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Chapter 18 of 25
皇极经世
Qian is One. The transformation of the fifth line of Qian produces Great Wealth (Dayou), corresponding to the number three hundred sixty. The transformation of the fourth line of Qian produces Small Gathering (Xiaoxu), corresponding to the number one hundred twenty-nine thousand six hundred. The transformation of the third line of Qian produces Treading (Lu), corresponding to the number 16,796,160,000. The transformation of the second line of Qian produces ** Fellowship with Others (Tongren)**, corresponding to the number 28,211,099,074,560,000,000,000. The transformation of the first line of Qian produces Encountering (Gou), corresponding to the number 7,958,661,099,464,884,391,936,000,000,000,000. This is called the number of division. When great numbers are divided into small numbers, they all operate from above downward, and therefore correspond to the yang numbers. 🔥
From One arises Two as Breakthrough (Guai), corresponding to the number twelve. From Two arises Four as Great Strength (Dazhuang), corresponding to the number 4,320. From Four arises Eight as Peace (Tai), corresponding to the number 559,872,000. From Eight arises Sixteen as Overseeing (Lin), corresponding to the number 944,369,691,520,000,000,000. From Sixteen arises Thirty-two as Return (Fu), corresponding to the number 26,528,870,366,488,294,797,312,000,000,000,000,000. From Thirty-two arises Sixty-four as Kun, corresponding to the number of the Infinite. This is called the number of growth. When small numbers grow into great numbers, they all arise from below upward, and therefore correspond to the yin numbers. 🌊
Heaven governs its substance, therefore after eight transformations it terminates at sixteen. Earth distributes its function, therefore after six transformations it terminates at twelve. Heaven begins at One and terminates at 7,958,661,099,464,884,391,936,000,000,000,000. Earth begins at twelve and terminates at 204,698,073,815,491,849,907,200,000,000,000,000.
After Earth exists, there is Two. After Two exists, there is the distinction of day and night.
Two and Three mutually transform, interweaving in complexity. Therefore the Yi generates numbers by Two and raises numbers by Twelve, and One itself is not a number—yet though it is not a number, numbers depend upon it for their completion. Heaven operates without ceasing and originally has no division of day and night; humans, dwelling upon Earth, thereby perceive day and night. Thus the numbers of Earth serve as the function of humans. 🌱
Heaven from Overseeing (Lin) upward, and Earth from Teacher (Shi) upward, belong to the numbers of cosmic epochs. Heaven from Fellowship with Others (Tongren) downward, and Earth from Retreat (Dun) downward, belong to the numbers of years. The numbers of cosmic epochs reside in Heaven; the numbers of years reside in Earth. Heaven from Adorning (Bi) upward, and Earth from Keeping Still (Gen) upward, belong to the numbers of function. Heaven from Darkening of the Light (Mingyi) downward, and Earth from Standstill (Pi) downward, belong to the numbers of interaction. Heaven from Thunder (Zhen) upward, and Earth from Advancing (Jin) upward, belong to the numbers of existence. Heaven from Increase (Yi) downward, and Earth from Contentment (Yu) downward, belong to the numbers of nonexistence. Heaven's numbers of existence arise from Qian and terminate at Zhen; what remains enters into the numbers of nonexistence, because the celestial stems are invisible. Earth removes One and arises from Twelve, because the fire of Earth is always hidden. Therefore Heaven takes substance as its foundation and constantly conceals that foundation, while Earth takes function as its root and constantly conceals its operation.
One double-hour terminates at three months, one month terminates at thirty days—both discard the number of hours. Therefore among the sixty-four hexagrams, eight do not transform, while seven can transform. Seven times eight equals fifty-six, and its meaning derives from this. ⛰
The yang lines are the numbers of day; the yin lines are the numbers of night. Heaven and Earth depend upon each other; yin and yang interpenetrate. Therefore day and night are intermingled, and hardness and softness alternate.
In spring and summer yang predominates, so the numbers of day exceed the numbers of night. In autumn and winter yin predominates, so the numbers of night exceed the numbers of day.
The measures of the substance-number amount to 384. Removing the measures of the four hexagrams Qian, Kun, Kan, and Li, one obtains the number of function, 360. The function of the substance-number is 270. Removing the measures of Qian and of Kan and Li, one obtains the function of the function-number, 252.
Among the 270 of the function of the substance-number, 156 are yang and 114 are yin. Removing the measures of the Li hexagram, one obtains 152 yang and 112 yin—these are the numbers of actual function.
