I Ching Numerology Hidden Diagrams

三衢劉氏牧《鉤隠圗序》曰：夫《易》者，陰陽氣交之謂也。若夫陰陽未交，則四象未立；八卦未分，則萬物安從而生哉？是故兩儀變易而生四象，四象變易而生八卦，重卦六十四卦，於是乎天下之能事畢矣。夫卦者，聖人設之，觀於象也；象者，形上之應。原其本，則形由象生，象由數設，舍其數則無以見四象所由之宗矣。是故仲尼之贊《易》也，必舉天地之極數以明成變化而行鬼神之道，則知《易》之為書，必極數以知其本也。詳夫註疏之家，至於分經析義，妙盡精研，及乎解釋天地錯綜之數，則語惟簡略，與《繫辭》不偶，所以學者難曉其義也。今採摭天地奇偶之數，自太極生兩儀而下，至於復卦，凡五十五位，點之成圗，於逐圗下各釋其義，庶覽之者易曉耳。夫《易》道淵邈，雖往哲難窺於至賾，牧也蕞生祖述，誠媿其狂簡，然象有定位，變有定數，不能妄為之穿鑿耳，博雅君子試為詳焉。

Three衢 person Liu Mu, in his "Preface to the Diagrams of the Numbers in the Book of Changes," says: The so-called "Book of Changes" discusses the mutual interaction of yin and yang energies. If yin and yang do not interact, the four images cannot be established; if the eight trigrams are not differentiated, how could all things come into being? Therefore, the two principles (yin and yang) change to produce the four images, the four images change to produce the eight trigrams, and the eight trigrams overlap to form the sixty-four trigrams, thus encompassing all things and principles under heaven. The trigrams are established by the sages, derived from observing physical phenomena; the images are the responses of the transcendent. Tracing back to its origin, form arises from the images, and the images are determined by numbers. Without numbers, one cannot see the origin of the four images. Therefore, Confucius praised the "Book of Changes," necessarily citing the extreme numbers of heaven and earth to explain the principle of "producing change and acting like ghosts and spirits." This shows that the "Book of Changes" must be thoroughly studied in terms of numbers to understand its fundamental nature. Looking closely at the commentaries and annotations by scholars throughout history, they are indeed detailed and precise in distinguishing the text and analyzing the meaning; but when it comes to explaining the numbers of heaven and earth, they are vague and do not match the "Great Treatise," making it difficult for scholars to grasp the meaning. Now I have collected the odd and even numbers of heaven and earth, starting from the generation of the Taiji into the two principles, down to the Fu trigram, totaling fifty-five positions. I have drawn diagrams, and for each diagram, I have explained its meaning, hoping that readers will find it easy to understand. The way of the "Book of Changes" is profound; even the sages of previous generations found it difficult to reach its deepest aspects. I, Liu Mu, a humble person, dare to trace back and follow in the footsteps of the ancients, and I feel deeply ashamed of my own simplicity and lack of caution. However, the images have fixed directions, and changes have fixed numbers; I have not fabricated or invented them at will. I humbly request that learned and refined gentlemen examine and verify my work carefully.

晁氏《讀書志》曰：《易解》十五卷，劉牧長民撰。仁宗時，言數者皆宗之。慶歴初，吳秘獻其書於朝，優詔奬之，田況為序。又有《鉤隠圗》三卷，皆《易》之數也，凡五十五圗，並《遺事》九，有歐陽永叔序，而其文殊不類。

Chao Gongwu in "Juzhai Reading Notes" says: "The "Yi Xie" consists of fifteen volumes, written by Liu Mu (style name Changmin). During the reign of Emperor Renzong of the Song dynasty, those who discussed the principles of figures and numbers all held him in high esteem. In the early years of the Qingle era, Wu Mi presented this book to the court, and Emperor Renzong issued a commendatory edict. Tian Kuang wrote a preface for it. Liu Mu also wrote "Yi Shu Gou Yin Tu" in three volumes, which discusses the numbers of the "Zhou Yi", containing fifty-five diagrams, plus "Yi Wei Shi" with nine sections. The beginning of the volume has a preface by Ouyang Xiu (style name Yongshu), but the style of the text does not resemble Ouyang Xiu's handwriting."

渭按：歐公與劉牧同時，位尊望重，不信圗書，乃祖述《鉤隠》者之所憂，故其序託名歐公以欺世。千載而下，黨同作偽者之肺肝，猶如見之也。

Hu Wei said: Ouyang Xiu and Liu Mu were contemporaries, holding high positions and enjoying great prestige, yet they did not believe in the Yellow River Chart and the Luo Writing. This was precisely what those who followed the "Yi Shu Gou Yin Tu" were worried about. Therefore, that preface falsely attributed to Ouyang Xiu in order to deceive the world. A thousand years later, the hearts and minds of those who formed factions, fabricated texts, and conspired still appear as clearly as if seen with one's own eyes.

三百年來，學者唯知有《本義》卷首所列之圗書，而不復問其原委，故餘詳考《龍圗》及《鉤隠》，以著謬種所自出，使學者參觀而猛省焉，非為其有當於聖人之《易》而錄之也。

For three hundred years, scholars have only known the two diagrams of the Yellow River Chart and the Luo Writing listed at the beginning of Zhu Xi's The Original Meaning of the Zhou Changes, but they no longer inquire into their origins. Therefore, I have carefully examined The Dragon Diagram and The Hidden Numbers of the Zhou Changes to reveal the source of this absurd origin, so that scholars may compare and see it suddenly with clarity; it is not because they conform to the sage's Zhou Changes that they are included.

劉氏曰：昔虙犧氏之有天下，感龍馬之瑞，負天地之數出於河，是為龍圗者也。戴九履一，左三右七，二與四為肩，六與八為足，五為腹心，縱橫數之皆十五，葢《易繫》所謂「參伍以變，錯綜其數」者也。太皥乃則而象之，遂因四正定五行之數。以陽氣肇於建子，為發生之源；陰氣萌於建午，為肅殺之基。二氣交通，然後變化，所以生萬物焉、殺萬物焉。且天一生坎，地二生離，天三處震，地四居兊，天五由中，此五行之生數也。且孤陰不生，獨陽不發，故子配地六，午配天七，卯配地八，酉配天九，中配地十。既極五行之成數，遂定八卦之象，因而重之，以成六十四卦、三百八十四爻，此聖人設卦觀象之奧旨也。

Liu Mu said: In ancient times, when Fuxi ruled the world, he was inspired by the auspicious omen of the dragon horse. The dragon horse emerged from the Yellow River bearing the numbers of heaven and earth, which is called the Dragon Diagram. Its head was adorned with nine, its feet with one, its left side with three, and its right side with seven. The numbers two and four were on its shoulders, six and eight on its feet, and five in the center of its body. When added vertically and horizontally, they all totaled fifteen. This is what the Great Treatise refers to as "three and five intermingling to produce change, and their numbers being interwoven." Fuxi then took inspiration from it and established the numbers of the Five Phases according to the four cardinal directions. Because yang energy begins in the first month of the rat (Jian Zi), which is the source of all things; and yin energy begins in the seventh month of the horse (Jian Wu), which is the basis for the withering of all things. The interaction of these two energies leads to transformation, thus enabling the generation and nurturing of all things, as well as their destruction. Furthermore, the first heavenly number is Kan (Water), the second earthly number is Li (Fire), the third heavenly number is Zhen (Thunder), the fourth earthly number is Dui (Lake), and the fifth heavenly number is in the center. These are the generating numbers of the Five Phases. Moreover, isolated yin cannot produce life, and isolated yang cannot develop. Therefore, the hour of Zi (rat) is paired with the number six on earth, the hour of Wu (horse) with seven on heaven, the hour of Mao (rabbit) with eight on earth, the hour of You (rooster) with nine on heaven, and the center is paired with ten on earth. Once the numbers of the Five Phases are fully established, the images of the Eight Trigrams are determined. These trigrams are then overlapped to form the sixty-four hexagrams and the 384 lines, which is the profound meaning of the sages' method of setting up hexagrams and observing their images.

今龍圗其位有九，四象八卦皆所包蘊，且其圗縱橫皆合天地自然之數，則非後人能假偽而設之也。

Now this dragon diagram has nine positions, encompassing the four symbols and the eight trigrams. Moreover, both its horizontal and vertical additions conform to the natural numbers of heaven and earth. Clearly, it could not have been fabricated by later people.

河圖兩儀（此即龍圖天地已合之上位而虛其中也）。河圖四象（此即龍圖天地已合之下位而虛其中也）。河圖八卦（關子明則圖畫卦之說與此無異）。

The Yellow River Chart two principles diagram (this is the upper part of the Dragon Diagram "the position where heaven and earth have already joined," but with the middle left empty). The Yellow River Chart four images diagram (this is the lower part of the Dragon Diagram "the position where heaven and earth have already joined," also with the middle left empty). The Yellow River Chart Eight Trigrams diagram (Guan Lang, whose courtesy name is Ziming, followed the method of drawing trigrams based on the Yellow River Chart, and this diagram is essentially the same as this one).

劉氏曰：原夫八卦之宗，起於四象。四象者，五行之成數也。水數六，除三畫為坎，餘三畫布於亥上成乾；金數九，除三畫為兊，餘六畫布於申上成坤；火數七，除三畫為離，餘四畫布於已上成巽；木數八，除三畫為震，餘五畫布於寅上成艮。此所謂四象生八卦也。

Liu Mu said: Investigating the origin of the Eight Trigrams, they arise from the Four Symbols. The Four Symbols refer to the completed numbers of the Five Phases. The number for water is six; subtracting three lines for the Kan trigram, the remaining three lines are arranged at the亥 position to form the Qian trigram; the number for metal is nine; subtracting three lines for the Dui trigram, the remaining six lines are arranged at the申 position to form the Kun trigram; the number for fire is seven; subtracting three lines for the Li trigram, the remaining four lines are arranged at the巳 position to form the Xun trigram; the number for wood is eight; subtracting three lines for the Zhen trigram, the remaining five lines are arranged at the寅 position to form the Gen trigram. This is what is called "the Four Symbols give birth to the Eight Trigrams."

