Volume One: Notes on Collation

＃1『者終』，原作『終者』，據四庫本乙正。 ＃2『至元丙戌』，四庫本作『至元丙辰』，疑兩說皆誤。查《中國歷代紀年手冊》，元順帝至元年問，無『丙戌』和『丙辰』年，唯至正年問有『丙戌』年。 ＃3『切至』，四庫本作『精切』，疑當作『精切至』為宜。 ＃4『斯文宗』，四庫本作『斯文宗主』。 ＃5『但』，四庫本作『特』。 ＃6『祇』，原作『祇』，據四庫本改。

The first point: the two characters 「者终」originally read 「终者」, and the order has been corrected according to the Siku Quanshu edition. The second point: 「至元丙戌」in the Siku Quanshu edition reads 「至元丙辰」, and it is suspected that both versions may be incorrect—checking the Handbook of Chinese Dynastic Years, there was no year of Bingxu or Bingchen in the Zhiyuan period of Yuan Shundi, but only the Zhiyuan period had a year of Bingxu. The third point: 「切至」in the Siku Quanshu edition reads 「精切」, and it is suspected that 「精切至」would be more appropriate. The fourth point: 「斯文宗」in the Siku Quanshu edition reads 「斯文宗主」. The fifth point: the character 「但」in the original text is actually 「特」. The sixth point: the character 「祇」originally had a distorted form, and it has been corrected according to the Siku Quanshu edition.

天地之理，未始不有數行乎其中。然或有餘於數，不足於數。唯其餘、不足而為之中制，故雖陰陽奇耦之數，有分有合，有虛有實，有進有退，有自然互相生成之中道焉。

The principles of heaven and earth have numbers operating within them everywhere. However, numbers sometimes have excess and sometimes deficiency. Because of this excess and deficiency, it is necessary to establish a central regulation. Therefore, although the numbers of yin and yang, odd and even, have separation and union, emptiness and reality, advancement and retreat, there is naturally a middle path among them, a mutual generation that is harmonious.

一陰一陽之謂道者，陰陽分一以為天地，上下合一以為乾坤，天地上下分為四方，而中元以合，乾坤陰陽合為四時，而中有可分，是同一道也。

What is meant by "One yin and one yang is called the Way": When yin and yang separate the "one," it becomes heaven and earth; when up and down combine the "one," it becomes Qian and Kun. Heaven and earth divided into the four directions, the central position cannot be combined again; Qian and Kun combined with yin and yang form the four seasons, yet the central position can still be divided further. These two are actually the same principle.

天地以形言，則南北為經，上天下地；而水、火、木、金之為形者，既可以圖而分，故日法象莫大乎天地，廣大配天地也。陰陽以氣論，則東西為緯，乾陽坤陰；而冬、夏、春、秋之為氣者，亦可以理而合，故日變通莫大乎四時，變通配四時也。於斯二者，元以異也。

Heaven and earth refer to physical forms, so north and south are the meridians, with heaven above and earth below. Water, fire, wood, and metal, being tangible substances, can be arranged separately on a diagram. Therefore, the Great Treatise says, "The model of all things is none greater than heaven and earth," and also says, "Greatness and breadth correspond to heaven and earth." Yin and yang refer to qi, so east and west are the parallels, with Qian (Heaven) belonging to yang and Kun (Earth) to yin. Winter, summer, spring, and autumn, as climatic phenomena, can also be logically combined. Therefore, the Great Treatise says, "Change and adaptability are none greater than the four seasons," and also says, "Change and adaptability correspond to the four seasons." These two aspects, in reality, have no distinction.

四方有中，中元定體，而四象八卦以之立；四時有五，五元定著，而四時八節以之行。孔子著《大傳》，不存河圖，而特存天地之數者，示其有得於數，則圖在其中矣。

In the center of the four directions there is a central position, which has no fixed form, and the four symbolic figures and the eight trigrams depend on it to be established. In the midst of the four seasons there is a five, which has no fixed place, and the four seasons and eight festivals depend on it to function. Confucius wrote the Great Treatise, without preserving the diagram of the River Chart, but specifically preserved the numbers of heaven and earth, intending to show that if one understands the numbers, the River Chart is naturally contained within them.

圖之數生以天一者，成以地六；生以地二者，成以天七；生以天三者，成以地八；生以地四者，成以天九，至謂天五地十之生成，則何自而求哉？按圖索駿也。

The numbers of the River Chart are such that whatever is generated by the Heavenly Stem Yi is completed by the Earthly Branch Wei; whatever is generated by the Earthly Branch Yin is completed by the Heavenly Stem Geng; whatever is generated by the Heavenly Stem Jia is completed by the Earthly Branch You; whatever is generated by the Earthly Branch Mao is completed by the Heavenly Stem Ren. As for the Heavenly Stem Wu and the Earthly Branch Shi, how they are generated and completed, and where they come from, it is like searching for a thousand-mile horse according to a map, which is a futile effort.

故制五於八卦之虛中，猶可容其數；而制十於其中，則元所容其數矣。是知數之有生有成，而不知其所以生且成者，徒得其貌，而元以得古聖人之所以為天地立心，為生民立極者，千載可為之長太息，則古猶今也。

Therefore, placing the number five in the empty center of the Eight Trigrams can still accommodate it; but if we place the number ten in the center, there would be no room left for it. Those who only know that numbers have origin and formation, but do not understand why they can originate and form, merely grasp a superficial outline and cannot comprehend the intention of the ancient sages to 'establish the heart for heaven and earth, and set the standard for the people.' A thousand years later, it is truly worth sighing about—this was the case in ancient times, and it is still the case today.

夫奇數皆天，耦數皆地，生以天者成以地，生以地者成以天，其為天若地，特陰陽之進退、出入，奇耦之體用，分合而已。《說卦》曰：「參天兩地而倚數，觀變於陰陽而立卦，發揮於剛柔而生爻。」《大傳》日：「參伍以變，錯綜其數。通其變，遂成天地之文；極其數，遂定天下之象。非天下之至變，孰能與於此？」其曲盡乎至妙之用者，河圖法也。

Odd numbers belong to Heaven, and even numbers belong to Earth; those generated by Heaven numbers are completed by Earth numbers, and those generated by Earth numbers are completed by Heaven numbers. What is called Heaven and what is called Earth are merely the advance and retreat, going out and coming in of yin and yang, and the substance and function, division and combination of odd and even. The "Sayings of the Trigrams" says: "By taking three from Heaven and two from Earth, the numbers are supported. Observing changes in yin and yang, the trigrams are established. Through the flexibility of strong and weak, the lines are generated." The "Great Treatise" says: "By combining three and five, changes are made, and the numbers are interwoven and combined. By comprehending the changes, the text of Heaven and Earth is formed; by reaching the extreme of the numbers, the images of the world are determined. If it is not the most changeable thing in the world, who could reach this level?" The method of the River Chart is precisely what can fully and intricately express this profound function.

