Yi Tu Tong Bian (易圖通變)

經名：《易圖通變》。原題「臨川道士雷思齊學」。五卷。底本出處：《正統道藏》太玄部。參校版本：文淵閣《四庫全書》本經部易類（簡稱「四庫本」）。

The title of this book is "Yitu Tongbian." The original manuscript is attributed to "Linchuan Daoist Lei Siqi." The entire work consists of five volumes. The edition used for collation comes from the Ming Dynasty's "Zhengtong Daozang" Taixuan Department. Additionally, the "Wenyuan Ge Siku Quanshu" Jingbu Yi Leiju edition was consulted, and this version is referred to as the "Siku Ben" in the following text.

《道德》數千言，吾教之所獨尊，古今未有能廢之者？然傳注層出，渺茫叢惑，莫適指歸，徒見多岐之紛紛也。

The Dao De Jing consists of only a few thousand words, yet it is the sole revered classic of our Taoist tradition. From ancient times to the present, no one has been able to discard it. However, through the ages, layers of interpretations and commentaries have continuously emerged, leading to vague and uncertain explanations, with many doubts arising. No one can point out a definite conclusion, and what readers encounter is only a multitude of divergent paths, with no clear resolution.

雷思齊嗜學有要，精研是書，探竅本旨，為之傳釋，合儒、老之所同，歷詆其所異，條分緒別，終始一貫，不翅入老氏之室，避之席以相授受也。

Lei Siqi was diligent in his studies and skilled at grasping the essentials. He thoroughly researched the "Dao De Jing," seeking out its key principles and providing an interpretation. He integrated wherever Confucianism aligned with Laozi's teachings, and clarified differences where they diverged. His entire work was well-organized, clear in its structure, and consistent throughout. It was not just that he entered Laozi's hall of learning, but rather as if Laozi himself had stepped aside to personally teach him face to face.

其將學是者，終究其說，知其玄之玄，而不昧其所向，傳之將來，庶幾於吾教非小補也。至元丙戌，嗣天師簡齋張宗演序。

Those who study this text in the future, if they thoroughly understand its meaning, will be able to grasp the essence of Laozi's saying "darkness within darkness," and will not lose their way. Passing it on to future scholars will be a significant contribution to our Taoist tradition. This preface was written in the third year of the reign of Zhiyuan, by Zhang Zongyan, the successor to the Han Celestial Master, also known as Jianjia.

雷先生思齊，字齊賢者，臨川之高士也。遭宋亡，獨居空山之中，著《易圖筮通變義》、《老子本義》、《莊子旨義》凡數十卷，《和陶詩》三卷。去儒服，稱黃冠師，與故淳安令曾公子良、今翰林學士昊公澄相友善。四方名士、大夫慕其人，往往以書、疏自通，或聞其講學，莫不爽然自失。

Master Lei Siqi, whose courtesy name was Qixian, was a man of high moral character from Linchuan. After the fall of the Southern Song Dynasty, he withdrew to live in seclusion in an empty mountain, composing several works including "The Comprehensive Meaning of the Diagrams and Divination in the Book of Changes," "The Original Meaning of the Tao Te Ching," and "The Meaning of the Zhuangzi," totaling dozens of volumes, as well as three volumes of "Poems in Accord with Tao Yuanming." He removed his Confucian robes and called himself "Master of the Yellow Crown," meaning he became a Taoist priest. He was on friendly terms with the late County Magistrate of Chun'an, Zeng Zibiang, and the present Grand Secretary of the Hanlin Academy, Wu Cheng. Scholars and officials from all over the country admired his character, often sending letters and messages to communicate with him. Those who had heard him lecture were left in a state of confusion and regret, realizing their own understanding was far inferior to his.

故翰林侍講學士袁公桶，博雅君子也，稱其所著書，援據切至感厲，奮發合神以窮變，盡變以翼道。且日：「知齊賢不如是，齊賢之意不明。」方今天下稱為斯文宗，莫先昊公；天下稱善著書，莫先昊公。亦曰：「與談《老子》甚契。」又稱其詩精深工緻，豪健奇傑，有杜、韓風。蓋皆以為知言。

The late Hanlin Shidejiangshi Yuan Jue was a learned and refined gentleman, who praised the books written by Leixian, citing arguments that were concrete and had the power to inspire and invigorate, enabling one to strive to explore changes and use them to support the great Dao. Yuan Jue also said, "Unless one understands Qixian to this extent, one cannot properly explain the original meaning of Qixian." At present, there is no one recognized as the master of essays who can surpass Wu Cheng; nor is there anyone recognized as the most skilled in writing books who can surpass Wu Cheng. Wu Cheng also said, "Discussing the Book of Dao with Qixian was extremely in harmony." He also praised his poetry as profound and skillful, vigorous and unique, possessing the style of Du Fu and Han Yu. These comments can all be considered as knowing the true meaning of words.

嗚呼，古稱虞卿非窮愁不能著書，太史公世掌天官，使不遇禍，《史記》不作。夫求雷先生之志，讀雷先生之書，遑與世俗道哉。遑與世俗道哉。余讀其所著之書，想其為人幾五十年。今又得其詩文二十卷，於其徒孫傅性真與周惟和傳入京師來，因識以辭。至順三年三月揭俱斯書。

Alas! The ancients said that Yu Ke of the Warring States period would not have written his book had he not met with poverty and distress; Sima Qian of the Han dynasty, whose family had long held the office of celestial officials, would not have written the "Records of the Grand Historian" had he not suffered the calamity of Li Ling. To understand the aspirations of Le Xun and to comprehend his writings, how could one explain it to the common people? How could one explain it to the common people? I have read his writings and tried to imagine his character for nearly fifty years. Now I have again obtained twenty volumes of his poetry and prose, which were sent to the capital by his grand-disciple Fu Xingzhen and Zhou Weihe. Therefore, I have written these few words as a preface to the book. Written in the third month of the third year of Zhisun. Jie Xi Si.

昔世祖皇帝既定江南，首召三十六代天師入朝。未幾，天師奉旨掌道教。還山，遂禮請先生為玄學講師，以訓迪後人。餘時雖幼，而有志於學，遂受學於先生。

When the Yuan Shizu Kublai Khan pacified the Jiangnan region, he first summoned the 36th generation heavenly master Zhang Zongyan to the capital. Soon after, the heavenly master was entrusted with overseeing Taoist affairs. Upon returning to Longhu Mountain, he respectfully invited Master Lei Siqi to serve as a lecturer in the study of metaphysics, to guide and enlighten future scholars. Although I was still young at that time, I had already developed a strong interest in learning, and thus I studied under Master Lei Siqi.

先生嘗誨余曰：「文章於道一技耳。人之為學，將以明斯道也。不明斯道，不足以為聖賢之學矣。」餘由是日知所省，益自奮發。及弱冠，入侍先師開府公於朝，遂不得事先生以終學焉。

Master had once instructed me, saying, "For the great way, articles are merely a skill. People study in order to explain this way. If one does not understand this way, it is not enough to be called the learning of the sages." From then on, I gradually learned self-reflection and became more diligent. When I reached twenty and performed the coming-of-age ceremony, I entered the court to serve Master Zhang Liushun, and thus could no longer continue my studies under Master Lei.

先生嘗注《易圖筮通變義》、《老子本義》、《莊子旨義》凡數十卷，詩文二十卷。至順二年秋，先生之徒弟傅性真，遣徒孫周惟和持所注諸書，來京師示余，且日：「先生所注之書，雖有黃公震、曾公子良、昊公澄為之序，而未得公一言以發明之，敢以為請。」

Master Wu Cheng once annotated several volumes including The Meaning of Change in the Book of Changes, The Original Meaning of the Tao Te Ching, and The Meaning of the Zhuangzi, as well as twenty volumes of poetry and prose. In the autumn of the second year of the Zhi Shun era, his disciple Fu Xingzhen sent his disciple Zhou Weihe with these annotated works to the capital to show me, and said, "The books annotated by Master, although already introduced by three masters, Huang Zhen, Zeng Zilang, and Wu Cheng, have not yet received your words to explain their meaning. Therefore, we have dared to make this request."

餘日：「嗟夫，先生之歿，迨三十年矣。今不可得見，得見所注之書斯可矣。然先生之學，豈所注之書可能盡哉？後之觀者，考諸書則知先生之學，誠不止若是也。」餘今且老，久留於朝，常欲集先生諸文而序之，未果也。

I said, "Alas, the master has been gone for nearly thirty years now. He is no longer seen, but seeing his annotated works is enough. However, the master's learning cannot be fully exhausted by these few annotations. Later readers, if they study these works carefully, will realize that the master's learning was indeed far beyond these works." I myself am now also growing old, and having stayed in court for a long time, I have often thought of collecting the master's various writings to write an introduction, but have never yet completed it.

今諸文既已成編，又豈敢嘿嘿無一言以發明先生之學哉？故續為之序。先生諱思齊，字齊賢，學者尊之日空山先生，撫之臨川人也。至順三年，歲在壬申六月旦日，昊全節序。

Now that these writings have already been compiled, how could I dare remain silent and say nothing to promote the learning of the master? Therefore, I have written this preface in continuation. The master's name is Siqi, his courtesy name is Qixian, and scholars respectfully call him "Master Kongsan." He was from Lincun in Fuzhou. On the first day of the sixth month in the year Renshen of the third year of Zhi Shun, Wu Quanjie wrote this preface.

河圖，八卦是也。圖之出，聖人則之。包犧氏仰觀象，俯觀法，近取身，遠取物，以通神明之德，類萬物之情，始因之畫八卦以作《易》者也。孔子謂其則之，豈欺我哉？圖之數以八卦成列，相蕩相錯，參天兩地，參伍以變，皆自然而然。

The so-called "He Tu" refers to the Eight Trigrams. After the appearance of the He Tu, sages took it as their model. Fuxi, also known as Baoxi, looked up to observe the phenomena in the heavens and looked down to study the principles on the earth. He took the human body as a close model and all things as a distant model, thereby connecting with the virtue of the divine and comparing the forms of all things. Thus he drew the Eight Trigrams and composed the Book of Changes. Confucius said in the Great Treatise that sages "modeled themselves after it," meaning they took the He Tu as their model. Could this be a deception? The numbers of the He Tu arranged the Eight Trigrams in order, causing them to interact and intersect. The phrase "three to heaven and two to earth" means pairing three with heaven and two with earth to establish numbers. The phrase "three and five interacting to produce change" refers to the alternating interactions of three and five, all of which occur naturally, not by human arrangement.

後世不本其數，實惟四十，而以其十五會通於中，乃妄計天地之數五十有五，以意增制於四十之外，以求其合幸其中，故愈說愈迷，紛紛迄今。餘因潛心有年，備討眾說，獨識先聖之指歸，遂作《通變》，傳以與四方千載學《易》者，同究於真是焉。兼筮法亦乖素旨，附見後篇，求古同志宜能明其非敢誣也。大元大德庚子九月，臨川道士雷思齊齊賢序。

Later generations have not pursued the origin of the numbers in the River Chart, not knowing that its actual number is only forty, but instead have combined the fifteen numbers used in a vague way in the middle. They have recklessly calculated according to the fifty-five numbers of heaven and earth, and added their own interpretations outside of forty, hoping to match up and make a lucky guess. Thus, the more they explain, the more confused they become, and this confusion has continued up to the present day. Therefore, I have devoted many years to careful study, thoroughly examining the views of various schools, and have uniquely recognized the true intention of the ancient sages. I have thus written this book, "Tongbian," to pass it on to people in all directions and to those who study the Book of Changes in future generations, so that they may jointly investigate the true right and wrong of it. In addition, the current divination methods also deviate from the original intent of the ancient methods. I have attached my corrections in the following chapters, hoping that those who value the ancient methods will understand: these words are not something I dare to fabricate out of thin air. Written in the ninth month of the Gengzi year of the Great Yuan Dynasty's Dade era, by Lei Siqi, a Daoist from Linchuan, also known as Qixia.

河圖：河圖本數兼四方四維，共四十，員布為體，以天五地十虛用以行其四十，故合天地之數五十有五。

The first diagram is titled "He Tu". The He Tu includes the eight directions, namely the four cardinal directions and the four intermediate directions, totaling forty numbers, arranged in a circular form as its shape. Then, using the numbers "Heavenly Five" and "Earthly Ten", which are abstract and not associated with any specific direction, to operate upon these forty numbers, thus combining to form the fifty-five numbers mentioned in the Great Treatise.

河圖四十徵誤之圖：舊圖改作方體，且實以天五於中，故不可推用。今首出圓體，先征元誤，覽者詳審後圖，則自可見也。

The second diagram is titled "Diagram of the River Chart's Forty Errors." The old transmission of the River Chart was redrawn in a square form, and the "Heavenly Five" was actually placed right in the center of the diagram, making it impossible to explain or use. Now I first present the original circular version of the chart, pointing out the errors present in the old diagram. Readers only need to carefully examine the following diagrams, and they will naturally understand clearly.

河圖參天兩地倚數之圖、參伍以變錯綜數圖、參兩錯綜會變總圖。

The following three diagrams are respectively titled "The Diagram of the River Chart with Heavenly and Earthly Numbers," "The Diagram of Interchanging and Comprehensive Numbers," and "The Diagram of Combined Interchanging and Comprehensive Changes." The first diagram explains how numbers are established by pairing three with heaven and two with earth. The second diagram discusses the transformation of numbers through mutual interaction and comprehensive combination. The third diagram combines the methods of "three pairs" and "interchanging and comprehensive" to fully illustrate the changes. Only the titles of the diagrams are provided, and the actual figures are not included here.

則河圖以作《易》，其數之所由起乎？數之起，不過一陰一陽之道而已。《易》道之所以一陰一陽者，不過以奇耦之數互為分且合，以生且成而已。

The sage derived the principles of the Book of Changes from the River Chart. So where do the numbers of the Book of Changes originate? The origin of numbers is simply the principle that "yin and yang together are called the Way." The Way of the Book of Changes manifests as yin and yang because it uses odd and even numbers to separate and combine with each other, thus giving rise to "origin numbers" and "completed numbers."

陽奇陰耦，變而通之，繩繩兮雜而不亂者，不過《大傳》所謂天一、地二，以至天五且地十而已。《易》有太極；極，中也，一也；中自一也。是生兩儀；儀，匹也，二也；匹而二也。兩儀生四象，一、一不三、四分之以為四生數；四象生八卦，則六、七、八、九合之以為四成數也。

Yang belongs to the odd numbers, Yin to the even numbers. By transforming and connecting them, they form a continuous and intricate pattern that is neither chaotic nor disorderly. This is based solely on the "Great Treatise" which states that Heaven is one and Earth is two, continuing up to Heaven five and Earth ten. The "Great Treatise" says, "The Book of Changes has the Taiji." Here, "Taiji" refers to the center, which is one, and the center is originally one. It also says, "From the Taiji come the two principles." Here, "principles" refer to pairs, which are two, forming a pair being two. "The two principles give rise to the four images," which divides one, two, three, and four as the four "generative numbers." "The four images give rise to the eight trigrams," which combines six, seven, eight, and nine as the four "completed numbers."

四奇為陽，陽雖有生成之異，而各列於四方之正；四耦為陰，陰亦有生成之異，而同均於四維之偏，由正生偏，由偏成正也。一與三為奇，為陽之生數，而必待於六與八之陰數以為成。二與四為耦，為陰之生數，亦必賴於七與九之陽數以為成也。

There are four odd numbers which belong to yang, and although yang numbers are distinguished as "generative numbers" and "completing numbers," they are each arranged in the four cardinal directions of southeast, northeast, northwest, and southwest. There are also four even numbers which belong to yin, and although yin numbers are also distinguished as generative and completing, they are all evenly distributed in the four corners, the four oblique positions. The cardinal positions generate the oblique positions, and the oblique positions complete the cardinal positions. One and three are odd numbers, yang generative numbers, but they must rely on six and eight, the yin numbers, to complete them. Two and four are even numbers, yin generative numbers, but they also depend on seven and nine, the yang numbers, to complete them.

自北而東，一而三，乃陽生之進數；而其自八而六，東而北，陰成之退數者，固已寓乎其中。自西而南，二而四，乃陰生之進數；而其自九而七，南而西，陽成之退數者，亦兼具乎其中矣。四方者，各以其陽奇居於正；四維者，各以其陰耦附於偏。

Moving from the north to the east, advancing from one to three, this represents the order of the yang generating numbers moving forward; while moving from eight back to six, and returning from the northeast to the northwest, this represents the order of the yin completed numbers retreating backward, which is inherently included within it. Moving from the west to the south, advancing from two to four, this represents the order of the yin generating numbers moving forward; while moving from nine back to seven, and retreating from the south to the west, this represents the order of the yang completed numbers retreating backward, which is also included within it. The four cardinal directions each have their yang odd numbers in the central position; the four cardinal directions also each have their yin even numbers in the lateral positions.

然天數之有五，地數之有十，均合於陰陽之奇耦，而同謂之生成，乃獨無所見於四方之位，何也？四象元五，八卦元十故也。然《易》之所以範圍不過，曲成不遺者，正由假此天五與地十之虛數，以行其實用；於四象、八卦而成河圖者，特不當效世習置五虛點於圖中心，而附以十點，謬雲五十，而不知其用也。

But the five from the heavenly numbers and the ten from the earthly numbers also conform to the principles of yin and yang and odd and even, and are also called the generating numbers and the completing numbers. Yet they are not seen in the four cardinal directions. Why is this? Because the four trigrams originally do not include five, and the eight trigrams originally do not include ten. However, the Book of Changes is able to "encompass heaven and earth without exceeding, and thoroughly form all things without omission," precisely by borrowing these two abstract numbers, heavenly five and earthly ten, to carry out its practical functions. In the matter of matching the four trigrams and eight trigrams to form the River Chart, it should not imitate the common practice of placing five abstract points in the center of the diagram and adding ten points, mistakenly claiming that this is fifty. In reality, people do not understand its true use.

坎以一始於正北，而一五為乾六於西北；坤以二分於西南，而二五為兌七於西；震以三出於東，而三五為艮八於東北；巽以四附於東南，而四五為離九於正南。故陽得五而陰，耦得五而奇，陰得五而陽，奇得五而耦，是生數之所以成，成數之所以生者也。

The Trigram Kan begins with the number one at the direction of Due North. Adding five to one results in six, which corresponds to the Trigram Qian in the Northwest. The Trigram Kun has the number two residing in the Southwest. Adding five to two results in seven, which corresponds to the Trigram Dui in the West. The Trigram Zhen has the number three in the East. Adding five to three results in eight, which corresponds to the Trigram Gen in the Northeast. The Trigram Xun has the number four in the Southeast. Adding five to four results in nine, which corresponds to the Trigram Li in the South. Therefore, when the yang number five is added, it transforms into yin; when the even number five is added, it transforms into odd; when the yin number five is added, it transforms into yang; when the odd number five is added, it transforms into even. This is precisely why the generating numbers can form the numbers and why the formed numbers can generate the numbers.

生數少於五，元所待以制中也，故陽數之一與三，自為中於乾六艮八成數之外。陰眾而陽孤，坎故謂陷也。成數多於五，不能自適於中也，故陰數之二與四，取五於離九兌七成數之陽之內。陽正而陰偏，離故謂麗也。陷必止而求動，故帝出乎震。動所以取合，由分而也。麗必順而相說，故說言乎兌。說所以致分，由合而分也。

The generated numbers are all less than five, and there are no other numbers to rely on for establishing the middle position. Therefore, the yang numbers one and three can only seek the middle on their own outside the formed numbers of Qian six and Gen eight. At this time, yin is abundant while yang is isolated, so the "Commentary on the Hexagrams" refers to Kan as "陷" (falling into a pit). The formed numbers are all greater than five, and they cannot establish themselves in the middle by themselves. Therefore, the yin numbers two and four must take the number five from within the yang formed numbers of Li nine and Dui seven. At this time, yang occupies the central position while yin is in a secondary position, so the "Commentary on the Hexagrams" refers to Li as "丽" (adhesion), meaning attachment. A fall into a pit must first come to a stop before seeking movement, hence it is said "The Emperor arises from Zhen." Movement is about achieving union, moving from separation toward union. Adhesion must be harmonious and mutually joyful, hence it is said "Speech is expressed in Dui." Joy leads to separation, moving from union toward separation.

陽之進數坎，獨為生數之始，由北順行，東交於巽四；陰之進數坤，亦為生數之次，由西南逆行，東合於震三，是陰陽之生數三、四俱前而合。考於時，今四月為乾月，卦六陽之交；坤月，卦六陰之承，故有巽元乾，帝出震而齊乎巽也。

The yang progression starts with Kan one, acting as the beginning of the generating numbers, moving from the north eastward, meeting the southeast Wind four. The yin progression starts with Kun two, as the second generating number, moving from the southwest eastward, combining with the east Thunder three. Thus, the generating numbers three and four of yin and yang meet ahead. Observing the seasons: now in the fourth month, it is the "Heavenly Month," when all six lines are pure yang; in the tenth month, it is the "Earthly Month," when all six lines are pure yin. Therefore, in this passage, we see Wind but not Heaven, which matches the order in "Shuo Gu" that "The Emperor arises from Thunder, and unites with Wind."

