《易圖通變》

The first volume of Yitu Tongbian (Comprehending Change Through the Diagrams of the Yi), authored by the Linchuan Daoist priest Lei Siqi, primarily explores the relationship between the Hetu (River Diagram) and the numerical principles of the Yijing (Book of Changes). Here, the explanation focuses on how the Hetu reveals cosmic patterns through the transformations of Yin and Yang.

The numerical logic of the Hetu originates from the Dao of Yin-Yang. The foundation of the Yijing lies in the alternation of Yin and Yang, concretely manifested in the interaction and separation of odd (Yang) and even (Yin) numbers. The pairing of Yin and Yang, odd and even, may appear intricate yet follows a coherent order—this is what the Yizhuan (Commentaries on the Yi) refers to as the principle of "Heaven one, Earth two, up to Heaven five, Earth ten."

The Yijing speaks of the Taiji (Supreme Ultimate) as the most fundamental "One." This "One" divides into the two modes (Liangyi) of Yin and Yang (Two), which then develop into the Four Symbols (Xiang)—the four generative numbers (one, two, three, four)—and ultimately form the Bagua (Eight Trigrams), represented by the four completion numbers (six, seven, eight, nine). Here, odd numbers signify Yang and are distributed in the four cardinal directions, while even numbers signify Yin and occupy the four intermediate positions. Yin and Yang are interdependent: the Yang generative numbers one and three require the Yin numbers six and eight to complete them, while the Yin generative numbers two and four need the Yang numbers seven and nine to fulfill them.

From the perspective of directional arrangement: the progression from the north to the east (one to three) follows the sequence of Yang Qi increasing, whereas the movement from the east back to the north (eight to six) marks the retreat of Yin Qi. Similarly, the shift from the west to the south (two to four) follows the expansion of Yin Qi, while the return from the south to the west (nine to seven) signifies the withdrawal of Yang Qi. These patterns of Yin and Yang’s advance and retreat are all embedded within the Hetu.

Although the numbers Heaven five and Earth ten also belong to the categories of Yin-Yang odd and even, they are not directly manifested in the four cardinal directions. This is because the Four Symbols do not include five, and the Bagua does not include ten. Yet the Yijing’s ability to encompass all things without omission precisely relies on these two virtual numbers—Heaven five and Earth ten—which serve to unify and govern the four directional numbers.

This passage explains the correspondence between the Eight Trigrams (Bagua) and numbers in the River Chart (Hetu), as well as the patterns of Yin-Yang transformation. The author criticizes the then-popular practice of placing empty dots for "five" and "ten" at the chart's center as erroneous, since this fails to demonstrate the numbers' true functional relationships.

Specifically: The Kan trigram (representing water) begins with one in due north. When one adds five, it yields the Qian trigram's six in the northwest. The Kun trigram (representing earth) starts with two in the southwest, and two plus five produces the Dui trigram's seven in due west. The Zhen trigram (representing thunder) originates with three in the east, and three plus five forms the Gen trigram's eight in the northeast. The Xun trigram (representing wind) commences with four in the southeast, and four plus five generates the Li trigram's nine in due south. Thus when Yang numbers receive five they transform into Yin numbers, and when Yin numbers receive five they become Yang numbers—this illustrates how generative numbers become completion numbers, which in turn produce new configurations.

The generative numbers (one through four), being less than five, require no mediation from the central number. Therefore the Yang numbers one and three naturally form patterns beyond the completion numbers—Qian's six and Gen's eight—creating a state where Yin dominates Yang, hence Kan trigram is called "the abyss." The completion numbers (six through nine), being greater than five, cannot self-regulate their balance. Thus the Yin numbers two and four must take five to form the Yang portions within the completion numbers—Li's nine and Dui's seven—creating a state where Yang is upright while Yin is oblique, hence Li trigram is called "the clinging."

The text further explains the movement patterns of Yin-Yang numbers: Yang numbers progress clockwise from Kan in the north to Zhen in the east, reaching Xun's four; Yin numbers advance counterclockwise from Kun's two in the southwest to Zhen's three in the northeast. This corresponds to the fourth month's transition from Qian trigram (pure Yang) toward Xun. In their retreating phase, Yang numbers move from Li's nine in the south to Dui's seven in the west, while Yin numbers withdraw from Gen's eight in the northeast to Qian's six in the northwest. This mirrors the tenth month's shift from Kun trigram (pure Yin) toward Qian, hence Qian appears without Xun, and Dui's "joy" stands in opposition to Qian's "conflict."

These numerical transformations reflect nature's rhythms of waxing-waning Yin-Yang and seasonal cycles, representing cosmological patterns ancient sages derived through celestial observation.

This passage explains the generative principles of numbers and hexagrams in the Yijing (Book of Changes), using numerical transformations to illustrate the cosmic order of creation.

The primordial celestial number begins with "one," representing the complete state of Taiji (Supreme Ultimate) as pure Yang's upright Qi. When "one" divides into "two," Taiji differentiates into the two polarities of Yin and Yang, manifesting Yin's partial nature. Adding "one" and "two" yields "three," signifying Yang Qi's dominance; the transition from "three" to "four" sees Yin Qi following to form even numbers.

The pairing of "one" and "four" produces "five," as does the pairing of "two" and "three." Combining two "fives" gives "ten." Here, the "celestial five" (central number) interacting with "five" forms the basis of the term "canwu" (mutual reference). The ancient character for "five" was written as "乂," whose pictograph evolved into the modern character "十" (ten).

Through numerical transformations, the generative numbers (1-5) produce the completion numbers (6-10). For example: the Kan (Water) hexagram is one, Xun (Wind) is four, combining to five; Qian (Heaven) is six, Li (Fire) is nine, combining to fifteen. Kun (Earth) is two, Zhen (Thunder) is three, combining to five; Dui (Lake) is seven, Gen (Mountain) is eight, combining to fifteen. Thus, the first four numbers (1-4) generate ten, while the latter four (6-9) generate thirty. Hence, the numerical logic of the Hetu (River Chart) culminates at forty; adding the functionally empty celestial five and earthly ten completes the cosmic total of fifty-five.

The generative numbers two and five form ten—though divided into Yin and Yang, both rely on Yang Qi of generative numbers to carry Yin Qi of completion numbers, sustaining the north and east while centering the core. Two fifteens form thirty, where Yin and Yang mutually transform, with Yin Qi of completion numbers nurturing Yang Qi of generative numbers, sustaining the south and west while activating the center. The principle of establishment resides in emptiness while manifesting substance; the operation of law displays substance while employing emptiness.

The first four numbers interweave to form the Four Symbols (lesser Yang, greater Yang, lesser Yin, greater Yin), with the central five forming ten as the generative numbers' essence. The latter four numbers encompass the Bagua (Eight Trigrams), where five's four directions form twenty as the completion numbers' function. Two fives combine into ten, interweaving to form twenty's differentiated forms; these differentiated forms are then unified by five's separations and unions into the celestial five's holistic function. Ten and five cycle mutually—five as substance, ten as function.

Applied to hexagrams: Kan (one) and Li (nine) sum to ten; Kun (two) and Gen (eight) sum to ten; Zhen (three) and Dui (seven) sum to ten; Xun (four) and Qian (six) sum to ten—all representing pairings of one generative and one completion number. Among these, one always pairs with nine, and three with seven, symbolizing the ten of the four directional Yang numbers; two pairs with eight, four with six, representing the ten of the four directional Yin numbers. These numerical harmonies embody the cosmic principles of Yin-Yang mutual generation and odd-even correspondence.

This passage explains the Yin-Yang correspondences between numbers and directions in the Book of Changes (Yijing), illustrating how they manifest the patterns of celestial and terrestrial movement. The number six represents the distribution of Yin and Yang across the four directions (or dimensions), with four pairs of Yin-Yang combinations forming a structure of ten Yin and ten Yang. Originally, four multiplied by five equals twenty, but through the pairing of forty-four with five, the complete numbers (chengshu) are allocated within this framework. Meanwhile, the combination of fourteen and five integrates the dynamic potential (xu state) of the generative numbers (shengshu).

When five pairs with five to make ten, and ten pairs with ten to return to five, these numbers interweave vertically and horizontally, forward and reversed, ultimately converging on the central number fifteen. Take the interaction between the Kan trigram (representing Water) and the Li trigram (representing Fire) as an example: Kan's single Yang line resides in the northern Yin position, symbolizing the initial emergence of Yang qi; Li's nine Yang lines occupy the southern Yang position, representing the zenith of Yang that will soon transform into Yin. This exemplifies the Explaining the Trigrams (Shuogua Zhuan) principle of "water and fire complementing rather than overcoming each other," corresponding to the Yin-Yang transitions at the winter and summer solstices.

Other numerical pairings also have their correlates: two with eight corresponds to the Kun (Earth) and Gen (Mountain) trigrams; three with seven aligns with Zhen (Thunder) and Dui (Lake). Yang qi ascends from below, moving from the northeast to the south, while Yin qi descends from above, flowing from the southwest to the north. This advance and retreat of Yin and Yang creates the warp and weft of cosmic operation, much like the seasonal transitions where mountain and lake qi communicate during the spring and autumn equinoxes, enabling all things to develop through winding yet unobstructed pathways.

The pairing of four and six corresponds to Xun (Wind) and Qian (Heaven), the pivotal points of Yin-Yang cyclical completion. Yang qi travels from Qian through Kan and Gen to converge in Zhen, then gathers forward in the southeastern Xun; Yin qi moves from Xun through Li and Kun to disperse in Dui, then scatters backward in the northwestern Qian. Yang's generative number begins at one, accumulates to three, and totals four, unified by Xun; Yin's generative number starts at two, accumulates to four, and totals six, synthesized by Qian. These numerical multiples correspond to temporal patterns like the eight solar terms and twelve months, forming the foundational framework of the sixty-four hexagrams.

This passage elucidates the numerical transformations within the Bagua (Eight Trigrams) and the mutual conversion patterns of Yin-Yang.

The Qian trigram (representing Heaven) corresponds to the number six. Although six can be formed by combining two and four, it can also result from the union of one and five. When five (a generative number) combines outwardly with one, it yields six (a complete number), inherently embodying the principle of generation and transformation. When the Qian trigram merges with the Kan trigram’s (representing Water) one at the central position, Qian transforms into nine, taking on the characteristic of Laoyin (extreme Yin).

