I Ching启蒙

聖人現象以畫卦，揲蓍以命爻，使天下後世之人，皆有以決嫌疑，定猶豫，而為迷於吉凶悔吝之途，其功可謂盛矣。然其為卦也，自本而乾，自乾而枝，其勢若有所迫而自不能已；其為蓍也，分合進退，從橫逆順，亦無往而不相值焉，是豈聖人心思智慮之所得為也哉。特氣數之自然，形於法象，見於圖書者，有以啟於其心而假手焉爾。

Zhu Xi's preface to "Yixue Qimeng" says: The sages observed the forms of things and thus drew the trigrams. They divided the chrysanthemum grass to determine the positions of the lines, enabling people in later generations to resolve doubts and settle hesitations, without getting lost in the path of fortune, misfortune, regret, and hardship (the original text mistakenly reads "and be lost" instead of "not be lost"). This achievement is indeed great. However, the origin of the trigrams is like a main trunk growing from the root, and branches growing from the trunk. The momentum is as if something is pushing from behind, making it impossible to stop. The calculation with the chrysanthemum grass involves separating and rejoining, advancing and retreating, moving vertically and horizontally, pushing forward and backward, and every aspect is perfectly aligned. Could this be something the sages arranged with their thoughts and wisdom? It is merely the natural principle of qi and numbers, manifesting as a pattern, revealed in the Yellow River Chart and the Luo Writing, which inspired the sages' minds. The sages merely used their hands to draw it out.

近世學者，類喜談《易》而不察乎此，其專於文義者，既支離散漫，而無所根著，其涉於象數者，又皆牽合傅會，而或以為出於聖人心思智慮之所為也，若是者予竊病焉，因與同志，頗輯舊文，為書四篇，以示初學，使毋疑於其說雲。淳熙丙午暮春既望魏氏了翁曰：朱文公《易》，得於邵子為多。蓋不讀邵《易》，則茫不知《啟蒙》、《本義》之所以作。

Recent scholars tend to talk a lot about the I Ching, but they have not paid attention to this point: those who focus solely on the meaning and words of the text speak in a scattered and confused manner, lacking a solid foundation; those who involve the symbolic numbers and figures are all forced and artificial, even going so far as to think that these things are entirely the inventions of the sage's mind. I have privately felt deeply uneasy about this situation, so I have compiled with like-minded friends the old views of previous scholars into four sections, and presented them to beginners so that they will no longer be confused about these views. Written on the sixteenth day of the late spring of the year Bingwu in the Chunxi period. Collected Sayings: Wei Le翁 said that Zhu Wen Gong's commentary on the I Ching was mostly derived from Shao Yong's studies; if one does not read Shao Yong's I Ching studies, then one would be completely confused as to why the I Ching启蒙 and The Original Meaning of the Zhou Changes were written in that way.

《易大傳》曰：“河出圖，洛出書，聖人則之。”孔安國云：“河圖者，伏羲氏王天下，龍馬出河遂則其文，以畫八卦。洛書者，禹治水時，神龜負文而列於背，有數至九，禹遂因而第之，以成九類。

The Great Treatise of the Book of Changes, specifically the Upper Part of the Great Treatise, says: "When the river Huang appeared with a chart and the Luo river appeared with a book, the sage took them as models." Kong Anguo explained that the so-called Yellow River Chart appeared during the time when Fuxi ruled over the world. A dragon horse emerged from the Yellow River, and Fuxi took the patterns on its body and used them to draw the Eight Trigrams. As for the Luo Writing, it appeared during the time when Yu the Great controlled the floods. A divine tortoise carried a pattern on its back, arranged in numbers up to nine, and Yu followed this arrangement to establish the Nine Categories of the Great Plan.

集說 朱子答袁樞曰：以《河圖》、《洛書》為不足信，自歐陽公以來，已有此說，然終無奈《顧命》、《繫辭》、《論語》皆有是言，而諸儒所傳二圖之數，雖有交互而無乖戾，順數逆推，縱橫曲直皆有明法，不可得而破除也。至如《河圖》與《易》之天一至 劉歆云：伏羲氏繼天而王，受《河圖》而畫之，八卦是也。禹治洪水，錫《洛書》法而陳之，九疇是也。《河圖》、《洛書》相為經緯、八卦九章相為表裡。

Collected Commentaries: In a letter to Yuan Shu, Zhu Xi said that since Ouyang Xiu, there has been the view that the Yellow River Chart and the Luo Writing are not reliable. However, it is difficult to dismiss them because the Great Treatise, the Book of Documents' Gu Ming section, and the Analects all contain such statements. The number of the two diagrams transmitted by various scholars, although differing in their interpretations, do not contradict each other. Whether counted in order, reversed, vertically, horizontally, or diagonally, there are clear rules, making it impossible to disregard them entirely. As for the Yellow River Chart's heavenly one to... the text here is missing. Zhu Xi's words are incomplete, and the text directly continues with Liu Xin's words. Liu Xin said: Fuxi inherited the mandate of heaven and became king, receiving the Yellow River Chart and drawing it, which became the Eight Trigrams; Yu the Great, when he controlled the floods, received the Luo Writing from heaven and based his laws upon it, which became the Nine Categories of the Book of Documents. The Yellow River Chart and the Luo Writing form the meridians and the looms, while the Eight Trigrams and the Nine Categories form the outer and inner aspects.

關子明云：《河圖》之文，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。集說 朱子書《河圖》、《洛書》曰：讀《大戴禮》書，又得一證甚明。其《明堂》篇有“二九四七五三六一八”之語，而鄭氏注云：“法龜文也”，然則漢人固以九數者為《洛書》也。

Guan Lang said: The numbers on the Yellow River Chart are seven in front, six behind, eight on the left, and nine on the right; the numbers on the Luo Writing are nine in front, one behind, three on the left, seven on the right, four in the upper left corner, two in the upper right corner, eight in the lower left corner, and six in the lower right corner. Here, "front" refers to the south, "behind" refers to the north, "left" refers to the east, and "right" refers to the west. Collected comments: Zhu Xi in his essay "After Reading the Yellow River Chart and Luo Writing" said that by reading the Dai Dai Li Ji, he obtained a very clear evidence: the "Bright Hall" chapter contains the phrase "two nine four, seven five three, six one eight." Zheng Xuan's commentary says "this is based on the patterns on the turtle's back." This shows that people in the Han dynasty originally regarded the diagram with nine numbers as the Luo Writing.

