The Yijing Numerical Hidden Diagrams

經名：易數鈞隱圖。原題三衢劉牧撰三卷。底本出處：《正統道藏》洞真部靈圖類。校版本：文淵閣四庫全書本經部易類（簡稱四庫本）。

This book is recorded in the Daozang under the title "Yishu Gouyun Tu." The original title attributes it to Liu Mu of Sanqu, divided into three volumes: upper, middle, and lower. The base text used for collation comes from the Ming Dynasty's Zhengtong Daozang, Dongzhen Bu Lingtu Leiyu; the collation text is a version included in the Wenyuan Ge Siku Quanshu, Jingbu Yi Leiyu, which is hereafter referred to simply as the "Siku Ben."

三衢劉牧撰。

This book was written by Liu Mu, who was from Sanqu (now Quzhou, Zhejiang Province).

夫《易》者，陰陽氣交之謂也。若夫陰陽未交，則四象未立；八卦未分，則萬物安從而生哉？是故兩儀變易而生四象，四象變易而生八卦、重卦六十四卦，於是乎天下之能事畢矣。

The Book of Changes discusses the interaction of yin and yang energies. If yin and yang had not yet interacted, the four images would not have arisen; if the eight trigrams had not yet been differentiated, how could all things have come into being? Therefore, the two principles (yin and yang) transform into the four images, the four images transform into the eight trigrams, and the eight trigrams combined in pairs form the sixty-four hexagrams. At this point, all the principles and phenomena that can be explored are fully established.

夫卦者，聖人設之觀於象也。象者，形上之應。原其本，則形由象生，象由數設；舍其數，則無以見四象所由之宗矣。是故仲尼之贊《易》也，必舉天地之極數，以明成變化而行鬼神之道。則知《易》之為書，必極數以知其本也。

A hexagram is something established by the sages for people to observe the forms and appearances of things. The 'form' refers to the 'supra-formal' principle that corresponds with all things in the universe. Investigating its origin: tangible things arise from forms, and forms are established through numbers; once numbers are set aside, one cannot perceive the origin from which the four trigrams emerge. Therefore, Confucius praised the principles of the Book of Changes, necessarily citing the extreme number of fifty-five to explain the principle of 'achieving transformation and movement, governing spirits and ghosts'. From this we can understand that the Book of Changes must thoroughly investigate numbers in order to recognize its fundamental nature.

詳夫註疏之家，至於分經析義，妙盡精研，及乎解釋天地錯綜之數，則語惟簡略，與《擊辭》不偶，所以學者難曉其義也。今採摭天地奇偶之數，自太極生兩儀而下，至於《復》卦，凡五十五位，點之成圖。於逐圖下，各釋其義，庶覽之者易曉耳。

If we look closely at the various scholars who have commented on and explained the text, they have carefully divided the passages, analyzed the meanings, and studied them in great detail. However, when it comes to explaining the part where the numbers of heaven and earth change and intermingle, their explanations become extremely brief, not matching the original text of the Great Treatise (originally written as "ji ci," a miswriting of "xi ci"). Therefore, scholars find it difficult to understand the meaning. Now I have collected various numbers of odd and even, starting from "the Taiji produces the two principles" down to the hexagram Fu, totaling fifty-five numbers. I have arranged them into a diagram using dots. Each diagram is followed by an explanation of its meaning, hoping that readers will find it easy to understand.

夫《易》道淵邈，雖往哲難窺於至跡。牧也，蕞生祖述，誠魄其狂簡，然則象有定位，變有定數，不能妄為之穿鑿耳。博雅君子，試為詳焉。

The Way of the Book of Changes is profound and difficult to measure. Even the sages of previous generations found it hard to grasp its most subtle aspects. I, Liu Mu, am but an insignificant scholar, able only to transmit the old teachings of others. I am truly ashamed that my understanding is crude and reckless. However, the nature of the images has a fixed order, and change follows a definite number; therefore, I dare not fabricate or impose my own interpretations. Esteemed and learned gentlemen, please examine my words carefully.

太極無數與象。今以二儀之氣，混而為一以畫之，蓋欲明二儀所從而生也。

The Taiji itself has no number and no image to describe. The Taiji diagram drawn here combines the two principles (Wu Wei) into one whole, representing it as an undivided unity. The intention is to show clearly from where the two principles originate.

