The fifth section on trigrams.

《易》之為書，有天道焉，有地道焉，有人道焉。故《易》三畫而成卦。立天之道，曰陰與陽。立地之道，曰柔與剛。立人之道，曰仁與義。分陰分陽，迭用柔剛。故《易》六位而成章。三畫之卦，因而重之。為六畫之卦六十有四。《周禮》“太卜掌三《易》之法，《連山》、《歸藏》、《周易》，經卦皆八，其別皆六十有四”。經卦即乾坎艮震巽離坤兌三畫之卦，別卦則六畫之卦。三《易》法雖不同，而經卦別卦之數皆同。可見文王以前，早有六畫之卦。安得至《周易》始有後天八卦之方位乎？三代而後，師說紛歧，因卦有正反及上下左右相易，而皆別成一卦。故類別日多，稱名互異，後人幾無所適從。爰擇要疏錄，並各舉其例。以便初學。其名異而實同，或名同而各家之說互異者，名從其朔。要皆由別卦所孽乳，故署曰“卦別”。

The book of Change contains the principles of Heaven, Earth, and humanity, so the book of Change uses three lines to form one trigram. To establish the principle of Heaven, it is called "Yin" and "Yang"; to establish the principle of Earth, it is called "flexibility" and "rigidity"; to establish the principle of humanity, it is called "benevolence" and "righteousness." By distinguishing Yin and Yang, and alternating flexibility and rigidity, the book of Change requires six lines to form a complete hexagram. By stacking the three-line trigrams, they become six-line hexagrams, totaling sixty-four. The Zhou Li records that the Grand卜 (diviner) was in charge of three methods of the book of Change, namely the Lianshan, Guicang, and Zhouyi; their fundamental trigrams are all eight, and their compound trigrams are all sixty-four. The fundamental trigrams refer to the eight trigrams of Qian, Kan, Gen, Zhen, Xun, Li, Kun, and Dui, each consisting of three lines; the compound trigrams refer to the hexagrams with six lines. Although the three methods of the book of Change differ, the number of fundamental and compound trigrams is exactly the same, indicating that the hexagrams with six lines existed before King Wen. How could the Later Heaven Trigrams with their specific orientation have first appeared in the Zhouyi? After the Xia, Shang, and Zhou dynasties, different schools of thought diverged: because each trigram has a positive and a negative aspect, and there are various changes involving upper and lower, left and right reversals, each change produces a new trigram, leading to an increasing number of categories and differing names, making it difficult for later generations to follow. Here, I have selected and arranged the key points, with examples given for beginners. Where names differ but refer to the same thing, or where names are the same but different schools have different interpretations, the earliest terminology is used as the standard. All these terms ultimately originate from the compound trigrams, so the title is "Trigram Differentiation."

重三為六畫，合兩卦成一卦。故六十四卦，每卦均有內外之分。卦畫由下生，故以下卦為內，上卦為外。泰否之“內陰”而“外陰”，“內柔”而“外剛”，即《象傳》之自舉其例也。　泰　乾內坤外。故《彖》曰“內陽而外陰，內健而外順，內君子而外小人”。　否　坤內乾外。故《彖》曰“內陰而外陽，內柔而外剛，內小人而外君子”。　觀泰否兩《彖》，內卦外卦之別，顯然易明。惟卦有以內外言者，有不必以內外言者。通變無常，各適其當，未可泥一端而論也。

When a three-line trigram is overlaid to form a six-line diagram, it combines two three-line trigrams into one, so each of the sixty-four hexagrams has an inner and outer trigram. The hexagram is formed from bottom to top, so the lower three-line trigram is called the inner trigram, and the upper three-line trigram is called the outer trigram. The phrases such as "inner... and outer..." and "inner softness and outer hardness" in the hexagrams of Tai and Pi are examples given by the Commentary on the Judgment itself. The hexagram Tai has the trigram Qian inside and the trigram Kun outside, so the Commentary on the Judgment says, "inner yang and outer yin, inner strength and outer compliance, inner gentleman and outer petty person." The hexagram Pi has the trigram Kun inside and the trigram Qian outside, so the Commentary on the Judgment says, "inner yin and outer yang, inner softness and outer hardness, inner petty person and outer gentleman." By looking at these two commentaries on Tai and Pi, the distinction between inner and outer trigrams is clearly evident. However, among the sixty-four hexagrams, some can be explained using the concept of inner and outer, while others do not require it. There are no fixed rules, and each case should be considered according to its own context, without being rigidly confined to one approach.

《繫傳》曰：“陽卦多陰，陰卦多陽。陽卦奇，陰卦偶。”此專指六子之卦言也。後天以用為先，故以少者為主。一陽之震坎艮為陽卦，一陰之巽離兌為陰卦。而先天以體立言，則取多者為主。故兌巽以二陽之卦，合為澤風大過，言陽過乎中也。震艮以二陰之卦，合為雷山小過，言陰過乎中也。乾坤坎離，各得陰陽之中，皆三畫卦也。

The Great Treatise says: "Yang hexagrams contain more yin lines, and yin hexagrams contain more yang lines; yang hexagrams are odd, and yin hexagrams are even." This refers specifically to the six hexagrams derived from Qian and Kun. Later Heaven prioritizes function, so it takes the side with fewer lines as the main focus: Zhen, Kan, and Gen, which each contain one yang line, are considered yang hexagrams; Xun, Li, and Dui, which each contain one yin line, are considered yin hexagrams. Earlier Heaven, however, speaks in terms of essence, and takes the side with more lines as the main focus: Dui and Xun are both considered hexagrams with two yang lines, and together they form the hexagram Zhefeng (Great Excess), which discusses yang exceeding the center; Zhen and Gen are both considered hexagrams with two yin lines, and together they form the hexagram Le Shan (Small Excess), which discusses yin exceeding the center. As for Qian, Kun, Kan, and Li, each possesses the middle of yin and yang—Qian is purely yang, Kun is purely yin, Kan and Li each have half yang and half yin. All of the above still refer to the three-line hexagrams.

若六畫之卦，其陰陽之分，例至不一。有以卦言者，則乾與三男為主者稱陽卦，坤與三女為主者稱陰卦。有以爻為主者，則以奇爻為主者稱陽卦，偶爻為主者稱陰卦。自宋以後，諸說紛歧。有以邵子陽儀陰儀立論者，有以闢卦陰陽所生分別者，有以《序卦》之先後分陰陽者，如屯陽蒙陰需陽訟陰之類。雖各言之成理，要皆有當有不當。蓋乾坤為《易》之門，六子生自乾坤。八純而外，餘五十六卦。雖各有陰陽之分，必先明其體用，而後有分陰分陽之可言。以體用交互，體陽者用必陰，體陰者用必陽。準此以觀，諸家之圖說，自可瞭然矣。

As for the hexagram composed of six lines, the method of dividing yin and yang is highly inconsistent. Some classify based on the trigrams: those with Qian (Heaven) and three male lines are considered yang trigrams, while those with Kun (Earth) and three female lines are considered yin trigrams. Others classify based on the lines themselves: those with an odd number of lines (yang lines) are considered yang trigrams, and those with an even number of lines (yin lines) are considered yin trigrams. After the Song Dynasty, various schools of thought became more divergent: some based their views on Shao Yong's "Yang Instrument" and "Yin Instrument," others on the division of yin and yang according to the twelve great trigrams, and still others on the sequence of the "Sequence of Trigrams" text, such as Zu (Emergence) being yang and Meng (Awakening) being yin, Xu (Need) being yang and Song (Contention) being yin, and so on. Although each school presents its reasoning, some explanations make sense while others do not. To understand this, we must remember that Qian and Kun are the gateways of the Book of Changes, and the six subsidiary trigrams all originate from Qian and Kun. Outside the eight pure trigrams, there are fifty-six other trigrams, each with its own division of yin and yang. However, before discussing yin and yang, it is essential to first clarify their nature and function. Because nature and function are interdependent: if the nature is yang, the function must be yin; if the nature is yin, the function must be yang. By following this principle, the various diagrams and explanations of different schools will become clear at a glance.