又偶讀《漫記》曰：子華子論《河圖》之二與四抱九而上躋，六與八蹈一而下沈，五居其中，據三持七，巧亦甚矣，唯其甚巧，所以知其非古書也。案 鄭注《大戴禮》是確證，至子華子，則位置雖明，但錯以《洛書》為《河圖》，故朱子疑其非古書。

Zhu Xi also wrote in a note on reading: The "Zi Hua Zi" discusses the "Heavenly River Chart," saying that two and four embrace nine and ascend upward, six and eight step down, and five is in the center, with three on the left and seven on the right. This explanation is remarkably accurate. However, because it is too accurate, Zhu Xi concluded that it was not an ancient book from the Pre-Qin period. Note: The annotation by Zheng Xuan on the "Da Dai Li Ji" is solid evidence. As for the "Zi Hua Zi," although its discussion of directions is clear, it mistakenly took the "Luo Writing" as the "Heavenly River Chart," so Zhu Xi suspected it was not an ancient text.

邵子曰：圓者星也，曆紀之數，其肇於此乎。曆法合二始以定剛柔，二中以定律曆，二終以紀閏餘，足所謂曆紀也。方者土也，畫州井地之法，其仿於此乎。州有九，井九百畝，是所謂畫州井地也。蓋圓者《河圖》之數，方者《洛書》之文，故羲文因之而造《易》，禹箕敘之而作《範》也。

Shao Yong said: The circular form represents the starry heavens, and the calculation of years and the calendar system probably originated here. The calendar combines the two initial numbers to determine whether a day is yang or yin, combines the two central numbers to determine the pitch and the calendar numbers, and combines the two final numbers to record the leap years. This is called "the calendar record." The square form represents the land, and the method of dividing the states and managing the well-field system probably imitates this. There are nine states, and one well-field is nine hundred mu, which is called "drawing the states and the well-field." In general, the circular form belongs to the numbers of the Yellow River Chart, and the square form belongs to the writings of the Luo Writing. Therefore, Fuxi and King Wen used them to create the Zhou Changes, and Da Yu and Ji Zi used them to arrange the Hong Fan.

蔡元定曰：古今傳記，自孔安國劉向父子、班固皆以為《河圖》授羲，《洛書》錫禹。關子明、邵康節，皆以十為《河圖》，九為《洛書》。蓋《大傳》既陳“天地五十有五”之數，《洪範》又明言“天乃錫禹洪範九疇”，而九宮之數，戴九履一，左三右七，二四為肩，六八為足，正龜背之象也。唯劉牧臆見，以九為《河圖》，十為《洛書》，託言出於希夷，既與諸儒舊說不合，又引《大傳》以為二者皆出於伏羲之世，其《易》置圖書，並無明驗，但謂伏羲兼取圖書，則《易》、《範》之數誠相表裡，為可疑耳。

Cai Yuanding said: The biographies of ancient and modern times, from Kong Anguo, Liu Xiang and his son Liu Xin, up to Ban Gu, all hold that the Yellow River Chart was given to Fuxi and the Luo Writing was bestowed upon Da Yu. Guan Lang and Shao Yong both regarded the diagram with ten numbers as the Yellow River Chart and the one with nine numbers as the Luo Writing. Since the Great Treatise has already stated "the numbers of heaven and earth total fifty-five," and the Hongfan chapter clearly says "Heaven gave Da Yu the Hongfan nine categories," the numbers of the nine palace diagram are arranged with nine on top, one beneath, three on the left, seven on the right, two and four as the shoulders, and six and eight as the feet, forming exactly the shape of a tortoise shell. Only Liu Mu made up his own theory subjectively, taking the nine-numbered diagram as the Yellow River Chart and the ten-numbered one as the Luo Writing, and even claimed that his source was Chen Tuan. This contradicts the previous views of the scholars of earlier generations. Moreover, he cited the Great Treatise to argue that both diagrams originated in the time of Fuxi, but he reversed the names of the two diagrams, which lacks solid evidence. However, if one says that Fuxi used both the diagram and the writing at the same time, then the numbers of the Zhou Changes and the Hongfan indeed complement each other, and this point is worth considering.

其實天地之理，一而已矣，雖時有古今先後之不同，而其理則不容於有二也。故伏羲但據《河圖》以作《易》，則不必豫見《洛書》，而已逆與之合矣。大禹但據《洛書》以作《範》，則亦不必追考《河圖》，而已暗與之符矣。其所以然者何戰？誠以此理之外，無復它理故也。

In fact, the principle of the universe is one: although times have distinctions of ancient and modern, the principle does not allow for two different systems. Therefore, Fuxi composed the Book of Changes solely based on the Yellow River Chart, without needing to have seen the Luo Writing beforehand, yet the Book of Changes was already in harmony with the Luo Writing. Similarly, the Great Yu composed the Huang Fan solely based on the Luo Writing, without needing to refer back to the Yellow River Chart, yet the Huang Fan was already in accord with the Yellow River Chart. Why is this so (the original text mistakenly says 'why fight' instead of 'why so')? It is really because there is no other principle apart from this one.

