Great Center,” the central position. In this way it opens the beginning of the generating numbers and completes the end of the completed numbers. It is deeply hidden and impossible to fathom; it also appears in the method of reckoning called “three for Heaven and two for Earth,” serving as the basis for pairing one with ten. This is what is called the “Center.”

The “Center” means gathering together numbers that cannot center themselves and bringing them to meet at the central position, thereby harmonizing numbers that have strayed from the center. Beginning with the one of the Kan trigram in the north, the numbers are first governed by the central position. In the second step, the center is divided out from seven, belonging to the Dui trigram in the west; in the third, from three, belonging to the Zhen trigram in the east; and in the fourth, from nine, belonging to the Li trigram in the south. At each location the center is set at ten: the numbers are first brought together to make ten, and then the center is divided out. This is how the “one” is held in reserve.

Only the one of the Kan trigram itself has no number from which a portion can be divided out. It stands alone at the center and acts through the fundamental “one.” If there is a function whose very operation is that of the “Center,” would its efficacy not be immense? Thus, although the one of the Kan trigram does not take part in the specific calculations of “three for Heaven and two for Earth” or “joining numbers in threes and fives,” it establishes the methods by which these numbers are reckoned. For this reason, “three for Heaven” does not begin with the Kan trigram, but with the Gen trigram.

Here, “joining in threes” is the same as the “transformations of one and two” mentioned earlier. The “one” first divides from itself and becomes another “one.” Set against one another, the two reveal “two,” while the original “one” is no longer apparent. This “one” then differentiates out from the “two,” which is why the process is called “joining in threes.” This is the initial form of that operation; all transformations involving joining in twos, threes, and fives begin from it. It is used here to explain the basic principle of “joining.”

This concludes Volume Two of Yitu Tongbian. The text that follows is a collation note explaining differences in wording among various editions, as well as words supplied or corrected on the basis of those editions.

Volume Three of Yitu Tongbian, written by the Daoist priest Lei Siqi of Linchuan.

“Transmission of the Yellow River Diagram” is the title of the next section of this volume, indicating that it continues the account of the Yellow River Diagram’s transmission.

Daoist thought often uses the sun and moon to explain “Yi,” the Way of change. At first, people merely repeated an old interpretation and did not regard it as definitive. But a closer examination of the discussions in the Commentary on the Changes shows that this interpretation does have a basis. The Commentary says: “Are Qian and Kun not, perhaps, the substance of the Changes? With Qian and Kun side by side, the Way of change moves between them. If Qian and Kun were to disappear, the Way of change could no longer be seen; if the Changes could not be seen, Qian and Kun might be close to coming to rest.” Even so, this does not yet make clear what the “Yi” that appears actually is.

The Commentary on the Changes also says: “Are Qian and Kun not, perhaps, the gateways of the Changes? Qian belongs to Yang, and Kun to Yin. When Yin and Yang combine, hardness and softness each take form. In this way, the transformations of Heaven and Earth are expressed, and the Virtue of the spirits is brought into connection.” This suggests that the Changes have a visible form, which can be understood through the relationship between Yin and Yang, hardness and softness.

The Commentary on the Changes further says: “Opening a gate is called Qian; closing it is called Kun. Opening and closing in turn is called change; coming and going without end is called free passage. What appears is called an image; what takes on a concrete form is called an instrument. Making and using things in accordance with the patterns is called a method. When use is fitting, allowing the myriad things to move in and out while remaining available to the people, it is called the mysterious.” Here, the “substance” of the Changes also reveals its function. Once Heaven and Earth have established their positions, the Way of change moves between them.

The Commentary on the Changes also says that nothing is greater than Heaven and Earth as models and manifestations of all things’ images; nothing makes the flow of change clearer than the four seasons; and nothing stands out more among the bright images hung in the sky than the sun and moon. The vastness of Heaven and Earth corresponds to Heaven and Earth themselves; the flow of change corresponds to the four seasons; and the principle of Yin and Yang corresponds to the sun and moon. Since all these passages use the sun and moon to explain the Changes, there is ample basis for interpreting Yi through the sun and moon.

The Yi accords with the patterns of Heaven and Earth and can encompass and connect the Dao of both. Looking up, one observes the signs in the sky; looking down, one examines the principles of the earth. Thus Qian, which manifests the celestial signs, is associated with the sun, while Kun, which embodies earthly principles, is associated with the moon. The sun corresponds to the number of the Li trigram and reaches its limit at nine; the moon corresponds to the number of the Kan trigram and begins at one. Every nine must be matched with six. The moon is said to begin at one because one combines with Heaven’s number, five, to make six. The one of Kan and the six of Qian symbolize the generation of water, and so they are used to explain the moon.

Heaven and Earth are the substance of Qian and Kun, while Qian and Kun are the workings of Heaven and Earth. Heaven and Earth each have their fixed place, and the Qi and form above and below are clearly distinguished. Once Qian and Kun are set apart, the substance and workings of Yin-Yang become manifest as the numbers of change. Heaven and Earth have a constant substance, but people cannot see how it works; Yin-Yang has constant workings, but people can scarcely fathom its substance. Only the substance and workings of the sun and moon are both plainly visible, so they are used in particular to explain Yin-Yang.

The Yi can regulate Heaven and Earth without going beyond their bounds, allowing all things to come to fruition without leaving anything out, because it understands the patterns of change between day and night. The Dao of day and night is not difficult to understand; people simply experience it every day without noticing it. Here are a few examples. Qian belongs to Yang, and the sun is its counterpart; Kun belongs to Yin, and the moon is its counterpart. As Yin and Yang take their places and harmonize with one another, the four seasons proceed in order, and the sun and moon each shine forth.

Opening a door is called “opening the gate” and symbolizes Qian. On a short timescale, this means the sun rising in the morning and the moon reaching its first quarter; over a longer cycle, it means the spring equinox, when day and night are equal and Yang Qi begins to open. Closing a door is called “closing the gate” and symbolizes Kun. On a short timescale, this means the sun setting in the evening and the moon reaching its last quarter; over a longer cycle, it means the autumn equinox, when day and night are equal and Yin Qi closes in. Looking up at the signs in the sky, one sees the sun shining by day, the full moon, and the long daylight hours of summer; then, looking down at the principles of the earth…

Look up at the celestial phenomena and you can see the sun by day, the moon at its fullest around the fifteenth day of the month, and the longest daylight in summer. Then look down at the patterns followed by the myriad things: night and day each have their appointed limits, the new moon begins at the start of the month, and daylight is shortest in winter. When the sun sets in the west, the moon rises; when the moon sets in the west, the sun rises. The sun and moon alternate in their courses, and from this comes light. Cold passes and heat arrives; heat passes and cold returns. The alternation of cold and heat gives rise to the years.

Change is how things advance and recede; firmness and yielding are how day and night alternate. The sun and moon move, and cold and heat cycle in turn. Thus it is said: Heaven and Earth move according to constant patterns, the sun and moon shine with constant brightness, and the movement of everything under Heaven takes the One as its root. The boundaries between the seasons and their alternation all embody this One. The text singles out the One as the root of all movement under Heaven because the number of the Kan trigram is one.

