the Luo Writing

戴九履一，左三右七，二四為肩，六八為足，五在中央。

The "Luo Writing" arranges nine numbers in the following way: nine is placed at the top in the southern position, like the crown on one's head; one is placed at the bottom in the northern position, like the shoes on one's feet; three is in the left eastern position, and seven is in the right western position, standing side by side on both sides; two is in the southwest, and four is in the southeast, living apart in the upper two corners, like the two shoulders of a person; six is in the northwest, and eight is in the northeast, living apart in the lower two corners, like the two legs of a person; five is alone in the center, serving as the hub of the numbers of the eight directions. The entire image resembles a person standing upright: nine is the head, one is the feet, three and seven are the left and right arms, four and two are the two shoulders, eight and six are the two legs, and five stands in the position of the heart and abdomen. After this arrangement, looking horizontally, we have four, nine, two; three, five, seven; and eight, one, six. Looking vertically, we have four, three, eight; nine, five, one; and two, seven, six. Looking diagonally, we have four, five, six; and two, five, eight. In each case, the sum of the three numbers is fifteen. The pairs facing each other, nine and one, three and seven, two and eight, four and six, when added together, each total ten.

河圖洛書，同一數也，而象異，以體用之分也。天地生化萬物，自其先言之，則為不易者也；自其後言之，則日易者也。故有體用之殊焉。體者一成不易，示其極也；用者隨事而易，明其變也。天地之大，萬物自生，變化自出，體用之間為氣數之所至。明其體而後知其本源，明其用而知其變化。

The Yellow River Chart and the Luo Writing both use the same set of numbers from one to ten, but the images they present are different. This is because they have the distinction of "body" and "function." The transformation of heaven and earth to generate all things, when viewed from the source before all things come into being, is a constant and unchanging aspect. However, when viewed from the aspect of things already existing and in motion, it is a constantly changing aspect. Therefore, the same set of numbers can manifest in two different forms: body and function. The "body" once established does not change, and is used to indicate the ultimate principles of heaven and earth. The "function," on the other hand, varies with specific things, and is used to explain the changes in heaven and earth. Heaven and earth are so vast, and all things naturally grow and change, and the distinction between body and function lies exactly where the movement of qi and numbers reach. Understanding the body allows one to know the origin of all things; understanding the function allows one to know the changes of all things.

蓋氣數同而所以氣數不同也。河圖者氣數之大體，洛書者氣數之變化，以不息之德，成無盡之用。此變化之例，有不可盡言，而洛書則舉其大用耳。洛書之謂象，亦猶河圖，生於二氣，分於五行，本於天地之數，成於陰陽之類，而名一貫實異也。

The principles underlying the two diagrams are essentially the same, but the reasons for their formation differ. The "Yellow River Chart" presents the fundamental structure of the numbers and energies, while the "Luo Writing" shows their specific transformations. Through the ceaseless virtue of heaven and earth, these numbers and energies achieve infinite functions. The rules of transformation are numerous and cannot all be explained, so the "Luo Writing" merely highlights the most important aspects. The "images" referred to in the "Luo Writing" are, like those in the "Yellow River Chart," generated by the dual forces of yin and yang, divided according to the "Five Phases," rooted in the numbers from one to ten of heaven and earth, and manifested in the categories of yin and yang odd and even. Though their names are connected, their actual contents differ.

故河圖之數重列五方，而洛書則平列九位。河圖之物，陰陽合得；而洛書則互為推移。其象之殊，即其取義之異；其物之別，即其致用之差。由是以合於河圖，則體用以全；以證於易，則變化以見。苟非聖人，焉足以明之哉。洛書象數，由河圖變化而來。其陽數自一而三，方位不變；自七而九，則互易矣。陰數則全與河圖易其方位。蓋陽體陰用，洛書為明用之圖，故陰數大易。

Therefore, the numbers of the Yellow River Chart are arranged in overlapping positions on the five cardinal directions—east, south, west, north, and center—with both the generating numbers and the completed numbers residing in the same direction. The Luo Writing, on the other hand, spreads the numbers out, distributing them across nine palace positions. The things associated with the Yellow River Chart are paired in yin and yang, fitting together mutually; the things associated with the Luo Writing, however, involve mutual shifting and exchanging of positions. Since the images differ, their meanings naturally differ as well; since the associated things differ, their applications naturally differ. When the Luo Writing and the Yellow River Chart are viewed together, both the essence and the application are complete. When used to verify the Zhou Changes, the principles of change become evident. Without the sages, who could explain this principle? The numerical patterns of the Luo Writing originate from the Yellow River Chart: for the yang numbers, from one to three, the positions remain unchanged; from seven to nine, their positions are exchanged. For the yin numbers, they are entirely exchanged in position with the Yellow River Chart. This is because yang represents the essence and yin represents the application, and the Luo Writing is originally a diagram that explains the concept of application. Therefore, the yin numbers undergo the greatest change.

然自其次序觀之，則一之與六，以重列變為先後平行；三之與八，四之與九，二之與七亦然，皆平行也。以一居北，而六在西北；三居東，而八在東北；九居南，而四在東南；七居西，而二在西南。其所合仍相得也。五之在中，本於中極不變之理，此所以允執厥中為至道也。體固如是，用亦不違，非中極之道，不足以喻之也。

From the perspective of numerical order: in the Yellow River Chart, one and six were originally located in the north, overlapping above and below, but in the Luo Writing they became positioned front and back in parallel. Similarly, three and eight, four and nine, and two and seven also changed from being stacked to being aligned side by side. Specifically, one is located in the central north, while six moves to the northwest; three is in the central east, while eight moves to the northeast; nine is in the central south, while four moves to the southeast; seven is in the central west, while two moves to the southwest. Although their positions have changed, their complementary relationship remains valid. As for five, it remains in the center, following the principle that the center is unchanging, which is also the reason why the phrase "uphold the middle path" from the Book of Documents, Great Yu's Mandate is regarded as the highest principle. The structure is like this, and the application cannot contradict it; without this principle of the central point, it would be impossible to explain.

就書象言之，數之變化在於動。以北方之一，東方之三，皆原位不動者，非不動也。先天之生化首為水，後天之生化首為木，其所變即在其位，故動而不動也。至南方之七，變而居西；西方之九，變而之南。以後天金火互用，與先天恰反其位，故動而易其地也。

In terms of the image of the Luo Writing, the changes in numbers all revolve around the concept of "movement." The number one in the north and the number three in the east appear to remain in their original positions without moving, but in reality, they are not truly stationary. The generation and transformation of the Earlier Heaven begin with "water," while those of the Later Heaven begin with "wood." The changes occur precisely through their steadfast adherence to their respective positions, hence movement that appears as stillness. As for the number seven in the south moving to the west and the number nine in the west moving to the south, this is because in the Later Heaven, metal and fire exchange their functions, which is the opposite of the Earlier Heaven's directional arrangement. Therefore, this is movement accompanied by a change in position.

以數之序言，則一三者順行北而東仍前，而七九者由西而南，則與前異，此由變化而分其用也。陰數之序，則東南之四，而西南之二，與東北之八，而西北之六，其序已與河圖反行；而其合乎陽數，則正相得。蓋陽由一三而七九分作兩途，一由北之東，一由西之南，以明四正之氣，猶相偶合也。而陰數則與之成逆行，一由東南之西南，一由東北之西北，亦分兩途，以與陽合。其所行之序雖變，而所合之數不差，蓋亦天地自然之變化，而成其用者也。

From the sequence of numbers, one and three move from north to east, following the original order of the Yellow River Chart; seven and nine, however, move from west to south, differing from the original order: this is precisely how differentiation occurs due to change, leading to their respective functions. The sequence of yin numbers is such that four from the southeast moves to two in the southwest, and eight from the northeast moves to six in the northwest, with their direction of movement now opposite to that of the Yellow River Chart; yet when they are paired with yang numbers, they still perfectly match. Overall, the yang numbers are divided into two groups: one group moves from north to east, and the other from west to south, indicating that the qi of the four cardinal directions remains paired in pairs. The yin numbers, moving in the opposite direction to the yang numbers, are also divided into two groups: one group moves from southeast to southwest, and the other from northeast to northwest, to be paired with the yang numbers. Although their sequence of movement has changed, the numbers obtained from their pairing remain exactly the same. This is also a natural change of heaven and earth, used to accomplish their functions.