然不特此爾，律呂有五聲十二律，而其相乘之數，究於六十，日名有十干十二支，而其相乘之數，亦究於六十，二者皆出於《易》之後，其起數又各不同，然與《易》之陰陽策數多少自相配合，皆為六十者無不合若符契也。下至運氣參同太乙之屬雖不足道，然亦無不相通，蓋自然之理也。

Moreover, this is not only one matter. The five tones and twelve pitch pipes, when multiplied together, result in a total of sixty; the ten heavenly stems and twelve earthly branches, when multiplied together, also result in a total of sixty. These two systems appeared after the Zhou Changes, and their methods of calculation differ from each other, yet they correspond with the number of yin and yang counts in the Zhou Changes, and wherever the number sixty is reached, they match perfectly like a seal and its counterpart. Further, when discussing the theory of the Five Elements, the Zhou Changes Combined with the Great Ultimate, and the Taiyi numerology, although they are not worth much emphasis, none of them can be separated from them—these are all natural phenomena.

假令今世復有圖書者出，其數亦必相符，可謂伏羲有取於今日而作《易》乎？《大傳》所謂“河出圖，洛出書，聖人則之”者，亦汛言聖人作《易》作《範》，其原皆出於天之意，如言“以卜筮者尚其占”，與“莫大乎蓍龜”之類，《易》之書豈有龜與卜之法子？亦言其理無二而已爾。

Suppose such a book were to appear again in the present age, its numbers would certainly correspond with them; could one say that Fuxi derived his material from today's books to compose the Book of Changes? The passage in the Great Treatise says "the river produced a diagram, the Luo produced a book, and the sage followed it," which is only a general statement about the sage composing the Book of Changes and the Book of Hongfan, whose origin lies in heaven, just like it says "those who use divination should value the divination" and "nothing is greater than the tortoise shell and yarrow stalks." Where in the Book of Changes is there really a method of scorching tortoise shells and using them for divination (the original text mistakenly has "hu" instead of "zi")? It is merely saying that the principles of divination and casting lots are essentially the same.

此一節，夫子所以發明《河圖》之數也。天地之間，一氣而已，分而為二，則為陰陽，而五行造化，萬物始終，無不管於是焉。故《河圖》之位，一與六共宗而居乎北，二與七為朋而居乎南，三與八同道而居乎東，四與九為友而居乎西，五與十相守而居乎中。蓋其所以為數者，不過一陰一陽，以兩其五行而已。

The text of "Qimeng" says: The passage from "Tian Yi Di Er" to "This is why there are changes and the activities of ghosts and spirits" in the "Great Treatise" is used by Confucius to explain the numbers of the Yellow River Chart. Between heaven and earth there is only one qi; when it is divided into two, it becomes yin and yang. The transformations of the Five Phases and the beginning and end of all things are all governed by this. Therefore, the directions of the Yellow River Chart are: one and six share the same origin and reside in the north, two and seven are companions and reside in the south, three and eight follow the same path and reside in the east, four and nine are friends and reside in the west, five and ten keep each other and reside in the center. The reason it takes this form is simply that one yin and one yang are paired with each of the Five Phases, forming two sets for each.

所謂天者，陽之輕清而位乎上者也。所謂地者，陰之重濁而位乎下者也，陽數奇，故一三五七九皆屬乎天，所謂“天數五”也。陰數偶，故二四六八十皆屬乎地，所謂“地數五”也，天數地數，各以類而相求，所謂“五位之相得”者然也。

The term "Heaven" refers to the light and clear aspect of yang energy that resides in the upper position; the term "Earth" refers to the heavy and turbid aspect of yin energy that resides in the lower position. The numbers of yang are odd, so one, three, five, seven, and nine belong to Heaven, which is what the Great Treatise calls "Heaven's number is five." The numbers of yin are even, so two, four, six, eight, and ten belong to Earth. Heaven's numbers and Earth's numbers each seek their corresponding counterparts within their own categories, which is what is meant by "the five positions correspond to each other."

天以一生水，而地以六成之，地以二生火，而天以七成之，天以三生木，而地以八成之，地以四生金，而天以九成之，天以五生土，而地以十成之，此又其所謂“各有合”焉者也。積五奇而為二十五，積五偶而為三十，合是二者而為五十有五，此《河圖》之全數，皆夫子之意，而諾儒之說也。至於《洛書》，則雖夫子之所未言，然其象其說，已具於前，有以通之，則劉歆所謂經緯，表裡者可見矣。

Heaven produces water with the first odd number, and Earth gives it form with the sixth even number; Earth produces fire with the second odd number, and Heaven gives it form with the seventh odd number; Earth produces wood with the third odd number, and Heaven gives it form with the eighth even number; Earth produces metal with the fourth odd number, and Heaven gives it form with the ninth odd number; Earth produces earth with the fifth odd number, and Heaven gives it form with the tenth even number. This is what the Great Treatise refers to as 'each has its corresponding number'. Adding up the five odd numbers gives twenty-five, and adding up the five even numbers gives thirty; together they total fifty-five, which is the full number of the Yellow River Chart. The part about the numbers of Heaven and Earth is the original meaning of Confucius, while the part about the generation of the Five Phases is the view of scholars from the Han dynasty onwards (the original text mistakenly writes '诺儒' instead of '诸儒'). As for the Luo Writing, although Confucius did not mention it, its diagram and related explanations have already been presented earlier. By connecting both sides, the idea that Mencius said they are interwoven like meridians and collaterals, and interdependent as inside and outside, becomes clear.

案 中間述《大傳》處，是夫子之意，天一生水之類，則是諸儒之說，蓋諸儒舊說，皆以五行說圖書，故朱子於《啟蒙》、《本義》，因而仍之，它日又曰：《河圖》、《洛書》於八卦九章不相著，未知如何也。然則朱子之意，蓋疑圖書之精蘊，不盡於諸儒之所云者爾。

Note: The passage above mentions that the part quoting the Great Treatise is Confucius' original meaning; phrases like "Heaven produces water" are the views of scholars from the Han dynasty onwards. These scholars used the Five Phases to explain the Yellow River Chart and the Luo Writing, so Zhu Xi in his works The Awakening of the Mind to the I Ching and The Original Meaning of the Zhou Changes also followed this approach. However, he later said: "The Yellow River Chart and the Luo Writing do not closely correspond with the Eight Trigrams and the Nine Chapters; it is unclear how to explain this." From this, Zhu Xi's meaning is that he doubted whether the subtle implications in the Yellow River Chart and the Luo Writing had been fully explained by the scholars.

