The Great Changes (Yi) Diagram of Figures and Numbers Revealing Hidden Meanings

經名：大易象數鈞隱圖。元張理撰。底本未題撰人。三卷。底本出處：《正統道藏》洞真部靈圖類。參校版本：文淵閣四庫全書本經部易類（簡稱四庫本）。

This book is titled in the Daozang as "Da Yi Xiangshu Gouyin Tu," written by Zhang Li of the Yuan dynasty. The base text does not bear the author's name, and the entire work is divided into three volumes: upper, middle, and lower. The base text comes from the "Zhengtong Daozang" under the category of Lingtu (Spiritual Diagrams) in the Dongzhen Department. During the collation, another edition from the "Wenyuan Ge Siku Quanshu" under the category of Yi (Divination) in the Jing Department was consulted, and it is hereafter referred to as the "Siku Ben."

卷上：太極貫一之圖、易有太極圖、舊有此圖、太極函三自然奇耦之圖、德事相因皆本奇耦之圖、說卦八方之圖、乾知太始、坤作成物、大易象數鉤深圖目錄、天尊地卑、參天兩地圖、日月為易、河圖數圖、洛書數圖、河圖四象之圖、河圖始數益洛書成數圖、河圖八卦圖、乾元用九坤元用六圖、天地之數乾坤之策、河圖天地十五數圖、其用四十有九圖、乾坤六子圖、渾天六位圖、六位三極圖、伏羲先天圖、方圓相生圖、仰觀天文圖、俯察地理圖、伏羲八卦圖、八卦取象圖、文王八卦圖、八卦象數圖。

The first volume contains a total of thirty-two diagrams, arranged in the following order: The diagram of the Taiji encompassed in one, titled "The Diagram of Taiji Encompassed in One"; the diagram illustrating the sentence "The Book of Changes has the Taiji" from the Great Treatise, titled "The Diagram of 'The Book of Changes has the Taiji'"; an old diagram traditionally passed down from old editions; the diagram showing the Taiji containing three lines, naturally distinguishing odd and even, titled "The Diagram of Taiji Containing Three Lines, Naturally Distinguishing Odd and Even"; the diagram explaining the mutual dependence of virtue and affairs, both originating from odd and even, titled "The Diagram of Virtue and Affairs Being Mutually Dependent, Both Originating from Odd and Even"; the diagram illustrating the eight directions from the Sayings on the Hexagrams, titled "The Diagram of the Eight Directions from the Sayings on the Hexagrams"; the diagram explaining "The Heaven knows the beginning" (from the text); the diagram explaining "The Earth completes the things" (from the text) (here inserted is a line of the book's title in the printing block: "Catalogue of the Diagrams of the Great Changes: The Deep Insights of the Figures and Numbers"); the diagram explaining "Heaven is exalted and Earth is humble"; the diagram explaining "Participating in the heavens with two and five, leaning on the numbers," titled "The Diagram of Participating in the Heavens with Two and Five, Leaning on the Numbers"; the diagram explaining the formation of the character "Yi" from the combination of "Sun" and "Moon," titled "The Diagram of Sun and Moon Forming Yi"; the "He Tu Number Diagram"; the "Lu Shu Number Diagram"; the "He Tu Four Images Diagram"; the diagram explaining the combination of the generating numbers of He Tu with the completed numbers of Lu Shu, titled "The Diagram of the Starting Numbers of He Tu Enhanced by the Completed Numbers of Lu Shu"; the "He Tu Eight Trigrams Diagram"; the diagram explaining the use of all nines in the hexagram Qian and all sixes in the hexagram Kun; the diagram explaining the numbers of Heaven and Earth and the numbers of the trigrams Qian and Kun; the "He Tu Heaven and Earth Fifteen Numbers Diagram"; the diagram explaining the use of forty-nine in the Great Derivation; the diagram explaining the generation of the six subordinate trigrams from Qian and Kun; the "Holistic Heaven Six Positions Diagram"; the "Six Positions Three Poles Diagram"; the "Fu Xi Anterior Heaven Diagram"; the diagram explaining the mutual generation of the square and circular diagrams, titled "The Diagram of Square and Circle Mutually Generating"; the "Observing the Celestial Phenomena Diagram"; the "Observing the Terrestrial Phenomena Diagram"; the "Fu Xi Eight Trigrams Diagram"; the "Diagram of the Symbolic Meanings of the Eight Trigrams"; the "Wen Wang Eight Trigrams Diagram"; the "Diagram of the Symbolic Numbers of the Eight Trigrams."

卷中：四象合律圖、八卦納甲圖、剛柔相摩圖、八卦相蕩圖、六爻三極、五位相合、帝出震圖、曹卦之德、序上下經圖、三變大成圖、重易六爻圖、六十四卦天地數圖、十四卦萬物數圖、卦爻律呂圖、運會曆數圖、乾坤大父母圖、復妮小父母圖、八卦生六十四卦圖、八卦變六十四卦圖、陽卦順生陰卦逆生、復娠臨逛泰否六卦生六十四卦圖、六十四卦反對變圖。

The first half of the volume contains the following diagrams in order: the "Four Images and Twelve Tones" diagram explaining the combination of the Four Images with the Twelve Tones, the "Eight Trigrams and Ten Heavenly Stems" diagram explaining the assignment of the Heavenly Stems to the Eight Trigrams, a diagram illustrating "the interaction of firm and yielding," a diagram illustrating "the mutual transformation of the Eight Trigrams," a diagram explaining the division of the six lines into the three extremes of Heaven, Earth, and Humanity, a diagram explaining the pairing of the numbers of the five directions, the "Five Positions and Their Mutual Harmony" diagram, a diagram illustrating the "Imperial Departure from Zhen" from the "Commentary on the Trigrams," the "Deeds of the Divining Stalk and the Trigrams" diagram (mistakenly listed as "Deeds of the Cao Trigrams" in the catalog), a diagram arranging the order of the Upper and Lower Classics, the "Three Transformations Forming One Line" diagram, the "Repeating the Trigrams to Form Six Lines" diagram, the "Heaven and Earth Numbers of the Sixty-Four Trigrams" diagram, the "Numbers of All Things in the Sixty-Four Trigrams" diagram (missing the character "six" in the catalog), the "Trigram and Line and Twelve Tones" diagram, the "Cycle of Yuan, Hui, Yun, and Shi" diagram, a diagram taking the Trigrams Qian and Kun as the major parents, a diagram taking the Trigrams Fu and Guo as the minor parents (the catalog mistakenly writes "Ni" instead of "Guo"), a diagram explaining the generation of the sixty-four trigrams from the Eight Trigrams, a diagram explaining the transformation of the sixty-four trigrams from the Eight Trigrams, a diagram explaining the orderly generation of Yang trigrams and the reverse generation of Yin trigrams, a diagram explaining the generation of the sixty-four trigrams from the six trigrams Fu, Guo, Lin, Duan, Tai, and Poi (the catalog mistakenly writes "Shen" and "Gang" instead of "Guo" and "Duan"), and the "Opposition and Transformation of the Sixty-Four Trigrams" diagram.