卦之有貞悔，亦如爻之有動靜也。悔古作每卜。論卦體，則內貞而外悔。言占卜，則靜貞而動悔。坤之六三曰“含章可貞”，乾之上九曰“亢龍有悔”，此內貞外悔，經文之自舉其例也。《左傳》：“秦伯伐晉，卜徒父筮之，其卦遇蠱。曰：蠱之貞風也，其悔山也。”亦內貞外悔之一例也。《國語》：“秦伯召公子重耳於楚，楚子厚幣以送公子於秦。公子親筮之曰：尚有晉國。得貞屯悔豫皆八。”此靜貞動悔之例也。

A hexagram has both "Zhen" and "Hui," just as a line has both stillness and movement. The character "Hui" was written in ancient times with "Mei" above and "Bo" below. In terms of the hexagram structure, the inner trigram is Zhen and the outer trigram is Hui; in terms of divination, the unchanging line is Zhen and the changing line is Hui. The hexagram "Kun" says "Containing virtue, it is proper to be Zhen," and the hexagram "Qian" says "The dragon soaring high brings regret," which is an example of inner Zhen and outer Hui. The "Zuo Zhuan" records that Duke Mu of Qin attacked Jin, and Bo Tu Fu cast a divination for him, obtaining the hexagram "Gu," and said, "The Zhen of 'Gu' is wind (the lower trigram Xun), and its Hui is mountain (the upper trigram Gen)," which is another example of inner Zhen and outer Hui. The "Guoyu" records that Duke Mu of Qin summoned Prince Chong'er from Chu, and King Chu prepared rich gifts to send Prince Chong'er to Qin; Prince Chong'er himself cast a divination, praying, "May I be able to obtain Jin," and obtained "Zhen 'Zun' and Hui 'Yu' both eight," which is an example where the unchanging is Zhen and the changing is Hui.

《左傳正義》曰：“伏羲作十言之數，曰乾坤震巽坎離艮兌消息。”虞仲翔氏，即本此十言立說。《荀九家易注》：“泰卦曰陽息而升，陰消而降。”虞氏十二消息，復臨泰大壯夬乾為息卦，姤遁否觀剝坤為消卦。《易》曰：“君子尚消息盈虛，天行也。”又曰：“天地盈虛，與時消息。”故陰陽往復，此息則彼消，此消則彼息，亦卦義相傳之最古者也。卦例圖說詳下《卦候》章。

The "Commentary on the Spring and Autumn Annals with Justification" says: "Fuxi created the system of ten words, which are Qian, Kun, Zhen, Xun, Kan, Li, Gen, and Dui, plus Xiao and Xi." Yu Fan of the Three Kingdoms period based his theories on these ten words. The "Commentary on the Nine Schools of the Book of Changes by Xun Shuang" explains the Tai hexagram as: "Yang energy grows and rises, while Yin energy declines and descends." Yu Fan's twelve message hexagrams are: Fu, Lin, Tai, Da Zhuang, Guai, and Qian as "Xi hexagrams," meaning the growth of Yang energy month by month; Gou, Duan, Pih, Guan, Po, and Kun as "Xiao hexagrams," meaning the growth of Yin energy month by month. The Book of Changes says, "A gentleman values the waxing and waning of fullness and emptiness, which is the way of Heaven," and also says, "Heaven and Earth are full and empty, changing with the seasons." Therefore, yin and yang alternate endlessly; where one grows, the other declines, and where one declines, the other grows. This is one of the oldest transmitted meanings in the hexagrams. Specific examples of the hexagrams and their diagrams are detailed in the following chapter "Hexagram Seasons."

易道陰陽往來，乾圓往屈，坤方來信，往來屈伸。而六十四卦陰陽，與時消息，循環不窮。數往者順，知來者逆。《序卦》首乾坤終既未濟。乾往卦，坤來卦。屯往卦，蒙來卦。至既濟往卦，而未濟來卦，一往一來，各以其序。順逆之數，皆天地自然之法象也。

The principles of the Book of Changes are based on the movement of yin and yang. The trigram Qian (Heaven) is round in form, and its movement is characterized by bending; the trigram Kun (Earth) is square in form, and its movement is characterized by extension (the term 'xin' means 'extension'). One movement and one return correspond to one bending and one extension. The yin and yang in the sixty-four hexagrams change with the seasons, circulating endlessly. The "Commentary on the Hexagrams" states, "To understand the past is to follow the sequence, to foresee the future is to use reverse calculation." Calculating the past follows the order of progression, while predicting the future uses reverse numbers. The "Sequence of Hexagrams" begins with Qian and Kun and ends with "Ji Ji" and "Wei Ji": Qian is a hexagram of movement, Kun is a hexagram of return; "Zun" is a hexagram of movement, "Meng" is a hexagram of return; this continues until "Ji Ji" is a hexagram of movement and "Wei Ji" is a hexagram of return. One movement and one return follow their respective order, and this sequence of forward and reverse numbers reflects the natural patterns of heaven and earth.

對卦（旁通、錯卦、類卦）者，陰陽相對。如乾對坤，坎對離，屯對鼎，蒙對革，頤對大過，中孚對小過之類。虞氏謂之旁通，來知德氏謂之錯卦，《周易指》謂之類卦，皆對卦也。六十四卦相對者八，皆自為一卦。不相對者二十八卦，反之別為一卦，得五十六卦。然不相對者，亦各有其相對之卦。略例如右：以上八卦，陰陽相對，反覆不衰，有對而無反者也。餘卦類推，有對而又有反者也。

The concept of "opposite hexagrams" (also called "parallel," "opposite hexagrams," and "class hexagrams") refers to two hexagrams whose six lines are completely opposite in yin and yang. For example, the hexagram Qian (Heaven) is opposite to Kun (Earth), Kan (Water) is opposite to Li (Fire), Zun (Sprouting) is opposite to Ding (Tripod), Meng (Confusion) is opposite to Ge (Revolution), Yi (Nourishment) is opposite to Da Guo (Great Excess), Zhong Fu (Sincere Trust) is opposite to Xiao Guo (Small Excess), and so on. Yu Fan called it "parallel," Lai Zhide called it "opposite hexagrams," and Du Mu Guo Hu called it "class hexagrams," all referring to opposite hexagrams. Among the sixty-four hexagrams, there are eight hexagrams that remain the same when reversed, and each of these forms a hexagram on its own. The remaining twenty-eight hexagrams, when reversed, form new hexagrams, totaling fifty-six. However, those hexagrams that can form new ones when reversed also have their corresponding opposite hexagrams. A general example is as follows: these eight hexagrams have opposite yin and yang, but when the entire hexagram is reversed, its shape remains unchanged, belonging to the category of having "opposite" but not "reversal." The rest of the hexagrams follow the same pattern, all having both "opposite" and "reversal."

