《天原發微》Vol. 16

This passage traces the lineage of ancient Chinese mathematics and Yi studies (the study of Changes). The origins can be traced back to Fuxi, who created the Bagua (Eight Trigrams) by observing the Hetu (River Chart)—a mystical pattern said to have been carried on the back of a dragon-horse emerging from the Yellow River. Over seven hundred years later, during Yu the Great's flood control efforts, he obtained the Luoshu (Luo Writing)—a divine diagram borne by a turtle from the Luo River. Another eleven centuries passed before Jizi, a wise minister of the Shang dynasty, expounded the nine categories of numbers from the Luoshu in the "Hong Fan" chapter.

Confucius was born in the twenty-first year of King Ling of Zhou's reign, five hundred and seventy years after Jizi's era. At seventy, Confucius studied the Yi Jing (Book of Changes), unveiling the mystery of the number fifty-five from the Hetu. Fifteen centuries later, Shao Yong (styled Kangjie) composed the Huangji Jingshi (Supreme Principles Governing the World). These mathematical principles never degenerated into mere divination techniques because they all originated from Fuxi's innate Bagua diagram. Fuxi transmitted mathematics through the trigrams, Confucius developed the Dayan numbers (the fifty yarrow stalks method for divination), and Shao Yong established the Huangji Jingshi system.

The text mentions Yang Xiong's Taixuan Jing (Canon of Supreme Mystery), Wei Yuansong's Yuanbao Jing (Canon of Primordial Unity), and Sima Guang's Qianxu (Hidden Void), noting that these works grasped only partial aspects of Yi studies without reaching its highest realization. True mathematical principles are deeply rooted in the Qi (vital energy) of heaven and earth—their transformations are too vast to calculate exhaustively, making comprehension through principle more essential. Confucius taught people to "exhaust principles and fulfill nature to reach destiny," emphasizing that understanding principles allows mastery over numerical configurations.

Zhu Xi regarded Taiji (Supreme Ultimate) as principle (Li), Yin-Yang as Qi, and movement/stillness as the pivot of transformation—where Qi operates, principle follows. Cai Yuanding (styled Jitong) stated that Qi is number; even in primordial chaos, numbers were already latent within it. The author argues that before Taiji's activation, Qi remained unmanifested and numbers invisible, yet they contained the principle of "zhenyi hansan" (holding unity while encompassing trinity—the cosmic law). When Taiji stirred to generate Yang Qi, the embryonic form of numbers became faintly discernible, though not yet fully revealed.

The Tianyuan Fawei scripture primarily elucidates the patterns of Yin-Yang transformation and the sages' comprehension of celestial laws.

It first explains the principles of Yin-Yang alternation: When movement reaches its extreme, stillness emerges, creating the state of "two"; when stillness culminates, movement regenerates, forming "three"; when movement again peaks, it returns to stillness as "four." The Yang numbers (odd sequences) through the progression of one, three, and nine generate all phenomena, while the Yin numbers (even sequences) through two, four, and eight manifest concrete forms. Those who understand this truth cultivate themselves at the subtle junctures of Yang transformation, and when the formative Yin numbers seven, eight, nine, and six appear, they achieve precise mastery, producing auspicious outcomes.

Regarding human nature: The innate Yang Qi (benevolent nature) bestowed by heaven is often obscured by Yin Qi (temperamental disposition). The noble person's task is to expand and illuminate this Yang virtue. The reason sages refrain from discussing destiny is their focus on heavenly principles to educate humanity, though ultimately they cannot completely transcend destiny's influence.

The second section discusses the five great paradigms in the Yijing (Book of Changes): Heaven-Earth, four seasons, sun-moon, divination tools (millet stalks and tortoise shells), plus the sage—totaling five. Heaven-Earth manifests forms, seasons display transformations, sun-moon reveals illumination, and divination tools disclose the future—all demonstrating numerical patterns. Only sages comprehend these mysteries: Why does Heaven-Earth divide into three Yang and two Yin? Why do seasons correspond to numbers seven, eight, nine, and six? Why is the sun counted as one and the moon as two? Why does the River Chart contain ten numbers while the Luo Writings show nine? Why use fifty millet stalks but only forty-nine for divination? Through creating instruments, establishing systems, and benefiting all under heaven, the sage occupies the central position—emulating Heaven-Earth, modeling phenomena, and formulating laws.

Finally, it explains the Yijing's origins: Before Fu Xi, there were no diagrams, only the natural manifestation of Change's principles in Heaven-Earth (hence calling the River Chart "heaven-and-earth-drawn Change"). Change means the alternating principle of Yin-Yang, where principle and number are inseparable, both spontaneously arising. In primordial chaos, humanity was simple yet profoundly wise. Heaven displayed numbers through sun, moon, and stars; Earth revealed numbers through the River Chart and Luo Writings. Fu Xi first unveiled these numerical patterns to people, later transmitted by sage-kings like Yao and Shun.

This scripture primarily elucidates two core themes: first, the transmission lineage of Chinese civilization from ancient sage-kings to Confucius; second, the pivotal role of the number "five" in cosmic operations.

Regarding the transmission lineage: From Yu the Great, Tang of Shang, King Wen of Zhou, King Wu of Zhou, the Duke of Zhou to Confucius, the Dao of sages was passed down through generations. Like a great tree growing branches from its roots while allowing the roots to be traced through its branches, though its manifestations varied endlessly, its essence remained constant. The text mentions how Fuxi (one of the Three Sovereigns of antiquity) innovatively created the Bagua (Eight Trigrams) and devised writing to replace knot-tying records; during the era of the Yellow Emperor (first of the Five Emperors), his minister Da Nao explored the essence of the Five Phases (Wuxing) to establish the Heavenly Stems and Earthly Branches, while Rong Cheng formulated calendars and Li Shou invented arithmetic. It also records the legend of using bamboo pipes to measure phoenix calls for establishing musical tones—the male phoenix's six calls creating the six Yang pitches, the female's six calls forming the six Yin modes.

Concerning the mystery of the number "five": Five occupies the central position, symbolizing the harmonizing Qi between heaven and earth. In celestial patterns, five resides at the North Star position governing the four seasons; geographically, five sits at the center of the Hetu (River Chart) and Luoshu (Luo Writing), commanding the four directions. Just as the Gong note anchors the pentatonic scale, or how the fundamental Huangzhong pitch remains constant while resonating with the twelve monthly variations. The numerical pattern of heaven and earth totals fifty-five, with the Dayan number (Great Expansion) being fifty—the central five representing the undifferentiated primordial state (Taiji). When chaos first divided, primordial Qi first transformed into Water (number one) in the north, then Fire (number two) in the south, revealing the mathematical principles of cosmic generation.

Note: The "Hetu and Luoshu" mentioned are essential Daoist texts, said to have emerged from the Yellow River and Luo River respectively, recording the mysteries of cosmic operations; the "Dayan number" originates from the Yijing (Book of Changes), referring to the fifty foundational numbers for deducing celestial and terrestrial transformations.

This passage explains the relationship between numbers and the Five Phases (Wood, Fire, Earth, Metal, Water) in ancient mathematical philosophy, along with the calculation method of the Dayan numbers (divinatory numerology).

The number two represents Fire, as it is derived from the number one through two transformations. The number three represents Wood (associated with the East), produced by three transformations of one. The number four represents Metal (associated with the West), formed through four transformations. Though the numbers two, three, and four are all derived from one and are called "generating numbers," they fundamentally originate from the number five. Five represents Earth—the harmonizing Qi—acting as the sovereign of the Five Phases, capable of both externally generating other numbers and internally completing them.

The number five is unique in that it can both generate and fulfill itself. All numerical combinations ultimately return to five or ten (two fives), just as in the Luoshu (an ancient mystical numerical diagram), no matter how the numbers are calculated vertically or horizontally, they always revolve around the numbers three and five. This is why five can participate in all transformations without losing itself and can accomplish all things without perishing.

Regarding the Dayan numbers (a numerical system used in divination): There are fifty yarrow stalks in total, but only forty-nine are used in practice. The calculation for the Qian hexagram (Yang) yields 216, while the Kun hexagram (Yin) yields 144, totaling 360—symbolizing the number of days in a year. The sum of all hexagram calculations is 11,520, corresponding to the number of all things in existence.

Shao Yong (a Song Dynasty Yijing scholar) stated that the Dayan numbers are the source of all calculation methods. These methods are based on four forms: square, round, curved, and straight. Yang numbers do not use ten, and Yin numbers do not use one. Multiplication generates numbers, while division reduces them—all calculations adhere to this principle. Scholar Zhang (possibly another commentator) added that Yin and Yang are merely waxing and waning transformations, and calculations are nothing more than multiplication and division. Using the Four Images (Lesser Yang, Greater Yang, Lesser Yin, Greater Yin) as the foundation, the ascending and descending interplay of Yin and Yang within them produces six (the number of Greater Yin).

Finally, from the perspective of the Earlier Heaven Bagua: Heaven belongs to the round, with a base number of three, transforming as 3 → 6 → 7 → 9. Earth belongs to the square, with a base number of four, transforming as 4 → 8 → 6 → 9. Heaven and Earth mutually transform, their numbers interweaving in endlessly complex variations. In the Yijing, the Qian hexagram uses the Greater Yang number nine, the Kun hexagram uses the Greater Yin number six, and the Dayan method employs the Four Images' number four. Multiplying four by nine yields thirty-six.