乾坤易簡之圖、屯象之圖、蒙象養正圖、需須之圖、訟象之圖、師比御眾圖、大小畜吉凶圖、履虎尾之圖、否泰往來圖、同人之圖、大有守位圖、謙象之圖、豫象之圖、隨卦繫失圖、蠱象之圖、臨象之圖、觀國光之圖、噬嗑身口象圖、賁天文之圖、剝為陽氣種圖、復七日圖、元妄本中孚圖、頤靈龜圖。

In the latter half of the volume, the focus shifts to individual hexagrams, with diagrams presented in sequence: the diagram explaining the simplicity of the Heaven and Earth in the "Diagram of the Simplicity of Heaven and Earth," the symbolic diagram of the "Zun" hexagram, the diagram illustrating the principle of nurturing righteousness in the "Meng" hexagram, the diagram explaining the meaning of waiting in the "Xu" hexagram, the symbolic diagram of the "Song" hexagram, the diagram illustrating the governance of the "Shi" and "Bi" hexagrams, the diagram discussing the fortune and misfortune of the "Da Xu" and "Xiao Xu" hexagrams, the diagram of the "Lv" hexagram depicting "Walking on the Tiger's Tail," the diagram of the "Po" and "Tai" hexagrams showing their mutual transition, the diagram of the "Tong Ren" hexagram, the diagram of the "Da You" hexagram illustrating the maintenance of position, the symbolic diagram of the "Qian" hexagram, the symbolic diagram of the "Yu" hexagram, the diagram of the "Sui" hexagram depicting attachment and loss, the symbolic diagram of the "Gu" hexagram, the symbolic diagram of the "Lin" hexagram, the diagram of the "Guan" hexagram illustrating "Observing the Splendor of the State," the diagram of the "Shi He" hexagram symbolizing the mouth and teeth, the diagram of the "Ben" hexagram representing celestial phenomena, the diagram of the "Bo" hexagram showing the single yang line as the seed of yang energy, the diagram of the "Fu" hexagram illustrating "Recovery after Seven Days," the diagram of the "Wu Wang" hexagram emphasizing trust based on the "Zhong Fu" hexagram (the catalog entry "Yuan Wang" refers to "Wu Wang," with the character changed due to taboo in the Daozang edition), and the diagram of the "Yi" hexagram depicting "Abandoning Your Auspicious Turtle."

卷下：大過楝隆橈圖、習坎行險圖、離繼明圖、咸朋從圖、恆久之圖、逐象之圖、大壯羊藩圖、晉康侯之圖、明夷箕子圖、家人象圖、睽卦象圖、賽往來之圖、解出坎險圖、損益用中圖、央央之圖、娠遇之圖、萃聚之圖、升階之圖、困羨華葛萬株木圖、井鼎水火二用之圖、革卦爐精鼓鑄圖。

In the following section, the lower part of the text continues with the explanation of each hexagram, accompanied by diagrams. The previous entries in order are: the diagram of the hexagram "Da Guo" illustrating the image of a beam that may be raised or bent, the diagram of "Xi Kan" showing movement amidst danger, the diagram of "Li" illustrating the continuation of brightness and illumination in all directions, the diagram of "Xian" depicting "Friendship follows your thoughts," the diagram of "Heng" illustrating the principle of constancy, the diagram of "Dun" (with the note that the catalog entry "Zhu" is a mistake for "Dun"), the diagram of "Da Zhuang" showing a male sheep encountering a fence, the diagram of "Jin" illustrating "Kanghou used a gift of many horses," the diagram of "Ming Yi" depicting Ji Zi's internal hardship, the diagram of "Jia Ren," the diagram of "Kui," the diagram of "Jian" showing movement forward and backward (with the note that the catalog entry "Sai" is a mistake for "Jian"), the diagram of "Jie" illustrating escape from the peril of Kan, the diagram of "Sun" and "Gen" illustrating the principle of moderation, the diagram of "Gua" depicting the image of decisive action (with the note that the catalog entry "Yangyang" is a mistake for "Gua"), the diagram of "Gou" showing meeting, the diagram of "Cui" illustrating gathering, the diagram of "Sheng" depicting ascending steps, the diagram of "Kun" showing the images of thornbushes, grapevines, and stumps, the diagram of "Jing" and "Ding" illustrating the functions of water and fire, and the diagram of "Ge" depicting the process of smelting in a furnace.