Within a single day, Yang Qi begins to stir at midnight; this may be said to take the One as its root. Within a year, Yang Qi stirs anew at the winter solstice, at the turn of the year and the beginning of the month, making this principle even clearer. Kan has the number one, and Li the number nine; together, Kan and Li combine as Qian. Li represents the sun, and Kan represents the moon. The sun and moon meet on the first day of each month, forming the shuo, or new-moon day. The classic speaks of the moon having a “shuo,” but not the sun; it speaks of the north having a “shuo,” but not the south. This is because “shuo” refers to the moon moving counter to the sun’s course and meeting it, hence the term. The meaning of the character “shuo” makes clear that it refers to the moon’s counter-motion. The “Discussion of the Trigrams” says, “The Changes proceeds by reckoning in reverse.” The Book of Documents also says, “Establish and examine the new moon.” The principles described in these two classics are in fact connected. Seen in this light, how the sun and moon embody the principles of the Changes—and how clearly these celestial phenomena make them visible—becomes plain.

When the moon reaches the fifteenth day or so, it is full; this corresponds to the numbers seven and eight of Lesser Yin and Lesser Yang. At the end of the month, it wanes; this corresponds to the numbers nine and six of Greater Yin and Greater Yang. In general, a month follows a thirty-day cycle based on the Earth numbers. This, too, is a pattern embodied in the River Chart, a diagram that represents the patterns of Heaven and Earth through numbers.

A complete circuit of the heavens is 365 and one-quarter degrees. The sun moves one degree each day, reaching its southernmost position in winter and its northernmost position in summer; these are the turning points of the changes between cold and heat. The moon moves a little more than thirteen degrees each day and meets the sun about twelve times, forming twelve new-moon days—the first day of each month, when the moon’s phase shifts from waning to new. A year consists of six yin months, six yang months, and twelve months in all. The heavens and earth thereby acquire a rhythm, and the four seasons take shape. The moon’s course is arduous.

In general, reckoning by the terrestrial number “thirty,” a month comes to an end. This is also the principle of the Hetu. The Kan trigram is called the “trigram of toil” not simply because it represents water. The number “one” associated with Kan is not included in the numerical arrangements of “three and two” or “three and five.” Instead, it specifically marks the beginning of movement and the initial position at the center.

In general, Heaven and Earth are form, while Yin and Yang are function. Form is outward shape, and Qi (vital energy) is the force through which it acts; the movement of Qi, in turn, allows form to take shape. Heaven is formed from Qi, so its form arches down over Earth. Earth receives and makes use of Heaven’s Qi through its own form, so Earth’s Qi rises up into the sky. In other words, Heaven’s Qi acts fully through Earth’s form, and Earth’s form makes full use of Heaven’s Qi. This is also expressed by the “system of three and two,” a principle that uses three and two to explain the relationship between Heaven and Earth, Yin and Yang.

“The beginning of one is matched by three, and the completion of three is also matched by three” both refer to Heaven; yet both lie below the level of the level ground. For example, the Qian trigram represents mature Yang, the state in which Yang Qi has reached its utmost fullness. It is associated with “six Yin” among the completed numbers, lies in the northwest, the Yin region, and is close to the endpoint of the descending sequence. The Gen trigram represents young Yang, newly arisen but still weak. It is associated with “eight Yin” among the completed numbers, lies in the northeast, the Yang region, and is close to the starting point of the ascending sequence. It, too, lies below the level ground.

Only the central Yang line in the Kan trigram falls outside the workings of the “three and two.” It is the naturally arising first “one,” standing alone at the boundary between the ending and beginning of Yin and Yang. It opens the beginnings of the year and the month, embodying the principle of Yi (change and generation), through which the myriad things continually come to life. This is enough to show that Earth’s form makes full use of Heaven’s Qi.

The two modes distinguished at the beginning and the two modes arrived at in the end both belong to Earth, yet they are also arranged together upon the level ground. Take the Xun trigram, for example. It represents Lesser Yin. Although it corresponds to the Yin position paired with Earth’s generating number, “four,” it lies in the southeast, where Yang energy has risen past the halfway point and continues to advance. The Kun trigram represents Greater Yin. Although it corresponds to the Yin position paired with Earth’s generating number, “two,” it still lies in the southwest, where Yang energy has not yet descended to the midpoint and continues to recede.

Moreover, these two earthly positions together support the center. When they are considered alongside the extreme point of the heavenly position, corresponding to the highest point in the sky and the Sun’s passage across the center, they can explain the workings of Heaven and fully represent them on Earth. From this we can see that the principle of combining three and two to infer the changes of Heaven and Earth cannot be clearly understood without the Sun to establish day and night and their lengths, or the Moon to show the new and full phases and its waxing and waning. How else could we investigate the Qi and form of Heaven and Earth, their functions, the changes and generation of Yin and Yang, or the patterns and destinations of ascent and descent, advance and retreat, division and union, and coming forth and entering?

I therefore tried using this method to examine how the hexagram images in the Yijing (Book of Changes) come into being—in other words, the source of numerical change. The “one” of the Heaven numbers begins in the north and moves to the center; the “two” of the Earth numbers is placed in the southwest. At this point, we cannot yet say that “Heaven and Earth take their proper positions,” because Yang is below in its proper position, while Yin is above in a displaced position. “Two” is called Kun, the hexagram that symbolizes Earth and Yin, but “one” is not called Qian, the hexagram that symbolizes Heaven and Yang, because “one” originates in Kan, the hexagram that symbolizes water and danger.

When “one” joins with “two,” they form “three,” which is then advanced toward the east. This makes clear that human beings arise through their participation between Heaven and Earth, and that this process begins in Zhen, the hexagram that symbolizes thunder and movement. When “two” is divided in half to form “four,” it is paired with Xun, the hexagram that symbolizes wind and entry, which enters in accordance with the natural course; its hexagram image is then posited. The “image” spoken of here is not a physical entity. It takes Heaven, Earth, and humanity as its images, and carries to its fullest extent the principle that male and female are paired and that all things begin from this.

Yang, represented by “one” and “three,” naturally occupies the central positions in each case. “Two” and “four” have not yet reached the center. Is there no way to set them right? The answer is to bring together the completed numbers, the numbers used to form the hexagram images, so that their positions can be corrected. Although “five” occupies the center only as an empty position, it has a corresponding number that can be used to complete the hexagram images.

“One” joins with “five,” and is paired with “one,” forming the number “six” of Qian. “Two” joins with “five,” and is paired with “two,” forming the number “seven” of Dui, the hexagram that symbolizes marsh and joy. “Three” joins with “five,” and is paired with “three,” forming the number “eight” of Gen, the hexagram that symbolizes mountain and stillness. “Four” joins with “five,” and is paired with “four,” forming the number “nine” of Li, the hexagram that symbolizes fire and brightness. Only at this point can we speak of “completed images.” Once the images are complete, they are called Qian; what takes Qian as its model and responds to it is called Kun. Only then can we say that Heaven and Earth have each attained their proper position.