《經》日：《易》有太極，是生兩儀。太極者，一氣也。天地未分之前，元氣混而為一，一氣所判，是日兩儀。《易》不云乎天地，而云兩儀者，何也？蓋以兩儀則二氣始分，天地則形象斯著；以其始分兩體之儀，故謂之兩儀也。

The Great Treatise says: "The Book of Changes contains the Supreme Ultimate, from which the two principles arise." The Supreme Ultimate refers to a unified vital energy. Before heaven and earth were differentiated, the original qi was undivided; once this qi was divided, it became the two principles. Why does the Book of Changes not directly say "heaven and earth," but instead say "the two principles"? Because when called the two principles, the two energies had just begun to separate; when called "heaven and earth," the forms and phenomena were already distinct. It takes the state when they had just split into two forms, hence it is called the two principles.

何以明其然？略試論之：夫氣之上者輕清，氣之下者重濁。輕清而圓者，天之象也。重濁而方者，地之象也。茲乃上下未交之時，但分其儀象耳。若二氣交，則天一下而生水，地二上而生火，此則形之始也。五行既備，而生動植焉，所謂在天成象，在地成形也。則知兩儀乃天地之象，天地乃兩儀之體爾。

How can we prove this? Let me briefly explain: the part of the qi that rises is light and clear, while the part that sinks is heavy and turbid. The light and clear part, which is round in shape, represents the image of heaven; the heavy and turbid part, which is square in shape, represents the image of earth. This is exactly when the upper and lower qi have not yet interacted, and only two forms of images are distinguished. When the two qi interact, the heavenly qi of the first kind descends to form water, and the earthly qi of the second kind ascends to form fire. This is the beginning of tangible things. After the five elements are fully present, animals and plants then come into being. This is what the Great Treatise says: "In heaven, images are formed; in earth, forms are created." From this we can understand that the two principles are the "images" of heaven and earth, while heaven and earth are the "substances" of the two principles.

今畫天左旋者，取天一、天三之位也。畫地右動者，取地二、地四之位也。分而各其處者，蓋明上下未交之象也。

This diagram depicts heaven rotating to the left, taking the positions of the odd numbers (yang numbers) 天一 and 天三; and earth rotating to the right, taking the positions of the even numbers (yin numbers) 地二 and 地四. By separating them and having each occupy one side, it shows the image of the two energies, heaven and earth, before they interact.

天一、地二、天三、地四，此四象生數也。至於天五，則居中而主乎變化，不知何物也，強名日中和之氣，不知所以然而然也。交接乎天地之氣，成就乎五行之質，彌綸錯綜，無所不周。三才之道既備，退藏於密，寂然無事，茲所謂陰陽不測之謂神者也。

The numbers one, two, three, and four are the "generative numbers" of the Four Symbols. As for the number five, it is located in the center and governs all changes. It is unclear what it really is, so it is merely named "the vitalizing energy of harmony and balance." It arises naturally without a clear cause. It connects the energies of heaven and earth, forms the physical manifestations of the Five Phases, pervades and interconnects everything, and is present everywhere. When the principles of heaven, earth, and humanity are all complete, it retreats to a hidden place, remaining still and inactive. This is precisely the "spirit" mentioned in the Great Treatise, which refers to "the indescribable nature of yin and yang." This diagram places the five points in the center, with empty space around them, depicting precisely this central number five that governs all changes.

《經》雖云四象生八卦，然須三、五之變易，備七、八、九、六之成數，而後能生八卦，而定位矣。

The Great Treatise says only that "the four images give rise to the eight trigrams," but in between there must be "combining three and five to produce change"—the interaction of the generating numbers with the number five of heaven, leading to the four completed numbers of seven, eight, nine, and six. Only then can the eight trigrams come into being, and their directions can be determined.