震動心邊之圖、艮背象之圖、鴻漸南北圖、歸妹君娣之袂圖、豐日見斗之圖、旅次舍圖、巽林下圖、兌象之圖、渙躬之圖、節氣之圖、中孚小過卵翼生成圖、既濟未濟合律之圖、六十四卦卦氣圖、日月運行一寒一暑卦氣之圖、十三卦取象圖、三陳九卦之圖、參伍以變圖、十有八變圖、一陰一陽圖、先甲後甲圖、陰陽君民。

The next set includes: a diagram explaining the hexagram Zhen (Thunder) which moves in the heart and is manifested in external forms (the directory entry "bian" is a mistake for "ji" meaning "trace"), a diagram explaining the hexagram Gen (Mountain) "Gen qi bei" (keeping still on the back), a diagram explaining the hexagram Jian (Gradual Progress) with the image of wild geese flying south and north, a diagram explaining the hexagram Gui Mei (Return of the Maiden) with the image of the ruler and the maiden's sleeves overlapping, a diagram explaining the hexagram Feng (Brightness) "ri zhong jian dou" (the star Dou visible at noon), a diagram explaining the hexagram Lu (Traveler) with the image of lodging during a journey, a diagram explaining the hexagram Xun (Wind) "Xun zai chuang xia" (wind beneath the bed) (the directory entry "lin" is a mistake for "chuang" meaning "bed"), a diagram of the hexagram Dui (Joy) with its symbolic imagery, a diagram explaining the hexagram Huan (Dispersal) "Huan qi gong" (dispersing oneself), a diagram explaining the hexagram Jie (Moderation) with its alignment to the twenty-four solar terms, a diagram explaining the hexagrams Zhong Fu (Sincerity) and Xiao Guo (Minor Excess) with the image of bird eggs hatching and generating life, a diagram explaining the hexagrams Ji Ji (Completion) and Wei Ji (Non-Completion) with their alignment to the five tones, the diagram of the sixty-four hexagram qi (energy) chart, a diagram explaining the movement of the sun and moon, the alternation of cold and heat, and their correspondence with the hexagram qi, a diagram explaining the thirteen hexagrams mentioned in the Great Treatise with their symbolic imagery and the creation of tools, a diagram explaining the three times the Great Treatise discusses the nine hexagrams, a diagram explaining the concept of "san wu yi bian" (three and five transformations), a diagram explaining the process of eighteen transformations forming one hexagram, a diagram explaining the statement "yi yin yi yang zhi wei dao" (the interplay of yin and yang is called the Dao), a diagram explaining the hexagram Gu (Restlessness) "xian jia san ri, hou jia san ri" (three days before and after the first day of the month), and the diagram of "Yin Yang Jun Min" (Yin and Yang, Ruler and Subject).

陰陽奇耦、二儀得十變化、十日五行相生、大衍之數圖、搽曹之法圖、河圖百六數、八卦司化圖、類聚群分圖、通乎晝夜圖、陽中陰、陰中陽、序卦圖、雜卦圖、太玄準易卦名圖、太玄準易卦氣圖、皇極經世全數圖、邵氏皇極經世圖、溫公潛虛擬玄圖、潛虛性圖、說卦配方圖、古今易學傳授圖。

The final set includes: "Yin and Yang Odd and Even" explaining that odd trigrams represent yang and even trigrams represent yin; a diagram illustrating how the two principles generate ten and thus produce change; a diagram showing the relationship between the ten heavenly stems and the Five Phases; a diagram explaining the number of the Great Derivation; a diagram illustrating the method of divining by counting the yarrow stalks, with the catalog entry "cha cao" being a mistake for "che zhi"; a diagram of the River Chart with the extreme number 106; a diagram showing each trigram governing a particular type of transformation; a diagram illustrating the principle of "classifying things by their nature and grouping objects by their kind"; a diagram explaining the concept of "understanding the principle of day and night"; a diagram of "Yang containing Yin"; a diagram of "Yin containing Yang"; a diagram illustrating the sequence of the "Sequence of Trigrams"; a diagram showing the reversal in the "Miscellaneous Trigrams"; a diagram comparing the names of the trigrams in the Book of Changes with those in Yang Xiong's Tai Xuan; a diagram comparing the trigram energies with those in the Tai Xuan; a diagram of all the numbers in Shao Yong's Huang Ji Jing Shi; the "Shao氏 Huang Ji Jing Shi Diagram"; a diagram of Sima Guang's Qian Xu imitating the Tai Xuan; a diagram from Qian Xu discussing the nature; a diagram assigning the "Say of Trigrams" to directions; and a diagram recording the transmission of the study of the Book of Changes through generations.

右《太極圖》，周敦實茂叔傳二程先生。茂叔曰：無極而太極。太極動而生陽，動極而靜，靜極復動，一動一靜，互為其根，分陰分陽，兩儀立焉。陽變陰合而生水、火、木、金、土。五氣順布，四時行焉。五行一陰陽也。陰陽一太極也。太極本無極也。五行之生也，各一其性。無極之真，二五之精，妙合而凝。乾道成男，坤道成女，二氣交感，化生萬物。萬物生生，而變化無窮焉。

The diagram on the right is the Taiji Diagram passed down by Zhou Dunyi (also known as Zhou Dunshi, whose courtesy name was Maoshu) to Cheng Hao and Cheng Yi. Maoshu said: the "wuji" (formless and imageless) is the ultimate principle, which is also called "taiji." When taiji moves, it produces yang; when movement reaches its extreme, it turns into stillness; when stillness reaches its extreme, it once again begins to move. Movement and stillness are each the foundation of the other, thus differentiating into yin and yang, and the two principles of heaven and earth are established. Yang qi moves and changes, while yin qi condenses and combines, thus producing water, fire, wood, metal, and earth. These five elements spread in order, and the four seasons of spring, summer, autumn, and winter come into being. In the end, the five elements are merely yin and yang, and yin and yang are ultimately taiji, which itself is originally wuji. When each of the five elements is generated, it possesses its own nature. The true essence of wuji, combined with the two qi of yin and yang and the essence of the five elements, mysteriously converges and condenses: the path of Qian gives rise to male, the path of Kun gives rise to female, and the two qi interact and harmonize, giving birth to all things. All things continuously generate and transform from generation to generation, and change has no end.

太極未有象數，惟一氣耳。一氣既分，輕清者上為天，重濁者下為地，太極生兩儀也。兩儀既分，則金、木、水、火四方之位列，兩儀生四象也。水數六，居坎而生乾；金數九，居兌而生坤；火數七，居離而生巽；木數八，居震而生艮，四象生八卦也。

In the Taiji stage, there was no form or number, only a single mass of primordial qi. When this mass divided, the light and clear part rose to become Heaven, and the heavy and turbid part sank to become Earth. This is called "Taiji gives birth to the Two Principles." Once the Two Principles were separated, metal, wood, water, and fire each occupied the directions of east, south, west, and north respectively. This is called "the Two Principles give birth to the Four Symbols." Water has the number six and is located in the Kan position, giving rise to Qian; metal has the number nine and is located in the Dui position, giving rise to Kun; fire has the number seven and is located in the Li position, giving rise to Xun; wood has the number eight and is located in the Zhen position, giving rise to Gen. This is called "the Four Symbols give birth to the Eight Trigrams."

