The teachings of the Master Zeng, concerning the clarification of qi through numbers and the sequence of generation in the Yellow River Chart.

夫子前講河圖，為天地自然之數；其始終包萬物之生成，萬事之吉凶成敗；故易本此而示人以修養省存，趨避逆順之道，蓋莫非氣為之也。天地為萬物父母，言萬物必自天地始；而萬物之生成，得天地之氣，其數亦分天地之數。河圖溯物之生成，明五行所出；二氣所運，而以數示之，其旨深矣。

The Master previously spoke of the Yellow River Chart, saying that it is the natural numbers of heaven and earth, covering from beginning to end the generation of all things and the fortune, success and failure of all affairs. Therefore, the Book of Changes uses it to guide people in cultivating their bodies and minds, examining and nurturing themselves, and understanding how to seek good fortune and avoid misfortune, as well as how to choose and reject things according to the right path. All of this ultimately depends on the principle of "qi." Heaven and earth are the parents of all things; when speaking of all things, one must begin with heaven and earth. Since the generation of all things comes from the qi of heaven and earth, their numbers are derived from the numbers of heaven and earth. The Yellow River Chart traces the origin of all things, explains where the Five Phases come from, and how the two qi of yin and yang operate, expressing them through numbers. Its intention is profound.

蓋氣無形而難言，數有名而可紀；故以數代氣，數之所在，即氣之所至；數之幾分，即氣之幾分；以數之分合，而知氣之變化；以數之順逆，而知氣之吉凶；其言數，皆言氣也。氣有精粗，有動靜；數有奇偶，有多寡；由數明氣，則生化之跡，無不知之；故河圖實以數示造化之本者也。

The qi has no form and is difficult to describe with words; numbers have names and can be recorded, so numbers are used to represent qi. Where the numbers are, qi is there as well; how much the numbers are, so much is the qi. By the combination and division of numbers, we can understand the changes of qi; by the order and disorder of numbers, we can know the fortune and misfortune of qi. Therefore, whenever numbers are discussed, it is actually about qi. Qi has both refined and coarse aspects, as well as moving and still aspects; numbers have odd and even, as well as more and less. By using numbers to understand qi, the process of generation and transformation becomes fully known. Thus, the Yellow River Chart is actually using numbers to reveal the fundamental principles of creation and transformation.

數之為用，自一至十；為一切數之本，無論何數，皆自此生化；故一至十之數，實包無盡數也；知其變化，不可盡於用；不知，則不得其用；數者能示變化，而通鬼神；夫數既出於氣，氣有陰陽，故數亦有陰陽；陽奇陰偶，陽屬天，陰屬地；以二者之分合，而萬有以生成變化；不獨五行也。

The use of numbers begins from one to ten, which are the fundamental numbers from which all other numbers are generated and transformed. Therefore, the numbers from one to ten actually contain an infinite number of other numbers. Understanding their transformations allows for endless application; without understanding, one cannot utilize them. Numbers can reveal changes, hence they are connected to spirits and deities. Since numbers originate from qi, and qi has yin and yang, numbers also have yin and yang: odd numbers are yang and even numbers are yin; yang belongs to heaven, and yin belongs to earth. Through the separation and combination of these two, all things are generated and transformed, not only in the Five Phases.

而五行為生變之始，故舉之以例萬物；五者生變，由數知之；必將數之奇偶，位之順逆；相合而生成，變化乃見；圖中之一三五七九，奇數順行；四二十八六，偶數逆行；而其合也，遂成一與六，二與七，三與八，四與九，五與十，各當其位；以生成水火木金土，依次列北南東西中；以示五者之所自出，本乎數之相得有合也；亦氣之相生有成焉耳。

The Five Phases are the beginning of generation and transformation, so they are used as a model for all things. To understand how the Five Phases generate and transform, one must refer to numbers. The odd and even numbers must be combined with the direction of movement, clockwise or counterclockwise, so that the process of generation can be seen and transformation can be demonstrated. On the Yellow River Chart, the odd numbers one, three, five, seven, and nine move counterclockwise; the even numbers four, two, ten, eight, and six move clockwise. When these two sets meet, they form pairs: one with six, two with seven, three with eight, four with nine, and five with ten. Each pair occupies its proper direction, thus generating water, fire, wood, metal, and earth, arranged in order in the north, south, east, west, and center. This shows the origin of the Five Phases, which arise from the mutual combination of numbers, indicating the mutual generation of energies leading to their formation.

故知一之與六，兩數相得，而水之生成，得天地之氣若幾分也；火木金土皆然，皆由天地之氣，奇偶之數，恰相得而合也；故易曰：「一陰一陽謂之道」。數之順逆偕行，以循環始終；於是生化之道見焉。以奇偶之數，一陰一陽猶男女也；而夫婦之道，即人生之始；人以男女配偶，而生息無盡；數以奇偶配合，而生化無窮；此天地造化之妙也。

From this we can see that the numbers one and six complement each other. The generation and formation of water come from taking a portion of the qi from heaven and earth. Fire, wood, metal, and earth are also like this, all formed by the proper combination of the qi from heaven and earth and the even and odd numbers. Therefore, the Great Treatise says, "One yin and one yang is called the Way." Numbers can move in a forward or reverse direction, and they rotate together in a cycle, thus revealing the principles of generation and transformation. In terms of odd and even numbers, one yin and one yang are like one man and one woman; the way of husband and wife combining is the beginning of human life. Humans continue their lineage through male and female partners, while numbers generate infinite transformations through the combination of odd and even. This is the marvelous creation of heaven and earth.

故數之一三五七九，皆陽皆奇；二四六八十，皆陰而隅。二者相得有合，而五行生成；陽道虛，陰道實；陰統於陽。陽包其陰；皆自然配偶，各有其序；故奇數合偶數，仍為奇；奇數合奇數，反為偶；偶數合偶數，仍為偶；以見陽之統攝陰，而不可亢；陰之順承陽，而不可離；亢則害，離則乖；唯相得有合，而後成其生化也。

Therefore, the numbers one, three, five, seven, and nine are all yang and odd numbers; two, four, six, eight, and ten are all yin and even numbers. These two aspects combine to form the Five Phases. Yang is empty and yin is substantial; yin belongs to yang, and yang contains yin. They are naturally complementary and have their own order. Thus, when an odd number is added to an even number, the result is still an odd number; when two odd numbers are added together, the result is even; when two even numbers are added together, the result is still even. Therefore, yang controls yin but should not be excessive, and yin follows yang but should not deviate. Excessiveness brings calamity, deviation leads to disorder. Only when they complement and combine with each other can they achieve the function of generating and transforming life.

