Cross (Christian symbol)

泰西之十字架，相傳以為耶穌代眾人受刑釘死於十字架上，故尊奉之，以為耶穌流血之紀念。此宗教家附會之說，不足信也。其實十字架者，乃數學之交線也。數不交不生，如兩線平行，各不相交，雖引之至於極長，縱環繞地球一周，仍為兩平行線而已，不生數也。惟兩線相交，成十字形，動則為圓，靜則為矩，而三角勾股八線，皆由此生焉。此乃幾何原本之原本，實數學之初祖。

The cross in the West is said to have originated because Jesus suffered crucifixion on a cross-shaped scaffold, and believers revere it as a memorial of Jesus' sacrificial death. This is a fanciful explanation by religious people, not reliable. In reality, the original meaning of the cross is the intersection of two lines in mathematics. Numbers cannot be formed without intersection; for example, two parallel lines go their separate ways and never meet, even if extended infinitely around the Earth, they remain just two parallel lines, producing no numbers. Only when two lines intersect, forming a cross, can they rotate to create a circle, and when stationary, form a right angle ruler. Triangles, the Pythagorean theorem, and trigonometric functions such as sine and cosine all originate from this. Therefore, this cross is the fundamental basis of Euclid's Elements, truly the ancestor of mathematics.

我國相傳之兩儀圖（今通俗皆稱為“太極圖”，其實此圖分陰分陽，有黑有白不可謂之太極，當名為兩儀圖，校為愜當。北方亦有稱之為陰陽魚兒者，魚字亦儀字之傳訛也），天然之配偶也。兩儀為理學之祖，由兩儀生四象，四象生八卦，變化無窮，莫不肇端於此一陰一陽。一陰一陽之謂道。道也者，形而上者也。

The traditional Chinese "Two Principles Diagram" (now commonly called the "Taiji Diagram") is actually a natural counterpart to the Western cross. One represents numbers, the other principles; one from the West, one from the East, perfectly complementary. The Two Principles are the origin of Neo-Confucianism. From the Two Principles come the Four Symbols, from the Four Symbols come the Eight Trigrams, and the transformations are endless, all originating from the interplay of one yin and one yang. The "Great Treatise" states, "Yin and yang together are called the Dao," and "Dao" belongs to the realm of the ultimate reality.

交線成勾股成三角八線，推衍無盡，莫不導源於此一縱一橫。一縱一橫數也。數也者，成器之所先也。形而下者也。故道運於虛，而數徵諸實。我國數千年來，專尚儒家以空言談經，鄙術數為小道，崇虛黜實，末流之弊，舉國皆無用之學。所謂形而上者，幾墜於地矣。

As for the intersection of lines, it gives rise to right angles, triangles, and trigonometric functions, and its derivations are endless, yet all originate from a single vertical and a single horizontal line. The vertical and horizontal lines belong to 'numbers'. Numbers are the prerequisite for making objects and belong to the level of the material world. Therefore, 'Dao' operates in the abstract, while 'numbers' must be verified through concrete facts. For thousands of years in our country, we have only praised Confucian scholars who talk about classics in abstract terms, and regarded numerical methods as a minor and inferior discipline. We have favored the abstract and rejected empirical verification, leading to the consequence that the whole nation's learning has become useless. Even their own proclaimed metaphysical Dao has almost fallen into oblivion, with no one to carry it forward.

泰西之學，則不尚空談，立一說必徵諸實驗，制一器必極其效用，不以理想為止境，不以無成而中輟，千迴百折，精益求精，於是科學功能，幾俟造化；器物利用，無阻高深；形學發達，於斯為盛。然極其弊，則謂世界將可以力爭，強權兒足以勝天演。物慾無限，而生人者適以殺人，殺人者即以自殺。物質之文明，浸成□焉不可終日之勢。此倚重於數之一偏，與倚重於理之一偏，各趨極端。其末流之失，亦正相等也。

Western scholarship does not emphasize empty talk; when establishing a theory, it must be verified by experiment, and when making an object, it must fully utilize its function, not taking imagination as the end, nor abandoning it halfway due to lack of immediate success. Through repeated trials and continuous refinement, the function of science almost rivals nature. The use of tools is not hindered by their complexity, and geometry thus flourishes greatly. However, pushing this to an extreme brings about drawbacks: people come to believe that worldly matters can be decided by force, and that power alone can overcome the laws of natural evolution. Unlimited desires, originally means for sustaining life, are now used for killing; the means for killing ultimately become self-destruction. When material civilization reaches this stage, it gradually becomes precarious, unsustainable. This is the extreme of the aspect of 'numbers,' while our country's emphasis on 'principles' also leads to extremes, and the errors of their respective extremes are equally severe.