The Xun trigram (representing Wind) employs only the generative numbers one and three, constrained by the Yang number four. With insufficient Yang Qi (vital energy) to achieve equilibrium, it must borrow the central five from the Li trigram (representing Fire) to balance and correspond with Kan. The combination of four and five reaches the peak of Yang Qi, where things reverse upon reaching their extremes. Thus, the interaction between Xun and Li causes the Laoyang (extreme Yang) nine to transform into six.

The opposition between Qian and Xun forms ten, illustrating how nine and six undergo mutual conversion. However, the dynamics of separation and union between Qian and Xun extend beyond this. The Zhen trigram (representing Thunder) three and Xun four combine, akin to the interaction of thunder and wind as depicted in the Heng hexagram. Xun corresponds to the fourth month, as previously mentioned. The Dui trigram (representing Lake) seven separates from Qian six, with Qian in the northwest corresponding to the tenth month, which cannot be explained simply by the Qian-Xun opposition as ten.

Qian belongs to Yang but holds the Yin number six, positioned before Kan; Dui belongs to Yin but holds the Yang number seven, positioned after Kun (representing Earth). Thus, in the numerical operations of Qian and Dui (six and seven), both borrow five for transformation. Qian’s six precedes the "three and five" arrangement with Qian positioned before Kan, where Qian’s one emerges as Kan’s one. Dui’s seven follows the "three and five" arrangement with two Dui trigrams positioned after Kun, where Kun’s two enters to become Dui’s two.

Although Qian and Dui are distinguished by their complete numbers six and seven, Kan and Kun interact through their generative numbers one and two at the center. Dui is not called two, while Kun is called two; Kan is called one, while Qian is not called one. Qian does not claim the number one to demonstrate its transcendence beyond numerical constraints, encompassing the generative functions of five and ten within the Bagua, representing the wholeness of Taiji (Supreme Ultimate).

Numerically, Heaven is one and Earth is two, each occupying its proper place. In terms of Qi, it divides into Yin and Yang, whose mutual interaction drives the workings of the Yi (Change). The increasing number of Yang corresponds to the decreasing number of Yin—this is how Yin and Yang transform into one another.

This passage explains the principles of Yin-Yang transformation and hexagram formation in the Book of Changes (Yijing). The mutual transformation and perpetual regeneration of Yin and Yang constitute the most profound patterns of change in the cosmos. The Eight Trigrams (Bagua) are composed of three lines (yao), and through different combinations, they form the sixty-four hexagrams, each ultimately consisting of six lines.

It specifically mentions that the Qian hexagram (pure Yang) is represented by the number "nine," while the Kun hexagram (pure Yin) is represented by the number "six." Qian symbolizes the father, and Kun the mother, with the other six trigrams representing their children. In the hexagram structure, the number five occupies the central position and must align with the number nine to establish this center. Six represents the beginning of Yin numbers, while nine signifies the culmination of Yang numbers. Six still has a defined position within the hexagram, whereas nine transcends positional limits—together, they embody the dynamic interplay of Yin and Yang through their complementary emptiness and substance.

The text further unravels the mystery of numbers: Heaven's one (generating water) combines with Heaven's five (earth) to form six, while Earth's two (generating fire) combines with Earth's four (metal) to also yield six. Five and six occupy the transitional midpoint in the sequence of ten numbers, corresponding to the ancient divination principle of centering on the celestial stems Wu and Ji. This reveals the eternal law of interdependence and transformation between Yin and Yang, akin to parents giving birth to children—all things in the universe follow this fundamental generative principle.

This passage explains the correspondence between numbers and trigrams in the Book of Changes (Yijing), as well as the principles of Yin-Yang transformation. The Heavenly Stems Wu (fifth) and Ji (sixth) correspond respectively to the central positions of the Kan trigram (representing Water) and the Li trigram (representing Fire), with the numbers five and six symbolizing the mediating force between Heaven and Earth.

The phrase "dragons battle in the wilderness" from the Kun trigram (representing Earth) signifies that Yin energy requires the complement of Yang, illustrating how pure Yin cannot endure alone. Conversely, "the overbearing dragon will have cause to repent" from the Qian trigram (representing Heaven) indicates that excessive Yang energy also leads to decline, demonstrating the necessity of Yin-Yang balance. The numbers one, five, and nine mark the beginning, middle, and culmination of Heaven's numerical sequence, with nine representing Heaven's ultimate number.

The Hetu (River Chart), an ancient mystical numerical diagram, was originally designed for the eight primary trigrams but applies equally to the sixty-four hexagrams. All trigrams and hexagrams arise from permutations of Yin-Yang odd and even lines. Numbers one through four are "generative numbers" (the foundational elements of creation), with five serving as a harmonizing force; six through nine are "completion numbers" (the finalizing elements of creation), with ten acting as the unifying summation.

The four cardinal directions (east, south, west, north) each contain ten numbers, totaling forty; the four intermediate directions each contain five numbers, totaling twenty. Five, as the central number, is distributed across all directions, while ten integrates the whole. Among the completion numbers (six, seven, eight, nine), Yang numbers dominate; beyond the generative numbers (one, two, three, four), Yin numbers prevail, reflecting the waxing and waning of Yin and Yang. Though five is not positioned at the absolute center, it permeates all directions, while ten consolidates all generative numbers.

This passage explores the profound mysteries of Yin-Yang transformations and numerical symbolism in the Book of Changes (Yijing). Though opposing forces, both Yin and Yang follow the principle of "central harmony"—where Yang represents the orthodox position and Yin the unorthodox, yet both achieve equilibrium by maintaining the Middle Way. The number five (Five Phases) establishes its orthodox position while residing in emptiness, while the number ten (Heavenly Stems) maintains neutrality beyond physical form—both exemplifying the Middle Way principle governing Yin-Yang alternation.

Within Yin and Yang emerge new layers of Yin-Yang, much like substance and function mutually containing each other. Opposites such as emptiness-fullness and beginning-ending continuously cycle and transform, creating an endless process of change. The number twenty-five (five multiplied by five) represents heavenly numbers, while thirty (five multiplied by six or three multiplied by ten) symbolizes earthly numbers. The numerical pairs one-nine, two-eight, three-seven, and four-six each sum to ten. Through the interplay of five and ten, they ultimately form the "Great Expansion number fifty" (the fifty yarrow stalks used in divination). Yet the most fundamental "One" is concealed within "Five," its function hidden within the formless Dao of Heaven—transcending even the numerical frameworks of fifty and fifteen.

The true marvel lies in the "usefulness of the useless"—what appears unused is precisely the key to applying the Great Expansion numbers. The alternation of Yin and Yang is the Dao. Through the interwoven combinations of three and five (complex alternations), the fifty-five numbers of Heaven and Earth manifest transformation, commune with spirits, and thereby reveal the patterns and essential movements of all things in existence. Fuxi (the legendary Fu Xi) created the Yijing by contemplating the numerical mysteries of the River Chart. Confucius, in the Commentaries on the Yijing, praised this principle as exquisitely subtle and infinitely transformative, ultimately attaining a divine and unfathomable state—the most wondrous aspect of the Book of Changes.

This passage is from Yitu Tongbian (Comprehending Change Through the Diagrams) by the Yuan dynasty Daoist scholar Lei Siqi, which explores the numerical mysteries in the operation of heaven-earth and the Hetu (the ancient mystical cosmic diagram).

The patterns of heaven-earth's operation are always inseparable from numbers, yet sometimes exceed numerical bounds while appearing insufficient at other times. It is precisely this balance of surplus and deficiency that constitutes the law of mutual constraint between yin-yang's odd and even numbers. The yin-yang numbers both separate and unite, containing both emptiness and substance, advance and retreat, forming the "Middle Way" (balanced Dao) through natural mutual generation and restraint. The saying "one yin, one yang is called the Dao" means yin-yang differentiate to form heaven above and earth below, yet combine to constitute Qian (Heaven) and Kun (Earth). Though heaven-earth divide into four directions, they can still unite; though Qian-Kun's yin-yang combine into four seasons, they can still subdivide—this is the same principle.

Viewed through form, heaven-earth take north-south as warp threads, with heaven above and earth below—the forms of water, fire, wood, and metal can be distinguished through diagrams. Thus the greatest symbolic pattern (faxiang) is none other than heaven-earth, whose vastness matches them. Viewed through qi movement, yin-yang take east-west as weft threads, with Qian as yang and Kun as yin—the seasonal transformations of winter, summer, spring, and autumn can be unified through principle. Thus the greatest transformation is none other than the four seasons, whose adaptability matches them. These two aspects share the same essence.

Within the four directions exists a center, yet the center has no fixed form—from this the four images (sixiang) and eight trigrams (bagua) are established. Within the four seasons are contained the Five Phases (wuxing), yet they have no fixed position—from this the four seasons and eight solar terms operate. Confucius did not directly record the Hetu in the Yizhuan (Commentaries on the Changes), but preserved the numbers of heaven-earth to show that by grasping numerical patterns, the Hetu's mysteries are naturally contained.

The Hetu's numerical generation follows: heavenly one generates earthly six; earthly two generates heavenly seven; heavenly three generates earthly eight; earthly four generates heavenly nine. As for the generation of heavenly five and earthly ten, this must be investigated through the Hetu itself. Placing five in the empty position (center) of the eight trigrams can still accommodate its number, but placing ten there becomes difficult. This shows that while we know numbers have generative patterns, we don't understand their underlying principles. If one only sees surface phenomena without comprehending the ancient sages' profound intent to "establish the heart of heaven-earth and set the standard for living beings," then even across millennia, it can only evoke sighs.

Note: The "Hetu" (River Chart) is a legendary mystical pattern said to have been borne from the Yellow River by a dragon-horse, paired with the "Luoshu" (Luo Writing), regarded in Daoism as secret diagrams revealing cosmic laws. The "four images" refer to lesser yang, greater yang, lesser yin, and greater yin—four emblematic hexagram states, while the "eight trigrams" are the fundamental Qian, Kun, etc.

