Earlier Heaven Taiji

清容袁氏桷《謝仲直易三圗序》曰：「上饒謝先生遯於建安，番昜吳生蟾往受易，而後出其圗焉。建安之學為彭翁，彭翁之傳為武夷君，而莫知所受，或曰托以隠秘，故謂之武夷君焉。始晁以道紀傳易統緒，截立疆理，俾後無以偽。至荊州袁溉道潔，始受於薛翁而易復傳，袁乃以授永嘉薛季宣士龍。始薛授袁時，嘗言河洛遺學多在蜀漢間，故士大夫聞是說者爭陰購之。」

Qingrong Ju Shi Yuan Jue wrote in "Xie Zhongzhi Yi San Tu Xu" that in Shangrao, the scholar Xie Sir lived in Jian'an. A man named Wu Chan from Poyang went to study the Book of Changes with him, and later Xie Sir revealed the three diagrams. The scholarly tradition of Jian'an came from Peng翁, who learned from the Master of Wuyi, but no one knows who the Master of Wuyi's teacher was. Some say it was a pseudonym to keep it secret, hence the name "Master of Wuyi." At first, Chao Shuozhi (styled Yidao) recorded the transmission of the Book of Changes, setting boundaries so later people could not fabricate. Later, Yuan Gai (styled Daojie) from Jingzhou learned from Xue翁, and thus the study of the Book of Changes was revived. Yuan Gai then passed it on to Xue Jixuan (styled Shilong) from Yongjia. When Xue翁 taught Yuan Gai, he said that the remaining teachings of the Yellow River Chart and the Luo Writing were mostly preserved in Sichuan and Shu Han regions. Therefore, when scholars heard this, they all competed to secretly spend money to acquire them.

「後有二張，曰行成，精象數；曰演，通於玄。最後朱文公屬其友蔡季通如荊州，復入峽，始得其三圗焉。或言洛書之傳，文公不得而見。今蔡氏所傳書訖不著圗，藏其孫抗，秘不復出。臨卭魏了翁氏嘗疑之，欲經緯而卒不可得。季通宋武夷，今彭翁所圗疑出蔡氏，惜彭不具本始。謝先生名字今不著，其終也世能道之。」

Yuan Jue continued to say: Later there were two men surnamed Zhang. One was Zhang Xingcheng, who was proficient in the principles of figures and numbers. The other was Zhang Yan, who understood the profound meanings. Finally, Zhu Xi entrusted his friend Cai Yuan ding (whose name was Jitong) to go to Jingzhou and further into the Three Gorges, only then obtaining those three diagrams. Some people say that Zhu Xi did not see the transmitted version of the Luo Writing. Now the book passed down by the Cai family does not include the diagrams, which are kept by Cai Kang, his grandson, and are not shown to others. Wei Le翁 from Linqiong once doubted this, wanting to systematically organize the diagrams and their explanations, but ultimately failed to do so. Cai Yuan ding was a person from the Wuyi area in the Song Dynasty. Now the diagram drawn by Peng翁 is suspected to have come from the Cai family, but unfortunately Peng翁 did not explain its origin. The name of Xie先生 is not written now, and his final fate is something that people can still talk about.

渭按：清容博雅君子也，君子之言信而有徵，故首著之。季通所得三圗，一為先天太極圗無疑矣，其二葢九宮圗與五行生成圗，而希夷未嘗名之曰洛書，故「或言洛書朱子不得見」也。（謝枋得字君直，信州弋陽人，宋末以江西招諭使知信州事，國亡變姓名遯入建陽，其後人稍識之，被徵不就。福建行省參政魏天佑送至大都，遂不食而死，事具《宋史》本傳。仲直即君直也。清容以謝拒元命為時所忌，故隠其名，復更其字。）

Hu Wei's commentary: Yuan Jue was a learned and refined gentleman. A gentleman speaks with credibility and evidence, so he first recorded Yuan Jue's words here. Cai Yuanding obtained three diagrams, one of which was the "Earlier Heaven Taiji Diagram," which is without doubt; the other two were probably the "Nine Palace Diagram" and the "Five Phases Generation Diagram." Chen Tuan himself did not call them the "Luo Writing," so Yuan Jue said, "Some people say that the transmission of the Luo Writing was not seen by Zhu Xi." (Xie Fangde, whose name was Junzhi, was from Xinyang in Xingzhou, and in the late Southern Song dynasty, he served as an envoy to pacify Jiangxi and administered the affairs of Xingzhou. After the fall of the Song, he changed his name and surname and fled to Jianyang. Later, the local people gradually recognized him, and the Yuan court summoned him, but he refused to take office. The provincial governor of Fujian, Wei Tianyou, sent him to Dadu, where he starved himself to death. His deeds are recorded in the "Biography of the Song History." The term "Zhongzhi" in the text refers to "Junzhi." Yuan Jue concealed Xie Fangde's name because he refused to accept the Yuan court's summons, and he also altered his given name.)

趙氏撝謙《六書本義》曰：「天地自然之圗，虙戲氏龍馬負圗出於榮河，八卦所由以畫也。《易》曰『河出圗，聖人則之』，《書》曰『河圗在東序』是也。此圗世傳蔡元定得於蜀之隠者，秘而不傳，雖朱子亦莫之見，今得之陳伯敷氏。嘗熟玩之，有太極函陰陽、陰陽函八卦之妙。」（撝謙字古則，餘姚人，宋宗室，別號老古先生。《名山藏》作趙謙，雲洪武初聘修《正韻》。）

Zhao Huiqian in his work "The Original Meaning of the Six Scripts" says: this "Diagram of Heaven and Earth in Nature" is the same diagram that appeared when the dragon horse carried it out of the Xing River during the time of Fuxi, upon which the Eight Trigrams were drawn. The "Great Treatise" of the "Zhou Changes" says "the diagram appeared from the Yellow River, and the sage took it as a model." The "Book of Documents, Gu Meng" says "the diagram was placed in the eastern hall," referring to this very diagram. This diagram is said to have been obtained by Cai Yuanding from a hermit in Sichuan, and it was kept secret and not passed on outside. Even Zhu Xi did not see it. Now I have obtained it from Chen Boyufu. I have pondered it repeatedly, and found the subtle aspect of the Taiji containing yin and yang, and yin and yang containing the Eight Trigrams. (Hu Wei's note: Zhao Huiqian, whose courtesy name was Guzhe, was from Yuyao, a descendant of the royal family of the Song dynasty, and was also known as Old Master Gugong. The "Name Mountains Collection" writes his name as Zhao Qian, stating that he was summoned in the early years of Hongwu to participate in compiling the "Hongwu Correct Pronunciation." )

楊氏時喬《周易全書》曰：「趙氏圗書，世競傳之為真圗書，靈寶許公誥、郴陽何公孟春尤篤信之。獨季彭山本云：朱子與蔡氏無書不講明，豈有秘不與言之理？」渭按：蔡氏所得之三圗，清容不言其形象，未知何如。據古則所傳以為蔡氏之所得，葢三圗之中此居其一，名曰先天圗，亦曰太極圗，取《參同契》之月體納甲、二用三五與九宮八卦混而一之者也。朱子發云「陳摶以先天圗授种放，三傳而至邵雍」，則康節之學實出於希夷，其所演以為先天古易者悉本此圗，可知也。

Yang Shi Qiao said in "The Complete Book of the Zhou Yi": Zhao Huiqian's diagram has been widely circulated among people, who claim it to be the true River Chart and Luo Writing. Xu Guo, a person from Lingbao, and He Mengchun, a person from Chenyang, especially believe in it wholeheartedly. Only Ji Ben, who was known as Pengshan, based on the original version said: Zhu Xi and Cai Yuan ding had discussions on every book, so how could there be any secret knowledge that they did not share with each other? Hu Wei's commentary: The three diagrams that Cai Yuan ding obtained, Yuan Jue did not describe their shapes, so we cannot know what they looked like. If we take Zhao Huiqian's diagram as one of the three diagrams that Cai Yuan ding obtained, then this diagram would be one of the three, called the "Earlier Heaven Diagram," also called the "Taiji Diagram." In reality, it is a combination of the "Cantongqi" moon body stem assignment, the two uses of three and five, and the nine palace trigrams. Zhu Zhen said, "Chen Tuan passed down the Earlier Heaven Diagram to Zhongfang, and it passed through three generations to Shao Yong." This shows that Shao Yong's learning indeed came from Chen Tuan. All the system he developed and called the "Earlier Heaven Ancient Yi" was based on this diagram, which can be definitely concluded.

