The Earlier Heaven Diagram Explanation · Zhaozi's Sayings and Li Guangdi's Annotations

乾之策，二百一十有六。坤之策，百四十有四，凡三百有六十，當期之日。二篇之策，萬有一千五百二十，當萬物之數也。

The Great Treatise says: The six lines of the Qian hexagram total 216 measures, and the six lines of the Kun hexagram total 144 measures. Together they make 360, corresponding to the number of days in a year. The total measures of the sixty-four hexagrams in the upper and lower parts amount to 11,520, corresponding to the number of all things.

“二篇”者，上下經六十四卦也，其陽爻百九十二，每爻各三十六策，積之得六千九百一十二，陰爻百九十二，每爻二十四策，積之得四千六百八，又合二者，為力‘有一於五百二十也。若為少陽，則每爻二十八策，凡五千三百七十六，少陰則每爻三十二策，凡六千一百四十四，合之亦為萬一千五百二十也。

Zhu Xi explained that the term "two chapters" refers to the upper and lower classics of the Zhou Changes, which contain sixty-four hexagrams. The sixty-four hexagrams consist of three hundred and eighty-four lines, with yin and yang each accounting for exactly half: one hundred and ninety-two yang lines. Each yang line is calculated as thirty-six strategies, resulting in a total of six thousand nine hundred and twelve. Similarly, one hundred and ninety-two yin lines, each calculated as twenty-four strategies, total four thousand six hundred and eight. When these two totals are combined, they amount to exactly eleven thousand five hundred and twenty. If instead, the calculation is based on lesser yang and lesser yin, each lesser yang line is twenty-eight strategies, totaling five thousand three hundred and seventy-six; each lesser yin line is thirty-two strategies, totaling six thousand one hundred and forty-four. When these two totals are combined, they also amount to exactly eleven thousand five hundred and twenty, with no discrepancy.

是故四營而成易，十有八變而成卦，八卦而小成。引而伸之，觸類而長之，天下之能事畢矣。

The Great Treatise says: Thus, four lines are arranged to form one transformation, and eighteen transformations form one hexagram. The first three lines forming the eight trigrams are called the small completion. From this, by drawing connections and extending the principles, one can apply them universally, encompassing all things under heaven.

“四營”者，四次經營也。“分二”者，第一營也；“掛一”者，第二營也；“揲四’’者，第三營也；“歸奇”者，第四營也，《易》變易也，謂揲之一變也，四營成變，三變成爻。一變而得兩儀之象，再變而得四象之象，二變而得八卦之象。一爻而得兩儀之畫，二爻而得四象之畫，三爻而得八卦之畫，四爻成而得其”卜六者之一，五爻成而得其三十二者之一，至於積七十二二營而成十有八變，則六爻見，而得乎六十四卦之一矣。

Zhu Xi explained that the term "four camps" refers to the four procedures involved in the process of divination. "Dividing into two" is the first camp, "attaching one" is the second, "dividing four" is the third, and "returning the odd" is the fourth. "Change" refers to one change in the process of divining with yarrow stalks. Four camps make up one change, and three changes make up one trigram. One change produces the image of the two principles, two changes produce the image of the four images, and three changes produce the image of the eight trigrams. When one trigram is drawn, the image of the two principles is formed; when two trigrams are drawn, the image of the four images is formed; when three trigrams are drawn, the image of the eight trigrams is formed; when four trigrams are drawn, one of the sixteen is formed; and when five trigrams are drawn, one of the thirty-two is formed. Each trigram requires three changes, and each change requires four camps. With six trigrams, there are a total of eighteen changes. When the process reaches seventy-two camps and the eighteen changes are completed, all six trigrams are revealed, and one of the sixty-four hexagrams is obtained.

然方其三十六而九變也已得三畫，而八卦之名可見，則內卦之為貞者立矣，此所謂“八卦而小成”者也，自是而往，“引而伸之”，又三十六營九變以成三畫，而再得小成之封者也，則外卦之為悔者亦備矣。六爻成，內外卦備，六十四卦之別可見，然後視其爻之變與不變，而觸類以長焉，則天下之事，其吉凶悔吝，皆不越乎此矣。

However, when the operations reach thirty-six camps and nine changes, the three lines have already been drawn, and the names of the eight trigrams become visible. The inner trigram, referred to as "Zhen" (the initial part), is then established, which is what the classic text refers to as "the eight trigrams form a small completion." From this point onward, "extending and elaborating," after another thirty-six camps and nine changes to draw the three lines again, another small completion trigram is formed, and the outer trigram, referred to as "Hui" (the latter part), is also complete. "Zhen" refers to the inner trigram, and "Hui" refers to the outer trigram, which are old terms from the "Book of Documents · Hongfan." When all six lines are drawn and both inner and outer trigrams are complete, the distinctions of the sixty-four trigrams become clear. Then, by observing the changes and unchanges of the lines and extending this understanding, all the matters under heaven, their fortune, misfortune, regret, and hardship, are all encompassed within this framework.

顯道神德行，是故可與酬酢，可與祐神矣。道因辭顯，行以數神，“酬酢”者，言幽明之相應，如賓主之相交也。“佑神”者，言有以佑助神化之功也。

The Great Treatise says: The virtue of the divination with the yarrow stalks can manifest the great way and the profound virtue, so it can be used to respond and interact, and can assist the divine. Zhu Xi explained: The principles are made manifest through the words of the hexagram lines, and virtue is made profound through the numbers of the yarrow stalk divination. The term 'chou zu' refers to the mutual response between the hidden divine and the visible human affairs, like the interaction between guest and host. The term 'you shen' refers to the divination method assisting the divine in its creative function.