In fact, “one” still joins with “one,” “two” with “two,” “three” with “three,” and “four” with “four”; it is only their joining with “five” that brings about change. Thus, when the single Yang line of Kan joins, it changes into the Yin corresponding to Qian’s number six. When the two Yin lines of Kun join, they change into the Yang corresponding to Dui’s number seven. When the three Yang lines of Zhen join, they change into the Yin corresponding to Gen’s number eight. The joining of Xun’s “four” with “five” likewise brings about change.

Thus, the single yang line of Kan combines with the six of Qian; the two yin lines of Kun combine with the seven of Dui; the three yang lines of Zhen combine with the eight of Gen; and the four yin lines of Xun combine with the nine of Li. What originally occupied the correct position is supplemented through combination to support what is out of position; what was originally out of position is brought into the correct position through combination. In this way, the Eight Trigrams take shape, and their images are contained within them.

Only the “one” emerges from the center. The Kan trigram alone remains as it was and is not included among these numbers. Thus, the Hetu’s principle of “three for Heaven and two for Earth”—matching three with Heaven and two with Earth—and the rules of interweaving and mutual transformation can be applied here.

Therefore, since the system of “three for Heaven” always uses nine, we know that nine is the number used by the Qian trigram. Since the system of “two for Earth” always uses six, we know that six is the number used by the Kun trigram. Qian initially combines with Gen’s eight, but the sum is still insufficient. So the “one” of Zhen is added to the middle of nine, bringing it to the fullest yang number and completing its numerical form. Because the remainder is still insufficient, the combination continues from the end of the nine-count: the three in excess beyond the midpoint of Dui is taken to make up the final six of Qian.

The starting number in “two nines” begins with the two at the center of Zhen, used to make up the deficiency in Xun’s four. The ending number in “two sixes” begins with the two originally present in Kun, used to match the four in excess at the midpoint of Dui, making six. This explains the sequence in which “three for Heaven,” “two for Earth,” and the numbers nine and six are applied.

Some say that Kun’s two and Dui’s seven combine to make nine, while the six obtained at Qian’s own position can likewise yield nine and six respectively. This fails to recognize that the system of “two for Earth” always ends with the number above the Earth, while the system of “three for Heaven” always ends with the number below the Earth. How could their positions be exchanged? Moreover, Qian is inherently nine, yet the number used for it is now reduced to six; Kun is inherently six, yet its number is now increased to nine. This further shows that six, the number of old Yin, changes into nine, and nine, the number of old Yang, changes into six. The transformations of the trigram lines arise from this as well.

The Kun hexagram originally consists of six lines. Raising the count to nine better illustrates how the number of old Yin, six, transforms into nine, and how the number of old Yang, nine, transforms into six. Thus, hexagram lines are counted as either nine or six. The Way of Heaven takes multiples of three, and its number of operation is nine; the Way of Earth takes multiples of two, and its number of formation is six. The opening of “The Great Treatise, Part Two” says: “When the Eight Trigrams are set out, the images are contained within them; when the Eight Trigrams are doubled, the lines are contained within them.” I do not believe the later view that the doubling of the hexagram figures in the Classic of Changes did not begin with Fuxi, the ancient sage traditionally credited with creating the Eight Trigrams.

The Discussion of the Trigrams says that speech flourishes in Dui and that things reach completion in Gen. It repeatedly speaks of “mountain and lake exchanging Qi,” referring to the Xian hexagram. The Image Commentary on the Xian hexagram says: “The Yin and Yang energies respond to one another and join together; the heart is settled and can rejoice.” Heaven and Earth respond to one another, and all things are transformed, nurtured, and brought into being. The sages transform people’s hearts, and peace comes to the world. By observing how things respond, we can see the true condition of all things between Heaven and Earth.

The phrase “thunder and wind press against each other”—thunder and wind stirring one another—refers to the Heng hexagram. Its Image Commentary says: “Thunder and wind work together, moving in accord.” The Way by which Heaven and Earth operate is constant and enduring, never ceasing. The sun and moon follow the Way of Heaven and so can shine for a long time. The seasons change without interruption and so can complete the work of transformation and nurture. When sages follow their Way over time, the world is educated and brought into good order. By observing what endures, we can see the true condition of all things between Heaven and Earth.

If we then examine the beginning of the human Way in the Lower Canon, the layered hexagrams discussed in the Discussion of the Trigrams—formed by placing two of the eight basic trigrams together to make the sixty-four hexagrams—do not depart from this meaning in the least. I therefore believe that the layered hexagrams originated with Fuxi. Otherwise, when the Book of Rites says that each of the three versions of the Changes has eight trigrams and sixty-four hexagrams, was it meant to deceive us? This is a good place to begin seeking the principles of the Changes.

The Discussion of the Trigrams says: “In ancient times, the sages created the Changes. They took three as the number of Heaven and two as the number of Earth to establish the system of numbers; they observed the transformations of Yin and Yang and created the images of the hexagrams; they made clear the principles of firmness and yielding, and the line statements came into being.” In light of this, the sages’ reason for taking three and two can indeed be confirmed by the Yellow River Diagram. The chapter beginning “The Di moves forth in Zhen” is especially a profound elaboration of that diagram. Starting with spring, which corresponds to the Zhen trigram, it proceeds around in sequence and ends at the Gen trigram, signifying the cycle of the year according to the Xia calendar. Thus it says: “In bringing all things to an end and setting them in motion again, nothing is greater than Gen.” It also says: “Gen is the trigram of the northeast. Here all things complete their ending, and here they complete their beginning.” The general meaning is that all things begin to grow again from the point represented by “The Di moves forth in Zhen.” In fact, Confucius explains the matter specifically through the succession and changes of the four seasons. Here he does not yet discuss in detail the Way that runs through day and night and how to understand its pattern.

This probably also expresses what is meant by “beginning at the Zhen position.” Confucius draws specifically on the changing seasons here to explain the principle, but has not yet fully explained the principle embodied in the alternation of day and night.

The system represented by “two” begins from level ground and operates in conjunction with the “three” at the center. It advances gradually from spring to autumn, finally reaching the summer solstice, which symbolizes daytime. The “three” of beginning and ending all lie below ground; they advance gradually from autumn to spring, finally reaching the winter solstice, which symbolizes night.

We can understand this through the changes of everyday life. When all things “emerge from Zhen,” it is like people beginning their work in the daytime. The ancients called Zhen “one of the sun,” using it to verify this change. “Becoming orderly at Xun” means that all things become active and interact with one another. “Coming into view at Li” means that their interactions reach their height. “Devoted to service at Kun” means that activity reaches its limit and gradually turns to stillness. “Speaking with delight at Dui” means resting and preparing to enter the night. From Zhen to Dui, we see the course and function of daytime, something everyone can observe.

At “struggling at Qian,” the arrangement follows the number three at the time of ending. The number is nine: the remaining three at the Dui position are retained, while the other six return to the Qian position as the sky darkens, bringing the process to its close. This is the time when Yin and Yang press against each other, and all things lie hidden and inactive. Yet even at the end, there is still a form and a function to it, both of which everyone can see.

