Zhou Yi Hang Shi Xue (周易杭氏學) · Volume 1

《易》注自宋以前未嘗有圖也。逮周廉溪傳陳希夷《太極圖》而為之說，遂開理學之宗。但《圖》與《易》猶不相屬也。至朱子《本義》，取邵子河洛、先後天八卦、大小方圓各圖，與其改訂之卦變圖，弁諸經首，歷代宗之。自是圖之與《易》，相為附麗。後之說《易》者，無不有圖。

Before the Song dynasty, those who interpreted the Zhou Changes did not use diagrams. It was not until the Northern Song dynasty when Zhou Dunyi received the Taiji Diagram from Chen Tuan and wrote the Taiji Diagram Explanation that the school of Neo-Confucianism began to take shape; however, at that time, the Taiji Diagram and the Zhou Changes text still spoke separately, and they were not yet connected. It was only when Zhu Xi wrote The Original Meaning of the Zhou Changes that he included the river chart and the Luo square, the positions of the Earlier Heaven and Later Heaven trigrams, and the large and small circular diagrams, along with his own revised diagram of changing hexagrams, all placed at the beginning of the text. Since then, diagrams and the Zhou Changes have become interdependent, and later scholars who interpreted the Zhou Changes have all used diagrams.

漢學家雖力攻河洛先後天之名，而其為說，仍不能廢陰陽四象及五行生成、九宮變化之義。而為之圖者，且層出不窮。蓋數理繁賾，卦爻錯綜，表之以圖，說乃易明。故學《易》者先辨圖書，識其陰陽生化之原，奇偶交變之義，而後觀象玩辭，有所準的。不致眩惑歧誤而靡所適從，亦事半功倍之一道焉。

In the Qing dynasty, scholars who specialized in Han learning strongly criticized terms such as "Heavenly River Chart," "Luo Writing," "Earlier Heaven," and "Later Heaven." However, when they formulated their own theories, they could not do away with the concepts of yin and yang, the Four Symbols, the numerical principles of the Five Phases, and the changes in the Nine Squares system. In fact, those who illustrated these principles with diagrams became more numerous in each generation. This is because the principles of numbers and reason are complex and intertwined, and using charts helps clarify the meaning. Therefore, those studying the Book of Changes should first clearly distinguish the types of diagrams such as the Heavenly River Chart and Luo Writing, understand the fundamental principles of yin and yang transformation, and the meaning of the alternation between odd and even numbers. Only then can they observe the hexagrams and contemplate the line statements, having a clear standard in their minds, and avoid being confused by various conflicting interpretations. This is also an efficient way to achieve greater understanding.

《繫傳》曰“《易》有太極”，又曰“六爻之動，三極之道也”。極者，至極而無對之稱。三極者，天極地極人極也。故曰“太極”者，所以異於三極也。極既無對，極而益之曰“太”，則更無有可以並之而尚之者矣。

The Great Treatise says, "The Book of Changes contains the Supreme Ultimate," and also says, "The movement of the six lines represents the principles of the three extremes." The term "extreme" refers to the highest and ultimate, having nothing else to be its counterpart. The term "three extremes" refers to the heavenly extreme, the earthly extreme, and the human extreme. Confucius deliberately called it the "Supreme Ultimate" in order to distinguish it from the three extremes of heaven, earth, and humanity. Originally, the term "extreme" already had nothing to be its counterpart. By adding the character "tai" (great) to "extreme," it indicates that nothing else can be placed on the same level or above it.

是以太極者，立乎天地之先，超乎陰陽之上，非言詞擬議所可形容。蓋狀之以言則有聲，有聲非太極也。擬之以形則有象，有象亦非太極也。《詩》曰“上天之載，無聲無臭”，庶或似之。然無字為有字之對，有對亦非太極也。

Therefore, the Taiji exists prior to the generation of heaven and earth, and transcends the dual forces of yin and yang. It cannot be described by words or discourse. To describe it with words would involve sound, and anything that involves sound is not Taiji. To compare it with forms would involve images, and anything that involves images is not Taiji. The Classic of Poetry, Great Yamen, attributed to King Wen, says, "The heavens carry on their work silently and without odor," perhaps coming closer to describing the workings of heaven without sound or smell. However, the character "wu" (nothingness) only makes sense in contrast to "you" (somethingness). Anything that involves a contrast or duality is still not Taiji.

孔子於無可形容擬議之中，而形容擬議之曰“太極”，可謂聖人造化之筆，更無他詞足以附益而增損之矣。然而有太極之名，似亦非太極之真諦。乃無礙其為太極者，則以“太極”二字，均無物質無精神可言，更無其他之詞義足以相併相對。可以謂之名，亦可以謂之非名。此聖筆之神化，所以不可思議也。

In a place where there is no way to describe or compare, Confucius managed to use the term "Taiji" to express it. This can be considered the extraordinary skill of a sage in participating in the workings of the universe; there is no other word that can add to or subtract from it. However, once the term "Taiji" is established, it seems that it is no longer the true essence of Taiji. Yet it remains Taiji, because the two characters "Taiji" cannot be classified as either material or spiritual, nor can another term be found to stand in contrast or parallel to it. It can be regarded as a name, or it can be said that it is not a name at all. This is precisely the marvelous and inconceivable skill of the sages' writing.

夫焉能又為之圖？自周子而後，相傳之圖有三。於是“太極圖”三字，流播環寰宇，幾於婦孺皆知。以論傳論，恐學者認《太極圖》為太極，則是非混淆（“《易》有太極，是生兩儀”。《易》曰“辨是與非”，即辨諸此。另有《圖說》見後。）。差以毫釐，謬以千里，而象數終無以明矣。爰詮其義，並列後出諸圖，以資辨別。

So if that is the case, how can we still draw a diagram of the Taiji? Since the time of Zhou Dunyi, there have been three transmitted versions of the Taiji diagram, and the term "Taiji Diagram" has become widely known throughout the world, almost familiar to every man and woman. Following the previous interpretations, the greatest danger for scholars is that they take the "Taiji Diagram" itself as Taiji, thereby confusing the concepts of "is" and "is not." The Great Treatise says, "The Book of Changes has Taiji, from which the two principles arise," and the Book of Changes also says, "distinguish between 'is' and 'is not.'" What needs to be distinguished is precisely this point. There is also a "Diagram Explanation" mentioned later. A slight error at the beginning leads to a great mistake in the end, and thus the study of figures and numbers can never be properly explained. Therefore, first we should explain the meaning of "Taiji," and then list all the various diagrams that have appeared in later generations, for readers to distinguish.

老子曰“有物無形，先天地生”，即謂太極也。以孔子《十翼》告成，老子已出關西去，故未知孔子有此假定之名，而曰“無以名之，強名之曰道”。究竟“道”字，實未能妙合無間，老子亦無可如何，而強名之耳。使老子得見孔子“《易》有太極”一語，必舍其名而從之。《道德經》更可省卻無數語言矣。

Laozi said, "There is something that exists without form, coming into being before heaven and earth," referring to the Taiji. By the time Confucius completed the Ten Wings, Laozi had already left for the western pass of Hangu. Therefore, he did not know that Confucius had already given this concept the name "Taiji." Laozi could not give it a proper name, so he said, "It is impossible to name it, so we must give it a name, calling it Dao." In fact, using the term "Dao" to refer to it can never fully capture its boundless nature. Laozi himself had no other way, so he merely gave it a provisional name. If Laozi had seen Confucius' statement "The Book of Changes contains Taiji," he would have abandoned his own term and adopted this one instead. The "Dao De Jing" would then have saved countless words of argumentation.

《說文》：“極，棟也。”《易·大過》：“棟隆，本末弱也。”棟亦訓極。“建中立極”，故極亦訓中。棟從東從木，木生於東，得木之正，故以室中主幹之巨木謂之棟。蓋上棟下宇，初創宮室之時構造皆甚簡單，以巨木支柱正中而四周下垂，與今日營帳之制略同。故棟必在室之正中，與極字之訓，義可以相通。若以今日之所謂棟者例之，則與“極”字之義不相蒙矣。此亦言太極者不可不知者也。

The "Sayings of the Classic of Characters" explains the character "ji" as "a central beam of a house," referring to the main beam in the center of a building. The "Zhou Yi" (Book of Changes) mentions "dong long" in the "Da Guo" hexagram, and the "Commentary on the Judgment" interprets it as "both beginning and end are weak." This shows that "dong" can also be interpreted as "ji." Ancient sayings also mention "jian zhong li ji," meaning "establishing the center as the standard," so "ji" can also mean "center." The character "dong" is composed of "dong" and "mu," meaning east and wood. Wood is strong in the east and receives the correct energy of wood, so the main central beam of a house is called "dong." In ancient times, when people first built houses, their structures were simple, using only one large wooden beam placed in the center as a support. The covering around it would naturally slope downward, similar to the structure of today's tents. Therefore, the beam was always in the center of the house, which connects with the meaning of "ji" as "center." If we compare it to today's concept of "dongliang" (beam), it would be unrelated to the original meaning of "ji." This is a point that those who discuss the Taiji (Supreme Ultimate) should not be unaware of.