數始於一，而中於五，而終於十。中於五者，分其四之偏；終於十者，合其八之正。蓋一之始，其全體元餘不足，不可得而用，而獨得於中。既以其一寄生於五，遂以其一寄成於十以終之，故四象有五，八卦有十，皆有數而元所定其位，以五生其偏，以十成其正，而所謂一者，遂莫知所尋，獨見於五，見於十之為中者以上齊焉耳。

Numbers begin with one and end with ten, with five in the center. The reason five is in the center is to assign the four cardinal directions their respective positions. The reason ten is the end is to combine the four cardinal directions into their proper positions. Because one is at the beginning, the whole is neither excessive nor deficient, so it cannot be used, and can only occupy the central position. It relies on five to speak of "generation" and relies on ten to speak of "completion," thus concluding. Therefore, the four trigrams are associated with five, and the eight trigrams are associated with ten. These are numbers without fixed directions. Five is used to generate the secondary positions, and ten is used to complete the main positions. The so-called "one" can no longer be found, and people can only see it together with five and ten in the center.

故嘗求其所以然而莫之得，適記兒時於牧堅問見所謂八格戲者，其局不過口中加十之文而已。時極厭薄之，以為至鄙、至賤，未嘗加之意也。不謂年瑜七十，乃知其然。所謂百姓日用而不知，終身由之而不知其道者，古之人豈欺我哉？

I once pursued the underlying reasons but failed to grasp them. At that time, I suddenly recalled a game called "Eight Sections" that I had seen as a child among shepherds. Its board was simply a square with a cross in the middle. At that time, I thought it very lowly and despised it, never giving it a second thought. Unexpectedly, at the age of seventy, I finally understood that it was precisely about this principle. The "Great Treatise" says, "The common people use things daily without knowing their meaning." Mencius also said that one follows a path all his life without knowing its principles. Could our ancestors have deceived us?

蓋今謂即古之○，今之五即古文之，今之十猶古文之十，而畫為十於中，此十五之法由之以有所見矣。況又十下畫一，則土文之為中者，汰可知已，故云中土也。

In general, the circle mentioned today is the ancient character 「○」; the modern character 「五」 is the ancient 「乂」; the modern 「十」 is still the ancient 「十」, and by drawing a cross in the square, the principles of five and ten become visible. Moreover, adding one horizontal stroke below 「十」 forms the character 「土」, which clearly shows why earth is considered the center, hence the term 「中土」.

其中直筆之，即可見南北為經，天尊地卑，天地設位；橫筆之－，則可見東西為緯，分陰分陽，陰陽成列。從而久之則四維之偏，各得其均；四方之正，亦隨以定其氣之互進互退，交相生成，率由五與十以立中道也。

This cross shape, with its vertical stroke, shows the north-south direction as the meridian, with heaven above and earth below, each occupying its proper position. The horizontal stroke indicates the east-west direction as the latitude, dividing yin and yang, and arranging them in order. Adding the diagonal strokes forming the character "乂", the four cardinal directions' secondary positions are balanced. The four main directions are thus established. The movement of energy advancing and retreating, mutually generating, is all determined by the numbers five and ten to establish the middle path.

故河圖之十五，大衍之五十，皆虛有其十與五之數，以為之用，而實元其位之體也。欲知十、五之道，所以為中，始於一，而中於五，而終於十者，十即一也，以十求一，則其一於此固亦可以目存而心得於中矣。

Therefore, the fifteen of the He Tu and the fifty of the Da Yan are merely hypothetical numbers set up to function, actually having no specific directions or forms associated with them. To understand why the numbers ten and five can become the center, just observe that it begins with one, reaches the center at five, and ends at ten—ten is actually one. By going from ten back to seek one, this one can then be seen with the eyes and understood with the mind.

因參稽自擬《易》以來，其會十久之為中為土者，惟司馬氏《潛虛》似為得之，然《港虛》乃以久著為基，十著為冢，而分列為五十五數以為名，則是其中者已先實之矣，非虛用也。與河圖虛五寄十之理，自不相入也。河圖之數，雖五十有五，而其實體之數正只四十，所餘十有五，特虛以用於四十之中，不過通為之數，而實非有體位者也。

Therefore I examined thoroughly all the ancient works that imitated the Book of Changes. Only Sima Guang's "Potent Hidden Emptiness" seems to have grasped the meaning of "ten" and "yi" as representing the center and earth. However, "Potent Hidden Emptiness" refers to the form of "yi" as "ji" and the form of "ten" as "jia," and arranges them into fifty-five numbers with names, which is equivalent to fixing the center position beforehand, rather than using it in a virtual way. This does not align with the principle of the River Chart, which virtually sets five and entrusts ten. Although the River Chart's numbers are said to be fifty-five, its actual numerical count is only forty; the remaining fifteen are merely virtual numbers functioning within these forty, merely a general count, without actual form or direction.

其曰參、日兩、日參五者，通其數之所變以用也。參也者，一二之所以變也。由一自分其一以為二，起自為之對，則見其二而不見其一矣。其一又自參出於二之中，故三也。於參之法是即為一小成，是三畫之立見人參於天地之中以生也。積三小成，參其三而合於乾元用九，是參之始制，即陽奇之所謂天者所由然也。

The so-called "参" (participation), the so-called "两" (two), and the so-called "参五" (participation of five) are all ways of applying the changes of numbers. The "参" refers to the reason why one and two can change. One divides itself into another to form two, which arises to be opposite to itself, so people only see two and not one anymore; that one then emerges again from the two, thus becoming three. According to the principle of "参," this is a small achievement, which is the formation of the trigrams with three lines, showing how humans participate in the workings of heaven and earth. Accumulating three small achievements, multiplying three by three to align with the "用九" (all nines) in the Hexagram Qian, this is the initial establishment of "参," which is why the yang odd number is called "heaven."

兩也者，由一生二，起而對，並以立，既以象兩儀，因之兩其二以象四象。夫一本自一而二，而參，其一以為三，況二本一之對，豈不得二其三而反對之乎？故汰而三其二以成六，即合於坤元用六，是又兩之始制，兼三才而兩之，故六因之六畫成卦，六位成章，即陰耦之所謂地者亦由以然也。

The concept of "two" arises from one producing two, which then stand in opposition and are placed side by side. Since it is used to symbolize the two principles (yin and yang), it is then doubled again to symbolize the four images. Originally, one becomes two, and then another one is added to make three; and since two is the opposite of one, can it not be doubled again to form six? Therefore, further development leads to three twos combining to form six, which corresponds to the "use of six" in the hexagram of "Kun." This is again the original principle of "two": combining the three elements and doubling them, hence six. From this comes the six lines in a hexagram and the six positions in a chapter, which is the reason why the even number six, representing yin, is called "earth."