After ending comes beginning again. “Toiling at Kan” means that at midnight, Qi begins to stir once more, in keeping with the pattern in the Book of Changes that “after seven days, return comes.” “Bringing things to completion at Gen” means continuing from afar the three numbers remaining at the Dui position. Dui symbolizes joy; the Qi of mountain and lake communicates, responds, and joins together. Though all things rest in their proper places, they still move when the time comes. This is the beginning of human affairs rising anew.

Changes symbolize advance and retreat; firmness and yielding symbolize daytime and night. In my view, from the time of the ancient sage Xi and King Wen onward, the alternation of Yin and Yang shown in the River Chart has continued to flow without pause through advance and retreat, rising and descending, and cyclical return. This pattern governs everything from winter and summer over the course of a year to day and night within a single day.

I can follow the Dao in my everyday conduct and need not seek it outside myself. To investigate the Dao of Heaven and Earth, one begins with the Gen trigram—one of the Eight Trigrams in the Classic of Changes, symbolizing a mountain and stillness. That is precisely the point.

The Taixuan uses 729 as the number for its deductions, drawing comparisons with the Classic of Changes to expound the Dao. The Huangji makes its deductions through successive levels—yuan, hui, yun, and shi—and illustrates its reckoning with the years, months, days, and hours within a cycle. Today, if we wish to understand how things change and fathom their numerical principles, we can glimpse them in the Yellow River Chart, a legendary diagram said to embody the patterns of Heaven and Earth. Is this not a way to grasp the Dao through simplicity? Is this not a realm of utmost subtlety, unfathomable change, and wondrous mystery?

The “Discourses on the Yellow River Chart” states that, according to legend, in ancient times a pattern appeared on the Yellow River. Fuxi based his drawing of the Eight Trigrams on it, and King Wen of Zhou later wrote the statements for the hexagrams on that basis. Later generations treasured and passed down this account. Yet in the “Announcement of the Duke of Zhou” chapter of the Book of Documents, the place where the Yellow River Chart was displayed is called only the “Eastern Hall.” On what grounds did later writers identify it as a particular kind of “chart”? Who can say for certain?

From the Book of Rites’ accounts of ancient events to later debates over whether the Yellow River Chart appeared in the “Announcement of the Duke of Zhou,” all the various claims are conjectures, not eyewitness accounts. After the Qin dynasty burned the books, the relevant account was found only in Kong Anguo’s Commentary on the Book of Documents. Although Han dynasty bibliographies catalogued claims about charts and texts alongside the classics, they regarded them only as complementary to the Book of Documents. The Book of Changes apocrypha, writings from the Han dynasty that interpreted the Book of Changes, added many elaborate and tangential claims, gradually straying from the original meaning. Thus the original form of the Yellow River Chart remained difficult to establish, even as late as the end of the Tang and the Five Dynasties period.

In the early Song dynasty, Chen Tuan, whose courtesy name was Tunan, began to organize and explain the ancient study of images and numbers: the use of hexagram images and numbers to expound the principles of the Book of Changes. He lamented that this field had been used by Yin-Yang specialists to devise rules for divination, and had been obscured by methods in works such as the Qian zuo du, which used the number “Great One” to arrange the Nine Palaces. He therefore created and circulated the “Dragon Chart.” I was fortunate enough to read his complete work. Seeing its discussions of the transformations in numerical combinations, I felt that Chen Tuan seemed to have probed deeply into the original intentions of Fuxi and King Wen.

Although Chen Tuan did not fully grasp the intentions of Fuxi and King Wen, he also drew a chart using the number fifty-five, separate from the existing diagrams, and labeled it the “Luo River Writing.” The author considers this his first mistake.

The principles of crisscrossing, interweaving, integration, division and reunion, and substance and function are all contained in this diagram. But once the number fifty-five was split apart and drawn separately as another diagram, later generations lost sight of the root and chased after secondary matters, coming to doubt that the Hetu and Luoshu had appeared at the same time. This claim is misleading: it fails to investigate the underlying principles and instead gives rise to many disputes.

These ideas were later passed on to Liu Mu, whose courtesy name was Changmin. On their basis, he drew many diagrams—fifty-five in all—and called them the Gouyin Tu. This only added further disagreements. Li Guan, whose courtesy name was Taibo, criticized Liu Mu’s errors, but said he had retained three of the diagrams. Of these three, the first is what is now called the Hetu; the second separately depicts the fifty-five numbers and arbitrarily labels them the Luoshu; the third shows only the positions of the Eight Trigrams.

Liu Mu did not grasp the fundamental principles, so his work was naturally mistaken. Li Guan, too, failed to understand that these three diagrams were essentially the same, and presumptuously divided them into three kinds. From then on, the accounts grew increasingly distorted, and the explanations increasingly far-fetched. Some even doubted that the Great Commentary—a section of the Commentary on the Changes discussing the Hetu and Luoshu—was written by Confucius, because it says, “The River brought forth the Diagram; the Luo brought forth the Writings.”

They did not know that the Hetu was never the Luoshu, and the Luoshu was never the Hetu. Worse still, some mistakenly called the diagram of fifty-five numbers the Hetu, while claiming that the Hetu transmitted by Tunan, the person who passed it down, was the Luoshu. The confusion and reversal had no end.

This shows a failure to understand the matter clearly. The Eight Trigrams and the number fifty-five are plainly set out in the Hetu, referring to “five numbers of Heaven and five numbers of Earth.” The numbers run from one for Heaven and two for Earth through five for Heaven and ten for Earth, adding up to fifty-five. The principles of Yin-Yang, emptiness and fullness, beginning and end, and the like are all fully displayed in a single diagram. Why, then, seek elsewhere for a correspondence with the Luoshu? And where does the Luoshu itself ever say that it consists of fifty-five numbers?

The diagram now used by Tunan contains both abstract and concrete numbers. It presents, in a single diagram, the whole course of things from their beginning to their end. Why, then, seek to make it conform to the Luoshu? Did the Luoshu ever claim to contain fifty-five numbers?

Tunan has established a separate set of principles, rules, and explanations, singled out the fifty-five numbers in the diagram, and called it the Luoshu. This is precisely where doubts about its authenticity began. To infer his original intention, it was probably because Han Confucians followed the sequence of the Five Phases in the Hongfan, assigning Fire to the second position and Metal to the fourth. The Taixuan, taking the Yijing as its standard, likewise followed this account, matching two with Fire and the south, and four with Metal and the west. The Hetu we have today, however, places two in the southwest and four in the southeast, thereby reversing the order of Fire and Metal.

Since Tunan did not dare alter the Hetu directly, he separated out its fifty-five numbers and drew them as the Luoshu. He then placed the Hongfan sequence, with Fire second and Metal fourth, in the south and west, and made seven follow two and nine follow four, thereby switching the arrangements in the south and west. This is why Changmin, unaware of the reasoning behind it, said that the positions of Fire and Metal had been switched.

The number of the Li trigram is nine; it occupies the due south and belongs to Fire. The number of the Dui trigram is seven; it occupies the due west and belongs to Metal. This sequence of numbers arose naturally in Heaven and Earth through the changing patterns of joining and interweaving, and must never be altered at will. How could Han Confucians’ assignment of two and four to Fire and Metal, based on the Hongfan, justify moving their positions? Moreover, the Hongfan gives only the numbers of the Five Phases. What basis is there for now adding seven and nine, making seven follow two and nine follow four?

