《天原發微》Vol. 13

This passage primarily discusses the relationship between the Hetu (River Chart) and Luoshu (Luo Writing) with the origins of the Yijing (Book of Changes), as well as the subsequent development of the Xiangshu (Image-Number) school.

The Hetu and Luoshu are two mysterious diagrams from ancient times. Legend holds that when Fuxi ruled the world, a dragon-horse emerged from the Yellow River bearing the chart on its back. Fuxi then drew the eight trigrams (Bagua) based on the patterns of the Hetu. During Yu the Great's flood control efforts, a divine turtle appeared carrying the Luoshu, which displayed numbers arranged up to nine. Yu used this to establish the nine categories of governance. These two diagrams were regarded as auspicious signs formed naturally by heaven and earth, not created by human hands.

Zhu Xi (Master Zhu) explained that heaven and earth themselves cannot speak, so sages like Fuxi and Yu were needed to interpret their principles. The Hetu and Luoshu are like the Yijing drawn directly by heaven and earth. By observing the transformations of all things, Fuxi discovered their patterns perfectly matched what the Hetu revealed, thus using it to create the trigrams.

Regarding the development of the Xiangshu school: Confucius's disciple Shang Qu transmitted the study of the Changes, passing through scholars like Ding Kuan until it reached Meng Xi and Jing Fang—a lineage traceable even to the Tang dynasty. The Tang monk Yi Xing synthesized the theories of these two schools, including concepts like hexagram qi (guaqi), stem-branch correspondence (najia), and the Five Phases (Wuxing), all of which originated from the Xici Zhuan (Commentary on the Appended Phrases) and Shuogua Zhuan (Discussion of the Trigrams) in the Zhouyi. Later scholars like Ma Rong, Zheng Xuan, Xun Shuang, and Yu Fan each developed their own interpretations, though none strayed too far from the original principles of image-number. However, during the Wei period, Wang Bi and Zhong Hui jointly studied the Changes, completely abandoning prior scholarship. They blended in Laozi and Zhuangzi's ideas, focusing solely on literary embellishment while neglecting the Yijing's foundational texts. This caused the study of heavenly patterns (tiandao) and human affairs (rendao) to diverge permanently.

The author also notes that some (like the compilers of the Huainanzi and Chunqiu Wei) offered overly fantastical interpretations of the Hetu and Luoshu. The passage emphasizes the interdependence of images (hexagram symbols) and numbers (Five Phases numerology) in Yijing studies, asserting that both are indispensable and mutually illuminating.

This passage primarily discusses the numerical arrangements and Yin-Yang Five Phases relationships in the Hetu (River Chart) and Luoshu (Luo Writing)—two ancient Chinese mystical diagrams believed to contain the fundamental principles of heaven and earth.

Regarding the Luoshu (said to be the pattern carried on the back of a divine turtle), Gu Yewang considered its core to consist of eighteen characters including "农用敬用" (agricultural use and reverent use), while Ban Gu held that the original text comprised sixty-five characters from "初一" (first beginning) to "六极" (six extremes). Kong Yingda of the Tang Dynasty believed that the will of heaven was concise and not so elaborate, thus favoring Kong Anguo's interpretation, a view later adopted by Zhu Xi.

The passage then explains the numerical arrangement of the Hetu (said to be the pattern carried on the back of a dragon-horse): the heavenly numbers (odd) 1, 3, 5, 7, and 9 sum to 25, while the earthly numbers (even) 2, 4, 6, 8, and 10 sum to 30. Zhu Xi stated that this was Confucius's revelation of the numerical mysteries within the Hetu. The specific arrangement is as follows: 1 and 6 are positioned below, 2 and 7 above, 3 and 8 to the left, 4 and 9 to the right, with 5 at the center as the foundational number and 10 as the derived number.

The numbers are divided into four groups corresponding to the Four Symbols (Lesser Yang, Greater Yang, Lesser Yin, Greater Yin), with Greater Yang and Greater Yin positioned in the northwest and Lesser Yang and Lesser Yin in the southwest. The numerical pairings—formed by matching odd and even numbers—create order: 1 paired with 2, 3 with 4, and so on, resembling the ordering of brothers; while 1 and 6 combine to form Water, 2 and 7 Fire, 3 and 8 Wood, 4 and 9 Metal, and 5 and 10 Earth, akin to the union of husband and wife. These pairings exhibit both sequence and harmony.

The Ten Heavenly Stems (Jia, Yi, Bing, Ding, etc.) follow a similar logic: Jia and Yi correspond to Wood, while Jia and Ji also transform into Earth; Bing and Ding correspond to Fire, while Bing and Xin transform into Water, and so forth. These pairings maintain the attributes of the Five Phases while allowing mutual transformations, forming relationships of mutual generation and restraint. The entire system, through the ingenious interplay of numbers and Yin-Yang, reveals the patterns of generation and transformation in all things between heaven and earth.