或曰：《河圖》、《洛書》之位與數其所以不同何也？曰《河圖》以五生數統五成數而同處其方，蓋揭其全以示人而道其常，數之體也。《洛書》以五奇數統四偶數而各居其所，蓋主於陽以統陰而肇其變，數之用也。

Someone asked: Why do the directions and numbers in the Yellow River Chart and the Luo Writing differ from each other? Answer: The Yellow River Chart uses five generating numbers to govern five forming numbers, with generating and forming numbers sharing the same direction. This is presenting the whole to people, discussing the constant principle, belonging to the 'essence of numbers'. The Luo Writing uses five odd numbers to govern four even numbers, with odd and even numbers each residing in their own positions. This is using yang to govern yin, thereby initiating change, belonging to the 'function of numbers'.

集說 趙氏汝楳曰：一對二，三對四，而五居中，六七合一二，八九合三四，而十合五，奇偶數對，陰陽有合，而數之體以立。聖人所謂“陰陽合德而剛柔有體”者，此其類也。

Collected interpretations: Zhao Ru Mei said, one and two are opposites, three and four are opposites, while five is in the center; six and seven correspond with one and two, eight and nine correspond with three and four, and ten corresponds with five. Odd and even numbers form pairs, yin and yang correspond with each other, thus establishing the nature of numbers. The "Great Treatise" states, "Yin and yang combine in virtue, and hardness and softness have form," which refers precisely to this kind of principle.

體立矣不變則數不行，故陽以三左行，陰以二右行，三其一為三，而居東，三其三為九，而居南，三其九為二十七，而七居西，三其二十七為八十一，而一復居北，等而上之，至於億兆，其餘數之位皆然。二其二為四而居東南，二其四為八而居東北，二其八為十六，而六居西北，二其十六為三十二，而二復居西南，上而億兆亦然。八位既列，五仍居中，而數之用以通，聖人所謂“參伍以變，錯綜其數”者，此其類也。

The structure is already established, but if it does not undergo change, the numbers will not function. Therefore, the yang numbers progress by multiplying by three and moving left, while the yin numbers progress by multiplying by two and moving right. Three times one is three, which resides in the East; three times three is nine, which resides in the South; nine times three is twenty-seven, of which seven remains and resides in the West; twenty-seven times three is eighty-one, of which one remains and returns to the North. Proceeding in this way layer by layer up to the number of myriad myriads, the remaining numbers will also reside in their respective directions. On the yin side: two times two is four, which resides in the Southeast; four times two is eight, which resides in the Northeast; eight times two is sixteen, of which six remains and resides in the Northwest; sixteen times two is thirty-two, of which two remains and returns to the Southwest; proceeding upward to the number of myriad myriads follows the same pattern. The eight directions are thus arranged, with five remaining in the center, and thus the function of numbers is established. The passage in the Great Treatise says, "By combining three and five, numbers change and are interwoven," referring precisely to this principle.

鮑氏云龍曰：以《洛書》變數推之，陽以三左行，天圓徑一圍三。三，天數也。一在北：一而三之，三在東；三其三為九，而居南；九而三之，三九二十七而居西；三其二十七為八十一，而一復居於北。北而東，東而由，南而西，西而復北，循環不窮，有以符；天道左旋之義。

Bao Yunlong said: Using the changing numbers of the Luo Writing to calculate, the yang numbers progress to the left in multiples of three, because the heavens are round, and when the diameter is one, the circumference is three, and three represents the number of heaven. One is in the north; one multiplied by three gives three, which is in the east; three multiplied by three gives nine, which is in the south; nine multiplied by three gives twenty-seven, and taking the remainder seven, which is in the west; twenty-seven multiplied by three gives eighty-one, and taking the remainder one, which returns to the north. From north to east, from east to south (the original text mistakenly writes 'east and by' instead of 'east and south'), from south to west, and from west back to north, continuing in a cycle, which corresponds to the heavenly way of rotating to the left.

地方徑一圍四，兩其二也，蓋以地上之數起於二，而陰資以為始，位在西南而右行，二而二之為四，而居東南，二而四之為八，而居東北：二其八為十六，而居西北，二其十六為三十二，而二復居西南本位。西南而東南，東南而東北，東北而

The earth is square in shape, with a diameter of one and a circumference of four. Four is exactly twice two. Since the numbers of the earth begin with two, the yin energy starts from it, so two is located in the southwest and moves to the right. Twice two is four, which is located in the southeast; twice four is eight, which is located in the northeast; twice eight is sixteen, and the remainder six is located in the northwest; twice sixteen is thirty-two, and the remainder two returns to the original position in the southwest. From the southwest to the southeast, from the southeast to the northeast, from the northeast downward... The original text here is missing some words, and Bo's passage is incomplete. The following directly continues with Li Guangdi's commentary.

