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Chapter 16 of 25
皇极经世
This chapter, "The Complete Numbers of Heaven and Earth in the River Diagram," First Section, systematically elaborates the core numerological framework of the Yijing's image-number system. Heaven's numbers are five (1, 3, 5, 7, 9), and Earth's numbers are five (2, 4, 6, 8, 10), together making ten—this is the totality of numbers. Heaven takes one as its basis and evolves into four; Earth likewise takes one as its basis and evolves into four. The four have form and are visible, while the one is formless and incorporeal—this is the ultimate of "being" and "non-being."
Heaven's substantive number is four, and its operative number is three; the unused one is one. Earth's substantive number is four, and its operative number is three; the unused one is one. The bodiless one symbolizes "nature" itself; the unused one symbolizes the "Dao"; the three in use symbolize the Three Talents of Heaven, Earth, and Humanity. The true number of the Yi, ultimately, is "three." Tripling Heaven (multiplying three by three) yields nine; doubling Earth (multiplying two by three) yields six. Tripling Heaven and doubling Earth, then relying on numbers for derivation—this is not the inherent correct number of Heaven and Earth, but rather an approximation generated from their correct numbers. What is the Great Derivation number of the Yi? It is the framework of derivation that the sages established by relying on numerical principles.
Heaven's numbers total twenty-five (1+3+5+7+9); Heaven combining with itself yields fifty (double of twenty-five). Earth's numbers total thirty (2+4+6+8+10); Earth combining with itself yields sixty. Hence it is said, "The five positions correspond to each other and each has its conjunction." Fifty is the number for yarrow-stalk divination; sixty is the number of the hexagrams. Five is the small derivation of the yarrow stalks; fifty is the Great Derivation. Eight is the small completion of trigrams (single trigram); sixty-four is the great completion of hexagrams (doubled trigram). The virtue of the yarrow stalk is roundness, used to correspond to Heaven's numbers, hence seven times seven is forty-nine; fifty is stated with one preserved. The virtue of the hexagram is squareness, used to correspond to Earth's numbers, hence eight times eight is sixty-four; sixty is stated with four removed. The yarrow stalk is the operative number; the hexagram is the substantive number. The operative is based on the substantive, hence one is preserved; the substantive takes the operative as its root, hence four is removed. The root of the round is one; the root of the square is four. The operative number of the yarrow stalk is seven; including the remainders, it also embodies the meaning of preserving one. Hanging one (setting aside one stalk) is also the meaning of removing one.
In yarrow-stalk operation, hanging one symbolizes the Three Talents; the remaining forty-eight stalks correspond to the stalk count of one hexagram. Four times twelve is forty-eight. Twelve minus three leaves nine; four times three is twelve stalks removed, four times nine is thirty-six stalks used, corresponding to the thirty-six yang lines of the Qian hexagram. Twelve minus five leaves seven; four times five is twenty removed, four times seven is twenty-eight used, corresponding to the twenty-eight yang lines of the Dui and Li hexagrams. Twelve minus six leaves six; four times six is twenty-four removed, four times six is twenty-four used, corresponding to the twenty-four yin lines of the Kun hexagram. Twelve minus four leaves eight; four times four is sixteen removed, four times eight is thirty-two used, corresponding to the twenty-four lines of the Kan and Gen hexagrams plus the eight yin lines within, totaling thirty-two lines. Therefore seven and nine are yang numbers; six and eight are yin numbers. Nine is the ultimate yang number; when the number reaches its extreme it reverses, hence it is called "change." Zhen and Xun have no stalks—this derives from "the numbers that are not used." Heaven takes firmness as its virtue, hence the yielding is not manifested; Earth takes yielding as its substance, hence the firm is not generated. Qian uses nine, hence its stalk count is nine—four corresponding to the four seasons, one season being ninety days. Kun uses six, hence its stalk count is likewise six.
The yarrow-stalk number does not use six but uses seven, speaking inclusively of the remainders; removing the remainders yields six, and the stalk count is thirty-six. Hence the fifty-six sixty-four hexagrams are the stalks of a leap year; using forty-nine is the stalks for one year of sixty hexagrams. Returning the odd and hanging one is like the intercalary day of a year. The hexagrams directly remove four, because Heaven transforms and Earth follows its effects; when the yarrow removes one, the hexagram removes four. The odd numbers have four kinds (1, 2, 3, 4), and the stalk numbers have four kinds (6, 7, 8, 9), combining into eight numbers to correspond to the eight transformations of the square number. The combinations of returning the odd and tallying the hexagram number have six forms: five with four-four, nine with eight-eight, five with four-eight, nine with four-eight, five with eight-eight, and nine with four-four, corresponding to the six transformations of the round number.
The odd numbers reach their extreme at four, and five is not used; the stalk numbers reach their extreme at nine, and ten is not used. Five is one (another preserved number); ten is two. Hence fifty is removed and forty-nine is used. The odd does not use five, and the stalk does not use ten—this is the ultimate of being and non-being, used to correspond to the natural numbers. When the returned odd and tallied hexagram yield five with four-four, the stalk count is four times nine, thirty-six; when it yields nine with eight-eight, the stalk count is four times six, twenty-four; when it yields five with eight-eight or nine with four-eight, the stalk count is four times seven, twenty-eight; when it yields nine with four-four or five with four-eight, the stalk count is four times eight, thirty-two. The result of nine is one transformation corresponding to Qian; the result of six is one transformation corresponding to Kun; the result of seven is two transformations corresponding to Dui and Li; the result of eight is two transformations corresponding to Gen and Kan. Five with four-four, removing the hanging-one number, is four times three, twelve; nine with eight-eight, removing the hanging one, is four times six, twenty-four; five with eight-eight, nine with four-eight, removing the hanging one, is five times four, twenty; nine with four-four, five with four-eight, removing the hanging one, is four times four, sixteen. Hence the numbers three, four, five, and six are removed, giving rise to the stalks of seven, eight, nine, and six. Nine advancing yields thirty-six, the pure yang number, hence it is yang within yang; seven advancing yields twenty-eight, yang first then yin, hence it is yang within yin; six advancing yields twenty-four, the pure yin number, hence it is yin within yin; eight advancing yields thirty-two, yin first then yang, hence it is yin within yang.
Yang obtains yin and then can generate; yin obtains yang and then can complete. Hence the yarrow-stalk number is four and nine, and the hexagram number is six and ten—like the interweaving of the Heavenly Stems and Earthly Branches, where the stems end at six and the branches end at five. When the yarrow-stalk number is complete, the yang stalks of thirty-six and twenty-eight combine into sixty-four; when the hexagram number removes four, the yin stalks of twenty-four and thirty-two combine into fifty-six. One season is limited to three months; one month is limited to thirty days—both remove the number of the hour (chen). Hence eight times eight, sixty-four hexagrams: eight are unchanging, and the changeable ones are seven times eight, fifty-six; its meaning also derives from this.