乾三畫而為天者，以一含三也。坤六畫而為地者，耦三而為六也。天一地二之本數，天奇地耦之本畫，不待較而可知，然妙理在乎一含三、二含六耳。乾一含三，故索為三男，而皆奇；坤二含六，故索為三女，而皆耦。此天地生成之理，豈不妙哉？

Heaven is represented by three lines because "one" contains "three"; Earth is represented by six lines because "three" doubled becomes "six." The fundamental numbers for Heaven and Earth are one and two respectively, and the fundamental lines for Heaven are odd while those for Earth are even, which is clear without comparison. The truly profound principle lies in "one contains three" and "two contains six." Heaven, by containing three, combines with Earth to produce the three male trigrams, Zhen, Kan, and Gen, whose main lines are all odd; Earth, by containing six, combines with Heaven to produce the three female trigrams, Xun, Li, and Dui, whose main lines are all even. This is the principle by which Heaven and Earth generate all things. Is it not marvelous?

震為雷，雷出於地下，故一陽在下。坎為水，水畜於地中，故一陽在中。艮為山，山形於地上，故一陽在上。然陽動陰靜，以動為基者，故動震是也。以靜為基者，故止艮是也。動者在中，非內非外，故或流或止，或動或靜焉。此坎所以為水。

The Trigram Zhen represents Thunder, which originates from beneath the earth, so the single Yang line in Zhen is placed at the bottom. The Trigram Kan represents Water, which accumulates in the earth, so the single Yang line in Kan is placed in the middle. The Trigram Gen represents Mountain, which stands upright on the earth, so the single Yang line in Gen is placed at the top. However, Yang is active and Yin is passive: those based on movement are the "movement" of Zhen; those based on stillness are the "stillness" of Gen. As for the Yang line in the middle, it is neither inside nor outside, so it sometimes flows and sometimes stops, sometimes moves and sometimes remains still—this is why Kan is symbolized by Water.

巽為木。木發生於地下，故一陰亦在下。離為火，火出於木中，故一陰在中。兌為澤，澤鍾於地上，故一陰在上。然陰柔而陽剛，故木也始弱而終強，陽在末也。陽明而陰晦，故火也外明而內晦，陽在外也。陽燥而陰潤，故澤也外潤而內燥，陽在內也。或問：澤內燥，何也？愚日：內燥則能生金，外潤則能鍾水，金所以能生水，土所以能生金者，即澤而知之也。聖人豈苟之哉？

The Trigram Xun represents Wood, which emerges from beneath the earth, so the yin line in Xun is placed at the bottom. The Trigram Li represents Fire, which arises from Wood, so the yin line in Li is placed in the middle. The Trigram Dui represents a Lake, whose water gathers on the surface, so the yin line in Dui is placed at the top. Yin is soft and Yang is strong, so Wood starts soft and becomes strong, with Yang at the top. Yang is bright and Yin is dark, so Fire is bright on the outside and dark inside, with Yang on the outside. Yang is dry and Yin is moist, so a Lake is moist on the outside and dry inside, with Yang inside. Someone asked: If the inside of a Lake is dry, why is that? My answer is: The dryness inside allows for the generation of Metal, while the moisture outside gathers Water. The principle that Metal generates Water and Earth generates Metal is clearly shown through the image of the Lake. Could the sages have chosen these symbols by chance?

一陽生於子，二陽在丑，三陽在寅，四陽在卯，五陽在辰，六陽在巳，而乾位在西北，居子之前，故日：乾知太始。言乾以父道始天地也。

The first yang line arises in the month of Zi, the second in Chou, the third in Yin, the fourth in Mao, the fifth in Chen, and by the time of Si month, all six yang lines are complete. The trigram Qian is located in the Northwest direction, right before the position of Zi. Therefore, the Great Treatise states, "Qian knows the beginning of the Great Ultimate," referring to Qian embodying the fatherly principle that initiates the creation of heaven and earth.

一陰生於午，二陰在未，三陰在申，四陰在酉，五陰在戌，六陰在亥。而坤位在西南，蓋西南方，申也；物成於正秋，酉也。坤作於申，成於酉，故日：坤作成物。

The first yao of yin is born in the month of Wu, the second yin in the month of Wei, the third yin in the month of Shen, the fourth yin in the month of You, the fifth yin in the month of Xu, and by the month of Hai all six yins are complete. The position of the K'un hexagram is in the southwest. The earthly branch corresponding to the southwest is Shen. All things mature in the beginning of autumn, and the earthly branch corresponding to the beginning of autumn is You. Therefore, K'un begins its work in the position of Shen and achieves completion in the position of You. Hence, the Great Treatise says, "K'un brings things to completion."

自一至十，天尊於上，地卑於下。尊者乾之位，故乾為君，為父，為夫。卑者坤之位，故坤為臣，為母，為婦。皆出於天尊地卑之義也。故日：天尊地卑，乾坤定矣。

Counting from one to ten, the numbers of Heaven are superior and occupy the upper position, while the numbers of Earth are inferior and occupy the lower position. The superior position belongs to the Trigram Qian (Heaven), so Qian is taken as symbolizing the ruler, the father, and the husband. The inferior position belongs to the Trigram Kun (Earth), so Kun is taken as symbolizing the subject, the mother, and the wife. These symbolic associations all arise from the principle of "Heaven is exalted and Earth is humble." Therefore, the Great Treatise begins by saying: "Heaven is exalted and Earth is humble; thus, Qian and Kun are established."

乾元用九，參天也。坤元用六，兩地也。故日：參天兩地而倚數。九、六者，止用生數也。

The "Use of Nine" in the Qian hexagram takes "three heavens," meaning the heavenly number three is tripled; the "Use of Six" in the Kun hexagram takes "two earths," meaning the earthly number two is doubled. Therefore, the "Commentary on the Hexagrams" says: "Triple the heavens and double the earths to rely on numbers." The numbers nine and six are derived solely from the five original numbers one to five, and there is no separate basis for them.