卷內蔡氏說，為奇者三，為偶者二，蓋凡初揲，左手餘一餘二餘三皆為奇，餘四為偶。至再揲三揲，則餘三者亦為偶，故曰奇二而偶二也。

This passage refers to Cai Yuan ding's statement that "the odd count is three, and the even count is two." This is because, on the first round of counting the stalks, if there is one, two, or three left over in the left hand, they are all considered "odd." Only when four are left over is it considered "even." However, on the second and third rounds of counting, the case of three left over is considered "even" as well. Therefore, it is said later that "odd has two types and even has two types."

《乾》卦用九“見群龍無首，吉”。《象》曰：“用九，天德不可為首也。 ”《坤》卦用六“利永貞”。《象》曰：“用六永貞，以大終也。”其所占者名爻，不謂六爻皆九六也。及其筮也，七八常多，而九六常少，有無九六者焉，此不可以不釋也。六十四斟皆然，特於《乾》、《坤》見之，則餘可知耳。

The hexagram Qian has "Yong Jiou" (All Nines), with the phrase saying "Seeing a group of dragons without a leader, it is auspicious." The Image Commentary says "Yong Jiou, the Heavenly Virtue cannot be the leader." The hexagram Kun has "Yong Liu" (All Sixes), with the phrase saying "It is advantageous to maintain constancy." The Image Commentary says "Yong Liu maintaining constancy leads to a great conclusion." Here, what is referred to is the changing line, not that all six lines are nines or sixes. In actual divination, getting seven or eight is common, while getting nine or six is rare. Sometimes, a hexagram may have neither a nine nor a six. This point must be clearly explained. All sixty-four hexagrams are like this, but they specifically mention the hexagrams Qian and Kun, and the rest can be inferred similarly.

愚案此說發明先儒所未到，最為有功。其論七八多而九六少，又見當時占法，三變皆卦，如一行說。

Zhu Xi commented that the above discussion has clarified points that previous Confucian scholars did not address, and it has made the greatest contribution. It discusses the cases of getting seven and eight more frequently than those of getting nine and six, which also shows that the commonly used method of divination at that time required all three changes to be attached, consistent with the approach proposed by the monk Yixing.

凡卦六爻皆不變，則占本卦彖辭，而以內卦為貞，外卦為悔。彖辭為卦下之辭，孔成子筮立衛公子元，遇《屯》，曰“利建侯”。秦伯伐晉筮之遇《蠱》，曰貞風也，其悔山也。

Whenever a hexagram has all six lines unchanging, one should consult the彖辞 of the primary hexagram, taking the inner trigram as 'zhen' and the outer trigram as 'hui'. The彖辞 is the passage that follows the hexagram diagram. For example, when Kong Chengzi cast a divination for establishing Wei State's Prince Yuan, he encountered the hexagram Zhun, and based on the彖辞, he concluded 'favorable for establishing feudal lords'. When Qin Duke cast a divination for attacking Jin State, he encountered the hexagram Gu, and he stated that the zhen trigram was wind and the hui trigram was mountain, using the two trigrams to explain the meaning.

一爻變，則以本卦變爻辭占。沙氏程氏曰，畢萬遇《屯》之《比》，初九變了也；蔡墨遇《乾》之《同人》，九二變也；晉文公遇《大有》之《睽》，九三變也；陳敬仲遇《觀》之《否》，六四變也；南蒯遇《坤》之《比》六五變也；晉獻公遇《歸妹》之《睽》，上六變也。

If there is only one changing line, use the line statement of that changing line in the primary hexagram for divination. Ancient texts refer to this as "a hexagram changing into another hexagram," indicating the specific line in the primary hexagram that changes. Shashui Cheng Jing says: Bi Wan's divination resulted in the hexagram "Zun" changing into "Bi," which refers to the line "Initial Nine" changing; Cai Mo mentioned the hexagram "Qian" changing into "Tong Ren," referring to the line "Nine in the Second" changing; Jin Wen Gong's divination resulted in "Da You" changing into "Kui," referring to the line "Nine in the Third" changing; Chen Jingzhong's family divination resulted in "Guan" changing into "Po," referring to the line "Six in the Fourth" changing; Nan Kuan's divination resulted in "Kun" changing into "Bi," referring to the line "Six at the Top" changing; and Jin Xian Gong's divination for a marriage resulted in "Gui Mei" changing into "Kui," referring to the line "Six at the Top" changing.

集說 胡氏一佳曰：《啟蒙》謂一爻變，則以本卦變爻辭占，其下引畢萬所筮，以今觀之，未嘗不取之卦，且不特論一爻，兼取貞悔卦體，似可為占者法也，觀陳宣公筮公子完之生，尤可見矣。

Collective Commentary: Hu Yigui said: The "Easy启蒙" discusses the case of one line changing, using the line statement of the changed line in the primary hexagram for divination. It cites the example of Bi Wan's divination. From today's perspective, that divination did not merely rely on the line statement of the changed line, but also took into account the hexagram formed by the change, and considered both the original and the changed aspects of the hexagram. This seems to establish a rule for the diviner. Looking further at the divination concerning the birth of Gongzi Wan during the time of King Chen Xuan, it becomes even clearer.

二爻變，則以本卦二變爻辭占，仍以上爻為主。經傳無文，今以例推之當如此。集說 胡氏一桂曰：案陳摶為宋太祖占，亦旁及諸爻與卦體。

Whenever two lines change, one should use the line statements of these two changing lines from the primary hexagram for divination, still taking the upper line as the main reference. This rule is not explicitly stated in the classic texts, but according to the logical reasoning of the tradition, it should be so. Collected Commentaries mention Hu Yigui's statement: According to Chen Tuan's divination for Song Taizu, he also considered other lines and the entire hexagram structure, not rigidly adhering to a single line.