If, following the Hetu’s way of assigning numbers, seven were attached after two and nine after four, then “Deliberation on Doubts,” represented by seven, would have to follow the “Five Matters,” while the “Five Blessings,” represented by nine, would have to follow the “Five Calendrical Regulations.” What kind of order of principles would that produce? If two, seven, four, and nine are regarded as numbers alone, they have no fixed form or direction, so one could move them about as one pleased.

Li and Dui each have a fixed direction, while Fire and Metal each have their own nature and function. Could the creative processes of Heaven and Earth really change according to human arrangements? Tunan fails to see that the numbers in the Hetu and Luoshu are plainly different. Instead, he tries to force the two, though they originally conflict, into a supposed agreement and draw a rigid boundary between them. This is like recognizing the finger but forgetting the direction it points.

Moreover, the appearance of the Hetu was not merely to display the symbols of the Four Images and Eight Trigrams; it also embodies the nature and function of Heaven and Earth. Li occupies the south and symbolizes Heaven. Its middle line is Yin, so Yin Qi descends. Kan occupies the north and symbolizes Earth. Its middle line is Yang, so Yang Qi rises. The first line of Zhen is Yang, symbolizing Yang Qi sprouting and rising like Wood. The top line of Dui is Yin, symbolizing Yin Metal descending toward the west.

Qian and Kun embody Heaven and Earth, while Yin and Yang bring the creative processes into operation. Yet the Qian and Kun trigrams are placed in the southwest and northwest. Judged by ordinary directions, this would make Heaven and Earth seem to be upside down. There is more: divination assigns Zhen and Xun to Wood, Qian and Dui to Metal, and Kun and Gen to Earth, while Kan belongs exclusively to Water and Li to Fire. Thus Kan and Li each govern Water and Fire, but Qian and Kun cannot each govern Heaven and Earth exclusively. This also shows that the properties of Fire and Metal cannot be determined by rigidly adhering to fixed directions.

Kun lies in the southwest and has the number two, symbolizing that Earth’s number two generates Fire. Xun lies in the southeast and has the number four, symbolizing that Earth’s number four generates Metal. Both are Yin positions. Dui lies in the west and has the number seven, while Li lies in the south and has the number nine; both are Yang positions. This shows that the generating numbers all belong to Yin and must be paired with Yang numbers to form complete numbers. Earth’s generating numbers are arranged from west to south, so Kun comes first, followed by Xun.

The numbers of completion shift from the south toward the west, so the Dui trigram follows the Li trigram. Observe how a blazing fire turns to ashes: we may say that fire has undergone change and, through this, a wondrous transformation. Metal, though molten, still retains its material form; we may say that metal has changed, but its essential substance has not. Fire and metal can transform into one another because they share certain qualities and differ in others. Thus the numbers one, seven, four, and nine interact and combine in complex transformations. We cannot confine them to fixed directions in order to seek their ultimate principle.

This diagram of the fifty-five numbers places one, two, three, and four within the four directions, then assigns six, seven, eight, and nine to the directions outside them. It is certainly possible to count according to the directions, but how can this arrangement be made to proceed in sequence, return, and circulate? Moreover, five and ten are merely numbers attached to the positions of the four directions and used as abstract values. Although five and ten are called Earth numbers, Five Phases theory, when discussing Earth, must distribute it among the four seasonal nodes of Chen, Xu, Chou, and Wei. In medicine, Earth corresponds to the spleen. According to the account of the movement of the Five Qi, each phase lasts seventy-two days; together, the five phases make a cycle of three hundred and sixty days. The seventy-two days of the Earth phase are divided into four periods of eighteen days, placed at the end of each season and called the time when the spleen governs.

Medical theory is most closely connected to the human body. Physicians assign the days governed by Earth to the end of each season and use this arrangement as a standard for examination and practice. Did they ever calculate Earth’s number, five, on its own, place it outside the four directions and apart from each of them, then set up another five as the number of the center?

Moreover, divination takes four operations to form one line, and eighteen changes to form one hexagram. Each change comprises four operations, so a hexagram requires seventy-two operations in all. This broadly accords with the physicians’ account. Yet this diagram sets five apart and deliberately places another five in the center, designating it as the central position. In that case, forming one line through four operations and one hexagram through eighteen changes…

Only after four rounds of calculation is the whole hexagram formed. A single hexagram requires seventy-two rounds in all, which also accords with what physicians say about the channels and vessels. In the diagram used today, the five is split apart and placed in different positions, yet another “five” is deliberately placed in the center and identified as the center. If so, how is this supposed center actually to be used?

I therefore believe that Tunan (a person’s name) was already mistaken from the outset when he set up fifty-five numbers of his own, marked them as different forms, and called them the Luo Writing. Each of the Nine Categories in the Book of Documents has its own number. For instance, the first category, the Five Phases, and the second, the Five Matters, can each be explained in terms of the numbers of the Five Phases and the five kinds of matters. But if one insists on applying this method of interpretation, then by the fifth category, the Great Standard, one can no longer say precisely what thing or matter it refers to. What is more, the ninth category, the Five Blessings, includes the Six Extremes, the six kinds of calamity. Should this then be counted as nine numbers, or should the Five Blessings and Six Extremes be counted together?

The Great Standard refers to the principle of utmost greatness and centrality. Can the “extreme” in the Six Extremes also be considered “central”? The Great Standard is not a concrete thing or matter, which is why it can be described as “central.” But how can it make sense to claim that five dots represent the center? The Nine Categories run from one to nine, each clearly distinct and with its own original number. I cannot see how they could be combined to make fifty-five. To insist that they fit together is merely to force an unfounded interpretation. Later, people went further still, turning the Book into the Diagram and the Diagram into the Book.

I want to make one point especially clear: the Diagram has numerical principles that can be worked out; the Book has categories that can be counted, but they cannot be arranged into a diagram. I adhere strictly to the River Diagram, not because I claim to have seen with my own eyes the basis on which Fuxi and King Wen created the Book of Changes. Rather, it is because the numbers five and ten in the River Diagram, together with the ways things generate and complete one another across horizontal, diagonal, adjacent, and cardinal directions; their advances, retreats, increases, and decreases; and the divisions and combinations of Yin-Yang, odd and even numbers, as well as the changes between substance and function in the eight bodies and two applications, all seem to run through it. It appears to contain the wondrous, profound principles of creation, far beyond anything human beings could devise.

This phenomenon seems to arise almost spontaneously from Heaven’s workings. It must have been the fundamental basis on which Fuxi and King Wen created the Changes. I once pondered it privately and pursued the matter as far as I could. Then, reasoning outward from analogous things and tracing how change runs through them, I entered the subject by way of its fundamentals and explored it in depth. Believing that I had grasped, on my own, a complete principle of substance and function, I ventured to set down these remarks.

Yet I would not, like some Han dynasty Confucians, force every matter into the framework of the Five Phases (Wood, Fire, Earth, Metal, Water) in order to explain the portents and disasters recorded in the Spring and Autumn Annals. Nor would I insist that the number fifty-five must be the Luo Writing. I simply cannot accept that.

