The Luo Writing arranges numbers in a way that positions add up to ten and when multiplied together they form hundreds.

一與九對成十，十自乘其積一百。九自乘八十一；一自乘一；一乘九、九乘一，俱為九，共十八；合之一百，與十自乘積同。二與八對成十。八自乘六十四；二自乘四；二乘八、八乘二，俱十六，共三十二；合之一百。

One and nine are opposite and combined to make ten. Ten multiplied by itself gives one hundred. Nine multiplied by itself gives eighty-one, and one multiplied by itself gives one. One multiplied by nine and nine multiplied by one each give nine, totaling eighteen. Adding eighty-one, one, and eighteen together gives exactly one hundred, the same as ten multiplied by itself. Two and eight are opposite and combined to make ten. Eight multiplied by itself gives sixty-four, and two multiplied by itself gives four. Two multiplied by eight and eight multiplied by two each give sixteen, totaling thirty-two. Adding sixty-four, four, and thirty-two together also gives exactly one hundred.

三與七對成十。七自乘四十九；三自乘九；三乘七、七乘三，俱二十一，共四十二；合之一百。四與六對成十。六自乘三十六；四自乘十六；四乘六、六乘四，俱二十四，共四十八；合之一百。中五含五成十。五自乘二十五；又五自乘二十五；又五互乘各二十五，共五十。合之一百。

Three and seven are opposite and together make ten: seven multiplied by itself gives forty-nine, three multiplied by itself gives nine; three multiplied by seven and seven multiplied by three each give twenty-one, the two results together make forty-two; adding the three numbers together makes exactly one hundred. Four and six are opposite and together make ten: six multiplied by itself gives thirty-six, four multiplied by itself gives sixteen; four multiplied by six and six multiplied by four each give twenty-four, the two results together make forty-eight; adding the three numbers together makes exactly one hundred. The central five contains one five and together makes ten: one five multiplied by itself gives twenty-five, the other five multiplied by itself also gives twenty-five; the two fives multiplied together each give twenty-five, totaling fifty; adding the three numbers together also makes exactly one hundred.

勾三，股四，弦五。勾九，股十二，弦十五。勾二十七，股三十六，弦四十五。勾八十一，股一百零八，弦一百三十五。此《洛書》四隅合中方，而寓四勾股之法者，推之至於無窮法皆視此。

Three, four, five; nine, twelve, fifteen; twenty-seven, thirty-six, forty-five; eighty-one, one hundred and eight, one hundred and thirty-five. This is how the four corners and the center of the Luo Writing correspond with the square, and it contains four sets of the Pythagorean theorem. In each subsequent set, the numbers are three times those of the previous set, while the ratio of the base, the height, and the hypotenuse remains three to four to five. Continuing in this manner indefinitely, the method remains the same.

《河圖》之數，五十有五，《洛書》之數，四十有五，合為一百，此天地之全數也，以一百之全數，為斜界而中分之，則自一至十者，積數五十有五，自一至九者，積數四十有五，二者相交，而成河洛數之兩三角形矣。凡積數自少而多，必以三角，而破百數之全方，以為三角，其形不離乎此二者，下諸圖之根，實出於此。

The total numbers of the Yellow River Chart amount to fifty-five, and those of the Luo Writing amount to forty-five. When added together, they make exactly one hundred, which is the complete number of heaven and earth. If the number one hundred is arranged into a square, and then divided diagonally from the center, one side adds up from one to ten, resulting in a total of fifty-five; the other side adds up from one to nine, resulting in a total of forty-five. These two totals interlock to form the triangular shapes of the Yellow River Chart and the Luo Writing. Whenever numbers accumulate gradually from small to large, they necessarily form a triangle. When the complete square of one hundred is divided into two triangles, the resulting shapes can only be these two. The fundamental principles of the following diagrams actually originate from here.

《河圖》之數，自一至十；《洛書》之數，自一至九。象之已分者也。圖則生數居內，成數居外，書則奇數居正，偶數居偏，位之已變者也。如前圖破全方之百數，以為河洛二數，又就點數十位，中涵冪形之九層，以為河洛合一之數，則雖其象未分，其位未變，而陰陽相包之理，三極互根之道，已粲然默寓於其中矣。故為分析以明之，如後點數應《河圖）十位周圍三角，分三重，中一重九，次內一重二九一十八，外一重三九二十七，除中心，凡五十四。若自上而下作三層，亦如之。中含六角，亦分三重，中一重六，次內一重二六一十二，外一重三六一一十八，除中心，凡三十六。若自十而下作三層，亦如之。

The numbers of the Yellow River Chart range from one to ten, while the numbers of the Luo Writing range from one to nine. This refers to the case where the figures are already separated. The Yellow River Chart places the generating numbers one, two, three, four, and five inside and the forming numbers six, seven, eight, nine, and ten outside. The Luo Writing places the odd numbers at the four cardinal points and the even numbers at the four corners. This refers to the case where the positions have already changed. If we take the entire square of one hundred numbers and divide it into the numbers of the Yellow River and Luo Writing, and also arrange the points into a triangle of ten positions, with the inner triangle containing nine layers of square shapes, this represents the combined numbers of the Yellow River and Luo Writing. In this case, the figures are still not separated and the positions have not changed, yet the principle of yin and yang being interwoven and the principle of the three supreme entities—Heaven, Earth, and Humanity—being interdependent are already clearly and silently contained within. Therefore, further analysis is needed to explain it, and the diagram is presented as follows. Using the point numbers to correspond with the ten positions of the Yellow River Chart: arranged in a triangle around the perimeter, divided into three layers. The innermost layer has nine points, the next layer has eighteen points (two times nine), and the outermost layer has twenty-seven points (three times nine). Removing the central point, the total is fifty-four points. If instead we count from top to bottom in three layers, the result is the same. The hexagon contained in the center is also divided into three layers: the innermost layer has six points, the next layer has twelve points (two times six), and the outermost layer has eighteen points (three times six). Removing the central point, the total is thirty-six points. If instead we count from top to bottom in three layers, the result is the same.