河圖之數，天地之本數也；其用自其變化而生。五行之物，萬物之根本也；其生化自其分合而成。蓋有天地而有萬物，有天地之大數，而後有萬物之數可紀；故有序有類有名，而後成其德；萬物並育而不相害，道並行而不相悖；此德之見也。此「道」字，即指一陰一之道，言二氣並行而不亂其序，則由有數也。

The numbers of the Yellow River Chart are the fundamental numbers of heaven and earth, and their function arises from their own changes. The Five Phases are the fundamental elements of all things, and their generation and transformation come from their division and combination. Only with heaven and earth can there be all things; only with the great numbers of heaven and earth can the numbers of all things be recorded. Thus there is order, classification, and names, and then the "de" of things is established. The Zhongyong says, "All things grow together without harming each other, and the ways run parallel without contradicting each other," which is the manifestation of "de." Here the term "Dao" refers to the Dao of "one yin and one yang," meaning that the two energies of yin and yang operate together without disorder, precisely because there is "number" within them.

猶物之陳於室，高下遠近，各如其分；此數之所見也。人之智愚賢否，貴賤壽夭，各有其數，以定其分；而能守其分，安其數，是為知命；以命自天，知命者，安於天命之數，而不失其序；以順承天生我之德，而養育其中和之氣；是為明道。道不離二氣，知乎數而和於氣，則成道之士也。故易者，言數以明道，達命以知天者也。

This is like objects arranged in a room, each placed at its proper height and distance, which is the manifestation of "number" in daily life. A person's intelligence or foolishness, virtue or vice, wealth or poverty, longevity or short life, each has its own number, thus determining its proper position. To hold fast to one's position and be content with one's number is called "knowing one's destiny." Destiny comes from Heaven; those who know their destiny accept the number set by Heaven's mandate without disturbing its order, thus following the virtue bestowed by Heaven and cultivating the harmonious energy within oneself, which is called "understanding the Way." The Way cannot be separated from the dual energies of yin and yang. Those who understand number and can harmonize with the energies are the ones who achieve the Way. Therefore, the Book of Changes is about using number to explain the Way and using destiny to understand Heaven.

夫易之言數，自河圖始；河圖之數，即易之數；如水火木金土五者，易之乾坤離坎震兌也；由用言之，則坎離坎震兌，分合水火木金四氣之象；而艮為土，此後天之卦也；若在先天，則乾坤震兌為四方，而坎離在中；其氣數各有所合也。以一六之水，為天地二氣初化；二七之火，三八之木，四九之金，五十之土，皆依次而生成者；陽生則陰成者，陰生則陽成。

The Book of Changes begins with numbers from the Yellow River Chart. The numbers of the Yellow River Chart are the numbers of the Book of Changes: for example, water, fire, wood, metal, and earth correspond to the trigrams Qian, Kun, Li, Kan, Zhen, and Dui in the Book of Changes. In practical application, the trigrams Kan, Li, Zhen, and Dui correspond to the images of water, fire, wood, and metal respectively, while Gen corresponds to earth, which is the arrangement of the Later Heaven trigrams. In the Earlier Heaven arrangement, Qian, Kun, Zhen, and Dui are located in the four directions, while Kan and Li are in the center, and their energies and numbers are matched accordingly. The number one combined with six forms water, which is the initial generation of the energies of heaven and earth. The number two combined with seven forms fire, three with eight forms wood, four with nine forms metal, and five with ten forms earth, all generated in sequence. All yang numbers are responsible for generation, and they are completed by yin numbers; all yin numbers are responsible for generation, and they are completed by yang numbers.

二氣更互以生以成，五行於是出，萬物於是生；其生化之道，不離乎此圖之序；其賦與之命，不外乎此圖之數；數之所在，即二五之所分合；凡二五之所分合；凡屬後天之生物，莫不由此二五之分化也。數有終始，運行有序；命有善惡，道有先後；皆氣之所偏全有所合也。不合不生，故必合而有得也；此天地生化之妙用，亦世界自無入有之次第也。

Yin and yang energies interact with each other; one is responsible for generation, and the other for completion. The Five Phases arise from this interaction, and all things are born from it. The pathways of their generation and transformation cannot depart from the order established by the Yellow River Chart. The destiny they receive is also beyond the numbers listed in the Yellow River Chart. Wherever numbers exist, there is the division and combination of yin and yang energies and the Five Phases. Everywhere where these energies and phases combine, all things born under the Later Heaven originate from the differentiation of 'two' and 'five'. Numbers have a beginning and an end, and their movement follows a certain order. Destiny has good and evil, and the Way has precedence and succession. These are all due to the partiality and completeness of the energies, each having its own coordination. Without combination, there is no generation, so combination is essential for results. This is the subtle function of heaven and earth in generating things, and also the sequence by which the world moves from 'nothing' to 'something'.

數之所合，易卦而外，則乾支為最易知；以其習見習聞也。天乾之數，甲乙為木，丙丁為火，戊已為土，庚辛為金，壬癸為水，其數同也；地支之數，寅卯為大，巳午為火，申酉為金，亥子為水，辰戌丑未為土，其數合也。皆以五行分配，而相生相制；以成其生化之用，與圖相同；故易卦有納甲之義。

The combination of numbers is easiest to understand besides the Yi hexagrams, because people are familiar with the heavenly stems and earthly branches. The numbers of the heavenly stems: Jia and Yi belong to wood, Bing and Ding to fire, Wu and Ji to earth, Geng and Xin to metal, and Ren and Gui to water; they all have the same numbers. The numbers of the earthly branches: Yin and Mao belong to wood, Si and Wu to fire, Shen and You to metal, Hai and Zi to water, and Chen, Xu, Chou, and Wei, the four branches, belong to earth; their numbers also correspond with each other. Both heavenly stems and earthly branches are assigned according to the Five Phases, and they generate and control each other, thus fulfilling their functions of creation and transformation. This is consistent with the Yellow River Chart, so the Yi hexagrams have the system of "Najia" (stem assignment).