夫陽之與陰，為萬物生化之本，有其一必有其二。如陽數順分而變位，陰數勢必逆分而變方，以便合陽而成其生化之用。故一之三、七之九，為生成數之經然；而四之二、八之六，為生成數之緯然。經緯相得，生成之功見矣。且合數之，自北而東，自西而南，陰陽之數，各得其半，即由兩以生四，兩儀生四象是也。陽由一三七九之一序，分為一三與七九兩繫；陰由四二八六之一繫，分而為四二與八六兩繫，以成其四也。

Yin and yang are the fundamental principles of the generation and transformation of all things. Where there is yang, there must be yin to correspond to it. Therefore, the numbers of yang proceed in order and divide into two lines to change their positions, while the numbers of yin necessarily reverse their order and divide into two lines to change their direction, so as to match with the numbers of yang and achieve the function of generation and transformation. The numbers from one to three and seven to nine form the 'jings' or longitudinal lines of the generating and completing numbers; the numbers from four to two and eight to six form the 'weis' or transverse lines of the generating and completing numbers. When the jings and weis interweave and correspond, the effects of generation become evident. If we combine them together, from north to east is one line, and from west to south is another line. Yin and yang numbers each occupy half, which is how the 'two principles' (two yi) give rise to the 'four images' (four xiang). This is exactly what the 'Great Treatise' of the 'Zhou Changes' (I Ching) says: 'The two principles give rise to the four images.' Originally, the numbers of yang are in the order of one, three, seven, and nine, now divided into the lines of one-three and seven-nine. Originally, the numbers of yin are in the order of four, two, eight, and six, now divided into the lines of four-two and eight-six. When combined, they form a total of four.

故陽有太陽少陽，陰有太陰少陰，則一與九為太陽，三與七為少陽，四與六為太陰，二與八為少陰。以在數之始終為太，中為少也。以此相分合，而變化成、生成見，萬物自是出，萬數自是成矣。合此兩繫之數觀之，為一八三四，此自北而止於東南也；為六七二九，此由西北而止於南也。兩數相錯而正相得，以符天地生成之用，陰陽均平之數故也。

Therefore, the yang is divided into "Taiyang" and "Shaoyang," while the yin is divided into "Taiyin" and "Shaoyin": one and nine are Taiyang, three and seven are Shaoyang, four and six are Taiyin, and two and eight are Shaoyin. The criterion for this division is that the numbers at the ends of the sequence are called "Tai," while those in the middle are called "Shao." According to this division and combination, change is formed, generation is manifested, and thus all things come into being, and all numbers are established. If we combine the two series of numbers: one, eight, three, four form a group, starting from the north and ending at the southeast; six, seven, two, nine form another group, starting from the northwest and ending at the south. These two groups are interwoven and precisely matched, which conforms to the function of heaven and earth in generating things. This is also because the numbers of yin and yang inherently require balance.

蓋左右兩行，亦陰陽之道路，二者必求其平均。左之八與三，右之七與二，雖原來火木生成之數，而二與八則互換其位以相合。若不換位，則三與二、七與八，皆相失，而生成之功不能見矣。故二八易位而後陰陽之數以平，生成之功不改，此則天地之自然之妙，不可不知者也。

The two rows on either side of the diagram represent the channels through which yin and yang interact. These two must be balanced. On the left, the numbers eight and three, and on the right, the numbers seven and two, are originally the numbers representing wood and fire, which are generative elements. However, in order for them to harmonize with each other, the numbers two and eight must be exchanged. If they are not exchanged, then three and two, as well as seven and eight, will not match, and the generative effect will not be evident. Therefore, after exchanging the positions of two and eight, the numbers of yin and yang become balanced, yet the generative effect remains unchanged. This is the subtle and ingenious aspect of the natural transformation of heaven and earth, which should not be overlooked.

因此互換，如是相對之位，皆合而成十，即一與九、三與七、二與八、四與六，莫不成土之成數。洛書中央用五而不見十，不知十之數，恰藏於八方，以相合而後得之。河圖之合皆五，洛書之合皆十，明五十為天地之心，土為天地之中、萬物之本，其義至深切矣。夫洛書之象，明用者也，天地大用，在此圖中見之。

Because of this exchange, any two opposite positions add up to ten: one and nine, three and seven, two and eight, four and six, all combining to form the number ten, which represents "earth." The center of the Luo Shu only uses five and does not show ten, so people think ten is absent from the chart. In fact, the number ten is hidden within the eight directions; it can only be obtained by combining the numbers of opposite positions. In the He Tu, the generating numbers and the formed numbers are paired, each separated by five and combined with it; in the Luo Shu, the numbers of opposite positions add up to ten. This shows that five and ten are the core of heaven and earth, "earth" being the central position of heaven and earth and the fundamental basis of all things. The meaning is profound. The Luo Shu diagram illustrates the concept of "use," the great function of heaven and earth, which can all be seen in this diagram.

即易之卦象，亦於此圖明之。南九北一，由乾坤變為離坎也；東三西七，由離坎變為震為兌也；東北之八、西南之二，由震巽變為艮坤也，而艮坤易其位焉；東南之四、西北之六，由兌艮變為巽乾也，而巽乾首明其用焉。此皆易象所自仿也。蓋河圖者體，合伏羲之卦，而含文王卦之變者也；洛書者用，合文王之卦，而合文王之卦，而存伏羲之卦之體者也。

Even the positions of the hexagrams in the Zhou Yi depend on this diagram. The nine in the south and one in the north result from the transformation of the Earlier Heaven trigrams, where Qian (Heaven) was originally in the south and Kun (Earth) in the north, becoming Li (Fire) in the south and Kan (Water) in the north in the Later Heaven arrangement. The three in the east and seven in the west come from the Earlier Heaven trigrams Li (Fire) in the east and Kan (Water) in the west, changing to Zhen (Thunder) in the east and Dui (Lake) in the west in the Later Heaven arrangement. The eight in the northeast and two in the southwest result from the Earlier Heaven trigrams Zhen (Thunder) and Xun (Wind) changing to Gen (Mountain) and Kun (Earth) in the Later Heaven arrangement, with Gen and Kun swapping positions. The four in the southeast and six in the northwest come from the Earlier Heaven trigrams Dui (Lake) and Gen (Mountain) changing to Xun (Wind) and Qian (Heaven) in the Later Heaven arrangement, where Xun and Qian first display their functions. All these are the sources from which the Zhou Yi hexagrams are derived. The He Tu (Yellow River Chart) governs the essence and corresponds with Fuxi's Earlier Heaven Trigrams, while also containing the seeds of movement for Wen Wang's Later Heaven Trigrams. The Luo Shu (Luo Writing) governs application and corresponds with Wen Wang's Later Heaven Trigrams, yet still preserves the essence of Fuxi's Earlier Heaven Trigrams.

二者必互觀之，方明其義。以天地之生化萬物，由一而眾，故由太極而兩儀，而四象，而八卦；即由二氣而五行，而九宮，皆由一而二，三而四，自少分多，自寡化眾，此生化自然之象，而圖書所示之例也。以生化之始，不離於一，故生化之眾，不失其中。有此中極，則千萬變化，不亂其序，此中央之五永不易者也。而其它象數，則隨氣數所至，二五分合而成其變。變成而天地之大用見，萬物之生化出。其理其事，固顯然如指掌矣！

The Yellow River Chart and the Luo Writing must be viewed together in order to understand their meaning. The generation of all things in heaven and earth begins from one and then produces many. Therefore, from the Taiji to the two principles, from the two principles to the four images, and from the four images to the eight trigrams; that is, from the two energies of yin and yang to the Five Phases, and then to the nine palaces, all proceed from one to two, from three to four, from few to many, from few to numerous. This is the natural manifestation of generation and transformation, which is exactly the pattern demonstrated by the Yellow River Chart and the Luo Writing. Since the beginning of generation does not depart from one, even though the generation is numerous, it never loses its central point. With this central point, myriad changes do not disrupt the order, which is why the central five remains constant. As for the images and numbers in other areas, they change according to the movement of the energies, arising from the combination and separation of yin and yang energies and the Five Phases. Once change occurs, the great function of heaven and earth is revealed, and the generation and transformation of all things spread out. The principles and facts in between are clear and distinct, as clear as pointing to one's own palm.

洛書言數，有明用者，有明變者，有與道同者，有與人事同者，大別之有三。（一）圖中之數與河圖異：河圖為五十五，洛書則四十五。此由天地生化，陰陽消長之數使然也。河圖言體，陰陽平均，此合乎道者也。洛書則重在用，用者以陰從陽，以陽統陰，陽主陰從，天包地而生化方就其序。故陽數二十五不變，陰數以三十變為二十，明陽道常長，陰道常消，而以立天地之極。

The numbers in the Luo Shu explain different aspects. Some are used to clarify function, some to clarify change, some connect with the heavenly way, and some connect with human affairs. Generally, they can be divided into three categories. First, the total number in the Luo Shu differs from that in the He Tu. The He Tu has a total of fifty-five, while the Luo Shu has forty-five. This is due to the numerical principles of the generation and transformation of heaven and earth, and the waxing and waning of yin and yang. The He Tu discusses the "body" or essence, with yin and yang equally balanced, which conforms to the heavenly way. The Luo Shu focuses on the "use" or application, which requires yin to follow yang and yang to govern yin. Yang is primary and yin is secondary, with heaven encompassing earth, allowing generation and transformation to proceed in order. Therefore, the total of yang numbers in the Luo Shu remains twenty-five, while the total of yin numbers decreases from thirty to twenty, indicating that the way of yang is constantly increasing while the way of yin is constantly decreasing, thereby establishing the standard of heaven and earth.