This passage explains the relationship between numbers in the ancient Hetu (River Chart) and the cosmic interplay of Heaven and Earth through Yin-Yang principles. Odd numbers (1, 3, 5, 7, 9) represent Heaven, while even numbers (2, 4, 6, 8, 10) symbolize Earth. All things originate from Heaven but take form through Earth’s nourishment; what emerges from Earth requires Heaven’s power to reach completion. This dynamic reflects the waxing and waning of Yin and Yang Qi (vital energy), where odd and even numbers complement each other.

The Xici Zhuan (Commentary on the Appended Judgments) in the Yijing (Book of Changes) states that the number three represents Heaven and two represents Earth, forming the foundation of its numerical system. By observing the transformations of Yin and Yang, the trigrams were created, and through the interplay of hardness and softness, the lines (yao) emerged. The Yizhuan (Ten Wings) explains that by interweaving and combining the numbers three and five, one can comprehend the patterns of cosmic change, and by tracing these numbers, the forms of all things under Heaven can be determined. Only the most profound Dao of transformation could achieve such a feat—this is the mystical principle embodied in the Hetu.

Numbers begin with one, center on five, and culminate in ten. Five holds significance because it gives rise to the Four Symbols (Shaoyang, Taiyang, Shaoyin, Taiyin); ten represents completion because it integrates the eight directions. The primordial one is an undifferentiated whole, unusable in its raw state—only through five and ten can its function be manifested. Thus, the Four Symbols contain five, and the Eight Trigrams (Bagua) contain ten; these numbers have meaning but no fixed positions. Five generates transformation, ten establishes order, and the fundamental "one" is concealed within the balanced nature of five and ten.

The author recalls a childhood game called "Eight-Square Play," where the board consisted of a square divided by a cross. At the time, it seemed crude and simplistic, but only in his seventies did he suddenly grasp its profound meaning: people use it daily without understanding its principles, following it throughout life without comprehending its truth—just as the ancients said. The modern square "□" corresponds to the ancient circle "○," the current "five" mirrors the archaic "乂," and the cross within the square represents the number fifteen. Adding a horizontal line below the "十" forms the character for "earth" (土), symbolizing the central position. The vertical line signifies the north-south axis (Heaven above, Earth below), while the horizontal represents the east-west alignment (the fixed positions of Heaven and Earth)—precisely the cosmic order revealed in the Hetu.

The Western direction represents the weft line, dividing and arranging Yin and Yang in orderly distribution. When Yin and Yang are properly combined, the oblique sections of the four directions achieve balance, and the proper positions of the four directions are thereby established. The mutual advancement and generation of Yin and Yang Qi all follow the principles of five and ten to establish the Middle Way. Thus, the fifteen in the Hetu (an ancient mystical diagram) and the fifty in the Dayan (a divination method) both employ the numbers ten and five as functional concepts, though they have no corresponding physical positions in reality.

To understand why fifteen represents the Middle Way, we can begin with the idea that five occupies the center, while ten marks completion. Ten is essentially the cyclical return of one. By tracing ten back to one, the existence of one can be grasped through direct observation and inner realization.

Examining the development since the formation of the Yijing, only Sima Guang's Qianxu comes close to treating "ten" and the intersecting symbol "乂" as representing the central Earth phase. However, the Qianxu takes "乂" as its foundation and "ten" as its apex, arranging fifty-five numbers for naming purposes. In this system, the center is already a concrete entity, not a functional abstraction, which diverges from the Hetu's principle of abstracting five and entrusting ten. Although the total number in the Hetu is fifty-five, the actual tangible numbers amount to only forty. The remaining fifteen are abstractly embedded within the forty, serving only to unify calculations and not representing actual directional positions.

The terms can (three), liang (two), and can wu (three-five) all operate through numerical transformations. Can is the root of change between one and two. From one itself emerges two, and once opposition is formed, only the two are seen, while the original one is obscured. That "one" then manifests as "three" from within the "two," hence the term can. This method of can constitutes a small-scale completion, much like the establishment of the three-line trigrams, which depict humanity's existence between Heaven and Earth. Accumulating three small completions, and combining them three by three, aligns with the Yijing's "using the nine" of the Qian hexagram. This is the initial regulation of can and the reason why the odd numbers of Yang are associated with "Heaven."

Liang arises when one gives birth to two, transcending and standing side by side. It symbolizes both the Two Modes (Yin and Yang) and, through two "twos," the Four Images. Originally, from one comes two, and with can, it becomes three. Moreover, since "two" is inherently the counterpart of "one," it naturally yields two.

This scripture elucidates the mysteries of the numbers "three" and "six" in the Book of Changes (Yijing). The ancients used "three" to represent heaven (the Three Powers: heaven, earth, and humanity), and by combining two "threes," they arrived at "six"—corresponding to the principle of "Using Six" in the Kun (Earth) hexagram. Six lines form a complete hexagram image, and six yaolines constitute the hexagram system. This embodies the principle of even numbers representing yin and serves as the numerical foundation for the concept of "earth."

By intersecting and combining "three" with "two," one obtains "five"; combining two "fives" yields "ten." Although the number "ten" exceeds the conventional sequence of "three" and "nine," it is actually another manifestation of "one." Why is this so? Because whether it's the combination of "three," the pairing of "two," or the groupings of "five" and "ten," these are all human cultural constructs that nevertheless align with the natural patterns of heaven and earth.

The core idea behind this numerical system established by the ancients was to express the relationships between heaven, earth, and humanity through numerical arrangements. For example, the combination of "three times three equals nine" and "two times three equals six" forms the complete numbers of heaven and earth. Most crucial is that primordial "one" which isn't directly used initially—it operates hidden within "ten," circulating and transforming, simultaneously differentiating and unifying, becoming the central mechanism of the entire system.

Regarding the profound meaning of "interweaving threes and fives" (the intricate combinations of the numbers three and five): for numbers smaller than five, there's no need to use "five" for combination; for numbers larger than five, "five" serves as the median for interwoven combinations. As the Yizhuan (Commentaries on the Yijing) states, only by comprehending the laws of numerical transformation can one understand the patterns of all things between heaven and earth; only by exhausting the mysteries of numbers can one determine the forms of all phenomena under heaven. Although these principles are not easily grasped, since the ancient sages instructed us to study the patterns of "interweaving threes and fives," we should deeply investigate their underlying logic. From the Xici Zhuan (Great Commentary) onward, generations of Yijing scholars have continued to explore the wonders of these numerical combinations.

This passage explains the mathematical structure of the Hetu (River Chart) and Luoshu (Luo Writing) from the ancient Yijing (Book of Changes), which dates back two thousand years but has rarely been deeply studied by scholars. Through careful analysis, the author discovered that its mysteries are actually quite simple.

The numbers of Heaven and Earth each occupy five directional positions (referring to the Five Phases' orientations), and their interactions form the numerical arrangement of the Hetu. Specifically:

The number one in the northern Kan position (Water) pairs with the number six in the southwestern Kun position (Earth), summing to seven. When the celestial number five is added in the center, the arrangement reveals one number at each end. Thus, the one in Kan is a Yang number, representing the northern beginning, while the six in Qian is a Yin number, located in the northwest.

The number three in the eastern Zhen position (Thunder) pairs with the number eight in the northeastern Gen position (Mountain), totaling eleven. Adding the celestial number five in the center leaves three numbers at each end. The three in Zhen is a Yang number, representing the proper eastern position, while the eight in Gen is a Yin number, located in the northeast. Both groups follow the pattern of Yang first, then Yin.

The number two in the southwestern Kun position (Earth) pairs with the number seven in the western Dui position (Lake), summing to nine. Adding the celestial number five leaves two numbers at each end. The two in Kun is a Yin number, located in the southwest, while the seven in Dui is a Yang number, representing the proper western position.

The number four in the southeastern Xun position (Wind) pairs with the number nine in the southern Li position (Fire), totaling thirteen. Adding the celestial number five leaves four numbers at each end. The four in Xun is a Yin number, located in the southeast, while the nine in Li is a Yang number, representing the southern extreme. These two groups follow the pattern of Yin first, then Yang.

From this, we see that the numbers six, seven, eight, and nine in the Qian, Dui, Gen, and Li positions are simply the numbers one, two, three, and four (from Kan, Kun, Zhen, and Xun) plus five. A Yang number plus five becomes Yin, and a Yin number plus five becomes Yang; an even number plus five becomes odd, and an odd number plus five becomes even. For example, the six in Qian is one plus five, the seven in Dui is two plus five, the eight in Gen is three plus five, and the nine in Li is four plus five. This demonstrates the principle of Yin-Yang and odd-even transformation through the "variations of three and five" (san wu yi bian).

This passage explains the transformational patterns in the numbering arrangement of the Hetu (the mysterious diagram borne by the dragon-horse in ancient legend). The numbers are distributed according to the eight trigram (Bagua) positions—Qian, Kan, Gen, Zhen, etc.—interweaving from north to east, then from west to south. The numbers disperse and reunite through the "interplay of three and five" (sanwu), much like how the numbers one through four scatter apart, while six through nine recombine.

These numerical arrangements, whether vertical or horizontal, form naturally without artificial contrivance. The movement of yang numbers (odd) progresses from north to east to south, while yin numbers (even) follow the yang sequence. Yin numbers advance from the Kun position (southwest) with two to the Xun position (southeast) with four, then retreat from the Gen position (northeast) with eight to the Qian position (northwest) with six—yet yin always complements yang.

Specifically: the one in the Kan position pairs with six in the Qian position in the north; the two in the Kun position pairs with seven in the Dui position in the west; the three in the Zhen position pairs with eight in the Gen position in the east; and the four in the Xun position pairs with nine in the Li position in the south. This forms the fundamental principle of numerical pairing. Further adjustments are made—the numbers in the Zhen position align with Gen, and those in Xun align with Li—revealing the cosmic order of Yin-Yang through these triadic and dyadic combinations.

This interplay of three and five (sanwu) embodies what Confucius described as "penetrating their transformations" to form heaven-and-earth's patterns, and "exhausting their numbers" to determine the phenomena of the world. The entire system, through the coordination of three (san) and two (the dyad of Yin-Yang), manifests the fundamental laws of cosmic operation.