Because yang discards Li and employs Qian, while yin discards Kun and employs Kan. Therefore the yang measures of Heaven are 112 (removing its yin), and the yin measures of Earth are 112, with 40 yang measures (removing the yang of its south and north). At the extreme south is great heat; at the extreme north is great cold [translator's note: this view does not accord with modern astronomy and geography—in fact, both the Earth's north and south poles are extremely cold; Shao Yong's statement was limited by the geographical knowledge of his time], and the ten thousand things cannot grow there, so they are removed. Those 40 are the surplus fractions of Heaven. Yang encroaches upon yin; day encroaches upon night. Therefore they reside within Earth. Combined, they yield 152 yang and 112 yin.
Yang removes the measures of Qian; yin removes the measures of Kan, yielding 144 yang and 108 yin—these are the function of the function-number.
Yang is 36; multiplied by three it becomes 108. Yin is also 36; multiplied by three it also becomes 108. Three yang and three yin—yin and yang each at half. The surplus fraction of yang is 36. Combining them yields 144 yang and 108 yin. Therefore the function of the substance-number is 270, but those actually functioning are 264, and the function of the function-number is 252.
Among the sixty-four hexagrams, only thirty-six are functional. Among the 384 lines, only 216 are functional. Dividing sixty-four yields 256. Therefore a single hexagram with its first and top lines removed also yields 256—these are the numbers that generate beings. Thus Li and Kan are the masters of generating beings, because Li has four yang and Kan has four yin. Hence that which generates beings must be four. Yang is 112, yin is 112. Removing the lines of Li and Kan yields 216. The 40 of yin and yang together yield 256. Therefore the eight hexagrams employ six lines, with Qian and Kun as their masters; the six lines employ four positions, with Li and Kan as their masters. Thus Heaven's twilight and dawn do not generate beings, but the middle of the day generates them; Earth's south and north do not generate beings, but the center does. 🌿
What is the substance-number for? It is for generating beings. What is the function-number for? It is for operating. That which operates is Heaven; that which generates beings is Earth.
Heaven operates in solitude, therefore it multiplies the function-number by itself, and uses the function of the function-number as the time of generating beings. Earth pairs and generates, therefore it employs the function of the substance-number—yang multiplied by yin—as the number of generating beings.
The number of Heaven is three, therefore six times six and again six yields the measures of Qian as 216. The number of Earth is two, therefore twelve times twelve yields the measures of Kun as 144. Qian uses nine, therefore three times eight is twenty-four, and multiplied by nine it also yields 216; two times eight is sixteen, and multiplied by nine it also yields 144. Kun uses six, therefore three times twelve is thirty-six, and multiplied by six it also yields 216; two times twelve is twenty-four, and multiplied by six it also yields 144.
Kun takes three of twelve, four of sixteen, and one-half of six as the surplus fraction of Qian. Then Qian obtains 252 and Kun obtains 108.
Among the four yang hexagrams, in twelve lines there are eight yang and four yin. Multiplying the yang by 36 and the yin by 24 yields 384.
The substance has 384, but the function only reaches 360—why? Because the four hexagrams Qian, Kun, Kan, and Li are not employed. That Qian, Kun, Kan, and Li are not employed is precisely to accomplish the function of 360. Therefore the ten thousand things transform, but these four do not transform. Only by not transforming can transformation be accomplished. The function stops at 360 yet has 366—why? It is the surplus of the number. What is the surplus of the number used for? It is the complete functioning of Qian. If Qian and Kun are not employed, then Li and Kan employ only half. Why does Qian function completely? Yang governs surplus. Why are Qian and Kun not employed? Solitary yang cannot generate; solitary yin cannot complete. Why do Li and Kan employ only half? Li is in the east and Kan is in the west; they occupy the half of yin and yang, serving as the gateway of spring-autumn and day-night. Why sometimes employ Qian and sometimes employ Li and Kan? From the perspective of governing yang, therefore Qian is employed. From the perspective of governing the surplus fractions, then yang encroaches upon yin and day encroaches upon night, therefore Li and Kan are employed. Yang governs surplus, therefore Qian functions completely. Yin governs emptiness, therefore Kun does not function at all. Yang encroaches upon yin and yin encroaches upon yang, therefore Kan and Li function by half. Therefore Heaven's south is entirely visible and its north entirely invisible, while east and west are each half visible. Li and Kan are the boundaries of yin and yang. Therefore Li corresponds to Yin (3 a.m.–5 a.m.) and Kan corresponds to Shen (3 p.m.–5 p.m.), and the numbers always surpass them, because of the overflow of yin and yang. Yet the function-number does not surpass Yin, and the interaction-number does not surpass Shen.