This scripture primarily elucidates the profound connection between all things in heaven and earth with the number "five," as well as the patterns of Yin-Yang transformation embedded in the Hetu (River Chart) and Luoshu (Luo Writing)—two ancient mystical numerical diagrams.

It first explains how the five fundamental human relationships (ruler-subject, father-son, elder-younger brothers, husband-wife, and senior-junior) are interconnected like the Five Phases (Metal, Wood, Water, Fire, Earth), forming an inseparable whole. The numerical combinations between heaven and earth also revolve around "five" as their core: the odd numbers 1, 3, 5, 7, 9 constitute the heavenly numbers, while the even numbers 2, 4, 6, 8, 10 form the earthly numbers. Their combined sum of 55 invariably returns to "five" as the central number. "Five" represents the Dao of harmonious balance, capable of generating infinite transformations.

The text then deciphers the numerical arrangements of Hetu and Luoshu: In the Hetu, numbers 1-5 are "generative numbers" (symbolizing growth) belonging to Yang in the inner circle, while 6-10 are "completion numbers" (representing maturity) belonging to Yin in the outer circle. In the Luoshu, odd numbers 1, 3, 7, 9 occupy the four cardinal directions corresponding to Yang trigrams, whereas even numbers 2, 4, 6, 8 reside at the four corners aligned with Yin trigrams. These two configurations manifest the relationships of Yin-Yang hierarchy and ruler-subject precedence.

Citing perspectives from Zhu Xi and Shao Yong, the text notes that the Hetu governs completion numbers through generative numbers, while the Luoshu dominates even numbers through odd numbers—an arrangement that simultaneously embodies eternal principles and possibilities for change. Just as celestial phenomena and calendars originate from simple numerical sequences, the operational laws of all creation can be glimpsed within these numerical combinations. This numerical philosophy fundamentally differs from later superstitious tales of ghosts and deities, instead revealing heaven and earth's fundamental operating principles through rigorous mathematical relationships.

This passage explores the relationship between the Hetu (River Chart) and Luoshu (Luo Writing) with ancient Chinese astronomy and calendrics. The ancients believed heaven was round and earth was square, with the Hetu (a mystical pattern said to have been carried from the Yellow River by a dragon-horse) representing the celestial sphere, while the Luoshu (a pattern borne from the Luo River by a divine turtle) symbolized the square earth.

In astronomical terms, the seven stars of the Big Dipper, combined with the sun, moon, and five planets (Venus, Jupiter, Mercury, Mars, and Saturn) make seven celestial bodies. Together with the twenty-eight lunar mansions (divided into four groups of seven) and the six stars of the Southern Dipper, they total fifty-five - corresponding exactly to the numerical sum of the Hetu. The calendrical system was based on five (the Heavenly Stems) and six (the Earthly Branches), with the pentatonic scale and hexatonic pitch standards mirroring the movements of heaven and earth. Northern Song philosopher Shao Yong observed that the numbers of celestial bodies and stems-branches (ten stems and twelve branches) perfectly reflected the yin-yang balance between five and six.

Geographically, the Luoshu manifests as a nine-square grid, symbolizing the nine provinces demarcated after Yu the Great's flood control efforts. The well-field system's division of land into nine hundred mu parcels directly corresponds to the Luoshu's nine numbers. Upon receiving the Luoshu, Yu established the "Nine Categories of Hongfan" (a foundational political document), instituting tribute systems and water conservancy projects.

The Hetu and Luoshu respectively represent heaven's yang and earth's yin: the Hetu's outer circle and inner square symbolize heaven's roundness with a base-ten numerical system, while the Luoshu's outer square and inner circle represent earth's squareness with a base-nine system. Fuxi created the Eight Trigrams based on the Hetu, while Yu and Jizi (a Shang dynasty sage) composed the Hongfan according to the Luoshu, forming the cosmological concept of "three representing heaven and two representing earth." Originally, heaven and earth had form but no numerical representation - it was only with the appearance of the Hetu and Luoshu that this mathematical framework became established.

This passage elucidates the numerical relationships and symbolic meanings of the Hetu (River Chart) and Luoshu (Luo Writing), two ancient mystical diagrams. The Hetu arranges numbers from one to ten, with its empty center symbolizing the principle of change in the "Yi" (Book of Changes). It contains twenty-five heavenly numbers (odd) and thirty earthly numbers (even), totaling fifty-five. Removing ten yields Luoshu's forty-five, while removing five corresponds to the Dayan number fifty used in divination. The circular Hetu represents heaven, with its numerical sequence illustrating the Five Phases' (Wood, Fire, Earth, Metal, Water) generative cycle: Water → Wood → Fire → Earth → Metal → Water. Yet it simultaneously conceals restraining relationships—like Water overcoming Fire or Metal overcoming Wood—demonstrating how cosmic balance requires both generation and restraint.

