The Earlier Heaven Diagram Explanation · Zhaozi's Sayings and Li Guangdi's Annotations

又曰：坎離者，陰陽之限也，故離當寅，坎當申，而數常逾之者，陰陽之溢也。然用數不過乎中也。此更宜思。離當卯，坎當酉，但以坤為子半可見矣。

Shao Yong also said: "The trigrams Kan and Li are the boundaries of yin and yang respectively, so they should be paired with the twelfth earthly branches, with Li paired with Yin and Kan paired with Shen. However, when actual calculations are made, the numbers often exceed this boundary, which is due to the excess of yin and yang energy overflowing. Yet the true numbers used will never exceed the middle path. This point requires further careful reflection. If another method of pairing is used, Li should be paired with Mao and Kan with You. This can be understood by looking at the half of the Zi hour where the trigram Kun is located in the Earlier Heaven diagram."

又曰：先天學，心法也，故圖皆自中起，萬化萬事，生於心也。又曰：圖雖無文，吾終日言而未嘗離乎是，蓋天地萬物之理，盡在其中矣。

Shao Yong also said: "The study of Earlier Heaven deals with the mind method, so all the diagrams of Earlier Heaven start from the center of the diagram. All the myriad changes and things under heaven are generated from this one mind. Shao Yong also said: Although there are no characters written on the diagram of Earlier Heaven, I speak of it all day long, and what I speak of does not deviate from it at all, because the principles of the universe and all things are contained in this diagram."

案自孔子既沒，易道失傳，義理既巳差訛，圖象尤極茫渺，唯《大傳》“帝出乎震”一條，所載八卦方位，顯然明白，故學者有述焉。其餘如“卦氣”“月候”之屬，皆漢儒傅會，非聖人本法也，至宋康節邵子，乃有所謂“先天圖”者，其說有六十四卦生出之序，則今之橫圖，自一畫至六畫，一每分二者是已，有八卦方位，則今之小囤圖，乾南坤北離東坎西者是已，有六十四卦方位，則今之大圓圖，始《復》、《姤》、終《乾》、《坤》者是巳，大圓圖中自方圖，又所以象天地之相函也。

Li Guangdi remarked: Since Confucius passed away, the transmission of the Yi Jing has been interrupted. The principles and meanings have already become distorted and erroneous, and the diagrams and symbolic representations are even more obscure and difficult to verify. Only the passage in the Great Transmission, "The Emperor arises from Zhen," (from the Discourse on the Trigrams) clearly records the positions of the Eight Trigrams, so later scholars still have something to rely on. Other things like "trigram qi" and "monthly cycles" were all artificially fabricated by Han dynasty scholars, and are not part of the original rules of the sage. It was not until the Song dynasty that the scholar Shao Yong introduced the "Earlier Heaven Diagram." In his system, the one that explains the sequence of the sixty-four trigrams is the "horizontal diagram" currently known, which starts with one line and proceeds to six lines, with each line divided into two. The one that explains the positions of the Eight Trigrams is the "small circular diagram" currently known, with Qian in the south, Kun in the north, Li in the east, and Kan in the west. The one that explains the positions of the sixty-four trigrams is the "large circular diagram" currently known, starting from Fu and Gou and ending with Qian and Kun. Inside the large circular diagram there is also a "square diagram," which is used to symbolize the mutual inclusion and containment of heaven and earth.

諸圖之義，廣大高深，信非聖人不能造作，然當邵子之時，伊川程子則未之見，龜山楊氏見而未之信，唯明道程子，稍見其書，而括以加倍之一言，然則當時知邵子者，明道一人而已。南渡之後，如林栗袁樞之徒，攻邵者尤眾，雖象山陸氏，亦以為先天圖非聖人作《易》本指。獨朱子與蔡氏，闡發表章，而邵學始顯明於世，五百年來雖復有為異論者，而不能奪也。

The meaning of these diagrams is vast and profound, indeed not something that could be created except by the sages. However, during Shao Yong's lifetime, Cheng Yi did not see these diagrams, and Yang Shi of Guishan saw them but did not believe in them. Only Cheng Hao of Mingdao briefly read Shao's writings and summarized them with the phrase "the method of doubling." Thus, it appears that only Cheng Hao truly understood Shao Yong at that time. After the Southern Migration of the Song Dynasty, people like Lin Li and Yuan Shu attacked Shao Yong's ideas more fiercely, even Lu Jiuyuan of Xiangshan claimed that the "Earlier Heaven Diagram" was not the original intention of the sages in composing the Book of Changes. Only Zhu Xi and his son Cai Yuanding explained and promoted Shao's teachings, thereby making Shao's school of thought prominent in the world. Although there were still continuous objections over five hundred years, they could not ultimately shake its foundation.

顧朱子之意，以為孔於之後諸儒不能傳授，而使方外得之，故其流為丹灶小術，至康節然後返之於易道。今以《參同契》諸書觀之，其六卦月候，蓋即“納甲”之法，其十二闢卦主歲，蓋即“卦氣”之流，所為始於震復者，與先天偶同爾，似未足為先天傳受之據。

However, Zhu Xi's meaning was that after Confucius, the scholars of the Confucian tradition were unable to pass on this body of knowledge, instead allowing it to fall into the hands of those outside the orthodox schools, leading it to become reduced to minor techniques such as alchemy and furnace work. It was only until Shao Yong that this was brought back into the realm of the Yi Jing. Now, looking at works such as the Zhou Yi Cang Tong Qi, which discusses the pairing of the six hexagrams with the lunar months, this is actually the method of 'Najia' assignment; when it speaks of the twelve dividing hexagrams governing a year, this is merely a branch of the 'hexagram qi' theory; as for its starting with the hexagrams Zhen and Fu, this merely coincidentally aligns with the Earlier Heaven diagram, and thus is not sufficient evidence to claim that the Earlier Heaven diagram has a genuine lineage of transmission.

唯揚雄作《太玄》，其法始於三方，重於九洲，又重於二十七部，又重於八十一家，則與先天極儀象卦加倍之法相似也。流行之序，始於中羨從，中於更晬廓，終於減沈成，則與先天始《復》終《乾》，始《姤》終《坤》之序相似也。首用九九，策閒六六，則與先天卦用八八，策用七七之數相似也。

Yang Xiong composed the "Tai Xuan," which first established the concept of "three directions," then overlapped them to form "nine states," and further overlapped to form "twenty-seven parts," and again overlapped to form "eighty-one families." This method of overlapping is similar to the Earlier Heaven diagram, which starts from the Taiji, then the two principles, then the four images, then the eight trigrams, dividing one into two and doubling at each level. The "Tai Xuan" also follows the order of movement and change starting with "Zhong," "Xian," and "Cong," proceeding through "Geng," "Zui," and "Kuo," and ending with "Jian," "Shen," and "Cheng." This sequence is similar to the Earlier Heaven diagram, which begins with the hexagram "Fu" and ends with "Qian," and also begins with "Gou" and ends with "Kun." Furthermore, the "Tai Xuan" uses the number 81 for its initial structure and the number 36 for its counting method, which is similar to the Earlier Heaven diagram's use of 64 for the hexagrams and 49 for the counting method.

意者康節讀揚雄之書而心悟作《易》之本與，然非揚雄之時，《易》傳未泯，則雄亦無自而依仿之，故康節深服《太玄》，以為見天地之心，蓋其學所啟發得力處也。然白邵書既出，則《太玄》為僭經，為汨陰陽之敘，與邵書迥乎如蒼素之不相侔矣。

By this reasoning, perhaps the Master Kangjie read Yang Xiong's writings and thus came to understand the fundamental principles of the Sage's composition of the Book of Changes. However, if during Yang Xiong's time the transmission of the Book of Changes had already been lost, Yang Xiong himself would have had no model to imitate. Therefore, Shao Yong greatly admired the Tai Xuan, believing it had seen the heart of heaven and earth. This was probably where his learning was inspired and developed. But since Shao Yong's writings appeared, the Tai Xuan became regarded as an act of僭拟 (imitating) the classics, a book that disturbed the order of yin and yang. Compared to Shao Yong's work, the difference is as great as that between blue and white, which are fundamentally incomparable.

觀明道程子之意，蓋以為康節能自得師，故於希夷之傳，揚雄之書，皆有取焉，而其純一不雜，汪洋浩大，則非揚陳之所能及也，故曰堯夫之數，似玄而不同。又曰：穆李皆得之希夷者，而其言與行事，概可見矣，堯夫特因其門戶而入者爾，程子之言至當，後之學者，欲考先天之傳，不可以不知。

Now looking at the meaning of Cheng Hao's Mingdao, it is roughly that he thought Kangjie was able to find his own lineage of teachers, so he took some elements from Chen Tuan's Xixi lineage and from Yang Xiong's writings. However, the purity and comprehensiveness of Kangjie's learning, its vastness and grandeur, surpassed those of Yang Xiong and Chen Tuan. Therefore, Cheng Hao said: "Yao Fu's mathematics resembles the Tai Xuan but is different from it." Cheng Hao also said: "Mu Xiu and Li Zhi Cai all received their teachings from Chen Tuan; their words and deeds can be roughly seen, and Yao Fu merely entered through this path." These few words of Cheng Hao are most appropriate. Later scholars who wish to investigate the transmission and lineage of the Earlier Heaven studies cannot fail to know this.

