Water and fire also have two.

八卦播五行於四時，木金土各二，惟水與火各一。震糞木有二也，兌乾金有二也，坤艮土有二也，惟離火坎水各一。先儒謂離坎居中，中不可有二，故水火均一也。又謂木金土皆有剛柔，惟水火無剛柔可分，故不能有二。然以十干分配五行，則水火木金土各有二。以十二辰分屬五行，則水火木金各有二，而土有四，則水火不二之說又不可通。

The eight trigrams distribute the five phases across the four seasons, with wood, metal, and earth each having two trigrams, while water and fire each have only one: The trigrams Zhen and Xun both belong to wood, hence wood has two; Dui and Qian both belong to metal, hence metal has two; Kun and Gen both belong to earth, hence earth has two; only Li belongs to fire and Kan belongs to water, each having only one trigram. Earlier scholars have offered two explanations for this: one says that Li and Kan are in the central position, and since the center cannot have two, water and fire can only each have one; the other says that wood, metal, and earth can be divided into yin and yang, but water and fire cannot be divided into yin and yang, hence they cannot have two. However, when using the ten heavenly stems to match the five phases, water, fire, wood, metal, and earth each have two; when using the twelve earthly branches to match the five phases, water, fire, wood, and metal each have two, while earth has four. In this case, the idea that water and fire cannot have two becomes inconsistent.

辛齋以為八卦於水火，亦各有二，與木金土無異焉。離為火，震為雷，雷亦火也。坎為水，兌為澤，澤亦水也。震為雷，震之一陽出於坎。陰根於陽，《內經》所謂“龍雷之火”，乃真火也。故於十干屬丙，而離火屬丁。兌之一陰麗於離，故曰麗澤兌。陽根於陰，其義取明水於月，乃真水也。於十干屬壬，而坎水屬癸，水火同源，陰陽互根，皆歸本於太一。俗儒未察五行之原理，以卦只有八，而五行之分陰分陽，其數有十，遂無可措置。曲為之解，遂有水火不二之說，而不自知其不可通也。

Hang Xinzhai believes that the Eight Trigrams also contain two instances of water and fire, just like wood, metal, and earth. The Trigram Li represents fire, and the Trigram Zhen represents thunder, which is also fire. The Trigram Kan represents water, and the Trigram Dui represents marsh, which is also water. The Trigram Zhen represents thunder, and its one yang line comes from Kan, which is yang arising from yin, exactly as the "Yellow Emperor's Inner Canon" describes "dragon fire and thunder fire," which is true fire, so it is paired with the heavenly stem Bing. The Trigram Li fire is paired with Ding. The Trigram Dui has one yin line attached to Li, so the Trigram Dui's "Great Image" says "beautiful marsh," which is yin arising from yang, taking the ancient practice of using a vessel to collect moonlight to obtain "bright water," which is true water, so it is paired with the heavenly stem Ren, while the Trigram Kan water is paired with Gui. Water and fire originate from the same source, and yin and yang are interdependent, ultimately returning to "Taiyi." Common scholars fail to grasp the principles of the Five Phases, seeing only eight trigrams, while the Five Phases have ten numbers divided into yin and yang. Therefore, they are unable to reconcile them, and they resort to forced interpretations, creating the idea of "water and fire not being two," without realizing their reasoning is flawed. His argument is based on the fact that the lines of the Trigrams Zhen and Dui come from Kan and Li respectively, so thunder can be paired with Bing fire and marsh with Ren water, thus making the Eight Trigrams and ten heavenly stems correspond perfectly.

八卦坎一，離九，震三，兌七，乾六，巽四，坤二，艮八，乃九宮之數，即洛書數也。後人不得其解，異說紛紜，各執一是，而互相駁難，究莫能得其要領。此由於不知九宮之真諦，無怪其開口便錯也。九宮本於太一，以一行九，如乾坤之周流六虛，分陰分陽，循環無端。一九三七二四六八，即陰陽太少之四位焉。

The Eight Trigrams are paired with numbers: The Trigram Kan is one, Li is nine, Zhen is three, Dui is seven, Qian is six, Xun is four, Kun is two, and Gen is eight. This is the number system of the Nine Palaces, also known as the number system of the Luo Writing. Later scholars failed to understand its principles, leading to various conflicting interpretations, each holding their own view and contradicting others, yet never grasping the essence. This is because they did not understand the true meaning of the Nine Palaces. It is no wonder they made mistakes as soon as they spoke. The Nine Palaces originally come from "Taiyi." With the number one moving around the Nine Palaces, it is like the energies of Qian and Kun circulating through the six lines, distinguishing yin and yang, cycling endlessly without beginning or end. The pairs one and nine, three and seven, two and four, six and eight correspond to the four forms of yin and yang: Yang, Lesser Yang, Lesser Yin, and Great Yin.

陽為氣，陰為形，氣與形有純駁，而太與少別也。一九太陽，純乎氣者也。故坎離居之。二八少陰，純乎形者也。故坤艮居之。三七少陽，氣與形相兼者也。故震兌居之。四六太陰，純乎形亦純乎氣者也，故乾巽居之。(巽為柔木，得氣之初，成形最早。乾為金剛，得氣最純，成形最堅。故柔木燒之而無炭，金剛化之而無質。此所以為太陰，陰極生陽，而玄牝為萬化之根也。)支支節節而解之，一卦一象以求之，其泥而不能通也宜矣。

Heavenly Stems: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui Earthly Branches: Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai Trigrams: Qian (Heaven), Dui (Lake), Li (Fire), Zhen (Thunder), Xun (Wind), Kan (Water), Gen (Mountain), Kun (Earth) Hang Xinzhai explains in his own notes that the last point refers to the fact that Xun is a flexible wood, which receives energy first and takes form first; Qian is a diamond, which receives energy most purely and takes form most firmly. Therefore, flexible wood, when burned, leaves no charcoal, and diamond, when melted, leaves no substance. This is why they belong to the category of Taiyin. When Yin reaches its extreme, it gives rise to Yang, and thus "Xuanpu" becomes the fundamental source of the transformation of all things. If one were to explain each section separately and rigidly match each hexagram to a single image, it would inevitably lead to being confined and unable to understand properly.

《上經》首乾坤，《下經》首咸恆。咸恆天地合體之卦也。泰否損益，亦天地合體之卦也。乾坤合德，見於咸恆。泰否交構(泰否為天地交)，見諸損益。故“天地絪縕，萬物化醇。男女媾精，萬物化生”，贊諸損之六三。天地絪縕，陰陽首交，而物以氣化。男女媾精，雌雄尾接，而物以形化矣。形能奪氣，物既成形，專於形化，而氣不復化，亦致一也。

The Upper Classic begins with the hexagrams Qian and Kun, while the Lower Classic begins with Xian and Heng. Xian and Heng are hexagrams representing the union of heaven and earth. Tai, Pinyin, Shen, and Yi are also hexagrams representing the union of heaven and earth. The harmonious virtue of Qian and Kun is expressed in Xian and Heng; the mutual interaction of Tai and Pinyin (both representing the intermingling of heaven and earth) is expressed in Shen and Yi. Therefore, the passage in the Great Treatise, "Heaven and earth blend together, and all things are transformed into purity; the union of male and female leads to the generation of all things," is the commentary on the sixth line of the third hexagram, Shen. The blending of heaven and earth's energies is the initial meeting of yin and yang, leading to the transformation of all things through energy; the union of male and female is the meeting of male and female forms, leading to the generation of all things through form. Form can take away the function of energy: once things have taken form, they proceed solely through transformation by form, and energy transformation ceases, which is the reason for becoming specialized.

是故天下之物，其初生者無不以氣化，天地之始合也。天地終合，萬物畢具，形成名立，氣為形奪，而氣化者少矣。然終則復始，氣機之流行，仍無少異。即氣化之用終不可絕，不但形之微者仍歸氣化(如無骨之蟲類是也)，即形化者亦或感於氣。而蛻化，而變化(如雀化蛤，魚化雀，沙魚化鹿化虎，人亦有時化虎之類)。生生化化，神妙莫測，皆資始於乾元，資生於坤元，實資於天元一(數學以立天元一御無窮不盡之數，亦此理也)。

Therefore, all things in the world, when they first came into being, were all transformed from qi. That was the stage when heaven and earth first came together. When heaven and earth were fully united, all things had fully formed, their forms were established, and their names were set. At that time, qi was taken by form, and the process of transformation through qi became less evident. However, the cycle continues, and the movement of qi remains the same as before. The function of transformation through qi never ceases. Not only do minute forms still rely on the transformation of qi (for example, creatures without bones), but even those that have already taken form may sometimes be transformed and changed by qi (for example, the ancient saying that a sparrow turns into a clam, a fish turns into a sparrow, a sandfish turns into a deer, then into a tiger, and even humans are said to occasionally turn into tigers). Thus, the continuous process of generation and transformation, which is mysterious and unfathomable, all originate from the beginning of the Trigram Qian, and from the generation of the Trigram Kun. In reality, they all draw upon the "Heavenly Origin One" (the concept of "Heavenly Origin One" in mathematics is used to manage infinite numbers, and it is the same principle).

天元一不用(以不用為用，故其用不窮)，用地元二(一二天地合，三三生萬物)。天元一子，地元二丑，子丑天地始合。絪縕氣化之候，在卦子一復丑二臨。復一小不用(復小而辨於物)，用臨二大(臨下兌。兌數十。十數盈數也。故大。)。故臨之初九、九二，皆曰“咸臨”(咸，乾在坤中，以陰含陽。陽動陰中，陰陽合德。資始資生，二而一矣。)。而乾坤二《彖傳》，孔子皆以一“咸”字形容之。

"The Heavenly Stem 'Jia' (天元一) is not used in itself (using non-use as use, thus its utility is boundless), but it uses 'Yi' (地元二). The combination of one and two represents the union of heaven and earth, and when three and three are multiplied, all things are generated. The Heavenly Stem 'Jia' is paired with the Earthly Branch 'Zi', and the Earthly Stem 'Yi' is paired with the Earthly Branch 'Chou'. 'Zi' and 'Chou' are precisely the beginning where heaven and earth first come together. During the process of mingling and generating qi, in the trigrams, this corresponds to the trigram 'Fu' (复) at the 'Zi' position and the trigram 'Lin' (临) at the 'Chou' position. 'Fu' is one, and its trigram is small and not used (as stated in the 'Great Treatise' (系辞传), 'Fu' is small but discerning in things). What is used is 'Lin', the two, which is called 'great' (the lower trigram of 'Lin' is 'Dui' (兑), and the number of 'Dui' is ten, which is the number of fullness, hence it is called 'great'). Therefore, the line statements of the first and second lines of the trigram 'Lin' both say 'Xian Lin' (咸临). 'Xian' refers to the 'Qian' trigram within the 'Kun' trigram, with yin containing yang; yang moves within yin, and yin and yang combine in harmony, thus the two functions of origin and generation are combined. Both the 'Commentary on the Judgment' (彖传) of the trigrams 'Qian' and 'Kun' also use the character 'Xian' to describe them."