正陰陽之位，俾下毋僭上，小毋陵大，並育不害並生不賊，而全其用，以至無窮也。乃為陽位者五，為陰位者四，陽居四正，陰居四隅，陽正位中宮，陰環行四極，而後名正事順，體建用周。雖陰陽似失其平衡，實則如是方得其平衡也。蓋陰氣不可長，中極不可亂，以五之陽正立中宮，而四圍二氣，仍自平均，未嘗偏頗，此其為數以四十五為制也。

To properly arrange the positions of yin and yang, so that the lower does not exceed the upper, the weak does not oppress the strong, and they grow together without hindrance, coexist in mutual nourishment without mutual harm, thus preserving their functions to infinity. Thus, the positions of yang are five, and those of yin are four: yang occupies the four cardinal directions—east, south, west, and north—while yin resides in the four corners. Yang is located at the central palace, while yin circulates around the four extreme directions. In this way, the distinctions are clear, the principles are harmonious, the essence is established, and the functions are fully extended. At first glance, yin and yang may seem unbalanced, but actually, this arrangement achieves true balance. Because yin should not grow unchecked, and the central position should not be disturbed, five yang elements are placed in the central palace, while the yin and yang energies around the four directions remain balanced and impartial. This is the established principle of the Luo Writing, which totals forty-five.

（一）洛書之數，陰陽平列，而與河圖位異序殊。與河圖數同而象二者何也？則河圖明天地生成者，所生成為五行，其位限於五。又河圖明天地偶合者，其合以順逆，故序唯依陽順陰逆之例。洛書則由河圖而推其變化以極其用，所象在生成之後，自五行推至九宮，以明生化之序，由二五而可推至無窮也，故數位分而為九。

The numbers in the Luo Shu are arranged with yin and yang laid out side by side, differing from the He Tu in both direction and order. It is the same set of numbers, yet why do they appear in two different forms? Because the He Tu explains the origin of heaven and earth, and what is generated is the Five Phases, so its directions are limited to five. The He Tu also explains the mutual pairing of heaven and earth's yin and yang, with pairing methods that can be in order or reverse order, so its sequence follows the principle of yang moving in order and yin moving in reverse. The Luo Shu, however, is derived from the He Tu to show its transformations and exhaust its functions. The images it takes are after generation, expanding from the Five Phases to the Nine Palaces, to explain the sequence of generation and transformation, showing that from the two energies of yin and yang and the Five Phases, one can proceed to infinity. Therefore, the numbers are divided into nine.

又洛書明生化之例，本於二氣之交錯。二氣分合，生化以眾，則數序之行，亦循環往復以至無盡。而其序或順或逆，或移彼就此，或分此合彼，互相推移，成其變化，此亦天地生化自然之序，萬類消長一定之例，有不易而易，不變而變，而變仍不失，易仍不亂之道存焉。故與河圖同異也。

Furthermore, the Luo Shu explains the principles of generation and transformation, originating from the intermingling of yin and yang energies. These two energies continuously divide and combine, leading to an ever-increasing diversity of generation and transformation. Thus, the sequence of numbers operates in a cyclical manner, continuing endlessly. Its order sometimes proceeds forward, sometimes backward, sometimes moving elements from one side to the other, sometimes separating one side and merging it with the other, shifting back and forth, thereby producing change. This is precisely the natural order of generation and transformation in heaven and earth, the fixed pattern of the increase and decrease of all beings. Within it lies the principle that what is unchanging can become changed, what remains constant can undergo transformation, even when changed it does not lose its constancy, and even when transformed it does not disrupt its order. Therefore, the Luo Shu and the He Tu have both similarities and differences.

如陽數東北原位，以水木之生成，先後天之所始。言先天之生成者首水，以天之一氣切化也；言後天之生成者首木，以天之二氣再化也。故卦象以六為坤之用，定爻之數；以三為卦之數；以八之自乘，為全卦之數，皆取乎水木之數，以領先後天一切生成之象也。

For instance, the yang numbers remain in their original positions in the east and north, because the generation of water and wood marks the starting points of "Earlier Heaven" and "Later Heaven" respectively. The generation of "Earlier Heaven" begins with water, as water is produced from the first layer of heavenly qi. The generation of "Later Heaven" begins with wood, as wood is produced from the second layer of heavenly qi. Therefore, in the hexagram structure, the number six is used for the "Kun" trigram to determine the six lines in a hexagram. The number three is used for the three lines in a trigram. The number eight multiplied by itself gives sixty-four, which represents the total number of hexagrams. All these numbers are derived from water's one and six, and wood's three and eight, to unify the imagery of generation in both "Earlier Heaven" and "Later Heaven."

若西南之七九，則先天後天之生化各異。先天之火在南，數為七；金在西，數為九。而後天之九屬火非金，七屬金非火，二者恰易其數，故變其位。蓋火之與金本相剋也，非同水木之相生。既克則有勝有復，火之勝金於先天，則金必復於後天。若勝而不復，則生化將窮，更何後天變化可言哉。故在洛書火退而西，金進而南，恰得其反，以成其序。

As for the numbers seven and nine in the west and south, the generation and transformation of the Earlier Heaven and Later Heaven differ. In the Earlier Heaven, fire is in the south and corresponds to the number seven; metal is in the west and corresponds to the number nine. In the Later Heaven, however, nine belongs to fire rather than metal, and seven belongs to metal rather than fire. Thus, the numbers associated with these elements have exchanged places, and so have their positions. This is because fire and metal are naturally opposing, unlike water and wood which are mutually generating. Since they are opposing, there is a concept of 'victory' and 'recovery': in the Earlier Heaven, fire overcame metal, so metal must necessarily recover in the Later Heaven. If there were only victory without recovery, the process of generation and transformation would reach its end, and how could there be any change in the Later Heaven? Therefore, in the Luo Writing, the number seven, originally associated with fire in the Earlier Heaven, moves to the west, while the number nine, originally associated with metal in the Earlier Heaven, moves to the south. This is exactly the opposite of the Earlier Heaven, thus establishing its order.

蓋五行之生，火必因土而後生金。既互易其位，乃得受中央之土氣，而後不相剋而相生，生克之間，即在此易位數耳。且因變化之例，必二氣之平衡，且相得有合也。以火移西，以金移南，一則九與一對，一則七與三對，皆合十而受土氣，以自成後天之化。

According to the order of mutual generation in the Five Phases, fire must first generate earth, and then earth generates metal. After the numbers seven and nine exchange positions, they are exactly able to support the central earth energy. Thus, the originally opposing fire and metal transform into a mutual generation relationship. The distinction between mutual generation and mutual control lies precisely in this exchange of positions and numbers. Furthermore, following the principle of change, it is necessary to maintain balance between yin and yang energies and ensure they are compatible. Moving the number associated with fire to the west and the number associated with metal to the south, one side becomes nine and one facing each other, while the other side becomes seven and three facing each other. Both sides add up to ten and can support the central earth energy, thus completing the Later Heaven transformation on their own.

即以陰數言之，其變例亦同。以二之接七，而仍不失火之生成；以二之對八而合十，以受土氣，此二之必移西南與西七接也。其它四六八皆然。既以此互易，而後相得，則其原位必依此變易而後成用，此固自然之理也。

In terms of the yin numbers, the rules of change are the same. Two and seven connect with each other, yet the meaning of fire's generation is not lost; two and eight oppose each other to form ten, and they can also receive the earth's energy: this is why two must move to the southwest to connect with seven in the west. Similarly, the other numbers four, six, and eight are all like this. Since mutual exchange is necessary for compatibility, their original positions must be adjusted accordingly in order to achieve their functions. This is naturally the way things are.