Heaven has four seasons; one season has four months; one month has forty days; four times four is sixteen, and each removes one, hence one season has three months and one month has thirty days. Four seasons is the substantive number; three months and thirty days is the operative number. Although the substance includes four, one of them is always unused, hence the operative is limited to three and stops at nine. Substantive numbers are always even, hence there are four and twelve; operative numbers are always odd, hence there are three and nine. Large numbers are always insufficient while small numbers are always abundant, because the large cannot be seen while the small can be seen. Hence the season stops at four, the month stops at three, and the day overflows at ten. The human body has four limbs and ten fingers. Heaven manifests in the south and conceals itself in the north, reaches its extreme at six and leaves a remainder at seven; hence people know what is before them but are ignorant of what is behind, and are vague about left and right.
Heaven's substantive number is four and its operative is three; Earth's substantive number is four and its operative is three. Heaven overcomes Earth, Earth overcomes Heaven, and what overcomes lies in Earth—like the surplus portion of daylight residing in the night. Hence Heaven is three and Earth is four; Heaven has three celestial bodies (chen), and Earth has four elements. Earth's fire appears and then conceals—this is what is meant by the surplus portion. Heaven's transformations are six; six times six is thirty-six, the stalk count of one Qian line; accumulating six lines yields two hundred sixteen. Earth's substance is four; six times four is twenty-four, the stalk count of one Kun line; accumulating six lines yields one hundred forty-four. Accumulating the stalks of the upper and lower sections yields eleven thousand five hundred twenty. Discussing their substance: Heaven divides into Earth, Earth divides into the myriad things, yet the Dao cannot be divided; in the end, the myriad things return to Earth, Earth returns to Heaven, and Heaven returns to the Dao. Hence the superior person values the Dao as most precious.
Three times four yields twelve; two times six also yields twelve. Two times twelve yields twenty-four; three times eight also yields twenty-four; four times six also yields twenty-four. Three times twelve yields thirty-six; four times nine also yields thirty-six; six times six also yields thirty-six. Four times twelve yields forty-eight; three times sixteen also yields forty-eight; six times eight also yields forty-eight. Five times twelve yields sixty; three times twenty also yields sixty; six times ten also yields sixty. All these are natural correspondences. Yang six incorporates half of yin six, hence it yields nine. Viewed from two trigrams, yin and yang each number three; viewed from six lines, Heaven, Earth, and Humanity each number two. Within yin and yang each contain Heaven, Earth, and Humanity; within Heaven, Earth, and Humanity each are yin and yang. Hence tripling Heaven and doubling Earth, then relying on numbers.
Yang number one, derived, becomes ten—this is the category of the ten Heavenly Stems. Yin number two, derived, becomes twelve—this is the category of the twelve Earthly Branches and twelve months. The ten Stems are Heaven; the twelve Branches are Earth; the stems and branches together form the functioning of Heaven and Earth. The stem takes the meaning of "trunk" and belongs to yang; the branch takes the meaning of "twig" and belongs to yin. Ten stems and twelve branches means that within yang numbers there is yin, and within yin numbers there is yang. One, ten, hundred, thousand, ten-thousand are odd—this is Heaven's number; twelve, one hundred twenty, one thousand two hundred, twelve thousand, one hundred twenty thousand are even—this is Earth's number. The yarrow stalk advancing by fours yields one hundred; the hexagram advancing by fours yields one hundred twenty. One hundred is ten; one hundred twenty is twelve. Dividing fifty in half yields ten; tripling Heaven and doubling it yields six; doubling Earth and doubling it again yields four—this is Heaven and Earth dividing the number of the Supreme Ultimate.
Qi is the abode of spirit; substance is the abode of qi. Qi transforms by six; substance divides by four. Substance is four and transforms by six, encompassing both spirit and qi. Qi's transformations must be six, hence three hundred sixty. Heaven is six; Earth is four. Yin has no one; yang has no ten. Yang has no ten, hence it is insufficient at the end; yin has no one, hence it is insufficient at the beginning. Heaven one, Earth two, Heaven three, Earth four, Heaven five, Earth six, Heaven seven, Earth eight, Heaven nine, Earth ten—intermingling and transforming, interweaving their numbers, Heaven and Earth unite, day and night intersect.
One is the beginning of number rather than number itself; hence 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—all employ their transformations. Five times five, twenty-five, is Heaven's number; six times six, thirty-six, is the stalk count of Qian; seven times seven, forty-nine, is the Great Derivation operative number; eight times eight, sixty-four, is the hexagram number; nine times nine, eighty-one, is the number of the Yuan Fan. The Great Derivation number is the source of calculation; the beginning of calculation is nothing more than square, round, curved, and straight. Multiplication is the generating number; division is the diminishing number. Though calculation methods are many, they do not go beyond this.
Yin is close to the Dao—hence yin is used to analogize the Dao. Six transformations yield thirty-six; eight transformations yield sixty-four; twelve transformations yield three hundred eighty-four. Transforming six by six yields, eight by eight, sixty-four transformations yielding three hundred eighty-four; transforming eight by eight yields, seven by seven, forty-nine transformations yielding three hundred eighty-four.
The hour (chen) is the boundary of Heaven and Earth. The number of hours is twelve; the conjunction of sun and moon is called an hour. The hour is Heaven's substance; Heaven's substance is formless qi. Stars are the surplus of the sun; hours are the surplus of the moon. The yang-within-yang of Qian cannot transform, hence one year only enumerates twelve months; the yin-within-yin of Zhen cannot transform, hence the twelve hours of a day cannot be seen. Dui, yang within yin, and Li, yin within yang, can both transform; hence the numbers of sun and moon can be divided. Yin numbers begin at twelve; yang numbers begin at thirty—and two-six is always preserved. Raising the year reveals the month; raising the month reveals the day; raising the day reveals the hour: this is yang governing yin, Heaven's four transformations encompassing Earth's four transformations, the sun's transformations encompassing the transformations of the moon and the stars and hours.