案 朱子此條，己盡圖書之大義。蓋以生數統成數而同處其方者，自五以前為方生之數，自五以後為既成之數。陰生則陽成，陽生則陰成，陰陽二氣，相為終始，而未嘗相離也。以奇數統偶數而各居其所者，四正之位，奇數居之；四維之位，偶數居之。陰統於陽，地統於天，天地同流，而定分不易也。揭其至以示人，而道其常者數至十而始全，缺一則不全矣。故曰數之體主於陽以統陰，而肇其變者，始於一，終於九，所以起因乘歸除之法，故曰數之用。然生成之理則明矣，而正維之位所自定者，唯趙氏鮑氏之說，為能推明其義，諸家皆不及也。

The commentary says: Master Zhu's explanation of the Yellow River Chart and the Luo Writing has already covered their main principles. The phrase "using the generating numbers to unify the completed numbers and placing them in their respective directions" means that the numbers one, two, three, and four before five are generating numbers, while the numbers six, seven, eight, nine, and ten after five are completed numbers; when yin is generated, it is accomplished by yang, and when yang is generated, it is accomplished by yin. Yin and yang energies alternate as beginning and end, never separating. The phrase "using odd numbers to unify even numbers and placing them in their respective positions" means that the four cardinal directions are occupied by odd numbers, while the four intermediate directions are occupied by even numbers; yin is governed by yang, and earth is governed by heaven. Although heaven and earth flow together in one energy, the distinction of respect and hierarchy remains unchanged. "Revealing the whole to people and explaining the constant" is because numbers only become complete when they reach ten; lacking one number makes them incomplete, so they are called "the essence of numbers." "Taking yang as the main principle to unify yin and initiating change" is because it begins with one and ends with nine, from which the methods of cause, multiplication, return, and division are derived, so they are called "the application of numbers." The principle of generation has been clearly explained, but regarding how the four cardinal directions and four intermediate directions are determined, only the interpretations of Zhao Ruyi and Bao Yunlong can elaborate on the meaning, while the other schools are unable to match them.

曰：其皆以五居中者何也？曰：凡數之始，一陰一陽而已矣。陽之象圓，圓者徑一而圍三，陰之象方，方者徑一而圍四，圍三者以一為一，故參其一陽而為三。圍四者以二為一，故兩其一陰而為二，是所謂“參天兩地”者也。二二之合，則為五矣，此《河圖》、《洛書》之數，所以皆以五為中也。案 三二之合，五也，一四之合，亦五也，一一二二之積，又五也，三二四四之積，又五之積也，此五所以為數之會而位之中與。

Question: Why do both diagrams have five in the center? Answer: The beginning of numbers is only one yin and one yang. The image of yang is round, and when the diameter is one, the circumference is three. The image of yin is square, and when the diameter is one, the circumference is four. For the circumference of three, taking one as a unit, one yang is combined with three to form three; for the circumference of four, taking two as a unit, one yin is doubled to form two — this is what the "Say卦Transmission" refers to as "参天两地" (three for heaven, two for earth). When three and two are combined, it becomes five (the original text mistakenly writes "二二" instead of "三二"). This is why both the Yellow River Chart and the Luo Writing have five in the center. Note: When three and two are combined, it is five; when one and four are combined, it is also five. The numbers one, one, two, two, when multiplied together, do not exceed five. The numbers three, two, four, four, when multiplied together, are only multiples of five. This is why five becomes the convergence of all numbers and the center of directions.

然《河圖》以生數為主，故其中之所以為五者，亦具五生數之象焉。其下一點，天一之象也，其卜一點，地二之象也，其左一點，天三之象也，其右一點，地四之象也，其中一點，天五之象也。

However, the He Tu focuses on the generating numbers, so the five points in the center also embody the forms of the five generating numbers: the point below represents the image of Tian Yi; the point above (the original text mistakenly writes 'bu' instead of 'shang') represents the image of Di Er; the point on the left represents the image of Tian San; the point on the right represents the image of Di Si; and the central point represents the image of Tian Wu.

《洛書》以奇數為主，故其中之所以為五者，亦具五奇數之象焉。其下一點，亦天一之象也，其左一點，亦天三之象也，其中一點，亦天五之象也，其右一點，則天七之象也，其上一點，則天九之象也，其數與位，皆三同而二異，蓋陽不可易，而陰可易，成數雖陽，固亦生之陰也。

The Luo Writing is based mainly on odd numbers, so the five points in the center also embody the forms of five odd numbers: the point below represents the image of Tian Yi (Heavenly One); the point on the left represents the image of Tian San (Heavenly Three); the central point represents the image of Tian Wu (Heavenly Five); the point on the right represents the image of Tian Qi (Heavenly Seven); and the point above represents the image of Tian Jiū (Heavenly Nine). The numbers and positions of the five central points in the two diagrams are the same in three places and different in two. This is because the yang is unchangeable, while the yin can be changed; although seven and nine are yang numbers that have been formed, they are still on the side of being formed after one and three, which are the original numbers.

曰：中央之五，既為五數之象矣，然其為數也，奈何？曰：以數言之通乎一圖，由內及外，固各有積實可紀之數矣，然《河圖》之一二三四各居其五象本方之外，而六七八九十者，又各因五而得數，以附於其生數之外。《洛書》之一三七九，亦各居其五象本方之外，而二四六八者，又各因其類以附於奇數之側，蓋中者為主，而外者為客，正者為君，而側者為臣，亦各有條而不紊也。

Question: Since the central five are the images of five numbers, how should we explain them as numbers themselves? Answer: If we consider them as numbers alone, looking at the whole diagram, from the center outward, each has a real countable number. However, in the Yellow River Chart, the numbers one, two, three, four are each located outside the central five's corresponding directions, while six, seven, eight, nine, and ten each add to the central five and are attached to the side of their respective generating numbers. In the Luo Writing, the numbers one, three, seven, and nine are each located outside the central five's corresponding directions, while two, four, six, and eight are each attached to the side of the odd numbers. In summary, the central is the main, the outer is the guest, the one in the central position is the ruler, and the one in the side position is the minister, each has its order and does not become disorderly.