咸臨於初爻，初應四四曰“至臨無咎”，即“至哉坤元”之“至”也。咸臨於二爻二應五，五曰“知臨大君之宜”，即“大哉乾元”之“大”也。乾大坤至，天地之氣，充滿流行，密合無間。天地絪縕，萬物化醇也。一二合生三，天地合生人。一氣而二形一精而二純，子丑二而寅三為人道。在卦為泰，泰為通。天地之氣，以人而通。乾道成男，坤道成女。乾交於坤，坤交於乾，而咸恆夫婦之道立。損益盈虛，相為消息，男女媾精，萬物化生也。故孔子於損之六三爻言之。

"Xian Lin" in the first line, the first line corresponds with the fourth line, and the sixth line of the fourth says "Zhi Lin, Wu Jiu," where "Zhi" is the same as "Zhi Zai Kun Yuan" in the "Zhi." "Xian Lin" in the second line, the second line corresponds with the fifth line, and the sixth line of the fifth says "Zhi Lin, Da Jun Zhi Yi," where "Da" is the same as "Da Zai Qian Yuan" in the "Da." The "Qian" hexagram represents "Da," and the "Kun" hexagram represents "Zhi." The energies of Heaven and Earth are fully present and flow continuously, closely combined without interruption, which is "Tian Di Yin Yun, Wan Wu Hua Chen." One and two combine to produce three, and Heaven and Earth combine to produce human beings; one is energy and two is form, one is essence and two is purity. Zi and Chou represent two, while Yin represents three, and Yin is the way of human beings. In the hexagram, this is the "Tai" hexagram, which means communication: the energies of Heaven and Earth pass through human beings to connect. The "Qian" way produces male, and the "Kun" way produces female; "Qian" interacts with "Kun," and "Kun" interacts with "Qian," thus establishing the way of husband and wife as established by "Xian" and "Heng." "Sun" and "Yi" represent the mutual increase and decrease of fullness and emptiness, which is exactly "Nan Nu Gou Jing, Wan Wu Hua Sheng," so Confucius attached this sentence to the sixth line of the third line in the "Sun" hexagram.

二與四，三與五，全《易》之重要關鍵也。不明乎此，則《象》《彖》終莫得而解。故孔子於“原始要終”一章，特詳言之。然此章文義深奧，學者莫明其妙，甚或不能句讀，因之異說蜂起。易傳燈甚至疑為偽作。是二三四五之說終不能明，故特將此章全文詮釋之。而後剛柔變易之道，始可得言也。《繫傳》曰：“《易》之為書也，原始要終，以為質也。六爻相雜，唯其時物也。其初難知，其上易知。本末也。初辭擬之，卒成之終”。此一節明設卦觀象之法。

The relationship between the second line and the fourth line, as well as between the third line and the fifth line, is a crucial point in the entire Zhou Yi. Without understanding this, the explanations in the Image Commentary and the Commentary on the Judgment cannot be properly understood. Therefore, Confucius particularly elaborates on this in the chapter "Original and Final" of the Great Treatise. However, this chapter is very profound and difficult to grasp, and those studying the Yi often fail to understand its subtleties, even unable to correctly punctuate the sentences, leading to various conflicting interpretations. Even the book "Yi Transmission Lamp" doubts that this chapter is a forgery. As a result, the concepts of the second, third, fourth, and fifth lines remain unclear. Therefore, Hang Xinzhai specially interprets the entire chapter, only then can the principles of change between yin and yang be discussed. The Great Treatise says: "The Yi is a book that traces the origin and considers the end, taking this as its essence. The six lines are mixed together, corresponding to the time and things. At the beginning, it is difficult to know; at the end, it is easy to know. This is the beginning and the end. The initial words are used to imitate, and the final outcome is formed." This passage explains the method of setting up hexagrams and observing their images.

《易》之為書，上下二篇，皆原始要終。《上經》以乾坤始，坎離終。《下篇》以咸恆始，既未濟終。即全《易》以乾坤始而坎離終(乾坤坎離為先天，南北東西之中。乾坤純陽純陰，其體始終不易。坎離為陰陽之中，其體亦始終不易。故後天以離坎代乾坤之位。六十四卦原始要終，皆乾坤坎離而已。六十四卦有其始終，一卦亦各有其始終。乾坤坎離以為之體，而離變其終則為震。震反則為艮坎，變其始則為兌。兌反為巽而離，變其始亦為艮。坎變其終亦為巽，巽艮反仍為兌。震則六爻相雜矣。六爻相雜，唯其時之不同，而陰陽之變化，其成象各異。其初難知，變之始也。其上易知，物之終也。初立其本終得其末。初率其辭，而揆其方。既有典常，卒能成其終。而一卦之體用可見矣。)，而一卦六爻相雜，唯其時物，亦各自有其始終。

The book of The Zhou Changes consists of two parts, both of which trace the origin and the end. The Upper Classic begins with the hexagrams Qian and Kun and ends with Kan and Li, while the Lower Classic begins with Xian and Heng and ends with Ji Ji and Wei Ji. When viewed together, the entire book begins with Qian and Kun and ends with Kan and Li. Hang Xinzhai added a long explanatory note here: Qian, Kun, Kan, and Li are the Earlier Heaven hexagrams, located in the center of the four cardinal directions. Qian and Kun are entirely yang and entirely yin, respectively, and their structures remain unchanged throughout. Kan and Li are the middle of yin and yang, and their structures also remain unchanged. Therefore, the Later Heaven positions use Li and Kan to replace the positions of Qian and Kun. The origin and end of the sixty-four hexagrams ultimately depend on the four hexagrams Qian, Kun, Kan, and Li. The entire book has its beginning and end, and each hexagram also has its own beginning and end: taking Qian, Kun, Kan, and Li as the foundation, when Li changes its end it becomes Zhen, and when Zhen is reversed it becomes Gen. When Kan changes its beginning it becomes Dui, and when Dui is reversed it becomes Xun, which connects with Li, and also becomes Gen when changing its beginning. When Kan changes its end it becomes Xun, and when Xun and Gen are reversed, they return to Dui. When reaching Zhen, the six lines become mixed. The six lines being mixed is due to the different times they occupy, and the images formed by the changes of yin and yang differ accordingly. The first line is difficult to know, because it is the beginning of change; the top line is easy to know, because it is the end of a matter. The first line establishes the foundation, and the top line represents the conclusion. By following the words of the first line to infer the direction, there is a regular pattern to follow, and ultimately it can reach its end. Thus, the nature and function of a hexagram become clear. Returning to the main text: the six lines of a hexagram are intermingled with each other, depending on the time they occupy and the things they represent, and each has its own beginning and end.

初難上易，有本末之分。知其所先後，則擬之於初者，卒能成其終。此就全卦六爻而言之也。然初者物之始，而用未著。上者時已過，而用或窮。故又曰“若夫雜物撰德，辨是與非，則非其中爻不備”。(物者，陰物陽物之總名。雜物則陰陽迭居其位而文生矣。撰者有選撰擬議之意。因陰陽之雜居，其德不同，則是非可辨矣。)

The first line is difficult to know, while the top line is easy to know; here there is a distinction between the beginning and the end. Knowing the order of precedence, the intention proposed on the first line can ultimately achieve its completion. This is speaking in terms of the entire hexagram's six lines. However, the first line is the beginning of things, and its function has not yet manifested; the top line is when the time has passed, and its function may have already reached its limit. Therefore, the Great Treatise says: "If one is to deal with various things and compose virtue, distinguish right from wrong, then it is not the middle lines that are complete." Hang Xinzhai's own commentary says: "The term 'things' refers to the general term for yin and yang things; 'miscellaneous things' means that yin and yang alternate in occupying the positions of the lines, and the beauty of the text arises from this. 'Compose' includes the meaning of selection and proposal; because yin and yang dwell together, the virtue of each line differs, so right and wrong can be distinguished."

又曰：“噫！亦要存亡去凶，則居可知矣。知者觀其彖辭，則思過半矣。”(此又詠歎以長言之，而申明上文意也。來知德氏以亦要為句，而要讀腰，謂即指中爻，似屬牽強。亦要者，即大要云爾猶言約舉其大概也。居者，剛柔迭居屯見而不失其居，即二四三五是也。)二四三五，已過全卦之半，此四爻皆謂之中者。以二五乃上下卦之中，三四又二與四，三與五之中也。

The Great Treatise continues by saying: "Alas! The main point is to preserve existence, prevent destruction, and avoid misfortune; this can be known by observing the positions of the lines. Those who understand will observe the meaning of the hexagram's lines and have already thought through more than half of it." This means that the main idea is simply to preserve existence, prevent destruction, and avoid misfortune; one can know this by looking at the positions of the lines. Hang Xinzhai's own note says that this uses a sighing, long sentence to further explain the previous meaning. A Ming dynasty scholar, Lai Zhide, divided "yi yao" as a separate sentence and read "yao" as "yao," suggesting it refers to the middle lines, which seems forced. "Yi yao" simply means "the main point," equivalent to saying roughly summarizing the main idea. "Ju" refers to the positions of the lines where yin and yang alternate, gathering and manifesting while each remaining in its proper place, that is, the second, fourth, third, and fifth lines. The second, fourth, third, and fifth lines have already exceeded half of the hexagram, and can all be called "zhong" (center): because the second and fifth lines are the centers of the lower and upper trigrams respectively, while the third and fourth lines are the middle of the pairs of second and fourth, and third and fifth lines.

故又曰：“二與四同功而異位，其善不同。二多譽，四多懼。近也，柔之為道，不利遠者。其要無咎，其用柔中也。”此即所謂下中爻也。二至四合成三畫卦一。既成一卦，又自有其上下之位。既分上下，其善與不善，便有不同。統觀六十四卦，凡二爻每多佳譽，而四爻每多恐懼。何哉？因四之位近於五，而四為柔爻。柔之為道，逼近於剛者不利，故多懼。而二雖同為柔爻，則遠於五矣。遠者大要可以無咎，況二又為下卦之中，柔而得用，此其所以多譽也。

Therefore, the Great Treatise also says: "The second and fourth lines have the same function but different positions, hence their good outcomes differ: the second line is often praised, the fourth line is often fearful, because it is close. The way of the flexible is not favorable to the distant; its main point is to avoid blame, for it uses flexibility in the center. " Here it refers to the middle lines in the lower trigram. The second line to the fourth line together form a three-line trigram; since they form a trigram, they themselves have a distinction between upper and lower. Once there is a distinction between upper and lower, good and bad differ. Observing all sixty-four trigrams, the second line often has praise, while the fourth line often has fear. Why is this? Because the fourth line is close to the fifth line, and the fourth line is a flexible line. The way of the flexible is not favorable when it approaches the strong, hence it is often fearful. The second line, although also a flexible line, is farther from the fifth line; the distant is generally without blame, and moreover, the second line is in the center of the lower trigram. Flexible yet properly used, this is why it is often praised. In other words, the good or bad of the same flexible line does not depend on the line itself, but on how close it is to the nine-five line and whether it is in the center and properly positioned.

又曰：“三與五同功而異位。三多凶，五多功，貴賤之等也。”此即所謂上中爻也。三至五，亦合成三畫卦一。而六十四卦之三爻，多凶，而五爻多功。何哉？則以五貴而三賤。五得位得中，而三則非所論也。綜二四三五而言之，二與四柔爻也，三與五剛爻也。故又曰：“其柔危，其剛勝耶？”二四雖同為柔，惟四獨危。三五雖同為剛，而五獨勝。此總論二四三五之位。而成卦以後，則剛柔雜居。柔者或以柔而益見其危，剛者或過剛而更著其凶。存亡凶吉，觀其居，則可知矣。

The Great Treatise also says: "The third and fifth lines have the same function but different positions; the third line is often凶 (dangerous) while the fifth line is often功 (beneficial), which reflects the distinction between the noble and the humble." Here it refers to the middle lines in the above group. The third line to the fifth line also form a three-line trigram together. In the sixty-four trigrams, the third line is often dangerous while the fifth line is often beneficial. Why is this so? Because the fifth line is noble and occupies the central position, while the third line does not have these advantages. If we consider the second, fourth, third, and fifth lines together, the second and fourth lines are yin positions, while the third and fifth lines are yang positions. Therefore, the Great Treatise concludes: "Is it the yin that is dangerous, and the yang that prevails?" The second and fourth lines, although both yin, are particularly dangerous; the third and fifth lines, although both yang, are particularly successful. This is a general discussion of the positions of the second, fourth, third, and fifth lines. Once the trigram is formed, the yin and yang lines are mixed in their positions. The yin lines may become even more dangerous due to their yin nature, while the yang lines may become even more凶 (dangerous) due to their excessive yang. Whether something is present or not, whether it is凶 or吉 (beneficial), can be determined by the position it occupies. This shows that the second, fourth, third, and fifth lines are the key to determining the survival, destruction, danger, or benefit of a trigram. This is also the reason why Hang Xinzhai carefully explains this entire passage in detail.