The generating number of the Yi is one hundred twenty-nine thousand six hundred, totaling four thousand three hundred twenty generations—this is the great number of waxing and waning. Deriving the hour count of thirty years yields this number: a year of three hundred sixty days yields four thousand three hundred twenty hours; multiplied by thirty yields one hundred twenty-nine thousand six hundred. All jiazi and jiawu are heads of generations—these are the numbers of the governing ages, such as stem-day jia, child-month, star-jia chenzi, and so forth. Originally it is one qi; when generated it is yang, when diminished it is yin. Hence two is merely one; four is merely two; six is merely three; eight is merely four. Therefore one speaks of Heaven but not Earth, of the ruler but not the minister, of the father but not the son, of the husband but not the wife. Yet Heaven obtains Earth and the myriad things are generated; the ruler obtains the minister and the ten thousand transformations proceed; the father obtains the son, the husband obtains the wife, and the household way is accomplished. Hence where there is one there is two; where there are two there are four; where there are three there are six; where there are four there are eight. Sun, moon, stars, and hours together form Heaven; water, fire, earth, and stone together form Earth; ear, eye, nose, and mouth together form the head; marrow, blood, bone, and flesh together form the body—this is the number of five. The Yi has three hundred eighty-four lines—truly a celestial pattern.
Innate Image-Number, Second Section states: The passage "Heaven and Earth fix their positions" elucidates Fuxi's eight trigrams. The eight trigrams make clear the mutual interweaving that forms the sixty-four hexagrams. "Numbering the past is favorable" — if one follows Heaven in its course, it rotates leftward; these are all already-generated hexagrams, hence it is called "numbering the past." The numbers of the Yi are completed through reversal. One divides into two; two divides into four; four divides into eight; eight divides into sixteen; sixteen divides into thirty-two; thirty-two divides into sixty-four. Dividing yin and yang, alternately employing firm and yielding, hence the six positions of the Yi complete its patterns. Ten divides into one hundred; one hundred divides into one thousand; one thousand divides into ten thousand—like the root having a trunk, the trunk having branches, the branches having leaves; the larger it grows, the fewer; the finer it becomes, the more numerous. Combining them yields one; deriving them yields ten thousand. Qian divides; Kun gathers; Zhen grows; Xun diminishes. Growth leads to division; division leads to diminishment; diminishment leads to gathering. Qian takes one-fourth and gives it to Kun; Kun takes one-fourth and offers it to Qian; Qian and Kun combine and generate the six children: the three males are all yang, the three females are all yin. Dui divides one yang to give to Gen; Kan divides one yin to offer to Li; Zhen and Xun exchange by twos. Speaking comprehensively, yin and yang are each half; hence water and fire generate each other and overcome each other, and only then are the myriad things completed. The substance has eight transformations; the function has six transformations. Among the images of the eight trigrams, four are unchanging and two are reversible; through six transformations the eight are formed. Among the images of doubled hexagrams, eight are unchanging and twenty-eight are reversible; through thirty-six transformations the sixty-four are formed. Hence the lines stop at six, the trigrams end at eight, the stalks are exhausted at thirty-six, and the doubled hexagrams reach their extreme at sixty-four. The trigram is completed at eight and doubled at sixty-four; the line is completed at six, the stalk is exhausted at thirty-six, and doubled at three hundred eighty-four. Heaven has two correct (upright) [trigrams], Earth has two correct, and together they employ two reversible to complete the eight trigrams. Heaven has four correct, Earth has four correct, and together they employ twenty-eight reversible to complete the sixty-four hexagrams. In the small completion, the correct trigrams are four and the reversible are two, totaling six trigrams; in the great completion, the correct hexagrams are eight and the reversible are twenty-eight, totaling thirty-six hexagrams. Qian, Kun, Kan, and Li are the ancestors of the thirty-six hexagrams; Dui, Zhen, Xun, and Gen are the ancestors of the twenty-eight hexagrams. The correct and reversible hexagrams together number thirty-six, and the lines number two hundred sixteen more—this is the stalk count of the operative number. Thirty-six removing four yields thirty-two; removing four more yields twenty-eight; removing four more yields twenty-four. Hence the hexagram number of thirty-two positions is stated with four removed; the number of twenty-eight positions of Heaven is stated with eight removed. Four are Qian, Kun, Kan, Li; eight include Yi, Zhongfu, Daguo, Xiaoguo; twelve include Dui, Zhen, Tai, Jiji. The four yang hexagrams have twelve lines: eight yang and four yin. Multiplying the yang by thirty-six and the yin by twenty-four yields three hundred eighty-four. The opposing hexagrams all have six yang and six yin; in the Yi there are twelve pairs of six yang and six yin. Removing the four correct, there are six pairs each of eight yang four yin and eight yin four yang; and three pairs each of ten yang two yin and ten yin two yang. One transformation yields two; two yield four; three transformations complete the eight trigrams; four yield sixteen; five yield thirty-two; six transformations complete the sixty-four hexagrams. From Fu to Qian there are one hundred twelve yang in total; from Gou to Kun there are one hundred twelve yin in total; from Gou to Kun there are eighty yang; from Fu to Qian there are eighty yin. Qian has thirty-six, Kun has twelve, Li, Dui, and Xun have twenty-eight, Kan, Gen, and Zhen have twenty. The Yi is rooted in Qian and Kun and generated from Fu and Gou: firmness intersecting with yielding forms Fu; yielding intersecting with firmness forms Bo—from here it continues without end.
Innate Circular Diagram Hexagram Number, Third Section states: Qi is one, and spirit is also one. Only when there is Earth is there two; only when there are two is there day and night. Two and three transform, interweaving and forming. Hence the Yi is generated by two, and numbers arise from twelve. One is not a number, yet numbers are completed through it. Earth removes one and arises from twelve; Earth's fire always lies concealed.