或問日：天地之數，何以由天五而生變化？答日：天地之生數足，所以生變化也。天地之數十有五，自天一至天五，凡十五數也。天一、天三、天五成九，此陽之數也，故《乾》元用九。地一不地四成六，此陰之數也，故《坤》元用六。兼五行之成數四十，合而為五十有五，備天地之極數也，所以能成變化而行鬼神。

Someone asked: Why does the number of heaven and earth require the number five of heaven to produce change? The answer is: When the number five of heaven is reached, the 'generative numbers' of heaven and earth are complete, so change can arise. The combined generative numbers of heaven and earth total fifteen: from the number one of heaven to the number five of heaven, adding one plus two plus three plus four plus five, the total is fifteen, which is exactly the number of points marked in this diagram. Among these, the three odd numbers, heaven one, heaven three, and heaven five, add up to nine, which is the number of yang, so the "Use of Nine" is mentioned in the Qian hexagram. The two even numbers, earth two and earth four, add up to six (the original text says "earth one not earth four," where "one not" is a mistake for the character "two"), which is the number of yin, so the "Use of Six" is mentioned in the Kun hexagram. Adding the completed numbers of the Five Phases, six, seven, eight, nine, and ten, which total forty, the total becomes fifty-five. At this point, the extreme numbers of heaven and earth are complete, so they can "accomplish change and operate the spirits."

《經》曰：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。義日：參，合也；伍，為偶配也，為天五合配天一，下生地六之類是也。以通其變化，交錯而成四象、八卦之數也。

The Great Treatise says: "By combining three and five, changes are produced; their numbers are interwoven. By comprehending these changes, the text of heaven and earth is formed; by reaching the limit of these numbers, the images of the world are established." These words mean that "参" refers to combination, and "伍" refers to pairing into even numbers—for example, the heavenly number five combines with the heavenly number one, giving rise to the earthly number six below, which is an instance of this. By relying on such combinations, the changes are connected, and they intermingle to form the four basic images and the eight trigrams. The above four diagrams illustrate this passage: the heavenly number one receives the heavenly number five and gives rise to the earthly number six below; the earthly number two receives the heavenly number five and gives rise to the heavenly number seven above; the heavenly number three receives the heavenly number five and gives rise to the earthly number eight on the left; the earthly number four receives the heavenly number five and gives rise to the heavenly number nine on the right. Each of the four generating numbers, when added to the central heavenly number five, becomes a completed number.

成天地之文者，為陰陽交而著其文理也。極其數者，為極天地之數也。天地之極數，五十有五之謂也。遂定天地之象者，天地之數既設，則象從而定也。

The phrase "forming the patterns of heaven and earth" means that after the interaction of yin and yang, their patterns and order became evident. "Exhausting the numbers" refers to investigating the numbers of heaven and earth to their utmost limit. The extreme number of heaven and earth refers to the total of fifty-five. "Thus establishing the images of heaven and earth" means that once the numbers of heaven and earth are arranged, their images are thereby determined.

《經》曰：兩儀生四象。孔氏《疏》謂：金、木、水、火稟天地而有，故云兩儀生四象。土則分王四季，又地中之別惟雲四象也。且金、木、水、火有形之物，安得為象哉？孔氏失之遠矣。

The Great Treatise says: "The two principles give rise to the four images." Kong Yingda's commentary in The Correct Meaning of the Zhou Changes explains that metal, wood, water, and fire are endowed with the qi of heaven and earth, hence they are called the four images. However, earth is strong in the last months of the four seasons and is a special category of the earth's center, so it is not mentioned among the four images. But metal, wood, water, and fire are all tangible things; how can they be called "images"? Kong's explanation is far from correct.

又云：《易》有四象，所以示者。莊氏云：四象謂六十四卦之中有實象，有假象，有義象，有用象也。今於釋卦之處已破之矣。

The Great Treatise also states: "The Book of Changes has four images, so as to show [things]." Zhuang explained that these four images refer to four types among the sixty-four hexagrams: real images, false images, meaningful images, and practical images. This view I have already refuted in my explanation of the hexagram structure.