雷氏《易圗通變》曰：究核諤昌之取用，不過循納甲之緒餘，及五子歸庚之殊向，又謂天上八卦，坎離對中之外，移置乾兊坤於東，艮震巽於西。不謂五行之說多起於《易》後，而反引五行以為定卦之原，此又其敢於創異之大端也。是宜長民不獨増以五十五圗，又因諤昌坎離震兊四正之外，而以四成數同於四方，謂坎六退本卦三數，以餘三數三畫為乾；離七退本卦三數，以餘四數四畫為巽；震八退本卦三數，以餘五數五畫為艮；兊九退本卦三數，以餘六數六畫為坤，皆以數為畫，標為河圗，是不揣本而齊末。夫八卦各三畫，以剛柔生爻，未聞本卦止用三畫，而以其餘畫之多反分為別卦也。所陳之數抑配偶然，且以坎離震兊為四象，則尤非也。象本在未成卦之先，故曰四象生八卦也。

Lei Siqi in his book "Yitu Tongbian" says: Upon careful examination, what Fan Echang used was merely following the remnants of the Najia system and another peculiar approach called "Wu Zi Gui Geng." He also spoke of heavenly trigrams, placing Kan and Li in the center facing each other, while moving Qian and Dui to the east and Gen, Zhen, and Xun to the west. He did not consider that the theory of the Five Phases mostly emerged after the Zhou Yi, yet he used the Five Phases as the fundamental principle for determining the hexagrams, which was one of his bold innovations. It's no wonder Liu Mu not only added fifty-five diagrams but also followed Fan Echang's idea of placing Kan, Li, Zhen, and Dui in the four cardinal directions, assigning the four completed numbers to the four directions. He said that Kan six subtracts the three numbers from the original hexagram, using the remaining three numbers and three lines to form Qian; Li seven subtracts the three numbers from the original hexagram, using the remaining four numbers and four lines to form Xun; Zhen eight subtracts the three numbers from the original hexagram, using the remaining five numbers and five lines to form Gen; Dui nine subtracts the three numbers from the original hexagram, using the remaining six numbers and six lines to form Kun. He used numbers to represent lines and called it the Yellow River Chart, which was not considering the fundamental principles but only seeking to make the details neat. Each trigram has three lines, which are generated by the combination of yin and yang. There is no record of using only three lines for the original hexagram, yet he separated the extra lines into other hexagrams. The numbers he presented were merely forced together. Moreover, by treating Kan, Li, Zhen, and Dui as the Four Symbols, he made an even greater mistake. The images originally existed before the hexagrams were formed, so the Great Treatise says, "The Four Symbols give rise to the Eight Trigrams."

按：希夷天地自然之圗宗《參同契》，用乾南坤北、離東坎西之位，而《鉤隠》仍以坎離震兊居四正，乾坤艮巽居四隅，即此一端，亦足以證《龍圗》之本不出於希夷矣。

Hu Wei remarked: Chen Tuan (Xiyi) 's "Diagram of Heaven and Earth in their Natural State" followed the "Threefold Concordance" of the Zhou Changes, adopting the orientation of Qian in the south, Kun in the north, Li in the east, and Kan in the west. However, the "Diagram of Hidden Calculations of the Changes" still placed Kan, Li, Zhen, and Dui in the four cardinal directions, and Qian, Kun, Gen, and Xun in the four corners. Based on this alone, it is sufficient to prove that the "Dragon Diagram" was not originally created by Chen Tuan.

洛書五行生數。洛書五行成數（以上二圖合之，即是羲皇復位五行生成之數）。

The Luo Writing Five Phases Generating Numbers Diagram. The Luo Writing Five Phases Forming Numbers Diagram (the above two diagrams together are the Five Phases Generating Numbers as reorganized by Fuxi according to Fan Echanghai's account).

劉氏曰：或問：洛書一曰水，二曰火，三曰木，四曰金，五曰土，則與龍圗五行之數之位不偶者何也？答曰：此謂陳其生數也。雖則陳其生數，乃是已交之數，下篇分土王四季，則備其成數矣。且夫洛書九疇，惟出於五行之數，故先陳其已交之生數，然後以土數足之，乃可見其成數也。

Liu Mu said: Someone asked: The Luo Writing says "one is water, two is fire, three is wood, four is metal, five is earth," but this does not match the positions of the five elements' numbers in the Dragon Figure. Why is that? The answer is: Here it refers to the arrangement of the initial numbers. Although these are initial numbers, they are already numbers after yin and yang have combined; in the next section, which discusses the rule of earth over the four seasons, the completed numbers are supplemented. Moreover, the nine categories of the Luo Writing are derived from the numbers of the five elements, so first arranging the initial numbers that have already combined yin and yang, and then supplementing with the numbers of earth, allows one to see the completed numbers.

《書》之九疇，惟五行是包天地自然之數，餘八法皆是禹參酌天時人事類之耳，非龜所負之文也。今詳《洪範五行傳》，凡言災異，必推五行為之宗；又若「鯀無聖德，汨陳五行，是以彝倫攸斁」，則知五行是天垂自然之數，其文負於神龜，餘八法皆大禹引而伸之，猶龍圗止負四象八純之卦，餘重卦六十四皆伏羲仰觀俯察，象其物宜，伸之以爻象也。或曰：既云龍圗兼五行，則五行已具於龍圗矣，不應更用龜書也。答曰：雖兼五行，有中位而無土數，惟四十有五，是有其象而未著其形也，唯四象八卦之義耳；龜書乃具生成之數五十有五矣。《易》者包象與數，故聖人資圗書而作之也。

Liu Mu also said: The nine categories in the "Nine Categories" of the "Book of Documents: Hongfan" only the category of the Five Phases contains the natural numbers of heaven and earth. The other eight categories were classified and developed by Da Yu according to the heavenly timing and human affairs, and they were not originally on the turtle's back. Looking closely at the "Hongfan Wu Xing Zhuan," whenever it discusses calamities and anomalies, it always traces back to the Five Phases. For example, "Gun lacked virtue and recklessly manipulated the Five Phases, thus causing the constant decline of the way." This shows that the Five Phases are the natural numbers revealed by heaven. This passage is what the divine turtle carried, while the other eight categories were elaborated by Da Yu. This is like the Dragon Chart which only carries the Four Symbols and the eight pure trigrams, while the other sixty-four trigrams formed by overlapping are all derived by Fuxi observing the heavens above and examining the earth below, imitating the suitability of all things, and using the yao symbols to elaborate further. Someone asked: If the Dragon Chart includes the Five Phases, then the Five Phases are already contained within the Dragon Chart, so why use the turtle script? The answer is: Although the Dragon Chart includes the Five Phases, it only has the central position and lacks the earth numbers, making the total number forty-five. This is having the image but not the form, possessing only the meaning of the Four Symbols and the Eight Trigrams. The turtle script fully contains the generating numbers, fifty-five in total. The "Zhou Yi" includes both the images and numbers, so the sages created it based on the Yellow River Chart and the Luo Writing.

雷氏《易圗通變》曰：書之九疇，各疇自有成數，如一五行、二五事，猶或得以其數從而強推引之；至於五皇極，則已不可指實之為何物何事，而甚則九五福而附以六極，則將計九乎？計五福而兼計六極乎？皇極謂大中，而六極者，其極又可謂中乎？皇極本非物非事，故可指之為中，今徒實以五點，而五點者乃遂得為中乎？疇自一至九，界界然各存本有之數，不知何自而可以合於五十有五之數，強謂合者，葢其人之妄也。若其後而至於以書謂圗、以圗謂書者，又妄人中之妄人也。此餘特謂圗則有數可通，而書則有疇類可數，而不可布之以為圗也。

Lei Siqi in "Yi Tu Tong Bian" says: The nine categories of the "Hong Fan" each have their own fixed numbers. For example, the first category is the Five Phases, the second is the Five Affairs, and perhaps one can roughly infer them by their numbers. However, the fifth category, the Great Center, is already unclear as to what it refers to, whether it is an object or an event. Even more serious is the ninth category, which talks about the Five Blessings and adds the Six Extremes. Should this be counted as nine, or as five blessings plus six extremes? The meaning of "Great Center" is "Great Center," but the "extreme" in "Six Extremes," can it also be called "Center"? Originally, the Great Center was neither an object nor an event, so it could be called "Center." Now, however, people force it to be represented by five points. Can five points be called "Center"? The nine categories, from one to nine, each have clear original numbers. But how can they be combined to make fifty-five? To say they can be combined is the delusion of the person. Even worse, later people have confused the Luo Book with the Yellow River Chart and the Yellow River Chart with the Luo Book. These are the delusions of deluded people. Therefore, I specially say: the Yellow River Chart has numbers that can be connected, but the Luo Book can only be counted in categories, and cannot be arranged into a diagram.

按：劉牧謂洛書與河圗並出於伏羲之世，兼則之以畫卦，而五行之數未顯，故禹復法之以陳九疇。然一為五行，二為五事，以至九十為福極，禹何以知之？故又為之說曰「惟五行是天垂自然之數，餘八者皆禹自類之」，意謂五事以下禹從五行推演而得之也。然五事、皇極、庶徵、五福、六極，劉向嘗以此附會於五行，猶可通也，其餘則絶無交涉矣。六十四卦不離乎八卦，河圗具八卦之象，則六十四卦包在其中；若九疇五事以下，未見五行中具有此義也。禹乃鑿空而増之，以綴於五行之後，則幾同駢拇枝指矣，是亦不可以已乎？且經云「天錫禹九疇」，不言「錫禹五行」，九疇皆天之所設，非人之所為，謂八者禹自類之，妄也。

Hu Wei said: Liu Mu believed that the Luo Writing and the Yellow River Chart both originated in the time of Fuxi, and Fuxi combined both to draw the trigrams. However, the numbers of the Five Phases were not yet clear, so Yu the Great later took the Luo Writing as a model and explained the nine categories. But the first category was the Five Phases, the second was the Five Actions, and the ninth dealt with blessings and extremes. How did Yu the Great know to divide it this way? Therefore, Liu Mu tried to explain that "only the Five Phases are the natural numbers revealed by Heaven, while the other eight categories were classified by Yu the Great himself." This meant that the Five Actions and below were derived by Yu the Great from the Five Phases. However, Liu Xiang once connected these five items to the Five Phases, and this could bewith difficulty explained. The rest, however, had nothing to do with the Five Phases. The sixty-four trigrams are based on the eight trigrams, and the Yellow River Chart contains the images of the eight trigrams, so the sixty-four trigrams must be included within it. But the nine categories, from the Five Actions onwards, do not seem to be contained within the Five Phases. Yu the Great added them out of nowhere and attached them after the Five Phases, which is like having extra toes on the feet or extra fingers on the hands. Such a claim should be abandoned. Moreover, the classic text says "Heaven bestowed the nine categories upon Yu the Great," not "the Five Phases." The nine categories were all set by Heaven, not by human effort. To say that eight of them were classified by Yu the Great himself is pure fabrication.