（一）洛書中宮之數五，河圖之數十五，何也？則以河圖土之生成數全，陰陽之體，不可少也。洛書以明用為主，土之德在五，言五已賅十。前講河圖已詳其義。洛書之僅舉五者，又不獨以土德用五之例。蓋洛書明生化之變，為後天之象，非河圖可比。後天之例，以生化為用，生化之道雖本於五行，而非止於五行，故其象皆變河圖之原例，而成新例。

Third, the central palace of the Luo Shu contains only the number five, while the He Tu's central palace includes both five and ten, making fifteen. This is because the He Tu aims to fully list the "birth number" of earth, which is five, and the completed number ten. Yin and yang are both essential in their essence. The Luo Shu focuses mainly on illustrating function, and the virtue of earth is expressed through five, so mentioning five already includes ten. This point was explained in detail when discussing the He Tu earlier. However, the Luo Shu only presents five, not merely the principle that "earth's virtue is expressed through five." The Luo Shu explains the process of generation and transformation, which belongs to the "Later Heaven" pattern, and thus cannot be compared with the He Tu. In the Later Heaven pattern, the function of generation and transformation is emphasized, and although the way of generation and transformation originates from the Five Phases, it does not stop there. Therefore, all its patterns have changed from the original rules of the He Tu and established new ones.

後天者以生化無盡為用，故各方之數皆由內而外，由寡而眾以分合變化示生化者也。生化既眾，象數益繁，而並生不害、並育不亂者以有中極也。中極止於一，此數用五而去十，明其不二也。天地之道，一言可盡，為物不二，則生物不測。此即堯舜執中之道，以本天地之用，而見於洛書者也。

The concept of "Later Heaven" is based on the endless process of generation and transformation, so the numbers in each direction spread outward from the center, increasing from small to large, using division and combination to illustrate the process of generation and transformation. Since the process of generation and transformation is complex, the images and numbers also become increasingly intricate. Yet all things can grow together without conflict and be nurtured together without disorder, relying on the principle of "the central point." The central point is unique, so here only five numbers are used instead of ten, to indicate that the central point is singular and undivided. The way of heaven and earth can be fully expressed in one sentence. The "Zhongyong" says: "If a thing is not dual, then its ability to generate life is boundless." This is precisely the meaning of the transmission from Yao and Shun, "Yun zhi jue zhong," which originates from the function of heaven and earth and is manifested in the "Luo Book."

天地以五為中心，五行以土為根本，萬物萬事以中一為立極，皆自此象見之。以天道言之，則一切生化之數，不離於中五，而用以廣，德以大，萬物所倚，萬數所持，皆此五也。故曰：天地心，亦陰陽之中氣，五行之中道也。

Heaven and Earth take the number five as their center, the Five Phases take earth as their foundation, and all things and events take maintaining the center and unity as their highest principle. These can all be seen from the image of the Luo Writing. In terms of the Way of Heaven, all numbers of generation and transformation depend on the central five, its function is widely extended because of it, and its virtue is great because of it. What all things rely upon and all numbers depend on is this five. Therefore, it is the core of Heaven and Earth, the harmonious qi of yin and yang, and the upright path of the Five Phases.

易傳曰顯諸仁，藏諸用。此二語即釋洛書者也。洛書之中五，即仁也。而曰藏用者，則言洛書用五藏十也。夫陰陽二道，相倚而成用，而陰必藏，陽必顯。一藏一顯，其用始大。五與十皆土數，即土德也。一屬於仁陽也，一屬於智陰也。用仁而藏智，仁顯而智不顯。藏也者非去而不用，言藏之於內，不以顯也。

The Zhou Changes, Great Treatise says: "Manifest it in benevolence, conceal it in function." These two sentences can be used to explain the Luo Writing. The central number five in the Luo Writing is what is called "benevolence." Saying "conceal it in function" refers to the Luo Writing clearly using five but secretly containing ten. Yin and Yang depend on each other to achieve function, and Yin must be concealed while Yang must be manifest. One concealment and one manifestation allow function to be great. Both five and ten are numbers of earth, which is the virtue of earth: five belongs to benevolence, which is Yang; ten belongs to wisdom, which is Yin. Manifesting benevolence and concealing wisdom, benevolence is revealed externally while wisdom is not revealed. What is called "concealment" does not mean abandoning it, but rather keeping it within, not letting it appear externally.

蓋天下之道，必首德行。仁為人之全德，亦天地之全德也。若智者，以成仁之用也，不可獨用。仁顯其德，而智全其用，則用不顯而自大，是中庸成已成物之功，所謂不慮而中，不思而得者也，唯聖與至誠則之。以人之生本乎仁，仁之德無不全也，而智成之，即陰陽之道也。顯仁藏智，智在仁中，即扶陽抑陰之道也。

The principles of the world always begin with virtue. "Ren" is the complete virtue of a person, as well as the complete virtue of heaven and earth. As for "zhi" (wisdom), it is used to complete the function of ren, and cannot be used independently. Ren reveals its virtue, while zhi completes its function. Thus, although the function is not outwardly displayed, it naturally becomes great. This is the practice of "cheng ji cheng wu" (perfecting oneself and others) mentioned in the Zhongyong. It is also the state of naturally conforming to the middle way without deliberation and naturally attaining something without thinking. Only sages and those of utmost sincerity can achieve this. Human life originates from ren, and this virtue is present in all things. It is through zhi that this virtue is completed. This is precisely the way of yin and yang complementing each other. Making ren evident while concealing zhi, with zhi contained within ren, is the principle of "fu yang yi yin" (supporting yang and restraining yin).

此義則本於洛書。洛書顯五而藏十非無十也。四正四偶，所合皆十，即所謂藏也。藏之於內，而獨顯五，以明立極之義，實為天地微妙之道所見者也。天地之功，見於生成皆仁德也。不見其智，而功莫與大，此即能藏其智以成其用也。人若忘乎此義，不顯其仁而用智焉，是悖天逆道，無不敗矣，尚何成哉。故洛書用五之義至深切矣。洛書明變化之例，示生化之用，其所包者廣，此略舉其大者耳。要將河圖之象數，與之一一對參，而求其變化之跡，則可漸悟天地生物之則，聖人易教之心矣夫。

This principle is based on the Luo Writing. The Luo Writing reveals five but conceals ten, not lacking ten entirely. The sum of the numbers opposite each other in the four cardinal directions and the four corners is ten, which is called "concealment." By concealing ten within it, it only reveals five, thus expressing the meaning of establishing the central point. This is indeed a manifestation of the subtle way of heaven and earth. The achievements of heaven and earth are reflected in generating all things, which are all expressions of benevolence. Although its wisdom is not visible, its achievements are the greatest. This is precisely how wisdom is concealed to complete its function. If people forget this principle, and do not manifest their benevolence but instead rely solely on wisdom, they are going against the way of heaven and reversing the correct principles, and there is no way they can succeed, let alone achieve anything. Therefore, the Luo Writing's use of the number five is truly profound. The Luo Writing explains the principles of change and demonstrates the functions of generation and transformation, containing a vast range of content. Here, only the main points are briefly mentioned. Scholars should compare and study the images and numbers of the Heavens Chart with the Luo Writing, seeking the traces of change, and thus gradually understand the principles by which heaven and earth nurture all things, as well as the intentions of the sages in establishing the Zhou Changes (I Ching) as a teaching.

河圖負於龍馬，洛書呈於元龜，以其所出見在河洛也，故名河圖、洛書。圖之出雖以馬龜，象之征則符運數，蓋天地自然之象，顯於物者也。既出見於物，得聖人解之，以造成人文之始，而易象乃具。是河洛二圖者，天地之文顯於物，而用於人者也。

The Yellow River Chart appeared through a dragon horse, and the Luo Writing came through a great turtle; since they appeared at the Yellow River and Luo Water, they are called the Yellow River Chart and the Luo Writing. Although these charts emerged through a dragon horse and a divine turtle, the symbols they represent correspond with the numbers of the heavens and earth's vital energies. They are originally the natural phenomena of heaven and earth, merely manifested through specific objects. Once manifested through objects, they needed sages to explain them, thus initiating human civilization. The hexagrams of the Zhou Changes (I Ching) were thereby established. Clearly, these two charts, the Yellow River Chart and the Luo Writing, are manifestations of the principles of heaven and earth through objects, available for human use.

唯天地之神，能自成之，唯物顯之，唯人用之。蓋合天地人物四者之靈，而後見其功，成其名，以廣其德，著其道，其為物豈可輕言哉？觀易象之所本，六經之所仿，百度之所制，一切政教之所創，其為人文之袓，文化之源，固無他物可與方矣！

Only the mystery of heaven and earth can naturally produce it, only specific things can manifest it, and only humans can apply it. To combine the spirit of heaven, earth, human beings, and things, one can see its effectiveness and achieve its reputation, thus spreading its virtue and revealing its principles. Such a thing is not to be regarded lightly. The hexagrams of the Zhou Changes come from it, the Six Classics of Poetry, Book, Rites, Music, Changes, and Spring and Autumn Annals derive from it, all systems and institutions rely on it, and all political and educational measures originate from it. As the ancestor of human culture and the source of civilization, there is nothing else that can be compared with it.