三爻變，則占本卦及之卦之彖辭，而以本卦為貞，之卦為悔，前十卦主貞，後十卦主悔。凡三爻變者。通二十卦有圖在後。沙氏程氏曰，晉公子重耳筮得國，遇《屯》悔《豫》皆八，蓋初與四五凡三爻變也，初與五用九變，四用六變。其不變者二二上，在兩卦皆為八，故云皆八，而司空季子占之曰“皆利建侯。”集說 胡氏一桂曰：案《啟蒙》但云占本卦之卦彖辭，然以晉侯《屯》、《豫》之占，則並占卦體可見。

Whenever three lines change, one should consult the meaning of both the primary hexagram and the changing hexagram, taking the primary hexagram as "Zhen" and the changing hexagram as "Hui." If the hexagram is among the first ten listed, the primary hexagram is considered the main one; if it is among the last ten, the changing hexagram is considered the main one. There are a total of twenty hexagrams that involve three-line changes, and their list is provided later. Shashui Cheng Jing said: When Duke Wen of Jin, Prince Chong'er, sought divination about whether he could reclaim the state, he encountered "Zhen 'Zun' and Hui 'Yu' both as eight." This indicates that the first, fourth, and fifth lines changed, making three lines in total. The first and fifth lines changed due to the use of "Yong Ji" (all nines), and the fourth line changed due to the use of "Yong Liu" (all sixes). The second, third, and top lines remained unchanged. In both hexagrams "Zun" and "Yu," these lines are all nines, hence the statement "both as eight." Therefore, Si Kong Jizi interpreted it as "both hexagrams are favorable for establishing a feudal lord." Collected interpretations: Hu Yigui said: According to "Ming Tong," it only mentions consulting the meaning of the primary hexagram and the changing hexagram. However, from the divination of Duke Wen of Jin's "Zun" and "Yu," it is clearly evident that the structure of the hexagram was also considered in the interpretation.

熊氏朋來曰：七八皆不變爻，何以罕言七而專言八。曰：七七，蓍數也，八八，卦數也。

Collective commentary by Xiong Pengxue says: Both seven and eight are unchanging lines. Why did ancients rarely mention seven and focus on eight? The answer is: Seven times seven is forty-nine, which is the number of the divination stalks; eight times eight is sixty-four, which is the number of the hexagram lines. Since what is recorded is the hexagram and not the stalks, it is natural to use the number eight, which belongs to the hexagram, to indicate it.

四爻變，則以之卦二不變爻占，仍以下爻為主。經傳亦無文，今以例推之當如此。

If the fourth line changes, use the two unchanging lines from the 'zhī guà' to divinate, still taking the line below as the main one. This rule is not explicitly stated in the classic texts, but according to the established conventions, it should be interpreted this way.

五爻變，則以之卦不變爻占。穆姜往東宮，筮遇《艮》之八，史曰“是謂艮之隨”，蓋五爻皆變，唯二得八，故不變也，法宜以“係小子失丈夫”為占，而史妄引《隨》之彖辭以對，則非也。

If the fifth line changes, one should use the unchanging line from the 'changed hexagram' to make the divination. When Duke Mu Jiang moved to the Eastern Palace, he cast a divination and encountered 'The Mountain (Gen) changing to Eight'. The historian said, 'This is called The Mountain changing to Following'. That was when all five lines changed, except the second line which remained as eight. According to the rule, one should use the line statement of the sixth line in the second position of the Following hexagram, 'Binding the young, losing the old', to make the divination. However, the historian incorrectly cited the彖 statement of the Following hexagram instead, which was wrong.

六爻變，則《乾》、《坤》占二用，餘卦占之卦彖辭。蔡墨曰，《乾》之《坤》曰，“見群龍無首吉”是也，然“群龍無首”，即《坤》之“牝馬”“先迷”也，《坤》之“利永貞”，即《乾》之“不言所利”也。

Whenever all six lines of a hexagram change, if the hexagram is either Qian or Kun, then the interpretation refers to the lines "Yong Jiux" and "Yong Liux"; for other hexagrams, it refers to the interpretation of the resulting hexagram. Cai Mo said, "The hexagram Qian changing into Kun, the interpretation is 'Seeing a group of dragons without a leader, which is auspicious,' " using exactly this method. However, "a group of dragons without a leader" is actually the meaning of Kun's "female horse" and "first being lost"; and Kun's "advantageous to maintain perseverance in uprightness" is actually the meaning of Qian's "not speaking of advantage." The two are originally connected.

於是一卦可變六十四卦，而四千九十六卦在其中矣，所謂“引而仲之，觸類而長之，大下之能事畢矣”，豈不信哉。今以六十四卦之變，列為三十二圖，最初卦者，自初而終，自一亡而下，得末卦者，自終而初，自下而卜，變在第三十二卦以前者，占本卦爻之辭，變在第三十二卦以後者，占變卦爻之辭。凡言初終上下者，據圖而言，言第幾卦前後以上三十二圖，反覆之則為六十四圖，圖以一卦為主，而各具六十四卦，凡四千九十六卦，與焦贛《易林》合，然其條理精密，則有先儒所未發者，覽者詳之。

In this way, one hexagram can transform into all sixty-four hexagrams, and the total of four thousand and nine十六 hexagrams are included. As the saying goes, "By drawing out and extending, by touching similar things and increasing them, all the abilities in the world are completed," which is indeed true. Now the changes of the sixty-four hexagrams are arranged into thirty-two figures: those that belong to the initial hexagram are read from the first line to the top line, looking down from top to bottom; those that belong to the final hexagram are read from the top line back to the first line, looking up from bottom to top. If the change falls before the thirty-second hexagram, it uses the line statements of the primary hexagram; if it falls after the thirty-second hexagram, it uses the line statements of the resulting hexagram. Whenever terms like initial, final, upper, or lower are mentioned, they refer to the positions on the figure; when terms like before or after a certain hexagram are mentioned, they refer to the order of the figures. The above thirty-two figures, when reversed, form the sixty-four figures. Each figure takes one hexagram as its main one and contains all sixty-four hexagrams within itself, totaling four thousand and nine十六 hexagrams, which exactly matches the number of hexagrams in Jiao Gong's "Yi Lin." However, it is more systematic and precise, containing aspects that previous Confucian scholars did not discover. Those who study the figures should carefully reflect on them.