This passage explains the calculation method for the numerical values (ce 策) of the Qian (Heaven) and Kun (Earth) hexagrams in the Yijing (Book of Changes). Each yang line (yao) in the Qian hexagram (representing yang) corresponds to 36 ce. Multiplying by its six lines yields 216 ce (36×6=216), which is the total value for the Qian hexagram. The same result can be obtained by multiplying 36 by 6.

For the Kun hexagram (representing yin), each yin line corresponds to 24 ce. Multiplying by six lines gives 144 ce (24×6=144), the total value for Kun. This can likewise be verified by multiplying 24 by 6.

The text then explains the calculation for all 64 hexagrams of the Yijing: The 32 yang hexagrams (comprising 192 yang lines at 36 ce each) are calculated as 32×216. The detailed computation breaks down as: 32×200=6400, 32×10=320, and 32×6=192, summing to 6912 ce. Similarly, the 32 yin hexagrams (192 yin lines at 24 ce each) are calculated as 32×144: 32×100=3200, 32×40=1280, and 32×4=128, totaling 4608 ce. Combining both gives the significant Yijing number of 11,520 ce (6912+4608).

The author notes that alternatively, one can multiply Qian's 216 ce by 32 and Kun's 144 ce by 32 before summing them to conveniently arrive at 11,520. This represents one of the Yijing's fundamental computational principles. Confucius reportedly appended this method to the Yijing, though later scholars failed to explain it clearly. Even the great Neo-Confucian scholar Shao Yong (courtesy name Kangjie), who learned this method from Chen Tuan (Daoist name Xiyi), once forgot it and had to be reteached in a dream—demonstrating its difficulty to master.

The author, conscious of advancing age, deliberately records these detailed calculations to help future students comprehend this important Yijing principle. The "ce" values emphasized here represent the basic counting units used in ancient divination, through which hexagram transformations could be mathematically derived.

For those who study the Book of Changes (Yijing), understanding this principle makes numerical calculations straightforward.

The numbers operating beneath the earth are particularly challenging to deduce. While the sixteen phenomena of Yin-Yang transformations and all things between heaven and earth can be analogically derived, only the subterranean numbers remain concealed and unfathomable. Thus we say that all phenomena under heaven, all forms upon earth, and even the spirits and ghosts of the netherworld follow numerical patterns—though these remain invisible to ordinary perception. Like the principles of musical pitch that manifest when operating above ground, during the summer solstice when the sun rises from the Tiger position (Yin) and sets at the Dog position (Xu), during the three earthly branches of Hai (9-11 PM), Zi (11 PM-1 AM), and Chou (1-3 AM) when the sun travels underground, these numerical patterns become invisible.

Master Cheng Yi once remarked: Shao Yong (styled Yaofu) calculated that flavors undergo 28,600 variations—far beyond what human effort could possibly concoct. Colors likewise manifest 28,600 variations that no artificial dyeing could replicate, all emerging spontaneously from nature. The numerical principles of sound reveal only half their transformations because Yang sounds manifest solely when the sun is above the horizon, becoming incalculable after sunset. This follows natural logic—where there's form there's shadow; when the form disappears, how can the shadow persist? The transformations of hexagrams operate similarly: the numbers one through four circulate underground, emerging at five when human beings and all things begin to sprout and show signs of life, becoming fully manifest at six through nine.

The sages established their numerical systems by supporting Yang and restraining Yin according to heaven and earth's inherent patterns. The heavenly number twenty-five and earthly number thirty constitute the fundamental numerical framework. Though earthly numbers exceed heavenly numbers by five, the actual divination number used is fifty (Dayan number). Why then do heavenly numbers ultimately surpass earthly numbers by ten? This reflects the sages' principle of elevating Yang and suppressing Yin: the earthly numbers two-eight and four-six occupy only two portions, while the heavenly numbers one-nine, three-seven, and five-five claim three portions totaling thirty—thus heavenly numbers prevail. Yang numbers advance three places using thirty (calculated generously), while Yin numbers four and six retreat using twelve (calculated sparingly). From this we derive: heaven's three and earth's two combine to make five; heaven's six and earth's four combine to make ten. The Qian (Heaven) hexagram's nine and Kun (Earth) hexagram's six combine as fifteen. Qian manifests thirty-six while Kun manifests twenty-four. Qian employs six-seven to yield two hundred fifty-two, symbolizing flourishing vitality; Kun uses only three-six to yield one hundred eight, representing contraction and stillness. The sages established these numerical principles by observing the operating patterns of heaven and earth's creative transformations—truly profound and mysterious!

This passage discusses the relationship between the symbolic imagery of the Yijing (Book of Changes) and the Luoshu (the mysterious diagram said to have been carried on the back of a divine turtle during Yu the Great's flood control). Zhen Dexiu (styled Xishan) recorded Cai Yuanding's (styled Yinjun) view: The hexagrams of the Yijing manifest the patterns of Heaven and Earth's movement, while the numbers of the Luoshu record their transformations.

Numbers begin with one, with odd numbers symbolizing movement and change; hexagrams are composed of Yin and Yang lines, with even numbers symbolizing stability and form. Thus, the Two Modes (Yin-Yang) generate the Four Images, and the Four Images generate the Eight Trigrams (Bagua); multiplying three by three yields nine, forming the numerical system of the Luoshu's nine palaces. When the Eight Trigrams are extended to six-line hexagrams, they produce 4,096 possible variations; when the Luoshu's numbers are extended to the fourth power of nine, they yield 6,561 numerical permutations.

The Yijing, refined by four sages, has a complete system of hexagram imagery; the Luoshu, bestowed upon Yu the Great, had its numerical transmission interrupted. Later scholars, failing to grasp the origin of image-number theory, either forced numbers into hexagram interpretations or retroactively applied hexagrams to numbers, distorting the natural logic and obscuring its clarity.

Cai Shen (styled Zhongmo) said: The Luoshu is the source of numbers. Without understanding numbers, one cannot comprehend hexagrams; without understanding hexagrams, one cannot grasp numbers—the two are inseparable. Hexagrams belong to Yin (even numbers), where harmony brings auspiciousness; numbers belong to Yang (odd numbers), where conflict brings misfortune. The opposition of Yin and Yang forms hexagrams, while their alternation generates numerical changes.

One is the origin of numbers, and nine is the basis of eighty-one. The year begins at the winter solstice and ends at the next, with numbers cycling from one to nine corresponding to eight solar terms: Beginning of Spring (2), Spring Equinox (3), Beginning of Summer (4), Summer Solstice (5), Beginning of Autumn (6), Autumn Equinox (7), Beginning of Winter (8), and Winter Solstice (9). When numbers reach nine, they return to one, symbolizing the alternation of Yin and Yang, the cycle of day and night, and the succession of the four seasons. The beginning and end of all things in Heaven and Earth are reflected in the numerical patterns of the Luoshu and the Hetu.

From one to nine, multiplying nine by nine yields eighty-one, extending further to seven hundred twenty-nine, and ultimately forming 6,561 numerical permutations. The movement of odd numbers and the symbolism of even-numbered hexagrams are interdependent—hence the Eight Trigrams (images) and the Nine Chapters (numbers) are two sides of the same reality.

When asked about the numerical principles of the Pre-Heaven Bagua, Zhu Xi replied: The Yizhuan (Commentaries on the Yijing) already explains it clearly. The Kun hexagram (representing Earth) corresponds to the number six (the extreme of Yin numbers), demonstrating the correspondence between numbers and hexagram imagery.

This passage explores the numerical mysteries of the Yi Jing (Book of Changes) and their connection to trigrams. The Hetu (River Chart) and Luoshu (Luo Writing)—legendary ancient diagrams—are considered the origin of all numbers and the fundamental basis for the sages' creation of the trigrams. While Taiji (Supreme Ultimate) represents the cosmological principle, these diagrams serve as the progenitor of numerical systems.

Specifically: When Fu Xi drew the eight trigrams based on the Hetu, he employed the principle of "five heavenly numbers and ten earthly numbers," totaling fifty-five numerals. Confucius later explained that the "Great Expansion number fifty" refers to the yarrow stalks used in divination. The three hundred sixty lines of the Qian (Heaven) and Kun (Earth) hexagrams correspond to the days in a year, while the number 11,520 represents all things in existence—all demonstrating pre-existing numerical patterns. The Hetu-Luoshu system encompasses these universal numerical relationships.

King Wen's arrangement of the Zhou Yi placed Qian and Kun at the beginning because the distinction between yang lines (odd-numbered) and yin lines (even-numbered) forms their foundation. Fu Xi's original sequence started with Qian and ended with Kun, containing sixty-two intermediate hexagrams, as all numerical patterns governing worldly phenomena are contained within these two primal hexagrams.

Qian symbolizes heaven, beginning numerically with one. One contains three, and three threes make nine. All odd numbers (1,3,5,7,9)—the yang numbers—are governed by Qian, which generates the masculine principle manifested through the Zhen (Thunder), Kan (Water), and Gen (Mountain) trigrams distributed through the four seasons. Both humans and all creatures receive Qian's yang energy.