Perhaps people today have not yet understood the principle involved. Some say that this Diagram of the Yellow River comes from the Apocrypha of the Changes: The Qian Zao Du, and that the Great One (Taiyi), the supreme source in ancient philosophy and numerology, uses its numbers to move through the Nine Palaces, the nine divisions of space or direction. The numbers in the four cardinal and four intermediate directions each add up to fifteen. Zheng Kangcheng (Zheng Xuan), in his commentary, also cites the Great One’s movement through the Nine Palaces. As for the empty space at the center, the claim was never that a “five” was physically there; rather, “emperor” was taken to mean the fifth palace at the center.

These apocryphal texts and commentaries are not reliable to begin with. Their authors also failed to understand that five and ten are merely posited numbers, used to encompass odd and even numbers and the generation of change. Though these numbers exist, the August Origin itself occupies no particular position.

But the “Great Unity” they speak of is not the same concept as the Supreme Ultimate (Taiji). To see this, consider the Book of Rites: “Ritual must take the Great One as its root. The Great One divides into Heaven and Earth, turns into Yin-Yang, transforms into the four seasons, and arranges the spirits and deities.” The principle here is plainly no different from that of the Supreme Ultimate as the origin; the two may be used to illuminate one another without doubt. This passage from the Book of Rites is therefore part of the canonical text, so I have appended it after the Commentary as corroborating evidence.

End of Volume Four of Yitu Tongbian

1. The original text has “xian” for the character “ji”; this has been corrected on the basis of the Siku Quanshu edition.

2. The Siku Quanshu edition has “mang” for the character “mang.”

3. The Siku Quanshu edition has “Taiyi” for “Dayi.”

4. The original text has “wei” for the character “sui”; this has been corrected on the basis of the Siku Quanshu edition.

5. The original text has “gouyin” for “gouyin”; this has been corrected on the basis of the Siku Quanshu edition.

6. The original text includes the three characters “qitu”; they are absent from the Siku Quanshu edition.

7. The Siku Quanshu edition has “shi” for the character “shuo.”

8. The Siku Quanshu edition has “jiaoran” for the two characters “xiaoran.”

9. The original text has “cunbian” for “weibian”; this has been corrected on the basis of the Siku Quanshu edition.

10. The original text has “wangzhi” for “wangren”; this has been corrected on the basis of the Siku Quanshu edition.

11. The original text omits the character “wei”; it has been supplied on the basis of the Siku Quanshu edition.

12. The original text omits the character “wei”; it has been supplied on the basis of the Siku Quanshu edition.

Volume Five of Yitu Tongbian

Written by the Daoist Lei Siqi of Linchuan

“An Inquiry into the Lost Tradition of the Yellow River Diagram”: In ancient times, the sages ruled the world and began to establish…

The ancients spoke of “the legacy teachings of the River Chart.” They meant that the ancient sages, in governing the world, established fundamental principles meant to endure for all ages. How could such principles have been invented on someone’s own authority, based only on personal insight? They must have been bestowed by Heaven; that much is certain.

Thus, when the River Chart—a diagram said to have been bestowed by Heaven—appeared in the Yellow River, Fuxi followed it in drawing the Eight Trigrams (Bagua). When the Luo Writing—said to be characters or a diagram bearing Heaven’s intent—appeared in the Luo River, Yu the Great followed it in establishing the Nine Categories of the Great Plan (Hongfan), the nine guiding principles for governing the world. In the “Appended Statements” of the Commentary on the Changes, Confucius says that Heaven brings forth wondrous things and that the sages act in accordance with them. He includes the River Chart and the Luo Writing among these things, showing that the sages took both diagrams as their models. Confucius did not speak of strange occurrences, violence, rebellion, or spirits and ghosts. Could he really have failed to witness what happened, yet preferred to invent fantastic stories to deceive later generations?

After Fuxi became king, he looked up to observe the patterns of the heavens and down to examine the features of the earth. He also observed the markings on birds and beasts and the conditions suited to the land. Taking models from himself and from things both near and far, he began to create the Eight Trigrams, through which to understand the Virtue of the spirits and to classify the forms of the myriad things. Could all this have been devised by sages of successive generations through their own shallow understanding? The vastness of the Changes can be matched with Heaven and Earth; its workings and transformations with the four seasons; and its principles of Yin-Yang with the sun and moon. The book of Changes is vast and complete, beginning and ending with Heaven and Earth without limit. If it had not been bestowed by Heaven, how could it have achieved this? So when Confucius encountered accounts of the River Chart and Luo Writing being handed down, he said only that the sages took them as their models and offered no further explanation.

As for the River Chart, its earliest recorded appearance dates to the reigns of Kings Cheng and Kang. The “Counsels of the Duke of Zhou” in the Book of Documents records that it was displayed afterward in the Eastern Hall, the hall on the eastern side of the palace. There are no further records about it after that. Confucius also lamented that he had not been able to see the River Chart, sighing, “That is all.”

In the early Han dynasty, Kong Anguo explained the term “River Chart” in the “Counsels of the Duke of Zhou” in his Commentary on the Book of Documents. He said that when Fuxi was king, a dragon horse emerged from the Yellow River carrying the River Chart on its back, and that Fuxi followed the patterns on the chart to create the Eight Trigrams. When explaining the words in the Great Plan that “Heaven bestowed the Nine Categories on Yu,” he said that Heaven gave the Luo Writing to Yu the Great: a divine tortoise emerged from the Luo River carrying characters on its back, arranged as the numbers one through nine. Yu arranged them accordingly to form the nine categories; “category” here also means “principle.” These two explanations appear under the relevant passages in the “Counsels of the Duke of Zhou” and the Great Plan. He offered no further account.

By the time of Zheng Kangcheng (Zheng Xuan), scholars had begun citing the Spring and Autumn Weft texts, Han dynasty works of portent interpretation, to explain the River Chart and Luo Writing in the Commentary on the Changes. He said that the Yellow River opened the Way of Heaven and revealed heavenly omens, while the Luo River carried out the Way of Earth and brought forth earthly auspices; the River Chart appeared through a dragon, while the Luo Writing was confirmed through a tortoise. He also said that the River Chart consisted of nine chapters and the Luo Writing of six. This was the beginning of speculative elaboration. Even so, he still cited Kong Anguo’s commentary, holding that the River Chart corresponded to the Eight Trigrams and the Luo Writing to the Nine Categories.

After this, neither Wang Fusi (Wang Bi) nor Han Kangbo commented on the passage. Although Kong Yingda copied Zheng Xuan’s commentary in full in his Commentary on the Changes, he could not explain where Wang Fusi’s view had come from. This suggests that he probably did not accept Zheng Xuan’s commentary.

When Kong Yingda wrote his commentary on the “Great Plan” chapter of the Book of Documents, he again quoted the Appended Statements: “A diagram appeared from the Yellow River, and a writing appeared from the Luo River; the sages modeled their laws on them.” Each of the nine categories had its own written text, which is why it was called a “writing.” The chapter’s statement that Heaven bestowed something on Yu refers to the Luo Writing.