《易》柔乘剛之爻，皆二三五。始於屯之六二，陰陽始交而難生。難由於六二之乘剛也。(乘非二乘初之謂，乃坤六二之柔爻乘乾九二之剛爻。以柔敵剛，剛為所覆，而伏於下，故曰“乘剛”。)凡柔乘剛有其漸，始則柔遇剛。(姤，遇也，柔遇剛也。)遇而不以為異，漸而柔乘剛。(履，柔履剛也)履而不止，漸而柔乘剛。乘剛而不止，終之即柔變剛。(剝，柔變剛也)至柔變剛而乾爻二二五，均為柔乘。乃柔乘五剛。六十四卦終於柔變剛，乃剛決柔矣。

In the Zhou Changes, instances where a yao of yin (soft) overpowers a yao of yang (hard) occur at the second, third, and fifth positions. This example begins with the second line of the Zun hexagram: when yin and yang first meet, difficulty arises, and this difficulty comes from the yin line six in the second position overpowering the yang line nine in the second position. Hang Xinzhai explains: here, the term "multiplying over" does not refer to the second line multiplying over the first line, but rather to the yin line six in the second position of the Kun hexagram multiplying over the yang line nine in the second position of the Qian hexagram. This is a confrontation between soft and hard, with the hard being covered and hidden below, hence called "multiplying over the hard." Generally, when a soft line multiplies over a hard line, it is a gradual process: at first, the soft and hard merely meet (the Gou hexagram's Commentary on the Judgment says, "Gou means meeting, and it is the meeting of the soft with the hard"); when they meet, there is no noticeable difference, and gradually the soft steps on the hard (the Liu hexagram's Commentary on the Judgment says, "the soft steps on the hard"); when the soft steps on the hard and does not stop, it gradually becomes the soft dominating the hard; if the soft continues to multiply over the hard without stopping, it finally changes the hard into soft (the Po hexagram's Commentary on the Judgment says, "the soft changes the hard"). When the soft has changed the hard, the second, third, and fifth lines of the Qian hexagram, which are all hard lines, are all multiplied over by the soft, forming a situation where one soft line multiplies over five hard lines. Among the sixty-four hexagrams, the process ends with the soft changing the hard, and finally the hard comes to decisively overcome the soft.

始柔遇剛，而姤柔壯，壯震起，不艮止之，而震起者皆乘剛爻矣。二三五乘剛爻，天地三人爻睽不同人，遇剛敵剛而乘剛，莫凶於三爻矣。是故乘剛，始《屯·六二》“乘馬”。屯“乘馬”出之井，井二三五剛爻不可柔乘。柔乘則剛掩，剛掩則困。故乘剛莫凶於困之六三爻也。是以《震·六二》，乘剛也；《噬嗑·六二》，乘剛也。天地乘剛爻皆於六二。乘剛難始，即難在三五。三五天地日月候。三柔乘，困剛掩。五柔乘，豫貞疾。皆日月掩食象也。

The beginning is of soft meeting hard, but the softness in the hexagram Gou once becomes strong it turns into Dazhuang. The lower trigram of Dazhuang is Zhen, which signifies movement. If there is no Gen to stop it, any movement becomes a yao that rides on hardness. The second, third, and fifth lines are all positions of riding on hardness. In the three elements of heaven, earth, and humanity, the human yao is in the third position, its image is the hexagram Kui, which signifies discord, not the hexagram Tongren which signifies harmony. Meeting hardness, opposing hardness, and riding on hardness, there is no position more dangerous than the third line. Therefore, when discussing riding on hardness, it begins with the line six two in the hexagram Zun, which says "riding on horses, hesitating." The image of "riding on horses" in Zun comes from Jing. The second, third, and fifth lines in Jing are all yang lines, which cannot be ridden upon by yin. Once ridden upon by yin, yang is obscured, and when yang is obscured, it becomes Kunn. Therefore, the danger of riding on hardness is no greater than the line six three in Kunn. In addition, the line six two in Zhen is riding on hardness, and the line six two in Shike is also riding on hardness. All instances of riding on hardness in heaven and earth fall on the line six two. The difficulty of riding on hardness begins with the second line, but the real difficulty lies in the third and fifth lines. The third and fifth lines are the times when heaven and earth and sun and moon move. When the third line is ridden upon by yin, it is like the yang being obscured in Kunn; when the fifth line is ridden upon by yin, it is like the line "Zhen Ji" in Yu, which is like the sun and moon being obscured leading to an eclipse.

柔乘剛，則坤有尤，而乾有悔。故《易》無尤之爻，恆在二四五。尤，異也。睽，異也。(睽三五柔)西南坤柔道，以物見異為有尤，乃乘剛也。東北艮終止之，以終止坤柔，不見異，不乘剛，而終以無尤。故於天地西南(離兌)睽異，而終於東北蹇難，艮止之。二三五震起，而柔乘剛者，皆二四五。反震而艮止之，則終無尤矣。

Softness once takes over strength, and the line of the Earth (Kun) has "you," while the line of Heaven (Qian) has "regret." Therefore, in The Zhou Changes, the line without "you" is always in the second, fourth, or fifth positions. "You" means "difference," and the hexagram of Kui is one of divergence. The third and fifth lines of Kui are all soft. The southwest belongs to Kun, which follows the path of softness. When encountering things, it produces divergence, which is called "having you," which is precisely taking over strength. The northeast belongs to Gen, which can stop it—stopping the softness of Kun, so there is no divergence, and thus no taking over strength, ultimately achieving "no you." Therefore, in the directions of heaven and earth, the southwest (Li, Dui) is the source of divergence, and it ends with the difficulty of Jian in the northeast, which is stopped by Gen. The second, third, and fifth lines are stirred up by Zhen. All instances of softness taking over strength occur in these positions. By turning Zhen into Gen and stopping it, ultimately there is no you.

是以屯二，乘剛也。屯震反艮，屯成蹇，蹇二終無尤也。噬嗑二，乘剛也。噬嗑震反艮，噬嗑成旅。旅二，終無尤也。豫五，乘剛也。豫震反艮，豫成剝，剝五終無尤也。柔剛二三五終無尤，二四五天地中，三四反類，故困三乘剛也。困反類賁，賁四終無尤也。是以無尤皆二四五柔以艮止，乘剛皆二三五柔以震起也。

By following this method step by step: In the hexagram Zhun, the line six in the second is riding on a yang line. The Zhun hexagram, with its Zhen trigram reversed, becomes Gen, thus transforming Zhun into Jian. In Jian, the line six in the second is "finally without blame." In the hexagram Shihe, the line six in the second is riding on a yang line. The Shihe hexagram, with its Zhen trigram reversed, becomes Gen, thus transforming Shihe into Lu. In Lu, the line six in the second is "finally without blame." In the hexagram Yu, the line six in the fifth is riding on a yang line. The Yu hexagram, with its Zhen trigram reversed, becomes Gen, thus transforming Yu into Bo. In Bo, the line six in the fifth is "finally without blame." The reason why the yin lines in positions two, three, and five can finally be without blame is that positions two, four, and five are in the center of heaven and earth. However, positions three and four must instead take their kind from the opposite, so in the hexagram Kun, the line six in the third is riding on a yang line. The Kun hexagram, by reversing its kind, becomes Ben, thus transforming Kun into Ben. In Ben, the line six in the fourth is "finally without blame." This shows that all instances of "without blame" are due to the yin lines in positions two, four, and five being stopped by Gen. All instances of "riding on a yang line" are due to the yin lines in positions two, three, and five being stirred by Zhen.

二三五不柔乘，二四五終無尤，於是坤無尤而乾悔亡，為貞吉悔亡。故《易》貞吉悔亡爻，恆為二四五。乃坤柔不乘剛，乾剛不柔乘，是以坤貞吉而乾悔亡。坤貞吉，貞於坎；乾悔亡，悔於離。天地南北離坎，《易》日月一二明始，故正北方卦子丑，坎艮蹇貞吉；南方卦午未，離坤晉悔亡；而晉以下子丑二初貞吉，上午未三五日月明悔亡。

Two, three, and five are not being multiplied by the soft, so two, four, and five ultimately have no blame. Thus, the hexagram of Kūn has no blame and the hexagram of Qián has no regret, combining to form the line statement "Zhen Ji" and "Hui Wang." Therefore, in the Zhou Changes, the lines that say "Zhen Ji" and "Hui Wang" always fall on two, four, and five. This is because the soft of Kūn does not multiply the strong, and the strong of Qián is not multiplied by the soft, so Kūn attains "Zhen Ji" and Qián attains "Hui Wang." The "Zhen Ji" of Kūn is based on Kan; the "Hui Wang" of Qián is based on Li. The north and south of heaven and earth correspond to Li and Kan. The Zhou Changes take the sun and moon, and the numbers one and two as the beginning of brightness. Therefore, the position of Zi and Chou in the north, where Kan and Gen form Jian, speaks of "Zhen Ji." The position of Wu and Wei in the south, where Li and Kūn form Jin, speaks of "Hui Wang." From Jin onwards, all hexagrams that belong to Zi and Chou have the lines two and one often speaking of "Zhen Ji," while those that belong to Wu and Wei have the lines three and five, due to the brightness of the sun and moon, speaking of "Hui Wang."

內子丑一二，家人內也。外午未七八，睽外也。家人二貞吉而初悔亡，內也。睽五悔亡，外也。天地內外陰陽，陰不可過乎陽，內陰壯不可也。故家人內有恆不可有壯。是以家人二貞吉，而恆二悔亡。恆二悔亡，爻止二字；即大壯二悔亡，爻止二字。為內陰不可壯而止之。於是正北方坎卦內正，而南方離卦無悔。

The inner trigram corresponds to Zi and Chou, and to the first and second lines, taking the image of "inner" from the hexagram Family. The outer trigram corresponds to Wu and Wei, and to the seventh and eighth lines, taking the image of "outer" from the hexagram Kui. The line statement of line six-two in Family speaks of "uprightness and good fortune," and the line statement of line initial nine in Family speaks of "regret disappearing," both belonging to the inner. The line statement of line six-five in Kui speaks of "regret disappearing," belonging to the outer. Heaven and earth have a distinction between inner and outer, and yin and yang have a boundary: yin cannot surpass yang, so the yin in the inner trigram is not allowed to be strong. Therefore, the "inner" represented by Family can only have the constancy of the hexagram Hex, but not the strength of the hexagram Great Strength. Thus, the line statement of line six-two in Family says "uprightness and good fortune," while the line statement of line nine-two in Hex only says "regret disappearing," with the line statement consisting of only these two words. The line statement of line nine-two in Great Strength also only says "regret disappearing," with the line statement consisting of only these two words, in order to restrain the yin in the inner trigram from becoming strong. In this way, the hexagram Kan in the direction of true north gains its proper inner aspect, and the hexagram Li in the southern direction is free from regret.

