The origin of the great derivation of the round and square principles.

凡方圓可為比例，唯徑七者，方周二十八，圓周二十二，即兩積相比例之率也，用其半，故若十四與十一。合二十八與二十二，共五十，是大衍之數，函方圓同徑兩周數。

In general, the square and the circle should be taken in proportion, and the most harmonious ratio is when the diameter is seven. A square with a side length of seven has a perimeter of twenty-eight, while the circumference of a circle with a diameter of seven is approximately twenty-two. These two numbers represent the ratio of the areas of the square and the circle. Taking half of each gives fourteen to eleven. Adding twenty-eight and twenty-two together gives fifty, which is the number of the great derivation. This shows that the number fifty includes the total of the perimeters of the square and the circle with the same diameter.

勾三，其積九。股四，其積十六。弦五，其積二十五。合之五十，是大衍之數，函勾股弦三面積。

In a right-angled triangle, the shorter side called "gou" is three units long, and when squared, it gives an area of nine. The longer side called "gu" is four units long, and when squared, it gives an area of sixteen. The hypotenuse called "xian" is five units long, and when squared, it gives an area of twenty-five. When the areas of the three squares are added together, they total fifty, which is also known as "da yan zhi shu." This shows that the number fifty includes the areas of the three squares formed by the sides "gou," "gu," and "xian."

蓍策之數，必以七為用者，蓋方圓之形，唯以徑七為率，則能得周圍之整數。勾股之形，亦唯以三四為率，則能得斜弦之整數。徑七，固七也，勾三股四之合亦七也，是故論方圓周圍之合數則五十，論勾股弦之合積亦五十，此大衍之體也。因而開方，則不盡一數。而止於四十九，此大衍之用也。開方而不盡一數，則瞢策之虛一者是已，方面之中，函八勾股，而又不盡一數，則蓍策之掛一者是已，唯老陽老陰之數，與此密合，故作圖以明之。

The numbers used in divining with the yarrow stalks must always be based on seven because the shape of a circle and square can only yield an integer circumference when the diameter is seven. Similarly, the right-angled triangle can only yield an integer hypotenuse when the sides are three and four. Seven is indeed seven, and three plus four also equals seven. Therefore, the combined numbers of the circumference of a circle and square total fifty, and the combined numbers of the three sides of a right-angled triangle also total fifty. This is the essence of "Da Yan" (Great Transformation). Since fifty cannot be completely extracted as a square root, a remainder must be left, and only forty-nine can be taken, which is the function of "Da Yan." The remainder of one when taking the square root corresponds exactly to the initial step of removing one stalk from the yarrow stalks (the original text mistakenly writes "meng ce" instead of "shi ce"). Within a square, there are eight right-angled triangles, and a remainder remains in the middle, which corresponds exactly to the initial step of suspending one stalk. Only the numbers of the old yang and old yin are perfectly aligned with this calculation, hence the special diagram is drawn to illustrate this.

全方四十九。中含大方六六三十六，為過揲之數。小角一一如一，一六五互乘為十二，併成十三，為掛扐之數。此與前《洛書》以自乘互乘為積方之法同，但《洛書》用對數，如一與九之類足也，大衍用合數，則一與六是也。

The entire large square has an area of forty-nine and a side length of seven. Inside it is a six by six square, equaling thirty-six, which represents the count of the old yang "over counting" (过揲); the small squares at the corners are one by one, equaling one, plus two long strips formed by the mutual multiplication of one and six, totaling twelve, making altogether thirteen, which is the count of the old yang "hanging up" (挂扐) (the original text "一六五互乘" should be interpreted as the mutual multiplication of one and six twice, totaling twelve). This method is the same as the calculation used in the Luo Writing, where numbers are combined through self-multiplication and mutual multiplication to form the square's total. However, in the Luo Writing, the numbers are relative, such as one and nine opposing each other; whereas in the "Great Derivation," the numbers are complementary, meaning one and six combine with each other.

全方四十九。勾三股四，其積六，四因之得二一卜四，為過揲之數。弦五，其積二十五，為掛扐之數。弦實亦函四勾股積，而多勾股較一。

The total area of the large square remains forty-nine. The right-angled triangle formed by the sides of three and four has an area of six. Multiplying this by four gives twenty-four, which is the count of the old yin "over counting" (the original text "twenty-one plus four" is taken as "twenty-four"). The length of the hypotenuse is five, and squaring it gives twenty-five, which is the count of the old yin "hanging and separating." Within the square on the hypotenuse, there are also four such right-angled triangles, but there is an additional small square called "the difference between the leg and the base," which is the remainder after subtracting three from four.

十數之中，除一一不變，自二二至十十，皆可成方，然唯三三則五數居其中，七七則二十五數居其中，此二者為能得天地之中數，餘則不能也。蓋三二者《洛書》之數也，七七者菁策之數也，《洛書》之數，五居其中矣，而其四方，則又成四勾股之數，而以中萬為弦之法焉。

Among the numbers from one to ten, except for one multiplied by one which remains unchanged, the products from two multiplied by two up to ten multiplied by ten can all form squares. However, only in the case of three multiplied by three does the number five exactly occupy the center, and in the case of seven multiplied by seven, the number twenty-five exactly occupies the center. Only these two cases can attain the central number of heaven and earth. The three by three is exactly the number of the Luo Writing (the original text reads 'san er' but should be 'san san'), and the seven by seven is exactly the number of the divining stalks (the original text reads 'jing ce' but should be 'zhi ce'). In the nine numbers of the Luo Writing, five is located at the center, and its four sides form four right-angled triangles, with five at the center serving as the hypotenuse (the original text reads 'zhong wan' but should be 'zhong wu').

蓍策之數，二十五居其中矣，而其四方、則又具四勾股之積而即以二十五為弦之實焉，是故卦數之八，合乎《河圖》之四也，為其虛五十者同一根也。蓍數之七，合乎《洛書》之三也，為其用中五者同一根也，聖人因心之作，與天地自然之文，其相為經緯者如此。

The number of yarrow stalks used in divination is forty-nine, with twenty-five located at the center. The four sides also contain the area of four right-angled triangles, which precisely correspond to the square of the hypotenuse being twenty-five. Therefore, the number 'eight' in the Eight Trigrams corresponds with the number 'four' in the Yellow River Chart, as both discard the central numbers five and ten, sharing the same fundamental principle. The number 'seven' in the yarrow stalks corresponds with the number 'three' in the Luo Writing, as both take the central number five as their shared foundation. The sage, based on the principles within his heart, created the system of laws, which intertwines with the natural patterns of heaven and earth, forming a network of meridians and threads.