集說 胡氏一桂曰；焦延壽卦變法，以一卦變為六十四卦，六十四卦通變四千九十六卦，而卦變之次，本之文王序卦，且如以《乾》為本卦，其變首《坤》，次《屯》、《蒙》，以至《未濟》，又如以末一卦《未濟》為本卦，其變亦首《乾》，次《坤》、《屯》，以至《既濟》，每一卦變六個三卦，通本卦成六十四卦，紫陽夫子以爻變多寡，順而列之，以定一卦所變之序，又以《乾》、《卦》所變之次，引而伸之，為六十四卦所變相承之序，案 朱子三十二圖，其次第最為詳密，而後學之疑義有二：一曰筮法用“九”、“六”不用“七”、“八”，今四爻五爻變者，用之卦之不變爻占，則是兼用“七”、“八”也；二曰周公未繫爻之先，則彖辭之用，有所不周也。三代筮法，既不盡傳，今唯以經傳為據而推之，則“用九”“用六”，經文甚明，而用“七”、“八”者，諸書皆無明文，唯杜預以為夏商用之，先儒已摘其非矣。

Collective Commentary: Hu Yigui said: Jiao Yanshou's method of changing hexagrams involves transforming one hexagram into all sixty-four hexagrams, resulting in a total of four thousand and nine十六 hexagrams. The order of transformation follows the sequence of hexagrams established by King Wen. For example, if the primary hexagram is Qian (Heaven), the first transformed hexagram is Kun (Earth), followed by Zun (Sprouting) and Meng (Youth), continuing up to Wei Ji (Not Yet Completed). Similarly, if the last hexagram Wei Ji is taken as the primary, the first transformed hexagram is Qian, followed by Kun, Zun, and so on, up to Ji Ji (Already Completed). Each hexagram transforms into sixty-three others, combining with the original to form the sixty-four hexagrams. Zhu Xi, the Purple Yang Master, arranged the order of transformation based on the number of changing lines, using this to determine the sequence of transformation for each hexagram. He also extended the sequence of transformation from Qian to establish the order of transformation between the sixty-four hexagrams. Li Guangdi remarked: Zhu Xi's thirty-two diagrams are the most detailed and systematic. However, later scholars raised two questions. First, divination only uses 'nine' and 'six' and does not use 'seven' and 'eight'. Now, when there is a change in the fourth line, fifth line, or top line, the unchanging lines of the 'changed hexagram' are also used for divination, which involves the use of 'seven' and 'eight'. Second, before King Zhou annotated the line statements, divination relied solely on the hexagram statements, which might be incomplete. Since the divination methods of the Three Dynasties have not been fully transmitted, we can only rely on the classics to infer: 'Use nine' and 'use six' are clearly stated in the texts. However, the use of 'seven' and 'eight' is not mentioned in any text. Only Du Yu claimed they were used in the Xia and Shang dynasties, but earlier scholars have already criticized his view.

考之《春秋》內外傳，蓋無論變與不變，及變之多寡，皆論卦之體象與其彖辭。即一爻變者，雖占爻辭，而亦必先以卦之體象與其彖辭為主，則知古人占法，未有爻辭之先，即彖辭而已周於用，既有爻辭之後，則但以專動考占，而初亦不離乎彖辭以為斷也，唯其一卦可變為六十四，則兩卦相參，而可以盡事物之理。故卦之有變者，意主於生卦，不主於成爻。爻之有變者，專動則有占，雜動則無占，如是則傳記之文皆合，而學者之疑可釋矣。

Using the Spring and Autumn Annals and the Guoyu, whether the lines change or not, and whether they change a lot or little, when making a divination, one must refer to the image of the hexagram and its zhuang ci. Even if only one line changes, although one may refer to the line statement of that line, one must first rely on the image of the hexagram and its zhuang ci. From this we can understand the ancient method of divination: before the line statements appeared, the zhuang ci alone was sufficient for use; when the line statements appeared, one only used the particular line that changed for divination, but initially one still relied on the zhuang ci to make a judgment. Because a single hexagram can change into sixty-four hexagrams, by comparing the original hexagram with the changed one, one can fully understand the principles of things. Therefore, the purpose of changing a hexagram is to produce new hexagrams, not to complete a particular line; the purpose of changing a line is that only a single, specific line has a meaning, while multiple, chaotic changes have no meaning. In this way, all the records in the transmission are explained, and scholars' doubts can be resolved.

至內外傳言得八者三：一曰《泰》之八，則不變者也；一曰貞《屯》悔《豫》皆八，則三爻變者也；一曰《艮》之八為《艮》之《隨》，則五爻變者也。諸儒以八為不動之爻，考之文意，似未符協。蓋三占者，雖變數不同，然皆無專動之爻，則其為用卦一也，卦以八成，故以八識卦，猶之爻以“九”“六”成，則以“九”“六”識爻云爾，觀朱子之圖者，更須以《左傳》、《國語》諸書互相參考。

In the "Zuo Zhuan" and "Guoyu," there are three instances mentioning "eight" ("ba"). One says "the eight of Tai," which refers to a hexagram with no lines changing. Another says "the divination of Zun and the regret of Yu are both eight," which refers to a hexagram with the third line changing. The third says "the eight of Gen," which is "Gen" changing into "Sui," referring to a hexagram with the fifth line changing. Various Confucian scholars interpret "eight" as a line that does not change, but when checked against the text, this seems inconsistent. Probably, although the number of changing lines differs in these three instances, none of them has a single line that changes exclusively, so they all use the entire hexagram. A hexagram is formed by combining eight lines, so "eight" is used to indicate the hexagram, just as "nine" and "six" are used to indicate the lines, since lines are formed by "nine" and "six." Those who study Zhu Xi's diagrams should also refer to the "Zuo Zhuan" and "Guoyu" for cross-referencing.

