The Supreme Principles Governing the World · The Extended Interpretations of the Chapter on Observing Things Beyond (attributed to Shao Yong, with extended interpretations by Zhang Xingcheng of the Song dynasty)

天數五，地數五，合而為十，數之全也。天以一而變四，地以一而變四，四者有體也，而其一者無體也，是謂有無之極也。天之體數四而用者三，不用者一也。地之體數四而用者三，不用者一也。是故無體之一以況自然也，不用之一以況道也，用之者三以況天地人也。

The numbers belonging to Heaven are five, and those belonging to Earth are also five. When combined, they make ten, which is the complete form of "number." Heaven begins from "one" and develops into four, and Earth also begins from "one" and develops into four. These four are tangible entities, while the "one" that gives rise to them is intangible and cannot be pointed to, thus it is called the extreme of "being" and "non-being." The "number of form" for Heaven is four, but only three are actually active, with one remaining unused. Similarly, the "number of form" for Earth is also four, with three being active and one remaining unused. Therefore, the intangible "one" is used to represent the natural state before Heaven and Earth were separated. The unused "one" is used to represent the "Dao" that generates all things without personally acting. The actual three that are in use are used to represent the three principles of Heaven, Earth, and Humanity.

體者八變，用者六變，是以八卦之象不易者四，反易者二，以六卦變而成八也。重卦之象不易者八，反易者二十八，以三十六變而成六十四也。故爻止於六，卦盡於八，策窮於三十六，而重卦極於六十四也。卦成於八，重於六十四，爻成於六，策窮於三十六而重於三百八十四也。

The system of the "body" undergoes eight transformations, while the system of the "use" undergoes six transformations. Therefore, among the eight trigrams, four remain unchanged when viewed upside down, namely Qian (Heaven), Kun (Earth), Kan (Water), and Li (Fire). Two pairs of trigrams become each other when reversed, namely Zhen (Thunder) and Gen (Mountain), Xun (Wind) and Dui (Lake). Altogether, these six trigrams generate the eight trigrams through transformation. The sixty-four hexagrams follow the same principle: eight remain unchanged when viewed upside down, and twenty-eight pairs become each other when reversed. Altogether, these thirty-six hexagrams generate the sixty-four hexagrams. Therefore, the number of lines (yao) does not exceed six, and the number of single trigrams does not exceed eight. The number of "counting units" (ce shu) for a single line does not exceed thirty-six, and the number of hexagrams does not exceed sixty-four. A single trigram is composed of three lines to form eight trigrams, and when combined, they form sixty-four trigrams. A hexagram consists of six lines, and the maximum number of "counting units" for a single line is thirty-six. Since each of the sixty-four hexagrams has six lines, the total number of lines is three hundred and eighty-four.

天有四時，一時四月，一月四十日。四四十六而各去其一，是以一時三月，一月三十日也。四時體數也，三月三十日用數也。體雖具四而其一常不用也，故用者止於三而極於九也。體數常偶，故有四有十二。用數常奇，故有三有九。

If calculated according to "body numbers," a year has four seasons, each season should have four months, and each month should have forty days. Four multiplied by four gives sixteen, but each layer must subtract one portion, so in reality each season has only three months, and each month has only thirty days. Four seasons, four months, and forty days belong to "body numbers," while three months and thirty days belong to "used numbers." Although body numbers are complete with four parts, one part is always left unused, so only three parts are actually used, leading to nine when multiplied to the extreme. Body numbers are always even, hence four and twelve; used numbers are always odd, hence three and nine.

大數不足而小數常盈者何也？以其大者不可見而小者可見也。故時止乎四，月止乎三，而日盈乎十也。是以人之肢體有四而指有十也。天見乎南而潛乎北，極於六而餘於七，是以人知其前，昧其後，而略其左右也。

Why larger units often seem insufficient while smaller ones frequently appear abundant? Because larger units are distant and hard to perceive, while smaller units are clear and countable. Thus, "time" only reaches four, "month" only reaches three, yet "day" fills up to ten. The human body follows this principle: there are only four limbs, yet ten fingers. Celestial bodies appear in the south and disappear in the north; the boundary between day and night reaches six parts but slightly exceeds, becoming seven parts. Therefore, people can only see clearly in front of them, cannot see behind, and can only roughly attend to the sides.

天體數四而用三，地體數四而用三，天克地地克天而克者在地，猶晝之餘分在地也，是以天三而地四，天有三辰，地有四行也。然地之火且見且隱，其餘分之謂耶？

Heaven's fundamental number is four but is used as three, while Earth's fundamental number is four and is also used as three. Heaven controls Earth, and Earth controls Heaven; they mutually control each other. The result of being controlled falls upon the Earth, just as the extra part of daylight is accounted for on the ground. Therefore, Heaven uses three and Earth uses four: Heaven has three luminous celestial bodies, the sun, moon, and stars, while Earth has four material elements, water, fire, soil, and stone. However, fire on Earth appears sometimes and disappears sometimes, unlike water, soil, and stone which are always present. Perhaps fire is that extra portion?

乾七子，兌六子，離五子，震四子，巽三子，坎二子，艮一子，坤至陰，故無子。乾七子，坤六子，兌五子，艮四子，離三子，坎二子，震一子，巽剛，故無子。

According to the order of the trigrams in theEarlier Heaven (Heavenly Origin) arrangement, each trigram produces a number of "sons" or descendants: Qian has seven sons, Dui has six, Li has five, Zhen has four, Xun has three, Kan has two, Gen has one; Kun, being the extreme of yin, has no sons. Another arrangement states: Qian has seven sons, Kun has six, Dui has five, Gen has four, Li has three, Kan has two, Zhen has one; Xun, being of a yang nature, has no sons. The first view is based on the principle that pure yin cannot generate independently, while the second view is based on the principle that pure yang cannot generate independently. Each interpretation takes a different perspective.

天有二正，地有二正，而共享二變以成八卦也。天有四正，地有四正，共享二十八變以成六十四卦也。是以小成之卦正者四，變者二，共六卦也。大成之卦正者八，變者二十八，共三十六卦也。乾坤坎離為三十六卦之祖，兌震巽艮為二十八卦之祖也。

Heaven has two unchanging trigrams, which are Qian and Li; Earth also has two unchanging trigrams, which are Kun and Kan. In addition, there are two pairs of trigrams that are reversed and combined, making up the Eight Trigrams. When forming compound hexagrams, Heaven has four trigrams, Earth has four trigrams, and there are twenty-eight pairs of reversed trigrams, making up the sixty-four hexagrams. Therefore, the Eight Trigrams, referred to as the 'small completion,' consist of four original trigrams and two reversed ones, totaling six. The sixty-four hexagrams, referred to as the 'great completion,' consist of eight original trigrams and twenty-eight reversed ones, totaling thirty-six. Qian, Kun, Kan, and Li are the ancestors of the thirty-six hexagrams, while Dui, Zhen, Xun, and Gen are the ancestors of the twenty-eight hexagrams.

乾坤七變，是以晝夜之極不過七分也。艮兌六變，是以月止於六，共為十二也。離坎五變，是以日止於五，共為十也。震巽四變，是以體止於四，共為八也。

Heavenly Trigrams (Qian and Kun) each have seven changes, so the extremes of day and night are limited to seven parts. At the summer solstice, day is seven parts and night is three parts; at the winter solstice, it is reversed. The trigrams Gen and Dui each have six changes, so the number of months is limited to six. When yin and yang combine, they form twelve months. The trigrams Li and Kan each have five changes, so the number of days is limited to five. When yin and yang combine, they form ten days, which are the ten Heavenly Stems. The trigrams Zhen and Xun each have four changes, so the number of forms is limited to four. When yin and yang combine, they form eight, which are the Eight Trigrams.