蓋卦象人或不知，而乾支則易知；乾支固無異於卦，皆以示氣之運行，二五之化合；或以卦示其象，或以乾支繫其名；其實皆出河圖，亦猶數算之代名也。因氣數不易記，故以之代，而便於習究也；如甲乙屬木，即震巽也；丙丁屬火，即離；庚辛屬金，即兌；壬癸屬水，即坎；戊已屬土。即坤艮也。

The principles of divination are not easily understood by ordinary people, but the Heavenly Stems and Earthly Branches are familiar to everyone. In fact, the Heavenly Stems and Earthly Branches are essentially the same as the trigrams, both used to represent the movement of qi, the combination of yin and yang, and the Five Phases. Some use trigrams to show their symbolic meanings, while others use the Heavenly Stems and Earthly Branches to name them. In the end, both originate from the Yellow River Chart, just like symbols in mathematics. Since the movement of qi is difficult to remember, the Heavenly Stems and Earthly Branches are used as substitutes, making it easier to study and examine. For example, Jia and Yi belong to wood, corresponding to the trigrams Zhen and Xun; Bing and Ding belong to fire, corresponding to the trigram Li; Geng and Xin belong to metal, corresponding to the trigram Dui; Ren and Gui belong to water, corresponding to the trigram Kan; Wu and Ji belong to earth, corresponding to the trigrams Kun and Gen.

而其為用則與此稍異；以乾納甲壬，示陽氣之初動；坤納乙癸，示陰氣之偕行；震巽以木而納庚辛，示木受制而後成用也；坎離納戊己，以水火之用成於土也；兌艮納丙丁，以金受火制，土為火生，各得其用也。故其數與原位異，此先後天之殊，體用之不同也；然此所納亦有異同，則用者取義不一也。

But the application of the Najia system differs slightly from the above five-phase attributions: the Qian trigram is assigned Jia (Ren), indicating the initial emergence of yang energy; the Kun trigram is assigned Yi (Gui), indicating the concurrent movement of yin energy. The Zhen and Xun trigrams, which belong to wood, are assigned Geng and Xin (metal stems), indicating that wood must first be restrained by metal before it can function properly. The Kan and Li trigrams are assigned Wu and Ji (earth stems), because the functions of water and fire depend on earth to be fulfilled. The Dui and Gen trigrams are assigned Bing and Ding (fire stems), because metal requires fire for refinement and earth is nurtured by fire, allowing each to function properly. Therefore, the numbers assigned in the Najia system differ from the original five-phase directional attributions. This is the distinction between Earlier Heaven and Later Heaven, as well as the difference between form and function. However, different schools have slight variations in their assignments, due to differing interpretations by those who apply them.

如歸藏異乎連山，周易異乎歸藏；皆後天之用，非關先天之體用也；其義詳洛書中。河圖洛書一體一用，各有其指也；故在圖之數，方位相當；次序無亂，與伏羲之卦象同；而為一切數之根本，有不易者也。圖中之數，雖自一至九；而五十二數，卻在中心；故分數時，應分陽數一三七九，與陰數二四六八相對以合；另提五十二數為中心立極。

For example, the method of incorporating numbers in the "Guicang Yi" differs from that in the "Lianshan Yi," and the method in the "Zhou Yi" differs again from that in the "Guicang Yi." These all belong to the later heaven usage and have nothing to do with the earlier heaven's principles. The reason for this will be explained in detail when discussing the Luo Shu. The He Tu and the Luo Shu each have their own focus; one represents the fundamental structure, and the other represents the application. Therefore, the numbers on the He Tu are arranged in a way that their positions correspond with each other, and their order remains clear and orderly, consistent with the hexagrams drawn by Fuxi. The He Tu is the root of all numbers and cannot be altered. Although the numbers on the He Tu range from one to nine, the numbers five and ten are placed at the center. Therefore, when arranging the numbers, the yang numbers one, three, seven, and nine should be paired with the yin numbers two, four, six, and eight respectively. The numbers five and ten should be separated and placed at the center to establish the central point.

依陰數逆行之例，與陰數之始四終六；則一三七九，與四二八六適恰相當；而生成之功見，生化之序明；而五十二數亦正相當。若不如是，則位與數失，陰與陽乖；不克生成萬物矣！故一三七九陽數，必順次行；四二八六陰數，必顛倒而逆行；此必然之理。人知陽數順行，而不知陰數逆行；遂不明數之大用，此惑也。

According to the established practice of the yin numbers moving in reverse order, with the yin numbers starting from 'four' and ending at 'six', the numbers one, three, seven, and nine correspond one by one with four, two, eight, and six: one corresponds with four, three with two, seven with eight, and nine with six. Thus, the function of generation becomes evident, and the sequence of generation and transformation becomes clear. The numbers five and ten in the center also each take their proper positions. If they were not arranged in this way, the positions and numbers would be misaligned, yin and yang would be discordant, and thus, the generation of all things would not occur. Therefore, the yang numbers one, three, seven, and nine must necessarily proceed in order, while the yin numbers four, two, eight, and six must necessarily move in reverse. This is an inevitable principle. People only know that yang numbers proceed in order, but do not know that yin numbers move in reverse, thus failing to understand the great function of numbers, which is a great confusion.

夫使陰數亦如陽之順序而行，則一與二，三與四，五與六，七與八，九與十，相對而論，皆不能合；更何生成之功可言？蓋必一順一逆，一依次，一倒行；方能合其位，而成其用；而後一與四合，二與三合，七與八合，九與六合，五與十合，相得而皆有合；以成中極之數皆五也。故曰：「天數五，地數五。」五位相得而有合也。

Suppose the yin numbers were arranged in the same order as the yang numbers, then one would pair with two, three with four, five with six, seven with eight, and nine with ten. When compared in pairs, none of them would produce the central number. How could there be any generative power then? It is necessary for one to proceed in order and the other to go in reverse, one to follow the sequence and the other to go backwards, so that each can be matched with its proper position and fulfill its function. In this way, one and four combine to make five, two and three combine to make five, seven and eight combine to make fifteen, nine and six combine to make fifteen, and five and ten combine to make fifteen. Each combination produces the central number, and all of them are based on the number five. Therefore, the Great Treatise says, "Heavenly numbers are five, earthly numbers are five. The five positions interact and each has its corresponding match."

言合乎五，即合天地之心；在道言之，即合中和之德，示人執中用之道也；蓋數十而位五，唯五乃中；不得其五，即不得其中；不得其中，必有所偏；偏則害生，安能致生成之用？通天地之心哉。故五位之合，必依陰數逆行，而後得之也。

The concept of "harmony with the number five" refers to aligning with the heart of heaven and earth; from the perspective of the Dao, it means aligning with the virtue of moderation and harmony. It is indicating that people should adhere to the middle path and use the method of the middle. There are a total of ten numbers, yet only five directions exist. Only the number five is in the center; without this five, one cannot attain the middle; without the middle, there will certainly be some bias. Once there is a bias, calamity arises, how then can one achieve the accomplishment of creation and connect with the heart of heaven and earth? Therefore, the combination of the five directions must follow the rule of reversing the yin numbers, only then can it be attained.