朱子之作《啟蒙》，蓋因以象數言《易》者，多穿穴而不根，支離而無據。然《易》之為書，實以象數而作，又不可略焉而不講也，且在當日言圖書卦畫蓍數者，皆創為異論以毀成法，師其獨智而訾先賢，故朱子述此篇以授學者，以為欲知《易》之所以作者，於此可得其門戶矣。今摭圖書卦畫蓍數之所包蘊，其錯綜變化之妙，足以發朱子未盡之意者凡數端，各為圖表而繫之以說，蓋所以見圖書為天地之文章，立卦生蓍為聖神之制作，萬理於是乎根本，萬法於是乎權輿，斷非人力私智之所能參，而世之紛紛撰擬，屑屑疑辨，皆可以熄矣。

Zhu Xi wrote the "Yixue Qimeng" because those who used images and numbers to explain the Yi Jing were mostly superficial and lacked solid foundations, being fragmented and without proper basis. However, the Yi Jing itself was originally created through images and numbers, and it could not be ignored. Moreover, at that time, those who discussed the River and Lake diagrams, the hexagram illustrations, and the divination numbers all proposed different views, rejecting established methods, showing off their own private wisdom, and criticizing previous scholars. Therefore, Zhu Xi composed this text to pass on to scholars, believing that if one wished to understand why the Yi Jing was created, one could find the entrance from here. Now, by collecting the subtle transformations contained in the River and Lake diagrams, hexagram illustrations, and divination numbers, there are several aspects that can further explain Zhu Xi's unexpressed intentions. Each of these aspects is presented in a chart with an explanation. This is to show that the River and Lake diagrams are the natural writings of heaven and earth, the establishment of hexagrams and the creation of the divination numbers are the divine works of sages. All principles and methods originate from here, and they are not something that human effort or private wisdom can participate in. Thus, all the chaotic conjectures and trivial doubts in the world can be put to rest.

《大傳》言《河圖》，曰一二，曰三四，曰五六，曰七八，曰九十，則是以兩相從也。《大戴禮》言《洛書》，曰二九四，曰七五三，曰六一八，則是以三相從也。是故原《河圖》之初，則有一便有二，有三便有四，至五而居中；有六便有七，有八便有九，至十而又居中，順而布之，以成五位者也。原《洛書》之初，則有一二三，便有四五六，有四五六，便有七八九，層而列之，以成四方者也。

The Great Treatise explains the Yellow River Chart, saying "Heaven is one, Earth is two; Heaven is three, Earth is four; Heaven is five, Earth is six; Heaven is seven, Earth is eight; Heaven is nine, Earth is ten," which are paired two by two. The Dai Dai Li Ji explains the Luo Writing, saying "two, nine, four; seven, five, three; six, one, eight," which are grouped three by three. Therefore, tracing back to the original form of the Yellow River Chart: having one implies having two, having three implies having four, and five is in the center; having six implies having seven, having eight implies having nine, and ten is again in the center; arranging them in order like this forms five directions, including the center and the four cardinal points. Tracing back to the original form of the Luo Writing: having one, two, and three implies having four, five, and six; having four, five, and six implies having seven, eight, and nine; arranging them in layers like this forms the shape of the four cardinal directions.

若以陽動陰靜而論，則數起於上，故《河圖》之一二本在上也，三四本在右也，六七本在下也，八九本在左也，《洛書》之一二三，四五六，七八九，本自上而下也，於是陽數動而變易，陰數靜而不遷，則成《河圖》、《洛書》之位矣。如以陽靜陰動而論，則數起於下，故《河圖》之一二本在下也，三四本在左也，六七本在上也，八九本在右也。《洛書》之一二三，四五六，七八九，本自下而上也，於是陽數靜而不遷，陰數動而交易，則又成《河圖》、《洛書》之位

If we consider yang to be active and yin to be still, the numbers start from above: therefore, in the Yellow River Chart, the numbers one and two are at the top, three and four on the right, six and seven at the bottom, and eight and nine on the left; in the Luo Writing, the numbers one, two, and three, four, five, and six, seven, eight, and nine are arranged from top to bottom. Thus, yang numbers, being active, change their positions, while yin numbers, being still, do not move, resulting in the current arrangement of the Yellow River Chart and the Luo Writing. If we consider yang to be still and yin to be active, the numbers start from below: therefore, in the Yellow River Chart, the numbers one and two are at the bottom, three and four on the left, six and seven at the top, and eight and nine on the right; in the Luo Writing, the numbers one, two, and three, four, five, and six, seven, eight, and nine are arranged from bottom to top. Thus, yang numbers, being still, do not move, while yin numbers, being active, change their positions, and this also results in the current arrangement of the Yellow River Chart and the Luo Writing. There is a missing text here, with the title of the diagram and the following sentence written together.

自《洛書》以三三積數，為數之原，而自四以下，皆以為法焉，何則？三者天數也，故其象圓，如前圖，居四方與居四偶者，或動或靜，居中者一定不易而各成縱橫皆十五之數矣。四者地數也，故其象方，如後圖，居中居四偶與居四方者，或動或靜，亦各成縱橫皆三十四之數矣。自五五以下，皆以三三圖為根，自六六以下，皆以四四圖為根，而四四圖，又實以三三圖為根，赦《洛書》為數之原，不易之論也！今附四四圖如左（即後二圖），以相證明，其餘具數學中，不悉載。

Starting from the product of three threes in the Luo Writing, all numbers below four take this as their pattern. Why is that? Three represents the heavenly number, so its image is circular, as shown in the previous diagram: the squares in the four cardinal directions and the four corners either move or remain still, while the central square remains fixed, and the numbers in each row and column add up to fifteen. Four represents the earthly number, so its image is square, as shown in the following diagram: the central square, the four corners, and the four cardinal directions either move or remain still, and each row and column adds up to thirty-four. Numbers from five downwards take the three-three diagram as their basis, and numbers from six downwards take the four-four diagram as their basis, and the four-four diagram itself is based on the three-three diagram. Therefore, the Luo Writing is the fundamental origin of numbers, an unchangeable conclusion. Now the four-four diagram is appended below the two diagrams for mutual verification; the rest are included in specialized mathematical texts and are not listed here.