史稱黃帝迎口椎策，所謂策者，蓋即神蓍也。推衍策數以候日月，故曰“迎日推策”。考之後代，譚卦畫者多以曆法推配，然孔子未嘗言也，唯於“大衍之數”，則曰“象四時”“象閏”，又曰“當期之日”，則蓍策之與曆法相表裡也，可見矣。

Historical records say the Yellow Emperor "welcomed the sun and calculated the divination rods" (the original text "welcome mouth zhuo ci" should be "welcome the sun and calculate the divination rods"). The so-called "ci" probably refers to the divine divination rods used in divination. By using the method of calculating the divination numbers to predict the movement of the sun and moon, this is called "welcoming the sun and calculating the divination rods." When examining later generations, those who explain the trigrams often compare them with the calendar system, but Confucius did not do so. He only mentioned "the image of the four seasons" and "the image of the leap month" when explaining "the great derivation number," and also mentioned "the day of the appointed period." With just these few sentences, one can see that the numbers of the divination rods and the calendar system were originally complementary to each other.

顧有以理言之而肖似者，有以數推之而密合者，以理言而肖似者，孔子《大傳》所陳是也。蓋四十九算，排列成方，以勾股之數求之，則零一者歸於中而為心；以開方之法求之，則零一者歸於

However, some of them are similar in terms of principle, while others are precisely calculated to match exactly. Those that are similar in terms of principle are the ones Confucius discussed in the Great Treatise. He arranged forty-nine counting sticks into a square, using the Pythagorean method to calculate, and the extra stick was placed in the center to form the square's core. Using the method of square root calculation, the extra stick was placed at the bottom. The text here is missing the following part, and then proceeds to another calculation.

一年三百六十五日四分日之一。每日百分。凡三萬六十五百二十五分。以天數二十五除之，得一千四百六十一分，為日數。又以地數三個除日數，得四十八容七分，為月數。是為大衍用數。

A year consists of three hundred sixty-five and one fourth days. Each day is divided into one hundred parts, making a total of thirty-six thousand five hundred and twenty-five parts for the whole year (the original text reads 'three ten thousand six hundred and fifty hundred and twenty-five parts' with a missing 'thousand' character). Dividing this by twenty-five days gives one thousand four hundred and sixty-one parts, which is the 'day count'. Then dividing this day count by thirty (the original text mistakenly reads 'three' instead of 'thirty') gives forty-eight and seven parts, which is the 'month count' (the original text mistakenly reads 'forty-eight contains seven parts' instead of 'forty-eight and seven parts'). These two numbers are the numbers used in the Great Transformation.

《大傳》言蓍數，而以《河圖》之數首之，故一年全數，以二十五除之得日數者。日有曉午昏夜凡四限，四分期日，為一千四百六十一也；以三十除之得月數者，月有朔望，上下弦凡四限，四分歲月，每月三十日算，為四十八零七分也，與大衍用數相應。

The Great Treatise explains the numbers of the divination stalks. It begins by using the numbers of Heaven and Earth from the Yellow River Chart. Therefore, to calculate the total number of days in a year, we divide by twenty-five to obtain the "day count": a day is divided into four parts—morning, noon, evening, and midnight. Dividing the year into four parts results in 1,461 days. To calculate the "month count," we divide by thirty: a month is divided into four parts—new moon, full moon, waxing moon, and waning moon. Dividing the year into four parts and calculating each month as thirty days results in forty-eight and seven-tenths months. These results correspond with the numbers used in the great derivation as previously mentioned.

揲策合左右共四十八，應四十八弦，每弦七日半，為期日歲月之經數，三百六十。掛策一，應氣盈之餘數，五日四分日之一。以初變為主。日法十。揲策應弦，每弦以十分為率。掛策應氣盈五日四分日之一，於日法為十分弦之七。

When divining with yarrow stalks, the total number of stalks divided between the left and right hands is forty-eight, corresponding to the forty-eight phases of the moon in a year—each month has four phases: new moon, full moon, upper crescent, and lower crescent. Each phase lasts seven and a half days, adding up to exactly 360 days, which is the regular cycle of the sun and moon. The stalk left hanging corresponds to the extra time of "qi ying" (the surplus of the solar term), which is one and one fourth days. This calculation is based on the first division. Using the method of dividing by ten as one day: the number of stalks corresponds to each phase, with each phase being divided by ten; the hanging stalk corresponds to the extra one and one fourth days of "qi ying," which is converted into tenths of a phase, amounting to seven tenths of a phase.

扐策合陰陽共十二，得少則四為陽，得多則八為陰，應十二朔，每朔二十九日，九百四十分日之四百九十九，為一歲之實數，三百五十四日，九百四十分日之三百四十八。掛策一，應朔虛之餘數，十日，九百四十分日之八百二十七。亦以初變為主，月法十九。肋策應朔，每朔以十九分為率。掛策應朔虛十日九百四十分口之八百二十七，於月法為十九分朔之七。

The divination rods held between the fingers total twelve, corresponding to the yin and yang, with four being yang and eight being yin. This corresponds to the twelve lunar months in a year. Each lunar month is twenty-nine days and four hundred and ninety-nine parts of a thousand, making the total days in a year three hundred and fifty-four days and three hundred and forty-eight parts of a thousand. The rod that is hung corresponds to the excess of the new moon, which is ten days and eight hundred and twenty-seven parts of a thousand. This system also follows the first change as a standard, dividing nineteen parts into one month: the divination rod held corresponds to each new moon, with each new moon being measured by nineteen parts; the hung rod corresponds to the excess of the new moon, ten days and eight hundred and twenty-seven parts of a thousand, which when converted according to the nineteen parts per month, equals seven parts of a new moon.

以初變之揲策枘策計之，揲策四十八，以應四十八弦之整數，其“掛一，’餚，以應氣盈五日四分口之一也；抽策十二，以應十二朔之實數，其，掛一者，以應朔虛十日八百二十七分也。據四分曆法，每口九百四十分，故一歲之氣盈，有五日二百三十五分。

Take the first change, combining the number of yao obtained from the divination with the number of yao obtained from the handling: the number of yao from the first change is forty-eight, corresponding to the integer of forty-eight strings; the one yao hanging corresponds to the surplus of five and one fourth days. The number of yao from the handling is twelve, corresponding to the actual number of twelve lunar months; the one yao hanging corresponds to the deficiency of ten days and eight hundred and twenty-seven parts. According to the algorithm of the four fractions calendar, each day is converted into nine hundred and forty parts. Therefore, the surplus of a year is converted into five days and two hundred and thirty-five parts.

一歲之朔虛，此合氣盈總算，有十日八百二十七分，每弦七日四百七十分，如日法十分弦之七，則為五日二百三十五分矣；每朔二十九日四百九十九分，如月法十九分朔之七，則為十日八百二十七分矣。

The length of a lunar month from new moon to new moon, together with the interval of the full moon, totals ten days and eight hundred twenty-seven parts. Each half-month, or crescent, is seven days and four hundred seventy parts. By taking seven tenths of a half-month according to the method of dividing a day into ten parts, it exactly equals five days and two hundred thirty-five parts. Each synodic month is twenty-nine days and four hundred ninety-nine parts. By taking seven ninths of a synodic month according to the method of dividing a month into nineteen parts, it exactly equals ten days and eight hundred twenty-seven parts. The remaining fractions in both cases fall on the number seven.