覆卦（反卦、綜卦）者，一卦覆之而又別成一卦者也。如屯之覆為蒙，需之覆為訟，師之覆為比。六十四卦，除乾坤坎離等八卦，餘五十六卦皆有覆卦也。漢人亦曰反卦，來知德氏謂之綜卦。（來氏謂綜者如織布扣經之綜，一上一下者也，故名反覆之卦為綜。然覆實上下相倒置，非一上一下之謂，綜之名殊未確合，故非議者甚多。）如震反為艮，巽反為兌。而震之與巽，艮之與兌，又為對卦也。餘卦類推。若泰否既未隨蠱漸歸妹等卦，對而兼覆，所謂反覆不衰者也。

The concept of "reversed hexagram" (also called "inverted hexagram" or "complementary hexagram") refers to turning a hexagram completely upside down to form another hexagram. For example, the hexagram Zun (屯) reversed becomes Meng (蒙), Xu (需) reversed becomes Song (讼), and Shi (师) reversed becomes Bi (比). The term "reversed" means turning the order of the six lines from top to bottom completely upside down. Among the sixty-four hexagrams, eight hexagrams—Qian (乾), Kun (坤), Kan (坎), and Li (离)—remain unchanged when reversed. The remaining fifty-six hexagrams each have their own reversed hexagrams. The Han dynasty referred to this as "inverted hexagram," and Lai Zhide called it "complementary hexagram." Lai Zhide explained that "complementary" is like the shuttle in weaving, which alternates between raising and lowering the warp threads, hence the term "complementary" for the reversed hexagram. However, "reversed" actually refers to flipping the entire hexagram upside down, not just alternating one line up and down. Therefore, the term "complementary" is not very accurate, and many have criticized it. For example, the trigram Zhen (震) reversed becomes Gen (艮), and Xun (巽) reversed becomes Dui (兑). Zhen and Xun, as well as Gen and Dui, are mutual complementary hexagrams. The other hexagrams follow the same pattern. As for hexagrams such as Tai (泰) and Po (否), Ji Ji (既济) and Wei Ji (未济), Sui (随) and Gu (蛊), Zhen (渐) and Gui Mei (归妹), they are both complementary and reversed, which is called "reversed and complementary without decline."

交卦（兩象易、上下易）者，本卦內外兩象，交相易位。內卦出外，外卦入內。虞氏謂之兩象易，亦有謂上下易者，向無定稱。今以其內外交易，故名之曰交卦，取便演講時之辨識，非敢云確當也。（來氏謂綜者一上下，以名此卦，或尚相近。惜數一百年來稱名久混，不能用也。）交卦之義，互見於經傳甚多。略舉如左：如天澤履與澤天夬，內外兩象，交相易位者也。故履上九曰“夬履，貞厲”。又如雷風恆與風雷益，內外兩象，交相易位者也。故恆曰“立不易方”，益曰“為益無方”。各卦以此類推，《彖》、《象》之相互見義者，厥例正多，不勝縷指也。

The concept of "intersecting hexagrams" (also called "two images" or "upper and lower reversal") refers to exchanging the positions of the inner and outer trigrams of the primary hexagram, moving the inner trigram to the outside and the outer trigram to the inside. Yu Fan called this "two images," and others referred to it as "upper and lower reversal." There has never been a fixed name for it. Now, since it involves the mutual exchange of inner and outer positions, I have named it "intersecting hexagram," in order to make it easier to distinguish when teaching, and I do not claim that this name is particularly accurate. (Lai Zhide referred to this as "comprehensive," meaning one above and one below, which might be a more appropriate term for this type of hexagram. However, due to long-standing confusion in terminology over the past few centuries, this term can no longer be used.) The meaning of "intersecting hexagrams" is supported by many mutual confirmations in the classic texts. Here are a few examples: for instance, the hexagram of Heaven above and Lake below (Treading) and the hexagram of Lake above and Heaven below (Decisive Action) are a pair formed by exchanging the inner and outer trigrams. Therefore, the upper ninth line of the Treading hexagram says, "Decisive Treading, firmness brings danger." Another example is the hexagram of Thunder above and Wind below (Perseverance) and the hexagram of Wind above and Thunder below (Benefit), which are also a pair formed by exchanging the inner and outer trigrams. Thus, the Treading hexagram says, "Standing firm without changing direction," while the Benefit hexagram says, "Benefiting without fixed direction." Other hexagrams can be similarly analyzed. There are many examples in the Commentary on the Judgment and the Image Commentary where their meanings are clarified through mutual reference, and it would be impossible to list them all.

半對者，本卦之內外兩象，有一象易為對卦，（如乾易坤、震易巽之類）或內或外，均與所易之卦，象義相關。其例如下：如雷澤歸妹，外卦之雷易為風，成風澤中孚，是上半之對卦也，故歸妹曰“月幾望”，中孚亦曰“月幾望”。又如山水蒙，外卦之山易為澤，成澤水困，亦上半之對卦也。故蒙曰“困蒙”。

The concept of "half opposition" refers to a situation in which one of the two trigrams in the primary hexagram is replaced by its opposite trigram, such as replacing Qian with Kun or Zhen with Xun. Whether the trigram replaced is the inner or outer trigram, the resulting hexagram will have a relationship in both form and meaning with the original hexagram. For example, in the hexagram "Thunder over Lake" (Gui Mei), if the outer trigram Thunder (Zhen) is replaced by Wind (Xun), it becomes "Wind over Lake" (Zhong Fu), which is a hexagram of half opposition. Therefore, the line six in the fifth position of Gui Mei says "the moon is almost full," and the line six in the fourth position of Zhong Fu also says "the moon is almost full." Another example is the hexagram "Mountain over Water" (Meng), where if the outer trigram Mountain (Gen) is replaced by Lake (Dui), it becomes "Lake over Water" (Kun), which is also a hexagram of half opposition. Therefore, the line six in the fourth position of Meng says "trapped confusion."