此以十六數自左而右自上而下列之第一圖，其居中與居四偶者不易，而居四方者變易，則成縱橫皆三十四之數第二圖。若居四方者不易，而居中與居四偶者變易，亦成縱橫皆三十四之數第三圖。

This arranges the numbers one through sixteen in sequence from left to right and top to bottom, forming the first figure. If the four central squares and the four corner squares are kept fixed, while the eight squares on the sides are swapped, the result is a figure where each row and column adds up to thirty-four, forming the second figure. Conversely, if the squares on the sides are kept fixed, and the central and corner squares are swapped, the same result is achieved, with each row and column adding up to thirty-four, forming the third figure.

此以十六數自右而左，自下而上列之第一圖，用前法變為兩圖第二圖第三圖。並得縱橫皆三十四之數，但其不易者，即前之變易者，而其變易者，即前之不易者此第二圖同前第三圖，此第三圖同前第二圖，蓋亦陰陽互為動靜之理雲。

This arranges the sixteen numbers in sequence from right to left and from bottom to top, forming the first diagram; then using the same method to produce two more diagrams, which are the second and third diagrams. The results are the same, with the numbers in each row and column adding up to thirty-four. In this group, what was fixed in the previous group is now in motion, and what was in motion in the previous group is now fixed. The second diagram of this group is the same as the third diagram of the previous group, and the third diagram of this group is the same as the second diagram of the previous group. This is precisely the principle of yin and yang, where each is in motion and stillness relative to the other.

一 三 九 七用中兩率，三九相乘為二十七，以一除之得二十七，以二十七除之得一。若用一與二十七相乘，以三除之得九。以九除之得三。二 四 八 六用中兩率，四八相乘為三十二，以二除之得十六，以十六除之得二。若用二與十六相乘，以四除之得八，以八除之得四。

One, three, nine, seven: taking the middle two numbers, three multiplied by nine gives twenty-seven; dividing twenty-seven by one gives twenty-seven, and dividing twenty-seven by twenty-seven gives one. If instead taking the first and last numbers, one multiplied by twenty-seven gives twenty-seven; dividing twenty-seven by three gives nine, and dividing twenty-seven by nine gives three. Two, four, eight, six: taking the middle two numbers, four multiplied by eight gives thirty-two; dividing thirty-two by two gives sixteen, and dividing thirty-two by sixteen gives two. If instead taking two and sixteen, two multiplied by sixteen gives thirty-two; dividing thirty-two by four gives eight, and dividing thirty-two by eight gives four.

《大傳》曰：“天一地二，天三地四，天五地六，天七地八，天九地十。”天地之數，皆自少而多，多而復還於少，此加減之原也。又曰：“參天兩地而倚數。”天數以三行，地數以二行，此乘除之原也，是故《河圖》以一二為數之體之始，《洛書》以三二為數之用之始。然《洛書》之用，始於參兩者，以叄兩為根也，實則諸數循環互為其根，莫不寓乘除之法焉，而又皆以加減之法為之本。今推得洛書加減之法四，乘除之法十六，積方之法五，勾股之法四，各為圖表以明之如左（即如下）。

The Great Treatise says: "Heaven is one, earth is two; heaven is three, earth is four; heaven is five, earth is six; heaven is seven, earth is eight; heaven is nine, earth is ten." The numbers of heaven and earth all begin with small numbers and increase gradually, and when they reach their maximum, they return to small numbers again. This is the origin of addition and subtraction. The Great Treatise also says: "By combining three from heaven and two from earth, numbers are calculated." The numbers of heaven are calculated by three, and the numbers of earth are calculated by two. This is the origin of multiplication and division. Therefore, the Yellow River Chart begins with one and two as the origin of numbers, while the Luo Writing begins with three and two as the origin of the application of numbers. However, the Luo Writing starts with three and two as its foundation, using three and two as the root. In reality, all numbers cycle and interrelate, each being the root of the other, and all contain multiplication and division within them, yet they are all based on addition and subtraction. Now, we have derived four rules of addition and subtraction, sixteen rules of multiplication and division, five rules of multiplication methods, and four rules of the Pythagorean theorem for the Luo Writing, which are presented in charts below for explanation.

一用奇數左旋相加，得相連之偶數。一加三為四，三加九為十二。九加七為十六，七加一為八。若用奇數減左旋相連之偶數，得右旋相連之奇數。七域十六為九，一減八為七。

The first method: take the odd numbers from the Luo Writing and add them in the order of left rotation, one pair at a time. The result is the even number that lies between them (if the sum is more than ten, only the last digit is taken). One plus three gives four; three plus nine gives twelve, taking the last digit gives two; nine plus seven gives sixteen, taking the last digit gives six; seven plus one gives eight. Conversely, if you subtract one of the odd numbers from the even number obtained by the left rotation addition, you get the odd number connected to it in the right rotation direction: sixteen minus seven gives nine, eight minus one gives seven.

一用偶數左旋相加，得相連之偶數。二加六為八，六加八為十四。八加四為 I一二，四加二為六。若用偶數減左旋相連之偶數，得右旋相連之偶數。六減八為二，八減[一四為六。四減十二為八，二減六為四。

The second method: take the even numbers at the four corners and add them in the order of left rotation, which is two, six, eight, four. Add them pairwise in this sequence, and the result will be the even number connected to them (if the sum exceeds ten, only the last digit is taken). Two plus six equals eight; six plus eight equals fourteen, take the last digit which is four; eight plus four equals twelve, take the last digit which is two; four plus two equals six. Conversely, if you subtract the left-rotated added even number from one of the even numbers, you will get the even number connected to it in the right-rotated direction: eight minus six equals two, fourteen minus eight equals six, twelve minus four equals eight, six minus two equals four.