可大壯往外，是為大壯內二貞吉，而外四五貞吉悔亡而無悔也。天地坤始柔不可壯，壯則乘剛，坤有尤。乾後剛不可不壯，不壯剛柔乘剛，乾有悔。是以大壯於坤柔壯止，於乾剛晉進。大壯正大，天地之情可見，而二四五貞吉悔亡而無悔可也。《易》於南北離坎，乃未濟上下卦。大壯二四五剛柔爻，即未濟離坎二四五剛柔爻，故大壯二貞吉。坎貞吉，大壯四貞吉悔亡，坎離貞吉悔亡，大壯五無悔，離貞吉無悔，故南北乃天地未濟離坎六十四卦終也。故二三五二四剛柔中爻，為全《易》至要關鍵。不明乎此，《易》不可得而言也。

Only when strength is vigorous and outwardly expressed can it be effective. This is why in the Inner Trigram of "Great Strength" (大壮), the line "Nine in the Second" says "Zhen Ji" (贞吉), while the lines four and five of the Outer Trigram respectively say "Zhen Ji Hui Wang" (贞吉悔亡) and "Wu Hui" (无悔). In the space between Heaven and Earth, the Trigram "Kun" (坤) comes first, and its softness cannot be strong; once strength is applied, it must ride upon the strong, and "Kun" then becomes "You" (有尤). The Trigram "Qian" (乾) comes later, and its strength cannot be weak; if it is not strong, then its strength is reversed by the softness, and "Qian" then becomes "You Hui" (有悔). Therefore, the hexagram "Great Strength" (大壮) stops the vigorous growth of "Kun" and allows the strength of "Qian" to advance like in "Jin" (晋). The "Commentary on the Judgment" (彖传) of "Great Strength" says, "Righteousness and grandeur, and the truth of Heaven and Earth can be seen." The lines two, four, and five of "Great Strength" speaking of "Zhen Ji," "Zhen Ji Hui Wang," and "Wu Hui" are exactly in line with this meaning. The "Zhou Yi" (周易) uses the Trigrams "Li" (离) and "Kan" (坎) in the north and south as the upper and lower trigrams of "Wei Ji" (未济). The lines two, four, and five of "Great Strength" which are yao (爻) of yin and yang are the same as the lines two, four, and five of "Li" and "Kan" in "Wei Ji." Therefore, the line "Nine in the Second" of "Great Strength" saying "Zhen Ji" corresponds with the "Zhen Ji" of "Kan"; the line "Nine in the Fourth" of "Great Strength" saying "Zhen Ji Hui Wang" corresponds with the "Zhen Ji Hui Wang" of both "Kan" and "Li"; and the line "Six in the Fifth" of "Great Strength" saying "Wu Hui" corresponds with the "Zhen Ji Wu Hui" of "Li." Therefore, the north and south trigrams are the "Li" and "Kan" of the "Wei Ji" and also the final hexagrams of the sixty-four. From this we can see that the middle lines two, three, and five, and the lines two and four of yin and yang are the most important elements in the entire "Zhou Yi." Without understanding this, one cannot even begin to speak of the "Yi."

乾陽二三五在井，(乾初九天一勿用，用地二陰陽合德。故乾陽不曰一三五，曰二三五。)故井初下也，時舍也。與乾初下也時舍同也。三多凶，惟井三王明受福。(坤陰晉二“受茲介福”，《既濟·九五》“東鄰殺牛不如西鄰之禴祭實受其福”。《象》曰“實受其福吉大來也”。《困·九五·象傳》“利用祭祀受福也”。經傳言福者，只此四爻而已。)坤陰二三五在噬嗑，故曰“噬嗑合也”，言陰與陽相合也。先王分疆畫界取諸井，日中為市，取諸噬嗑。燮理阻陽，以養萬民，得二四三五之用也。井反為困，二與五同也。而三則變九為六。《彖傳》曰“困剛掩也”，即井九三之陽為柔所掩，故曰“剛掩”。困數三十六。天地數窮，窮則變，變則通，而通在井二三五。乾陽出用，俱在三四人爻反覆。故二與四三與五內重要之關鍵，尤在三四也。

The yang of the Qian hexagram falls on the second, third, and fifth lines, and its image is seen in the Jing hexagram. The first line of the Qian hexagram is the heavenly one, "the dragon is hidden and not used," and the line used is the earthly two, where yin and yang combine. Therefore, the yang of the Qian hexagram is not called one, three, and five, but rather two, three, and five. Thus, the first line of the Jing hexagram, which is at the bottom and "abandons the time," is the same principle as the first line of the Qian hexagram, which is at the bottom and "abandons the time." The third line is usually dangerous, but in the Jing hexagram, the third line says, "the king is wise and receives his blessing." Hang Xinzhai's self-commentary lists the references to "blessing" in the classics: the yin of the Kun hexagram is seen in the Jin hexagram's line two, "receives this intermediate blessing"; the Ji Ji hexagram's line five says, "the eastern neighbor sacrifices an ox, but the western neighbor's simple sacrifice is actually blessed"; the Image Commentary says, "actually receives the blessing, which is a great coming"; and the Kun hexagram's line five in the Image Commentary says, "using worship brings blessing." The references to "blessing" in the classics are only in these four lines. The yin of the Kun hexagram falls on the second, third, and fifth lines, and its image is seen in the Shi He hexagram. Therefore, the Commentary on the Judgment says, "Shi He means union," which refers to the union of yin and yang. The Great Treatise says that the ancient kings divided territories by taking the image of the Jing hexagram, and the market at noon by taking the image of the Shi He hexagram, both of which regulate yin and yang to nourish the people. These are based on the use of the second and fourth lines, and the third and fifth lines. When the Jing hexagram is reversed into the Kun hexagram, the second and fifth lines remain unchanged, but the third line changes from nine to six. The Commentary on the Judgment says, "Kun means being covered by strength," which is exactly the yang of the Jing hexagram's third line being covered by yin. The number of the Kun hexagram is thirty-six, and the numbers of heaven and earth reach their limit here. When the limit is reached, change occurs, and change leads to communication, and communication again comes back to the second, third, and fifth lines of the Jing hexagram. The function of the yang of the Qian hexagram is revealed in the repeated changes of the second and fourth lines, and the third and fifth lines. Therefore, among these key pairs, the most important are the third and fourth lines.

卦用七八，爻用九六。七八其體，九六其用。故六十四卦無一非九六之用。孔子首以乾坤發明之，以示六十四卦無一非九六之用，實無一非乾坤之用也。然坤以從乾，地道無成，坤之用皆乾之用，六之用皆九之用，故特於《乾·文言》一再申明其義，曰“乾元用九，天下治也”，“乾元用九，乃見天則”，又曰“天德不可為首也”。

The hexagram itself uses seven and eight, while the lines use nine and six: seven and eight represent the essence, while nine and six represent the application. Therefore, none of the sixty-four hexagrams are without the use of nine and six. Confucius first used the hexagrams of Qian and Kun to illustrate this principle, indicating that since none of the sixty-four hexagrams are without the use of nine and six, none are without the use of Qian and Kun. However, Kun's function is to follow Qian, and the way of the earth does not claim success for itself. Therefore, the application of six is actually the application of nine, and the use of six is actually the use of nine. Hence, Confucius particularly emphasized in the Wenyan Zhuan of the Qian hexagram, saying, "The primordial power of Qian uses nine, and the world is governed," and again, "The primordial power of Qian uses nine, and thus the principle of heaven is revealed," and also said, "The virtue of heaven cannot be first."

先儒於用九用六二語，解釋不一。或以占變，或以剛柔，或泛論君臣之道，或比附天地之氣，或以“群龍無首”四字望文生義，或以“用六永貞”一言互明其旨，而皆未有確切不移之說，為至當不易之論者。揆厥原因，實由蔡墨之一語，曰乾之坤“見群龍無首吉”。故主占變之說者，皆以用九為純乾變純坤，用六為純坤變純乾。其他泛言義理言君臣，更悠悠無可捉摸矣。

Previous scholars have offered various interpretations of the phrases "use nine" and "use six," with no consensus. Some have explained them from the perspective of changing lines in divination, others from the concepts of yin and yang, some have discussed them in general terms regarding the relationship between sovereign and minister, others have compared them to the energies of heaven and earth, and still others have taken the phrase "a multitude of dragons without a leader" literally, or used the phrase "use six and maintain purity" to elaborate on the meaning. None of these interpretations have been definitive. The reason for this is that the "Zuo Zhuan" records a statement by Cai Mo, who said that the divination phrases for the hexagrams "Qian" and "Kun" are "a multitude of dragons without a leader, auspicious." Therefore, those who advocate for the theory of change in divination interpret "use nine" as the transformation of pure "Qian" into pure "Kun," and "use six" as the transformation of pure "Kun" into pure "Qian." As for those who discuss the meaning in general terms or in terms of the relationship between sovereign and minister, their interpretations are even more vague and difficult to grasp.

夫經文既明明曰九曰六，則釋經者自當先從九六二字研究，得其確詁，然後推究用九用六及所以用九用六之原，而證之以經，方為有據。端木鶴田氏曰：“用九者，天一不用，用地二至地十。”誠為最扼要之論矣。夫先聖既以九六命爻，自無爻非用九用六也。既稱之曰九曰六，則必根據於天一地二之數可知也。

The text clearly states "nine" and "six," so those interpreting the text should first study the characters "nine" and "six" to obtain an accurate explanation. Then they should investigate the meanings of "using nine" and "using six," as well as the reasons for using nine and six, and verify these interpretations with the text itself. Only then can their interpretations be considered valid. Qing dynasty scholar Du Menghu (styled Hetaian) said: "所谓用九" means "heavenly one is not used, while earthly two up to earthly ten are used." This is indeed a concise and accurate statement. Since the ancient sages used "nine" and "six" to name the lines, it is certain that no line is without the use of nine or six. Since they are called "nine" and "six," they must have been based on the system of heavenly one and earthly two, which can be definitely concluded.

今以天一地二之數推之，惟一與九不變。一不變者體也，(以一乘十乘百皆仍為一，而乘三乘四仍為三四。與二合則生三，而一之本數乃不見，故一不用。)九不變者用也。(凡以九遍乘他數，無一非九。如二九十八，一八九也。三九二十七，二七亦九也。至九九八十一，八一亦九也。)故自九乘一以至十，其積數為三百八十四，乃全《易》六十四卦之爻數也。而自一以後遞加至十，積數為九十九，加至百為九千九百九十九，而皆虛其一。故一不用，不為首，而《易》之全數皆用九之數。

Now using the system of "Heavenly One and Earthly Two," only one and nine remain unchanged. The unchanging nature of one is in terms of substance (when one is multiplied by ten, hundred, or any multiple, it remains one; when multiplied by three or four, it results in three or four; when combined with two, it produces three, and the original number one disappears, so one is not used). The unchanging nature of nine is in terms of function (when nine is multiplied by any other number, the result always adds up to nine: two times nine is eighteen, one plus eight is nine; three times nine is twenty-seven, two plus seven is nine; until nine times nine is eighty-one, eight plus one is still nine). Therefore, multiplying nine by one up to nine by ten and adding them all together results in three hundred and eighty-four, which is exactly the total number of lines in the sixty-four hexagrams of the Zhou Changes. Adding from one up to ten gives ninety-nine; adding up to one hundred gives nine thousand nine hundred and ninety-nine, both of which are one digit short of a full number. This shows that one is not used, does not occupy the first position, while the entire numerical system of the Zhou Changes is based on nine. This calculation provides solid evidence for the concept of "All Nines" (用九): not relying on divination or abstract reasoning, but proving through numbers that nine is used and one is the substance.

