Loading...
Chapter 17 of 25
皇极经世
The number of Heaven is five, the number of Earth is five, and when combined they constitute ten—this is the totality of number. Heaven generates "four" from "one," and Earth generates "four" from "one." These "four" have physical form, while that "one" is formless—this is the ultimate of "being" and "non-being." The substance-number of Heaven is four, yet of its functions only three are used; one of them is not employed. The substance-number of Earth is four, yet of its functions only three are used; one of them is not employed. Therefore, that "one without substance" is used to analogize Nature, that "one not employed" is used to analogize the Dao, and the "three that are employed" are used to analogize the three powers of Heaven, Earth, and Humanity.
The transformations of substance are eight, and the transformations of function are six. Therefore, among the eight trigrams, four are unchanging, and two pairs reverse into each other—through the transformations of six trigrams the eight trigrams are accomplished.
Among the doubled trigrams (the sixty-four hexagrams), eight are unchanging and twenty-eight reverse into each other—through thirty-six transformations the sixty-four hexagrams are accomplished.
Thus, the lines stop at six, the trigrams are completed at eight, the calculating stalks are exhausted at thirty-six, and the doubled hexagrams reach their limit at sixty-four. Trigrams are completed at eight and doubled into sixty-four; lines are completed at six, stalks are exhausted at thirty-six, and doubled into three hundred eighty-four lines.
☀️ Heaven has four seasons, one season has four months, and one month has forty days. Four fours make sixteen; from each one is removed, so one season has three months and one month has thirty days. The four seasons constitute the substance-number; the three months and thirty days constitute the function-number. Though the substance possesses four, one of them is always unused, so the function stops at three and culminates at nine. The substance-number is always even—hence there are four and twelve; the function-number is always odd—hence there are three and nine.
The great numbers fall short while the small numbers constantly exceed—why is this? Because the great are invisible while the small are visible. Thus the seasons stop at four, the months stop at three, and the days exceed at ten. Therefore the human body has four limbs but ten fingers.
Heaven appears in the south and lies hidden in the north, reaches its limit at six and has a remainder at seven. Thus humans know what lies before them but remain unclear about what lies behind, and also neglect what lies to left and right.
The substance-number of Heaven is four and its function three; the substance-number of Earth is four and its function three. Heaven overcomes Earth, Earth overcomes Heaven, and what is overcome resides in Earth—just as the residual fraction of daylight returns to Earth. Therefore Heaven is three and Earth is four: Heaven has three celestial bodies (sun, moon, stars) and Earth has four elements. Yet Earth's fire (🔥) appears and disappears by turns—is this perhaps the manifestation of that "residual fraction"?
Qian has seven sons, Dui six sons, Li five sons, Zhen four sons, Xun three sons, Kan two sons, Gen one son; Kun is the utmost yin, hence has no sons.
Qian has seven sons, Kun six sons, Dui five sons, Gen four sons, Li three sons, Kan two sons, Zhen one son; Xun is the rigid, hence has no sons.
Heaven has two correct [trigrams], Earth has two correct [trigrams]; together they use two transformations to form the eight trigrams. Heaven has four correct [hexagrams], Earth has four correct [hexagrams]; together they use twenty-eight transformations to form the sixty-four hexagrams. Therefore, among the lesser completed trigrams (the eight trigrams), four are correct and two are transformative—six in total; among the greater completed hexagrams (the sixty-four), eight are correct and twenty-eight are transformative—thirty-six in total. Qian, Kun, Kan, Li are the ancestors of the thirty-six hexagrams; Dui, Zhen, Xun, Gen are the ancestors of the twenty-eight hexagrams.
Qian and Kun undergo seven transformations, so the extreme number of day and night does not exceed seven parts. Gen and Dui undergo six transformations, so the month stops at six, totaling twelve. Li and Kan undergo five transformations, so the day stops at five, totaling ten. Zhen and Xun undergo four transformations, so the body stops at four, totaling eight.
The correct and transformative hexagrams together number thirty-six, and the lines additionally number two hundred sixteen—this is the stalk-number of the function-number. Thirty-six minus four gives thirty-two; minus another four gives twenty-eight; minus another four gives twenty-four. Thus the hexagram-number of thirty-two positions is speaking after removing four; the Heaven-number of twenty-eight positions is speaking after removing eight; the Earth-number of twenty-four positions is speaking after removing twelve. What is called "four" refers to Qian, Kun, Kan, Li; what is called "eight" refers to adding Yi, Zhongfu, Daguo, Xiaoguo; what is called "twelve" refers to further adding Dui, Zhen, Tai, Jiji.