後人謂之天地自然之圗，又謂之太極真圗。其環中為太極，兩邊白黑回互，白為陽，黑為陰。陰盛於北而陽起薄之，故邵子曰「震始交陰而陽生」，自震而離而兊以至於乾，而陽斯盛焉。震東北，白一分黑二分，是為一奇二偶；兊東南，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。

Later people called it the "Diagram of Heaven and Earth in Their Natural State," also known as the "True Diagram of the Taiji." In the center of the diagram is the Taiji, with black and white swirling around it, white representing Yang and black representing Yin. Yin energy is strong in the north, and Yang energy rises to approach it, so Shao Yong said, "The Zhen hexagram begins to interact with Yin, and Yang energy arises." From Zhen to Li to Dui up to Qian, Yang energy becomes increasingly strong. Zhen is in the northeast, with white taking one part and black taking two parts, which is one odd and two even; Dui is in the southeast, with white taking two parts and black taking one part, which is two odds and one even; Qian is in the due south, entirely white, which is three odds representing pure Yang; Li is in the due east, taking the black dot in the western white as two odds containing one even, so it is said, "the single point of Yin is in the center."

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 strong in the south, and yin energy comes to meet it, so Shao Yong said, "The Xun trigram begins to diminish yang energy and yin energy arises." From Xun to Kan to Gen and finally to Kun, yin energy becomes more powerful. Xun is in the southwest, with black occupying one part and white occupying two parts, which is one even and two odd; Gen is in the northwest, with black occupying two parts and white occupying one part, which is two even and one odd; Kun is in the due north, entirely black, which is three even and pure yin; Kan is in the due west, taking the white dot from the eastern black as two even containing one odd, so it is said, "the yang dot is in the middle opposite."

坎離為日月，升降於乾坤之間而無定位，納甲寄中宮之戊巳，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，天工乎？人巧乎？其自然而然之妙，非竊窺造化陰陽之秘者亦不能為也，但不可指以為伏羲之河圗耳。

The trigrams Kan and Li symbolize the sun and moon, rising and falling between Qian and Kun, without fixed positions. When applying the Najia stem assignment, they are寄托ed on the Central Palace's Wu Ji earth. Thus, they alternate between east and west, differing from the other six trigrams. The odd and even lines of the three-line trigrams correspond in order to the black and white dots on the chart; is this the work of heaven's skill or human ingenuity? This natural and subtle artistry could not be achieved without secretly understanding the secret of yin and yang from the creation. Yet we cannot simply call it the River Chart of Fuxi.

或問：「朱子謂希夷之學源出《參同契》，何以知其然乎？」曰：即其陰陽盛衰之數以推晦朔弦望之氣，而知其理有若合符節者矣。陽氣生於東北而盛於正南，震、離、兊、乾在焉，即望前三候陽息陰消之月象也；陰氣生於西南而盛於正北，巽、坎、艮、坤在焉，即望後三候陽消陰息之月象也。

Someone asked: How do we know that Chen Tuan's learning indeed comes from the "Zhou Yi Cang Tong Qi"? The answer is: By examining the proportions of yin and yang in the diagram and using them to deduce the climatic changes of new moon, full moon, waxing moon, and waning moon, one will find that the principles align perfectly like a matching seal. Yang energy originates in the northeast and reaches its peak in due south, with the four trigrams Qian, Dui, Li, and Zhen arranged on this side, corresponding exactly to the three days before the full moon when yang increases and yin decreases. Yin energy originates in the southwest and reaches its peak in due north, with the four trigrams Xun, Kan, Gen, and Kun arranged on this side, corresponding exactly to the three days after the full moon when yang decreases and yin increases.

陰極於北而陽起薄之，陰避陽，故回入中宮而黑中復有一點之白；陽極於南而陰來迎之，陽避陰，故回入中宮而白中復有一點之黑。葢望夕月東日西，坎離易位，其黑中白點即是陽光，白中黑點即是陰魄，東西正對，交注於中，此二用之氣所以納戊巳也。舉《參同》千言萬語之玄妙而括之以一圗，微而著，約而賅，丹家安得不私之為秘寶而肯輕出示人耶？

When yin reaches the north and becomes most vigorous, yang energy arises and approaches it, so yin retreats from yang, thus turning back into the Central Palace, resulting in a small white spot within the black. When yang reaches the south and becomes most vigorous, yin energy comes to meet it, so yang retreats from yin, thus turning back into the Central Palace, resulting in a small black spot within the white. Generally, on the full moon evening, the moon is in the east and the sun is in the west, with the trigrams Kan and Li exchanging positions. The white spot in the black is the sunlight, and the black spot in the white is the moon's essence. East and west are directly opposite, and their energies intersect and flow into the center, which is why the energies of Kan and Li are said to include the earth elements Wu and Ji. The myriad profound meanings of the "Supplement to the Secret Teachings" are all summarized in a single diagram, being both subtle and clear, concise and comprehensive. How could the alchemists not regard it as a closely guarded treasure, and be willing to easily reveal it to others?

自种放之後，儒者受此圗皆有所變通恢廓，而非復希夷之舊，唯蜀之隠者得其本真，私相授受以為丹家之要訣，篾叟、醬翁之徒是也。故雖朱子之博洽亦不得見，而必屬季通入峽求之，葢即酷愛《參同契》之意，以為坐談龍肉不如吃豬肉而飽也。其終不以為河圗而列諸經首者，葢以圗出希夷，源自伯陽，不若根柢《大傳》五十有五之數為得其正耳，非季通秘之而不與言也。

Since Zhong Fang, the figure who first introduced the diagram, scholars of the Confucian tradition who accepted it each interpreted it in their own way, so it was no longer in its original form as presented by Chen Tuan. Only those hermits in Sichuan province retained the original version, passing it on secretly among themselves as a key secret of alchemy. Even people like the old man selling bamboo strips or the old man selling sauce would possess it. Therefore, even a person as knowledgeable as Zhu Xi was unable to see it, and he had to entrust Cai Yuan Ding to search for it deep in the Three Gorges. This was actually due to his deep affection for the "Compendium of the Supreme Ultimate," thinking that merely talking about dragon meat was not as satisfying as actually tasting a piece of pork. In the end, Zhu Xi did not include this diagram among the Yellow River Chart in the beginning of his "Original Meaning of the Zhou Changes," because the diagram originated from Chen Tuan and was rooted in Wei Boyang, which was not as pure as basing his argument on the "Great Treatise" saying "the numbers of heaven and earth total fifty-five." It was not because Cai Yuan Ding concealed the secret from him.

