皇极经世：观物外篇衍义卷三

Huangji Jingshi — Observing Things: Outer Chapters, Volume 3

Integral Paraphrase

Why does the yarrow-stalk divination take the number seven rather than six? Because seven contains the "remainder fraction" 🌱. If the remainder is removed, it becomes six, and thus the number of strategies for six is thirty-six. The number fifty corresponds to the strategies of the sixty-four hexagrams plus the intercalary days of a year; while the actual use of forty-nine corresponds to the strategies of the sixty hexagrams for a single year. Returning the odd stalks and hanging one, is like the intercalary day in a year. ☀️

A circular object with a diameter of one has a circumference of three, and doubling it gives six; a square object with a side of one has a perimeter of four, and doubling it gives eight. Trimming the square into a circle is the way of Heaven's motion; dividing the great into the small is the way of Earth's transformation. Therefore Heaven uses six transformations, and Earth uses four. ⛰️

The law of numerical trimming: One and eight combine to make nine, trimmed to seven; eight trimmed to six; sixteen trimmed to twelve; twenty-four trimmed to eighteen; thirty-two trimmed to twenty-four; forty trimmed to thirty; forty-eight trimmed to thirty-six; fifty-six trimmed to forty-two; sixty-four trimmed to forty-eight. The other direction is "division": one divided into four, eight divided into thirty-two, sixteen divided into sixty-four, and so on up to ninety-six divided into three hundred and eighty-four. 🌊

The number of motion (the cycle of Heaven): one generates six, six generates twelve, twelve generates eighteen, eighteen generates twenty-four, twenty-four generates thirty, thirty generates thirty-six; from this expansion extends further, sixty transformations generate three hundred and sixty. This is the number of motion for one year. The number of generation (the transformation of Earth): four generates twelve, twelve generates twenty, twenty generates twenty-eight, twenty-eight generates thirty-six. Thus:

The transformations of the circular (Heaven): one transformation generates six, removing one leaves five; two transformations generate twelve, removing two leaves ten; three transformations generate eighteen, removing three leaves fifteen; four transformations generate twenty-four, removing four leaves twenty; five transformations generate thirty, removing five leaves twenty-five; six transformations generate thirty-six, removing six leaves thirty. Thus "preserving them" gives six times six, thirty-six; "removing them" gives five times five, twenty-five. 🔥

The transformations of the square (Earth): one transformation becomes four; four generates eight, and adding four gives twelve; eight generates twelve, and adding eight gives twenty; twelve generates sixteen, and adding twelve gives twenty-eight; sixteen generates twenty, and adding sixteen gives thirty-six. Four times four is sixteen — this is the meaning of each of the four images containing its own four images. 🌍

What is the Great Derivation number of the Changes? It is the sages relying on numbers to perform their calculations. The number of Heaven is twenty-five, combined to make fifty; the number of Earth is thirty, combined to make sixty. Therefore it is said: "The five positions mutually correspond, and each has its union." Fifty is the number of the yarrow stalks; sixty is the number of the hexagrams. Five is the small derivation of the yarrow stalks, so fifty is the Great Derivation. Eight is the small completion of the hexagrams; sixty-four is the great completion. 🔢

The virtue of the yarrow stalks is circular, used to model the number of Heaven, hence seven times seven is forty-nine; saying fifty is "preserving one when speaking of it." The virtue of the hexagrams is square, used to model the number of Earth, hence eight times eight is sixty-four; saying sixty is "removing four when speaking of it." The yarrow stalks are the number of function; the hexagrams are the number of substance. Function must be based on substance, hence preserving one; substance must take function as its root, hence removing four. The circular has one as its root; the square has four as its root. Therefore the yarrow stalks preserve one, and the hexagrams remove four. The yarrow stalks use seven, together with their remainder fraction — this too is the meaning of "preserving one"; hanging the one stalk — this too is the meaning of "removing one." 🔄