夫理與數，本不可以須臾離。故聖人倚數，必參天而兩地。故形上之道，與形下之器，雖相生相成，無偏重亦無偏廢。舍道而言器，則器為虛器。離器而言道，則道盡空談。三代傳統，一揆諸道（故曰“道統”），不但禮樂政刑，悉本度數（度生於律，律本黃鐘，子丑一二），即德行言語（言語即文章也），亦無不各有其典則。（《孝經》“先王之法言法行”。法者，即合於度數者也。）故節以制度，以議德行，而《大學》治國平天下之道，必本於絜矩也。

Reason and numbers have never been separated for a moment. Therefore, the sage established numbers by following the principle of "participating in heaven and dividing by two," using the heavenly number three and the earthly number two to match. Although the formless Dao and the formed objects (qi) are interdependent and complementary, neither should be overemphasized nor neglected. If one discards Dao and talks only about objects, the objects become empty shells; if one leaves objects and talks only about Dao, Dao becomes mere empty talk. The traditions of the Xia, Shang, and Zhou dynasties all return to the root of Dao (hence called the "Dao tradition"), not only because rites, music, politics, and punishment are based on measurements and numbers (measurements come from the pitch, which is based on the Huangzhong tone, matching the number of Zi one and Chou two), but even virtues and speech (speech is literature) have their own rules and standards. (The "Xiao Jing" says, "The methods of the former kings' words and deeds," where "method" refers to conformity with measurements and numbers.) Therefore, the "Great Treatise" says to "regulate by institutional rules and discuss virtue," and the "Great Learning" also states that the way to govern the country and pacify the world must be based on "the principle of the square measure."

道不可見，故聖人示之以象。象無可稽，故聖人又準之以數。數與象合，而道無不可見。制器尚象，而器以立。載道以器，而道不虛。理象數一貫之道，皆出諸《易》。自王弼以玄理說《易》，後世畏象數之繁，因靡然從之，創掃象之說。（弼以玄理說《易》，運實於虛，歸有於無。芻狗天地，糟粕仁義，更何有於象？後儒既主其說，乃闢其玄談，是買櫝而還珠，亦非弼之所及料也。）自是象數與《易》，又離為二。

The Way is invisible, so the sage used "xiang" to display it; since "xiang" cannot be examined, the sage then used "shu" to define it. When "shu" and "xiang" are combined, the Way becomes visible. By using "xiang" to make objects, objects can be formed; by using objects to carry the Way, the Way does not become empty. The study that integrates principle, xiang, and shu all comes from the Book of Changes. Since Wang Bi explained the Book of Changes with metaphysical principles, later people feared the complexity of xiang and shu, and followed him blindly, creating the doctrine of "sao xiang" (rejecting xiang). (Wang Bi used metaphysical principles to explain the Book of Changes, transforming the concrete into the abstract and the existent into non-existence, viewing heaven and earth as straw dogs and benevolence and righteousness as dregs, leaving no room for xiang. Later Confucian scholars adopted his views but attacked his metaphysical discussions, which is like buying the box and returning the pearl, something Wang Bi could not have foreseen.) From then on, xiang and shu have been separated from the Book of Changes.