陽之退數離，獨得成數之中，由南順行，西終於兌七；陰之退數艮，實始之由東北逆行，既北而西，究於乾六，是陰陽之成數六、七俱後而分。考於時，今十月為坤月，卦六陰之終；乾月，卦六陽載始，故有乾而元巽，說言乎兌，戰乎乾也。

The retreat number of Yang is Li nine, which alone occupies the central position of the formed numbers. It proceeds from the south towards the west, ending at the western Dui seven. The retreat number of Yin is Gen eight, actually starting from the northeast and moving backwards, passing through the north and then west, finally arriving at the northwest Qian six. This is why the formed numbers six and seven of Yin and Yang are both located at the rear and separate. Observing the seasonal periods: now October is the "Kun month," the end of the six pure Yin lines; April is the "Qian month," the beginning of the six pure Yang lines. Therefore, during this period, only Qian is visible and Xun is not seen, which corresponds to the order in "Shuo Gu" that says "it speaks of Dui, and fights with Qian."

天數始於一，則太極之全也，陽之正也；一析而二，則太極之分也，陰之偏也。一二參而三，則陽既唱而奇；一三轉而四，則陰遂隨而耦。由一與四唱於前，交對而為五，而其二與三隨於後，亦交對而為五，是兩其五。兩五相伍，則十也。參以天五，是參伍也。古文之五久，古今文之十皆十，是其象也。變而通之，生數之所以成，成數之所以生也。

The numbers begin with one, which represents the completeness of the Taiji and the proper position of Yang; when one splits into two, it signifies the differentiation of Taiji and the secondary position of Yin. One added to two makes three, so Yang leads and forms the odd numbers; one added to three becomes four, so Yin follows and forms the even numbers. One and four in front interact and harmonize to form five; two and three behind also interact and harmonize to form five, resulting in two fives. Two fives combined make ten. Then incorporating 'heavenly five,' this is called 'sanwu' (three and five). The ancient character for 'five' was written as '乂,' and the character for 'ten' has always been written as '十,' which is exactly their symbolic representation. By transforming and connecting these numbers, we understand the reason why generating numbers can come into being and why formed numbers can be generated.

坎一、巽四而五，故乾六、離九而十五也，合之而二十。坤二、震三而五，故兌七艮八而十五也，合之亦二十。是一、二、三、四之十成六、七、八、九之三十，故河圖之數止於四十，而虛用天五與地十，而為天地之數五十有五也。

Heavenly Stem Jia combined with Earthly Branch Yin gives five, so the corresponding Heavenly Stem Qian combined with Earthly Branch Li gives fifteen, and together they total twenty. Heavenly Stem Kui combined with Earthly Branch Zhen gives five, so the corresponding Heavenly Stem Dui combined with Earthly Branch Gen also gives fifteen, and together they total twenty. This is how the numbers one, two, three, and four form ten, generating the numbers six, seven, eight, and nine forming thirty. Therefore, the actual numbers in the River Chart only go up to forty. Then, by hypothetically adding Heaven Five and Earth Ten for use, it totals fifty-five as mentioned in the Great Treatise.

生數之二五而十，雖分陰分陽，而俱以生數之陽載成數之陰，以維北若東，而中由以立。兩其十五而三十，遂互陰互陽，而又以成數之陰奉生數之陽，以維南若西，而中所以行也。立者，寄於虛以體其實；行者，布其實以用其虛也。

In the numbers of generation, two fives combine to make ten. Although there is a distinction between yin and yang, both are carried by the yang of the generating numbers to sustain the north and the east, thus establishing the central position. Two fifteens combine to make thirty, and then yin and yang exchange roles, with the yin of the completed numbers serving the yang of the generating numbers to sustain the south and the west, thus enabling the central position to move. The term 'standing' refers to placing the center upon the void number to give it substance; the term 'moving' refers to spreading out the real numbers to fulfill the function of the void number.

一、二、三、四，交羅四象，而四互其虛五以為十，而為生數之體；六、七、八、九並包八卦，而四分其實五以為二十，而為成數之用；兩五之合十，固已旁正而交為二十之分體。二十之分五，乃離合而統為天五之全用；十交五體，五周十用也。

One, two, three, and four are arranged alternately to form the four images, these four numbers each combine with the hypothetical five to make ten, forming the essence of the generating numbers; six, seven, eight, and nine encompass the eight trigrams, these four numbers each take part of the practical five to make twenty, forming the function of the completed numbers. The ten formed by the combination of two fives originally extends to the four directions and occupies the four cardinal points, interacting to divide into the twenty parts; the five derived from the twenty then disperses and combines again, returning to the comprehensive function of "heavenly five." This is the interaction between the body of ten and the five, with the five circulating in the function of ten.

坎一、離九而十，坤一不艮八而十，震三一兌七而十，巽四、乾六而十，皆一生一成也。一必九，三必七者，四方四奇之十陽十也。二必八，四必六者，四耦四維之十陰十也。是四五本二十，今則四十四。伍而什，皆所以分中於成數之實十四；什而伍，又所以合中於生數之虛五也。是五五而十，十什而五，參伍錯綜，其數從橫斜正，倒之顛之，元不十五者也？

The Heavenly Stem Jia (1) pairs with the Earthly Branch Wu (9) to make ten, the Earthly Branch Ji (2) pairs with the Earthly Branch Gen (8) to make ten, the Earthly Branch Zhen (3) pairs with the Earthly Branch Dui (7) to make ten, and the Earthly Branch Xun (4) pairs with the Earthly Branch Qian (6) to make ten. Each pair consists of a generating number and a completing number. One must necessarily pair with nine, three must necessarily pair with seven; these are the four odd numbers from the four directions forming the number ten, which is the yang ten. Two must necessarily pair with eight, four must necessarily pair with six; these are the four even numbers from the four cardinal directions forming the number ten, which is the yin ten. Thus, the four fives, which were originally twenty, now become forty. Using five to combine into ten is to distribute the central position into the body of the completing numbers; using ten to combine into five is to gather the central position back into the empty five of the generating numbers. Therefore, five multiplied by five gives ten, ten combined with ten gives five, and the numbers three and five alternate and combine in all directions, vertically, horizontally, diagonally, and straight. No matter how they are arranged, they all amount to fifteen.

伍之而十，均十也。一與九之十，維坎維離。坎一居陰北，陽之始生，陰中之陽，陰極而生陽也。陽之一初而獨立於中。離九居陽南，陽之終極，陽、中之陽，陽極生陰，麗於巽四、坤二生數之陰。陰不專立，而當九陽之盛際，陽旱合而制中者也。《說卦》謂水火相逮而不相射者，冬、夏之至也。冬至坎之自中，夏至陽極而陰中也。

Two numbers combined to make ten, each group forms ten. One and nine combine to make ten, associated with Kan and Li. Kan one is located in the northern direction, which belongs to yin, being the place where yang begins to arise, yang within yin, precisely where yin reaches its extreme and yang emerges. This initial yang is independent in the central position. Li nine is located in the southern direction, which belongs to yang, being the ultimate of yang, yang within yang, where yang reaches its extreme and yin emerges, attached to the two yin numbers, Xun four and Kun two. Yin cannot exist alone; at the moment when nine, the extreme of yang, is present, yang has already combined with it and constrained the central position. The "water and fire approach each other without conflicting" mentioned in "Shuo Gu" refers to the winter solstice and summer solstice: the winter solstice is when Kan is in the center, and the summer solstice is when yang reaches its extreme and yin is in the center.

二與八之十，維坤維艮；三與七之十，維震維兌。陽自下而升，由北東而極於南；陰自上而降，由南、西而極於北。此其交進交退以適於中者，陰陽東西之所緯，往復啟閉之所塗，否泰通塞之期，而出震致坤，說言成言，而山澤通氣以為春秋之分，所以曲而暢、旁而通之也。

The number ten formed by two and eight is associated with the trigrams Kun and Gen; the number ten formed by three and seven is associated with the trigrams Zhen and Dui. Yang energy rises from below upwards, moving from the north through the east to the southern extreme; Yin energy descends from above downwards, moving from the south through the west to the northern extreme. This alternating advance and retreat, converging towards the central position, represents the meridians of yin and yang in the east-west direction. It is the path of continuous movement and transformation, the way of opening and closing, and the time of change between blockage and smooth flow. Thus there is the sequence: "The Emperor arises from Zhen," "He exerts his influence on Kun," "He expresses joy through Dui," "He completes his expression through Gen." Mountains and marshes communicate with each other, forming the vernal and autumnal equinoxes. This is precisely where the Way of the Book of Changes is winding yet unimpeded, reaching out and connecting thoroughly.

至於四與六之十，維巽維乾，此又陰陽之始終。終始，天地之窮變，變通莫不由此制用。何以明之？陽自乾、坎，而艮、震自後而前，以合於東南之巽。陰自巽、離，而坤、兌自前而後，而分於西北之乾。

The number ten, formed by four and six, is associated with Xun and Qian. This marks the end and beginning of yin and yang. Having both an end and a beginning, it represents the ultimate transformation of heaven and earth. All changes and adaptations are based on this principle. How can we prove this? Yang energy originates from Qian and Kan, progresses through Gen and Zhen from back to front, and converges at Xun in the southeast. Yin energy originates from Xun and Li, progresses through Kun and Dui from front to back, and disperses at Qian in the northwest.

陽之生數一而三，總而四，而巽兼之；陰之生數二而四，總而六，而乾兼之。四而倍分之，則八節，與經卦所以序者，類而長之也。六而倍分之，則六陰六陽，十有二月，與重卦之所以列者，引而伸之也。

The yang generating numbers are one and three, which together make four, and the Xun trigram possesses this number. The yin generating numbers are two and four, which together make six, and the Qian trigram possesses this number. Doubling four and dividing it results in eight solar terms, which corresponds to the order of the eight fundamental trigrams expanded. Doubling six and dividing it results in six yin and six yang, making twelve months in a year, which corresponds to the order of the sixty-four compound trigrams extended.

況乾之六雖二與四合而成，然以一合五亦成也。由五外交乎一而六，五猶生數，六已成數矣。乃自有生成之道焉。況合乎坎之一即中，故乾特以老陰而變九也。若巽但以一與三之生數而限之以四，陽氣不足於數，元以自達於中，故借數於離九之中五，乃始得中而與坎對。中以四合五，乃至於陽之極位，極而必反，是巽、離兼數，所以九為老陽而變六也。

Moreover, although the number six of Qian is formed by the combination of two and the Four Ministers, using one plus five can also result in six. Five, when combined with one externally, produces six. Five is still a number of generation, but six has already become a number of completion, and there is an inherent principle of generation and completion in this. Furthermore, since it combines precisely with Kan one, which is the center, Qian specifically uses the number six, the "old yin" number, to transform into nine. As for Xun, it is limited to the two generative numbers one and three, and thus is fixed at four. The yang energy is insufficient in quantity to reach the center on its own, so it must borrow the number five from Li nine's center in order to attain the center and correspond with Kan. After attaining the center, combining four with five leads to the extreme position of yang. When something reaches its extreme, it inevitably turns into its opposite, so Xun and Li both take their numbers, and nine becomes the "old yang" number and transforms into six.

是特以乾、巽對十，而九六之變為然。至於乾、巽之互為分合、始終，則有不止此。震三、巽四合而雷風相薄，見於《怛》卦。巽當四月，則己如前說。若兌七乾六分而西北，乾當十月，則又未易以乾、巽對十為說而止者。

The above explanation only discusses how nine and six change based on the fact that Qian and Xun form a total of ten. However, the relationship between Qian and Xun, where they both separate and combine, and are beginning and end to each other, goes far beyond this. When Zhen three combines with Xun four, it forms "Thunder over Wind," which is seen in the Hexagram of Constancy. Xun is paired with the fourth month, as previously mentioned. As for Dui seven and Qian six, they are located in the west and northwest respectively, with Qian paired with the tenth month. This cannot be fully explained merely by the fact that Qian and Xun total ten.

乾陽也，數乃陰六，而位於坎前；兌陰也，數乃陽七，而位於坤後。故於乾、兌六、七之中，各借五為之乘除，而變乾之六而於參伍之前，有一乾在坎前，是乾之一出而為坎之一；兌之七而於參伍之後，有二兌在坤後，是坤之二入為兌之二也。乾、兌雖以成數之六、七互分，而坎、坤乃以生數之一、二互起於中矣。

Heavenly Stem Qian is originally yang, but its number is the yin number six, and it comes before Kan. Dui is originally yin, but its number is the yang number seven, and it comes after Kun. Therefore, in the numbers Qian six and Dui seven, each borrows a five for addition and subtraction. When Qian's six is adjusted before "San Wu," it results in Qian being placed before Kan, which means Qian's one emerges to become Kan's one. When Dui's seven is adjusted after "San Wu," it results in Dui being placed after Kun, which means Kun's two enters to become Dui's two. Although Qian and Dui are separated by the numbers six and seven, Kan and Kun are connected by the numbers one and two, which arise from them.

然而兌不名二，而坤名二；坎名一，而乾不名一。乾之所以不自名其一者，蓋以寓其不入於數，而並包八卦之五與十生成之用，而為太極之全也。此以數言，則天一、地二，天地設位。以氣言，則分陰分陽，陰陽相薄。而《易》所以行乎其中者，始由於是，則陽生之進數所以為陰成之退數，陰生之進數所以為陽成之退數，乃互進互退，迭生迭成，此其生生之易，一陰一陽之道，元所終窮也。豈不至精至變而至神者與？

However, Dui is not called two, but Kun is called two; Kan is called one, but Qian is not called one. The reason Qian does not call itself one is to indicate that it does not fall within numbers, but instead encompasses the generative functions of five and ten from the Eight Trigrams, thus becoming the complete Whole of the Taiji. In terms of numbers, this is expressed as Heaven one and Earth two, with Heaven and Earth each occupying their proper position; in terms of qi, this is the division of yin and yang, with yin and yang approaching each other closely. The reason the Book of Changes can operate within this is because the generative numbers of yang advance precisely when the completed numbers of yin retreat, and the generative numbers of yin advance precisely when the completed numbers of yang retreat. Thus, they advance and retreat from each other, alternating in generation. This is the ever-renewing Book of Changes, the way of alternating yin and yang, endless and boundless. Is this not what the Great Treatise calls the most refined, the most changeable, and the most mysterious?

又況《易》以一其三畫，因得析而生八卦；二其三畫，遂得錯而成八八之卦。若其卦之成畫，則止於六矣。乃復於乾獨三，其三以至謂之九，何也？凡卦畫如圖，以五為中，必有九而後中，乃在五既中五之後。六為上爻者，蓋其卦氣之餘也。生數既終，成數攸始，此九、六虛實之所以寄也。

The Book of Changes uses three lines as a group, thus giving rise to the Eight Trigrams. By doubling the three lines to form six lines, they intersect to create the sixty-four trigrams. When considering the number of lines in a single trigram, it ends at six. However, in the case of the Qian trigram, the three lines are tripled and called nine. Why is this so? Because the positions of the trigrams are arranged like the River Diagram, with five in the center. Only with nine can the central position be clearly shown, and this center appears after five has already been placed in the middle. The sixth line is called the top line because it belongs to the remaining part of the trigram's energy. The generation numbers end here, and the completion numbers begin here. This is where the concepts of nine and six, as well as emptiness and reality, are established.

坤起於成數之六而實，乾極於成數之九而虛。六乃坤數成始，附於圖位，則以寄其乾九。九乃乾數成終，附於卦位，則以同於坤六。其六居退數之極，而進居乎坎中之前；九居進數之極，而退附乎離麗之後，故六於卦猶有位，九於卦已無位，而並寄位於六。而迭為九六者，九虛六實，因兼著其體用也。

The number six, representing the Earth Trigram, begins with the number six and is substantial. The number nine, representing the Heaven Trigram, reaches its peak at nine and is virtual. Six is the starting point of the Earth Trigram's number, and when it is placed on the position of the River Chart, it is used to symbolize the Heaven Trigram's nine. Nine is the end of the Heaven Trigram's number, and when it is placed in the position of the hexagram lines, it becomes the same as six. Six is at the end of the decreasing numbers, yet it advances to be before the坎 (Water) Trigram. Nine is at the end of the increasing numbers, yet it retreats to be after the position associated with the离 (Fire) Trigram. Therefore, six still has a position within the hexagram, while nine has no position within the hexagram and must be combined with six. The alternation of nine and six is due to nine being virtual and six being substantial, thus revealing both the essential and functional aspects.

且乾老陽也，坤老陰也，父母之謂也。少陰、少陽，則六子男女之謂也。少陰、少陽之數，則七、八也。乾元用九，坤元用六，合六十四卦，元有七、八，而唯有九、六。是男女者，亦由父母之所變，而隨以變者也。凡物之產，莫不由於母，而莫名其父之德之在，故《坤》猶見於六畫之體。而《乾》之九有其九用，而莫究其體之由，是所以神元方而《易》元體者也。

Moreover, Qian is old yang and Kun is old yin, referring to the parents; the lesser yin and lesser yang refer to the six children and the various hexagrams of male and female. The numbers of lesser yin and lesser yang are seven and eight. However, the hexagram Qian says "use nine" and the hexagram Kun says "use six." When considering all sixty-four hexagrams together, there is never a seven or eight, only nine and six. This shows that the hexagrams of male and female are also derived from the two parents' hexagrams, changing accordingly. All things are born from the mother's body, yet the father's virtue cannot be clearly identified. Therefore, the hexagram Kun can be seen in its six lines, while the nine in Qian has its nine uses, yet its origin cannot be traced. This is exactly the meaning of the Great Treatise that says "the spirit has no fixed form and the Book of Changes has no fixed body."

天一兼天五而六，地二合地四而六，五與六當十數生成互中之地，納甲以戊己為中，納戊以中乎坎，納己以中乎離。戊己，即五六也，置於坎、離之中，所以見其互中也。

Heavenly Stem one plus Heavenly Stem five equals six, and Earthly Branch two plus Earthly Branch four also equals six. Five and six are exactly at the midpoint within the number ten, where the generating numbers and the formed numbers are in mutual center positions. The Han dynasty's method of 'Najia' uses Wu and Ji as the central stems, placing Wu within the Kan trigram to occupy its center, and Ji within the Li trigram to occupy its center. Wu and Ji correspond to five and six, being placed within the Kan and Li trigrams respectively, which is used to demonstrate their mutual central position.

六合二四，為地數之成始，見於卦之《坤》曰：「龍戰于野。」即戰乎乾。疑於陽而嫌於元陽，見陰之有待於陽也，故其道窮。而《坤》之用六，不過日：「利永貞。」天數之一、五、九，是所謂始、中、終。九為天數之成終，見於卦之《乾》日：「亢龍有悔。」盈不可久，而不可為首；而《乾》之用九，則天下治，而見天則是見其行健，元所待於陰也。

Six is formed by two and four, representing the beginning of the earth number generation. This is manifested in the hexagram as the line statement of the Upper Sixth line of the Hexagram Qian, which says "the dragon contends in the wild," corresponding to the statement in "Shuo Guai" that "it contends with Qian." At this point, Yin is so strong that it resembles Yang, yet it still lacks Yang, clearly showing that Yin depends on Yang. Thus, its path has reached its end. The "use of Six" in the Hexagram Qian merely states "it is advantageous to maintain constant uprightness." In the heavenly numbers, one, five, and nine represent the beginning, middle, and end respectively. Nine is the end of the generation of heavenly numbers, manifested in the hexagram as the line statement of the Upper Ninth line of the Hexagram Qian, which says "the dragon reaches its zenith and incurs regret," indicating that fullness cannot last, hence it should not be the leader. As for the "use of Nine" in the Hexagram Qian, it says "the world is well governed" and "then the heavenly principle becomes evident," clearly showing its vigorous and strong movement, which does not depend on Yin.

一與九合為乾，則其數十，而十乃寄於天九之外，元所致其用矣。地道元成，而代有終，又有在於是，是築論九、六之大凡，可以意討也。河圖之列，只為經卦設，今而並及於重卦之體用也。經卦、重卦元往而不一陰一陽，由奇耦之分合，以生始成終者也。

One and nine combine to form Qian, but its number is ten, which can only be expressed through the nine of Heaven and cannot fully exert its function. The saying "the way of Earth has no completion but is succeeded by another end" is exactly this meaning. The above briefly discusses the general idea of nine and six; the rest can be inferred from this. The arrangement of the He Tu was originally for the eight fundamental trigrams, but now it also covers the principles and applications of the sixty-four combined trigrams. Whether it is the eight fundamental trigrams or the sixty-four combined trigrams, nowhere is there not a combination of yin and yang. All beginnings and endings arise from the division and combination of odd and even numbers.

一、二、三、四而五同為生數，一至四，其實體；五，其虛用也。六、七、八、九而十同為成數，六至九，其實體；十，其虛用也。四象元五，八卦元十，坦然明白矣。

One, two, three, four, and five all belong to the category of generating numbers: one to four are their actual forms, while five is their virtual use. Six, seven, eight, nine, and ten all belong to the category of completing numbers: six to nine are their actual forms, while ten is their virtual use. In this way, the number five is absent from the four yao, and the number ten is absent from the eight trigrams, which becomes clear.