坤之用六曰“利永貞”，天下之道貞夫一者也，則正與九對。由此推之，自二至八，其五位之相得相合，與卦位爻數之相得相合，均可次第而明。然後驗之於日月，征之於四時，考之於聲音律呂，發而為禮樂政治，人倫道德，皆一以貫之。而八卦六十四卦三百八十四爻，亦一以貫之。持論皆有實據，而非徒託之空言矣。(九六之用皆出於數理之自然，非人力可能加減。參看《易數偶得》各圖自明。)

The hexagram "Kun" says "it is advantageous to pursue uprightness forever" under the use of six. The "Great Treatise" explains that the way of the world returns to "uprightness as one," which corresponds precisely to the nine. From this, we can proceed downward, from two to eight, and understand how these numbers relate to each other, as well as how the positions of the hexagrams and the numbers of the lines correspond. Then, using the sun and moon for verification, the four seasons for confirmation, and the tones and pitch-pipes for examination, we can extend this to the rites, music, politics, and human relationships, all of which can be unified. Thus, this argument has solid foundation, not empty talk. Hang Xinzhai's own note: the use of nine and six comes from the natural order of numbers, not from human intervention. Refer to his work "Accidental Gains on the Numbers of the Book of Changes" for further clarification.

《大學》、《中庸》，皆本於大《易》。以象證之，幾無一句無一字，不與卦義卦數相合。數始於天一，卦始於坎子。(八卦之位，坎北方子，其數一；艮東北丑寅，其數二三；震東方卯，其數四；巽東南辰巳，數五六；離南方午，其數七；坤西南未申，其數八九；兌西方酉，其數十；乾西北戌亥，其數無。乾無也，於文天屈西北為無。)子天一不用，乾初勿用，用始丑地二。(用地二至地十而一勿用，是為乾元用九)子天一復，復小。丑地二臨，臨大。坎習教事，(《坎·象》曰：“君子以常德行習教事。”)教者斆也。臨內卦兌，兌為學。子一至兌十，一始十終，而艮成終成始，念終始與於學，故曰“大學”。

The chapters "Great Learning" and "Meaning of the Middle Way" both originate from the "Zhou Yi" or "I Ching." When verified through hexagram symbols, there is hardly a sentence or a single character that does not align with the meanings and numbers of the hexagrams. The numbers begin with the heavenly stem "Jia" (天一), and the hexagrams start from the position of the hexagram "Kan" (坎) which corresponds to the position "Zi" (子) in the north. Hang Xinzhai himself annotated the positions and numbers of the eight trigrams: "Kan" is in the north, at "Zi" position, with the number one; "Gen" (艮) is in the northeast, at "Chou" and "Yin" positions, with numbers two and three; "Zhen" (震) is in the east, at "Mao" position, with the number four; "Xun" (巽) is in the southeast, at "Chen" and "Si" positions, with numbers five and six; "Li" (离) is in the south, at "Wu" position, with the number seven; "Kun" (坤) is in the southwest, at "Wei" and "Shen" positions, with numbers eight and nine; "Dui" (兑) is in the west, at "You" position, with the number ten; and "Qian" (乾) is in the northwest, at "Xu" and "Hai" positions, with the number "wu" (nothing) — "Qian" is "wu" (nothing), as the character "tian" (heaven) is curved in the northwest, forming the character "wu." The position "Zi" with the number one is not used, just as the first line of the hexagram "Qian" (乾) says "the hidden dragon does not act." The actual beginning of use is from the position "Chou" with the number two (using the number two up to ten, but not using one, which is why it is called "Qian Yuan Yu Ji" (the origin of heaven uses nine)). The position "Zi" with the number one corresponds to the hexagram "Fu" (复), which is called "small"; the position "Chou" with the number two corresponds to the hexagram "Lin" (临), which is called "great." The hexagram "Kan" governs "learning and teaching" (as the "Great Image" of "Kan" says, "the gentleman maintains constant virtue and practices teaching and learning"). "Teaching" is also "xiao" (敩), which includes both teaching and learning. The inner trigram of "Lin" is "Dui" (兑), which represents learning. From the number one at the position "Zi" to the number ten at the position "You" (兑), one is the beginning and ten is the end. "Gen" (艮) can both complete the end and begin again, constantly focusing on learning without separating the beginning and the end, hence it is called "Great Learning."

大學終始在艮，即在於《乾·九二》。(乾初一子勿用，用地二丑。地二丑乃坤二之數，乾坤合德，故坎子隱伏丑二。兌見故乾二曰“利見大人”。)九二“君子學以聚之，問以辨之，寬以居之，仁以行之”，大人之學也。大人之學，由於謹小而慎微。庸言之信，庸行之謹，閒邪存其誠，善世而不伐，德博而化，乃龍德而正中者也。故曰“中庸”。故《大學》、《中庸》皆本於《易》，皆始於乾之九二。九二乃坎離爻。坎離南北正中，君子中道而行。《大學》由離而至坎，《中庸》由坎而至離。離坎上下，水火既濟。聖功王道，備於此矣。

The beginning and end of the Great Learning are based on the Gen trigram, which corresponds to the Nine in the Second line of the Qian hexagram. Hang Xinzhai explains: The first line of the Qian hexagram is one, which corresponds to the Zi branch, and it is not suitable for use. What is used is the second line of the Earthly Branch, which is Chou. The second line of Chou corresponds to the second line of the Kun hexagram. The Qian and Kun trigrams are in harmony, so the Kan trigram is hidden within the second line of Chou. The Dui trigram has already appeared, so the Nine in the Second line of the Qian hexagram says, "It is advantageous to see a great man." The Wenyan commentary on the Nine in the Second line says, "A gentleman studies to gather knowledge, asks questions to distinguish truth, lives with broad-mindedness, and acts with benevolence." This is the learning of a great man. The learning of a great man is achieved through careful attention to small details: in ordinary speech, one must be sincere; in ordinary actions, one must be cautious; one must guard against evil to preserve sincerity; one must benefit the world without boasting; one must have broad virtue and be able to influence others. This is called "dragon virtue and correct center," hence it is called "the Mean." Thus, both the Great Learning and the Doctrine of the Mean are based on the Zhou Changes and originate from the Nine in the Second line of the Qian hexagram. The Nine in the Second line is the line of the Kan and Li trigrams. The Kan and Li trigrams are located at the center of the north and south respectively, so the gentleman follows the middle path. The Great Learning proceeds from the Li trigram to the Kan trigram, and the Doctrine of the Mean proceeds from the Kan trigram to the Li trigram. The Li and Kan trigrams, one above the other, form the hexagram of Water and Fire in Completion. The achievement of sainthood and the way of the sovereign are both complete here.

大學之道，道本於天。(乾為天戌亥數無。以無出有，乾知大始，冒天下之道，貞下起元，而元在坎子天一。)天一坎數，由一生二，陰陽合德，故曰“一陰一陽之謂道”。子一丑二，於天象為日月，子與丑合，(一二天地始合)天地合德，日月合明。明也者，離也。離坤(八卦離之右為坤，火地晉)乃晉卦，《晉·象》曰“君子以自昭明德”。“自昭明德”者，乃君子戒謹恐懼慎獨之功。其象著於西南離坤，而其本則仍在東北。故坎與艮，水山蹇，君子以反身修德。艮與坎，山水蒙，君了以果行育德。坎艮反覆，蒙以養正，而聖功基於此矣。然君子之德，非獨善其身己焉。己立立人，己達達人，重離繼照。《象》日“明兩作離，大人以繼明照于四方”。故曰“在明明德”。

The "Dao" in "The Great Learning" originates from Heaven. Hang Xinzhai explains: The trigram "Qian" represents Heaven, which resides in the positions of Xu and Hai, and its number is "wu" (nothing). From nothingness comes being. "Qian" governs the great beginning of all things, covering the world, and through "zheng xia qi yuan" (establishing the beginning and initiating the cycle), this "yuan" is found in the trigram "Kan" (Water), specifically the "tian yi" (Heavenly One). "Tian yi" is the number of "Kan." From one comes two, and yin and yang combine to form virtue, hence the "Great Treatise" states, "One yin and one yang is called Dao." The trigram "Zi" represents one, and "Chou" represents two, corresponding to the sun and moon in the celestial phenomena. "Zi" and "Chou" combine (one and two are the initial union of heaven and earth), and heaven and earth combine their virtue, while the sun and moon combine their brightness. This brightness is represented by the trigram "Li" (Fire). "Li" connects with "Kun" (Earth) in the eight trigrams, forming the hexagram "Jin" (Progress). The "Great Image" of "Jin" says, "The gentleman cultivates his own brightness and virtue." "Cultivating one's own brightness and virtue" is the practice of the gentleman being cautious and respectful, carefully self-cultivating. This image is evident in the southwest trigrams "Li" and "Kun," but its root lies in the northeast. Thus, "Kan" and "Gen" combine to form the hexagram "Jian" (Obstruction), and the "Great Image" says, "The gentleman returns to himself and cultivates virtue." "Gen" and "Kan" combine to form the hexagram "Meng" (Youth), and the "Great Image" says, "The gentleman acts decisively and nurtures virtue." "Kan" and "Gen" repeatedly combine, cultivating virtue in the state of ignorance, thus laying the foundation for becoming a sage. However, the virtue of the gentleman is not only about self-cultivation but also about establishing others when oneself is established, and guiding others when oneself is successful. Just as two "Li" trigrams overlap, the light continues, and the "Great Image" says, "Two lights shining, the great man continues the brightness and illuminates the four directions." Therefore, the "Great Learning" says, "To manifest the bright virtue."

離南方之卦，在先天乃乾之位也。乾與離，天火同人。同人親也。乾君坤民，乾施坤受。離以乾體包孕坤陰，其親也至矣。故曰“在親民，在止於至善”。善者，乾之元也。(《文言》曰“元者善之長也”)至者，坤之元也。(《坤·彖》曰“至哉坤元”)乾元一，坤元二。乾一勿用，用在坤二。乾坤合德，則至善也。一生二，二生三，三乃艮之數。艮，止也，止於至善者也。(一二三三生萬物，子丑寅天地人。建子一復丑二臨寅三泰，泰通而天地人之道通，萬物咸備於我。此所以為君子立德之本，三者為《大學》之綱領焉。)

The hexagram Li is the southern hexagram. In the Earlier Heaven arrangement, it originally occupied the position of the hexagram Qian. The hexagram Qian and Li combine to form the hexagram Tongren, which signifies intimacy. Qian represents the ruler, and Kun represents the people; Qian gives and Kun receives. Li contains the yao of Kun within its structure, indicating a profound intimacy. Therefore, the Great Learning says, "To care for the people, to attain the highest good." The term "good" refers to the original principle of Qian (as stated in the Wenyan Zhuan: "The original principle is the beginning of goodness"). The term "highest" refers to the original principle of Kun (as stated in the Commentary on the Judgment of the hexagram Kun: "The original principle of Kun is indeed profound"). The original principle of Qian is one, and the original principle of Kun is two. The one of Qian is not used; the two of Kun is used. When Qian and Kun share the same virtue, it is called "the highest good." One gives rise to two, and two gives rise to three. Three corresponds to the number of Gen. Gen signifies stopping, which is "to attain the highest good." Hang Xinzhai notes: one, two, and three, and from three all things arise. Zi, Chou, and Yin correspond to Heaven, Earth, and Humanity. The beginning of Zi is the hexagram Fu, the beginning of Chou is the hexagram Lin, and the beginning of Yin is the hexagram Tai. When Tai is harmonious, the way of Heaven, Earth, and Humanity is all harmonious, and all things are complete within me. This is the fundamental basis for a gentleman to establish virtue. To illuminate the bright virtue, to care for the people, and to attain the highest good are the three main principles of the Great Learning.