震，東方也。齊乎巽，巽，東南也。齊也者，言萬物之潔齊也。離也者，明也，萬物皆相見，南方之卦也。聖人南面而聽天下，嚮明而治，蓋取諸此也。坤也者，地也，萬物皆致養焉，故曰致役乎坤。兌，正秋也，萬物之所說也，故曰說言乎兌。戰乎乾，乾，西北之卦也，言陰陽相薄也。

The *Sayings on the Trigrams* continues to explain: "Zhen belongs to the East." It says "qi hu Xun," meaning Xun is in the Southeast; "qi" refers to the fact that at this time all things have grown clean and orderly. "Li" means brightness; at this time all things become visible to each other, and it is the trigram of the South. The sage faces south to govern the world, directing his rule toward the direction of brightness, which is probably derived from this symbolism. "Kun" represents the earth, which nurtures all things, so it is said "zhi yi hu Kun," meaning all things serve and are nourished here. "Dui" corresponds to the time of the autumn equinox, when all things have matured and are joyful, hence it is said "shuo yan hu Dui." As for "zhan hu Qian," "Qian" is the trigram of the Northwest, referring to the struggle between yin and yang forces here.

坎者，水也，正北方之卦也，勞卦也，萬物之所歸也，故曰勞乎坎。艮，東北之卦也，萬物之所成終而所成始也，故曰成言乎艮。神也者，妙萬物而為言者也。動萬物者莫疾乎雷，撓萬物者莫疾乎風，燥萬物者莫熯乎火，說萬物者莫說乎澤，潤萬物者莫潤乎水，終萬物始萬物者莫盛乎艮。故水火相逮，雷風不相悖，山澤通氣，然後能變化既成萬物也。

"The trigram 'Kan' represents water, and it is the trigram of the north. It is also called the 'Labor trigram,' as all things labor throughout the year and here they gather and rest. Therefore, it is said 'Labor in Kan.' The trigram 'Gen' is the trigram of the northeast. Here, all things have completed the end of one year and begin a new cycle of growth, so it is said 'Completion is expressed in Gen.' The term 'spirit' refers to its subtle ability to transform all things. Nothing moves things more swiftly than thunder, nothing shakes things more powerfully than wind, nothing dries things more intensely than fire, nothing delights things more than a marsh, and nothing nourishes things more than water. Nothing concludes and initiates things more greatly than 'Gen.' Therefore, water and fire interact without destroying each other, thunder and wind clash without contradiction, mountains and marshes communicate through their energies, and thus they transform and move, giving rise to all things."

邵子曰：此一節明文王八卦也。又曰：至哉文王之作《易》也，其得天地之用乎。故乾坤交而為《泰》，坎離交而為《既濟》也。乾生於子，坤生於午，坎終於寅，離終於申，以應天之時也。置乾於西北，退坤於西南，長子用事，而長女代母，坎離得位，而兌艮為偶，以應地之方也。王者之法，文王也其盡於是矣。

Shao Yong said: The section of "Shuo Gua Zhuan" discusses the Eight Trigrams of King Wen, which represent the Later Heaven directions. Shao Yong also said: What a great achievement! King Wen's composition of the Book of Changes must have grasped the functions of heaven and earth. Therefore, the trigrams "Qian" and "Kun" intersect to form the hexagram "Tai," and the trigrams "Kan" and "Li" intersect to form the hexagram "Ji Ji." "Qian" originates from the position of Zi, "Kun" originates from the position of Wu, "Kan" concludes at the position of Yin, and "Li" concludes at the position of Shen. This is to correspond with the sequence of heaven's time. "Qian" is placed in the Northwest, "Kun" retreats to the Southwest, the eldest son "Zhen" comes forth to take charge, and the eldest daughter "Xun" takes over the duties of the mother. "Kan" and "Li" each occupy their proper positions, while "Dui" and "Gen" face each other as pairs. This is to correspond with the directions of the earth. The system of governance for a ruler to manage the world has been fully exhausted by King Wen.

此言文王改《易》伏羲卦圖之意也。蓋自乾南坤北而交，則乾北坤南面為《泰》矣。自離東坎西而交，則離西坎東而為《既濟》矣。乾坤之交者，自其所已成，而反其所由生也。故再變則乾退乎西北，坤退乎西南也。坎離之變者，東自上而西，西自下而東也。故乾坤既退，則離得乾位，而坎得坤位也。震用事者，發生於東方，巽代母者，長養於東南也。

Zhu Xi explained that this is about King Wen's intention in altering Fuxi's trigram diagram. Originally, in the Earlier Heaven diagram, Qian was in the south and Kun in the north. When they were exchanged, Qian moved to the north and Kun to the south, which corresponds to the image of the Tai hexagram with Kun below and Qian above. Originally, Li was in the east and Kan in the west. When they were exchanged, Li moved to the west and Kan to the east, which corresponds to the image of the Ji Ji hexagram with Kan below and Li above. The reason Qian and Kun are exchanged is to return from their formed positions to their original sources. Therefore, after another change, Qian retreats to the northwest and Kun to the southwest. The movement of Li and Kan is that Li moves from the east towards the west, while Kan moves from the west towards the east. Since Qian and Kun have retreated to the corners, Li takes the position that was originally Qian's southern position, and Kan takes the position that was originally Kun's northern position. Zhen emerges to take charge, initiating vitality in the east; Xun takes over in the southeast, nurturing all things.

案 邵子言乾坤交而為“泰”者，釋“先天”變為“後天”之指也。先天之位，乾南坤北，今變為乾北坤南，故曰“交”。然邵子盲乾生於子，坤生於午，今按圖考之，則乾在西方，乃亥而非子，坤在西南，乃未而非午，其故何也？曰：陽自靜以之動，故氣肇於子。然自亥月而已聯兆胚胎，故古人以亥為陽月，言天道於是始也。陰自動以之靜，故功著於午。然至未而後育養蕃庶，故古人以未為中央，言土德於是，王也。

Li Guangdi remarked: Shao Yong said, "The trigrams Qian and Kun interact to form the hexagram Tai," which is explaining the meaning of the transformation from the Earlier Heaven diagram to the Later Heaven diagram. In the Earlier Heaven, the direction of Qian is south and that of Kun is north; now, in the Later Heaven, Qian is north and Kun is south, so it is called "interaction." However, Shao Yong also said, "Qian is born from Zi, and Kun is born from Wu." But when we check according to the Later Heaven diagram, Qian is clearly in the northwest, paired with the earthly branch Hai instead of Zi; and Kun is in the southwest, paired with Wei instead of Wu. Why is this so? The answer is: Yang energy moves from stillness to activity, so the vital energy begins at Zi; however, starting from the month of Hai, the embryo has already begun to take shape in secret, so ancients called Hai the "month of Yang" and said the heavenly way begins here. Yin energy moves from activity to stillness, so its function becomes evident at Wu; however, it is only after the month of Wei that it truly nurtures and flourishes, so ancients regarded Wei as the central position and said the earth's virtue rules here.

亥字從草為蓑，從木為核，皆聯兆胚胎之意。未從日為昧，言日於是始向昧谷，而萬物將西成也。樂律黃鐘子為天統，然自應鐘亥而陽氣已應於內，故曰應鐘。林鐘未為地統，故班固引“西南得朋”釋之，下至納甲星命淺術，亦以亥為天門，未為坤始，疑皆本於後天以為說也。若乃火雖始於東而盛於南，水雖始於西而盛子北，雷霆之氣，雖動於寅，

The character 「亥」 with a grass radical becomes 「荄」, and with a wood radical becomes 「核」, both meaning the initial sprout or embryo. The character 「未」 with a sun radical becomes 「昧」, indicating that the sun begins to turn toward the west at this point, and all things will mature in the west. In the system of musical tones, the Huangzhong tone is paired with Zi, called 「天统」 (Heavenly Order); however, from Yingzhong paired with Hai, the yang energy has already begun to respond internally, so this tone is called 「应钟」 (Responsive Bell). The Linzhong tone is paired with Wei, called 「地统」 (Earthly Order), so Ban Gu cites the 「Kun」 hexagram's statement 「西南得朋」 (friendship is gained in the southwest) to explain it. Down to more superficial systems such as the method of incorporating stems (纳甲) and star destiny (星命), they all regard Hai as 「天门」 (Heaven's Gate) and Wei as the beginning of Kun, suggesting they all base their explanations on the Later Heaven diagram. As for the fact that fire originates in the east but flourishes in the south, and water originates in the west but flourishes in the north, and the energy of thunder and lightning originates in Yin but this passage is missing some text; the commentary below is incomplete.