月每日行十二度十九分度之七，故以十，九為法，日月之法不同，而其餘分皆七。故漢儒卦氣，每卦直六口，尚餘七分，每卦直六門七分者，門以八十分為法也。

The moon moves twelve degrees and seven parts of nineteen each day, so nineteen is taken as the denominator (the original text has a comma between 'ten' and 'nine', which is an addition). Although the method of division used by the sun and moon differs, the remaining fraction is seven in both cases. Therefore, Han dynasty scholars discussing 'hexagram qi' stated that each hexagram corresponds to six days and seven parts, because they calculated based on a system where eighty parts made up one day (the characters 'mouth' and 'door' in the original text are actually the character for 'day').

蓋歲數二百六十五日四分日之一，四乘而三除之，為四百八十七，故四百八十七者，歲策也，，每卦直六日，六八四十八，得四百八個分，又餘七分，歲策之根也。積六十卦，直三百六十日，餘分之積，共四百二十分，以日法除之，為五日四分日之一。古今曆法，一章之內，有七閏月昔，法由茲起也。其在蓍數，則何以見掛一之策，為餘七之算乎？蓋亦以生菁之法而知之爾。

The number of days in a year is three hundred sixty-five and one fourth day (the original text mistakenly says "two hundred sixty-five" but should be "three hundred sixty-five"). Multiplying by four and dividing by three gives four hundred eighty-seven, so four hundred eighty-seven is called the "annual measure" (sui ce). Each trigram corresponds to six days, and six times eighty gives four hundred eighty, with an additional seven parts, making a total of four hundred eighty-seven, which is the root of the annual measure. Accumulating sixty trigrams totals three hundred sixty days, and the remaining parts add up to four hundred twenty parts. Dividing these by eighty parts per day results in five and one fourth day. Ancient and modern calendars determine seven leap months within nineteen years, and this calculation comes from here. Then, in the context of divination with yarrow stalks, how do we know that the remaining seven parts correspond to the "remainder seven" calculation? This is understood from the method of dividing the yarrow stalks.

過揲為正策，乾策三卜六，合六爻二百一十有六。坤策二十四，合六爻百四十有四。凡三百有六個，當一期之日數。掛扐為餘策，乾策一十二，合六爻七十八，坤策二十五，合六爻百五十。凡二百二十有八，當一章之月數。正策以三十為進退之法，故其合皆六十，餘策以十九為進退之法，故其合皆三十八，三十者，日法也，十九者，朔法也。

After divining, the number of lines held in the hand is called "zheng ce" (正策). For the trigram Qian (乾), one line corresponds to thirty-six ce (策), and six lines total two hundred and sixteen. For the trigram Kun (坤), one line corresponds to twenty-four ce, and six lines total one hundred and forty-four. Together, they total three hundred and sixty, which corresponds to the number of days in a year (the original text mistakenly writes "three hundred and six" instead of "three hundred and sixty"). The remaining ce, called "yu ce" (余策), are those held in the hand and between the fingers. For Qian, one line corresponds to thirteen ce, and six lines total seventy-eight. For Kun, one line corresponds to twenty-five ce, and six lines total one hundred and fifty. Together, they total two hundred and twenty-eight, which corresponds to the number of months in nineteen years. The number of ce for the lines in zheng ce is counted by thirty, so the total ce for the lines of Qian and Kun together is sixty. The number of ce for the lines in yu ce is counted by nineteen, so the total ce for the lines of Qian and Kun together is thirty-eight. Thirty is the day cycle, and nineteen is the lunar cycle.

“二篇之策”為全策，陽爻百九十二，得六乾九百一十二，陰爻百九十二，得四千六百零八。凡萬有一千五百二十，當閏終之總數。

The "two chapters of divination rods" mentioned in the Great Treatise refer to the total number of divination rods. The upper and lower classics, comprising sixty-four hexagrams, contain a total of 192 yang lines. Each yang line corresponds to 36 rods, resulting in 6,912 rods. Similarly, there are 192 yin lines, each corresponding to 24 rods, totaling 4,608 rods. When these two numbers are combined, they sum up to 11,520, which corresponds exactly to the total number of rods at the end of a leap cycle.

此因《大傳》之說而推備之者。歲者，正數也，太陽主之；閏者，餘數也，太陰主之。故《堯典》始而殷正四時，則曰“日中”“日永”“日短”，此以太陽為主者也，終則曰以閏月定四時成歲，此以太陰為主者也。

The above is a complete derivation following the explanation in the Great Treatise. One "year" is the main count, governed by the sun; "leap month" is the remainder, governed by the moon. Therefore, the beginning of the Book of Documents, the Yao Dian, establishes the four seasons, mentioning "noon," "long day," and "short day," which refer to the equinox in spring when day and night are equal, the summer solstice when the day is longest, and the winter solstice when the day is shortest; this is governed by the sun. At the end, it also mentions using the leap month to determine the four seasons and complete a year, which is governed by the moon.

蓍策之正數三百有六十，當一期之日，蓋日周大而為一期，故為太陽所主也。其餘數二百一二十有八，當一章之月，蓋氣朔分齊而為一章，故為太陰所主也，其全數萬有一千五百二十，當閏終之總數，蓋三卜二月而閏一月，其辰萬有一千五百二十；三十二年而閏一年，其日萬有一千五百二十。此則日月正餘會終，蓍卦齊同之數也。

The standard number of the yarrow stalks is three hundred and sixty, corresponding to the number of days in a year: the sun completes one orbit in a period, so it is governed by the sun. Its remainder is two hundred and twenty-eight, corresponding to the number of months within a chapter (the original text mistakenly writes "two hundred and ten and twenty-eight" instead of "two hundred and twenty and eight"): the conjunction of the solar terms and the new and full moons completes a chapter, so it is governed by the moon. Its total number is eleven thousand five hundred and twenty, corresponding to the total number at the end of the leap year cycle: a leap month is inserted every thirty-two months, and the number of hours in that period is eleven thousand five hundred and twenty; a leap year is inserted every thirty-two years, and the number of days in that period is also eleven thousand five hundred and twenty. This is where the day and the month, the standard number and the remainder, converge, and where the yarrow stalk numbers and the hexagram numbers become aligned.