卦之正變共三十六，而爻又有二百一十六，則用數之策也。三十六去四則三十二也，又去四則二十八也，又去四則二十四也，故卦數三十二位，去四而言之也。天數二十八位，去八而言之也。地數二十四位，去十二而言之也。四者乾坤坎離也，八者並頤中孚大小過也，十二者兌震泰既濟也。

The original hexagram and the changing hexagram together total thirty-six. These thirty-six hexagrams have two hundred and sixteen lines, which is exactly the "number used" or "counting method" mentioned as the count of the hexagram Qian (Heaven) being two hundred and sixteen in the Great Treatise. Subtracting four from thirty-six gives twenty-eight, then subtracting another four gives twenty-four, and subtracting another four gives twenty. Therefore, the statement "the number of hexagrams is thirty-two" refers to subtracting four from thirty-six; "the number of Heaven is twenty-eight" refers to subtracting eight; and "the number of Earth is twenty-four" refers to subtracting twelve. The four subtracted refer to the four trigrams Qian, Kun, Kan, and Li. The eight subtracted include these four trigrams plus Yi (Repose), Zhongfú (Faithful Son), Daguò (Great Overcoming), and Xiaoguo (Small Overcoming). The twelve subtracted include these eight trigrams plus Dui (Lake), Zhen (Thunder), Tai (Great Passage), and Ji Ji (Completion).

日有八位而用止於七，去乾而言之也。月有八位而用止於六，去兌而言之也。星有八位而用止於五，去離而言之也。辰有八位而用止於四去震而言之也。日有八位而數止於七，去泰而言之也。月自兌起者，月不能及日之數也。故十二月常餘十二日也。

The sun is paired with the positions of the eight trigrams, but only seven are used, excluding the trigram Qian; the moon is paired with eight positions, but only six are used, excluding the trigram Dui; stars are paired with eight positions, but only five are used, excluding the trigram Li; the hour is paired with eight positions, but only four are used, excluding the trigram Zhen; the sun again is paired with eight positions, but the count only reaches seven, excluding the trigram Tai. The moon's position starts from Dui, which clearly indicates that the moon's movement is slower than the sun's. Therefore, after twelve lunar months, there are often twelve days left over, which is the reason for the need to intercalate a month in the calendar.

乾陽中陽，不可變，故一年只舉十二月也。震陰中陰，不可變，故一日之十二時不可見也。兌陽中陰，離陰中陽，皆可變，故日月之數可分也。是以陰數以十二起，陽數以三十起，而常存二六也。舉年見月，舉月見日，舉日見時，陽統陰也。是天四變含地四變，日之變含月與星辰之變也。是以一卦含四卦也。

Heavenly Stem Jia is yang within yang, pure and unchangeable, so it is only mentioned once a year with twelve months. Zhen is yin within yin, pure and unchangeable, so the twelve hours in a day are invisible. Dui is yang within yin, and Li is yin within yang, both of which can change, so the numbers of days and months can be divided. Therefore, the count of yin starts from twelve, and the count of yang starts from thirty, while the numbers two and six remain constant. Mentioning "year" shows "month," mentioning "month" shows "day," and mentioning "day" shows "hour," which reflects the principle of yang governing yin. In other words, the four changes of heaven include the four changes of earth, and the changes of day include those of the moon and stars, so one hexagram contains four hexagrams.

日一位，月一位，星一位，辰一位。日有四位，月有四位，星有四位，辰有四位，四四十有六位，盡此一變而日月之數窮矣。天有四變，地有四變，變有長也，有消也，十有六變而天地之數窮矣。日起於一，月起於二，星起於三，辰起於四，引而伸之，陽數常六，陰數常二，十有二變而大小之運窮矣。

The sun occupies one position, the moon one, the stars one, and the celestial stems one; beneath the sun there are four positions, beneath the moon four, beneath the stars four, and beneath the celestial stems four. Four multiplied by four gives sixteen positions. After completing this cycle of changes, the numbers of the sun and moon are exhausted. Heaven has four changes, and earth has four changes. Among these changes, there is growth and there is decline. After sixteen changes, the numbers of heaven and earth are exhausted. The sun begins with one, the moon with two, the stars with three, and the celestial stems with four. Following this pattern, the yang numbers always result in six, and the yin numbers always result in two. After twelve changes, the numbers of the great cycle and the small cycle are exhausted.

三百六十變為十二萬九千六百，十二萬九千六百變為一百六十七億九千六百一十六萬，一百六十七億九千六百一十六萬變為二萬八千二百一十一兆九百九十萬七千四百五十六億。以三百六十為時，以十二萬九千六百為日，以一百六十七億九千六百一十六萬為月，以二萬八千二百一十一兆九百九十一萬七千四百五十六億為年，則大小運之數立矣。

Multiply 360 by itself once to get 12,9600; multiply 12,9600 by itself once to get 1,679,616,0000; multiply 1,679,616,0000 by itself once to get 28,211,990,7456,0000. After this progressive scaling, take 360 as one "shi," 12,9600 as one "ri," 1,679,616,0000 as one "yue," and 28,211,990,7456,0000 as one "nian." The numbers for "da yun" and "xiao yun" are thus established.

二萬八千二百一十一兆九百九十一萬七千四百五十六億分而為十二，前六限為長，後六限為消，以當一年十二月之數，而進退三百六十日矣。一百六十七億九千六百一十六萬分而為三十，以當一月三十日之數，隨大運消長而進退六十日矣。十二萬九千六百分而為十二，以當一日十二時之數，而進退六日矣。三百六十以當一時之數，隨小運之進退以當晝夜之時也。

Divide the number of years, two million eight hundred and twenty-one thousand one hundred and eleven trillion nine hundred and ninety-one billion seven hundred and forty-five million six hundred, into twelve parts, the first six parts representing growth and the last six parts representing decline, corresponding to the twelve months of a year, with a total of three hundred and sixty days of movement. Divide the number of months, one hundred and sixty-seven billion nine thousand six hundred and sixteen million, into thirty parts, corresponding to the thirty days of a month, with a total of sixty days of movement according to the great cycle. Divide the number of days, twelve million nine thousand six hundred, into twelve parts, corresponding to the twelve hours of a day, with a total of six days of movement. Then take three hundred and sixty as the number for an hour, corresponding to the moments within day and night according to the small cycle.

十六變之數去其交數，取其用數，得二萬八千二百一十一兆九百九十一萬七千四百五十六億，分為十二限，前六限為長，後六限為消，每限得十三億九千九百六十八萬之一百六十七億九千六百一十六萬。每一百六十七億九千六百一十六萬年開一分，進六十日也。六限開六分，進三百六十日也。猶有餘分之一，故開七分，進三百六十六日也。其退亦若是矣。

The total number obtained from sixteen changes should subtract the overlapping numbers called 「交数」 and only take the effective numbers called 「用数」, resulting in 28,211,991,745,600. Divide this into twelve periods, the first six periods belong to growth and the last six to decline. Each period corresponds to 1,399,680,000,000,000. Every 167,961,600,000 years, one part is opened, advancing sixty days. After six periods, six parts are opened, advancing three hundred and sixty days. There is also a remaining part, so it is opened to seven parts in total, advancing three hundred and sixty-six days. The declining half is also calculated in the same way but in reverse.