取日月二字交配而成，如篆文日下從月，是日往月來之義，故日：陰陽之義配日月。

The character "Yi" is formed by the intermingling of the characters "sun" and "moon," just as in the seal script, "Yi" has the sun above and the moon below. This signifies the continuous alternation between day and night. Therefore, the Great Treatise states: "The principle of yin and yang corresponds to the sun and moon."

戴九履一，左三右七，二四為肩，六八為足，五為腹心，總四十五，縱橫數之，皆十五也。天五居中央，聖人觀之，遂定八卦之象。

This diagram has nine at the top and one at the bottom, three on the left and seven on the right, two and four on the shoulders, six and eight on the feet, and five in the center. The nine numbers together total forty-five. Whether counted horizontally, vertically, or diagonally, each row adds up to fifteen. The number of heaven, five, is in the center. The sage observed this diagram and thus established the images of the eight trigrams. Zhang Li followed the naming convention of Liu Mu from the Song dynasty, calling this diagram of the forty-five numbers in the nine palace arrangement the "He Tu" or "River Chart."

河圖之數四十五，蓋聖人損去天、地二、天三、地四，凡十數。獨天五居中而主乎土，至洛書則有土十之成數，故水、火、金、木成形矣。

The total number of the River Chart is forty-five, because the sage subtracted the four generating numbers, heaven one, earth two, heaven three, and earth four, which sum up to ten, from the fifty-five number. What remains is the number five of heaven in the center, governing earth. Only when the Luo Book appears does the number ten of earth become complete, allowing water, fire, metal, and wood to take form. Here, "Luo Book" refers to the diagram of fifty-five numbers, also the term used by the school of Liu Mu.

《繫辭》日：《易》有太極，是生兩儀，兩儀生四象，四象生八卦。其四象在乎天一、地二，天三、地四，天五。天一居北方坎位，為水；地二居南方離位，為火；天三居束方震位，為木；地四居西方兌位，為金，此在四正之位，而為生數也；天五居中央，則是五土數也。土無定位，然後分王四正之方，能生萬物，故北方水一得土五而成六，南方火二得土五而成七，束方木三得土五而成八，西方金四得土五而成九，此謂之四象也。

The Great Treatise says: "The Book of Changes contains the Taiji, from which the two principles arise, from which the four images arise, and from which the eight trigrams arise." Here, the four images refer to the five generating numbers: heaven one, earth two, heaven three, earth four, and heaven five. Heaven one is located in the northern position of Kan, which belongs to water; earth two is located in the southern position of Li, which belongs to fire; heaven three is located in the eastern position of Zhen, which belongs to wood; and earth four is located in the western position of Dui, which belongs to metal. These four are situated in the four cardinal directions of southeast, northwest, south, and north. They are all generating numbers. Heaven five is located in the center, which is the number five of earth. Earth does not have a fixed direction; it must be strong in the four cardinal directions to nurture all things. Therefore, the water one in the north combines with the earth five in the center to form six; the fire two in the south combines with the earth five to form seven; the wood three in the east combines with the earth five to form eight; and the metal four in the west combines with the earth five to form nine. These four groups of generated numbers are called the four images.

河圖有天一、地二，天三、地四，為象之始。至於天五，則居中而主乎土變化，但未能成形謂之四象矣。至於洛書，有土十之成數，故水、火、金、木皆相奇耦而成形矣。故河圖合四象之數，可以定八方之位；洛書有五行之數，可以備八卦之象也。是象生其卦，必俟天之變，而備於洛書土十之成數，而後成八卦矣。二者相胥，方能成卦。

The River Chart contains Tian Yi (Heavenly One), Di Er (Earthly Two), Tian San (Heavenly Three), and Di Si (Earthly Four), which mark the beginning of the Four Symbols. As for Tian Wu (Heavenly Five), it resides in the center and governs the changes of Earth, but at this stage, the four elements are still not formed into definite shapes and can only be called the "Four Symbols." When the Lo Shu appears, the number ten for Earth is completed, allowing water, fire, metal, and wood to pair with odd and even numbers to form definite shapes. Therefore, the River Chart combines the numbers of the Four Symbols to determine the directions of the eight cardinal points; the Lo Shu includes the numbers of the Five Phases, enabling the formation of the trigrams of the Eight Trigrams. This shows that the emergence of trigrams depends on the change of heavenly numbers, and only after the numbers of Earth ten in the Lo Shu are properly matched can the Eight Trigrams be established. The River Chart and the Lo Shu depend on each other, and only then can the trigrams be formed.

此八象者，始由四象所生也。伏羲氏先按河圖有八方，將以八卦位焉；次取洛書土十之成數，將以八卦象焉。乃觀陰陽而設奇偶二畫，觀天、地、人而設上、中、下三位，以三畫純陽則為乾；以六畫純陰則為坤；以一陽處二陰之下，不能屈於柔，以動出而為震；以一陰處二陽之下，不能犯於剛，乃復入為巽；以一陽處二陰之問，上下皆弱，罔能相濟，以險難而為坎；以一陰處二陽之中，上下皆強，足以自記；乃麗而為離；以一陽處二陰之上，剛以馳下，則止而為艮，以一陰處二陽之上，柔能撫下，則說而為兌。

The eight trigrams originated from the four images. Fuxi first arranged the eight trigrams according to the eight directions indicated by the River Chart. Then he used the number ten from the Lo Shu square to determine the forms of the trigrams. Observing yin and yang, he established the odd and even lines as the basic components. Observing heaven, earth, and humanity, he established the upper, middle, and lower positions of the lines. Three lines all yang form Qian (Heaven), and six lines all yin form Kun (Earth). One yang line situated below two yin lines, unwilling to yield to softness, thus vibrates and becomes Zhen (Thunder). One yin line situated below two yang lines, unable to challenge strength, thus retreats and becomes Xun (Wind). One yang line situated between two yin lines, with both above and below weak, unable to support each other, thus falls into danger and becomes Kan (Water). One yin line situated between two yang lines, with both above and below strong, sufficient for self-reliance, thus adheres and becomes Li (Fire). One yang line situated above two yin lines, using strength to control the yin below, thus stops and becomes Gen (Mountain). One yin line situated above two yang lines, using softness to comfort the yang below, thus becomes joyful and becomes Dui (Lake).