The Luoshu arranges numbers one to nine with a solid center representing order. Its twenty-five heavenly numbers and twenty earthly numbers sum to forty-five. The square Luoshu symbolizes earth, displaying the Five Phases' restraining cycle: Water → Fire → Metal → Wood → Earth → Water. However, it equally contains generative sequences like Metal producing Water or Wood fueling Fire. Though differing in form, both diagrams manifest the cosmic progression from Taiji (Supreme Ultimate) to Liangyi (Two Modes), Sixiang (Four Symbols), and Bagua (Eight Trigrams). Their numbers interconvert, jointly forming a complete model of celestial-terrestrial operations.

The text emphasizes that without restraint, generation alone could never give form to things; without generation, restraint alone would prevent humanity's continuity. This equilibrium of creation and control constitutes the universal truth the Hetu and Luoshu reveal—where opposing forces dynamically sustain cosmic harmony.

This scripture primarily discusses the central position of the number five in the Hetu (River Chart) and Luoshu (Luo Writing) and their generative-conquest relationships. It first points out that the principles of mutual generation and conquest among the Five Phases (Wuxing) are actually contained within the "Supreme Polarity" (Huangji, the fundamental cosmic law). Though this ultimate law is not directly manifested as a specific number, it governs the nine great categories (jiuchou). Just as the Five Phases, five affairs, and other categories each have their own numerical correspondences, the Supreme Polarity alone resides at the center, overseeing all.

The text then elaborates on the numerical patterns of the Hetu and Luoshu: both feature the number five at their center. Five represents the fundamental cosmic structure of "three heavens and two earths" (sanyang eryin, three yang and two yin). The Hetu primarily focuses on the generative numbers (one to five), where the number five itself embodies the characteristics of all five generative numbers—the lower dot symbolizes Heavenly One (number one), the left dot symbolizes Heavenly Three (number three), the right dot symbolizes Earthly Four (number four), and the central dot symbolizes Heavenly Five (number five). Adding five to a generative number transforms it into a completion number (six to nine). For example, one plus five yields six, two plus five yields seven, and so forth.

The Luoshu, on the other hand, emphasizes odd numbers, with the number five similarly encompassing the characteristics of all five odd numbers: the lower dot is Heavenly One (number one), the left dot is Heavenly Three (number three), the central dot is Heavenly Five (number five), the right dot is Heavenly Seven (number seven), and the upper dot is Heavenly Nine (number nine). The odd numbers occupy the four cardinal directions, while the even numbers are distributed at the four corners, forming corresponding relationships through the addition of five. This demonstrates that the mysteries of the Hetu’s fifty-five numbers and the Luoshu’s forty-five numbers are all contained within the arrangement and combinations of the central five dots.

This passage explains the patterns of Yin-Yang transformation in the numerical arrangement of the Luoshu (Luo River Writing). The numbers are divided into odd and even: one, three, five, seven, and nine are odd (Yang numbers), while two, four, six, and eight are even (Yin numbers). Though these numbers appear scattered in a crisscross pattern, they conceal an underlying order. All forty-five numbers of the Luoshu are embedded within the central arrangement of five points.

Zhu Xi (Master Zhu) proposed that Yin and Yang mutually contain each other. The Hetu (River Chart), another mystical numerical diagram, pairs one, two, three, and four with six, seven, eight, and nine, totaling forty. When combined with the central five and the hidden five, the sum becomes fifty-five. Dong Panjian elaborated on this: the number one represents the position of Greater Yang (Lao Yang), while six represents the value of Greater Yin (Lao Yin). Yet within one lies nine (Greater Yang), and within four lies six (Greater Yin). Yin and Yang depend on each other—if this mutual containment were lost, the natural order of creation would cease.

Seven and nine are Yang numbers, symbolizing expansion. From Lesser Yang (seven), it grows to eight, ultimately reaching the peak of nine (Greater Yang). At this extreme, Yang transforms into Yin. Six and eight are Yin numbers, representing contraction. From Lesser Yin (eight), it diminishes to seven, eventually reaching the extreme of six (Greater Yin). At this limit, Yin reverts to Yang. This cyclical interplay of Yin and Yang is the ceaseless rhythm of the cosmos. Yang numbers always expand (increase), while Yin numbers always contract (decrease), much like the Yijing (Book of Changes) states: "Heaven and Earth wax and wane with the seasons." Understanding this principle is to grasp the Dao.

In the Luoshu, the sum of numbers in any row or column is fifteen, symbolizing the ebb and flow of Yin and Yang. The pairing of one, two, three, and four with seven, eight, nine, and six represents the four manifestations (Greater Yang, Lesser Yin, Lesser Yang, Greater Yin), from which the eight trigrams emerge. Shao Yong, quoting the Yizhuan (Commentary on the Changes), explained: the Zhen trigram symbolizes growth, the Qian trigram distribution, the Xun trigram decline, and the Kun trigram closure—this is the core principle of Yin-Yang transformation. Beginning with Zhen, Yin and Yang interact to generate Yang energy, which fully unfolds at Qian.