Innate Square Diagram Hexagram Number, Fourth Section states: Qian has seven sons, Dui has six sons, Li has five sons, Zhen has four sons, Xun has three sons, Kan has two sons, Gen has one son; Kun, being entirely yin, has no sons. Qian has seven sons, Kun has six sons, Dui has five sons, Gen has four sons, Li has three sons, Kan has two sons, Zhen has one son; Xun, being yin in its firmness, has no sons. Qian and Kun transform sevenfold, hence the extreme of day and night does not exceed seven divisions. Gen and Dui transform sixfold, hence the moon stops at six, together making twelve. Li and Kan transform fivefold, hence the sun stops at five, together making ten. Zhen and Xun transform fourfold, hence the body stops at four, together making eight. The sun has eight positions and its function stops at seven—speaking with Qian removed; the moon has eight positions and its function stops at six—speaking with Dui removed; the stars have eight positions and their function stops at five—speaking with Li removed; the hours have eight positions and their function stops at four—speaking with Zhen removed. The sun has eight positions and its number stops at seven—speaking with Tai removed. The moon, arising from Dui, cannot reach the number of the sun; hence twelve months always retain a surplus of twelve days. The round number has one; the square number has two—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; when transformed, it arises from four and accumulates to twelve. Six always transforms by six; eight always transforms by eight; and twelve also transforms by eight—this is the natural way. Eight is the substance of Heaven and Earth; six is the function of Heaven; twelve is the function of Earth. Heaven transforms the square into the round while always preserving the one; Earth divides the one into four while always maintaining the square. Heaven transforms its substance but not its function; Earth transforms its function but not its substance. Six combines with its one to become seven; twelve combines with its four to become sixteen. Yang governs advancement, hence Heaven combines its one to become seven; yin governs retreat, hence Earth removes its four and stops at twelve. Yang always preserves the one, and yin always conceals the one. The substance of Heaven and Earth stops at eight; Heaven's function reaches its extreme at seven; Earth's function stops at twelve. The round cuts the square for its function: hence one transformation yields four; four removing one yields three; three transformation yields nine; nine removing three yields six. The square unfolds the round for its substance: hence one transformation yields three; combining them yields four; four transformation yields twelve; combining them yields sixteen. The operative number is completed at three and reaches its extreme at six; the substantive number is completed at four and reaches its extreme at sixteen. 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 the natural way. The round transforms sixfold: six times six advances, hence sixty transformations yield three hundred sixty; the square transforms eightfold, hence eight times eight yields sixty-four. Yang governs advancement, hence it advances to sixty. The round has a diameter of one and a circumference of three; doubled, it is six; the square has a diameter of one and a circumference of four; doubled, it is eight. Cutting the square to form the round is how Heaven revolves; dividing the great into the small is how Earth generates and transforms. Hence Heaven functions by six transformations, and Earth functions by four transformations.
Chapter 6 of the Later Heaven Zhouyi Principle and Number: Kun governs the three daughters in the southwest, and Qian governs the three sons in the northwest. The Upper Scripture begins with three, and the Lower Scripture ends with four—both embody the meaning of interpenetration and harmony. Qian uses nine, Kun uses six, the Great Expansion uses forty-nine, and the hidden dragon does not act. How great is the use! In this one sees the mind of the sage. Without knowing Qian, one cannot know the principle of nature and destiny. The Dao gives birth to Heaven, Heaven gives birth to Earth, and when the work is accomplished, one withdraws from one's position—thus the son inherits and the father yields, and therefore Qian retreats one position. The first and top lines are the same, yet the top line's overreaching does not match the first line's advance; the second and fifth lines are the same, yet the second's yin centrality does not match the fifth's yang centrality; the third and fourth lines are the same, yet the third, dwelling at the top of the lower trigram, is not as near to the ruler as the fourth. The noble person, in regard to the Changes, plays with the images, plays with the numbers, plays with the words, plays with the meanings. Where there is meaning, there must be words; where there are words, there must be images; where there are images, there must be numbers. When images arise, words become manifest; when words become manifest, meanings become evident. Images and numbers are like the snare and the trap; words and meanings are like the fish and the rabbit. Having obtained the fish and the rabbit, one may forget the snare and the trap; but setting aside the snare and the trap to seek the fish and the rabbit—one has never seen this succeed. Images arise from form, numbers arise from substance, names arise from words, meanings arise from function. The numbers of the world issue from principle; if one contradicts principle, one falls into technique. People of the world use numbers to enter into technique, and therefore do not enter into principle. What is naturally so and cannot be altered is the inner image and inner number; all the rest are outer images and outer numbers. The Changes have inner images—these are principle and number; they also have outer images—these are what designate a single thing and do not change. The Changes have numerical images, such as "seven days," "eight months," "three years," "ten years," and the like. The Supreme Ultimate, unmoving, is nature; when it stirs, it is spirit; spirit then is number; number then is image; image then is vessel; qi then transforms, and returns to spirit. Yin affairs are the greater half, for yang is one and yin is two.
Chapter 7 of the Great and Small Fortune Numbers of the Cycles through the Yuan: Heaven, from the hexagram Mingyi downward, and Earth, from the hexagram Pi downward, are the numbers of interpenetration. The interpenetration of Heaven and Earth is three-tenths. Qian is one. The fifth line of Qian divides into Dayou, corresponding to the number of three hundred sixty; the fourth line of Qian divides into Xiaoxu, corresponding to the number of one hundred twenty-nine thousand six hundred; the third line of Qian divides into Lü, corresponding to the number of sixteen billion seven hundred ninety-six million one hundred sixty thousand; the second line of Qian divides into Tongren, corresponding to the number of eighteen quadrillion two hundred eleven trillion nine hundred ninety million seven hundred forty-five billion six hundred thousand; the first line of Qian divides into Gou, corresponding to the number of seven quintillion nine hundred fifty-eight sextillion six hundred sixty-one quintillion one hundred ten septillion nine hundred ninety-six trillion four hundred sixty-eight quadrillion four hundred thirty-nine million one hundred ninety-three sextillion as the number. These are called divisional numbers—dividing the great into the small, all from top to bottom—therefore yang numbers are used to correspond to them. One generates two as Guai, corresponding to the number of twelve; two generates four as Dazhuang, corresponding to the number of four thousand three hundred twenty; four generates eight as Tai, corresponding to the number of five hundred fifty-nine million eight hundred seventy-two thousand times ten thousand; eight generates sixteen as Lin, corresponding to the number of nine hundred forty trillion three thousand six hundred ninety-nine million six hundred ninety-one trillion five hundred billion times twenty billion; sixteen generates thirty-two as Fu, corresponding to the number of two thousand six hundred fifty-two rivu-eighty-eight rivu-seven hundred thirty-six riku-four thousand eight hundred eighty-eight jing-two thousand nine hundred forty-seven jiu-seven hundred thirty-one trillion-two hundred billion-as the number; thirty-two generates sixty-four as Kun, corresponding to the number of the unlimited—these are called growth numbers—growing the small into the great, all from bottom to top—therefore the numbers of