歷代之歷，歲分消長不同，故有五日四分日之一而有餘者，亦有五日四分日之一而不足者，然舉其中者以演其變者，則四分為常法，故顓頊曆《周髀經》皆用之，而司馬遷曆書述焉，蓋古法也。

The calendars used in various dynasties have different lengths for a year, some being longer than 365 and 1/4 days, and some shorter. However, to calculate the average and predict changes, the method of using "four parts" has been commonly adopted. Therefore, the颛顼 calendar and the Zhou Bi Suan Jing both used this method. Sima Qian also recorded this method in his Records of the Grand Historian, which was originally a traditional calculation passed down from ancient times.

以地平線分周天之度為二，各一百八十度，日出入朦景昏旦各十八度，共二十六度， 《大傳》曰：“乾坤之策，凡三百有六十，當期之日”，故各一百八十者，寒暑晝夜並行之體數也。

This diagram divides the celestial sphere along the horizon into two equal parts, each measuring 180 degrees. The dawn and dusk shadows before sunrise and after sunset each account for 18 degrees, totaling 36 degrees at both ends (the original text says "twenty-six degrees," which should be "thirty-six degrees"). The Great Treatise states: "The measures of Heaven and Earth amount to a total of 360, corresponding to the cycle of a year." Therefore, the division of 180 degrees above and below corresponds to the simultaneous movement of cold and heat as well as day and night.

然陽生而陰殺，陽明而陰暗，故陽饒而陰乏，陽盈而陰虛。今以晝夜平分推之，其自然之數如此，若一歲寒暑之候，則若邵子之說，開物於寅末，是亦先十八日也，閉物於戌初，是亦後十八日也，以故萬物之數，萬有一千五百二十，其從陽者六千九百一十二，其從陰者四千六百八，生氣常盛，則為豐年，善類常多，則為治世，其消息盈虛之理，亦若是而已矣。

However, yang represents generation and birth, while yin represents肃杀 (harshness and decline). Yang belongs to brightness, while yin belongs to darkness. Therefore, yang is always expansive and abundant, while yin is always narrow and deficient. Yang is always full, while yin is always empty. Now, according to the calculation of day and night being equal, this is the natural pattern. As for the seasonal changes of cold and heat in a year, as Shao Yong said: the beginning of all things occurs at the end of the寅 (Yin) hour, which is eighteen days earlier; the closing of all things occurs at the beginning of the戌 (Xu) hour, which is eighteen days later. Because of this, among the total number of ten thousand one thousand five hundred and twenty, those following yang amount to six thousand nine hundred and twelve, while those following yin are only four thousand six hundred and eight. The vital energy of growth is often strong, thus becoming a bountiful year; the virtuous people are often numerous, thus becoming a well-governed era. The principle of increase and decrease, fullness and emptiness in the universe is no more than this.

此圖用加一倍法，如第二層兩一，生第三層中位之二，並左右兩一成四，是倍二為四也，第三層一二各生第四層中位之三，並左右兩一成八，是倍四為八也，下放此，出於數學中，謂之開方求廉率，其法以左一為方，右一為隅，而中間之數，則其廉法也。第三層為平方，第四層為立方，第五層六層七層，為三乘四乘五乘方，於成卦之理，亦相肖合、伺則？

This diagram uses the method of "doubling each time." For example, the two ones in the second layer produce the two ones on either side of the two in the third layer, combining with the two ones to make a total of four, which is doubling the two to make four; the one and the two in the third layer each produce the three in the middle of the fourth layer, combining with the two ones on either side to make a total of eight, which is doubling the four to make eight; and so on (the original text "下放此" should be "下仿此"). This method comes from mathematics and is called "finding the 'lian rate' through square roots": taking the one on the left as the "fang" (square), the one on the right as the "yu" (corner), and the numbers in the middle as the "lian fa" (oblique rate). The third layer corresponds to squaring, the fourth layer to cubing, and the fifth, sixth, and seventh layers correspond to third power, fourth power, and fifth power respectively. It also resembles the principles of hexagram formation, which is why this is so. (the original text "伺则" should be "何则")

陽大陰小，陽如方，陰如隅，分居兩端，陰陽合則生小問之兩象。如平：廳之方隅合而生兩廉，其長如方，其廣如隅也。又乘則生中間之六卦，如立方之方隅合而生六廉，三平廉根於方，而其厚如隅，二長廉根於隅，而其長如方也。

Yin is small and Yang is great. Yang is like 'fang' and Yin is like 'yu', each occupying the two extremes. When Yang and Yin combine, they produce two middle images. This is similar to the numbers in square measurement, where 'fang' and 'yu' combine to produce two 'lian'. The length of the 'lian' is the same as 'fang', and the width is the same as 'yu'. When multiplied again, they produce the six central trigrams. This is similar to the numbers in cube measurement, where 'fang' and 'yu' combine to produce six 'lian'. Three 'ping lian' have 'fang' as their root and thickness the same as 'yu'; two 'chang lian' have 'yu' as their root and length the same as 'fang'.

故開方之法，雖相乘至於無窮，莫不依方隅以立算，成卦之法，雖相加至於無窮，莫不根陰陽以定體。成卦之始，一陰一陽，每每相加而已，及卦成而分析觀之，則自一畫至六畫，唯純陰純陽者常不動，其餘則方其為四象也；中間一陰一陽者二，方其為八卦也；中間一陰二陽者三，一陽二陰者三，方其為四畫也；

Therefore, the method of square root calculation, even when multiplied infinitely many times, does not depart from the principles of "fang" and "yu" in setting up the calculations; similarly, the method of forming hexagrams, even when added infinitely many layers, does not depart from yin and yang as the fundamental basis for determining their nature. At the beginning of forming a hexagram, it is merely yin and yang added layer by layer. When the hexagram is completed, if we look at it separately, from one line to six lines, only the pure yin and pure yang remain unchanged at the extremes. The other middle lines increase progressively with each layer: when it becomes the four trigrams, there are two instances of one yin and one yang in the middle; when it becomes the eight trigrams, there are three instances of one yin and two yangs, and three instances of one yang and two yins; when it becomes a hexagram with four lines,

中間一陰三陽者四，一陽三陰者四，二陰二陽者六，方其為五畫也；中間一陰四陽者五，一陽四陰者五，二陰三陽者十，二陽三陰者十，及其六畫之既成也；中間一陰五陽者六，一陽五陰者六，二陰四陽者十五，二陽四陰者十五，三陰三陽者二十。朱子卦變之圖，以此而定也，蓋其倍法同於畫卦，而其多寡錯綜之數，則卦變用之。

In the case of three yang and one yin, there are four; in the case of one yang and three yin, there are four; and in the case of two yin and two yang, there are six. When it becomes a five-line trigram, there are five instances of one yin and four yang, five instances of one yang and four yin, ten instances of two yin and three yang, and ten instances of two yang and three yin. When it becomes a six-line hexagram, there are six instances of one yin and five yang, six instances of one yang and five yin, fifteen instances of two yin and four yang, fifteen instances of two yang and four yin, and twenty instances of three yin and three yang. Zhu Xi's diagram of changing hexagrams is based on these numbers: its method of doubling is the same as drawing the hexagrams, and the numbers of various lines and yins and yangs, arranged in a comprehensive manner, are exactly suitable for the changing hexagrams. When adding up all the categories in the six-line hexagrams along with the two extremes of pure yin and pure yang, the total is exactly sixty-four hexagrams.