This scripture primarily elucidates the patterns of waxing and waning between Yin and Yang, as well as the relationship between the Hetu (River Chart), Luoshu (Luo Writing), and the Bagua (Eight Trigrams).

Regarding the fluctuations of Yin and Yang, Yang begins its growth from seven, progresses through eight, and reaches its zenith at nine. Yin starts its decline from eight, passes through seven, and culminates at six. The number seven represents the expansion of Yang, while six signifies the contraction of Yin. Numerical increase and decrease symbolize the waxing and waning of these forces: increasing numbers denote "growth" (长), while decreasing numbers indicate "decline" (消). For instance, when six advances to nine, nine grows while six diminishes; when nine recedes to six, nine declines while six expands. These transformations correspond to directional relationships across all four cardinal points.

Concerning the connection between the Hetu-Luoshu and the Bagua, the text cites Zhu Xi's interpretation. Fuxi derived the Bagua based on the Hetu, as referenced in the "Yijing" (Book of Changes) statement "Taiji (Supreme Ultimate) generates the Two Modes (Liangyi)"—where the central five and ten concealed in the Hetu's palace represent Taiji. The "Two Modes" refer to the odd numbers one, three, seven, and nine constituting the Yang mode, while the even numbers two, four, six, and eight form the Yin mode. The "Four Images" (Sixiang) emerge through the pairing of generative numbers (一至四) with consummate numbers (六至九).

Numerologically, nine is the complement of one, eight of two, seven of three, and six of four. These numbers interweave in intricate combinations, generating diverse transformations of Yin-Yang dynamics. The entire system revolves around the central number five, with all directional sums equaling fifteen—a manifestation of balanced Yin-Yang philosophy.

This passage explains the generative principle and arrangement of the Eight Trigrams (Bagua) — Qian (Heaven), Kun (Earth), Li (Fire), Kan (Water), Gen (Mountain), Dui (Lake), Zhen (Thunder), and Xun (Wind). The ancients assigned the numbers seven, six, eight, and nine to Qian, Kun, Li, and Kan respectively, positioning them at the four cardinal directions (east, south, west, north), while using one, two, three, and four for Gen, Dui, Zhen, and Xun to fill the four intercardinal corners (northeast, northwest, southeast, southwest). It was based on this system that Fuxi drew the original Bagua diagram.

The number "one" represents the Supreme Ultimate (Taiji), the primordial source of the cosmos. From Taiji emerged the two polarities of Yin and Yang (Liangyi), which in turn generated the Four Symbols (Shaoyang, Taiyang, Shaoyin, Taiyin). These Four Symbols then produced the Eight Trigrams, which further multiplied into sixty-four hexagrams in an unending generative process. The number "one" symbolizes the beginning of Yang, represented by the solid line (Yang Yao) of the Qian trigram, while "two" signifies the origin of Yin, depicted by two broken lines (Yin Yao) forming the Kun trigram. The remaining six trigrams are all combinations of these fundamental Yin and Yang lines.

For instance, Zhen trigram follows the "one-two" pattern (a solid Yang line below two broken Yin lines above), Kan appears as "two-one", and Gen as "two-two-one". Similarly, Xun manifests as "two-one-one", Li as "one-two-one", and Dui as "one-one-two". King Wen of Zhou later expanded this system by stacking trigrams to develop the sixty-four hexagrams.

In this arrangement, the northwest position is occupied by the purely Yang Qian trigram (three solid lines), while the purely Yin Kun trigram (three broken lines) resides in the southwest. The Gen trigram (two Yin lines above one Yang line) occupies the northeast, and the Xun trigram (one Yin line beneath two Yang lines) fills the southeast. The entire Bagua configuration follows the sequential logic of "one" and "two" (Yang and Yin). Fuxi originally designed the Prenatal Bagua based on the River Chart (Hetu), the mythical dragon-horse's celestial diagram. King Wen later developed the Postnatal Bagua system, just as Yu the Great governed floods according to the Nine Categories (fundamental administrative principles), and Jizi referenced the Five Phases (Wood, Fire, Earth, Metal, Water) when explaining the "Great Plan" chapter of the Classic of Documents. All these systems ultimately derive from the cosmic principle of "Heaven's one and Earth's two". Without the foundational River Chart as their source, none of these profound doctrines could have been formulated.