又曰：《易》者，一陰一陽之謂也。震兌始交者也，故當朝夕之位，坎離交之極者也，故當子午之位，巽艮不交而陰陽猶雜也，故當用中之偏，乾坤純陽純陰也，故當不用之位也。

Shao Yong also said: The "Yi" refers to the mutual interaction between yin and yang. The trigrams "Zhen" and "Dui" represent the initial meeting of yin and yang, so they are placed in the east and west positions facing each other. The trigrams "Kan" and "Li" represent the extreme meeting of yin and yang, so they are placed in the south and north positions facing each other. The trigrams "Xun" and "Gen" represent yin and yang that have not yet properly met but are still mixed, so they are placed in positions that are in the center but slightly off-center. The trigrams "Qian" and "Kun" represent pure yang and pure yin, with no interaction, so they are placed in positions that are not in the active center.

又曰：兌離巽，得陽之多者也，艮坎震，得陰之多者也，是以為天地用也。乾極陽，坤極陰，是以不用也，、又曰：震兌橫而六卦縱，《易》之用也。

Shao Yong also said: "The trigrams Dui, Li, and Xun have more yang lines, while the trigrams Gen, Kan, and Zhen have more yin lines. Because yin and yang have such differences in quantity, they can become tools for the operation of heaven and earth. Qian is yang reaching its extreme, and Kun is yin reaching its extreme; they are pure and unadulterated, so they are not suitable for active use. Shao Yong also said: in the diagram, the trigrams Zhen and Dui are arranged opposite each other horizontally, while the other six trigrams are paired vertically, which is exactly where the application of the Yi lies."

嘗考此圖而史為之說曰：震東兌西者，陽主進，故以長為先而位子左，陰主退，故以少為貴而位乎右也。坎北者，進之中也，離滷者，退之中也，男北而女南者，互藏其宅也。四者皆當四方之止位，而為用事之卦，然震兌妨而坎離終，震兌輕而坎離重也。

Zhu Xi once examined this diagram and explained it saying: "Zhen is located in the east, Dui in the west, because yang is dominant in advancing, so the older is placed first on the left; yin is dominant in retreating, so the younger is esteemed and placed on the right. Kan is located in the north, which is yang advancing halfway; Li is located in the south, which is yin retreating halfway. The male trigram is in the north while the female trigram is in the south, indicating that they mutually contain each other's dwelling, with yang containing yin and yin containing yang. These four trigrams are each in the central positions of the four directions, being the trigrams in charge at the time; however, Zhen and Dui are at the beginning while Kan and Li are at the end, with Zhen and Dui being lighter in weight and Kan and Li heavier."

乾西北坤西南者，父母既老而遲居不用之地也，然母親順父尊，故砷猶半用頁乾全下用也。艮東北巽東南者，少男進之後而長女退之先，故亦皆不用也，然男未就傅，女將在行，故巽稍向用，而艮全未用也。四者皆居四隅不正之位，然居東者耒用而居西者不復用也。故下文歷舉“六子”而不數乾坤，至其水火雷風山澤之相偶，則又用伏羲卦云。

The trigram Qian is located in the Northwest, and the trigram Kun is in the Southwest, indicating that the parents have grown old and have retired from active roles. However, the mother represents obedience and the father represents dignity, so Kun still partially engages in affairs, while Qian is entirely inactive. The trigram Gen is in the Northeast, and the trigram Xun is in the Southeast. Gen represents the young son, who is at the very end of the sequence, and Xun represents the elder daughter, who is at the very beginning of the retreat. Therefore, neither should be in active use. However, the young son has not yet left to study under a teacher, while the elder daughter is about to marry and take on responsibilities, so Xun is slightly more active, while Gen is entirely inactive. These four trigrams are located in the corners, not in the central positions. Those located in the East have not yet come to their turn, and those in the West have already passed their time of activity and are no longer active. Therefore, the "Sayings on the Trigrams" later lists the six sons but does not include Qian and Kun. When discussing water and fire, thunder and wind, mountain and marsh in pairs, this follows the method of the Earlier Heaven trigrams as used by Fuxi.

集說 邵子曰：乾統三男於東北，坤統三女於西南。

Collected Explanations: Shao Yong said: In the Later Heaven diagram, "Qian" (Heaven) governs the three male trigrams "Zhen" (Thunder), "Kan" (Water), and "Gen" (Mountain), which are all arranged in the Northeast area; "Kun" (Earth) governs the three female trigrams "Xun" (Wind), "Li" (Fire), and "Dui" (Lake), which are all arranged in the Southwest area.

案 邵子之言，可蔽圖之全義，《周易》、《坤》，《蹇》、《解》諸卦彖辭，皆出於此也，大抵先天則以東南為陽方，西北為陰方，故自陽儀而生之卦，皆居東南，自陰儀而生之卦，皆居西北也。後天則以北東為陽方，南西為陰方，故凡屆陽之卦，皆居東北，屬陰之卦，皆居西南也。然先天陽卦雖起於東，而其重之以敘卦氣，則所謂“復見天地心”者。仍以北方為始，後天陽卦雖起於北，而其播之以合歲序，則所謂“帝出乎震”者，仍以東方為先。蓋兩義原不可以偏廢，必也參而互之，則造化之妙，《易》理之精，可得而識矣。

Li Guangdi remarked: Shao Yong's two sentences can summarize the meaning of the entire diagram. In the "Zhou Yi" (I Ching), the "Kun" hexagram, "Jian" hexagram, and "Jie" hexagram all mention the advantages and disadvantages of the directions "southeast" and "northeast" in their "tuan ci" (commentary on the hexagram). These ideas all originate from here. In general, the Earlier Heaven diagram takes the southeast as the yang direction and the northwest as the yin direction. Therefore, all hexagrams generated from the yang trigrams are located in the southeast, and those generated from the yin trigrams are located in the northwest. In the Later Heaven diagram, the north and east are considered the yang directions, while the south and west are the yin directions. Therefore, all yang hexagrams are located in the northeast, and all yin hexagrams are located in the southwest. However, although the yang hexagrams in the Earlier Heaven diagram originate from the east, when arranging the hexagram energies by overlapping them, the phrase "Fu hexagram reveals the heart of heaven and earth" still begins with the north. Similarly, although the yang hexagrams in the Later Heaven diagram originate from the north, when spreading them out to align with the seasonal cycle of a year, the phrase "The Emperor emerges from Zhen" still begins with the east. This shows that these two aspects cannot be neglected separately; they must be referenced and compared with each other. Only then can the subtle workings of nature and the profound principles of the Yi (Book of Changes) be truly understood.

歲始於東，終於北，而西南在其間，後天圖意主乎陽以統陰，故自震而坎而艮者，以陽終始歲功也，自巽而離而兌者，以陰佐陽於中也。震陽生，故直春生之令，以始為始也，乾則以終為始，而莫得其端，乃《傳》所謂“大始”者也，所謂“不可為首者”也，兌陰成，故畢西成之事，陰功之終也，坤則致役以終事，而不居其成，乃《傳》所謂“作成”者也。所謂“無成而代有終”者也，是故陽居終始，而陰在中間，乃天地萬物之至理。

A year begins in the east and ends in the north, with the southwest located in between. The intention behind the Later Heaven diagram is to use yang to govern yin, so moving from "Zhen" to "Kan" and then to "Gen" represents using yang energy to mark the beginning and end of a year's function; moving from "Xun" to "Li" and then to "Dui" represents using yin energy to assist yang in the middle. "Zhen" signifies the initial emergence of yang energy, thus corresponding to the spring season when things begin, marking the beginning as the beginning. "Qian," however, takes the ending as the beginning, making its origin indistinguishable, which is what the Great Treatise says "Great Beginning," also known as "Not to be the leader" in the "Yao" of the "Qian" hexagram. "Dui" signifies the completion of yin's function, thus concluding the autumn harvest in the west, marking the end of yin's function. "Kun," however, bears the labor to complete the task without claiming the merit, which is what the Great Treatise says "Completing things," also known as "No achievement but taking over the completion" in the "Wenyan" of the "Kun" hexagram. Thus, yang energy resides at both ends while yin energy is in the middle, which is the fundamental principle of heaven and earth and all things.