《易乾鑿度》曰：陽起而進，陰動而退，故陽以七、陰以八為彖。《易》一陰一陽，合而為十五之謂道，陽變七之九，陰變八之六，亦合於十五，則彖變之數若一。陽動而進，變七之九，象其氣之息也；陰動而退，變八之六，象其氣之消也。故太一取其數以行九宮，四正四維皆合於十五。五音、六律、七始，由此作焉。

The Yi Wei · Qian Zao Du says: When the yang energy begins to rise and move forward, and the yin energy begins to move and retreat, therefore seven is used to represent yang and eight to represent yin as the彖 numbers. The Zhou Yi speaks of one yin and one yang, combining to make fifteen, which is called "the Dao." Yang changes from seven to nine, and yin changes from eight to six, combining also to make fifteen, showing that the彖 numbers and the changing numbers are consistent. When yang energy moves forward and increases, changing from seven to nine, it symbolizes its energy growing; when yin energy moves backward and decreases, changing from eight to six, it symbolizes its energy diminishing. Therefore, the Taiyi uses these numbers to move through the nine palaces, with the four cardinal directions and the four corners adding up to fifteen. The five tones, six laws, and seven beginnings are all made based on this.

《隋經籍志》：宋大明中始禁圗讖，梁天監以後又重其制，及髙祖受禪，禁之愈切。煬帝即位，乃發使四出，搜天下書籍與讖緯相涉者皆焚之，為吏所糾者至死，自是無復其學，秘府之內亦多散亡。今按志有鄭氏注《易緯》七卷，而無《乾鑿度》；唐《藝文志》有宋均注《易緯》九卷，亦無《乾鑿度》。孔穎達義疏、章懷《後漢書注》所引《乾鑿度》，葢即秘府散亡之餘，僅存於《易緯》九卷中者也，是為可信。宋時復有《乾鑿度》二卷，晁氏云「舊題蒼頡修古籀文，鄭氏注」。案唐四庫書目有鄭玄注《詩》《書》緯及宋均注《易緯》，而無此書，其中有不可曉者，獨九宮之法頗明。

The Sui Shu · Jing Ji Zhi records that during the reign of Emperor Ming of the Southern Song Dynasty, the government began to ban diagrams and prophecies. After the reign of Emperor Wen of Liang, the ban was intensified, and by the time Emperor Gaozu of Sui took the throne, the prohibition became even stricter. When Emperor Yang of Sui ascended the throne, he sent envoys to search everywhere for any books connected with prophecies and had them all burned. Those accused by officials were even executed. From then on, this field of study ceased to be passed down. The books kept in the imperial library were mostly lost. Now, checking the entries in the Sui Zhi, there is a record of Zheng Xuan's commentary on the Yi Wei with seven volumes, but there is no entry for the Qian Zao Du. The Xin Tang Shu · Yi Wen Zhi records that there were nine volumes of the Yi Wei with commentary by Song Jun, but again there is no entry for the Qian Zao Du. The commentary on the Yi by Kong Yingda in the Zhou Yi Zhengyi and the annotations by Prince Li Xian of Changhui on the Later Han Shu both cite the Qian Zao Du. These are probably the remaining parts of the nine volumes of the Yi Wei after the imperial library's collection was dispersed. These are considered reliable. Later, in the Song Dynasty, another version of the Qian Zao Du with two volumes appeared. Zhao Gongwu said that it was "originally attributed to Cang Jie, written in ancient cursive script, with commentary by Zheng Xuan." Checking the catalog of the four libraries of the Tang Dynasty, there are entries for Zheng Xuan's commentaries on the Shi Wei and Shu Wei, and for Song Jun's commentary on the Yi Wei, but there is no entry for this book. The text has some parts that are unclear, but the section on the nine palace method is quite clear.

愚謂此贗書無疑。圗緯興於哀平之際，《乾鑿度》縱出其先，亦當在漢世，而題曰蒼頡修，將誰欺乎？「太一取其數以行九宮」，章懷所引鄭注盡之矣，無七八九六進退消息、彖易彖變之說也，是必蒼頡二卷中語。葢其時《洞極經》出，十圗九書早已萌芽，故劉牧之徒偽造《乾鑿度》二卷，以《參同契》七八九六之文竄入於其中，以見此河圗之象出自西漢，逺有端緒，使人不敢動搖。陳氏所云「殘闕不完，於偽之中又有偽焉」者也。而昧者更題云蒼頡修，若此書為黃帝繼伏羲而作，適以自獻其偽，可不謂大愚乎？

I am certain that this must be a forgery. Diagrams, prophecies, and esoteric writings emerged during the reigns of Emperor Ai and Emperor Ping of the Han dynasty. Although "The Qian Zao Du" appeared even earlier, it should have originated in the Han dynasty, yet it is attributed to "Cang Jie," which is meant to deceive whom? The statement "The Taiyi uses these numbers to operate within the nine palaces" has already been fully explained by Li Xian's quotation of Zheng Xuan's commentary. There is no mention of the concepts of seven, eight, nine, and six advancing and retreating, nor the ideas of "tuan shu" and "tuan bian" in his explanation. This indicates that these statements must have been added later in the two-volume forgery attributed to Cang Jie. Probably at the time when "The Dong Ji Jing" was published, the idea of "the River Chart has ten numbers, the Luo Writing has nine numbers" had already begun to take shape. Therefore, the school of Liu Mu fabricated "The Qian Zao Du" in two volumes, inserting the words about seven, eight, nine, and six from "The Zhou Yi Cang Tong Qi" into it, in order to show that the image of the River Chart originated in the Western Han dynasty, thus appearing to have a long and continuous source, making people hesitate to doubt it. This is exactly what Chen Zhenzun referred to as "incomplete and further forged within the realm of forgeries." Furthermore, the ignorant ones even further attributed it to "Cang Jie's revision," as if the book was composed in the Yellow Emperor's era, following Fuxi. This is precisely how they expose their own forgery, which is not a great stupidity?

《子華子》曰：二與四抱九而上躋也，六與八蹈一而下沈也。戴九而履一，據三而持七，五居中宮，數之所由生；一從一橫，數之所由成。胃之實也，神氣之守也。故曰：天地之數，莫中於五，莫過於五。

The Zhihua Zi says: Two and four embrace nine and rise upward, while six and eight step on one and descend downward. The head bears nine, the feet rest on one, the left holds three, the right holds seven, and five resides in the Central Palace. This is where numbers originate. One vertical and one horizontal intersect, which is the method by which numbers are formed. It is like the fullness within the abdomen, the dwelling place guarded by spirit and energy. Therefore, it is said that the numbers of heaven and earth have none more central than five, nor any more transcendent than five.

晁氏《讀書志》曰：《子華子》十卷，其傳云子華子程氏名本，晉人也，劉向校定其書。按《莊子》稱子華子見韓昭侯，陸德明以為魏人，既不合；又《蓺文志》不錄，且多用字說，謬誤淺陋，殆元豐以後舉子所為耳。

Chao Gongwu in "Junzai Dushu Zhi" says: "The book "Zihua Zi" has ten volumes. The preface states that Zihua Zi was a man surnamed Cheng, named Ben, from the state of Jin. Liu Xiang once corrected this book. However, according to the "Zhuangzi," Zihua Zi visited Han Zhaohou, and Lu Deming thought he was from the state of Wei, which conflicts with the statement that he was from Jin. Moreover, the "Han Shu · Yiwen Zhi" did not record this book, and the book often uses the method of interpreting characters as in Wang Anshi's "Zishuo," which is erroneous and superficial. Therefore, it is likely that this book was written by candidates taking the imperial examinations after the Yuanfeng period of Emperor Shenzong of the Song dynasty."

朱子《偶讀漫記》曰：會稽官書版本有《子華子》，雲是程本字子華者所作，孔子所與傾葢而語者。以予觀之，其詞故為艱澀而理實淺近，其體務為髙古而氣實輕浮，但如近年後生巧於摹擬變換者所為，不惟決非先秦古書，亦非百十年前文字也。如論河圗，巧亦甚矣，惟其巧甚，所以知其非古書也。或云王銍性之、姚寛令威多作贗書，二人皆居越，恐出其手。又曰：但觀其書數篇與前後三序皆一手文字，其前一篇托為劉向，而殊不類向它書。

Zhu Xi wrote in his "Evening Notes on Reading": In the official printed editions of books from Kuaiji, there is a text called "Zi Hua Zi", which is said to have been written by Cheng Ben (whose courtesy name was Zi Hua), the person with whom Confucius once stopped his carriage to talk. In my opinion, its language is deliberately written in a difficult and obscure style, yet its meaning is shallow. Its style strives to be ancient and refined, but its character is actually light and superficial. It is merely like the works of recent young men who are good at imitating and changing styles. It is not at all an ancient text from the Pre-Qin period, nor is it from a hundred years ago. For example, the text's discussion on the Yellow River Chart is very clever; precisely because it is too clever, we know it is not an ancient text. Some people say that Wang Zhi (whose courtesy name was Xingzhi) and Yao Kuan (whose courtesy name was Lingwei) often fabricated false texts. Both of them lived in the region of Yue. This text is probably the work of one of them. Zhu Xi also said: Just look at the main body of this text and the three prefaces before and after, all written by the same person. Among them, the first preface falsely attributes it to Liu Xiang, but it does not resemble Liu Xiang's other works.

周氏《涉筆》曰：其書多解字義，吾嘗疑其三經後此書方出，故信《字說》而主老莊。

The Zhou氏's "Ink Brush" says: This book contains much content explaining the meanings of characters. I once suspected that it appeared after Wang Anshi's "Three Classics New Interpretations," so I came to believe in the "Character Explanation" and took the teachings of Laozi and Zhuangzi as my main principles.

陳氏《書錄解題》曰：《家語》有孔子遇程子傾葢事，而莊生亦載子華子見昭僖侯一則。莊生固寓言，而《家語》亦未可考信，班固《古今人表》亦無之。

In the Southern Song dynasty, Chen Zhenwen's "Zhai Zhai Shu Lu Jie Ti" says: The "Analects of Confucius" records an incident where Confucius met Master Cheng on the road and stopped his carriage to talk with him. The "Zhuangzi" also mentions a passage about Zi Huazi visiting Han Zhaoxi Hou. The "Zhuangzi" book itself is mostly made up of parables and not reliable. The book "Analects of Confucius" is also not verifiable. As for Ban Gu's "Book of Han: Table of Ancient and Modern People," there is no mention of Zi Huazi at all.

王氏《困學紀聞》曰：《子華子》後序謂鬼谷子之師，水心銘鞏仲至所謂程子即此書也。朱文公謂詞艱而理淺，近世巧於模擬者所為，決非先秦古書。

In the Southern Song dynasty, Wang Yinglin in his work "Kunxue Jiwen" said that the postscript of the book "Zihua Zi" claims that Zihua Zi was the teacher of Guiguzi. The "Chengzi" mentioned in the inscription written by Ye Xie (also known as Shuixin) for Gong Zhongzhi refers to this book. Zhu Xi criticized it for having obscure language and shallow meaning, saying it was the work of modern people who were good at imitating ancient texts, and not an ancient book from the Warring States period.