趙氏仲全《道學正宗》曰：「古太極圗，陽生於東而盛於南，陰生於西而盛於北，陽中有陰，陰中有陽，而兩儀、而四象、而八卦，皆自然而然者也。」按：潛溪宋氏濂曰：「新安羅端良願（羅願字端良）作陰陽相含之象，就其中八分之以為八卦，謂之河圗；用井文界分九宮，謂之洛書。言出於青城山隠者，然不寫為象。」今觀趙氏此圗，正所謂陰陽相含、就中八分之以為八卦者，青城隠者之所授當亦如此，然不著陰陽分數，視古則為疎略。其不曰河圗而謂之古太極圗，何也？葢其時既從《啟蒙》以五十五數為河圗，而濂溪又自有所為太極圗者，故不名河圗，曰太極圗而加「古」以別之。

Zhao Zhongquan said in "Dao Xue Zheng Zhong": This "ancient Taiji diagram" shows that yang is born in the east and reaches its peak in the south, while yin is born in the west and reaches its peak in the north. Yang contains yin and yin contains yang, thus giving rise to the two principles, the four images, and the eight trigrams, all naturally formed. Hu Wei's commentary: Qianxi Xian Shen (Song Lian) said that Luo Yuan (Luo Yuan, whose courtesy name was Duanliang) created an image called "yin and yang containing each other." He divided it into eight parts as the eight trigrams, calling it the Yellow River Chart; then he divided it into nine sections with a grid pattern, calling it the Luo Writing. It is said that this came from a hermit in Qingcheng Mountain, but Luo Yuan did not draw it as a diagram. Now looking at Zhao Zhongquan's diagram, it is exactly the kind of image called "yin and yang containing each other," dividing it into eight parts as the eight trigrams. The hermit in Qingcheng Mountain must have also used this method, but he did not indicate how much yin and yang each occupied. Compared to Zhao Huiqian's diagram, it is more rough and simplified. It is not called the Yellow River Chart but "ancient Taiji diagram." Why is that? Probably because at that time, people had already followed Zhu Xi's "Yi Xue Ming Lun" to regard the number fifty-five as the Yellow River Chart, and Zhou Dunyi had his own "Taiji Diagram." Therefore, it is not called the Yellow River Chart, but instead called the Taiji Diagram, and adding the word "ancient" to distinguish it.

先天圗雖丹家修煉之訣，然亦必得其人而傳之，非其人則不傳也，故宋初唯种放、穆修受希夷之學，而他無聞焉。其後穆修授李之才，之才授邵雍，而天下始知有象數之學，即上所列二圗是也，亦曰太極圗，或謂之河圗。（有蔣公順字得之者，魏鶴山之門人也，著論以先天圗為河圗，五行生成數為洛書，戴九履一為太一下行九宮圗，鶴山云「此亦是一說」。）

The Earlier Heaven diagram is although a secret teaching for alchemists' cultivation, it must be transmitted only to suitable persons; it is not passed on to unsuitable ones. Therefore, at the beginning of the Song dynasty, only Zong Fang and Mu Xiu received Chen Tuan's learning, and there is no record of others. Later, Mu Xiu passed it on to Li Zhi Cai, who passed it on to Shao Yong. Thus, the world became aware of what is called the study of figures and numbers, referring to the two diagrams mentioned above, also called the Taiji diagram. Some people call it the Yellow River Chart. There was a Master Jiang Shun, whose name was De Zhi, a disciple of Wei Le翁 He Shan Master. He wrote articles advocating that the Earlier Heaven diagram should be regarded as the Yellow River Chart, the Five Phases generation numbers as the Luo Writing, and the "Wearing Nine and Wearing One" as the diagram of the Supreme Ultimate descending into the nine palaces. Wei Le翁 said, "This is also a kind of interpretation."

希夷之所授受盡於此矣，而說者謂此外別有河圗洛書，种放得之以傳李溉及許堅，不亦謾乎？溉惟有卦氣圗（見漢上《周易卦圗》，雲其說源於易緯《類是謀》），而堅無所著，不知其說云何。今觀范諤昌、劉牧之言，則皆祖述偽龍圗者也，與希夷之學相去徑庭矣。

Chen Tuan's lineage of transmission ends here, but some scholars say there are also the Yellow River Chart and the Luo Writing. Zheng Zong obtained them and passed them on to Li Gai and Xu Jian. Is this not a lie? Li Gai only had one "Diagram of Hexagram Qi" (see Zhu Zhen's "Zhou Yi Gu Tu," which says its theory comes from the Yi Wai text "Lei Shi Miao"). Xu Jian had no writings, so we do not know what his theory was. Now looking at the words of Fan Echang and Liu Mu, they all trace back to that forged "Longtu," which differs from Chen Tuan's learning by ten thousand eight thousand miles at least.

葢自种放既沒，天禧以後龍圗託名希夷，當世翕然宗之。邵子之書雖得真傳，而變通恢廓多所自得，良工不示人以樸，人莫測其所從來；其流傳蜀漢間者，又不過二三隠淪私為養生之訣，儒者無過而問焉，非若劉牧之徒造作文字、更相標榜以簧鼔天下也。故朱子雖力辨劉牧之非，而終不能脫龍圗之窠臼，逮乎晚年始覺其妄，遂令季通入峽購得三圗耳。蔡氏秘不肻出，及元末明初復見於世，雖無當於聖人之易，而源出《參同》，猶勝龍圗之怪誕。彼疑清容、二趙之言為不足信者，得吾說而思之，夫亦可渙然釋矣。

In general, since Zhongfang passed away, after the Tianxi period, the "Longtu" (Dragon Diagram) was falsely attributed to Chen Tuan. At that time, people universally regarded it as the standard. Although Shao Yong's writings received the true transmission, he made many variations and interpretations, expressing his own insights, like a master craftsman who does not show his unprocessed raw materials to others, so others could not understand its origin. As for the branch that spread in Sichuan and Hanzhong regions, it was merely a few reclusive scholars who used it as a method for health preservation, and Confucian scholars did not even pay attention to it, unlike Liu Mu and his group who fabricated texts and boasted to deceive the world. Therefore, although Zhu Xi strongly refuted Liu Mu's errors, he could not escape the framework of the "Longtu," and only in his later years did he realize its absurdity, so he sent Cai Yuanding to go to the Three Gorges to purchase the three diagrams. The Cai family kept them secret and did not reveal them, until they reappeared again in the Yuan and early Ming dynasties. Although these diagrams have nothing to do with the sages' "Yi," they originate from the "Cantongqi" (The Secret of the Supreme Ultimate), which is still better than the absurdity of the "Longtu." Those who doubt the reliability of Yuan Jue and Zhao Huiqian, Zhao Zhongquan, should think again after reading my discussion, and then they will understand clearly.

右論希夷先天圗。

The above is a discussion on Master Chen Tuan Xiyi's "Earlier Heaven Diagram."

希夷陳先生《龍圗序》曰：「且夫龍馬始負圗，出於羲皇之代，在太古之先也。今存已合之位尚疑之，況更陳其未合之數邪？然則何以知之？答曰：於夫子三陳九卦之義探其旨，所以知之也。況夫天之垂象，的如貫珠，少有差則不成次序矣，故自一至於盈萬，皆累累然如繫之於縷也。且若龍圗便合，則聖人不得見其象；所以天意先未合而形其象，聖人觀象而明其用。是龍圗者，天散而示之，伏羲合而用之，仲尼黙而形之。」

Chen Tuan, the Master Xiyi, in his "Preface to the Dragon Chart," said: The dragon horse first appeared carrying the chart in the time of Fuxi, even before the ancient times. Now the "already combined positions" that are preserved are still doubted by people, how much more so the "uncombined numbers"? Then how can we know it? The answer is: by exploring its meaning from Confucius' three explanations of the nine trigrams. Therefore, we come to know it. Moreover, the celestial phenomena shown by heaven are orderly like a string of pearls, and any slight error would disrupt the sequence. Therefore, from one to ten thousand, they are connected one after another like beads strung with a thread. Furthermore, if the Dragon Chart was originally combined, the sages would not have seen its image. Therefore, heaven first shows its image in an uncombined form so that the sages can observe it and understand its use. Thus, the Dragon Chart is something that heaven first breaks apart to show people, Fuxi combines it and applies it, while Confucius does not explicitly state it but reveals its meaning.

「始龍圗之未合也，惟五十五數。上二十五，天數也，中貫三五九，外包之十五，盡天三、天五、天九，並十五之位，後形一六無位，又顯二十四之為用也，茲所謂天垂象矣。」

The Dragon Diagram sequence continues to say: At first, when the Dragon Diagram had not yet been completed, there was only the number fifty-five. Above was the number twenty-five, representing the heavenly number; in the middle ran through three, five, and nine, while outside was surrounded by fifteen, thus occupying all the positions of three, five, nine, and fifteen in the heavens; afterwards, the numbers one and six were found to have no place, and then the number twenty-four appeared as a function — this is what is called "the heavens revealing the star patterns."