Kun represents earth, beginning numerically with two. Two contains six, with twelve lines at its core. All even numbers (2,4,6,8,10)—the yin numbers—are governed by Kun, which generates the feminine principle manifested through the Xun (Wind), Li (Fire), and Dui (Lake) trigrams distributed through the seasons. Both humans and all creatures receive Kun's yin energy.

The pairing of odd and even constitutes yin-yang, with feminine and masculine principles combining to produce numerical sequences like 49, 46, 47, and 48—constantly generating dynamic transformations. Whether counting forward or backward, these numerical patterns remain inescapable. Heaven's Dao moves leftward in harmonious progression, making the Five Phases (Wood, Fire, Earth, Metal, Water) mutually generate; moving rightward in reverse makes them mutually restrain. Only through restraint can completion occur—hence the Yi Jing's numerical system achieves completion through reverse movement.

The Shuogua Zhuan states: "The Changes' numbers proceed inversely," and further explains: "Calculating the past follows the sequence, but foreseeing the future requires inversion." Shao Yong (Northern Song philosopher) remarked: "When human thought has not yet arisen, and even spirits remain unaware—upon whom should this principle depend if not me?" Cheng Yi (Master Yichuan) acknowledged that Shao Yong (Kangjie) perfected the philosophical understanding of these numerical principles, a view later affirmed by Liu Yunzhuang.

The hexagrams of the Yijing (Book of Changes) originate from the Taiji (Supreme Ultimate, the cosmic source), thus containing the most subtle principles in the world. The evolution of hexagrams traces back to the Hetu and Luoshu (ancient mystical diagrams), making their numerical transformations the most profound mysteries. These principles must have foundations—while distinct from the numerical patterns of Hetu-Luoshu, they nevertheless rely on these fundamental calculations.

Here we further explain the mystery of the number "ten": Ten can be viewed as two fives combined, or three threes multiplied plus one. Subtracting three from ten yields seven; subtracting two gives eight; subtracting one makes nine; subtracting four results in six. Conversely, six plus four, nine plus one, seven plus three—no matter how the numbers increase or decrease, they ultimately resonate with the fundamental number ten. The essence of seasonal changes (the Four Symbols) also follows this decimal principle. Viewed fractionally, one divides into two, two into four, four into sixteen, sixteen into thirty-two, then sixty-four—this process reflects the differentiation of Yin-Yang and alternation of hardness-softness. From ten to hundred, thousand, ten-thousand, and onward to yi (100 million), zhao (trillion), jing (10^16), and gai (10^20)—each order of magnitude follows base-ten, totaling fifteen levels, all adhering to the root number ten. Like a tree with trunk, branches, and leaves, division brings refinement, and refinement brings flourishing. Yang Qi begins growing from the Zhen hexagram (Thunder), reaches its peak at Qian (Heaven), then starts dividing; Yin Qi sprouts from Xun (Wind) and completes its gathering at Kun (Earth). This process of growth, division, and convergence fundamentally operates through the number ten. Thus Cheng Yi stated, "The union of two and five accomplishes the transformations of Yin-Yang" (referring to how Heaven-Earth's dual Qi combines with the Five Phases to create all things).

Shao Yong (Northern Song Neo-Confucianist) proposed: The four hexagrams Qian, Kun, Kan, and Li form the foundation for the other thirty-six hexagrams, while Xun, Dui, Gen, and Zhen govern twenty-eight variant hexagrams. Among these, Qian, Kun, Kan, Li plus Zhongfu, Yi, Daguo, and Xiaoguo—eight hexagrams—are immutable fundamentals that generate thirty-six derived hexagrams (eighteen in each Yijing section). Meanwhile, Dui, Xun, Zhen, and Gen produce twenty-eight variants through pairwise combinations. When observing all fifty-six hexagram forms, why emphasize thirty-six? Because Qian corresponds to number one and Kun to eight—within the hundred-number system, sixty-four hexagrams are selected as the substance (ti), while thirty-six serve as function (yong). From one to ten forms the foundation; ten to hundred completes the system. Kun's position in the upper layer yields thirty-six as function, while in the lower layer it manifests sixty-four as substance (referring to the even-numbered Yin lines' foundational role).

This scripture discusses the mathematical principles governing the transformations of the Eight Trigrams (Bagua) in ancient numerology, primarily concerning the correspondence between the sixty-four hexagrams of the Book of Changes (Zhouyi) and astronomical calendrics. It employs numerical combinations to reveal the mysteries of celestial and terrestrial movements:

First, it explains the mathematical patterns of the Eight Trigrams overlapping to form sixty-four hexagrams. Within these sixty-four hexagrams (8×8=64), multiple numerical layers are concealed: for instance, the thirty-six hexagrams (4×9=36) contain the pattern of seven (4×7=28), while the twenty-eight hexagrams further embed the pattern of eight (7×8=56). This nested numerical relationship embodies the Yijing philosophy of "moving from simplicity to complexity, then returning complexity to simplicity."

Next, it describes how the line numbers of the Qian (Heaven) and Kun (Earth) hexagrams correspond to celestial phenomena. Each line of the Qian hexagram contains thirty-six counts, totaling 216 counts across its six lines (6×36). Each line of the Kun hexagram holds twenty-four counts, totaling 144 counts (6×24). Their sum equals 360, symbolizing the days of a year. The phrase "opening at Yin and closing at Xu" refers to the ancient practice of correlating the twelve two-hour periods of the day with earthly branches: Yin (3–5 a.m.) marks sunrise (opening), while Xu (7–9 p.m.) marks sunset (closing).

Finally, it outlines the operational principles of celestial and terrestrial numbers: celestial (Yang) numbers take seven as their functional base and three as their transformative pivot, while terrestrial (Yin) numbers take eight as their functional base and four as their transformative pivot. The Qian hexagram's original 216 counts, when calculated using the "sevenfold rule," become 252 counts. The Kun hexagram's original 144 counts, when reduced to one-third to harmonize with Qian, leave 108 counts—corresponding to the three periods of Hai, Zi, and Chou (9 p.m.–3 a.m.). These numbers align with the movements of the sun, moon, and year while subtly resonating with the systems of heavenly stems and earthly branches.

(Note: The term "nine categories" refers to the nine grand governance principles recorded in the Hongfan chapter of the Book of Documents, with "six calamities and five blessings" being its discourse on fortune and misfortune. These numerical models reflect the ancient practice of using hexagram lines to calculate celestial timings.)

This passage discusses the numerical principles in ancient Chinese calendrical astronomy, primarily concerning the mathematical relationships between the waxing and waning of Yin-Yang, seasonal transitions, and their correspondence with hexagram lines.

First, it explains the correlation between day-counts (Yang) and celestial-hour counts (Yin): 110 corresponds to 150, while 80 corresponds to 240; 120 corresponds to 270, and 90 corresponds to 240. This reveals that the calculations of day-counts (symbolizing heaven) and celestial-hour counts (symbolizing earth) follow specific proportional relationships.

Regarding the fullness and emptiness of Yin-Yang: At the peak of Yang energy during the summer solstice, there is an excess of 2,100 celestial-hours, equivalent to 6 days. Each monthly solar term contains this accumulated Yang energy. Six accumulations form 12 days, or 120 celestial-hours. Similarly, at the peak of Yin energy during the winter solstice, there is a deficit of 2,106 celestial-hours. The monthly new moon (first day of the lunar month) also shows a 6-day deficiency, with six accumulations forming a 12-day (120 celestial-hour) Yin recession. The combined Yin-Yang cycle totals 240 celestial-hours, corresponding to the grand cycle of the 24 solar terms. Every 60 days constitutes a major cyclical phase, with intercalary days accumulating every 6 days and distributed across the 24 solar terms.

The text introduces the "substance-function" (ti-yong) relationship: The substance-number (384) corresponds to the total lines of the 64 hexagrams, with 360 lines serving as the function-number (symbolizing the days of a year). The 24 lines of the Qian (Heaven), Kun (Earth), Kan (Water), and Li (Fire) hexagrams govern the 24 solar terms, while the remaining 360 lines serve as the annual function-number. The 90 lines represent the substance-number (corresponding to 15 hexagrams), while 270 lines constitute the heaven-earth function-number (a cycle opening at the Tiger hour and closing at the Dog hour). The 100 lines represent the substance-number (18 hexagrams), while 252 lines constitute the earthly function-number (cycles from Tiger to Rooster months plus intercalation). The hexagrams of the Pig, Rat, and Ox palaces are excluded, preserving 108 days to manifest the interpenetration of substance and function.

Finally, it notes that the practical function-number 264 serves as the foundation for musical temperament and the generation of all things, while 256 represents the earthly substance-number through which the Kan and Li hexagrams transform and generate all things. These numerical systems embody the cosmological view of Taiji generating heaven and earth, which in turn generate all things. Through precise correspondences between hexagram lines and calendrical cycles, they construct a mathematical model of cosmic resonance between heaven and humanity.

This passage elucidates the mathematical principles governing the transformations of hexagrams in the Book of Changes (Yijing), exploring the relationship between Yin-Yang line mutations and the growth of all things.

The ancients used numerical calculations to deduce the patterns of heaven and earth: twelve plus four yields six, and 256 days correspond to 3,072 two-hour periods—these are critical cycles for the growth of all things. Since all life depends on Yang Qi, calculations primarily employ the numerical values of Yang lines. The Qian (Pure Yang) hexagram has 36 transformations, while the Dui, Li, and Xun hexagrams collectively have 84. The Kun (Pure Yin) hexagram has 12, and the Zhen, Kan, and Gen hexagrams total 60. The eight trigrams sum to 192 Yang lines, and when added to the foundational 64, they arrive at the key number 256.