周子《太極圖說》曰“無極而太極。太極動而生陽，動極而靜。靜而生陰，靜極復動。一動一靜，互為其根。分陰分陽，兩儀立焉。陽變陰合，而水火木金土五氣順布，四時行焉。五行一陰陽也，陰陽一太極也，太極本無極也。五行之生也，各一其性。無極之真，二五之精，妙合而凝。乾道成男，坤道成女。二氣交感，化生萬物，萬物生生而變化無窮焉。

The Taiji Tu Shuo by Zhou Dunyi says: From the state of no extreme, Taiji arises. When Taiji moves, yang qi is generated; movement reaches its extreme and turns into stillness; when stillness arises, yin qi is generated; stillness reaches its extreme and turns back into movement. Movement and stillness are the root of each other, and thus yin and yang are separated, establishing the two principles. Yang qi moves and transforms, while yin qi condenses and combines, leading to the sequential arrangement of the five elements: water, fire, wood, metal, and earth. The four seasons, spring, summer, autumn, and winter, follow accordingly. In the end, the five elements are merely one yin and one yang, and yin and yang in the end are merely one Taiji, and Taiji itself is originally the state of no extreme. Each of the five elements generates itself and possesses its own nature. The true essence of the state of no extreme, together with the two yin and yang energies and the essence of the five elements, skillfully combine and condense: the Qian principle generates male, the Kun principle generates female. These two energies interact and respond, giving rise to all things. All things continuously generate and transform, with no end to their changes.

惟人也得其秀而最靈。形既生矣，神知發矣。五性感動，而善惡分萬事出矣。聖人定之以中正仁義，而主靜立人極焉。故聖人與天地合其德，日月合其明，四時合其序，鬼神合其吉凶。君子修之吉，小人悖之凶。故曰‘立天之道曰陰與陽，立地之道曰柔與剛，立人之道曰仁與義’，又曰‘原始反終，故知死生之說’。大哉《易》也！斯其至矣。”

Only those who possess the most refined and spiritual aspect among the Five Phases can attain true virtue. Once the physical form is formed, the spirit and perception become active. The five human qualities—benevolence, righteousness, propriety, wisdom, and faithfulness—are stirred by external influences, leading to the distinction between good and evil, and the origin of all things. The sage establishes principles of the middle, correctness, benevolence, and righteousness to govern these, and advocates stillness as the main approach, thereby establishing the "human extreme." Thus, the sage's virtue harmonizes with heaven and earth, his brightness with the sun and moon, his seasonal changes with the four seasons, and his discernment of good and evil with the spirits. The gentleman follows this principle in self-cultivation and attains good fortune, while the petty person who violates it brings about misfortune. Therefore, the "Commentary on the Changes" states, "The way of heaven is called yin and yang, the way of earth is called softness and hardness, and the way of humanity is called benevolence and righteousness." The "Great Treatise" also says, "To trace back to the beginning and return to the end, thus understanding the doctrine of life and death." How great is the "Zhou Yi"! This is its ultimate meaning.

周子此圖，出自希夷，宋儒諱之甚深。然希夷亦非自作也，實本諸《參同契》。彭曉注《參同契》，有《明鏡圖訣》一卷。（《序》稱：廣政丁未為蜀孟昶廣政十年，漢天福十二年也。彭號真一子，字秀川，蜀人。）

The diagram of the Taiji diagram originated from Chen Tuan, and Song dynasty Confucian scholars kept this origin very secret. However, Chen Tuan himself did not create it; it actually came from the Eastern Han dynasty alchemist Wei Boyang's "Zhouyi Cangtongqi." In the Five Dynasties period, Peng Xiao wrote annotations on the "Cangtongqi" and also composed a separate work called "Mingjing Tu Jue" (his "Preface" is dated "Guangzheng Dingwei," which refers to the tenth year of Guangzheng in the reign of Meng Chang, the ruler of the Later Shu, also known as the twelfth year of Tianfu in the Later Han dynasty). This line of transmission traces the original source of the Taiji diagram back to the alchemical tradition.

毛氏奇齡曰：“《參同契》諸圖，自朱子注後（朱子注《參同契》，隱姓名為鄒欣，號空同道人。），學者多刪之。”徐氏注本已亡（按：徐名景休，東漢人，官青州從事，注《參同契》。桓帝時授淳于叔通。），他本龐雜不足據。惟彭本有《水火匡廓圖》、《三五至精圖》、《斗建子午圖》、《將指天罡圖》、《昏見圖》、《晨見圖》、《九宮八卦圖》、《納甲圖》、《舍元播精圖》、《三五歸一圖》。

In the Qing dynasty, Mao Qiling said that the original set of diagrams accompanying the "Supplement to the Supreme Truth" had, since Zhu Xi made annotations (Zhu Xi annotated the "Supplement to the Supreme Truth" but concealed his real name, signing himself as Zou Xin, with the courtesy name Kongtong Daoren), been mostly deleted by scholars. The annotated edition by Xu Shi of the Eastern Han dynasty has already been lost (Hang noted: Xu Shi was named Jingxiu, a person of the Eastern Han dynasty, who once served as a minor official in Qingzhou, and he made annotations on the "Supplement to the Supreme Truth." During the reign of Emperor Huan, he passed on this discipline to Chunyu Shutong). Other editions are too miscellaneous to be reliable. Only the edition by Peng Xiao still preserves the "Diagram of Water and Fire Encircling," "Diagram of the Three and Five Ultimate Essence," "Diagram of the Celestial Stem and Earthly Branches," "Diagram of Pointing to the Heavenly Gang," "Diagram of the Setting Sun," "Diagram of the Rising Sun," "Diagram of the Nine Palaces and Eight Trigrams," "Diagram of the Stem Assignment," "Diagram of the Dispersal of Essence," and "Diagram of the Three and Five Returning to One." These ten diagrams are the only physical evidence available for tracing the origin of the Taiji Diagram.

今周子之黑白分三層者，即《水火匡廓圖》也。其中間之水火金木土，即《三五至精圖》也。惟圖式雖同，尚末有太極之名也。考唐《真元妙經品》，有《太極先天圖》，合三輪五行為一，而以三輪中一○五行下一○為太極，又加以陰靜陽動男女萬物之象，凡四大○。陰靜在三輪之上，陽動在三輪之下，男女萬物皆在五行之下，則與周子之圖名義皆同，但多“先天”二字耳。

The part of the diagram in Zhou Dunyi's illustration today that shows black and white intermingled divided into three layers is the Peng edition's "Water and Fire Enclosure Diagram." The part in the middle where the five phases of water, fire, metal, wood, and earth are connected is the "Three and Five Ultimate Essence Diagram." Although the diagram format is the same, the term "Taiji" was not yet used at that time. Further examination of the Tang dynasty "Zhenyuan Miaojing Pin," which contains an illustration called the "Taiji Earlier Heaven Diagram," shows that the three-layered circle and the five phases are combined into one. It uses a circle in the middle of the three circles and a circle below the five phases to represent Taiji, and adds images of yin stillness, yang movement, and male and female as well as all things, making a total of four large circles: yin stillness above the three circles, yang movement below the three circles, and male and female as well as all things arranged below the five phases. This shows that it is entirely consistent with Zhou Dunyi's diagram, differing only in the addition of the two characters "Earlier Heaven."

然則此圖，自道家傳出，已無疑義。周子但為之說，並將上下次序略有修改而已。首曰“無極而太極”，終有語病。當時陸梭山已有疑義，與朱子往反辯論累數萬言。朱子雖曲為迴護，並於《太極圖說》注中申明謂“非太極之上覆有無極”，但其圖明明確太極之上有無極，其說終不可通也。其作《本義》，取邵子先天諸圖，而不以此圖列諸卷首，殆亦有所悟歟！

This suggests that the diagram originated from Daoist tradition, leaving no doubt. Zhou Dunyi merely wrote an explanatory commentary on it and slightly adjusted the order of the sections. His opening sentence, "From the boundless void comes the Supreme Ultimate," inevitably left a logical flaw. At the time, Lu Jiushao (Suoshan) raised a question, and they debated back and forth for tens of thousands of words with Zhu Xi. Although Zhu Xi tried to defend it, he still explicitly stated in the annotations to "The Diagram of the Supreme Ultimate" that "there is not another boundless void above the Supreme Ultimate." However, the diagram clearly shows a layer of boundless void above the Supreme Ultimate, making his statement ultimately incoherent. Later, when Zhu Xi wrote "The Original Meaning of the Zhou Changes," he only included Shao Yong's Earlier Heaven diagrams at the beginning, but did not include this diagram of the Supreme Ultimate. Perhaps he himself had some realization about it.