冪形應《洛書》九位周圍三角分三重，中一重九，次內一重三九二十七，外一重五九四十五，凡八十一。若自上而下作三層，亦如之。中含六角，亦分三重，中一重六，次內一重三六一十八，外一重五六三十，凡五十

Using hexagonal shapes to correspond to the nine positions of the Luo Square: arranged in a triangle around the center, divided into three layers. The innermost layer is nine, the next layer is three times nine, which is twenty-seven, and the outermost layer is five times nine, which is forty-five. In total, this sums to eighty-one. If instead we count from top to bottom in three layers, the result is the same. The hexagon contained within also divides into three layers: the innermost layer is six, the next layer is three times six, which is eighteen, and the outermost layer is five times six, which is thirty. The total should be fifty-four — however, the original text here has the character 'four' missing, and it is written as 'a total of fifty'.

以上諸圖，本同一根，雖積數若異，而其為九六之變則一也。九六可分為內外中之三重，亦可分為上下中之三層，就每重每層論之，則九為天而包地，六為地而涵於天，心為人而主乎天地。統三重而論之，則外為天，內為地，而中為人也。統三層而論之，則上為天，下為地，面中為人也。

The diagrams mentioned above all originate from the same root; although they appear different when viewed in terms of numbers, they are all variations of the numbers nine and six, which remain consistent. The numbers nine and six can be divided into three layers: inner, outer, and central, or into three levels: upper, lower, and central. When examining each layer or level, nine represents heaven and encompasses the earth, while six represents the earth and is contained within heaven. The central point represents human beings, who govern heaven and earth. When considering the three layers as a whole, the outer layer is heaven, the inner layer is earth, and the central part is human. When considering the three levels as a whole, the upper level is heaven, the lower level is earth, and the central level is human.

又合而論之，則九六者，在天為陰陽，在地為剛柔，在人為陰陽剛柔之會，而其心則天地人之極也，以上下分者，其心有三，所謂三極之道，三才各具一太極也；以內外分者，其心惟一，所謂人者天地之心，三才經體一太極也。

When combined, the numbers nine and six in the heavens represent yin and yang, on the earth they represent strength and flexibility, and on the human body they represent the convergence of yin and yang, strength and flexibility. The center in this context is the ultimate point of heaven, earth, and humanity. When divided from top to bottom, there are three centers, known as the 'three ultimate principles,' with each of the three elements—heaven, earth, and humanity—possessing its own Taiji. When divided by internal and external, there is only one center, known as 'humanity as the heart of heaven and earth,' with the three elements interwoven within the same Taiji.

此圖之中，渾具理象數之妙者如此，故分而為圖，則應乎陰陽剛柔之義，根於極而迭運不窮，“聖人則之”，“易有太極，是生兩儀”，陽九陰六，命爻衍策者此也。分而為書，則應乎三才之義，主於人而成位其中，“聖人則之”，皇極既建，彝倫攸敘，參天貳地，垂範作疇者此也。

This diagram comprehensively embodies the profound principles of li (principle), xiang (image), and shu (number). When divided into the Yellow River Chart, it corresponds to the meaning of yin and yang, firmness and flexibility, rooted in the Taiji and continuously interchanging without end, which is precisely what is meant by "the sage models it." It is also exactly what is expressed by "the Book of Changes begins with the Taiji, from which the two principles (yin and yang) arise." Yang is numbered nine and yin six, which is used to determine the names of the lines (yao) and to deduce the principles of the divination method (shi ce). When divided into the Luo Writing, it corresponds to the meaning of the three elements (tian, di, ren), with humans at the center between heaven and earth. It is also "the sage models it." Once the Huajie (Imperial Standard) is established, the proper order of conduct (yi lún) is set, harmonizing with heaven and comparing with earth, thus establishing the principles and composing the "Hongfan" (Nine Categories) as explained.

或曰：《河圖》、《洛書》，出於兩時，分為兩象，今以一圖括之可乎？曰：十中涵九，故數終於十，而位止於九，此天地自然之紀，而圖書所以相經緯而未嘗相離也。非有十者以為之經，則九之體無以立，非有九者以為之緯，則十之用無以行，不知圖書之本為一者，則亦不知其所以二矣。

Someone asked: The Yellow River Chart and the Luo Writing appeared in different eras and are divided into two forms of imagery. Now, can they be combined into a single diagram? The answer is: Among ten there is nine, so the number ends at ten while the position goes only up to nine. This is the natural order of heaven and earth, and also the reason why the diagram and the writing form the meridians and threads of each other without ever separating. Without ten as the meridian, the essence of nine would not stand; without nine as the thread, the function of ten would not unfold. Not understanding that the diagram and the writing are originally one will prevent one from understanding why they are divided into two.

或曰：《河圖》、《洛書》，有定位矣，今以為有末變者何與？曰：《易大傳》之言《河圖》也，曰：“天一地二，天三地四，天五地六，天七地八，天九地十”，順而數之，此其未變者也。又曰：“天數五，地數五，五位相得，而各有合”，分而置之，此其定位者也，如《易》卦一每生二，以至六十有四，則其未變者也，乾南坤北，離東坎西，則其定位者也，不知未變之根，則亦不足以識定位之妙矣。

Someone asked: The "Heavenly River Chart" and "Luo Writing" each have a fixed direction, yet now it is said they have another state of "unchanged" form. What is the reason for this? The answer is: The "Great Transmission" of the "Book of Changes" discusses the "Heavenly River Chart," saying "Heaven is one, Earth is two; Heaven is three, Earth is four; Heaven is five, Earth is six; Heaven is seven, Earth is eight; Heaven is nine, Earth is ten." Counting in order, this is the form of the chart in its unchangeable state. It also says "Heavenly numbers are five, Earthly numbers are five; the five positions correspond and each has its match," when they are separated and arranged, this is the form after they have been fixed in position. It is like the hexagrams of the "Book of Changes" which start with one line and each time two lines are added, continuing until sixty-four hexagrams, which is the form in its unchangeable state; while the positioning of Qian in the south and Kun in the north, Li in the east and Kan in the west, is the form after they have been fixed in position. Without understanding the root of the unchangeable state, one cannot fully comprehend the subtlety of the fixed positions.