In practical application, the 64 hexagrams exclude the initial and top lines, focusing only on the middle four, which also yields 256. This is because "four" represents the fundamental structure of the earth: the Kan hexagram has four Yin lines, and the Li hexagram has four Yang lines, making four the basis for all growth. Qian (Heaven) and Kun (Earth) are fixed above and below, while Kan (Water) and Li (Fire) intersect in the center—this is the root of life's gestation. The initial and top lines are omitted because dawn, dusk, and the polar extremes are unfavorable for growth, whereas midday is most conducive. If the Li hexagram does not maintain its four Yang lines, it cannot receive Kun's Yin Qi; if the Kan hexagram does not retain its four Yin lines, it cannot accommodate Qian's Yang Qi.

The initial line represents the earth's breath and the foundation of life. In the Prenatal Bagua arrangement, 32 Yang and 32 Yin lines remain immutable. The top line embodies the divine nature and origin of heaven, so the eight pure hexagrams undergo five transformations into "wandering soul" hexagrams, yet their top lines remain unchanged.

Master Shao Yong (a Northern Song Neo-Confucian scholar) stated: "The Book of Changes follows patterns of transformation, where hexagrams shift into one another. Heavenly numbers encompass three layers: three yielding six, three times three yielding nine, and nine times nine yielding 81—the limit of Yang numbers, at which point Yang turns to Yin. Earthly numbers double four to yield eight, and eight times eight yields 64—the limit of Yin numbers, at which point Yin turns to Yang." Hexagram mutations occur in three forms:

1. Qian transforms into Zhen, Kan, and Gen; Kun transforms into Xun, Li, and Dui.

2. Qian shifts from Guai to Tai, while Kun shifts from Bo to Pi.

3. The three male hexagrams (Zhen, Kan, Gen) and three female hexagrams (Xun, Li, Dui) mutually transform.

Divination-derived transformations: encountering "old Yang" (9) or "old Yin" (6) leads to Qian or Kun; encountering "young Yang" (7) or "young Yin" (8) preserves the forms of Zhen, Kan, Gen or Xun, Li, Dui. Line mutations, as in the Kun hexagram, begin with the initial line and proceed until all six lines change.

This passage explains the transformational patterns of hexagrams in the Book of Changes (Yijing), along with the principles of mutual correspondence in the Prenatal Bagua Diagram (a cosmological diagram created by Song dynasty scholar Shao Yong).

First, the hexagram transformation rules are explained: A single changing yang line generates six hexagrams (Fu, Shi, Qian, Yu, Bi, and Bo). Two changing yang lines produce fifteen hexagrams, all derived from the Lin hexagram. Three changing yang lines yield ten hexagrams, all originating from the Tai hexagram. Starting from the Qian (Heaven) hexagram, changing the first line produces Gou, and when all six lines change, it becomes Kun (Earth). The same logic applies to yin line changes: A single changing yin line generates five hexagrams (Gou, Fu, Lü, Xiaochu, and Dayou); two changing yin lines produce fifteen hexagrams derived from Dun; three changing yin lines create ten hexagrams from Pi.

Through progressive transformations, the eight trigrams expand into sixty-four hexagrams, and further into 4,096 possible combinations. When twelve-line hexagrams (composed of two stacked six-line hexagrams) are extended to twenty-four lines, they yield over 16 million variations. In the Prenatal Diagram, one changing line produces two hexagrams, two changing lines generate four, and so on until six changing lines fully manifest all sixty-four hexagrams.

Next, the correspondence principles in the Prenatal Square-Circle Diagram are explained: Shao Yong observed that internal transformations correspond to external ones, and lower changes mirror upper changes. For instance, the twenty-eight yang lines in the Xun, Li, and Dui trigrams correspond to the twenty-eight yin lines in Kan, Gen, and Zhen. When Qian, Dui, Xun, and Kan appear above, Li, Zhen, Gen, and Kun manifest below. These relationships mirror celestial phenomena: Heaven's leftward rotation corresponds to the sun's rightward path (Qian-Li pairing), lunar orbits (Dui-Zhen pairing), or fire emerging from earth (Kun-Kan pairing). Natural analogies extend to animal habitats (Li-Gen representing birds perching on trees; Zhen-Kun depicting beasts resting in grass) and human anatomy (Qian-Xun matching heart-lungs; Dui-Kan aligning liver-gallbladder). These systematic correspondences embody the fundamental principle of resonant interaction among the sixty-four hexagrams in the Yijing.

This passage explains the patterns of cosmic Yin-Yang transformations and the operations of the Eight Trigrams (Bagua) in modern terms:

In the hexagrams, the first line corresponds with the fourth, the second with the fifth, and the third with the sixth—these correspondences are often complementary opposites. Heaven and Earth embrace each other like male and female attracting one another. Heaven belongs to Yang, Earth to Yin, with Heaven's patterns resonating with Earth's harmonics.

Specifically: The four trigrams of Qian (Heaven), Dui (Lake), Li (Fire), and Zhen (Thunder) positioned in the northwest govern the movement of Earth's sixteen hexagrams (numbers 5-8) in the southeast, while simultaneously interacting with sixteen hexagrams (including Pi, Dun, Song, and Gou) in the southwest. These configurations correspond to seasonal heat and cold, day-night alternation, animal movements, plant growth, and other natural phenomena. Through sixteen dynamic numerical transformations, they ultimately form 256 hexagram positions encompassing four transformation states. All creatures' successes and failures manifest through these symbolic changes.

Conversely, Earth's harmonics respond to Heaven: The four trigrams Kun (Earth), Gen (Mountain), Kan (Water), and Xun (Wind) in the southeast coordinate with Heaven's sixteen hexagrams (numbers 1-4) operating in the northwest, while interacting with sixteen hexagrams (including Tai, Lin, Yi, and Fu) in the northeast. These correspond to weather patterns (wind, rain, dew, thunder) and vegetation cycles. Similarly, through sixteen vegetative numerical changes, they form 256 hexagram positions. All plants' flowering, fruiting, flourishing and withering manifest through these transformations.

Southwestern hexagrams rotate left-to-right, observing animal life characteristics where vitality originates in the head, supported by Heaven's Yang Qi emerging below. Among the lower thirty-two hexagrams, all develop upward, indicating life force ascending. Hence humans have heads atop, while birds and beasts grow horizontally.

Northeastern hexagrams rotate top-to-bottom, observing plant life characteristics where vitality roots below, supported by Earth's Yin Qi emerging above. Among the upper thirty-two hexagrams, all develop downward, indicating life force descending. Thus human kidneys reside below, while plants grow inverted (roots downward).

Further matching sound correspondences: The voices of sun, moon and stars resonate with 112 celestial hexagrams, while sounds of water, fire, earth and stone resonate with 152 terrestrial hexagrams.

Finally, it notes: Heaven and Earth share sixteen transformations totaling 672 waxing-waning relationships. Heaven's eight transformations spiral left—from midnight (Zi) to mid-morning (Si) representing day, totaling 336 growth patterns. Earth's eight transformations spiral right—from noon (Wu) to midnight (Hai) representing night, totaling 336 decline patterns. The complete sixteen transformations show Heaven governing form through eight changes culminating in sixteen, with Qian (Heaven) as sovereign. Beginning with Guai's operation, every two hexagrams correspond to one transformation—Tongren matching the eighth, Gou the sixteenth.

This passage explains the correspondence between the Eight Trigrams (Bagua) and numerical values in ancient Yijing studies, along with the computational principles of waxing and waning Yin-Yang. The phrase "each earthly [trigram] divides into eight transformations" refers to the eight variations of the trigrams, each corresponding to the number 336, detailing the calculation method for the 56 hexagrams' line counts.

The specific calculation proceeds as follows: Among the 64 hexagrams, after excluding the eight hexagrams—Qian, Kun, Kan, Li, Daguo, Xiaoguo, Zhongfu, and Yi (totaling 24 lines)—the remaining 48 hexagrams contain 336 lines. Yin and Yang each account for half, with the Yang count exceeding by six days and the Yin count falling short by six days. Combined, this yields the total of 672, embodying the cyclical pattern of Yin-Yang transformation.

Next, the passage explains the generative numbers of the trigrams: Qian is 1, Dui is 2, Li is 3, Zhen is 4, Xun is 5, Kan is 6, Gen is 7, and Kun is 8. The Cai scholar (likely Cai Yuanding) elaborates: Yang generating Yin is calculated as 2 × 6 = 12, while Yin generating Yang is 3 × 10 = 30. Starting from Qian as 1, Dui becomes 12, Li becomes 12 × 30 = 360, and Zhen becomes 12 × 360 = 4,320. The subsequent numbers from Xun to Kun are derived through the alternating pattern of odd and even generation.

Here, temporal units serve as metaphors: Qian symbolizes the year (1), Dui symbolizes months (2 × 6 = 12 months), Li symbolizes days (12 × 30 = 360 days), and Zhen symbolizes hours (360 × 12 = 4,320 double-hours). The first four numbers—1, 2, 3, 4—represent the primordial generative numbers of heaven and earth, reflecting Yang's precedence over Yin. After Zhen (4) comes Xun (5), marking the inception of Yin Qi, calculated as the larger figure 129,600, followed by Kan (6) continuing this computational system.