四方各具十數，總而四十，四隅各包五數，共為二十。二十則四其五，而虛用者各居其半；四十者八其五，而實體者兼會其全，故四象合八卦，而總之則四十也。

Each of the four squares has the number ten, totaling forty; each of the four corners contains the number five, totaling twenty. Twenty is four fives, and the virtual numbers each take up half of it; forty is eight fives, and the actual numbers encompass all of it. Therefore, the four trigrams combined with the eight trigrams result in a total of forty.

五為立中之體而生，十為行中之用而成也；且五分於成數六、七、八、九之內，皆陰數少而陽數多。陰固偏，陽固正。五不得中而寄於四隅之偏，十合於生數一、二、三、四之外，皆陽數少而陰數多。陰亦偏，而陽亦正。十各成中而居四方之正，然則正則中，中則正也。

Five establishes the fundamental principle of the central position, thus governing 'birth'; ten carries out the function of the central position, thus governing 'completion'. Moreover, five is distributed within the four numbers of completion, which are six, seven, eight, and nine. These four numbers have fewer yin numbers and more yang numbers. Yin originally resides in the biased position, and yang originally resides in the central position, so five cannot attain the central position and can only rely on the biased positions of the four corners. Ten, on the other hand, is outside the four numbers of birth, which are one, two, three, and four. These four numbers have fewer yang numbers and more yin numbers. Yin also resides in the biased position, and yang also resides in the central position, so ten becomes the central position and resides in the central positions of the four directions. Thus, residing in the central position means attaining the middle, and attaining the middle means residing in the central position.

五始既所以正位於虛用之中，十中又所以中立於實用之外，皆所以為一陰一陽之中道也。由陰陽中既復有陰陽，此體用中所以復有體用虛實，實虛始終，終始變化，生成相與為元所終窮者也。

Five is already properly positioned in the beginning within the realm of emptiness, and ten is able to stand independently beyond practical use in the middle. This is all to achieve the middle path of yin and yang. Since yin and yang each contain their own yin and yang, there is also a distinction between body and function, as well as between emptiness and reality within the body and function. Reality and emptiness, beginning and end, change continuously, beginning again after the end, with generation and completion complementing each other, never ceasing.

又況五其五，而二十有五，亦天數也。六其五，五其六，三其十，而十其三，而三十，亦地數也。一且九，二且八，三且七，而四且六之各十，與本數虛用之十，以之伍其什，什其五，斯則大衍之數五十矣。而其兆始之一，寄中之五，藏其用於元形之天，則又出乎五十、十五之外，大衍所不得而用也。

Moreover, five fives make twenty-five, which corresponds to the heavenly numbers mentioned in the Great Treatise. Six fives, five sixes, three tens, and ten threes all add up to thirty, which corresponds to the earthly numbers. One pairs with nine, two with eight, three with seven, and four with six, each pair totaling ten. Adding the ten that is used in the middle, these tens are combined into fives, and the fives are combined into tens, resulting in the fifty numbers of the Great Derivation. As for the initial one and the five that occupies the central position, their function is concealed within the invisible heavens, thus going beyond fifty and fifteen, and are not used in the method of the Great Derivation.

又惡知不用之用，乃所以用大衍者，以是見一陰一陽之謂道，參伍而錯綜之。凡天地之數五十有五，所以成變化而行鬼神，有以見天下之跡，見天下之動，而知神之所為者，斯羲皇所以則河圖而作《易》者也。孔子贊《易》，謂其至精至變以至於至神。籲，斯其至神者矣。

How could one know that precisely this "useless use" is the fundamental principle driving the great derivation of numbers. Thus, "the way is the interplay of yin and yang" is fully expressed through the combination and interaction of a few and a few. The number of heaven and earth is fifty-five, used to accomplish transformation and movement, to serve ghosts and spirits, enabling people to see the forms and movements of the world, and thus understand the workings of the divine—this is the reason why Fuxi modeled the River Chart to compose the Book of Changes. Confucius praised the Book of Changes, saying it is most refined, most changeable, and even most divine. Alas, this can truly be called the most divine.

＃1『者終』，原作『終者』，據四庫本乙正。 ＃2『至元丙戌』，四庫本作『至元丙辰』，疑兩說皆誤。查《中國歷代紀年手冊》，元順帝至元年問，無『丙戌』和『丙辰』年，唯至正年問有『丙戌』年。 ＃3『切至』，四庫本作『精切』，疑當作『精切至』為宜。 ＃4『斯文宗』，四庫本作『斯文宗主』。 ＃5『但』，四庫本作『特』。 ＃6『祇』，原作『祇』，據四庫本改。

The first point: the two characters 「者终」originally read 「终者」, and the order has been corrected according to the Siku Quanshu edition. The second point: 「至元丙戌」in the Siku Quanshu edition reads 「至元丙辰」, and it is suspected that both versions may be incorrect—checking the Handbook of Chinese Dynastic Years, there was no year of Bingxu or Bingchen in the Zhiyuan period of Yuan Shundi, but only the Zhiyuan period had a year of Bingxu. The third point: 「切至」in the Siku Quanshu edition reads 「精切」, and it is suspected that 「精切至」would be more appropriate. The fourth point: 「斯文宗」in the Siku Quanshu edition reads 「斯文宗主」. The fifth point: the character 「但」in the original text is actually 「特」. The sixth point: the character 「祇」originally had a distorted form, and it has been corrected according to the Siku Quanshu edition.

天地之理，未始不有數行乎其中。然或有餘於數，不足於數。唯其餘、不足而為之中制，故雖陰陽奇耦之數，有分有合，有虛有實，有進有退，有自然互相生成之中道焉。

The principles of heaven and earth have numbers operating within them everywhere. However, numbers sometimes have excess and sometimes deficiency. Because of this excess and deficiency, it is necessary to establish a central regulation. Therefore, although the numbers of yin and yang, odd and even, have separation and union, emptiness and reality, advancement and retreat, there is naturally a middle path among them, a mutual generation that is harmonious.

一陰一陽之謂道者，陰陽分一以為天地，上下合一以為乾坤，天地上下分為四方，而中元以合，乾坤陰陽合為四時，而中有可分，是同一道也。

What is meant by "One yin and one yang is called the Way": When yin and yang separate the "one," it becomes heaven and earth; when up and down combine the "one," it becomes Qian and Kun. Heaven and earth divided into the four directions, the central position cannot be combined again; Qian and Kun combined with yin and yang form the four seasons, yet the central position can still be divided further. These two are actually the same principle.

天地以形言，則南北為經，上天下地；而水、火、木、金之為形者，既可以圖而分，故日法象莫大乎天地，廣大配天地也。陰陽以氣論，則東西為緯，乾陽坤陰；而冬、夏、春、秋之為氣者，亦可以理而合，故日變通莫大乎四時，變通配四時也。於斯二者，元以異也。

Heaven and earth refer to physical forms, so north and south are the meridians, with heaven above and earth below. Water, fire, wood, and metal, being tangible substances, can be arranged separately on a diagram. Therefore, the Great Treatise says, "The model of all things is none greater than heaven and earth," and also says, "Greatness and breadth correspond to heaven and earth." Yin and yang refer to qi, so east and west are the parallels, with Qian (Heaven) belonging to yang and Kun (Earth) to yin. Winter, summer, spring, and autumn, as climatic phenomena, can also be logically combined. Therefore, the Great Treatise says, "Change and adaptability are none greater than the four seasons," and also says, "Change and adaptability correspond to the four seasons." These two aspects, in reality, have no distinction.

四方有中，中元定體，而四象八卦以之立；四時有五，五元定著，而四時八節以之行。孔子著《大傳》，不存河圖，而特存天地之數者，示其有得於數，則圖在其中矣。

In the center of the four directions there is a central position, which has no fixed form, and the four symbolic figures and the eight trigrams depend on it to be established. In the midst of the four seasons there is a five, which has no fixed place, and the four seasons and eight festivals depend on it to function. Confucius wrote the Great Treatise, without preserving the diagram of the River Chart, but specifically preserved the numbers of heaven and earth, intending to show that if one understands the numbers, the River Chart is naturally contained within them.

圖之數生以天一者，成以地六；生以地二者，成以天七；生以天三者，成以地八；生以地四者，成以天九，至謂天五地十之生成，則何自而求哉？按圖索駿也。

The numbers of the River Chart are such that whatever is generated by the Heavenly Stem Yi is completed by the Earthly Branch Wei; whatever is generated by the Earthly Branch Yin is completed by the Heavenly Stem Geng; whatever is generated by the Heavenly Stem Jia is completed by the Earthly Branch You; whatever is generated by the Earthly Branch Mao is completed by the Heavenly Stem Ren. As for the Heavenly Stem Wu and the Earthly Branch Shi, how they are generated and completed, and where they come from, it is like searching for a thousand-mile horse according to a map, which is a futile effort.

故制五於八卦之虛中，猶可容其數；而制十於其中，則元所容其數矣。是知數之有生有成，而不知其所以生且成者，徒得其貌，而元以得古聖人之所以為天地立心，為生民立極者，千載可為之長太息，則古猶今也。

Therefore, placing the number five in the empty center of the Eight Trigrams can still accommodate it; but if we place the number ten in the center, there would be no room left for it. Those who only know that numbers have origin and formation, but do not understand why they can originate and form, merely grasp a superficial outline and cannot comprehend the intention of the ancient sages to 'establish the heart for heaven and earth, and set the standard for the people.' A thousand years later, it is truly worth sighing about—this was the case in ancient times, and it is still the case today.

夫奇數皆天，耦數皆地，生以天者成以地，生以地者成以天，其為天若地，特陰陽之進退、出入，奇耦之體用，分合而已。《說卦》曰：「參天兩地而倚數，觀變於陰陽而立卦，發揮於剛柔而生爻。」《大傳》日：「參伍以變，錯綜其數。通其變，遂成天地之文；極其數，遂定天下之象。非天下之至變，孰能與於此？」其曲盡乎至妙之用者，河圖法也。

Odd numbers belong to Heaven, and even numbers belong to Earth; those generated by Heaven numbers are completed by Earth numbers, and those generated by Earth numbers are completed by Heaven numbers. What is called Heaven and what is called Earth are merely the advance and retreat, going out and coming in of yin and yang, and the substance and function, division and combination of odd and even. The "Sayings of the Trigrams" says: "By taking three from Heaven and two from Earth, the numbers are supported. Observing changes in yin and yang, the trigrams are established. Through the flexibility of strong and weak, the lines are generated." The "Great Treatise" says: "By combining three and five, changes are made, and the numbers are interwoven and combined. By comprehending the changes, the text of Heaven and Earth is formed; by reaching the extreme of the numbers, the images of the world are determined. If it is not the most changeable thing in the world, who could reach this level?" The method of the River Chart is precisely what can fully and intricately express this profound function.

數始於一，而中於五，而終於十。中於五者，分其四之偏；終於十者，合其八之正。蓋一之始，其全體元餘不足，不可得而用，而獨得於中。既以其一寄生於五，遂以其一寄成於十以終之，故四象有五，八卦有十，皆有數而元所定其位，以五生其偏，以十成其正，而所謂一者，遂莫知所尋，獨見於五，見於十之為中者以上齊焉耳。

Numbers begin with one and end with ten, with five in the center. The reason five is in the center is to assign the four cardinal directions their respective positions. The reason ten is the end is to combine the four cardinal directions into their proper positions. Because one is at the beginning, the whole is neither excessive nor deficient, so it cannot be used, and can only occupy the central position. It relies on five to speak of "generation" and relies on ten to speak of "completion," thus concluding. Therefore, the four trigrams are associated with five, and the eight trigrams are associated with ten. These are numbers without fixed directions. Five is used to generate the secondary positions, and ten is used to complete the main positions. The so-called "one" can no longer be found, and people can only see it together with five and ten in the center.

故嘗求其所以然而莫之得，適記兒時於牧堅問見所謂八格戲者，其局不過口中加十之文而已。時極厭薄之，以為至鄙、至賤，未嘗加之意也。不謂年瑜七十，乃知其然。所謂百姓日用而不知，終身由之而不知其道者，古之人豈欺我哉？

I once pursued the underlying reasons but failed to grasp them. At that time, I suddenly recalled a game called "Eight Sections" that I had seen as a child among shepherds. Its board was simply a square with a cross in the middle. At that time, I thought it very lowly and despised it, never giving it a second thought. Unexpectedly, at the age of seventy, I finally understood that it was precisely about this principle. The "Great Treatise" says, "The common people use things daily without knowing their meaning." Mencius also said that one follows a path all his life without knowing its principles. Could our ancestors have deceived us?

蓋今謂即古之○，今之五即古文之，今之十猶古文之十，而畫為十於中，此十五之法由之以有所見矣。況又十下畫一，則土文之為中者，汰可知已，故云中土也。

In general, the circle mentioned today is the ancient character 「○」; the modern character 「五」 is the ancient 「乂」; the modern 「十」 is still the ancient 「十」, and by drawing a cross in the square, the principles of five and ten become visible. Moreover, adding one horizontal stroke below 「十」 forms the character 「土」, which clearly shows why earth is considered the center, hence the term 「中土」.

其中直筆之，即可見南北為經，天尊地卑，天地設位；橫筆之－，則可見東西為緯，分陰分陽，陰陽成列。從而久之則四維之偏，各得其均；四方之正，亦隨以定其氣之互進互退，交相生成，率由五與十以立中道也。

This cross shape, with its vertical stroke, shows the north-south direction as the meridian, with heaven above and earth below, each occupying its proper position. The horizontal stroke indicates the east-west direction as the latitude, dividing yin and yang, and arranging them in order. Adding the diagonal strokes forming the character "乂", the four cardinal directions' secondary positions are balanced. The four main directions are thus established. The movement of energy advancing and retreating, mutually generating, is all determined by the numbers five and ten to establish the middle path.

故河圖之十五，大衍之五十，皆虛有其十與五之數，以為之用，而實元其位之體也。欲知十、五之道，所以為中，始於一，而中於五，而終於十者，十即一也，以十求一，則其一於此固亦可以目存而心得於中矣。

Therefore, the fifteen of the He Tu and the fifty of the Da Yan are merely hypothetical numbers set up to function, actually having no specific directions or forms associated with them. To understand why the numbers ten and five can become the center, just observe that it begins with one, reaches the center at five, and ends at ten—ten is actually one. By going from ten back to seek one, this one can then be seen with the eyes and understood with the mind.

因參稽自擬《易》以來，其會十久之為中為土者，惟司馬氏《潛虛》似為得之，然《港虛》乃以久著為基，十著為冢，而分列為五十五數以為名，則是其中者已先實之矣，非虛用也。與河圖虛五寄十之理，自不相入也。河圖之數，雖五十有五，而其實體之數正只四十，所餘十有五，特虛以用於四十之中，不過通為之數，而實非有體位者也。

Therefore I examined thoroughly all the ancient works that imitated the Book of Changes. Only Sima Guang's "Potent Hidden Emptiness" seems to have grasped the meaning of "ten" and "yi" as representing the center and earth. However, "Potent Hidden Emptiness" refers to the form of "yi" as "ji" and the form of "ten" as "jia," and arranges them into fifty-five numbers with names, which is equivalent to fixing the center position beforehand, rather than using it in a virtual way. This does not align with the principle of the River Chart, which virtually sets five and entrusts ten. Although the River Chart's numbers are said to be fifty-five, its actual numerical count is only forty; the remaining fifteen are merely virtual numbers functioning within these forty, merely a general count, without actual form or direction.

其曰參、日兩、日參五者，通其數之所變以用也。參也者，一二之所以變也。由一自分其一以為二，起自為之對，則見其二而不見其一矣。其一又自參出於二之中，故三也。於參之法是即為一小成，是三畫之立見人參於天地之中以生也。積三小成，參其三而合於乾元用九，是參之始制，即陽奇之所謂天者所由然也。

The so-called "参" (participation), the so-called "两" (two), and the so-called "参五" (participation of five) are all ways of applying the changes of numbers. The "参" refers to the reason why one and two can change. One divides itself into another to form two, which arises to be opposite to itself, so people only see two and not one anymore; that one then emerges again from the two, thus becoming three. According to the principle of "参," this is a small achievement, which is the formation of the trigrams with three lines, showing how humans participate in the workings of heaven and earth. Accumulating three small achievements, multiplying three by three to align with the "用九" (all nines) in the Hexagram Qian, this is the initial establishment of "参," which is why the yang odd number is called "heaven."

兩也者，由一生二，起而對，並以立，既以象兩儀，因之兩其二以象四象。夫一本自一而二，而參，其一以為三，況二本一之對，豈不得二其三而反對之乎？故汰而三其二以成六，即合於坤元用六，是又兩之始制，兼三才而兩之，故六因之六畫成卦，六位成章，即陰耦之所謂地者亦由以然也。

The concept of "two" arises from one producing two, which then stand in opposition and are placed side by side. Since it is used to symbolize the two principles (yin and yang), it is then doubled again to symbolize the four images. Originally, one becomes two, and then another one is added to make three; and since two is the opposite of one, can it not be doubled again to form six? Therefore, further development leads to three twos combining to form six, which corresponds to the "use of six" in the hexagram of "Kun." This is again the original principle of "two": combining the three elements and doubling them, hence six. From this comes the six lines in a hexagram and the six positions in a chapter, which is the reason why the even number six, representing yin, is called "earth."

由是參其兩二，而交久以合之，是伍也。兩其二伍，則又什矣。十特見於參、九之外，則又一之所由以寄者也。何以明之？參之，兩之，而伍而什之，皆人文之為。而自然與天地合者，參文之血斗，兩文之雨仄，至維伍維什，皆立人於維中維旁者，斯可默識古人立心立極之制，有在於是矣。

By combining two dual elements and crossing them into the shape of 「乂」, they form the number five; two fives then form ten. The number ten appears only in the characters 「参」 and 「九」, which is precisely where the number one is寄托ed. How can we prove this? By combining and pairing, forming five and ten, these are all human-made textual forms, yet they naturally correspond with heaven and earth. The structure of the character 「参」, the form of 「两」, as well as the writing of 「五」 and 「十」, all place the character 「人」 in the center or beside it. From this, one can silently recognize that the principles and standards of the ancients' hearts and ultimate standards are embodied in these forms.

必參其三三，兩其三二，是參兩之一小成矣。故必又參其三三，而參之為參天；兩其三二，而兩之為兩地，是天地之數元汰而合且分矣。

It is necessary to have three instances of the triple combination and three instances of the double combination, which constitutes a small achievement of the principle of '参两'. Therefore, it is also necessary to have three instances of the triple combination to form '参天' (reaching the heavens), and three instances of the double combination to form '两地' (two lands). In this way, the numbers of heaven and earth do not need to be added separately, and harmony and distinction will naturally arise.

乃以其初不入於用之一數，寄於十以行其中，而為之分且合，以成其中制者，是一也。而求於圖之數，如環元端，其一莫見何在，而所見惟十爾。夫參其三三，與兩其三二，合而為大。參兩，斯有以見天地渾成一始終者所以然矣。至所謂參伍錯綜，則未究也。

Thus, the initial "one" that was not used in the process is entrusted to the ten, and it operates within it, dividing and combining it, thereby achieving the principle of moderation in the center. This is precisely due to the "one." However, when searching for numbers in the River Chart, the numbers form a circular ring without a beginning or end, and the "one" itself cannot be seen; only the ten is visible. Three threes interrelate, and three twos interrelate in pairs, and together they form the whole. With the concept of "three interrelations and two pairings," it is sufficient to understand why the heavens and earth are unified and continuous throughout. As for the so-called "interweaving of three and five," this has not yet been fully explained.

其謂之參伍錯綜者，五以前之數少於五，元所用伍；五以後之數多於五，故參以五。伍於其中，錯且綜之，而行其制中之法也。然《大傳》於參伍以變，錯綜其數，亦必謂通其變，然後能成天地之文；極其數，然後能定天下之象，則知其未易知也。

The reason it is called "San Wu Cuo Zhong" is that all numbers before five are smaller than five and cannot be combined with five; therefore, a five must be introduced. By combining five with them, then crossing and integrating them, we implement the method of maintaining balance and restraint in the center. However, the "Great Treatise" states "San Wu yi bian, cuo zong qi shu," meaning that one must first understand its changes and then achieve the brilliance of heaven and earth; one must first exhaust its numbers and then determine the forms of all things under heaven. This principle is clearly not easy to comprehend.

然既教以參五錯綜，則安得不研求其所以然哉。因參稽自擊《易》以來，已幾二千年矣，學者元探索及之者，至於龍圖續始亦元發明，深切異之。心港力索，極深研幾，適然省悟，元元它事也。《大傳》日：「天數五，地數五，五位相得而各有合。」即圖之數矣。

Since the sages have already taught us "San Wu Cuo Zhong" (threefold combination and intermingling), how could we not investigate its underlying principles? I have thoroughly examined the interpretations of the Book of Changes since Confucius composed the Great Treatise, nearly two thousand years ago, and yet no scholar has explored this aspect. Even Chen Tuan's transmission of the Longtu (Dragon Chart), which discusses the continuation and beginning, has not uncovered this point. I find this deeply surprising. Therefore, I have devoted myself wholeheartedly to the study, exhaustively investigating the subtle and profound aspects, and suddenly I have gained an insight. In fact, there is no other way. The Great Treatise states: "The heavenly numbers are five, the earthly numbers are five, the five positions interconnect and each has its corresponding combination." This refers precisely to the numbers of the River Chart.