知止而後有定，(乾為知，艮為止，知止為乾坎。艮三位，艮反身修德，反諸坎。所謂含三為一是也。)知止反身修德，止於至善，則陰陽合德而既濟定，(既濟定也。既濟六爻皆當位，皆一陰一陽，此之謂道。)定而後能靜，(乾動坤靜，止於至善。以乾合坤，由動而返諸靜矣。)靜而後能安，(坤先迷後得主，後順得常。安貞之德，應地無疆。)安而後能慮，(由艮而返諸坎者，復由坎而出諸艮。坎為思慮。《傳》曰“能說諸心”，坎也。“能研諸侯之慮”，艮也。)慮而後能得。(乾坤合德，乾以易知，坤以簡能，易簡天下之理得矣。)

"Knowing where to stop leads to determination" —— The hexagram "Qian" represents knowledge, and "Gen" represents stopping. Knowing where to stop combines "Qian" and "Kan"; "Gen" is in the third position, representing self-reflection and cultivating virtue, which is returning to "Kan," that is, the concept of "containing three as one." Knowing where to stop and returning to self-cultivation, stopping at the highest good, leads to the harmonization of yin and yang, achieving the stability of "Ji Ji" (The Completed). The "彖传" of "Ji Ji" says "stability"; all six lines of "Ji Ji" are in proper positions, each consisting of one yin and one yang, which is called the Way. "After determination comes stillness" —— "Qian" represents movement, and "Kun" represents stillness; stopping at the highest good is combining "Qian" with "Kun," returning from movement to stillness. "After stillness comes peace" —— The "Kun" hexagram says first being lost and then gaining mastery, later being harmonious and attaining constancy, the virtue of maintaining steadfastness should be in harmony with the vastness of the earth. "After peace comes contemplation" —— Returning from "Gen" to "Kan," and then from "Kan" back to "Gen"; "Kan" represents thought, and the "Great Treatise" says "being able to express in the heart" is "Kan," "being able to examine the thoughts of feudal lords" is "Gen." "After contemplation comes attainment" —— "Qian" and "Kun" harmonize in virtue; "Qian" knows through simplicity, "Kun" acts through ease, simplicity and ease encompass all the principles of the world. Hang Xinzhai's method of matching each sentence with a hexagram is to explain that the five stages of practice in the "Great Learning" —— "determination, stillness, peace, contemplation, attainment" —— are not merely arbitrary sequences, but each stage has a corresponding hexagram and position that can be clearly identified.

物有本末，(天地之數，子一丑二寅三，三生萬物，皆始於丑二。丑屬牛，牛大物也。故物從牛星，紀起於牽牛。本末皆從木，木下為本，上為末。寅與卯屬木，《易》象三才天地人，乾天坤地艮人，三才之數不齊，皆齊於巽木。故巽也者齊也。齊之以巽，巽震成益，木道乃行。象也者，材也。材成輔相而天地，人之用以彰。)

"The phrase 'things have their origins and their ends' is explained by Hang Xinzhai as follows: the numbers of heaven and earth, with Zi representing one, Chou two, Yin three. Three generates all things, and everything begins with Chou two. Chou belongs to the ox, which is a large creature, so the character 'wu' (thing) includes the radical 'niu' (ox). The star system also begins with the constellation Niutou (the Ox). The characters 'ben' (origin) and 'mo' (end) both contain the radical 'mu' (wood); the lower part of 'mu' is 'ben' and the upper part is 'mo'. Yin and Mao both belong to wood. The Zhou Changes take their imagery from the three elements of heaven, earth, and humanity. The Qian hexagram represents heaven, the Kun represents earth, and the Gen represents humanity. The numbers of the three elements are uneven, so they rely on the Xun hexagram, which represents wood, to harmonize them. Therefore, the 'Say on the Hexagrams' states that Xun is the one that harmonizes. Using Xun to harmonize, Xun and Zhen combine to form Yi, and thus the way of wood is established. 'Xiang' (image) is also 'cai' (material), and once the material is complete, it assists heaven and earth, and thus the function of humanity is revealed."

事有終始，(坤為有，乾為事。坤以從乾，故坤曰從事，從乾之事也。乾以統坤，故乾用格物，格坤之物也。乾始坤終，一始十終，而成終成始於艮。則終則有始，物無盡而事無窮。)知所先後。(有本末終始則有先後。先者體也，後者用也。先後一德。明德體也，在所先也。明明德，用也。在所後也。艮其止，止其所也。故所先所後皆在艮，而知其所先後則在乾。乾知太始。)知所先，則先天而天不違。知所後，則後天而奉大時。則庶幾近乎道矣。

The sentence "things have beginning and end" is explained in the self-commentary as follows: The hexagram "Kun" represents having, and the hexagram "Qian" represents things; "Kun" is responsible for following "Qian," so "Kun" discusses "engaging in things," which means engaging in the matters of "Qian." "Qian" is responsible for governing "Kun," so "Qian" discusses "investigating things," which means investigating the things of "Kun." "Qian" represents the beginning, and "Kun" represents the end. One represents the beginning, ten represents the end, and both the completion of the end and the beginning are found in "Gen." Thus, after the end comes the beginning again, and things have no end, and matters have no end. The sentence "knowing what comes first and what comes later" is explained in the self-commentary as follows: Once there is beginning and end, there is also order of precedence and succession; what comes first belongs to the essence, and what comes later belongs to the function. Precedence and succession are actually the same virtue. "Mingde" is the essence, which belongs to what comes first; "Mingmingde" is the function, which belongs to what comes later. The "Commentary on the Judgment" of the hexagram "Gen" says, "Gently stopping, stopping at its place," so both what comes first and what comes later are found in "Gen." Knowing the order of precedence is found in "Qian," because "Qian" governs the great beginning of all things. Knowing what comes first allows one to be as the "Commentary of the Literary Meaning" says, "acting before heaven does not contradict heaven"; knowing what comes later allows one to be "acting after heaven and following the timing of heaven," thus approaching the Dao.

古之慾明明德於天下者，(乾坤一古今。乾以先坤，古以鑑今。乾鎮而靜，性也。感於坤而動性之慾也。人各有欲，欲貴知止。欲能知止，知止於至於善，足以自昭明德矣。)明明德者，明兩作離，大人以繼明照于四方，是所謂由離而到坎者也。知所先，必先治其國。由離而巽(南方離，東南巽)鼎，鼎通屯，屯建侯，鼎凝命，治其國也。由巽而震，巽為齊，震巽恆，夫婦之道。上承巽離風火家人，齊其家也。

The Great Learning says, "In ancient times, those who wished to manifest their bright virtue throughout the world," and the commentary explains: The trigrams Qian and Kun represent ancient and modern times, with Qian preceding Kun, just as ancient times serve as a mirror for the present. Qian is deep and still, representing nature; when it feels the movement of Kun, it becomes active, which is the desire arising from nature. Everyone has desires, but what is precious is knowing where to stop; when desire knows where to stop, stopping at the highest good, it is sufficient to manifest one's bright virtue. What is called "manifesting bright virtue" is "the two lights shine together, the great man continues the bright illumination to the four directions," which is the transition from Li to Kan mentioned earlier. Once the order of precedence is understood, one must first govern the state: from Li to Xun (South is Li, Southeast is Xun) forms the Ding trigram, which is connected to Tun. Tun speaks of "favorable for establishing a feudal lord," while Ding speaks of "correct position and receiving mandate," which is governing the state. Then from Xun to Zhen, Xun represents order, and together with Zhen, they form Heng, which is the way of husband and wife; above them, Xun and Li form the wind and fire family, which is harmonizing the family.

震反為艮，反身修德，修其身也。修身為本，自離至艮，由天下國家而反諸身，德之由用而歸之體也。更由艮而坎，坎維心亨，正於天一，正其心者也。由坎而乾，意出於心。乾為言，而艮成之。從成從言，誠也。誠其意者也。知者徹始徹終，聞一知十。由乾而兌，乾知大始，智至至之。從坎一至兌十，致其知者也。由兌而坤，坤為物，與艮對。艮手格之，故曰“致知在格物”。自艮而至坤終焉。此德之體也，皆知所先也。

The hexagram Zhen turned upside down becomes Gen. The Gen hexagram in the Great Image says "reverse oneself and cultivate virtue," which refers to cultivating oneself. Cultivating oneself is the fundamental aspect: from Li to Gen, moving from the world and state back to oneself, virtue moves from its application back to its essence. Then from Gen to Kan, the Kan hexagram's Commentary on the Judgment says "maintaining the heart is smooth," aligning with the position of Heaven's first principle, which is to correct one's heart. From Kan to Qian, the mind originates from the heart, Qian governs speech, and Gen makes it complete; the character "cheng" (sincerity) is composed of "cheng" (completion) and "yan" (speech), which is precisely to sincere one's intention. A wise person can carry through consistently, understanding ten things from one: from Qian to Dui, Qian governs knowing the great beginning of all things, and wisdom reaches there, from the number one in Kan to the number ten in Dui, which is to attain knowledge. Then from Dui to Kun, Kun represents things, opposite to Gen, which represents the hand; using the hand to investigate it, hence the Great Learning says "attaining knowledge lies in investigating things." From Gen to Kun, this path concludes. All of the above belong to the essence of virtue, all being the aspect of "knowing what comes first."

知所先，則知所後。物格而後知至，(至者坤也。)知至而後意誠，意誠而後心正，心正而後身修，身修而後家齊，家齊而後國治，國治而後天下平。是由本而及於末，由體而達諸用。於八卦從坤而兌，兌而乾，乾而坎，坎而艮。艮為修身，修身齊家之始，成終成始。而震而巽而離，更復於坤。所謂全體大用，無不明矣(坤為地，天之下也。坤體承乾坎，平也，亦由終而反始也。)。故大學之道，首明明德。明兩作離也。終天下平，祗既平坎也。坎離正中，由體達用，則水火既濟功成，剛柔正而位當，知止而後有定(既濟定也。)，《易》道備矣。

Knowing the beginning, one also knows the end. "When things are thoroughly understood, then knowledge is complete" (the term "to reach the utmost" belongs to the Hexagram 63, the Earth). "When knowledge is complete, then the mind is sincere; when the mind is sincere, then the heart is upright; when the heart is upright, then the person is cultivated; when the person is cultivated, then the family is harmonious; when the family is harmonious, then the state is well governed; when the state is well governed, then the world is at peace." This is a process from the fundamental to the external, from the essence to the application. In the Eight Trigrams, it proceeds from the Hexagram 63 (Earth) to the Hexagram 58 (Lake), then to the Hexagram 1 (Heaven), then to the Hexagram 64 (Water), then to the Hexagram 19 (Mountain); the Hexagram 19 governs self-cultivation, which is the beginning of family harmony, capable of both concluding and beginning again. Continuing from the Hexagram 19 to the Hexagram 21 (Thunder), then to the Hexagram 22 (Wind), then to the Hexagram 31 (Fire), and finally returning to the Hexagram 63 (Earth). This is called the complete essence and full application, and at this point, everything is clear. The Hexagram 63 (Earth) is the Earth, beneath the Heaven; the essence of the Hexagram 63 receives the influence of the Hexagram 1 (Heaven) and the Hexagram 64 (Water), embodying the meaning of "peace," and thus returning from the end to the beginning. Therefore, the Way of the Great Learning begins with "brightening the bright virtue," symbolizing the image of "two lights shining, the Fire"; it ends with "the world is at peace," which is simply the image of "the Water" after peace has been achieved. The Hexagrams 64 (Water) and 31 (Fire) are in the central position, moving from essence to application, forming the Water and Fire Containing Each Other (Hexagram 60), where success is achieved, and the strong and the weak each hold their proper position. "When one knows where to stop, then one has stability" (the Commentary on the Judgment of the Water and Fire Containing Each Other says "stability"), and the Way of the Book of Changes is thus complete. Clearly, the sequence from investigating things to achieving world peace in the Great Learning corresponds exactly to a full cycle through the Eight Trigrams, returning to the starting point, with the beginning and end corresponding precisely to the Hexagrams 64 (Water) and 31 (Fire).