十二萬九千六百，去其三者，交數也，取其七者，用數也。用數三而成於六，加餘分故有七也。七之得九萬七百二十年，半之得四萬五千三百六十年，以進六日也。日有晝夜，數有誂朒，以成十有二日也。每三千六百年進一日，凡四萬三千二百年進十有二日也。二千一百六十年以進餘分之六，合交數之二千一百六十年，共進十有二分以為閏也。故小運之變凡六十，而成三百六十有六日也。

This number, twelve thousand nine hundred and sixty, if we remove three tenths of it, results in the 'interleaving number' or 'cross number'; if we take seven tenths of it, that is the 'used number'. The used number is based on three and formed into six, then adding a little remainder, making it seven. Taking seven tenths gives nine thousand and seventy-two years, taking half of that gives four thousand five hundred and sixty years, which is used to advance six days. Days have two aspects, day and night, and numbers have two tendencies, surplus and deficiency. Together they form twelve days. Every three thousand six hundred years advances one day, so four thousand three hundred and twenty years advance twelve days. Another method uses two thousand one hundred and sixty years to advance the remainder of six, and adding the two thousand one hundred and sixty years occupied by the cross number, totaling twelve parts, which is used to accommodate the intercalary remainder. Therefore, the changes of the small cycle total sixty times, exactly forming three hundred and sixty-six days.

六者三天也，四者兩地也。天統乎體而托地以為體，地分乎用而承天以為用。天地相依，體用相附。

"Six" comes from "three" multiplied by one and added to itself, and "four" comes from "two" multiplied by one and added to itself. This is the "method of doubling" referred to in the study of theEarlier Heaven (primordial) principles. Heaven encompasses the "form," but it needs to be embodied on earth to manifest its shape; Earth takes charge of the "function," but it needs to receive the movement of Heaven to have function. Heaven and Earth rely on each other, and form and function depend on each other. Without one side, neither can stand alone.

乾為一，乾之五爻分而為大有，以當三百六十之數也。乾之四爻分而為小畜，以當十二萬九千六百之數也。乾之三爻分而為履，以當一百六十七億九千六百一十六萬之數也。乾之二爻分而為同人，以當二萬八千二百一十一兆九百九十萬七千四百五十六億之數也。乾之初爻分而為姤，以當七秭九千五百八十六萬六千一百十垓九千九百四十六萬四千八京八千四百三十九萬一千九百三十六兆之數也。是謂分數也。分大為小，皆自上而下，故以陽數當之。

In the Earlier Heaven sequence, the Trigram Qian (Heaven) is first. The fifth line of Qian changes to produce the hexagram Dali (Great Possession), corresponding to the number 360; the fourth line changes to produce Xiaoxu (Small Accumulation), corresponding to 129,600; the third line changes to produce Lu (Footwear), corresponding to 167,961,600; the second line changes to produce Tongren (Harmonious People), corresponding to 28,211,990,745,600; and the first line changes to produce Gu (Meeting), corresponding to 7,958,661,109,946,480,843,919,360. This is called 'fraction': dividing the large into the small, all derived from top to bottom, hence using the yang numbers to match it.

一生二為夬，當十二之數也。二生四為大壯，當四千三百二十之數也。四生八為泰，當五億五千九百八十七萬二千之數也。八生十六為臨，當九百四十四兆三千六百九十九萬六千九百一十五億二千之數也。十六生三十二為復，當二千六百五十二萬八千八百七十垓三千六百六十四萬八千八百京二千九百四十七萬九千七百三十一兆二千萬億之數也。三十二生六十四為坤，當無極之數也。是謂長數也，長小為大，皆自下而上，故以陰數當之。

From the perspective of the Earth (Kun), it is the method of doubling each time: one becomes two, forming Gua, corresponding to the number twelve; two becomes four, forming Da Zhuang, corresponding to the number four thousand three hundred twenty; four becomes eight, forming Tai, corresponding to the number five hundred fifty-nine million eight hundred seventy-two thousand; eight becomes sixteen, forming Lin, corresponding to the number nine hundred forty-four trillion three thousand six hundred ninety-nine million six hundred ninety-one thousand five hundred twenty; sixteen becomes thirty-two, forming Fu, corresponding to the number twenty-six million five hundred twenty-eight thousand eight hundred seventy myriads three thousand six hundred sixty-four million eight hundred eighty thousand two thousand nine hundred forty-seven million nine hundred seventy-three thousand one hundred twenty; thirty-two becomes sixty-four, forming Kun again, corresponding to an endless number of the Supreme Ultimate. This process is called the "long number": it grows the small into the large, proceeding from the lower to the upper, so it is paired with the yin numbers.

天統乎體，故八變而終於十六。地分乎用，故六變而終於十二。天起於一，而終於七秭九千五百八十六萬六千一百一十垓九千九百四十六萬四千八京八千四百三十九萬一千九百三十六兆。地起於十二，而終於二百四垓六千九百八十萬七千三百八十一京五千四百九十一萬八千四百九十九兆七百二十萬億也。

Heaven encompasses the "body," so it undergoes eight transformations and ends at sixteen; Earth governs the "function," so it undergoes six transformations and ends at twelve. The Heaven sequence starts from one and continues up to seven thousand nine hundred eighty-six million six hundred eleven thousand nine hundred forty-six thousand eight hundred forty-three million nine hundred thirty-six trillion. The Earth sequence starts from twelve and continues up to two hundred forty gai six thousand nine hundred eighty million seven hundred thirty-eight thousand one jing five thousand four hundred ninety-one million eight thousand four hundred ninety-nine trillion seven hundred twenty trillion.

有地然後有二，有二然後有晝夜。二三以變，錯綜而成，故易以二而生數，以十二而起，而一非數也，非數而數以之成也。天行不息，未嘗有晝夜，人居地上以為晝夜，故以地上之數為人之用也。

Heavenly Stems and Earthly Branches are the fundamental elements used in divination. The concept of "two" arises from the Earth, and with "two" comes the distinction between day and night. The interaction between "two" and "three" leads to the myriad phenomena, hence the Book of Changes begins counting from "two" and establishes positions based on twelve. The number "one" itself is not counted, yet all numbers depend on it for their existence. The heavens move ceaselessly without inherent day and night; humans on Earth observe the rising and setting of the sun and thus establish the names of day and night. Therefore, the calendar system uses numbers based on what is observed on Earth for practical human use.

天自臨以上，地自師以上，運數也。天自同人以下，地自遯以下，年數也。運數則在天者也，年數則在地者也。天自賁以上，地自艮以上，用數也。天自明夷以下，地自否以下，交數也。天自震以上，地自晉以上，有數也。天自益以下，地自豫以下，無數也。天之有數起乾而止震，餘入於無者，天辰不見也。地去一而起十二者，地火常潛也，故天以體為基而常隱其基，地以用為本而常藏其用也。

In the diagram of the primordial trigrams, the heavenly side starts from the trigram of Qian above the trigram of Lin, and the earthly side starts from the trigram of Shi above the trigram of Shi, which belongs to 'movement and number'. The heavenly side from Tongren downwards and the earthly side from Duan downwards belong to 'yearly number'. Movement and number are on the heavenly side, while yearly number is on the earthly side. The heavenly side from Ben upwards and the earthly side from Gen upwards belong to 'used number'; the heavenly side from Mingyi downwards and the earthly side from Pinyin downwards belong to 'intersecting number'. The heavenly side from Zhen upwards and the earthly side from Jin upwards belong to 'existing number'; the heavenly side from Yi downwards and the earthly side from Yu downwards belong to 'non-number'. The existing number of heaven starts from Qian and ends at Zhen, while the rest fall into the 'non' side, because the heavenly stars are dark and invisible. The numbers of the earth do not start from one but from twelve, because the fire in the earth is often hidden. Therefore, heaven takes 'body' as its foundation but often hides its foundation, while the earth takes 'use' as its fundamental aspect but often conceals its function.