如草木之種實，陽也；華葉，陰也。人類之父子，陽也；妻妾，陰也。始於植種，終於成實，而其間華葉盛焉，始於有父，終於有子，而其間嫡媵繁焉，實生於華，子生於母，此陰佐陽之驗，然而實成、則為來歲之種矣，子生則為它日之父矣，此又所謂以終為始者，而元陽之生生不已，其首尾端倪，真不可得而窺矣。謝氏良佐論一起於震，發生也。又曰：一起於乾，探本也，其有得於後天之精意者與。

For example, in plants, seeds and fruits belong to yang, while flowers and leaves belong to yin; in humans, father and son belong to yang, while wife and concubine belong to yin. From sowing seeds to the completion of fruit, the middle period is when flowers and leaves are most flourishing; from the existence of a father to the completion of a son, the middle period is when the main wife and concubines multiply. Fruits arise from flowers, sons arise from mothers, which is a clear proof that yin supports yang. However, once the fruit is formed, it becomes the seed for the next year; once a son is born, he becomes the father of the future. This is what is called taking the end as the beginning. The original yang is ever-reproducing; its beginning and end are truly beyond comprehension. Xie Liangzuo discussed the Later Heaven diagram, saying "everything starts with Zhen," referring to the point of origin; and also saying "everything starts with Qian," referring to the root. He probably grasped the subtle intention behind the Later Heaven diagram.

程子曰：凡陽在下者動之象，在中者陷之象，在上止之象，陰在下者入之象，在中者麗之象，在上說之象。

Cheng Yi said: Whenever a single yang line is in the lowest position of a three-line trigram, it takes the image of "movement," which is "Zhen"; when a single yang line is in the middle position, it takes the image of "being trapped," which is "Kan"; when a single yang line is in the upper position, it takes the image of "resting," which is "Gen." Whenever a single yin line is in the lowest position, it takes the image of "entering," which is "Xun"; when a single yin line is in the middle position, it takes the image of "adhering," which is "Li"; when a single yin line is in the upper position, it takes the image of "pleasure," which is "Dui."

乾為馬，坤為牛，震為龍，巽為雞，坎為豕，離為雉，艮為狗，兌為羊。此遠取諸物之象。

The Sayings on the Trigrams states: "Qian takes its image from the horse, Kun from the ox, Zhen from the dragon, Xun from the chicken, Kan from the pig, Li from the wild chicken, Gen from the dog, and Dui from the sheep." Zhu Xi explains: this passage refers to the sages taking the images of the trigrams from outside themselves, from birds, beasts, and all things. Qian represents strength and ceaseless movement, hence the horse; Kun represents flexibility and bearing weight, hence the ox; Zhen represents vigorous movement and rising, hence the dragon; the other trigrams also take their images from things whose nature corresponds to the virtues of the trigrams.

乾為首，坤為腹，震為足，巽為股，坎為耳，離為目，艮為手，兌為口。此近取者身之象。

The Sayings on the Trigrams states: "Qian is taken as the image of the head, Kun as the image of the abdomen, Zhen as the image of the feet, Xun as the image of the thighs, Kan as the image of the ears, Li as the image of the eyes, Gen as the image of the hands, and Dui as the image of the mouth." Zhu Xi comments: this section takes the trigrams' images from the human body itself. The head is at the top and round in shape, so it belongs to Qian; the abdomen is capable of containing, so it belongs to Kun; the feet are active, so they belong to Zhen; the hands are mainly used for holding and stopping, so they belong to Gen; the rest can also be inferred in a similar way.

集說 《朱子語類》云：伏羲畫八卦，只此數畫該盡天下萬物之理，學者於言上會得者淺，於象一亡會得者深，王輔嗣伊川皆不信象，伊川說象只似譬喻樣說，郭子和云，不獨是天地雷風水火山澤謂之象，只是卦畫便是象，亦說得好。鄭東卿專取象，如以鼎為鼎，革為爐，小過為飛鳥，亦有義理，但盡欲如此牽合傅會便疏脫，學者須先理會得正當道理，然後於此等另碎處收拾以相資益，不為無補。

Collected Commentaries: Master Zhu said in The Words of the Master: Fuxi drew the Eight Trigrams, and with just these few lines he encompassed the principles of all things under heaven. If scholars only grasp the meaning through words, they will gain little; but if they understand the trigrams, they will gain much. Wang Bi (styled Fusi) and Cheng Yi did not believe in the trigrams, and even when Cheng Yi spoke of the trigrams, he regarded them merely as metaphors. Guo Zhongxiao (styled Zihe) said: Not only the heavens, earth, thunder, wind, water, fire, mountains, and marshes are called images, but the trigram diagrams themselves are images — this is a good saying. Zheng Dongqing specialized in taking images, for example, seeing the Trigram of the Cauldron as a cauldron, the Trigram of Change as a furnace, and the Trigram of Small Crossing as a bird. This is not without meaning; however, if everything is forced into such associations, it will inevitably lead to omissions and superficiality. Scholars must first clearly understand the fundamental principles, and then collect and select these minor details to complement each other, which is not without benefit.

乾，天也，故稱乎父。坤，地也，故稱乎母。震一索而得男，故謂之長男；巽一索而得女，故謂之長女。坎再索而得男，故謂之中男；離再索而得女，故謂之中女，艮三索而得男，故謂之少男；兌三索而得女，故謂之少女。

The *Sayings on the Trigrams* states: "Qian" represents Heaven, so it is called the father; "Kun" represents Earth, so it is called the mother. "Zhen" is the first inquiry yielding a male, so it is called the Eldest Male; "Xun" is the first inquiry yielding a female, so it is called the Eldest Female. "Kan" is the second inquiry yielding a male, so it is called the Middle Male; "Li" is the second inquiry yielding a female, so it is called the Middle Female. "Gen" is the third inquiry yielding a male, so it is called the Youngest Male; "Dui" is the third inquiry yielding a female, so it is called the Youngest Female.

今按坤求於乾，得其初九而為震，故曰“一索而得男”，乾求於坤，得其初六而為巽，故曰“一索而得女”。坤再求而得乾之九二以為坎，故曰“再索而得男”。乾再求而得坤之六二以為離，故曰“再索而得女”。坤三求而得乾之九三以為艮，故曰“三索而得男”。乾三求而得坤之六三以為兌，故口“三索而得女”。

Zhu Xi's commentary says: When the Earth (Kun) seeks the Heaven (Qian), it takes the first line of Qian, which is a single yang, and places it at its own lowest line, forming the Trigram Zhen (Thunder), hence saying "one search and a male is obtained." When Heaven seeks the Earth, it takes the first line of Kun, which is a single yin, and places it at its own lowest line, forming the Trigram Xun (Wind), hence saying "one search and a female is obtained." The Earth seeks again, taking the second line of Qian, which is a yang, and placing it at its middle line, forming the Trigram Kan (Water), hence saying "a second search and a male is obtained." Heaven seeks again, taking the second line of Kun, which is a yin, and placing it at its middle line, forming the Trigram Li (Fire), hence saying "a second search and a female is obtained." The Earth seeks a third time, taking the third line of Qian, which is a yang, and placing it at its upper line, forming the Trigram Gen (Mountain), hence saying "a third search and a male is obtained." Heaven seeks a third time, taking the third line of Kun, which is a yin, and placing it at its upper line, forming the Trigram Dui (Lake), hence saying "a third search and a female is obtained."

凡此數節，皆文王觀於已成之卦，而推其未明之象以為說，邵子所謂後天之學，人用之位者也。

The passages "Qian represents a horse," "Qian represents the head," and "Qian is called the father" are all interpretations by King Wen, who observed the hexagrams he had already drawn and sought out the unexplained phenomena within them to establish his explanations. This approach is what Shao Yong referred to as "Later Heaven studies," belonging to the category of directional applications related to human affairs.

案 邵子既以“天地定位”一章為先天之《易》，因以“帝出乎震”以下為後天之《易》，先羲後文，其序既可信，而先天圖易簡渾涵，得畫卦自然之妙，後天圖精深切至，於《周易》義例合者為多，其理尤可信也：，然後天所以改置先天之意，朱子之說頗略，其見於答袁樞書者，可以得先賢慎重之盛心矣。

Li Guangdi remarked: Shao Yong regarded the chapter "Heaven and Earth are positioned" in the "Shuo Gua Zhuan" as the Earlier Heaven Yi, and then regarded the section "The Emperor arises from the Zhen trigram" and following as the Later Heaven Yi. The order of first Fuxi and then King Wen is credible. Moreover, the Earlier Heaven diagram is simple and straightforward, encompassing the natural emergence of the hexagrams; the Later Heaven diagram is profound and practical, and aligns with the principles of the Zhou Yi most closely, making it more credible in terms of reasoning. However, the reason why the Later Heaven diagram altered the positions of the Earlier Heaven diagram is not clearly explained by Zhu Xi, who was quite brief in his explanation. The words he used in his letter to Yuan Zhu clearly show the cautious attitude of previous scholars, who were unwilling to make assumptions without evidence.