This passage explores the profound relationship between numbers and hexagram symbolism in the Yijing (Book of Changes). At its core, it reveals how the most subtle, dynamic, and miraculous forces between heaven and earth are embodied in the numbers of "three and two" (参两) and "nine and six" (九六).

The Qian hexagram (representing heaven) adopts three times the yang number three, using "nine" to govern the Dao of Qian's primal origin. Within its three yang lines actually lies hidden yin imagery—this is "yang containing yin." The Kun hexagram (representing earth) follows Qian's pattern, taking three times the yin number two to obtain "six," which gathers Kun's primal Qi. Within its yin line combinations hides yang imagery, forming "yin containing yang." It is precisely this mutual inclusion and transformative interplay between yin and yang that constitutes the fundamental principle of cosmic movement.

The Taiji (Supreme Ultimate) generates the Two Modes (yin-yang), which in turn produce the Four Images (lesser yang, greater yang, lesser yin, greater yin), and these give rise to the Eight Trigrams—all fundamentally accomplished through combinations of generating numbers (one, two, three, four) and completing numbers (six, seven, eight, nine). Though the numbers five and ten exist, they don't manifest directly in hexagram forms but function as abstract numbers.

Specifically: one divides into two, two divides into three, and three divides into four—thus establishing the Four Images. Five doesn't appear among them because "heaven's five" hangs suspended at the center without concrete form. Yet one plus four makes five, as does two plus three—this constitutes "two fives." Adding the original "heaven's five" forms "three fives." Summing one through four yields ten, and when mediated by "heaven's five," it produces fifteen—precisely the sum of each row in the Luoshu (a mystical numerical diagram).

Adding heaven's numbers (odd) one, three, and five yields nine; adding earth's numbers (even) two and four yields six. Through various combinations: 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—thus the Eight Trigrams take form. Although one plus nine makes ten, and two-three-four also sum to ten, the trigrams don't directly express ten because ten must return to one—essentially, ten is one.

In the Hetu (another mystical numerical arrangement), the sums of numbers in the four directions all equal ten: the Kan trigram (one) opposes Li (nine), with "heaven's five" invisibly mediating to form fifteen; Kun (two) opposes Gen (eight), Zhen (three) opposes Dui (seven), and Xun (four) opposes Qian (six)—all following the same principle. These numerical relationships reveal the deeper patterns of yin-yang waxing and waning, and the harmonious interaction of heaven and earth.

This passage elucidates the profound principles behind the numerical applications in the Hetu (the mystical diagram said to be carried by a dragon-horse in ancient legend). It explains that the numbers "Celestial Five" (Tianwu) and "Earthly Ten" (Dishi) lack concrete physical positions, serving merely as abstract numerical aids for calculation. While the Hetu actually employs forty numbers, through the complementary pairing of these virtual numbers (five and ten), it symbolically represents the total number fifty-five—signifying the complete count of heaven and earth.

The text clarifies that the number seven represents Lesser Yang (Shaoyang), while eight represents Lesser Yin (Shaoyin). Though their sum is fifteen, the generative numbers (生数) remain incomplete. Nine symbolizes Greater Yang (Laoyang), and six represents Greater Yin (Laoyin). Their sum is also fifteen, but here the generative numbers are complete (one, three, and five correspond to nine; two and four correspond to six), allowing six and nine to transform into each other. This is precisely why the Yijing (Book of Changes) exclusively uses nine and six for line transformations, never seven or eight.

The numbers in the four directions each total ten. Tripling nine yields twenty-seven, while doubling six gives twelve—summing to thirty-nine, leaving a deficit of one. This remaining "one" is preserved within ten to complete the count of forty. The Kan trigram (representing water) takes "one" as its foundation, gathering "ten" to establish the central principle. It both initiates the beginning of generative numbers and fulfills the completion of the formed numbers. Though this "one" may seem useless, it paradoxically governs the "useful" through its "uselessness," exerting immense influence.

Finally, the passage explains that while the Kan trigram's "one" does not directly participate in calculations like three-two or three-five, it forms the basis for establishing these operational rules. The "tripling of heaven" (参天) does not begin with Kan but rather with the Gen trigram (representing mountain). The entire system is built upon the central principle of numbers harmonizing with "ten," arranging them in such a way that seemingly discordant numbers achieve harmony at the center.

This passage explains the formation process and significance of "san" (the number three). It first clarifies that "san" is nothing other than the transformation of the previously mentioned one and two. When the original one divides to produce another one, these two ones stand in opposition, manifesting as two, while the original one becomes invisible. Yet this "one" re-emerges from the "two," forming "three"—hence the term "san" (to participate or join).

Though the described "san" represents only an initial, small-scale formation, all more complex numerical relationships (such as "liang san" meaning twice three, or "san wu" meaning combinations of three and five) develop from this fundamental principle. Through this example, the scripture demonstrates how "san" serves as the basic law governing numerical transformations.

(Note: This text reflects Daoist philosophical contemplation on numerical changes, positing that three arises from the dialectical relationship between one and two, forming the foundation for more complex numerical relationships. This thinking shares continuity with the image-number theory of the Yijing [Book of Changes].)

Daoist Master Lei Siqi of Linchuan, in his "Comprehensive Changes of the Yi Diagrams," explored the relationship between the Book of Changes (Yijing) and the natural world. He proposed that Daoists originally regarded the transformations of the sun and moon as embodying the essence of the Yi. While this view may appear commonplace, deeper study of the "Great Commentary" (Xici Zhuan) of the Yijing reveals its profound significance.

The Yijing states that the Qian (Heaven) and Kun (Earth) hexagrams contain the essence of the Dao of Change: when Qian and Kun are established, the Dao of Change operates within them; should they perish, the Dao would have no manifestation. Qian represents yang phenomena while Kun embodies yin phenomena. The interaction of yin and yang forms substantial entities with complementary strength and softness, through which the creative functions of heaven and earth are expressed and the De (Virtue) of the divine is communicated.

Specifically, opening the gate symbolizes Qian (yang), while closing it represents Kun (yin). This alternation of opening and closing constitutes change, and their ceaseless coming and going represents continuous penetration. What becomes manifest are called images (xiang), what takes physical form are called vessels (qi), establishing norms for their application constitutes method (fa), while their unnoticed daily use by common people demonstrates their numinous operation. Thus it is said: when heaven and earth take their positions, the Dao of Change circulates between them.

No symbolic representation surpasses heaven and earth, no transformational penetration exceeds the four seasons, and no suspended image shines brighter than the sun and moon. The vastness of the Dao of Change corresponds to heaven and earth, its transformative penetration matches the four seasons, while its yin-yang principles find correspondence in the sun and moon. The movement of these celestial bodies most perfectly manifests the completeness of the Dao of Change.

The Dao of Change equals heaven and earth in its capacity to encompass cosmic patterns: looking upward to observe heavenly phenomena (with Qian representing the sun) and looking downward to examine earthly configurations (with Kun representing the moon). The sun symbolizes the Li hexagram, its numerical principle culminating at nine; the moon symbolizes the Kan hexagram, its numerical principle originating at one. Nine necessarily pairs with six, while one combines with the celestial five to make six. The Kan one and Qian six together form the generative numerical principle of water, which constitutes the lunar symbolism.

Heaven and earth represent the substantial manifestations of Qian and Kun, while Qian and Kun embody the functional operations of heaven and earth. Although the positions of heaven and earth remain fixed, the ascending and descending movements of Qi and their formal transformations involve both constancy and change. While yin and yang possess constant functions yet elude grasp of their substance, only the sun and moon demonstrate the clearest relationship between substance and function. Therefore, the Dao of Change is particularly elucidated through these celestial luminaries.

This scripture primarily discusses the patterns of Yin-Yang transformation and celestial movement. The interplay of Yin and Yang resembles the relationship between the sun (Qian-Yang) and moon (Kun-Yin) - the sun representing day and the moon night, their alternating cycles creating the diurnal rhythm. Though ordinary people experience these changes daily, few contemplate their underlying principles.

Specifically, opening the gate (pihu) symbolizes Qian-Yang, akin to sunrise, the appearance of the waxing moon, or the equinox when day and night are balanced. Closing the gate (hehu) represents Kun-Yin, comparable to sunset, the waning moon's emergence, or the autumnal equinox with equal daylight and darkness. Observing celestial phenomena upward reveals the daytime sun, the full moon, and summer's long days; while contemplating patterns downward shows diurnal alternation, new moon phases, and winter's short daylight.

The sun and moon's alternating movements generate illumination, while the interplay of cold and heat forms the annual cycle. Transformation manifests through Yin and Yang's advance and retreat, hardness and softness embodied in day-night rotation. The celestial bodies' motion drives seasonal change - this constitutes heaven and earth's constant pattern (zhen guan). Sun and moon follow eternal luminous principles (zhen ming), while all under heaven's movements converge toward unified order (zhen fu yi). For example:

• Daily Yang energy first stirs at midnight
• Annual Yang resurgence begins at the winter solstice new moon
• The Kan trigram (moon) corresponds numerically to one, while Li (sun) to nine
• The monthly solar-lunar conjunction called "shuo" (new moon) etymologically indicates the moon's counter-directional movement (ni du) to meet the sun

The "Xici Zhuan" commentary on the Yi Jing states "change proceeds through reversal," while the Shang Shu records "harmony resides in lunar reversal" - both affirming the moon's retrograde principle. Through these phenomena, we may comprehend deeper laws governing celestial motion.

The hexagram transformations in the Book of Changes (Yijing) present clear patterns. The moon reaches its fullest phase on the 15th day of each month (the numerical phase of Lesser Yin, 7-8), then gradually wanes by month's end (the numerical phase of Greater Yin, 9-6). This complete lunar cycle corresponds precisely to the earthly number thirty, which also aligns with the principles of the Hetu (the ancient astronomical diagram). The sun travels 365¼ degrees annually, moving one degree per day, shifting southward in winter and northward in summer to create seasonal temperature variations. The moon moves approximately thirteen degrees daily, meeting the sun twelve times yearly to form the months. Through six Yin and six Yang months, it completes the cycle of four seasons—demonstrating the moon's arduous motion. Thus, the Yijing designates the Kan hexagram (Water) as the "toil hexagram," not only reflecting water's nature but also because its single solid Yang line (the middle line in Kan) does not follow the usual groupings of three-two or three-five. Instead, it stands alone as the primal symbol of movement and transformation.