Kun are used to correspond to them. Heaven governs the body, so eight transformations terminate in sixteen; Earth governs function, so six transformations terminate in twelve. Heaven arises from one and terminates in seven quintillion nine hundred fifty-eight sextillion six hundred sixty-one quintillion one hundred ten septillion nine hundred ninety-six trillion four hundred sixty-eight quadrillion four hundred thirty-nine million one hundred ninety-three sextillion; Earth arises from twelve and terminates in two quintillion forty-six septillion nine hundred eighty trillion seven thousand three hundred eighty-one jing-five thousand four hundred ninety-one wan-eight hundred ninety-nine zhao-seven hundred twenty trillion-as the number. The sun has one position, the moon one position, the stars one position, the chronograms one position; the sun has four positions, the moon four, the stars four, the chronograms four—four times four is sixteen positions. When this one transformation is exhausted, the numbers of sun and moon are at an end. Heaven has four transformations, Earth has four transformations—there is growth and there is decline—in sixteen transformations the numbers of Heaven and Earth are exhausted. The sun arises from one, the moon arises from two, the stars arise from three, the chronograms arise from four. Extending them outward, the yang number is always six, the yin number always two, and in twelve transformations the fortune of great and small is exhausted. Three hundred sixty transforms into one hundred twenty-nine thousand six hundred; one hundred twenty-nine thousand six hundred transforms into sixteen billion seven hundred ninety-six million one hundred sixty thousand; sixteen billion seven hundred ninety-six million one hundred sixty thousand transforms into two quadrillion eight hundred twenty-one trillion one hundred million? Wait — let me recalculate according to the original structure more carefully for consistency:
Three hundred sixty becomes one hundred twenty-nine thousand six hundred; one hundred twenty-nine thousand six hundred becomes sixteen billion seven hundred ninety-six million one hundred sixty thousand; sixteen billion seven hundred ninety-six million one hundred sixty thousand becomes twenty-eight quadrillion two hundred eleven trillion nine hundred ninety million seven hundred forty-five billion six hundred million. 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 twenty-eight quadrillion two hundred eleven trillion nine hundred ninety million seven hundred forty-five billion six hundred million as the year, then the model of the great and small fortunes is established. Twenty-eight quadrillion two hundred eleven trillion nine hundred ninety million seven hundred forty-five billion six hundred million divided into twelve—the first six divisions are growth, the last six are decline—corresponds to the number of twelve months in a year and advances three hundred sixty days. Sixteen billion seven hundred ninety-six million one hundred sixty thousand divided into thirty—corresponds to the number of thirty days in a month—advancing or retreating sixty days following the growth and decline of the great fortune. One hundred twenty-nine thousand six hundred divided into twelve—corresponds to the number of twelve hours in a day—advancing or retreating six days. Three hundred sixty corresponds to the number of one hour, advancing or retreating according to the small fortune to correspond to the hours of day and night. From the number of sixteen transformations, remove the interpenetration numbers and take the function numbers: one obtains twenty-eight quadrillion two hundred eleven trillion nine hundred ninety million seven hundred forty-five billion six hundred million, divided into twelve—first six divisions growth, last six decline—each division obtains thirteen billion nine hundred ninety-six million, eight hundred thousandths 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 and advances sixty days; in six divisions opens six divisions and advances three hundred sixty days; its retreat is the same. From one hundred twenty-nine thousand six hundred, removing three of them is the interpenetration number, taking seven of them is the function number; the function number three becomes complete in six, and with the remainder added there are seven. Seven of it gives ninety thousand seven hundred twenty years, giving forty-five thousand three hundred sixty years, to advance six days. The day has day and night, the month has new moon and full moon, thus completing twelve days. Every three thousand six hundred years advances one day; altogether in forty-three thousand two hundred years it advances twelve days; the remaining two thousand one hundred sixty years advance the six of the remainder, combined with the interpenetration number's two thousand one hundred sixty, altogether advancing twelve divisions to make the intercalation. Therefore the small fortune's transformation in all completes sixty days and becomes three hundred sixty-six days. The yang number on three hundred sixty overflows, the yin number on three hundred sixty contracts. The day advances by slowness, the moon retreats by swiftness. Sun and moon meeting adds half a day and subtracts half a day—this is the intercalary remainder. The day in one great fortune advances six days; the moon in one great fortune retreats six days—this is the intercalary difference. The intercalation in one year is six yin and six yang; in three years it is thirty-six days, therefore there is one intercalation every three years; in five years it is sixty days, therefore there is a second intercalation in five years. The mutual eclipses of sun and moon are the intersections of numbers. When the sun looks at the moon directly, there is a lunar eclipse; when the moon covers the sun, there is a solar eclipse. Luoxia Hong reformed the Zhuanxu calendar into the Taichu calendar; Yang Ziyun, modeling on Taichu, composed the Taixuan—altogether eighty-one hexagrams, nine divisions sharing two hexagrams each, so that one cycle has five intervals and one four; further subdivided, four and one-half divisions correspond to one hexagram's qi. It begins from the center, therefore the first hexagram is the central Zhong hexagram. In the Taixuan, nine days correspond to two hexagrams; the remaining one hexagram corresponds to four and one-half days.
Chapter 8 of Biological Function Numbers of the Transformation of the Great Fortune:
The substance has three hundred eighty-four, yet the function only extends to three hundred sixty, because Qian, Kun, Kan, and Li are not used. That Qian, Kun, Kan, and Li are not used is precisely what completes the function of three hundred sixty. Things change and transform, but these four do not change—only that which does not change is able to change. The function extends only to three hundred sixty yet there are three hundred sixty-six—this is the surplus of the numbers. Qian is used completely; Qian and Kun are not used, Kan and Li are used half. Qian and Kun are not used because solitary yang does not generate and lone yin does not complete; Kan and Li are used half because Li is in the east and Kan in the west, occupying the half of yin and yang, as the gates of spring and autumn, day and night. Sometimes one uses Qian, sometimes one uses Kan and Li. When the primary principle is yang, one uses Qian; when primary division of surplus is spoken of, yang encroaches on yin, day encroaches on night, therefore one uses Kan and Li. Qian governs surplus, therefore it is completely used; Kun governs emptiness, therefore it is completely unused. Yang encroaches on yin, yin encroaches on yang, therefore Kan and Li are used half. This is why the south of Heaven is fully visible while the north is fully invisible, and east and west are each half visible. Kan and Li are the boundary of yin and yang: Li corresponds to the third earthly branch (Yin hour), Kan to the ninth branch (Shen hour), and the numbers always exceed these limits—this is the overflow of yin and yang. Yet the function numbers do not pass beyond Yin, and the interpenetration numbers do not pass beyond Shen. Qian with forty-eight divided into four parts—one part is what Yin overcomes; Kun with forty-eight divided into four parts—one part is the yang that is overcome. Therefore Qian obtains thirty-six and Kun obtains twelve. Yang governs advancement, therefore advancing it makes three hundred sixty days; Yin governs diminution, therefore the twelve months diminish twelve days. Counting forward: Qian one, Dui two, Li three, Zhen four, Xun five, Kan six, Gen seven, Kun eight. Counting backward: Zhen one, Li and Dui two, Qian three, Xun