此圖左方注者，本數也，自一至九而用數全矣。中列注者，加數也，一加二為三，二加三為五，至於八加九而為十七，皆以本數遞加，而每層之冪積如之。右方注者，乘數也，一自乘一，其冪積一，二自乘四，其冪積合一三兩層而為四，至於九自乘八十一，

The numbers on the left are called "base numbers," ranging from one to nine, which provides all the necessary numbers. The numbers in the middle column are called "added numbers": one plus two is three, two plus three is five, continuing up to eight plus nine being seventeen. These are calculated by adding the base numbers layer by layer, and the products of each layer also increase in this sequence. The numbers on the right are called "multiplied numbers": one multiplied by itself gives one, and its product is one; two multiplied by itself gives four, and its product is formed by combining the first and third layers to make four; this continues up to nine multiplied by itself giving eighty-one—here the original text is missing some words, and the explanation is incomplete.

凡有數則有象，象不離乎數也，萬象起於方圓，而測方圓者以三角，此勾股所以為算之宗也。圓者天象，方者地象，三角形者人象，何則？天之道如環無端，故其象圓也。地之道奠定有常，故其象方也。人受性於天，受形於地，猶三角之形，其心則圓之心，其邊則方之邊也。今就九數而三分之？則一者圓之根也，而十數之內，唯六角八角，為有法之圓形，其自十以後，角愈多以至於無角者視此矣，此一六八所以為圓象之數也。

Whenever there is a number, there is also an image, and the image cannot be separated from the number. All images originate from square and circle, and measuring square and circle depends on the triangle, which is why the method of the Pythagorean theorem is considered the foundation of mathematics. The circle represents the image of heaven, the square represents the image of earth, and the triangle represents the image of humanity. Why is this so? The way of heaven is like a circle, having no beginning and no end, so its image is circular. The way of earth is stable and follows constant principles, so its image is square. Humans receive their nature from heaven and their form from earth, just like the triangle, whose center is circular and whose edges are square. Now, divide the nine numbers into three parts: the first part is the fundamental of the circle. Among numbers within ten, only the hexagon and octagon are considered to conform to the standard of a circle. From ten onwards, the more angles there are, until they approach having no angles at all, they should all be viewed according to this principle. This is why one, six, and eight are considered the numbers that represent the image of the circle.

先天之陽卦，曰震離兌乾；其陰卦，曰巽坎艮坤。後天之陽卦，曰乾震坎艮，其陰卦，曰坤巽離兌，不同何也？蓋先天分陰陽卦，自兩儀而分之，由陽儀以生者，皆陽卦也；由陰儀以生者，皆陰卦也。後天分陰陽卦，自爻畫以定之，其以陽為主者，皆陽卦也；其以陰為主者，皆陰卦也。先天則因乎畫卦之序而中分之，後天則卦之已成，觀其爻畫之多寡而命之也，其理如何？曰：陽儀上有陰卦，此所謂“立天之道曰陰與陽”也。

In the Earlier Heaven diagram, the yang hexagrams are "Zhen," "Li," "Dui," and "Qian," while the yin hexagrams are "Xun," "Kan," "Gen," and "Kun." In the Later Heaven diagram, the yang hexagrams are "Qian," "Zhen," "Kan," and "Gen," and the yin hexagrams are "Kun," "Xun," "Li," and "Dui." Why are they different? Originally, in the Earlier Heaven diagram, the division of hexagrams into yin and yang is based on the two principles: those derived from the yang principle are considered yang hexagrams, and those derived from the yin principle are considered yin hexagrams. In the Later Heaven diagram, the division into yin and yang hexagrams is based on the number of yin and yang lines: those with more yang lines are considered yang hexagrams, and those with more yin lines are considered yin hexagrams. The Earlier Heaven diagram divides the hexagrams in the order of drawing, splitting from the center, while the Later Heaven diagram already has the hexagrams drawn, and their classification is based on the number of lines. What is the underlying principle here? The answer is that the yang principle also gives rise to yin hexagrams, which is exactly what the "Sayings of the Hexagrams" states: "The way of Heaven is called yin and yang."

陰儀上有陽卦，此所謂“立地之道曰柔與剛”也。其法象之自然者如何？曰：火之炎熱光明，其為陽也，明矣。澤者水之積溼，為陽氣所驅，以滋潤萬物者也，是亦陽也，水之幽暗寒肅，其為陰也，明矣。山者土之隆起，與地為一體者也，是亦朋也。是放先天之卦，陰陽之象之正也，其變而後天，則火與澤從風而俱為陰，水與山從雷而俱為陽，蓋有由矣。

Above the Yin principle, the Yang trigrams arise, which is exactly what the "Commentary on the Hexagrams" says: "The way of establishing the earth is called flexibility and firmness." So how should we explain this in terms of natural phenomena? The answer is: fire is hot and bright, and it clearly belongs to Yang; a marsh is a gathering of water vapor, driven by Yang energy to moisten all things, so it also belongs to Yang. Water is dark and cold, and clearly belongs to Yin; a mountain is formed by earth rising, and is essentially part of the earth, so it also belongs to Yin. Therefore, the division of Yin and Yang in the Earlier Heaven hexagrams is the most accurate representation of the Yin and Yang phenomena. But when it comes to the Later Heaven, fire and marsh are counted as Yin along with wind, and water and mountain are counted as Yang along with thunder. There is a reason for this.

凡陰陽之氣，未有不合而成者也，然有感應先後之別焉，先有陽而遇陰者屬陽，先有陰而遇陽者屬陰，有陽氣在下將發而遇陰壓之，則奮而為雷矣，有陽氣在中將散而遇陰包之，則鬱而為雨矣，有陽氣直騰而上而遇陰承之，則止而為山矣，此皆主於陽而遇陰，所以皆為陽卦也。有陰在內，陽氣必入而散之，觀之陰霾盡而後風息可見也，有陰在中，陽氣必附而散之，觀之薪芻盡而後火滅可見也；有陰在外，陽氣必敷而散之，觀之溼潤盡而後澤竭可見也。

All yin and yang energies must first combine to form things; however, there is a distinction in the order of their interaction. If yang comes first and then meets yin, it belongs to yang. If yin comes first and then meets yang, it belongs to yin. When yang energy is about to rise from below and meets yin energy pressing upon it, it struggles upward and becomes thunder, which is the trigram Zhen. When yang energy is about to dissipate in the middle and meets yin energy enclosing it, it becomes condensed into rain, which is the trigram Kan. When yang energy rises continuously upward and meets yin energy holding it back, it stops and becomes mountain, which is the trigram Gen. These three cases are all instances where yang is dominant and then meets yin, so they are all yang trigrams. Conversely, if yin is inside, yang energy must penetrate it and scatter it. When the haze is completely dispersed and the wind calms down, this can be understood as the trigram Xun. If yin is in the middle, yang energy must attach to it and burn it away. When the firewood is completely burned and the fire dies out, this can be understood as the trigram Li. If yin is outside, yang energy must spread over it and evaporate it. When the dampness is completely evaporated and the marsh dries up, this can be understood as the trigram Dui.