且陽順陰逆，氣之自然；逆之雲者，對陽順言；其實不過一往一復，一左一右而已。陰氣原與陽對行，陽左則陰右，陽先則陰後，陽升則陰降，自然而然，非圖之有異也；如晝夜寒暑，互相推移，而日歲以成；若同一途，則一氣獨用，不復有生化之功矣！故言數不明可證於象，以象之明見於目也。

Heavenly stems move in a positive direction while earthly branches move in a negative direction, which is natural for the energies. The term "reverse" merely refers to the contrast with the positive movement of heavenly stems; in reality, it is simply a back and forth, left and right movement. Yin energy naturally moves in the opposite direction of yang: when yang moves left, yin moves right; when yang is ahead, yin is behind; when yang rises, yin descends. All of this happens naturally, without any special arrangement in the Yellow River Chart. Just as day and night, cold and heat alternate to form a day and a year; if they followed the same path, there would only be one kind of energy acting alone, and there would be no process of generation and transformation. Therefore, when numbers cannot be fully explained, we can find confirmation through images, because images are clearly visible before us.

今就圖象證之，亦自可見；陽自一左旋至九，序位皆順；陰自四右旋至六，序位倒逆；而恰合一六、三八、二七、四九、五十，之五行數，不待移易；即此可知天地大數，不離陰陽生化之功，不外陰陽順逆之用；而萬物以生，萬事以立，其微妙固非難測者矣！

Now referring to the Yellow River Chart for verification, it becomes clear: the yang numbers start from "one" and rotate left until "nine," with their positions all in order; the yin numbers start from "four" and rotate right until "six," with their positions all in reverse. In this way, exactly five sets of the Five Phases numbers are formed: one and six, three and eight, two and seven, four and nine, five and ten. There is no need for further movement or adjustment. From this alone, we can understand that the great numbers of heaven and earth cannot be separated from the generative power of yin and yang, nor can they go beyond the use of yin and yang in order and reverse; all things are generated from this, and all affairs are established upon this. The subtle nature within is not something difficult to comprehend.

天一生水，地六成之；地二生火，天七成之；天三生木，地八成之；地四生金，天九成之；天五生土，地十成之。

Heavenly numbers "one" produces water, and earthly numbers "six" complete it; earthly numbers "two" produce fire, and heavenly numbers "seven" complete it; heavenly numbers "three" produce wood, and earthly numbers "eight" complete it; earthly numbers "four" produce metal, and heavenly numbers "nine" complete it; heavenly numbers "five" produce earth, and earthly numbers "ten" complete it. These five groups of numbers live apart in the five directions: water from one and six is in the north, fire from two and seven is in the south, wood from three and eight is in the east, metal from four and nine is in the west, and earth from five and ten is in the center. Each group consists of one odd and one even number, one heavenly and one earthly, thus enabling the five phases to be generated.

易傳曰：河出圖，洛出書，聖人則之。此言河圖洛書所由來也。天地之間，運行者氣，紀量者數，顯示者象，而主其中御之以合度者道。道兼神形，氣數則神之見也，象則形之呈也。凡天文地理，可紀可指可名者，皆不外此三類，而皆不離乎道。

The Great Treatise says, "The Yellow River produced the chart, and the Luo River produced the writing; the sage followed them," which explains the origin of the Yellow River Chart and the Luo Writing. In the universe, what is constantly in motion is "qi," which is used to measure as "numbers," and what is manifested is "images." The principle that governs and controls these, bringing them into harmony with order, is "the Dao." The Dao encompasses both the spiritual and the physical: qi and numbers are manifestations of the spiritual, while images are expressions of the physical. All that can be recorded, identified, or named in astronomy and geography falls within these three categories, yet none can exist apart from the Dao.

河圖者，天地自然氣數所賦，而成其自然之象也。蓋天地既奠，神形全具，有其氣者有其數，有其數者有其象。河圖之出，以象之顯也；其象之所示則數也，其數之所在則氣也。氣有陰陽，名曰兩儀，言既生成之後，必有此兩者之質，以為其成物之本也。兩儀化合，遂分為水火木金土五者之氣，名曰五行。

The Yellow River Chart is a natural phenomenon formed by the inherent energies and numbers of heaven and earth. Once heaven and earth were established, both spirit and form were complete: there is energy, and thus there is number; there is number, and thus there is image. The Yellow River Chart appeared because the image became visible; the image indicates the number, and the number exists where the energy is. Energy is divided into yin and yang, called "the Two Principles" — this means that after heaven and earth were formed, there must necessarily be yin and yang as the fundamental qualities that transform all things. The Two Principles combine with each other, thus differentiating into the five energies of water, fire, wood, metal, and earth, called "the Five Phases."

言兩儀既化，必有是五者流行其間，以辨其性而別其類，亦物之始成所秉者也。兩儀既見，五行乃名；二五構精，萬有始生，此屬天地萬物生生之本，莫之能外者也。五行之氣，因二為五，雖五仍二，然言其本質，必自五者始。後人或云為四，或云為三，其意大同，究不若別之為五，適合天地自然之氣與數。試證之河圖，可見其旨矣。

This refers to the fact that once the two principles (yin and yang) combine, there must be the five elements circulating among them, used to distinguish the nature of things and differentiate their categories. This is also the nature that all things receive at their initial formation. After the two principles manifest, the five elements gain their names; when the yin and yang energies interact with the essence of the five elements, all things begin to arise. This is the fundamental principle of the continuous generation and transformation of all things in the universe, and nothing is exempt. The energies of the five elements are divided from the two (yin and yang). Although they are called five, they are still rooted in two. However, when discussing the essence of things, one must start with these five. Later scholars have proposed dividing them into four or three, with similar intentions, but ultimately, dividing them into five is more accurate, precisely conforming to the natural energies and numbers of heaven and earth. By referring to the Yellow River Chart, this meaning becomes clear.