天一起於坎，地六附於坤，是一六之合，綜之則七也。參以天五錯伍於中，並列而求之，其首尾各一也。故坎之一數為陽，立極於北方之始；而乾之六數為陰，附於西北之維也。天三出於震，地八附於艮，是三八之合，綜之則十有一也。參以天五錯伍於中，並列而求之，其首尾各三也。故震之三數為陽，立於束方之正；而艮之八數為陰，而附束北之維也。是二者，先陽後陰。

Heavenly Stem One starts from the Kan trigram, and Earthly Branch Six is attached to the Northwest dimension. This is the meeting of One and Six, combining to form Seven. Then Heavenly Stem Five is interwoven and placed in the middle. When examined in sequence, the beginning and end each receive One. Therefore, the number One of Kan is yang, establishing its center in the beginning of the North; the number Six of Qian is yin, attached to the Northwest dimension. Heavenly Stem Three appears from the Zhen trigram, and Earthly Branch Eight is attached to the Gen trigram. This is the meeting of Three and Eight, combining to form Eleven. Then Heavenly Stem Five is interwoven and placed in the middle. When examined in sequence, the beginning and end each receive Three. Therefore, the number Three of Zhen is yang, established in the central position of the East; the number Eight of Gen is yin, attached to the Northeast dimension. These two groups both follow the order of yang first and then yin.

地二分於西南，為坤；天七列於西，為兌，是二七之合，綜之則九也。參以天五錯伍於中，並列而求之，其首尾各二也。坤之二數為陰，先處於西南之維；而兌之七數為陽，始立於西方之正也。地四偏於巽，天九正於離，是四九之，綜之而十有三也。參以天五錯伍於中，並列而求之，其首尾各四也。巽之四數為陰，退守束南之維；離之九數為陽，正南面而立於極位也。是二者，先陰後陽。

The second earthly branch is located in the southwest, which corresponds to the trigram坤 (Kun); the seventh heavenly stem is positioned in the west, which corresponds to the trigram兑 (Dui). This is the meeting of two and seven, combining to form nine. Incorporating the fifth heavenly stem, which is interwoven in the center, when examined in sequence, the ends each receive two. The number two of坤 belongs to yin and is first situated in the southwest direction; the number seven of兑 belongs to yang and is established in the central west position. The fourth earthly branch is located in the southeast, corresponding to the trigram巽 (Xun); the ninth heavenly stem is positioned in the south, corresponding to the trigram离 (Li). This is the meeting of four and nine, combining to form thirteen. Incorporating the fifth heavenly stem, which is interwoven in the center, when examined in sequence, the ends each receive four. The number four of巽 belongs to yin and retreats to the southeast direction; the number nine of离 belongs to yang and stands directly facing south in the extreme position. Both groups follow the pattern of yin first and then yang.

由是知乾、兌、艮、離之所以六、七、八、九，即坎、坤、震、巽之一、一不三、四也。故陽得五而陰，陰得五而陽；耦得五而奇，奇得五而耦。乾之六，乃一五而六也；兌之七，乃二五而七也；艮之八，乃三五而八也；離之九，乃四五而九也。

From this we can see that the numbers six, seven, eight, and nine assigned to Qian, Dui, Gen, and Li are precisely the numbers one, two, three, and four assigned to Kan, Kun, Zhen, and Xun. Therefore, when the yang numbers reach five, they turn into yin; when the yin numbers reach five, they turn into yang. When even numbers reach five, they turn into odd; when odd numbers reach five, they turn into even. The number six of Qian is formed by adding one and five; the number seven of Dui is formed by adding two and five; the number eight of Gen is formed by adding three and five; the number nine of Li is formed by adding four and five.

又由是知參伍以變，參乾、坎，參艮、震，自北而束；參兌、坤，參離、巽，自西而南。參而分，伍而合，是謂參伍以變。錯而分之，見其參；綜而合之，見其位，是謂錯綜其數。而錯綜者，即參伍也。散而錯之，如一、二、三、四之重；分斂而綜之，乃六、七、八、九之復合也，是謂五位相得而各有合，豈嘗如後世別置五虛點於圖心，而實謂天五哉？

From this we can understand what is meant by "参伍以变": combining Qian with Kan, and Gen with Zhen, moves from north to east; combining Dui with Kun, and Li with Xun, moves from west to south. When things are combined, they separate; when they are joined together, they unite. This is called "参伍以变." Interweaving and separating allows us to see "参"; integrating and gathering allows us to see the respective positions, which is called "错综其数." So-called "错综" is actually "参伍." When spread out and interwoven, they resemble one, two, three, and four overlapping; when gathered together and integrated, they form the composite numbers six, seven, eight, and nine. This is called "五位相得而各有合." How is this different from later practices that place five empty points in the center of the diagram and call it "天五" in a very concrete way?

故由是知河圖之數縱橫順逆，莫匪自然而然，不待人為牽合傅會以使然也。其進數之升，自北束極乎南；退數之降，自南西終極乎北，皆陽數行而陰自隨之。若陰數之進，則自坤二而巽四，退則自艮八而乾六，然陰不過奉陽而已，雖萬變亦一陰一陽而已爾。

Therefore, from this we can see that the numbers of the River Chart, whether arranged horizontally or vertically, in order or in reverse, nowhere are they not naturally formed, without the need for artificial connections or forced interpretations. Its advancing numbers rise from the north, passing through the east, until reaching the southernmost point; its retreating numbers descend from the south, passing through the west, and finally return to the northernmost point. These are all the movements of the Yang numbers, with the Yin numbers naturally following. As for the movement of the Yin numbers, they advance from the second position of the Earthly Branch (Kun) to the fourth position of the Wind Trigram (Xun); and when retreating, they go from the eighth position of the Mountain Trigram (Gen) to the sixth position of the Heaven Trigram (Qian). However, Yin is merely following Yang. Even with myriad changes, it remains simply the interplay of one Yin and one Yang.

是故伍坎一以為乾六於北，伍坤二以為兌七於西，伍震三以為艮八於束，伍巽四以為離九於南，則見制數之元自參伍矣。於是又損震後之一以退，益艮之全八而九，三而三之，始參之制也。析震中前之二以合巽之全，四分而六，則兩之繼始也。惟離之全數自得於九，且當始終之中，三而三之，中參之制也。兼坤之全二，以貫兌中前之四合而六，則兩之代終也。絕兌後之三，以補乾之全六，三而三之，終參之制也。

Therefore, by combining the number one of Kan with five, it becomes the number six of Qian in the Northwest; by combining the number two of Kun with five, it becomes the number seven of Dui in the West; by combining the number three of Zhen with five, it becomes the number eight of Gen in the Northeast; and by combining the number four of Xun with five, it becomes the number nine of Li in the South. Thus, it can be seen that the numbers were originally derived from 'San Wu'. Then, by subtracting one from Zhen and adding it to the full eight of Gen, it becomes nine. Three multiplied by three, this is the beginning of 'San'. By separating the front two from Zhen and combining them with the full number of Xun, it forms six, which is the beginning of 'Liang'. Only the full number of Li naturally becomes nine, and it is exactly in the middle of the beginning and end. Three multiplied by three, this is the middle section of 'San'. By combining the full two of Kun with the front four of Dui, it forms six, which is the ending of 'Liang'. By cutting off the three at the end of Dui and adding it to the full six of Qian, three multiplied by three, this is the final section of 'San'.

是則參天兩地之綱領，於此亦可見。或錯而分，綜而合，參之兩之，參兩之制，亦必伍也。是則參伍之大通變、大極數者也。孔子之謂通其變，成天地之文；極其數，定天下之象。非天下之至精之至變之至神，孰能與於此。誠哉言也。

Thus, the principle of "three for heaven, two for earth" is clearly seen here. Either they are alternately divided or combined, either they are interrelated or paired, but the establishment of principles involving both interrelation and pairing must necessarily involve the number five. This is the greatest variation and the greatest extremity of "three and five." Confucius said, "To understand its variations and form the patterns of heaven and earth; to reach its utmost numbers and determine the images of the world." If this is not the most subtle, most changeable, and most mysterious aspect of the world, who could reach such a level? This statement is indeed accurate.

故夫參天者成象謂乾，於是參陽數之三三，以統乾元用九，而參中有兩，陽中見陰；兩地者效法謂坤，於是兩陰數之二二，效法於乾，遂參其二而六之，以會坤元用六，而兩中用參，陰中見陽，以是見一陰一陽，變而通之於天地之問，所以必參兩也。參之，兩之，所以必九六也。

Therefore, "参天" refers to the statement in the Great Treatise that "forming the image is called Qian." Thus, the yang number three is tripled to encompass the "use of all nines" in the Qian hexagram. The term "参" includes two, which signifies the presence of yin within yang. "两地" refers to "emulating as called Kun," so the yin number two is doubled to emulate Qian, and then the two are tripled to form six, which is used to combine with the "use of all sixes" in the Kun hexagram. The term "两" here requires "参," indicating the presence of yang within yin. Thus, the interplay of one yin and one yang pervades the heavens and the earth. Therefore, it is necessary to use "参两." And the combination of one "参" and one "两" inevitably results in nine and six.

且太極而兩儀，所以四象而八卦，不過生數之一、二、三、四，成數之六、七、八、九而已。至於五與十，雖有其數，特存虛用，未嘗列於卦象之實體者也。蓋自一分一而二，一二而三，一三而四，是四象立矣。四象元五也，天五特虛中而元體，然一四而五，二三亦五，是兩伍矣。具以本數之天五，又參伍矣。況一、二、三、四自然總而十數，中以天五，參伍而十五也。

From the Taiji to the two yin-yang, from the four images to the eight trigrams, it is merely the generating numbers one, two, three, and four, and the formed numbers six, seven, eight, and nine. As for the numbers five and ten, although they are also present, they are merely hypothetical and not included in the actual forms of the trigrams. Because one splits into one to form two, one added to two forms three, one added to three forms four, and thus the four images are established. The four images originally do not include five; heaven five is only a virtual position in the center without a physical form. However, one added to four makes five, and two added to three also makes five, which are the two fives. When combined with the original number of heaven five, it becomes "San Wu" (three and five). Moreover, the numbers one, two, three, and four naturally form the number ten, and when the heaven five is introduced in between, the combination of San Wu results in fifteen.

天數之一、三、五則九，地數之一不四則六，而一五亦六也。一六而七，二五亦七也。一七而八，三五亦八也。一八而九，四五亦九也。則八卦成列矣。一九則十，一、二、三、四自然亦十也。八卦則元自而十也，十故元所容其體，以一用其一不三、四，則其十亦特九也。十之為九，乃所以復歸於一，是十即一也。

In the heavenly numbers, one, three, and five together make nine, while in the earthly numbers, two and four together make six. One plus five is also six. One plus six is seven, and two plus five is also seven; one plus seven is eight, and three plus five is also eight; one plus eight is nine, and four plus five is also nine. Thus the eight trigrams are arranged in order. One plus nine is ten, and one, two, three, and four naturally add up to ten. However, the eight trigrams themselves cannot add up to ten, so there is no place for ten to take form. Using one to operate on two, three, and four, ten is only equivalent to nine. The reason ten becomes nine is that it returns to one, showing that ten is actually one.

河圖之數，四方各十，故坎一與離九相對而十，中虛天五之用而十五；坤二與艮八相對而十，中虛天五之用而十五；震三與兌七相對而十，中虛天五之用而十五；巽四與乾六相對而十，中虛天五之用而十五，以是知天五與地十皆元其體與位之正，特虛用其數，以相久十而什伍也。斯所以參伍也，參伍所以參兩也，故河圖之數正實四十，而以天五地十之虛數通用之以計，凡天地之數五十有五者也。

The numbers of the River Chart are distributed such that each of the four directions has ten: the number one of Kan and the number nine of Li face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen; the number two of Kun and the number eight of Gen face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen; the number three of Zhen and the number seven of Dui face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen; the number four of Xun and the number six of Qian face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen. From this we can see that the number five of Heaven and the number ten of Earth do not have their own forms or fixed positions; they are merely set as numbers for use. They rely on the '乂' shape and the '十' shape crossing each other to form five and ten together. This is called '参伍'; only with '参伍' can there be '参两'. Therefore, the actual numbers of the River Chart amount to forty. When the two imaginary numbers of Heaven five and Earth ten are added in, the total becomes fifty-five, the numbers of Heaven and Earth.

七為少陽，八為少陰，其合固成十五，然其生數不備，不能變。九為老陽，六為老陰，其合亦雖只十五，而生數之一、三、五而九，二、四而六，故六之變則九，九之變則六，此通重《易》之爻不見七八，而元非九六，而參伍以變，參天兩地，莫非是也。

Seven is the young yang, and eight is the young yin. Although their sum is also fifteen, their generating numbers are incomplete, so they cannot change. Nine is the old yang, and six is the old yin. Although their sum is also fifteen, nine is formed by the generating numbers one, three, and five, while six is formed by the generating numbers two and four. Therefore, six changes into nine, and nine changes into six. This is why, when examining the line statements of all sixty-four hexagrams, we never see seven and eight, but only nine and six. Also, the principles of 'three combinations to produce change' and 'three heavens and two earths' are all based on this reasoning.

四方之數各十，合參其九而二十有七，兩其六而十有二，總之則三十有九，而特遺其一，則其一將何所用乎？故以其一存諸十，則四十矣。故坎之立極於一，斂其十以自歸，以建大中，以生生數之始，以成成數之終。退藏於密，而莫之究極，見於參天兩地倚數之一以為十，則所謂中也。中也者，合數之所不得其中者，而使之交合於中，所以中其不中者也。

The numbers of the four directions each amount to ten. Multiplying nine by three gives twenty-seven, and doubling six gives twelve. Adding them together gives thirty-nine, yet there remains an extra one unaccounted for. This one is placed in the ten, making forty. Therefore, the trigram Kan is established with one as its central point, gathering the ten and bringing it back to itself, thus establishing the great central way, the beginning of the generating numbers and the end of the completing numbers. It retreats into secrecy, beyond the reach of human inquiry. It appears only when "the heavens are counted by three and the earth by two," using one to represent ten. This is called the "center." The center refers to gathering together those numbers that cannot naturally reach the central position, bringing them together at the center, so that even those not in the center can attain the center.

由坎北之一始制於中，二效之而分兌西之七以為中，三效之而分震束之三以為中，四效之而分離南之九以為中，凡中皆立於十，皆以數合於十而中分之者，蓋以寄其一云爾。惟坎之一，獨元所可用，特立於中而以元用之用而用，有用之用則中之用，不既大矣哉。故坎之一，雖不在參兩參五之用，而用之以制參天兩地參伍之制焉。故參天不自坎而自艮始也。

The first one from the northern Kan position first established the central position. The second followed it and took seven from the western Dui to form the center. The third followed it and took three from the eastern Zhen to form the center. The fourth followed it and took nine from the southern Li to form the center. All centers are based on ten, using numbers to make up ten and then dividing it evenly in the middle, in order to embody the principle of one. Only the one from Kan has no specific number to use, standing alone in the center, functioning through the use of something that appears to be of no use. This kind of useful function is precisely the function of the center, which is indeed great. Therefore, although the one from Kan is not included in the applications of the two and five, it is precisely through it that the principles of "three heavens and two earths" and "three and five" are established. Hence, "three heavens" do not start from Kan, but from Gen.

參亦非它，即前所謂一二之變也。一自分其一以為一，起而相對，則見其二，而不見一；其一又自參出於二之中，故曰參也。此參制之一小成，然參兩參伍，莫不於是焉起，斯故示其參之凡也。

The concept of "参" is not anything else but the change between one and two as previously mentioned. One splits off a part of itself to form another one, resulting in two opposing entities, so people only see two and not one. Then that original one re-emerges from the two, which is called "参." This is a small manifestation of the principle of "参," yet all instances of "参" and "参伍" originate from this. Therefore, here we first briefly explain the general idea of "参."

#1『也』，原脫，據四庫本補。 #2『謂』，四庫本作『之』。 #3『二其三』，四庫本作『三其二』。 #4『所』，據其上文義例補。 #5『莫匪』，四庫本作『莫非』。 #6『大極』，四庫本作『太極』。

The first point: the character 「也」 was originally missing, and has been added according to the Sanzhuo edition. The second point: the character 「谓」 in the Sanzhuo edition is 「之」. The third point: the phrase 「二其三」 in the Sanzhuo edition is 「三其二」. The fourth point: the character 「所」 has been added according to the meaning and style of the preceding text. The fifth point: the phrase 「莫匪」 in the Sanzhuo edition is 「莫非」. The sixth point: the term 「大极」 in the Sanzhuo edition is 「太极」.

道家以日月為易，始蹈常襲故，未始謂然，卒究於《大傳》則未始不然。日：「乾、坤，其《易》之蘊邪？乾、坤成列，而《易》立乎其中矣。乾、坤毀，則元以見《易》。《易》不可見，則乾、坤或幾乎息矣。」是猶未識所見之《易》為何物也。

Daoist scholars explain the character "Yi" by combining the sun and the moon. At first, I thought this was just following old interpretations and sticking to conventional views, and I did not agree with it. But after examining the Great Treatise, I realized that it might indeed be the case. The Great Treatise says: "The trigrams Qian and Kun are probably the essence of Yi? When Qian and Kun are arranged in order, Yi is established within them. If Qian and Kun were destroyed, we would not be able to see Yi; if we could not see Yi, then Qian and Kun would almost cease to exist." On their own, these words do not yet reveal what exactly the Yi is.

則又日：「乾、坤，其《易》之門邪？乾，陽物也；坤，陰物也。陰陽合德，而剛柔有體，□以體天地之撰，以通神明之德。」是《易》之為物有體可以見矣。則又日：「闢戶謂之乾，闔戶謂之坤，一闔一闢謂之變，往來不窮謂之通。見乃謂之象，形乃謂之器，制而用之謂之法，利用出入、民咸用之謂之神。」是《易》之謂體者，又見其有用矣。

The Great Treatise also says: "Qian and Kun are perhaps the gateways of the Book of Changes? Qian is the yang substance, and Kun is the yin substance. Yin and yang combine their virtues, and the firm and the yielding each have their forms, used to manifest the workings of heaven and earth, and to connect with the virtues of the divine." This shows that the Book of Changes, as a thing, has a visible form. The Great Treatise also says: "Opening the door is called Qian, closing the door is called Kun. One closing and one opening is called change. Coming and going without end is called communication. Manifesting is called image, taking form is called object, making and using it is called method, entering and exiting, taking and using it, with the people all using it is called spirit." This shows that the body of the Book of Changes also has function.

故曰：「天地設位，而《易》行乎其中矣。」日：「法象莫大乎天地，變通莫大乎四時，縣象著明莫大乎日月。」曰：「廣大配天地，變通配四時，陰陽之義配日月。」凡是數端，莫不舉日月以徵，則信其為易也完矣。

Therefore, the Great Treatise says: "Heaven and Earth each take their proper position, and the Book of Changes operates within them." It also says: "There is no greater model for taking inspiration than Heaven and Earth, no greater transformation and penetration than the four seasons, and no greater display of suspended images than the sun and moon." It also says: "Vastness and breadth match Heaven and Earth, transformation and adaptability match the four seasons, and the meaning of yin and yang matches the sun and moon." In all these instances, the sun and moon are cited as evidence. Therefore, the idea that the Book of Changes is associated with the sun and moon can be fully trusted.

夫《易》與天地準，能彌綸天地之道，仰觀象於天，俯觀法於地，則成象謂乾者日也，效法謂坤者月也。日象離數，極乎九；月象坎數，始乎一。凡九必對六，今雲一者，一合天五為六，坎一乾六為水生成，所以為月者也。

The Book of Changes corresponds with heaven and earth, encompassing the principles of the cosmos. Looking up to observe celestial phenomena and looking down to study earthly patterns, the phrase "the image is called Qian" refers to the sun, and "the pattern is called Kun" refers to the moon. The sun takes its image from Li (Fire), and its number reaches its extreme at nine; the moon takes its image from Kan (Water), and its number begins at one. By reasoning, nine should necessarily correspond with six. However, here the moon is said to be one because one added to the heavenly five makes six. The number one of Kan and the number six of Qian represent the generative number and the completed number of water, so they are used to refer to the moon.

且天地乃乾坤之體，乾坤乃天地之用，惟天地定位，而上下氣形有不易之分，故乾坤成列，而陰陽體用有變易之數。天地有常體，而莫見其用；陰陽有常用，而莫究其體，惟日月之體用兼著，故特著其義以配陰陽云爾。

Moreover, heaven and earth are the forms of Qian and Kun, while Qian and Kun are the functions of heaven and earth. Because heaven and earth each occupy their fixed positions, there is an unchangeable distinction between the upper and lower aspects in terms of qi and form. Therefore, Qian and Kun are arranged in order, and the nature and function of yin and yang thus have the capacity for change. Heaven and earth have constant forms, yet their functions are not visible; yin and yang have constant functions, yet their forms cannot be discerned. Only the sun and moon have both forms and functions clearly evident, so the Great Treatise especially highlights their significance to correspond with yin and yang.