周子傳太極圖（實即兩儀，以為繼述道統之圖），道果在是矣，而器已無存（禮樂政刑皆器也），則道亦不亡而亡矣。諺曰“皮之不存，毛將焉附”，其是之謂歟？是故三代而後，易學晦盲，數學流於西方（西人於借根方名之曰“東來法”），理學顯於有宋。即此一縱一橫之十字架，分陰分陽之兩儀圖，足以為東西近世學術源流之代表矣。

The Taiji diagram passed down by Zhou Dunyi is actually the diagram of the two principles (two yi), using it as a model to inherit the transmission of the Dao. The Dao is indeed present here, but the instruments have completely disappeared (rituals, music, politics, and law are all instruments). Therefore, the Dao has only a name but no real existence. The proverb says, "If the skin is gone, where can the hair attach?" This is exactly the situation described. Therefore, after the Three Dynasties, the study of the Book of Changes became obscure and unclear. Mathematics was passed on to the West (Westerners call algebraic methods such as the method of solving equations by radicals the 'Eastern method,' showing their origin in the East). Meanwhile, Neo-Confucianism became prominent only during the Song Dynasty. On one side there is only numbers, on the other only principles. This vertical and horizontal cross and the diagram of the two principles (yin and yang) can serve as two representative symbols of the development of modern scholarship in both Eastern and Western traditions.

近西人極物質文明之益，即倦而知返，更探其原於哲學。我受理學空疏之害，尤備嘗苦痛力求自拔於沉淪。是東西人心，已同有覺悟之機。所謂通其變使民不倦，《易》窮則變變則通通則久，其在斯乎！故合中西之學，融會而貫通之。以此所有餘，助彼所不足；截此之所長，補彼之所短，一轉移間而各劑其平，各得其當，非《易》道又烏能貫而通之！

In recent times, Westerners have enjoyed the benefits of material civilization to the fullest, then become weary and turn to philosophy to seek the fundamental principles. We, on the other hand, have suffered greatly from the emptiness of Neo-Confucianism and are striving desperately to free ourselves from thispredicament. It is evident that the hearts of people in both East and West have now reached a moment of awakening. The Great Treatise says, "To understand the changes, so that the people will not grow weary," and "The Book of Changes reaches an end, then changes; changes lead to smoothness; smoothness leads to longevity." These words precisely describe the current situation. Therefore, we must integrate Eastern and Western learning, combine them and make them coherent. Use the excess of one side to compensate for the deficiency of the other, take the strengths of one to make up for the weaknesses of the other. With a single turn, both sides will achieve balance and appropriateness. If it is not the Way of the Book of Changes, what else could unify them?

夫十字架與兩儀圖，各產生於數千年前。一東一西，不相謀面。自西教東漸，於是天然配偶之兩代表，乃日相接近。又迭經幾許之歲月，始得消除種種之障害隔閡，而日即親洽，今殆去自由結婚之期會不遠矣。結婚以後，必能產生新文明之種子，為我全世界放一異彩。吾將摻券期之，拭目俟之矣。

The cross and the diagram of the Two Principles originated thousands of years ago, one in the East and one in the West, never having met before. Since the spread of Western religion to the East, these natural partners have gradually drawn closer. After many years, they have progressively overcome various obstacles and misunderstandings, becoming more and more intimate and harmonious. Now, the day when they can freely marry is probably not far off. After the marriage, they will certainly produce seeds of a new civilization, bringing a brilliant light to our whole world. I wish to hold this paper marriage contract as evidence, waiting for that day to come, watching with eager eyes.

宋以前言《易》者，未有圖也，而太極更不可圖。自周濂溪始為太極圖，又於太極之上添一無極，曰“無極而太極”。後儒因之，遂以太極為第二位，似乎太極之前，更有一無極之圖恍然在於心目間者，而不知其說之不可通也。

Before the Song dynasty, those who explained the Book of Changes never drew diagrams, let alone the Taiji (Supreme Ultimate). After Zhou Dunyi drew the Taiji diagram, he added the concept of "Wují" (Ultimate Void) above the Taiji, saying "Wují gives rise to Taiji." Later Confucian scholars followed this idea, placing Taiji in second position, as if there was another diagram of Wují floating vaguely in the mind before Taiji. But they did not realize that this notion is fundamentally incoherent.