劉牧之徒偽撰《乾鑿度》以自固其學，而猶未已也。葢緯書出於西漢，恐不足以厭服天下之心，故又造《子華子》，其人為與孔子傾葢而語者，以「戴九履一，據三持七」為河圗，始可以屈《洞極經》而伸吾之說。然格致凡近，辭義淺陋，序雲劉向作，而《漢蓺文志》無之。人皆知其出於元豐後《三經》《字說》盛行之時，竟何益哉？故曰：作偽心勞日拙。

Liu Mu's school forged the "Qian Zao Du" to strengthen their own doctrine, yet they were not satisfied. Probably because the writings of the Western Han dynasty were not enough to silence the doubts of the people, they then created another work called "Zi Hua Zi", claiming that the person named Zi Hua in the book was someone who had a deep conversation with Confucius when they stopped their carriage and talked heart to heart. By using this person's mouth, they attributed the "nine palace numbers" of "dai jiu lu yi, ju san chi qi" to the Yellow River Chart. In this way, they could surpass the "Dong Ji Jing" and promote their own views. However, the book's discussion on the study of things and the attainment of knowledge is extremely ordinary, and its language and reasoning are very shallow. The preface claims that it was written by Liu Xiang of the Western Han dynasty, but the "Han Shu · Yi Wen Zhi" did not record it. Everyone can see that it was written in the period after the Yuanfeng era of Emperor Shenzong of Song, when Wang Anshi's "San Jing Xin Yi" and "Zi Shuo" were popular. So what use is there in forging such a work? Therefore, the more effort a forger puts into his deception, the more obvious the flaws become.

旴江李氏覯《刪定易圗論序》曰：世有治易根於劉牧者，其說日不同。因購牧所為《易圗》五十有五觀之，則甚復重，乃刪其圗而存之者三，所謂河圗也、洛書也、八卦也。於其序解之中撮舉而是正之，諸所觸類亦復詳說，成六論。別有一冊黃黎獻為之序者，頗増多誕謾，自鄶以下無譏焉。[論作於仁宗景佑三年。王湜《易學》曰：吳秘之《通神》、黃黎獻之《畧例》《隠訣》、徐庸之《易縕》皆本劉氏。逮鮮于侁稍辨其非，其後論易者交攻之，而九圗十書宋之群儒恆主其說。]

In the "Preface to the Revised Diagrams of the Book of Changes" written by Li Gou, a native of Xujiang during the Northern Song Dynasty, he said: Those who study the Book of Changes based on Liu Mu's teachings have varying interpretations each day. I therefore bought Liu Mu's fifty-five diagrams of the Book of Changes and examined them carefully, finding them repetitive and redundant. I deleted the superfluous parts and retained only three diagrams: the Yellow River Chart, the Luo Writing, and the Eight Trigrams Diagram. I also summarized and corrected his original preface and explanations, elaborating further wherever analogical reasoning could be applied, and wrote six essays. Additionally, there is another version with a preface by Huang Lixian, which contains even more absurd elements. That would be like the fifteen states of the Book of Songs starting from the state of Hui, which is not worth commenting on. (This essay was written in the third year of Jingyou of Emperor Renzong of Song. Wang Shi's "Studies on the Book of Changes" states: Works such as Wu Mi's "Communicating with the Divine," Huang Lixian's "General Principles" and "Secret Methods," and Xu Yong's "Essence of the Book of Changes" all base themselves on Liu Mu's teachings. It was only until Xian Yu Shen that someone began to slightly clarify his errors. After that, those who discussed the Book of Changes started to criticize him. However, the idea that "nine refers to diagrams and ten refers to writings" remained a common view among scholars of the Song Dynasty.)

谷水林氏至《易裨傳·極數篇》曰：夫子於《繫辭》言數者三：曰「天地之數五十有五」，曰「天一終於地十」，曰「參天兩地」而已。雖曰「河出圗，洛出書」，初未嘗曰某為圗、某為書也。至劉牧出《鉤隠圗》，始以四十五為河圗而五十五為洛書，且以為出於希夷之所傳授，始有「戴九履一，左三右七，二四為肩，六八為足，縱橫十五，總四十有五」之說，前此未之見也。孔穎達曰：「龜負洛書，先無此事，見之緯候之書。」牧不信聖人之言而主緯候之說，何邪？大抵聖人闡易，有象則有數，有數則有變。象則太極生兩儀、四象、八卦者也；數則五十有五奇耦之生成者也；變則揲蓍倚數生爻者也。要不出於三者而已。如牧之為圗，巧則巧矣，於易何取焉？且其數曰縱橫十五而已，所以生卦成卦無聞焉，不知何與於易哉？

In the Southern Song dynasty, Lin Zhi (also known as Huashui) wrote in his "Yi Bi Zhuan · Ji Shu Pian" that Confucius mentioned "number" three times in the "Great Treatise." The first was "the number of heaven and earth is fifty-five," the second was "the beginning is heaven one... ending with earth ten," and the third was "three for heaven, two for earth." Although it was said that "the Yellow River produced the chart and the Luo River produced the book," there was never a specific mention of which one was the chart and which was the book. It was only when Liu Mu wrote "Gou Yin Tu" that he fixed forty-five as the Yellow River Chart and fifty-five as the Luo River Book, claiming that this knowledge was passed down by Chen Tuan. Thus, the idea of "nine at the top, one at the bottom, three on the left, seven on the right, two and four as shoulders, six and eight as feet, with the vertical and horizontal sums all being fifteen, and the total number forty-five" came into being. This was not seen before. Kong Yingda said, "The tortoise did not carry the Luo Book before, it was only mentioned in the writings of the 'waihou' class." Why did Liu Mu not believe the words of the sages but instead follow the 'waihou' theory? In general, the sages explained the Way of the Book of Changes, with images, numbers, and transformation. Images are the generation of the Taiji into two principles, four images, and eight trigrams. Numbers are the odd and even numbers that form fifty-five. Transformation is the process of divination with yarrow stalks, which produce the lines according to numbers. These three aspects are the main points. Liu Mu's making of diagrams was indeed skillful, but how does it help with the Book of Changes? Moreover, his numbers only discussed the vertical and horizontal sums being fifteen, but he gave no explanation of how the hexagrams were generated or formed. It is unclear how this relates to the Book of Changes.

又曰：緯候之家不過借河圗洛書之說以神其事，至牧反覆言之，而世之論易者多宗其說，易之本原愈以不明。本朝惟歐陽公以圗書為怪妄之甚，為說以黜之，然一人之言不能勝久習之溺也，特學易者不深察之耳。今所謂極數者，本之天地五十有五之數而黜圗書之妄，雖得罪於君子不辭也。

Lin Zhi also said: The school of Weihou merely uses the terminology of the Yellow River Chart and the Luo Writing to make their own theories seem mysterious and supernatural; when Liu Mu took up this idea and elaborated on it repeatedly, most scholars who discussed the Book of Changes followed him as their master, and thus the original nature of the Book of Changes became increasingly obscure and unclear. In our dynasty, only Ouyang Xiu considered the theories of the Yellow River Chart and the Luo Writing to be utterly absurd, and he specifically wrote treatises to oppose them; however, the words of one person cannot overcome the long-standing habits of the scholarly community. It is merely that those who study the Book of Changes are unwilling to examine deeply. Now, what I am explaining as 'ji shu' (extreme numbers) is based on the number fifty-five of heaven and earth, and I discard the false claims of the Yellow River Chart and the Luo Writing. Even if this leads me to be condemned by the virtuous, I would not hesitate.

按：五十有五即自一至十之數，林氏因錯簡而誤析為二耳。《繫辭》言數者三：天地之數一也，參伍錯綜二也，參天兩地三也。此皆主蓍而言，於圗書無涉。宋人不知此義，而以數為圗書，易道之榛蕪至此而極。雖命世大儒如伊川，不能廢五行之說，而郭子和非之，以為歴數之學；如考亭，不能廢圗書之說，而林德久斥之，以為緯候之流。嗚呼，之二子者，豈非豪傑之士哉！

Hu Wei said: "Fifty-five" is originally the total sum from one to ten, and Lin Zhi mistakenly split it into two parts because the text of the Great Treatise was disordered. The Great Treatise discusses "number" in three places: the number of heaven and earth, "three and two intermingled to produce change, and the numbers interwoven," and "three for heaven, two for earth." These three instances all refer to the method of divination using the yarrow stalks, and have nothing to do with the Yellow River Chart or the Luo Writing. The Song scholars did not understand this, and they took these numbers as part of the diagrams. The confusion in the study of the Book of Changes has reached its peak here. Even among the great scholars of their time, Cheng Yi did not abandon the theory of the Five Phases combined with the Book of Changes, but Guo Yong (whose name was Zihe) criticized him, saying that it was the study of astronomy and mathematics. Zhu Xi did not abandon the Yellow River Chart and the Luo Writing, but Lin Zhi (whose name was Dejiu) criticized him, saying that it belonged to the category of esoteric doctrines. Alas, these two men who dared to correct the great scholars, are they not truly heroic men!

雷氏《易圗通變》曰：龍圗流傳未逺，知者亦鮮，至劉牧乃増至五十五圗，名以《鉤隠》，師友自相推許，更為倡述，各於易有註釋：曰《卦德論》，曰《室中語》，曰《記師說》，曰《指歸》，曰《通微》，亦總謂之《周易新注》。每欲自神其事，及跡而究之，未見其真能有所神奇也。時則有李覯泰伯者，著六論以駁之，刪其復重，止存三圗。自後愈傳愈失，甚者反以五十五數為河圗，而圗南所傳者為洛書，顛倒迷謬，靡所底止。

In the Yuan dynasty, Lei Siqi in his work "Yitu Tongbian" said that Chen Tuan's "Longtu" did not circulate for a long time and few people understood it. When it came to Liu Mu, it was expanded to fifty-five diagrams, named "Gouyin Tu." They and their disciples and friends praised each other, continuously wrote, and each annotated the "Yi," producing works such as "Gude Lun," "Shizhong Yu," "Jishi Yue," "Zhiui," and "Tongwei," collectively called "Zhouyi Xin Zhu." They tried everywhere to make their works seem very mysterious, but when tracing their origins, there was nothing really miraculous. At that time, Li Gou (whose name was Taibai) wrote six essays to refute them, deleting repeated diagrams and keeping only three. Later, the transmission became more distorted, even reversing the number fifty-five as the Yellow River Chart, and taking Chen Tuan's transmission as the Luo Writing, causing confusion and disorder without end.