自天子以至於庶人，一是皆以修身為本，言人類階級至不齊焉，惟道能齊之。名類大小至不一焉，惟道能一之。齊之一之，故曰壹是以修身為本。艮為身，艮止，止其所所先所後，無不歸於至善矣。其本亂而末治者否矣。修身為本，身不修，既濟道反，初吉終亂(既濟“初吉終亂”)，本末易位。二之五亦五之二，為天地否矣。否泰反覆損益，益下不厚事，而損錯雷風相薄，天地變化，治亂之消息甚微。君子明乎消息盈虛，知所先後，明乎厚薄(剝上“以厚下安宅”則反覆。)，安土敦仁(修身)，而治可保矣。故曰“其所厚者薄而其所薄者厚，未之有也”。

"From the Son of Heaven down to the common people, all should take self-cultivation as the foundation," meaning that human ranks are extremely uneven, and only the Way can bring them into order; the distinctions of names and categories are also extremely inconsistent, and only the Way can unify them. By ordering them and unifying them, it is said "all should take self-cultivation as the foundation." The Hexagram of Gen represents the body, and Gen also signifies stillness. Stillness at the proper place ensures that what comes first and what comes later is all directed toward the highest good. As for "if the foundation is disorderly and the branches are orderly, it is impossible," self-cultivation is the foundation. If the body is not cultivated, the Way of Ji Ji reverses, becoming "initially auspicious but eventually disorderly" (the Hexagram Ji Ji's statement indeed says "initially auspicious but eventually disorderly"). The foundation and the branches are reversed; the second line moves to the fifth, and the fifth line moves to the second, turning into the Hexagram of Po. The alternation of Po and Tai and the mutual dependence of Shen and Yi: the Great Image of Yi says "benefiting the lower does not involve heavy affairs," while Shen, which is opposite to Yi, is thunder and wind colliding, thus causing changes in heaven and earth. The rise and fall of order and disorder are actually extremely subtle. The gentleman understands the waxing and waning of things, knows the order of precedence and succession, and understands the boundaries of thickness and thinness (the upper ninth line of the Hexagram Bao says "by being generous to the lower and securing the house," which is the meaning of reversal). Being settled in one's place and being earnest in benevolence is self-cultivation, and thus peace can be maintained. Therefore, the Great Learning says "if what is esteemed is made light and what is made light is esteemed, there has never been such a thing."

此謂知本，此謂知之至也。古本《大學》經文之末有此二句，今本移此二句於“大畏民志此謂知本”之下，作第五章。與“親民”之為“新民”，皆程子所改定者也。實則“親民”親字，含有教養二義。改作“新”字，僅有教之一義。則《傳》所稱如保赤子及同好同惡諸義，均贅文矣。證之於《易》，親字非訛尤確而可信，故並及之。

"These words, 'This is knowing the root, this is knowing thoroughly,' are located at the end of the classical text of the Great Learning. However, in the present version, they have been moved below 'Great fear of the people's will, this is knowing the root,' and are made into a fifth chapter. This change, like the alteration of 'care for the people' to 'renew the people,' was made by Cheng Yi. In fact, the character 'qin' in 'care for the people' includes both teaching and nurturing. Changing it to 'xin' leaves only the aspect of teaching. Thus, the meaning of 'as caring for a newborn child' and the idea of sharing the people's joys and sorrows in the transmission text become redundant. Using the Zhou Changes as a reference, the character 'qin' is not an error, which is even more convincing. Therefore, I mention this together here."

“中庸”之中，即離坎中正之中。庸者從庚。陰陽之義，用始乙庚(故蠱先甲後甲，巽先庚後庚。)。後天震出東方，首出庶物，萬象更新，故納甲以震納庚。而庚之本位則屬西方，西正秋兌，震仁兌義，立人之道。故庸字之義，乃合震兌二象，兼仁義之用者也。程子曰“不易之謂庸”，朱了曰“庸平常也”，均非確詁。《乾·九二》“庸言之信庸行之謹”，兌言震行，取象尤極顯明。

The "zhong" in "Zhongyong" refers to the middle and upright nature of the trigrams Li (Fire) and Kan (Water). The character "yong" contains the radical "geng." In terms of yin and yang principles, it is used in relation to Yi and Geng (thus the hexagram Gu explains "three days before Jia and three days after Jia," and the hexagram Xun explains "three days before Geng and three days after Geng"). In the Later Heaven arrangement, the trigram Zhen (Thunder) is located in the East, where all things first emerge, marking the beginning of renewal, so the method of Najia assigns Geng to Zhen. However, Geng's original position belongs to the West, which corresponds to the trigram Dui (Lake) and the autumn season. Zhen represents Ren (benevolence), and Dui represents Yi (righteousness), which are the two fundamental aspects of human conduct. Therefore, the meaning of "yong" is to combine the images of Zhen and Dui and to incorporate the functions of Ren and Yi. Cheng Yi said "yong refers to something unchanging," and Zhu Xi said "yong means ordinary," but these interpretations are not accurate. The "Wenyan" of the second line of the hexagram Qian states "ordinary words are trustworthy, ordinary actions are cautious." "Words" belong to Dui, and "actions" belong to Zhen, and the symbolic meaning is especially clear.

天命之謂性，惟天之命，於穆不已。天者，乾也。天命則乾元資始，而始於坎，即坎中之一陽，所謂性也。孔子贊《易》，利貞者性情也。又曰“一陰一陽之謂道，繼之者善也，成之者性也”，又曰“成性存存道義之門”，又曰“窮理盡性以至於命”。言簡意賅，源委分明，直截了當。後之言性理者，累千百萬言，枝枝葉葉，糾紛纏繞，其本根反不易尋覓矣。

The Zhongyong says, "Heaven's mandate is called nature." The Book of Poetry, Zhou Songs, says, "Only Heaven's mandate, profound and unceasing." Here, the term "Heaven" refers to the Qian hexagram. Heaven's mandate means the Qian hexagram's original yang line, which is the source of all things. This beginning falls on the Kan hexagram, specifically the central yang line of the Kan hexagram, which is referred to as "nature." Confucius wrote the Commentary on the Yi, and in the Wenyan, it says, "Beneficial constancy is the nature and disposition." In the Great Treatise, it says, "One yin and one yang is called the Way; what continues it is goodness, and what completes it is nature." It also says, "Maintaining nature consistently is the gate to the Way and righteousness," and again, "Exhausting principles, completing nature, and reaching the mandate." These words are concise and to the point, clearly distinguishing the beginning and end, the source and flow. Later scholars who discussed nature and principle wrote millions of words, but their discussions became tangled and complicated, making the fundamental principles difficult to find.

率性之謂道。率者率也，即成性存存也。率性之謂道，非性之謂道也。道本於天，性亦受於天。人能不失天賦之性，即能合於天之道矣。於《易》象坎為性，性隱伏不可見者也。坎常德行，習教事，教人明於天之道而察於民之故，凡皆所以成其性也。故曰修道之謂教。

The Zhongyong continues saying, "To follow one's nature is called the Way." "Follow" means to adhere to, which is exactly what the Great Treatise says, "To realize one's nature and maintain it steadfastly." One should note that it says "to follow one's nature is called the Way," not "nature is called the Way." The Way originates from Heaven, and human nature also receives its origin from Heaven. When a person does not lose his innate nature, he can harmonize with the Way of Heaven. In the symbolic imagery of the Zhou Changes, the Kan hexagram represents nature, which is hidden and not visible. The Great Image of the Kan hexagram says, "To practice constant virtue and to become accustomed to teaching and learning," which teaches people to understand the Way of Heaven and to observe the affairs of the people. All these are for the purpose of realizing one's nature. Therefore, the Zhongyong continues, "To cultivate the Way is called teaching."

性不可見，感於物而動，則情也。情亦不可見，而見乎色(喜怒哀樂見於色)，見乎辭，見乎行。性貞天一者也。動則情見乎色見乎辭見乎行。其為喜怒哀樂者，則不一矣。於是由艮而震，而以巽齊之。則不一者一，而合乎道。故道也者，不可須臾離也，可離非道也。此象之由坎而至於離者也。

Human nature itself is invisible; when it is touched by external things, it becomes emotion. Emotion is also invisible, but it appears in expressions (joy, anger, sorrow, and happiness are seen on one's face), in speech, and in actions. Human nature is fixed and constant as heaven's one principle. Once it moves, emotion appears in expressions, in words, and in actions, manifesting as joy, anger, sorrow, and happiness, which differ greatly. Thus, from the hexagram of "Stillness" (Gèn) it moves to "Thunder" (Zhèn), then uses "Wind" (Xùn) to regulate it, bringing disorder back to order and aligning with the Way. Therefore, the "Doctrine of the Mean" says, "The Way cannot be separated for a moment; if it can be separated, it is not the Way." This passage in terms of hexagram imagery corresponds precisely to the transition from "Water" (Kǎn) to "Fire" (Lí).

戒慎乎其所不睹(離目睹)，恐懼乎其所不聞(坎耳聞)，莫見乎隱，莫顯乎微者，皆坎為心，隱而顯微而著者也。坎心天地中，故喜怒哀樂之未發謂之中。坎一發而十至(兌十)。坎兌為節，故曰發而皆中節。先天乾坤南北中，後天震兌東西和(萬物出震)，故曰“致中和天地位焉，萬物育焉”(《易》與天地準，東震西兌，準平震卯兌西。日月出入，天象半顯半隱。正北坎，西北乾，東北艮，皆隱而不見也。天地之氣，始於亥子之間微也。)。

The Zhongyong says, "Be cautious in what is not seen" (the Li trigram represents the eye, which governs sight), "and be fearful in what is not heard" (the Kan trigram represents the ear, which governs hearing). "Nothing is more visible than the hidden, and nothing is more evident than the subtle" — all of this is because the Kan trigram governs the heart, which is hidden yet can become visible, subtle yet can become prominent. The heart governed by Kan is located in the center of heaven and earth, so it says, "When joy, anger, sorrow, and joy have not yet been expressed, this is called the middle." The Kan trigram is one, and when it is expressed, it reaches ten (the Dui trigram is ten); since Kan and Dui combine to form the Jie trigram, it says, "When expressed, they all conform to the proper measure." In the Earlier Heaven arrangement, the Qian and Kun trigrams are located in the north and south, forming the center; in the Later Heaven arrangement, the Zhen and Dui trigrams are located in the east and west, forming harmony (all things originate from Zhen). Therefore, it says, "Achieving the middle and harmony, heaven and earth are in their proper places, and all things flourish." Hang Xinzhai's own note: The Zhou Changes correspond to heaven and earth. The east is Zhen, the west is Dui; this perfectly matches Zhen corresponding to Mao and Dui corresponding to You. The rising and setting of the sun and moon, heavenly phenomena are always half visible and half hidden — the Kan trigram in the due north, the Qian trigram in the northwest, and the Gen trigram in the northeast are all hidden and not visible. The vital energy of heaven and earth arises at the intersection of Hai and Zi, which is the most subtle.