The functional number of the yarrow stalks: hanging one to symbolize the three powers, the remaining forty-eight stalks are the strategy number of one hexagram. Four times twelve equals forty-eight. Specific calculation:

Thus seven and nine are yang; six and eight are yin. Nine is the extreme number of yang; six is the extreme number of yin. When numbers reach their extreme, they reverse, hence they become the transformations of the hexagrams. Zhen and Xun have no strategies, representing the numbers that are not used. Heaven takes firmness as its virtue, so the yielding is not manifested; Earth takes softness as its substance, so the firm is not generated. Therefore Zhen and Xun have no strategy numbers. ⚡

Qian uses nine, so its strategies are nine; multiplied by four corresponds to the four seasons, each season being ninety days. Kun uses six, and its strategies are also six.

The odd numbers are four: one, two, three, four. The strategy numbers are four: six, seven, eight, nine. Combined they make eight numbers, corresponding to the eight transformations of the square number. The returned odd and hung numbers have six cases, corresponding to the six transformations of the circular number. Qian together with the remainder seven forms the seven luminaries. 🌟

The odd numbers are extreme at four, and five is not used; the strategy numbers are extreme at nine, and ten is not used. Five returns to one; ten returns to two. Hence removing fifty and using forty-nine. The odd numbers do not use five; the strategies do not use ten — these are the extremes of being and non-being, used to model the natural numbers.

The hexagrams number sixty-four, but the use stops at sixty, because sixty hexagrams have three hundred and sixty lines, just as the sexagenary cycle stops at sixty — six cycles of ten Heavenly Stems exhaust the cycle of Heaven. Therefore the strategy numbers respond accordingly: thirty-six and twenty-four combine to make sixty; thirty-two and twenty-eight also combine to make sixty. Distribution of strategies for each hexagram:

The above totals sixty. The yarrow stalk number is full, hence the yang strategies thirty-six and twenty-eight combine to make sixty-four; the hexagram number removes four, hence the yin strategies twenty-four and thirty-two combine to make fifty-six.

Nine advanced becomes thirty-six, all being yang numbers, hence it is yang within yang; seven advanced becomes twenty-eight, yang first then yin, hence it is yin within yang; six advanced becomes twenty-four, all being yin numbers, hence it is yin within yin; eight advanced becomes thirty-two, yin first then yang, hence it is yang within yin.

Heaven one, Earth two, Heaven three, Earth four, Heaven five, Earth six, Heaven seven, Earth eight, Heaven nine, Earth ten. Alternate these in transformations, interlock and interweave their numbers. One is the beginning of number, yet itself is not a transformative number. Thus two times two is four, three times three is nine, four times four is sixteen, five times five is twenty-five, six times six is thirty-six, seven times seven is forty-nine, eight times eight is sixty-four, nine times nine is eighty-one, while one cannot be transformed. One hundred returns to ten; ten returns to one — neither can be transformed. Hence number removes one and is extreme at nine, all using its transformations.

The Great Derivation number is the origin of computational methods. The rise of arithmetic is nothing more than the square, the circle, the curved, and the straight. Multiplication generates numbers; division diminishes numbers. Though algorithms are many, they do not depart from this principle. 📐

Yin has no one; yang has no ten. Yang gains from yin to be born; yin gains from yang to be completed. Hence the yarrow stalk number is four and uses nine; the hexagram number is six and uses ten. This is like the interweaving of the Heavenly Stems and Earthly Branches: the Stems end at six, the Branches end at five.