《繫傳》曰“《易》有太極”。極者，至極而無對之稱。物各有極，故曰“天極”，曰“地極”，曰“人極”。陽有陽極，陰有陰極，所以別之曰“太極”。太者，至尊無上之稱，原以對三極及其他諸極而言也。即曰太極，則太字之上，又何能更有所加？況無者，有之對也。既可名之曰無，則無之對便是有。若云無極而有極則可，無極而太極，則文不當而辭不順也。

The *Great Treatise* says, "The *Book of Changes* has the Supreme Ultimate." "Ultimate" refers to the utmost limit, something without a counterpart. All things have their own ultimate, so there is a "heavenly ultimate," a "terrestrial ultimate," and a "human ultimate"; yang has its yang ultimate, and yin has its yin ultimate. In order to distinguish these, it is called the "Supreme Ultimate." "Tai" means supreme and exalted, and originally it is a name established in contrast to the three ultimates and other various ultimates. If it is already called the Supreme Ultimate, what can be added above the character "Tai"? Moreover, "wu" (nothingness) is in contrast to "you" (being). Since it can be called nothingness, then the opposite of nothingness is being. If one says "from nothingness to the ultimate of being," the reasoning is still coherent. But to say "from nothingness to the Supreme Ultimate" is inconsistent in terms of meaning and also awkward in expression.

夫太極者，渾淪無端，立乎天地之先，無名可以名之，無象可以象之。老子所謂“有物無形，先天地生，無以名之，強名之曰道”者，庶乎其近之。《易》道元亨利貞，而太極者，乃元之所自出。今之言幾何學者，曰點線面體，此物質之元亨利貞也。故物之形皆起於一點，謂之起點。而以精神上言之，則必有以立乎此起點之前者。

The Taiji is a single, undivided entity, without beginning or end, existing before heaven and earth. It cannot be named or imaged. Laozi said, "There is something that has substance but no form, born before heaven and earth. It cannot be named, so it is called Dao." This is almost the same meaning. The Book of Changes speaks of four virtues: Yuan, Heng, Li, and Zhen. The Taiji is the source from which Yuan originates. Today, those who study geometry say that points, lines, surfaces, and solids are four elements, which correspond to the four virtues in the material world. Therefore, any form begins with a point, called the "starting point." But from the spiritual perspective, there must be something that exists before this starting point.

言能得此起點，譬有二人於此，甲繪平圓圖，而乙繪立方圖。平圓立方，各有其起點，起點之形式同也。而繪成以後，則圓者自圓，方者自方。則因甲於起點之前，其胸中之所蘊蓄者，已完具一平圓之形象。乙於起點之前，其胸中之所蘊蓄者，已完具一立方之形象。此蘊蓄者，即為平圓與立方之太極也。然當其未落筆以前，又烏從而見之？烏從而聞之？

This principle can be explained from the starting point. For example, here are two people: Jia draws a circle, and Yi draws a square. Both the circle and the square have their own starting points, and the form of the starting point is exactly the same. However, after they are drawn, the circle remains naturally circular, and the square remains naturally square. This is because before Jia begins to draw, the complete image of a circle already exists in his mind; before Yi begins to draw, the complete image of a square already exists in his mind. This stored image in the mind is the Taiji of the circle and the square respectively. However, before they start to draw, where can this stored image be seen or heard? How can it be perceived? Clearly, although it is real, it cannot be sought after in terms of sound or form.

所以雲萬事萬物各有其太極，而《易》有太極，則在兩儀未判陰陽無始以前，先天渾沌之中，自有此肫然穆然、孕育萬物，具足萬理之渾淪元氣，以立乎天地之先，而為造化之主宰者，無形可象，無名可稱。孔子稱之為太極，已至矣盡矣。更何能於此外再加一字，再著一筆乎？加一字，著一筆，即非太極矣。

Therefore, all things and phenomena have their own Taiji. As for the Taiji mentioned in the Book of Changes, it exists before the two principles (yin and yang) have been differentiated and before the beginning of yin and yang, within the primordial chaos of the Earlier Heaven. There is inherently a single, sincere and tranquil primordial qi that nurtures all things and contains all principles. It stands before heaven and earth, as the master of creation. It cannot be depicted in form or named. Confucius gave it the name 'Taiji', which already expresses the ultimate and the extreme. How could there be another character or stroke added to this name? Once a character is added or a stroke is made, it falls into words and forms, and thus is no longer Taiji.