且明教以範圍天地而不過，曲成萬物而不遺者，特通乎晝夜之道而知也。夫晝夜之道，豈難知哉？百姓日甩而不知爾。請略舉其凡：乾，陽物者，日其類也；坤，陰物者，月其類也。陰陽成列而合德者，四時合序，日月合明，亦類也。

Moreover, the Great Treatise clearly tells us: therefore it can "encompass heaven and earth without excess, and skillfully complete all things without omission," simply because it understands the principle of day and night. Is the principle of day and night difficult to comprehend? It is nothing more than something that the common people use every day without realizing it. Let me briefly illustrate its general meaning: Qian is a yang element, and the sun is its counterpart; Kun is a yin element, and the moon is its counterpart. When yin and yang are arranged in order and harmonized in their virtues, it is like the four seasons following one another in sequence, and the sun and moon combining their brightness, all belonging to the same category.

闢戶謂乾，近則日之出旦，月之上弦；遠則春之日夜分而謂啟也。闔戶謂坤，近則日之入夕，月之下弦；遠則秋之日夜分而謂閉也。仰觀其成象，則日之晝，月之望，夏之景長；而至俯察其效法，則夜之分，月之朔，冬之景短而至也。

"Opening the door is called Qian," near at hand it refers to the sun rising in the east and the moon waxing; far away it refers to the spring equinox when day and night are equal, which the ancients called "QI." "Closing the door is called Kun," near at hand it refers to the sun setting in the west and the moon waning; far away it refers to the autumn equinox when day and night are equal, which the ancients called "BI." Looking up to observe its "manifestation" aspect, it is day, full moon, and summer solstice when the shadow is longest; looking down to examine its "emulation" aspect, it is midnight, new moon, and winter solstice when the shadow is shortest.

是以日往則月來，月往則日來，日月相推而明生焉。寒往則暑來，暑往則寒來，寒暑相推而成歲焉。變化者，進退之象也；剛柔者，晝夜之象也。日月運行，一寒一暑故日：「天地之道，貞觀者也；日月之道，貞明者也；天下之動，貞夫一者也。」

Therefore, when the sun has passed, the moon comes, and when the moon has passed, the sun comes; the sun and moon alternate and thus light is generated. When cold has passed, heat comes, and when heat has passed, cold comes; cold and heat alternate and thus a year is formed. What is called change is the image of advance and retreat; what is called hardness and softness is the image of day and night. The sun and moon move, alternating between cold and heat, so the Great Treatise says: "The way of heaven and earth is to manifest through correctness; the way of the sun and moon is to shine through correctness; all movements in the world return to the one through correctness."

貞夫一也者，四時分至之中皆一也。而此特謂天下之動貞夫一，則正由坎之一也。一日之陽氣初動於夜分之際，固謂貞夫一也。一年之陽，氣復動於冬朔之始，尤見其貞夫一者也。

The phrase "Zhen Fu Yi" refers to the fact that the two solstices and two equinoxes within the four seasons all return to the principle of one. Here, however, it specifically says "the movement of the world is Zhen Fu Yi," which is referring to the "one" in the trigram Kan. Within a day, the yang energy first begins to emerge at the midpoint of the night, which is naturally called "Zhen Fu Yi." Within a year, the yang energy re-emerges at the beginning of the winter solstice and the first day of the month, which more clearly demonstrates its "Zhen Fu Yi."

坎之數一，離之數九，坎、離以一、九為乾。離為日，坎為月，日月必合於月之初一為朔。而日不云朔，月乃云朔；南不云朔，北乃云朔。夫朔逆度也，逆與日會，故謂之朔。以朔文觀之，明著月之逆度者也。《說卦》日：「《易》，逆數也。」《書》亦日：「平在朔。」《易》理實同也。

The number of Kan is one, and the number of Li is nine. Kan and Li combine the numbers one and nine to form Qian. Li represents the sun, and Kan represents the moon. The sun and moon must meet on the first day of each month, which is called "Shuo." The sun does not have a "Shuo," only the moon does. The south does not have a "Shuo," only the north does. The term "Shuo" refers to the retrograde movement of the moon, which moves backward to meet the sun, hence it is called "Shuo." From the structure of the character "Shuo," it clearly shows the retrograde movement of the moon. The "Say卦" says, "The Book of Changes is about reverse calculation." The "Shangshu · Yao Dian" also says, "Peace is in the month of Shuo," which refers to the same principle.

由是以進乎日月之為易，其縣象得不謂著明矣乎？故月之三五而盈，少陰、陽之七、八；三五而闕，老陰陽之九、六。總之，以地數三十而月盡，又河圖法也。

Thus, looking further, the sun and moon are the "Book of Changes" (Yi). They hang high and display clear signs, are they not the most evident? Therefore, the moon, after three periods of five days, becomes full, corresponding to the seven of lesser yin and the eight of lesser yang; then, after another three periods of five days, it wanes, corresponding to the nine of old yin and the six of old yang. In summary, using the earth number thirty, a month is completed, which is also the method of the River Chart (He Tu).

周天三百六十五度四分度之一，日每行一度，冬南、夏北為寒暑之中極，月行每十有三度而奇，與日之合者十有二朔，而歲常以備六陰六陽十有二月，天地節而四時成，是月之運行勞亦甚矣。其稱坎為勞卦者，不徒謂水也，此坎之一數不入於參兩、參伍之制，而特著其動之始也，中之始也。

The celestial sphere comprises 365 and 1/4 degrees. The sun moves one degree each day, and at the winter solstice it is偏 south, while at the summer solstice it is偏 north, marking the two central points of cold and heat. The moon moves thirteen degrees more than a day, and in a year it meets the sun twelve times, forming twelve new moons, thus making the year always have six yin and six yang, and twelve months. Heaven and earth have order and measure, giving rise to the four seasons. Thus, the moon's movement is indeed laborious. The "Say Divination" text refers to Kan as the "labor" trigram, not merely because it belongs to water. This is because the number one in Kan does not participate in the establishment of "three and two" or "three and five," but is specifically used to indicate the beginning of movement, that is, the beginning of the center.

合而言之，天地者體也，陰陽者用也。體者形，所以用其氣；用者氣，所以體其形。天以氣體其形，故其形下包乎地；地以形用其氣，故其氣上行乎天。是天之氣，全體乎地之形；而地之形，全用乎天之氣，此又參兩之制得以明之也。

In summary, heaven and earth are the "body," and yin and yang are the "function." The body refers to form, which is used to circulate its qi; function refers to qi, which is used to fill its form. Heaven uses its qi to fill its form, so its form envelops the earth downward; the earth uses its form to circulate its qi, so its qi ascends to heaven. Thus, the qi of heaven is entirely based on the form of the earth as its body, and the form of the earth is entirely based on the qi of heaven as its function—this can also be explained by the principle of "two principles."

一之始參，三之終參，皆天也，而並位乎平地之下。如：乾者，老陽也，乃附於成數之六陰，位乎陰地西北，偏極乎退數之終。艮者，少陽也，乃附於成數之八陰，位乎陽地東北，偏立乎進數之始。而特以坎中之一陽，不在參兩之用者，因其始生自然之一，介然特立於陰陽終始之際，起歲之朔，月之朔，以體生生之謂易，斯足以明地之體全用乎天矣。

The "San" (trinity) of the beginning and the "San" of the end both belong to Heaven, yet they are all situated below the horizon. For example, Qian is old Yang, yet it is attached to the even number six, which is a yin number, and it is located in the northwest, which belongs to yin, at the end of the retreating numbers. Gen is young Yang, yet it is attached to the even number eight, which is a yin number, and it is located in the northeast, which belongs to yang, at the beginning of the advancing numbers. However, the single Yang in Kan is not included in the use of San and San, because it is the original one that naturally arises, standing alone and distinct at the boundary where yin and yang begin and end, initiating the beginning of the year and the beginning of the month, thus embodying the idea that "the continuous generation is called the Book of Changes." This sufficiently demonstrates that the form of the earth entirely uses the qi of Heaven.

一始之兩，二終之兩，皆地也，而兼列於平地之上。如：巽，少陰也，雖以附地生數之四陰，位乃在東南，偏陽氣升已過中之進數。坤，老陰也，雖以附地生數之二陰，位猶在西南，偏陽氣降未及中之退數。且兼兩地並以奉中，參天位之極，當天之極高，日之極中，斯足以明天之用，全體乎地矣。

The "two" of the starting point of a structure and the "two" of the ending point both belong to the earth, yet they are both arranged above the horizon. For example, Xun is the Lesser Yin, although it is attached to the earth's generating number four, which is a yin number, its position is in the Southeast, leaning toward the side of the advancing number where the yang energy has risen past the midpoint. Kun is the Old Yin, although it is attached to the earth's generating number two, which is also a yin number, its position remains in the Southwest, leaning toward the side of the retreating number where the yang energy has not yet reached the midpoint. Moreover, they both take the numbers of the earth and jointly honor the center, assisting and complementing the extreme position of heaven, precisely at the highest point of heaven and the most central point of the sun. This sufficiently explains that the function of heaven is entirely based on the earth as its form.

以是知參兩之制，非有日以定其晝夜長短，有月以更其朔望贏縮，則天地之氣形體用，陰陽之變化生成，其升降進退之變通分合，出入之歸宿，則惡風而討其真是哉。

From this we can see that the system of "San Liang" depends on the sun to determine the length of day and night, and on the moon to mark the waxing and waning of the moon. Without these, how could we investigate the true nature of the qi and form, the function and application, the changes and generation of yin and yang, as well as the transformations of rising and falling, advancing and retreating, the combination and separation, and the return and movement of all things?

故嘗由是以參之制，數之元矣。天一之始，自北而中，而分地二於西南，不得謂天地定位者，陽正而下，陰偏而上也。二得名坤，一不得名乾者，一起於坎也。由一參二以為三，以前而東，蓋所以見人參於天地之中以生，而動於震，因兩其二而四，附以順而相入之巽，虛生其象以為四。然象非實體也，象天地與人，極男女之相配以始者爾。

Therefore, I have used the method of "参" to trace back the numbers. Heaven begins with one and starts from the north, moving to the central position, while Earth two is placed in the southwest. This still cannot be called "Heaven and Earth taking their positions," because Yang is in the proper position but below, and Yin is in the secondary position but above. The number two can be called Kun, but one cannot be called Qian, because this one originates from Kan. By combining one with two, we arrive at three, moving forward to the east, which shows that human beings are born in the center of Heaven and Earth, thus initiating movement at the position of Zhen. Then doubling two to make four, and adding the following and entering one of Xun, setting up its image virtually to form four. However, the image is not the actual thing; it is merely taking the form of Heaven and Earth and human beings, exhausting the meaning of the union of male and female as the beginning.

□一與三之陽，自然已各據其中，而二與四之未中，未有以正之可遂已乎？故有待取用於成數之交合，以成其正也。五雖虛位於中，而有其數可制其用。一交五，而以其一相配而成六於乾；二交五，而以其二相配而成七於兌；三交五，而以其三相配而成八於艮；四交五，而以其四相配而成九於離，然後始謂之成象。成象則謂之乾，而效法則謂之坤矣，乃得謂之天地定位也。

One and three, these two yang numbers, have already each occupied their central positions; but two and four have not yet reached the center. Is there no way to correct their positions, or must they remain this way? Therefore, it is necessary to use the combination of the completed numbers to establish their proper positions. Although five is merely a virtual occupant of the center, it has a number that can be used to set a standard: when one combines one with five, it pairs with one to form six in Qian; when two combines with five, it pairs with two to form seven in Dui; when three combines with five, it pairs with three to form eight in Gen; and when four combines with five, it pairs with four to form nine in Li. This is called "forming the image." With "forming the image is called Qian" and "emulating the image is called Kun," only then can it be said that heaven and earth are positioned.

其實一自合一，二自合二，三自合三，四自合四，特以交五而變爾。於是坎一之陽合而成乾六之陰，坤二之陰合而成兌七之陽，震三之陽合而成艮八之陰，巽四之陰合而成離九之陽，其始正者合而輔於偏，始偏者合而歸之正，於是八卦成列，而象在其中矣。亦惟一之自中，坎獨自若而不入於數，而河圖參天兩地，參伍錯綜之變，得以行乎其中也。

In fact, one corresponds with one, two with two, three with three, and four with four. It is only because they interact with five that changes occur. Thus, the yang of Kan one combines with the yin of Qian six, the yin of Kun two combines with the yang of Dui seven, the yang of Zhen three combines with the yin of Gen eight, and the yin of Xun four combines with the yang of Li nine. Those originally in the central position combine to assist the偏 position; those originally in the偏 position return to the central position. Thus, the Eight Trigrams are arranged in order, and the forms of things are contained within them. It is precisely because the number one remains in the central position that Kan remains unchanged and does not fall into the sequence of numbers, allowing the transformation of the River Chart "参天两地" (three heavens, two earths) and "参伍错综" (three and five interwoven and complex) to operate within it.

故由參天之制皆用九，而知九為《乾》之用，由兩地之制皆用六，而知六為《坤》之用。蓋乾之始參於艮八，而不足於數，則取震之一而九之中參，當陽極而體數之全；元餘不足，則自九之終參，則取兌過中有餘之三，以奉乾六之終。而九之兩之始起數於震中之二，以合巽四之不足。而六之兩之終起數於坤之本二，以合兌中有餘之四而六之，是明參兩九六所用之盤也。

Therefore, from the method of "ascending to heaven" which uses all nines, we know that nine is used for the Trigram Qian. From the method of "two places" which uses all sixes, we know that six is used for the Trigram Kun. Generally, Qian starts from the eight of Gen and ascends, but the number is not yet complete, so it takes one from Zhen to make up nine. The middle one, when combined, is exactly at the peak of yang and its number is complete, neither excessive nor lacking. The final one takes three from Dui, which exceeds the center, to complement the end of Qian's six. As for "two," the initial two starts from the two in Zhen to complement the lack of Xun four; the final two starts from the two inherent in Kun to combine with the excess four in Dui to form six. This explains clearly the calculations used in the参两 and 九六.

或有謂坤二與兌七而九，乾位自得六數亦自成九、六也，曾不識兩之制皆終地上，參之制皆終地下，又焉得而易之哉？況乾本九也，今數乃損而六；坤本六也，今數乃增之而九者，尤足以明老陰之六變而之九，老陽之九變而之六，而卦爻之所以成九、六也。

Perhaps someone might say: "The second line of the Earth (Kun) plus the seventh line of the Lake (Dui) equals nine, and the Heaven (Qian) position originally has the number six. Thus, it naturally forms nine and six." This view completely fails to understand that the principles of "two" are completed above the horizon, while the principles of "three" are completed below the horizon. How can they be arbitrarily exchanged? Moreover, Heaven (Qian) originally is nine, but now its number is reduced to six; Earth (Kun) originally is six, but now its number is increased to nine. This further demonstrates that old yin (six) changes to nine, and old yang (nine) changes to six. The reason why the lines of the hexagram are only called nine and six is precisely because of this.

由參天之用必九，兩地之體必六，《下繫》之首日：「八卦成列，象在其中矣。因而重之，爻在其中矣。」而後世乃謂《易》畫之重，非由羲始，吾不信也。

Because the function of "shantian" (reaching the heavens) must necessarily be nine, and the nature of "lidi" (two lands) must necessarily be six, therefore the beginning of the "Great Treatise" (Xici) states: "When the eight trigrams are arranged in order, the images are already contained within them. Then by stacking them again, the lines (yao) are already contained within them." However, later people have even claimed that the stacking of trigrams in the Book of Changes was not started by Fuxi, but I do not believe this.

且《說卦》謂說言乎兌，成言乎艮。凡再言其山澤通氣者，即《咸》卦也。《咸》之《象》日：「二氣感應以相與。止而說。天地感而萬物化生，聖人感人心而天下和平：觀其所感，而天地萬物之情可見矣。」

Moreover, the "Shuo Gua" says, "It speaks of Dui, and completes with Gen," and mentions "mountains and marshes communicate their energies" twice, which refers to the "Xian" hexagram. The "彖传" of the "Xian" hexagram states: "Yin and yang energies mutually respond and interact; Gen represents stillness while Dui represents joy. Heaven and Earth interact and thus all things are generated; the sage moves people's hearts and brings peace to the world. By observing where the response occurs, the conditions of heaven, earth, and all things can be understood."

其雷風相薄者，即《怛》卦也。《怛》之《象》日：「雷風相與，巽而動。天地之道，恆久而不已也。日月得天而能久照，四時變化而能久成，聖人久於其道而天下化成：觀其所怛，而天地萬物之情可見矣。」

As for "Thunder and Wind interacting," this refers to the Hexagram of Constancy. The Commentary on the Judgment of the Hexagram of Constancy says: "Thunder and Wind interact, with Wind being yielding and Thunder causing movement. The way of Heaven and Earth is constant and unceasing. The sun and moon follow the way of Heaven and Earth and thus can shine for a long time, the four seasons change according to the way of Heaven and Earth and thus can sustain all things, and the sage follows the way for a long time and thus achieves success in governing the world. By observing its constancy, the nature of Heaven and Earth and all things can be understood."

迦求於下經之人道攸始，則《說卦》重卦其旨一毫不悖也。故愚謂重《易》斷自羲始。不然，《禮》謂三《易》其經卦皆八，其別皆六十四，其欺我哉？求《易》之道，試以是進之。

Taking the lower part of the Book of Changes, which begins with human affairs, as evidence, it is clear that what is discussed in the "Shuo Gua" does not contradict the meaning of the compound hexagrams. Therefore, I believe the creation of the compound hexagrams definitely began with Fuxi. If not, then why does the "Zhou Li" say that the simple trigrams were eight and the compound hexagrams sixty-four for the three dynasties of Xia, Shang, and Zhou? Those who seek to understand the Way of the Changes should start from this point.

《說卦》日：「昔者聖人之作《易》也，參天兩地而倚數，觀變於陰陽而立卦，發揮於剛柔而生爻。」因參稽其所以參兩者，其得於河像素問然矣。且帝出乎震一章，尤河圖之大講明也。特截自震起於春中，環周而訖於艮，寓夏正之歲更爾。

The Sayings on the Trigrams say: "Former sages composed the Book of Changes, using three to correspond to Heaven and two to correspond to Earth, thus establishing numbers. They observed the changes of yin and yang and established the trigrams, and displayed the roles of firmness and flexibility to produce the lines." Investigating the basis of their statement "three for Heaven, two for Earth," it is originally derived from the image of the River Chart. Moreover, the passage "The Emperor arises from Zhen" further elaborates on the River Chart. It merely takes a section starting from Zhen, beginning at the middle of the spring equinox, and proceeds in a circle until it reaches Gen, thus symbolizing the order of the new year in the traditional lunar calendar.

故曰：「終萬物，始萬物者，莫盛乎艮。」日：「艮，東北之卦，萬物之所成終而所成始。」意蓋謂又始於出乎震者也。此實孔子特以變通四時言也，而於通乎晝夜之道，而知則未及悉也。

Therefore, the "Shuo Gua" says: "There is nothing more profound than Gen in bringing about the completion of all things and initiating all things." It also says: "Gen is the hexagram of the Northeast, the place where all things complete their end and begin anew." This means that after completion, one must begin again from "Chu Hu Zhen." This is indeed what Confucius particularly emphasized regarding the transformation of the four seasons. As for the aspect of "understanding the way of day and night," it has not been explained in detail.

夫兩之制，並起於平地，以奉中參者，自春而秋以分，必極於夏之至者，亦晝之象也。始終之參，皆在地中者，自秋而春以分，必極於冬之至者，亦夜之象也。

The method of "two" begins from the horizon, used to honor the central "San" (Three), which moves from the spring equinox to the autumn equinox, necessarily reaching its peak at the summer solstice, which represents the image of daylight. The starting and ending "San" are both below the horizon, moving from the autumn equinox to the spring equinox, necessarily reaching its peak at the winter solstice, which represents the image of darkness.

請以日用常行變而通之，則出乎震者，人之始作於日一也。古謂震日一所以驗也。齊乎巽者，動而相與之交也。相見乎離，則交際之極也。致役乎坤，則動極徐靜也。說言乎兌，休且入也。自震至兌，此晝之體用，人所同見。

Take everyday life and understand it through practical application: "Emerging from Zhen" means people rise and work at sunrise. The ancients said Zhen represents "sunrise," which confirms this. "Harmonizing with Xun" means becoming active and interacting with others. "Meeting at Li" means socializing and engaging in affairs at their peak. "Employing K'un" means movement reaches its peak and then gradually turns to stillness. "Speaking at Dui" means resting and preparing to return home. From Zhen to Dui, this represents the body and function of the day, something everyone can observe.

至於戰乎乾，卒取終參之制。其數之九，則存兌餘之三，嚮晦入息於乾六以終也。陰陽相薄，閉塞不用之時也。然終則有始，至於勞乎坎，則夜分之氣復動，準於七日之復則同也。成言乎艮，則遠接兌餘之三，說言乎兌也。山澤通氣，感而相與，雖止於其所而時行則行，斯人事之又興始者也。

As for "fighting in Qian," this refers to using the last "San" according to its method. Its number is nine, so it retains the three remaining from Dui. When people return indoors at night to rest, it concludes with Qian's six and comes to an end. At this time, yin and yang approach each other closely, which is a time of closure and inactivity. However, if there is an end, there must be a beginning. When it reaches "exerting effort in Kan," the energy of the middle of the night begins to revive again, which is similar to the principle of the hexagram "Fu" that "after seven days, the energy returns." When it reaches "forming words in Gen," it connects far back to the three remaining from Dui, which is "speaking in Dui." The energies of mountain and marsh communicate with each other, interact, and associate. Although they temporarily stop at their respective places, once the right moment comes, they resume their movement, which is precisely the beginning of human affairs once again.