「下三十，地數也，亦分五位，皆明五之用也。十分而為六，形地之象焉；六分而成四象，地六不配。在上則一不配，形二十四；在下則六不用，亦形二十四。」

The *Longtu Xu* also says: The number thirty below refers to the earth numbers, which are also divided into five positions, all showing the function of the number 'five'. Dividing ten into six parts forms the image of the earth; six is then divided again into four images, and the earth number six has no corresponding match. Above is the number 'one' without a match, showing twenty-four; below is the number 'six' that is not used, also showing twenty-four.

「後既合也，天一居上為道之宗，地六居下為器之本，三乾地二地四為之用（本注：參一三乂天數合九，乾元用九也；兩二四地數合六，坤元用六也）。三若在陽則避孤陰，在陰則避寡陽（本注：成八卦者三位也。上則一三五為三位，二四無中正，不能成卦，為孤陰；下則六八十為三位，七九無中正，不能成卦，為寡陽。三皆不處，若避之也）。大矣哉，龍圗之變，岐分萬塗，今畧述其梗槩焉。」

The Dragon Diagram sequence also says: later, when combined, the number one of the heavenly numbers resides above, becoming the master of the "Dao"; the number six of the earthly numbers resides below, becoming the foundation of the "objects" (things). The number three, together with the second and fourth earthly numbers, fulfills their functions (original note: one and three combine to form nine, which is the "use of all nines" in the Qian trigram; two and four combine to form six, which is the "use of all sixes" in the Kun trigram). The number three, if it is on the yang side, avoids isolated yin; if it is on the yin side, it avoids isolated yang (original note: the three positions that can form the eight trigrams are as follows. The top positions one, three, and five are the three positions; the positions two and four lack the proper central and upright positions, and cannot form trigrams, called "isolated yin." The bottom positions six, eight, and ten are the three positions; the positions seven and nine lack the proper central and upright positions, and cannot form trigrams, called "isolated yang." The number three does not stay on either side, as if avoiding them). How great! The transformation of the Dragon Diagram branches out into countless paths; now we can only roughly outline its main points.

仲尼黙示三陳九卦：履，德之基（《序卦》次十，明用十，示人以辨上下）；謙，德之柄（次十五，明用十五，示人以裒多益寡）；復，德之本（次二十四卦，示氣變之始）；恆，德之固（下經次二卦，示形化之始）。

Confucius' secret revelation of the "Three Presentations and Nine Trigrams" includes: the Guan trigram represents the foundation of virtue (it is the tenth trigram in the "Sequence of Trigrams," indicating the number 'ten,' which guides people to discern the distinction between above and below); the Qian trigram represents the handle of virtue (it is the fifteenth trigram, indicating the number 'fifteen,' which guides people to reduce excess and increase deficiency); the Fu trigram represents the root of virtue (it is the twenty-fourth trigram, showing the beginning of the transformation of qi); the Heng trigram represents the stability of virtue (it is the second trigram in the lower section, showing the beginning of the generation and nurturing of form).

損，德之修；益，德之裕（此二卦示人以盛衰之端）。困，德之辨；井，德之地（此二卦示人以遷通之義）。巽，德之制（巽以行權，權者聖人之大用也，因事制宜、隨時變易之義備矣）。

The hexagram Sun indicates the cultivation of virtue, and the hexagram Yi indicates the abundance of virtue (these two hexagrams indicate the signs of prosperity and decline). The hexagram Gun indicates the discernment of virtue, and the hexagram Jing indicates the place of virtue (these two hexagrams indicate the principles of movement and adaptability). The hexagram Xun indicates the judgment of virtue (Xun is used for flexible action, "flexibility" is the greatest function of the sage, and the meaning of adapting to circumstances and changing with time is fully contained here).

按：李邯鄲淑《書目》有《易龍圗》一卷，陳摶撰，朱子以為假書，其序則錄於《宋文鑑》。劉靜修云：「龍圗之說未必出於劉牧之前。」（牧受能圗於范諤昌，諤昌傳自李溉、許堅，則其書固先牧而出。）呂伯恭從而誤信之，猶張敬夫為戴師愈所欺也。希夷未聞有書。（《宋史·隠逸傳》：陳摶好讀易，著《指玄篇》，言導養及還丹之事，而無所謂龍圗者。）

Hu Wei's commentary: The book "Book Catalogue" by Li Shu of Handan records a one-volume work titled "Yi Longtu," attributed to Chen Tuan. Zhu Xi considered it a forgery, and its preface is included in the "Song Wenjian." Liu Yin (Jingxiu) said: "The idea of 'Longtu' may not have appeared before Liu Mu." (Liu Mu received the Yellow River Chart from Fan Echang, who in turn received it from Li Gai and Xu Jian, so the book "Longtu" originally appeared before Liu Mu.) Lü Zuguan (Bogong) therefore mistakenly believed in it, just like Zhang Xi (Jingfu) was deceived by Dai Shiyu. Chen Tuan was not known to have written any books. (The "Song Shi · Yinyeti Zhuan" records that Chen Tuan liked to read the "Yi," and wrote "Zhi Xuan Pan," which discusses guiding breathing and alchemy, but there was no "Longtu.")

今觀其序之荒謬，則有不可勝言者。未合之數以為探三陳九卦之旨而得之，夫三陳九卦於「河出圗」之義有何干涉？比儗不倫，殊為可笑。

Now looking at the absurdity of that preface, it is simply beyond words. It claims that "the number of uncombined elements" is derived from the meaning of the three teachings and nine trigrams, but how does the three teachings and nine trigrams relate to the meaning of the Yellow River producing a map? This comparison is completely inappropriate and truly laughable.

又云「若龍圗本合，則聖人不得其象，故天必先散而示之」。夫以聖人之智，天即合而示之，聖人豈不能見其所分？且使天合而聖人果不能見其所分，當其散也，聖人又豈能合而用之邪？狂瞽之談，不足深辨。而宋景濂謂「序非圗南不能作」，甚矣儒者之易愚，非獨一呂伯恭也。餘姚黃先生云：「河出未合之圗，聖人合而用之，是伏羲畫卦又畫圗也。」一言破的，此書之偽妄灼然可覩矣。

The preface also says, "If the Dragon Chart was originally closed, the sage would not be able to perceive its image, so heaven must have first scattered it to show people." With the wisdom of a sage, even if heaven had closed it for him to see, could the sage not see how it could be separated? Moreover, if heaven had closed it and the sage truly could not see how to separate it, then when it was scattered, how could the sage use it again? Such reckless and foolish words are not worth discussing in detail. Yet Song Lian (Jinglian) even said, "This preface could not have been written by Chen Tuan (Tunan)." This shows how easily scholars are deceived; not only Lu Zuxian was deceived. Master Huang Zongxi from Yuyao said, "The Yellow River produced an open chart, and the sage later used it to close it again. That would mean Fuxi both drew the trigrams and the chart." With one sentence, he exposed the core issue, making the falseness and absurdity of this book clearly evident.

三陳九卦之義，其有合於龍圗者不可曉。雖然，彼既妄言之，吾不妨妄解之：葢天地未合之數、已合之位，一陳也；龍馬負圗，二陳也，所謂「散而示之」者也；伏羲復位五行生成之數、地上八卦之體，三陳也，所謂「合而用之」者也（詳見後）。仲尼不明言，而於《大傳》三陳九卦以微示其意，故曰「黙而形之」。作偽者之肺腸，容或如此。舊注泥九卦立解，則穿鑿無理甚矣。

The meaning of "San Chen Ji Guà" (Three Arrangements of Nine Trigrams) in relation to the "Longtu" (Dragon Chart) is really unclear. However, since the forgers have said it this way carelessly, I will also explain it carelessly: perhaps the "uncombined numbers" of heaven and earth and the "combined positions" are the first arrangement; the dragon horse carrying the chart out is the second arrangement, which is called "breaking it down to show to people"; and Fuxi reordering the numbers of the Five Phases and the forms of the eight trigrams on earth is the third arrangement, which is called "combining them for application" (see details later). Confucius did not clearly state this, but he mentioned the nine trigrams three times in the "Great Treatise" and subtly revealed this meaning, hence saying "silently shaping it." The forgers' thoughts may be like this. Old interpretations that fixate on the nine trigrams to make their theories are completely forced and meaningless.