一時止於三月，一月止於三十日，皆去其辰數也。是以八八之卦六十四而不變者八，可變者七。七八五十六，其義亦由此矣。陽爻晝數也，陰爻夜數也，天地相銜，陰陽相交，故晝夜相雜，剛柔相錯。春夏陽多也，故晝數多夜數少，秋冬陰多也，故晝數少夜數多。

At one time, only three months were calculated, and each month was considered to have thirty days, with the portion belonging to 'Chen' removed. Therefore, among the sixty-four hexagrams formed by the eight trigrams overlapping, there are eight that remain unchanged when reversed, while only seven types can change. Seven multiplied by eight gives fifty-six, and the reasoning comes from this. A yang line represents the number of days, while a yin line represents the number of nights. Heaven and Earth connect with each other, and yin and yang intermingle, so day and night are mixed together, and the qualities of yang and yin alternate. In spring and summer, yang energy is more prevalent, so the number of days is greater and the number of nights is less; in autumn and winter, yin energy is more prevalent, so the number of days is less and the number of nights is greater.

體數之策三百八十四，去乾坤坎離之策為用數三百六十。體數之用二百七十，去乾與坎離之策為用數之用二百五十二也。體數之用二百七十，其一百五十六為陽，一百十四為陰，去離之策，得一百五十二陽，一百一十二陰，為實用之數也。

The total number of counts in "ti shu" is 384. After subtracting the 24 counts of the four trigrams Qian, Kun, Kan, and Li, the remaining 360 counts are called "yong shu". Among the counts used in "ti shu", actually 270 are in use. After subtracting the counts of Qian and Kan and Li, the remaining 252 are called "yong shu of yong". Within the 270 counts used in "ti shu", 156 belong to yang and 114 belong to yin. After removing the counts of Li, we get 152 yang and 112 yin, which are the actual counts used.

蓋陽去離而用乾，陰去坤而用坎也。是以天之陽策一百十二，去其陰也。地之陰策一百十二，陽策四十，去其南北之陽也。極南大暑，極北大寒，物不能生，是以去之也。其四十為天之餘分耶？陽侵陰，晝侵夜，是以在地也。合之為一百五十二陽，一百十二陰也。陽去乾之策，陰去坎之策，得一百四十四陽，一百八陰，為用數之用也。

This is because on the yang side, the trigram Li (Fire) is excluded and only Qian (Heaven) is used, while on the yin side, the trigram Kun (Earth) is excluded and only Kan (Water) is used. Therefore, the yang count of Heaven is 112, which excludes the yin; the yin count of Earth is also 112, while the yang count is 40, which excludes the yang at the northern and southern extremes. The southern extreme is extremely hot, and the northern extreme is extremely cold, where all things cannot grow, so they are excluded. The remaining 40, perhaps, is the leftover yang of Heaven. When yang enters yin and day enters night, this portion falls to the earth. Combined, it remains 152 yang and 112 yin. If we further exclude the trigram Qian from the yang side and the trigram Kan from the yin side, we get 144 yang and 108 yin, which is called "the use of the numbers."

陽三十六，三之為一百八。陰三十六三之為一百八。三陽三陰，陰陽各半也。陽有餘分之一為三十六，合之為一百四十四陽，一百八陰也，故體數之用二百七十，而實用者二百六十四，用數之用二百五十二也。

The number of a yang line is thirty-six, and three times that makes one hundred and eight. A yin line also takes thirty-six, and three times that also makes one hundred and eight. Three yang lines and three yin lines are paired together, so yin and yang each take up half. There is an additional thirty-six for yang, which when combined makes one hundred and forty-four yang and one hundred and eight yin. Therefore, the total number of the body is two hundred and seventy, but the actual number used is two hundred and sixty-four. The number of the use is two hundred and fifty-two.

卦有六十四而用止乎三十六，爻有三百八十四而用止乎二百一十六也。六十四分而為二百五十六，是以一卦去其初上之爻亦二百五十六也，此生物之數也，故離坎為生物之主，以離四陽，坎四陰，故生物者必四也。陽一百一十二，陰一百一十二，去其離坎之爻則二百一十六也，陰陽之四十共為二百五十六也。是以八卦用六爻，乾坤主之也。六爻用四位，離坎主之也。故天之昏曉不生物而日中生物，地之南北不生物而中央生物也。

There are sixty-four hexagrams but only thirty-six are actually used. There are three hundred and eighty-four lines but only two hundred and sixteen are actually used. Each of the sixty-four hexagrams is divided into four parts, resulting in a total of two hundred and fifty-six. Each hexagram has six lines, but the first and sixth lines are excluded, leaving only the middle four lines. Thus, the sixty-four hexagrams also result in two hundred and fifty-six. This is called the 'number of generation'. Therefore, the trigrams Li (Fire) and Kan (Water) are the masters of generation. Li trigram has four yang lines when stacked, and Kan trigram has four yin lines when stacked, showing that the number of generation must be based on four. There are one hundred and twelve yang lines and one hundred and twelve yin lines. Removing the lines of the trigrams Li and Kan leaves two hundred and sixteen. Adding the remaining forty yin and forty yang lines respectively brings the total to two hundred and fifty-six. Therefore, the eight trigrams must use all six lines, with Qian (Heaven) and Kun (Earth) as the main ones. When using only the middle four lines, Li and Kan are the main ones. Hence, Heaven does not generate life during the early morning and evening but does so at noon, and Earth does not generate life at the north and south ends but does so at the center.

體數何為者也？生物者也。用數何為者也？運行者也，運行者天也，生物者地也。天以獨運，故以用數自相乘，而以用數之用為生物之時也。地偶而生，故以體數之用陽乘陰，為生物之數也。

"The body number" is used for nurturing all things. "The functional number" is used for continuous movement. The aspect of movement belongs to heaven, and the aspect of generating life belongs to earth. Heaven moves by itself, so the functional number is multiplied by itself, and the "function of the functional number" is used to determine the timing of generation. Earth needs to pair and mate to produce offspring, so the yang number from the function of the body number is multiplied by the yin number to determine the quantity of living things.

天數三，故六六而又六之，是以乾之策二百一十有六。地數兩，故十二而十二之，是以坤之策百四十有四也。乾用九，故三其八為二十四，而九之亦二百一十六，兩其八為十六，而九之亦百四十有四。坤用六，故三其十二為三十六，而六之亦二百一十六，兩其十二為二十四，而六之亦百四十有四也。坤以十二之三，十六之四，六之一與半為乾之餘分，則乾得二百五十二，坤得一百八也。陽四卦，十二爻八陽四陰。以三十六乘其陽，以二十四乘其陰，則三百八十四也。

Heaven's number is three, so six multiplied by six and three gives thirty-six, which multiplied by six results in two hundred and sixteen for the trigram Qian. Earth's number is two, so twelve multiplied by twelve gives one hundred and forty-four for the trigram Kun. For Qian using all nines, three eights make twenty-four, which multiplied by nine gives two hundred and sixteen; two eights make sixteen, which multiplied by nine gives one hundred and forty-four. For Kun using all sixes, three twelves make thirty-six, which multiplied by six gives two hundred and sixteen; two twelves make twenty-four, which multiplied by six gives one hundred and forty-four. Then, Kun gives up three from twelve, four from sixteen, and half from six to supplement Qian's share, so Qian gets two hundred and fifty-two and Kun gets one hundred and eight. The four trigrams of yang have twelve lines, with eight yang lines and four yin lines. Multiplying thirty-six by the yang lines and twenty-four by the yin lines together gives exactly three hundred and eighty-four.

