The title of the book and the note on the original manuscript.

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.