《東都事略·儒學傳》：陳摶讀《易》，以數學授穆修，以象學授种放，放授許堅，堅授范諤昌。

The Dongdu Shi Lue: Ru Xu Transmission records that Chen Tuan studied the Zhou Changes, passing on the branch of knowledge concerning numbers to Mu Xiu, and the branch concerning images to Zhong Fang. Zhong Fang then passed it on to Xu Jian, and Xu Jian to Fan Echang. This is the transmission lineage of the scholarly study of the Zhou Changes in the Song dynasty.

按：馬令《南唐書》，許堅不知其家世，或曰晉長史穆之裔。形陋而怪，或寓廬阜白鹿洞，幘巾芒屩，短襴至骭，亦無齎裝，唯自負布囊，常括不解。每沐浴不脫衣，就澗浴，出而暵之。癖嗜魚，得大魚則全體而烹，不加酰鹽，熟而啖之。後或居茅山，或居九華，適意往返，人不能測。舊與錢若水善，若水後因轉挽江南，遇堅於簡寂觀，勉之以仕，則顰蹙不答。堅嘗至陽羨，人不之識。一日涉西津，臨波闊步若平地然，眾始神之，不知其所往雲。此亦是神仙面目，絶無儒者氣象，其自托於希夷之徒，有以也。

Hu Wei said: Ma Ling's "Southern Tang Book" says that Xu Jian's family background is unknown, and some people say he was a descendant of Xu MuZhi, a chief of the Eastern Jin Dynasty. He had an ugly and strange appearance. He once lived in the White Deer Cave on Mount Lu, wearing a cloth headscarf and straw shoes, wearing a short robe that only reached his knees, and had no luggage, only carrying a single cloth bag on his back, which was always tied tightly and never untied. Every time he bathed, he would not take off his clothes and go directly to the mountain stream to wash, and after washing, he would let it dry naturally. He had a strong fondness for fish, and when he caught a big fish, he would cook it whole, without adding vinegar or salt, and eat it when it was cooked. Later, he sometimes lived in Mao Mountain and sometimes in Jiuhua Mountain, wandering freely, and others could not understand his whereabouts. In his early years, he was friendly with Qian Ruoshui. Later, Qian Ruoshui was sent to Jiangnan to supervise the transportation of grain, and met Xu Jian at Jianji Guan, and advised him to take up an official post, but Xu Jian only frowned and did not answer. Xu Jian once went to Yangxian, where the local people did not recognize him. One day, he crossed the Xijin River, walking on the waves with large steps, as if walking on flat ground. Everyone then realized that he was extraordinary, and after that, his whereabouts were unknown. This was entirely the appearance of a immortal, with no trace of a Confucian scholar; he claimed to be a disciple of Chen Tuan (Xiyi), which was not without reason.

《宋史·儒林傳》：朱震有《漢上易解》，云：种放以河圗、洛書授李溉，溉傳許堅，堅傳范諤昌，諤昌傳劉牧，牧陳天地五十有五之數。其論圗書授受源委如此，葢莫知其所自雲。

The "Song History · Confucian Scholars Biography" records that Zhu Zhen wrote "Han Shang Yi Jie," in which he says that Zhong Fang passed on the Yellow River Chart and Luo Writing to Li Gai, who passed them on to Xu Jian, who passed them on to Fan Echao, who passed them on to Liu Mu. Liu Mu then deduced the fifty-five numbers of heaven and earth from them. Zhu Zhen's account of the transmission of the Yellow River Chart and Luo Writing is like this, but perhaps even he did not know where this account originally came from.

晁氏《讀書志》：《易證墜簡》一卷，天禧中毘陵從事范諤昌撰，自謂其學出於湓浦李處約、廬陵許堅。（陳氏《書錄解題》又有《易源流圖》一卷。）

Chao Gongwu's "Juzhai Reading Notes" records: "Yizheng Zuijian" in one volume, written by Fan Echang, an official in Piling during the Tianxi period of the Song Zhenzong emperor. He claimed his learning came from Li Chuyue of Penu, and Xu Jian of Luling. (Chen Zhenzun's "Zhaizhai Shulü Jieti" also records Fan Echang's "Yiyuanliutu" in one volume.)

按：《東都事略》言陳摶以象學授种放，放授許堅，象學者，河圗、洛書也。而朱震云：放以圗書授李溉，溉傳許堅，堅傳范諤昌，諤昌傳劉牧。晁公武云：諤昌自謂其學出於李處約、許堅。其說互異。溉與處約不知是一是二，諤昌又不言處約傳自誰氏，中間授受不甚分明，識者疑之。昔孟喜得《易》家候陰陽災變書，詐言師田生且死時枕喜厀，獨傳喜。又蜀人趙賓為《易》，持論巧慧，非古法，雲受孟喜，喜為名之。賓死，喜因不肻仞。（見《漢書·儒林傳》。）葢曲學授受之際，往往多依託隠諱，不可考究。李、許之學自附於种放，其亦田生獨傳孟喜不仞之類乎？

Hu Wei said: The "Eastern Capital Affairs" states that Chen Tuan passed on the "Xiangxue" to Zhong Fang, who passed it on to Xu Jian. The so-called Xiangxue refers to the Yellow River Chart and the Luo Writing. However, Zhu Zhen said that Zhong Fang passed the books to Li Gai, who passed them on to Xu Jian, then to Fan Echang, and finally to Liu Mu. Chao Gongwu also said that Fan Echang claimed his learning came from Li Chuyue and Xu Jian. These three accounts are contradictory. It is unclear whether Li Gai and Li Chuyue are the same person or two different individuals. Furthermore, Fan Echang does not mention from whom Li Chuyue learned his knowledge, so the transmission between generations is not clear. Even discerning people have long been suspicious. In the past, Meng Xi obtained a book on divination and yin-yang omens from the School of Changes, and he falsely claimed that his teacher Tian Wangsun had placed his head on his knee as he was dying, and passed the knowledge solely to him. There was also a man from Shu, Zhao Bin, who studied the School of Changes and presented his arguments in a clever and novel way, which did not conform to ancient methods. He claimed to have learned it from Meng Xi, and Meng Xi even supported him. But after Zhao Bin died, Meng Xi refused to acknowledge this. (See "The Book of Han: The School of Scholars." ) Generally speaking, in the transmission of esoteric learning, there are often fabrications and concealments, making it difficult to investigate. The learning of Li Gai and Xu Jian, who claimed to have inherited from Zhong Fang, might also be a trick similar to Meng Xi's claim of being the sole inheritor of knowledge from Tian Wangsun, which he later denied.