古太極圖，亦名“天地自然之圖”。趙氏撝謙《六書本義》曰：“天地自然之圖，伏羲氏龍馬負之出於滎河，八卦所由以畫也。世傳蔡季通得於蜀隱者，秘而不傳。趙氏得之於陳伯敷氏，熟玩之有太極函陰陽，陰陽函八卦之妙。自明洪武以後，此圖遂盛傳於世。”

The ancient Taiji diagram is also known as the "Diagram of Heaven and Earth in Their Natural State." In his work "Six Classics: Original Meaning" from the Ming dynasty, Zhao Huiqian said that this diagram of heaven and earth is said to have been brought out by a dragon horse from the river Xinghe during the time of Fuxi, and the Eight Trigrams were drawn based on it. It is said that in the Southern Song dynasty, Cai Yuan ding (Jitong) obtained it from a hermit in Sichuan and kept it secret, not passing it on. Zhao Huiqian himself obtained it from Chen Bofu, and after repeatedly contemplating it, he realized the wonderful aspect that Taiji contains yin and yang, and yin and yang contain the Eight Trigrams. Since the Ming dynasty's Hongwu period, this diagram has been widely circulated.

（按：蔡氏晚年得此圖未久即病卒，故朱子亦未之見也。蔡死後秘藏於家，至其孫始傳佈之，已在宋亡之後。胡元一代，尚鮮稱述。至明初劉青田取以繪入八卦之中，遂風行海內，幾於家喻戶曉，無人不知有太極圖矣。）

(Hang氏按：Cai Yuan ding died soon after obtaining this diagram, so Zhu Xi did not see it. After Cai's death, the diagram was kept secret in his family and was only made public by his grandchildren, which was already after the fall of the Southern Song dynasty. During the entire Yuan dynasty, few people mentioned it. It was not until Liu Ji (Qingtian) of the early Ming dynasty included it within the Eight Trigrams that the diagram became widespread throughout the country, almost known to everyone.)

朱子發云：“陳摶以《先天圖》授种放，三傳而至邵雍。則康節之學，實出自希夷。所演《先天圖》，陰陽消長，亦與此圖悉合，故又謂之《太極真圖》。其環中為太極，兩邊黑白回互。白為陽，黑為陰。陰盛於北，而陽起而薄之，故邵子曰‘震始交陰而陽生’。自震而離而兌以至於乾，而陽斯盛焉。

In the Song dynasty, Zhu Zhen (Zi Fa) said: Chen Tuan passed on the "Earlier Heaven Diagram" to Zhong Fang, who passed it on to three generations until it reached Shao Yong. This shows that Shao Yong's learning actually came from Chen Tuan. The "Earlier Heaven Diagram" that Shao Yong deduced discusses the waxing and waning of yin and yang, and it matches the ancient Taiji diagram in every aspect. Therefore, this diagram is also called the "True Taiji Diagram." The center of the diagram is Taiji, with black and white circles interlocking on both sides: white represents yang, and black represents yin. Yin is strongest in the north, and yang arises from here, gradually approaching yin, so Shao Yong said, "Zhen begins to meet yin and yang is born." From Zhen to Li to Dui, and then to Qian, yang gradually becomes stronger.

震東北，白一分，黑二分，是為一奇二偶。兌東南，白二分，黑一分，是為二奇一偶。乾正南全白，是為三奇純陽。離正東，取西之白中黑點，為二奇含一偶，故云‘對過陰在中也’。

Now looking at the scores on the diagram: Zhen is in the northeast, with white occupying one part and black occupying two parts, forming the Zhen hexagram of one odd line and two even lines; Dui is in the southeast, with white occupying two parts and black occupying one part, forming the Dui hexagram of two odd lines and one even line; Qian is in the due south, entirely white, forming the Qian hexagram of three odd lines, all yang; Li is in the due east, where one must take the black dot from the western white part to form two odd lines with one even line, hence it is said 'the yin is in the middle of the opposite,' meaning the yin is contained within oneself, opposite the other side.

陽盛於南，而陰來迎之，故邵子曰‘巽始消陽而陰生’。自巽而坎而艮以至於坤，而陰斯盛焉。巽西南，黑一分，白二分，是為一偶二奇。艮西北，黑二分，白一分，是為二偶一奇。坤正北全黑，是為三偶純陰。坎正西，取東之黑中白點，為二偶含一奇，故云‘對過陽在中也’。

Yang energy is strongest in the south, and Yin energy comes from there to meet it, so Shao Yong said, "Xun begins to reduce Yang and Yin is generated." Starting from Xun, then Kan, then Gen, and finally Kun, Yin energy gradually becomes stronger. Xun is located in the southwest, with black occupying one part and white occupying two parts, making a Xun hexagram with one even and two odd lines; Gen is located in the northwest, with black occupying two parts and white occupying one part, making a Gen hexagram with two even and one odd line; Kun is located in the due north, entirely black, making a Kun hexagram with three even lines and pure Yin; Kan is located in the due west, and one white dot from the east is taken from the black to form two even lines containing one odd line, so it is said, "Yang is in the center opposite."

坎離為日月，升降於乾坤之間，而無定位。納甲寄中宮之戊己，故東西交易，與六卦異也。八方三畫之奇偶，與白黑之質，次第相應，深得陰陽造化自然而然之妙。但既有黑白之分，備陰陽之用，已非太極之真相，不當稱之為《太極圖》。今北方俗呼此圖為‘陰陽魚兒’，‘魚’字實‘儀’字之誤。稱之曰‘陰陽儀圖’，或曰‘兩儀圖’，斯名實相符矣。”

The trigrams Kan (Water) and Li (Fire) symbolize the sun and moon, rising and setting between Qian (Heaven) and Kun (Earth), and originally have no fixed direction. In terms of "Najia" stem assignment, they are associated with the stems Wu and Ji in the Central Palace. Therefore, the eastern and western positions must exchange their symbolic meanings, differing from the method used for the other six trigrams. The odd and even numbers of the three strokes in each of the eight directions correspond precisely with the number of black and white blocks, layer by layer, clearly demonstrating the natural workings of yin and yang. However, once black and white are distinguished and the functions of yin and yang are established, it is no longer the true nature of the Taiji. Therefore, it should not be called the "Taiji Diagram." Now, in northern folk, this diagram is commonly called the "Yin-Yang Fish," but the word "fish" is actually a miswriting of "instrument." It should be called the "Yin-Yang Instrument Diagram" or the "Two Principles Diagram" to match the actual meaning.

此圖流傳甚古，蘊蓄宏深，決非後人所能臆造。大抵老子西出函關，必挾圖書以俱行，故遺留關中，為道家之秘藏。至於唐宋而後，始逐漸傳佈，要皆為三代以上之故物，無可疑也。

This diagram has a very ancient origin, containing profound meaning, and could not have been invented by later people. The general account is that when Laozi left the pass of Hangu, he must have taken these books with him, so such texts remained in the area of Hangu and were secretly kept by the Taoist school. It was only after the Tang and Song dynasties that these texts gradually spread throughout the world. In the end, they are all ancient relics from before the Xia, Shang, and Zhou dynasties, and there is no doubt about this. From this we can conclude that the ancient Taiji diagram dates back far earlier than the Confucian scholars of the Song dynasty, and the Song people merely inherited the old diagram.

惟圖像既顯分黑白，是已生兩儀。分之為四，即成四象。分之為八，即成八卦。可謂之“兩儀生四象，四象生八卦之圖”。但流傳既久且遠，世俗已無人不認此為太極圖者。所謂習非勝是，辨不勝辨。惟學者宜詳究其義理，因名責實，而求得真諦。斯源頭不誤，自能清澈見底。不可以習見而忽之，反為流俗所誤也。

The image, once clearly divided into black and white, signifies the emergence of the Two Principles. Dividing each of black and white again into two results in the Four Symbols; dividing again into eight produces the Eight Trigrams. Therefore, this image can at most be called the "Diagram of the Two Principles generating the Four Symbols, and the Four Symbols generating the Eight Trigrams," not the Taiji Diagram. However, due to its long and widespread transmission, the common people have come to recognize it as the Taiji Diagram. This is what is called "accumulating errors into truth," and no amount of argument can change it. Scholars should only carefully examine its meaning, follow the name to investigate the reality, and thus seek the true essence. If the source is correct, the following will naturally be clear. One should not become careless merely because it is familiar, and be misled by popular misconceptions.