Naturally corresponding numbers: three times four is twelve, two times six is also twelve; twice twelve is twenty-four, three times eight is also twenty-four, four times six is also twenty-four; three times twelve is thirty-six, four times nine is also thirty-six, six times six is also thirty-six; four times twelve is forty-eight, three times sixteen is also forty-eight, six times eight is also forty-eight; five times twelve is sixty, three times twenty is also sixty, six times ten is also sixty. All are naturally corresponding. ✨

Four nines are thirty-six; six sixes are thirty-six. Yang six, and moreover including half of yin six, therefore it is nine. Hence considered as two hexagrams, yin and yang are three each; considered as six lines, Heaven, Earth, and Humanity are two each. Within yin and yang there are each Heaven, Earth, and Humanity; within Heaven, Earth, and Humanity there are each yin and yang. Hence "three Heaven and two Earth, relying on the numbers." ☯️

The yang number one, derived becomes ten, like the ten Heavenly Stems. The yin number two, derived becomes twelve, like the twelve Earthly Branches and the twelve months.

One transformation to two, two transformations to four, three transformations and the eight trigrams are completed. Four transformations to sixteen, five transformations to thirty-two, six transformations and the sixty-four hexagrams are complete. This is the expansion of powers of two: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64. 🌿

The Changes have true numbers — only three. Three Heavens: three times three is nine; two Earths: doubling three is six. Three Heaven and two Earth relying on numbers does not refer to the actual proper numbers of Heaven and Earth themselves, but to "emulation" — emulating the proper numbers of Heaven and Earth to generate them. ⚠️

The generating number of the Changes is one hundred and twenty-nine thousand six hundred (129,600), totaling four thousand three hundred and twenty generations (each generation 30 years). This is the great measure of waxing and waning. A year has 360 days, yielding 4,320 double-hours (12 per day), and multiplying by 30 yields this number. All Jiazi and Jiawu years mark the beginning of an era. This is the number of the world-governing eras, beginning with Day Jia, Month Zi, Star Jia, Hour Zi.

One, ten, hundred, thousand, ten-thousand, hundred-thousand are the odd numbers of Heaven. Twelve, one hundred twenty, one thousand two hundred, twelve thousand, one hundred twenty thousand, one million two hundred thousand are the even numbers of Earth.

Dividing fifty gives ten; if taking three Heaven and doubling it gives six; doubling two Earth again gives four. These are the numbers of Heaven and Earth dividing the Supreme Ultimate.

From the Fu hexagram to the Qian hexagram, there are altogether one hundred and twelve yang lines (the process of yang qi growth from winter solstice to summer solstice); from the Fu hexagram to the Kun hexagram, there are eighty yang lines. From the Fu hexagram to the Kun hexagram, there are altogether one hundred and twelve yin lines (the process of yin qi growth from summer solstice to winter solstice); from the Fu hexagram to the Qian hexagram, there are eighty yin lines. The waxing and waning of yin and yang expand and contract alternately over the three hundred and sixty days. 🌞🌙

🧠 Deep Understanding

Core Concepts 💡

This chapter centrally elaborates the numerological-philosophical framework of the Huangji Jingshi, encompassing several key ideas:

I. The Distinction of Circle and Square, and the Numbers of Heaven and Earth

Shao Yong builds upon "the circle with diameter one has circumference three; the square with side one has perimeter four" to construct the cosmic numerological model of round Heaven and square Earth, Heaven six and Earth four. Heaven uses six transformations (expansion in multiples of 6), Earth uses four transformations (expansion in multiples of 4). These are not merely geometric observations but mathematical models of cosmic operating rhythms. The circle symbolizes the cycle of Heaven's Way and the flow of time; the square symbolizes Earth's Way of bearing and the stillness of space.

II. The "Preservation" and "Removal" in the Art of Function

The yarrow stalks use forty-nine (removing one); the hexagrams use sixty (removing four). Shao Yong emphasizes the "substance-function" relationship: function must be grounded in substance, and substance must take function as its foundation. This is not a simple numerical game but expresses a profound philosophical proposition: any "function" (utility, practice, application) does not exist in isolation but must rely on "substance" (essence, structure, foundation), and vice versa.