Kong Yingda then cited Liu Xin’s account in the Treatise on the Five Phases in the Book of Han. When Fuxi rose to power as king, a Yellow River Diagram appeared, and Fuxi modeled the Eight Trigrams on it. When Yu controlled the floods, Heaven bestowed the Luo Writing, and Yu used it to set out the nine categories of the Great Plan—the nine principles of governance. The Yellow River Diagram and the Luo Writing complemented one another, just as the Eight Trigrams corresponded to the nine chapters, the nine principles of the Great Plan. Most scholars of earlier generations held this view.

The classic itself does not say that a divine turtle emerged bearing the Luo Writing on its back. The Central Signs and various apocryphal writings, which attached predictions of auspicious signs to the classics, often say, however, that the Yellow Emperor, Yao, Shun, Yu, Tang, King Wen, and King Wu received diagrams and writings: the diagram emerged carried on a dragon’s back, and the writing on a turtle’s back. The authors of these apocryphal works are unknown. After examining them, learned scholars concluded that they were probably fabricated during the reigns of Emperor Ai and Emperor Ping. Although the apocryphal writings appeared only toward the end of the Western Han, scholars may already have been circulating stories that ancient rulers had received diagrams and writings.

Kong Yingda therefore held that the written text of the Great Plan’s nine categories had emerged on the back of a divine turtle, arranged there with numbers from one to nine. After seeing the text, Yu organized its contents in order and formed the nine principles. The classic attributes the work of arranging them to Yu because Heaven’s divine will would be conveyed concisely, not laid out in such detail category by category. Kong Yingda thus concluded that Yu himself had arranged them into a sequence.

In his commentaries on the phrase “the Yellow River Diagram, bound and arranged” in the “Announcement of the End of the Mandate” chapter of the Book of Documents, and on the “horse emerging with a diagram” passage in the Book of Rites, Kong Yingda varied the level of detail slightly, but the basic meaning was much the same: the Yellow River Diagram corresponded to the Eight Trigrams, and the Luo Writing to the nine categories of the Great Plan. He did not change this interpretation.

From the Han dynasty through the Tang, the Yijing circulated with its canonical text, commentaries, and annotations, but no one had placed diagrams and charts before the text. In the early Song, Chen Tuan (style name: Tunan) began to develop and explain the principles of images and numbers in the Yijing. He said that by repeatedly reflecting on the meaning of Confucius’s three expositions of the Nine Trigrams, he had grasped their profound intent. He then wrote a new book, which he called the Longtu (Dragon Chart).

Chen Tuan’s diagrams changed and combined in many ways, each connecting with the others. There were more than twenty diagrams in all. They made use of the numbers in the Yizhuan—“Heaven one, Earth two” through “Heaven five, Earth ten,” totaling fifty-five—and also incorporated Najia, a method of matching the Heavenly Stems and Earthly Branches with hexagrams and their lines, to connect the principles of the Yijing. One diagram arranged in nine palaces, and another called the Luoshu, both omitted the five generating numbers and listed only “Earth six” through “Earth ten.” Chen Tuan explained that ten represents the functioning of numbers and provides the basis for the formation of physical forms. Thus, the numbers discussed in the Hongfan are only those used in the operation of the Five Phases.

At the end of the book, Chen Tuan included two more diagrams. The Nine Palaces diagram remained unchanged and was labeled “Hetu”; the other simply arranged the fifty-five numbers from the Yizhuan across the four directions and the center, but was separately labeled “Luoshu.” The relevant commentaries contain no explanation of this arrangement. Chen Tuan merely moved two and seven to the south, and four and nine to the west. The old method behind this arrangement is no longer known. Perhaps he did so because the Hongfan associates two with Fire and four with Metal among the Five Phases, while the Taixuan, modeled on the Yijing, also places two in the south and four in the west.

This system of diagrams, however, did not circulate for long, and few people knew of it. Five generations after Chen Tuan, it passed into the hands of Liu Mu (style name: Changmin), who expanded the diagrams to fifty-five and titled the book Gouyin (Exploring the Hidden).

Teachers and friends held one another in high regard and exchanged accounts of their readings of the Changes. Each wrote commentaries on the Changes, with titles such as Treatise on the Virtue of the Trigrams, Words Spoken Indoors, Record of the Master’s Teachings, Pointing to the Ultimate Meaning, The Subtle and Profound, and Communing with the Spirit. Together, these works were also called New Commentaries on the Zhou Changes. They often tried to present their doctrines and experiences as wondrous and extraordinary, but careful examination reveals nothing truly supernatural in them.

At the time, Li Guan, styled Taibo, wrote the Six Discourses to refute these claims. He argued that these people distorted the original meaning, misled students, and undermined moral instruction in the world. He also removed repeated material from their diagrams, retaining only three: the Yellow River Chart, the Luo River Writing, and the Eight Trigrams. Yet the correctness of the many diagrams these people drew was difficult to judge in the first place, and Taibo did not see that these three diagrams shared the same origin. The Yellow River Chart originally arranged the Eight Trigrams, and its numbers totaled fifty-five.

Later, these people began to overturn the old account of the Luo River Writing. They argued that it contained more than what is described in the Great Plan; before Xi (Fuxi), there must have been another ancient book, which Fuxi obtained and used as the basis for making the Changes. Taibo held the same view. I truly do not understand why they would say this.

I once carefully examined this passage in the “Appended Statements, Part One” of the Commentary on the Changes. It speaks of taking Heaven and Earth as models, following the changing seasons, and observing the movements of the sun and moon; it then mentions milfoil and the tortoise shell, both used in divination. Thus, “Heaven produces the numinous things, and the sages take them as models” refers to milfoil. The Changes uses milfoil for divination, and the statement “The virtue of milfoil is round and numinous; it silently assists the spirits and brings forth milfoil” also refers to milfoil. How could the Changes be the sun and moon? The tortoise is mentioned because both milfoil and the tortoise shell can be used for divination, so they are named together. After speaking of taking Heaven and Earth as models and following the changing seasons, the text says, “Heaven and Earth transform, and the sages emulate them.” This, too, serves to make the same point.

The passage above says that celestial signs such as the sun and moon are suspended high in the sky. It then explains further: “Heaven above displays all kinds of signs, and the sages model themselves on them.” As for Heaven’s creation of wondrous things, the text offers further examples: the Yellow River produced the River Chart, and the Luo River produced the Luo Writing; the sages, too, took these as models.

After the River Chart appeared in the Yellow River, Fuxi used it to draw the trigrams. Later, after the Luo Writing appeared in the Luo River, Yu the Great used it to establish the Nine Categories of the Great Plan, the nine guiding principles for governing the state. What is wrong with comparing these two events in this way? Must we insist that the numbers, meanings, and patterns of the River Chart and the Luo Writing were exactly the same? Must we insist that they appeared at the same time?

It was not only Han-dynasty Confucians who linked the contents of the Great Plan with the Luo Writing. Even Tunan, in his earliest explanation of the Luo Writing’s form, said only that “ten” was its function and that “ten” gave it a complete structure. Since the Great Plan describes the functions of the Five Phases, he explicitly identified it as a discussion of the Luo Writing. People like Changmin, unaware of the origins of these accounts and departing from the teachings they had inherited, claimed that the Luo Writing had not appeared in Yu the Great’s time. This has left later generations even more confused, unsure what to rely on. Why is that?