集說 翁氏泳曰：《河圖》東北陽方，則主之以奇，而與合者偶，西南陰方，則主之以偶，而與合者奇。吳氏曰慎曰：陽始北而終西，一三陽尚微，故居內。七九陽盛而著於外也，必實其中而後能著乎外，故五居中。陰始南而終東，二四陰尚微，故居內。六八陰盛而凝於外也，必堅乎外而後能實其內，故十居中。自中而外，陽之生長，自外而中，陰之收藏，觀於草木之枝葉果實，亦可見矣。

Collected Explanations: Wang Yong said, the East and North belong to Yang, so they are governed by odd numbers, and they are paired with even numbers. The West and South belong to Yin, so they are governed by even numbers, and they are paired with odd numbers. Wu Yashen said, Yang begins from the North and ends at the West. One and three are still weak in Yang energy, so they are located in the inner layer. Seven and nine are strong and prominent in Yang energy, so they are located in the outer layer. One must first strengthen the inner part before the outer part can be expressed, so five is in the center. Yin begins from the South and ends at the East. Two and four are still weak in Yin energy, so they are located in the inner layer. Six and eight are strong and concentrated in Yin energy, so they are located in the outer layer. One must first solidify the outer part before the inner part can be filled, so ten is in the center. Moving from the center outward is the growth of Yang energy; moving from the outside inward is the gathering of Yin energy. Observing the process of plants first sprouting branches and leaves and then bearing fruit, this can also be understood.

五，生數之終，十，成數之終，而藏於中，此“太和”之所以“保合”深固，而生機之所以充實於內也。案 此段即與上生數統成數奇數，統偶數一段相發明。以生數統成數者，生數常居內而為主，成數常居外而為客。如一歲之寒暑往來，一月之陰魄死生，一日之晝夜進退，

Wu Yueshen further explains: Five is the last of the generating numbers, and ten is the last of the completing numbers. Both are contained within the center. This is precisely what the Commentary on the Judgment of the Qian hexagram says: "Preserving the great harmony" is why it is profound and firm, and why the vital energy is full within. Note: This passage complements the previous section on "using generating numbers to unify completing numbers, and using odd numbers to unify even numbers." What is meant by "using generating numbers to unify completing numbers" is that generating numbers are usually internal and primary, while completing numbers are external and secondary, just as the alternation of cold and heat in a year, the birth and death of the moon's essence in a month, and the advance and retreat of day and night in a day... The original text here is missing some words, and the note is incomplete. The text then directly continues with the question and answer from the main body of the启蒙 (Awakening).

曰：其多寡之不同何也？曰：《河圖》主全，故極於十，而奇偶之位均，論其積實，然後見其偶贏而奇乏也。洛書主變，故極於九，而具位與實，皆奇贏而偶乏也。必皆虛其中也，然後陰陽之數，均於二十而無偏耳。案 此段亦與上段數之體、數之用相發明。

Question: Why do the two diagrams have different numbers? Answer: The Yellow River Chart focuses on completeness, so it goes up to ten. The odd and even positions each have five numbers, making them equal. Only when the actual numbers in each position are added together does it become clear that the even numbers total thirty and the odd numbers total twenty-five, with the even numbers being more. The Luo Writing focuses on change, so it goes up to nine. Whether considering the positions or the actual numbers, the odd numbers occupy five positions totaling twenty-five, while the even numbers occupy four positions totaling twenty, with the odd numbers being more. Only when the central numbers of both diagrams are excluded, the numbers on the yin and yang sides each total twenty, making them equal and balanced. Note: This passage also complements the previous one regarding the "nature of numbers" and "use of numbers."

曰：其序之不同何也？曰《河圖》以生出之次言之，則始下次上次左次右以復於中，而又始下也。以運行之次言之，則始東次南次中次西次北，左旋一周而又始於東也。其生數之在內者，則陽居下左而陰居上右也，共成數之在外者，則陰居下左，而陽居上右也。《洛書》之次，其陽數，則首北次東次中次西次南，其陰數，則首西南次東南次西北次東北也。合而言之，則首北次西南次東次東南次中次西北次西次東北而究於南也。

Question: How do the sequences of the two diagrams differ? Answer: The sequence of the Yellow River Chart, if considered according to the order of generation, starts with the number one in the north, then the number two in the south, followed by the number three in the east, then the number four in the west, returning to the center with the number five. Then it starts again from the number six in the north. If considered according to the order of the Five Phases, it begins with wood in the east, then fire in the south, then earth in the center, then metal in the west, and finally water in the north, rotating to the left and starting again from the east. The inner numbers, which are yang numbers, are located below and to the left, while the yin numbers are located above and to the right. The outer numbers, which are the completed numbers, have yin numbers below and to the left, and yang numbers above and to the right. The sequence of the Luo Writing, in terms of yang numbers, starts with one in the north, then three in the east, five in the center, seven in the west, and nine in the south. In terms of yin numbers, it starts with two in the southwest, four in the southeast, six in the northwest, and eight in the northeast. When combined and generally described, the sequence begins with the north, then the southwest, then the east, then the southeast, then the center, then the northwest, then the west, then the northeast, and ends in the south.

其運行，則水克火，火克金，金克木，木克土，右旋一周，而土復克水也，是亦各有說矣。曰：其七八九六之數不同何也？曰《河圖》六七八九，既附於生數之外矣，此陰陽老少進退饒乏之正也，其九者，生數一三五之積也，故自北而東，自東而西，以成於四之外。其六者，生數二四之積也，故自南而西，自西而北，以成於一之外。七則九之自西而南者也，八則六之自北而東者也，此又陰陽老少、互藏其宅之變也。

As for the movement of the Luo Shu, it involves water overcoming fire, fire overcoming metal, metal overcoming wood, wood overcoming earth, and earth overcoming water, rotating clockwise in a cycle. Each of these has its own explanation. Question: Why are the numbers seven, eight, nine, and six different between the two charts? Answer: In the He Tu, the numbers six, seven, eight, and nine are placed outside the original numbers, representing the standard example of yin and yang's progression and regression, and the increase and decrease of old and young. The number nine is the sum of the original numbers one, three, and five, so it forms a shape outside the number four, moving from north to east and then from east to west. The number six is the sum of the original numbers two and four, so it forms a shape outside the number one, moving from south to west and then from west to north. The number seven is nine moved from the west to the south, and the number eight is six moved from the north to the east—this is another example of yin and yang's old and young mutually hiding within each other's positions.