III. The Hierarchical Classification of Yin and Yang Strategies

Shao Yong classifies yang strategies into "yang within yang" (Qian, 36) and "yin within yang" (Dui, Li, Xun, 28); and classifies yin strategies into "yin within yin" (Kun, 24) and "yang within yin" (Zhen, Kan, Gen, 32). This four-tier structure corresponds to the Four Images: Great Yang ⚡, Lesser Yang 🌱, Great Yin 🌊, Lesser Yin 🌬. Zhen and Xun, due to undetermined yin-yang, have no independent strategies, embodying the state of "primordial begins of Heaven and Earth, qi not yet distinguished."

IV. The Dialectical Law of Numbers Reaching Their Extreme and Reversing

"Nine is the extreme number of yang; six is the extreme number of yin. When numbers reach their extreme, they reverse, hence becoming the transformations of the hexagrams." This is the precise mathematical expression of the Yijing principle that things reaching their extreme must reverse. Nine is Old Yang, at its peak about to transform into yin; six is Old Yin, at its peak about to transform into yang. This perfectly accords with the divination method of using nine and using six in the Book of Changes.

V. The Inherent Harmony of Natural Numerological Relations

Through numerous numerical equalities (such as three fours are twelve, two sixes are twelve; three twelves are thirty-six, four nines are thirty-six, six sixes are thirty-six, etc.), Shao Yong demonstrates a kind of beauty inherent in mathematics: different computational paths can reach identical values — this is not coincidence but "natural correspondence," reflecting the inner unity and harmony of the cosmos.

Modern Interpretation 🌟

I. Mathematical Modeling: Ancient Systems Thinking

Shao Yong's numerical system constitutes a form of early systems theory. He uses numerical relationships to model cosmic structures and changes, sharing much in common with mathematical models in contemporary science. His use of "six transformations" and "four transformations" to represent the Way of Heaven and the Way of Earth can be understood as two differing generative rules or recursive functions — one advancing by sixes, the other by fours — whose interaction generates the complexity of all things in the universe.

II. The "Remove One" Principle and Resource Management

The yarrow stalks use forty-nine rather than fifty, which Shao Yong calls "preserving one." This wisdom of "leaving room to spare" is embodied in modern management as redundancy design: systems should not operate at their extreme limits but should retain a certain buffer. In practical life, this translates to: maintaining savings economically, leaving margin in time management, and preserving room for development in one's capabilities. "Removing four" embodies another wisdom: appropriate simplification — the sixty-four hexagram system operates with sixty (corresponding to 360 days); not every potential possibility needs activation; efficient system operation requires judicious trimming.

III. The Modern Metaphor of Yin-Yang Classification

"Yang within yang," "yin within yang," "yin within yin," "yang within yin" provides a four-quadrant analytical framework, employing logic similar to the SWOT analysis (Strengths, Weaknesses, Opportunities, Threats) common in modern management or the four-type personality classifications in psychology. Nothing is purely yang or purely yin — within yang there is yin, and within yin there is yang. This framework reminds us: when judging people, affairs, and things, we should avoid simple binary labels.

IV. Multiplicative Thinking and Exponential Growth

"One transformation to two, two to four, three and the eight trigrams are complete" demonstrates the ancient recognition of exponential growth (2ⁿ). Shao Yong already perceived the power of doubling — corresponding today to concepts like compound interest, network effects, and viral propagation. Meanwhile, his "trimming the square to make the circle" and "dividing the great into the small" also embody division-analytical thinking — how to scale down principles of large systems to smaller applications (such as scaling yearly cyclical patterns down to months and days).

V. Numbers Reaching Extremes Reversing, and Cycle Theory

"Nine is the extreme of yang; six is the extreme of yin; numbers at their extreme reverse" is highly consistent with modern oscillation system theory and economic cycle theory. When any system (economy, ecosystem, individual state) reaches a certain extremum, the momentum for reversal begins to accumulate. This reminds us: remain vigilant at the height of good fortune; maintain hope in the depth of adversity — for extremes mark turning points.