乾陽之位，共十二畫，謂乾三爻，震、坎、艮各一爻，巽、離、兌各二爻，共十二畫也。坤陰之位，其二十四畫，謂坤六畫，巽、離、兌各一畫，震、坎、艮各二畫，計二十四畫也。陽爻，君道也，故得兼之，計有三十六畫，所以四九三十六畫，陽爻則稱九也。坤，臣道也，不得僭上，故四六二十四畫，所以陰爻則稱六也。故乾三畫兼坤之六畫，成陽之九也。陽進而《乾》元用九矣，陰退而《坤》元用六矣。合此餘九、六者，蓋天地剛柔之性也。

The yang lines belonging to the trigram Qian total twelve lines in the Eight Trigrams: Qian itself has three yang lines, Zhen, Kan, and Gen each have one yang line, Xun, Li, and Dui each have two yang lines, adding up to twelve. The yin lines belonging to the trigram Kun total twenty-four lines in the Eight Trigrams: Kun itself has six yin lines, Xun, Li, and Dui each have one yin line, Zhen, Kan, and Gen each have two yin lines, totaling twenty-four. Yang lines represent the way of the sovereign, so they can encompass yin lines, totaling thirty-six lines. Four multiplied by nine equals thirty-six, so yang lines are called "nine." Kun represents the way of the minister, who cannot exceed the sovereign, so it only has four multiplied by six, twenty-four lines, hence yin lines are called "six." Thus, the three lines of Qian combined with the six lines of Kun form the number nine for yang. Since yang is progressive, the beginning of the Book of Changes (Yijing) states "use nine" for Qian. Since yin is recessive, the beginning of the Book of Changes states "use six" for Kun. In summary, the numbers nine and six reflect the nature of heaven being strong and earth being yielding.

此天地之數，十有五也，或統而取之，其旨有六，蓋合自然之數也。且其一者，始就河圖縱橫之，皆十五數也。其二者，蓋天一、地二、天三、地四、天五，凡十五數也。其三者，以其太極未分，混而為一，即是一也。一氣判而為儀，見三也；二儀分而為象，見七也；四象演而生八卦，見十五數也。

The so-called heavenly and earthly numbers, when summed up, total fifteen. If viewed as a whole, they contain six layers of meaning, all consistent with natural numbers. The first layer: according to the River Chart, whether counted horizontally or vertically, the total is fifteen. The second layer: the five generating numbers, namely heaven one, earth two, heaven three, earth four, and heaven five, when added together, amount to fifteen. The third layer: before the Taiji splits into yin and yang, it is unified as 'one'; when this single qi splits into two yin and yang, one plus two gives 'three'; when the two yin and yang further divide into four images, three plus four gives 'seven'; when the four images evolve into the eight trigrams, seven plus eight results in the number 'fifteen'.

其四者，謂五行之生數，是一水、二火、三木、四金、五土，見十五數也。其五者，將五行之數中分為之，以象陰陽，則七為少陽，八為少陰，亦見其十五數也。其六者，以少陰少陽不動，則不能變，亦且八為少陰，陰動而退，故此少陰動而為盛陰，所以退稱六也，七為少陽，陽動而進，故少陽動而為盛陽，所以進稱九也；六與九合，亦見其十五也。斯蓋一、三、五陽位為一不四間，而五居中，然配王四方也。如六、七、八、九合而周，以為四象也。

The fourth layer: The generating numbers of the Five Phases are one for water, two for fire, three for wood, four for metal, and five for earth. When added together, they total fifteen. The fifth layer: By splitting the numbers of the Five Phases to symbolize yin and yang, seven is considered lesser yang and eight is lesser yin. Adding seven and eight also results in fifteen. The sixth layer: If lesser yin and lesser yang do not move, they cannot change. Eight is lesser yin, and when yin moves, it retreats, so lesser yin moving becomes full yin, which retreats and is called 'six'. Seven is lesser yang, and when yang moves, it advances, so lesser yang moving becomes full yang, which advances and is called 'nine'. Adding six and nine still results in fifteen. Generally, the three yang numbers one, three, and five are arranged in sequence without mixing, with five in the center, allowing them to be distributed to the four directions. As for the four numbers six, seven, eight, and nine combined, they circulate through one cycle, forming the Four Symbols.

天一下降，與地六合，而生水於北。地二上駕，與天七合，而生火於南。天三左旋，與地八合，生木於束。地四右轉，與天九合，而生金於西。天五冥運，與地十合，而生土於中。以奇生者成而耦，以耦生者成而奇。天，陽也，故其數奇。地，陰也，故其數耦。奇偶相合，而五十有五之數備。

Heavenly numbers have been continuously decreasing and have combined with the earthly number six, producing water in the north. Earthly number two has been rising and has combined with heavenly number seven, producing fire in the south. Heavenly number three has been rotating left and has combined with earthly number eight, producing wood in the east. Earthly number four has been rotating right and has combined with heavenly number nine, producing metal in the west. Heavenly number five has been secretly moving in the center and has combined with earthly number ten, producing earth in the center. All things generated by odd numbers have even numbers as their completion; all things generated by even numbers have odd numbers as their completion. Heaven belongs to yang, so heavenly numbers are all odd; earth belongs to yin, so earthly numbers are all even. Odd and even numbers are paired together, and the total number of fifty-five is complete.

大衍之數，臧其五者，五行之用也；虛其一者，元氣之本也。蓋天五為變化之始，散在五行之位，故中無定象。天始生一，肇其有數也，而後生四象、五行之數。今焉虛而不用，是明元氣為造化之宗，居尊不動也。

The reason the number of the Great Derivation is fifty-five, with five subtracted, is because that five belongs to the part of the Five Phases that initiates transmission. Therefore, a 'one' is left in an empty position without being used, because one is the root of the original qi. The number of Heaven is five, which is the beginning of change, spread throughout the five phases. Therefore, in the center, there is no fixed form. Heaven first produces one, thereby initiating the concept of numbers, and then produces the four images and the numbers of the Five Phases. Now, by leaving this one in an empty position without using it, it is to indicate that the original qi is the master of creation, occupying the supreme position without moving.