由是參其兩二，而交久以合之，是伍也。兩其二伍，則又什矣。十特見於參、九之外，則又一之所由以寄者也。何以明之？參之，兩之，而伍而什之，皆人文之為。而自然與天地合者，參文之血斗，兩文之雨仄，至維伍維什，皆立人於維中維旁者，斯可默識古人立心立極之制，有在於是矣。

By combining two dual elements and crossing them into the shape of 「乂」, they form the number five; two fives then form ten. The number ten appears only in the characters 「参」 and 「九」, which is precisely where the number one is寄托ed. How can we prove this? By combining and pairing, forming five and ten, these are all human-made textual forms, yet they naturally correspond with heaven and earth. The structure of the character 「参」, the form of 「两」, as well as the writing of 「五」 and 「十」, all place the character 「人」 in the center or beside it. From this, one can silently recognize that the principles and standards of the ancients' hearts and ultimate standards are embodied in these forms.

必參其三三，兩其三二，是參兩之一小成矣。故必又參其三三，而參之為參天；兩其三二，而兩之為兩地，是天地之數元汰而合且分矣。

It is necessary to have three instances of the triple combination and three instances of the double combination, which constitutes a small achievement of the principle of '参两'. Therefore, it is also necessary to have three instances of the triple combination to form '参天' (reaching the heavens), and three instances of the double combination to form '两地' (two lands). In this way, the numbers of heaven and earth do not need to be added separately, and harmony and distinction will naturally arise.

乃以其初不入於用之一數，寄於十以行其中，而為之分且合，以成其中制者，是一也。而求於圖之數，如環元端，其一莫見何在，而所見惟十爾。夫參其三三，與兩其三二，合而為大。參兩，斯有以見天地渾成一始終者所以然矣。至所謂參伍錯綜，則未究也。

Thus, the initial "one" that was not used in the process is entrusted to the ten, and it operates within it, dividing and combining it, thereby achieving the principle of moderation in the center. This is precisely due to the "one." However, when searching for numbers in the River Chart, the numbers form a circular ring without a beginning or end, and the "one" itself cannot be seen; only the ten is visible. Three threes interrelate, and three twos interrelate in pairs, and together they form the whole. With the concept of "three interrelations and two pairings," it is sufficient to understand why the heavens and earth are unified and continuous throughout. As for the so-called "interweaving of three and five," this has not yet been fully explained.

其謂之參伍錯綜者，五以前之數少於五，元所用伍；五以後之數多於五，故參以五。伍於其中，錯且綜之，而行其制中之法也。然《大傳》於參伍以變，錯綜其數，亦必謂通其變，然後能成天地之文；極其數，然後能定天下之象，則知其未易知也。

The reason it is called "San Wu Cuo Zhong" is that all numbers before five are smaller than five and cannot be combined with five; therefore, a five must be introduced. By combining five with them, then crossing and integrating them, we implement the method of maintaining balance and restraint in the center. However, the "Great Treatise" states "San Wu yi bian, cuo zong qi shu," meaning that one must first understand its changes and then achieve the brilliance of heaven and earth; one must first exhaust its numbers and then determine the forms of all things under heaven. This principle is clearly not easy to comprehend.

然既教以參五錯綜，則安得不研求其所以然哉。因參稽自擊《易》以來，已幾二千年矣，學者元探索及之者，至於龍圖續始亦元發明，深切異之。心港力索，極深研幾，適然省悟，元元它事也。《大傳》日：「天數五，地數五，五位相得而各有合。」即圖之數矣。

Since the sages have already taught us "San Wu Cuo Zhong" (threefold combination and intermingling), how could we not investigate its underlying principles? I have thoroughly examined the interpretations of the Book of Changes since Confucius composed the Great Treatise, nearly two thousand years ago, and yet no scholar has explored this aspect. Even Chen Tuan's transmission of the Longtu (Dragon Chart), which discusses the continuation and beginning, has not uncovered this point. I find this deeply surprising. Therefore, I have devoted myself wholeheartedly to the study, exhaustively investigating the subtle and profound aspects, and suddenly I have gained an insight. In fact, there is no other way. The Great Treatise states: "The heavenly numbers are five, the earthly numbers are five, the five positions interconnect and each has its corresponding combination." This refers precisely to the numbers of the River Chart.

天一起於坎，地六附於坤，是一六之合，綜之則七也。參以天五錯伍於中，並列而求之，其首尾各一也。故坎之一數為陽，立極於北方之始；而乾之六數為陰，附於西北之維也。天三出於震，地八附於艮，是三八之合，綜之則十有一也。參以天五錯伍於中，並列而求之，其首尾各三也。故震之三數為陽，立於束方之正；而艮之八數為陰，而附束北之維也。是二者，先陽後陰。

Heavenly Stem One starts from the Kan trigram, and Earthly Branch Six is attached to the Northwest dimension. This is the meeting of One and Six, combining to form Seven. Then Heavenly Stem Five is interwoven and placed in the middle. When examined in sequence, the beginning and end each receive One. Therefore, the number One of Kan is yang, establishing its center in the beginning of the North; the number Six of Qian is yin, attached to the Northwest dimension. Heavenly Stem Three appears from the Zhen trigram, and Earthly Branch Eight is attached to the Gen trigram. This is the meeting of Three and Eight, combining to form Eleven. Then Heavenly Stem Five is interwoven and placed in the middle. When examined in sequence, the beginning and end each receive Three. Therefore, the number Three of Zhen is yang, established in the central position of the East; the number Eight of Gen is yin, attached to the Northeast dimension. These two groups both follow the order of yang first and then yin.

地二分於西南，為坤；天七列於西，為兌，是二七之合，綜之則九也。參以天五錯伍於中，並列而求之，其首尾各二也。坤之二數為陰，先處於西南之維；而兌之七數為陽，始立於西方之正也。地四偏於巽，天九正於離，是四九之，綜之而十有三也。參以天五錯伍於中，並列而求之，其首尾各四也。巽之四數為陰，退守束南之維；離之九數為陽，正南面而立於極位也。是二者，先陰後陽。

The second earthly branch is located in the southwest, which corresponds to the trigram坤 (Kun); the seventh heavenly stem is positioned in the west, which corresponds to the trigram兑 (Dui). This is the meeting of two and seven, combining to form nine. Incorporating the fifth heavenly stem, which is interwoven in the center, when examined in sequence, the ends each receive two. The number two of坤 belongs to yin and is first situated in the southwest direction; the number seven of兑 belongs to yang and is established in the central west position. The fourth earthly branch is located in the southeast, corresponding to the trigram巽 (Xun); the ninth heavenly stem is positioned in the south, corresponding to the trigram离 (Li). This is the meeting of four and nine, combining to form thirteen. Incorporating the fifth heavenly stem, which is interwoven in the center, when examined in sequence, the ends each receive four. The number four of巽 belongs to yin and retreats to the southeast direction; the number nine of离 belongs to yang and stands directly facing south in the extreme position. Both groups follow the pattern of yin first and then yang.