This passage discusses the correspondence between the Hetu (River Chart) and Luoshu (Luo Writing)—two ancient mystical diagrams said to have emerged from the Yellow River and Luo River respectively—and their relationship with the Bagua (Eight Trigrams) and Wuxing (Five Phases). Zhu Xi, the Southern Song Dynasty Neo-Confucian scholar, interpreted this as follows:

The Hetu's total number of 55 represents a fixed foundational pattern, while the Luoshu's 45 symbolizes transformation. The Hetu embodies the principles of the "Nine Categories" (Jiuchou, the nine governing principles from Yu the Great's flood control), whereas the Luoshu manifests the operational logic of the Five Phases. The Hetu signifies eternal laws, while the Luoshu represents dynamic principles. The "generating numbers" (1-5) govern the "completing numbers" (6-10) to form stable structures, while odd numbers dominate even numbers to achieve transformative functions.

Specifically: The Hetu takes generating numbers as its core, while the Luoshu prioritizes odd numbers. Outwardly, the Hetu can deduce the Eight Trigrams, while internally it corresponds to the Nine Categories; conversely, the Luoshu outwardly aligns with the Nine Categories while internally deriving the Eight Trigrams. The Hetu combines Yin-Yang through generating-completing number pairs, maintaining unity with hierarchy, whereas the Luoshu distinguishes Yin-Yang through odd-even differentiation, appearing oppositional yet concealing profound cyclical transformation.

Regarding directional correspondences: In the Hetu, numbers and Five Phases maintain fixed positions—1 and 6 belong to Water in the North, 2 and 7 to Fire in the South, 3 and 8 to Wood in the East, 4 and 9 to Metal in the West, and 5 and 10 to Earth at the center. This forms the foundational framework, while the Luoshu reveals its transformational patterns. Notably, Earth, Water, and Wood maintain identical positions in both diagrams, representing eternal principles: the central Heaven's 5 generating Earth symbolizes Taiji (Supreme Ultimate); the northern Heaven's 1 transforming into Water; and the eastern Heaven's 3 transforming into Wood—these reflect the fundamental laws of Yang energy's winter concealment and spring emergence, corresponding to the intrinsic nature of generating numbers.

As for the shifting positions of Fire and Metal: While 2 and 7 in the South normally represent Fire's position, during summer the Qian Metal (Yang Metal) manifests at the Si position (southeast), representing the unique state where Yang energy reaches its zenith to nourish all things. This variation demonstrates dynamic regulation within fixed patterns.

This passage primarily discusses the relationship between the Hetu (River Chart) and Luoshu (Luo Writing)—two ancient mystical diagrams symbolizing the operation of heaven and earth—and the transformations of yin and yang in the cosmos.

Just as the sovereign governs from the Bright Hall when the sun rises, the Lesser Yang (nascent yang energy) emerges from the southeast (Xun position). When the Water trigram (Kan, representing north) manifests through the numbers two and seven, it must yield to the supreme position of the Heaven trigram (Qian, representing the celestial father) and retreat to the west. By autumn, the Qian and Dui trigrams return to the west, while the Xun and Kan trigrams revert to the south, each following their inherent cyclical patterns.

The fundamental principle of heaven and earth is to support yang and restrain yin, reflecting the cosmos' inherent care for life. The Hetu (symbolizing the static structure of heaven and earth) takes ten as its foundation—where yang dominates but odd numbers are insufficient—forming complementary counterparts between heaven and earth. The Luoshu (symbolizing dynamic transformation) takes nine as its foundation—where odd numbers dominate but even numbers are insufficient—showing heaven enveloping earth in functional manifestation. Thus, yin numbers (even) are restrained to form twenty, while yang numbers (odd) are supported to manifest externally. Seven and nine represent mutable numbers, while six and eight represent immutable numbers—through these, heaven and earth reveal their mysteries in the Hetu and Luoshu.

Master Cai Jiezhai (a Song Dynasty scholar of the Yijing) stated: "The Book of Changes emphasizes trigram imagery, while the Hongfan (a chapter in the Book of Documents) emphasizes numerical patterns." The Hetu employs even numbers representing complementary opposition, thus based on yin and yang energies. The Luoshu employs odd numbers representing fluid transformation, thus based on the Five Phases (Wood, Fire, Earth, Metal, Water).

Even numbers belong to stillness but function through movement—hence the Hetu's numerical pairs always sum to odd numbers (1+6, 2+7, etc.). Odd numbers belong to movement but function through stillness—hence the Luoshu's numerical pairs always sum to even numbers (1+9, 2+8, etc.). Therefore, the auspicious or inauspicious outcomes in the Book of Changes arise from transformation—the Hetu's movement originates from the self, enabling foresight. The Hongfan's outcomes manifest through stillness yet require movement for realization—the Luoshu's movement follows celestial patterns.

Between heaven and earth exist yin and yang energies, whose operation manifests as the Five Phases: 1-3-5-7-9 represent yang's movement (totaling 25), while 2-4-6-8-10 represent yin's movement (totaling 30). Without the Five Phases, transformation cannot occur; without yin and yang energies, autonomous operation is impossible. The key to determining yin-yang transformations and communicating with spiritual forces lies in the number "five"—the Five Phases—which constitutes the fundamental principle of reciprocal operation between celestial and terrestrial yin-yang.