Qian is 48, and one-fourth of it is overcome by yin. Kun is 48, and one-fourth is the yang overcome by it. [translator's note: just as in the eight trigrams and the periodic table of elements, about one-fourth of the portions between lesser yin and lesser yang are mutually exchanged] Therefore Qian obtains 36 and Kun obtains 12. Yang governs advance, therefore it advances to become the 360 days. Yin governs diminishment, therefore the twelve months diminish by twelve days.
Counting forward: Qian 1, Dui 2, Li 3, Zhen 4, Xun 5, Kan 6, Gen 7, Kun 8. Counting backward: Zhen 1, Li/Dui 2, Qian 3, Xun 4, Kan/Gen 5, Kun 6.
Qian is 48, Dui is 30, Li is 24, Zhen is 10. Kun is 12, Gen is 20, Kan is 36, Xun is 40. Qian is 36, Kun is 12, Li/Dui/Xun is 28, Kan/Gen/Zhen is 20.
The circular number has One; the square number has Two—this is the meaning of odd and even. Six is One; twelve is Two.
Heaven is round and Earth is square. The round number arises from One and accumulates to Six; the square number arises from One and accumulates to Eight. Transforming them, they arise from Four and accumulate to Twelve. Six always transforms by six, Eight always transforms by eight, and Twelve also transforms by eight—this is the way of nature. Eight is the substance of Heaven and Earth, Six is the function of Heaven, and Twelve is the function of Earth. Heaven transforms the square into the round while always preserving its One. Earth divides One into Four while always maintaining its squareness. Heaven transforms its substance but does not transform its function; Earth transforms its function but does not transform its substance. Six combining with its One becomes Seven; Twelve combining with its Four becomes Sixteen. Yang governs advance, therefore Heaven combines with its One to become Seven. Yin governs retreat, therefore Earth removes its Four and stops at Twelve. Therefore yang always preserves One and yin always obscures One. Thus the substance of Heaven and Earth stops at Eight, the function of Heaven reaches its limit at Seven, and the function of Earth stops at Twelve. The round trims the square to create function; therefore One transforms to Four, and Four removed of its One is Three. Three transforms to Nine, and Nine removed of its three is Six.
The square unfolds the round to create substance; therefore One transforms to Three, combined to become Four, and Four transforms to Twelve, combined to become Sixteen.
Therefore the function-number is completed at Three and reaches its limit at Six, while the substance-number is completed at Four and reaches its limit at Sixteen. Therefore the round has a diameter of One and a circumference of Three, arising from One and accumulating to Six. The square divides One into Four, and divides Four into Sixteen—all are the way of nature.
One employs Two to generate Three; Three removed of its One is Two. Three generates Nine; Nine removed of its One is Eight, and removed of its Three is Six. Therefore One employs Three, and Three in turn employs Two. Three employs Nine, and Nine in turn employs Eight and Six. Therefore Two generates Four, Eight generates Sixteen, and Six generates Twelve. Three combined with One is Four; Nine combined with Three is Twelve; Twelve combined with Four is Sixteen. Therefore Four takes One as its foundation and Three as its function. Twelve takes Three as its foundation and Nine as its function. Sixteen takes Four as its foundation and Twelve as its function.
The round transforms by six; six times six advances, therefore sixty transformations yield three hundred sixty. The square transforms by eight, therefore eight times eight yields sixty-four. Yang governs advance, therefore it advances to become sixty.
This chapter is one of the most central chapters of numerical deduction in Shao Yong's system of image-number Yi learning, and its core concepts can be summarized as follows:
I. Numbers of Division and Numbers of Growth: Two Paths of Cosmic Generation
Shao Yong takes the transformation of the six lines of the Qian hexagram from above downward as the number of division (transforming the great into the small), and the transformation of the six lines of the Kun hexagram from below upward as the number of growth (accumulating the small into the great). These two paths model the two fundamental directions of cosmic evolution: first, the differentiation from the holistic "One" into the myriad things (from above downward); second, the accumulation from a minuscule starting point into a grand whole (from below upward). Qian, with yang numbers, governs division; Kun, with yin numbers, governs growth. Yin and yang complement each other, constituting a complete numerical cosmology.
II. Substance-Number, Function-Number, and Numbers that Generate Beings
Shao Yong strictly distinguishes three layers of numerical structure:
| Layer | Value | Meaning |
|---|---|---|
| Substance-number | 384 | The complete number of the myriad things (64 hexagrams × 6 lines) |
| Function-number | 360 | The number of operation (384−24 measures of Qian, Kun, Kan, Li) |
| Function of the function-number | 252 | The number that generates beings (further removing the measures of Qian, Kan, and Li) |
The substance-number is static and potentially complete; the function-number is dynamic, being the number that actually participates in operation; the function of the function-number is the core number that directly produces life. This three-layer structure is analogous to the "total resources—mobilizable resources—actually utilized resources" relationship in modern systems theory.