變化者，進退之象也；剛柔者，晝夜之象也。吾見自羲、文以來，河圖之一陰一陽，進退升降，往復循環，周流不息；開古迄今，遠之一歲之冬夏，邇之一日之晝夜，莫不由斯道也。

What is called change is the image of advance and retreat; what is called yin and yang is the image of day and night. I have observed that since Fuxi and King Wen, the River Chart has shown the alternation of yin and yang, the movement of advance and retreat, the rising and falling, the continuous cycle and circulation, which flows without ceasing. From ancient times until today, in the distant case of the seasons of winter and summer, and in the near case of day and night, there is nothing that does not follow this principle.

吾日用常行，不在他求，參天之道自艮始者，於是乎在其視《太玄》以七百二十九，贊之準《易》，《皇極》以元、會、運、世之旁通，各以一期之歲、月、日、時為數例，視今通其變、極其數於河圖，日可見之行也。豈不易簡而理得哉？斯非至精變而神者與？

The Way is present in our daily activities and ordinary actions, and we need not seek it elsewhere. The principle of "reaching the heavens" begins with the trigram Gen (Mountain), and this is precisely the reason. Comparing this to Yang Xiong's "Tai Xuan" which uses 729 verses to approximate the "Yi," and to Shao Yong's "Huang Ji Jing Shi" which employs the concepts of Yuan, Hui, Yun, and Shi to interconnect and use cycles of years, months, days, and hours as numerical examples, we can now observe the changes on the River Chart and exhaustively explore its numbers. This is truly a practical way that can be seen every day. Is this not both simple and easy to understand yet in harmony with reason? Is this not the most refined, the most changeable, and the most mysterious?

#1『亦類也』，此三字四庫本缺。 #2『爻』，原作『××』，據四庫本改。 #3『毫』，原作『豪』，據四庫本改。 #4『春中』，四庫本作『泰中』。 #5『他求』，四庫本作『它求』。

The first point: the three characters "亦类也" are not present in the Siku Quanshu edition. The second point: the character "爻" was originally written with two ink blots, but it is now corrected according to the Siku Quanshu edition. The third point: the character "毫" was originally written as "豪", but it is now corrected according to the Siku Quanshu edition. The fourth point: "春中" in the original text is "泰中" according to the Siku Quanshu edition. The fifth point: "他求" in the original text is "它求" according to the Siku Quanshu edition.

河之有圖，處犧則之以畫八卦，文王因之以繫卦辭，歷代寶而傳之。自見於《書》之《顧命》陳之束序者，而今乃謂其元，誰敢哉？然自《記》以思古，則云其出語之傷，今則云其不出《顧》已皆意之，而非其真見矣。秦火之後，獨見於《書傳》孔安國之雲而已。

The River Chart appeared on the Yellow River. Fuxi took inspiration from it and drew the Eight Trigrams. King Wen based his Hexagram Statements on it. Through the ages, it has been regarded as a precious treasure passed down from generation to generation. It was clearly mentioned in the Book of Documents, Gu Ming, as being displayed in the Eastern Hall. Now, however, some people even claim that such a thing never happened. Who dares to make such a statement? Yet since the Rites of Zhou reflected on ancient matters, it expressed a sorrowful sentiment about the River Chart not appearing. Modern people say it was never mentioned in Gu Ming. These are all speculations based on personal opinion, not genuine evidence. After the Qin Dynasty burned books, the idea of the River Chart was only found in one sentence from Kong An Guo's Commentary on the Book of Documents.

《漢志》雜以圖書，而徒謂其與《書》經緯表裹，《易緯》則徒又枝辭蔓引，愈違本旨，以至芒乎莫之究其有元形似之真為何如者，訖於唐五季也。

The Book of Han's Catalogue of Arts and Literature discusses the River Chart and the Lo Book together, merely stating that they are interrelated with the Book of Documents as vertical and horizontal threads; books like the Yijing Wei further complicate matters by adding speculative interpretations, further deviating from the original intent. As a result, there has been confusion, with no one able to investigate whether the River Chart actually has a form and, if so, what its true appearance is. This situation continued until the late Tang and Five Dynasties period.

及宋之初，陳傳圖南始創古推明象數，閔其賤用於陰陽家之起例，而蕪沒於《乾鑿度》大一取其數以行九宮之法，起而著為龍圖，以行於世。愚幸及其全書，觀其離合出入具於制數之說，誠若剖心而有以求羲、文之心者也，然皇有不得於羲、文之心者。

In the early Northern Song dynasty, Chen Tuan (styled Tunan) began to develop his own original ideas and clarify the principles of numbers and forms. He lamented that the River Chart had been misused by the yin-yang school as an example for divination, thereby diminishing its true value, and had also been obscured by the method described in the "Yi Wei · Qian Zao Du" that "the Great One takes its numbers to move through the nine palaces." Therefore, he wrote the "Longtu" and passed it down to later generations. I had the fortune of seeing his complete work, and observed how he discussed the coming together and separation of numbers, which indeed seemed to reflect his heartfelt pursuit of the original intentions of Fuxi and King Wen. However, there were still aspects that did not fully align with the original intentions of Fuxi and King Wen.

於本圖之外，就以五十有五之數別出一圖，自標之以為形洛書者已，是其初之失也。雖其縱橫、錯綜、分合、體用之意，皆在本圖，而五十有五之數，既離而別出，世遂舍本逐末，因疑圖書並出，雖是若非，元以究其詣極，而徒長紛紛之論。

He also drew another diagram with the number fifty-five, labeling it as the shape of the Luo Shu, which was his initial mistake. Although the meanings of vertical and horizontal, reversal and combination, division and integration, and the distinction between form and function are still present in this diagram, once the number fifty-five is separated from this diagram to form another one, people tend to abandon the original and focus on the secondary. As a result, they doubt that the He Tu and Luo Shu appeared at the same time, leading to confusion and endless disputes that are difficult to resolve, thus unnecessarily fueling continuous controversies.

至其傳及劉牧長民，因之泛出五十五圖，名以《鉤隱》，則又以增異。而時則李觀泰伯即駁其非是，然又自謂存其三圖。及見此三圖，則其首即所傳今謂河圖者，其二即所別出五十五數、漫標謂形洛書者，其三又止於八卦方位而已。是長民不揣其本，其事固非，而泰伯亦元不識此之三本之則一，而妄析以為三爾。

It came to Liu Mu (style name Changmin), who drew fifty-five pictures based on it, naming his work The Diagrams of Hidden Numbers in the Book of Changes. This led to the proliferation of strange theories. At that time, Li Gou (style name Taibo) wrote an article to refute his mistakes, but Li Gou also claimed he retained three of the diagrams. When he saw these three diagrams: the first one was said to be the so-called River Chart of today, the second was another set of fifty-five numbers, arbitrarily labeled as the shape of the Luo Book, and the third was merely the diagram of the eight trigrams' positions. This shows that Liu Mu did not investigate the fundamentals, and his action was indeed wrong. Meanwhile, Li Gou also did not understand that these three diagrams were originally one, yet he mistakenly split them into three.

自時已後，愈傳愈失，愈說愈鑿，至有因河出圖、洛出書見於《大傳》，而並致疑《大傳》非孔子所作者。不知圖本非書，書本非圖。至其甚者，以五十五數之圖，乃妄謂之河圖，而以圖南所傳之河圖反謂之洛書，顛倒迷繆，靡所底止。

From this point on, the transmission became increasingly distorted, and the interpretations became more forced. Even some people, because the phrase "the river produced the diagram and the Luo produced the book" appears in the Great Treatise, began to doubt the Great Treatise itself, claiming it was not composed by Confucius. They did not understand that diagrams are not books and books are not diagrams. Worse still, they arbitrarily called the diagram with the number fifty-five the River Diagram, and mistakenly referred to the River Chart passed down by Chen Tuan as the Luo Book, thereby reversing and confusing the two, leading to endless confusion.

殊不明河圖八卦明著五十五數，實指以天數五、地數五；而其數自天一、地二以至天五、地十，合而五十有五，有體有用，有虛有實，原始要終，而盡發於一圖，豈嘗別求合於洛書哉？洛書亦豈嘗謂五十五數哉？

They had no idea that the River Chart and the Eight Trigrams originally clearly contain the number fifty-five, which actually refers to "five heavenly numbers and five earthly numbers." These numbers, from heavenly one and earthly two up to heavenly five and earthly ten, altogether make fifty-five. Within them there is substance and function, emptiness and reality, origin and conclusion, all fully displayed in a single diagram. Why then would they need to combine it with the Lo Shu? Has the Lo Shu ever claimed to be about fifty-five?

今圖南既別元義例辭說，誤以圖之五十五數，別標一圖以為洛書，是其傳疑之始也。原其初意，蓋由漢儒襲傳《洪範》初一之五行，其二曰火，四日金。《太玄》準《易》實本之，亦以二為火，為南；四為金，為西。今河圖乃置二於西南，置四於束南，是火、金改次矣。

Now that Chen Tuan has not given any further explanations or examples, yet mistakenly took the fifty-five numbers of the River Chart and marked them in another diagram as the Lo Shu, this has become the beginning of a doubtful case. Investigating his original intention, it is probably because the Han scholars had passed down the passage from the "Huang Fan" in the Classic of Documents, which says "the first is the Five Phases," where the second is fire and the fourth is metal. Yang Xiong's "Tai Xuan" imitated the "Yi," and actually based itself on this, also taking two as fire located in the south and four as metal located in the west. However, the River Chart placed two in the southwest and four in the southeast, thus changing the order of fire and metal.

既不敢遂改河圖，乃別以其五十五數析為洛書，而以《洪範》二火次於南，四金次於西，且以七隨二，九隨四，而易置其南西焉。以故長民不識其由，至謂火、金易位也。

He did not dare to alter the He Tu directly, so he instead took out the fifty-five numbers and used them as the Luo Shu. According to the Hong Fan, he placed the two fire elements in the south and the four metal elements in the west. He also arranged the seven following the two and the nine following the four, swapping the positions of the south and the west. Because of this, Liu Mu did not understand the reason behind it and mistakenly claimed that fire and metal had exchanged their positions.

夫離之數九，居正南，為火；兌之數七，居正西，為金，乃天地自然參五以變之數，斷斷元以易之，豈容以漢儒任意比校《洪範》火、金之二、四，得而移易之乎？況《洪範》只有五行之數，今增以七、九，又以七隨二，九隨四，又何所本哉？若校以河圖之例，七附二，九附四，是七稽疑當隨二五事，九五福當隨四五紀，則成何義類也？

One must know that the number of Li is nine, it resides in the direction of Due South, and belongs to the element of fire; the number of Dui is seven, it resides in the direction of Due West, and belongs to the element of metal: this is the natural number of heaven and earth, "interwoven by five and three," which is absolutely unchangeable. How can one allow Han dynasty scholars to arbitrarily associate it with the "Five Constants" in the "Hóngfàn" where fire is assigned the number two and metal the number four, and then move them around? Moreover, the "Hóngfàn" only has the order of numbers for the five elements. Now, however, the numbers seven and nine are added, and seven is made to follow two, nine to follow four. What is the basis for this? If we check according to the pattern of the He Tu, seven is attached to two, nine to four, then the seventh category "Kexi" would follow the second category "Wushì," and the ninth category "Wufu" would follow the fourth category "Wujì." What kind of reasoning or classification is this?

二、七、四、九，徒論其數，元形象，元方位可定指，空移易之則可；離、兌之有方所，火、金之有體用，豈天地之造化，亦遂肯依附人之作為，亦為之變移乎？今圖南不謂圖書之數，校然不可相同，而欲以背理之人為比而同之，截截自分界限，是得指失肩背矣。

Two, seven, four, and nine, if considered merely as numbers, have no definite form or direction, so changing them around is not a problem. However, Li (Fire) and Dui (Lake) each have fixed directions, and fire and metal each have their own substance and function. Can the workings of heaven and earth be so accommodating as to change according to human arrangements? Now Chen Tuan refuses to acknowledge that the numbers of the River Chart and the Lake Chart are clearly different, yet he tries to force them into agreement by artificial means, and arbitrarily draws boundaries. This is like being able to take care of one finger but losing the whole shoulder.

且河圖之出，非徒謂四象、八卦之具文也，天地之體用固猶是也。離南象天，離中乃陰，陰所以降；坎北象地，坎中有陽，陽所以升。震陽在初，所以象腸木之束升；兌陰在上，故以象陰金之西降。乾、坤所以體天地，陰、陽所以用造化，而二卦乃寄於西南、西北之維，習於常見，是天地易位矣，不惟是也。

The appearance of the River Chart is not merely presenting the ready-made patterns of the Four Symbols and the Eight Trigrams. The essence and function of heaven and earth are also contained within it. The Trigram Li resides in the south and represents heaven, yet its middle line is yin, which is why yin governs descent; the Trigram Kan resides in the north and represents earth, yet its middle line is yang, which is why yang governs ascent. The Trigram Zhen has its yang line at the very bottom, thus symbolizing the rising yang wood from the east; the Trigram Dui has its yin line at the very top, thus symbolizing the descending yin metal from the west. The Trigrams Qian and Kun are originally used to represent heaven and earth, while yin and yang are originally used to govern the workings of the universe. Yet these two trigrams are placed in the southwest and northwest directions. People have become accustomed to this and do not notice, but in reality, this already signifies a reversal of heaven and earth. Moreover, the issue is not limited to this.

筮法又以震、巽為木，乾、兌為金，坤、艮為土，而坎獨為水，離獨為火，則坎、離得專水、火，而乾、坤乃不得專天、地，亦猶火、金之不可以執方求定也。坤二位於西南，乃地二生火；巽四位於束南，乃地四生金，皆陰也。而兌七位於西，離九位於南，皆陽也。是生數皆陰，必待陽數以成之也。

Divination methods also assign Zhen and Xun to wood, Qian and Dui to metal, Kun and Gen to earth, while Kan is alone assigned to water and Li alone to fire. In this way, Kan and Li can exclusively represent water and fire, whereas Qian and Kun cannot exclusively represent heaven and earth. This is similar to how fire and metal cannot rigidly be confined to a single direction for identification. Kun, with its two positions, is located in the southwest, which corresponds to 'earth two produces fire'; Xun, with its four positions, is located in the southeast, which corresponds to 'earth four produces metal'. Both of these belong to yin. Dui, with its seven positions, is located in the west, and Li, with its nine positions, is located in the south; both of these belong to yang. Thus, the generating numbers belong to yin and must await the completion by the yang numbers.

地之生數自西而南，故坤而後巽；成數自南而西，故離而後兌。觀於火既熾而灰滅，是火既變而神化；金雖銷而質存，是金既化而精猶未變也。此火、金之所以互變化者，同而不同，故一不七、四、九之數，自是參伍錯綜之變，不可以執方究其竟也。

The earth's generating numbers move from west to south, so first comes Kun and then Xun; the completed numbers move from south to west, so first comes Li and then Dui. When fire burns extremely bright and then the ashes are extinguished, this shows that fire has transformed and its subtle nature is hard to find; although metal is melted, its substance remains, showing that metal has transformed but its essential quality has not changed. This is exactly where fire and metal transform into each other. They appear similar but are actually different. Therefore, the numbers two, seven, four, and nine are originally the "three and five interwoven and complex" changes, and one cannot rigidly pursue them to a single direction.

今五十五數之圖，以一、二、三、四置於四方之內，而以六、七、八、九隨置其外者，按其方而數之則可也。不知將何以循序迴環，以運行之乎？況不知五與十者，特有數寄於四方之位而虛用之也。

Now that diagram with fifty-five numbers places one, two, three, and four within the inner circle of the four directions, and six, seven, eight, and nine following them in the outer circle. Although counting them one by one according to the directions can be done, how exactly do they proceed in sequence, circulate, and flow? Moreover, the person who created the diagram did not understand that five and ten are merely numbers placed in the positions of the four directions as placeholders, waiting to be used.

且五與十雖謂土數，五行家於土，必以分王於四方辰、戌、丑、未之位；醫家謂土為脾，以五氣之運，每運七十二日，總三百六十為期之例，特以土之七十二日，四分之各十八日於四時之附末，謂為脾之主事。醫於人至為切已，以土之日四分而試之以為常驗，亦豈嘗專以土之五總之，而特設五於四方之外，與四方分位，而別立之五以為中五哉？

The numbers five and ten are called earth numbers, but when the Five Phases school talks about earth, it always says that earth rules the four directions, namely the months of Chen, Xu, Chou, and Wei. The medical school considers earth as the spleen, and according to the system of five cycles, each cycle lasts seventy-two days, totaling three hundred and sixty days. To conform to this system, they specially divide the seventy-two days of earth into four parts, each part being eighteen days, and attach them to the end of each of the four seasons, saying that this is the time period governed by the spleen. Medicine is most relevant to the human body, yet it still divides the days of earth into four parts for verification and regards it as a standard practice. How could it specifically place the number five of earth in a separate position outside the four directions, and set up a so-called 'central five' to stand equal with the four directions?

況筮法四營而成《易》，以十有八變而成卦；每變四營，總卦之成，凡七十有二營，亦與醫說胳合。今圖乃分五離立，而特設異五於中位，指以為中，則為此中者，又將孰適於用乎？故餘以為圖南之別出五十五數，標異謂之形洛書者，是其初作已自失之矣。

Moreover, the method of divination says "four operations form the Book of Changes, and eighteen transformations form a hexagram," with each transformation involving four operations, thus forming a hexagram through seventy-two operations, which coincides with the medical tradition. Now, this diagram separates the number five and places it independently, especially setting a different five in the central position, pointing to it as the "center." So, what practical use can this so-called center have? Therefore, I believe that Chen Tuan's separation of fifty-five numbers and his distinctive naming of it as the form of the Luo Book was already incorrect from the beginning.

《書》之九疇，各疇自有成數，如一五行，二五事，猶或得以五行、五用之數；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？

The Nine Categories of the "Nine Categories" in the "Book of Documents · Hongfan" each have a fixed number: for example, the first category is "Five Phases," the second is "Five Affairs," and perhaps the numbers of "Five Phases" and "Five Affairs" can be loosely connected. However, by the fifth category, "Huangji," it is already impossible to clearly identify what it refers to. Worse still, after the ninth category "Wufu" (Five Blessings), there is also "Liuji" (Six Evils) attached, so the question arises: should this be counted as nine, or should it be considered as five blessings combined with six evils?

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？強謂其合者，蓋其人之妄也心。

"The term 'Huangji' refers to the concept of 'Great Center,' but the 'ji' in 'Liuji' can also be called 'center'? The original 'Huangji' was neither a specific thing nor a specific matter, so it could be called 'center.' Now, if we just draw five points on the ground, can these five points become 'center'? The nine categories range from one to nine, clearly defined, each maintaining its original number. How can they be combined to form fifty-five? Those who insist they can be combined are merely speaking without basis."

若其後而至於以書謂圖，以圖謂書者，又妄人中之妄人也。此餘特謂圖則有數可通，而書則有疇類可數，而不可布之以為圖也。

Later, some people even claimed that the Luo Shu was the He Tu and the He Tu was the Luo Shu, which is the talk of the most foolish. Therefore, I specifically want to point out: the He Tu can be understood through numbers, while the Luo Shu can only be counted through its categories and cannot be arranged into a single diagram.

凡餘所以專守河圖者，非敢自謂親見羲、文所以本之而作《易》者，而其五且十之數，橫、斜、旁、正相生相成之進、退、贏、縮，一陰一陽之奇、耦、分、合，八體二用之虛、實、變、通，殆有造化神明莫窮之蕙，非人之所能為；而殆乎天機之自然者，宜其為羲、文所以作《易》之本原也。

I devote myself exclusively to the River Chart not because I dare to claim to have personally seen the original object upon which Fuxi and King Wen based the Book of Changes, but because its numbers five and ten, arranged horizontally, diagonally, sideways, and straight, mutually generate and complement each other, advancing and retreating, expanding and contracting. They embody yin and yang, with odd and even numbers, division and union. The Eight Trigrams form the body, while the numbers nine and six serve as the function, with emptiness and reality, change and communication. It almost contains infinite and boundless meanings of creation and divine wisdom, not something human could fabricate, but close to the natural workings of heaven. It should be the original source from which Fuxi and King Wen derived the Book of Changes.

竊嘗心潛力索，觸類引伸，變而通之，由門入蘊，謂獨得一全之體用，僭著其說。然與漢諸儒事事必強推五行，以求合於《春秋》災異，傅會五十五數以為洛書者，實不敢謂然，不敢同也。

I have privately devoted myself wholeheartedly to exploration, drawing connections and making extensions, transforming and integrating, entering from the gate into the hall, believing myself to have attained a complete system of principles and applications, and thus dared to write these statements. However, regarding the Han dynasty scholars who insisted on forcibly linking everything to the Five Phases in order to associate with the Spring and Autumn Annals' omens and disasters, and those who forced the number fifty-five into the Lo Shu diagram, I really do not agree, and I dare not approve of such views.