此皆主於陰而遇陽，所以皆為陰卦也。總而淪之，唯乾純陽，坤純陰，不可變也。雷陽動之始，凡陰生之始，亦不可變也，火溫暖，澤發散，故以用言之則陽，然火根於陰之燥，澤根於陰之溼，故以體言之則陰；水寒涼，山凝固，故以用言之則陰，然水根於陽之噓而流，山根於陽之矗而起，故以體言之則陽。

These trigrams are all dominated by yin lines and meet yang lines, so they are generally regarded as yin trigrams. The editors of "Zhezhong" summarize (the original text mistakenly writes "lun" instead of "lun", which should be "lun") that among the eight trigrams, only "Qian" (Heaven) is entirely yang and "Kun" (Earth) is entirely yin. Their nature is simple and pure, so they are always fixed and unchangeable. "Zhen" (Thunder) represents thunder, which is the beginning of yang energy's movement, while "Xun" (Wind), mistakenly written as "Fan" in the original, represents wind, which is the beginning of yin energy's emergence. These two trigrams mark the starting points of yin and yang, and are also fixed and unchangeable. As for "Li" (Fire), its nature is warm and gentle, and "Dui" (Lake), which represents a marsh, has a nature that spreads out. In terms of function, they belong to yang. However, the root of fire lies in the dryness of yin, and the root of a marsh lies in the moistness of yin, so in terms of essence, they belong to yin. "Kan" (Water) has a nature that is cold and cool, and "Gen" (Mountain) has a nature that is solid and unmoving. In terms of function, they belong to yin. However, water flows due to the movement of yang energy, and mountains rise due to the firmness of yang energy, so in terms of essence, they belong to yang.

先天之象，著其用也，後天之象，探其根也，正如仁之發生為陽，而其柔和亦可以為陰，義之

The symbolic positions of the Earlier Heaven Trigrams reveal the functions of each trigram; the symbolic positions of the Later Heaven Trigrams trace the fundamental nature of each trigram. This is similar to "ren" (benevolence), which appears as yang when viewed from its aspect of generating all things, yet appears as yin when viewed from its aspect of gentleness and包容 (embrace). As for "yi" (righteousness), it can also be viewed from both aspects. The text here ends abruptly, with a section missing, followed by the title of the next diagram.

造化所以為造化者，天地水火而已矣，《易》卦雖有八而實唯四，何則？風即天氣之吹噓而下交於地者也，山即地形之隆起而上交於天者也，雷即火之鬱於地中而搏擊奮發者也，澤即水之聚於地上而佈散滋潤者也。道家言天地日月，釋氏言地水火風，西人言水火土氣，可見造化之不離乎四物也。

This diagram explains that the origin of creation and transformation in heaven and earth ultimately depends on four elements: heaven, earth, water, and fire. Although the Classic Trigrams of the I Ching consist of eight, in reality they are based on four. Why is this so? The trigram "Xun" (Wind) represents the movement of heavenly air descending and meeting the earth. The trigram "Gen" (Mountain) represents the rising form of the earth meeting the heavens. The trigram "Zhen" (Thunder) represents the accumulation of fire energy beneath the earth, bursting forth forcefully. The trigram "Dui" (Lake) represents the gathering of water on the surface and its spreading to nourish. Daoism speaks of heaven, earth, sun, and moon; Buddhism speaks of earth, water, fire, and wind; Westerners speak of water, fire, earth, and air. It is evident that all discussions of creation and transformation revolve around these four elements.

故先天以南北為經，而天地居之體也，以東西為緯，而水火居之用也。後天則以天地為體，而居四維，以水火為用，而居四正。雷者火之方發，故動於春；及火播其氣，則王於夏矣；澤者水之未收，故散於秋；及水歸其根，則王於冬矣。水火為天地之用，故居四正以司時令也。

Therefore, the Earlier Heaven diagram uses the north-south line as the meridian, with Heaven (Qian) at the northern end and Earth (Kun) at the southern end, representing the 'body'; the east-west line serves as the latitude, with Fire (Li) at the eastern end and Water (Kan) at the western end, representing the 'function'. In the Later Heaven diagram, Heaven and Earth are regarded as the 'body', placed at the four corners (Northwest Qian, Southwest Kun, Northeast Gen, Southeast Xun), while Water and Fire, as the 'function', are placed at the four cardinal directions (South Li, North Kan, East Zhen, West Dui). Thunder represents the state when Fire is about to act, hence Zhen (Thunder) is active in spring; when Fire spreads its energy, Li (Fire) becomes dominant in summer. Marsh represents the state when Water has not yet gathered, hence it spreads in autumn; when Water returns to its source, Kan (Water) becomes dominant in winter. Water and Fire are the functions of Heaven and Earth, hence they are placed at the four cardinal directions to govern the four seasons.

天氣聯兆於西北，至東南而下交於地，《易》所謂’天下有風”《娠》也。故乾巽相對而為天綱，地功緻役於西南，至東北而上交於天，《易》所謂“天在山中”《大畜》也，故坤艮相對而為地紀。天地為水火之體，故居四維以運樞軸也，天地水火，體用瓦根，以生成萬物，此先後天之妙也。

The vital energy of Heaven originates in the Northwest at the position of Qian and proceeds to the Southeast at the position of Xun, then descends to meet the Earth. This is what the Book of Changes refers to as "Heaven has Wind," which represents the image of the Gou hexagram (the original character "娠" is a variant form of "姤"). Therefore, Qian and Xun are opposite each other, forming the framework of Heaven. The function of Earth begins at the Southwest at the position of Kun and proceeds to the Northeast at the position of Gen, then ascends to meet Heaven. This is what the Book of Changes refers to as "Heaven in the Mountain," which represents the image of the Daxu hexagram. Therefore, Kun and Gen are opposite each other, forming the framework of Earth. Heaven and Earth are the fundamental substances of Water and Fire, so they reside at the four corners to rotate the central axis. Heaven, Earth, Water, and Fire are interdependent in their essence and function, thus being able to generate all things. This is precisely the harmonious complementarity between the Earlier Heaven diagram and the Later Heaven diagram.