The sun has eight positions yet its function stops at seven—this is speaking after removing Qian. The moon has eight positions yet its function stops at six—this is speaking after removing Dui. The stars have eight positions yet their function stops at five—this is speaking after removing Li. The celestial stems have eight positions yet their function stops at four—this is speaking after removing Zhen. The sun has eight positions yet its number stops at seven—this is speaking after removing Tai.
The moon arises from Dui, because the moon's number cannot catch up to the sun's number; thus twelve months constantly retain a remainder of twelve days.
Qian is yang within yang and cannot transform, so a year merely enumerates twelve months. Zhen is yin within yin and cannot transform, so the twelve hours of a day are not visible. Dui is yang within yin, and Li is yin within yang—both can transform, so the numbers of sun and moon can be divided. Therefore yin numbers arise from twelve, yang numbers arise from thirty, and two and six are constantly preserved. (Zhou Wenjian's note: This is the basis for Shao Yong's system of Yuan-Hui-Yun-Shi using twelve and thirty as periods.)
To lift up the year reveals the month; to lift up the month reveals the day; to lift up the day reveals the hour—this is yang governing yin. Therefore the four transformations of Heaven contain the four transformations of Earth, and the transformations of the sun contain the transformations of the moon and the stars. Thus one hexagram contains four hexagrams.
The sun occupies one position, the moon one position, the stars one position, the celestial stems one position. The sun has four positions, the moon four positions, the stars four positions, the celestial stems four positions. Four times four makes sixteen positions; when this one transformation is exhausted, the numbers of sun and moon are completed.
Heaven has four transformations, Earth has four transformations; transformations have growth and decline. When sixteen transformations are complete, the numbers of Heaven and Earth are exhausted.
The sun arises from one, the moon from two, the stars from three, the celestial stems from four. Extending from this, yang numbers are always six and yin numbers always two. When twelve transformations are complete, the numbers of the greater and lesser cyclical movements are exhausted.
Three hundred sixty transformations give one hundred twenty-nine thousand six hundred; one hundred twenty-nine thousand six hundred transformations give sixteen billion seven hundred ninety-six million one hundred sixty thousand; sixteen billion seven hundred ninety-six million one hundred sixty thousand transformations give two hundred eighty-two thousand one hundred ten trillion, nine hundred ninety million seven hundred forty-five thousand six hundred billion. Taking three hundred sixty as the hour, one hundred twenty-nine thousand six hundred as the day, sixteen billion seven hundred ninety-six million one hundred sixty thousand as the month, and two hundred eighty-two thousand one hundred ten trillion, nine hundred ninety million seven hundred forty-five thousand six hundred billion as the year, the numbers of the greater and lesser cycles are established.
Two hundred eighty-two thousand one hundred ten trillion, nine hundred ninety million seven hundred forty-five thousand six hundred billion, divided into twelve parts—the first six limits constitute growth, the last six limits constitute decline—corresponds to the number of twelve months in a year, advancing and retreating three hundred sixty days.
Sixteen billion seven hundred ninety-six million one hundred sixty thousand, divided into thirty parts, corresponds to the number of thirty days in a month, advancing and retreating sixty days according to the growth and decline of the greater cycle.
One hundred twenty-nine thousand six hundred, divided into twelve parts, corresponds to the number of twelve hours in a day, advancing and retreating six days.
Three hundred sixty corresponds to the number of one season, advancing and retreating according to the lesser cycle to correspond to the hours of day and night.
From the number of sixteen transformations, removing its "intercourse-number" and taking its "function-number," one obtains two hundred eighty-two thousand one hundred ten trillion, nine hundred ninety million seven hundred forty-five thousand six hundred billion. Divided into twelve limits, the first six limits are growth and the last six are decline. Each limit obtains one-thirteenth billion nine hundred ninety-six million-eight-thousandth of sixteen billion seven hundred ninety-six million one hundred sixty thousand. Each sixteen billion seven hundred ninety-six million one hundred sixty thousand years opens one division, advancing sixty days. Six limits open six divisions, advancing three hundred sixty days. There remains yet one part of "residual fraction," so it opens seven divisions, advancing three hundred sixty-six days. Its retreat is likewise.
From one hundred twenty-nine thousand six hundred, removing three is the intercourse-number; taking seven is the function-number. The function-number, being three, is completed at six; adding the residual fraction gives seven. Seven times this gives ninety thousand seven hundred twenty years; half of this gives forty-five thousand three hundred sixty years, advancing six days. The day has day and night; the number has yin and yang—thus twelve days are accomplished. Every three thousand six hundred years advances one day; in forty-three thousand two hundred years, twelve days are advanced. Two thousand one hundred sixty years advance the residual six; combined with the intercourse-number of two thousand one hundred sixty, twelve divisions are advanced altogether as an intercalary [month]. Therefore the transformations of the lesser cycle number sixty in all, completing three hundred sixty-six days.