體有三百八十四而用止於三百六十何也？以乾坤坎離之不用也。乾坤坎離之不用所以成三百六十之用也。故萬物變易而四者不變也，夫惟不變，是以能變也。用止於三百六十而有三百六十六何也？數之贏也。數之贏則何用也？乾之全用也。乾坤不用則離坎用半也。乾全用者何也？陽主贏也。乾坤不用者何也？獨陽不生，專陰不成也。離坎用半何也？離東坎西，當陰陽之半，為春秋晝夜之門也。或用乾或用離坎何也？主陽而言之，故用乾也。主贏分而言之，則陽侵陰晝侵夜，故用離坎也。陽主贏，故乾全用也。陰主虛，故坤全不用也。陽侵陰陰侵陽，故坎離用半也。是以天之南全見而北全不見，東西各半見也。離坎陰陽之限也，故離當寅，坎當申，而數常踰之者，蓋陰陽之溢也。然用數不過乎寅，交數不過乎申。

The fundamental numbers amount to 384, but the numbers actually used only reach 360. Why is this so? Because the four trigrams Qian, Kun, Kan, and Li are not used in practice. It is precisely because these four trigrams are not used that the numbers used amount to 360. Therefore, although all things change ceaselessly, these four remain constant. Only by being constant can other changes be established. The numbers used amount to 360, yet there is also 366. What is the extra number? It is the surplus of numbers. What is the use of the surplus? It is the full use of Qian. Since Qian and Kun are not used, Kan and Li are only used half. Why is Qian considered to be fully used? Because yang is associated with surplus. Why are Qian and Kun not used? Because isolated yang cannot produce, and pure yin cannot form. Why is Kan and Li only used half? Because Li is located in the east and Kan in the west, exactly at the midpoint of yin and yang, the gate of spring equinox and autumn equinox, and the boundary of day and night. Why sometimes Qian is said to be used and sometimes Kan and Li? When considering yang alone, Qian is used; when considering the surplus part, yang intrudes into yin, day into night, so Kan and Li are used. Yang is associated with surplus, so Qian is fully used; yin is associated with depletion, so Kun is fully not used; yang intrudes into yin and yin also intrudes into yang, so Kan and Li are each used half. Therefore, the southern part of heaven is entirely visible, the northern part entirely invisible, while the eastern and western parts are each visible half. Kan and Li are the boundaries of yin and yang, so Li is at the position of寅 and Kan at the position of申. However, actual numbers often cross these boundaries, which is due to the overflow of yin and yang. Nevertheless, the numbers used never exceed the position of寅, and the numbers of transition never exceed the position of申.

乾四十八，而四分之一分為陰所克。坤四十八，而四分之一分為所克之陽也。故乾得三十六而坤得十二也。陽主進，是以進之為三百六十日。陰主消，是以十二月消十二日也。順數之，乾一兌二離三震四巽五坎六艮七坤八。逆數之，震一，離兌二，乾三，巽四，坎艮五，坤六也。

The number of measures for Qian is forty-eight, with a quarter of it being overcome by yin; the number of measures for Kun is also forty-eight, with a quarter of it being the part that is overcome by yang. Therefore, Qian actually has thirty-six, and Kun actually has twelve. Yang represents advancement, so it progresses to form three hundred and sixty days; yin represents retreat, so twelve days are subtracted over the course of twelve months. Counting the Earlier Heaven trigrams in order, they are Qian one, Dui two, Li three, Zhen four, Xun five, Kan six, Gen seven, Kun eight; counting them in reverse order, Zhen is one, Li and Dui are both two, Qian is three, Xun is four, Kan and Gen are both five, and Kun is six.

乾四十八，兌三十，離二十四，震十。坤十二，艮二十，坎三十六，巽四十。乾三十六，坤十二，離兌巽二十八，坎艮震二十。圓數有一，方數有二，奇偶之義也。六即一也，十二即二也。

Each trigram is associated with a certain number of策 (counting units): Qian (Heaven) has 48, Dui (Lake) has 30, Li (Fire) has 24, Zhen (Thunder) has 10; Kun (Earth) has 12, Gen (Mountain) has 20, Kan (Water) has 36, Xun (Wind) has 40. If we consider only the yang numbers, Qian has 36, Kun has 12, Li, Dui, and Xun each have 28, while Kan, Gen, and Zhen each have 20. The circular numbers take one as their fundamental basis, while the square numbers take two as their fundamental basis; this is the distinction between odd and even. Six is merely derived from one, and twelve is merely derived from two.

天圓而地方，圓者數之起一而積六，方者數之起一而積八，變之則起四而積十二也。六者常以六變，八者常以八變，而十二者亦以八變，自然之道也。八者天地之體也，六者天之用也，十二者地之用也。天變方為圓而常存其一，地分一為四而常執其方。天變其體而不變其用也，地變其用而不變其體也。六者並其一而為七，十二者並其四而為十六也。陽主進，故天並其一而為七。陰主退，故地去其四而止於十二也，是陽常存一而陰常晦一也，故天地之體止於八，而天之用極於七，地之用止於十二也。圓者裁方以為用，故一變四，四去其一則三也。三變九，九去其三則六也。

Heaven is round and Earth is square. The number of the round comes from one and accumulates to six, while the number of the square comes from one and accumulates to eight. When changing and transforming, they start from four and accumulate to twelve. Those belonging to six change in cycles of six, those belonging to eight change in cycles of eight, and those belonging to twelve change along with eight. This is the natural principle. Eight is the form shared by Heaven and Earth, six is the function of Heaven, and twelve is the function of Earth. Heaven transforms its form into roundness but retains the 'one'; Earth divides one into four but always keeps its squareness. Heaven changes its form but does not change its function, while Earth changes its function but does not change its form. Adding the retained one to six makes seven, and adding the divided four to twelve makes sixteen. Yang is active and advances, so Heaven includes the one to make seven; Yin is passive and retreats, so Earth subtracts the four and stops at twelve. This is why Yang always retains a part and Yin always hides a part. Therefore, the form of Heaven and Earth stops at eight, the function of Heaven reaches seven, and the function of Earth stops at twelve. The round path must transform the square to achieve its function, so one becomes four, and four minus the one becomes three; three becomes nine, and nine minus the three becomes six.

方者展圓以為體，故一變三，並之四也，四變十二，並之十六也。故用數成於三而極於六，體數成於四而極於十六也。是以圓者徑一而圍三，起一而積六。方者分一而為四，分四而為十六，皆自然之道也。

The square's path must spread out in a circle to form its body, so one becomes three, and including itself, it totals four; four becomes twelve, and including itself, it totals sixteen. Therefore, the number used to establish the three reaches its extreme at six, and the number of the body established at four reaches its extreme at sixteen. Thus, the diameter of a circle is one while its circumference is three, starting from one, it accumulates to six; the square divides one into four, then divides four into sixteen: all of this follows naturally.

一役二以生三，三去其一則二也。三生九，九去其一則八也，去其三則六也。故一役三，三復役二也。三役九，九復役八與六也。是以二生四，八生十六，六生十二也。三並一則為四，九並三則為十二也，十二又並四則為十六。故四以一為本，三為用。十二以三為本，九為用。十六以四為本，十二為用。圓者六變，六六而進之，故六十變而三百六十矣。方者八變，故八八而成六十四矣。陽主進，是以進之為六十也。

One drives two and produces three; three minus one returns to two. Three produces nine; nine minus one is eight, and nine minus three is six. Therefore, one drives three, and three in turn drives two; three drives nine, and nine in turn drives eight and six. Thus, two produces four, eight produces sixteen, and six produces twelve. Three plus one is four, nine plus three is twelve, and twelve plus four is sixteen. Therefore, four is based on one and uses three, twelve is based on three and uses nine, and sixteen is based on four and uses twelve. The circle has six transformations, and six multiplied by six gives thirty-six; the square has eight transformations, and eight multiplied by eight gives sixty-four. Yang represents advancement, so advancement leads to sixty.