若以卦畫論之，則震即離也，一“陰閉之於上則為震，兌即坎也，一陽敷之於下則為兌，巽即乾也，一陰行於下則為巽，艮即坤也，一陽亙於上則為艮，是以六十四卦始《乾》、《坤》，中《坎》、《離》，而終於《既濟》、《末濟》，，則知造化之道，天地水火盡之矣。

If we look at them in terms of hexagram diagrams: Zhen is a transformation of Li, by changing the top line of Li from yang to yin and closing it at the top, it becomes Zhen; Dui is a transformation of Kan, by changing the bottom line of Kan from yin to yang and spreading it out at the bottom, it becomes Dui; Xun is a transformation of Qian, by changing the bottom line of Qian to yin, it becomes Xun; Gen is a transformation of Kun, by changing the top line of Kun to yang, it becomes Gen. Therefore, the sixty-four hexagrams begin with Qian and Kun, have Kan and Li in the middle, and end with Ji Ji and Wei Ji (the original text mistakenly writes 'mèi jì' instead of 'wèi jì'). From this we can see that the principles of nature are fully expressed through the four elements of heaven, earth, water, and fire.

此圖先天凡四變而為後火也，蓋火之體陰也，其用則陽，而天用之，故乾中畫與坤

This diagram shows that the Earlier Heaven Trigrams undergo four transformations to become the Later Heaven Trigrams (the original text mistakenly writes 'after fire' instead of 'Later Heaven'). The general idea is that fire's essence belongs to yin, but its function belongs to yang, and heaven uses this functional aspect. Therefore, the line in the Qian trigram should be exchanged with the line in the Kun trigram. The original text then omits the following explanation and proceeds to the title of another diagram.

圖之左方，陽內陰外，即先天之震離兌乾，陽長而陰消也；其右方，陰內陽外，即先天之巽坎艮坤，陰長而陽消也，蓋所以象二氣之交運也。

The left half of this diagram has the yang numbers inside and the yin numbers outside, corresponding exactly to the sequence of trigrams Zhen, Li, Dui, and Qian in the Earlier Heaven arrangement, representing a phase where yang energy gradually increases while yin energy gradually decreases. The right half has the yin numbers inside and the yang numbers outside, corresponding exactly to the sequence of trigrams Xun, Kan, Gen, and Kun in the Earlier Heaven arrangement, representing a phase where yin energy gradually increases while yang energy gradually decreases. Using the inner and outer numbers from the Yellow River Chart to match the Earlier Heaven trigram positions is meant to symbolize the rhythmic interaction and mutual transformation of the two energies, yin and yang, where one grows while the other wanes.

直列《洛書》九數，而虛其中五以配八卦。陽上陰下，故九數為乾，一數為坤，因自九而逆數之，震八坎七艮六。乾生二陽也，又自一而順數之，巽二離三兌四，坤生三陰也，以八數與八卦相配，而先天之位合矣。

This diagram arranges the nine numbers of the Luo Shu in a vertical sequence, leaving the central number five empty, and uses the remaining eight numbers to correspond with the Eight Trigrams. Yang is above and Yin is below, so the largest number nine is paired with Qian, and the smallest number one is paired with Kun. Counting down from nine, eight is paired with Zhen, seven with Kan, and six with Gen, which are the three Yang trigrams generated from Qian (the original text mistakenly says "two Yang" instead of "three Yang"). Then counting up from one, two is paired with Xun, three with Li, and four with Dui, which are the three Yin trigrams generated from Kun. The eight numbers correspond one by one with the eight trigrams, thus aligning the positions of the Earlier Heaven trigrams with the numbers of the Luo Shu. As for the previous diagram titled "Later Heaven Trigrams Corresponding to the River Chart," the original text only retains the title, while the diagram and explanatory text have been lost.

火上水下，故九數為離；一一數為坎，火生燥土，故八次九而為艮；燥土生金，故七六次八而為兌為乾；水生溼土，故二次一而為坤；溼土生木，故三四次二而為震為巽：以八數與八卦相配，而後天之位合矣。

In the Later Heaven arrangement, the numbers are arranged with fire above and water below: the number nine is paired with the trigram Li (Fire), and the number one is paired with the trigram Kan (Water) (the original text mentions '一一数' which is interpreted as a single 'one'). Fire produces dry earth, so eight follows nine and is paired with Gen (Mountain); dry earth produces metal, so seven and six follow eight and are paired with Dui (Lake) and Qian (Heaven); water produces moist earth, so two follows one and is paired with Kun (Earth); moist earth produces wood, so three and four follow two and are paired with Zhen (Thunder) and Xun (Wind). The eight numbers are thus paired with the eight trigrams, making the positions of the Later Heaven trigrams correspond with the numbers of the Luo Writing.

《洛書》之左邊，本一二三四也；其右邊，本九八七六也。然陰陽之道，丑未之位必交，《洛書》之二與八，正東北西南之維，丑未之位，此其所以互《易》也。以此類之，則先天圖之左方，坤巽離兌；其右方，乾震坎艮，以震巽互而成先天也。後天圖之左方，坎坤震巽；具才；方。離艮兌乾，以艮坤互而成後天也。

The left half of the Luo Writing originally consists of the numbers one, two, three, and four; the right half originally consists of nine, eight, seven, and six. However, according to the principle of yin and yang, the positions of Chou and Wei must be exchanged. In the Luo Writing, the numbers two and eight are located in the Northeast and Southwest corners, which are the positions of Chou and Wei. This is the reason why two and eight must be exchanged. Following this principle, in the Earlier Heaven diagram, the left half originally consists of Kun, Xun, Li, and Dui, while the right half originally consists of Qian, Zhen, Kan, and Gen. The positions of Zhen and Xun must be exchanged to form the standard Earlier Heaven directions. In the Later Heaven diagram, the left half originally consists of Kan, Kun, Zhen, and Xun, while the right half originally consists of Li, Gen, Dui, and Qian. The positions of Gen and Kun must be exchanged to form the standard Later Heaven directions.