卦之序也，雜也，皆出於文王也，其所以序之雜之必有深意，亦必有略例。至夫子為之傳，乃因其次第，而發明陰陽相生相對之義，以見易道之無窮，蓋文王之立法至精，而夫子之見理至大，二者皆不可以不知也。

The ordering of the sixty-four trigrams and the disorderly arrangement in the Zage are both attributed to King Wen. He must have had a profound intention in arranging them in this way, and there must be a general principle underlying his arrangement. When Confucius later composed commentaries on them, he followed the original order, explaining the principles of mutual generation and mutual opposition between yin and yang, to illustrate the boundless nature of the Way of the Changes. King Wen's establishment of principles was extremely precise, and Confucius' understanding of the principles was extremely comprehensive. Both aspects are essential to know.

韓孔諸儒，疑卦序若如夫子所言，則不應卦皆反對，故《程傳》於卦下既述夫子之意，又為上下篇義以繹其未盡之指，至歐陽修諸人，直斥《序卦》為非孔子之書者，妄也。若《雜卦》則乾坤之後，繼以《比》、《師》，其次敘又與《序卦》無一同者，是豈無義存焉？而諸儒皆莫之及，唯元儒胡氏於篇終微發其端，未竟其緒也，今因程胡之說而詳推二篇之所以類序錯綜者，目曰明義以附焉。

Han Kangbo and Kong Yingda doubted that if the order of the hexagrams were truly as Confucius described in the "Xugua Zhuan," then each pair of hexagrams should not be opposites in reverse order. Therefore, Cheng Yi wrote the "Yi Zhuan," summarizing Confucius's meaning under each hexagram and also composing the "Shangxia Pianyi" to elaborate on the ideas not fully explained in the "Xugua." As for Ouyang Xiu and his followers, who directly claimed that the "Xugua" was not Confucius's work, their view was absurd. Regarding the "Zagua," after the hexagrams of Heaven and Earth come "Bi" and "Shi," their order is entirely different from that in the "Xugua." How could there be no meaning to be found in this? Previous scholars did not discuss this, but a Yuan dynasty scholar named Hu only slightly hinted at it at the end of the text without completing the thread. Now, following the views of Cheng and Hu, I will carefully investigate the reasons for the classification and ordering of the "Xugua" and "Zagua," as well as the interwoven structure of their texts, and title this explanation "Mingyi," which I will append here.

程子有上下篇義，今祖其意而詳推之。上篇，陽也，天道也，故凡天道之正，陽卦陽爻之盛，及陰陽長少先後有序者，皆上篇之卦也。下篇，陰也，人事也，故凡人事之交，陰卦陰爻之盛，及陰陽交感雜亂，長少先後無序者，皆下篇之卦也。故以八卦而論，乾坤，陰陽之純也，坎離，陰陽之中也，皆正中之正，放為陽。震巽，陰陽始交也，艮兌，交之極也，皆正中之交，故為陰。

Cheng Yi wrote the "Explanations of the Upper and Lower Sections." Today, following his meaning, we elaborate further. The Upper Classic belongs to yang, discussing the way of heaven. Therefore, all matters concerning the proper way of heaven, the predominance of yang trigrams and yang lines, and the orderly sequence of yin and yang, seniority and juniority, and precedence and succession, are included in the Upper Classic. The Lower Classic belongs to yin, discussing human affairs. Therefore, all matters concerning human interactions, the predominance of yin trigrams and yin lines, and the disorderly intermingling of yin and yang, as well as the disorder of seniority and juniority and precedence and succession, are included in the Lower Classic. Regarding the Eight Trigrams: Qian and Kun are the pure forms of yin and yang; Kan and Li are the intermediate forms of yin and yang. They occupy the four cardinal positions in the Earlier Heaven diagram, so they are considered the true center of the true. Therefore, they are classified as yang. Zhen and Xun are the beginning of the interaction between yin and yang; Gen and Dui are the extreme of such interaction, so they are considered the true center of the interaction. Therefore, they are classified as yin. The term "true center" refers to Qian, Kun, Kan, and Li occupying the four cardinal positions in the Earlier Heaven diagram.

以八卦之交而論，唯《否》、《泰》天地之交，交中之正也，故為陽，《咸》、《恆》、《損》、《益》，《既濟》、《未濟》六子之交，交中之交也，故為陰。又乾交陽卦。凡六，《需》、《訟》、《無妄》，《大畜》皆為陽盛，唯以爻畫參之，則《大壯》為陽過中，《遯》為陰浸長，故雖陽卦而居陰也。坤交陰卦凡六，《晉》、《明夷》、《萃》、《升》皆為陰盛，唯《臨》則陽浸長，《觀》則陰過中，故雖陰卦而居陽也。

In terms of the interaction between the Eight Trigrams, only the hexagrams 《否》 and 《泰》 represent the interaction between Heaven and Earth, which is considered the correct form of interaction and thus belongs to the yang category. The hexagrams 《咸》, 《恒》, 《损》, 《益》, 《既济》, and 《未济》 represent the interaction between the six subsidiary trigrams, which is considered the intersecting form of interaction and thus belongs to the yin category. Specifically, 《咸》 and 《损》 are paired with the trigrams 《艮》 and 《兑》, 《恒》 and 《益》 are paired with 《震》 and 《巽》, and 《既济》 and 《未济》 are paired with 《坎》 and 《离》. When Heaven interacts with yang hexagrams, there are six such hexagrams: 《需》, 《讼》, 《无妄》, and 《大畜》 are considered to have strong yang energy. However, according to the lines of the hexagrams, 《大壮》 has yang energy that has passed the center, and 《遯》 has yin energy that is gradually growing, so although they belong to the yang category, they are placed in the lower section of the yin position. Similarly, when Earth interacts with yin hexagrams, there are also six such hexagrams: 《晋》, 《明夷》, 《萃》, and 《升》 are considered to have strong yin energy. However, 《临》 has yang energy that is gradually growing, and 《观》 has yin energy that has passed the center, so although they belong to the yin category, they are placed in the upper section of the yang position. This shows that the division into sections is not only based on the hexagram itself but also takes into account the waxing and waning of the lines.