圖書之象簡，而包舉天地之間，生成之德，變化之道，盡氣數之妙，示形神之全，固以一而賅萬，中而立極，有同易道者矣！其所明用，在順自然之序，而推其不息之功；在本生化之原，而溯其不二之道，猶天地之覆載，人物之生育，而莫外於至一至中之極者也。然考其用之大例，有數則焉：

The images of the Yellow River Chart and the Luo Writing are extremely simple, yet they encompass entirely the virtues generated between heaven and earth, and the principles of change. They exhaust the subtle aspects of the vital energy and demonstrate the completeness of form and spirit. Indeed, they unify all things through one principle, standing at the center and establishing the ultimate standard, which is exactly the same as the way of the Zhou Changes. Their purpose is to follow the natural order and investigate the continuous process of generation and renewal, and to return to the origin of creation and trace back to the single, ultimate Dao. This is like heaven covering and earth supporting, with people and things flourishing in between, all arising from the ultimate principle of "absolute unity and absolute centrality." Now, examining their major applications, they can be summarized as follows.

（一）河圖者，明位之用，位無動也；洛書者，明時之用，時有變也。河圖立天地之極，而見其體，是明位不動，有其生成之序也。洛書列天地之序，而著其變，是明時之時遷，有其復之情之也。二者大例，一體一用，一守中而立極，一循行而不窮，是天地生成始終，道德氣數往復遷變，可見可微者也。

The Yellow River Chart explains the function of "position," which is fixed and unmoving; the Luo Writing explains the function of "time," which is ever-changing. The Yellow River Chart establishes the principles of heaven and earth and reveals their essence, showing that although position is fixed, there is an inherent order of generation within it. The Luo Writing arranges the sequence of heaven and earth and highlights their changes, showing that although time is in constant movement, there is a cyclical pattern of return and reversal within it. These two great principles represent one as the main subject and the other as the main function; one remains centered and establishes the standard, while the other cycles continuously without end. The beginning and end of the generation of heaven and earth, as well as the ebb and flow of moral virtue and destiny, whether revealed or hidden, can all be understood from here.

（一）河圖者，明天地之生成也；洛書者，明萬物之變化也。河圖自陰陽二氣之生成，而五行出；洛書自五方之推遷，而九宮見。五行為萬物之本，數簡而位定；九宮為萬物之變，數繁而行周。此生化由簡而繁，氣數由正而變之例。故河圖之數，自一至十，不離五行，而成生成之母；洛書之數，自一至九，化分無盡，而明往復之理。

The second point is that the Yellow River Chart explains the origin of heaven and earth, while the Luo Writing explains the transformation of all things. The Yellow River Chart begins with the generation of yin and yang, thus revealing the Five Phases. The Luo Writing begins with the movement of the five directions, thus revealing the nine palaces. The Five Phases are the fundamental elements of all things, with simple numbers and fixed positions; the nine palaces represent the changes of all things, with numerous numbers and widespread movement. This is precisely the common pattern of generation moving from simplicity to complexity and the movement of qi and numbers moving from order to change. Therefore, the numbers in the Yellow River Chart range from one to ten, always connected with the Five Phases, becoming the mother of the generation of all things; the numbers in the Luo Writing range from one to nine, transforming infinitely, used to explain the principle of continuous cycles.

其例如：一六之水，則化為一與六之北，與西北二宮；四九之金，則變為九與四之南，與東南二宮；餘數皆然。雖水火之數不易而位移，火金之數互易而位改，在河圖可名為水火金木，在洛書則稱為四正四偶各宮。正猶易之兩儀，一變為四象，再化為八卦，其數已由合而分，其物亦由寡而眾。此天地生化自然之例，而易之所以為易也，若執河圖以釋洛書，則將滯而不通矣！

For example, in the Yellow River Chart, the numbers one and six both belong to water and are located in the north, but in the Luo Writing, one is placed in the north and six in the northwest. Similarly, in the Yellow River Chart, the numbers four and nine both belong to metal and are located in the west, but in the Luo Writing, nine is placed in the south and four in the southeast. The other numbers are also arranged in this way. Although the numbers representing water and fire remain the same but their positions change, and the numbers representing fire and metal are exchanged with their positions altered, in the Yellow River Chart they are called water, fire, metal, and wood, but in the Luo Writing they are simply referred to as the four cardinal directions and the four intermediate directions. This is similar to the Zhou Changes, which starts with the two principles (two yi), then transforms into the four images (four xiang), and further into the eight trigrams (eight gua). The numbers first merge and then divide, and the elements go from few to many. This is the natural pattern of generation and transformation in heaven and earth, and it is precisely why the Zhou Changes is called the "Book of Changes." If one rigidly uses the Yellow River Chart to explain the Luo Writing, there will be many obstacles and inconsistencies.

（一）河圖者：以數紀二五之本質；洛書者，以數名九宮之循環。河圖不動而自生成，洛書以動而形變化。河圖象天地初奠，物類自生之時；洛書象萬有具陳，氣數並行之例。其合於易，則圖如伏羲之卦，定位明序，而立其極；書如文王之卦，變化錯綜，而推其用。二者各有其所象，亦不外造化自然之理數。

Third, the River Chart uses numbers to record the essence of yin and yang and the Five Phases, while the Luo Square uses numbers to indicate the cyclic pattern of the nine palaces. The River Chart is unchanging and naturally arises, while the Luo Square emerges through movement and shows change. The River Chart represents the time when heaven and earth were just established and things naturally emerged; the Luo Square represents the time when all things were already arranged and the principles of qi and numbers proceeded in harmony. When paired with the Zhou Yi, the River Chart is like Fuxi's Earlier Heaven Trigrams, which establish directions and clarify order, thereby setting standards; the Luo Square is like King Wen's Later Heaven Trigrams, which display complex transformations, thereby deriving their functions. Each has its own symbolic representation, yet both conform to the natural principles and numbers of creation.

可征於物，可驗於形，可候於時令，可證於天地之文，可通於人物之生變遷者也。故河圖之象，不易而易，如天地之道也；洛書之象，易而不易，如天地之德也。就河圖推而上之，可明天地之自始；就洛書推而化之，可知萬物之將來。而天道以明，人道以建，天人以達，性命斯順；氣數不乖，行止斯定，此易之所以立教，為象天道以定人道者也。

They can be verified through specific objects, confirmed through form, observed according to seasons, verified through the patterns of heaven and earth, and understood through the changes of things and beings. Therefore, the imagery of the Yellow River Chart is "unchanging yet changing," akin to the way of heaven and earth; the imagery of the Luo Writing is "changing yet unchanging," akin to the virtue of heaven and earth. By following the Yellow River Chart upward, one can understand the beginning of heaven and earth; by following the Luo Writing downward, one can know the destination of all things. Thus, the way of heaven becomes clear, the way of humanity is established, heaven and humanity are connected, and nature and destiny are harmonious. The balance of qi and numbers is not violated, and movement and stillness become fixed. This is why the Zhou Changes (I Ching) can establish teachings, by following the way of heaven to determine the way of humanity.

夫河圖之數變，而洛書之象以成；其屬於天道者，萬物生化自然之象也；屬於人道者，則扶陽抑陰；中極不動，顯仁藏用，中和以成之義也。蓋天道遠，人道邇；天道精微，聖人則之；人道明易，眾人能之；必以人道為言，而後眾人可通天道也。河圖之屬天道者多，洛書則寓人道；以後天之世，唯以人道為急務也。然人生自天命，人道出於天道；非天道之外有人道，乃天道之見於人事，而宜取法者；故名之人道，明人之所當由者也。

The numbers of the Yellow River Chart once change, the images of the Luo Writing then take form. Among these, those belonging to the Way of Heaven are the images of the natural generation and transformation of all things; those belonging to the Way of Humanity are the meaning of "supporting Yang and restraining Yin," the central point remaining unmoved, "manifesting benevolence and concealing function," and achieving all things through harmony and balance. The Way of Heaven is distant, while the Way of Humanity is near; the Way of Heaven is profound and subtle, known only to the sages; the Way of Humanity is clear and easy to understand, accessible to all people. One must begin with the Way of Humanity, so that people may thereby reach the Way of Heaven. The Yellow River Chart mainly discusses the Way of Heaven, while the Luo Writing embodies the Way of Humanity, because in the Later Heaven world, the Way of Humanity is the urgent matter. However, human life comes from the Mandate of Heaven, and the Way of Humanity originates from the Way of Heaven. It is not that the Way of Humanity exists apart from the Way of Heaven, but rather that the Way of Heaven is manifested in human affairs, the part that should be followed by people. Therefore, it is called the "Way of Humanity," to indicate the path people should follow.