六位以二、三、四、五為經，七、八、九、六為緯者，象天文也。天以龍、虎、烏、龜為經，鎮、歲、熒惑、太白為緯。六位之經，元玄武之數者，北望惟見斗樞、辰極也。故斗謂之北斗、辰謂之北辰。雖名日北，而實天中，是一、六太極之數，潛宿於五也。河圖六、一居亥、子之北，五居中央，是中央與北皆得太極之數也。

The diagram of the Six Positions of the浑天 is arranged with the numbers two, three, four, and five as the meridians, and seven, eight, nine, and six as the parallels, based on the imagery of the heavens. In the sky, the four celestial beings, the Azure Dragon, White Tiger, Vermilion Bird, and Black Tortoise, are considered the meridians, while the planet Jupiter (岁星), Mars (荧惑), Venus (太白), and Saturn (镇星) are the parallels. The meridian lines of the Six Positions do not include the number corresponding to the Black Tortoise (玄武), because looking north one can only see the Dipper (斗) and the Northern Pole (辰极). Therefore, the Dipper is called "North Dipper" (北斗), and the Northern Pole is called "North Star" (北辰). Although the term "North" is used, in reality it is at the center of the sky. This is the number one and six, the two poles of the Taiji, hidden and contained within the number five. In the River Chart (河图), the numbers six and one are located in the north, at the positions of亥 and子, while the number five is at the center. Thus, both the center and the north have received the Taiji numbers.

六位之緯，元鎮星之數者，鎮星二十八載一周天，歲、熒惑、太白、辰，皆歷鎮星所鎮之宿而行，如陰陽家所謂土居中宮，王四季之說也。故《易》爻凡三百八十四，上契天象之數；而吉凶之變，占於七、八、九、六，如五星之變焉。

The six positions along the meridian do not include the number for Zhèn Xīng, because Zhèn Xīng takes twenty-eight years to circle the sky. When the Sui Xīng, Fēng Mò, Tàibái, and Chénxīng (Water Star) move, they all pass through the star mansions guarded by Zhèn Xīng. As the Yin-Yang school explains, the Earth occupies the Central Palace and is strong in all four seasons. Therefore, the sixty-four lines in the Book of Changes total three hundred and eighty-four, exactly matching the numbers of celestial phenomena. The changes of good and bad fortune depend on the numbers seven, eight, nine, and six, just as the movements of the five planets change their positions.

右伏羲八卦圖，王豫傳於邵康節，而堯夫得之。《歸藏》初經者，伏羲初畫八卦，因而重之者也。其經初乾初爽坤，初艮初兌，初犖坎初離，初釐震初巽，卦皆六畫，即此八卦也。八卦既重，爻在其中。

The eight trigrams on the right side of the diagram were passed down to Shao Kangjie by Wang Yu, and thus Shao Yaofu obtained it. The so-called "Initial Classic" of the "Guicang" refers to the part of the classic where Fuxi first drew the eight trigrams and then transformed them into hexagrams. The hexagrams listed in that classic are called Initial Qian, Initial Shu (which is actually Kun, the original text mistakenly writes it as "Shuang"), Initial Gen, Initial Dui, Initial Luo (which is actually Kan), Initial Li, Initial Li (which is actually Zhen), and Initial Xun. Each hexagram consists of six lines, which is exactly what is referred to here as the eight trigrams. Once the eight trigrams were transformed into hexagrams with six lines, the six yao (lines) were naturally included within them.

鄭氏云：古先天圖，楊雄《太玄經》、關子明《洞極》、魏伯陽《參同契》、邵堯夫《皇極經世》而已。惜乎，雄之《太玄》、子明之《洞極》仿《易》為書，泥於文字，後世忽之，以為屋上架屋，頭上安頭也。伯陽之《參同契》，意在於鍛鍊，而入於術；於聖人之道，又為異端也。堯夫擺去文字小術而著書，天下又不願之，但以為律曆之用，難矣哉。四家之學，皆先於古先天圖。先天圖，其《易》之源乎？復無文字解注，而世亦以為無用之物也。今予作方圓圖，腳註比之四家為最簡易。而四家之意，不出於吾圖之中，於《易》之學為最要。

Zheng's view is that among ancient scholars who could follow the Heavenly Diagram, only Yang Xiong's The Classic of Supreme Mystery, Guan Ziming's The Classic of Ultimate Void, Wei Boyang's The Concordance of the Three, and Shao Yaqian's The Imperial Ultimate Classic were able to do so. Unfortunately, Yang Xiong's The Classic of Supreme Mystery and Guan Ziming's The Classic of Ultimate Void imitated the Book of Changes, being overly concerned with words, so later generations regarded them with contempt, saying they were like building another roof on top of a house or adding another head to a head. Wei Boyang's The Concordance of the Three focused on alchemical cultivation, thus leading into the realm of alchemy; from the perspective of the Way of the Sages, it became considered as heresy. Shao Yaqian wrote his book by setting aside words and minor techniques, yet the world did not sincerely accept his teachings, only treating it as a tool for the study of the calendar and music. How truly difficult it is. The learning of these four schools originated before the ancient Heavenly Diagram. The Heavenly Diagram, perhaps, is the source of the Book of Changes? Only because it lacks textual explanations, the world has regarded it as useless. Now I have drawn this diagram of mutual generation between the square and the circle, adding notes beside it, which is more concise and clear than those four schools. The intent of those four schools does not go beyond the scope of this diagram, and for the study of the Book of Changes, it is the most essential.

伏羲仰觀天文，以畫八卦，故日月星辰之行度運數，十日四時之屬，凡麗於天之文者，八卦無不統之。

Fuxi looked up to observe the patterns of heaven and stars, and based on these he drew the Eight Trigrams. Therefore, the paths and cycles of the sun, moon, and stars, as well as the ten Heavenly Stems and the four seasons, all things connected to the heavenly patterns, are comprehended by the Eight Trigrams without exception.