Practical Value ⚡

I. The Principle of Decision-Making with Reserved Margin

Shao Yong's "preserving one" principle can be translated into the principle of built-in slack in daily decision-making: never mobilize 100% of resources to pursue a single goal. Leaving at least 2% (1/50) of buffer endows the system with necessary resilience. This applies to time management (don't fill your schedule to the brim), financial planning (never invest everything), and interpersonal relations (don't play all your cards at once).

II. Four-Quadrant Analysis Method

Applying "yang within yang, yin within yang, yang within yin, yin within yin" to problem analysis:

• Yang within yang: matters that are completely clear, to be advanced with full effort
• Yin within yang: generally positive but requiring attention to specific risks
• Yang within yin: seemingly unfavorable but harboring opportunities
• Yin within yin: matters to be completely avoided or postponed

In team management, this model can also be used to identify member traits: some people appear positive outwardly yet are firm inwardly (yang within yang), some appear positive outwardly but hesitate inwardly (yang within yin), some appear passive outwardly yet have strong opinions inwardly (yin within yang), and some are exhausted both inwardly and outwardly (yin within yin).

III. Cyclical Awareness and Rhythm Management

The model from Fu to Qian with 112 yang lines, and from Fu to Kun with 112 yin lines, corresponds to the rhythms of yang expanding while yin contracts, and yin expanding while yang contracts, across the course of a year. Modern people likewise need cyclical awareness: the human body has circadian rhythms, monthly rhythms, and seasonal rhythms. Work and life arrangements should cooperate with—rather than resist—these natural cycles. For example, creative work is best scheduled during periods of rising personal energy, while review and restoration are better suited to periods of decline.

IV. The Principle of Simplicity in Numbers

The extensive set of equivalent numerical expressions Shao Yong demonstrates (three fours make twelve = two sixes make ten; four nines make thirty-six = six sixes make thirty-six, etc.) implies a mathematical aesthetic of diverse paths converging on the same outcome: when facing complex problems, different routes may lead to identical answers. In solving practical problems, one should not cling to a single approach, but rather recognize the validity of multiple paths and choose the route best suited to the current context.

Philosophical Reflections 🤔

I. The Deep Paradox of the Substance–Function Relationship

Shao Yong states, "Function is grounded in substance, hence the one is preserved; substance is rooted in function, hence the four are removed." This reveals a mutually recursive philosophical relationship: substance is the foundation of function, and function is the root of substance. If substance is completely preserved, function has no avenue to manifest; if function is entirely favored, substance will be lost. This thought transcends the simple notion of "substance and function share a single source," pointing toward a dynamic equilibrium ontology—existence (substance) and activity (function) are not two separate things, but two facets of one whole, which must maintain tension through the continuous process of "preserving" and "removing."

II. The Immutability of One

"One is the beginning of numbers, yet not itself a number" and "one cannot be transformed." Shao Yong regards "one" as transcending number—it is not the first number, but the a priori condition of number. This resonates with Laozi's idea that "the Way generates one," and parallels the discussion of the "First Cause" in Western philosophy. Within the world of numbers, all numbers can be transformed through operations; only "one" perpetually remains itself—it is the frame of reference for change, the foundation of identity. In modern terms, this can be analogized to the indispensable constant or axiom within any system—it cannot be derived, yet it must exist, otherwise the system cannot operate.

III. The Ultimate of Presence and Absence: Unused Existence

"The odd does not use five, the strategies do not use ten—this is the ultimate of presence and absence." Shao Yong points out that the true "ultimate" is not the highest number, but rather the number that remains unused. Though five and ten exist, they lie outside the operational scope—they exist as a boundary condition or potentiality. This idea carries profound philosophical implications: a system is constituted not only by the elements it actually employs, but also delimited by those it does not use. Just as silence in language defines the meaning of speech, blank space defines the structure of writing. That which is "unused" is not waste, but rather the boundary of meaning.