世恐未悉，況此河圖或謂由《易緯·乾鑿度》謂太一取其數以行九宮，四正四維皆合十五。鄭康成又注引其所以行九宮者，至於環中之虛位，本非謂五，而乃謂皇為五之中宮。固其《緯》、《注》俱妄，又不知其為五若十，皆虛用以制其奇耦生成，雖有其數，皇元其位也。

People may not yet understand this, and some even claim that the River Chart comes from the Yi Zhuan · Gan Zao Du, saying that the number of Taiyi is used to move through the Nine Palaces, with the four cardinal directions and the four intermediate directions adding up to fifteen. Zheng Xuan also made annotations and cited the method of the movement of the Nine Palaces, mentioning the empty position in the center, which originally does not refer to five. Yet he called it the 'Huang', meaning the central palace of five. Both the Yi Zhuan and Zheng's annotations are baseless. They also do not understand that whether it is five or ten, these numbers are merely hypothetical and used to constrain the generation of odd and even numbers. Although there is such a number, there is no actual direction associated with it.

然其所謂之太乙者，與太極則元以異也。苟未識其然，《記》日：「夫禮必本於太一，分而為天地，轉而為陰陽，變而為四時，列而為鬼神。」其理固與太極元以異也，自可旁通而元問然者；《記》經之屬也，因附諸《傳》後以徵。

However, what they refer to as "Taiyi" is essentially the same as "Taiji." If one still fails to recognize this, one may refer to the passage in the "Book of Rites: The Ritual Propriety" which says: "Ritual must originate from Taiyi, which differentiates into heaven and earth, rotates into yin and yang, transforms into the four seasons, and arranges into ghosts and spirits." This principle is fundamentally the same as Taiji, so it can be understood through analogy without further doubt. The "Book of Rites" is a classic text, so this passage is appended after the "River Chart Transmission" as a verification.

#1『擊』，原作『縣』，據四庫本改。 #2『芒』，四庫本作『茫』。 #3『大一』四庫本作『太一』。 #4『雖』，原作『維』，據四庫本改。 #5『鉤隱』，原作『勾隱』，據四庫本改。 #6『其三圖』，四庫本空缺此三字。 #7『說』，四庫本作『失』。 #8『校然』，四庫本作『較然』。 #9『未變』，原作『存變』，據四庫本改。 #10『妄人』，原作『妄之』，據四庫本改。 #11『為』，原脫，據四庫本補。 #12『為』，原脫，據四庫本補。

First: The character 「击」 was originally written as 「县」, but has been corrected according to the Siku Quanshu edition. Second: The character 「芒」 in the Siku Quanshu edition is written as 「茫」. Third: The phrase 「大一」 in the Siku Quanshu edition is written as 「太一」. Fourth: The character 「虽」 was originally written as 「维」, but has been corrected according to the Siku Quanshu edition. Fifth: The phrase 「钩隐」 was originally written as 「勾隐」, but has been corrected according to the Siku Quanshu edition. Sixth: The three characters 「其三图」 are missing in the Siku Quanshu edition. Seventh: The character 「说」 in the Siku Quanshu edition is written as 「失」. Eighth: The phrase 「校然」 in the Siku Quanshu edition is written as 「较然」. Ninth: The phrase 「未变」 was originally written as 「存变」, but has been corrected according to the Siku Quanshu edition. Tenth: The character 「妄人」 was originally written as 「妄之」, but has been corrected according to the Siku Quanshu edition. Eleventh: The character 「为」 was originally missing, but has been added according to the Siku Quanshu edition. Twelfth: The character 「为」 was originally missing, but has been added according to the Siku Quanshu edition.

古聖人之王天下，創始公至理，以惠啟千萬世者，豈易易然率己見以自擅制作哉？此有天授，斷可識矣。故河之出圖，伏羲因之以則而畫八卦；洛之出書，大禹因之以則而叔九疇。孔子於《大傳》叔天生神物，聖人則之，而並及圖書，而謂其兼則之也。子不語怪、力、亂、神，豈元所見於真實，而自樂為虛誕，以愚欺天下後世哉？

In ancient times, the sage rulers governed the world, establishing universal principles to benefit and enlighten countless generations. Would they lightly and arbitrarily create their own private views and fabricate things on their own? There is certainly a heavenly mandate involved, which can be clearly recognized. Therefore, when the Yellow River produced a diagram, Fuxi took it as a model and created the Eight Trigrams; when the Luo River produced a book, Yu the Great took it as a model and established the Nine Categories. In the Great Treatise of the I Ching, Confucius explained, "Heaven produces divine objects; sages take them as models." He also mentioned the River Chart and the Luo Book, meaning that sages took both as models. Confucius never spoke of strange phenomena, violence, disorder, or ghosts and spirits. Could he, without seeing real evidence, willingly fabricate false stories to deceive and mislead the world and future generations?

自稱庖羲氏之王天下，仰則觀象於天，俯則觀法於地，觀烏獸之文，與地之宜，近取諸身，遠取諸物，始作八卦，以通神明之德，以類萬物之情。是豈先聖、後聖各以私意淺識得措乎其間哉？《易》之廣大配天地，變通配四時，陰陽之義配日月，其為書廣大悉備，與乾坤相為終始而元弊，非有天授，能為是乎？此孔子有見於圖書之出，特謂則之而已，元它說也。

The Great Treatise says that when Fu Xi ruled over the world, he looked up to observe the phenomena in the heavens, looked down to observe the principles on the earth, observed the patterns of birds and beasts and the suitability of the land, took the image of himself nearby and the images of all things far away. Only then did he create the Eight Trigrams, to connect with the virtue of the divine and to compare the forms of all things. How could it be that the sage ancestors and later sages could each act on their own private views and shallow insights? The Book of Changes is vast and corresponds with heaven and earth, changeable and corresponds with the four seasons, and the meaning of yin and yang corresponds with the sun and moon. It is vast and includes everything, and corresponds with the beginning and end of the principles of heaven and earth without any faults. Could one achieve this without receiving it from heaven? Therefore, Confucius indeed saw the records about the River Chart and the Lo Book, and thus only said "the sage followed it," without any other words.

況圖之自見於成、康，《顧命》陳之束序之後，元所復聞。孔子固已自悲其不出，而有已夫之嘆。漢儒之始，孔安國乃由《書傳》於河圖束序之下，謂伏羲王天下，龍馬負圖出河，遂則其文，謂之河圖；

Moreover, the River Chart has been recorded since the time of King Cheng and King Kang of the Zhou dynasty. After it was mentioned in the "Gu Meng" as being displayed in the Eastern Hall, there is no further record of it. Confucius himself felt sorrowful about its disappearance and sighed, "Alas!" It was not until the early Han dynasty that Kong An Guo, a Han scholar, noted in his "Shangshu Zhuan" that when Fuxi ruled the world, a dragon horse emerged from the Yellow River carrying the chart, and Fuxi took its patterns as a model, which is called the River Chart.

於天乃錫禹《洪範》九疇之下，謂天錫禹洛書，神龜負文而出，列於背，有數汰一至九，禹因而第之以成九類，類即疇也。是二說自分載於《洪範》、《顧命》各篇本語之下，亦未之有它說也。

It also notes below the sentence "Heaven bestowed the Huafan Nine Categories to Yu" that Heaven gave Yu the Luo Book, which was a pattern on the back of a divine turtle, arranged in numbers from one to nine. Yu compiled these into nine categories, with "category" meaning the same as "category" in the text. These two accounts are originally recorded separately in the texts of Huafan and Gu Ming, and there are no other accounts elsewhere.

至鄭康成始引《春秋緯》合注《大傳》之圖書云：「河以通乾出天苞，洛以流坤吐地符。河圖龍發，洛書龜感。《河圖》有九篇，《洛書》有六篇。」已是詭說之開端；然仍引安國之注，以為河圖八卦是也，洛書九疇是也。其後王輔嗣、韓康伯於此並元所註釋，惟孔穎達《易疏》雖全篇寫鄭注，亦謂輔嗣之義未知何風，是必未以鄭注為然也。

Zheng Xuan was the first to cite the "Chunqiu Wei" in commenting on the "Great Treatise" mentioned in the "Zhou Changes," saying: "The He River rises to the Heaven Trigram and produces the heavenly bundle, while the Luo River flows down to the Earth Trigram and emits the earthly symbol. The He Diagram is initiated by the dragon, and the Luo Book is responded to by the tortoise. The He Diagram consists of nine chapters, and the Luo Book consists of six chapters." This already marks the beginning of obscure and strange interpretations. However, he still cited Kong An Guo's commentary, believing that the He Diagram is the Eight Trigrams and the Luo Book is the Nine Categories. Later, Wang Bi and Han Kangbo did not comment on this passage, while Kong Yingda's "Zhou Changes Justice" although fully quoting Zheng's commentary, also stated that Wang Bi's meaning was unknown in origin, indicating that he certainly did not endorse Zheng's commentary.

及其疏《書》之《洪範》，乃又引《繫辭》云：「河出圖，洛出書，聖人則之。」九類各有文字，即是書也。而云天乃錫禹，知此天與禹者，即洛書。就引《漢書·五行志》劉飲以為伏羲擊天而王，河出圖，則而畫之，八卦是也；禹治洪水，錫洛書，法而陳之，《洪範》是也。河圖、洛書，相為經緯；八卦、九章，相為表裹。

When Kong Yingda commented on the "Hongfan" section of the "Shangshu," he cited the "Great Treatise" saying, "The river produces a map, the Luo produces a book, and the sage follows them." He explained that the nine categories each have their own characters, which are referred to as "book." The text says, "Heaven bestowed the Book of Hongfan upon Yu," indicating that what Heaven gave to Yu was the Luo Book. He then cited Liu Xing's account from the "Hanshu · Wuxing Zhi," stating that Fuxi inherited the mandate of Heaven and became king; when the Yellow River produced a map, he followed it and drew the Eight Trigrams. Yu controlled the floods, and Heaven bestowed the Luo Book upon him, which he followed and recorded as the "Hongfan." The River Map and Luo Book form the vertical and horizontal threads, while the Eight Trigrams and Nine Chapters form the inner and outer parts.

先達共為此說。龜負洛書，經無是事。《中候》及諸《緯》多說黃帝、堯、舜、禹、湯、文、武受圖書之事，皆云龍負圖，龜負書。緯候之書，不知誰作，通人討竅，謂偽起哀、平，雖復前漢之末始有此書，已前學者必相傳此說，故孔以為九類是神龜負文而出，列於其背，有數仄一至九，禹見其文，因而第之以成九類法也。

Eminent scholars of the past all hold this view. In fact, the story of the divine turtle carrying the Luo Book is not mentioned in the canonical texts. The Shangshu Zhonghou and various ying books often talk about the Yellow Emperor, Yao, Shun, Yu, Tang, King Wen, and King Wu receiving the Tu and Shu. They all say that the dragon carried the Tu and the turtle carried the Shu. These ying and hou texts are unknown in origin, and learned scholars have investigated them, believing they were fabricated during the reigns of Emperor Ai and Emperor Ping of the Western Han dynasty. Even if these texts appeared only in the late Western Han, scholars before that time must have passed down this story orally. Therefore, Kong Anguo thought that the nine categories were formed when the divine turtle carried the patterns and arranged them on its back, with numbers from one to nine. Yu saw these patterns and thus arranged them into the nine categories.

言禹第之者，以天神言語必當簡要，不應曲有次第，丁寧如此，故以為禹第之也。

This is why it is said the order was compiled by Yu the Great, because the words of heavenly spirits must necessarily be concise and should not be arranged in a roundabout way or explained so meticulously. Therefore, it is concluded that this arrangement was made by Yu the Great.

其疏《顧命》之河圖束序，及疏《記》之出馬圖，詳略雖小有殊同，指意大校若此，是則所謂河圖為八卦，所謂洛書為九疇，並無改議也。

Kong Yingda, in his commentary on the sentence "The River Chart is in the eastern hall" from Gu Ming, and also in his commentary on the sentence "The River does not produce the Map of the Horse" from the Li Ji, although there are slight differences in detail, the general meaning is the same. This shows that the so-called River Chart refers to the Eight Trigrams, and the so-called Luo Book refers to the Nine Categories. This interpretation has never been changed by anyone throughout history.

由漢而唐，《易經》行世，凡經、傳、疏、釋之外，未有及於圖、書之文刊列經首者。迨故宋之初，陳搏圖南始創意推明象數，自謂因玩索孔子三陳九卦之義，得其遠旨，新有書述，特稱龍圖，離合變通，圖餘二十，是全用《大傳》天一地二至天五地十、五十有五之數，雜以納甲，貫穿《易》理。

From the Han dynasty to the Tang dynasty, the Book of Changes was widely circulated, and apart from the canonical text, commentaries, interpretations, and annotations, no one ever placed diagrams such as the River Chart and the Lo Book at the beginning of the text. It was not until the early Song dynasty that Chen Tuan, who was known by the courtesy name Tu Nan, innovatively emphasized the principles of figures and numbers. He claimed that after repeatedly pondering Confucius' three mentions of the nine trigrams in the Great Treatise, he grasped the profound meaning, and thus composed a new book, specially naming it The Dragon Diagram. The book discusses the combination, separation, transformation, and adaptability, containing more than twenty diagrams. These diagrams entirely adopted the fifty-five numbers from the Great Treatise, starting from Heaven One and Earth Two up to Heaven Five and Earth Ten, and incorporated the Han dynasty method of assigning stems, known as Najia, to connect the principles of the Book of Changes.

內一圖謂形九宮，附一圖謂形洛書者，則盡去其五生數，祇起地六至地十。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，再出兩圖，其一形九宮者，元無改異，標為河圖；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。

Among them there is a diagram called the Nine Palace Form, and another called the Luo Book Form, which removes all five generating numbers and starts from the earth number six up to the earth number ten. His own explanation is merely a few words: "Ten is used, ten forms the shape, so the 'Great Plan' states the numbers used for the Five Phases." At the end of the entire text, he lists two more diagrams: one of the Nine Palace Form remains unchanged from before but is now labeled as the He Tu; the other simply arranges the fifty-five numbers from the 'Great Treatise' in the four directions and the center, yet he boldly names it the Luo Book without any traditional rationale or explanation.

特移二、七於南，四、九於西，莫可知其何所祖法而作，而標以此名，大抵因如前說《洪範》五行之二火、四金，《太玄》準《易》因之為二南、四西而然。然其流傳未遠，知者亦鮮。

He merely moved the numbers two and seven to the south, and four and nine to the west, but no one knows what principle he used to create this diagram or why he gave it this name. Perhaps, as mentioned earlier, it is because the Five Phases in the "Hóngfàn" associate two with fire and four with metal, and Yang Xiong's "Tàxuán" follows the "Yì" by adopting this same system, placing two in the south and four in the west. Therefore, he arranged it this way. However, this system did not spread widely, and few people knew about it.

自圖南五傳而至劉牧長民，乃增至五十五圖，名以《鉤隱》。師友自相推許，更為唱述，各於《易》問有註釋，日《卦德論》，日《室中語》，日《記師說》，日《指歸》，曰《精微》，曰《通神》，亦總謂《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。

From Chen Tuan, the tradition passed down through five dynasties to Liu Mu, who was known by the name Changmin, and it was then increased to fifty-five diagrams, titled The Diagrams of Hidden Numbers in the Book of Changes. They mutually praised and responded to each other's teachings, passing them on, and each person had their own commentary on the Book of Changes. The titles of their works included The Discourse on the Virtues of the Hexagrams, The Words Within the Room, The Record of the Teacher's Teachings, The Return to the Origin, The Subtleties, and The Communication with the Divine. Collectively, they were also called The New Commentary on the Book of Changes. They always tried to present these matters and their actions as highly mysterious and divine, but upon close examination, there was not really anything truly miraculous to be found.

時則有李觀泰伯，著《六論》以駁其非是，至謂恩其志誤學子，壞爨世教，而刪其圖之復重存之者三焉：河圖也，洛書也，八卦也。夫長民之多為圖畫，固未知其是，而泰伯亦無未識此圖之三本之則一爾。河圖本列八卦，而數五十有五也。

At that time there was Li Gou, who took the name Tai Bo, and he wrote the "Six Discussions" to refute their errors, even saying that these people intended to mislead students and corrupt the world's teachings. Therefore, he deleted the repeated diagrams and kept only three: the River Chart, the Lo Shu, and the Eight Trigrams. In fact, Liu Mu's addition of more diagrams was not necessarily correct; yet Li Gou also did not understand, for these three diagrams were originally just one diagram—the River Chart itself already arranged the Eight Trigrams, and its numbers totaled fifty-five.

及長民輩始破洛書古說，謂非只是《洪範》，必別有書出於羲之上世，羲乃得而並則之以作《易》也。泰伯之見，則又與之同，此愚又莫能知何為其然也。

By the time of Liu Mu, the old views about the Luo Shu were first broken, and they believed that the Luo Shu was not only the Hong Fan but must have been another book that appeared in the time before Fuxi. Fuxi was then able to take both the Luo Shu and the He Tu as models to compose the Yi. Li Gou's view was also the same as theirs, and I really don't understand why they should see it this way.

蓋嘗夷考《大傳》此章，自有法象天地，變通四時，縣象日月，而又有及於蓄龜，故因謂之天生神物，聖人則之者，正謂曹也。《易》惟曹是用，所謂曹之德員而神，幽贊於神明而生曹，《易》豈有龜事哉？因其均可以占，故牽聯及龜爾。

I once carefully examined this chapter of the Great Treatise. It originally discusses the principles of the cosmos, the transformation of the four seasons, and the celestial phenomena of the sun and moon. Then it goes on to mention the yarrow stalks and the tortoise shell. Therefore, it says, "Heaven produces divine objects; sages follow them." Here, the term "divine object" clearly refers to the yarrow stalks. The Book of Changes only uses the yarrow stalks, as it says, "The virtue of the yarrow stalks is round and divine," and also states, "It is in the profound communion with the divine that the yarrow stalks arise." Where in the Book of Changes is there mention of the tortoise shell? It is only because the yarrow stalks and the tortoise shell can both be used for divination that the tortoise shell is mentioned by association.

以上文謂法象天地，變通四時，故申之曰：「天地變化，聖人效之。」上文謂縣象日月，故申之日：「天垂象，聖人象之。」至於天生神物，乃申謂之如河出圖，洛出書，聖人亦得以則之也。

Because the previous text discussed the principles of law and phenomena mirroring heaven and earth, and transformation in accordance with the four seasons, it then proceeds to explain: "Heaven and earth change, and the sage follows their example." Since the previous text mentioned the suspension of celestial phenomena such as the sun and moon, it then continues: "Heaven displays signs, and the sage imitates them." As for the phrase "Heaven produces divine objects," the explanation that follows means that events such as the Yellow River producing a map and the Luo River producing a book are also things that the sage can take as a model.

蓋由河之出圖，羲前既得以則而畫卦，因及於洛之出書，禹後復得以則而叔疇爾。又豈害於比類而互言之哉？豈嘗直欲以圖書之數、之義、之條貫比而同之哉？豈嘗直謂圖書並出於一時哉？

In general, this is because the Yellow River produced the diagram, and Fuxi came first, having taken it as a model to draw the trigrams. Thus, it naturally leads to the discussion of the Luo River producing the text, with Yu coming later, who took it as a model to establish the nine categories. By comparing and interweaving these two, what harm is there? Confucius never necessarily required that the numbers, meanings, and connections of the River Diagram and the Luo Book be entirely the same. Nor did he ever say that the diagram and the text appeared at the same time.

且不惟漢儒引《洪範》以為洛書，雖圖南之初謂形洛書者，亦不過謂十為用，十為成形，故《洪範》陳五行之用也者，是明指洛書為《洪範》矣。不知長民輩不本其初，故倍其師傳，而謂洛書非出於禹之時，益使後之人迷亂而失所據依，何也？

Moreover, not only did Han dynasty Confucian scholars use the "Hongfan" chapter to explain the Luo Shu, but even Chen Tuan's original statement about the "form of the Luo Shu" merely said, "Ten is used, ten forms the shape, so the 'Hongfan' chapter explains the use of the Five Phases." This clearly indicates that the Luo Shu is the same as the 'Hongfan'. It is surprising that Liu Mu and his contemporaries did not trace back to the source, but instead deviated from the teachings passed down by their predecessors, claiming that the Luo Shu did not originate in the time of Great Yu. This further confuses later generations and causes them to lose their basis. Why is this so?

又不謂孔子之謂聖人之則之也。非特去羲之時將幾世、幾年，雖祇去禹之世固已千數百年矣。其稱書之出洛，以比羲之圖之出河，於理惡乎而不可？長民故為此說，是且並孔子之《大傳》不之信邪。

They do not think about what Confucius meant by "the sage follows it." Confucius lived many generations and many years after the time of Fuxi; even if we count only from the time of the Great Yu, it would have been thousands of years. He speaks of the book emerging from the Luo River, comparing it with the map emerging from the Yellow River during Fuxi's time. How is that unreasonable? Liu Mu insists on this claim, but is he not then refusing to believe Confucius' Great Treatise?