伏羲俯察地理，以畫八卦，故四方九州島、烏獸草木、十二支之屬，凡麗於地之理者，八卦元不統之。

Fuxi looked down at the patterns on the ground and used them to draw the Eight Trigrams. Therefore, things such as the four directions, the nine provinces, birds and beasts, plants and trees, and the twelve earthly branches, all things related to the earth's patterns, were all included within the Eight Trigrams.

乾陽居上，坤陰居下，乾自震而左行，坤自巽而右行，天左地右，故曰剛柔相摩。

Heaven's yang energy resides above, and Earth's yin energy resides below. Heaven begins from the Zhen position and moves to the left, while Earth begins from the Xun position and moves to the right. The way of Heaven turns to the left, and the way of Earth turns to the right. They move in opposite directions and rub against each other, so the Great Treatise says, "Yang and Yin rub against each other."

震盪艮，兌蕩坤，離蕩巽，坎蕩乾，八卦往來，迭相推蕩。京房日：蕩陰入陽，蕩陽入陰。

Zhen overcomes Jie Tang Gen, Dui overcomes Jie Tang Kun, Li overcomes Jie Tang Xun, Kan overcomes Jie Tang Qian; the Eight Trigrams move towards each other, taking turns to push and stir each other. Jing Fang of the Han dynasty said: to stir Yin into Yang and to stir Yang into Yin.

曹之數，七也；七而七之，其用四十九，故其德圓。卦之數，八也，八而八之，為別六十四，故其德方。圓者運而不窮，可以逆知來物。方者其體有定，可以識乎既往，故圓象神，方象知。

The number of the yarrow stalks is seven. When seven is multiplied by itself three times, the actual number used is forty-nine, so the nature of the yarrow stalks is round. The number of the hexagrams is eight. When eight is multiplied by itself three times, sixty-four hexagrams are produced, so the nature of the hexagrams is square. Round things are ever turning and never cease, thus they can foresee what is to come. Square things have fixed forms, thus they can discern what has already passed. Therefore, roundness symbolizes the "spirit" mentioned in the Great Treatise, and squareness symbolizes the "knowledge" mentioned in the Great Treatise.

本《乾》九二爻變成《同人》，次變《無妄》，次變《益》，次變《頤》，終其變至《離》而止；本《坤》六二爻變成《師》，次變《升》，次變《恆》，次變《大過》，終其變至《坎》而止，故上經始於《乾》、《坤》，終於《坎》、《離》焉。

Starting from the second line of the Qian hexagram, changing one line transforms it into the Tongren hexagram, then into the Wuwang hexagram, then into the Yi hexagram, then into the Yi hexagram, continuing until reaching the Li hexagram. Starting from the second line of the Kun hexagram, changing one line transforms it into the Shi hexagram, then into the Sheng hexagram, then into the Heng hexagram, then into the Daguover hexagram, continuing until reaching the Kan hexagram. Therefore, the Upper Canon begins with Qian and Kun and ends with Kan and Li.

本《咸》六二爻變成《大過》，次變《困》，次變《坎》，次變《師》，次變《蒙》，而終於《未濟》；本《恆》初六爻變成《大壯》，次變《豐》，次變《震》，次變《復》，次變《屯》，而入《既濟》，故下經始於《咸》、《恆》，終於《既濟》、《未濟》焉。

Starting from the second line of the Hexagram 咸 (Xian), changing one line leads to 大过 (Da Guo), then to 困 (Kun), then to 坎 (Kan), then to 师 (Shi), then to 蒙 (Meng), and finally reaching 未济 (Wei Ji). Starting from the first line of the Hexagram 恒 (Heng), changing one line leads to 大壮 (Da Zhuang), then to 丰 (Feng), then to 震 (Zhen), then to 复 (Fu), then to 屯 (Zun), and finally reaching 既济 (Ji Ji). Therefore, the lower section begins with 咸 (Xian) and 恒 (Heng), and ends with 既济 (Ji Ji) and 未济 (Wei Ji).

#1『坤』，原脫，據四庫本補。#2『定位』，原作『宅位』，據四庫本改。#3『相胥』，四庫本作『相需』，其義同。#4『自記』，四庫本作『自托』。#5『馳下』，四庫本作『馭下』。#6『而』，據四庫本補。#7『為』，據四庫本補。#8『未』，據四庫本補。

The first point: The character 「坤」 was originally missing in the base text and has been added according to the Siku Quanshu edition. The second point: The term 「定位」 was originally written as 「宅位」 in the base text and has been corrected according to the Siku Quanshu edition. The third point: 「相胥」 in the base text is the same as 「相需」 in the Siku Quanshu edition, meaning the same thing. The fourth point: 「自记」 in the base text is 「自托」 in the Siku Quanshu edition. The fifth point: 「驰下」 in the base text is 「驭下」 in the Siku Quanshu edition. The sixth point: The character 「而」 has been added according to the Siku Quanshu edition. The seventh point: The character 「为」 has been added according to the Siku Quanshu edition. The eighth point: The character 「未」 has been added according to the Siku Quanshu edition.

#9『演』原作『言』，據四庫本改。#10『動』，據四庫本補。#11『退』，據四庫本補。#12『鎮』，四庫本作『辰』。疑『鎮』字誤。#13『天象』，原作『上象』，據四庫本改。#14『堯夫』，原作『鄭夫』，據四庫本改。#15『初突』，四庫本作『初奭』。#16『未濟』，疑底本脫，據句義增補。

Article 9: The character 「演」 in the original text was 「言」, and it has been corrected according to the Siku Quanshu edition. Article 10: The character 「动」 has been added according to the Siku Quanshu edition. Article 11: The character 「退」 has been added according to the Siku Quanshu edition. Article 12: The character 「镇」 in the Siku Quanshu edition is 「辰」, and it is suspected that 「镇」 is a mistaken character. Article 13: The original text had 「上象」, which has been corrected to 「天象」 according to the Siku Quanshu edition. Article 14: The original text had 「郑夫」, which has been corrected to 「尧夫」 according to the Siku Quanshu edition. Article 15: The original text had 「初突」, but the Siku Quanshu edition has 「初奭」. Article 16: The two characters 「未济」 are suspected to be missing in the original text, and have been added according to the meaning of the sentence.