又乾交陰卦凡六，《小畜》、之履》、《同人》。《大有》皆五陽而一陰，陽之盛也，唯以爻畫參之，則《夬》為陽已亢，《姤》為陰始生，故不得為陽而為陰也。坤交陽卦凡六，《師》、《比》、《謙》、《豫》、《剝》、《復》皆五陰而一陽，凡陽有主陰之義，陰雖多，不為盛而為役，陽雖少，不為衰而為主，故皆不為陰而為陽也。

The trigrams Qian (Heaven) and Yin (Earth) combine to form six trigrams. The trigrams Xiao Xu (Small Accumulation), Li (Following), Tong Ren (Union with Others), Da You (Great Possession) all consist of five yang lines and one yin line, belonging to the category of yang reaching its peak. However, when examining the lines, the trigram Guai (Decision) has five yang lines decisively removing one yin line, indicating yang has already reached its extreme. The trigram Gou (Meeting) has one yin line initially meeting five yang lines, indicating yin has just begun to emerge, at the moment when extreme yang is about to turn into yin. Therefore, these two trigrams are not considered yang but yin. Similarly, the trigrams Kun (Earth) and Qian (Heaven) combine to form six trigrams. The trigrams Shi (Army), Bi (Comparison), Qian (Humility), Yu (Joy), Po (Decrease), Fu (Return) all consist of five yin lines and one yang line. Generally, one yang line has the meaning of governing many yin lines. Although yin is numerous, it is not considered flourishing but rather subordinate; although yang is few, it is not considered weak but rather dominant. Therefore, these six trigrams are not considered yin but yang.

又陽卦相交凡六，《屯》、《蒙》、《頤》長少先後以序者也，故為陽：《蹇》、《解》、《小過》失序者也，故為陰。又陰卦相交凡六，獨《大過》為《頤》之對，又得其序，故亦為陽；《家人》、《睽》、《革》、《鼎》、《中孚》皆陰也，《革》、《鼎》得序，故猶為陰中之陽也。

Another set of six trigrams formed by the mutual interaction of yang trigrams are: 《Zun》, 《Meng》, and 《Yi》, which represent the eldest son, middle son, and youngest son in order, thus they are considered yang. The trigrams 《Jian》, 《Jie》, and 《Xiao Guo》, however, have the younger sons preceding the elder ones, thus they are considered yin—this disorder in sequence refers to the youngest or middle son coming before the eldest son. Similarly, among the yin trigrams, there are also six trigrams formed by mutual interaction: 《Da Guo》 and 《Yi》 are paired and maintain the correct order, so they are also considered yang. The trigrams 《Jia Ren》, 《Kui》, 《Ge》, 《Ding》, and 《Zhong Fu》 are considered yin, among which 《Ge》 and 《Ding》 have the correct order of elder and younger, so they are still considered as yang within yin.

又陰陽相交之卦凡十有二，其得序者六，《隨》、《蠱》、《噬嗑》、《賁》為陽中之陰，《井》田為陰中之陽；其失序者六，《漸》、《歸妹》、《豐》、《旅》、《渙》、《節》陰小之陰也。二篇之分既定，其逐節逐卦次第先後，則以陰陽盛衰消長之義次之，如後論。

There are twelve trigrams that are mutually intersecting in yin and yang: among them, the six that follow the proper order are The Following, The Pestilence, The Snap, and The Ornament, which belong to yin within yang; The困 and The Well belong to yang within yin (the original text mistakenly says 'The Well Field' and should be 'The困 and The Well'). The six that lose the proper order are The Gradual, The Return of the Maiden, The Abundance, The Traveler, The Dispersal, and The Regulation, which belong to yin within yin (the original text mistakenly says 'yin small yin' and should be 'yin within yin'). Once the division into upper and lower parts is determined, the order of each section and each trigram is arranged according to the principle of the waxing and waning of yin and yang, as detailed in the following discussion.

《乾》，《坤》、《屯》、《蒙》、《需》、《訟》、《師》、《比》、《小畜》、《履》右（上）陽卦第一節。《泰》、《否》、《同人》；《大有》、《謙》、《豫》《隨》、《蠱》、《臨》、《觀》、《噬嗑》、《賁》、《剝》、《復》右（上）陽卦第三節。

The ten hexagrams listed above, namely 《Qian》, 《Kun》, 《Zun》, 《Meng》, 《Xu》, 《Song》, 《Shi》, 《Bi》, 《Xiao Xu》, and 《Lu》, constitute the first section of the upper canon's yang hexagrams. The next six hexagrams, 《Tai》, 《Pou》, 《Tong Ren》, 《Da You》, 《Qian》, and 《Yu》, originally belong to the second section of yang hexagrams. The eight hexagrams 《Sui》, 《Gu》, 《Lin》, 《Guan》, 《Shi He》, 《Ben》, 《Bao》, and 《Fu》 are actually the third section of yang hexagrams. The original text here omitted the heading 「阳卦第二节」, causing the names of the hexagrams from both sections to be listed together, leaving only the heading 「阳卦第三节」. In the original book, such headings were written as 「右某卦第几节」, and when arranged vertically, 「右」 referred to the series of hexagrams listed above.

《無妄》、《大畜》、《頤》、《大過》、《坎》、《離》右（上）陽卦第四節。《咸》、《恆》《遯》、《大壯》、《晉》、《明夷》、《家人》、《睽》、《蹇》、《解》右（上）陰卦第一節。

The six hexagrams listed above, namely 《Wu Wang》, 《Da Xu》, 《Yi》, 《Da Guo》, 《Can》, and 《Li》, form the fourth section of the Yang hexagrams in the Upper Canon; together with the previous three sections, the thirty hexagrams of the Upper Canon are now completed. The following ten hexagrams, namely 《Xian》, 《Heng》, 《Dun》, 《Da Zhuang》, 《Jin》, 《Ming Yi》, 《Jia Ren》, 《Kui》, 《Jian》, and 《Jie》, constitute the first section of the Yin hexagrams in the Lower Canon; since the Lower Canon begins with the principle of the relationship between husband and wife, it is led by 《Xian》 and 《Heng》.

《損》、《益》、《央》、《舶》、《萃》、《升》右（上）陰卦第二節。《困》、《井》、《革》、《鼎》、《震》、《艮》、《漸》、《歸妹》，《豐》、《旅》、《巽》、《兌》右（上）陰卦第三節。《渙》、《節》、《中孚》，《小過》、《既濟》、《未濟》右（上）陰卦第四節。

The six hexagrams above, namely 《Zhen》, 《Yi》, 《Guai》, 《Gou》, 《Cui》, and 《Sheng》, are the second section of the yin hexagrams (the original text mistakenly writes 《Yang》 as 《Guai》 and 《Bo》 as 《Gou》). The following twelve hexagrams, namely 《Kun》, 《Jing》, 《Ge》, 《Ding》, 《Zhen》, 《Gen》, 《Jian》, 《Gui Mei》, 《Feng》, 《Lv》, 《Xun》, and 《Dui》, are the third section of the yin hexagrams. The next six hexagrams, namely 《Huan》, 《Jie》, 《Zhong Fu》, 《Xiao Guo》, 《Ji Ji》, and 《Wei Ji》, are the fourth section of the yin hexagrams; the lower section of the sixty-four hexagrams ends here. The entire book of sixty-four hexagrams is thus divided into upper and lower sections, each consisting of four parts.