This scripture primarily explains the patterns of Yin-Yang transformation and Fuxi's process of creating the Primordial Eight Trigrams (先天八卦) based on the River Chart (河图).

Regarding the numerical principles of Yin-Yang: Among the numbers one through ten, the odd numbers (1,3,5,7,9) belong to Yang, while the even numbers (2,4,6,8,10) belong to Yin. Arranged sequentially, the first five numbers ("generating numbers") represent Yang, while the latter five ("completing numbers") represent Yin. Numbers flow within Yin-Yang transformations before taking form, whereas phenomena (象) manifest only when Yin-Yang has solidified. The Taiji (Supreme Ultimate) discussed in the Yijing contains the principles of Yin-Yang but lacks concrete form—Yin governs stillness while numbers govern Yang's movement. Taiji itself transcends movement and stillness yet primarily embodies stillness. Prior to Taiji, phenomena and numbers were undifferentiated; when Taiji moves to generate Yang, numbers emerge, and when it stills to generate Yin, phenomena take shape. The division and interaction of Yin-Yang exhibit distinct characteristics: from the perspective of essence, stillness manifests as opposition (twofold), while movement manifests as flow (singular); from the perspective of operation, stillness unites Yin-Yang, whereas movement follows the sequence of Yang preceding Yin.

Regarding Fuxi's creation of the trigrams: Fuxi derived the Primordial Horizontal Diagram (平面八卦图) from the River Chart. This horizontal arrangement follows the sequence 1-2-3-4-5-6-7-8 from right to left, differing from the circular diagram's counterclockwise and clockwise directions. The eight trigrams divide into four pairs: Greater Yang (乾 Qian and 兑 Dui) in the first group, Lesser Yin (离 Li and 震 Zhen) in the second, Lesser Yang (巽 Xun and 坎 Kan) in the third, and Greater Yin (艮 Gen and 坤 Kun) in the fourth—arranged top to bottom. Viewed conversely from bottom to top and left to right, Greater Yin (艮坤) corresponds to 1-6, Lesser Yang (巽坎) to 2-7, Lesser Yin (离震) to 3-8, and Greater Yang (乾兑) to 4-9. Generating one solid (odd) and one broken (even) line on Greater Yang produces 乾兑; doing so on Lesser Yin yields 离震; on Lesser Yang creates 巽坎; and on Greater Yin forms 艮坤. This horizontal diagram inherently follows the 1-8 sequence of trigram generation.

Fuxi also created the Primordial Circular Diagram (圆形八卦图) based on the River Chart, though the text does not elaborate on its specific construction method.

In explaining the arrangement of the sixty-four hexagrams of the Zhouyi (Book of Changes), Zhu Xi (Master Zhu) noted that in the horizontal diagram, the Zhen (Thunder), Xun (Wind), Fu (Return), and Gou (Encounter) hexagrams occupy the central positions. By folding and flipping the middle section, a circular arrangement emerges—starting from Zhen and Fu in reverse order toward Qian (Heaven), then proceeding from Xun and Gou in forward order toward Kun (Earth). This forms a circular hexagram diagram. Such an arrangement fully manifests the cyclical patterns of the four seasons (spring, summer, autumn, winter), the waxing and waning of the moon (new moon, first quarter, full moon, last quarter), and the alternation of day and night (dawn, noon, dusk). This is the core principle behind constructing hexagram diagrams. He further pointed out that while horizontal diagrams follow sequential order, the transformation into circular diagrams requires reverse sequencing—only half of the circular diagram is arranged in reverse (details can be found in the Left and Right Chapters).

Regarding the correspondence between the Xiantian (Earlier Heaven) Bagua (Eight Trigrams) and the Luoshu (Luo River Writing, an ancient mystical numerical diagram):

• Qian (Heaven) corresponds to the Greater Yang number nine in the due south,
• Dui (Lake) corresponds to number four in the southeast,
• Li (Fire) corresponds to the Lesser Yin number three in the due east,
• Zhen (Thunder) corresponds to number eight in the northeast,
• Xun (Wind) corresponds to the Lesser Yang number two in the southwest,
• Kan (Water) corresponds to number seven in the due west,
• Kun (Earth) corresponds to the Greater Yin number one in the due north,
• Gen (Mountain) corresponds to number six in the northwest.

In the circular arrangement, the four Yang trigrams—Qian, Dui, Li, Zhen—are placed on the left, while the four Yin trigrams—Xun, Kan, Gen, Kun—are on the right. The Luoshu similarly positions Qian, Kun, Kan, and Li at the four cardinal directions, while Zhen, Dui, Xun, and Gen occupy the four corners. The Xiantian Bagua and Luoshu largely align in their trigram placements, with the only difference being the interchange of Greater Yang (nine) and Lesser Yang (two) in the southwest, reflecting a transformation of Yin and Yang.