河圖之中，點畫成象，自一至十，各相連而不混，相離而不乖，其數之自然，足見其氣之自在也。氣之動也，以二氣之分合而後化，故生成之數，必屬於天地，天地者，二氣之所生也。物之生化，以五者為類，而水火木金土，以辨以名，以成其五行。水火木金土者，五行之所主也。故河圖之數，生成自天地，而名類止於水火木金土，而天下萬物莫不盡於是。

In the Yellow River Chart, numbers are formed by dots, arranged from one to ten, connected yet not confused, separated yet not discordant. The arrangement of numbers is so natural, showing the free flow of qi. When qi moves, it relies on the division and combination of yin and yang to generate things. Therefore, the numbers of generation and completion must belong to heaven and earth, and heaven and earth themselves are produced by the yin and yang. The generation and transformation of all things are categorized into the five phases, namely water, fire, wood, metal, and earth, used for distinguishing and naming, thus forming the Five Phases. Water, fire, wood, metal, and earth are precisely the elements each of the Five Phases governs. Therefore, the numbers in the Yellow River Chart originate from heaven and earth, yet their categories are limited to these five elements, yet they encompass all things under heaven.

是故河圖之象，即天地生化終始之圖；其氣與數，即天下萬有分合變化之本，斯所以為天道之所見也。而聖人則之，因以明天道，以立天下萬物有生成變化之規，而述之為易，則河圖者易之所自仿也。易傳曰：「天一地二，天三地四，天五地六，天七地八，天九地十。天數五，地數五，五位相得而各有合。天數二十有五，地數三十，天地之數五十有五。」雲者，即河圖中之數也。

Therefore, the image of the River Chart represents the beginning and end of the generation and transformation of heaven and earth; its qi and numbers are the fundamental principles of the division, combination, and change of all things under heaven, which is the way of the heavenly Dao made manifest. The sage took it as a model, used it to explain the heavenly Dao, established rules for the generation and transformation of all things in the world, and recorded it as the Book of Changes. Thus, the River Chart is the model that the Book of Changes imitates. The Great Treatise says: Heaven has one, Earth has two; Heaven has three, Earth has four; Heaven has five, Earth has six; Heaven has seven, Earth has eight; Heaven has nine, Earth has ten. The numbers of Heaven are five in total, and the numbers of Earth are five in total. These five numbers correspond to each other in the five directions, each having its own pairing. The total of the Heaven numbers is twenty-five, and the total of the Earth numbers is thirty; altogether, the numbers of Heaven and Earth total fifty-five. What it says is precisely the numbers found in the River Chart.

河圖以天地統全數，以水火木金土分配各數，而建立天地萬物萬數之本，以明天下之物，莫不有數，莫非生成於天地，而得其氣與數以成其氣數。自河圖言之，則氣數者自然成形；自物言之，則氣與數皆出自天地。

The Yellow River Chart uses the categories of "Heaven" and "Earth" to encompass all numbers, distributing each number among the elements of water, fire, wood, metal, and earth, thereby establishing the fundamental numbers of all things in the universe. It explains that nothing in the world lacks a number, and nothing is not generated from Heaven and Earth, nor does it not receive the qi and numbers from Heaven and Earth to form its own qi and numbers. From the perspective of the Yellow River Chart, qi and numbers naturally take form; from the perspective of all things, qi and numbers originate from Heaven and Earth.

蓋天地亦在此氣數中也。未有天地，氣數已具；既有天地，氣數周流，以成萬物。故氣數者，自天地以下，莫之能外也。今言河圖氣數溯始於天地，蓋自有象言之，即自天地建立以後，而推溯萬物生化所本、萬有氣數所由而已。

Because heaven and earth themselves are included in this cycle of qi and numbers. Before heaven and earth came into being, the cycle of qi and numbers was already present; once heaven and earth appeared, the cycle of qi and numbers flowed continuously to transform all things. Therefore, this cycle of qi and numbers, starting from heaven and earth, includes everything without exception. Now, when we say that the cycle of qi and numbers in the Yellow River Chart should be traced back to heaven and earth, it refers to the aspect of "having form," meaning from the time heaven and earth were established, we go back to investigate the fundamental basis for the generation of all things and the origin of the cycle of qi and numbers.

圖中自天一至地十，為數之根本，亦生成之根源。天一生水，地六成之，則言水之生成之由來：以天之一，合地之六，而後水之生成始全；分之則水者得天之一以生，地之六以成。天生地成，是其氣之備天地也。

In the diagram from "Tian Yi" to "Di Shi," these numbers represent the fundamental principles and the source of generation. "Tian Yi Sheng Shui, Di Liu Cheng Zhi" explains the origin of water: water is formed by combining the heavenly number "one" with the earthly number "six." Separately, water is generated by the heavenly number "one" and completed by the earthly number "six." Heaven gives birth and earth completes; thus, the energy of water encompasses both heaven and earth.

天一地六，是其數之賅天地也，而水以成，此五行之一所自生也。火也、木也、金土也亦然，而土為其中心，故五十在中，水火木金四者，皆依乎土以立也。如一加五為六，故水為一六；二加五為七，故火為二七；三加五為八，故木為三八；四加五為九，故金為四九；土則五加五為十，故土為五十，此數所自合也。

Heavenly One and Earthly Six show that the number for water includes both Heaven and Earth, thus forming water. This is the origin of one of the Five Phases. Fire, Wood, Metal, and Earth are similar, with Earth occupying the center. Therefore, the numbers five and ten are placed in the center. Water, Fire, Wood, and Metal all depend on Earth to stand firm. For example, one plus five equals six, so water is paired with one and six; two plus five equals seven, so fire is paired with two and seven; three plus five equals eight, so wood is paired with three and eight; four plus five equals nine, so metal is paired with four and nine; Earth is five plus five equals ten, so Earth is paired with five and ten. This is the origin of how each primary number combines with another. Each primary number must be added to the central Earth number five to form the corresponding secondary number.

天之數五，一三五七九；地之數五，二四六八十；生之數五，一二三四五；成之數五，六七八九十。其分也皆五，五則中央之數，中而不偏，動而能和，中和天地之道，土之德也。故五行皆以五為其根，皆麗於土以為生，此自然之序也。天數之累為二十五，即一三五七九之和也；地數累為三十，即二四六八十之和也。

Heavenly numbers consist of five: one, three, five, seven, and nine; earthly numbers also consist of five: two, four, six, eight, and ten. The generating numbers are five: one, two, three, four, and five; the forming numbers are also five: six, seven, eight, nine, and ten. No matter how they are divided, each group contains five numbers, and the number five is the central number—residing in the middle without bias, capable of movement and harmony. Middle harmony is the way of heaven and earth, and it is also the virtue of earth. Therefore, the five phases all take five as their root, and they depend on earth to grow, which is the natural order. The total of heavenly numbers is twenty-five, which is the sum of one, three, five, seven, and nine; the total of earthly numbers is thirty, which is the sum of two, four, six, eight, and ten.