These numerical calculations reflect the ancient mindset of correlating the trigrams with temporal cycles, using numerical transformations to elucidate the cosmic principles of Yin-Yang fluctuation.

This passage from Tianyuan Faming primarily elucidates the numerical derivation patterns in Daoist cosmology and the corresponding relationships between heaven-earth and yin-yang.

First, it explains the temporal calculations of the Xun trigram (one of the Eight Trigrams (Bagua), representing wind): The Xun trigram corresponds to the number five, and calculating the months yields 1,555,200 months. Next, the Gen trigram (representing mountain) corresponds to the number seven, and further calculating the days yields 46,656,000 days. The hour count of the Xun trigram corresponds to the Kun trigram (representing earth) with the number eight, deriving 559,872,000 hours. These numbers—five, six, seven, and eight—represent the ultimate numerical principles through which heaven and earth generate all things, embodying the pattern of yin energy receiving and continuing yang energy.

It then discusses the generational calculations of the Qian trigram (representing heaven) and the Dui trigram (representing lake): The generational count of the Qian trigram is 4,320, deriving 559,872,000; the generational count of the Dui trigram is 51,840, deriving 67,184,640,000. From the calculations of the Qian, Dui, Li, and Zhen trigrams, it can be seen that the patterns of the Xun, Kan, Gen, and Kun trigrams are already contained within them.

Regarding the division of time: The six two-hour periods from Zi (11 PM–1 AM) to Si (9–11 AM) belong to the Qian trigram and are termed "pre-heaven" (xiantian); the six periods from Wu (11 AM–1 PM) to Hai (9–11 PM) belong to the Kun trigram and are termed "post-heaven" (houtian). The post-heaven always follows the patterns of the pre-heaven in its operations. Thus, it is said that the Qian trigram accomplishes all phenomena, while the Kun trigram emulates the Qian trigram—a principle that becomes clear through derivation.

The second part explains the application of the numbers twelve and thirty:

• Twelve months in a year equate to one year multiplied by twelve.
• Three hundred sixty days in a year equate to twelve months multiplied by thirty days.
• Four thousand three hundred twenty hours in a year equate to 360 days multiplied by twelve hours.
• Thirty years in a generation (shi) equate to one year multiplied by thirty.
• Three hundred sixty months in a generation equate to thirty years multiplied by twelve months.
• Ten thousand eight hundred days in a generation equate to 360 months multiplied by thirty days.
• One hundred twenty-nine thousand six hundred hours in a generation equate to 10,800 days multiplied by twelve hours.

Finally, it discusses the derivation of the yin number two: From two, twelve is derived, corresponding to the twelve Earthly Branches, twelve two-hour periods, twelve months, twelve generations, and twelve trigrams. The Kun trigram has twelve strokes, and the human body has twelve key parts. Earthly numbers begin with twelve, and twelve two-hour periods constitute the monthly count. The periods from Zi to Si belong to yang, while those from Wu to Hai belong to yin, each occupying six periods.

These numerical derivations reflect the Daoist mathematical understanding of cosmic operational laws. Through the mutual correspondences of yin-yang, the Eight Trigrams (Bagua), and temporal divisions, they construct a comprehensive system for measuring time.

This scripture primarily discusses the Daoist correspondences between Yin-Yang, numerical patterns, and temporal cycles, as well as the application of Qian (Heaven) and Kun (Earth) within this system.

The text first explains the division of Yin-Yang: the time periods from Zi (11pm-1am) to Xu (7-9pm) belong to Yang, while Chou (1-3am) to Hai (9-11pm) belong to Yin—making twelve two-hour periods equally divided between both principles. Using odd and even numbers yields the same balanced division. The number twelve can be split into two parts or four sections of three, corresponding to the four seasons. Twenty-four corresponds to the twenty-four solar terms and twelve months. The eight major solar terms (including Start of Spring, Spring Equinox, etc.) each contain three qi phases, while each season contains six qi phases.

The total line count of the Prenatal Bagua (Eight Trigrams) is forty-eight, corresponding to the four directions and four phases (Metal, Wood, Water, Fire). The abstracted twelve in the center represents Earth, resonating with the four directions. Within the sixty Jiazi cycles, twelve are preserved—during divination, the hexagram count is one, with Kun (Earth) hexagram reserving twelve Yang lines and Qian (Heaven) hexagram reserving twelve Yin lines. Subtracting twelve from forty-eight leaves thirty-six as Lao Yang (supreme Yang, symbolizing Heaven), applying the "remove one from three" principle from Yin to Xu periods. Subtracting four and six from twenty-four yields fourteen as Lao Yin (supreme Yin, symbolizing Earth), with halves used alternately from Mao to Shen periods.

Numbers like twenty-eight and thirty-two each have specific applications, where active numbers typically exceed inactive ones, symbolizing the dynamic interplay of Qian and Kun. Twelve holds particular significance, corresponding to solar and lunar movements: the sun's daily cycle and the moon's waxing and waning. When Yang advances for twelve days, qi becomes abundant; when Yin recedes for twelve days, qi becomes deficient. The annual sixty Jiazi cycles operate on this foundation—from small six-period cycles to major sixty-year cycles.

Finally, the text explains Qian-Kun applications: Qian employs nine, Kun employs six, with alternating progression among seven, eight, nine, and six. Qian encompasses four forms (directions), while Kun contains six functions (hexagram lines). Thirty-six can transform through four or six variations, as can twenty-four through four variations. These numerical transformations all manifest the operating principles of Heaven and Earth.

(Note: "Yao" refers to the Yin-Yang lines in hexagrams; "Lao Yang/Lao Yin" denotes pure Yang/Yin states in trigrams; the "Prenatal Bagua" is a fundamental Daoist cosmological diagram.)

This passage primarily explains the mathematical principles governing the transformations of Yin and Yang. The ancients used numbers to symbolize the waxing and waning of these forces: twenty-four represents the extreme of Yin (polar Yin), which increases by six to become thirty (central Yang), then thirty-six (polar Yang). Conversely, decreasing by six from thirty-six returns to the Yin state. When using four as the incremental unit, twenty-four (polar Yin) progresses through additions of four to become twenty-eight, thirty-two, and ultimately thirty-six (mature Yang), with the reverse process returning to Yin through subtraction.

The Qian (Heaven) and Kun (Earth) hexagrams embody the pivotal numbers seven, eight, nine, and six, symbolizing the flourishing and decline of Yin-Yang. The other six hexagrams (Zhen, Xun, Kan, Li, Gen, Dui) cannot transform through six like Qian and Kun. When Yang emerges from Yin, the numbers grow from six to nine to reach full vigor; when Yin recedes from Yang, they decline from nine to six. These transformations always involve the Four Symbols (Great Yang, Lesser Yin, Lesser Yang, Great Yin), hence the use of four as the foundational number layered upon nine, eight, seven, and six.

Among the sixty-four hexagrams of the Yijing (Book of Changes), sixty correspond to the sexagenary cycle, while four represent the seasons. Though Yin and Yang follow distinct patterns of maturity and youth, Yin can never precede Yang. The Qian hexagram inherently contains six Yang lines but incorporates half of Kun to become nine. Multiplying three by nine yields eighty-one—this demonstrates how the number of Great Yang begins with three and culminates in nine's perfection. The numerical variations of Great Yin, Lesser Yang, and Lesser Yin all operate within this system. Thus, from the perspective of Qi's totality, everything can be encompassed under "Qian."

Later sections mention the Zhouyi's use of dual eighteens forming the significant number thirty-six. While King Wen's arrangement of hexagrams and Confucius' Ten Wings provided thorough explanations, none since the Han dynasty have fully clarified this. Shao Yong's (Jingshi) system employs dual one-hundred-eights to form the cyclical number three hundred sixty, within which all transformations—from grand cycles to yearly, monthly, daily, and hourly fluctuations—find expression.

Zhang Guanwu, a Song Dynasty scholar of the Yijing (Book of Changes), proposed twelve fundamental numerical systems derived from the waxing and waning interactions of Yin and Yang qi (vital energies).

First, let's clarify key concepts: The "substance number" (ti shu, representing fundamental reality) is 384, while the "function number" (yong shu, representing active application) is 360. Among these, 10 and 72 are commonly used numbers, and 8 represents transformative interactions. Yang numbers (odd) typically exceed by 6, while Yin numbers (even) fall short by 6, reflecting the natural rhythm of Yin-Yang expansion and contraction. The odd numbers are 1, 3, 5, 7, 9; the even numbers 2, 4, 6, 8, 10. "Growth numbers" describe developmental progression from small to large (as seen in the Fu hexagram evolving toward Qian), whereas "fraction numbers" represent differentiation from large to small (as in Gou transitioning toward Kun). A single yang line in the Qian hexagram can differentiate into six hexagram manifestations - Dayou, Xiaoxu, Lü, Tongren, Gou, etc. - each retaining Qian's essential qualities.