IV. The Epistemology of Approximating Numbers

"To lean upon is to approximate—approximating the true numbers of heaven and earth into being." Shao Yong acknowledges that the numbers he uses are a simulation of "the true numbers of heaven and earth," not those numbers themselves. This reflects a humble and sober epistemological stance: human mathematical systems approximate the natural order rather than equate with it. In today's age of highly advanced science, this idea remains valid—any scientific model is an approximate description of reality, not a complete replication of it. The "leaning" (approximate) nature of models reminds us to remain vigilant about the limits of theory.

📚 Related Knowledge

Associated Concepts

• The Great Expansion Number: The mathematical basis for divination using fifty yarrow stalks, originating from Chapter Nine of the Great Commentary on the I Ching: "The number of the Great Expansion is fifty; its use is forty-nine." In this chapter, Shao Yong provides a philosophical explanation for "using forty-nine" grounded in "preserving the one."

• Triple Heaven and Double Earth: From the Shuo Gua Zhuan: "Triple heaven and double earth, and thereby lean upon numbers." Shao Yong interprets "triple heaven" as three threes making nine, and "double earth" as two twos making four (or doubling three to make six), used to explain the generative proportions of heavenly and earthly numbers.

• Using Nine and Using Six: The special line texts of the Qian and Kun hexagrams, originating from the divination system in which nine represents old yang and six represents old yin; when numbers reach their extreme, they transform. Shao Yong here correlates them with the strategy-number system (four nines make thirty-six; four sixes make twenty-four).

• The Number of World Governance: The core number 129,600, used in the Huangji Jingshi to construct the historical cycle theory of "yuan, hui, yun, shi." One yuan equals 129,600 years, divided into twelve hui (each 10,800 years), thirty yun (each 360 years), and twelve shi (each 30 years), forming a homomorphic mapping with the twelve months of a year and three hundred sixty days.

Further Reading

• I Ching Commentary, Great Commentary, Section One, Chapter Nine: the original source of the Great Expansion number
• I Ching Commentary, Shuo Gua Zhuan, Chapters One to Two: positioning of heaven and earth and mathematical foundations
• Zhouyi Benyi · Divination Ritual: Zhu Xi's standardized description of the Great Expansion method
• Meihua Yishu · Eight Trigrams Divination by Myriad Things: correspondence between trigram numbers and natural phenomena
• Huangji Jingshi · Inner Chapters on Observing Things: Shao Yong's philosophical meditations on numbers
• Huangji Jingshi · Expanded Meaning of Outer Chapters on Observing Things, Volume Two: the precursor sections of the relevant mathematical derivations

Modern Research

• Li Shen's Ancient Chinese Philosophy and Natural Science: Analyzes Shao Yong's "mathematical cosmology" from the perspective of the history of science, arguing that the Huangji Jingshi system is essentially one of the earliest attempts to employ mathematics as the language of the cosmos.

• Feng Youlan's A New History of Chinese Philosophy: Positions Shao Yong's image-number philosophy as a "mathematized expression of objective idealism," holding that although the system originates from ancient divination studies, it developed a new cosmology independent of the Confucian tradition.

• Kidder Smith et al.'s Sung Dynasty Uses of the I Ching (Princeton University Press, 1990): Explores the influence of Shao Yong's mathematical philosophy on Song dynasty literati modes of thinking, particularly his methodological innovation of "substituting number for image."

• Algorithms and Yinyang: Contemporary scholars have compared Shao Yong's binary subdivision method—"one transforms into two, two transforms into four"—with computer binary code, pointing out that it presciently touched upon the fundamental logic of information generation. Leibniz was inspired by Chinese divination studies when inventing binary code, and Shao Yong's diagram of the sequential arrangement of the sixty-four hexagrams served as a significant intellectual bridge in this connection.