惟周子之所謂太極者，即有其圖，圖中有黑有白，有陰有陽，是即兩儀四象，實已不可謂之曰太極。故又加一無極於其上。蓋為中人以下說法，非有跡象可指，不能以言語形容，乃繪此圖以明一陰一陽之道所由來，使人可一覽而得，其用心亦良苦矣。

Zhou Dunyi said that since the Taiji had already been represented in a diagram, which contained black and white, yin and yang, it was actually the two principles and four images, and could no longer be called Taiji. Therefore, he added the term "wuji" above it. He did this primarily for those with average or lower intellectual capacity: since Taiji has no form or trace and cannot be described with words, he drew this diagram to explain the origin of the principle of yin and yang, so that people could understand it at a glance. His intention can be said to have been very thoughtful.

但以本無跡象之太極，以跡象求之，終有語病，無論如何設辭，終覺似是而非。（太極沖溟無形，可以意會，不可以言傳。言傳則有聲，有聲非太極也。著筆則有象，有象非太極也。）若讀者不善體會，以辭害意，是將使太極之真相，終無由瞭解。所謂“差以毫釐，謬以千里”矣。可不慎哉！

But to seek the formless Taiji with forms and traces inevitably leaves linguistic flaws. No matter how one phrases it, it always seems somewhat ambiguous. Taiji is vast and empty, indistinct and formless; it can only be understood by the mind and not transmitted through words. Once expressed in words, it has sound, and once it has sound, it is no longer Taiji; once written down, it has form, and once it has form, it is no longer Taiji. If readers are not good at grasping the meaning, they may be hindered by the words, and thus the true nature of Taiji will forever remain unknown. This is precisely what is meant by "a slight error at the beginning leads to a great mistake later." How can we not be cautious!

生生之謂《易》，故《易》道遏惡而揚善，好生而惡殺，此所以扶陽而抑陰也（陽以象善，陰以象惡也）。自專制帝王，借經術以愚民，而頭巾腐儒，又逢惡助長，曲借經文為乾祿固寵之具，於是正義湮晦。古聖救世之經文，幾反為賊民之愛書矣。尊君而抑臣，貴君而賤民，受誣之甚，莫過《易》與《春秋》。蓋《春秋》亦原本於《易》者也。

The Great Treatise says, "To produce life is called the Book of Changes." Therefore, the principles of the Book of Changes restrain evil and promote goodness, favor life and detest killing. This is why it supports Yang and restrains Yin (Yang symbolizes goodness, and Yin symbolizes evil). Since autocratic emperors have used Confucian classics to deceive the people, and those corrupt scholars in square hats have flattered the authorities and encouraged their wickedness, twisting the classics and using them as tools for seeking official positions and securing favor, the righteous principles within the classics have become obscured and forgotten. The ancient sages' writings meant to save the world have almost become evidence for harming the people. The doctrines of honoring the sovereign and suppressing the ministers, valuing the sovereign and despising the people, have been most misinterpreted and slandered, none more so than the Book of Changes and the Spring and Autumn Annals—the Spring and Autumn Annals itself was originally derived from the Book of Changes.

來瞿塘氏《易注》，尚為善本。乃其釋乾元用九，亦云為人君者體春生之元，用秋殺之元，一張一弛，為天道之法則。道千古人君以殺人者，皆此等曲說階之厲也。夫春生秋殺，《易》固無此明文。後儒以此明天地之氣，盈虛消息，亦如陽死陰生，陰死陽生之類。非春以生之者，至秋必殺之，天地預存一必殺之心也。

Lai Zhide's "Collected Commentaries on the Zhou Changes" is still considered a good edition, but he explains the term "Yuan" and "Yong Ji" in the Qian hexagram, saying that a ruler should observe the "Yuan" of spring's generation and use the "Yuan" of autumn's severity, alternating between relaxation and restraint, which is the principle of the Way of Heaven. Throughout history, those who have provided rulers with paths to kill people have all been caused by such twisted interpretations. In fact, the phrase "spring generates, autumn kills" is not explicitly stated in the "Zhou Changes." Later Confucian scholars used it to explain the waxing and waning of the energies of heaven and earth, similar to the idea of yang dying and yin being born, or yin dying and yang being born. It does not mean that because something is generated in spring, it must necessarily be killed in autumn, as if heaven and earth had a predetermined intention to kill.