何氏謂：天生神物，聖人則之，一也；天地變化，聖人效之，二也；天垂象，見吉凶，三也；河出圖，洛出書，聖人則之，四也。今謂此四事，聖人《易》外別有其功，非專《易》內之物。稱《易》有四象，且又云：《易》有四象，所以示也；擊辭焉，所以告也。然則象與辭，相對之物。辭既爻卦之下辭，象謂爻卦之象也。

He Fei regarded the four things listed in the Great Treatise as the four trigrams: the birth of divine objects such as the yarrow stalks and tortoise shells, which the sage took as a model, as the first; the transformation of heaven and earth, which the sage imitated, as the second; the heavenly signs appearing in physical forms to indicate auspiciousness and misfortune, as the third; and the emergence of the River Diagram and the Lake Book from the Yellow River and Luo River, which the sage took as a model, as the fourth. I believe these four things are the achievements of the sage outside the Book of Changes, and not exclusively internal to the Book of Changes, and thus cannot be used as evidence for the statement that "the Book of Changes has four trigrams." Moreover, the text also says: "The Book of Changes has four trigrams, so that they may be shown; and explanations are attached, so that they may be told." This clearly shows that "trigrams" and "explanations" are two distinct things: since "explanations" refer to the words attached below the hexagrams and lines, "trigrams" must refer to the images inherent in the hexagrams and lines themselves.

上兩儀生四象，七、八、九、六之謂也。諸儒有謂七、八、九、六，今則從以為義也。且疏家以七、八、九、六之四象，為所以示之四象，則駁雜之甚也。何哉？夫七、八、九、六，乃少陰、少陽、老陰、老陽之位，生八卦之四象，非《易》之所以示四象也。

The four images mentioned in the previous text "Two Principles give rise to Four Images" refer to the numbers seven, eight, nine, and six. Among various schools of thought, some have already proposed the idea that these four numbers represent the four images. I will now follow this interpretation. However, commentators have confused the group of four images consisting of seven, eight, nine, and six with the group of four images mentioned in "so as to illustrate," making the explanation very messy. Why? The numbers seven, eight, nine, and six correspond to the numbers of Young Yang, Young Yin, Old Yang, and Old Yin, respectively. They are used to generate the eight trigrams, and they are not the four images that the Book of Changes uses to convey its teachings to people.

略試論之：且夫四象者，其義有二：一者謂兩儀所生之四象，二者謂《易》有四象，所以示之四象，若天一、地一不天三、地四所以兼天五之變化，上下交易，四象備其成數，而後能生八卦矣。於是乎，坎、離、震、兌居四象之正位。不云五象者，以五無定位，舉其四則五可知矣。

Let me briefly discuss: the term "four images" has two meanings. The first refers to the four images generated from the two principles, and the second refers to the "four images of the Book of Changes, so as to show them." In the first sense, the four generating numbers, namely heaven one, earth two, heaven three, and earth four, each combine the transformation of heaven five. They interact with each other above and below, and the four images complete their numbers, thus giving rise to the eight trigrams. Therefore, the four trigrams Kan, Li, Zhen, and Dui occupy the central positions of the four images. The reason we do not say "five images" is that five has no fixed direction; by mentioning these four, five is naturally included among them.

夫五上駕天一，而下生地六；下駕地二，而上生天七；右駕天三，而左生地八；左駕地四，而右生天九，此河圖四十有五之數耳。斯則二儀所生之四象。

Heavenly Five moves upward to control Heavenly One, and generates Earthly Six below; it moves downward to control Earthly Two, and generates Heavenly Seven above; it moves right to control Heavenly Three, and generates Earthly Eight on the left; it moves left to control Earthly Four, and generates Heavenly Nine on the right—this is the number of forty-five points in the He Tu diagram (the numbers one through nine add up to forty-five). All of the above belong to the group of four images generated from the two principles.

所謂《易》有四象，所以示者，若《繫辭》云：吉凶者，失得之象，一也；悔吝者，憂虞之象，二也；變化者，進退之象，三也；剛柔者，晝夜之象，四也。且孔氏《疏》云：象之與辭，相對之物。辭既爻卦之下辭，象謂爻卦之象也。又上句云：《易》有四象，所以示也。下句云：擊辭焉，所以告也。

As for the four images mentioned in the phrase "The Book of Changes has four images, so as to show," these should be determined according to what is said in the Great Treatise. The first is jiu xiong, which refers to the image of gain and loss; the second is hui lin, which refers to the image of worry; the third is bian hua, which refers to the image of advancing and retreating; the fourth is gang rou, which refers to the image of day and night. Moreover, Kong Yingda's commentary itself states that images and words are two different things. Since words refer to the textual expressions attached to the hexagrams and lines, images refer to the symbolic representations of the hexagrams and lines. The previous sentence says "The Book of Changes has four images, so as to show," and the next sentence immediately follows with "attached are words, so as to inform."