此來瞿塘氏之圖，亦從《古太極圖》研索而出者也。以居中之黑白二線，代兩點，以象陽方盛而陰已生，陰方盛而陽已生，有循環不絕之義。留中空者以象太極，其陰陽之由微而顯，由顯而著，亦悉合消息之自然。與河圖陰陽之數，由微而著，由內而外，亦適相合。來氏另有一圖，以河圖陽數佈於白中，陰數佈於黑中，又以配八卦四時十二辰，及古今治亂盛衰之象。圖多，茲不贅錄。

This is a diagram created by Lai Zhide of the Ming dynasty (Qutang), derived from the ancient Taiji Diagram. He used a single black and white line in the center to replace the original black and white dots, symbolizing that when Yang is at its peak, Yin has already begun to emerge, and when Yin is at its peak, Yang has already begun to emerge, conveying the meaning of continuous cycles. The blank space in the diagram symbolizes the Taiji. The transition of Yin and Yang from hidden to manifest, and from manifest to prominent, fully conforms to the natural principle of waxing and waning. This is consistent with the arrangement of the numbers of Yin and Yang in the Yellow River Chart, which progresses from subtle to prominent and from inner to outer. Lai Zhide also created another diagram in which the Yang numbers of the Yellow River Chart are arranged on the white side and the Yin numbers on the black side. He used this to associate with the Eight Trigrams, the four seasons, the twelve earthly branches, and the changes of prosperity and decline in the world throughout history. There are many such diagrams, but they are not listed here one by one.

來氏此圖，蓋亦悟太極之非圖所可狀，非言語可能形容，故留其外之黑白回互者為兩儀，而空其中以為太極。然“太極”二字之下繫之以圖，終有所難通也。以上三圖，皆稱太極。周子之圖，僅見於性理諸書，習之者今已不多。來氏之圖，傳亦不廣。惟第二圖風行宇內，幾與八卦並傳。圖像雖精，與孔子“太極”二字之義，實不相符。此為學易推原象數之第一步，不可不深思明辨，而求得其至當者也。

Lai Zhide's diagram probably also realized that the Taiji cannot be depicted by a diagram or described by words, so he left the outer black and white interlocking parts as the two principles, and left the center empty as the Taiji. However, beneath the two characters "Taiji," there is still a diagram attached, which remains difficult to explain. The three diagrams mentioned above are all called Taiji: Zhou Dunyi's diagram appears only in books discussing nature and principles, and now few people study it; Lai Zhide's diagram is not widely circulated either; only the second ancient Taiji diagram is widely circulated throughout the world, almost as popular as the Eight Trigrams. Although its image is exquisite, it does not conform to the meaning of "Taiji" as mentioned by Confucius. This is the first step in studying the Book of Changes concerning its symbolic representations and numerical patterns. It is essential to think deeply and discern clearly, seeking the most appropriate conclusion.

太極既不可以圖，然非圖又不足明陰陽顯化之妙，是惟有舍太極而圖兩儀。兩儀既明，則太極自立乎其先。無象之象，亦因象而顯矣。

The Taiji cannot be drawn as a diagram, yet without a diagram it is insufficient to explain the subtle transformations of yin and yang. Therefore, we must abandon the Taiji and instead draw the two principles (liang yi). Once the two principles are clearly drawn, the Taiji naturally stands before them; the formless form is thus revealed through the tangible two principles.

此端木國瑚氏之圖也，於文日正為是。下正北方正而上直日，日中影正。影正，則左右自分為兩。左陽而右陰。故《易》於乾之初爻曰“不見是”，未濟之上爻曰“有孚失是”。全《易》六十四卦，三百八十四爻，以此一“是”為始終，其故可深長思也。

This is a diagram created by Du Mengguo Hu in the Qing Dynasty. In terms of character form, adding "zheng" under "ri" forms the character "shi." The diagram's lower part faces due north directly, while its upper part faces the sun directly. The sun is in the center, and its shadow is straight; when the shadow is straight, left and right naturally divide into two sides, with the left being yang and the right being yin. Therefore, in the Qian hexagram's initial line of the Zhou Yi, the Wenyan Zhuan states, "Not seeing it but without distress." In the Weiji hexagram's upper line, the line statement says, "Having sincerity but losing the correct path." All sixty-four hexagrams and 384 lines in the entire Zhou Yi are unified by this single character "shi." The reason behind this is worth deep contemplation.

有丨畫中央，便分兩列。《繫傳》曰：“乾坤成列，而易立乎其中矣。”是則乾坤左陽儀，右陰儀。從丨生出，是太極，是兩儀。一至而二分（東西分，南北至。），是一是二，易陰陽生，不待再言“太極生兩儀”，此為“《易》有太極，是生兩儀”。

In the diagram, there is a vertical line 「丨」 standing in the center, dividing the space into two columns on either side. The Great Treatise says, "When Heaven and Earth are arranged in order, the Book of Changes stands in the midst." This precisely corresponds to the arrangement here: Heaven (Qian) is on the left as the Yang symbol, and Earth (Kun) is on the right as the Yin symbol. All of this originates from the central vertical line, which is both the Taiji and the two principles (two yi). One "Zhi" (solstice) gives rise to two "Fen" (equinoxes), with the spring equinox and autumn equinox dividing east and west, and the winter solstice and summer solstice marking the extremes of north and south. It is both one and two, and thus the yin and yang of the Book of Changes arise naturally. There is no need to add another sentence such as "Taiji gives rise to two yi," for this is precisely the original meaning of "The Book of Changes has Taiji, which gives rise to two yi."

端木氏此圖，似較舊說更為明顯。不名曰“太極圖”，更無語病。學者與前三圖合而觀之，“太極”二字之義，可瞭然矣。“太極”之上，無能更加以字者。惟孔子曰“《易》有太極”。“《易》有”二字，必須重讀，與《序卦》“有天地”“有萬物”諸“有”字，皆一氣貫注。孔子贊易，皆是從“有”立說，以示與老子之以“無”立說者不同。

The diagram of Duanshu Guohu seems clearer than previous interpretations; since it is not called the "Taiji Diagram," there is no grammatical error. Scholars can view this diagram together with the previous three diagrams, and the meaning of "Taiji" becomes clear. No other words can be added above "Taiji," for only Confucius said "The Book of Changes contains Taiji." The two characters "has" in "has Taiji" must be read with emphasis, and they are connected with the series of "has" in the "Xugua Zhuan" such as "has heaven and earth" and "has all things." Confucius praised the Book of Changes, always starting from the concept of "has," to show the difference from Laozi's approach which started from "nothingness."

故將“有”字直提至“太極”之上，曰“《易》有太極”，而全《易》之生生不已，皆由此“有”字以發生者也。繼之曰“是生兩儀”，此“是”字上貫“《易》有太極”四字，而下澈全章。全《易》六十四卦，皆以此一字作骨，故特於初、終二爻發明之。觀此知孔子贊《易》筆法，一字一義，無不與全書精神貫注，脈絡相通，未可滑口讀過也。

Therefore, Confucius elevated the character "有" (to have, to exist) directly above "太极" (the Ultimate Ultimate), saying "The Book of Changes has the Ultimate Ultimate." All the ceaseless generative principles of the entire Zhou Changes are generated from this single character "有." Then he says "from which the two principles arise." The character "是" (this, that) above inherits the meaning of the previous four characters "The Book of Changes has the Ultimate Ultimate," and below it connects with the entire passage. All sixty-four hexagrams of the Book of Changes take this character as their backbone. Therefore, Confucius deliberately pointed it out in the first line of the Qian hexagram and the top line of the Wei Ji hexagram, which mark the beginning and the end respectively. From this we can see the style of Confucius in commenting on the Book of Changes: each character has its own meaning, and none is disconnected from the spirit of the whole book or the logical flow. When reading, one must not pass it by carelessly.