《洛書》之縱橫十五而七八九六，迭為消長，虛五分十，而一含九，二含八，三含七，四含六，則參伍錯綜，無適而不遇其合焉，此變化無窮之所以為妙也。曰：然則聖人之則之也，奈何？曰：則《河圖》者虛其中，則《洛書》者總其實也。《河圖》之虛五與十者，太極也，奇數二十偶數二十者，兩儀也。以一二三四為六七八九者，四象也，析四方之合，以為乾坤離坎，補四隅之空，以為兌震巽艮者，八卦也。

The Luo Shu has the sum of its rows and columns equal to fifteen, with the numbers seven, eight, nine, and six alternating in their increase and decrease. If the central number five is removed and the remaining numbers are distributed evenly, then one and nine are paired, two and eight are paired, three and seven are paired, and four and six are paired. These numbers are interwoven in a complex arrangement, and every combination adds up to ten. This is the marvelous and boundless nature of the Luo Shu. Question: Then, how did the sage take inspiration from both the He Tu and the Luo Shu? Answer: Taking inspiration from the He Tu, the sage removed its central number; taking inspiration from the Luo Shu, the sage summarized its actual numbers. Removing the central numbers five and ten from the He Tu leaves the Taiji. The odd numbers twenty and the even numbers twenty form the two principles. Pairing one, two, three, and four with six, seven, eight, and nine forms the four images. Splitting the numbers at the four cardinal directions and combining them to form Qian, Kun, Li, and Kan, then adding the four corners to complete the eight trigrams, which are Dui, Zhen, Xun, and Gen.

《洛書》之實，其一為五行，其二為五事，其三為八政，其四為五紀，其五為皇極，其六為三德，其七為稽疑，其八為庶徽，其九為福極，其位與數尤曉然矣。

The actual numbers summarized in the Luo Writing correspond to the following: one corresponds to the Five Phases in the Hongfan, two to the Five Affairs, three to the Eight Policies, four to the Five Records, five to the Imperial Center, six to the Three Virtues, seven to the Inquiry into Doubts, eight to the Various Manifestations (the original text mistakenly writes 'micro' instead of 'manifestation'), and nine to the Five Blessings and Six Evils. The correspondence between directions and numbers is particularly clear here.

集說 《朱於語類》云：《洛書》本文，只四十五點。班固云：六十五字，皆《洛書》本文，古字畫少，恐或有模樣，但今無所考。漢儒此說未是，恐只是以義起之，不是數如此，蓋皆以天道人事參互言之：五行最急，故第一；五事又參之，故第二；身既修，可推之於政，故八政次之；政既成，又驗之於天道，故五紀次之；

Collected Commentaries: In the "Sayings of Master Zhu" (the original text has "zi" mistakenly written as "yu"), it is said that the original text of the "Luo Writing" contains only forty-five dots. Ban Gu claimed that the sixty-five characters in the "Hongfan" chapter are all the original text of the "Luo Writing," and since ancient characters were simpler and strokes were fewer, perhaps they were actually written in such a way. However, we can no longer verify this today. The view of the Han scholars may not be correct. The order of the Nine Categories is probably arranged according to meaning and principle, rather than according to the numbers in the "Luo Writing" itself. The Nine Categories generally discuss the interaction between heavenly principles and human affairs: the Five Phases are most essential, so they are listed first; the Five Actions must be referenced against the Five Phases, so they are listed second; once one has cultivated oneself, one can then apply this to governance, so the Eight Affairs are listed third; once governance is established, one must verify it through the heavenly signs, so the Five Records are listed fourth.

又繼之以皇極居五，蓋能推五行，正五事，用八政，修五紀，乃可以建極也，六三德，乃是權衡此皇極者也，德既修矣，稽疑庶徵繼之者，著其驗也，又繼之以福極，則善惡之效，至是不可加矣，皇極非大中也，皇乃天子，極乃極至，言皇建此極也。

The next is the fifth position of the Supreme Principle. It must be able to clarify the Five Phases, regulate the Five Affairs, implement the Eight Policies, and maintain the Five Records, before it can be considered as establishing the ultimate standard. The sixth, the Three Virtues, are used to weigh and adjust this Supreme Principle, adapting it according to the circumstances. Once the virtues are cultivated, they are followed by Investigating Doubts and Observing Signs, to demonstrate their effectiveness. Then comes the Five Blessings and Six Curses, so the effects of good and evil are fully expressed here, with nothing more to add. The term 'Supreme Principle' does not mean 'Great Center': 'Huang' refers to the Son of Heaven, and 'Ji' refers to the ultimate standard. It refers to the Son of Heaven establishing this standard.

吳氏曰慎曰：《河圖》虛中宮以象太極，故周子曰“無極而太極”，《洛書》主中五以為皇極，故曰皇建其有極。陰陽皆自內始生，窮外而盡，觀四時之寒暑相推，萬物之榮枯生死可見。《河圖》 《洛書》上三數象天，中三數象人，下三數象地，人能參天地，贊化育，建中和，故歸重於五皇極焉。

Wu Yashen said: The "Yellow River Chart" leaves out the Central Palace to symbolize the Taiji, so Zhou Dunyi said "From the state of no extreme comes the Taiji." The "Luo Writing" establishes the Emperor's Pole based on the number five in the center, so the "Hongfan" says "The Emperor establishes the standard of the extreme." Yin and Yang both originate from within and extend to the outermost layers; by observing the successive changes of the four seasons' cold and heat and the flourishing and withering of all things, one can understand this. The "Yellow River Chart" ... the "Luo Writing" (the original text here is missing, the passage between the two book names is lost). The three numbers above symbolize Heaven, the three in the middle symbolize humanity, and the three below symbolize Earth. Humanity can stand alongside Heaven and Earth as a trinity, assist Heaven and Earth in their nurturing of all things, and establish the virtue of harmony between extremes, so the nine categories are especially emphasized on the fifth Emperor's Pole.