詳其吉凶、悔吝、變化、剛柔四者之象，既《擊辭》所陳，則與爻卦正協其義也。而又孔氏復引二儀所生之四象，舉七、八、九、六之數，則其義非也，不亦失之甚乎？

Carefully examining the images of auspiciousness, misfortune, hesitation, and transformation, since all these are stated in the Great Treatise, they naturally correspond with the hexagrams and yao lines, and their meanings are coherent. However, the Master Kong introduced the four images generated by the two principles, bringing in the numbers seven, eight, nine, and six, which is incorrect. Is this not a serious mistake?

五行成數者，水數六，金數九，火數七，木數八也。水居坎而生乾，金居兌而生坤，火居離而生巽，木居震而生艮。巳居四正而生乾、坤、艮、巽，共成八卦也。

The completed numbers of the Five Phases are: water's completed number is six, metal's is nine, fire's is seven, and wood's is eight. Water resides in the position of Kan and gives rise to the Qian trigram, metal resides in the position of Dui and gives rise to the Kun trigram, fire resides in the position of Li and gives rise to the Xun trigram, and wood resides in the position of Zhen and gives rise to the Gen trigram. Once Kan, Li, Zhen, and Dui have occupied the four cardinal positions (the original text's "巳" is a mistake for "已"), each of these four cardinal positions gives rise to one trigram, resulting in the four trigrams of the corners: Qian, Kun, Gen, and Xun. Adding the four cardinal trigrams to the four corner trigrams, the Eight Trigrams are formed. This diagram distributes the numbers six, seven, eight, and nine among the four cardinal positions, and then draws one trigram from each cardinal position to the four corner positions.

此乃五行生成數，本屬洛書，此畫之者欲備天地五十五數也。

This picture depicts the pairing of the generating numbers and the completed numbers of the Five Phases—the one with six, two with seven, three with eight, four with nine, and five with ten, which originally belong to the Luoshu diagram. Here, it is drawn to make up the total number of fifty-five for heaven and earth.

內十五，天地之用，九六之數也。兼五行之數四十，合而為五十有五，備天地之數也。

These three diagrams arrange the odd numbers of Heaven: one, three, five, seven, and nine, and the even numbers of Earth: two, four, six, eight, and ten, as well as the complete set of numbers combining both. The fifteen dots in the inner layer represent the sum of the Heaven and Earth generating numbers, which are the numbers based on the hexagram Qian's "All Nines" and Kun's "All Sixes." Adding the five elements' completing numbers makes a total of forty dots, resulting in a total of fifty-five, completing the numbers of Heaven and Earth.

《經》曰：凡天地之數五十有五，此所以成變化而行鬼神也。又日：大衍之數五十，則臧天地之數五也。韓氏日：演天地之數，所賴者五十也，則不言臧五之數。所以孔氏《疏》以為五十有五，乃天地陰陽奇偶之數，非是文演天地之策也。

The Great Treatise says: "All the numbers of heaven and earth total fifty-five, which is why they can produce changes and act as ghosts and spirits." It also says: "The great derivation involves fifty numbers." This shows that the number of the great derivation is obtained by subtracting five from the numbers of heaven and earth (the original text "臧" is a mistake for "减," meaning "to subtract"). Han Kangbo's commentary says that the derivation of the numbers of heaven and earth is based on fifty, but he does not explain why five is subtracted. Therefore, Kong Yingda's commentary argues that fifty-five is merely the total number of yin and yang odd and even numbers in heaven and earth, and not the actual number used by the sage in calculating the divination sticks (this sentence is missing a character in the original text, and the same is true in the Siku Quanshu edition).

且諸儒分大衍之數，分而為二之義中，則述天地之數五十有五之用，末則陳四營成易、十有八變而成卦之理。此豈可同乎本末，而異其中之數也？瓦乎蝶曹之數以象天地，豈可舍其數而求其象乎？斯亦疏家之失，不求天五退藏於密之義也。

Now, when various schools explain the "Great Derivation Number," they refer to the part about "dividing into two" based on the number fifty-five of heaven and earth. But when they reach the end, "four operations form the Book of Changes, and eighteen transformations form the hexagram," they rely on another set of numbers. Since the beginning and end both refer to the number of heaven and earth, how can the numbers in between be based on a different system? Moreover, the number used in divination with yarrow stalks is originally meant to symbolize heaven and earth (the original text's "蝶曹" is a mistake for "揲蓍"), so how can one ignore its numerical aspect and seek only its symbolic meaning? This is also a mistake in the commentary. The root of the problem lies in not investigating the meaning of "five of heaven hidden in secrecy."