諸家以五行為說者，亦有條理，然今即八卦之象求之，則唯坎水離火巽木坤土，合於本象耳。金者乾之一象，而不足以盡乾也。蒼筤竹者震之一象，而不足以盡震也。艮山之為土，猶可假借，兌則絕無為金之義也，況《易》之為書，不言五行，而《說卦》解釋圖體，亦與五行生剋，邈不相涉，

Each school of thought uses the Five Phases to explain the directions of the Eight Trigrams, each having its own system; however, when examining the original images of the Trigrams themselves, only four correspond directly to their respective elements: "Kan" represents water, "Li" represents fire, "Xun" represents wood, and "Kun" represents earth. Metal is merely one of the many images associated with "Qian," and thus cannot fully represent "Qian." Green bamboo is only one of the many images associated with "Zhen," and thus cannot fully represent "Zhen." "Gen" is considered earth as a mountain, which can be loosely accepted; but "Dui" has no reason to be associated with metal. Moreover, the book of Changes does not discuss the Five Phases throughout its entire text, and the "Say on the Trigrams" which explains the structure of the trigrams' diagrams has nothing to do with the generation and overcoming of the Five Phases.

大衍之數五十。

The Great Treatise says: "By deducing the numbers of heaven and earth, the method of divination with yarrow stalks is carried out, using a total of fifty stalks." This sentence serves as the general principle for the entire method of forming a hexagram through divination with yarrow stalks. The subsequent sections, such as "subtract one," "divide into two," "hang one," "draw four," and "return the odd," all originate from this number of fifty.

《河圖》、《洛書》之中數皆五，衍之而各極其數以至於十，則合為“五十”矣。《河圖》積數五十五，其“五十”者，皆因五而後得，獨五為“五十”所因，而自無所因，故虛之則但為“五十”。又五十五之中，其四十者，分為陰陽老少之數，而其五與十者無所為，則又以五乘十，以十乘五，而亦皆為“五十”矣。《洛書》積數四十五，而其四十者，散佈於外，而分陰陽老少之數，唯五居中而無所為，則亦自含五數，而併為“五十”矣。

Zhu Xi explained that the number five in both the Yellow River Chart and the Luo Writing is at the center. By extending this five, each reaches its extreme as ten. When five and ten are multiplied together, they form the number fifty. The total of all numbers in the Yellow River Chart is fifty-five, of which fifty comes from the presence of the central five first. The five itself is the fundamental support of fifty, but it has no support of its own, so it is left out, leaving fifty. Furthermore, among the fifty-five, forty are used to represent yin, yang, old and young, while the numbers five and ten are not involved. Multiplying five by ten or ten by five results in fifty. The total of all numbers in the Luo Writing is forty-five, of which forty are distributed around the perimeter to represent yin, yang, old and young, while the central five is not involved. However, this central five contains another five within it, making two fives in total, which again results in fifty.

案 《洪範》曰：“卜五占用二衍忒。”衍者，推衍也。忒者，過差也。卜筮所以推衍人事之過差，故揲蓍之法，謂之“大衍”。大音太，如太卜太筮之比，乃尊之之稱，非如先儒小衍大衍之說也。五十之數，說者不一，唯推本於圖書者得之，《河圖》之數則贏五，數之體也。《洛書》之數則虛五，數之用也。大衍者，其酌河洛之數之中，而兼體用之理之備者與。

Li Guangdi remarked: The Book of Documents, Hongfan, says, "The divination involves five, and the use of two, with extension and error." "Extension" refers to the process of inference, and "error" refers to mistakes or deviations. Divination and卜筮 are used to infer the gains and losses in human affairs, so the method of divining with yarrow stalks is called "Da Yan." "Da" is read as "Tai," which is used in the same way as "Tai Bu" and "Tai Shi," indicating a respectful term. It is not like the earlier Confucian scholars who distinguished between "Xiao Yan" and "Da Yan." As for the number fifty, different commentators have various interpretations. Only those who trace back to the Yellow River Chart and the Luo Writing have touched upon the root. The number in the Yellow River Chart is five more than fifty, representing the essence of numbers; the number in the Luo Writing is five less than fifty, representing the function of numbers. The term "Da Yan" probably refers to balancing the numbers of the Yellow River Chart and the Luo Writing, and encompassing both the essence and function of numbers.

大衍之數五十，而菁一根百莖，可當大衍之數者二，故揲菁之法，取五十莖為一握，置其一不用，以象太極，而其當用之策，凡四十有九，蓋兩儀體具而未分之象也。

Zhu Xi explained that the number of great derivation is fifty, and one stalk of yarrow grass can produce a hundred stems, which exactly corresponds to two great derivation numbers. Therefore, the method of divination with yarrow sticks involves taking fifty stems in the hand, first removing one stem and setting it aside, using it to symbolize the Taiji. The remaining forty-nine stems are used for divination, which represents the form of the two principles (Wu Wei) that have already been formed but have not yet been separated.

集說 崔氏憬曰：“其用四十有九”者，法長陽七七之數也，六十四卦，既法長陰八八之數，故四十九蓍，則法長陽七七之數，蓍圓而神象天，卦方而智象地，陰陽之別也，舍一不用者，以象太極虛而不用也。

Collected Commentaries: Cui Jing said: The "Great Treatise" states "its use is forty-nine," which refers to the number forty-nine derived from the old yang number seven times seven. Since the sixty-four hexagrams are based on the old yin number eight times eight to make sixty-four, the forty-nine stalks of yarrow are based on the old yang number seven times seven to make forty-nine. The yarrow stalks are round and subtle, symbolizing heaven; the hexagram structure is square and wise, symbolizing earth: this is the distinction between yin and yang. Removing one stalk and setting it aside is used to symbolize the Taiji (Supreme Ultimate) being empty and quiet without personally taking action.

邵子曰：蓍之用數，“掛一以象三”，其餘四十八，則一卦之策也。四其十二為四十八也，十二去三而用九，四三十二，所去之策也，四九三十六，所用之策也；十二去五而用七，四五二十，所去之策也，四七二十八，所用之策也；十二去六而用六，四六二十四，所去之策也，四六二十四，所用之策也；十二去四而用八，四四十六，所去之策也，四八三十二，所用之策也。是故七九為陽，六八為陰，九者陽之極數，六者陰之極數，數極則反，故為卦之變也。

Shao Yong said: The method of using the yarrow stalks begins with "hanging one to represent three," leaving forty-eight stalks, which are the number of stalks used to form a hexagram. Four multiplied by twelve equals forty-eight. From twelve, subtract three and use nine: four multiplied by three equals twelve, which is the number of stalks removed; four multiplied by nine equals thirty-six, which is the number of stalks used, this is old yang. From twelve, subtract five and use seven: four multiplied by five equals twenty, which is the number of stalks removed; four multiplied by seven equals twenty-eight, which is the number of stalks used, this is young yang. From twelve, subtract six and use six: four multiplied by six equals twenty-four, which is the number of stalks removed; four multiplied by six still equals twenty-four, which is the number of stalks used, this is old yin. From twelve, subtract four and use eight: four multiplied by four equals sixteen, which is the number of stalks removed; four multiplied by eight equals thirty-two, which is the number of stalks used, this is young yin. Therefore, seven and nine belong to yang, and six and eight belong to yin; nine is the extreme number of yang, and six is the extreme number of yin. When numbers reach their extremes, they turn back, so old yang and old yin become the points where the lines of the hexagram change.

又曰：奇數極於四而五不用，策數板於九而十不用，故去五十而用四十九也。

Shao Yong also said: The remaining odd numbers after divination are at most four, and five is not used; the number of counts for each line is at most nine, which is the nine in four nines making thirty-six, and ten is not used. Because of this, when taking one count out of fifty, actually only forty-nine is used.

分而為二以象兩，掛一以象三，揲之以四，以象四時，歸奇於扐以象閏，五歲再閏，故再扐而後掛。

The Great Treatise says: Divide the forty-nine divination rods at random into two parts to symbolize the two principles of heaven and earth. Take one rod and hold it between your fingers to symbolize the three elements of heaven, earth, and humanity. Count them in groups of four to symbolize the four seasons of a year. Place the remaining rods between your fingers to symbolize a leap month. Since there is a leap month every five years, after having placed two times the remaining rods, you should rehang the rods for the next change.