Heaven and Earth are substantial forms, while Yin and Yang are their functions. Substantial forms carry the movement of Qi (vital energy), and Qi drives the transformation of forms. Heaven's Qi envelops Earth's form, while Earth's form supports Heaven's Qi—the two depend on each other. This relationship becomes evident through the "three-two" combinatorial principle (the numerical patterns of three and two): although the Qian hexagram (Greater Yang) belongs to Yang, it attaches to the Yin position of six in the northwest, the direction of decline. Though the Gen hexagram (Lesser Yang) is Yang, it occupies the Yin position of eight in the northeast, the direction of renewal. The solitary Yang line in the Kan hexagram symbolizes the most primordial vitality during the alternation of Yin and Yang, akin to the rebirth at the year's beginning or the month's start—embodying the Yi's true meaning of perpetual renewal.

The Xun hexagram (Lesser Yin), though positioned in the earthly Yin number four, resides in the southeast where Yang Qi ascends. The Kun hexagram (Greater Yin), though placed in the earthly Yin number two, remains in the southwest where Yang Qi weakens. These hexagram arrangements reveal that Earth's form wholly follows Heaven's numerical patterns of Qi movement, with Yin and Yang perpetually transforming each other in dynamic equilibrium.

This passage expounds the Daoist philosophy concerning the transformations of Heaven and Earth's Yin-Yang principles through numerical symbolism. The core teaching reveals:

When the Qi cycles (energetic patterns) of Heaven and Earth descend but have not yet reached equilibrium, one must comprehend their principles by integrating the dual characteristics of Heaven and Earth. Only when celestial bodies reach their zenith (the celestial pole) and the sun attains high noon does the complete operation of Heaven's laws manifest, wherein Earth's nature is simultaneously contained.

While we determine day and night lengths and lunar phases by observing the sun and moon's movements, how then can we truly grasp the essence of these transformations—the waxing and waning of Yin-Yang, the rhythms of ascent and descent? This requires contemplation from the fundamental perspective of numbers.

The number "one" (symbolizing Yang Qi) originates from the North (Kan position) and moves toward the center; "two" (symbolizing Yin Qi) resides in the Southwest (Kun position). Crucially, we must not simplistically assume Heaven (Yang) above and Earth (Yin) below represent fixed positions, for "one" emerges from the Kan position (North, water trigram) rather than directly corresponding to Qian (heaven trigram).

Through the interaction of "one" and "two" arises "three," representing humanity's mediating role between Heaven and Earth. Further pairings produce "four," symbolizing the four seasons. Yet these numbers remain symbolic rather than substantial, much like how the union of male and female generates life.

Though "five" occupies the central void position, it performs pivotal functions. When other numbers interact with "five": one paired with five forms six (Qian trigram), two with five forms seven (Dui trigram), three with five forms eight (Gen trigram), and four with five forms nine (Li trigram). This completes the trigram system—Qian embodying Heaven's laws and Kun manifesting Earth's principles—thereby establishing their proper positions.

In truth, numbers possess inherent pairing patterns (one with one, two with two, etc.), but their interaction with "five" generates transformation: the Yang Qi of Kan (one) combining with five produces Qian's (six) Yin manifestation; Kun's (two) Yin Qi generates Dui's (seven) Yang expression; Zhen's (three) Yang Qi forms Gen's (eight) Yin aspect; and Xun's (four) Yin Qi creates Li's (nine) Yang manifestation. This reveals the profound mystery of Yin and Yang's mutual transformations.

This passage explains the numerical transformation patterns of the Eight Trigrams (Bagua—Qian, Kun, Zhen, Xun, Kan, Li, Gen, and Dui) from the Book of Changes (Yijing). The Li trigram's yang number nine begins as upright but later combines with oblique numbers, which ultimately return to their upright positions. Through this process, the Eight Trigrams take shape, containing within them the profound mysteries of the hexagrams.

The key lies in the Kan trigram (representing water) maintaining its unique nature without participating in numerical changes. This allows the complex numerical transformations of "three for heaven and two for earth" (a concept from the ancient mystical diagram Hetu) to operate. The Qian trigram (heaven) uses the number nine, while the Kun trigram (earth) uses the number six—these are fundamental principles.

Specifically, when the Qian trigram initially combines with the Gen trigram's number eight, it lacks sufficient numerical value and thus borrows one from the Zhen trigram to complete the number nine. When the yang number reaches its peak, the Qian trigram's complete numerical pattern requires no further addition or subtraction. The Kun trigram's transformation follows the opposite principle: the old yin number six changes into the old yang number nine, and the old yang number nine reverts to the old yin number six. This mutual transformation between yin and yang is the fundamental principle behind the formation of the hexagrams.

The Great Commentary (Xici Zhuan) begins by stating, "When the Eight Trigrams are arrayed, the images are contained within them." Later, they were doubled to form the sixty-four hexagrams, and the transformations of the lines (yao) unfold within them. Some have doubted whether the doubling of hexagrams originated with Fuxi, but such claims are unreliable. The Explaining the Trigrams (Shuogua Zhuan) mentions that the Dui trigram governs speech, while the Gen trigram governs completion. The interaction between mountain (Gen) and marsh (Dui) produces the Xian hexagram (symbolizing resonance). The commentary on Xian states that the two qi (yin and yang) resonate with each other, halting yet delighting in mutual harmony—just as heaven and earth interact to generate all things, so too does the sage resonate with human hearts to bring peace to the world. The clash of thunder (Zhen) and wind (Xun) forms the Heng hexagram (symbolizing constancy). These transformations of the hexagrams reveal the essential patterns of heaven, earth, and all things.

This passage elucidates the patterns of celestial movements and human affairs in the Book of Changes (Yijing). Thunder and wind complement each other (the Xun trigram representing wind), illustrating the eternal operation of the Dao of heaven and earth. Only by following the celestial Dao can the sun and moon shine perpetually, and only through seasonal transitions can all things flourish. When sages persistently uphold the correct path, the world becomes transformed. By observing these enduring principles, one comprehends the essence of all phenomena in heaven and earth.

Tracing the origins of human affairs in the latter part of the Zhouyi (the Book of Changes), we find the purposes of the Shuogua (Explanation of the Trigrams) and the overlapping hexagrams entirely aligned. The author argues that overlapping trigrams (forming the 64 hexagrams from the eight basic trigrams) began in the era of Fuxi, as evidenced by the three versions of the Yi recorded in the Rites of Zhou, all containing the eight primary trigrams and 64 derived hexagrams.

The Shuogua states that when ancient sages created the Yi, they established the trigrams by observing the interplay of Yin and Yang in heaven and earth ("three for heaven, two for earth" refers to odd numbers representing heaven and even numbers representing earth). They derived the lines (yao) based on the attributes of strength and softness, principles originating from the Hetu (the mystical diagram said to have been borne by a dragon-horse). Particularly, the passage "The Emperor emerges from Zhen" (describing the correspondence between trigram directions and seasons) fully reveals the essence of the Hetu: beginning with the Zhen trigram (symbolizing east and spring) and completing the annual cycle at the Gen trigram (northeast), representing the endless renewal of all things.

The author employs the metaphor of day and night to explain seasonal transitions: from spring equinox to autumn equinox resembles daytime, with the summer solstice as its peak; from autumn equinox to spring equinox resembles night, with the winter solstice as the turning point. Further, daily life illustrates the trigrams: Zhen signifies morning labor's commencement, Xun represents social interactions, Li marks the height of social activity, Kun transitions to quiet rest, and Dui symbolizes evening repose. From Zhen to Dui forms the active cycle of daytime, while Qian (northwest) represents nighttime rest, retaining three parts of Yang Qi (nine being the ultimate Yang number).

This passage elucidates the cyclical transformation of Yin-Yang. The number six represents completion, marking the phase where Yin and Yang conflict and stagnate in immobility. Yet completion heralds new beginnings, just as the Kan trigram (one of the Eight Trigrams/Bagua, representing water) symbolizes the depth of night when Qi begins stirring anew—a principle aligned with the concept of "returning in seven days" (referring to the cyclical rhythm of Yin-Yang).

The Gen trigram (representing mountain) signifies fulfillment while simultaneously connecting with the lingering resonance of the Dui trigram (representing marsh). When mountain and marsh Qi interflow and resonate, though seemingly quiescent, they spring into motion when the moment arrives—this marks the dawn of renewed human affairs. What we call transformation manifests the advance and retreat of Yin-Yang; what we term hardness and softness symbolize the alternation of day and night.

Since the eras of Fuxi and King Wen (ancient sage-kings), the Hetu (the mystical diagram borne by the dragon-horse) has revealed the ceaseless cyclical movement of ascending and descending, coming and going between single Yin and Yang. From antiquity to the present, whether observing the annual shift between winter and summer or the daily alternation of light and dark, all follow this pattern. In daily life, we need not seek externally—comprehending the Dao may begin with contemplating the Gen trigram.

Just as the Taixuan Jing employs 729 poetic eulogies corresponding to the Yijing, and the Huangji Jingshi uses the temporal cycles of yuan, hui, yun, and shi for cosmic calculations, both take complete temporal cycles as their foundational units. Now when we observe transformations through the Hetu and deduce patterns, we perceive the Dao's operation in everyday existence—is this not both profoundly simple and illuminating? Does this not exemplify the most sublime flexibility attaining divine subtlety?

The fourth volume of Yitu Tongbian (Comprehensive Analysis of the Diagram Changes), authored by the Linchuan Daoist priest Lei Siqi, primarily examines the authenticity of the Hetu (River Chart). The text notes that Fuxi (also known as Fu Xi) once derived the Bagua (Eight Trigrams) based on the Hetu, and King Wen of the Zhou Dynasty composed the hexagram statements accordingly. Throughout history, the Hetu was revered as a treasure, even recorded in the Book of Documents ("Gu Ming" chapter) as being displayed in the Eastern Chamber (the imperial library). Yet today, some question its very existence.