陽卦第一節《乾》、《坤》者，眾卦之宗，故居篇首。先儒謂《周易》首《乾》，則此是文王所定，不可易也。《乾》、《坤》之外，三男為尊。《屯》、《蒙》者，二男之卦也，而皆長少先後不失其序，得陽道之正，故次《乾》、《坤》焉。《需》、《訟》上下皆陽卦，二五皆陽爻，陽之盛之，故次《屯》、《蒙》焉，《師》、《比》皆以一陽為眾陰主，而居二五中位，亦陽之盛也，故次《需》、《訟》焉。《小畜》、《履》五陽一陰，陽既極多，而二陰又退居三四之偏位，皆陽盛之卦也，故次《師》、《比》焉。

First comes the section on the yang trigrams. The trigrams 乾 (Qian) and 坤 (Kun) are the main trigrams of all others, so they are placed at the beginning of the entire text. Previous Confucian scholars said that the Zhou Changes (I Ching) begins with 乾, which was determined by King Wen and cannot be changed. Outside of 乾 and 坤, the three yang trigrams 震 (Zhen), 坎 (Kan), and 艮 (Gen) are considered most esteemed. The trigrams 屯 (Zun) and 蒙 (Meng) are formed by the repetition of two yang trigrams, and their order of seniority and sequence is correct, following the proper path of yang, so they are placed after 乾 and 坤. The trigrams 需 (Xu) and 讼 (Song) have both upper and lower trigrams that are yang, and their second and fifth lines are also yang lines, indicating the full strength of yang, so they come after 屯 and 蒙. The trigrams 师 (Shi) and 比 (Bi) each have one yang line as the master over the many yin lines, and that single yang line is in the middle positions of second and fifth, also showing the full strength of yang, so they are placed after 需 and 讼. The trigrams 小畜 (Xiao Xu) and 履 (Li) each have five yang lines and one yin line. Since yang is already very strong, the single yin line in both trigrams is in the secondary positions of third and fourth, not in the central position. Therefore, these two trigrams are also considered to be of strong yang, and they are placed after 师 and 比. Since the upper section of the Zhou Changes focuses on heavenly principles, the first ten trigrams are arranged in order according to the amount of yang and whether it is in the central or secondary positions.

陽卦第二節《泰》、《否》者，《乾》、《坤》之合體，義同《乾》、《坤》者也，然以其僦》、《坤》之交，故亞於《乾》、《坤》，《同人》、《大有》義反《師》。《比》。然以其剛多極盛，故同《小畜》、《履》而亞於《師》、《比》，《謙》、《豫》義反《小畜》、《履》，然陽為卦主，故同《師》、《比》而亞於《小畜》、《履》，此六者併為陽盛之次也。

Now discussing the second section of the yang hexagrams. The hexagrams of Tai and Po are composed of the trigrams Qian and Kun, and their meanings are similar to those of Qian and Kun; however, they are hexagrams where Qian and Kun intersect, so they come after Qian and Kun. The hexagrams of Tongren and Dàyǒu have meanings opposite to those of Shī and Bǐ—wherein one yang line is the master of many yin lines, here one yin line is the destination of many yang lines; although there are many yang lines and they are extremely prosperous, they are similar to Xiaoxù and Lǚ and come after Shī and Bǐ. The hexagrams of Qian and Yu have meanings opposite to those of Xiaoxù and Lǚ; however, the single yang line in each hexagram is the main line, so they are similar to Shī and Bǐ and come after Xiaoxù and Lǚ. These six hexagrams are grouped together as the second level of yang prosperity.

陽卦第三節以上二節，除《屯》、《蒙》為三男純卦，餘則皆有《乾》、《坤》為主，未嘗有男女之交也，故曰陽盛。至《隨》、《蠱》、《噬嗑》，《賁》然後有男女之交，是陰始生也，然而長少先後皆不失序，故猶為陽中之陰，《隨》、《蠱》之後，繼以《臨》、《觀》、《噬嗑》，《賁》之後，繼以《剝》、《復》，則陽又盛矣。

Now let us discuss the third section of the yang trigrams. In the previous two sections, the trigrams except for 《Zun》 and 《Meng》, which are three yang trigrams stacked upon each other, are all based on 《Qian》 and 《Kun》, and have not yet shown the image of male and female union. Therefore, they are called yang in excess. It is only in 《Sui》, 《Gu》, 《Shi He》, and 《Ben》 that male and female union appears, indicating the beginning of yin. However, in these, the order of seniority and sequence is still maintained, so they are still considered yin within yang. After 《Sui》 and 《Gu》 come 《Lin》, 《Guan》, 《Shi He》, and 《Ben》, and after 《Ben》 come 《Bo》 and 《Fu》. 《Lin》 represents two yang trigrams growing, 《Guan》 still has two yang trigrams above, and 《Bo》 and 《Fu》 both have one yang trigram as the main element. Therefore, it is said that the yang energy again becomes strong.

陽卦第四節《無妄》、《大畜》，《乾》與陽卦合體，義同《需》、《訟》，然二五不皆陽爻，故亞於《需》、《訟》。《頤》、《大過》男女類，分長少先後，義同《屯》、《蒙》。然二卦不皆陰卦第一節下篇主人事之交，故以夫婦之道始。男女之合，少則情專，老則誼篤，故《咸》為首，《恆》次之，《遯》、《大壯》陰長陽過，陰之盛也，故次《咸》、《恆》。《晉》、《明夷》上下皆陰卦，二五皆陰爻，義反陽之《需》、《訟》。《家人》、《睽》三陰之卦也，而又長少失序，陰道也，義反陽之《屯》、《蒙》，故四卦次《遯》、《大壯》。《蹇》、《解》本三陽之卦，而亦長少失序，義反《屯》、《蒙》，故從《家人》、《睽》焉。

Now discussing the fourth section of the yang trigrams. The hexagrams 《Wu Wang》 and 《Da Xu》 have the trigram Qian as their outer trigram and combine with the yang trigrams Zhen and Gen, their meaning is similar to 《Xu》 and 《Song》; however, the second line and fifth line are not necessarily yang lines, so they come after 《Xu》 and 《Song》. 《Yi》 and 《Da Guo》 belong to the categories of male and female respectively, each having distinctions of age and order, their meaning is similar to 《Zun》 and 《Meng》; however, these two hexagrams are not all— the original text here is incomplete, and the next part is the first section of the lower canon's yin trigrams. First section of the yin trigrams. The lower canon focuses on human relationships, so it begins with the way of husband and wife. The union of male and female, young love is devoted, old love is sincere and deep, so 《Xian》 (a young man and young woman feeling each other) comes first, and 《Heng》 (an elder man and elder woman growing old together) comes next. 《Dun》 and 《Da Zhuang》 represent yin gradually growing while yang has passed its middle point, belonging to the peak of yin, so they come after 《Xian》 and 《Heng》. 《Jin》 and 《Ming Yi》 have both their upper and lower trigrams as yin trigrams, and the second and fifth lines are also yin lines, their meaning is exactly opposite to 《Xu》 and 《Song》 in the yang trigrams. 《Jia Ren》 and 《Kui》 are formed by three female trigrams stacked together, and they also have disorder in age and order, which fits the nature of yin, their meaning is opposite to 《Zun》 and 《Meng》 in the yang trigrams, so these four hexagrams come after 《Dun》 and 《Da Zhuang》. 《Jian》 and 《Jie》 are originally formed by three male trigrams stacked together, but they also have disorder in age and order, their meaning is opposite to 《Zun》 and 《Meng》, so they follow 《Jia Ren》 and 《Kui》.