案 吳氏三條，於圖書卦疇，深有發明。所謂無極有極雲者，則《易》、《範》之第一義也，其以先天圖合《河圖》，語尤真切。聖人所謂則之者，為其理之符契耳，豈必規規於點畫方位而求密合哉！

The case in point is Wu Yueshen's three points, which have made significant contributions to understanding the relationship between the Yellow River Chart, the Luo Writing, and the Eight Trigrams and the Nine Categories. The terms "wuji" and "youji" he mentions are precisely the fundamental principles of the Zhou Changes and the Hongfan. His comparison of the Earlier Heaven diagram with the Yellow River Chart is especially apt. The sages' statement "to follow it as a model" merely refers to taking its principles of correspondence, not necessarily requiring exact numerical matches or rigid alignment in every detail.

《洛書》以四正之參數象天，四隅之兩數象地，中宮之合數象人，吳氏分三重者，似亦本於《大戴禮》子華子之說，然今以《洪範》考之，蓋始於一二三，中於四五六，終於七八九，而各以相天道，建主極，協民居，為之先後次第，自日用飲食修己治人之近，層累增高，至於上下同流而後已焉，皆所謂得其理而不規規於點畫方位以求密合者。大抵《易》卦以八為節，其根起於兩儀也，《範疇》以九為節，其根起於三才也，知《易》、《範》所起之根，則知“圖書”所蘊之妙矣。

The Luo Shu uses the odd numbers in the four cardinal directions to symbolize Heaven, the even numbers in the four corners to symbolize Earth, and the number formed in the Central Palace to symbolize Humanity. Wu Shi divides it into three layers, which seems to be based on the ideas from the Da Dai Li Ji and the Zi Hua Zi. However, when compared with the Hong Fan today, the nine categories generally start with one, two, and three, reach the middle with four, five, and six, and end with seven, eight, and nine. Each follows the order of assisting Heaven's way, establishing the supreme authority of the ruler, and harmonizing the people's dwellings. It begins from close aspects such as daily life and self-cultivation and governance, then gradually ascends until it merges with Heaven and Earth. These all belong to interpreting the meanings and principles rather than strictly matching numbers and directions. Generally speaking, the hexagrams of the Yi Jing take eight as a segment, originating from the two principles; the categories of the Hong Fan take nine as a segment, originating from the three elements. Understanding the origin of numbers in the Yi Jing and the Hong Fan enables one to grasp the profound mysteries contained in the He Tu and the Luo Shu.

曰：《洛書》而虛其中，則亦太極也，奇偶各居二十，則亦兩儀也，一二三四而含九八七六，縱橫十五而互為七八九六，則亦四象也，四方之正，以為乾坤離坎，四隅之偏，以為兌震巽艮，則亦八卦也。《河圖》之一六為水，二七為火，三八為木，四九為金，五十為土，則固《洪範》之五行，而五十有五者，又九疇之子目也，是則《洛書》固可以為《易》，而《河圖》亦可以為《範》矣。且又安知“圖”之不為書，“書”之不為“圖”也邪！

Question: If the Luo Writing is made empty in the center, it becomes the Taiji; if the odd and even numbers each total twenty, it becomes the two principles; if one, two, three, and four each contain nine, eight, seven, and six, and when added horizontally and vertically they all total fifteen, and seven, eight, nine, and six are interchanged, it becomes the four images; if the four cardinal directions are assigned to Qian, Kun, Li, and Kan, and the four corners are assigned to Dui, Zhen, Xun, and Gen, it becomes the eight trigrams. As for the Yellow River Chart, one and six represent water, two and seven represent fire, three and eight represent wood, four and nine represent metal, and five and ten represent earth, which is exactly the Five Phases of the Hongfan; and the total number fifty-five is precisely the number of each item under the nine categories. Thus, it can be seen that the Luo Writing can lead to the Zhou Changes, and the Yellow River Chart can also lead to the Hongfan. Then how can we say that the chart is not the book, and the book is not the chart!

曰：是其時雖有先後，數雖有多寡，然其為理則一而已。但《易》乃伏羲之所先得乎“圖”，而初無所待於“書”，《範》則大禹之所獨得乎“書”，而未必追考於“圖”耳。且以《河圖》而虛十，則《洛書》四十有五之數也，虛五則大衍五十之數也，積五與十，則《洛書》縱橫十五之數也，以五乘十，以十乘五，則又皆大衍之數也。

Answer: Although the two diagrams appeared in different periods and have different numbers, the principle they embody is the same. The Zhou Changes (I Ching) was first derived by Fuxi from the Yellow River Chart, without relying on the Luo Writing. The Hongfan (Book of Documents) was derived by the great Yu solely from the Luo Writing, and he may not have gone back to examine the Yellow River Chart. Moreover, if the five voids are subtracted from the Yellow River Chart's ten, the remaining number corresponds exactly to the forty-five numbers of the Luo Writing. If the five voids are subtracted from the fifty numbers of the Great Derivation, the remaining number also corresponds exactly to the Great Derivation's fifty numbers. The sum of five and ten is fifteen, which matches the fifteen numbers obtained by adding the numbers across the rows and columns of the Luo Writing. Multiplying five by ten or ten by five gives fifty, which is exactly the number of the Great Derivation.

《洛書》之五，又自含五而得十，而通為大衍之數矣，積五與十，則得十五，而通為《河圖》之數矣，苟明乎此，則橫斜曲直，無所不通，而《河圖》、《洛書》，又豈有先後彼此之間哉！

The number five in the center of the Luo Shu contains another five, making ten, which combined with forty forms the total of fifty numbers of the Great Derivation; when five and ten are added together, they make fifteen, and when the eight directions are added together, they form the numbers of the He Tu. If this layer is understood clearly, then calculating horizontally, diagonally, curving, or straightly, there is no path that cannot be followed; how could there still be a distinction between the He Tu and the Luo Shu in terms of sequence or mutual relationship?