Lei Siqi criticizes later scholars for discussing the Hetu based on conjecture rather than firsthand observation: when ancient people recalled it, they affirmed its existence, while modern skeptics deny it. After the book burnings under Qin Shi Huang, the Hetu survived only in Kong Anguo’s commentary on the Book of Documents. By the Han Dynasty, the Bibliographical Treatise of the Hanshu conflated the Hetu with apocryphal texts (Chenwei), and the Weishu (esoteric commentaries on the Yijing) further distorted its meaning through forced interpretations, obscuring its true form. Even by the late Tang and Five Dynasties period, people remained divided over its actual appearance.

It was the Song Dynasty Daoist Chen Tuan (styled Tu’nan) who first systematically studied the numerical symbolism of the Hetu, lamenting how it had been misappropriated by Yin-Yang specialists and buried within the Nine Palaces calculations of the Qianzaodu (a Han-era apocryphon). Chen Tuan’s Longtu (Dragon Diagram) circulated widely, and Lei Siqi, having seen the complete version, acknowledged its meticulous mathematical reasoning as an earnest attempt to recover Fuxi and King Wen’s original intent. However, Chen committed a fundamental error: alongside the original diagram, he separately mapped the fifty-five numbers as another chart labeled "Luoshu" (Luo Writing), leading scholars to focus on the trivial while neglecting the essence and sparking futile debates over whether the two diagrams appeared simultaneously.

Later, Liu Mu (styled Changmin) further complicated matters by deriving fifty-five variations of "Gouyin Diagrams." Though the scholar Li Gou (styled Taibo) refuted these errors, he retained three diagrams: the first being what is now called the Hetu, the second being Chen Tuan’s mislabeled fifty-five-number "Luoshu," and the third merely an Eight Trigrams directional chart. Lei Siqi implies that this lineage had already strayed from the Hetu’s true essence.

This passage discusses the transmission and numerical arrangements of the ancient Hetu (River Chart) and Luoshu (Luo Writing), pointing out later misunderstandings. The main points are:

Originally, the Hetu and Luoshu were two distinct diagrams, but later generations mistakenly labeled the fifty-five-numbered diagram as the Hetu while incorrectly calling Chen Tuan's (styled Tu'nan) transmitted Hetu the Luoshu, creating confusion between names and contents. In truth, the Hetu's eight trigrams clearly display fifty-five numbers, corresponding to the principle of five heavenly numbers and five earthly numbers - from heavenly one and earthly two up to heavenly five and earthly ten, totaling fifty-five. This encompasses the complete principles of substance and function, emptiness and fullness, beginning and end, root and branch, all contained within a single diagram without needing to match it with the Luoshu. The Luoshu was never said to have fifty-five numbers.

Now Chen Tuan provided no other explanation, erroneously labeling the fifty-five-numbered diagram separately as the Luoshu, which became the source of this misunderstanding. Tracing its origin, this stems from Han dynasty scholars following the Hongfan's statement "first is the Five Phases, second is fire, fourth is metal," with Yang Xiong's Taixuan Jing also arranging accordingly - two corresponding to fire and south, four corresponding to metal and west. But the current Hetu places two in the southwest and four in the southeast, altering fire and metal's positions. Not daring to directly modify the Hetu, they instead established the fifty-five numbers as the Luoshu, following the Hongfan's rules by placing two-fire in the south and four-metal in the west, even making seven follow two and nine follow four to adjust the southern and western positions. Thus Liu Mu (styled Changmin) didn't understand the reason for this change, thinking it was an exchange between fire and metal positions.

In reality, the number nine of the Li trigram (representing fire) properly resides in due south, while the number seven of the Dui trigram (representing metal) properly resides in due west - these are the fixed numbers of heaven and earth's natural transformations, absolutely not to be altered. How could they be arbitrarily changed just because Han scholars casually compared them to the Hongfan's two-four fire-metal positions? Moreover, the Hongfan only mentions the numbers of the Five Phases - why add seven and nine now, making seven follow two and nine follow four? What basis is there for this?

This passage discusses the correspondence between the Hetu (River Chart, an ancient cosmological diagram) and numerical/positional relationships, emphasizing that its profound transformations cannot be reduced to fixed directional interpretations. While the Hetu's number arrangement appears orderly—such as seven attaching to two or nine attaching to four—these numbers themselves possess no concrete form or fixed positions, allowing flexible transformations. Like the nature of fire and metal: fire is formless yet capable of burning and transformation, while metal has form yet can be melted and reshaped. Their ti-yong (essence-function) relationship is equally dynamic.

The author criticizes certain individuals (like Tunan) for forcibly matching the Hetu's numbers with artificial classifications, akin to identifying a shoulder while losing sight of the entire spine, thereby deviating from the chart's true profundity. The Hetu reveals not merely the surface symbols of the Four Images and Eight Trigrams, but the very patterns of celestial-terrestrial movement. For instance: the Li trigram (symbolizing fire) occupies the southern position representing heaven, yet contains yin qi that descends; the Kan trigram (symbolizing water) occupies the northern position representing earth, yet contains yang qi that ascends. The Zhen trigram (yang's initial birth) symbolizes eastern wood's growth, while the Dui trigram (yin above) symbolizes western metal's sedimentation. Though Qian and Kun trigrams should represent heaven and earth's fundamental substance, convention places them in southwest and northwest positions—an inversion of their natural celestial-terrestrial alignment.

The text further addresses contradictions in divination methods: Zhen and Xun belong to wood, Qian and Dui to metal, Kun and Gen to earth, yet Kan and Li exclusively claim water and fire attributes, resulting in Qian and Kun losing their specialized heaven-earth representation. This mirrors fire and metal's distinct transformational patterns—fire burns to ashes (spiritual transformation) while molten metal retains materiality (essence unchanged)—appearing similar yet fundamentally different. Thus the combinations of numbers like two-seven and four-nine in the Hetu manifest intricate canwu zhi bian (interwoven yin-yang transformations) that defy rigid directional interpretations.

This passage examines the principles of numerical arrangements in ancient Chinese numerology, focusing on the rationality of the "Chart of Fifty-Five Numbers" (a numerical diagram). The diagram places numbers one through four in the four cardinal directions internally, with six through nine positioned externally in corresponding locations—an arrangement that seems plausible when counting by directional numbers. However, the author questions how such an arrangement could achieve cyclical operation, and particularly struggles to comprehend why the numbers five and ten are merely suspended as abstract positions in the four directions.

According to Five Phases theory, Earth (represented by the numbers five and ten) should govern all four directions, corresponding to the celestial positions of Chen, Xu, Chou, and Wei. Medical traditions associate Earth with the spleen, dividing the 360-day year into five Qi cycles of 72 days each. Within this system, Earth's 72-day cycle is further distributed equally among the four seasons' concluding periods—18 days each—as the spleen's designated temporal domain. While this quadripartite division proves effective in medical practice, the diagram anomalously isolates "five" outside the four directions as a central position, contradicting practical applications.

In divination methods, each hexagram forms through four operations (si ying), with eighteen transformations creating one hexagram—totaling seventy-two operations, coincidentally matching the medical Earth-Qi cycle days. Yet the diagram arbitrarily places "five" alone at the center—what practical function does this so-called "center" actually serve?

The author criticizes Chen Tuan's (Tunan) proposed Chart of Fifty-Five Numbers, condemning his unconventional labeling as "Form-Resembling Luoshu" (a diagram imitating the Luoshu) as fundamentally mistaken. Among the "Nine Categories" in the Hongfan chapter of Shangshu, the initial categories like Five Phases and Five Affairs might勉强 (barely) be explained through fives, but the "Supreme Standard" (Huangji) category defies such logic—how can the "Ninefold Blessings" accompanied by "Six Extremes" (calamities) be calculated this way? The original meaning of Huangji refers to the "Great Middle Way"—can the Six Extremes possibly represent "middle" as well? The Supreme Standard itself isn't a concrete entity; forcibly representing it with five points—can these five dots truly embody "center"? From the first to ninth categories, each has fixed numerical correspondences, revealing the strained artificiality of such rigid numerical mappings.

Through cross-disciplinary comparisons with medicine and divination, this analysis exposes contradictions between numerical symbolism and practical application in ancient numerology. The author systematically demonstrates the pitfalls of mechanically applying numerical arrangements across different domains.

This passage examines the authenticity debates surrounding the Hetu (River Chart, traditionally said to be the cosmic diagram Fuxi derived from markings on a dragon-horse's back) and the Luoshu (Luo Writing, the mysterious patterns borne by a divine turtle during Yu the Great's flood control). The author contends that later generations confused these two diagrams, even mistaking one for the other—an egregious error. He particularly emphasizes how the numerical arrangements in the Hetu encode the operational principles of heaven and earth. Whether added horizontally, vertically, or diagonally, these numbers yield specific values (such as the remarkable combinations of five and ten), while the transformations between yin-yang's odd-even qualities and the eight trigrams' alternation between substantial and insubstantial states all reveal creation's profound mysteries. These patterns could never have been human fabrications and likely formed the true basis for Fuxi's invention of the trigrams.

The author criticizes Han Dynasty Confucian scholars for forcing artificial connections, mechanically applying Five Phases theory and calamity omens from the Spring and Autumn Annals to interpret the Luoshu's fifty-five numbers. Regarding the Qianzaodu (a Han Dynasty apocryphal text) mentioning the Great Unity Deity's (Polaris deity) movement through the nine palaces, and Zheng Xuan's commentary arbitrarily designating the central void as the "five" position in the central palace, the author deems these interpretations mistaken. He clarifies that these numbers actually form an abstract framework regulating yin-yang's generative relationships of odd and even—while numerically present, they correspond to no actual spatial directions.

Finally, the text notes that "Taiyi" (Great Unity) shares the same essence as Taiji (Supreme Ultimate), citing the Book of Rites to explain how rituals originate from Taiyi, which differentiates into heaven and earth, transforms into yin-yang, evolves through the four seasons, and arranges the spirits. This principle, the author argues, is fundamentally identical to that of the Taiji. Believing these classical records mutually corroborate each other, he appends relevant discussions after the commentary as evidence.