道路也；由之則可行；人生猶行路也，由道則全其生；大則治平之功，小則孝弟之事；遠則聖賢之德，近則日用之常；皆必由是而達焉。此人道之所以為重，而次於天道，以先於天道者也。洛書之象，由河圖而變；即由先天而為後天，亦即由天道而入人道；此可征之於其數也。

The original meaning of "Dao" is a path, and by following it one can proceed smoothly. A person's life is like walking, and by following the Dao one can preserve one's life. Whether it is great achievements in governing the country and pacifying the world, or small acts such as filial piety towards parents and respect for elder brothers in daily life; whether it is the virtue of sages and virtuous persons, or the routine of eating, drinking, and living, all must be achieved through this path. This is why the human Dao is regarded as important: in terms of order, it comes after the Heavenly Dao, but in practice, it precedes the Heavenly Dao. The diagram of the Luo Book originates from the He Diagram, meaning it transitions from the Earlier Heaven to the Later Heaven, from the Heavenly Dao to the Human Dao. This can be verified through its numbers.

五行雲者，非僅指水火木金土也，凡物與事之同列而有五者，皆是；亦如兩儀也；故在時，有春夏秋冬四季也；在地，有東南西北中央也；在人，有心肝脾肺賢也；在味，以酸鹹辛苦甘也；在音，有宮商角徵羽也；在德，有仁義禮智信也；在倫，有君臣父子兄弟夫婦朋友也；其類正多，要皆五行也。

The concept of "Five Phases" does not merely refer to the five elements of water, fire, wood, metal, and earth; any set of things that are grouped into exactly five categories falls under the scope of the Five Phases, just as "Two Principles" does not specifically refer to only two things. Therefore, in terms of seasons, there are spring, summer, autumn, and winter; in terms of geography, there are east, south, west, north, and the center; in terms of the human body, there are the five organs: heart, liver, spleen, lung, and kidney; in terms of taste, there are sour, salty, pungent, bitter, and sweet; in terms of music, there are the five tones:宫, 商, 角, 徵, 羽; in terms of virtue, there are the five constants: benevolence, righteousness, propriety, wisdom, and faithfulness; in terms of human relationships, there are the five bonds: ruler and subject, father and son, siblings, husband and wife, and friends. There are many such examples, and ultimately they all belong to the Five Phases.

行者，言其周流往復循環而行，不相害也；故一年四時象之，年以四時成歲，而土旺於四季；如河圖以水火木金定四方，而以五十之土立中極也；至洛書則方之四者變八，中之五者終不移，則可知天道之行，有化少為眾，變合為分之例；而中極不易也。去其十者，抑陰而扶陽也，顯仁而藏用也；此即人道所始仿也。

The character "行" refers to its flowing and circulating nature, moving in cycles without hindrance. Therefore, the four seasons of a year take their form from this principle: a year is formed by the four seasons, and "earth" (土) is strongest at the end of each season. Just as the Yellow River Chart uses water, fire, wood, and metal to establish the four cardinal directions, and uses five and ten to establish the central position, so too in the Luo Writing, the four directions become eight, yet the central five remains unchanged. From this we can see that the operation of heaven follows a general principle of transforming few into many and uniting into division, while the central remains constant. The Luo Writing omits ten, which is "restraining yin and supporting yang," and "manifesting benevolence while concealing its function," which is precisely the principle that human beings first emulated.

人之於生，有德業焉，有事功焉；不得徒生也。故天命之性以成其德，仁是也；天賦之知以立其功，智是也；人之以仁成性，則道成而生全；以智立功，則事成而名貴；二者人之所以為人也，然必顯其一；而藏其一，以合乎洛書之象，而契乎天地之德；若不能顯仁藏智，則有功無德，失主之本。

A person living in the world has virtue and learning, has achievements and accomplishments, and cannot live a life in vain. Therefore, Heaven gives a person his nature to accomplish his virtue, which is "ren" (benevolence); and gives a person his knowledge and ability to establish his achievements, which is "zhi" (wisdom). A person relies on ren to complete his nature, thus the Way is realized and life is preserved; relies on zhi to establish his achievements, thus affairs are accomplished and reputation is esteemed. These two aspects are precisely what make a person a human being. However, it is necessary to manifest one and conceal the other, in accordance with the image of the Luo Writing, and to conform to the virtue of Heaven and Earth. If one cannot manifest ren and conceal zhi, there will be achievements without virtue, losing the fundamental role of being the master.

若不顯仁而徒用智，則反未為本；離性日遠，去道日甚；不獨無以全生，必反因以賈禍。蓋智者，用也，末也，必有本體為之先焉；失其仁而唯智是尚，是舍本逐末，失體求用；無根之花，無源之泉，何以自全而成用哉。

If one does not manifest benevolence but relies solely on wisdom, this is reversing the order, taking the end as the beginning, moving further and further away from one's true nature and the great way. Not only will this fail to preserve life, but it will inevitably bring about calamity. "Wisdom" belongs to the function, to the secondary aspect; before it, there must be a fundamental principle as its foundation. To abandon benevolence and solely pursue wisdom is to discard the root for the branches, to seek function without the essence, like a flower without roots or a spring without a source. How can one preserve oneself and achieve function in such a case?

用者，言有所用也；有所主宰而後成其用也，非獨用可成功者；故顯仁藏用，示天地之道；必先立德，明人道之本；必先修已，其與中極不易之義，為易教並重者也。仁之德，即生成也；天地以生成為本，故人以仁為先。仁、人也；有仁斯有生，安有忘其仁，而能生者？即安有忘其生，而能致用也？故用者，為仁用也；為用以成其仁也。

The term "use" refers to application; one must first have a master principle before it can be effectively applied, and it is not sufficient to rely solely on use to achieve success. Therefore, the Zhou Changes say "manifest in benevolence, conceal in use," revealing the way of heaven and earth. Humans must first establish virtue in order to understand the fundamental principle of human conduct; they must first cultivate themselves, a principle that is equally emphasized in the teaching of the Zhou Changes as "the center does not shift." Benevolence is the virtue that generates all things; heaven and earth take generation as their fundamental principle, so humans should take benevolence as their primary concern. The character "仁" (benevolence) includes "人" (person), meaning that with benevolence there is life; how could there be life without benevolence? How could there be application without life? Therefore, "use" is used for the sake of benevolence, and it is through application that benevolence is fulfilled.

分言之，則有諸德之名；約言之，即仁也；用之屬智，亦為仁也；苟無仁，何以名其智哉？譬如其生且不省者，焉得為智？此夫子有予智不能中庸之嘆也！河洛二圖所重，在變化之例；洛書者，河圖之變；河圖者，天地之變。河圖以二氣之變，而五行出；洛書則五行之變，而九宮分，此易見者也。其中微妙難知者，則五行之數，本二氣之數；

When discussed separately, virtue includes various names such as benevolence, righteousness, propriety, wisdom, and faithfulness; but when summarized, it is simply the character 'ren' (benevolence). The wisdom used is actually also a form of benevolence; if there is no benevolence, how can it be called wisdom? For example, if a person has not even examined his own life, how can he be considered wise? This is precisely what Confucius lamented in the "Doctrine of the Mean," about how people think they are clever but cannot maintain the mean for even a month. The two diagrams, the "Yellow River Chart" and the "Luo Writing," focus on the principles of change. The "Luo Writing" is a variation of the "Yellow River Chart," and the "Yellow River Chart" itself is a variation of heaven and earth. The "Yellow River Chart" originates from the changes of yin and yang to produce the Five Phases, while the "Luo Writing" derives the nine palaces from the changes of the Five Phases. This is easy to see. However, the subtle and difficult part is that the numbers of the Five Phases originally come from the numbers of yin and yang.

而九宮之數，即五行之數；推之則萬物之數，莫非九宮之數，亦莫非五行之數，二氣之數也，溯而上之，則九而五而二，歸於太極之一；推而下之，則五而九，至於萬物，以達無窮；一本而萬殊，萬殊歸於一本，如布之經緯，網之綱目；其為數殊，為絲一也；故自太極以至無窮，為數雖殊，為氣無二；而數之殊者，非殊數也。

The numbers of the Nine Palaces are the numbers of the Five Phases; from this derivation, the numbers of all things are none other than the numbers of the Nine Palaces, none other than the numbers of the Five Phases, and none other than the numbers of the two energies of yin and yang. Tracing back, from nine to five, from five to two, ultimately returning to the unity of the Taiji. Moving forward, from five to nine, until all things, reaching infinity. One fundamental principle gives rise to myriad differences, and these myriad differences return to the one fundamental principle, just as a fabric has warp and weft, and a net has its main strands and its meshes. Although the number of strands varies, the threads that weave them are the same. Therefore, from the Taiji to infinity, although the numbers differ in myriad ways, the energies remain unchanging. The differences in numbers are not due to another set of numbers altogether.