空山雷氏思齊《易圗通變》曰：由漢而唐，《易經》行世，凡經傳疏釋之外，未有及於圗書之文刋列經首者。迨故宋之初，陳摶圗南始創意推明象數，自謂因玩索孔子三陳九卦之義，得其逺旨，新有書述，特稱《龍圗》。離合變通，圗第一為龍馬圗，餘二十是全用《大傳》天地五十有五之數，雜以納甲，貫穿《易》理。內一圗謂形九宮，附一圗謂形洛書者，則盡去其五生數，秖起地六至地十，自釋十為用、十為成形，故《洪範》陳五行之用數語而已。及終其書，再出兩圗：其一形九宮者，元無改異，標為河圗；其一不過盡置列《大傳》五十有五之數於四方及中，而自標異謂為洛書，並無傳例言說。

Lei Siqi, the Dao Ren of Empty Mountain, in his work "The Communication of the Diagrams and Changes of the Book of Changes," said: From the Han dynasty to the Tang dynasty, the Book of Changes was popular in the world, and besides the classic text, the commentary and annotations, there was never any arrangement of the diagrams and writings in the beginning of the text. It was not until the early Song dynasty that Chen Tuan, who was called Tu Nan, began to create a new approach in explaining the images and numbers. He claimed that he had understood the meaning of the three times that Confucius mentioned the nine trigrams in the Great Treatise, and thus grasped the profound meaning. He then wrote a new work, and specially named it "Longtu." In the book, he discussed the combination, separation, and transformation of diagrams. The first diagram was the Dragon Horse Diagram, and the remaining twenty diagrams all used the number fifty-five of heaven and earth mentioned in the Great Treatise, combined with the method of assigning stems, to connect the meaning of the Zhou Changes. One of the diagrams was said to be a depiction of the Nine Palaces, and another was said to be a depiction of the Luo Writing, but he removed all five generating numbers, and only used the numbers from the earth six to the earth ten. He explained that ten was "use" and ten was "taking shape," so the "Discourses of the Duke of Yu" in the Book of Documents only briefly mentioned the use of the Five Phases. At the end of the whole book, there were two more diagrams: one was a depiction of the Nine Palaces, exactly as it was, but labeled as the Yellow River Chart; the other was simply arranging the fifty-five numbers from the Great Treatise in the four directions and the center, but he gave it a new name, the Luo Writing, without any explanation of the traditional system.

圗南之後，种放、許堅、李溉未及見其它有著述。若劉長民所親授之師如范諤昌，所著《大易源流》，其稱龍馬負圗出河，羲皇窮天人之際，復位五行生成之數、地上八卦之體，故老子自西周傳授孔子造《易》之原。天一正北，地二正南，天三正東，地四正西，天五正中央，地六配子，天七配午，地八配卯，天九配酉，地十配中寄於未，乃天地之數五十有五矣。因考其既以圗之前五數置於北南東西之正及中，復以後五數配子午卯酉及中，何也？夫子午卯酉非四方之正邪？地十配中雲寄於未，夫中抑有未邪？諦詳所置之數，正今圗所傳有四方而無四維者。是諤昌元不識圗南所以標異，特因《太玄》準《易》，取於《洪範》一水、二火、三木、四金、五土而然，鑿空無故，造端老子，増立怪論，以實圗南易置二七四九之位耳。

After Chen Tuan, Zhong Fang, Xu Jian, and Li Gai did not leave any other writings. As for Liu Mu (whose courtesy name was Changmin), who directly followed his teacher Fan Echang, in his book "The Origin of the Great Changes," he said: the dragon horse carried a diagram out of the Yellow River, and Fuxi thoroughly examined the relationship between heaven and humanity, reorganized the numbers of the Five Phases' generation and the structure of the eight trigrams on earth. Therefore, Laozi passed on the original principles of the "Zhou Yi" from the Western Zhou dynasty to Confucius. He also said that Tianyi (Heavenly One) is located in the due north, Di'er (Earthly Two) in the due south, Tian San (Heavenly Three) in the due east, Di Si (Earthly Four) in the due west, and Tian Wu (Heavenly Five) in the center. Di Liu (Earthly Six) corresponds to Zi (the rat), Tian Qi (Heavenly Seven) corresponds to Wu (the horse), Di Ba (Earthly Eight) corresponds to Mao (the rabbit), Tian Jiú (Heavenly Nine) corresponds to You (the rooster), and Di Shí (Earthly Ten) corresponds to the center but is assigned to Wei (the sheep). Altogether, these make up fifty-five numbers of heaven and earth. I therefore examined this: since he placed the first five numbers in the four cardinal directions and the center, why did he assign the remaining five numbers to Zi, Wu, Mao, You, and the center again? Are Zi, Wu, Mao, and You not the four cardinal directions? If he says Di Shí corresponds to the center but is "assigned to Wei," does the center have another position called Wei? Upon careful examination of his arrangement, it is exactly the same as the current diagram that has four cardinal directions but no four corners. This shows that Fan Echang completely misunderstood Chen Tuan's original intention. He merely followed Yang Xiong's "Tai Xuan" which compared the "Zhou Yi" and used the "Hong Fan" concept of one water, two fire, three wood, four metal, and five earth. He thus arranged them in the same way. He fabricated a baseless origin, traced the source back to Laozi, and added strange interpretations, merely to confirm Chen Tuan's change of the positions of Erqi (two seven) and Sijiu (four nine).

按：雷氏所言，則似親見《龍圗》之書者，然不著其形象，唯清江張氏《易象圗說》載之頗詳，謹列如左。

Hu Wei said: According to what Lei Siqi said, it seems he had personally seen the original book of the Longtu, but unfortunately he did not draw the shape of the diagram. Only Qingjiang Zhang Li's "Yi Xiang Tu Shuo" provides a fairly detailed account. Here follows a careful listing of it.

右龍圗天地未合之數。張氏曰：上位天數也。天數中於五，分為五位，五五二十有五；積一、三、五、七、九，亦得二十五焉。五位縱橫見三，縱橫見五；三位縱橫見九，縱橫見十五。序言「中貫三五九，外包之十五」者，此也。下位地數也。地數中於六，亦分為五位，五六凡三十；積二、四、六、八、十，亦得三十焉。序言「十分而為六，形地之象」者，此也。

The above is the "Number of Heaven and Earth Not Yet Combined" from the Dragon Diagram. Zhang Li says: The figure above represents the heavenly number. The heavenly number takes five as its center, divided into five positions. Five multiplied by five gives twenty-five; adding one, three, five, seven, and nine together also results in twenty-five. Five positions arranged in rows and columns show three, and arranged in rows and columns show five. Three positions arranged in rows and columns show nine, and arranged in rows and columns show fifteen. The "middle passes through three, five, and nine, and the outer encompasses fifteen" mentioned in the "Preface to the Dragon Diagram" refers to this. The figure below represents the earthly number. The earthly number takes six as its center, also divided into five positions. Five multiplied by six gives thirty; adding two, four, six, eight, and ten together also results in thirty. The "ten parts into six, forming the shape of the earth" mentioned in the "Preface to the Dragon Diagram" refers to this.

古龍圗天地已合之位。張氏曰：上位象也。合一、三、五為參天，偶二、四為兩地，積之凡十五，五行之生數也。即前象上五位：上五去四得一，下五去三得二，右五去二得三，左五去一得四，惟中□不動。序言「天一居上，為道之宗」者，此也。案《律歴志》云「合二始以定剛柔」，一者陽之始，二者陰之始。今則此圗，其上天○者，一之象也；其下地□者，之象也；其中天□者，四象五行也。左上一○太陽，為火之象；左上一○少陰，為金之象；左下一○少陽，為木之象；右下一○太陰，為水之象。土者，沖氣居中，以運四方，暢始施生，亦陰亦陽。右旁□三才之象，卦之所以畫三；左旁□四時之象，蓍之所以揲四。是故上象一二三四者，蓍數卦爻之體也。

This is the Dragon Diagram "the position where heaven and earth are united." Zhang Li says: the figure above refers to "the image." Combining the three odd numbers one, three, and five is called "ascending to heaven," while combining the two even numbers two and four is called "united earth." Adding them together gives fifteen, which is the generating numbers of the Five Phases. This corresponds to the five positions above the previous diagram: the five at the top minus four gives one, the five at the bottom minus three gives two, the five on the right minus two gives three, the five on the left minus one gives four, and only the center remains unchanged. The "heavenly one resides above, as the origin of the Dao" mentioned in the "Preface to the Dragon Diagram" refers to this. According to the "Book of Han · Laws and Calendars," it says "combine the two beginnings to determine hardness and softness." One is the beginning of yang, and two is the beginning of yin. Now looking at this diagram, the circle above belongs to heaven and represents the image of one; the square below belongs to earth and represents the image of two; the center square belongs to heaven and represents the four images and five phases. The circle above on the left represents the sun, symbolizing fire; the circle above on the left represents the lesser yin, symbolizing metal; the circle below on the left represents the lesser yang, symbolizing wood; the circle below on the right represents the great yin, symbolizing water. Earth is the harmonious qi in the center, used to transform the four directions, promote vitality, and distribute nourishment. It is both yin and yang. The square on the right represents the three elements, so the hexagram is drawn with three lines; the square on the left represents the four seasons, so divination with yarrow stalks counts four stalks at a time. Therefore, the numbers one, two, three, and four above represent the "body" of the yarrow stalks and the lines of the hexagram.