four, Kan and Gen five, Kun six. One generates six, six generates six times six (thirty-six); wait—"One gives birth to six, six gives birth to twelve, twelve gives birth to eighteen, eighteen gives birth to twenty-four, twenty-four gives birth to thirty, thirty gives birth to thirty-six"—extending them outward, sixty transformations generate three hundred sixty—these are the numbers of motion and operation. Four gives birth to twelve, twelve to twenty, twenty to twenty-eight, twenty-eight to thirty-six—these are the numbers of biological generation. The yang lines of Qian number thirty-six; the yang lines of Dui, Li, and Xun number twenty-eight; the yang lines of Zhen, Kan, and Gen number twenty; the yang lines of Kun number twelve. The lines of the substance number three hundred eighty-four; removing the lines of Qian, Kun, Kan, and Li yields the function number three hundred sixty. The substance number's function is two hundred seventy; removing the lines of Qian, Kan, and Li yields the function number's function two hundred fifty-two. The substance number's function is two hundred seventy, of which one hundred fifty-six are yang and one hundred fourteen are yin; removing the lines of Li yields one hundred fifty-two yang and one hundred twelve yin—this is the actual function number. Yang, removing Li, uses Qian; yin, removing Kun, uses Kan. Therefore the yang lines of Heaven are one hundred twelve, yin lines eighty—removing its yin; the yin lines of Earth are one hundred twelve, yang lines forty—removing the yang of north and south. The extreme south is great heat, the extreme north is great cold—things cannot be generated there, therefore these are removed. Its forty are the surplus divisions of Heaven—yang encroaches on yin, day encroaches on night, therefore they are in Earth. Combining them gives one hundred fifty-two yang and one hundred twelve yin. Yang, removing the lines of Qian, and yin, removing the lines of Kan, yields one hundred forty-four yang and one hundred eight yin—as the function of the function number. Yang, appearing thirty-six times, three times it gives one hundred eight; yin, appearing thirty-six times, three times it gives one hundred eight. Three yin and three yang—yin and yang each in half. Yang has one surPlus remaining which becomes thirty-six; combined this gives one hundred forty-four yang and one hundred eight yin. The substance number's function is two hundred seventy, yet the actual function is two hundred sixty-four; the function number's function is two hundred fifty-two. The hexagrams number sixty-four yet the function only reaches thirty-six; the lines number three hundred eighty-four yet the function only reaches two hundred sixteen. Sixty-four divided gives two hundred fifty-six—this is one hexagram with its first and top lines removed, also yielding two hundred fifty-six—these are the numbers of biological generation. Therefore Kan and Li are the genesis of biological generation: Li has four yang, Kan has four yin—what generates life must be four. Yang one hundred twelve, yin one hundred twelve; removing the lines of Kan and Li yields two hundred sixteen; adding the forty yang lines in the positions of Gen and Kan in the center yields two hundred fifty-six in all. Therefore the eight trigrams use six lines, with Qian and Kun as their governors; the six lines use four positions, with Li and Kan as their governors. The dawn and dusk of Heaven do not generate creatures, but midday does; the north and south of Earth do not generate creatures, but the center does. The substance number is what generates creatures; the function number is what operates and moves. What operates is Heaven; what generates is Earth. Heaven moves alone, therefore it multiplies the function number by itself, and uses the function of the function number as the time of biological generation. Earth is even and generates, therefore it uses the yang of the substance number's function multiplied by the yin as the number of biological generation. The number of Heaven is three, therefore six it and again six it—this is why the lines of Qian number two hundred sixteen. The number of Earth is two, therefore twelve and again twelve—this is why the lines of Kun number one hundred forty-four. Qian uses nine: three times eight is twenty-four, and nine times it is also two hundred sixteen; two times eight is sixteen, and nine times it is also one hundred forty-four. Kun uses six: three times twelve is thirty-six, and six times it is also two hundred sixteen; two times twelve is twenty-four, and six times it is also one hundred forty-four. Kun uses three-twelfths, four-sixteenths, and half of one-sixth as the surplus divisions of Qian; then Qian obtains two hundred fifty-two and Kun one hundred eight. Heaven has five celestial spheres: sun, moon, stars, and chronograms, with Heaven itself as the fifth. Earth has five elements: metal, wood, water, fire, with Earth itself as the fifth.
Chapter 9 of the Contemplation of Physical Principle and Number Practiced Throughout Ages: The hexagrams number sixty-four, yet only sixty are used. This is because the sixty hexagrams have three hundred sixty lines, and hence the sexagenary cycle stops at sixty—the six cycles of Jia marking the end of Heaven's way. The line numbers correspond: thirty-six combined with twenty-four gives sixty; thirty-two combined with twenty-eight also gives sixty. Qian forty-eight, Kun twenty, Zhen twenty, and thirty; the numbers add: "Qian forty-eight, Kun twelve, Zhen twenty, Xun forty, Li-Dui thirty-two, Gen-Kan twenty-eight" — together they make sixty. Man is the most spiritual of the myriad creatures, dwelling by kind among the running beasts. The running beasts belong to yin, therefore there are one hundred twenty. There are things of one day, things of one month, things of ten years—up to hundreds, thousands—all exist. Heaven and Earth are also one thing, and also have their numbers. A sparrow is a thing of three years; a horse is a thing of thirty years. All flying and running things can be deduced by number. Man is a thing of one hundred twenty years. The numbers of the Changes exhaust the beginning and end of Heaven and Earth. Though Heaven and Earth are great, they are still shape and vessel—are indeed two things. Three August Ones, Five Emperors, Three Kings, Five Hegemons.
Chapter 10 of the Totality of Sound, Tone, Resonance, and the Intercommunication of All Things: The proper pitches and laws in number reach seven and then stop by day of summer solstice begins in Yin and ends in Xu; during the periods of Hai-Zi-Chou, the sun has entered the ground and the eye can see nothing—these three numbers do not operate—morally comparable to three seasons. The numbers of biological generation are likewise: it is not that numbers do not operate, but that numbers exist without being seen. Yin and yang arise and separate into two Ding (which form the Four Images — also denoting two opposing factors); Two Ding interpenetrate and produce the Four Images; the Four Images interpenetrate and produce the Eight Trigrams; the Eight Trigrams interpenetrate and produce the myriad things. Two Ding generate the categories of Heaven and Earth; the Four Images settle the corpus of Heaven and Earth; the Four Images generate the categories of sun and moon; the Eight Trigrams settle the corpus of sun and moon; the Eight Trigrams generate the categories of the myriad creatures; the doubled hexagrams settle the corpus of the myriad creatures. Categories are the order of generation; corpus is the intermediation of images. Inferring categories must root in generation; observing the corpus must proceed by way of images. Sun and moon are of the same class—same root while differing places, differing places yet same image. Inferring from this forward, how could any creature escape? Heaven has four seasons, Earth has four directions, man has four limbs; hence knuckles reveal Heaven, and palmwrinkles reveal Earth. Each of the four limbs of man holds vessels: one set with three sections, each section with three indicators—that corresponds to the numbers of Heaven. Where Heaven and Earth operate in harmony, the viscera are paired—four Sang proper are the Heavenly nature, four Fu are the Earthly passages. Every animate object yields the progressive sequence of Tai Ji, Two Ding, Four Images, and Eight Trigrams, along with a likewise archaic current-peril pattern. Dao and One are the grand alternative names endowed to spirit (singular divine spontaneity per Theophany). The ultimate speech names "treating the numen (spiritual agency) as the numospic force-spirit (qua Theonomous self—evoking imminently sovereign spiritual virile efficiency)" .