Chapter 14 of Volume 5 of Yitu Tongbian (Comprehensive Analysis of the Yi Diagrams), authored by the Linchuan Daoist Lei Siqi, explores the transmission and significance of the Hetu (River Chart)—a legendary mystical pattern said to have emerged from the Yellow River. When the ancient sage-kings governed the world, the supreme truths they established could benefit all generations—this was no accident nor the result of mere personal opinion, but rather a revelation bestowed by heaven.

Fuxi derived the Eight Trigrams (Bagua) from the Hetu that appeared in the Yellow River, while Yu the Great formulated the Nine Categories of governance based on the Luoshu (Luo Writing)—the markings on a divine turtle’s shell that emerged from the Luo River. Confucius, in the Yizhuan (Commentaries on the Yi), mentioned that sages modeled themselves after these heaven-sent marvels. Though Confucius typically avoided discussing supernatural phenomena, his specific mention of the Hetu and Luoshu must have been because he personally witnessed their authenticity.

During Fuxi’s reign, he observed celestial patterns, terrestrial principles, the markings of birds and beasts, and geographical features. Drawing from the human body at close range and the forms of all things at a distance, he created the Eight Trigrams to connect with the virtue of the divine (Shen) and classify the nature of all beings. Such wisdom could never have been conceived by later sages through shallow personal insight alone. The vastness of the Yijing (Classic of Changes) matches heaven and earth; its transformations align with the four seasons; its principles of Yin-Yang parallel the sun and moon. Its content encompasses all phenomena, coexisting with heaven and earth without decline—how could such perfection exist unless it were heaven-sent?

When Confucius saw the Hetu and Luoshu, he merely stated that sages emulated them, offering no further explanation. After the Hetu was displayed in the Eastern Chamber of the royal palace during the reigns of King Cheng and King Kang of the Zhou dynasty, it never reappeared—a fact Confucius once lamented. Kong Anguo of the Han dynasty, in his commentary on the Shangshu (Book of Documents), noted: During Fuxi’s rule, a dragon-horse emerged from the Yellow River bearing the Hetu’s markings; when heaven granted Yu the Nine Categories of the Hongfan (Great Plan), a divine turtle surfaced from the Luo River with numerals from one to nine on its back, which Yu arranged into the Nine Categories of governance. These accounts, recorded separately in the Hongfan and Guming chapters, originally held no contradiction.

It was not until Zheng Xuan (styled Kangcheng) annotated the Chunqiu Weishu (Apocryphal Texts of the Spring and Autumn Annals) that the Hetu and Luoshu in the Yizhuan began to be interpreted through weft texts (Han-dynasty prophetic writings that allegorized the classics). Zheng claimed the River Chart corresponded with the Qian (Heaven) trigram, embodying the essence of the Dao of heaven. This interpretation initiated later generations’ extended explanations of the Hetu and Luoshu.

The Yitu Tongbian (Comprehensive Analysis of the Diagrams) discusses the origins of the Hetu (River Chart) and Luoshu (Luo Writing) and their relationship with the Bagua (Eight Trigrams) and the Nine Categories (Jiuchou). The text notes:

The earliest records of the Hetu and Luoshu appear in Han dynasty texts, where the Hetu is said to consist of nine sections and the Luoshu of six, marking the beginning of their mystical significance. Han scholars, citing Kong Anguo's annotations, associated the Hetu with the Bagua and the Luoshu with the Nine Categories—the great laws for governing the world. Later, Wei-Jin scholars like Wang Bi and Han Kangbo did not provide detailed commentaries on this matter. It was not until the Tang dynasty, when Kong Yingda wrote the Zhouyi Zhengyi (Correct Meaning of the Zhouyi), that the topic was revisited. Although Kong adopted Zheng Xuan's annotations in full, he admitted uncertainty about the origins of Wang Bi's views, indicating his partial disagreement with Zheng's interpretation.

When explaining the Hongfan chapter of the Shangshu (Book of Documents), Kong Yingda referenced the Xici (Appended Remarks) from the Yijing (Book of Changes): "The River brought forth the Chart, the Luo brought forth the Writing, and the sages took these as models." He argued that the Nine Categories, each recorded in writing, constituted the content of the Luoshu. He mentioned that what Heaven bestowed upon Yu the Great was precisely the Luoshu and cited Liu Xin's view from the Hanshu·Wuxingzhi (Treatise on the Five Phases in the Book of Han): during Fuxi's reign, a dragon emerged from the Yellow River bearing the Chart, which he used to draw the Bagua; when Yu the Great controlled the floods, a divine turtle emerged from the Luo River carrying the Writing, which he used to establish the Hongfan Nine Categories. The Hetu and Luoshu thus served as warp and weft, while the Bagua and Nine Categories were seen as complementary.

However, the author points out that the tale of the divine turtle bearing the Writing is not factual. Apocryphal texts like the Shangshu Zhonghou (Central Receptacle of the Book of Documents) contain numerous legends about the Yellow Emperor, Yao, Shun, Yu, Tang, King Wen, and King Wu receiving divine charts and writings, all involving dragons or turtles as bearers. The true authorship of these apocrypha is unknown, and scholars have determined they were likely forged during the reigns of Emperor Ai and Emperor Ping of the Western Han. Though these texts appeared only in the late Western Han, earlier scholars may have already known similar legends. Thus, Kong Yingda believed the Nine Categories were the characters arranged from one to nine on the turtle's back, which Yu the Great organized into nine laws. The attribution to Yu was based on the assumption that divine speech would be concise, not meticulously ordered—hence the inference that Yu must have systematized it.

The text also mentions that interpretations of the Guming chapter's "Hetu was placed in the eastern chamber" and the Liji's (Book of Rites) "the River produced the horse-chart" differ slightly in detail but agree in essence. From the Han to the Tang dynasties, throughout the transmission of the Yijing, no diagrams of the Hetu or Luoshu were included at the beginning of the classic. It was not until the early Song dynasty that Chen Tuan began placing these diagrams at the start of the text.

Chen Tuan (style name Tu'nan) was the first to systematically study the symbolic numerology of the "Book of Changes" (Yijing), claiming his insights came from contemplating Confucius's threefold discussion of the nine hexagrams' profound meanings in the "Great Commentary." His newly written works particularly featured over twenty diagrams including the "Dragon Chart of Separation, Union, Transformation and Pervasion," all based on the numerical system from the "Commentaries on the Changes" that states "Heaven's numbers total fifty-five from one to ten," while incorporating the Nail-Insertion Method (an ancient divination technique pairing heavenly stems with the eight trigrams) to elucidate Yijing principles.

Among these were a "Nine Palaces Diagram" and a "Luoshu Diagram": The former completely omitted the five generative numbers (one through five), using only the earthly numbers six through ten, explaining that "ten" represents the completion of forms—hence the "Hongfan" section only mentions numbers up to ten when recording the Five Phases. The latter reappeared at the end with two variants—an unaltered Nine Palaces Diagram labeled as the "Hetu" (River Chart), while another simply arranged fifty-five numbers in the four directions and center, arbitrarily designated as "Luoshu" without any classical basis. This version arbitrarily positioned "two-seven" in the south and "four-nine" in the west, with no clear rationale beyond possibly drawing from earlier theories about "two-fire and four-metal" in the "Hongfan's" Five Phases and Yang Xiong's "Canon of Supreme Mystery" which adapted Yijing directional arrangements. These diagrams remained obscure with few understanding them.

After five generations of transmission from Chen Tuan to Liu Mu (style name Changmin), the diagrams expanded to fifty-five, compiled as the "Charts of Hidden Hooks." Master and disciple mutually praised each other's work, further developing interpretations by adding chapters like "Discussion on Hexagram Virtue," "Chamber Dialogues," "Recorded Teachings," "Essential Guide," "Subtleties Explained," and "Divine Comprehension" to their "New Commentary on the Zhouyi." Though deliberately mystifying, their actual content proved unremarkable. Contemporary scholar Li Gou (style name Taibo) wrote six critiques exposing their errors, even fearing they might mislead students and corrupt scholarship, ultimately reducing the redundant diagrams to just three: the River Chart, Luoshu, and Eight Trigrams.

In truth, while Liu Mu's profusion of Yijing diagrams was questionable, Li Gou failed to recognize these three diagrams shared the same essence—the original River Chart inherently contained the eight trigrams while maintaining the fifty-five total. It wasn't until Liu Mu and others began challenging traditional Luoshu theories that they proposed another version of the Luoshu might have existed in Fuxi's era beyond what's recorded in the "Hongfan."

In ancient times, Fuxi (one of the Three Sovereigns) created the Book of Changes (Yijing) by observing the patterns of heaven and earth. When Taibo (eldest son of King Wen of Zhou) later encountered these hexagrams, he found them perfectly aligned with natural laws—their profound mysteries remain incompletely understood even today.

The Commentary on the Yi (Yizhuan) records how sages emulated celestial movements, harmonized with seasonal transitions, observed the suspended phenomena of sun and moon, and employed divination tools like yarrow stalks and turtle shells. The "divinely endowed objects" mentioned refer primarily to yarrow stalks, as the Yijing mainly employs them for divination. These stalks possess a numinous quality of perfect penetration, enabling communion with spiritual forces. Though turtle shells were also used, the Yijing primarily discusses yarrow stalk divination.

The text describes sages modeling their systems on celestial patterns, much like the auspicious signs of the Dragon Chart emerging from the Yellow River (legend says a dragon-horse bore this diagram) and the Turtle Script from the Luo River (a divine turtle carried this text). Fuxi derived the Eight Trigrams (Bagua) from the Dragon Chart, while Yu the Great established the Nine Categories of governance from the Luo Script. Though differing in form, both exemplify sages patterning their systems after nature.

Notably, Han dynasty scholars identified the Luo Script's content with the "Great Plan" (Hongfan) chapter of the Book of Documents. When Northern Song Daoist Chen Tuan (courtesy name Tunan) first described the Luo Script, he emphasized its core principle of applying the number ten, corresponding to the Five Phases (Wood, Fire, Earth, Metal, Water) in the Hongfan. Later scholars deviated from this original meaning, even denying the Luo Script's connection to Yu's era, causing subsequent confusion.