一用奇數右旋加偶數，得相連之奇數。一加六為七，七加二為九。九加四為十三，三加八為十一。若用奇數減相連之奇數，得相連之偶數。一減七為六，七減九為二。九減十三為四，三減十一為八。

Third method: Take the odd numbers and add them to the even numbers in the order of right rotation. The result obtained is exactly the connected odd number (if the sum exceeds ten, only the last digit is taken). One plus six gives seven; seven plus two gives nine; nine plus four gives thirteen, taking the last digit gives three; three plus eight gives eleven, taking the last digit gives one. Conversely, if the original odd number is subtracted by the connected odd number obtained in this way, the result is the even number between the two: seven minus one gives six, nine minus seven gives two, thirteen minus nine gives four, eleven minus three gives eight.

一用偶數右旋加奇數，得相對之奇數。二加九為十一，四加三為七。八加一為九，六加七為十三。若用奇數減相對之奇數，得相連之偶數。九減十一為二，三減七為四。一減九為八，七減十三為六。

The fourth method: take an even number and add it to an odd number in the order of right rotation; the result is the odd number opposite to the original odd number (if the sum is more than ten, only the last digit is taken). Two added to nine gives eleven, taking the last digit gives one, which is opposite to nine; four added to three gives seven, which is opposite to three; eight added to one gives nine, which is opposite to one; six added to seven gives thirteen, taking the last digit gives three, which is opposite to seven. Conversely, if the original odd number is subtracted from the opposite odd number, the result is the even number connected to them: eleven minus nine gives two, seven minus three gives four, nine minus one gives eight, thirteen minus six gives seven.

一用三左旋乘奇數，得相連之奇數。三二如九，尋九二十七。三七二十一，三一如三。一用八左旋乘偶數，得相連之偶數。八八六十四，八四三一卜二。八二一十六，八六四十八。

First method: Use three and multiply it by an odd number, rotating left, the result is the next odd number connected to it (if it exceeds ten, only take the last digit). Three times three is nine; three times nine is twenty-seven, the last digit is seven; three times seven is twenty-one, the last digit is one; three times one is three. Second method: Use eight and multiply it by an even number, rotating left, the result is the next even number connected to it. Eight times eight is sixty-four, the last digit is four; eight times four is thirty-two, the last digit is two; eight times two is sixteen, the last digit is six; eight times six is forty-eight, the last digit is eight.

一用三左旋乘偶數，得相連之偶數。三四一十二，三二如六。三六一十八，三八二十四。一用八左旋乘奇數，得相連之偶數。八三二十四，八九七十二。八七五十六，八一如八。

Third method: Using three and rotating left to multiply by even numbers, the result is the connected even number. Three times four is twelve, the last digit is two; three times two is six; three times six is eighteen, the last digit is eight; three times eight is twenty-four, the last digit is four. Fourth method: Using eight and rotating left to multiply by odd numbers, the result is the connected even number. Eight times three is twenty-four, the last digit is four; eight times nine is seventy-two, the last digit is two; eight times seven is fifty-six, the last digit is six; eight times one is eight.

一用二右旋乘偶數，得相連之偶數。二二如四，二四如八。一用七右旋乘奇數，得相連之奇數。七七四十九，七九六十二。七三二十一，七一如七。一用二右旋乘奇數，得隔二位之偶數。二九一十八，二三如六。二一如二，二七一十四。

Fifth method: Using two and multiplying it clockwise with even numbers, the result is the next even number: two times two is four, two times four is eight. Sixth method: Using seven and multiplying it clockwise with odd numbers, the result is the next odd number: seven times seven is forty-nine, the last digit is nine; seven times nine is sixty-three, the last digit is three; seven times three is twenty-one, the last digit is one; seven times one is seven. Seventh method: Using two and multiplying it clockwise with odd numbers, the result is the even number two places away: two times nine is eighteen, the last digit is eight; two times three is six; two times one is two; two times seven is fourteen, the last digit is four.

一用七右旋乘偶數，得相連之偶數。七二一十四，七四二十八。七八五十六，七六四十二。一用一乘奇數，得本位之奇數。一一如一，一三如三。一九如九，一七如七。一用六乘偶數，得本位之偶數。六六三十六，六八四十八。六四二十四，六二一十二。一用一乘偶數，得本位之偶數。一二如二，一四如四。一八如八，一六如六。

Eighth Method: Using seven to multiply an even number, the result is the connected even number: seven times two is fourteen, the last digit is four; seven times four is twenty-eight, the last digit is eight; seven times eight is fifty-six, the last digit is six; seven times six is forty-two, the last digit is two. Ninth Method: Using one to multiply an odd number, the result is still the original odd number: one times one is one, one times three is three, one times nine is nine, one times seven is seven. Tenth Method: Using six to multiply an even number, the result is still the original even number: six times six is thirty-six, the last digit is six; six times eight is forty-eight, the last digit is eight; six times four is twenty-four, the last digit is four; six times two is twelve, the last digit is two. Eleventh Method: Using one to multiply an even number, the result is still the original even number: one times two is two, one times four is four, one times eight is eight, one times six is six.

一用六乘奇數，得相連之偶數。六七四十二，六九五卜四。六三一十八，六一如六。一用四乘偶數，得相對之偶數。四四一十六，四六二十四。四二如八，四／＼二寸’二。一用九乘奇數，得相對之奇數。九九八十一，九一如九。九三二十七，幾七六十三。

Twelfth Method: Multiply six by an odd number, the result is the even number connected to it: six times seven is forty-two, the last digit is two; six times nine is fifty-four, the last digit is four; six times three is eighteen, the last digit is eight; six times one is six. Thirteenth Method: Multiply four by an even number, the result is the even number opposite to it: four times four is sixteen, the last digit is six; four times six is twenty-four, the last digit is four; four times two is eight; four times eight is thirty-two, the last digit is two. Fourteenth Method: Multiply nine by an odd number, the result is the odd number opposite to it: nine times nine is eighty-one, the last digit is one; nine times one is nine; nine times three is twenty-seven, the last digit is seven; nine times seven is sixty-three, the last digit is three.