III. The Pivotal Position of the Four Hexagrams Qian, Kun, Kan, and Li
The four cardinal hexagrams (Qian, Kun, Kan, and Li) occupy a special position in Shao Yong's system. They do not transform, and precisely because they do not transform, they are able to accomplish transformation. Qian is pure yang, Kun is pure yin, Li is yang containing yin, and Kan is yin containing yang. The four function like the four poles of a coordinate system, serving as the reference frame for all transformations. This concept embodies the deep wisdom of "responding to change through that which does not change" in Chinese philosophy.
IV. Quantitative Expression of Waxing and Waning of Yin and Yang
Through precise numerical operations, Shao Yong concretizes the abstract pattern of yin-yang waxing and waning into specific numerical ratios—such as the practical ratio of 152 yang to 112 yin, and the function-of-function ratio of 144 yang to 108 yin. These ratios are not arbitrarily set but derived through layers of logical deduction by "removing measures," embodying the dynamic equilibrium of yin and yang at different levels.
I. A Premodern Model of Systems Theory
Shao Yong's image-number system essentially constructs a mathematical model of cosmic operation. He uses the relationships between numbers to simulate the patterns of generation, operation, and transformation of heaven, earth, and the myriad things. This shares a remarkable affinity with the approach of modern systems theory and complexity science in understanding natural phenomena through mathematical models. Although his specific numerical derivations may not possess precision in the modern scientific sense, his mode of thinking—"explaining things through numbers"—deserves our attention.
II. The Philosophical Wisdom of "That Which Does Not Change Accomplishes Change"
Shao Yong points out: "The ten thousand things transform, but the four do not; only by not transforming are they able to bring about transformation." This reveals a profound philosophical proposition: within any system there exist necessary invariants. Physical constants, logical axioms, and natural laws are all foundations that "do not change yet accomplish change." In the business world, core values are invariants, while products, strategies, and organization may flexibly transform. In personal growth, a stable sense of self-identity serves as the spiritual anchor in a changing world.
III. The Ecological View of Yin-Yang Interpenetration
"Heaven and Earth depend upon each other; yin and yang interpenetrate; therefore day and night intermingle, and hardness and softness alternate." This reveals that yin and yang are not diametrically opposed poles but are mutually penetrating, interwoven, and coexisting dynamic relationships. This coincides remarkably with the concept of the "ecotone" in modern ecology—the boundary zones between different ecosystems often harbor the greatest biodiversity. In organizational management, the intersecting zones between different departments and cultural backgrounds are likewise frequently the most active sites of innovation.
IV. Modern Mapping of the Substance-Function Relationship
The distinction between "substance-number" and "function-number" can be mapped onto multiple dimensions of modern management:
Shao Yong's logic implies: not all substance can transform into function, and not all function can actually produce living beings (results). This layer-by-layer "convergence" mode of thinking presents an interesting contrast with modern society's logic of pursuing maximum efficiency—Shao Yong instead emphasizes that appropriate "non-use" is the very precondition of "use."
I. Identifying "Invariants" in Decision-Making
Shao Yong's idea that "the unchanging gives rise to change" can be transformed into a practical decision-making tool. When faced with complex and ever-changing situations, one must first identify which elements are immutable foundations (Qian, Kun, Kan, Li) and which are flexible variables that can be adjusted. Confusing the two leads either to a shaken foundation or to rigid inflexibility. For example:
II. The Wisdom of Resource Management: "Use and Non-Use"
Shao Yong emphasizes that the "non-use" of Qian, Kun, Kan, and Li is precisely what enables the "usefulness" of the three hundred sixty. This reminds us:
Not all resources should be "fully utilized."
Leaving room for flexibility and maintaining redundancy are necessary conditions for the long-term stable operation of any system. The safety redundancy in modern engineering, the reserve system in finance, and the blank zones in ecological conservation are all modern manifestations of "non-use that enables use."
III. The Rhythms of Yin-Yang, Day-Night, and Physical and Mental Health
"The Yang line is the number of daylight, the Yin line is the number of night" directly correlates Yin-Yang with day and night. "In spring and summer Yang predominates, so days are many and nights are few; in autumn and winter Yin predominates, so days are few and nights are many" reveals the rise and fall of Yin-Yang in natural rhythms. Modern society's extensive use of artificial lighting and electronic devices has severely disrupted the synchronization between the human body and natural day-night rhythms. According to this thinking, adjusting one's daily schedule in accordance with natural rhythms (rising early and being more active in spring and summer; sleeping early and resting more in autumn and winter) may be an effective approach to improving modern people's sleep disorders and sub-health conditions.