數之自加減乘除也。故數殊而不殊，雖由萬物至於無窮，仍不外此一二三四五六七八九之數；不過有其分合變化而已。故數之殊者，數自殊之；非數外有數，則由氣之不二也。因氣不離陰陽，雖有錯綜生化之殊，仍不失其為陰陽；故氣無殊，數無殊；數以紀氣，氣之所至則數成，數之所見則氣達，氣數無殊用也。

The difference is merely the result of addition, subtraction, multiplication, and division within the same set of numbers. Therefore, numbers appear different but are essentially the same: even when derived from all things to infinity, they still do not go beyond the nine numbers one, two, three, four, five, six, seven, eight, and nine. They are only differentiated through their combination and separation. The difference among numbers arises from their own differentiation, not from another set of numbers outside this system. The origin lies in the oneness and non-duality of qi. Qi cannot be separated from yin and yang; although there are complex and varied processes of generation and transformation, they are ultimately yin and yang. Therefore, qi has no distinctions, and numbers also have no distinctions. Numbers are used to record qi: where qi goes, numbers take effect there; where numbers manifest, qi flows there. In terms of function, qi and numbers are essentially the same.

河洛者，明氣數之用，示天地萬物之始終也；果明其變例，則天下之氣數，皆可推而知之；以其用因變而後見也。設無所變。即無所用；果無變無用，天地已歸澌滅；更何河洛之圖存哉？唯其變多則用神，用神則天地無極；人物無盡，莫不在氣數之中；即莫能出此圖象之外，而後河洛二圖，乃為人之所必明，世之所必重者矣！

The Yellow River Chart and the Luo Writing are used to explain the function of qi and numbers, and to display the beginning and end of all things in heaven and earth. If one can truly understand the principles of their changes, then the qi and numbers of the world can all be calculated and known, because function is revealed through change. If there were no change, there would be no function; if there were truly no change and no function, heaven and earth would have long since perished, and how could the Yellow River Chart and the Luo Writing still exist? It is precisely because of the many changes that function becomes profound; because function is profound, heaven and earth have no end, and human beings and things continue endlessly. All of this is contained within the qi and numbers, and thus lies within the scope of these two diagrams. In this way, the Yellow River Chart and the Luo Writing are indeed something that every person must understand and every generation must value.

河圖與洛書異同之處，即氣數變之跡，與天地萬物生化之例；其所列雖僅此九宮之位，五行之名；而其類推，可至無盡；欲明其變，可就易象證之也；所須知者，河洛之變何自來？其氣數分合何所本？此則求明天地生成之序，人物生化之數，所必留意者也；茲試論之：

The differences and similarities between the Yellow River Chart and the Luo Writing are traces of the movement of qi and numbers, and they also serve as principles for the generation and transformation of all things in heaven and earth. Although they only arrange nine palace positions and the names of the Five Phases, by analogy and deduction, they can be extended infinitely. To understand their changes, one can refer to the hexagrams of the Zhou Changes. It is essential to first clarify: where does the change between the Yellow River Chart and the Luo Writing originate? What is the basis for their division and combination of qi and numbers? This is precisely what must be taken into consideration when trying to understand the sequence of the generation of heaven and earth, and the numerical principles of the birth and transformation of human beings and things. Now I will attempt to discuss this.

洛書之數，因河圖而來；即萬物之生化，本天地五行所出；如五行生自二氣，即天地也；已見河圖。而九宮變自五行，即五方金木火水土也，則見洛書；而九宮雲者，非言九物，或九事也；以數盡於九，九宮足以包數之全；天下之物，盡在九數中，則莫逃乎九宮之外；故九宮者，萬物萬事之所隸屬，而萬數之所循環者也。

The numbers of the Luo Shu are derived from the He Tu, just as the generation and transformation of all things fundamentally originate from the Five Phases of heaven and earth. The Five Phases arise from the two energies of yin and yang, which represent heaven and earth. This principle is already shown in the He Tu. The nine palaces come from the Five Phases, which are the five elements—metal, wood, fire, water, and earth—located in the five directions. This aspect is demonstrated in the Luo Shu. The term "nine palaces" does not refer to nine separate things or nine different matters; since the number nine represents completion, the nine palaces are sufficient to encompass all numbers. All things in the world exist within the nine numbers, and none can escape the scope of the nine palaces. Therefore, the nine palaces form the framework to which all things belong, and they are also the place where all numbers cycle and operate.

無論生化之無盡，所推所衍，莫不盡於九數之中；由九而內則返於一，由九而外，則推至無量數；數雖無量，仍不外此九也；乾之用九，已盡天下生生之道；而大衍之用四十九，已極天下往來之數；其曰：大衍者，由中數之五，而十倍之也；衍者、推衍者：推其數之所至，而足以包舉一切數；故曰：大衍。以數之老陽在九，老陰在六；六九五十四，去中五，而得四十九；

No matter how infinite and boundless the transformations of all things may be, all calculations and deductions remain within the range of nine numbers. By contracting from nine inward, one returns to one; by expanding from nine outward, one reaches an immeasurable large number. Although the number is immeasurable, it still does not go beyond these nine numbers. The "Use of Nine" in the Qian hexagram of the Zhouyi has already explained the principle of continuous generation and transformation in the world. The passage in the Great Treatise says, "The great derivation consists of fifty numbers, but its use is forty-nine." This has already exhausted the numbers of all coming and going transformations. The term "great derivation" is called so because it comes from the central number five multiplied by ten. "Derivation" means deduction: investigating the limit that this number can reach, which is sufficient to include all numbers, hence it is called "great derivation." In terms of numbers, old yang is nine, old yin is six. Six multiplied by nine gives fifty-four. Subtracting the central five, we get forty-nine.

即大衍之用，此亦河圖洛書所有之數也；以其數之重在變也，則河圖之象變為洛書，所始在金火互易位也。以金之九，為先天數；若已變，則屬火；離代乾用，離為火；而乾為金，故其位隨乾離而易；此易見者也。

This is the number forty-nine used in the "Great Derivation" (dà yǎn), which is exactly the number found in both the Yellow River Chart (Hé tú) and the Luo Writing (Luò shū). Since these numbers emphasize the concept of change ("biàn"), the image of the Yellow River Chart transforms into that of the Luo Writing, beginning with the exchange of positions between metal and fire. The number nine originally belongs to metal in the Earlier Heaven (xiān tiān), but once it changes, it becomes associated with fire. This is because in the Later Heaven (hòu tiān), the Hexagram Li (Fire) replaces the Hexagram Qian (Heaven), and since Li represents fire and Qian represents metal, the position of the number nine changes along with the shift between Qian and Li. This is something that is easy to observe.

其間尚含深意，即以數之成，乃因氣之行；氣之行也，非置於一方而不動，必周流八方而無息；以位言之，則河圖之火與水對，金與木對；數隨五行之氣，而定其方，無所易也；若以時言之，則洛書北方之一，與東方之三，一屬太陽，一屬少陽；而後南方易以九，西方易以七；亦一太陽，一少陽；其位互易，而後正對；以其氣之既行，不復如河圖之初生也。

This contains a deeper meaning: numbers exist because qi is in motion; and since qi is in motion, it does not remain fixed in one direction but flows through all eight directions without ceasing. In terms of 'position,' the He Tu shows fire opposite water and metal opposite wood, with numbers determined by the Five Phases' qi and fixed in position without change. But in terms of 'time,' the Luo Shu has one in the north and three in the east, one belonging to the sun and the other to the young sun, so the south is assigned nine and the west seven, also belonging to the sun and the young sun. After their positions are exchanged, they become perfectly opposite. This is because qi is already in motion and circulation, no longer like the He Tu which only reflects the initial state.

河圖之氣，以陽生陰，以陰繼陽；故陽老於西方之九，而陰四繼之；陰老於北方之六，而陽一承之。陰陽互根於定位，而相得以合，而生成之功乃見，此其以氣之初生為象也；若在洛書，則陰陽已分，二氣環行；行者有序，太少必對；是以北方之一，南方之九；一始一終，皆太陽；東方之三，西方之七；一生一成，皆少陽也。

The qi of the Yellow River Chart arises from yang generating yin and yin receiving yang. Therefore, the yang number reaches nine in the west and becomes old, followed by the yin number four to take over. The yin number reaches six in the north and becomes old, followed by the yang number one to take over. Yin and yang mutually support each other in fixed directions, complementing and harmonizing with each other, thus producing effects. This is taking the form of qi in its initial state. In the Luo Writing, yin and yang have already separated, and the two energies circulate around. Since there is circulation, there must be order; the greater and the lesser must be paired accordingly. Therefore, one in the north and nine in the south, one is the beginning and the other is the end, both belonging to the sun; three in the east and seven in the west, one is the master of generation and the other is the master of completion, both belonging to the lesser sun.