按：劉牧之學，當時皆謂其源出於希夷，而不知希夷所傳者乃天地自然之圗、白黑回互之狀[見第三卷]，康節之所受而演之者也，於《龍圗》曷與焉？於《鉤隠》又曷與焉？葢自天禧之後，偽書盛行，而天地自然之圗隠矣。說者以劉牧之學為希夷之傳，是猶呂之代嬴、牛之易馬，世仍以秦、晉目之，而不知其血脈之已非也。李泰伯存其三圗，雷齊賢歸咎後人，亦尋常之見耳。《鉤隠》支離破碎，繳繞窒塞，真無一可取。譬諸田功，聖人之易，五穀也；希夷之易，荑稗也；牧之易，進不可窮理以盡性，退不可養生以盡年，徒為稂莠而已矣。

Hu Wei said: Liu Mu's learning, at that time, was said by everyone to have come from Chen Tuan (who was called Xiyi), but no one knew that what Chen Tuan passed down was actually the 'Diagram of Heaven and Earth in their Natural State', which is the one with black and white dots interconnecting in a circular manner (see Volume Three of this book). That was actually what Shao Yong inherited and further developed, so what connection does it have with the 'Dragon Diagram'? What connection does it have with the 'Hooked Hidden Diagram'? Probably since the time of Emperor Zhenzong of the Song Dynasty's Tianxi period, fake books became widespread, and the Diagram of Heaven and Earth in their Natural State gradually disappeared. Those who discussed Liu Mu's learning as being the direct transmission from Chen Tuan were like the Liu family replacing the Ying family, or the Niu family taking over the Sima family. The world still called them Qin and Jin, but they no longer had the original lineage. Li Gou only preserved three diagrams, and Lei Siqi (whose name was Qixian) blamed the later generations, which was just common opinion. The Hooked Hidden Diagram was fragmented and confused, with no part worth taking. Taking farming as an analogy: the sage's Book of Changes is the five grains, Chen Tuan's Book of Changes is the稗草, Liu Mu's Book of Changes could not reach the ultimate understanding of principles and human nature, nor could it help people live a long and healthy life, it was just weeds in the field.

右論易數鉤隠圗。

The previous section examines the origin and authenticity of Liu Mu's "Yishu Gouyun Tu."

西山蔡氏元定曰：古今傳記，自孔安國、劉向父子、班固，皆以為河圗授羲、洛書錫禹；闗子明、邵康節皆以十為河圗、九為洛書。葢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，正龜背之象也。惟劉牧意見以九為河圗、十為洛書，託言出於希夷，既與諸儒舊說不合，又引大傳以為二者皆出於伏羲之世，其易置圗書並無明驗。但謂伏羲兼取圗書，則易、範之數誠相表裡，為可疑耳。其實天地之理一而已矣，雖時有古今先後之不同，而其理則不容於有二也。故伏羲但據河圗以作易，則不必豫見洛書，而已逆與之合矣；大禹但據洛書以作範，則亦不必追考河圗，而已暗與之符矣。其所以然者何哉？誠以此理之外無復他理故也。

In the Southern Song dynasty, Cai Yuan ding, who was known as Xishan, said: The biographies of ancient and modern times, from Kong An guo, Liu Xiang and his son Liu Xin to Ban Gu, all believed that the Yellow River Chart was given to Fuxi and the Luo Writing was bestowed upon Da Yu. Guan Ziming and Shao Yong both regarded the number ten as belonging to the Yellow River Chart and the number nine as belonging to the Luo Writing. Since the Great Treatise has already listed the numbers of heaven and earth as fifty-five, and the Book of Documents, the Hongfan chapter clearly states that "Heaven thus bestowed upon Da Yu the Hongfan Nine Categories," and the nine palace numbers "with nine on top, one beneath, three on the left, seven on the right, two and four as shoulders, six and eight as feet," exactly resemble the pattern of a tortoise's shell. Only Liu Mu, based on his own speculation, took the number nine as the Yellow River Chart and the number ten as the Luo Writing, and falsely attributed this view to Chen Tuan. This contradicts the previous views of the scholars of earlier generations, and he also cited the Great Treatise to claim that both the Yellow River Chart and the Luo Writing originated in the time of Fuxi. His reversal of the association between the charts and writings lacks any solid evidence. However, his statement that Fuxi used both the charts and writings together does point out that the numbers of the Book of Changes and the Hongfan indeed complement each other, and this is something worth considering. In reality, the principles of heaven and earth are one, and although times have different sequences of ancient and modern, the principles cannot be divided into two different systems. Therefore, Fuxi only used the Yellow River Chart to draw the hexagrams and did not need to see the Luo Writing beforehand, yet naturally aligned with it; Da Yu only used the Luo Writing to compose the Hongfan and did not need to investigate the Yellow River Chart, yet naturally corresponded with it. This is because there is no other principle beyond this one.

右見《易學啟蒙》注。季通為朱子起藳，故不敢列已說為正文。然朱子河圗洛書之象數實由季通而定。初劉牧以四十有五為河圗、五十有五為洛書，實龍圗之本象，而季通以為牧之所易置，至此乃復其舊，且引闗、邵以相證。今按闗《易》明繫偽書，不可以為據；而邵子圎星方土之論，其意別有所在，未嘗以五十有五為河圗也。謹列其說如左，明者幸垂察焉。

The above passage appears in the annotations of the "Yixue Qimeng" (Introduction to the Study of the Changes). Cai Yuanding, who was given the courtesy name Jitong, was the one who drafted the "Qimeng" for Zhu Xi, so he did not dare to present his own opinions as the main text; however, the figures and numbers of the Yellow River Chart and the Luo Writing in Zhu Xi's book were actually determined by Cai Yuanding. Originally, Liu Mu had taken forty-five as the Yellow River Chart and fifty-five as the Luo Writing, which was already the original form of the "Longtu" (Dragon Chart). Cai Yuanding, however, believed that Liu Mu had reversed the two, and only here did it become "restored to its original appearance," and he cited the views of Guan Ziming and Shao Yong as evidence. Now, upon examination: Guan Ziming's "Yi Zhuan" (Commentary on the Changes) is clearly a forgery and cannot be used as evidence; and Shao Yong's statement "the round is the star, the square is the earth" originally had a different meaning and did not take fifty-five as the Yellow River Chart. I now list the views of various scholars below, and I hope discerning readers will carefully examine them.

《闗子明易傳》曰：河圗之文，七前六後，八左九右，聖人觀之以畫卦。是故全七之三以為離，竒以為巽；全八之三以為震，奇以為艮；全六之三以為坎，奇以為乾；全九之三以為兊，奇以為坤。正者全其位，隅者盡其畫。[此與《鉤隠》四象生八卦同。]洛書之文，九前一後，三左七右，四前左，二前右，八後左，六後右。後聖稽之為三象：一四七為天生之數，二五八為地育之數，三六九為人資之數。

The Guanzi Ming Yi Zhuan says: The diagram of the Yellow River Chart has seven in front, six behind, eight on the left, and nine on the right. The sage observed it to draw the trigrams. Therefore, taking the number seven, filling three strokes to form the trigram Li (Fire), and using the remaining odd number to form Xun (Wind); taking the number eight, filling three strokes to form Zhen (Thunder), and using the remaining number to form Gen (Mountain); taking the number six, filling three strokes to form Kan (Water), and using the remaining number to form Qian (Heaven); taking the number nine, filling three strokes to form Dui (Lake), and using the remaining number to form Kun (Earth). The trigrams in the four cardinal directions occupy their proper positions, while those in the four corners use up the remaining strokes. (This is consistent with Liu Mu's Gou Yin Tu, which says 'the four basic trigrams give rise to the eight trigrams.') The diagram of the Luo Writing has nine in front, one behind, three on the left, and seven on the right, four in the upper left, two in the upper right, eight in the lower left, and six in the lower right. Later sages examined it, dividing it into three groups of numbers: one, four, and seven are the numbers generated by Heaven; two, five, and eight are the numbers nurtured by Earth; three, six, and nine are the numbers relied upon by humans.

《邯鄲書目》[宋皇佑中李淑撰]：《闗子明易傳》一卷，唐趙蕤注。魏孝文使幷州刺史王虯與子明著《疑筮論》數十篇，蕤雲亡篇過半，今無能詮次，但隨文解義，庶學者觸類而長。阮逸詮次刋正《洞極元經傳》五卷[逸字天隠，宋仁宗時人]。闗子明以生、育、資為傳以釋其蘊，為經論十一篇[《玉海》云：子明《易傳》卜百年義第一，次以統言易義、大衍、乾坤策、盈虛、闔闢、理性、時變、動靜、神義，終於雜義第十一]，又為圗以序其象。《文中子》錄子明事：太和末，穆公與談易，言於孝文帝，帝曰：「且與卿就成筮論。」子明曰：「乾坤之策，陰陽之數，推而行之不過三百六十六，引而伸之不過三百八十四，天之道也。象生有定數，吉凶有前期，變而能通，故治亂有可易之理。」

The "Handan Book Catalogue" (compiled by Li Shu during the Huangyou period of Emperor Renzong of the Song dynasty) records: "The Commentary on the Changes by Guan Ziming" in one volume, with annotations by Tang dynasty person Zhao Rui. In the Northern Wei dynasty, Emperor Xiaowen sent the governor of Pingzhou, Wang Qiu, to collaborate with Guan Ziming in writing "Discourses on Divination" in dozens of chapters. Zhao Rui said that the lost chapters were more than half, and now it is impossible to compile them in order, so only the meaning of the text can be explained, hoping that scholars can understand the principles by analogy. Ruan Yi compiled and edited "The Transmission of the Transmission of the Deep Origin Classic" in five volumes (Ruan Yi, whose name was Tianyin, was a person of the Song dynasty during Emperor Renzong's reign). Guan Ziming explained the meaning of the book by using the three concepts of "birth, nurture, and support," composing eleven commentaries on the Classic (The "Zhyouzi" records Guan Ziming's deeds, saying: in the late Tahe period, Mu Gong talked about the Changes with him and recommended him to Emperor Xiaowen. Emperor Xiaowen said: "Let us first write the Discourses on Divination together." Guan Ziming replied: "The numbers of the two trigrams, Qian and Kun, and the numbers of yin and yang, when calculated, amount to no more than 366, and when extended, no more than 384. This is the way of heaven. The generation of things has a fixed number, and the fortune and misfortune have a predictable limit. Change can lead to understanding, so there is a principle for transforming between peace and chaos.")

晁氏《讀書志》曰：《闗子明易傳》一卷，魏闗朗撰，李邯鄲始著之目，王通贊易葢宗此也。

In the Southern Song dynasty, Zhao Gongwu's "Juzhai Reading Notes" states: "The "Gu Ziming Yijie" is one volume, written by Guan Lang of the Northern Wei dynasty. It was first included in the "Hendan Book Catalogue" by Li Shu. In the Sui dynasty, Wang Tong promoted the principles of the Book of Changes, probably taking this work as his main reference."