《繫傳》曰“形而上者謂之道，形而下者謂之器”，太極者實超乎道與器之上，而立乎其先者也。故分言之。形而上者有太極，形而下者亦未始無太極也。故曰“天地一太極”。萬物各有其太極。後儒以太極為形而上者，是與形而下者對待，實失“太極”之本義也（老子曰：“無以名之，強名之曰道。”此道字與形而上之道意義不同。）。

The Great Treatise says, "What is above form is called the Way, and what is below form is called the thing." But the Supreme Ultimate actually transcends both the Way and the thing, and exists prior to them. Therefore, when considered separately: the level above form contains the Supreme Ultimate, and the level below form also contains the Supreme Ultimate. Thus it is said, "Heaven and Earth are a single Supreme Ultimate," and all things also have their own Supreme Ultimate. Later Confucian scholars regarded the Supreme Ultimate solely as what is above form, thus making it a counterpart to what is below form, which actually misses the original meaning of "Supreme Ultimate" (Laozi says, "It cannot be named; force it to be named, call it the Way," but the "Way" here is not the same as the Way above form).

是以天下之事事物物，凡有對待者，皆太極所生之兩儀，非太極也（今世科學家曰精神，曰物質，相對待者也，太極實超乎精神物質之上。曰空間，曰時間，相交午者也，太極實超乎空間時間之外。哲學主唯心論者謂一切唯心所造，主唯物論者謂科學萬能物質不滅。太極則超乎唯心唯物之上，而不可以唯心唯物概之者也。）。

Therefore, all things and matters in the world, whenever they can be presented as opposing aspects that depend on each other, are the two principles (yin and yang) generated by the Taiji. They are not the Taiji itself. (Today's scientists refer to spirit and matter as a pair of opposing concepts; in reality, the Taiji exists beyond both spirit and matter. Similarly, space and time are two intersecting axes, but the Taiji exists beyond space and time. Philosophers who advocate idealism claim that everything is created by the mind, while those who advocate materialism claim that science is omnipotent and that matter is eternal; however, the Taiji exists beyond both idealism and materialism, and cannot be summarized by either of these two perspectives.)

以質言之曰剛與柔，而太極超乎柔剛之外。以氣言之曰陰與陽，而太極立乎陰陽之先。以事理言之曰動與靜，曰善惡吉凶，而太極實幾於動靜善惡吉凶之微，無有而無不有，無在而無不在。《序卦傳》曰“盈天地之間唯萬物”，謂盈天地之間惟太極可也。此天地一太極也。小而至一塵之微，極至於微而不可見之物，亦莫不各具有動靜生滅之機，即莫不有其太極，此物各一太極也。

In terms of quality, there is hardness and softness, but the Taiji exists beyond hardness and softness; in terms of qi, there is yin and yang, but the Taiji exists prior to yin and yang; in terms of principles, there is movement and stillness, good and evil, auspicious and inauspicious, but the Taiji lies hidden in the subtle beginnings of movement and stillness, good and evil, auspicious and inauspicious, before they manifest. It is said to have no existence yet it is present everywhere; it is said to be absent yet it is everywhere present. The "Xugua Zhuan" states, "The fullness of the universe is only the myriad things." If it is said that what fills the universe is only the Taiji, that is also acceptable, which is "the universe is one Taiji." Even as small as a single grain of dust, or as minute as something invisible to the naked eye, each possesses the mechanism of movement and stillness, birth and death, and thus each has its own Taiji, which is "each thing has its own Taiji."

宋儒言太極，不離乎動靜陰陽，已落言詮。牽及五行，則更遠矣。朱子注《太極圖說》，謂“無靜不成動，無動不成靜。譬如鼻息，無時不噓，無時不吸。噓盡則生吸，吸盡則生噓。理自如此，陰陽只是一氣。陰氣流行即為陽，陽氣凝聚即為陰”。其論雖精，總未達一間。程子曰：“動靜無端，陰陽無始。非知道者，孰能識之？”然動靜無端而自有其端，陰陽無始而自有其始也。識此端與始，以言太極，庶乎近矣。

The Song dynasty Confucian scholars discussed the Taiji, always relating it to movement and stillness, yin and yang, which already falls into verbal explanation. When they further involved the Five Phases, they became even more distant. Zhu Xi commented on the "Taisi Tu Shuo" saying: "Without stillness, there can be no movement; without movement, there can be no stillness. For example, the nose breathing, there is always inhalation and exhalation without cease; when inhalation reaches its end, exhalation arises; when exhalation reaches its end, inhalation arises. This is the inherent nature of things. Yin and yang are merely one breath: when yin qi spreads out, it becomes yang; when yang qi condenses, it becomes yin." Although this discussion is precise, it still lacks one layer. Cheng Yi said: "Movement and stillness have no beginning, yin and yang have no origin. Who, except those who truly understand the Dao, can recognize it?" However, although movement and stillness are said to have no beginning, they do have their beginning; although yin and yang are said to have no origin, they do have their origin. Recognizing this beginning and origin, and then discussing Taiji, one is already close to the truth.

《繫傳》曰：“河出圖，洛出書，聖人則之。”《書·顧命》曰：“河圖在東序。”《論語》曰：“河不出圖。”《禮記·禮運》曰：“山出器車，河出馬圖。”鄭康成《易注》引《春秋緯》：“河以通乾出天苞，洛以流坤吐地符。河龍圖發，洛龜書成。河圖有九篇，洛書有六篇。”揚雄《旡靈賦》曰：“大《易》之始，河序龍馬，洛出龜書。”

The Great Treatise says: "The Yellow River produced the chart, and the Luo River produced the writing, and the sage followed them." The Book of Documents, Gu Ming, records "the Yellow River Chart was in the eastern hall," indicating that it was once displayed in the eastern chamber of the royal palace. The Analects records Confucius lamenting "the Yellow River does not produce the chart." The Book of Rites, Li Yun, says "mountains produce vehicles and instruments, and the Yellow River produces horse figures." Zheng Xuan's Commentary on the Changes cites the Spring and Autumn Writings, saying: "The Yellow River, through the communication of Heaven, produces the heavenly symbol, and the Luo River, through the flow of Earth, produces the earthly symbol. The Yellow River Chart arises with the dragon horse, and the Luo Writing is completed with the turtle book. The Yellow River Chart has nine chapters, and the Luo Writing has six chapters." This means that the Yellow River Chart communicates with Heaven and produces the heavenly mandate, while the Luo Writing flows through Earth and produces the earthly symbol, and the two are originally a pair of Heaven and Earth. Yang Xiong's Wu Ling Fu also says: "The beginning of the great Changes is the Yellow River's dragon horse and the Luo River's turtle book."

《漢書·五行志》劉歆曰：“伏犧氏繼天而王，受河圖則而畫之，八卦是也。禹治洪水，錫洛書而陳之，洪範是也。聖人行其道而實其真，河圖洛書，相為經緯；八卦九章，相為表裡。昔殷道弛，文王演《周易》。周道敝，孔子作《春秋》。則乾坤之陰陽，效洪範之咎徵。天人之道粲然著矣。”

The Han Shu's "Five Phases Chapter" cites Liu Xin's words: Fuxi received the mandate of heaven and became king, accepted the Yellow River Chart, and based on it he drew the trigrams, which are the Eight Trigrams. When Da Yu controlled the floodwaters, heaven bestowed the Luo Writing upon him, and based on it he arranged the categories, which is the "Hong Fan" Nine Categories. The sages promoted their teachings and implemented their truths. The Yellow River Chart and the Luo Writing form the vertical and horizontal threads, while the Eight Trigrams and the Nine Categories form the inside and outside. In the past, during the Shang dynasty, when the government became lax, King Wen developed the Zhou Changes; during the Zhou dynasty, when the government declined, Confucius wrote the Spring and Autumn Annals. One drew upon the yin and yang of Qian and Kun, the other verified the omens of the Hong Fan, thus the principle of the connection between heaven and humanity became clearly evident.

河圖洛書之見於經傳者如此，而其內容如何，則無可考。至宋初陳希夷氏，始有龍圖之數。邵康節因之，以五十五、四十五兩數，分為河圖洛書，當時頗多爭議。如范諤昌、劉牧諸氏，以四十五為河圖，五十五為洛書。漢上朱氏因之。朱文公從蔡季通之議，定十為河圖，九為洛書，以冠《本義》之首。更詳演其數，以入《易學啟蒙》，相傳至今。

The accounts of the Yellow River Chart and the Luo Writing found in classical texts are limited to these references. As for their actual content, it has become impossible to verify. It was not until the early Northern Song dynasty that Chen Tuan introduced the concept of "Dragon Diagram" numbers. Shao Yong continued this tradition, assigning the number fifty-five to the Yellow River Chart and forty-five to the Luo Writing. At that time, there was considerable controversy. For example, Fan Echanchang and Liu Mu argued that forty-five belonged to the Yellow River Chart and fifty-five to the Luo Writing, a view also followed by Zhu Zhen, the Han Shang Master. Later, Zhu Xi adopted the opinion of Cai Yuan ding, determining that the number ten belonged to the Yellow River Chart and nine to the Luo Writing. He placed these two diagrams at the beginning of the first volume of The Original Meaning of the Zhou Changes, and elaborately explained their numerical patterns in his work The Awakening of the Study of the Changes, a tradition that has continued to the present day.