帝王亦人耳，以人殺人而謂之道，而謂之天道之法則，可謂誣聖蔑經之甚矣。孔子懼人之誤解也，故特於《繫辭傳》鄭重以明之，曰“吉凶與民同患”，又曰“其孰能與於此哉！古之聰明睿知神武而不殺者夫”！聖意之深遠，亦可謂周且至矣，而謬誤仍若此。諸注類是者，不勝枚舉。聊舉一二，以見讀注者，當具隻眼。以辭害意且不可，況經本無其辭，而以意出入之，其誤天下後世也大矣。

Emperors are merely human beings; to kill people and call it the Dao, or the principle of the heavenly Dao, is an extreme defamation of the sages and a desecration of the classics. Fearing that later generations might misunderstand, Confucius deliberately emphasized in the Great Treatise that "good fortune and misfortune are shared by the people," and also said, "Who could possibly be involved in this? It must be the ancient wise and enlightened ruler who was both divine and martial yet did not resort to killing." How profound and comprehensive the sage's meaning is, yet misunderstandings remain as prevalent as ever. There are countless examples of such misinterpretations in the annotations of various schools. Here I will mention just a couple, so that those reading the annotations of the Book of Changes may know that they must have their own independent perspective. Using words to obstruct the meaning of the classics is already unacceptable, and how much worse it is when the classics themselves do not contain such words, yet people add or subtract according to their own interpretations, thereby causing even greater confusion for future generations.

日光具七色光線，而此七色之光線，動而合成一色，則為白色，此近世泰西科學家所發明也。現時學校儀器中，有一種圓形之平面版，其上分繪七色。附一機輪，以皮帶聯之，搖其輪令此圓版之旋轉極速，則只見白色。（另以青黃二色之版轉動則成綠色，紅青二色者則成紫色，其他之色之配合亦各如其應得之色，而色之深淺則以兩色所占面積之多寡而分，與兩色之顏料相調合者無異。惟七色則只成白色。）即所以證明七色變白之說也。凡高等小學以上之學生，殆無不知有此者矣。

Sunlight contains seven colors: red, orange, yellow, green, blue, indigo, and violet. When these seven colors are mixed together, they form white, a discovery made by modern Western scientists. Nowadays, there is a round flat disc in school laboratories, with each side painted in one of the seven colors. It is connected to a wheel by a belt. When the wheel is turned rapidly, the seven colors overlap quickly, making it impossible to distinguish them, and the eye perceives only a single white color. If green and yellow are mixed, green is produced; if red and green are mixed, purple is produced. Similarly, other color combinations produce their respective colors. The brightness or darkness of the resulting color depends on the proportion of the areas of the two colors, just like mixing two pigments. However, when all seven colors are combined, none of them is visible, and only white is seen. This instrument is used to demonstrate the principle that seven colors combine to form white. Almost all students in higher elementary schools know about this.

然我國舊學家，固未有言之者也。不料《易經》於數千年以前，早已發明之。象義明晰，確切不移，與西人之說正相吻合。其象即見於山火賁之一卦，賁離內艮外，離為日，離居九位，九宮之色：一白，二黑，三碧，四綠，五黃，六白，七赤，八白，九紫。數雖九，而白居其三，並之仍為七色。離為日而居九位，則自當有七色，與西說正相同也。

However, our old scholars had never discussed this principle before. Unexpectedly, the Book of Changes revealed it thousands of years ago: the meaning of the hexagrams is clear and definite, consistent with the Western theory. This image is seen in the hexagram of "Brightness" (Zhen). In the hexagram of Zhen, the trigram Li is inside and the trigram Gen is outside. Li represents the sun. In the nine palaces, Li is in the ninth position. The colors corresponding to the nine palaces are: one white, two black, three green, four green, five yellow, six white, seven red, eight white, and nine purple. Although there are nine colors, white appears three times, making a total of seven colors. Since Li represents the sun and is in the ninth position, it naturally should have seven colors, which coincides exactly with the Western theory.