且夫五十有五，天地之極數也。大衍之數，天地之用數也。蓋由天五不用，所以大衍之數少天地之數五也。

Fifty-five is the extreme number of heaven and earth; the great derivation number fifty is the number used by heaven and earth. Since the number five of heaven is not directly involved in use, the great derivation number is five less than the number of heaven and earth.

或曰：天五不用，何以明其不用之由？答日：天五不用，非不用也，是用四象者也。且天一、地二、天三、地四，此四象生數也，天五所以斡四象生數而成七、九、六、八之四象，是四象之中皆有五也。則知五能包四象，四象皆五之用也。舉其四，則五在其中矣。故《易》但言四象，以示不言五象也。

Someone asked: If it is said that the number five of Heaven is not used, on what basis can we explain why it is not used? The answer is: The statement that the number five of Heaven is not used does not mean it is truly unused, but rather that it is used in the four trigrams. The numbers one and two of Heaven and Earth, three and four of Heaven and Earth are the generating numbers of the four trigrams. The number five of Heaven coordinates these four generating numbers, transforming them into the four completed numbers seven, nine, six, and eight. That is, two plus five equals seven, four plus five equals nine, one plus five equals six, and three plus five equals eight. This shows that the number five is present in each of the four trigrams. Therefore, the number five encompasses the four trigrams, and the four trigrams are all the result of the number five. By mentioning these four, the number five is already included. Hence, the Book of Changes only mentions the four trigrams to reveal to people, and does not mention the fifth trigram.

今搽蓄之義，以筮而尚占者也，以象天地之用數，所以大衍之數臧天地之數五也。

The purpose of using the divination method of counting the yarrow stalks is to emphasize divination and inquiry, drawing upon the symbolic representation of the numerical patterns of heaven and earth. Therefore, the number of the Great Transformation is five less than the number of heaven and earth.

大衍之數五十，其用四十有九。韓氏注日：衍天地之數，所賴者五十。其用四十有九，則其一不用也。不用而用以之通，非數而數以之成，斯《易》之太極也。四十有九，數之極也。夫元不可以元明，必因於有，固嘗於有物之極，而必明其所由之宗也。

The Great Treatise says: "The number of the great derivation is fifty, but only forty-nine are actually used." Han Kangbo's commentary explains that the number used to deduce the principles of heaven and earth is fifty, but only forty-nine are actually employed, meaning one stalk of yarrow is left unused. This "one" does not participate in the actual use, yet all the use depends on it to be connected; it is not a number, yet all numbers depend on it to exist. This is the Taiji of the Book of Changes. Forty-nine represents the ultimate of numbers. "Nothing" cannot be made evident by "nothing" itself; it must rely on "something." Therefore, the origin of things is revealed precisely where the form reaches its extremity. (In the original text, every occurrence of the character "yuan" is a mistake for "wu," and the same applies below.)

孔氏《疏》：京房云：五十者，謂十日、十二辰、二十八宿也。凡五十其一不用者，天之生黑將欲以虛求實，故用四十九焉。馬季長云：《易》有太極，謂北辰。北辰生兩儀，兩儀生日月，日月生四時，四時生五行，五行生十二月，十二月生二十四黑。北辰居位不動，其餘四十九運而用之也。

Kong Yingda cites several previous scholars in his "Justice." Jing Fang says: fifty refers to the total number of ten heavenly stems, twelve earthly branches, and twenty-eight constellations added together; one is omitted because the qi generated by heaven requires emptiness to seek reality, so only forty-nine are used. Ma Rong (style name Jichang) says: The "I Ching" has the Taiji, which is the North Star; the North Star gives rise to the two principles, the two principles give rise to the sun and moon, the sun and moon give rise to the four seasons, the four seasons give rise to the Five Phases, the Five Phases give rise to the twelve months, and the twelve months give rise to the twenty-four solar terms; the North Star remains in its place and does not move, while the other forty-nine are in motion to be used by people. (In the original text, all instances of the character "he" are errors for "qi.")