復經丁易東、張行成、熊良輔，及勝清江慎修、萬彈峰諸氏之推演，義蘊畢宣。所謂神變化而行鬼神者，無不與易義悉相貫通，而《彖》、《象》所不可解者，亦得以數象相證而通其義。雖未敢謂此即為古之河圖洛書，而數理之神化，則固建諸天地而不悖，質諸鬼神而無疑，百世以俟聖人而不惑者也。

Later scholars such as Ding Yidong, Zhang Xingcheng, Xiong Liangfu, and Qing dynasty figures Jiang Yong (Shenxiu) and Wan Nianchun (Tansheng) further developed these ideas, and it was through their efforts that the deeper meanings were fully revealed. The statement in the Great Treatise that "the divine changes and moves like ghosts and spirits" is fully consistent with the principles of the Book of Changes; even those parts of the Commentary on the Judgment and Image Commentary that were previously difficult to understand could be clarified through the mutual verification of numbers and images. Although we cannot definitively say that this is the original Yellow River Chart and Luo Writing from ancient times, this system of numerical and logical transformations indeed stands in harmony with the heavens and earth without contradiction, and it can be verified against ghosts and spirits without doubt. Even if a sage appears a hundred generations later, he would not be confused by it.

自明季以來，言漢學者雖盡力攻擊，但只能爭河洛之名。而於其數，則無能置喙焉。欲探易道無盡之蘊，發千古神秘之局者，端在於是。乃舍其實而鶩其名，不亦慎哉！茲仍從《本義》、《啟蒙》之名，以五十五者為河圖，以四十五者為洛書，以便稱引。至陳希夷龍圖之說，義甚膚淺，故朱子疑為偽造。然其所謂龍圖者，分合之數皆五十有五，且明明曰龍圖不曰龜圖。劉牧之以九為圖十為書，是顯畔其師說矣。

Since the Ming dynasty, those who study Han learning have vigorously attacked the River Chart and the Luo Square, but they could only dispute over the names; regarding their numbers, they had no say. To explore the boundless depth of the I Ching and to reveal the ancient mystery, the key lies in these numbers. To abandon the substance and chase after the names, should we not carefully reflect? Therefore, we still use the names of The Original Meaning of the Zhou Changes and The Illumination of the Study of Changes, taking fifty-five numbers as the River Chart and forty-five numbers as the Luo Square, for reference. As for Chen Tuan's theory of the Dragon Chart, its meaning is very shallow, so Zhu Xi suspected it was fabricated. However, what he called the Dragon Chart, when separated or combined, always totals fifty-five, and clearly called it the Dragon Chart, not the Turtle Chart. Liu Mu, on the other hand, took nine numbers as the River Chart and ten numbers as the Luo Square, which clearly contradicts his master's teachings.

揚子《太玄》曰：“一六為水，二七為火，三八為木，四九為金，五十為土。一與六共宗（北），二與七為朋（南），三與八成友（東），四與九同道（西），五與五相守（中）。”按：揚子不言十者，五與五即十，太玄用九數，故置十不言。且數止於九，至十則復為一。故河洛以十為盈虛，亦數象關鍵之一。學者宜深思之。

Yang Xiong's "Tai Xuan" says: "One and six belong to water, two and seven to fire, three and eight to wood, four and nine to metal, five and ten to earth. One and six share the same origin (in the north), two and seven are companions (in the south), three and eight are friends (in the east), four and nine are on the same path (in the west), and five and five keep each other (in the center)." Hang's commentary states: Yang Xiong does not mention "ten" because five plus five equals ten; "Tai Xuan" uses nine as the key number, so it omits ten. Moreover, the counting stops at nine, and ten returns to one. Therefore, the He Tu and Luo Shu treat ten as the key to fullness and emptiness: the He Tu includes ten in its total of fifty-five, while the Luo Shu does not include ten in its total of forty-five. The difference lies here. This is a crucial point in numerical symbolism, and scholars should think deeply about it.

鄭康成氏曰：“天地之氣各有五。五行之次：一曰水，天數也。二曰火，地數也。三曰木，天數也。四曰金，地數也。五曰土，天數也。此五者，陰無匹，陽無耦，故又合之地六為天一匹也，天七為地二耦也，地八為天三匹也，天九為地四耦也，地十為天五匹也。”又曰：“布六於北方以象水，布八於東方以象木，布九於西方以象金，布七於南方以象火。”

Zheng Xuan said: The energies of heaven and earth each have five kinds. The order of the Five Phases is: one is water, belonging to the heavenly numbers; two is fire, belonging to the earthly numbers; three is wood, belonging to the heavenly numbers; four is metal, belonging to the earthly numbers; five is earth, belonging to the heavenly numbers. These five generating numbers have no counterparts on the yin side and no pairs on the yang side, so each is given a complementary number: the earthly six pairs with the heavenly one, the heavenly seven couples with the earthly two, the earthly eight pairs with the heavenly three, the heavenly nine couples with the earthly four, and the earthly ten pairs with the heavenly five. He also said: place six in the north to symbolize water, place eight in the east to symbolize wood, place nine in the west to symbolize metal, and place seven in the south to symbolize fire. The arrangement of the completed numbers in the four directions of the He Tu chart originates from this.

朱子曰：“此言天地之數。陽奇陰耦，即所謂河圖者也。中五為衍母，次十為衍子。次一二三四為四象之位，次六七八九為四象之數。二老位於西北，二少位於東南。其數則各以其類，交錯於外也。”按：此非言大衍之數，不必以衍母衍子相牽涉。而四象之數之位，亦為揲蓍得卦之用。於此言之，其義反窄矣。

Zhu Xi said: This passage discusses the numbers of heaven and earth. Yang is odd, and Yin is even, which is what is referred to as the Yellow River Chart. The central five is the mother of the great derivation number, and the next ten is the son of the great derivation number; then one, two, three, and four are the positions of the four images, while six, seven, eight, and nine are the numbers of the four images. The two "old" (old Yang and old Yin) are located in the northwest, and the two "young" (young Yang and young Yin) are located in the southeast. Their numbers are arranged alternately in the outer circle according to their categories. Hang氏 remarked: This passage is not actually discussing the great derivation number, so there is no need to involve the concepts of the mother and son of the derivation; the numbers and positions of the four images have separate uses during the process of divining by means of the yarrow stalks, and placing them here actually narrows the meaning of their significance.

《繫傳》曰：“天一，地二。天三，地四。天五，地六。天七，地八。天九，地十。天數五，地數五，五位相得而各有合。天數二十有五，地數三十。凡天地之數五十有五，此所以神變化而行鬼神也。”揚子《太玄》及鄭注所演其方位與生成之數，均極明晰。雖未繪為圖，已與圖無異矣。陳邵但按其說而圖之耳。

The Great Treatise says: "Heaven is one, Earth is two; Heaven is three, Earth is four; Heaven is five, Earth is six; Heaven is seven, Earth is eight; Heaven is nine, Earth is ten. The numbers of Heaven are five, the numbers of Earth are five. The five positions are paired two by two and each has its union. The total of Heaven's numbers is twenty-five, the total of Earth's numbers is thirty. The total numbers of Heaven and Earth together are fifty-five. It is because of this number that it can produce mysterious transformations and the movement of ghosts and spirits." The directions and the numbers of generation explained in Yang Xiong's Tai Xuan and in Zheng Xuan's annotations are already very clear; although they were not drawn into a diagram, they are essentially the same as if there were a diagram. Chen Tuan and Shao Yong merely drew them according to these statements. This shows that the content of the diagram was not created out of nowhere by the Song people, but was already present in the original texts.

五行之次，始見於《洪範》。而坎水、離火、乾金、巽木，均備載於《說卦》。經傳之互見者，更不勝枚舉。故毛西河雖攻擊河圖洛書之說最力，終不能蔑去此數，謂應改名曰“天地生成圖”。然其數之體用自在，名稱之同異，抑其末耳。

The order of the Five Phases first appears in the Shangshu · Hongfan. The associations of Kan as water, Li as fire, Qian as metal, and Xun as wood are also fully recorded in the Shuogua Zhuan. Examples of mutual reference between the classic text and its commentary are numerous. Therefore, Mao Qilin (Xihé) although most vigorously attacked the theories of the Yellow River Chart and the Luo Writing, could not erase this system entirely, and could only advocate renaming it as the "Heaven and Earth Generating Chart." However, the essence and application of this system have always existed. Whether the name is the same or different is merely a minor detail.