陰卦第二節《損》、《益》二少二長之交，義同《咸》、《恆》，《夬》、《姤》陽極陰生，義同《遯》、《大壯》，《萃》、《升》、《坤》與陰卦交，義同《晉》、《明夷》，故六卦相繼，陰盛之次也。

The second section of the Yin trigrams. The hexagram "Zhen" (Loss) is formed by the combination of Gen (Mountain) and Dui (Lake), representing the young male and young female. The hexagram "Yi" (Benefit) is formed by Zhen (Thunder) and Xun (Wind), representing the elder male and elder female. Both pairings involve the combination of young with young and elder with elder, similar in meaning to "Xian" (Attraction) and "Heng" (Permanence). The hexagrams "Guai" (Decision) and "Gou" (Meeting) represent the situation where Yang has reached its peak and Yin begins to emerge again, similar in meaning to "Dun" (Retreat) and "Da Zhuang" (Great Strength). The hexagrams "Cui" (Gathering) and "Sheng" (Ascending) are formed by the combination of Kun (Earth) with Dui (Lake) and Xun (Wind), two Yin trigrams, similar in meaning to "Jin" (Progress) and "Ming Yi" (Bright Abduction). Therefore, these six hexagrams follow each other in sequence, forming the second order of Yin dominance.

陰卦第三節《困》、《井》男女交而以序，義同陽之《隨》、《蠱》，《噬嗑》、《賁》，陰中之陽也，《革》、《鼎》三陰之卦，同《家人》、《睽》，然長少以序，故從《困》、《井》，猶《大過》之從《頤》也。《震》、《艮》雖下經之主，然本陽卦也，故此六卦併為陰小之陽，《漸》、《歸妹》、《豐》、《旅》男女交而失序，與《困》、《井》、《革》、《鼎》反，《巽》、《兌》陰卦，與《震》、《艮》反，此六卦則又自陽而向乎陰矣。

The third section of the yin trigrams. The hexagrams "困" (Obstruction) and "井" (Well) represent the interaction between male and female with proper order. In "困", the trigram Kan (Water) representing the middle male is below, and Dui (Lake) representing the young female is above. In "井", the trigram Xun (Wind) representing the elder female is below, and Kan (Water) representing the middle male is above. Their combinations do not exceed the proper sequence, and their meanings are similar to the yang hexagrams "随" (Following), "蛊" (Restlessness), "噬嗑" (Breaking the Bones), and "贲" (Brightness), thus belonging to yang within yin. The hexagrams "革" (Revolution) and "鼎" (Tripod) are formed by three yin trigrams stacked together, which is similar to "家人" (Family) and "睽" (Strife), but because there is proper order of age, they follow "困" and "井", just as "大过" (Great Excess) follows "颐" (Nourishment). Although "震" (Thunder) and "艮" (Mountain) are the main trigrams of the lower section, they themselves are yang trigrams, so these six trigrams are grouped as yang within yin. The hexagrams "渐" (Gradual Progress), "归妹" (Return of the Maiden), "丰" (Great Excess), and "旅" (Traveler) represent the interaction between male and female with disorder, or the pairing of a young male with an elder female, or a young or middle female with an elder male, which is the opposite of "困", "井", "革", and "鼎". The trigrams "巽" (Wind) and "兑" (Lake) are yin trigrams, which is the opposite of "震" and "艮". These latter six trigrams thus shift from yang to yin.

陰卦第四節《漸》、《歸妹》、《豐》、《旅》、《渙》、《節》六卦，男女交而失序，相類也。然《漸》、《歸妹》兩卦，長男長女皆在焉，《豐》、《旅》有長男在焉，《渙》、《節》唯長女在焉，則《渙》、《節》者變之窮，陰道之極也。《中孚》、《小過》與上篇《頤》、《大過》相對，《大過》雖陰卦，以得其序而從《頤》，故《小過》雖陽卦，以失其序而從《中孚》，其義與《蹇》、《解》之從《家人》、《睽》者同，併為陰復盛之卦也。《既濟》、《未濟》終篇，所重在《未濟》，蓋三陽失位，男之窮也，陰盛之極也，然物不可窮也，故受之以《未濟》終焉。

The fourth section of the yin hexagrams. The hexagrams 《Gradual Progress》, 《Return of the Maiden》, 《Great Excess》, 《Traveler》, 《Dispersal》, and 《Restraint》 are all about the interaction between male and female elements that have lost their proper order, and they involve similar types. However, 《Gradual Progress》 and 《Return of the Maiden》 both contain the eldest son and eldest daughter, while 《Great Excess》 and 《Traveler》 only contain the eldest son, and 《Dispersal》 and 《Restraint》 only contain the eldest daughter. This shows that 《Dispersal》 and 《Restraint》 have reached the extreme, and the yin path has reached its peak. 《Middle Trust》 and 《Small Excess》 correspond with 《Yi》 and 《Great Excess》 from the upper section. Although 《Great Excess》 is a yin hexagram, it follows 《Yi》 because it maintains the proper order; similarly, although 《Small Excess》 is a yang hexagram, it follows 《Middle Trust》 because it has lost the proper order. This principle is the same as 《Jian》 and 《Jie》 following 《Family》 and 《Kui》, as they are all hexagrams with heavy yin energy that is turning back to strength. 《Already Completed》 and 《Not Yet Completed》 conclude the entire text, with emphasis on 《Not Yet Completed》: the three yang lines at positions nine two, nine four, and upper nine in 《Not Yet Completed》 are all in even positions and not in the correct place, symbolizing the hardship of men and the extreme dominance of yin. However, things cannot be entirely exhausted, so the 《Sequence of Hexagrams》 says, 'Things cannot be exhausted,' using 《Not Yet Completed》 to follow 《Already Completed》 and conclude the entire text. These six hexagrams in this section correspond exactly with the fourth section of the upper part.