由是知乾、兌、艮、離之所以六、七、八、九，即坎、坤、震、巽之一、一不三、四也。故陽得五而陰，陰得五而陽；耦得五而奇，奇得五而耦。乾之六，乃一五而六也；兌之七，乃二五而七也；艮之八，乃三五而八也；離之九，乃四五而九也。

From this we can see that the numbers six, seven, eight, and nine assigned to Qian, Dui, Gen, and Li are precisely the numbers one, two, three, and four assigned to Kan, Kun, Zhen, and Xun. Therefore, when the yang numbers reach five, they turn into yin; when the yin numbers reach five, they turn into yang. When even numbers reach five, they turn into odd; when odd numbers reach five, they turn into even. The number six of Qian is formed by adding one and five; the number seven of Dui is formed by adding two and five; the number eight of Gen is formed by adding three and five; the number nine of Li is formed by adding four and five.

又由是知參伍以變，參乾、坎，參艮、震，自北而束；參兌、坤，參離、巽，自西而南。參而分，伍而合，是謂參伍以變。錯而分之，見其參；綜而合之，見其位，是謂錯綜其數。而錯綜者，即參伍也。散而錯之，如一、二、三、四之重；分斂而綜之，乃六、七、八、九之復合也，是謂五位相得而各有合，豈嘗如後世別置五虛點於圖心，而實謂天五哉？

From this we can understand what is meant by "参伍以变": combining Qian with Kan, and Gen with Zhen, moves from north to east; combining Dui with Kun, and Li with Xun, moves from west to south. When things are combined, they separate; when they are joined together, they unite. This is called "参伍以变." Interweaving and separating allows us to see "参"; integrating and gathering allows us to see the respective positions, which is called "错综其数." So-called "错综" is actually "参伍." When spread out and interwoven, they resemble one, two, three, and four overlapping; when gathered together and integrated, they form the composite numbers six, seven, eight, and nine. This is called "五位相得而各有合." How is this different from later practices that place five empty points in the center of the diagram and call it "天五" in a very concrete way?

故由是知河圖之數縱橫順逆，莫匪自然而然，不待人為牽合傅會以使然也。其進數之升，自北束極乎南；退數之降，自南西終極乎北，皆陽數行而陰自隨之。若陰數之進，則自坤二而巽四，退則自艮八而乾六，然陰不過奉陽而已，雖萬變亦一陰一陽而已爾。

Therefore, from this we can see that the numbers of the River Chart, whether arranged horizontally or vertically, in order or in reverse, nowhere are they not naturally formed, without the need for artificial connections or forced interpretations. Its advancing numbers rise from the north, passing through the east, until reaching the southernmost point; its retreating numbers descend from the south, passing through the west, and finally return to the northernmost point. These are all the movements of the Yang numbers, with the Yin numbers naturally following. As for the movement of the Yin numbers, they advance from the second position of the Earthly Branch (Kun) to the fourth position of the Wind Trigram (Xun); and when retreating, they go from the eighth position of the Mountain Trigram (Gen) to the sixth position of the Heaven Trigram (Qian). However, Yin is merely following Yang. Even with myriad changes, it remains simply the interplay of one Yin and one Yang.

是故伍坎一以為乾六於北，伍坤二以為兌七於西，伍震三以為艮八於束，伍巽四以為離九於南，則見制數之元自參伍矣。於是又損震後之一以退，益艮之全八而九，三而三之，始參之制也。析震中前之二以合巽之全，四分而六，則兩之繼始也。惟離之全數自得於九，且當始終之中，三而三之，中參之制也。兼坤之全二，以貫兌中前之四合而六，則兩之代終也。絕兌後之三，以補乾之全六，三而三之，終參之制也。

Therefore, by combining the number one of Kan with five, it becomes the number six of Qian in the Northwest; by combining the number two of Kun with five, it becomes the number seven of Dui in the West; by combining the number three of Zhen with five, it becomes the number eight of Gen in the Northeast; and by combining the number four of Xun with five, it becomes the number nine of Li in the South. Thus, it can be seen that the numbers were originally derived from 'San Wu'. Then, by subtracting one from Zhen and adding it to the full eight of Gen, it becomes nine. Three multiplied by three, this is the beginning of 'San'. By separating the front two from Zhen and combining them with the full number of Xun, it forms six, which is the beginning of 'Liang'. Only the full number of Li naturally becomes nine, and it is exactly in the middle of the beginning and end. Three multiplied by three, this is the middle section of 'San'. By combining the full two of Kun with the front four of Dui, it forms six, which is the ending of 'Liang'. By cutting off the three at the end of Dui and adding it to the full six of Qian, three multiplied by three, this is the final section of 'San'.

是則參天兩地之綱領，於此亦可見。或錯而分，綜而合，參之兩之，參兩之制，亦必伍也。是則參伍之大通變、大極數者也。孔子之謂通其變，成天地之文；極其數，定天下之象。非天下之至精之至變之至神，孰能與於此。誠哉言也。

Thus, the principle of "three for heaven, two for earth" is clearly seen here. Either they are alternately divided or combined, either they are interrelated or paired, but the establishment of principles involving both interrelation and pairing must necessarily involve the number five. This is the greatest variation and the greatest extremity of "three and five." Confucius said, "To understand its variations and form the patterns of heaven and earth; to reach its utmost numbers and determine the images of the world." If this is not the most subtle, most changeable, and most mysterious aspect of the world, who could reach such a level? This statement is indeed accurate.

故夫參天者成象謂乾，於是參陽數之三三，以統乾元用九，而參中有兩，陽中見陰；兩地者效法謂坤，於是兩陰數之二二，效法於乾，遂參其二而六之，以會坤元用六，而兩中用參，陰中見陽，以是見一陰一陽，變而通之於天地之問，所以必參兩也。參之，兩之，所以必九六也。

Therefore, "参天" refers to the statement in the Great Treatise that "forming the image is called Qian." Thus, the yang number three is tripled to encompass the "use of all nines" in the Qian hexagram. The term "参" includes two, which signifies the presence of yin within yang. "两地" refers to "emulating as called Kun," so the yin number two is doubled to emulate Qian, and then the two are tripled to form six, which is used to combine with the "use of all sixes" in the Kun hexagram. The term "两" here requires "参," indicating the presence of yang within yin. Thus, the interplay of one yin and one yang pervades the heavens and the earth. Therefore, it is necessary to use "参两." And the combination of one "参" and one "两" inevitably results in nine and six.

且太極而兩儀，所以四象而八卦，不過生數之一、二、三、四，成數之六、七、八、九而已。至於五與十，雖有其數，特存虛用，未嘗列於卦象之實體者也。蓋自一分一而二，一二而三，一三而四，是四象立矣。四象元五也，天五特虛中而元體，然一四而五，二三亦五，是兩伍矣。具以本數之天五，又參伍矣。況一、二、三、四自然總而十數，中以天五，參伍而十五也。

From the Taiji to the two yin-yang, from the four images to the eight trigrams, it is merely the generating numbers one, two, three, and four, and the formed numbers six, seven, eight, and nine. As for the numbers five and ten, although they are also present, they are merely hypothetical and not included in the actual forms of the trigrams. Because one splits into one to form two, one added to two forms three, one added to three forms four, and thus the four images are established. The four images originally do not include five; heaven five is only a virtual position in the center without a physical form. However, one added to four makes five, and two added to three also makes five, which are the two fives. When combined with the original number of heaven five, it becomes "San Wu" (three and five). Moreover, the numbers one, two, three, and four naturally form the number ten, and when the heaven five is introduced in between, the combination of San Wu results in fifteen.