As for the correspondence between the Houtian (Later Heaven) Bagua and the Hetu (Yellow River Diagram, another ancient numerical diagram):

• Kan (Water) corresponds to numbers one and six in the north,
• Li (Fire) corresponds to numbers two and seven in the south,
• Zhen (Wood) corresponds to number three in the due east,
• Xun (Wind) corresponds to number eight in the southeast,
• Dui (Metal) corresponds to number four in the due west,
• Qian (Heaven) corresponds to number nine in the northwest,
• Gen (Mountain) corresponds to number five in the northeast,
• Kun (Earth) corresponds to number ten in the southwest, with both passing through the central Earth element.

The numerical distribution of the Houtian Bagua perfectly matches the Hetu system. Some scholars, seeing numbers two and seven in the south attributed to Li or numbers four and nine in the northwest attributed to Qian and Dui, mistakenly associate the number ten with the Luoshu. Others, observing number nine in the south as Qian or numbers two and seven in the southwest as Xun and Kan, erroneously claim that nine belongs to the Hetu, even suggesting that King Wen derived the Houtian Bagua from this, leading to confusion between the two diagrams. This mistaken view (promoted by Liu Mu) has been widely circulated, and scholars should carefully discern the truth.

Regarding the Yin-Yang odd-even patterns of the Hetu and Luoshu:

The Hetu combines one, two, three, and four with seven, eight, nine, and six to form ten.

The Luoshu pairs one, two, three, and four with seven, eight, nine, and six to also form ten.

Here, "ten" is essentially the combination of two and five (two times five equals ten).

This passage primarily discusses the generative relationship between the Hetu (River Chart) and Luoshu (Luo Writing) with the Bagua (Eight Trigrams) and Nine Categories, as well as the corresponding numerical principles of the Five Phases (Wuxing).

Regarding the correspondence between numbers and directions: The Greater Yang (pure yang) number one resides in the north, while its completed form nine actually manifests in the west; the Lesser Yin number two resides in the south, with its completed form eight appearing in the east; the Lesser Yang number three resides in the east, with its completed form seven manifesting in the south; the Greater Yin (pure yin) number four resides in the west, with its completed form six appearing in the north. These numbers mutually generate correspondences—one and nine oppose each other across north-south, three and seven oppose across east-west, forming the four cardinal directions. The four corners also correspond: northeast pairs with southwest as two and eight, southeast pairs with northwest as four and six, creating a crisscross pattern of warp and weft.

Both the Hetu and Luoshu conceal the number ten (representing the Five Phases centered), manifested through their mutual generation and completion. Dong Zhongshu (of the Han dynasty) interpreted this solely through the directional and numerical aspects of the Four Symbols, which remains a valid perspective.

Zhu Xi held that Fuxi first obtained the Hetu and thereby drew the Bagua, initially without needing the Luoshu; Yu the Great independently obtained the Luoshu to establish the Nine Categories (the nine great methods for governing the world), without needing to trace back to the Hetu. Zheng Xuan stated that sages based the Bagua on the Hetu and arranged the Nine Categories according to the Luoshu—this wasn't mere imitation but a convergence of principles. When Fuxi derived the trigrams from the Hetu, he didn't foresee the Luoshu, yet the numbers naturally aligned; when Yu established the Nine Categories from the Luoshu, he didn't trace back to the Hetu, yet the principles naturally converged. The Hetu corresponds to the Earlier Heaven Bagua, the Luoshu to the Later Heaven Bagua—though their directional sequences differ and their numbers vary between ten and nine, both fundamentally root in the Five Phases, unified by the central principle of "five."

Finally, the transmission lineage is noted: Chen Tuan (Master Xiyi) passed the numerical philosophy to Mu Xiu (styled Bochang), who transmitted it to Li Zhicai, who then passed it to Shao Yong (Shao Kangjie), who consequently composed the Huangji Jingshi Shu (Book of Supreme World Order). Mu Xiu also transmitted the Taiji Diagram to Zhou Dunyi (styled Lianxi). Zhu Zhen (styled Hanshang) recorded that Zhou Dunyi transmitted it to the Cheng brothers (Cheng Hao and Cheng Yi), while Zhang Zai also lectured under their school. This outlines the numerical-philosophical transmission system of Song dynasty Neo-Confucianism.

This passage primarily discusses the transmission and controversies surrounding the Hetu (River Chart) and Luoshu (Luo Writing), two mystical diagrams of paramount importance in Daoism. Tradition holds these diagrams were borne by divine creatures emerging from the Yellow River and Luo River respectively.

Historical records present multiple accounts of their transmission. Zhou Dunyi authored the Tongshu, while Cheng Yi composed the Yizhuan, both documenting related theories. This knowledge was later transmitted to Chong Fang, who passed it to Xu Jian of Lujiang, establishing a southern lineage. Some attribute the earliest comprehension of the innate square and circular diagrams to the Ma family, but Zhu Xi (Master Zhu) refuted this, asserting these diagrams were personally created by Fuxi—containing only trigrams without written characters. After Confucius and Mencius, these diagrams were secretly preserved by fangshi (esoteric practitioners) until Chen Tuan (Xi Yi) reintroduced them to the world.