以上為上卦對易者也，名曰“上對”。其下卦對易者，更舉例如左：如天澤履，內象之澤易為山，成天山遁，是下半之對卦也。故履曰“履虎尾”，遁曰“遁尾”。又如地水師，內象之水易為火，成地火明夷，亦下半之對卦也。故師曰“左次”，明夷曰“左股”。以上為下卦對易者也。無論上下卦對易，與所易之卦，象義必相聯貫。惟有見於《彖》、《象》者，有不見於《彖》、《象》者。然雖不見於《彖》、《象》，而其意義自在。潛心玩之，必有所得也。

The above discusses the case of exchanging the upper trigram, known as "upper opposition." As for the case of exchanging the lower trigram, let us take another example: take the hexagram Heaven and Lake (Zou), and replace the inner image Lake (Dui) with Mountain (Gen), resulting in Heaven and Mountain (Dun). This is a hexagram with lower opposition, so the hexagram Zou says "treading on the tiger's tail," while the hexagram Dun's initial line says "fleeing to the tail." Another example is Earth and Water (Shi), and replacing the inner image Water (Kan) with Fire (Li), resulting in Earth and Fire (Mingyi), which is also a hexagram with lower opposition. Therefore, the line four of hexagram Shi says "moving to the left," while the line two of hexagram Mingyi says "dimmed on the left thigh." The above discusses the case of exchanging the lower trigram. Whether it is upper opposition or lower opposition, the original hexagram and the transformed hexagram must be connected in terms of image and meaning. Some of these connections can be seen in the Commentary on the Judgment and Image Commentary, while others cannot be found there. However, even if they are not explicitly mentioned in the Commentary on the Judgment and Image Commentary, the meaning still exists. If one carefully contemplates and reflects, one will surely gain some understanding.

半覆卦者，與半對卦同例。或內或外，各以覆象所得之卦求之，其義自見。舉例如左：如水雷屯，內卦之雷覆為山，成水山蹇，即下半之覆卦也。故“屯難也”，“蹇難也”。又如天雷無妄，內卦之雷覆為山，成天山遁，亦下半之覆卦也。故無妄曰“無妄之疾”，遁曰“有疾厲”。無妄曰“行人得牛”，遁曰“執用黃牛”。

The method of forming a "half-covered hexagram" is the same as that of the "half-opposite hexagram": in a six-line hexagram, the inner and outer trigrams are each three lines. In this case, one trigram is completely reversed to form its "inverted image," while the other remains unchanged. The resulting new hexagram is naturally related in meaning to the original. For example, in the hexagram of "Obstruction" (Zhun), the inner trigram is Zhen (Thunder), and when reversed it becomes Gen (Mountain), making the whole hexagram become "Water and Mountain: Obstruction" (Jian). This is an example of taking the inverted image from the lower half. Therefore, the "Sequence of Hexagrams" states that Zhun represents "difficulty" and that Jian also represents "difficulty," both hexagrams establishing their meaning through the concept of difficulty. Another example is the hexagram of "No False Movement" (Wu Wang), where the inner trigram Zhen (Thunder) is reversed to become Gen (Mountain), resulting in the hexagram "Heaven and Mountain: Retreat" (Dun). This is also an example of taking the inverted image from the lower half. Thus, in "No False Movement" (Wu Wang), the line Nine in the Fifth speaks of "sudden illness," and in "Retreat" (Dun), the line Nine in the Third speaks of "serious illness," both referring to illness. In "No False Movement" (Wu Wang), line Six in the Third mentions a cow being led away and "travelers" obtaining it, while in "Retreat" (Dun), line Six in the Second speaks of "using the hide of a yellow cow," both using the image of a cow. The language of the two hexagrams corresponds with each other because they are related by the half-covered relationship.

以上下半之覆卦也，名曰“下覆”。其上卦易為覆卦者，為例亦同。更略舉如左：如山天大畜，外象之山覆為雷，成雷天大壯，即上半之覆卦也。故大畜“利貞”，大壯“利貞”。大畜曰“輿說輹”，大壯曰“壯于大輿之輹”。又如風天小畜，外象之風覆為澤，成澤天夬，亦上半之覆卦也。故小畜曰“惕出”，夬曰“惕號”。小畜曰“既雨”，夬曰“遇雨”。

All the previous examples are cases of taking the lower trigram as the covered trigram, named "lower covering." If we instead take the upper trigram as the covered trigram, the method remains the same. Here are a few more examples: In the hexagram of Da Xu (Great Accumulation), the outer trigram is Gen (Mountain), which when reversed becomes Zhen (Thunder), transforming the entire hexagram into Da Zhuang (Great Strength). This is an example of taking the upper trigram as the covered trigram. Therefore, the hexagram statement of Da Xu says "favorable to uprightness," and the hexagram statement of Da Zhuang also says "favorable to uprightness." Da Xu's Nine in the Second says "the carriage axle's wooden block falls off," and Da Zhuang's Nine in the Fourth says "strength in the carriage axle's wooden block," both using the image of the carriage axle's wooden block. Another example is the hexagram of Xiao Xu (Small Accumulation), where the outer trigram is Xun (Wind), which when reversed becomes Dui (Lake), transforming the entire hexagram into Guai (Breaking Up). This is also an example of taking the upper trigram as the covered trigram. Therefore, Xiao Xu's Six in the Fourth says "blood flows and alarm arises," and Guai's Nine in the Second says "alarm and crying out," both discussing caution. Xiao Xu's Nine at the Top says "rain has fallen and the place is settled," and Guai's Nine in the Third says "meeting rain as if soaked," both discussing encountering rain.

以上上半之覆卦也，名曰“上覆”。覆卦亦稱反易，故互見之義，往往相反。如地山謙下覆為地雷復，謙曰“利用征伐”，復曰“十年不克征”，其最顯著者也。昔人不解八卦相錯，即六十四卦相錯之義，每卦只於六爻中摸索，望文生義，宜無從索解者多矣。

All the above examples are cases of the upper section forming the covered hexagram, named "upper covering." The covered hexagram is also called "reverse transformation," because it involves turning a three-line trigram completely upside down. Therefore, the meanings of the two hexagrams often turn out to be exactly opposite. For example, the hexagram of "Modesty" (Qian), with the mountain above, when the mountain is reversed to become thunder, the whole hexagram becomes "Recovery" (Fu). The hexagram of "Modesty" says "it is advantageous to use military force" (the line six in the fifth says "it is advantageous to invade," and the line six at the top says "it is advantageous to march, to conquer states"). However, the line six at the top of "Recovery" says "it will take ten years before military action can be taken," meaning that military action cannot be taken for ten years. On one side it is advantageous to wage war, on the other it is not. This is the most obvious example of opposite meanings in covered hexagrams. In the past, people did not understand the principle that two trigrams are opposite to each other, and this principle extends to the sixty-four hexagrams, each of which is opposite to another. Therefore, when reading a hexagram, they only searched within the six lines of that hexagram and guessed the meaning from the words used, which is why there are so many passages in the Book of Changes that are difficult to understand and cannot be interpreted.

上下對易者，本卦上下兩象，自相對易。如泰否既未濟之類。象既對易，卦義必自相對。如泰否，上下自相對易者也。故泰曰“小往大來”，否曰“大往小來”，自相對舉。既未濟言上下例同，不贅。

The concept of "upper and lower trigrams being exchanged" refers to the primary hexagram's upper and lower trigrams being swapped. Examples include Tai and Pi, as well as Ji Ji and Wei Ji. Since the two trigrams have been reversed, their meanings must also be opposites. Taking Tai and Pi as an example: when the upper and lower trigrams of Di Tian Tai are exchanged, it becomes Tian Di Pi, which is exactly an exchange of upper and lower. Therefore, the statement of the Tai hexagram says "small things go out and great things come in," meaning the yin and flexible move outward while the yang and strong move inward; the statement of the Pi hexagram says "great things go out and small things come in," meaning the yang and strong move outward while the yin and flexible move inward. The two statements are directly opposite. The principle of exchanging upper and lower trigrams in Ji Ji and Wei Ji is the same, so there is no need to repeat it.