荀爽云：卦各有六爻，六八四十八，加《乾》、《坤》二用，凡五十。初九潛龍勿用，故用四十九也。鄭康成云：天地之數，五十有五者，以五行黑通於萬物，故臧五；大衍又臧一，故用四十九。姚信、董遇云：天地之數，五十有五者，其六以象六畫之數，故臧而用四十九也。顧懼云：立此五十數以數神，神雖非數，因子而顯，故虛其一數，以明不可言之義也。

Xun Shuang said: Each of the eight trigrams has six lines, six multiplied by eight gives forty-eight. Adding the two special lines, 'Use Nine' in the Qian hexagram and 'Use Six' in the Kun hexagram, makes fifty. However, since the initial line of the Qian hexagram is 'The Dragon Hidden, Not to Be Used,' the actual number used is forty-nine. Zheng Xuan (styled Kangcheng) said: The number of heaven and earth is fifty-five because the five phases' energies permeate all things, so five is subtracted. Then again, the Great Derivation subtracts one more, resulting in forty-nine. Yao Xin and Dong Yu said: The number of heaven and earth is fifty-five, where six is used to symbolize the six lines of a single hexagram, so six is subtracted, resulting in forty-nine. Gu Huan said: The number fifty is established to calculate numbers in relation to the divine. Though the divine is not a number, it is revealed through numbers. Therefore, one number is left out to indicate the meaning that cannot be spoken.

今詳諸家所釋，義有多端，雖各執其說，而理則未允。敢試論之：韓氏《注》以虛一為太極，則未詳其所出之宗也。何者？夫太極生兩儀，兩儀既分，天始生一，肇其有數也。而後生四象五行之數，合而為五十有五，此乃天地之極數也。

Looking carefully at the various interpretations, they are all confused and contradictory, each holding to its own view, yet none of the reasons are convincing. Please allow me to attempt my own explanation. Han Kangbo took the 'one' that is empty and not used as the Taiji, but he did not clarify where this Taiji comes from. Why is this so? Taiji gives rise to the two principles, and after the two principles are separated, the 'one' is born from Heaven, and thus numbers begin. Then from this, the four images and the numbers of the Five Phases are generated, totaling fifty-five, which is the extreme number of heaven and earth.

今若以太極為虛一之數，則是大衍當用五十有四也。不然，則餘五之數，元所設耳。況乎大衍衍天地之數也，則明乎後天地之數矣。大衍既後天地之數，則太極不可配，虛其一之位也明矣。

Now, if the Taiji is considered the empty and unused "one," then the Da yan should only subtract one from fifty-five, leaving fifty-four for practical use. If not calculated this way, the remaining five numbers would become idle numbers with nowhere to be placed. Moreover, Da yan is used to deduce the numbers of heaven and earth, indicating that it comes into being after the numbers of heaven and earth. Since Da yan comes after the numbers of heaven and earth, Taiji cannot be paired with it, and thus cannot serve as the empty and unused "one" position. This point is quite clear.

又元不可以元明，必因於有，是則以太極為元之稱。且太極者，元黑混而為一之時也。其黑已兆，非元之謂，則韓氏之《注》，義亦迂矣。

Han's commentary also says, "Nothing cannot be manifested by itself; it must rely on something existing." This equates the Taiji with the term "wu" (nothingness). However, the so-called Taiji refers to the stage where the primordial qi is unified as a whole; since the qi has already begun to manifest, it cannot be called "wu" anymore. In this view, Han's commentary appears to be overly convoluted.

或曰：韓氏之《注》，承輔嗣之旨。且輔嗣之《注》，獨冠古今，斐然議之，元乃不可乎？答曰：此必韓氏之寓言，非輔嗣之意也。且若愚以胸臆論之，是謂狂簡。今質以聖人辭，且《易》有太極，是生兩儀，《易》既言有，則非元之謂也。不其然乎？

Some people say: Han Kangbo's commentary follows the spirit of Wang Bi (styled Fusi). Wang Bi's "Commentary on the Yi" is unparalleled through the ages. How can you criticize it so extensively? The answer is: This statement must be Han's own way of expressing his views, not Wang Bi's original intention. Moreover, if I were to criticize merely based on my own opinion, that would be reckless and hasty. Now, I am using the words of the sages to challenge it — the "Great Treatise" says, "The Yi contains the Taiji, from which the two principles arise." The text says "has," not "does not have." Is it not so?