陰氣亦然，東北之八，西南之二，皆少陰；西北之六，東南之四，皆少陰；兩兩相當，以周行於八方；而成一陰一陽之道，遂不可不易其南西金火之位也。此例可以一歲證之，春與秋對，冬與夏對；一寒一暑，相當其令；而一歲之氣，周而復始，如環無端，此其用之可明者也。故洛書之象，象後天已定之氣數；陰陽迭興，數位周流；以成其往來不息之德，不獨歲令然也；即一人之生，一日之晝夜；無不如是，皆以循環為率；此即洛書以明變化之用為象者也。

Yin energy is also the same: the eight in the Northeast and the two in the Southwest both belong to the Lesser Yin, and the six in the Northwest and the four in the Southeast are also called Lesser Yin (according to the system they should be Tai Yin). They correspond in pairs and circulate through the eight directions, thus fulfilling the statement in the Great Treatise, "One Yin and one Yang are called the Way." Therefore, the positions of the South and the West, which are Fire and Metal respectively, must be exchanged. This principle can be verified by the cycle of a year: Spring and Autumn are opposite, Winter and Summer are opposite, one cold and one hot each take their turn; and the energy of a year circulates continuously, like a circular ring with no beginning or end. This is the most easily understood aspect of its function. Therefore, the image of the Luo Writing represents the established Later Heaven pattern of energies: Yin and Yang alternate, numbers circulate without rest, thus forming an unceasing virtue. This is not only true for the cycle of the year, but also for a person's life and the day's night cycle, all following the principle of circulation. This is precisely why the Luo Writing takes its image to explain the function of change.

且數之既變，有數例焉；陽之合陽，則反為陰；陰之合陽，則化為陽；故二陽相遇，則成陽中陰；二陰相遇，亦為陰中陽；此在太少之外，更有陰陽也。是一與九對而成十，則九為陽，一為陽之陰；三與七對亦成十，則三為陽，七為陽之陰；陰數亦然。故四隅陰數，東北西南為陽，相對為陰；其所以然者，合則變，分則化也；後天之數多雜，無純陽純陰之物，以其不能生化也。

Furthermore, when numbers change, there are certain rules: when two yang numbers combine, they become yin; when a yang and a yin combine, they become yang. Therefore, when two yang numbers meet, they form yang with yin within it; when two yin numbers meet, they form yin with yang within it. This is another layer of yin and yang distinction beyond the Taiyang, Shaoyang, Taiyin, and Shaoyin. For example, one and nine combine to make ten. Nine is yang, and one is yang with yin within it. Three and seven combine to make ten. Three is yang, and seven is yang with yin within it. The same applies to yin numbers. Therefore, the four corners of the numbers, the Northeast and Southwest are considered yang, while their opposites are considered yin. This is because when numbers combine, transformation occurs; when they separate, change occurs. Later Heaven numbers are mostly mixed, with no purely yang or purely yin elements, because pure yang and pure yin cannot transform or change.

洛書之數，參伍錯綜，各有其分合之用；故變化無盡，不獨火金之易；即二八之易位，亦其例也；天地生成，全依此陰陽之摩蕩；而後萬物生化。世界悠久，以迄無窮，此所謂造化之機也。其行也，則以往來為則，循環不已，以成其生生不盡之功；即至誠無息之德也。故氣之陽者陰之，陰者陽之；天者下之，地者上之；左者右之，右者左之；相互以生，相交以成；不獨金火之互換也；

The numbers in the Luo Writing are interwoven in a complex manner, each having its own way of combination and separation, thus leading to infinite changes. It is not only the exchange between fire and metal; even the exchange between two and eight follows the same principle. The generation of heaven and earth entirely depends on such mutual interaction between yin and yang, allowing all things to be generated and transformed, and the world to continue endlessly. This is referred to as the key mechanism of creation and transformation. Its operation follows the principle of coming and going, circulating endlessly, thus achieving the endless generation of things. This is the virtue described in the Zhongyong as "the utmost sincerity has no cessation." Therefore, qi that is originally yang turns to yin, and qi that is originally yin turns to yang; what is in heaven descends, and what is on earth ascends; what is on the left moves to the right, and what is on the right moves to the left. They generate each other through mutual interaction and form each other through mutual crossing, not only involving the exchange between metal and fire.

故東南西北，四方之序；不似河圖，自一至九，九宮之數；不同五行，而後見後天之道，明世間之數。如河圖始於北，而洛書首東；河圖陰逆陽順，以一繫行；洛書均分二繫以往復，此皆二者同異，可資考證者也；蓋先天之象，由無入有；其始生也，因氣之動；後天則已生成之後，萬物隨之生化；

Therefore, the order of the numbers one to nine in the Luo Shu, arranged in the four directions east, south, west, and north, differs from that of the He Tu. The Luo Shu employs the numbers of the Nine Palaces, which differ from the arrangement of the Five Phases. This distinction reveals the principle of the Later Heaven and explains the numerical principles of the world. For example, the He Tu begins with the north, while the Luo Shu starts with the east. The He Tu has a single system of operation, with yin moving in reverse and yang moving forward, while the Luo Shu divides into two systems that operate reciprocally. These differences and similarities between the two can be used for reference and verification. The image of the Earlier Heaven arises from nothingness into being, beginning with a single movement of qi. The Later Heaven, however, occurs after generation, with all things continuously transforming and evolving.

氣之流行不息，數以變化不窮；其所以異者，皆自氣合之；而氣者，內徵於道，即理也；外徵於象，即數也；理數之間，不可過求其跡，以有神也；神御其用，故天地有主宰之數不易；而神不可測，以其微也；是陰陽之事，不可實之以物也。唯神通之，氣感之；固聖智之事也。以易言之，神而明之，存乎其人，言非易明也；而氣數之常變，吉凶生焉，禍福降焉，則盡人所當知也。

The movement of qi never ceases, and thus numbers change endlessly. The differences among them are all harmonized by qi. As for "qi," when examined internally according to the Dao, it is "li" (principle); when examined externally according to phenomena, it is "shu" (number). Between principle and number, one should not overemphasize their forms, for there is another "shen" (spirit) involved. The spirit governs their function, so there is a constant number that serves as the ruler in the world. However, the spirit itself is indescribable because it is too subtle. Therefore, the affairs of yin and yang cannot be fixed to a particular thing; they must be connected by the spirit and responded to by qi. This is the concern of sages and the most wise. Regarding the Zhou Yi, the Great Treatise says, "To grasp the spirit and make it clear depends on the person," which means this principle is not easily explained. As for the constant and changing aspects of qi and number, which give rise to good and bad fortune, and thus bring about blessings and misfortunes, these are things that everyone should know.

故逢其合，適其宜；去其逆，塞其空；為趨吉凶之道，不可不講；此數雖止於九，而用可千萬；物雖止於五，而道將無窮；如水之數，遇金而旺，遇火而尊，遇木而退；遇土而死；其道有窮通也；他行皆然；以數合之而自卜之，則吉凶無遁情，禍福有定數；不必求之人也。是故數之為用，合道明生者也；知數知生，順數順道；其所指者義精意切，惜乎人之不講，而不明聖人立教之旨，斯可慨矣！

Therefore, when encountering something that harmonizes with oneself, one should follow it; when encountering something suitable, one should seize it. One should avoid what is contrary to oneself and make up for what is lacking. This is the method to seek good fortune and avoid misfortune, which is something not to be neglected. Although numbers only go up to nine, their functions can be varied in countless ways; although things are only categorized into the Five Phases, their principles are endless. For example, the number of water, when encountering metal, becomes strong; when encountering fire, it attains its proper position; when encountering wood, it yields; when encountering earth, it declines and dies. Its path has both obstructions and smooth passages. The other four phases are also like this. By combining numbers with each other and using them for divination, good and bad fortune cannot be concealed, and calamity and blessing have a fixed destiny, without the need to consult others. Therefore, the function of numbers lies in conforming to the Dao and understanding the principle of generation. To understand numbers is to understand generation, and to follow numbers is to follow the Dao. The principles it points out are subtle and essential, yet people are unwilling to study them, thus failing to comprehend the purpose of the sages' teachings. This is truly lamentable.