上下反易者，本卦上下兩象，自相反易，如頤如大過及中孚小過之類。象既反易，卦義亦往往見相反之意。舉例如下：頤，山雷頤，上下兩象，自相反易。故《象》曰“道大悖也”。悖字古文本從兩或字，正反相對。一字之微，其與卦象之適合精當至此，謂非造化之筆哉！

The phrase "upper and lower reversed" refers to the primary hexagram consisting of two three-line trigrams that are inverted relative to each other. For example, the hexagram Yi (颐), Da Guo (大过), Zhong Fu (中孚), and Xiao Guo (小过) all exhibit this relationship. Since the two trigrams are in a reversed position, the meaning of the hexagram often expresses a contrast between opposites. To illustrate, Yi (颐) is composed of the trigram Gen (山, mountain) above and Zhen (雷, thunder) below. When Zhen is inverted, it becomes Gen, and when Gen is inverted, it becomes Zhen. Thus, the upper and lower trigrams are directly opposite. Therefore, the Image Commentary of line six in the third position of Yi states, "the path is greatly contradictory." The character "悖" (contradictory) originally consisted of two "或" (or) characters, one upright and one inverted, facing each other. The form of the character itself embodies the concept of opposition. How could such a subtle detail of a character so precisely correspond to the hexagram's imagery? Is this not the brushwork of nature's artistry?

頤之名，取上下相合。而上止下動，非頤亦無以確肖其象。而象之上下，又為反易。玩《彖》、《象》六爻，無不各顯其義。神矣哉！又如大過，澤風大過，亦上下兩象自相反易。故象取枯楊生稊，枯楊生華，亦以示反常之意也。

The hexagram "Yi" derives its name from the image of the upper and lower jaws coming together to form the mouth and cheeks. The upper trigram is Gen (Mountain), representing stillness, and the lower trigram is Zhen (Thunder), representing movement. The upper jaw remains still while the lower jaw opens and closes, and there is no other image that can so accurately depict this. Moreover, the two trigrams are in a reciprocal relationship. Careful examination of the "Commentary on the Judgment," the "Image Commentary," and the statements of the six lines reveals that each passage conveys this meaning, truly marvelous! Similarly, in the hexagram "Da Guo" (Great Excess), which is composed of the upper trigram Dui (Lake) and the lower trigram Xun (Wind), Dui is inverted to become Xun, and Xun is inverted to become Dui. This also represents the mutual reversal of the two trigrams. Therefore, in the hexagram's imagery, it says "a withered poplar tree produces new sprouts," meaning a withered poplar tree unexpectedly sprouts new growth; it also says "a withered poplar tree produces flowers," meaning a withered poplar tree unexpectedly blooms, both used to express the meaning of abnormal reversal.

半對半覆以下，古人皆未嘗言之。觀象玩辭，發見各卦之互見其義，或互見其名者，於卦象無不相關。反覆推求，始知皆由於上下兩象，各有正對反對之故。《彖》、《象》經傳，不啻自舉其例，至為明顯。遍徵各卦，無不貫通。一旦豁然，如撥雲霧，而見天日。因廣對卦覆卦之意，類別其名，以便講習之稱引。然字累而文不雅馴，殊為歉憾。海內外專精小學名學之君子，能各撰一簡賅切當之名，畀垂久遠，則著者所寤寐求之，馨香祝之者焉。

From the categories of "half paired" and "half reversed" and the like, ancient scholars have never spoken. Observing the hexagrams and pondering the line statements, I found that the meanings of the hexagrams are interrelated, and even the names of the hexagrams are interrelated, and all of these are connected to the hexagram images. After repeated inquiry, I realized that all of this comes from the two three-line hexagrams above and below, each having two paths: "correct pairing" (complete yin-yang transformation) and "opposite pairing" (reversing upper and lower). The commentaries of the "Commentary on the Judgment" and "Image Commentary" are like setting forth the rules themselves, clear and explicit; examining all the hexagrams, they are all interconnected. Once suddenly enlightened, it is like removing the clouds to see the sun. Therefore, I extended the meaning of "paired hexagram" and "reversed hexagram," classifying them and giving them names for convenience in discussion. However, these names are cumbersome in wording and not sufficiently refined, which I find regrettable. If scholars throughout the world who are proficient in textual analysis, word meaning, and logic could each propose a simple and accurate name for them to be passed down, that would be exactly what I long for and pray for with incense.

《繫傳》曰：“爻也者，各指其所之。”之，往也，由此往彼也。《左傳》蔡墨曰“坤之乾，亦乾之坤”，言八卦陰陽相交，奇偶相易。虞仲翔氏專言之正，謂不當位之爻(如陽爻之二四六，陰爻之一三五。)，皆當之而當位。

The Great Treatise says: "A line refers to its movement toward a particular place." "Zhi" means going, moving from one place to another. The Zuo Zhuan records that the historian Cai Mo of the Jin state said, "The Earth Trigram transforms into the Heaven Trigram, and the Heaven Trigram transforms into the Earth Trigram," referring to the mutual interaction of the Eight Trigrams and the exchange between odd and even lines. Yu Fan of the Three Kingdoms period (with the courtesy name Zhongxiang) specialized in explaining "zhi zheng," believing that any line that is not in the correct position should change once to return to the proper position. For example, a yang line in an even position such as second, fourth, or sixth, or a yin line in an odd position such as first, third, or fifth, should change.

乾之二四六爻不當位，之坤之一三五則當位。坤之一三五不當位，之乾之二四六則當位。名曰“之正”，謂之而得其正也。惟乾之坤，坤亦之乾，則乾坤兩卦，成兩既濟。既濟定，陰陽消息，皆止而不行，乾坤幾乎息矣。故又以成兩既濟為凶。

The second, fourth, and top lines of the Qian hexagram are yang lines in even positions, which are considered improper. They should move to the first, third, and fifth lines of the Kun hexagram, which are odd positions, thus becoming proper. Similarly, the first, third, and fifth lines of the Kun hexagram are yin lines in odd positions, considered improper, and should move to the second, fourth, and sixth lines of the Qian hexagram, which are even positions, thus becoming proper. This is called "zhizheng," meaning that moving once leads to correctness. However, when Qian moves to Kun and Kun moves to Qian, both hexagrams become the Ji Ji hexagram. Once Ji Ji is formed, the structure of the hexagram becomes fixed, and the growth and decline of yin and yang come to a halt. As a result, Qian and Kun almost cease to exist. Therefore, Yu Shi judges "cheng liang ji ji" as an ominous outcome.

必乾初之坤四而上之坤三，然後旁通交變以周六十四卦。初上未動，而二五先行，亦為凶。焦理堂《易通釋》，又本此以演為《時行》、《失道》各圖。要皆一家之言，足備參考。若泥其例以言《易》，多窒礙難通，學者不可不知也。

Following the example of the Yu school, one must first move the initial line of Qian (Heaven) to the fourth line of Kun (Earth), and the top line of Qian to the third line of Kun, and then proceed to the lateral transformation and change, in order to circulate through all sixty-four hexagrams. If the first and top lines have not yet moved, but the second and fifth lines move first, this is considered凶 (unfavorable). Qing dynasty scholar Jiao Xun (styled Litan) in his work "Yi Tong Shi" further developed diagrams such as "Shi Xing" and "Shi Dao" based on this theory. These are merely the views of one school of thought and can only be considered as reference. If one rigidly adheres to this system when interpreting the Zhou Changes, there will be many obstacles and difficulties. Students of the Zhou Changes should not be unaware of this.