考圖南之為龍圖，雖自謂得於孔子三陳九卦之旨而作，然其序日：「龍圖者，天散而示之，羲合而用之，孔默而形之。」且明稱始圖之未合，惟五十五數，則是謂《大傳》天數二十有五，地數三十，合而言之，不知何以於其末改標之以為洛書，殆其始誤也。

Chen Tuan's work, the Dragon Chart, though he claimed to have understood the meaning of Confucius's three times nine hexagrams, his own preface says, "The so-called Dragon Chart is dispersed by Heaven and displayed by Earth, Fuxi combined it and applied it, and Confucius silently revealed it." Moreover, he clearly states that the original chart had not yet been combined, only the number fifty-five, which refers to the number mentioned in the Great Treatise, that Heaven's number is twenty-five and Earth's number is thirty, combined together. It is really unclear why he later changed it to the Luo Book at the end of the book, which is probably the original mistake.

長民輩乃承誤而益增其誤，遂真以為洛書，而有五十五圖以實其誤；以至泰伯已後，悉以圖南標異之五十五數為真洛書也。不思圖南標洛書之始偶謂用十，夫洛書所叔九疇，其十數當何汰而起哉？

Liu Mu and his contemporaries inherited this mistake and made it worse, actually treating it as the Luo Shu, and even drew fifty-five diagrams to confirm this error; so from Li Gou onwards, everyone regarded the fifty-five numbers that Chen Tuan introduced as the real Luo Shu. They never considered: when Chen Tuan first referred to the Luo Shu, he only casually mentioned 'using ten'; yet the number ten in the nine categories described in the Luo Shu, from where should this number ten be counted?

意豈謂《大傳》之河、圖既可出而示世，則洛書亦可以並出示之，因假《大傳》五十五數，托為《書》之九疇乎？九疇所言皆用縱比而同之謂五十五數，其用又安在哉？此不可之大者也。

Are they not thinking that since the River Chart mentioned in the Great Treatise can be presented to others, the Lo Shu should also be presented? Therefore, they have borrowed the number fifty-five from the Great Treatise and falsely attributed it to the Nine Categories of the Book of Documents? The Nine Categories discuss practical applications. Even if one forcefully equates them with the number fifty-five, where would this number be applied? This is the most illogical point of all.

故嘗因長民之事討其原，圖南之後，种放、許堅、李溉，未及見其它有著述。若其所親授之師，如範誇昌所著《太易源流》，其稱龍馬負圖出河，羲皇窮天人之際，復位五行生成之數，定地上八卦之體，故老子自西周傳授孔子造《易》之原，天一正北，地二正南，天三正束，地四正西，天五中央，地六配子，天七配午，地八配卯，天九配酉，地十配中，寄於末，乃天地之數五十有五矣。

So I once investigated the origin of Liu Mu's theory. After Chen Tuan, the scholars Zong Fang, Xu Jian, and Li Gai did not have any other writings. As for Liu Mu's direct teacher Fan Echang, in his work "The Origin of Taiyi," he said: the dragon horse carried the diagram from the Yellow River, and Fuxi explored the connection between heaven and humanity, reordering the numbers of the Five Phases' generation. He established the forms of the Eight Trigrams on earth. Therefore, Laozi passed down the original principles of the Book of Changes to Confucius during the Western Zhou dynasty. These principles are: 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, Tian Wu (Heavenly Five) in the center. Di Liu (Earthly Six) corresponds to Zi (the first Earthly Branch), Tian Qi (Heavenly Seven) corresponds to Wu (the fifth Earthly Branch), Di Ba (Earthly Eight) corresponds to Mao (the second Earthly Branch), Tian Jiug (Heavenly Nine) corresponds to You (the eighth Earthly Branch), and Di Shiy (Earthly Ten) corresponds to the center, with the position of the center being assigned to the last position. This is the total of fifty-five numbers of heaven and earth.

因考其既以圖之前五數置於北、南、束、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？地十配中雲寄於末，夫中抑有末邪？

I therefore examined this: since he placed the first five numbers in the north, south, east, west, and center, yet assigned the last five numbers to zi, wu, mao, you, and center, why is this so? Zi, wu, mao, you are not precisely the four cardinal directions? It is also said that the ten heavenly stems are paired with the center and reside in the last position; does "center" have another "end"?

諦詳所置之數，正今圖所傳有四方而元四維之數者，是誇昌已元不識圖南所以標異，特因《太玄》準《易》取於《洪範》一水、二火、三木、四金、五土，而然鑿空元故，造端老子，增立怪論，以實圖南易置二、七、四、九之位爾，然猶未嘗以九疇分配九宮也。

Carefully examining the numbers he arranged, it is exactly the present version of the diagram that has only four directions and not the four cardinal directions. This shows that Fan Gechang originally did not understand why Chen Tuan was innovative. He merely fabricated this without cause, simply because the Tai Xuan followed the Yi and adopted the order of the Hong Fan, which lists one as water, two as fire, three as wood, four as metal, and five as earth. He even fabricated a strange theory, attributing it to Laozi, in order to justify Chen Tuan's arrangement of the numbers two, seven, four, and nine in those positions. However, he did not yet distribute the nine categories into the nine palaces.

究竅誇昌之取用，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、兌、坤於束，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。

He examined the methods used by Fan Echang, finding them to be merely remnants of the Najia stem assignment and an alternative approach called "five sons returning to Geng." He also stated that among the eight trigrams in the heavens, apart from Kan and Li which are positioned centrally opposite each other, the rest should be moved outward, placing Qian and Dui in the east and Gen, Zhen, and Xun in the west. He failed to understand that the theory of the Five Phases largely originated from the Book of Changes, yet he used the Five Phases as the fundamental principle for determining the trigrams, which was another bold innovation on his part.

是宜長民不獨增以五十五圖，又因誇昌坎、離、震、兌四正之外，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；震八數退本卦三數，以餘五數五畫為艮；兌九數退本卦三數，以餘六數六畫為坤，皆以數為畫，標為河圖，是不揣本而齊末。

It's no wonder that Liu Mu not only added illustrations to make fifty-five pictures, but also followed Fan Ech Chang's method of placing Kan, Li, Zhen, and Dui in the four cardinal directions. He equated the four complete numbers with the four directions, saying: subtract the three numbers of the primary hexagram from the number six of Kan, and use the remaining three numbers and three lines to form Qian; subtract the three numbers of the primary hexagram from the number seven of Li, and use the remaining four numbers and four lines to form Xun; subtract the three numbers of the primary hexagram from the number eight of Zhen, and use the remaining five numbers and five lines to form Gen; subtract the three numbers of the primary hexagram from the number nine of Dui, and use the remaining six numbers and six lines to form Kun. He consistently used numbers to represent lines and called it the He Tu. This is truly not measuring the root, but only seeking uniformity at the periphery.

夫八卦各三畫，以剛柔生爻，未聞本卦只用三畫，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、兌為四象，則尤非也。象本在未成卦之先，故曰：「四象生八卦」也。

The eight trigrams each consist of three lines, which are generated through the principles of yin and yang. There has never been a record of using only three lines for the primary hexagram and then using the remaining lines to form another hexagram based on their quantity. The numbers he arranged are merely arbitrary. Moreover, his classification of Kan, Li, Zhen, and Dui as the four trigrams is incorrect. The trigrams originally existed before the hexagrams were formed, which is why the Great Treatise states, "The four trigrams generate the eight trigrams."

又云：洛書之九疇，本河圖自然之數，虛皇極於中，而以八疇分佈四正、四維，五行置於坎一，五事置於坤二，五紀置於巽四，五福置於離九，一以九疇之次叔，陳列於河圖之卦次。夫九疇謂禹次第之者，直自初一、次一不次三、四而以次用之也。今隨河圖十五縱橫而置之，則成亂次矣。

He also said that the nine categories of the Luo Square are based on the natural numbers of the He Diagram, with the Emperor's Pole set in the center and the remaining eight categories arranged in the four cardinal directions and the four intermediate directions. The Five Phases are placed in the position of Kan one, the Five Affairs in the position of Kun two, the Five Records in the position of Xun four, and the Five Blessings in the position of Li nine, all arranged in the order of the nine categories on the hexagram positions of the He Diagram. However, the reason the nine categories are said to have been arranged by Great Yu is that they should be used in sequence, starting from the first day, then the second, then the third, and so on. Now, if we place them according to the fifteen grid pattern of the He Diagram, the order becomes completely disrupted.

未暇一一辯詁，且以初一之口五行言之。既謂五行，自當分配五方，何得以五者限萃一方？不以推行，惡得謂五行哉？以至謂火、金易位，與河圖不同，其顛倒迷繆不可訓知若此。

I didn't have time to argue point by point, so let's just take the first category, "Five Phases," as an example. Since it is called the Five Phases, they naturally should be distributed among the five directions. How could these five elements be confined and gathered in one direction? Without allowing them to move and transform, how can they be called the Five Phases? It even goes so far as to say that fire and metal have exchanged positions, differing from the He Tu diagram. The confusion and disorder have reached such an extent that it is utterly illogical.

其後楊次公自著《洞極經》，託名於關子明，其叔本篇稱子曰：「河圖之本，七前六後，八左九右，聖人觀之以畫八卦，是故全七一之三以為離，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；全九之三以為兌，奇以為坤。正者全其位，偶者盡其畫。」而謂四象生八卦，此謬之尤者。

Later Yang Cigong himself wrote the "Dongji Jing," but falsely attributed it to Guan Ziming. When explaining this passage, he cited "Zi Yu" (the Master said): "The fundamental principle of the River Chart is that seven is in front, six is behind, eight is on the left, and nine is on the right. The sage observed this and drew the eight trigrams. Therefore, taking the full number of seven, three lines were used for Li (Fire); taking its remainder, three lines were used for Xun (Wind). Taking the full number of eight, three lines were used for Zhen (Thunder); taking its remainder, three lines were used for Gen (Mountain). Taking the full number of six, three lines were used for Kan (Water); taking its remainder, three lines were used for Qian (Heaven). Taking the full number of nine, three lines were used for Dui (Lake); taking its remainder, three lines were used for Kun (Earth). Those in the central position preserved their direction, while those in the even position used up their line numbers." And he said this was the "Four Symbols giving rise to the Eight Trigrams." This is truly the most erroneous statement.

乾、坤相索而生六子，今為男女者，反能生父母之卦邪？況泛言一不七為火，四、九為金，猶可饅所未識者。今至明言離九為七，兌七為九，是河圖一可變置矣而可乎？

Heavenly Stem and Earthly Branches are used to generate the six sons' trigrams, which are derived from the interaction between Heaven and Earth. Now, these six sons' trigrams, being male and female, can generate the parents' trigrams? Moreover, it is possible to generally say that numbers two and seven are fire, and four and nine are metal, which can deceive those who do not understand. But now it is clearly stated that Li nine is seven and Dui seven is nine, which is equivalent to saying that the He Tu can be arbitrarily rearranged. How can that be acceptable?

又謂洛書之文，五處其中，九前一後，三左七右，四前左二，前右八後，左六後右，後世稽之以為三象，因而九變之以二十七象。今按其說，惟五處其中一語，於洛書不悖，餘悉河圖之數，而河圖之數乃因洛書以五處中，而盡失其所以為五之理爾。

He also said that the pattern of the Luo Shu is such that the number five is in the center, nine is in front, one is behind, three is on the left, seven on the right, four is in the front left, two in the front right, eight in the back left, and six in the back right; later generations studied it and established three images, and through nine transformations, it became twenty-seven images. Now according to his statement, only the phrase "five is in the center" does not contradict the Luo Shu; the rest are all numbers from the He Tu. Once the number five in the He Tu is placed in the center according to the Luo Shu, it completely loses the meaning of five being five.

《書》之九疇，以皇極之五處中，固元別用，是可以定位也。至河圖之有天五，乃兼地十以為十五，其五乃四分於成數之內而合於十，豈嘗有五虛點而實置於圖之中位哉？圖之傳誤正由此五。

The nine categories of the *Shangshu* (Book of Documents) have the fifth category, the Supreme Center (huangji), in the middle, which originally had no other function, so it was positioned that way. As for the number five in the River Chart (hetu), it is meant to combine with the number ten of the earth to form fifteen. This number five is divided into four parts and scattered within the formed numbers, then combined with ten. How could there ever have been five empty dots actually placed at the center of the chart? The error in the transmission of the River Chart originates from this number five.

次公之假設此妄例，特以蓋其所祖者，乃楊子云三方、九州島而二十七部，私以為書耳。於斯二者，固已真易置圖、書矣。而朱元晦、蔡季通反祖之，信以為關子明作也。不計子明之《易傳》精詣，豈此摹擬者比，乃循此膚臆之見，因扶其說，亦謂河圖七前六後，八左九右；洛書之文，九前一後，三左七右，四前左，二前右，八後左，六後右。

Yang Cigong assumed this false example merely to cover up what he actually followed, which was actually Yang Xiong's "Ta Xuan" with its three directions, nine states, and twenty-seven parts, which he privately regarded as a classic. From these two points, he indeed completely reversed the River Chart and the Lo Shu. However, Zhu Xi (Zhu Yuanhui) and Cai Yuanding (Cai Jitong) instead followed him, believing that this was truly composed by Guan Ziming. They did not think about how profound Guan Ziming's "Yi Zhuan" was, which could not be compared to such imitative work; yet they followed this shallow and speculative view, further supporting it, and also explained that the River Chart has seven in front and six behind, eight on the left and nine on the right, while the Lo Shu has nine in front and one behind, three on the left and seven on the right, with four in front-left, two in front-right, eight in back-left, and six in back-right.

謂《大傳》既陳五十有五數，《洪範》又謂天乃錫禹《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。劉牧意見以九為河圖，十為洛書，託言出於希夷，與諸儒舊說不合；又引《大傳》以為二者皆出於伏羲之世，其易置圖、書並元明驗。

They say: Since the Great Treatise presents the number fifty-five, and the Hongfan states "Heaven gave Yu the Hongfan Nine Categories," the nine-square grid number is said to have the number nine on top, one at the bottom, three on the left, seven on the right, two and four as the two shoulders, and six and eight as the two feet, forming exactly the shape of a tortoise's back. As for Liu Mu's view, he took nine as the He Tu and ten as the Luo Shu, attributing them to Chen Tuan, which differs from the old views of the scholars. He also cited the Great Treatise to say both originated in the time of Fuxi, but his reversal of the He Tu and Luo Shu lacks clear evidence.

其朱、蔡之指斥又如此，而直以圖南始標誤之洛書為河圖，而以其初正指河圖，反以為洛書，則朱、蔡實自誤而反罪長民之先誤。專己自是，張其辯說，不克自反，一至於此。

Zhu Xi and Cai Yuan ding criticized Liu Mu in this way, yet they themselves directly took the Luo Book that Chen Tuan had mistakenly labeled as the He Tu, and instead took the one Chen Tuan originally correctly identified as the He Tu as the Luo Book. In this way, Zhu and Cai actually made their own mistakes, yet they blamed Liu Mu for having made the error first. They were stubborn and self-willed, believing themselves to be right, and they spread their arguments without self-reflection, reaching such a point.

惟程泰之《易原》時能理到，至鄭束卿輩尋邇依傍，一意誕饅，廣增圖說，皆非實指，徒使後之人失其真是，信其所不當信，疑其所不當疑，並與本來宗旨樊然擾亂，是其所非，而非其所是。

Only Cheng Dachang (style name Taizhi) of "The Origins of the Changes" (Yi Yuan) could consistently address the principles. As for those of the generation of Zheng Dongqing, they only knew how to attach themselves closely to existing interpretations, indulging in sheer absurdity and deceiving people. They added a great deal of illustrations and diagrams, but all were not genuine references. This caused later generations to lose the true meaning: they did not believe what should be believed, and doubted what should not be doubted. The original purpose was also thoroughly confused, mixing right and wrong, taking the wrong for the right and the right for the wrong.

跡究其始，不過由圖南用十之一語立一圖之誤，至於為此紛紛，始猶借河圖以相因依，而為洛書之說，久則舍圖取書，又久乃遂以書為圖，荒然莫知河圖為何物，所以有圖者，寫為何事，執其五十五數以為圖，不知其五十五數其始特以誤及洛書，今及執之為圖，是圖既非興；其謂書以為圖者，書益非其書者矣。

Tracing back the origin of all this, it was merely Chen Tuan's mistaken statement of "using ten" that led to this endless confusion. At first, they still relied on the River Chart to support the theory of the Lo Shu; over time, they abandoned the River Chart and focused solely on the Lo Shu; eventually, they even took the Lo Shu to be the River Chart, completely unaware of what the River Chart actually was and why it was originally created. They took the number fifty-five as representing the River Chart, but did not realize that these fifty-five numbers were originally mistakenly added to the Lo Shu. Now, using them as the River Chart, this "chart" is no longer the true chart, and the "book" that claims the Lo Shu is the River Chart is also no longer the original Lo Shu.

徒自言人人，殊使學者亡羊多歧，求馬唐肆，紛如聚訟，吾誰適仄？不啻三百年於此，故推原致誤之由，疏列莘疑之妄，直以河圖本數討正之，庶幾學者見之瞭焉，會心自斷，知圖南之謂其由三陳九卦而得者，類於饅；陰腸家襲於以一、六、八白取以為起例者，流於末，若《乾鑿度》謂太一取其數以行四正四維皆合十五者。《鑿》於《緯》之拘贅，亦不盡然。

Everyone has their own opinions, leaving scholars confused, like searching for a sheep in a crossroad or buying a horse in an empty stable. They argue endlessly, like in a lawsuit. Who should I believe? This situation has persisted for more than three hundred years. Therefore, I trace back the causes of these errors, list various questionable and false theories, and directly use the original numbers from the River Chart to correct them. I hope scholars can understand at a glance and make their own judgment. One must know that Chen Tuan claimed he derived his method from "San Chen Ji Guà" (Three Chens and Nine Hexagrams), which is almost a deception. The Yin-Yang school followed the idea of one white, six white, and eight white to serve as examples for divination, which is merely a superficial technique. As for what is said in the "Qian Zao Du" (The Book of the凿ing of the Heaven), that the Taiyi (Supreme Oneness) takes numbers to move through the nine palaces, and that the four cardinal directions and four secondary directions add up to fifteen, this is an overly rigid and cumbersome interpretation in the "Yi Wei" (Commentaries on the Changes), and not entirely reliable.

然從昔儒先莫不斷斷然謂非人所能偽作者，今以餘自本觀之，直謂不必它求，特以《說卦》帝出乎震一□一章，環中往復參究之，以天一至地四，地六至天九求其實體，天五與地十明其虛用，則參伍以變，參天兩地，錯綜倚數，莫不左右逢原，一皆胳合乎孔子繫《易》之初旨。

However, all the distinguished Confucian scholars throughout history have firmly stated that the River Chart could not have been forged by human beings. Now, from the source, I can simply say: there is no need to seek elsewhere. Just take the passage in the "Shuo Gu" chapter that says "The Emperor arises from Zhen," and examine it thoroughly by considering the circular repetition. Use the numbers from Heaven One to Earth Four and from Earth Six to Heaven Nine to seek its substance, and use Heaven Five and Earth Ten to clarify its function. Thus, "Three and Two to produce change," "Three heavens and two earths," and "Interweaving and reversing numbers" will all be naturally in harmony, fully aligning with the original intent of Confucius in interpreting the Book of Changes.

是之謂參。淆亂抓論，轟明；河之出圖，通古不妄，足以見《易》矣。因洞悉洛之出書，一汍疇之外，豈復有圖有數之妄作者，誠古今之一大快事也。

This is what is truly called "participation" (参). It clears up confusion and exposes fallacies, making the principles clearly evident: the emergence of the River Chart (黄河出图) throughout history is not baseless, and from this we can see the origin of the Book of Changes (《易》). Therefore, we can fully understand that the emergence of the Luo Book (洛水出书) involved only the Nine Categories (九畴), and there is no other fabricated diagram or number. This is indeed a great event for both past and present.

#1『平』，原作一乎』，據四庫本改。 #2『禹』，四庫本作『語』。 #3『祇』，原作一祇，據四庫本改。 #4『非特』，仰庫本誤把後文自『八疇分佈四正、四維』至『是可以定位也』一段倒置於此處之後。特此說明。 #5『太易』，四庫本作『大易』。 #6『謬』，原作『繆』，據四庫本改。 #7『不啻騙』，原作『不趨』，據四庫本改。

The first point: the character "平" was originally written as "乎", and has been corrected according to the Siku Quanshu edition. The second point: the character "禹" in the Siku Quanshu edition is written as "语". The third point: the character "祇" originally had a distorted form, and has been corrected according to the Siku Quanshu edition. The fourth point: the phrase "非特" has been misplaced in the original text, with the passage from "八畴分布四正、四维" to "是可以定位也" appearing after it, which is an error in the Siku Quanshu edition. The fifth point: "太易" in the Siku Quanshu edition is written as "大易". The sixth point: the character "谬" was originally written as "缪", and has been corrected according to the Siku Quanshu edition. The seventh point: the phrase "不啻" was originally written as "不趋", and has been corrected according to the Siku Quanshu edition.