二者雖分各半而數異，則即生有形之後，得天者不足，地者有餘也。天地之總和五十有五，即二十五加三十，以成天地生成全數，為大衍之數而餘五。五即中央之數，二氣之全，五行之本，為天地數之妙用，即河圖之中，天地之心也。明此五數之妙用，則通天地之數，直見天地之心，而萬物生化之序，無不明矣。

Although the numbers of Heaven and Earth each take up half, their total is not equal, which indicates that after the emergence of tangible things, they receive one part from Heaven but find it insufficient, while receiving one part from Earth and find it excessive. The total of the numbers of Heaven and Earth is fifty-five, composed of twenty-five and thirty, forming the complete number of the generation of Heaven and Earth. Compared to the "Great Derivation" mentioned in the "Great Treatise," which states that the number of great derivation is fifty, there is an extra five. This "five" is the number of the center, the complete form of yin and yang energies, the foundation of the Five Phases, and the key to the function of the numbers of Heaven and Earth. It is also the center of the Yellow River Chart, the heart of Heaven and Earth. Understanding the subtle function of this "five" allows one to grasp the numbers of Heaven and Earth, directly see the heart of Heaven and Earth, and thus have no difficulty in understanding the order of the generation and transformation of all things.

故一至十者，天地生成之數根；而五者，天地生成數之中極。此河圖四列各數，而獨以五十建立中極，其旨趣之微妙，有不盡矣。五位相得各有合者，言數各五而必合，始能致生化之用。如一天數也，生數也；六地數也，成數也，二數合而後水以成。

Therefore, the numbers from one to ten are the fundamental numbers of the generation of heaven and earth, while the number five is the central point of the numbers of heaven and earth generation. The Yellow River Chart arranges each number in the four directions, but uses five and ten alone to establish the central point. The subtle meaning behind this is endless. The phrase "the five positions interact and each has its complement" means that the numbers of heaven and earth each have five, and they must combine in pairs to exert the function of generation and transformation. For example, "one" is a heavenly number and also a number of generation; "six" is an earthly number and also a number of completion: these two numbers combine to form water.

以合則前數化，新數成。一六既合，天地之氣已化，而水乃成。此由數之合也。水之成分，得天之一，地之六，故水中所含者屬天者一，屬地者六。天地既化，水數乃見。凡五行之生化皆然，皆由二氣之合也。五行亦有合矣。

When numbers combine, the original numbers transform and new numbers are formed. Once one and six combine, the qi of heaven and earth is transformed, and water takes shape, resulting from the combination of numbers. The components of water come from heaven as 'one' and from earth as 'six', so the part of water that belongs to heaven is 'one', and the part that belongs to earth is 'six'. Only when the qi of heaven and earth has combined and transformed does the number of water become apparent. All the generation and transformation of the Five Phases are like this, resulting from the combination of yin and yang qi. Each of the Five Phases also combines with others in its own way.

如水與木，木與火、火與土、土與金、金與水，各以其合而生化萬物，則各以其數之合，而生成新數，故變化至無窮焉。然二氣之合與分，恆相得；五行則或得或否，如水與火、火與金、金與木、木與土、土與水，皆相害相制而不相得，則數之因亦有相消相除相加相乘之異。其相得也，則生成之；其制害也，則克化之。

For example, water and wood, wood and fire, fire and earth, earth and metal, metal and water, each combine according to the principle of mutual generation to produce all things. Similarly, they combine according to numerical relationships to generate new numbers, hence the endless changes. However, yin and yang, whether combined or separated, always complement each other. The Five Phases, on the other hand, sometimes complement and sometimes do not. For instance, water and fire, fire and metal, metal and wood, wood and earth, and earth and water all mutually harm and control each other, and do not complement. Therefore, in numerical operations, there are distinctions of mutual cancellation, division, addition, and multiplication: those that complement each other generate new results, while those that mutually harm or control each other are transformed or eliminated.

由其數驗之可知者也。故天一與地六合，一陰一陽，合乎道也。若天一與地二，則不則生成矣。若天一與天三，更無所生成，以其不相得而不能合也。故生化之序，必因相得而後合，合則生成之功見。不得則相反，反則變化之數成。

This can be known by verifying through numbers. Therefore, the heavenly number "one" combines with the earthly number "six," representing one yin and one yang, in accordance with the Dao. If the heavenly number "one" is paired with the earthly number "two," nothing is generated; if the heavenly number "one" is paired with another heavenly number "three," nothing is produced either, because they are not compatible and thus cannot combine. This shows that the order of generation must first involve compatibility before combination; once combined, the function of generation becomes evident. If they are not compatible, they turn to the opposite, and once they turn to the opposite, the numerical changes are formed thereby.

故天一至地十之數，實包舉天下生成變化之數，以至無盡，不可紀極，而莫外於合否二字。由是可見天地造化之源，萬物生成變化之序，皆在河圖中，展轉推求，以窮其氣數所至，則無不知之。如天一地十之數，實數之根；二氣者氣之本；五行者萬物生成之本，皆造化之始基。

Therefore, the ten numbers from Heaven One to Earth Ten actually encompass all numbers of generation and transformation in the world, extending infinitely and beyond counting. In the end, they come down to the two characters "he" (union) and "bu he" (non-union). From this we can see that the origin of heaven and earth's creative power and the order of generation and transformation of all things are entirely contained in the Yellow River Chart. By thoroughly investigating the places where the vital energies reach, there is nothing that remains unknown. Numbers such as Heaven One to Earth Ten are indeed the root of all numbers; yin and yang energies are the fundamental nature of qi; the Five Phases are the fundamental basis for the generation of all things—these are all the initial foundations of creation.

天地人物之原質，由一而二而三，而後萬物以生以成，其道固昭昭矣。二者兩儀，三者兩儀合化五行是也。二五構精，妙合而凝，天地因縕，萬物化醇，此由三而生萬物也。故河圖天地之數，生成之本數也；五行分合之數，萬物所自生化之本也。

The original material of heaven, earth, human beings and things comes from one to two, then from two to three, and only then can all things be generated and formed. This principle is clear and evident. The so-called "two" refers to the two principles (wuxiang), and the so-called "three" refers to the five phases formed by the combination and transformation of the two principles. The "Taiji Tu Shuo" says, "the essence of two and five combines and solidifies," and the "Xici Zhuan" says, "heaven and earth blend and interact, and all things are transformed into purity," which refers to the generation of all things from "three." Therefore, the numbers of heaven and earth on the He Tu are the fundamental numbers of generation; the numbers of the division and combination of the five phases are the fundamental principles for the generation and transformation of all things.