天數之一、三、五則九，地數之一不四則六，而一五亦六也。一六而七，二五亦七也。一七而八，三五亦八也。一八而九，四五亦九也。則八卦成列矣。一九則十，一、二、三、四自然亦十也。八卦則元自而十也，十故元所容其體，以一用其一不三、四，則其十亦特九也。十之為九，乃所以復歸於一，是十即一也。

In the heavenly numbers, one, three, and five together make nine, while in the earthly numbers, two and four together make six. One plus five is also six. One plus six is seven, and two plus five is also seven; one plus seven is eight, and three plus five is also eight; one plus eight is nine, and four plus five is also nine. Thus the eight trigrams are arranged in order. One plus nine is ten, and one, two, three, and four naturally add up to ten. However, the eight trigrams themselves cannot add up to ten, so there is no place for ten to take form. Using one to operate on two, three, and four, ten is only equivalent to nine. The reason ten becomes nine is that it returns to one, showing that ten is actually one.

河圖之數，四方各十，故坎一與離九相對而十，中虛天五之用而十五；坤二與艮八相對而十，中虛天五之用而十五；震三與兌七相對而十，中虛天五之用而十五；巽四與乾六相對而十，中虛天五之用而十五，以是知天五與地十皆元其體與位之正，特虛用其數，以相久十而什伍也。斯所以參伍也，參伍所以參兩也，故河圖之數正實四十，而以天五地十之虛數通用之以計，凡天地之數五十有五者也。

The numbers of the River Chart are distributed such that each of the four directions has ten: the number one of Kan and the number nine of Li face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen; the number two of Kun and the number eight of Gen face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen; the number three of Zhen and the number seven of Dui face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen; the number four of Xun and the number six of Qian face each other to make ten, and in the middle, the number five of Heaven is included, making fifteen. From this we can see that the number five of Heaven and the number ten of Earth do not have their own forms or fixed positions; they are merely set as numbers for use. They rely on the '乂' shape and the '十' shape crossing each other to form five and ten together. This is called '参伍'; only with '参伍' can there be '参两'. Therefore, the actual numbers of the River Chart amount to forty. When the two imaginary numbers of Heaven five and Earth ten are added in, the total becomes fifty-five, the numbers of Heaven and Earth.

七為少陽，八為少陰，其合固成十五，然其生數不備，不能變。九為老陽，六為老陰，其合亦雖只十五，而生數之一、三、五而九，二、四而六，故六之變則九，九之變則六，此通重《易》之爻不見七八，而元非九六，而參伍以變，參天兩地，莫非是也。

Seven is the young yang, and eight is the young yin. Although their sum is also fifteen, their generating numbers are incomplete, so they cannot change. Nine is the old yang, and six is the old yin. Although their sum is also fifteen, nine is formed by the generating numbers one, three, and five, while six is formed by the generating numbers two and four. Therefore, six changes into nine, and nine changes into six. This is why, when examining the line statements of all sixty-four hexagrams, we never see seven and eight, but only nine and six. Also, the principles of 'three combinations to produce change' and 'three heavens and two earths' are all based on this reasoning.

四方之數各十，合參其九而二十有七，兩其六而十有二，總之則三十有九，而特遺其一，則其一將何所用乎？故以其一存諸十，則四十矣。故坎之立極於一，斂其十以自歸，以建大中，以生生數之始，以成成數之終。退藏於密，而莫之究極，見於參天兩地倚數之一以為十，則所謂中也。中也者，合數之所不得其中者，而使之交合於中，所以中其不中者也。

The numbers of the four directions each amount to ten. Multiplying nine by three gives twenty-seven, and doubling six gives twelve. Adding them together gives thirty-nine, yet there remains an extra one unaccounted for. This one is placed in the ten, making forty. Therefore, the trigram Kan is established with one as its central point, gathering the ten and bringing it back to itself, thus establishing the great central way, the beginning of the generating numbers and the end of the completing numbers. It retreats into secrecy, beyond the reach of human inquiry. It appears only when "the heavens are counted by three and the earth by two," using one to represent ten. This is called the "center." The center refers to gathering together those numbers that cannot naturally reach the central position, bringing them together at the center, so that even those not in the center can attain the center.

由坎北之一始制於中，二效之而分兌西之七以為中，三效之而分震束之三以為中，四效之而分離南之九以為中，凡中皆立於十，皆以數合於十而中分之者，蓋以寄其一云爾。惟坎之一，獨元所可用，特立於中而以元用之用而用，有用之用則中之用，不既大矣哉。故坎之一，雖不在參兩參五之用，而用之以制參天兩地參伍之制焉。故參天不自坎而自艮始也。

The first one from the northern Kan position first established the central position. The second followed it and took seven from the western Dui to form the center. The third followed it and took three from the eastern Zhen to form the center. The fourth followed it and took nine from the southern Li to form the center. All centers are based on ten, using numbers to make up ten and then dividing it evenly in the middle, in order to embody the principle of one. Only the one from Kan has no specific number to use, standing alone in the center, functioning through the use of something that appears to be of no use. This kind of useful function is precisely the function of the center, which is indeed great. Therefore, although the one from Kan is not included in the applications of the two and five, it is precisely through it that the principles of "three heavens and two earths" and "three and five" are established. Hence, "three heavens" do not start from Kan, but from Gen.

參亦非它，即前所謂一二之變也。一自分其一以為一，起而相對，則見其二，而不見一；其一又自參出於二之中，故曰參也。此參制之一小成，然參兩參伍，莫不於是焉起，斯故示其參之凡也。

The concept of "参" is not anything else but the change between one and two as previously mentioned. One splits off a part of itself to form another one, resulting in two opposing entities, so people only see two and not one. Then that original one re-emerges from the two, which is called "参." This is a small manifestation of the principle of "参," yet all instances of "参" and "参伍" originate from this. Therefore, here we first briefly explain the general idea of "参."

#1『也』，原脫，據四庫本補。 #2『謂』，四庫本作『之』。 #3『二其三』，四庫本作『三其二』。 #4『所』，據其上文義例補。 #5『莫匪』，四庫本作『莫非』。 #6『大極』，四庫本作『太極』。

The first point: the character 「也」 was originally missing, and has been added according to the Sanzhuo edition. The second point: the character 「谓」 in the Sanzhuo edition is 「之」. The third point: the phrase 「二其三」 in the Sanzhuo edition is 「三其二」. The fourth point: the character 「所」 has been added according to the meaning and style of the preceding text. The fifth point: the phrase 「莫匪」 in the Sanzhuo edition is 「莫非」. The sixth point: the term 「大极」 in the Sanzhuo edition is 「太极」.