後儒有泥於“龍馬負圖”之文，謂此五十五數，馬體所生之旋毛。黑白五十五點，其陰陽匹耦如今圖，惟當繪圖點為旋毛形。於是有繪馬而寫圖於馬腹者，亦有將五十五點均改作旋毛形者。其拘泥附會，與漢學家之斷斷然徒爭其名稱者，同一蔽耳。更有泥圓方之說，將河圖一六二七三八四九之數，均作弧線成圓形，於義理並無出入，均可不必也。

Later scholars became fixated on the phrase "a dragon horse carried the diagram," claiming that the fifty-five numbers represented the spiraled hairs on a horse's body, with fifty-five black and white dots symbolizing yin and yang. They believed the diagram should be drawn in the form of these spiraled hairs. Some even drew a horse and inscribed the diagram on its belly, while others uniformly transformed the fifty-five dots into spiraled shapes. This kind of rigid and forced interpretation is no different from the Han scholars' insistence on names and arguments, which is a form of obstruction. Others were fixated on the idea of a round heaven and square earth, drawing the numbers 1 and 6, 2 and 7, 3 and 8, 4 and 9 in curved lines to form a circle. Although this does not contradict the meaning, it is entirely unnecessary to do so.

關子明曰：“洛書之文，九前一後，三左七右。四前左，二前右。八後左，六後右。”邵子曰：“圓者星也。曆紀之數，其肇於此乎？（《啟蒙》曰：曆法合三始以定剛柔，二中以定歷律，二終以紀閏餘。是所謂曆紀也。）方者土也，畫州井地之法，其仿於此乎？蓋圓者河圖之數，方者洛書之文，故羲文因之而造《易》，禹箕敘之而作《範》也。”

Guan Lang (Ziming) said: The arrangement of the Luo Book is nine in front, one at the back; three on the left, seven on the right; four in the front left, two in the front right; eight in the back left, six in the back right. Shao Yong said: The circular arrangement relates to celestial phenomena, and the numbers used in the calendar and year reckoning probably originated from this (The "Yixue Qimeng" says: The calendar combines the "three beginnings" to determine yin and yang, combines the "two middles" to determine the calendar and music, combines the "two endings" to record the leap months, which is what is called the calendar system). The square arrangement relates to the land, and the methods of dividing states and managing the well-field system probably imitate this. In general, the circular is the numbers of the He Tu, and the square is the text of the Luo Book, so Fuxi and King Wen used it to create the Book of Changes, and Great Yu and Ji Zi used it to compose the "Hong Fan."

《大戴禮·明堂篇》曰：“明堂者，古有之也，凡九室。二九四七五三六一八。”（其數及位均與此同。明堂九室，故亦稱九宮。）

The Great Dai Ritual Records · Bright Hall Chapter says: The Bright Hall system has existed since ancient times, consisting of nine rooms with numbers arranged in the order of two, nine, four, seven, five, three, six, one, and eight. (These nine numbers and their respective positions are exactly the same as the Luo Writing. Since the Bright Hall has nine rooms, it is also called the Nine Palaces.) This shows that the numbers of the Luo Writing were already embodied in the ancient architectural system.

《易乾鑿度》：“太一取其數以行九宮。”鄭注曰：“太一者，北辰神名也。下行八卦之宮，每四乃還於中央。中央者，北辰之所居，故謂之九宮。天數大分，以陽出，以陰入。陽起於子，陰起於午。

The "Yi Wei Qian Zao Du" says: "The Supreme One takes the numbers to move through the nine palaces." Zheng Xuan's commentary explains that the Supreme One is the name of the deity of the North Star, which moves downward through the palaces governed by the Eight Trigrams. Each time it completes four palaces, it returns to the center. The center is the dwelling place of the North Star, hence the term nine palaces. The heavenly numbers are generally divided into two paths: emerging with yang and returning with yin. Yang energy originates from the Zi position, while yin energy originates from the Wu position.

是以太一下九宮，從坎宮始。自此而從於震宮，自此而從於巽宮，所行半矣。還息於中央之宮，既又自此而從於乾宮，又自此而從於兌宮，又自此而從於艮宮，又自此而從於離宮，則周矣。上游息於太一之星，而反於紫宮。行起從於坎宮始，終於離宮也。”

Therefore, the Supreme One moves through the nine palaces, starting from the Kan palace (North 1), then moving to the Zhen palace (East 3), then to the Xun palace (Southeast 4). This covers half the journey, so it rests in the central palace (5). Then it moves to the Qian palace (Northwest 6), then to the Dui palace (West 7), then to the Gen palace (Northeast 8), then to the Li palace (South 9), completing the full circuit. Finally, it ascends to rest at the Supreme Star, returning to the Ziwei palace. The entire route begins at the Kan palace and ends at the Li palace.

九宮之色：一白，二黑，三碧，四綠，五黃，六白，七赤，八白，九紫，以次遞推。今《曆書》承用不廢，亦猶行古之道也。

The nine palaces are each associated with a specific color: one white, two black, three green, four green, five yellow, six white, seven red, eight white, and nine purple, following this sequence in a cyclic manner. Today's almanac still continues to use this system of flying stars' colors without abandoning it, which can be considered as following the ways of ancient people.

張平子曰：“臣聞聖人明審律曆，以定吉凶。重之以卜筮，雜之以九宮。經天驗道，本盡於此。且河洛六藝，篇錄已定。後人皮傅，無所容竄。律曆卦候，九宮風角，數有徵效，世莫肯學，而競稱不占之書。譬猶畫工惡圖犬馬而好作鬼魅，誠以實事難形，而虛偽不窮也。”觀此可見易象數理之學，在漢時已就式微。宜晉唐而後之言《易》者，失所師承。致悉棄實事，而專尚空談也。

Zhang Heng (Pingzi) submitted a memorial saying, "I have heard that sages carefully examine the laws and calendars, using them to determine auspicious and inauspicious signs, and also employ divination and the Nine Palaces; observing celestial phenomena and verifying the heavenly way, the fundamental principles are all contained here. Moreover, the River and Luo diagrams and the Six Arts have long been established, and later generations can only superficially interpret them, unable to insert false works. The studies of laws and calendars, hexagram atmospheres, seasonal changes, the Nine Palaces, and wind divination all have verifiable evidence and practical effects, yet people are unwilling to learn them, instead competing to praise those books that cannot predict events. This is like a painter who dislikes painting dogs and horses but prefers to paint ghosts and spirits, because real things are difficult to depict, while false things can be freely fabricated without limit." From this, we can see that the branch of the Yi school focusing on images and numbers had already declined in the Han dynasty; it is no wonder that after the Jin and Tang dynasties, those who studied the Yi lost their proper transmission, abandoning the practical aspects entirely and focusing solely on empty talk.

洛書之數，四十有五。與河圖相合，適為一百。凡一百之數，列為平方，對角分之，則一得五十有五，一得四十有五，此亦數理之自然，非人意可為增損其間也。故圖之與書，相為表裡，不能分析。或謂伏羲得河圖而畫卦，禹受洛書而演範，是言其取用之方。伏羲則圖畫卦，以圖為主。禹受書演範，以書為主。非伏羲時未嘗有書，大禹未嘗有圖也。

The total of the Luo Shu is forty-five, and when added to the fifty-five of the He Tu, they make exactly one hundred. If the number one hundred is arranged into a square with ten rows and ten columns, and then divided along the diagonals, one part will be fifty-five and the other forty-five. This is naturally the case in numerical reasoning, and human intention cannot alter it. Therefore, the He Tu and the Luo Shu are interdependent and cannot be considered separately. Some people say that Fuxi obtained the He Tu and used it to draw the trigrams, while Yu the Great obtained the Luo Shu and used it to elaborate the "Hong Fan" (the Nine Codes). This is merely referring to their respective emphasis: Fuxi mainly used the He Tu to draw the trigrams, taking the diagram as the main focus; Yu the Great mainly used the Luo Shu to elaborate the "Hong Fan," taking the text as the main focus. It does not mean that the Luo Shu did not exist in Fuxi's time or that the He Tu did not exist in Yu the Great's time.