Six is three Heavens (the embodiment of the Heaven-number of the three powers), and four is two Earths. Heaven governs substance yet relies on Earth as its substance; Earth distributes function yet receives Heaven as its function. Heaven and Earth depend on each other; substance and function adhere to each other.
The core of this chapter lies in establishing a "substance four, function three" cosmological-numerological model, from which the complete system from the eight trigrams to the sixty-four hexagrams, and from days and hours to Yuan-Hui-Yun-Shi, is derived. The thread of thought may be summarized as follows:
The "substance four, function three" principle: Any complete thing (substance) possesses four potential structural dimensions, but in actual operation only three are activated, with one left unused. This "unused one" is not superfluous but rather the symbol of the Dao and Nature—it is the latitude and source through which transformation becomes possible. Just as a year has four seasons, yet each season operates with three months (rather than four); a month operates with thirty days (rather than forty).
The nesting of numerical levels: From three hundred sixty (one season) → one hundred twenty-nine thousand six hundred (one day) → sixteen billion seven hundred ninety-six million one hundred sixty thousand (one month) → two hundred eighty-two thousand one hundred ten trillion, nine hundred ninety million seven hundred forty-five thousand six hundred billion (one year), a progressive structure of alternating multiplication by twelve and thirty is formed. This is precisely the numerical foundation of Shao Yong's Yuan-Hui-Yun-Shi system (one Yuan = twelve Hui, one Hui = thirty Yun, one Yun = twelve Shi, one Shi = thirty years).
Symmetry of growth and decline: The great number is divided into twelve limits—the first six are growth, the last six decline; advance and retreat correspond, like the twelve months of a year or the twelve hours of a day. Cosmic time is not linear and endless but characterized by periodic undulation.
Yang governs yin; substance and function depend on each other: Heaven (yang) contains the function-number of Earth (yin) within its substance-number; one hexagram contains four hexagrams; the transformations of the sun contain the transformations of the moon and stars. Heaven and Earth are inseparable; substance and function mutually adhere—this is the manifestation of monism in Shao Yong's philosophy of "observing things."
"Substance four, function three" can be viewed as a principle of "redundant design." Any system—whether a biological organism, social organization, or technological architecture—if all its potential were fully deployed (using all four), would have no elasticity and could not respond to sudden changes. Preserving the "unused one" precisely reserves room for change and adjustment. This resembles "safety redundancy" in modern engineering or the "reserve of genetic diversity" in biological evolution.
The cyclical structure of twelve and thirty is essentially a mathematical model that "encodes" large-scale time into small-scale cyclical units. This echoes the approach in modern astronomy of constructing calendars based on the tropical year, the synodic month, and the diurnal cycle. Shao Yong's Yuan-Hui-Yun-Shi is not merely historical periodization but an attempt to establish a computable topology of time.
The observation that "the great numbers fall short while the small numbers constantly exceed"—the great being invisible and the small visible—may be analogized to the cognitive difference between macroscopic and microscopic scales in modern physics: the macroscopic structure of the universe (great numbers) is difficult to observe and grasp directly, while the microscopic level (small numbers) is relatively easy to quantify. Shao Yong uses this to explain why "the seasons stop at four, the months at three, and the days exceed at ten"—and this possesses a reasonable core in epistemology.
The distinction between intercourse-number and function-number: In calculating the lesser cycle, Shao Yong distinguishes between the "intercourse-number" (the three parts removed) and the "function-number" (the seven parts taken), adding a "residual fraction" for precise calibration, ultimately deriving three hundred sixty-six days (including intercalation). This embodies the scientific attitude that the function-number must be combined with the intercourse-number, and theoretical models must be calibrated against observed reality.
Time management and energy allocation: Drawing on the wisdom of "substance four, function three," in personal time planning one should not fill all time completely (using all four); approximately one quarter of elastic space should be reserved for reflection, adjustment, and responding to the unexpected. This is a sustainable principle of rhythm management.
Organizational architecture design: In an efficient organization, there should exist an "unused one" between its formal structure (substance-number four) and daily operational mechanisms (function-number three)—perhaps an advisory role, a think-tank function, or informal communication channels. This is not redundancy but the source of organizational flexibility.
Modern application of cyclical thinking: Shao Yong's framework of growth and decline suggests that the development of all things follows a rhythm of "first six limits growth, last six limits decline." In investment, career planning, and project advancement, identifying the current phase of the cycle (growth or decline) helps avoid blind expansion during the "decline" phase or excessive conservatism during the "growth" phase.
The "hypothesis–action–review" methodology: Taking the numerical deduction of this chapter as a model, complex problems can be decomposed into two levels—"substance" (structural framework) and "function" (actual operation): first establish the complete framework, then determine which parts are deployed for actual use, and regularly review which "unused one" needs to be activated. This method can be applied to research project planning, team management, and personal growth alike.