道家以日月為易，始蹈常襲故，未始謂然，卒究於《大傳》則未始不然。日：「乾、坤，其《易》之蘊邪？乾、坤成列，而《易》立乎其中矣。乾、坤毀，則元以見《易》。《易》不可見，則乾、坤或幾乎息矣。」是猶未識所見之《易》為何物也。

Daoist scholars explain the character "Yi" by combining the sun and the moon. At first, I thought this was just following old interpretations and sticking to conventional views, and I did not agree with it. But after examining the Great Treatise, I realized that it might indeed be the case. The Great Treatise says: "The trigrams Qian and Kun are probably the essence of Yi? When Qian and Kun are arranged in order, Yi is established within them. If Qian and Kun were destroyed, we would not be able to see Yi; if we could not see Yi, then Qian and Kun would almost cease to exist." On their own, these words do not yet reveal what exactly the Yi is.

則又日：「乾、坤，其《易》之門邪？乾，陽物也；坤，陰物也。陰陽合德，而剛柔有體，□以體天地之撰，以通神明之德。」是《易》之為物有體可以見矣。則又日：「闢戶謂之乾，闔戶謂之坤，一闔一闢謂之變，往來不窮謂之通。見乃謂之象，形乃謂之器，制而用之謂之法，利用出入、民咸用之謂之神。」是《易》之謂體者，又見其有用矣。

The Great Treatise also says: "Qian and Kun are perhaps the gateways of the Book of Changes? Qian is the yang substance, and Kun is the yin substance. Yin and yang combine their virtues, and the firm and the yielding each have their forms, used to manifest the workings of heaven and earth, and to connect with the virtues of the divine." This shows that the Book of Changes, as a thing, has a visible form. The Great Treatise also says: "Opening the door is called Qian, closing the door is called Kun. One closing and one opening is called change. Coming and going without end is called communication. Manifesting is called image, taking form is called object, making and using it is called method, entering and exiting, taking and using it, with the people all using it is called spirit." This shows that the body of the Book of Changes also has function.

故曰：「天地設位，而《易》行乎其中矣。」日：「法象莫大乎天地，變通莫大乎四時，縣象著明莫大乎日月。」曰：「廣大配天地，變通配四時，陰陽之義配日月。」凡是數端，莫不舉日月以徵，則信其為易也完矣。

Therefore, the Great Treatise says: "Heaven and Earth each take their proper position, and the Book of Changes operates within them." It also says: "There is no greater model for taking inspiration than Heaven and Earth, no greater transformation and penetration than the four seasons, and no greater display of suspended images than the sun and moon." It also says: "Vastness and breadth match Heaven and Earth, transformation and adaptability match the four seasons, and the meaning of yin and yang matches the sun and moon." In all these instances, the sun and moon are cited as evidence. Therefore, the idea that the Book of Changes is associated with the sun and moon can be fully trusted.

夫《易》與天地準，能彌綸天地之道，仰觀象於天，俯觀法於地，則成象謂乾者日也，效法謂坤者月也。日象離數，極乎九；月象坎數，始乎一。凡九必對六，今雲一者，一合天五為六，坎一乾六為水生成，所以為月者也。

The Book of Changes corresponds with heaven and earth, encompassing the principles of the cosmos. Looking up to observe celestial phenomena and looking down to study earthly patterns, the phrase "the image is called Qian" refers to the sun, and "the pattern is called Kun" refers to the moon. The sun takes its image from Li (Fire), and its number reaches its extreme at nine; the moon takes its image from Kan (Water), and its number begins at one. By reasoning, nine should necessarily correspond with six. However, here the moon is said to be one because one added to the heavenly five makes six. The number one of Kan and the number six of Qian represent the generative number and the completed number of water, so they are used to refer to the moon.

且天地乃乾坤之體，乾坤乃天地之用，惟天地定位，而上下氣形有不易之分，故乾坤成列，而陰陽體用有變易之數。天地有常體，而莫見其用；陰陽有常用，而莫究其體，惟日月之體用兼著，故特著其義以配陰陽云爾。

Moreover, heaven and earth are the forms of Qian and Kun, while Qian and Kun are the functions of heaven and earth. Because heaven and earth each occupy their fixed positions, there is an unchangeable distinction between the upper and lower aspects in terms of qi and form. Therefore, Qian and Kun are arranged in order, and the nature and function of yin and yang thus have the capacity for change. Heaven and earth have constant forms, yet their functions are not visible; yin and yang have constant functions, yet their forms cannot be discerned. Only the sun and moon have both forms and functions clearly evident, so the Great Treatise especially highlights their significance to correspond with yin and yang.

且明教以範圍天地而不過，曲成萬物而不遺者，特通乎晝夜之道而知也。夫晝夜之道，豈難知哉？百姓日甩而不知爾。請略舉其凡：乾，陽物者，日其類也；坤，陰物者，月其類也。陰陽成列而合德者，四時合序，日月合明，亦類也。

Moreover, the Great Treatise clearly tells us: therefore it can "encompass heaven and earth without excess, and skillfully complete all things without omission," simply because it understands the principle of day and night. Is the principle of day and night difficult to comprehend? It is nothing more than something that the common people use every day without realizing it. Let me briefly illustrate its general meaning: Qian is a yang element, and the sun is its counterpart; Kun is a yin element, and the moon is its counterpart. When yin and yang are arranged in order and harmonized in their virtues, it is like the four seasons following one another in sequence, and the sun and moon combining their brightness, all belonging to the same category.

闢戶謂乾，近則日之出旦，月之上弦；遠則春之日夜分而謂啟也。闔戶謂坤，近則日之入夕，月之下弦；遠則秋之日夜分而謂閉也。仰觀其成象，則日之晝，月之望，夏之景長；而至俯察其效法，則夜之分，月之朔，冬之景短而至也。

"Opening the door is called Qian," near at hand it refers to the sun rising in the east and the moon waxing; far away it refers to the spring equinox when day and night are equal, which the ancients called "QI." "Closing the door is called Kun," near at hand it refers to the sun setting in the west and the moon waning; far away it refers to the autumn equinox when day and night are equal, which the ancients called "BI." Looking up to observe its "manifestation" aspect, it is day, full moon, and summer solstice when the shadow is longest; looking down to examine its "emulation" aspect, it is midnight, new moon, and winter solstice when the shadow is shortest.

是以日往則月來，月往則日來，日月相推而明生焉。寒往則暑來，暑往則寒來，寒暑相推而成歲焉。變化者，進退之象也；剛柔者，晝夜之象也。日月運行，一寒一暑故日：「天地之道，貞觀者也；日月之道，貞明者也；天下之動，貞夫一者也。」

Therefore, when the sun has passed, the moon comes, and when the moon has passed, the sun comes; the sun and moon alternate and thus light is generated. When cold has passed, heat comes, and when heat has passed, cold comes; cold and heat alternate and thus a year is formed. What is called change is the image of advance and retreat; what is called hardness and softness is the image of day and night. The sun and moon move, alternating between cold and heat, so the Great Treatise says: "The way of heaven and earth is to manifest through correctness; the way of the sun and moon is to shine through correctness; all movements in the world return to the one through correctness."