Several operational methods deserve explanation: "Factor numbers" involve doubling sequences (e.g., 2×12=24), "analysis numbers" employ binary division (1 divides into 2, then 4, then 16), "diminishing numbers" indicate reduction, while "multiplying numbers" signify expansion. Consider the mutual multiplication of Yin and Yang: 360×360=129,600, corresponding precisely to one cosmological cycle (yuan) in ancient Chinese calendrics (approximately 129,600 years).

Regarding hexagram substance-function relationships: Each hexagram possesses both inherent qualities and active applications. The "substance-through-function" principle describes how one Yin and one Yang develop into two Yin and two Yang, then four Yin and four Yang, ultimately forming the eight trigrams system. In the Prenatal Bagua arrangement, the outer trigrams emerge through eight yang and eight yin interactions, progressively generating 16, 32, and finally 64 hexagrams. Structurally, the inner palace's left side contains 64 yang and 32 yin lines, mirrored symmetrically on the right; the upper palace balances 48 yang and 48 yin lines on each side, embodying perfect Yin-Yang equilibrium.

The transformation from substance to function follows this sequence: 64→32→16→8→4→2→1, where the ultimate "One" represents Taiji (Supreme Ultimate). Taiji generates Liangyi (the Two Modes: Yin-Yang), which produce Sixiang (Four Phenomena: lesser yang, greater yang, lesser yin, greater yin), culminating in the Bagua (Eight Trigrams) - the foundational generative principle of the Yijing.

Shao Yong (Northern Song Neo-Confucian philosopher) further observed: Heaven and Earth share four substance numbers, with three manifesting actively while one remains concealed. Heaven's four transformations encompass Earth's four transformations, demonstrating their cosmic correspondence. These numerical systems ultimately reveal the fundamental laws governing universal Yin-Yang dynamics.

The operation of Heaven and Earth follows the principles of the Eight Trigrams (Bagua), yet its core mechanism can be simplified to four fundamental hexagrams. Heaven (Qian hexagram) takes movement as its foundation, governing Earth's operations; Earth (Kun hexagram) takes form as its basis, its functions aligning with Heaven's patterns. Heaven reduces the four manifestations to three (referring to Lesser Yang, Greater Yang, and Lesser Yin), using three as its active principle; Earth divides the whole into four (the four directions and seasons), taking four as its substantial form.

From the Yin hour (3-5 am) to the Xu hour (7-9 pm), there are three periods of activity and one of stillness, corresponding to the Greater Yang number 49 (36). From the Mao hour (5-7 am) to the Shen hour (3-5 pm), half the day is active and half still, matching the Greater Yin number 46 (24). The four cardinal directions and seasonal changes form sixteen fundamental states, while Heaven's four manifestations (Greater Yang, Lesser Yang, Greater Yin, Lesser Yin) appear as cold, heat, day and night. Earth's four manifestations transform into rain, wind, dew, and thunder—demonstrating Heaven's capacity to encompass all earthly phenomena.

Heaven's threefold number combines to six (3×2), calculated from one: including the base number yields seven, excluding it yields six. Thus Yang numbers always retain "one" as their generative force, ultimately forming thirty-six (6×6). Earth's number two operates through twelve (2×6), with twelve beginning from four: including the base yields sixteen, excluding it yields twelve. Therefore Yin numbers consistently conceal "one" as their diminishing principle, corresponding to twelve months reducing by twelve days. Consequently, Heaven's operating principle (Taiji) governs Earth from right to left, while Earth's primordial Qi (Yuanqi) follows Heaven from left to right.

Three signifies thirty-two, and two signifies six—this uses the annual count of Yin-Yang Qi to explain: a year divides into four seasons, each containing three months of three ten-day periods. By celestial stems timing, five days make one pentad, three pentads (fifteen days) form one Qi phase; by earthly branches timing, six days make one division, three or four two-hour periods (twelve hours) constitute a day. In macro-cycles, a year integrates 360 days (30×12); in micro-cycles, a month integrates 360 two-hour periods (30×12). Temporal units progress layer by layer: 30 days form one transformation, 30 transformations form one convergence, 30 cycles form one generation (30 years). Twelve two-hour periods compose a month, twelve months a year, thirty years a generation, twelve generations a cycle, thirty cycles a convergence, twelve convergences an eon (complete cosmic period).

Heaven's six transformations produce 360 as the operational base: first transformation generates six, second twelve, third eighteen, fourth twenty-four, fifth thirty, sixth thirty-six. Through this progression, the complete cyclical number of three hundred sixty is ultimately formed.

The operation of the cosmos originates from the fundamental principle of "One." From this "One" can be derived the transformations of "Six," without requiring any other foundation.

From the perspective of Earth, "One" transforms into "Six," and "Six" further transforms into "Three Hundred Sixty." This follows the patterns of celestial and terrestrial movement. The idea that "Five" generates "Six" means: each month corresponds to five trigrams (gua, symbols from the Yijing), and each trigram contains six lines (yao, the basic components of a trigram). Five multiplied by six equals thirty. Excluding the four cardinal trigrams (Zhen, Li, Dui, and Kan, representing the four directions), a year consists of sixty-six trigrams, each with six lines, totaling three hundred sixty lines. A year also cycles through sixty sexagenary stems-and-branches (an ancient calendrical system), and six multiplied by sixty similarly yields three hundred sixty days.

When "One" transforms to produce "Six," removing "One" leaves "Five." "Five" takes "One" as its root. In the second transformation, "Twelve" is produced; removing "Two" leaves "Ten," which takes "Two" as its root. Continuing through the third, fourth, fifth, and sixth transformations—each time removing the corresponding number while preserving the root—yields the numbers twenty, twenty-five, and thirty. Thus, it is said that "Five" and "Six" generate "Thirty," with thirty trigrams corresponding to the six celestial transformations, producing one hundred eighty. The Xiantian Tu (Diagram of Prior Heaven) follows this principle on both its left and right sides, each undergoing five transformations to produce thirty-two yang and thirty-two yin. "Five" and "Six" interact alternately, pairing odd and even—this is why they are considered the core numbers between Heaven and Earth. The movement of the four seasons aligns perfectly with them.

Counting trigrams by the twelve earthly branches (Zi, Chou, Yin, etc.), each cycle changes every six days; counting climatic phases by the ten heavenly stems (Jia, Yi, Bing, etc.), each cycle changes every five days. Sixty transformations produce three hundred sixty. Multiplying three hundred sixty by itself yields the Yuan (a Daoist cosmic time unit) of one hundred twenty-nine thousand six hundred.

From Earth's perspective, four transformations generate thirty-six, the number of living things. "Four" generates "Twelve," "Twelve" generates "Twenty," "Twenty" generates "Twenty-Eight," "Twenty-Eight" generates "Thirty-Six," and this progression extends to two hundred fifty-six. The numerical basis for Earth's nurturing of all things is rooted in "Four," with "Two" used to derive "Twelve." Decomposing "One" into "Four," "Four" into "Sixteen," "Sixteen" into "Sixty-Four," and "Sixty-Four" into "Two Hundred Fifty-Six" represents Earth's four transformations. "Sixteen" is Earth's number, and the first transformation of "Four" into "Four" is Earth's foundation. "Four" generating "Eight" means one "Four" produces two "Fours"; "Eight" generating "Twelve" means two "Fours" produce three "Fours"; "Twelve" generating "Sixteen" means three "Fours" produce four "Fours"—thus progressively increasing.

This passage elucidates the Daoist cosmological principles governing the numerical patterns of celestial movements and Yin-Yang transformations. All phenomena between heaven and earth are composed of the two vital energies (Qi) of Yin and Yang, and these numerical combinations reveal the profound mysteries of cosmic operation.

The heavenly number (Yang) is based on six, while the earthly number (Yin) is based on four. Heaven operates through the number 36 (6×6), while earth generates all things through the number 36 (4×9). The annual cycle of growth spans 256 days from vegetation sprouting to earth freezing solid.

The Eight Trigrams (Bagua: Qian, Kun, Zhen, Xun, Kan, Li, Gen, Dui) inherently possess eight configurations, but only six transformations are practically employed. While there exist 384 possible hexagram variations, only 216 fundamental transformations are utilized. The combination of 36 trigram variations and 216 line changes yields the 252 practical transformation patterns actually employed.

Yang energy manifests in six states (three Yang and three Yin phases), each lasting six ten-day cycles (60 days), totaling 36 ten-day periods. Within the 360-day year, Yang Qi dominates 180 days (30 days per Yang phase) while Yin Qi governs the remaining 180 days (30 days per Yin phase). The pure Yang Qian trigram employs the number 36 (4×9), while the pure Yin Kun trigram uses 24 (4×6), their union forming the complete 60-year Jiazi cycle.

The specific numerical relationships are: triple the Yang number 36 equals 108, and triple the Yin number 36 also yields 108. The three Yang and three Yin phases each account for half. Three 36s make 108, and two 108s form 216. When the Qian trigram dominates, it incorporates aspects of Kun's nature—thus in practice Qian governs six-sevenths (216) while Kun governs three-sevenths (108).

A day contains twelve two-hour periods (shichen), as a year contains twelve months. From Yin hour (3-5 AM) to Wu hour (11 AM-1 PM) comprises 108 Yang periods, while Wu hour to Xu hour (7-9 PM) constitutes 108 Yin periods. Within these exist three transitional periods of Yin-Yang alternation, all representing active phases of universal growth. The remaining 144 periods belong to pure Yin states, akin to the late Yin hour when Yin energy predominates.