由其位言之，則按乎天地之定序，南北東西方位，而相得者也。故一六以水居北，三八以木居東，二七以火居南，四九以金居西，五十以土建立中極，以應四方而通天地。此地為世界中心，憑之以測天地之生長萬物，所謂極也。天地之極，即數之中，即五也。

From the perspective of direction, the River Chart arranges numbers according to the fixed order of heaven and earth, pairing them with the four cardinal directions—north, south, east, and west—so that each number complements the others. Thus, one and six combine to form water, located in the north; three and eight combine to form wood, located in the east; two and seven combine to form fire, located in the south; four and nine combine to form metal, located in the west; and five and ten combine to form earth, establishing the center. This central position is the center of the world, used to measure the growth of all things in heaven and earth, which is referred to as the 'extreme' or 'pole.' The extreme of heaven and earth is the center of numbers, which is 'five.'

河圖言五十，洛書則僅言五，以五已足賅十。十即二五，天地根數，祇存純數，不必及其餘盈，故言五而土之數已足，不似水之必兼一六也。其所合也，往往因其生化見之，故一六之合，二七之合，皆孚自然之序。

The Yellow River Chart places both "five" and "ten" at its center, while the Luo Writing only marks a single "five," because five is sufficient to encompass ten. Ten is merely two fives, and the fundamental numbers of heaven and earth retain only the purest one, without needing to consider the extra part. Therefore, it is enough to mention "five," as the number for earth is already sufficient. Unlike water, which must be paired with one and six, the combination of numbers often becomes apparent through their process of generation and transformation. Thus, one and six combine, and two and seven combine, all in accordance with the natural order.

而五與二五之十，尤可見矣。若言五則十已見，雖三五四五，仍不失其五，故不必徵於成數，而已知土之數之為十五。蓋以天之五，地之十，合而言之為十五，去其十仍存五。此五數示土已足，其數雖一而內藏十矣。此為言數者所當知也。

The number "five" and the number "ten" formed by two fives are especially evident in this relationship. As soon as the number five is mentioned, the number ten is already included within it. Even with three fives or four fives, they still do not depart from the five. Therefore, there is no need to refer to the number of completion, and we already know that the number associated with earth is fifteen. This is because the heavenly number five combined with the earthly number ten makes fifteen. If we subtract the ten, five remains. Using the number five to represent earth is sufficient, as it appears as a single number on the surface but contains the ten within it. This is something those who study numbers should understand.

河圖之數，以二數合而成一象，故十數合為五，以成水火木金土，唯其合也。有變化見焉。以六為成數之始，又地數也；九為天數之終，又成數也，二數實天地之主數。故易乾用九，坤用六。乾天也，陽也；坤地也，陰也。陽數用九，陰數用六，此中亦有自然之序存焉。

The numbers of the River Chart combine two numbers into one image, so ten numbers are grouped into five sets to form water, fire, wood, metal, and earth. The key lies in the concept of 'combination'. 'Six' is the first of the completed numbers and belongs to the earth numbers; 'Nine' is the last of the heavenly numbers and also a completed number. These two numbers are truly the main numbers of heaven and earth. Therefore, the Qian hexagram in the Book of Changes is called 'Use of Nine', and the Kun hexagram is called 'Use of Six'. Qian represents heaven, which is yang; Kun represents earth, which is yin. Yang numbers take nine, and yin numbers take six, and there is a natural order within this.

試就河圖觀之，自北之陽數一始，轉東之三，南之七，西之九，與中央之五，其次自然順行，為一三五七九，是陽數始一終九。九則陽氣已極，曰老陽，亦曰純陽，故乾取之為用。再就圖之陰數觀之，陽終則陰始。陽數終於西之九，即交於同方之四；陰數自四始，轉而南之二，東之八，北之六，合中央之十，其次亦自然逆行，為四二十八六，是陰數始四終六。以六為至陰，陰氣亦極於六，故陰數用六。

If we examine the River Chart, the yang numbers begin at the north with 'one', move to the east with 'three', then to the south with 'seven', and finally to the west with 'nine'. Adding the central 'five', the sequence naturally proceeds in order as one, three, five, seven, nine. This shows that the yang numbers start with one and end with nine. When yang reaches nine, it has reached its extreme, called 'old yang' or 'pure yang', so the Qian hexagram uses it. Now looking at the yin numbers: when yang reaches its end, yin begins. The yang number ends at nine in the west, and the yin number starts at four in the same direction. The yin numbers move from four, to two in the south, then to eight in the east, then to six in the north, and adding the central 'ten', the sequence naturally proceeds in reverse order as four, two, ten, eight, six. This shows that the yin numbers start with four and end with six. Six is the most yin, and yin energy reaches its extreme at six, so the yin number uses six.

以與陽九相接而正相得。蓋以圖平列其數觀之，陽之數自一至九，陰之數自四至六，自成一環，不待移易。陽氣方終，陰氣即起；陰氣甫盡，陽氣即伸。陽老於西而陰即始於西，陰極於北而陽即始於北。二者相環相合，以成五行，周流無盡，以遂其生生不息之道。

Six is connected with the yang number nine, and they complement each other perfectly. If you lay out the numbers on the diagram flat, the yang numbers run from one to nine, and the yin numbers run from four to six, each forming a circular pattern without the need for adjustment. As soon as the yang energy completes its cycle, the yin energy begins; and as soon as the yin energy is exhausted, the yang energy renews itself. When the yang energy reaches its old age in the west, the yin energy begins exactly in the west; when the yin energy reaches its extreme in the north, the yang energy begins exactly in the north. They are interlocked and interdependent, thus forming the Five Phases, flowing endlessly to accomplish the way of continuous generation and renewal.

而天地以立，萬有以生，一切事物，於以循環變化，此則大氣自然之序，大道自存之規，莫之為而為，莫之致而至者也。唯聖人則之，而制為易以象其度。而因陰陽之盡，知乾坤之用在九與六，故取九六以代乾坤。全易無非由乾坤生，則全易之數無非自九六出。九在易即一，六在易即 ， 與 即九與六。

Heaven and Earth thus become established, and all things arise from this. All things then undergo cycles of change. This is the natural order of the atmosphere and the inherent principles of the Great Way. It is not done by anyone, yet it is naturally accomplished; it does not require anyone to promote it, yet it naturally comes to be. Only the sage takes inspiration from it, creating the Book of Changes to imitate its principles. Also, because yin and yang have each reached their extremes, it is known that the principles of Heaven and Earth are embodied in the numbers "nine" and "six." Therefore, "nine" and "six" are used to represent Heaven and Earth. The entire Book of Changes arises from Heaven and Earth, so the numbers in the entire Book of Changes arise from nine and six. In the Book of Changes, "nine" represents the continuous yang line, while "six" represents the broken yin line. These two lines, one yang and one yin, are precisely the numbers nine and six.