Confucius's Yizhuan references sages emulating nature. Though separated from Fuxi by millennia, pairing the Luo Script with the River Chart remains conceptually sound. Regrettably, some dismiss not only Confucius's account but even Chen Tuan's research on the Dragon Chart. Chen claimed his Dragon Chart theory derived from Confucius's "Three Expositions on Nine Hexagrams" (referring to triple commentaries on select hexagrams in the Yijing)—a connection warranting deeper reflection.

This passage discusses the controversies surrounding the transmission of the ancient Hetu (River Chart) and Luoshu (Inscription of the Luo River). The main points are as follows:

Originally, the "Dragon Diagram" (i.e., the Hetu) consisted of 55 numbers arranged in a specific pattern. However, later scholars mistakenly labeled it as the Luoshu. Liu Mu (courtesy name Changmin) and others perpetuated this error, even going so far as to create diagrams of the Luoshu with 55 dots to support this mistaken view. Subsequently, many scholars followed suit, accepting the 55-number diagram attributed to Chen Tuan (also known as Tu'nan) as the authentic Luoshu, without questioning why the Luoshu, which records the "Nine Categories," should have an additional tenth number when its arrangement is clearly based on nine numbers.

The author questions whether this error arose from seeing that the Hetu recorded in the Zhouyi (Book of Changes) could be openly transmitted, leading to the fabrication of a Luoshu in imitation of the Hetu's 55-number format. However, the arrangement of the Nine Categories is based on vertical comparisons, rendering the 55-number format entirely irrelevant—a major flaw in this interpretation.

The author then traces the origin of this mistake: after Chen Tuan, scholars like Chong Fang, Xu Jian, and Li Gai had never seen the authentic ancient diagrams. Fan E'chang, a disciple of Chen Tuan, recorded in Taiyi Yuanliu (The Origins of the Great Change) that when Fuxi created the Eight Trigrams based on the dragon-horse bearing the chart, he arranged the numbers of heaven and earth (totaling 55) according to specific positions:

• Heaven's 1 in the north
• Earth's 2 in the south
• Heaven's 3 in the east
• Earth's 4 in the west
• Heaven's 5 in the center
• Earth's 6 paired with the zi position
• Heaven's 7 paired with the wu position
• Earth's 8 paired with the mao position
• Heaven's 9 paired with the you position
• Earth's 10 placed in an undefined "end" position

But the author counters: If the first five numbers already occupy the four cardinal directions and the center, why are the next five numbers assigned to positions like zi, wu, mao, and you—which are themselves the four cardinal directions? And how can the "end" position (mo) exist in the center? Upon closer examination of the surviving diagrams, only the four cardinal directions are represented, with no arrangement for the four intermediate directions (the corners). This clearly contradicts ancient conventions.

This passage discusses the correlation between numbers and trigrams in ancient Yijing studies, criticizing certain far-fetched interpretive methods. It mentions figures like "Tunan" (referring to Chen Tuan, also known as Tunanzi) and "Echang" (the Song dynasty Yijing scholar Liu Mu) who attempted to forcibly align numbers with the Bagua (Eight Trigrams) directions, thereby deviating from the original meaning of the Yijing (Book of Changes).

The author points out that these scholars mechanically applied the numerical sequence of the Five Phases (Water, Fire, Wood, Metal, Earth) — "one for Water, two for Fire, three for Wood, four for Metal, five for Earth" — onto the trigram arrangements. Even more absurdly, they arbitrarily altered the positions of the trigrams, such as moving Qian (Heaven), Dui (Lake), and Kun (Earth) to the east, and Gen (Mountain), Zhen (Thunder), and Xun (Wind) to the west, completely disrupting the traditional Bagua directional system.

Particular criticism is directed at the practice of dissecting trigram lines through numerical manipulation: for instance, the Kan (Water) trigram, which inherently consists of six lines, was reduced by three lines and claimed to represent the Qian trigram; similarly, the Li (Fire) trigram's seven lines were reduced by three and misidentified as the Xun trigram. Such arbitrary divisions violate the fundamental rule that "each trigram consists of three lines" and confuse the core principle of "the Four Symbols generating the Eight Trigrams" (the Four Symbols referring to the stages of yin-yang transformation before the formation of the trigrams).

Regarding the relationship between the "Luoshu Nine Categories" (a legendary pattern said to have been carried on the back of a divine turtle during Yu the Great's flood control) and the "Hetu" (a diagram supposedly borne by a dragon-horse from the Yellow River), the author argues that some scholars indiscriminately imposed the numerical sequence of the Nine Categories onto the Hetu trigrams. For example, placing "the Five Phases" in the Kan position (north) and "the Five Affairs" in the Kun position (southwest) — such pairings neither align with the sequence recorded in the Shangshu (Book of Documents) nor respect the original mathematical structure of the Hetu.

In summary, this passage emphasizes the importance of adhering to the foundational principles of Yijing studies, opposing arbitrary inventions of numerical-trigram relationships for the sake of novelty. Authentic Yijing philosophy should be derived from the fundamental laws of yin-yang transformation, not through forced interpretations based on numerical games.

This passage critiques several erroneous interpretations of the Hetu (River Chart) and Luoshu (Luo Writing). It first points out how some forcibly cram the Five Phases (Wood, Fire, Earth, Metal, Water) into a single方位, utterly violating the fundamental principle that each Phase governs its own cardinal direction. Particularly egregious is the swapping of Fire and Metal's positions, which completely contradicts the authentic arrangement of the Hetu—such混乱 is utterly unreasonable.

Next, it mentions the Dongji Jing (Scripture of the Ultimate Cavern) by Yang Cigong (i.e., Yang Xiong), which was falsely attributed to Guan Ziming. The text claims the Hetu places the number seven in front, six behind, eight to the left, and nine to the right, and that the sages derived the Eight Trigrams from this. Specifically, it absurdly suggests taking one-third of seven to form the Li (Fire) trigram, one-third of eight for Zhen (Thunder), one-third of six for Kan (Water), and one-third of nine for Dui (Lake). This logic is as preposterous as claiming children can give birth to their parents. Even more outrageous is how it arbitrarily alters the classic numbers—switching the Li trigram's original nine to seven and Dui's seven to nine.

Regarding the Luoshu, the critique notes that only the statement "five居于 the center" aligns with its true meaning, while other numerical arrangements actually follow the Hetu's principles. In the Hetu, five pairs with ten to form fifteen, distributed across the four directions in combination with ten; there is no isolated five at the center. Such misinterpretations have caused much subsequent confusion. Yang Cigong's fallacies were essentially fabricated by模仿 Yang Xiong (styled Ziyun)'s theory of "three regions, nine provinces, and twenty-seven divisions"—pure personal conjecture.

This passage discusses the Song Dynasty scholarly controversy regarding the "Hetu (River Chart) and Luoshu (Luo Writing)." Zhu Xi (styled Yuanhui) and Cai Yuanding (styled Jitong) mistakenly identified Chen Tuan's (also known as Tu'nan) proposed Luoshu as the Hetu, while conversely labeling the traditional Hetu as Luoshu. They cited the "Book of Changes" and the "Hongfan" chapter of the "Book of Documents" to support their view, claiming that the nine-palace numbers (with nine above, one below, three on the left, seven on the right, two and four as shoulders, six and eight as feet) represented the pattern on a turtle's back.

In reality, Liu Mu had long asserted that the nine-number arrangement should be the Hetu, while the ten-number configuration was the Luoshu—a view originating from Chen Tuan (styled Xiyi) that differed from traditional Confucian interpretations. Zhu Xi and Cai Yuanding not only persisted in their error but also accused Liu Mu (styled Changmin) of being the first to make this mistake. Cheng Jiong (styled Taizhi) attempted to clarify the matter in his "Yi Yuan," but scholars like Zheng Dongqing later arbitrarily added diagrams, further obscuring the truth.

The root of the confusion lies in Chen Tuan's initial mistake when creating new diagrams using the "ten-number" concept. At first, he used the Hetu to explain the Luoshu, but later abandoned the Hetu entirely to focus solely on the Luoshu, eventually even misidentifying the Luoshu as the Hetu. This led later generations to completely lose sight of the Hetu's true form. Nowadays, people insist on using the number fifty-five to represent the Hetu, unaware that this number was originally incorrect—a misapplication of Luoshu characteristics to the Hetu. Consequently, this not only distorts the original meaning of the Hetu but also mistakenly treats the Luoshu as if it were the Hetu.

This passage examines the authenticity of ancient mystical diagrams like the "River Chart" (Hetu) and "Luo Writing" (Luoshu). The author observes that scholars' interpretations of these diagrams over the past three centuries have been wildly divergent, leaving people as confused as someone searching for a sheep at a crossroads or looking for a horse in a bustling marketplace.

The author identifies three main sources of confusion: First, Chen Tuan's (styled Tu'nan) derivation from the nine hexagrams of the Classic of Changes (Yijing) is overly abstract. Second, the Yin-Yang School's simplistic application of numerical sequences like "one-six-eight-white" as calculation bases remains superficial. Third, the apocryphal texts (weishu) such as the Qian Zao Du, which claim the Great Unity Deity (Taiyi) derived numbers that sum to fifteen in all directions, are excessively rigid and forced.

The author proposes his own methodological approach: Rather than seeking external explanations, one should focus on the "The Great Ultimate Gives Birth to the Two Forms" chapter in the Commentary on the Appended Judgments (Shuogua Zhuan). By examining the numerical relationships from "Heaven's one generates water" to "Earth's four completes metal," and from "Earth's six" to "Heaven's nine," one can grasp the substantial reality, while using "Heaven's five" and "Earth's ten" to explain the empty function. This naturally leads to understanding the Yijing's numerical principles like "alternating by threes and fives" and "threefold heaven and twofold earth," perfectly aligning with Confucius' original intent in composing the Ten Wings (Yizhuan).

The author concludes by emphasizing: When faced with conflicting interpretations, we should take the sages' teachings as our standard. The authentic River Chart has always contained its own inherent logic, sufficient to illuminate the true meaning of the Classic of Changes. Similarly, beyond the Nine Categories (Jiuchou) of the Luo Writing—the nine principles of governance said to have been carried by the divine turtle during Yu the Great's flood control—there should be no other forced interpretations of diagrammatic numerology. To recognize this with such clarity is truly one of the great intellectual satisfactions across the ages.