下位形也。九、八、七、六，金木水火之成數，中見地十土之成數也。即前象下五位，以中六分開，置一在上六而成七，置二在左六而成八，置三在右六而成九，惟下六不配而自為六。序言「六分而成四象，地六不配」者，此也。案七者蓍之圎，七七而四十有九；八者卦之方，八八而六十有四；九者陽之用，陽爻百九十二；六者陰之用，陰爻亦百九十二；十者大衍之數，以五乘十，以十乘五，而亦皆得五十焉。是故下形六七八九者，蓍數卦爻之用也。上體而下用，上象而下形，象動形靜，體立用行，而造化不可勝既矣。

The next part discusses "form." The numbers nine, eight, seven, and six represent the completed numbers of metal, wood, water, and fire respectively. In the middle, the number ten appears, which is the completed number of earth. This refers to the five positions below the previous diagram: by separating the six in the center, placing one on top of the six makes seven, placing two on the left of the six makes eight, placing three on the right of the six makes nine. Only the six at the bottom does not combine with other numbers and remains as six on its own. The "Dragon Diagram Preface" says, "Six divides into the four images, and the earth six does not combine," referring to this. Note: Seven is the circular number of the divining stalks; seven times seven equals forty-nine. Eight is the square number of the hexagram; eight times eight equals sixty-four. Nine is the function of yang; the total number of yang lines is one hundred ninety-two. Six is the function of yin; the total number of yin lines is also one hundred ninety-two. Ten is the number of the great derivation; five times ten and ten times five both result in fifty. Therefore, the six, seven, eight, and nine below represent the "function" of the divining stalks and the lines of the hexagram. The upper part is the body and the lower part is the function; the upper part is the image and the lower part is the form. The image initiates action while the form is static. Once the body is established, the function operates, thus the subtlety of creation is endless.

案：一二三四，天之象，象變於上；六七八九，地之形，形成於下。上下相重而為五行，則左右前後生成之位是也；上下相交而為八卦，則四正四隅九宮之位是也。今以前後圗參考，當如太乙遁甲陰陽二局圗：一二三四猶遁甲天盤在上，隨時運轉；六七八九猶遁甲地盤在下，布定不易。法明天動地靜之義，而前此諸儒未有能發其旨。

Zhang Li also explained: one, two, three, and four are the images of heaven, changing above; six, seven, eight, and nine are the forms of earth, solidifying below. The upper and lower layers combined form the Five Phases, which correspond to the positions where the generating numbers and the productive numbers are paired left, right, front, and back. The upper and lower layers interwoven form the Eight Trigrams, which correspond to the positions of the four cardinal directions and four intermediate directions in the nine palaces. Now, by comparing the front and back diagrams, it should be like the diagram of Taiyi Dunjia with yin and yang: one, two, three, and four are like the Heaven Plate of Dunjia above, moving with the sequence of time; six, seven, eight, and nine are like the Earth Plate of Dunjia below, fixed and unmoving. This method explains the principle of heaven moving and earth still, and none of the previous scholars among the Confucianists could grasp its meaning.

是故一在南，起法天象，動而右轉。初交，一居東南，二居西北，三居西南，四居東北，四陽班布居上右，四陰班布居下左，分陰分陽而天地設位。再交，一居東北，二居西南，三居東南，四居西北，則牝牡相銜而六子卦生。合是二變，而成先天八卦自然之象也。然後重為生成之位，則一六、二七、三八、四九陰陽各相配合，即邵子、朱子所述之圗也。三交，一居西北，二居東南，三居東北，四居西南，則剛柔相錯而為坎離震兊。四交，一居西南，二居東北，三居西北，四居東南，則右陽左陰而乾坤成列。合是二變，而成後天八卦裁成之位也。再轉則一復於南矣。《大傳》所謂「參伍以變，錯綜其數」，劉歆云「河圗洛書相為經緯，八卦九章相為表裡」，此其義也。

Therefore, the trigrams begin in the south, following the celestial patterns, moving and rotating to the right. The first transformation leads one to the southeast, two to the northwest, three to the southwest, and four to the northeast. The four yang numbers are arranged above and to the right, while the four yin numbers are placed below and to the left. Yin and yang are thus divided, and heaven and earth each take their proper positions. The second transformation leads one to the northeast, two to the southwest, three to the southeast, and four to the northwest. Thus, the male and female elements connect, and the six trigrams of Zhen, Kan, Gen, Xun, Li, and Dui are formed. Combining these two transformations, the natural appearance of the Earlier Heaven Trigrams is established. Then, by stacking them into the positions of generation, one and six, two and seven, three and eight, four and nine, yin and yang are paired accordingly, as described by Shao Yong and Zhu Xi in their diagram. The third transformation leads one to the northwest, two to the southeast, three to the northeast, and four to the southwest, resulting in the intermingling of firm and yielding elements to form Kan, Li, Zhen, and Dui. The fourth transformation leads one to the southwest, two to the northeast, three to the northwest, and four to the southeast, resulting in right yang and left yin, and the trigrams Qian and Kun being arranged in order. Combining these two transformations, the positions of the Later Heaven Trigrams are established. If the rotation continues, one returns again to the south. The "Great Treatise" says, "By combining three and five, transformations occur, and numbers are interwoven." Liu Xin says, "The Yellow River Chart and the Luo Writing form the meridians and parallels, while the Eight Trigrams and the Nine Chapters form the inside and outside." This is precisely the meaning.

右龍圗天地生成之數。張氏曰：此即前圗一二三四，天之象也，動而右旋；六七八九，地之形也，靜而正位。是故一轉居北而與六合，二轉居南而與七合，三轉居東而與八合，四轉居西而與九合，五十居中而為天地運行之樞紐。《大傳》言「錯綜其數」者，葢指此而言。錯者，交而互之，一左一右，三四往來是也；綜者，總而挈之，一低一昂，一二上下是也。分作二層看之，則天動地靜上下之義昭然矣。

The above is the Dragon Figure "The Number of Heaven and Earth Generating Each Other." Zhang Li says: This represents the numbers one, two, three, and four from the previous diagram, which belong to the heavenly aspect, moving and rotating to the right. The numbers six, seven, eight, and nine belong to the earthly aspect, being still and each occupying its proper position. Therefore, one moves to the north and combines with six, two moves to the south and combines with seven, three moves to the east and combines with eight, and four moves to the west and combines with nine. Five and ten are in the center, becoming the pivot of the movement of heaven and earth. The "错综其数" mentioned in the Great Treatise probably refers to this. "错" means crossing and exchanging; one on the left and one on the right, with three and four exchanging places. "综" means comprehending and lifting up; one low and one high, with one and two stacked vertically. If we divide it into upper and lower layers, the meaning of heaven moving and earth still, with the upper part representing the principle and the lower part the application, becomes clear at a glance.