掛者，懸於小指之間。揲者，以大指食指間而別之。奇，謂余數，扐者，抽於中三指之兩間也，蓍凡四十有九，信手中分，各置一手，“以象兩儀”；而掛右手一策於左手小指之間，以象三才，遂以四揲左手之策，“以象四時”；而歸其：餘數於左手第四指間，“以象閏”；又以四揲右手之策，而再歸其餘數於左手第三指間，“以象再閏”；五歲之象，掛一一也，揲左二也，扐左三也，揲右四也，扐右五也，是謂一變。其掛扐之數，不五即九。

Zhu Xi explained that "guà" refers to suspending one yān between the tip of the little finger and the ring finger; "duó" refers to separating the yān into groups with the thumb and index finger; "qí" refers to the remaining number after counting; and "tà" refers to inserting the remaining number between the gaps of the middle three fingers. There are a total of forty-nine yān. They are randomly divided into two parts, each held in one hand, to symbolize the two principles; then one yān is taken from the right hand and inserted between the tip of the left little finger and the ring finger, to symbolize the three elements. Next, the yān in the left hand are counted in groups of four to symbolize the four seasons; the remaining number is then inserted between the gap of the fourth finger on the left hand, to symbolize the leap month. Then the yān in the right hand are also counted in groups of four, and the remaining number is inserted between the gap of the third finger on the left hand, to symbolize the second leap month. The so-called five-year symbol is that "guà yī" is the first, "duó zuǒ" is the second, "tà zuǒ" is the third, "duó yòu" is the fourth, and "tà yòu" is the fifth. Completing this process is called "yī biàn." After one change, the combined number of "guà" and "tà" is either five or nine.

案 《河圖》之中宮，太極也。《洛書》之中宮，人極也。故大衍之數，其虛一者，既以象太極之無為，其掛一者，又以象人極之參贊，虛一之後，繼以分二者，明乎分陰分陽，造化之本也。掛一之後，繼以揲四歸奇者，明平定時成歲，人事之綱也。分二掛一，則天地設位，而人立焉，而二才之體具欠。揲四歸奇，則四氣交運，五行參差，百物生焉，萬事起焉，而三才之用行矣，大衍之數，所以為酌河洛之中，而兼體用之備者如此。

Li Guangdi remarked: The Central Palace of the Yellow River Chart contains both five and ten, which represents the "Taiji" or Supreme Ultimate. The Central Palace of the Luo Writing contains only five, which represents the "Renji" or Human Ultimate, also known as the "Huangji" mentioned in the Hongfan Classic. Therefore, the method of Da Yan, which omits one yao (count), symbolizes the inaction of Taiji; whereas hanging up one yao symbolizes the participation of Renji in assisting the transformation of heaven and earth. After omitting one, the next step is to divide into two, which makes clear that the differentiation of yin and yang is the fundamental principle of creation. After hanging one, the next step is to shake four and return the odd, which makes clear that establishing the four seasons and completing one year are the main principles of human affairs. Dividing into two and hanging one together represents heaven and earth each occupying their proper positions while humans stand in the middle, thus the forms of the three principles (heaven, earth, and humanity) are complete. Shaking four and returning the odd together represents the mutual interaction of the four seasons' energies and the intermingling of the five phases, from which all things are generated and all affairs originate, thus the functions of the three principles are carried out. The reason why the number of Da Yan is considered to encompass both the Yellow River and Luo Writing calculations and to include both principle and application is precisely because of this.

得五者三，所謂奇也。五除掛一即四，以四約之為一，故為奇，即兩儀之陽數也。一變之後，除前餘數，復合其見存之策，或四十，或四十四，分掛揲歸如前法，足凋再變，其掛仂者不四則八。

After one change, the remaining number is five, which can be divided in three ways, and this is called "qi." From five, subtracting the one initially set aside leaves four. Dividing four by two gives one, so it is called "qi," which is the yang number among the two principles. After one change, the leftover number from the previous division is set aside and not counted again. The remaining number is either forty or forty-four. Using the same method, divide two, hang one, shake four, and return the remainder, which is called "another change." The number obtained from hanging in the second change is either four or eight.

得四者二，所謂奇也。不去掛一，餘同前義。得八者二，所謂偶也。不去掛一，餘同前義。再變之後，除前兩次餘數，復合其見存之策，或四十，或三十六，或三十二，分掛揲歸如前法，是謂三變，其掛扐者如再變例。三變既畢，乃合三變，視其掛肋之奇偶，以分所遇陰陽之老少，是凋一爻。

After another change, the number obtained by hanging and separating is four, with two ways of division, which is called "qi"; this change does not require removing the one stick used for hanging, and the rest follows the same principles as before. After another change, the number obtained by hanging and separating is eight, with two ways of division, which is called "ou"; similarly, there is no need to remove the one stick used for hanging, and the rest follows the same principles as before. After the third change, the leftover numbers from the first two changes are set aside, and the remaining sticks are gathered again, which may be forty, thirty-six, or thirty-two. These are then divided again by the same method: dividing into two, hanging one, separating four, and returning the remainder, which is called "three changes." The number obtained by hanging and separating in the third change follows the same pattern as in the second change. After completing the three changes, the numbers obtained from the three hangings are combined and examined to determine whether they are odd or even, thus distinguishing whether the situation is yin or yang, old or young, and thus determining one line of the hexagram.

右（即左圖）三奇為老陽者凡十有二，掛扐之數十有三，除初掛之一為十有二，以四約而三分之，為一者三，一奇象圓而圍三，故三一之中各復有三。而積三三之數則為九，過揲之數三十有六，以四約之，亦得九焉。掛扐除一，四分四十有八而得其一也，一其十二而三其四也，九之母也。過揲之數四分四十八而得其三也，三其十二而九其四也，九之子也。皆徑一而圍三也，即四象太陽居一含九之數也。

The diagram above lists twelve methods of forming an old yang line through three transformations, all resulting in a "qi" (odd number). The number of the hanging and separating is thirteen, and after subtracting the initial one stick, it becomes twelve. Dividing twelve by four and then dividing into three parts, each part is one, making three ones in total. A "qi" is like a circle with a circumference of three, so each of these three ones contains three, and the total of three threes is nine. The number of the passing and separating is thirty-six, and dividing thirty-six by four also results in nine. After subtracting the initial one stick from the hanging and separating, the remaining twelve is obtained by dividing forty-eight into four equal parts and taking one part, which is one twelve and three fours, making nine the parent number. The thirty-six from the passing and separating is obtained by dividing forty-eight into four equal parts and taking three parts, which is three twelves and nine fours, making nine the child number. Both conform to the ratio of "a diameter of one and a circumference of three," meaning that among the four images, the sun is first and contains the number nine.

右（即下左圖）兩奇一偶，以偶為主，為少陰者凡二十有八，掛扐之數十有七，除初掛之一為十有六，以四約而三分之，為一者二，為二者一，一奇象圓而用其全，故二一之中各復有三。二偶象方而用其半，故一二之中復有二焉。而積二三一二之數則為八，過揲之數三十有二，以四約之亦得八焉，掛扐除一，四其四也，自一其十二者而進四也，八之母也，過揲之數，八其四也，自三其十二者而退四也，八之子也，即四象少陰居二含八之數也。

The diagram on the left shows that among the three transformations, two are "qi" (odd) and one is "ou" (even), with the even one taking precedence, thus forming the pattern of "shaoyin." There are a total of twenty-eight such divisions. The total number of transformations and manipulations is seventeen; after subtracting the initial one stick, it becomes sixteen. Dividing sixteen by four and then dividing it into three parts results in two ones and one two. An odd number is like a circle, using its full number, so each of the two ones contains three. An even number is like a square, using only half, so the two contains two. Adding together the two threes and the one two gives eight. The number after the "guo xie" (counting after manipulation) is thirty-two, which when divided by four also results in eight. After removing the initial one stick, the remaining sixteen is four fours, which is four more than twelve, making it the maternal number of eight. The thirty-two from "guo xie" is eight fours, which is four less than three twelves, making it the paternal number of eight. The maternal and paternal numbers correspond with each other, meaning that in the four elements, "shaoyin" is the second and contains the number eight.

右（即前右圖）兩偶一奇，以奇為主，為少陽者凡二十，掛扐之數二十有一，除初掛之一為二十，以四約而三分之，為二者二，為一者一，二偶象方而用其半，故二二之中各復有二。一奇象圓而用其全，故一三之中，復有三焉。而積二二一三之數則為七，過揲之數二十有八，以四約之亦得七焉。掛抽除一，五其四也，自兩其十二者而退四也，七之母也，過揲之數，七其四也，自兩其十二者而進四也，七之子也，即四象少陽居三含七之數也。

The diagram on the right shows that among the three transformations, two are even and one is odd, with the odd one being the main one, thus forming the Lesser Yang. There are a total of twenty methods. The total number of the three transformations and the hanging and removing is twenty-one, but after removing the initial one, it becomes twenty. Dividing by four and then dividing into three parts gives two twos and one one. The two evens resemble a square but only use half of it, so each two contains another two. The one odd resembles a circle and uses its full number, so the one contains three. Adding the two twos and the three together gives seven. The number after the shaking is twenty-eight, which when divided by four also gives seven. After removing the initial one, the remaining twenty is five fours, which is four less than two twelves, making seven the mother number. The twenty-eight from the shaking is seven fours, which is four more than two twelves, making seven the son number. The mother and son numbers correspond with each other, meaning that the Lesser Yang is in the third position among the four images and contains the number seven.