This scripture primarily discusses the Daoist cosmological system of temporal calculation and the principles of Yin-Yang transformation.

It begins by stating that a year consists of 360 days, divided into 18 "Xu" (time units), with each Xu further subdivided into 36 parts. The residual power of Yang Qi governs the growth processes of spring, summer, and autumn, while assisting the operational function of the Qian trigram (representing Heaven). Here, the crucial concept of "Yang being rooted in Yin" is introduced, illustrating how Yang Qi requires Yin Qi as its foundation.

The text then employs the numerical system from the Yijing (Book of Changes) to calculate time: The fundamental number 129,600 can be decomposed into 4,320 generations (one generation = 30 years). Through the accumulation of Yin-Yang transformations, a year is formed—360 days containing 4,320 shichen (time periods, 1 shichen = 2 hours). Using 30-year generations as the calculation basis, the Jiazi and Jiawu years serve as starting points for chronological records.

Heaven and Earth each possess five fundamental temporal units: Heaven's generative numbers are Yuan (Epoch), Hui (Cycle), Yun (Fortune), Shi (Generation), and Nian (Year); Earth's formative numbers are Yue (Month), Ri (Day), Shi (Hour), Fen (Minute), and Miao (Second). The largest temporal unit "Yuan" equates to one year, while the smallest "Miao" corresponds to one month. All calculations of the eight trigrams in the Qian palace are based on this grand cycle of Heaven and Earth.

Within each month, day, and hour, the Yin and Yang forces continuously wax and wane. The temporal units maintain nested relationships: seconds are contained within minutes, minutes within hours, hours within months, months within days, and days within years. This relationship grows more expansive as one moves toward microscopic scales and more refined toward macroscopic scales—so profound that even spirits and deities cannot fully comprehend its mysteries.

Specific calculations include: One shichen contains 30 fen, totaling 360 miao, corresponding to the Dayou hexagram; one day contains 12 shichen, totaling 4,320 miao, corresponding to the Dazhuang hexagram; one month contains.... These numbers all correspond to Yijing hexagrams, forming a complete system of spatiotemporal correlations.

This scripture employs the Daoist system of temporal calculation, using numerical precision to measure the cyclical movements of heaven and earth. The term "miao" (杪) refers to an ancient unit of time equivalent to a second, while "gua" (卦) denotes specific hexagrams from the sixty-four hexagrams of the Yijing (Book of Changes), each corresponding to distinct temporal cycles.

The fundamental conversion relationships are as follows: 160 shichen (two-hour periods) equal 18,000 fen (minutes), totaling 129,600 miao, corresponding to the Xiaoxu (Minor Accumulation) hexagram, symbolizing a year composed of 4,320 shichen. Thirty years constitute one "shi" (generation), corresponding to 129,600 shichen, equivalent to 3,888,000 fen or 46,656,000 miao, matching the Daxu (Major Accumulation) hexagram. Twelve shi form one "yun" (cycle), totaling 1,555,200 shichen. Thirty yun make up one "hui" (assembly), amounting to 46,656,000 shichen. Twelve hui complete one "yuan" (epoch), representing 129,600 years and corresponding to the Dui (Joy) hexagram.

The scripture particularly emphasizes that whether considering large temporal units like yuan, hui, yun, and shi, or everyday measurements such as years, months, days, and shichen, all adhere to the fundamental numerical principle of 129,600.

The latter section explains the operational principles of the celestial stems and earthly branches: the circular celestial chart corresponds to the ten heavenly stems (Jia, Yi, Bing, etc.), symbolizing the ten-day cycle, while the square terrestrial chart aligns with the twelve earthly branches (Zi, Chou, Yin, etc.), representing the twelve months and twelve shichen. Five days make one "hou" (small cycle), ten days form one "jia" (complete stem cycle), fifteen days constitute one solar term, and thirty days complete one month. Combinations of stems like Jia-Wu and Ji-Gui manifest the transformations of Yin-Yang, while branch pairings such as Wu-Hai and Zi-Si reflect the circulation of the six qi (climatic energies). One year encompasses thirty-six Jiazi cycles, totaling 360 yun, corresponding to the degrees of the celestial sphere. This computational system integrates Yijing hexagrams, astronomical calendrics, and the Daoist cultivation perspective on space-time into a unified framework.

This scripture primarily expounds Daoism's profound understanding of cosmic operations and temporal calculations. The text describes numerical computations for tracking transformations between heaven, earth, and all things: fifteen units on the left undergo two transformations to become thirty variations; three times sixty yields one hundred eighty, mirrored identically on the right side—totaling three hundred sixty. Thirty subdivisions constitute one shichen (two-hour period), with three hundred sixty subdivisions representing the smallest temporal unit. Accumulating four thousand three hundred subdivisions forms one day, while considering the moon as a year, two and a half days comprising thirty shichen equal one month.

The division of time reaches extraordinary precision: one-thirtieth of a shichen constitutes one day, each fraction further divided into twelve miao (even smaller units), where one miao equals one shichen. Through such layered segmentation, 129,600 years can be broken down into 16,796,160,000 miao. This supremely refined calculation method penetrates to the essence of all phenomena—ascending to commune with the azure heavens, descending to connect with the yellow springs, encompassing the primal Qi (vital energy) in its vastness while measuring the immeasurably minute.

The text references circular diagrams demonstrating historical patterns: the rise and fall of emperors and generals, cycles of order and chaos across ages, seasonal rhythms of creation, and celestial movements of sun, moon, and stars—all manifestations of Qiankun (heaven-earth) governance that illuminate Dao's operational principles. Square diagrams symbolize earth, rooted in the number four. Through quadruple-layer permutations, they analyze two hundred fifty-six variations. Heaven and earth resonate acoustically—with the six Lü pitch standards above and six Lü below—as all beings follow this pattern governed by Kan-Li (water-fire), where four Yin and four Yang transformations gestate earthly phenomena.

The diagram of Xiantian studies (cosmic foundational principles) radiates from the center, revealing nature's mysterious creativity. Shao Yong (Northern Song Neo-Confucian) devoted his life to contemplating these patterns. Cheng Yi (Song Dynasty Confucian master) noted that the Yijing (Book of Changes) constitutes reverse-calculation knowledge for predicting future events—mere retrospection wouldn't require computation. Yet mathematical principles are heavenly principles: patterns like "wealth dissipates within five generations" remain bound by cosmic order.

Since Qin-Han times, people divorced numerology from heavenly principles, sinking into divinatory superstitions for centuries. Only when Shao Yong (Kangjie) expounded mathematical cosmology were celestial-earthly operations reunited. As he proclaimed: the principles existed before the trigrams were drawn and remain immutable thereafter—mathematical truth is eternal. I maintain that only by exhausting heavenly principles and fully investigating innate nature can one comprehend celestial mandate, for cosmic order is fundamental.

This passage explores the relationship between celestial-terrestrial numerology, the transformations of yin-yang, and the principles of the Yi Jing (Book of Changes). Although Confucius seldom spoke of heavenly mandate, he once lamented "the river no longer produces the chart" (referring to the legendary auspicious sign of a dragon-horse emerging from the Yellow River with the Hetu diagram), indicating that without clarity in numerical principles, heavenly patterns remain obscure. Shao Yong (a Northern Song Confucian scholar) believed that scholarship failing to integrate the three powers of heaven, earth, and humanity couldn't be considered true learning, and noble persons should pursue the highest realms.

The author observes that all numerical principles under heaven originate from celestial patterns—to violate these principles is to degenerate into trivial divination techniques. Zhu Xi commented that while Shao Yong's theories differed in form, they shared the same essence as the Eight Trigrams (Bagua) drawn by Fu Xi. Lü Zuqian (a Southern Song scholar) recorded in historical texts that the Yi Jing exhaustively mapped the patterns of all creation: using odd-even yin-yang numerology (heaven as odd numbers, earth as even), 4,096 hexagram transformations, and 11,520 divination stalks to encompass celestial movements (sun, moon, stars), natural cycles (seasons, day-night), the birth-decay of all beings, and even historical rise-fall (from the Three Sovereigns to the Warring States period). Since the Qin-Han dynasties, only Shao Yong achieved such comprehensive integration of heaven-earth principles.

The latter section details the numerical relationships between heaven's circle and earth's square: celestial numbers are odd (1,3,5) with the Qian hexagram using nine; terrestrial numbers are even (2,4) with Kun hexagram using six. Heaven's circle symbolizes wholeness (diameter-one, circumference-three), corresponding to the eight directions; earth's square originates from four (seasons/cardinal directions), accumulating to twelve (e.g., twelve months). Yin-yang dynamics manifest as: celestial yang governs expansion (1 transforming to 7), terrestrial yin governs contraction (4 transforming to 12). Circular numbers adapt square forms—through transformations like 1→4, 4 minus 1 yielding 3, 3→9—demonstrating heaven-earth interactions. These numerical models formed the core framework for ancient cosmological understanding.

This passage explains the cosmological principles of numerical symbolism in Daoist numerology, illustrating how Heaven (representing Yang) and Earth (representing Yin) interact through specific numerical combinations to form the operational patterns of the universe.