《後山叢談》曰：世傳王氏《元經》、薛氏傳、《闗子明易傳》、《李衛公問對》，皆阮逸所著。逸以草示蘇明允，而子瞻言之。

In the Northern Song dynasty, Chen Shidao of the Houshan Congtan said that the works circulating in the world, such as Wang Tong's "Yuan Jing," Xue Shou's biographies, the "Gu Ziming Yi Zhuan," and the "Li Weigong Wen Dui," were all fabricated by Ruan Yi. Ruan Yi had shown the original manuscript to Su Xun, who was known by his courtesy name Mingyun. This statement was made by Su Shi, who was known by his courtesy name Zizhan.

《朱子語類》：鄒浩問李壽翁椿最好《闗子明易》。先生曰：是阮逸偽作，陳無已集說得分明。

The Shu Li Lu records that Zou Hao asked about Li Chun, who most highly praised the Guan Ziming Commentary on the Yi Jing. Zhu Xi replied that it was forged by Ruan Yi, and that Chen Shidao's collected writings clearly explain this.

姚氏曰：世所傳闗子明《洞極經》亦言河圗洛書如劉氏說，而兩易之，以五方者為圗、九宮者為書。按唐李鼎祚《易解》盡備前世諸儒之說，獨無闗氏；至本朝阮逸始偽作《洞極經》，見於《後山叢談》，則闗亦不足為證矣。[見《周易玩辭》]

Yao said: The world has been passing on Guan Ziming's "Dongji Jing," which discusses the He Tu and Luo Shu in the same way as Liu Mu's account, but reverses the two, taking the five directions' generating numbers as the He Tu and the nine palace numbers as the Luo Shu. Upon examination, during the Tang dynasty, Li Dingzuo's "Collected Explanations of the Zhou Changes" collected the views of previous scholars almost entirely, but omitted Guan Lang; it was only in this dynasty that Ruan Yi fabricated the "Dongji Jing," as recorded in "Houshan Congtan." Therefore, Guan Ziming's view is not sufficient as evidence. (This entry is found in "Zhou Yi Wan Ci")

雷氏《易圗通變》曰：楊次公自著《洞極經》，託名於闗子明[渭按：楊傑字次公，元豐中與范鎮論樂。《洞極經》果為傑所撰，則又出阮逸之後，恐非]。如其敘本篇稱「子曰：河圗之文，七前六後，八左九右，聖人觀之以畫八卦。是故全七之三以為離，奇以為巽；全八之三以為震，奇以為艮；全六之三以為坎，奇以為乾；全九之三以為兊，奇以為坤。正者全其位，隅者盡其畫」，而謂四象生八卦，此謬之尤者。乾坤相索而生六子，今為男女者反能生父母之卦邪？

In the Yuan dynasty, Lei Siqi's "Yi Tu Tong Bian" says: "The 'Dong Ji Jing' was written by Yang Cigong himself, but it was falsely attributed to Guan Ziming. (Hu Wei notes: Yang Jie, who was called Cigong, once discussed music theory with Fan Zhen during the Yuanfeng period of Emperor Shenzong of the Song dynasty. If the 'Dong Ji Jing' was indeed written by Yang Jie, it would be later than Ruan Yi, which seems unlikely.) For example, the text describes the origin of this chapter as: 'Confucius said: The diagram of the Yellow River Chart has seven in front, six behind, eight on the left, and nine on the right. The sage observed it to draw the Eight Trigrams. Therefore, taking the number seven, filling three strokes to form Li (Fire), and the remaining number forms Xun (Wind); taking the number eight, filling three strokes to form Zhen (Thunder), and the remaining number forms Gen (Mountain); taking the number six, filling three strokes to form Kan (Water), and the remaining number forms Qian (Heaven); taking the number nine, filling three strokes to form Dui (Lake), and the remaining number forms Kun (Earth). The ones in the central position occupy their own position, while those in the corner positions use up the remaining strokes.' According to this, the Four Symbols can generate the Eight Trigrams, which is the most absurd mistake. According to 'Shuo Gua,' the two trigrams Qian and Kun interact to produce the six subsidiary trigrams. Now it is claimed that the six subsidiary trigrams, which are the male and female trigrams, can generate the parents trigrams, which is illogical."

《文中子中說》三卷，隋王通之門人共集其語為是書，而晁氏擿其三事以為可疑。其一則闗朗在太和中見魏孝文，自太和丁巳至通生之年甲辰，葢一百七十年矣，而其書有問禮於闗子明，是以知其妄也。程子謂王通隠君子，當時有些少言語，後來為人傅會，不可謂全書，續經之類皆非其作，朱子亦以為然。王氏《揮麈錄》直以為阮逸偽作。今按《中說》宋有阮逸、龔鼎臣二本，其文多異，闗子明事葢逸所傅會，與《洞極經》論總出一手，彼此扶同，以欺天下後世也。

The *Zhong Shuo* in three volumes was compiled by Bao Tong's disciples, collecting his sayings. However, Chao Gongwu pointed out three things in it as doubtful. One of them is that Guan Lang visited Emperor Xiaowen of the Northern Wei during the Taihe period, from the Taihe Dingsi year to the Jiachen year when Wang Tong was born, spanning about 170 years. Yet the book records Wang Tong consulting Guan Ziming on rites, which shows it is fabricated. Cheng Yi said that Wang Tong was a reclusive gentleman, who left some sayings, but later they were fabricated and added to, so it cannot be considered a complete and reliable work; he called it "continuation of classics," which were not written by Wang Tong. Zhu Xi also agreed with this view. Wang Mingqing's *Hui Chu Lu* directly claimed that *Zhong Shuo* was forged by Ruan Yi. Now we examine that in the Song dynasty there were two versions of *Zhong Shuo*, by Ruan Yi and Gong Dingchen, with many differences in wording. The story of Guan Ziming is probably fabricated by Ruan Yi, and the arguments in *Dong Ji Jing* seem to come from the same person, connected together to deceive future generations.

宋仁宗朝，劉牧著《易數鉤隠》，以九宮為河圗、五合為洛書，其學盛行於世。同時有阮逸者，陰欲排之，乃撰《闗子明易傳》而兩易其名。季通不察，以為真古書也，遂引為圗十書九之證。季通無論已，獨怪朱子素斥《龍圗》為假書，又據《後山叢談》深信闗《易》為阮逸所造，而於此則曲從季通之說，何也？

During the reign of Emperor Renzong of Song, Liu Mu wrote the "Yishu Gouyun Tu," treating the nine-square grid numbers as the Yellow River Chart and the Five Phases generation numbers as the Luo Writing. This system became popular at that time. At the same time, there was a person named Ruan Yi who secretly wanted to surpass it, so he fabricated the "Guanziming Yixuan," swapping the names of the charts and writings. Cai Yuanping did not realize this and thought it was indeed an ancient text, thus using it to support the idea that the Yellow River Chart has ten numbers and the Luo Writing has nine numbers. Cai Yuanping was not a problem, but it is strange that Zhu Xi, who usually condemned the "Longtu" as a fake text, and who was convinced by the "Houshan Congtan" that the "Guanziming Yixuan" was written by Ruan Yi, yet here he deliberately followed Cai Yuanping's view. Why was that?

葢自漢魏以來，言河圗者皆主《乾鑿度》《參同契》，而此獨根柢大傳五十有五之數，其據地尊而取義正也。然河圗久已無傳，不知其形象若何，而強以五十五數當之，大傳既無明文，古注亦無可考。「七前六後，八左九右」之位不見於經，「天生地成，奇白偶黑」之點非易所有。至其所以則之者，謂伏羲分四正之餘數以成四隅之卦畫，支離舛錯，全無義理。藉令此書真出於古，猶不可據以說經，而況明明為阮逸之偽作乎？且此數但可以生蓍而不可以畫卦，九與十何擇焉？

Since the Han and Wei dynasties, those who have discussed the River Chart have mostly relied on the "Yi Wei Gan Zuo Du" and the "Zhou Yi San Tong Qi," but this book alone bases itself on the "Xì Cí" passage "the numbers of heaven and earth total fifty-five," thus occupying a lofty position and presenting a proper meaning. However, the River Chart has long been lost, and no one knows what its actual form was. Now, forcing the number fifty-five upon it is arbitrary, as the "Xì Cí" does not clearly state this, and ancient commentaries offer no reference. Statements such as "seven is in front, six is behind, eight is on the left, nine is on the right" are not found in the text; similarly, the idea that odd numbers are represented by white dots and even numbers by black dots is not original to the "Yi." As for its claim that sages derived the hexagrams from the River Chart, saying that Fuxi separated the remainders of the four cardinal numbers to form the four corner hexagrams, this is fragmented and confused, with no logical basis. Even if this book were indeed an ancient text, it could not be used to explain the "Yi," let alone the fact that it is clearly a forgery by Ruan Yi. Moreover, these numbers are only suitable for divination with the yarrow stalks, not for drawing hexagrams. So whether it is nine or ten, what difference does it make?

范諤昌《大易源流》言：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：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，地八配夘，天九配酉，地十配中寄於未，乃天地之數五十有五矣。雷氏云：正今圗所傳有四方而無四維者也，闗子明之河圗實本諸此。然諤昌但謂伏羲據龍圗以復位五行之數、八卦之體有如此者，未嘗直指此為河出之圗也。故劉牧《鉤隠》仍以九為圗而十為書，偽闗《易》竊其義，遂以「七前六後，八左九右」為河圗之文。人皆知其書之出於阮逸，而不知其數之本乎諤昌也。

Fan Echáng's "Great Origins of the Changes" says: The dragon horse carried a diagram emerging from the Yellow River, and Fuxi investigated the connection between heaven and humanity, using it to determine the numbers of the Five Phases and the positions of the eight trigrams on the earth. Heaven one resides in the due north, earth two in the due south, heaven three in the due east, earth four in the due west, heaven five in the center, earth six is paired with the position of the rat, heaven seven is paired with the position of the horse, earth eight is paired with the position of the rabbit, heaven nine is paired with the position of the rooster, and earth ten resides in the center and is assigned to the position of the sheep. Together they make up the total number of heaven and earth, which is fifty-five. Lei Siqi says: This is exactly the diagram that is passed down today, which only includes the four cardinal directions and not the four intermediate directions. The river chart mentioned by Guan Ziming actually originates from this. However, Fan Echáng only said that Fuxi used the dragon diagram to determine the numbers of the Five Phases and the positions of the eight trigrams in this way, without directly stating that this was the diagram that emerged from the Yellow River. Therefore, Liu Mu's "Hooking the Hidden Diagram" still regarded the nine numbers as the river chart and the ten numbers as the Luo writing. The forged "Yi Transmission by Guan Ziming" stole Fan's meaning, and thus claimed that "seven in front, six behind, eight on the left, nine on the right" was the diagram of the river chart. Everyone knows that this book was written by Ruan Yi, but few know that these numbers originated from Fan Echáng.