右（即左圖）三偶為老陰者四，掛扐之數二個有五，除初掛之一為二十有四，以四約而三分之，為二者三，二偶象方而用其半，故三二之中各復有二，面積三二之數則為六，過揲之數亦二十有四，有四約之亦得六焉，掛扐除一，六之母也，過揲之數，六之子也，四分四十有八而各得其二也，兩其十二而六其四也，皆圍四而用半也，即四象太陰居四含六之數也。

The diagram above shows that all three transformations result in "even" to form old yin, and there are only four ways of division. The number of the hanging and splitting is twenty-five, and after subtracting the initial one strand, it becomes twenty-four. Dividing by four and then dividing into three parts gives three twos. "Even" resembles a square but only uses half of it, so each of these twos contains another two, and adding the three twos together gives six. The number after passing through the divination is also twenty-four, and dividing by four gives the same result, six. After removing the initial one strand from the hanging and splitting, the remainder is the root number of six; the number after passing through the divination is the derived number of six. Both are dividing forty-eight into four equal parts and taking two parts from each, which is two twelves or six fours, both conforming to the ratio of "using half of four," meaning that in the four images, the great yin is in the fourth position and contains the number six.

集說 蔡氏元定曰：蓍之奇數，老陽十二，老陰四，少陽二十，少陰二十八，合六十有四。三十二為陽，老陽十二，少陽二十；三十二為陰，老陰四，少陰二：十八；其十六則老陽老陰也，老陽十二，老陰四；其四十八則少陽少陰也，少陽二十，少陰二十八。老陽老陰，乾坤之象也，二八也，少陽少陰，“六子”之象也，六八也。

Collective commentary: Cai Yuan Ding said: The method of dividing odd and even numbers obtained from three transformations and the hanging of the yarrow stalks results in twelve types of old yang, four types of old yin, twenty types of young yang, and twenty-eight types of young yin. In total, there are exactly sixty-four types, matching the number of the sixty-four hexagrams. Among these, thirty-two are yang: twelve old yang plus twenty young yang; and thirty-two are yin: four old yin plus twenty-eight young yin. Furthermore, sixteen of them are old yang and old yin, that is, twelve old yang plus four old yin; and forty-eight are young yang and young yin, that is, twenty young yang plus twenty-eight young yin. Old yang and old yin correspond to the images of the hexagrams "Qian" (Heaven) and "Kun" (Earth), which exactly match the number two multiplied by eight one to make sixteen. Young yang and young yin correspond to the images of the six sons, which exactly match the number six multiplied by eight four to make forty-eight.

凡此四者，皆以三變皆掛之法得之，蓋經曰“再抽而後掛”，又曰“四營而成易”，其指甚明，註疏雖不詳說，然劉禹鍋所記僧一行、畢中和、顧彖之說，亦已備矣。近世諸儒，乃有前一變獨掛，後二變不掛之說，考之於經，乃為六仂而後掛，不應“五歲再閏”之義，且後兩變又止三營，蓋已誤矣。

The four types of yin and yang, namely old yang, young yin, young yang, and old yin, are all derived according to the old method of "three changes with one hanging." The classic text says "after double handling, then hanging," and also says "four camps form the Book of Changes," whose meaning is very clear. Although the annotations and commentaries do not elaborate in detail, the explanations recorded by Liu Yuxi regarding the views of the monks Yixing, Bi Zhonghe, and Gu Tuan are already quite complete. However, some modern scholars have proposed that only the first change should be hung, while the following two changes should not. When compared with the classic text, this approach results in hanging after six extra counts, which does not conform to the meaning of "five years and then a leap month." Moreover, the remaining three camps in the latter two changes indicate that this view is incorrect.

且用舊法，則三變之中，又以前一變為奇，後二變為偶，奇故其餘五九，偶故其餘四八。餘五九者，五三而九一，亦圍三經一之義也。餘四八者，四八皆二，亦圍四用半之義也。三變之後，老者陽饒而陰乏，少者陽少而陰多，亦皆有自然之法象焉。蔡元定集說 蘇氏軾曰：唐一行之學，以為三變皆少，則乾之象也，乾所以為老陽，而四數其揲得九，故以九名之；三變皆多，則坤之象也，坤所以為老陰，而四數其揲得六，故以六名之；三變面少者一，則震坎艮之象也，震坎艮所以為少陽，而四數其揲得七，故以七名之；三變而多者一，則巽離兌之象也，巽離兌所以為少陰，而四數其揲得八，故以八名之。故七八九六者，因揲數以名陰陽，而陰陽之所以為老少者，不在乎是，而在乎三變之間，八卦之象也，此唐一行之學也。

Moreover, according to the old method, in the three changes, the first change is considered 'qi' (odd), and the last two changes are considered 'ou' (even). Because it is odd, the remainder is either five or nine; because it is even, the remainder is either four or eight. The remainders five and nine, those that result in five have three ways of division, and those that result in nine have only one way, which corresponds to the ratio 'surrounding three and diameter one'. The remainders four and eight, those that result in four and those that result in eight each have two ways of division, which corresponds to the ratio 'surrounding four with half'. After the three changes, the older path has more yang and less yin, while the younger path has less yang and more yin. In this, there are natural phenomena and principles. This is the view of Cai Yuanping. The collected interpretations state that Su Shi said: the theory of Yixing, a Tang dynasty Buddhist monk, holds that if all three changes result in 'shao' (few), it is the image of Qian, hence called old yang; using four to count the excess yao (lines) gives nine groups, hence naming it nine. If all three changes result in 'duo' (many), it is the image of Kun, hence called old yin; using four to count the excess yao gives six groups, hence naming it six. If only one change results in 'shao', it is the image of Zhen, Kan, and Gen, hence called young yang; using four to count the excess yao gives seven groups, hence naming it seven. If only one change results in 'duo', it is the image of Xun, Li, and Dui, hence called young yin; using four to count the excess yao gives eight groups, hence naming it eight. Thus, seven, eight, nine, and six are the numbers used to name yin and yang based on the excess count. However, the reason yin and yang are divided into old and young is not in these numbers, but in the images of the eight trigrams revealed in the three changes: this is the theory of Yixing, the Tang dynasty Buddhist monk.

《朱子文集》曰：初一變得五者三，得九者一，故曰餘五九者，五三而九一。後二變得四者二，得八者二，故曰餘四八者，四八皆二。三變之後，為老陽者十有二，老陰四，故臼陽饒而陰乏，少陽二十，少陰二十八，故曰陽少而陰多。

Master Zhu's writings state: In the first change, there are three ways to obtain a remainder of five, and only one way to obtain a remainder of nine, hence it is said 'the remainders five and nine, five accounts for three, nine accounts for one'. In the latter two changes, there are two ways to obtain a remainder of four, and also two ways to obtain a remainder of eight, hence it is said 'the remainders four and eight, both four and eight account for two each'. After three changes, there are twelve ways to form an old yang, and only four ways to form an old yin, hence it is said 'yang is more and yin is less'; there are twenty ways to form a young yang, and twenty-eight ways to form a young yin, hence it is said 'yang is less and yin is more'.

沈氏《筆談》云，《易》象九為老陽，七為少陽，八為少陰，六為老陰，其九七八六之數，皆有所從來，得之自然，非意之所配也，凡歸餘之數，有多有少，多為陰，如爻之偶，少為陽，如爻之奇。

Shen Ku's "Mengxi Bitan" says: The Yi hexagrams use nine as old yang, seven as young yang, eight as young yin, and six as old yin. These numbers nine, seven, eight, and six each have their origin, arising naturally from the method of divination with yarrow stalks, and not arbitrarily assigned by people. The numbers of the remaining stalks held between the fingers vary in quantity: those with a larger number belong to yin, just as the even number in the broken line of a yao; those with a smaller number belong to yang, just as the odd number in the unbroken line of a yao.