When Confucius said that the sages modeled themselves on these wondrous things, many generations and many years had already passed between Fuxi and Confucius. Even considering only the time between Yu the Great and Confucius, more than a thousand years had passed. What is unreasonable about Confucius placing the Luo Writing, which came from the Luo River, alongside Fuxi’s River Chart, which came from the Yellow River? If Changmin makes this claim, does he even believe Confucius’s Commentary on the Changes—that is, the commentary texts discussing the Changes?

Lei Siqi argued that Chen Tuan (courtesy name Tunan) called a diagram containing fifty-five numbers the “Dragon Diagram.” Although Chen claimed to understand the purpose of the Nine Hexagrams described by Confucius in the Commentary on the Changes, his preface to the diagram says: “The Dragon Diagram was sent down from Heaven to be shown to the world. Fuxi, the sage king of ancient legend, comprehended it and put it to use, while Confucius silently understood it and gave it expression.” Chen also stated explicitly that the original diagram had no arrangement or formed pattern, only fifty-five numbers. This suggests that the diagram referred to the twenty-five Heavenly numbers and thirty Earthly numbers described in the Commentary on the Changes, which together make fifty-five. At some later point, however, it was inexplicably retitled “Luo Writing” at the end of the diagram. The mistake may have been made from the outset.

Liu Changmin and others carried this error forward and added further conjectures, going so far as to treat the diagram as the Luo Writing and draw a fifty-five-number diagram to support their claim. After Shao Yong (courtesy name Yaofu), later scholars likewise took Chen Tuan’s fifty-five-number diagram, labeled “Luo Writing,” to be the genuine Luo Writing. Yet they failed to consider that Chen may initially have labeled the diagram “Luo Writing” simply because it “uses ten.” The Luo Writing corresponds to the Nine Categories in the Book of Documents’ “Great Plan,” which set out nine principles of governance. If it is named for “nine,” why would a tenth number appear in it? Did Chen Tuan believe that, since the Commentary on the Changes records the River Diagram appearing in the world, the Luo Writing might appear alongside it, and so borrow the fifty-five numbers described in the Commentary on the Changes, claiming that they were the Nine Categories in the Book of Documents? Each of the Nine Categories is explained through the combination of vertical and horizontal lines. What purpose, then, could the number fifty-five serve? This is the clearest problem with the claim.

For this reason, Lei Siqi once traced the origins of this account through the example of Liu Changmin. He held that, after Chen Tuan, no other related writings by figures such as Zhong Fang, Xu Jian, and Li Gai had yet come to light.

His own teacher, Fan Kuachang, states in Origins and Development of the Great Change (Taiyi Yuanliu): “A dragon-horse emerged from the Yellow River bearing the Yellow River Chart. Fuxi, the legendary ruler of high antiquity, investigated the relationship between the Way of Heaven and human affairs, revised the numbers governing the generation of the Five Phases, and established the forms of the trigrams on earth. From the Western Zhou onward, Laozi therefore transmitted to Confucius the fundamentals of making the Changes.” The number of Heaven, one, occupies the north; the number of Earth, two, the south; the number of Heaven, three, the east; the number of Earth, four, the west; and the number of Heaven, five, the center. The number of Earth, six, corresponds to the Zi position (north); the number of Heaven, seven, to the Wu position (south); the number of Earth, eight, to the Mao position (east); the number of Heaven, nine, to the You position (west); and the number of Earth, ten, to the center, where it is lodged in the final position. In this way, the numbers of Heaven and Earth total fifty-five.

Consider this account: why place the first five numbers in the north, south, east, west, and center, and then assign the next five to Zi, Wu, Mao, You, and the center? Are Zi, Wu, Mao, and You not precisely the directions of the four quarters? If the number of Earth, ten, is assigned to the center and lodged in the final position, is there really a “final position” in the center?

A close look at the arrangement of these numbers shows that it is exactly the diagram now in circulation: it marks the four quarters but gives no numbers for the four corners. Clearly, Fan Kuachang had no understanding of why Tunan, the Song dynasty scholar of the Changes, deliberately marked the distinctions in his diagram. He merely borrowed the method of Taixuan in imitating the Changes, used the numerical correspondences from the Great Plan—“one is Water, two Fire, three Wood, four Metal, five Earth”—and invented an explanation from scratch. Attributing it to Laozi, he advanced an eccentric theory to defend Tunan’s rearrangement of the positions of two, seven, four, and nine.

Yet Fan Kuachang did not go so far as to assign the Nine Divisions of the Great Plan—nine categories of principles for governing the state—to the nine palaces. The sources of his claims, on examination, amount only to remnants of the method of “receiving the stem” (najia), which assigns the heavenly stems and earthly branches to the trigrams to interpret change, and to divergent interpretations such as “the five Zi return to Geng.” He also described the celestial positions of the trigrams as follows: Kan and Li shifted to the outer sides of the center, Qian and Dui placed in the east, and Gen, Zhen, and Xun placed in the west. It did not occur to him that much of the Five Phases theory derives from the Changes in the first place.

He later turned things around again and used the Five Phases as the basis for determining the hexagram images. This was one of the main ways he dared to establish an unconventional theory. Thus, Changmin not only added fifty-five diagrams, but further exaggerated the significance of the four primary trigrams Kan, Li, Zhen, and Dui, and assigned the four completed numbers to the four directions. He said: the number of Kan is six; subtract the three lines of the original trigram, and three remain, so the number three is used to draw Qian. The number of Li is seven; subtract three lines, and four remain, so the number four is used to draw Xun. The number of Zhen is eight; subtract three lines, and five remain, so the number five is used to draw Gen. The number of Dui is nine; subtract three lines, and six remain, so the number six is used to draw Kun. He treated numbers as line strokes and labeled them the “Yellow River Diagram” (Hetu). This approach mistakes the secondary for the essential.

Each of the Eight Trigrams has three lines, from which yin and yang, firmness and yielding, give rise to the line patterns. No one has ever said that an original trigram uses only three lines, then takes the extra numbers and turns them into other trigrams. The numbers listed in the text also seem to have been matched together arbitrarily. Moreover, it is even less appropriate to call Kan, Li, Zhen, and Dui the “Four Images.” The images existed before the trigrams took shape; hence the saying, “The Four Images give rise to the Eight Trigrams.”

He also said that the Nine Categories of the Luoshu—the nine broad principles of governing—were numbers that arose naturally from the Hetu. He left the “Supreme Ultimate” empty in the center and distributed the other eight categories among the four cardinal and four intermediate directions: the Five Phases at Kan One, the Five Affairs at Kun Two, the Five Periods at Xun Four, and the Five Blessings at Li Nine, arranging them in the order of the Nine Categories at the positions of the Hetu trigrams. But the Nine Categories have an order because Yu arranged them one by one in sequence, starting with the first and proceeding downward; he did not skip the third and fourth before placing the others. Now, however, the Nine Categories are positioned according to the Hetu’s fifteen-pattern arrangement across and down, with the result that…