邵子《觀物外篇》曰：圎者星也，歴紀之數其肇於此乎[《易學啟蒙》注云：歴法合二始以定剛柔，二中以定律歴，二終以紀閏餘，是所謂歴紀也。按此說出唐一行《歴本議》]？方者土也，畫州井地之法其放於此乎[《啟蒙》注云：州有九，井九百畝，是所以畫州井地也]？葢圎者河圗之數，方者洛書之文，故羲文因之而造易，禹箕敘之而作範也[《啟蒙》引此段遺後四句，殊為疎謬]。

Shao Yong's "Outer Chapter of Observing Things" says: "The round shape refers to stars, and the numbers used in the calendar and year reckoning probably originated from this (the commentary in "Awakening to the Yi" says: The calendar combines the two beginnings to determine the yin and yang, combines the two middles to determine the musical scales and calendar numbers, and combines the two endings to record the leap years, which is called 'calendar record'. Note: This statement comes from Yixing's "Discussions on the Calendar"). The square shape refers to earth, and the methods of dividing states and managing the well-field system probably imitate this (the commentary in "Awakening to the Yi" says: There are nine states in total, and one well-field is nine hundred mu, which is the origin of dividing states and the well-field system). Generally speaking, the round shape belongs to the numbers of the Yellow River Chart, and the square shape belongs to the writings of the Luo Writing. Therefore, Fuxi and King Wen based the Yi on them, and Da Yu and Ji Zi used them to compose the "Hong Fan". (The commentary in "Awakening to the Yi" omitted the following four sentences, which is a serious omission and error)."

《朱子語類》曰：二始者，一二也，一奇故為剛，二偶故為柔；二中者，五六也，五者十干，六者十二辰也；二終者，九與十也，閏餘之法以十九歲為一章。姑借其說以明十數之為河圗耳。又曰：「圎者星也」，圎者河圗之數，言無那四角底，其形便圎。又曰：河圗既無那四隅，則比之洛書固亦為圎矣。「方者土也」，方者洛書之文，言畫州井地之所依仿而作者也。《書·禹貢》禹別九州島，冀北、揚南、青東、梁西、兗東北、雍西北、徐東南、荊西南、豫中也。孟子言周家井地之制，井九百畝，其中為公田，八家皆私百畝，同養公田，是皆法洛書之九數也。

Master Zhu Xi said: The term "two beginnings" refers to one and two. One is an odd number and thus belongs to the category of yang, while two is an even number and thus belongs to the category of yin. The term "two middles" refers to five and six. Five corresponds to the ten heavenly stems, and six corresponds to the twelve earthly branches. The term "two endings" refers to nine and ten. The method of intercalation uses nineteen years as a cycle. This is merely using this terminology temporarily to explain that the number ten is from the Yellow River Chart. He also said: "A circle represents the stars." A circular shape belongs to the numbers of the Yellow River Chart, because it has no four corners, and thus naturally appears round. He further said: since the Yellow River Chart does not have four corners, it is naturally round compared to the Luo Writing. "A square represents the earth." A square shape belongs to the text of the Luo Writing, because dividing the regions and planning the well-field system were based on it. In the "Book of Documents: Yu Gong," Yu the Great divided the nine states: Ji in the north, Yang in the south, Qing in the east, Liang in the west, Yan in the northeast, Yong in the northwest, Xu in the southeast, Jing in the southwest, and Yu in the center. Mencius discussed the well-field system of the Zhou dynasty, which was one well with nine hundred mu, with the central part being public land and each of the eight families having one hundred mu of private land, all working together on the public land. These were all based on the nine numbers of the Luo Writing.

鶴山魏氏了翁《題司馬子已易圗》曰：涑水司馬叔原覃思易理之學，著《已易》。朱文公以十為河圗、九為洛書，引邵子說辨析甚詳，叔原從之。乃邵子不過曰「圎者河圗之數，方者洛書之文」，今戴九履一之圗其象圎，五行生成之圗其象方，是九圎而十方也，安知邵子不以九為圗、十為書乎？故朱子雖力攻劉氏，而猶曰「易範之數誠相表裡」，又曰「安知圗之不為書、書之不為圗」，則朱子尚有疑乎此也。近世朱子發、張文饒精通邵學，而皆以九為圗、十為書，朱以《列子》為證，張以邵子為主。予嘗以《乾鑿度》及張平子傳所載太一下行九宮法考之，即所謂戴九履一者，則是圗相傳已久，安知非河圗也？紹定四年書。

In the Southern Song dynasty, Wei Liaoong (also known as He Shan) wrote in his "Commentary on the Diagram of the Book of Changes by Sima Ziyi" that Sima Shuyuan, a man from Sushui, devoted himself to the study of the principles of the Book of Changes, and wrote a work called "Yi" (the Book of Changes). Zhu Xi regarded ten as the numbers of the Yellow River Chart and nine as the numbers of the Luo Writing, and he cited Shao Yong's statements to explain them in detail, and Sima Shuyuan followed Zhu Xi's view. However, Shao Yong merely said, "The circular form represents the numbers of the Yellow River Chart, and the square form represents the text of the Luo Writing." Now, the diagram called "Daijiu Lüyi" (the diagram with nine at the top and one at the bottom) has a circular form, while the diagram of the Five Phases has a square form. This shows that nine is circular and ten is square. How can we be sure that Shao Yong did not mean that nine is the diagram and ten is the writing? Therefore, although Zhu Xi strongly criticized Liu Mu, he still said, "The numbers of the Book of Changes and the numbers of the Hongfan (the Nine Chapters) are indeed complementary." He also said, "How can we know that the diagram cannot be the writing or the writing cannot be the diagram?" This shows that Zhu Xi still had doubts about this point. In recent times, Zhu Zhen (also known as Zifā) and Zhang Xingcheng (also known as Wenrao) both were well-versed in Shao's teachings, yet they both regarded nine as the diagram and ten as the writing. Zhu Zhen cited the "Liezi" as evidence, while Zhang Xingcheng mainly followed Shao Yong. I once examined this using the "Qian Zao Du" and the account in the "Later Han Book" about "Taiyi moving through the nine palaces," which clearly shows the "Daijiu Lüyi" diagram. This indicates that this diagram has been passed down for a long time. How can we be sure it is not the Yellow River Chart? Recorded in the fourth year of Shao Ding.

靜修劉氏因《河圗洛書辨》曰：邵子但言方圎之象，而不指九十之數。若以象觀之，則九又圎於十矣。且其所謂方圎而前後乎此者，不過指陰陽剛柔奇偶而已。在此則星少陽而土少柔，其偶者固當為方而為陰，奇者固宜為圎而為陽矣。

In his work "Discernment of the Yellow River Chart and the Luo Writing" (He Tu Luo Shu Bian), Liu Yin of the Yuan dynasty (styled Jingshu) said that Shao Yong only discussed the images of square and circle, but did not specify whether it was nine numbers or ten numbers. If we look solely at the forms, nine numbers appear more circular than ten numbers. Moreover, the square and circle mentioned by him before were merely referring to yin and yang, hardness and softness, odd and even. In this context, stars belong to the category of less yang, while earth belongs to the category of less softness. Even numbers should be paired with square and yin, while odd numbers should be paired with circle and yang.

邵子之於圗書，言方圎而不言九十。蔡季通以圎星為十為圗、方土為九為書，而朱子從之，謂河圗無四隅，比洛書便圎；魏華父則以為九圎而十方，劉夢吉亦云九圎於十，葢皆以九為圗。說者以為物有八隅必作圎形，九宮是也；物止四面必成方形，五合是也。魏說較長，以理言之誠有然者。若摹寫為奇白偶黑、累累貫珠之狀，則九亦可觚稜之以為方，十亦可彎環之以為圎，安見九必圎而十必方乎？方圎之或九或十終無定論，而邵子埓之以星土，固有所自來，其數可考而知也。

Shao Yong discussed the Yellow River Chart and the Luo Writing, referring only to square and circle, without mentioning nine and ten. Cai Yuanping designated "the round as stars" as the number ten, assigning it to the Yellow River Chart, and "the square as earth" as the number nine, assigning it to the Luo Writing. Zhu Xi followed this approach, saying that the Yellow River Chart has no four corners, so it is naturally round compared to the Luo Writing. However, Wei Leiweng (whose name was Huafu) argued that nine is round and ten is square, while Liu Yin (whose name was Mengji) also said that nine is more round than ten. Generally, they all tended to associate nine with the Yellow River Chart. The reasoning given by the commentators was that an object with eight corners must be round, like the nine-square diagram; and an object with four sides must be square, like the Five Phases generation diagram. Wei Leiweng's view seems more reasonable, as it makes sense in terms of logic. However, if they are drawn with odd numbers represented by white dots and even numbers by black dots, arranged in a chain-like pattern, then nine can also be drawn with sharp angles to form a square, and ten can be curved to form a circle. How can we be certain that nine is always round and ten is always square? Whether square and circle belong to nine or ten remains undetermined. Shao Yong's pairing of stars with earth has its own origin, and the numbers involved can be examined and clarified.

友人徐敬可嘗謂余曰：《觀物外篇》所謂圎星方土，乃論積數自然之理，非論圗書也，連上文讀之即明。朱子斷章取義，載之《啟蒙》，遂不可解，而引一行二始二中二終之說以釋歴數肇於圎星之義，殊為牽強。今按此之上文曰：「天圎而地方，圎之數起一而積六，方之數起一而積八，變之則起四而積十二也。六者常以六變，八者常以八變，而十二者亦以八變，自然之道也。」又曰：「圎者六變，六六而進之，故六十變而三百六十矣；方者八變，故八八而成六十四矣。陽主進，是以進之為六十矣。」

My friend Xu Shan (style name Jingke) once told me that in the "Outer Chapter of Observing Things" it says, "Round things are stars, square things are earth," which refers to the natural principle of the accumulation of numbers, not the Yellow River Chart or the Luo Writing. As long as one reads the preceding text, it becomes clear. Zhu Xi took this passage out of context and included it in his "Introduction to the Study of the Changes," which makes no sense. He also used Yixing's idea of "two beginnings, two middles, and two ends" to explain "the numbers of the calendar begin with the round stars," which is forced. Now, examining the preceding text, it says: "The sky is round and the earth is square; the numbers of the round start from one and accumulate to six, while the numbers of the square start from one and accumulate to eight. When changes occur, they start from four and accumulate to twelve. Six always changes in multiples of six, eight always changes in multiples of eight, and twelve also changes in multiples of eight, which is a natural principle." It also says: "The round undergoes six changes, and six multiplied by six and then progresses, so it reaches sixty and then three hundred and sixty. The square undergoes eight changes, so eight multiplied by eight becomes sixty-four. Yang is dominant in advancement, so it progresses to sixty."