右為乾坤二卦之正，餘卦類推。惟占家言卦變亦曰之，則不以一爻為限。三爻變至五爻變，皆以所變之卦為之。如乾之剝，家人之睽。一卦變六十三卦，皆謂之曰之卦也。詳《學易筆談初集》。

The above discussion explains the situation of the Qian and Kun hexagrams "zhi zheng" (the correct form). The other hexagrams can be understood in the same way. However, when diviners discuss hexagram changes, they also use the word "zhi," which does not refer to a single line change only. From three-line change to five-line change, the hexagram that results from the change is called "zhi suogua" (to become a certain hexagram), for example, "Qian zhi Bao" and "Jiaoren zhi Kui." One hexagram can change into the other sixty-three hexagrams, all of which are referred to as "zhi gua." For details, see my work "Xue Yi Biren Cijie."

互卦(中爻、約象)者內外四爻，二至四，三至五，又各成三畫之卦一。兩卦交互，又成兩卦，故漢人謂之互卦。京氏曰：“會於中而以四為用，一卦備四卦者，謂之互。”崔氏子元曰：“中四爻雜合所主之事，撰集所陳之德，能辨其是非，備在中四爻也。”(京氏所謂一卦備四卦，以本卦內外兩象，更以二至四三者至五，又各得一卦，合為四卦也。)

The "Nuclear Hexagram" (also called "Middle Lines" or "Approximate Image"), is formed by taking four lines from the middle of a six-line hexagram, specifically the second to the fourth lines, which make up a three-line trigram. Similarly, the third to the fifth lines also form another three-line trigram. These two trigrams interact with each other, producing two additional hexagrams. Therefore, the Han dynasty scholars called it the "Nuclear Hexagram." Jing Fang said: "Meeting in the middle and using the fourth line, when one hexagram contains four hexagrams, it is called the Nuclear Hexagram." Cui Xing (styled Ziyuan) said: "The four middle lines contain the matters that the hexagram governs, and gather the virtues that the hexagram presents. They can distinguish right from wrong, and all these are fully included within these four lines." -- Original note: Jing Fang's statement that one hexagram contains four hexagrams refers to the fact that the original hexagram already includes two three-line trigrams, one from the inner and one from the outer. Then, by taking lines two to four and three to five, two more trigrams are formed, making a total of four hexagrams.

《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。若夫雜物撰德，辨是與非，則非其中爻不備。噫！亦要存亡吉凶，則居可知矣。”

The Great Treatise says: "The Zhou Changes is a book that traces the beginning of things and concludes their end, and this is its essence. The six lines are intermingled, but each follows its time and represents its thing. The meaning of the first line is difficult to see, while the meaning of the top line is easy to understand, because one is the origin and the other is the result. The statement of the first line merely imagines the beginning, while the top line finally completes it. As for intermingling all things, recording their virtues, distinguishing right and wrong, this cannot be fully understood without the four middle lines. Alas! To grasp the key of existence and non-existence, good fortune and misfortune, one can do so while sitting still."

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。三與五，同功而異位。三多凶，五多功，貴賤之等也。其柔危，其剛勝耶？”(阮氏《校勘記》：其用柔中也“用柔”下有“得”字。)

The Great Treatise also says: "The second line and the fourth line have the same function but different positions, so their goodness differs. The second line often receives praise, while the fourth line often incurs fear, because the fourth line is close to the position of the ruler. The way of flexibility and gentleness is not beneficial when it is far from the position of honor. The reason why the second line can achieve no fault is that it uses flexibility and is in the central position. The third line and the fifth line also have the same function but different positions: the third line often encounters danger, while the fifth line often achieves accomplishments, this is due to the difference in rank and status. Being in such positions, using flexibility is dangerous, and using strength is the way to be competent." — Original note: Yuan Yuan's "Notes on Collation" points out that in the phrase "its use of flexibility and centrality," there is an additional character "de" (to be able to) after "use of flexibility."

《繫傳》之言中爻，詳矣備矣。字字精實，孕育宏深，不可忽略。即以六爻言，初上及中四爻之時位象數，已劃然分明。後儒尚有疑互卦為非聖人之說者，則承掃象之餘習，不辨中爻二四三五之說，他更無論矣。《繫傳》既詳備至此，而《左傳》之說，尤可為三代言易不廢中爻之確證。為圖如下：

The Great Treatise explains the middle lines in detail, thoroughly and precisely, with profound implications, and cannot be easily overlooked. Considering only the six lines, the time, position, image, and numbers of the first line, the top line, and the four middle lines are clearly divided. Later Confucian scholars even doubted that the mutual hexagram is not the saying of the sage. This is due to their following Wang Bi's practice of "sweeping away images," failing to distinguish the middle lines as explained in the Great Treatise, namely the second and fourth lines, and the third and fifth lines. Other matters are even less understood. Since the Great Treatise has reached such a detailed level, and the divination examples in the Zuo Zhuan can serve as evidence that the Book of Changes was practiced in the Xia, Shang, and Zhou dynasties without neglecting the middle lines, the following diagram is provided:

《左氏傳》：“敬仲之將生也，周史有以《周易》見陳候者，陳候使筮之，遇觀之否(四爻動，偶變奇，風地觀變天地否)。曰：是謂觀國之光，利用賓于王。又曰：坤土也，巽風也，乾天也。風為天(風為天，猶云風變天，即巽變乾也)於土上，山也。”謂觀中爻三至五為艮，變否二至四亦為艮，此言互卦之最古者。

The *Zuo Zhuan* records: "When Jinzhong, the son of Chen Liji, was about to be born, the royal historian of the Zhou court came to Chen Hou with the *Zhou Yi* to divinate. Chen Hou asked him to cast the divination, and he obtained the hexagram *Guan* (Observation) changing to *Pi* (Opposition) —— note: the fourth line changed, an even line becoming an odd line, so *Guan* (wind on earth) transformed into *Pi* (heaven and earth opposing). The historian said: This is what is called 'Observing the brilliance of the state, useful for serving the king.' He also said: *Kun* is earth, *Xun* is wind, *Qian* is heaven. Wind becoming heaven —— note: 'Wind becoming heaven' means wind transforming into heaven, that is, *Xun* transforming into *Qian* —— and being above earth, this is the image of a mountain." Why did the historian suddenly conclude a 'mountain' from *Kun* (earth), *Xun* (wind), and *Qian* (heaven)? The key lies in the *Hugua* (intermediate hexagram): *Guan* (Observation) produces *Gen* (mountain) from the third to fifth lines; after transforming into *Pi* (Opposition), *Gen* (mountain) is still produced from the second to fourth lines. Therefore, he dared to make this statement. This is the earliest recorded explanation of *Hugua*.