朱子《啟蒙》曰：河圖洛書之位與數，其所以不同何也？曰：河圖以五生數，統五成數，而同處其方，蓋揭其全以示人，而道其常，數之體也。洛書以五奇數統四偶數，而各居其所。蓋主於陽以統陰，而肇其變，數之用也。

Zhu Xi's "Yixue Qimeng" presents a question and answer: Why do the Yellow River Chart and the Luo Writing differ in their directions and numbers? Answer: The Yellow River Chart uses five generating numbers to govern five completing numbers, with generating and completing numbers sharing the same direction (one and six are both in the north, two and seven both in the south, and so on). This presents the whole in one place for people to see, discussing the constant way, which belongs to the 'essence' of numbers. The Luo Writing uses five odd numbers to govern four even numbers, with odd and even numbers each occupying their own positions (odds are in the four cardinal directions and the center, evens in the four corners). This uses yang to govern yin, thus initiating change, which belongs to the 'function' of numbers.

曰：其皆五居中何也？曰：凡數之始，一陰一陽而已矣。陽之象圓，圓者徑一而圍三。陰之象方，方者徑一而圍四。圍三者以一為一，故參其一陽而圍三；圍四者以二圍一，故兩其一陰而為二，是所謂參天兩地者也。三二之合則為五矣，此河洛之數所以皆以五為中也。

Question: Why do both the Yellow River Chart and the Luo Writing take five as the central number? Answer: The beginning of numbers starts with one yin and one yang. The image of yang is round, with a diameter of one and a circumference of three; the image of yin is square, with a diameter of one and a circumference of four. For the circumference of three, taking one yang gives three; for the circumference of four, taking half of one yin gives two. This is what the "Sayings of the Hexagrams" refers to as "three in heaven and two on earth." When three and two are added together, it makes five, which is why both the Yellow River Chart and the Luo Writing take five as their central number.

數與位皆三同而二異。蓋陽不可易，而陰可易。成數雖陽，固亦生之陰也。河圖之一二三四，各居其五象本方之外。而二四六八者，又各因五而得數，以附於其生數之外。洛書之一三七九，亦居其五象本方之外。而二四六八者，又各因其類以附於奇數之側。蓋中者為主，而外者為實。正者為君，而側者為臣，亦各有條而不紊也。

The numbers and positions of the two systems have three points in common and two points that differ. This is because the yang cannot be changed, while the yin can be altered; the formed numbers, although belonging to yang, originally arise from the yin generating numbers. In the Yellow River Chart, the numbers one, two, three, and four each reside outside their respective five-element positions, and the numbers two, four, six, and eight are derived by adding to the central five, thus attaching themselves to the outer layer of the generating numbers. In the Luo Writing, the numbers one, three, seven, and nine also reside outside their respective five-element positions, while the numbers two, four, six, and eight are arranged according to their categories, attached to the sides of the odd numbers. The central numbers are primary, while the outer numbers are actual; those in the central positions are like a sovereign, and those in the side positions are like ministers, each with order and without confusion. This is the distinction between the Yellow River Chart emphasizing generation and the Luo Writing emphasizing odd and even.

曰：其多寡之不同何也？曰：河圖主全，故極於十，而奇偶之位均。論其積實，然後見其偶贏而奇乏也。洛書主變，數極於九，而其位於實，皆奇贏而偶乏也。必皆虛其中焉，然後陰陽之數，均於二十，而無偏耳。

Question: Why are the numbers different in quantity? Answer: The Yellow River Chart governs the whole, so the numbers go up to ten, with equal numbers of odd and even positions. However, when considering the actual total sum, even numbers have an excess while odd numbers are deficient (the sum of the even numbers two, four, six, eight, and ten is thirty, while the sum of the odd numbers one, three, five, seven, and nine is twenty-five). The Luo Writing governs change, so the numbers only go up to nine. When looking at the actual positions, odd numbers have an excess while even numbers are deficient. Only when the middle five voids are excluded do the numbers on both yin and yang sides each total twenty, achieving balance without excess or deficiency.

曰：其序之不同何也？曰：河圖以生出之次言之，則始下，次上，次左，次右，以復於中，而又始終下也。以運行之次言之，則始東，次南，次中，次西，次北。左旋一周，而又始於東也。其生數之在內者，則陽居左而陰居上右也。

Question: Why are the orders different? Answer: The Yellow River Chart, if considered according to the sequence of the generating numbers, starts with the lower (Northern one), then the upper (Southern two), then the left (Eastern three), then the right (Western four), then back to the center (five), and then starts again from the lower. If considered according to the order of movement, it begins with the East, then the South, then the Center, then the West, then the North, rotating to the left for a full cycle, and then starts again from the East. As for the generating numbers within, the Yang numbers are arranged on the left, while the Yin numbers are arranged on the upper and right sides.

洛書之次，其陽數，則首北，次東，次中，次西，次南。其陰數，則首西南，次東南，次西北，次東北也。合而言之，則首北，次西南，次東北，次中，次西北，次西，次東北，而究於南也。其運行，則水克火，火克金，金克木，木克土。右旋一周，而土復克水也。是亦各有說矣。

The order of the Luo Writing is based on the yang numbers: first north (one), then east (three), then center (five), then west (seven), and finally south (nine). Based on the yin numbers: first southwest (two), then southeast (four), then northwest (six), and finally northeast (eight). Combined, the order is first north, then southwest, then northeast, then center, then northwest, then west, then northeast, and finally south. As for its movement, water overcomes fire, fire overcomes metal, metal overcomes wood, wood overcomes earth, and earth then turns back to overcome water. This also has its own reasoning.

曰：其七八九之數不同何也？曰：河圖六七八九，既附於生數之外矣，以陰陽老少進退饒乏之正也。其九者，生數一三五之積也。故自北而東，自東而西，以成於四之外。其六者，生數二四之積也。故自南而西，自西而北，以成於一之外。七則九之自西而南者也，八則六之自北而東者也，此又陰陽老少互藏其宅之變也。

Question: Why are the numbers seven, eight, and nine different? Answer: Since the numbers six, seven, eight, and nine in the Yellow River Chart are attached outside the generating numbers, they represent the correct examples of yin and yang's old and young, advancing and retreating, abundance and scarcity. Among them, nine is the product of the generating numbers one, three, and five added together, so it moves from north to east, then from east to west, forming outside the number four. Six is the product of the generating numbers two and four added together, so it moves from south to west, then from west to north, forming outside the number one. Seven is nine coming from the west to the south, and eight is six coming from the north to the east. This is another example of yin and yang's old and young mutually hidden within each other's dwellings.

洛書之縱橫十五，而七八九六迭為消長。虛五分十，而一含九，二含八，三含七，四含六，則參伍錯綜，無適而不遇其合焉。此變化無窮之所以為妙也。又曰：乾無十者，有坤以承之。坤無一者，有乾以首之。（按：洛書偶數，當曰由東南次西南，次西北，次東北。朱子蓋未知陰數逆行，從四起也。）

The Luo Square has numbers that add up to fifteen both across and down, with the numbers seven, eight, nine, and six alternating in their increase and decrease. When the central number five is removed and the ten is split apart, it becomes one containing nine, two containing eight, three containing seven, and four containing six. Thus, the numbers three and five are interwoven, and no matter how they are combined, they can always form a harmonious number, which is the wonderful aspect of its infinite transformations. Zhu Xi also said: Heaven does not have ten, but Earth takes over; Earth does not have one, but Heaven starts it off. (Huang's note: The even numbers in the Luo Square should be said to go from Southeast to Southwest, then to Northwest, and then to Northeast. Zhu Xi probably did not know that the yin numbers move in reverse order, starting from four.)

蔡氏元定曰：古人傳記，自孔安國、劉向父子、班固，固皆以為河圖授羲，洛書錫禹。關子明、邵康節，皆以十為河圖，九為洛書。蓋《大傳》既陳“天地五十有五”之數，《洪範》又明言“天乃錫禹洪範九疇”。而九宮之數，戴九履一，左三右七，二四為肩，六八為足，龜背之象也。

Cai Yuanping said: The biographies of ancient scholars, from Kong Anguo, Liu Xiang and his son Liu Xin to Ban Gu, all held that the Yellow River Chart was given to Fuxi and the Luo Writing was bestowed upon Da Yu. Guan Lang and Shao Yong also regarded the number ten as representing the Yellow River Chart and the number nine as representing the Luo Writing. Since the Great Treatise states "the numbers of heaven and earth total fifty-five," and the Hongfan chapter clearly says "heaven bestowed upon Yu the Hongfan nine categories," while the nine palace numbers are arranged with nine at the top and one at the bottom, three on the left and seven on the right, two and four as shoulders, six and eight as feet, exactly resembling the pattern of a turtle's shell. There is supporting text on both sides, so it was determined that ten represents the chart and nine represents the writing.