The "one without substance" and "the Dao": Shao Yong correlates the "unused one" with "the Dao" and the "one without substance" with "Nature." This means that the true substance is invisible and unusable—it does not manifest directly in the phenomenal world, yet it is the precondition for all phenomena to unfold. This continues the line of thought from Laozi's "non-being provides usefulness": a cup holds water because of its emptiness; a house provides dwelling because of its hollow space. Shao Yong's contribution lies in mathematizing this philosophical insight, expressing it through the precise structure of "substance four, function three."
"Heaven appears in the south and lies hidden in the north" suggests an innate asymmetry of human cognition: we can only see the manifest aspect of things (south, front) and cannot directly grasp their hidden aspect (north, back). This is not a defect but the fundamental structure of existence. Shao Yong deduces from this that "humans know what lies before them but not what lies behind"—human knowledge is essentially incomplete, yet through numerical deduction we may infer those invisible parts.
"Heaven and Earth depend on each other; substance and function adhere to each other"—this proposition transcends simple dualistic opposition. Heaven is not absolutely transcendent; it "relies on Earth as its substance." Earth is not purely passive; it "receives Heaven as its function." This thought of mutual subjectivity finds resonance in contemporary ecological philosophy and systems theory: no system exists in isolation; its "substance" and "function" can only be realized in interaction with its environment.
The symmetry of "growth" and "decline" reveals a profound cosmological vision: growth and decline are not accidental, external events but the intrinsic rhythm of temporal structure. The first six limits are growth, the last six decline—this symmetrical structure suggests that the operation of the cosmos possesses an inherent balancing mechanism, where excess on one side triggers compensation on the other. This is fully continuous with the thought of the Yijing: "One yin and one yang constitute the Dao."
| Concept | Explanation | Relationship to This Chapter |
|---|---|---|
| Pre-Heaven Eight Trigrams Sequence | Qian 1, Dui 2, Li 3, Zhen 4, Xun 5, Kan 6, Gen 7, Kun 8 | The chapter's "Qian's seven sons, Dui's six sons..." is based on the Pre-Heaven Eight Trigrams Sequence |
| Yuan-Hui-Yun-Shi | 1 Yuan = 12 Hui, 1 Hui = 30 Yun, 1 Yun = 12 Shi, 1 Shi = 30 years | The chapter's progression from 360 → 129,600 → larger numbers derives its mathematical basis from Yuan-Hui-Yun-Shi |
| Substance Four, Function Three | Substance number is four, function number is three, with one held in reserve and unused | The core principle of this chapter, running through the derivation of Sun, Moon, Stars, and Celestial Bodies as well as the Eight Trigrams |
| Intercalary number | Numbers that must be excluded (not participating in transformation) during cyclical calculations | Contrasted with "function numbers," used in this chapter for precise calculation of leap numbers |
| Growth and Decline | The first six periods represent growth (increase), the last six periods represent decline (diminution) | The Grand Cycle is isomorphic with the twelve months of a year and the twelve double-hours of a day |
| Small-Completion Trigrams / Great-Completion Trigrams | Small completion refers to the Eight Trigrams, great completion refers to the Sixty-Four Hexagrams | The derivation framework for the numbers of regular and transformed hexagrams |
| Scheme number | The number of changes in yarrow stalks used in divination | Thirty-six schemes is the number used for function, paired with the line number of 216 |
Research on the Mathematical Structure of Shao Yong's Pre-Heaven Yixue: Contemporary scholars focus on how Shao Yong transformed the hexagram system of the Book of Changes into a rigorous mathematical model. The sixteen-transformation and twelve-period derivations in this chapter demonstrate the logical chain of Shao Yong's large-number calculations, making it an important text for the study of Song dynasty mathematical thought. Some scholars suggest that this method of derivation inspired later systematic thinking about the "accumulated years of the superior epoch" algorithm in traditional Chinese calendrical science.
A Comparison between Yuan-Hui-Yun-Shi and Modern Cosmological Time Scales: Shao Yong used 129,600 years as the time unit of one "day," layered with the products of multipliers of thirty and twelve, to construct a large-scale time framework spanning hundreds of millions of years. Some researchers have noted that this hierarchical construction of time scales bears structural resemblance in approach to the "Eon–Era–Period–Epoch" division in contemporary geology and the "Cosmic Calendar" in cosmology.
"Substance Four, Function Three" and the Principle of Redundancy in Systems Theory: Modern systems theory emphasizes the role of "requisite redundancy" in robust systems. Shao Yong's formulation that "although substance comprises four, one of them is perpetually unused" can be regarded as an early recognition in traditional Chinese philosophy of the value of systemic resilience and redundancy, meriting further interdisciplinary research.