故卦中有 者，名曰九，如初九、九二、九三、九四、九五、上九，是也。卦中有一者，名日六，如初六、六二、六三、六四、六五、上六，是也。人不知九六之來自河圖之數，而不明用九用六，與一之為九， 之為六何義，實由不習河圖，遂不解九六之為陽陰之主數。陽主九，陰主六，唯河圖中已明示之。易雖言九六，未指所出，後人莫能解。故河圖者，全易所本，欲明易象而忘河圖，宜其惝怳於文字間，而不得聖人取象之旨趣矣。

Therefore, in the hexagrams, any yang line is called "nine," such as initial nine, nine in the second, nine in the third, nine in the fourth, nine in the fifth, and nine at the top. Any yin line is called "six," such as initial six, six in the second, six in the third, six in the fourth, six in the fifth, and six at the top. People do not know that the numbers nine and six come from the Yellow River Chart, so they fail to understand what "use nine" in the hexagram Qian and "use six" in the hexagram Kun mean. They also do not understand why a single yang line is called nine and a single yin line is called six. This is indeed because they do not study the Yellow River Chart, so they do not realize that nine and six are the main numbers representing yang and yin respectively. Yang is primarily associated with nine, and yin is primarily associated with six. This point was clearly indicated by the Yellow River Chart long ago. The Book of Changes mentions nine and six, but does not point out their origin. Thus, later people have no way to understand them. Therefore, the Yellow River Chart is the fundamental basis of the entire Book of Changes. If one wishes to understand the symbolic meanings of the Book of Changes but ignores the Yellow River Chart, one can only wander through words and fail to grasp the intent of the sage in taking symbols.

河圖之數，因陰陽之分合變化無盡，而萬物之氣數由此以見，全易卦爻皆由此推移轉變而來，以象天下萬物萬事。其始則仍是二氣五形而已。圖中之象雖寥寥，而生化之物不可紀極。不僅此也，就圖數之次序，而推及其未來與其過去，則天地未生之前，與世界變遷之例，皆可由此數以得之。蓋數者推之千萬無盡也，加減乘除而已，或分、或合、或奇、或偶，各致其用，用無窮則所生化亦無窮，要在知其用耳。

The numbers of the Yellow River Chart change endlessly due to the combination and separation of yin and yang. The vital energies of all things are revealed through these numbers, and the hexagrams and lines of the entire Book of Changes are derived from them, used to symbolize all things under heaven. Tracing back to its origin, it is still no more than the two energies of yin and yang and the Five Phases. Although the images on the chart are few, the things generated from them are countless. Moreover, following the sequence of numbers on the chart, one can predict the future and trace back to the past. Even the state before the creation of heaven and earth and the patterns of change in the world can be deduced from these numbers. Numbers, when used for deduction, are endless, yet they are merely addition, subtraction, multiplication, and division. Sometimes they are divided, sometimes combined, sometimes odd numbers are used, sometimes even numbers, each serving its purpose. Because their use is endless, the things they generate are also endless. The key lies in whether one understands how to use them.

乾坤用九六為天地大用，亦數之總綱，由九六而生化者不可紀極，而不外乎中極，此圖中以五為中心也。試就陰陽各數直列之，一三五七九與四二十八六，兩者相偶而加之，其和皆五。此即天地之心，為陰陽二數之中極，亦陰陽二氣之中和。天以此立極，世以此立樞，萬物以此為紐，故並育不害，並行不悖，並生存往復不失其序。此神之所存，道之所見，而為易之中極者也。

Heavenly Stem and Earthly Branch are derived from nine and six, which represent the greatest function of Heaven and Earth, and also the overall principle of numbers. The myriad things generated from nine and six do not go beyond the scope of "central extremity," which is why the Yellow River Chart takes five as its center. Try to list the numbers of yin and yang separately: one, three, five, seven, nine and four, two, ten, eight, six. When paired and added together, the results are either five or fifteen, with the last digit always being five. This is the heart of Heaven and Earth, the central extremity of the two categories of numbers, and the harmonious balance of the two energies of yin and yang. Heaven relies on it to establish the central extremity, the world relies on it to establish its pivot, and all things rely on it as a connecting link, thus enabling them to grow together without conflict, to move together without contradiction, and to exist and return in cycles without disorder. This is where the "spirit" resides and where the "Dao" manifests, and it is precisely the central extremity of the Book of Changes.

故陽者順行而奇，陰者逆行而偶，而皆集於中五，以為主宰之區，以行鬼神，而通變化，此所謂位之相得而有合者，以其建中立極，而不失其次也。河圖言數本乎氣，通乎道，實天地之精華，自然流露，而垂象於人，以示其例，非偶然所值者，亦非怪誕難稽者。

Therefore, the yang numbers proceed in order to form odd numbers, while the yin numbers reverse to form even numbers. Yet all of them converge at the central "five," using it as the central point of governance, by which the movement of ghosts and spirits and the transformation of things are carried out. This is what the Great Treatise says: "The five positions interconnect and each has its corresponding element." Because it establishes the center, defines the extreme, and does not disrupt the order of each. The River Chart discusses numbers, whose origin lies in qi, and which connect to the Dao. It is truly the essence of heaven and earth naturally revealed, presenting the trigrams to mankind, to indicate principles. It is not accidental, nor is it absurd and unfounded.

聖人則之，而易象以成，言道者之所宗，言數者之所本，即一切方術，亦莫不由是以為推衍之準，實人文所自始也。聖人在世感於天神，而圖乃見，圖為聖人出也。故欲深究圖之妙義，必自聖人所作之易求之。易者依圖為象，推衍其用，其例固繁，而其初則河圖之所見而已。

The sages took it as their model, thus enabling the formation of the images in the Book of Changes. It is revered by those who speak of the Way and relied upon by those who deal with numbers; even all kinds of esoteric arts take it as their standard for deduction. Indeed, it is the starting point of human civilization. When the sages were alive, they communicated with heavenly spirits, and the River Chart appeared. One could say that the chart emerged for the sages. Therefore, to deeply explore the profound meaning of the River Chart, one must seek it in the Book of Changes composed by the sages. The Book of Changes established its images based on the chart, and deduced its applications. Although the rules are numerous, when tracing back to the beginning, they are no more than what the River Chart reveals.