然後繼以圎星方土之說，玩其辭旨，葢積數自然之理。凡以七圎物相聚，必以一為心而環之以六□，此形是起一而積六也；凡以九方物相聚，必以一為心而圍之以八□，此形是起一而積八也；凡以十二方物相聚，必以四為心而圍之以十二□，此形是起四而積十二也。

Then it goes on to say, "A circle represents a star, and a square represents earth." Delving into its meaning, it is speaking of the natural principle of accumulation of numbers. Whenever seven circular objects are gathered together, one must be in the center with six surrounding it (as shown in the diagram), which is "accumulating from one to six"; whenever nine square objects are gathered together, one must be in the center with eight surrounding it (as shown in the diagram), which is "accumulating from one to eight"; whenever twelve square objects are gathered together, four must be in the center with twelve surrounding them (as shown in the diagram), which is "accumulating from four to twelve."

圎者聚七之形，如眾星之麗乎天，以六變而為三十六，進之極於三百六十，當期之日，得氣朔盈虛之中，而星於是乎一周，是為治歴之本，故曰歴紀之數肇於此，二始二終曷與乎？方者聚九之形，象九州島，亦八家同井、中為公田象也。變之則為聚十六之形，象四邑為丘，丘容十六井；又加兩圍共六十四，乃四丘為甸之象，甸容六十四井；更進而四甸為縣，縣容六十四邑；更進而四縣為都，都容六十四丘，皆所謂八八而成六十四也，故曰畫州井地之法放於此也。

The round shape represents the form of gathering seven, just like countless stars attached to the sky: after six transformations it becomes thirty-six, and when pushed to the extreme it reaches three hundred and sixty, exactly matching the number of days in a year. Within this number are the average figures of the twenty-four solar terms, new and full moons, and the waxing and waning of the moon. Thus the stars complete one cycle, which is the fundamental principle of establishing a calendar. Therefore, it is said, "The numbers of the calendar originate here." Now, what connection does the concept of "two beginnings" and "two endings" mentioned by Yixing have with this? The square shape represents the form of gathering nine, symbolizing the nine provinces. It also represents the image of eight families sharing one well-field, with the central part being the public field. When this form expands, it becomes the shape of gathering sixteen, symbolizing four towns forming one hill, with one hill containing sixteen well-fields; adding two more layers around it makes sixty-four, representing four hills forming one town, with one town containing sixty-four well-fields; further expansion leads to four towns forming one county, with one county containing sixty-four hills; and further expansion leads to four counties forming one capital, with one capital containing sixty-four hills. These are all referred to as "eight eights forming sixty-four." Therefore, it is said, "The methods of dividing states and well-fields are modeled after this."

更推而廣之，古有河圗亦圎體也，羲文因之以造易，則圎星之類也；古有洛書亦方體也，禹箕敘之以作範，則方土之類也。此不過因論方圎之積數而牽綴及之，不專為圗書而發。方九猶可言，彼圎則何以知其為九與十也哉？

When extending this further, in ancient times there was the Yellow River Chart, which was round in form, and Fuxi and King Wen based the Book of Changes on it, which would fall under the category of "round stars"; in ancient times there was also the Luo Writing, which was square in form, and Yu the Great and Ji Zi based the Classic of Documents on it, which would fall under the category of "square land". This is merely a passing mention due to the discussion of the accumulation of numbers in square and round forms, and not specifically about the Yellow River Chart and the Luo Writing. The square form is determined by nine numbers, which can be somewhat explained, but how can we determine the round form as either nine or ten?

其下文則曰：「蓍數不以六而以七，何也？並其餘分也。去其餘分則六，故策數三十六也。是以五十者，六十四卦閏歲之策也；其用四十有九，六十四卦一歲之策也；歸奇掛一，猶一歲之閏也。卦直去四者何也？天變而地效之，是以蓍去一則卦去四也。」又曰：「蓍德圎以況天之數，故七七四十九也，五十者存一而言之也；卦德方以況地之數，故八八六十四也，六十者去四而言之也。」

Continuing the text: "The number of the divining stalks uses seven instead of six, why? Because the remainder is also counted in; if the remainder is removed, it becomes six, so the number of stalks for each line is thirty-six. Therefore, the number fifty is the number of the sixty-four hexagrams in a leap year; actually using forty-nine is the number of the sixty-four hexagrams in a regular year; the remainder is set aside, just like the leap month in a year. As for the hexagram numbers being reduced by four, why is that? Because heaven changes first and earth follows, so the number of stalks is reduced by one, and the number of hexagrams is reduced by four." It also says: "The nature of the divining stalks is round, used to compare with the number of heaven, so it is seven times seven, forty-nine, saying fifty includes the one that is kept but not used; the nature of the hexagrams is square, used to compare with the number of earth, so it is eight times eight, sixty-four, saying sixty is the number after removing the four."

然則天圎主蓍、地方主卦，蓍數以七相乘則為四十九，以六相乘則為三十六，無九位四十有五之數；卦數以八相乘為六十四，去其四則為六十，亦無十位五十有五之數。竊意邵子所指為河圗者，即希夷先天太極一圈之形，故謂之圎，與闗朗之河圗全無交涉；其所指為洛書者，即九宮之形，中起一而外積八，故謂之方，此則與闗朗闇合，而其所以為方之意，季通亦未之識也，故漫引以為圗十書九之證。從來憒憒，茲特為一闡明之。

Thus, it appears that the circular form represents the yarrow stalks and the square form represents the trigrams: the number of yarrow stalks, when multiplied by seven, gives forty-nine, and when multiplied by six, gives thirty-six, with no mention of the forty-five number of the nine squares; the number of trigrams, when multiplied by eight, gives sixty-four, and subtracting four gives sixty, with no mention of the fifty-five number of the ten positions. I privately infer that the river chart referred to by Shao Yong is the circular shape of Chen Tuan's Earlier Heaven Taiji, hence called "circular," and has nothing to do with the river chart mentioned by Guan Lang; the Luo Writing he refers to is the shape of the nine squares, with one at the center and eight surrounding it, hence called "square," which coincides with Guan Lang's view, though the reason for calling it "square" was not understood by Cai Yuanping, so he mistakenly used it as evidence for the "chart ten, writing nine" theory. Throughout history, this has been confused, and here I specially clarify it.

朱子《答劉君房書》曰：「《啟蒙》本欲學者且就大傳所言卦畫蓍數推尋，不須過為浮說，而自今觀之，如河圗洛書亦未免有剩語。」觀此言，知季通代斵不中繩墨，文公晚年亦深悔之，但未及改削耳。而近代諸儒奉《啟蒙》為不刋之書，不亦過乎！

Zhu Xi in his letter to Liu Junfang said: "The original intention of the 'Easy Enlightenment' was to have scholars investigate the diagrams of the hexagrams and the method of divination with yarrow stalks as explained in the Great Treatise, without engaging in excessive and superficial discussions; looking back now, contents such as the Yellow River Chart and the Luo Writing are also unnecessary. From this statement, we can see that Cai Yuanding, acting on behalf of others, wielded the axe without following the correct pattern; Zhu Xi himself deeply regretted this in his later years, but it was too late to make revisions. However, modern scholars have regarded the 'Easy Enlightenment' as an unchangeable definitive text, which is surely excessive."

《本義圗說》：《繫辭傳》曰「河出圗，洛出書，聖人則之」，又曰「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，此所以成變化而行鬼神也」，此河圗之數也。洛書葢取龜象，故其數戴九履一，左三右七，二四為肩，六八為足。

The Original Meaning of the Zhou Changes states in its preface that the Great Treatise says, "The Yellow River produced the chart, and the Luo River produced the writing; the sage took them as models." It also says, "Heaven has one, Earth has two; Heaven has three, Earth has four; Heaven has five, Earth has six; Heaven has seven, Earth has eight; Heaven has nine, Earth has ten. The numbers of Heaven are five in total, and the numbers of Earth are five in total. Each of the five directions corresponds and is matched accordingly. The total of the Heaven numbers is twenty-five, and the total of the Earth numbers is thirty. The total numbers of Heaven and Earth together are fifty-five, which is used to accomplish changes and the movement of ghosts and spirits." This is the numerical pattern of the Yellow River Chart. The Luo Writing is probably modeled after the pattern of a turtle's back, so its numbers are arranged with nine at the head, one at the feet, three on the left, seven on the right, two and four as the shoulders, and six and eight as the feet.

《易學啟蒙》曰：河圗之位，一與六共宗而居乎北，二與七為朋而居乎南，三與八同道而居乎東，四與九為友而居乎西，五與十相守而居乎中。葢其所以為數者，不過一陰一陽、一奇一偶，以兩其五行而已。所謂天者，陽之輕清而位乎上者也；所謂地者，陰之重濁而位乎下者也。陽數奇，故一三五七九皆屬乎天，所謂天數五也；陰數偶，故二四六八十皆屬乎地，所謂地數五也。天數地數各以類而相求，所謂五位之相得者然也。

The *Yixue Qimeng* says: In the Yellow River Chart, the number one and six share the same origin and are located in the north; the number two and seven are companions and are located in the south; the number three and eight are on the same path and are located in the east; the number four and nine are friends and are located in the west; the number five and ten are companions and are located in the center. Basically, the principle of the formed numbers is simply the combination of one yin and one yang, one odd and one even, to pair each of the Five Phases with two numbers. The term "heaven" refers to yang energy, which is light and clear and resides in the upper position; the term "earth" refers to yin energy, which is heavy and turbid and resides in the lower position. Yang numbers are odd, so one, three, five, seven, and nine belong to heaven, which is called "five heavenly numbers"; yin numbers are even, so two, four, six, eight, and ten belong to earth, which is called "five earthly numbers." Heavenly and earthly numbers each seek their own kind, and the phrase "five positions correspond" refers to this.

天以一生水而地以六成之，地以二生火而天以七成之，天以三生木而地以八成之，地以四生金而天以九成之，天以五生土而地以十成之，此又其所謂各有合焉者也。積五奇而為二十五，積五偶而為三十，合是二者而為五十有五，此河圗之全數，皆夫子之意而諸儒之說也。至於洛書，則雖夫子之所未言，然其象其說已具於前，有以通之，則劉歆所謂經緯表裡者可見矣。

Heaven produces water with the number one and Earth completes it with six; Earth produces fire with the number two and Heaven completes it with seven; Earth produces wood with the number three and Heaven completes it with eight; Earth produces metal with the number four and Heaven completes it with nine; Earth produces earth with the number five and Heaven completes it with ten. This is what is called 'each has its corresponding number.' The sum of the five odd numbers is twenty-five, and the sum of the five even numbers is thirty. Together they make fifty-five, which is the total number of the Yellow River Chart. This is the meaning intended by Confucius and the views of the various scholars. As for the Luo Writing, although Confucius did not mention it, its image and its meaning are already presented earlier. If one can connect them together, then the statement by Liu Xin that 'the Yellow River Chart and the Luo Writing are like the meridians and the parallels, one inside and one outside' can be clearly understood.