坎一復小，故小德川流。丑二臨大，故大德敦化。一二天地中，天下之大本也。天一勿用，用丑二艮成言。成言為誠，先坎後震。雲雷屯，經綸，故唯天下之至誠，為能經綸天下之大經，立天下之大本。天一勿用，用二至十。故凡為天下國家有九經，所以行之者一也。

The Trigram Kan is paired with one, and its hexagram is Fu, which is called small, so the Zhongyong says "small virtue flows like a river." Chou is paired with two, and its hexagram is Lin, which is called great, so it says "great virtue promotes transformation." One and two are located in the center of heaven and earth, which is exactly "the great root of the world." Heaven one is not used; instead, Chou two is used. Gen is called "formed words"; "formed words" is sincerity. The order is first Kan and then Zhen. Kan and Zhen together form the hexagram Zun, which is Cloud and Thunder. The Great Image of Zun says "the gentleman is skilled in planning and organizing," so the Zhongyong says "only those with the highest sincerity can plan and organize the great principles of the world and establish the great root of the world." Heaven one is not used, but two is used up to ten, so it says "for governing the world and state there are nine principles, and they are all carried out by one."

正北坎，西北乾，東北艮。乾天道，對巽為直。來者信也，由亥出子。坎一兌十，動而直，直而中節，唯聖者能之。其二屈往(往者屈也)，故曰其次致曲。本東方艮人道，艮成言，曲能有誠。艮寅泰通，震出用事，萬物甲坼。誠則形也，巽物絜齊。形則著也，離皆相見，著則明也，皆由坎以至離者也。明動變化，由離坤而兌乾，故曰“唯天下至誠為能化”。

Due north is Kan (Water), northwest is Qian (Heaven), and northeast is Gen (Mountain). Qian governs the heavenly way and is opposite to Xun (Wind), its nature being straight. "Coming" means extension (as stated in the Great Treatise, "Going is bending, coming is sincerity"). It originates from the亥 position and emerges at the Zi position. Kan represents one, Dui represents ten; movement enables straightness, and straightness conforms to measure, something only the sage can achieve. The next level is bending and going (going is bending), so the Zhongyong says "the next level is to pursue the subtle." This path originates from the human way governed by Gen in the east. Gen can "form words," even though it begins with bending, it can still be sincere. Gen is paired with寅 and connects with Tai (Peace), while Zhen (Thunder) emerges and takes action, causing all things to break forth from their shells. "Sincerity becomes form" is due to Xun making all things clean and orderly; "form becomes evident" is due to Li (Fire) causing all things to appear; "evident becomes clear"—these are all the path from Kan to Li. Further down, "clearness leads to movement, movement leads to change, change leads to transformation," which is the path from Li and Kun (Earth) to Dui and Qian, so the Zhongyong says "only the supremely sincere can bring about transformation."

誠者艮也，不誠無物，是故君子誠之為貴，與《大學》之壹是皆以修身為本，象義皆同，後先一貫者也。蓋艮寅為人位，曾子稱思不出其位，惟日三省，故獨得孔子一貫之傳。《大學》、《中庸》，皆端本於修身，即皆本於艮寅之一位，重乾爻辰二爻在寅，故《大學》、《中庸》之義，皆出於乾之九二也。顧孔子贊《易》，於立人之道，其樞機又在於三四兩爻何也？曰：此六爻之義也。論六爻三極，三四為人爻曰人極；而三四兩爻，居內外之際，合言之，則為兩卦之中；分言之又為內之終而外之始。六十四卦之反覆上下，無不由此。故《易》之言人道於三四兩爻。“中庸”“大學”，以卦位言之。故均以艮寅明修道之本，皆所謂同條而共貫者也。(誠者物之終始。艮者成終而成始也。)

"Cheng" belongs to the trigram Gen, and the "Zhongyong" says "Without Cheng, there is nothing," so the gentleman values Cheng above all; this is consistent with the "Daxue" saying "All things should be based on self-cultivation," and the symbolic meaning and rational principles are entirely the same, maintaining continuity. Gen corresponds to the branch Yin, which is the position where humans reside; Zengzi said "Thoughts should not go beyond one's position," and he also examined himself three times daily, so he alone received the heart transmission of Confucius. Both the "Daxue" and "Zhongyong" take self-cultivation as the fundamental starting point, meaning they both originate from the position of Gen and Yin; the emphasis on the second line of the trigram Qian, which falls exactly on Yin, so the principles of the "Daxue" and "Zhongyong" both come from the second line of the trigram Qian. However, when Confucius wrote the "Yi Zhuan" and discussed establishing the human way, the key was in the third and fourth lines, so why is that? The answer is: this refers to the six lines of a single trigram. Considering the three extreme lines, the third and fourth lines are called the human lines, referred to as the human extreme; and the third and fourth lines are located at the junction of the inner and outer trigrams. When viewed together, they are the middle of the upper and lower trigrams; when viewed separately, they are the end of the inner trigram and the beginning of the outer trigram. The reversal and inversion of the sixty-four trigrams all originate from here, so the "Yi" discusses the human way by focusing on the third and fourth lines. As for "Zhongyong" and "Daxue," they refer to the trigram positions, so they both use Gen and Yin to explain the fundamental of cultivating the Dao. In fact, they are originally the same principle and the same lineage (Cheng is the beginning and end of all things, while Gen becomes the end and also the beginning). In other words, when discussing the lines of a trigram, it leads to the third and fourth lines; when discussing the trigram positions, it leads to Gen and Yin. Although they seem different, they are actually talking about the same thing: the key to the human way lies within oneself, and the starting point of practice is self-cultivation.

誠為艮，而至誠則為乾。至誠之道，是反身修德。由艮而反諸坎，由三而反諸一，即所謂涵三為一也。涵三為一，太極之象。太極以無含有，自坎至乾，是即由有而入無(乾西北戌亥數無)。無思也，無為也。寂然不動，感而遂通天下之故，不疾而速，不行而至，乃天下之至神，故至誠之道可以前知。

"Cheng" belongs to the trigram Gen, while "Zhi Cheng" belongs to the trigram Qian. The way of Zhi Cheng is to turn inward and cultivate virtue: from Gen returning to Kan, from three returning to one, which is called "Han San Wei Yi." Han San Wei Yi is precisely the image of the Taiji. Taiji contains nothingness to encompass existence, from Kan to Qian, it moves from existence into nothingness (Qian resides in the Northwest, its number is nothingness). When reaching this stage, it is what the Great Treatise says: "Without thought, without action, still and unmoving, yet responsive to all things and understanding the causes of the world." "Without haste yet swift, without movement yet reaching," this is the most marvelous state in the world, so the Zhongyong says that the way of Zhi Cheng can foresee the future. The difference between "Cheng" and "Zhi Cheng" is the difference between Gen and Qian, between existence and nothingness.

西北乾天道，東北艮人道。天地人，子丑寅，三建用中，建中立極。丑艮為身，寅艮為庶，三庶也，故曰“本諸身，徵諸庶民”，子天建，丑地建，寅人建，為“三王”、“三統”，故曰“考諸三王而不謬”。於乾知天，故曰“質諸鬼神而無疑”。“於艮知人”，故曰“百世以俟聖人而不惑”。此天地子丑而本諸戌亥，君子以一誠終始之，即以艮終始之也。誠則形，形則著，著則明，明則動，動則變，變則化。

The trigram 乾 (Qian) in the Northwest represents the Way of Heaven, and the trigram 艮 (Gen) in the Northeast represents the Way of Humanity. Heaven, Earth, and Humanity are assigned the three branches 子 (Zi), 丑 (Chou), and 寅 (Yin). By using these three branches together and taking the middle one, the concept of the center is established, which defines the standard for human conduct. 丑 (Chou) corresponds to 艮 (Gen) as the body, and 寅 (Yin) corresponds to 艮 (Gen) as the common people. The number three represents the common people, so the Classic of Rites (Zhongyong) says, "Begin with the self, and verify with the common people." 子 (Zi) represents the Heaven's establishment, 丑 (Chou) represents the Earth's establishment, and 寅 (Yin) represents the Human's establishment, which corresponds to the Three Kings of Xia, Shang, and Zhou and the Three Dynasties. Therefore, it says, "Examining the Three Kings does not lead to error." By observing 乾 (Qian), one understands Heaven, so it says, "Consulting ghosts and spirits brings no doubt." By observing 艮 (Gen), one understands Humanity, so it says, "Waiting for a sage for a hundred generations brings no confusion." The fundamental principle of the Heaven and Earth and the branches 子 and 丑 lies in 戌 (Xu) and 亥 (Hai). A gentleman upholds the character of "Sincerity" throughout, which is to uphold 艮 (Gen) throughout. Therefore, the Classic of Rites (Zhongyong) says, "When sincerity is present, it takes form; when it takes form, it becomes evident; when it becomes evident, it becomes clear; when it becomes clear, it leads to action; when it leads to action, it brings change; when it brings change, it leads to transformation."

西北乾庸言庸行終之，乾戌遠，坤申近，知遠之近也。東南巽辰，風自乾亥，知風之自也。乾亥微，艮寅顯，知微之顯也。君子入德深藏乾亥下，人所不見，潛也。西北隅屋漏相在爾室，乾為斧為鉞，不怒而民威於斧鉞也。西北乾天至神道，丑用中本末盡以在之。乾天神道，於震無聲，於巽無臭，故曰“聲色之於以化民末也，上天之載無聲是無臭至也”。

Finally, it concludes with the northwest trigram 乾 (Qian) and its concept of 'ordinary words and actions'. 乾 (Qian) resides in戌 (Xu), representing distance, while 坤 (Kun) resides in申 (Shen), representing proximity. Therefore, the Zhongyong says 'knowing the distant as near'. The southeast trigram 巽 (Xun) is paired with 辰 (Chen), while wind originates from 亥 (Hai), where 乾 (Qian) resides. Hence, it says 'knowing the source of the wind'. 乾 (Qian) at亥 (Hai) is subtle and profound, while 艮 (Gen) at寅 (Yin) is manifest and clear. Therefore, it says 'knowing the subtle and making it manifest'. A gentleman in the initial stage of cultivating virtue hides deep beneath 乾 (Qian) at亥 (Hai), which is unseen by others, corresponding to the initial line of 乾 (Qian), 'qian' (hidden). The northwest corner is where the roof leaks, as the Book of Poetry says 'when you are in your room, do not feel ashamed even in the place where the roof leaks'. 乾 (Qian) represents axes and halberds, hence 'without anger, the people fear it more than axes and halberds'. The northwest trigram 乾 (Qian) represents the most divine way of Heaven. The position of 丑 (Chou) uses the middle, encompassing both beginning and end. The divine way of Heaven, in 乾 (Qian), is silent in 震 (Zhen) and odorless in 巽 (Xun). Therefore, the final chapter of Zhongyong says 'sound and color are not the way to transform people, they are the lesser things', 'Heaven's carrying on is silent and odorless, that is the ultimate'.

此“中庸”乾九二正中，庸言庸行，閒邪存其誠，善世而不伐，德博而化。《中庸》一篇，以化終之也。

Here the term 「中庸」 refers to the second line, nine two, of the Qian hexagram. Nine two is located in the middle of the lower trigram's third line, which is the position of 「正中」, from which the character 「中」 is derived. The Wenyan commentary on the Qian hexagram explains nine two as follows: 「Ordinary speech is trustworthy, ordinary conduct is cautious, guarding against evil thoughts and preserving sincerity, benefiting the world without self-praise, having broad virtue and being able to transform all things.」 Being trustworthy in ordinary speech and cautious in ordinary conduct is called 「庸言庸行」, from which the character 「庸」 is derived. Preventing evil thoughts to preserve sincerity, benefiting the world without boasting, and having broad virtue that can transform all things. The concepts of 「sincerity」 and 「transformation」 discussed throughout the Zhongyong text are derived from these statements, so its conclusion also ends with the character 「化」.