故《朱子語錄》曰：“互體如屯卦震下坎上，就中四爻觀之，自二至四，則為坤。自三至五，則為艮。互體漢儒多用之。《左傳》中說占得觀卦處亦分明，看來此說亦不可廢。”蓋其時言易者，皆莫明中爻，而反對互卦。故朱說云爾。

Therefore, the "Collected Sayings of Master Zhu" states: "The mutual trigrams, for example, the Zun hexagram has Zhen below and Kan above. If we look at the fourth line, from line two to four we get Kun, and from line three to five we get Gen. The concept of mutual trigrams was commonly used by Han dynasty scholars. The passage in the "Zuo Zhuan" discussing the Guan hexagram also explains it clearly, indicating that this approach should not be discarded." Probably, at that time, those who interpreted the Book of Changes did not understand the middle lines and一味 opposed the mutual trigrams, so Master Zhu spoke like this.

《繫傳》言中爻謂二五三四，初與上不與焉。因二與五為本卦上下兩象之中爻，三為二至四成一卦之中爻，四為三至五成一卦之中爻，故四者皆謂之中爻。然後人之言互卦者，則不僅二三四五，實合初上與二三四五言之。又自五至上，復反至初言之，皆謂之互。故京氏又謂之約象。觀《雜卦傳》兩卦對舉，中爻皆成互卦。而自頤以下八卦，皆首尾交互，返歸於乾坤坎離四卦。列互卦如左：

The "middle lines" mentioned in the Great Treatise refer to the second, fifth, third, and fourth lines, excluding the first and top lines. This is because the second and fifth lines are the central positions of the lower and upper trigrams of the primary hexagram, respectively; the third line is the central position of the hexagram formed by the second to fourth lines; and the fourth line is the central position of the hexagram formed by the third to fifth lines. Therefore, these four lines are all called middle lines. However, later scholars discussing the nuclear hexagram did not limit themselves to the second, third, and fourth lines, actually including the first and top lines as well. They also took the image from the fifth line to the top line and then back to the first line, all of which are called "mutual." Hence, Jing Fang also referred to this as "approximate image." Looking at the Miscellaneous Hexagrams, where two hexagrams are contrasted, the middle lines can generate new hexagrams. Starting from the颐 hexagram, the eight following hexagrams all interconnect at their beginning and end, ultimately converging on the four hexagrams of Qian, Kun, Kan, and Li. Here is a list of the nuclear hexagrams:

吳草廬曰：“自昔言互體者，只以六畫之四畫互二卦而已，未詳其法象之精也。今以先天圖觀之，互體所成十六卦，皆隔八而得(外一層隔八卦得二卦，即中一層互體之卦名)，縮四而一(內層一卦縮外層四卦)。

In the Yuan dynasty, Wu Cheng, who was known as Cao Lu, said: "Previously, those who discussed mutual trigrams only took four lines out of the six lines to form two trigrams, but they did not clearly understand the subtle principles of the method and imagery involved. Now, by looking at the Earlier Heaven trigram diagram, the sixteen trigrams formed by mutual trigrams are all obtained by taking every eighth trigram. The original note says: the outer layer produces two trigrams by taking every eighth trigram, which are the names of the trigrams formed by the mutual trigrams in the middle layer. Moreover, four trigrams are condensed into one trigram. The original note says: the trigram in the inner layer is formed by condensing the four trigrams in the outer layer."

圖之左邊，起乾歷八卦至睽歸妹(中層睽妹即接乾夬)，又歷八卦而至家人既濟(家人既濟即接睽妹，餘卦仿此)，又歷八卦至頤復。圖之左邊，起姤大過，歷八卦至未濟解，又歷八卦至蹇漸，又歷八卦至剝坤。左右各二卦互一卦，合六十四卦互體只成十六卦，又合十六卦互成四卦，即乾坤既未濟。學《周易》始乾坤終既未濟，以此歟！”

On the left side of the diagram, starting from the Qian trigram, going through the eight trigrams to reach the Kui and Gui Mei trigrams—note: the middle trigrams Kui and Gui Mei follow Qian and Guai—then going through the eight trigrams again to reach the Jia Ren and Ji Ji trigrams—note: Jia Ren and Ji Ji follow Kui and Gui Mei, the rest follow the same pattern—then going through the eight trigrams again to reach Yi and Fu. On the other side (the original text still refers to it as the left side, but according to the meaning it should be the right side), starting from Guo and Da Guo, going through the eight trigrams to reach Wei Ji and Jie, then going through the eight trigrams again to reach Jian and Jie, then going through the eight trigrams again to reach Po and Kun. Each pair of trigrams on the left and right forms one trigram, combining the mutual relationships of the sixty-four trigrams into sixteen trigrams; then by mutual inference among these sixteen trigrams, four trigrams are formed, namely Qian, Kun, Ji Ji, and Wei Ji. Studying the Zhou Yi begins with Qian and Kun and ends with Ji Ji and Wei Ji, the reason for this is probably here.

蓋統云互卦，不但為中四爻，二至上五至初，亦皆交互成兩卦。更以五至上，復由上反之初二三四，亦互成兩卦，所以又稱約象。《序卦》“剝窮上反下”，即由上反初之一例也。

In general, the term 'mutual hexagram' refers not only to the four central lines from the second line to the top line and from the fifth line to the first line, which can form two hexagrams through interaction. Further, by counting from the fifth line to the top line and then returning from the top line to the first, second, third, and fourth lines, two hexagrams can also be formed, hence it is also called 'approximate image'. The 'Sequence of Hexagrams' states that the hexagram of 'breaking up' is 'exhausting the upper and returning to the lower', which is an example of returning from the top line to the first line.

澤山咸。中爻為天風姤。五至初，互天山遁。二至上，互澤風大過。若由五至上，更反之初二三四，仍互澤雷隨。由四至上，更反至初二，互地天泰，澤山反為雷風，則中爻又與上下互亦各相反。

Taking the Hexagram Xian (泽山咸) as an example: the four central lines produce the Hexagram Gou (天风姤). Counting from the fifth line to the first line, the resulting hexagram is Dun (天山遁). Counting from the second line to the top line, the resulting hexagram is Da Guo (泽风大过). If counting from the fifth line to the top line and then returning to the first, second, third, and fourth lines, the resulting hexagram is Sui (泽雷随). Counting from the fourth line to the top line and then returning to the first and second lines, the resulting hexagram is Tai (地天泰). The Hexagram Xian consists of Dui above and Gen below; when the entire hexagram is reversed, it becomes Heng (雷风恒). This shows that the two groups of hexagrams formed by the middle lines and the upper and lower lines respectively have opposite relationships with each other.

地天泰。中爻為雷澤歸妹。五至初，互雷天大壯。二至上，互地澤臨。五至上，更反之一二三四，為澤山咸。四至上，更反至初二，為澤雷隨。地天泰，反為天地否。即上下易，中爻上下互，相反亦同。

Taking the hexagram of Tai (Heaven over Earth) as an example: the four central lines produce the hexagram of Gui Mei (Lake over Thunder). Counting from the fifth line to the first line, the hexagram of Da Zhuang (Thunder over Heaven) is formed. Counting from the second line to the top line, the hexagram of Lin (Lake over Earth) is formed. Counting from the fifth line to the top line and then returning to the first, second, third, and fourth lines, the hexagram of Xian (Lake over Mountain) is formed. Counting from the fourth line to the top line and then returning to the first and second lines, the hexagram of Sui (Thunder over Lake) is formed. Reversing the hexagram of Tai results in Pi (Earth over Heaven), which means the upper and lower trigrams are exchanged. The situation where the middle trigram is opposite to the ones formed by the upper and lower lines is the same as in the hexagram of Xian.