三少，乾也，故曰老陽，，九揲而得之，故其數九，其策三十六；兩多一少，則一少為之主，震坎艮也，故皆謂之少陽，少在初為震，中為坎，末為艮，皆七揲而得之，故其數七，其策二十有八；三多，坤也，故曰老陰，六揲而得之，故其數六，其策二十有四；兩少一多，則一多為之主，巽離兌也，故皆謂之少陰，多在初謂之巽，中為離，未為兌，皆八揲而得之，故其數八，其策三十有二。

If all three changes are 'few', it is 'Qian', so it is called old yang; its over-counted number is calculated by counting four times to get nine groups, so its number is nine and its count is thirty-six. If two changes are 'many' and one is 'few', it takes the 'few' as the main one, forming 'Zhen', 'Kan', and 'Gen', so they are all called young yang; the 'few' appearing in the first change is 'Zhen', in the second change is 'Kan', and in the third change is 'Gen'. All are calculated by counting four times to get seven groups, so their number is seven and their count is twenty-eight. If all three changes are 'many', it is 'Kun', so it is called old yin; its over-counted number is calculated by counting four times to get six groups, so its number is six and its count is twenty-four. If two changes are 'few' and one is 'many', it takes the 'many' as the main one, forming 'Xun', 'Li', and 'Dui', so they are all called young yin; the 'many' appearing in the first change is 'Xun', in the second change is 'Li', and in the third change is 'Dui'. All are calculated by counting four times to get eight groups, so their number is eight and their count is thirty-two.

諸家揲蓍說，唯《筆談》簡而盡。孔穎達非不曉揲法者，但為之不熟，故其言之易差，然其子大數亦不差也。畢中和視疏義為詳，柳子厚詆劉夢得以為膚末於學者，誤矣！畢論三揲皆掛一，正合“四營”之義，唯以三揲之掛仂分措於三指間為小誤，然於其大數亦不差也。其言餘一益三之屬，乃夢得立文太簡之誤，使讀者疑其不出於自然，而出於人意耳。此與孔氏之說，不可不正，然恐亦不可不原其情也。

Different schools have explained the method of divining with yarrow stalks, but only the "Dream Pool Essays" provides a clear and comprehensive account. Kong Yingda was not ignorant of the method, but he was not proficient enough, so his explanations were prone to errors. However, the large numbers he cited were still accurate. Bi Zhonghe's explanation is more detailed than that in the "Correct Meaning of the Zhou Changes." Liu Yuzhong criticized Liu Yuxi, saying his learning was only superficial, which was a mistake. Bi Zhonghe advocated that all three transformations should involve hanging one stalk, which aligns with the meaning of "four operations." However, he separated the stalks used in each transformation and placed them between three fingers, which is a minor flaw. Nevertheless, the large numbers he gave were still accurate. His statements about "remaining one added three" and similar expressions were actually errors caused by Liu Yuxi's overly concise writing, which led readers to suspect that these numbers were not naturally generated but artificially arranged. This point differs from Kong Yingda's explanation and requires correction. However, it is also necessary to consider their actual circumstances at the time.

蔡氏所謂以四十九蓍虛一分二掛一揲四者，蓋謂虛一外，止用四十八。分掛揲之餘，為奇偶各二，老陽老陰變數各八，少陰少陽變數各二十四，合為六十四，八卦各得八焉，然此乃奇偶對待加倍而得者，體數也，著天三地二衍而為五十者，用數也，蓋體數常均，用數則陽饒面陰乏也，此正造化之妙，若陰陽同科，老少一例，是體數，非用數也。

Cai Yuan Ding said, "Take forty-nine stalks of yarrow, subtract one, divide into two, hang one, and draw four," meaning that one stalk is set aside, and actually only forty-eight are used. After subtracting one, dividing into two, hanging one, and drawing four, there remain odd and even numbers in two categories. The number of changes for old yang and old yin is eight each, while the number of changes for young yin and young yang is twenty-four each, totaling sixty-four. Each trigram of the Eight Trigrams receives eight changes. However, this number is derived from odd and even numbers doubling at each level, and it belongs to "body numbers." The "used numbers," on the other hand, are derived by extending three and two into fifty through the method of "heavenly three and earthly two." Body numbers are always evenly divided, while used numbers have more yang and less yin, which reflects the subtle nature of the cosmos. If yin and yang are considered equal and old and young are treated as equal, then it is body numbers, not used numbers.

若用近世之法，則三變之餘，皆為圍三徑一之義，而無復奇偶之分。三變之後，為老陽少陰者皆二十七，為少陽者九，為老陰者一，又皆參差不齊，而無復自然之法象，此足以見其說之誤矣。至於陰陽老少之所以然者，則請復得而通論之。蓋四十九策，除

If one adopts the method of the modern practice, where the first change is counted and the following two are not, then the remainders of the three changes would all be interpreted as 'surrounded by three and diameter one', making it impossible to distinguish between odd and even numbers. After three changes, the numbers for old yang and young yin are both twenty-seven, the number for young yang is nine, and the number for old yin is one, all of which are uneven and no longer reflect natural phenomena. This clearly shows the error in that method. As for the reasons why yin and yang are divided into old and young in this way, I will now discuss it in detail. Generally speaking, there are forty-nine counting sticks. The text here is missing some words, and the following discussion does not continue.

集說 歸氏有光曰：九具於揲，則三奇見於餘；六具於揲，則三偶見於餘；七具於揲，則二偶一奇見於餘；八具於揲，則二奇一偶見於餘。不必反觀其在揲之數，而已舉其要矣！

Collective commentary: Gui Youguang said: If the number of divination sticks passed through is a multiple of nine, then the remaining number will show three 'odd' numbers; if it is a multiple of six, the remaining number will show three 'even' numbers; if it is a multiple of seven, the remaining number will show two 'even' and one 'odd'; if it is a multiple of eight, the remaining number will show two 'odd' and one 'even'. In this way, there is no need to go back to count the number of sticks used in the divination process, as the essential points have already been explained.

其曰“乾之策二百一十有六，坤之策百四十有四”“二篇之策萬有一千五百二十”，何也？掛扐雖舉其要，而七八九六之數，仍以在揲之策為正，七八九六者，自揲之以四而取也，若掛仂之策，因過揲而見者也。故曰“揲之以四以象四時”，又曰“當期之日”，而歸奇以象閏。

Then why does the text say, "The trigram Qian has a count of two hundred and sixteen, and the trigram Kun has a count of one hundred and forty-four," and also say, "The total count of the two chapters is eleven thousand five hundred and twenty?" Although the method of hanging and dividing can present the main points, the numbers seven, eight, nine, and six still need to be based on the actual count of the divination sticks. The numbers seven, eight, nine, and six come from the process of dividing by four; the count of the hanging and dividing is only a secondary result that appears due to the overcounting. Therefore, the text says, "Divide by four to symbolize the four seasons," and also mentions, "the day of the period," while the remaining odd sticks are only used to symbolize the leap month.

何氏楷曰：案翼言“揲四以象四時，歸奇以象閏”。四時，正也，閏餘也。下文云：“乾之策二百一十有六，坤之策百四十有四。凡三百有六十，當期之日，二篇之策，萬有一千五百二十，當萬物之數也。”皆以七八九六起數，明乎用正數而不用餘數矣。案 歸氏何氏之說，亦可與朱子相參酌。

He Ke says: According to the "Yi Zhuan," it says "the four yarrow stalks are used to represent the four seasons, and the odd numbers are placed between the fingers to represent the leap month." The four seasons represent the regular count of a year, while the leap month is the extra number beyond the regular count. Later it says "the number of lines for Qian is 216, and for Kun is 144, together making 360, which corresponds to the number of days in a year; the total number of lines for the upper and lower chapters is 11,520, which corresponds to the number of all things." These are all calculated starting from the numbers seven, eight, nine, and six, that is, based on the number of stalks used after the divination. This shows that the method of divination uses the regular count rather than the remainder. Li Guangdi says: The views of Gu Youguang and He Ke can be compared with Zhu Xi's interpretation and chosen accordingly.

邵子曰：五與四四，去掛一之數，則四三十二也，九與八八，去掛一之數，則四六二十四也，五與八八，九與四八，去掛一之數，則四五二十也，九與四四，五與四八，去掛一之數，則四四十六也，故去其三四五六之數，以成九八七六之策，此之謂也。一爻已成，再合四十九策，復分掛揲歸以成一變，每三變而成一爻，並如前法。

Shao Yong said: When three changes are made, if the numbers obtained are five and two fours, after removing the number of 'hang yi', the remaining is four multiplied by three, which equals twelve; if the numbers obtained are nine and two eights, after removing the number of 'hang yi', the remaining is four multiplied by six, which equals twenty-four; if the numbers obtained are five and one four and one eight, or nine and two fours, after removing the number of 'hang yi', the remaining is four multiplied by five, which equals twenty; if the numbers obtained are nine and two fours, or five and one four and one eight, after removing the number of 'hang yi', the remaining is four multiplied by four, which equals sixteen. Therefore, by removing the groups of three, four, five, and six, the numbers nine, eight, seven, and six are formed, which is the principle being explained. Zhu Xi then said: Once a single line is completed, the process is repeated by combining the forty-nine lines, dividing them into two, hanging one, shaking four, and returning the odd number to form one change. Each of the three changes forms one line, and the process is carried out in the same way as before.