Cai Yuanding (Xishan) noted that records from Kong Anguo and Liu Xin of the Han dynasty, through Guan Lang (Zi Ming) of the Northern Wei, to Shao Yong (Master Kangjie) of the Song dynasty all maintained consistent interpretations. However, Liu Mu later reversed their nomenclature, a mistake subsequently adopted by many scholars. Contemporary scholarship advocates restoring the original designations.

Ancient records from Kong Anguo, Liu Xiang and his son, to Ban Gu uniformly state the Hetu was bestowed upon Fuxi while the Luoshu was granted to Yu the Great. Both Guan Lang and Shao Yong identified the ten-number diagram as Hetu and the nine-number diagram as Luoshu. The nine-palace arrangement—"nine above, one below, three left, seven right, two and four as shoulders, six and eight as feet"—precisely mirrors a tortoise shell's pattern. Zhu Xi cited explicit evidence from the Da Dai Liji's "Mingtang Chapter," where Zheng Xuan's commentary confirms the numerical arrangement imitated tortoise shell markings, proving Han scholars indeed regarded the nine-number diagram as Luoshu.

Yet Liu Mu inverted this, declaring the nine-number diagram as Hetu and the ten-number as Luoshu. He claimed these diagrams originated from Chen Tuan without textual records, transmitted from Xu Jian to E Chang, then to himself. This interpretation gained wide circulation, appearing in Zhu Zhen's (Hanshang) Yizhuan, Zhang Xingcheng's Qiyi, and works by the Puyang Zheng family. Only through Zhu Xi and Cai Yuanding's meticulous scholarship was the traditional view—ten as Hetu, nine as Luoshu—restored and universally accepted. Nevertheless, Liu Mu's interpretation retained some influence, with later scholars occasionally referencing his views.

Liu Mu cited the Yizhuan to argue both diagrams originated in Fuxi's era, denying later transmission narratives. This debate reveals substantial divergence in ancient understandings of these mystical diagrams' provenance.

This passage examines the origins of the Eight Trigrams (Bagua) from the Book of Changes and the Nine Categories from the Hong Fan (ancient principles of statecraft). The author argues that while the sixty-four hexagrams were systematically arranged during King Wen of Zhou's era, the prototype of the Eight Trigrams existed much earlier, dating back to the time of Fu Xi. Similarly, though the Nine Categories were transmitted by Jizi, their core concepts were already evident when Emperor Shun established the nine official positions.

Regarding the controversial Hetu (River Chart) and Luoshu (Luo Writing)—legendary celestial diagrams—some scholars question whether the Luoshu truly emerged during Yu the Great's flood control efforts. However, the author notes that the Book of Documents explicitly states "Heaven bestowed upon Yu the Nine Categories of the Hong Fan," confirming their divine revelation as principles of governance during Yu's reign. The text mentions that during the imperial examinations of the Xinwei year, Liu Mengjian achieved top honors by supporting this view, while those adhering to traditional interpretations of the Hetu and Luoshu ranked lower.

The discussion then turns to the relationship between numbers and trigrams: the Hetu totals fifty-five, with forty-five being practically applied (as the sum of numbers in the four directions and center always equals ten); the Luoshu totals forty-five, with thirty-six in actual use (since each pair of trigrams sums to nine). The author emphasizes the principle of Yin-Yang harmony, noting that the four cardinal directions represent Yang (odd numbers) while the four corners embody Yin (even numbers), and only through their interaction can all things be generated.

Finally, the symbolic meaning of the number "one" is expounded: in heaven it represents the Supreme One (Taiyi), in the celestial stems it corresponds to Jia, in trigrams it is the central line of the Kan trigram, in solar terms it aligns with the winter solstice, among the Five Phases it belongs to Water, and in music it signifies the Yellow Bell pitch. The author stresses that "one" is the foundation of all things—no numerical operation can exclude it. Just as subtracting one from forty-nine leaves forty-eight, the "one" persists in new forms within the remainder, demonstrating its eternal nature as the fundamental principle.

This passage from Tianyuan Fawei discusses how ancient sages believed in a fundamental principle (holding the mechanism of heaven and earth) that, once grasped, could reveal the essence of all things. The author finds this view profoundly insightful and therefore deliberately records it in the concluding section on Xiangshu (the ancient study of using symbols and numbers to explain cosmic patterns). At that time, this theory was as popular as the Hetu and Luoshu (legendary mystical diagrams bestowed by heaven), though some mistakenly applied them in reverse order. The author cautions students of Daoist learning to remain mindful of this distinction, emphasizing the importance of understanding the correct underlying principles.