蓍數不以六而以七，何也？並其餘分也。去其餘分則六，故策數三十六也。是以五十者，六十四卦閏歲之策也。其用四十有九者，六十卦一歲之策也，歸奇掛一猶一歲之閏也。卦直去四者何也？天變而地效之，是以蓍去一則卦去四也。

Why does the use of numbers in divination with yarrow stalks take seven instead of six? It is because the remaining fraction is also included. If the remaining fraction is excluded, it would be six, so the number of stalks used for one line is thirty-six. Therefore, the number fifty in the Great Derivation is used to match the sixty-four hexagrams and the leap year in the calendar. Among these, the actual number used is forty-nine, which corresponds to the sixty hexagrams of a regular year. When counting the remaining stalks and setting aside one stalk during the divination process, it is analogous to inserting a leap month in a year. As for why the number is reduced by four directly when casting the hexagrams, it is because heaven changes first and earth follows, so reducing one stalk in the yarrow stalks corresponds to reducing four hexagrams.

圓者徑一圍三，重之則六。方者徑一圍四，重之則八也。裁方而為圓，天所以運行。分大而為小，地所以生化。故天用六變，地用四變也。

The diameter of a circle is one, and its circumference is three; doubling this gives six. The diameter of a square is one, and its circumference is four; doubling this gives eight. Carving a square into a circle represents how heaven can continuously move and function. Dividing a large number into smaller numbers represents how earth can give birth and nurture. Therefore, heaven uses six as its principle, and earth uses four as its principle.

一八為九，裁為七。八裁為六，十六裁為十二，二十四裁為十八，三十二裁為二十四，四十裁為三十，四十八裁為三十六，五十六裁為四十二，六十四裁為四十八也。一分為四，八分為三十二，十六分為六十四，以致九十六分為三百八十四也。

One and eight together make nine; if one part is cut off, it becomes seven. Following the ratio of subtracting one from four parts, we proceed: eight becomes six, sixteen becomes twelve, twenty-four becomes eighteen, thirty-two becomes twenty-four, forty becomes thirty, forty-eight becomes thirty-six, fifty-six becomes forty-two, and sixty-four becomes forty-eight. If we instead use the method of dividing, one becomes four, eight becomes thirty-two, sixteen becomes sixty-four, and so on up to ninety-six becoming three hundred eighty-four. This matches exactly the number of hexagrams (sixty-four) and lines (three hundred eighty-four) in the Yijing.

一生六，六生十二，十二生十八，十八生二十四，二十四生三十，三十生三十六，引而伸之，六十變而生三百六十矣，此運行之數也。四生十二，十二生二十，二十生二十八，二十八生三十六，此生物之數也。故乾之陽中策三十六，兌離巽之陽策二十八，震坎艮之陽策二十，坤之陽策十二也。

The sequence for the circle is: one generates six, six generates twelve, twelve generates eighteen, eighteen generates twenty-four, twenty-four generates thirty, and thirty generates thirty-six; continuing in this way, sixty changes generate three hundred and sixty, which is called "the number of movement." The sequence for the square is: four generates twelve, twelve generates twenty, twenty generates twenty-eight, and twenty-eight generates thirty-six; this is called "the number of generation." Therefore, the yang number of Qian is thirty-six, the yang numbers of Dui, Li, and Xun are twenty-eight, the yang numbers of Zhen, Kan, and Gen are twenty, and the yang number of Kun is twelve.

圓者一變則生六，去一則五也。二變則生十二，去二則十也。三變則生十八，去三則十五也。四變則二十四，去四則二十也，五變則三十，去五則二十五也。六變則三十六，去六則三十也。是以存之則六六，去之則五五也。五則四而存一也，四則三而存一也，三則二而存一也，二則一而存一也，故一生二，去一則一也。二生三，去一則二也。三生四，去一則三也。四生五，去一則四也。是故二以一為本，三以二為本，四以三為本，五以四為本，六以五為本。

The circle's path begins with one, which transforms into six; subtracting one leaves five. Two transforms into twelve; subtracting two leaves ten. Three transforms into eighteen; subtracting three leaves fifteen. Four transforms into twenty-four; subtracting four leaves twenty. Five transforms into thirty; subtracting five leaves twenty-five. Six transforms into thirty-six; subtracting six leaves thirty. Therefore, keeping the full number results in six sixes, while subtracting layer by layer results in five fives. Five is formed by four plus one, four is formed by three plus one, three is formed by two plus one, and two is formed by one plus one. Thus, one gives rise to two, and removing that one leaves one. Two gives rise to three, and removing that one leaves two. Three gives rise to four, and removing that one leaves three. Four gives rise to five, and removing that one leaves four. Therefore, two is based on one, three is based on two, four is based on three, five is based on four, and six is based on five, with each layer arising successively from the previous.

方者一變而為四，四生八，並四而為十二。八生十二，並八而為二十。十二生十六，並十二而為二十八。十六生二十，並十六而為三十六也。易之大衍何？數也。聖人之倚數也。天數二十五，合之為五十。地數三十，合之為六十。故曰五位相得而各有合也。五十者蓍數也，六十者卦數也。五者蓍之小衍，故五十為大衍也。八者卦之小成，則六十四為大成也。

The way of the square begins with one, which transforms into four; from four come eight, and combining the original four gives twelve; from eight come twelve, and combining the original eight gives twenty; from twelve come sixteen, and combining the original twelve gives twenty-eight; from sixteen come twenty, and combining the original sixteen gives thirty-six. What is the "Great Derivation" (大衍) mentioned in the Book of Changes? It refers to numbers, which are the numbers relied upon by the sage to establish the system. The heavenly numbers are one, three, five, seven, and nine, which add up to twenty-five; doubling this gives fifty. The earthly numbers are two, four, six, eight, and ten, which add up to thirty; doubling this gives sixty. Therefore, the Great Treatise says, "The five positions are in harmony and each has its corresponding element." Fifty is the number of the divination sticks, and sixty is the number of the hexagram lines. Five is the small derivation of the divination numbers, so fifty is called the Great Derivation; eight is the small completion of the hexagram numbers, so sixty-four is called the Great Completion.

蓍德圓，以況天之數，故七七四十九也，五十者存一而言之也。卦德方，以況地之數，故八八六十四也，六十者去四而言之也。蓍者用數也，卦者體數也。用以體為基，故存一也。體以用為本，故去四也。圓者本一，方者本四，故蓍存一而卦去四也。蓍之用數七，並其餘分，亦存一之義也。掛其一，亦去一之義也。

The nature of the yarrow stalks is round, used to represent the number of Heaven, so they are counted as forty-nine, which is seven times seven. It is said to be fifty because the one unused stalk is also included. The nature of the hexagrams is square, used to represent the number of Earth, so they are counted as sixty-four, which is eight times eight. It is said to be sixty because the four unused hexagrams are subtracted. The yarrow stalks belong to the category of 'used numbers,' while the hexagrams belong to the category of 'essential numbers.' The used must be based on the essential, so one stalk is retained. The essential must be based on the used, so four hexagrams are subtracted. The round is based on one, and the square is based on four, hence the yarrow stalks retain one and the hexagrams remove four. The used number of the yarrow stalks is seven, and the remaining fraction is included, which is exactly the meaning of 'retaining one.' When divining with the yarrow stalks, one stalk is first suspended, which is exactly the meaning of 'removing one.'