據先儒說，圖書出有先後，又或謂並出於伏羲之世，然皆不必深辨，先聖後聖，其揆一也，況天地之理，雖更萬年，豈不合契哉，《洛書》晚出，而其理不妨已具於《河圖》之中，是故以《易》象推配，亦無往而不合也。

According to previous Confucian scholars, the Yellow River Chart came before the Luo Writing. Some, however, say both appeared in the time of Fuxi. This does not need to be deeply debated. The principles used by the earlier sage and the later sage were originally the same. Moreover, the principles of heaven and earth, even if they had passed ten thousand years, would not differ from each other. Even if the Luo Writing appeared later, the principles it contains were already present in the Yellow River Chart. Because of this, using the hexagrams of the Book of Changes to match the numbers of the Yellow River Chart and the Luo Writing can always correspond correctly.

先天圖者，序卦之根也。序卦之法，以兩卦相對為義，有相對面翻覆不可變者，《乾》、《坤》、《坎》、《離》、《頤》、《大過》、《中孚》、《小過》是也 c有相對而翻覆可變者，《屯》、《蒙》以後，《既濟》、《未濟》以前，五十六卦皆是也。

The Earlier Heaven diagram is the origin of the order in the "Xugua" (Sequence of Hexagrams). The method of arranging the hexagrams in the "Xugua" is to take pairs of hexagrams that are opposite to each other. One category consists of hexagrams that are opposite to each other and remain unchanged when their structure is reversed. These eight hexagrams—Qian (Heaven), Kun (Earth), Kan (Water), Li (Fire), Yi (Abundance), Da Guo (Great Excess), Zhong Fu (Middle Trust), and Xiao Guo (Small Excess)—belong to this category (the original text here has a 'c', which is an extraneous symbol introduced during the reproduction process). The other category consists of hexagrams that are opposite to each other and become another hexagram when their structure is reversed. From Zun (Obstruction) and Meng (Youth) down to Ji Ji (Completion) and Wei Ji (Non-Completion), the fifty-six hexagrams all belong to this category.

就五一卜六卦之中，則翻覆而二體不《易》者十二卦，《需》，，，《訟》、《師》、《比》、《泰》、《否》、《同人》、《大有》、《晉》、《朋夷》、《既濟》、《未濟》也。翻覆而二體皆易者十二卦，《隨》、《蠱》、《咸》、《恆》、《損》、《益》、《震》、《艮》、《漸》、《歸妹》、《巽》、《兌》也。其翻覆而止於一體易者三十二卦，則自《屯》、《蒙》至《渙》、《節》皆是也。

Among these fifty-six hexagrams (the original text mistakenly says 'five-one-six' but should be 'fifty-six'), there are twelve hexagrams that, when reversed, keep both the upper and lower trigrams unchanged. These are: The Need, The Contentious, The Army, The Following, The Supremacy, The Opposition, The Fellowship, The Great Possession, The Progress, The Light Dimmed, The Completion, and The Non-Completion (the original text mistakenly says 'Feng Yi' but should be 'The Light Dimmed'). There are also twelve hexagrams that, when reversed, change both the upper and lower trigrams. These are: The Following, The Pestilence, The Attraction, The Constancy, The Reduction, The Increase, The Thunder, The Mountain, The Gradual Progress, The Maiden, The Wind, and The Lake. There are thirty-two hexagrams that, when reversed, change only one of the upper or lower trigrams, ranging from The Sprouting, The Obstruction up to The Dispersal, The Regulation. The term 'two trigrams' here refers to the upper and lower trigrams of a single hexagram. Adding these three categories together—twelve, twelve, and thirty-two—exactly accounts for all fifty-six hexagrams.

蓋翻覆而不可變者，法八卦之乾坤坎離也。翻覆而可變者，法八卦之震艮巽兌也。就翻覆可變之中，其二體不《易》者，又皆乾坤坎離相交者也。其一體不易者，亦皆交於乾坤坎離者也。唯震艮巽兌相交之卦，則二體皆易焉，《頤》、《中孚》、《大過》、《小過》，雖為震艮巽兌相交之卦，而翻覆不可變者，《頤》、《中孚》具離之象，《大過》、《小過》具坎之象也，故《序卦》以之附於《坎》、《離》、《既濟》、《未濟》，為其具離坎之象焉爾。

In general, for hexagrams that do not change when reversed, they are based on the four fundamental trigrams that remain unchanged when reversed: Qian (Heaven), Kun (Earth), Kan (Water), and Li (Fire). For hexagrams that do change when reversed, they are based on the four trigrams that change when reversed: Zhen (Thunder), Gen (Mountain), Xun (Wind), and Dui (Lake). Among those that change when reversed, if neither of the two trigrams changes, they are all formed by the mutual combination of Qian, Kun, Kan, and Li. If only one trigram changes, they are also all formed by the mutual combination of Qian, Kun, Kan, and Li. However, hexagrams formed by the mutual combination of Zhen, Gen, Xun, and Dui will change both trigrams when reversed. The hexagrams Yi, Zhongfou, Dagu, and Xiaoguo, although all formed by the mutual combination of Zhen, Gen, Xun, and Dui, do not change when reversed. This is because Yi and Zhongfou have an outer reality and inner emptiness, containing the image of Li (Fire); Dagu and Xiaoguo have an outer emptiness and inner reality, containing the image of Kan (Water). Therefore, the "Sequence of Hexagrams" places these four hexagrams separately after Kan, Li, and Ji Ji and Wei Ji, because they possess the images of Li and Kan.