一用四乘奇數，得隔二位之偶數。四九三十六，四七二十八。四一如四，四三十二。一用九乘偶數，得相對之偶數。九二一十八，九八七十二。九四三十六，九六五十四。凡除法，除其所得之數，得其所乘之數。

Fifteenth method: Multiply four by an odd number, and the result is the even number two places apart: four times nine is thirty-six, the last digit is six; four times seven is twenty-eight, the last digit is eight; four times one is four; four times three is twelve, the last digit is two. Sixteenth method: Multiply nine by an even number, and the result is the even number opposite in the middle: nine times two is eighteen, the last digit is eight; nine times eight is seventy-two, the last digit is two; nine times four is thirty-six, the last digit is six; nine times six is fifty-four, the last digit is four. For division, simply divide the result to get back the original number that was multiplied.

《洛書》乘除十六法，可約為八法，何則？五者河洛之中數，自此以上，由五以生， 數有合數，有對數，合數生於五，對數成於十，一六二七三八四九，此合數也，皆相減而為五者也。一九二八三七四六，此對數也，皆相併而為十者也。在《河圖》，則合數同方，而對數相連。在《洛書》，則合數相連，而對數相對。相合之相從者，六從一也，七從二也，八從三也，九從四也，如前乘除十六法。相對之相從者，九從一也，八從二也，七從三也，六從四也，女垢積方五法。

The sixteen methods of multiplication and division in the Luo Shu can actually be grouped into eight categories. Why is this so? The number five is the central number of both the He Tu and the Luo Shu. All numbers above five are generated from five. Numbers can be classified as "combined numbers" and "opposite numbers": combined numbers are generated from five, while opposite numbers are formed by ten. One and six, two and seven, three and eight, four and nine are combined numbers; when subtracted from each other, they both result in five. One and nine, two and eight, three and seven, four and six are opposite numbers; when added together, they both result in ten. In the He Tu, combined numbers are located in the same direction while opposite numbers are connected to each other; in the Luo Shu, combined numbers are connected to each other while opposite numbers are opposite each other in the center. Combined numbers follow each other in the order of six following one, seven following two, eight following three, and nine following four, which is similar to the sixteen methods of multiplication and division mentioned earlier. Opposite numbers follow each other in the order of nine following one, eight following two, seven following three, and six following four, which is similar to the five methods of square calculation mentioned later.

凡以合數共乘一數，所得之數必同。乘偶既同數，乘奇則同報。若各自乘焉，則又必合矣，如三三得九，八八六十四。以對數共乘一數，所得之數必對，如三三得九，七三二十一。若各自乘焉，則又必同矣，如一一得一，九九亦八十一，二二得四，八八亦六十四。

Whenever a pair of complementary numbers are each multiplied by the same number, the results will always correspond. When multiplied by an even number, the last digits of the two results will be exactly the same. When multiplied by an odd number, the last digits of the two results will differ by five and form a complementary pair. If each of the complementary numbers is multiplied by itself, the results will also form a complementary pair. For example, three multiplied by itself gives nine, and eight multiplied by itself gives sixty-four, whose last digit is four; nine and four differ by five. If a pair of complementary numbers are each multiplied by the same number, the results will always form a pair. For example, three multiplied by three gives nine, and seven multiplied by three gives twenty-one, whose last digit is one; nine and one add up to ten. If each of the complementary numbers is multiplied by itself, the results will be the same. For example, one multiplied by itself gives one, and nine multiplied by itself gives eighty-one, whose last digit is also one; two multiplied by itself gives four, and eight multiplied by itself gives sixty-four, whose last digit is also four.

是以自乘之數，相合之相從者，此得自數，則彼亦得自數也，如一得一，六得六。此得對數則彼亦得對數也，如四得六，九得一。此得連數，則彼亦得連數也。如三得九，八亦得四，二鍀四，七亦得九。相對之相從者，此得自數，則彼得對數也。如一得一，九亦得一，六得六，四亦鍀六。此得連數，則彼亦得連數也，如三得九，七亦撂九，二得四，八亦得四。要皆會於一六四九而齊焉。故開平方之自乘數，止於一六四九而《洛書》之位。一六四九居上下以為經，二七三八、居左右以為緯者，此也。

Therefore, looking at the numbers obtained by self-multiplication: in a group where numbers are paired together, one side gets the original number, and the other side also gets the original number, for example, one self-multiplied gives one, six self-multiplied gives thirty-six, and the last digit is still six; one side gets the number's counterpart, and the other side also gets the counterpart, for example, four self-multiplied gives sixteen, and the last digit is six (four and six add up to ten), nine self-multiplied gives eighty-one, and the last digit is one (nine and one add up to ten); one side gets a consecutive number, and the other side also gets a consecutive number, for example, three self-multiplied gives nine, eight self-multiplied ends with four, two self-multiplied gives four, seven self-multiplied gives forty-nine, and the last digit is nine. In a group where counterparts are paired together, one side gets the original number, and the other side gets the counterpart, for example, one self-multiplied gives one, and nine self-multiplied ends with one; six self-multiplied gives six, and four self-multiplied ends with six; one side gets a consecutive number, and the other side also gets a consecutive number, for example, three self-multiplied gives nine, and seven self-multiplied ends with nine; two self-multiplied gives four, and eight self-multiplied ends with four. In the end, all these numbers converge on the four numbers one, six, four, and nine, and they are consistent. Therefore, when calculating square roots, the numbers obtained by self-multiplication only fall on one, six, four, and nine; and the positions in the Luo Writing also correspond to one, six, four, and nine occupying the vertical positions, while two, seven, three, and eight occupy the horizontal positions, which is the reason for this.