蓍之用數，掛一以像三，其餘四十八則一卦之策也。四其十二為四十八也，十二去三而用九，四三十二所去之策也。四九三十六所用之策也。以當乾之三十六陽爻也。十二去五而用七，四五二十所去之策也，四七二十八所用之策也，以當兌離之二十八陽爻也。十二去六而用六，四六二十四所去之策也，四六二十四所用之策也，以當坤之二十四陰爻也。十二去四而用八，四四十六所去之策也，四八三十二所用之策也，以當坎艮之二十四爻，並上卦之八陰為三十二爻也。是故七九為陽，六八為陰也。九者陽之極數，六者陰之極數，數極則反，故為卦之變也。震巽無策者，以當不用之數。天以剛為德，故柔者不見。地以柔為體，故剛者不生，是以震巽無策也。

The divination method using the yarrow stalks begins by hanging one stalk to symbolize the Three Powers. The remaining forty-eight stalks are used for a single hexagram. Four times twelve equals forty-eight. From twelve, subtract three and use nine; four times three equals twelve, which is the number of stalks subtracted, and four times nine equals thirty-six, which is the number of stalks used, corresponding to the thirty-six yang lines of the Qian hexagram. From twelve, subtract five and use seven; four times five equals twenty, which is the number of stalks subtracted, and four times seven equals twenty-eight, which is the number of stalks used, corresponding to the twenty-eight yang lines of the Dui and Li hexagrams. From twelve, subtract six and use six; four times six equals twenty-four, which is the number of stalks subtracted, and four times six equals twenty-four, which is the number of stalks used, corresponding to the twenty-four yin lines of the Kun hexagram. From twelve, subtract four and use eight; four times four equals sixteen, which is the number of stalks subtracted, and four times eight equals thirty-two, which is the number of stalks used, corresponding to the twenty-four lines of the Kan and Gen hexagrams, plus the eight yin lines of the upper trigram, totaling thirty-two lines. Therefore, seven and nine belong to yang, while six and eight belong to yin. Nine is the extreme number of yang, and six is the extreme number of yin. When numbers reach their extremes, they must reverse, so nine and six become the changing lines of the hexagram. The Zhen and Xun hexagrams are not included in the count of stalks, serving as the unused portion. Heaven's virtue is firm and strong, so its flexible aspect does not appear; Earth's nature is gentle and yielding, so its strong aspect does not emerge, hence Zhen and Xun do not have a count of stalks.

乾用九，故其策九也，四之者以應四時，一時九十日也。坤用六，故其策亦六也。奇數四，有一有二有三有四，策數四，有六有七有八有九，合而為八數，以應方數之八變也。歸奇合掛之數有六，為五與四四也，九與八八也，五與四八也，九與四八也，五與八八也，九與四四也，以應圓數之六變也。乾並餘分七為七政？

Heavenly Stem Qian uses all nines, so its measure is based on nine, multiplied by four to correspond to the four seasons, each season being ninety days. Earthly Stem Kun uses all sixes, so its measure is based on six. The remaining odd numbers from divination with yarrow stalks can be one, two, three, or four; the numbers used to form lines can be six, seven, eight, or nine. Combined, these make eight numbers, corresponding to the eight transformations of the directions. When the remaining measures are combined with the hanging measures, there are six possible results: five paired with two fours, nine paired with two eights, five paired with four and eight, nine paired with four and eight, five paired with two eights, and nine paired with two fours, corresponding to the six transformations of the circle. Qian includes the remaining fraction to make seven, which is probably what is meant by the 'seven governing principles'.

奇數極於四而五不用，策數極於九而十不用，五則一也，十則二也，故去五十而用四十九也。奇不用五，策不用十，有無之極也，以況自然之數也。卦有六十四而用止於六十者何也？六十卦者，三百六十爻也。故甲子止於六十也，六甲而天道窮矣。是以策數應之。三十六與二十四合之則六十也，三十二與二十八合之亦六十也。

The number of odd numbers used in divination reaches four, and five is not included; the number of lines in a trigram reaches nine, and ten is not included. Five is actually one, and ten is actually two, so the great derivation uses fifty but actually only employs forty-nine. Odd numbers do not use five, and the lines do not use ten. This is the extreme of "having" and "not having," used to illustrate the natural and spontaneous numbers. There are sixty-four hexagrams, but in practice only sixty are used. Why is this so? Because sixty hexagrams contain three thousand six hundred lines, so one cycle of Jiazi reaches sixty, and after six cycles, the heavenly path completes one full cycle. The numbers of the lines correspond exactly to this: thirty-six plus twenty-four equals sixty, and thirty-two plus twenty-eight also equals sixty.

乾四十八，坤十二，震二十，巽四十，離兌三十二，坎艮二十八，合之為六十。蓍數全，故陽策三十六與二十八合之為六十四也。卦數去其四，故陰策二十四與三十二合之為五十六也。九進之為三十六，皆陽數也，故為陽中之陽。七進之為二十八，先陽後陰也，故為陽中之陰。六進之為二十四，皆陰數也。故為陰中之陰。八進之為三十二，先陰後陽也。故為陰中之陽。

The trigram Qian has 48, Kun has 12, Zhen has 20, Xun has 40, Li and Dui combined have 32, and Kan and Gen combined have 28. When paired two by two, they all add up exactly to 60. The number of stalks used in divination is not reduced, so the yang stalks, which are 36, added to 28 give 64. The number of hexagrams is reduced by four, so the yin stalks, which are 24, added to 32 give 56. The number nine is derived by adding four to four, resulting in 36, which is entirely yang, so it is called 'yang within yang'. The number seven is derived by adding four to four, resulting in 28, which starts with yang and then turns to yin, so it is called 'yang within yin'. The number six is derived by adding four to four, resulting in 24, which is entirely yin, so it is called 'yin within yin'. The number eight is derived by adding four to four, resulting in 32, which starts with yin and then turns to yang, so it is called 'yin within yang'.

蓍四進之則百，卦四進之則百二十，百則十也，百二十則十二也。歸奇合卦之數得五與四四，則策數四九也。得九與八八，則策數四六也。得五與八八，得九與四八，則策數皆四七也。得九與四四，得五與四八，則策數皆四八也。為九者一變以應乾也，為六者一變以應坤也，為七者二變以應兌與離也，為八者二變以應艮與坎也。五與四四去掛一之數則四三十二也，九與八八去掛一之數則四六二十四也，五與八八，九與四八去掛一之數則四五二十也，九與四四，五與四八去掛一之數則四四十六也。故去其三四五六之數以成九八七六之策也。

The number of heavenly stems is twenty-five, and using four to advance it results in one hundred. The number of trigrams is thirty, and using four to advance it results in one hundred twenty. One hundred reduces to ten, and one hundred twenty reduces to twelve. Combining the remaining sticks with those used in the casting: if five and two fours are obtained, the number of lines is four multiplied by nine; if nine and two eights are obtained, the number of lines is four multiplied by six; if five and two eights, or nine and four or eight, the number of lines is four multiplied by seven; if nine and two fours, or five and four or eight, the number of lines is four multiplied by eight. If nine is obtained, it is one change, corresponding to Heaven; if six is obtained, it is one change, corresponding to Earth; if seven is obtained, it is two changes, corresponding to Lake and Fire; if eight is obtained, it is two changes, corresponding to Mountain and Water. If five and two fours are obtained, removing the stick used in the casting results in four multiplied by three, which is twelve; if nine and two eights are obtained, removing the stick used in the casting results in four multiplied by six, which is twenty-four; if five and two eights, or nine and four or eight, removing the stick used in the casting results in four multiplied by five, which is twenty; if nine and two fours, or five and four or eight, removing the stick used in the casting results in four multiplied by four, which is sixteen. It is precisely by subtracting three, four, five, and six that the numbers nine, eight, seven, and six are formed.