先天圖八卦，兩兩相對，《序卦》之根也，乾與坤對，坎與離對，震與巽對，艮與兌對。相對而不相交，所以定《序卦》之體也，然既相對，則必相交，四正之卦相交，則雖翻覆而其體不易，四維之卦相交，則翻覆而其體遂易矣。

In the Earlier Heaven diagram, the eight trigrams are paired opposite to each other, which is the origin of the "Xu Gua" (Sequence of Trigrams). Qian (Heaven) opposes Kun (Earth), Kan (Water) opposes Li (Fire), Zhen (Thunder) opposes Xun (Wind), and Gen (Mountain) opposes Dui (Lake). The eight trigrams merely oppose each other but have not yet combined with each other; this aspect forms the "body" of the "Xu Gua." However, since they oppose each other, they must inevitably intersect. The trigrams in the four cardinal positions (Qian, Kun, Kan, Li) intersect with each other, and even though the resulting trigrams are inverted, the upper and lower parts remain unchanged. The trigrams in the four intercardinal positions (Zhen, Xun, Gen, Dui) intersect with each other, and when the resulting trigrams are inverted, both the upper and lower parts change.

若四正之卦、與四維之卦雜交，則易者生，不易者半，所以極《序卦》之用也，足故“天地定位，上經所以始於《乾》、《坤》，中於《否》、《泰》也，“山澤通氣，雷風相薄”，下經所以始於《咸》、《恆》，中於《損》、《益》也，“水火不相射”，上下經所以終於《坎》、《離》、《既濟》、《未濟》也。

If the four cardinal trigrams and the four corner trigrams are interwoven, then the resulting trigrams, when reversed, will have half that are changed and half that are unchanged (the original text says 「易者生」, which should be 「易者半」). This principle is used to exhaust the meaning of the 「use」 in the 「Sequence of Trigrams」. Therefore, the 「Say of Trigrams」 states 「Heaven and Earth are positioned」, so the Upper Canon begins with 《Qian》 and 《Kun》 and has 《Po》 and 《Tai》 in the middle; it says 「Mountains and marshes communicate, Thunder and Wind interact」, so the Lower Canon begins with 《Xian》 and 《Heng》 and has 《Sun》 and 《Yi》 in the middle; it says 「Water and Fire do not oppose each other」, so both the Upper and Lower Canons end with 《Kan》 and 《Li》 and 《Ji Ji》 and 《Wei Ji》. The three quoted sentences all come from the chapter in the 「Say of Trigrams」 discussing the earlier heaven arrangement.

後天圖者，雜卦之根也。雜卦，即互卦也，互卦之法，或上去一畫而下生一畫，或下去一畫而上生一畫，則其體遂變矣。互體所成，凡十六卦，其陽卦從陽卦，陰卦從陰卦者八，《乾》、《坤》，《頤》，《大過》、《蹇》、《解》、《家人》、《睽》也。

The Later Heaven diagram is the origin of the order in the "Zagua" (Miscellaneous Hexagrams). The term "Zagua" refers to the Nuclear Hexagram. The method of obtaining the Nuclear Hexagram is either by removing one line from the top and adding one line below, or by removing one line from the bottom and adding one line above, thus changing the structure of the hexagram. Using this method, a total of sixteen hexagrams are generated. Among them, eight hexagrams remain within their original categories of yin or yang, namely "Qian" (Heaven), "Kun" (Earth), "Yi" (Abundance), "Daguo" (Great Excess), "Jian" (Obstruction), "Jie" (Resolution), "Jiaoren" (Family), and "Kui" (Strife). The term "Hutǐ" (Nuclear Structure) originates precisely from this method of derivation.

其陽卦交陰卦，陰卦交陽卦者亦八，《剝》、《復》、《夬》、《姤》、《漸》、《歸妹》、《既濟》、《未濟》也，以交互之法求之，乾而上去一陽，下生一陽，或下去一陽，上生一陽，仍是乾矣。坤而上去一陰，下生一陰，或下去一陰，上生一陰，仍是坤矣。

The eight hexagrams formed by the mutual interaction of yin and yang trigrams are 《Bao》, 《Fu》, 《Guai》, 《Gou》, 《Jian》, 《Gui Mei》, 《Ji Ji》, and 《Wei Ji》. Applying the method of mutual interaction mentioned earlier, if one yang line is removed from the top of the Qian trigram and one yang line is added to the bottom, or one yang line is removed from the bottom and one is added to the top, the result is still Qian. Similarly, if one yin line is removed from the top of the Kun trigram and one yin line is added to the bottom, or one yin line is removed from the bottom and one is added to the top, the result is still Kun. This shows that no matter how Qian and Kun trigrams interact with each other, they cannot escape their own forms.

唯震而上去一陰，下生一陰，則變為坎；下去一陽，上生一陽，則變為艮。巽而上去一陽，下生一陽，則變為離；下去一陰，上生一陰，則變為兌。坎而上去一陰，下生一陰，則變為艮；下去一陰，上生一陰，則變為震。離而上去一陽，下生一陽，則變為兌：下去一陽，上生一陽，則變為巽。艮而上去一陽，下生一陽，則變為震；下去一陰，上生一陰，則變為坎。兌而上去一陰，下生一陰，則變為後天圖八卦，陰陽上下畫互變，雜卦之根也，何則？後天之卦，有各從其類以相變者焉，有各得其對以相變者焉，乾居西北，而三陽從之，坤居西南，而三陰從之，此各從其類者也。乾與巽對，坎與離對，艮與坤對，震與兌對，此各得其對者也，

Only the other six trigrams can change: if a yin line is removed from the top of the Zhen trigram and a yin line is added to the bottom, it becomes Kan; if a yang line is removed from the bottom and a yang line is added to the top, it becomes Gen. If a yang line is removed from the top of the Xun trigram and a yang line is added to the bottom, it becomes Li; if a yin line is removed from the bottom and a yin line is added to the top, it becomes Dui. If a yin line is removed from the top of the Kan trigram and a yin line is added to the bottom, it becomes Gen; if a yin line is removed from the bottom and a yin line is added to the top, it becomes Zhen. If a yang line is removed from the top of the Li trigram and a yang line is added to the bottom, it becomes Dui; if a yang line is removed from the bottom and a yang line is added to the top, it becomes Xun. If a yang line is removed from the top of the Gen trigram and a yang line is added to the bottom, it becomes Zhen; if a yin line is removed from the bottom and a yin line is added to the top, it becomes Kan. The Dui trigram, if a yin line is removed from the top and a yin line is added to the bottom, becomes—this sentence in the original text is incomplete and the passage is interrupted. The next part is another diagram explanation. In the Later Heaven diagram, the eight trigrams change their upper and lower lines with yin and yang, which is the origin of the "Zaji" (Miscellaneous Trigrams). Why is this so? The transformation of the Later Heaven trigrams has two paths: one is changing according to their own kind, and the other is changing with their opposite trigrams. Qian is located in the Northwest, and the three yang trigrams Kan, Gen, and Zhen follow it; Kun is located in the Southwest, and the three yin trigrams Xun, Li, and Dui follow it, which is changing according to their own kind. Qian opposes Xun, Kan opposes Li, Gen opposes Kun, and Zhen opposes Dui, which is changing with their opposite trigrams.