IV. Core Focus Through the "Numbers of Generation"
"The generator must be four" emphasizes that the four Yang and four Yin of Li and Kan are the core from which all things arise. In project management, this means that genuine output tends to concentrate on a few key elements. Identifying and focusing on those core generative elements yields more tangible results than applying effort uniformly across a broad front.
I. "One is not number, yet number is completed through it" — The Philosophy of "Zero"
Shao Yong states, "One is not number; it is not number, yet all numbers are completed through it," placing "one" in the foundational position of the numerical system — it is not number, yet all numbers become possible because of it. This forms a profound structural isomorphism with the special status of "zero" in mathematics, the foundational role of the "empty set" in set theory, and the philosophical concept of "Being" as the ground of all beings. A thousand years ago, Shao Yong had already touched upon issues at the level of "meta-mathematics": the construction of a numerical system requires a "non-numerical" foundation.
II. "Dividing the great into the small" and "multiplying the small into the great" — Two Cosmological Paradigms
The "top-down" nature of division numbers and the "bottom-up" nature of multiplication numbers map onto two different cosmological paradigms. The former resembles deduction — deriving specific phenomena from universal principles; the latter resembles induction — accumulating general laws from specific observations. Shao Yong assigns Yang and Yin attributes to each, suggesting the complementarity of deduction and induction as Yin-Yang counterparts. A complete knowledge system requires both: Yang-type deduction provides direction, while Yin-type induction provides the foundation.
III. "Extreme South has great heat, extreme North has great cold" — Historical Limitations and Intellectual Openness
Shao Yong's statements about the earth's north and south reflect the limitations of Song dynasty geographical knowledge. However, even where specific facts are erroneous, the underlying insight — the structural judgment that nothing is generated at extremes — remains universally valid. This reminds us that when interpreting ancient thought, we must distinguish between specific empirical claims (which may have been falsified) and formal structural insights (which may remain valid). This distinction helps us neither blindly worship the ancients nor wholesale reject tradition.
IV. "Yang always preserves one while Yin always obscures one" — The Philosophical Significance of Asymmetry
Shao Yong repeatedly emphasizes the asymmetry between Yang and Yin in numerical terms: Yang advances while Yin retreats, Yang fills while Yin empties, Yang constantly preserves one while Yin constantly obscures one. This worldview does not treat Yin and Yang as perfectly symmetrical poles but attributes a subtle priority to Yang. Although this asymmetry operates in a different domain than the modern physics discovery of parity violation, both remind us philosophically that: deep asymmetries may exist in the fundamental structure of nature, and symmetry breaking is often the precondition for the emergence of richness. In biology, symmetry breaking is a crucial step in developmental processes; in economics, information asymmetry is a core source of market complexity. Shao Yong's "Yang preserves one, Yin obscures one" provides a mathematized expression of this asymmetry.
| Concept | Explanation | Position in This Chapter |
|---|---|---|
| Division numbers (Fen Shu) | Numbers of top-down differentiation | The six lines of Qian successively transform into Dayou, Xiaoxu, Lü, Tongren, and Gou, corresponding to decreasing cosmic magnitudes |
| Multiplication numbers (Zhang Shu) | Numbers of bottom-up accumulation | The six lines of Kun successively transform into Guai, Dazhuang, Tai, Lin, Fu, and Kun, corresponding to increasing cosmic magnitudes |
| Body numbers (Ti Shu) | The total number of all things | 384 (64 hexagrams × 6 lines) |
| Function numbers (Yong Shu) | The numbers of operation | 360 (body numbers minus Qian, Kun, Kan, Li) |
| Function of the body numbers | The usable portion within the body | 270 |
| Function of the function numbers | The portion directly generating things within the function | 252 |
| Cyclic numbers (Yun Shu) | Numbers of the Heavenly Way's operation | Heaven from Lin upward, Earth from Shi upward |
| Annual numbers (Nian Shu) | Numbers of the Earthly Way's yearly cycles | Heaven from Tongren downward, Earth from Dun downward |
| Numbers of generation (Shengwu Shu) | Numbers that engender and nurture all things | 256 (sixty-four divided into two hundred fifty-six) |
| Circle/Square numbers | The numerical principles of round heaven and square earth | Circle begins with one accumulating six; square begins with one accumulating eight |