貞夫一也者，四時分至之中皆一也。而此特謂天下之動貞夫一，則正由坎之一也。一日之陽氣初動於夜分之際，固謂貞夫一也。一年之陽，氣復動於冬朔之始，尤見其貞夫一者也。

The phrase "Zhen Fu Yi" refers to the fact that the two solstices and two equinoxes within the four seasons all return to the principle of one. Here, however, it specifically says "the movement of the world is Zhen Fu Yi," which is referring to the "one" in the trigram Kan. Within a day, the yang energy first begins to emerge at the midpoint of the night, which is naturally called "Zhen Fu Yi." Within a year, the yang energy re-emerges at the beginning of the winter solstice and the first day of the month, which more clearly demonstrates its "Zhen Fu Yi."

坎之數一，離之數九，坎、離以一、九為乾。離為日，坎為月，日月必合於月之初一為朔。而日不云朔，月乃云朔；南不云朔，北乃云朔。夫朔逆度也，逆與日會，故謂之朔。以朔文觀之，明著月之逆度者也。《說卦》日：「《易》，逆數也。」《書》亦日：「平在朔。」《易》理實同也。

The number of Kan is one, and the number of Li is nine. Kan and Li combine the numbers one and nine to form Qian. Li represents the sun, and Kan represents the moon. The sun and moon must meet on the first day of each month, which is called "Shuo." The sun does not have a "Shuo," only the moon does. The south does not have a "Shuo," only the north does. The term "Shuo" refers to the retrograde movement of the moon, which moves backward to meet the sun, hence it is called "Shuo." From the structure of the character "Shuo," it clearly shows the retrograde movement of the moon. The "Say卦" says, "The Book of Changes is about reverse calculation." The "Shangshu · Yao Dian" also says, "Peace is in the month of Shuo," which refers to the same principle.

由是以進乎日月之為易，其縣象得不謂著明矣乎？故月之三五而盈，少陰、陽之七、八；三五而闕，老陰陽之九、六。總之，以地數三十而月盡，又河圖法也。

Thus, looking further, the sun and moon are the "Book of Changes" (Yi). They hang high and display clear signs, are they not the most evident? Therefore, the moon, after three periods of five days, becomes full, corresponding to the seven of lesser yin and the eight of lesser yang; then, after another three periods of five days, it wanes, corresponding to the nine of old yin and the six of old yang. In summary, using the earth number thirty, a month is completed, which is also the method of the River Chart (He Tu).

周天三百六十五度四分度之一，日每行一度，冬南、夏北為寒暑之中極，月行每十有三度而奇，與日之合者十有二朔，而歲常以備六陰六陽十有二月，天地節而四時成，是月之運行勞亦甚矣。其稱坎為勞卦者，不徒謂水也，此坎之一數不入於參兩、參伍之制，而特著其動之始也，中之始也。

The celestial sphere comprises 365 and 1/4 degrees. The sun moves one degree each day, and at the winter solstice it is偏 south, while at the summer solstice it is偏 north, marking the two central points of cold and heat. The moon moves thirteen degrees more than a day, and in a year it meets the sun twelve times, forming twelve new moons, thus making the year always have six yin and six yang, and twelve months. Heaven and earth have order and measure, giving rise to the four seasons. Thus, the moon's movement is indeed laborious. The "Say Divination" text refers to Kan as the "labor" trigram, not merely because it belongs to water. This is because the number one in Kan does not participate in the establishment of "three and two" or "three and five," but is specifically used to indicate the beginning of movement, that is, the beginning of the center.

合而言之，天地者體也，陰陽者用也。體者形，所以用其氣；用者氣，所以體其形。天以氣體其形，故其形下包乎地；地以形用其氣，故其氣上行乎天。是天之氣，全體乎地之形；而地之形，全用乎天之氣，此又參兩之制得以明之也。

In summary, heaven and earth are the "body," and yin and yang are the "function." The body refers to form, which is used to circulate its qi; function refers to qi, which is used to fill its form. Heaven uses its qi to fill its form, so its form envelops the earth downward; the earth uses its form to circulate its qi, so its qi ascends to heaven. Thus, the qi of heaven is entirely based on the form of the earth as its body, and the form of the earth is entirely based on the qi of heaven as its function—this can also be explained by the principle of "two principles."

一之始參，三之終參，皆天也，而並位乎平地之下。如：乾者，老陽也，乃附於成數之六陰，位乎陰地西北，偏極乎退數之終。艮者，少陽也，乃附於成數之八陰，位乎陽地東北，偏立乎進數之始。而特以坎中之一陽，不在參兩之用者，因其始生自然之一，介然特立於陰陽終始之際，起歲之朔，月之朔，以體生生之謂易，斯足以明地之體全用乎天矣。

The "San" (trinity) of the beginning and the "San" of the end both belong to Heaven, yet they are all situated below the horizon. For example, Qian is old Yang, yet it is attached to the even number six, which is a yin number, and it is located in the northwest, which belongs to yin, at the end of the retreating numbers. Gen is young Yang, yet it is attached to the even number eight, which is a yin number, and it is located in the northeast, which belongs to yang, at the beginning of the advancing numbers. However, the single Yang in Kan is not included in the use of San and San, because it is the original one that naturally arises, standing alone and distinct at the boundary where yin and yang begin and end, initiating the beginning of the year and the beginning of the month, thus embodying the idea that "the continuous generation is called the Book of Changes." This sufficiently demonstrates that the form of the earth entirely uses the qi of Heaven.

一始之兩，二終之兩，皆地也，而兼列於平地之上。如：巽，少陰也，雖以附地生數之四陰，位乃在東南，偏陽氣升已過中之進數。坤，老陰也，雖以附地生數之二陰，位猶在西南，偏陽氣降未及中之退數。且兼兩地並以奉中，參天位之極，當天之極高，日之極中，斯足以明天之用，全體乎地矣。

The "two" of the starting point of a structure and the "two" of the ending point both belong to the earth, yet they are both arranged above the horizon. For example, Xun is the Lesser Yin, although it is attached to the earth's generating number four, which is a yin number, its position is in the Southeast, leaning toward the side of the advancing number where the yang energy has risen past the midpoint. Kun is the Old Yin, although it is attached to the earth's generating number two, which is also a yin number, its position remains in the Southwest, leaning toward the side of the retreating number where the yang energy has not yet reached the midpoint. Moreover, they both take the numbers of the earth and jointly honor the center, assisting and complementing the extreme position of heaven, precisely at the highest point of heaven and the most central point of the sun. This sufficiently explains that the function of heaven is entirely based on the earth as its form.

以是知參兩之制，非有日以定其晝夜長短，有月以更其朔望贏縮，則天地之氣形體用，陰陽之變化生成，其升降進退之變通分合，出入之歸宿，則惡風而討其真是哉。

From this we can see that the system of "San Liang" depends on the sun to determine the length of day and night, and on the moon to mark the waxing and waning of the moon. Without these, how could we investigate the true nature of the qi and form, the function and application, the changes and generation of yin and yang, as well as the transformations of rising and falling, advancing and retreating, the combination and separation, and the return and movement of all things?