The numerical pattern of the Heavenly Dao (Tian Dao) is based on the number three, reaching its culmination at six. For example, in the unfolding transformations of a square: one whole divides into three parts, and when the original one is added back, it becomes four; four then divides into twelve parts, and when the original four is added back, it becomes sixteen. The foundational number (体数, tǐ shù) of the Heavenly Dao is four, with sixteen as its limit.

The yarrow stalks used in divination (ancient divination tools) number fifty, but one is deliberately left unused, symbolizing the operational law of the Heavenly Dao. The Earthly Dao (Di Dao), on the other hand, is based on the sixty hexagrams, but four are removed, as the Earthly Dao emphasizes "manifesting the formless within the formed." The sun’s movement through one day corresponds to the moon’s movement through thirty days, and the solar year exceeds the lunar year by six days (in reality, the solar year is 365 days, while the lunar year is 354 days).

A single line of the Qian hexagram (the Yang hexagram) encompasses two lines of the Kun hexagram (the Yin hexagram), combining to make three; the six lines of Qian and the six lines of Kun combine to make nine. A single Yang line in Qian contains thirty-six Yang numbers, while a single Yin line in Kun contains twenty-four Yin numbers, totaling sixty. The Earthly Dao takes one-third from the Heavenly Dao, just as 108 days are taken from 360 days.

The movement of Heaven and Earth follows the principle of "Yang advancing, Yin retreating": the Heavenly Dao is Yang-dominant, while the Earthly Dao involves mutual advancement and retreat of Yin and Yang. Heaven employs the number 360, while Earth uses 384. Heaven is round and dynamic, while Earth is square and substantial.

Heaven’s generative numbers are one, two, three, and four, while Earth’s completing numbers are six, seven, eight, and nine. Heaven’s six times six equals thirty-six, and Earth’s four times six equals twenty-four. Heaven divides into three parts, so a month has thirty days and a year has 360 days; Earth begins with two, so a year has twelve months, divided into twenty-four solar terms. Heaven’s operational number is three, culminating at six, while Earth’s foundational number is four, culminating at sixteen.

Finally, the passage mentions the four directional trigrams (震 Zhèn, 艮 Gèn, 兑 Duì, 巽 Xùn) in the Pre-Heaven (先天, Xiāntiān) and Post-Heaven (后天, Hòutiān) Bagua (Eight Trigrams), illustrating the directional laws of cosmic movement. These numerical and symbolic correspondences form the Daoist mathematical expression of the universe’s operational principles.

This passage elucidates the profound numerical principles of cosmic Yin-Yang relationships from the "Exploration of Celestial Origins" (Tianyuan Fami). The fundamental patterns of celestial motion can be summarized by the numbers "three" (for Heaven) and "two" (for Earth). The North Pole rises 36 degrees above the horizon while the South Pole sinks 36 degrees below, embodying the symmetrical balance of cosmic Yin-Yang.

In hexagram operations, the 64 hexagrams with their 384 lines (six per hexagram) manifest 256 variations when distributed across the four cardinal directions. The purely Yang Qian (Heaven) generates three masculine trigrams (Zhen, Kan, Gen), each containing 28 Yang aspects, while the purely Yin Kun (Earth) produces three feminine trigrams (Xun, Li, Dui), each holding 28 Yin aspects. Kun conceals 12 Yang elements that accumulate to form 36 Yang in Qian, just as Qian harbors 12 Yin elements that aggregate into 36 Yin in Kun.

Numerically, Qian employs 360 degrees, taking two as the "manifestation" number; Kun uses 24, preserving three as the "concealment" number. Heaven and Earth mutually contain each other: Heaven inevitably embraces Earth's number (two within three), and Earth necessarily contains Heaven's number (three within two). Qian encompasses Kun's even numbers as three, while Kun divides Yang numbers into six. The 36 hexagrams of Qian represent odd numbers, the 24 of Kun symbolize even numbers.

Regarding Yin-Yang waxing and waning: Qian's 36 Yang governs growth, accumulating 360 degrees annually; Kun's 12 Yin governs diminishment, corresponding to twelve monthly reductions. Yang numbers progress by threes—six times six yielding 36, forming 360 hexagrams that divide into eight structural variations to create 64 hexagrams with 4,920 permutations. Among these, 36 represent Heaven's functional hexagrams, while 64 correspond to Earth's structural hexagrams.

In the Primordial Bagua diagram: the left (Heaven) portion totals 336 divisions governing growth (Zhen, Li, Dui, Qian representing expansion; Xun, Kan, Gen, Kun representing contraction); the right (Earth) portion totals 332 divisions governing diminishment (Xun, Gen, Kan, Kun for expansion; Zhen, Li, Dui, Qian for contraction). Heaven's numbers slightly exceed Earth's, hence sages honor odd Yang numbers (Heaven three, Earth two). When Earth predominates, the Taiji's initial division takes Yang one and Yin two as foundation, subsequently generating combinations like Heaven five/Earth six, Heaven five/Earth ten, Heaven six/Earth four, and Heaven four/Earth six—where six serves as Heaven's functional number and twelve as Earth's functional number, revealing the subtle patterns of celestial Dao's transformations.

This scripture primarily elucidates the correspondences between the numerical principles of heaven (yang) and earth (yin). Through various transformative combinations, the numbers of heaven and earth form a complete cosmological operating system.

Heaven (yang) takes six as its base number, while earth (yin) takes eight. Heaven undergoes six transformation methods: the first transformation yields six, six transforms into thirty-six (6×6), ultimately forming the cyclical number three hundred sixty (corresponding to the days in a year). Earth undergoes four transformation methods: the first yields four, four transforms into sixteen (4×4), ultimately reaching two hundred fifty-six. Heaven can encompass earth's characteristics, hence using six transformations; earth cannot encompass heaven's characteristics, thus using four transformations.

Six represents the active number of yang nature, while eight represents the substantial number of yin nature. The thirty-six hexagrams (6×6) manifest heaven's function, while the sixty-four hexagrams (8×8) manifest earth's substance. Heaven and earth cooperate mutually: earth follows heaven by using the number five, while heaven follows earth by using six.

Each season corresponds to specific yin-yang proportions: During spring when the Qian (Heaven) and Dui (Lake) hexagrams dominate, yang numbers fifty-six while yin numbers forty; during autumn when Kan (Water) and Gen (Mountain) dominate, the proportions reverse. In summer when Qian and Xun (Wind) dominate, yang numbers sixty-four while yin numbers thirty-two; in winter when Kun (Earth) and Gen dominate, the proportions invert. The Zhen (Thunder), Kan, and Gen hexagrams collectively contain six yang and twelve yin lines, while Xun, Li (Fire), and Dui contain six yin and twelve yang lines.

The Qian hexagram symbolizes perfection, corresponding to the complete three hundred sixty-day cycle; the Kun hexagram symbolizes emptiness, being six days shorter than the full year. Heaven's numbers combine twenty-five to make fifty, expandable to one hundred; earth's numbers combine thirty to make sixty, expandable to one hundred twenty. The Qian hexagram obtains two hundred fifty-two (36 hexagrams × 7 lines), while Kun obtains one hundred eight (24 hexagrams × 4.5 lines).

The fifty yarrow stalks used for divination set one aside unused, dividing the remaining forty-nine into four groups for practice—this demonstrates the principle of "function containing substance." Beyond the sixty hexagrams, four numbers are preserved; these four constitute the substance of the sixty hexagrams, revealing the profound mystery of "substance containing function."

This passage explains the principles of celestial and terrestrial Yin-Yang transformations in ancient numerology, using numbers to symbolize the operating laws of the cosmos. The phrase "heaven seven earth eight" indicates that heaven employs the number seven while earth embodies the number eight. The eight trigrams (Qian, Kun, Zhen, Xun, Kan, Li, Gen, and Dui - the eight fundamental symbols of the Bagua) are based on the number six.

The number six represents "function" (yong), pertaining to the transformation of Yang lines, while the number eight represents "substance" (ti), characteristic of Yin trigrams. The Qian trigram (symbolizing heaven) initiates all things using the number eight, whereas the Kun trigram (symbolizing earth) completes all things using the number three. The celestial number is nine and the terrestrial number is ten. Adding heaven's five and earth's four makes nine, as does multiplying three by three. Nine times nine equals eighty-one; four times nine equals thirty-six. From the Zi hour (midnight) to the Si hour (morning), the numbers transform from six to nine; from the Wu hour (noon) to the Hai hour (night), they change from nine back to six.

The Qian trigram's number is nine, yet heaven functionally uses six - nine grows from six. The Kun trigram's number is six, yet earth substantially uses four - six originates from four. When Qian employs nine: multiplying three by eight and then by nine yields 216; multiplying two by eight and then by nine also produces Kun's 144. When Kun employs six: multiplying two by twelve and then by six gives 144; multiplying three by twelve and then by six equally generates Qian's 216.

The celestial numbers culminate in the terrestrial number nineteen, with ninety representing the beginning and end of heaven and earth. The numerical value of the eight trigrams is thirty-six. All paired sums (1+8, 2+7, 3+6, 4+5) equal nine. Regarding the number ten: it is essentially the even counterpart of nine. Yang numbers peak at nine, while Yin numbers conclude at ten. Heaven's ten is one plus two (with two fives making ten); earth's ten is two plus four (with two eights making ten). Fifty takes one as its foundation, with forty-nine as its function; sixty-four takes four as its foundation, with sixty as its function.