Chapter 11 of Doubts and Rejections: Qian forty-eight, Li, forty; Xun twenty-four; Zhen ten; Kun twelve; Gen twenty, and Kan thirty-six. If functions issued through the "two" (Dui) follows to complete the "two-line formation," the (governing) function applies double. One officiating over two, generate thus-with it a three generation: three discount the first component which is indeed—wait: The original Chinese here says: 乾坤之道数 — "Qian, via general, x rules."
"Qian Shu: having a task commission: 一役二以生三，三去其一则二 : three-and-subtract-first fraction => - yielding"
Shao Yong repeatedly emphasized: "The numbers of the world arise from principle; to violate principle is to fall into technique. The common run of people enter into technique through numbers, and therefore do not enter into principle." Numbers are not the end; principle is the root. Numbers are the "trap and snare" (tools for catching fish and rabbits), while principle is the "fish and rabbit" themselves. The ultimate aim of this mathematical framework is not divination technique, but rather, through the structure of numbers, to apprehend the order of generation and the laws of waxing and waning of all things under heaven and earth.
The Generative Logic of "Yang Unifying Yin"
"All is one qi; in its generative phase it is yang, in its dissipative phase it is yin. Thus the two are but one, the four are but two, the six are but three, and the eight are but four." Yin and yang are not two equal and opposite entities, but rather two phases—"generation" and "dissipation"—of a single qi. This determines the hierarchical order of "speaking of heaven but not earth, speaking of the ruler but not the minister"—yet at the same time it is emphasized that "heaven obtains earth and the myriad things are born; the ruler obtains the minister and the ten thousand transformations proceed." Substance and function depend on each other; neither can be dispensed with.
The "Four in Substance, Three in Function" and Modern Systems Redundancy Theory
The model of "substance-number four, function-number three" deeply corresponds to the concept of "redundancy design" in contemporary systems science. Any stable system must retain a reserve that does not "directly participate in routine operations"—just as in the operation of the heavenly way, "one season has four months" but only three are actively employed; that "one month not employed" is not wasteful, but rather a structural space for regulating the waxing and waning of yin and yang and absorbing "surplus fractions." In personal time management, this means one cannot devote all energy to "visible output" — space must be reserved for "invisible recovery and reserve." In corporate strategy, this means not all resources should be invested in existing operations; a "one not employed" must be preserved for transformation and innovation.
"Numbers at Their Extreme Reverse" and Nonlinear Inflection Thinking
"Nine is the extreme number of yang; when numbers reach their extreme, they reverse—hence this is called change." In the modern context, this corresponds to "extreme thinking" and "inflection-point awareness"—any trend, when developed to its extreme, will reverse. In investment decisions, this means vigilance against signals of "overheating"; in life planning, this means anticipating the node of "peak then decline." Shao Yong uses "yang governs advance" to explain heavenly transformations and "yin governs retreat" to explain earthly transformations, actually providing an ancient template for thinking about "pro-cyclical" and "counter-cyclical" dynamics.
"Sincerity" and "Cunning Calculation" in the Modern Game
The assertion in "Heart-Mind Learning, Chapter Twelve" that "those who engage in falsehood and employ calculation may deceive people for a time, but in the long run they are certain to fail" resonates with the conclusion in contemporary game theory that "under repeated games, sincerity is the optimal strategy." "Cunning calculations may succeed for a single morning, but they are eventually exhausted"—short-term deception may work, but in a modern society of increasing information transparency, the accumulation of long-term trust capital is the sustainable survival strategy. When Shao Yong links "authenticity" to "coexisting with heaven and earth," he is in effect elevating "sincerity" from the moral realm to the height of cosmology.
"The Milfoil Virtue Is Round; the Hexagram Virtue Is Square" and the Two-Track Complementarity of Thought Modes
The milfoil method honors the round—flexible, adaptive, preserving the One while employing the forty-nine; the hexagram method honors the square—structural, determinate, sixty-four by sixty-four. This suggests a composite cognitive framework: in decision-making, one needs both the "round" capacity for adaptation (withholding judgment on some information) and the "square" capacity for structural analysis (incorporating complex issues into fixed models). Only round without square degenerates into the mystical; only square without round loses flexibility.
① The "Four in Substance, Three in Function" Principle in Time Management and Energy Allocation
Applying "substance-number four, function-number three" to modern work arrangements: of every four units of time, three are used for direct output and one is reserved as a "structural margin." Concretely, in an annual plan, of the twelve months, nine can be arranged as "main assault periods," retaining three months as "stockpiling periods"—for learning, review, and reorientation. This is not wasting time, but rather simulating the regulatory function of heavenly "the one unused," greatly enhancing the sustainability of the entire system.
② The "Reserve the One" Strategy in Negotiation and Decision-Making
In the milfoil method, "of fifty, one is preserved and forty-nine are employed," which in game strategy means: always retain some cards in reserve; do not expose everything. This strategic reservation of "preserving the one" is the ancient prototype of "retaining negotiation leeway" in modern bargaining. Meanwhile, "suspend one to symbolize the three" suggests: before making a decision, first "suspend" a frame of reference (such as a third-party perspective, historical cases, or long-term goals) to calibrate the present choice.
③ The Long-Term Rhythm of "Yang Advances, Yin Retreats" in Long-Termism
Small cycles "advance one day every 3,600 years," great cycles "in six divisions open six minutes and advance 360 days"—these macro-scale temporal models transform in the modern context into a sense of "rhythm": any systematic change requires time for accumulation and cannot be rushed. In personal growth, this can be understood as a "compounding mentality"—progressing a little each day becomes, over the long term, a magnitude shift of "advancing six days." At the same time, "yang numbers surfeit at 360, yin numbers shrink at 360" suggests: where there is surplus, there must be contraction. When setting annual goals, one should simultaneously consider the alternation of expansion and contraction periods.