闢卦亦稱十二消息卦，與月卦另詳下《卦氣》章。

"Bi Gua" is also called "the Twelve Message Hexagrams," referring to the twelve hexagrams that represent the gradual increase and decrease of yin and yang: from Fu, Lin, Tai, Da Zhuang, Guai, Qian, to Gu, Dun, Po, Guan, Po, and Kun. The method of pairing them with the "Monthly Hexagrams" is explained in detail in the following chapter "Gua Qi," but here only the names are listed.

宋儒言《易》，又有包卦之名。包卦者，六畫之卦，以上下三畫，中包三畫。如咸恆為坤包乾，損益為乾包坤之類，亦由互卦推衍而得者也。包卦之名，始於林黃中氏，謂由一卦分兩卦，兩卦分四卦，一正一反，又得四卦。是即太極生兩儀，兩儀生四象，四象生八卦也。因朱晦庵力駁之，故其說不行。然言之成理，亦不可謂毫無意義也。後儒如紀大奎等，頗採其說。略舉其例，如右圖：

In the Song dynasty, Confucian scholars discussed the Book of Changes and established the concept of "baogua" or "encompassing hexagram." The idea of baogua involves considering the top single line and the bottom two lines (or the bottom single line and the top two lines) of a hexagram as the outer trigram, which encloses the middle trigram. For example, the hexagram Xian (Attraction) and Heng (Perseverance) are formed by the trigram Kūn (Earth) enclosing the trigram Qián (Heaven), while the hexagrams Shen (Loss) and Yi (Gain) are formed by the trigram Qián enclosing the trigram Kūn. This method of interpretation is derived from the concept of mutual trigrams. The term "baogua" was first introduced by Lin Li (styled Huangzhong), a Song dynasty scholar. He believed that a single hexagram could be divided into two trigrams, which in turn could be divided into four trigrams, and with one being positive and the other negative, four more trigrams could be obtained. This corresponds to the principle mentioned in the "Xici Zhuan" (Great Treatise) that the Taiji (Supreme Ultimate) gives rise to two yi (binary opposites), two yi give rise to four xiang (images), and four xiang give rise to the eight trigrams. However, because Zhu Xi (styled Hui'an) strongly criticized this view, it did not become widely accepted. Nevertheless, the reasoning behind it is coherent and cannot be entirely dismissed as meaningless. Later scholars, such as Ji DaKui of the Qing dynasty, adopted this approach. For example, as shown in the diagram on the right:

六十四卦，皆可以此類推。但咸恆坤包乾也，而咸亦可謂之坎包巽，恆亦可謂之坎包兌。損益乾包坤也，損亦可謂之離包震，益亦可謂之離包艮，其實即中爻之二與四三與五也。他卦上下相推，亦無不如此。特以陰陽內外言之，亦自有此一象。故存之。

The sixty-four hexagrams can all be deduced in the same way. However, the hexagrams of Xian and Heng, although said to be composed of Kūn encompassing Qián, can also be interpreted as Càn encompassing Xūn for Xian, and Càn encompassing Dui for Heng; similarly, the hexagrams of Shen and Yi, although said to be composed of Qián encompassing Kūn, can also be interpreted as Li encompassing Zhen for Shen, and Li encompassing Gen for Yi. In the end, these encompassed hexagrams are actually the two middle lines of the nuclear hexagram, namely lines two to four and three to five. For all other hexagrams, the same principle applies when considering their upper and lower parts. However, from the perspective of yin and yang encompassing each other, these constitute a distinct kind of hexagram image, so they are retained.

像卦者，合六爻之奇偶觀之，像三畫之一卦也。《繫傳》曰：“象也者像也。”故謂之像卦。來知德氏謂之大象，如大坎大離之類，實皆像卦也。於卦義均有關係，《彖》、《象》可證者甚多。舉例如下：頤像離，中孚像離，大過像坎，小過像坎。

The concept of "like hexagram" refers to viewing the combination of odd and even numbers of the six lines together, resulting in an overall shape resembling one of the three-line trigrams. The "Great Treatise" states: "An image is like a likeness." Therefore, it is called a like hexagram. The Ming dynasty scholar Lai Zhide referred to it as "great image," such as Great Water (Kan) and Great Fire (Li), which are actually like hexagrams. It is related to the meaning of the hexagram, and there are many examples that can be found in the Commentary on the Judgment and the Image Commentary. For example, the hexagram Yi has two yang lines at the top and bottom and four yin lines in the middle, resembling the trigram Li; the hexagram Zhongfu also has two yang lines at the top and bottom and two yin lines in the middle, resembling Li. The hexagram Da Guo has two yin lines at the top and bottom and four yang lines in the middle, resembling Kan; the hexagram Xiao Guo has four yin lines at the top and bottom and two yang lines in the middle, also resembling Kan.

此來氏所謂大象之坎離，故《序卦》以列於坎離既未濟之前也。餘卦仿此。重畫或單畫者例亦同。大壯像兌。雷天大壯無兌，而合全卦觀之，有兌象。兌為羊，故象辭云“羝羊”“喪羊”。

This is what Lai Zhide said about the "Great Image of Kan and Li." Therefore, the "Sequence of Hexagrams" places Yi and Da Guo before Kan and Li, and Zhong Fu and Xiao Guo before Ji Ji and Wei Ji. The other hexagrams can be similarly inferred. Whether the Great Image is formed by overlapping two lines or a single line, the format remains the same. For example, Da Zhuang resembles Dui: although the six lines of Da Zhuang do not contain the hexagram Dui, when viewed as a whole, the upper two yin lines and the lower four yang lines form a magnified image of Dui. Dui is associated with the sheep in the "Discourse on the Hexagrams," which is why the line statements of Da Zhuang mention "the ram strikes against the fence" and "losing the sheep in ease."

剝像艮。山地剝，合全體觀之，有艮象，故重艮，《象》云“孚于剝”。六爻合觀之象，無不各有取義。如咸恆損益，為乾坤互交，而又像坎離者也。故咸恆繼坎離為《下經》之首，而損益次十一十二，與泰否等，其義大可見矣。

The hexagram Bo resembles Gen: The hexagram Bo itself already has Gen on top, and when viewed as a whole, it has one yang line at the top and five yin lines below, forming a larger image of Gen, equivalent to Gen on top of Gen. Therefore, the Image Commentary of line six in the fifth position of Bo says, "Trust in the situation of Bo." When all six lines are considered together, the overall image derived has meaning in every aspect. For example, the hexagrams Xian, Heng, Shen, and Yi are all formed by the interaction of Kan and Li, and their overall images also resemble Kan and Li. Therefore, Xian and Heng follow Kan and Li and are placed at the beginning of the Lower Classic, while Shen and Yi are placed at the eleventh and twelfth positions of the Lower Classic, corresponding exactly to the positions of Tai and Po in the Upper Classic. The significance behind this arrangement is clearly evident.