天一地二天三地四天五地六天七地八天九地十，參伍以變，錯綜其數也，如天地之相銜，晝夜之相交也。一者數之始，而非數也。故二二為四，三三為九，四四為十六，五五為二十五，六六為三十六，七七為四十九，八八為六十四，九九為八十一，而一不可變也。百則十也，十則一也，亦不可變也。是故數去其一而極於九，皆用其變者也。五五二十五，天數也。六六三十六，乾之策數也。七七四十九，大衍之用數也。八八六十四，卦數也。九九八十一，玄範之數也。

Heavenly One, Earthly Two, Heavenly Three, Earthly Four, Heavenly Five, Earthly Six, Heavenly Seven, Earthly Eight, Heavenly Nine, Earthly Ten, intermingle and change in a complex manner, combining and interacting with each other, just as Heaven and Earth connect with each other and day and night intermingle. "One" is the beginning of numbers, but it is not counted as a number. Therefore, two multiplied by two is four, three multiplied by three is nine, four multiplied by four is sixteen, five multiplied by five is twenty-five, six multiplied by six is thirty-six, seven multiplied by seven is forty-nine, eight multiplied by eight is sixty-four, nine multiplied by nine is eighty-one. Only one multiplied by one remains one, and cannot change. One hundred reduces to ten, ten reduces to one, and also cannot change. Therefore, numbers must exclude one and reach up to nine, taking only the parts that can change. Five multiplied by five is twenty-five, which is the heavenly number. Six multiplied by six is thirty-six, which is the number of the trigram Qian. Seven multiplied by seven is forty-nine, which is the number used in the Great Changes. Eight multiplied by eight is sixty-four, which is the number of the hexagrams. Nine multiplied by nine is eighty-one, which is the number used in Yang Xiong's Tai Xuan and Hong Fan's nine categories.

大衍之數，其算法之源乎？是以算數之起，不過乎方圓曲直也。乘數生數也。除數消數也。算法雖多，不出乎此矣。陰無一，陽無十。陽得陰而生，陰得陽而成，故蓍數四而九，卦數六而十也，猶干支之相錯，乾以六終而支以五終也。

The fundamental numbers of the Great Derivation are the origin of arithmetic. Therefore, all calculations begin with the forms of square, circle, curve, and straight line. Multiplication belongs to the number of growth, while division belongs to the number of reduction; although there are many names for calculation methods, they all fall within this scope. The yin side does not have the number one, and the yang side does not have the number ten; yang needs to meet yin in order to generate, and yin needs to meet yang in order to form. Therefore, the number of the yarrow stalks is based on four and completes at nine, while the number of the hexagrams is based on six and completes at ten. This is similar to the interweaving of the heavenly stems and earthly branches, where the heavenly stems cycle six times and the earthly branches cycle five times, both concluding together at sixty.

三四十二也，二六亦十二也，二其十二二十四也，三八亦二十四也，四六亦二十四也。三其十二三十六也，四九亦三十六也，六六亦三十六也。四其十二四十八也，三其十六亦四十八也，六八亦四十八也。五其十二六十也，三其二十亦六十也，六其十亦六十也，皆自然之相符也。四九三十六也，六六三十六也，陽六而又兼陰六之半，是以九也。故以二卦言之，陰陽各三也。以六爻言之，天地人各二也。陰陽之中各有天地人，天地人之中各有陰陽，故參天兩地而倚數也。

Three times four is twelve, two times six is also twelve; two twelves make twenty-four, three times eight is also twenty-four, four times six is also twenty-four; three twelves make thirty-six, four times nine is also thirty-six, six times six is also thirty-six; four twelves make forty-eight, three times sixteen is also forty-eight, six times eight is also forty-eight; five twelves make sixty, three times twenty is also sixty, six times ten is also sixty: all these naturally correspond with each other. Four times nine is thirty-six, six times six is also thirty-six: yang originally has only six, but it also takes half of yin's six, thus forming nine. In terms of the upper and lower trigrams, yin and yang each occupy three lines; in terms of a single trigram with six lines, heaven, earth, and humanity each occupy two lines. Yin and yang each contain heaven, earth, and humanity, and heaven, earth, and humanity each contain yin and yang. Therefore, it is said, "Three lines for heaven, two lines for earth, and the numbers rely on this principle."

陽數一，衍之而十，十干之類是也。陰數二，衍之為十二，十二支十二月之類是也。一變而二，二變而四，三變而八卦成矣。四變而十有六，五變而三十有二，六變而六十四卦備矣。易有真數三而已矣，參天者三三而九，兩地者倍三而六。參天兩地而倚數，非天地之正數也。倚者擬也,擬天地正數而生也。

The number for Yang is one, which expands to ten, giving rise to the ten Heavenly Stems; the number for Yin is two, which expands to twelve, giving rise to the twelve Earthly Branches and the twelve months. One transforms into two, two transforms into four, and three transformations lead to the eight Trigrams. Four transforms into sixteen, five transforms into thirty-two, and six transformations result in the sixty-four Hexagrams. This is the method of successive doubling in the study of theEarlier Heaven (Earlier Heaven) system. The true fundamental numbers of the Book of Changes are only three: "参天" (partaking in Heaven) refers to three multiplied by three to make nine; "两地" (partaking in Earth) refers to three doubled to make six. "参天两地而倚数" (partaking in Heaven and Earth to derive numbers) does not refer to the original numbers of Heaven and Earth; "倚" (relying on) means comparison, referring to numbers derived by analogy with the original numbers of Heaven and Earth.

易之生數一十二萬九千六百，總為四千三百二十世，此消長之大數，衍三十年之辰數即其數也。歲三百六十日，得四千三百二十辰，以三十乘之，得其數矣，凡甲子甲午為世首，此為經世之數，始於日甲月子星甲辰子。又云。此經世日甲之數，月子星甲辰子從之也。一十百千萬億，為奇天之數也。十二百二十千二百萬二千億二萬，為偶地之數也。

The numbers generated by the Book of Changes amount to twelve thousand nine hundred and sixty, which totals four thousand three hundred and twenty generations, representing the great number of increase and decrease. By extending the numbers of the thirty-year cycle, one arrives precisely at this number. A year consists of three hundred and sixty days, totaling four thousand three hundred and twenty hours. Multiplying this by thirty yields the number. All generations begin with the years of Jiazi and Jiawu, which is called the "number governing the world." The beginning is marked by the day being Jia, the month being Zi, the star being Jia, and the hour being Zi. It is also said that this is the number of "Jia day" within the governing of the world, with the month Zi, star Jia, and hour Zi all following in sequence. The sequence of one, ten, hundred, thousand, ten thousand, and hundred million belongs to the odd numbers, which are the numbers of heaven; while the sequence of twelve, two hundred and twenty, one thousand two hundred, twenty thousand, and two million belongs to the even numbers, which are the numbers of earth.

五十分之則為十，若參天兩之則為六，兩地又兩之則為四，此天地分太極之數也。復至乾凡百有十二陽，姤至坤凡八十陽。姤至坤凡百有十二陰，復至乾凡八十陰。陽數於三百六十上盈，陰數於三百六十上縮。

The number fifty, when divided evenly, becomes ten. If we take the number three from "参天" and double it, we get six; if we take the number two from "两地" and double it, we get four: these are the numbers derived from the division of the Taiji by Heaven and Earth. In the diagram of theEarlier Heaven (Earlier Heaven) circle, from the Fu hexagram to the Qian hexagram there are 112 yang lines; from the Guu hexagram to the Kun hexagram there are 80 yang lines. From the Guu hexagram to the Kun hexagram there are 112 yin lines; from the Fu hexagram to the Qian hexagram there are 80 yin lines. The number of yang lines exceeds 360, while the number of yin lines is less than 360.