至於京、荀、馬、鄭眾賢之論，皆採摭天地名數，強配其義。□□□且若以天地之名數，強加配偶，則靡所不可。□然而天地之數，生成相因，理如貫珠，不可駢贅而設也。雖能強立其義，推而究之，則於所由之宗不會矣。試論於末篇。

As for the interpretations of sages such as Jing Fang, Xun Shuang, Ma Rong, and Zheng Xuan, they all selected existing names and numbers from heaven and earth, forcing them together to form a meaning. (The original text is missing several characters here, and the Siku Quanshu edition also notes this omission.) If names and numbers from heaven and earth can be forced into such pairings, then nothing would be impossible to pair. (The original text also has missing characters here.) However, the numbers of heaven and earth, the original numbers and the completed numbers, are interdependent and mutually generating, their order resembling a string of beads connected in sequence. They cannot be arbitrarily added like extra fingers or redundant warts. Although these interpretations maywith difficulty hold together a certain meaning, when pursued further, they ultimately fail to connect with the original source of the numbers. This point I will discuss in the final chapter.

天地之數十有五，居其內而外斡；五行之數四十也，今止用其四十九者，何也？蓋由天五為變化之始，散在五行之位，故中元定象。又天一居尊而不動，以用天德也。天德，九也。天一者，象之始也，有生之宗也，為造化之主，故居尊而不動也。

The numbers of Heaven and Earth amount to fifteen in total, located in the inner layer and rotating outward. The completed numbers of the Five Phases total forty, arranged in the outer layer. Now only forty-nine are used. This is because the number five of Heaven is the beginning of all changes, which has already been distributed among the five phases. Therefore, there is no fixed symbol in the center (the term "yuan" in "zhongyuan dingxiang" refers to "wu," meaning "nothing"). Furthermore, the number one of Heaven occupies the honorable position and remains still, used to express the virtue of Heaven. The virtue of Heaven is nine. The number one of Heaven is the starting point of symbols, the origin of all living things, and the master of creation, so it remains still in the honorable position.

惟天三、地二、地四之數，合而成九陽之數也。天三則乾之三畫，地二、地四則坤之六畫也。地道元成而代有終，陽得兼陰之義也。故乾之三兼坤之六，成陽之九，斡運五行成數，而通變化也，所以搽曹之義以象其數也。

Only when the numbers three from Heaven, two from Earth, and four from Earth are combined together can the yang number nine be formed. The three from Heaven corresponds to the three lines of the Qian hexagram; the two and four from Earth add up to six, corresponding to the six broken lines of the Kun hexagram. The Earth path itself achieves nothing, but can only take the place of Heaven to complete its affairs, which is precisely the reason why yang can encompass yin. Therefore, the three lines of Qian incorporate the six lines of Kun, forming the yang number nine, which governs the five phases' completed numbers and thus connects all changes. The method of divination with yarrow stalks is intended to symbolize this number.

或問日：《易》云《坤》元用六，今則乾三兼之，是坤之六元用乎？答日：非也，在其中矣。此蓋《易》舉其多數而言之也。數六是少數，舉其多則少可知矣。是知陽進而《乾》元用九，陽退則《坤》元用六也。亦由當期之日，惟合老陰、老陽之數，其少陰、少陽之數，則在其中。舉多兼少，《易》義皆然矣。

Someone asked: The Book of Changes clearly states that the hexagram Kūn has "use of sixes," but now it is said that the three lines of Qian have absorbed it. Then, is the six of Kūn not of any use? The answer is: Not at all useless, but it is already included within. This is how the Book of Changes speaks by emphasizing the greater aspect: six is a smaller number, so by mentioning the greater, the smaller is naturally understood. From this we can see that when yang energy advances, Qian has "use of nines," and when yang energy recedes, Kūn has "use of sixes." This is similar to the full count of 360 days in a year, which is composed only by the two extreme numbers, old yang (Qian's count of 216) and old yin (Kūn's count of 144). The counts of young yang and young yin are naturally included within. By emphasizing the greater and including the smaller, this is the consistent method used in the Book of Changes.