相從者除乾坤純陽純陰不變外，坎而上去一陰，下生一陰，則為艮；艮而上去一陽，下生一陽，則為震；震而上去一陰，下生一陰，則復為坎，此三陽相次之序也。巽而上去一一陽，下生一陽，則為離；離而上去一陽，下生一陽，則為兌；兌而上去一陰，下生一陰，則復為巽；此三陰相次之序也。

First, the path of transformation within the same class: Qian and Kun are purely yang and purely yin respectively; their mutual transformation remains the same, so they need not be considered. When one yin line is removed from the top of Kan, and one yin line is added to the bottom, it becomes Gen. When one yang line is removed from the top of Gen, and one yang line is added to the bottom, it becomes Zhen. When one yin line is removed from the top of Zhen, and one yin line is added to the bottom, it returns to Kan. This is a cycle of three yang trigrams in succession. When one yang line is removed from the top of Xun, and one yang line is added to the bottom, it becomes Li. When one yang line is removed from the top of Li, and one yang line is added to the bottom, it becomes Dui. When one yin line is removed from the top of Dui, and one yin line is added to the bottom, it returns to Xun. This is a cycle of three yin trigrams in succession.

相對者，乾而上去一陽，下生一陰，則為巽；坎而上去一陰，下生一陽，則為離；艮而上去一陽，丁生一陰，則為坤；震而上去一陰，下生一陽，則為兌；此四陽卦變為對位四陰卦之序也。巽而下去一陰，上生一陽，則為乾；離而下去一陽，上生一陰，則為坎；坤而下去一陰，上生一陽，則為艮；兌而下去一陽，上生一陰，則為震；此四陰卦變為對位四陽卦之序也。

The next type involves the transformation between opposite trigrams: When one yin line is removed from the top of the Qian trigram and one yang line is added to the bottom, it becomes Xun; when one yin line is removed from the top of the Kan trigram and one yang line is added to the bottom, it becomes Li; when one yang line is removed from the top of the Gen trigram and one yin line is added to the bottom, it becomes Kun; when one yin line is removed from the top of the Zhen trigram and one yang line is added to the bottom, it becomes Dui. This is the sequence of the four yang trigrams transforming into the four opposite yin trigrams. Conversely, when one yin line is removed from the bottom of the Xun trigram and one yang line is added to the top, it becomes Qian; when one yang line is removed from the bottom of the Li trigram and one yin line is added to the top, it becomes Kan; when one yin line is removed from the bottom of the Kun trigram and one yang line is added to the top, it becomes Gen; when one yang line is removed from the bottom of the Dui trigram and one yin line is added to the top, it becomes Zhen. This is the sequence of the four yin trigrams transforming into the four opposite yang trigrams.

然尋其對位相變之根，則又自父母男女長少而來，蓋四陰卦，兌為最少，離為中，巽為長，坤為老。四陽卦，艮為最少，坎為中，震為長，乾為老。凡變者自少而老。故兌而卜去一陰，下生一陽，則變為乾矣。

Going deeper into the root of the transformation between opposing trigrams, it comes from the order of parents and children, and the distinction of seniority and juniority: among the four yin trigrams, Dui is the young girl, Li is the middle daughter, Xun is the elder daughter, and Kun is the mother, being the oldest; among the four yang trigrams, Gen is the young boy, Kan is the middle son, Zhen is the elder son, and Qian is the father, being the oldest. All transformations proceed from the younger to the older. Therefore, the Dui trigram, by removing one yin line from above and adding one yang line below, becomes Qian. (The original text says '卜去' which should be read as '上去'.)

離而上去一陽，下生一陰，則變為坎矣。巽而上去一陽，下生一陰，則變為艮矣。坤而上去一陰，下生一陽，則變為震矣。四陽卦之變，自陰而來，故又變而為對位之四陰也。艮而下去一陰，上生一陽，則變為巽矣。坎而下去一陰，上生一陽，則變為離矣。震而下去一陽，上生一陰，則變為坤矣。乾而丁去一陽，上生一陰，則變為兌矣。四陰卦之變，自陽而來，故又變而為對位之四陽也。

The trigram Li (Fire) is formed by removing one yang line from the top of the trigram Qian (Heaven) and adding one yin line below. Similarly, the trigram Gen (Mountain) is formed by removing one yang line from the top of the trigram Xun (Wind) and adding one yin line below. The trigram Zhen (Thunder) is formed by removing one yin line from the top of the trigram Kun (Earth) and adding one yang line below. Since the four yang trigrams Qian, Kan, Gen, and Zhen are all derived from the four yin trigrams, they can also transform back into the four yin trigrams. Conversely, the trigram Xun (Wind) is formed by removing one yin line from the bottom of the trigram Gen (Mountain) and adding one yang line above. The trigram Li (Fire) is formed by removing one yin line from the bottom of the trigram Kan (Water) and adding one yang line above. The trigram Kun (Earth) is formed by removing one yang line from the bottom of the trigram Zhen (Thunder) and adding one yin line above. The trigram Dui (Lake) is formed by removing one yang line from the bottom of the trigram Qian (Heaven) and adding one yin line above. Since the four yin trigrams Xun, Li, Kun, and Dui are all derived from the four yang trigrams, they can also transform back into the four yang trigrams.

合而觀之，凡陽卦相變者，震變坎艮也，坎變震艮也，艮又變震坎也。凡陰卦相變

When considering the interaction among yang trigrams, the transformation involves Zhen changing into Kan and Gen, Kan changing into Zhen and Gen, and Gen changing into Zhen and Kan. These three trigrams reciprocally transform within this range. As for the interaction among yin trigrams, the original text is incomplete, with the sentence cut off. The following part is missing, and then comes the title of another diagram.