右河洛縱橫十五之象。張氏曰：《洪範》「初一曰五行，次二曰敬用五事，次三曰農用八政，次四曰協用五紀，次五曰建用皇極，次六曰又用三德，次七曰明用稽疑，次八曰念用庶徵，次九曰向用五福、威用六極」，漢儒以此六十五字為洛書本文。而希夷所傳，則以此為龍圗三變，以生成圗為洛書本文，葢疑傳寫之誤而啟圗九書十之辨。今以二象兩易其名，則龍圗龜書不煩擬議而自明矣。

The above is the "cross and vertical fifteen forms" of the River Chart and the Lo Book. Zhang Li said: "The first of the nine sections in the 'Book of Documents: Hongfan' is the Five Phases, the second is the use of five things with reverence, the third is the use of eight administrative measures in agriculture, the fourth is the coordination with five celestial phenomena, the fifth is the establishment of the supreme principle, the sixth is the use of three virtues again, the seventh is the use of inquiry to resolve doubts, the eighth is the use of various signs with careful consideration, and the ninth is the use of five blessings and the use of six calamities." These sixty-five characters were considered by Han dynasty Confucian scholars to be the original text of the Lo Book. However, Chen Tuan (Xiyi) passed down a different interpretation, treating this diagram as the third transformation of the Dragon Chart, and instead regarded the diagram of generation as the original text of the Lo Book. This was probably due to his suspicion that there were errors in transmission, thus giving rise to the debate over "the River Chart has nine and the Lo Book has ten." Now, if we simply exchange the names of these two diagrams, then the Dragon Chart and the Tortoise Book no longer need to be discussed at length, and their distinctions will become clear naturally.

按：以上四圗並見《易象圗說》內篇，清江張理仲純所著也。其第一為天地未合之數，上位以五五為天數二十有五，下位以五六為地數三十，葢漢《律歴志》云「天之中數五，地之中數六」，故依託為此圗也。第二為天地已合之位，上位一上、二下、四左、三右、五居其中，即劉牧所謂上下未交之象也。及其已交，則天一下生地六，地二上生天七，天三左生（天地未合之數，一變也；已合之位，二變也；龍馬負圗，三變也）地八，地四右生天九，故下位六七八九十皆以生數乘中五而得之，即劉牧之洛書五行成數也。其縱橫十五之象，本龍圗三變，劉牧所謂龍馬負圗是也，雷氏以為河圗，張氏易其名曰洛書；天地生成之數，即范諤昌所謂羲皇復位五行生成之數、地上八卦之體者也，雷氏以為洛書，張氏易其名曰河圗。

Hu Wei said: The above four diagrams are all found in the Inner Part of "Yi Xiang Tu Shuo," written by Zhang Li, a man from Qingjiang, whose name is Zhongchun. The first diagram is called "the number before heaven and earth are combined." The upper part uses five fives to represent the heavenly number, totaling twenty-five, while the lower part uses five sixes to represent the earthly number, totaling thirty. This is probably based on the statement in the "Hanshu · Lüli Zhì" that "the central number of heaven is five, and the central number of earth is six," and thus this diagram was created by relying on these two sentences. The second diagram is called "the position after heaven and earth have combined." The upper part has one at the top, two at the bottom, four on the left, three on the right, and five in the center, which is the image of "heaven and earth not yet intersecting" as Liu Mu described. When they have intersected, heaven one gives birth to earth six, earth two gives birth to heaven seven, heaven three gives birth to earth eight, and earth four gives birth to heaven nine. Therefore, the numbers six, seven, eight, nine, and ten in the lower part are obtained by multiplying the generating numbers by the central number five. This is the five generating numbers of the Five Phases as described by Liu Mu in the Luo Shu. As for the "image of fifteen in both horizontal and vertical directions," it originally is the third transformation of the Longtu, that is, the Longma Fu Tu described by Liu Mu. Lei Siqi considered it to be the Hetu, but Zhang Li changed its name to Luo Shu. The "number of heaven and earth generating each other" is the diagram that Fan Ech Chang mentioned as "Fu Xi reordering the numbers of the Five Phases and the forms of the eight trigrams on the earth." Lei Siqi considered it to be the Luo Shu, but Zhang Li changed its name to the He Tu.

今考之本書，縱橫十五之象，九數各居一位，故序曰「天散而示之」；天地生成之數，一六、二七、三八、四九、五十皆耦居一方，故序曰「伏羲合而用之」。河圗之數四十有五，洛書五十有五，本象元自如此。張氏宗《啟蒙》，恐啟圗九書十之辨，故疑為傳寫之誤而兩易其名。然劉牧師范諤昌，諤昌師李溉、許堅，三傳弟子，一脈相承，使圗書果如張氏所列，而牧輒兩易之，是入室而操戈也，其何以取信於當世而學者翕然宗之乎？西山謂圗九書十出於劉牧之意見，非也，而張氏宗之亦過矣。

Now examining the book of the Dragon Chart: the image consists of fifteen rows and columns, with nine numbers each occupying a position, so the "Preface to the Dragon Chart" says "Heaven scatters and reveals it." The numbers of heaven and earth generating the numbers one and six, two and seven, three and eight, four and nine, five and ten are paired and reside in the same area, so the "Preface to the Dragon Chart" says "Fuxi combined and used them." The numbers of the Yellow River Chart total forty-five, and the numbers of the Luo Writing total fifty-five, and these two images have always been like this. Zhang Li respects Zhu Xi's "Awakening to the Study of Changes," fearing the controversy of "the Yellow River Chart has nine numbers, the Luo Writing has ten numbers," so he suspects it is a mistake in copying and swaps the names of the two. However, Liu Mu studied under Fan Echang, who studied under Li Gai and Xu Jian, and the disciples passed down through three generations in a continuous line. If the Yellow River Chart and Luo Writing were indeed arranged as Zhang Li proposed, and Liu Mu still swapped them, then he would be reversing the order and using the teacher's weapons against the teacher. Then on what basis could he gain the trust of the contemporary world and have scholars universally follow him? Cai Yuanding (Xishan) said that "the Yellow River Chart has nine numbers, the Luo Writing has ten numbers" comes from Liu Mu's personal speculation, which is incorrect; and Zhang Li following this view is also excessive.

易圗以白為陽、黑為陰，自《參同》水火匡廓始，其後先天太極圗亦然。而《龍圗》則獨用奇白偶黑之點，序曰「天之垂象，的如貫珠」，自一至於盈萬，皆累累然如繫之於縷也。因於點間為墨絲以聮絡之，使若貫珠然，思之可發一笑。前此未有此狀，圗出希夷之後，是亦一證也。雷氏不知《龍圗》源出湓廬，非華山道士所作，故以為希夷必不如此，而歸其罪於諤昌，此亦莫須有之獄。至以復位五行生成之數為老子自西周傳孔子，不知出何典記，鑿空造端，増立怪論，誠有如雷氏所譏者，諤昌直一妄人耳。語曰「不知其形視其景」，景曲則形必曲，觀諤昌之言，則李、許之為人從可知矣。

The diagram of the I Ching uses white dots for yang and black dots for yin, a practice that began with the Water and Fire Enclosure Diagram from the "Zhou Yi Cang Tong Qi" and continued in the Earlier Heaven Taiji Diagram. However, the "Longtu" uses only odd numbers of white dots and even numbers of black dots. The "Longtu Xu" says, "The heavens manifest their signs, clear as beads strung on a thread," from one end to the other, densely arranged as if strung on a single thread. Therefore, lines were drawn between the dots with ink to connect them, making them look like a string of beads. This is truly amusing. Before this, there was no such style, indicating that this diagram came after Chen Tuan. This is another piece of evidence. Lei Siqi did not know that the "Longtu" originated from the lineage of Pengpu and Luling, not from Chen Tuan, a Taoist priest from Huashan. Therefore, he concluded that Chen Tuan would never have done such a thing, and blamed Fan Echang entirely. This is another case of an unjust accusation without evidence. As for claiming that "rearranging the numbers of the Five Phases" was passed down from Laozi to Confucius from the Western Zhou dynasty, there is no source in any classic to support this. This is indeed, as Lei Siqi mocked, a fabrication by a foolish person. As the saying goes, "If you don't know the shape, look at its shadow." If the shadow is curved, the shape must be curved. Looking at Fan Echang's remarks, one can easily imagine the character of Li Gai and Xu Jian.

右論龍圗。

The above is part of the examination and verification of the authenticity of the book "Longtu."