故嘗由是以參之制，數之元矣。天一之始，自北而中，而分地二於西南，不得謂天地定位者，陽正而下，陰偏而上也。二得名坤，一不得名乾者，一起於坎也。由一參二以為三，以前而東，蓋所以見人參於天地之中以生，而動於震，因兩其二而四，附以順而相入之巽，虛生其象以為四。然象非實體也，象天地與人，極男女之相配以始者爾。

Therefore, I have used the method of "参" to trace back the numbers. Heaven begins with one and starts from the north, moving to the central position, while Earth two is placed in the southwest. This still cannot be called "Heaven and Earth taking their positions," because Yang is in the proper position but below, and Yin is in the secondary position but above. The number two can be called Kun, but one cannot be called Qian, because this one originates from Kan. By combining one with two, we arrive at three, moving forward to the east, which shows that human beings are born in the center of Heaven and Earth, thus initiating movement at the position of Zhen. Then doubling two to make four, and adding the following and entering one of Xun, setting up its image virtually to form four. However, the image is not the actual thing; it is merely taking the form of Heaven and Earth and human beings, exhausting the meaning of the union of male and female as the beginning.

□一與三之陽，自然已各據其中，而二與四之未中，未有以正之可遂已乎？故有待取用於成數之交合，以成其正也。五雖虛位於中，而有其數可制其用。一交五，而以其一相配而成六於乾；二交五，而以其二相配而成七於兌；三交五，而以其三相配而成八於艮；四交五，而以其四相配而成九於離，然後始謂之成象。成象則謂之乾，而效法則謂之坤矣，乃得謂之天地定位也。

One and three, these two yang numbers, have already each occupied their central positions; but two and four have not yet reached the center. Is there no way to correct their positions, or must they remain this way? Therefore, it is necessary to use the combination of the completed numbers to establish their proper positions. Although five is merely a virtual occupant of the center, it has a number that can be used to set a standard: when one combines one with five, it pairs with one to form six in Qian; when two combines with five, it pairs with two to form seven in Dui; when three combines with five, it pairs with three to form eight in Gen; and when four combines with five, it pairs with four to form nine in Li. This is called "forming the image." With "forming the image is called Qian" and "emulating the image is called Kun," only then can it be said that heaven and earth are positioned.

其實一自合一，二自合二，三自合三，四自合四，特以交五而變爾。於是坎一之陽合而成乾六之陰，坤二之陰合而成兌七之陽，震三之陽合而成艮八之陰，巽四之陰合而成離九之陽，其始正者合而輔於偏，始偏者合而歸之正，於是八卦成列，而象在其中矣。亦惟一之自中，坎獨自若而不入於數，而河圖參天兩地，參伍錯綜之變，得以行乎其中也。

In fact, one corresponds with one, two with two, three with three, and four with four. It is only because they interact with five that changes occur. Thus, the yang of Kan one combines with the yin of Qian six, the yin of Kun two combines with the yang of Dui seven, the yang of Zhen three combines with the yin of Gen eight, and the yin of Xun four combines with the yang of Li nine. Those originally in the central position combine to assist the偏 position; those originally in the偏 position return to the central position. Thus, the Eight Trigrams are arranged in order, and the forms of things are contained within them. It is precisely because the number one remains in the central position that Kan remains unchanged and does not fall into the sequence of numbers, allowing the transformation of the River Chart "参天两地" (three heavens, two earths) and "参伍错综" (three and five interwoven and complex) to operate within it.

故由參天之制皆用九，而知九為《乾》之用，由兩地之制皆用六，而知六為《坤》之用。蓋乾之始參於艮八，而不足於數，則取震之一而九之中參，當陽極而體數之全；元餘不足，則自九之終參，則取兌過中有餘之三，以奉乾六之終。而九之兩之始起數於震中之二，以合巽四之不足。而六之兩之終起數於坤之本二，以合兌中有餘之四而六之，是明參兩九六所用之盤也。

Therefore, from the method of "ascending to heaven" which uses all nines, we know that nine is used for the Trigram Qian. From the method of "two places" which uses all sixes, we know that six is used for the Trigram Kun. Generally, Qian starts from the eight of Gen and ascends, but the number is not yet complete, so it takes one from Zhen to make up nine. The middle one, when combined, is exactly at the peak of yang and its number is complete, neither excessive nor lacking. The final one takes three from Dui, which exceeds the center, to complement the end of Qian's six. As for "two," the initial two starts from the two in Zhen to complement the lack of Xun four; the final two starts from the two inherent in Kun to combine with the excess four in Dui to form six. This explains clearly the calculations used in the参两 and 九六.

或有謂坤二與兌七而九，乾位自得六數亦自成九、六也，曾不識兩之制皆終地上，參之制皆終地下，又焉得而易之哉？況乾本九也，今數乃損而六；坤本六也，今數乃增之而九者，尤足以明老陰之六變而之九，老陽之九變而之六，而卦爻之所以成九、六也。

Perhaps someone might say: "The second line of the Earth (Kun) plus the seventh line of the Lake (Dui) equals nine, and the Heaven (Qian) position originally has the number six. Thus, it naturally forms nine and six." This view completely fails to understand that the principles of "two" are completed above the horizon, while the principles of "three" are completed below the horizon. How can they be arbitrarily exchanged? Moreover, Heaven (Qian) originally is nine, but now its number is reduced to six; Earth (Kun) originally is six, but now its number is increased to nine. This further demonstrates that old yin (six) changes to nine, and old yang (nine) changes to six. The reason why the lines of the hexagram are only called nine and six is precisely because of this.

由參天之用必九，兩地之體必六，《下繫》之首日：「八卦成列，象在其中矣。因而重之，爻在其中矣。」而後世乃謂《易》畫之重，非由羲始，吾不信也。

Because the function of "shantian" (reaching the heavens) must necessarily be nine, and the nature of "lidi" (two lands) must necessarily be six, therefore the beginning of the "Great Treatise" (Xici) states: "When the eight trigrams are arranged in order, the images are already contained within them. Then by stacking them again, the lines (yao) are already contained within them." However, later people have even claimed that the stacking of trigrams in the Book of Changes was not started by Fuxi, but I do not believe this.

且《說卦》謂說言乎兌，成言乎艮。凡再言其山澤通氣者，即《咸》卦也。《咸》之《象》日：「二氣感應以相與。止而說。天地感而萬物化生，聖人感人心而天下和平：觀其所感，而天地萬物之情可見矣。」

Moreover, the "Shuo Gua" says, "It speaks of Dui, and completes with Gen," and mentions "mountains and marshes communicate their energies" twice, which refers to the "Xian" hexagram. The "彖传" of the "Xian" hexagram states: "Yin and yang energies mutually respond and interact; Gen represents stillness while Dui represents joy. Heaven and Earth interact and thus all things are generated; the sage moves people's hearts and brings peace to the world. By observing where the response occurs, the conditions of heaven, earth, and all things can be understood."