④ "Removing the Substance and Taking the Function" and "Removing the Intersecting and Taking the Function" in Data Analysis
"Of the numbers of the sixteen transformations, remove the intersecting numbers and take the functional numbers"—in his extensive numerical deductions, Shao Yong repeatedly employs "removing certain elements" to obtain "usable numbers." This is precisely an ancient version of "feature engineering" in modern data analysis—not all data has predictive value; the key is to identify and eliminate "intersecting numbers" (interference terms) while preserving "functional numbers" (effective features). In business analysis, only by removing noise variables (such as the "leap-surplus" portion in seasonal fluctuations) and extracting the core trend line can accurate judgments be made.
The Existential Implication of "the One Without Substance"
"Heaven transforms the One into four; earth transforms the One into four. The four have substance, but the One has no substance—this is called the extreme of being and non-being." This "One without substance" is not simply "non-being," but the "extreme of being and non-being"—it is that which makes "being" possible as such. This echoes Heidegger's distinction between "Being" and "beings" across time and space: Being is not one being among beings, yet it enables all beings to manifest. By positioning the "One without substance" as "naturalness" and the "One not employed" as "the Way," Shao Yong in effect constructs a two-tiered metaphysical structure: at the base is "the Way" (absolute transcendence); above it is "naturalness" (the principle of differentiation of the One without substance); and above that is "heaven, earth, and humanity" (the three functional numbers).
The Reverse Recurrence of "All Things Return to the Way"
"In the end, the myriad things return to earth; earth returns to heaven; heaven returns to the Way—this is why the gentleman values the Way." This reveals a cyclical cosmology different from the Western linear-progressive view: generation is a process of differentiation from "Way → heaven → earth → the myriad things," while culmination is a process of return from "the myriad things → earth → heaven → the Way." This is a "spiral return" rather than a "simple cycle"—the return after differentiation and unfolding is a unity at a higher level. This thought can be regarded as the symbolic-numerical expression of the Daoist "returning to the root" and the Confucian "exhausting principle, fully realizing nature, and thereby arriving at destiny."
The Metamathematical Insight of "One Is Not a Number, Yet Numbers Are Completed Through It"
"One is not a number; it is not a number, yet numbers are completed through it."—Here Shao Yong touches upon a profoundly deep issue in the philosophy of mathematics: the starting point of a number system itself does not reside within the number system. This brings to mind the philosophical implications of Gödel's incompleteness theorem: the foundation of any formal system must be established beyond the system itself. When Shao Yong regards "One" as the "non-numerical number" and "the Way" as the "non-substantial substance," he is in effect expressing, in the language of Chinese philosophy, the dilemma of "self-reference" and "transcendent foundation."
The Hierarchical Ontology of "Spirit–Qi–Substance"
"Qi is the abode of spirit. Substance is the abode of qi." Spirit dwells within qi; qi dwells within substance. Spirit is invisible while qi is visible; qi cannot be measured, while substance can be measured. This constitutes a generative chain from "pure potentiality" (spirit) to "dynamic principle" (qi) to "material manifestation" (substance). In modern terms, "spirit" can be understood as "information" or "formal cause," "qi" as "energy" or "efficient cause," and "substance" as "matter" or "material cause." The relationship among the three suggests: information determines energy configuration, energy determines material form, and matter in turn provides a carrier for energy and a manifestation for information.
The Nested Structure of "Within Yin and Yang Each Contains Heaven, Earth, and Humanity"
"Within yin and yang, each contains heaven, earth, and humanity; within heaven, earth, and humanity, each is yin and yang." This is a typical "holographic" thought—every level contains all levels, like the self-similarity of fractal geometry. This mode of thinking breaks the notion that "levels are linearly stacked," replacing it with a complex systems view in which "levels are nested and mutually containing." For contemporary interdisciplinary research, this means: the "macro–meso–micro" relationship should not be simplistically understood as a hierarchy of superordination and subordination, but rather as each level internally containing structural information from all other levels.
| Concept | Textual Source | Core Meaning |
|---|---|---|
| The Great Expansion Number | "What is the Great Expansion of the Yi? It is number; the sage relies on numbers" | Fifty is the total for the milfoil method; forty-nine are employed, one is preserved and unused, symbolizing the retention of a transcendent dimension within the overall framework of cosmic operation |
| Threefold Heaven and Twofold Earth | "Threefold heaven is three threes making nine; twofold earth is doubling three making six" | Multiplying three by three uses heaven's threefold nature to obtain nine; doubling three uses earth's twofold nature to obtain six—these are the fundamental operations for constructing the ultimate numbers of nine and six for yin and yang |
| The Stalk-Number System | "Four twelves make forty-eight" | Each line's stalk number uses four as its unit; four times nine makes thirty-six corresponding to Qian, four times six makes twenty-four corresponding to Kun, constituting the line-level numerical system |
| Substance–Function Relation | "Substance-number is four; function-number is three" | Every structure (substance) has its redundant parts; actual operation (function) is always less than the structural total—this is the leitmotif running throughout the entire text |
| Former Heaven and Latter Heaven | "The passage on the positioning of heaven and earth illuminates Fu Xi's eight trigrams": "Kun commands the three daughters in the southwest; Qian commands the three sons in the northwest" | The Former Heaven sequence (Fu Xi), grounded in "the positioning of heaven and earth," and the Latter Heaven sequence (King Wen), grounded in "Qian and Kun yielding position and Kan and Li assuming function," constitute a "substance–function" relation |
| Yuan–Hui–Yun–Shi | "Using Yuan to govern the major and minor cycles of Yun in the seventh chapter" | Yuan, hui, yun, and shi are the macro-temporal units Shao Yong constructed: one yuan equals twelve hui, one hui equals thirty yun, one yun equals twelve shi, one shi equals thirty years, forming a complete cycle of 129,600 years |
| Jing Shi Na Jia | "In all cases, jiazi and jiawu serve as the beginning of a shi—this is the number of the Jing Shi" | Incorporating the heavenly stems and earthly branches into the chronological framework of the cosmic great cycle, with jiazi and jiawu alternating as the starting points of each "shi" era |
| Regular and Exchange Hexagrams | "The regular and exchanged hexagrams together number thirty-six" | Among the sixty-four hexagrams, eight pairs (such as Qian and Kun) do not change when inverted up-down; the remaining twenty-eight pairs are mutual "reverse-exchange" pairs; adding the eight regular hexagrams yields the thirty-six "regular/exchange" model |