由九三至六五，中爻為震。震，動也。震動而離之七色皆變為白，故六四曰“白馬翰如”（翰如即轉動極速之狀也），至上九曰“白賁無咎”。曰“白賁”，則全卦皆變白矣。與七色變白之新學說，完全吻合，無一絲一毫之勉強穿鑿。而“翰如”二字，則幾將近日學校所用之儀器，並其轉動之狀曲繪而出，可謂神妙不可思議矣。子曰“《易》之為書也，廣大悉備”，略指其一而已可見矣。惜我國讀《易》者，皆忽略讀過，更無科學之知識與眼光以研究之，宜乎其不可解者多矣。

Taking the lines nine in the third, six in the fourth, and six in the fifth to form the interacting structure, the middle layer of the hexagram is Zhen. Zhen means movement. When Zhen moves, the seven colors of Li turn completely white. Therefore, the line statement of six in the fourth says, "A white horse gallops swiftly" ("gallops swiftly" means moving very fast). By the time we reach the line statement of nine at the top, it says, "White purity without blame" — meaning the entire hexagram has turned white. This perfectly matches the new theory that seven colors turn white. There is not the slightest forced or artificial interpretation. The two characters "gallops swiftly" almost depict the instrument used in schools today and its spinning motion, in a winding and intricate way. It is truly amazing beyond measure. Confucius said, "The Book of Changes is vast and comprehensive," and merely citing this example from the hexagram of Ben shows that this praise is not empty. Unfortunately, those who study the Book of Changes in our country have easily overlooked it, and lack scientific knowledge and insight to investigate it, which is why so many parts remain difficult to understand.

（朱子注《易》，至理有不可通，輒以不可解或未詳二字了之。然尚能闕疑者也。後人或錯綜其象以求之，或為卦變、爻變以附會之。但求於象可以聯屬，不顧義理之是否可通，與本卦之才物是否應有此理者，蓋比比矣。如賁卦之白字，或以為巽為白。而本卦無巽，則曰艮錯兌，兌反巽也。或以六五變則上卦為巽，故言白也。而不顧巽之義為入為伏，與本卦之義固皆無當也。唯中爻為震，震下伏巽，動而為白，義尚較近。然不以新說參證之，又烏知古聖取象之妙，實與造化同工，斷非拘拘於一爻一卦之單辭所能悉解也。）

Zhu Xi annotated the Book of Changes, and when he encountered passages whose meanings did not make sense, he often simply used the words "inexplicable" or "unknown" to pass over them, which was still considered a way of preserving doubt. Later commentators, however, attempted to solve problems by misinterpreting the hexagram images, or by using changes in hexagrams and yao changes to force interpretations, seeking only to connect the images together, regardless of whether the meaning made sense or whether the subject matter of the original hexagram could logically have such a meaning. Such examples are common among commentators. For instance, the character "white" in the hexagram Bi is said by some to be associated with Xun, but since Xun is not present in the original hexagram, they argue that Xun and Gen are opposite, and that Xun is the reverse of Xun. Others say that the fifth line changes, making the upper trigram Xun, thus explaining "white." They all ignore the meaning of Xun, which is "entering" or "hidden," and the meaning of the original hexagram, which has nothing to do with it. Only those who take the middle line as Zhen, with Xun hidden beneath Zhen, and a movement resulting in "white," offer an interpretation that is somewhat closer to the meaning of the original hexagram. However, without referring to new theories, how could one know that the subtle imagery used by the ancient sages was truly in harmony with the workings of nature, and not something that could be fully understood by merely clinging to isolated phrases from a single line or hexagram?

西教士花之安氏，頗注意於中國之經籍，曾著《自西徂東》一書，謂畫卦之伏羲，乃巴比倫人。巴比倫高原，為西洋文化策源地。伏羲八卦，以乾為天，以坤為地，至今巴比倫人猶稱天為乾，地為坤，此一證也。又巴比倫亦有十二屬象，與中國之十二辰大略相同，其證二也。

The missionary Hua Zhi'an from the West paid close attention to Chinese classics and wrote a book called "From West to East," claiming that the Fuxi who drew the trigrams was a Babylonian. His reasoning was that the Babylonian highlands were the origin of Western culture; Fuxi's eight trigrams used Qian for Heaven and Kun for Earth, and the Babylonians still call Heaven "Qian" and Earth "Kun" today, which is the first piece of evidence. Babylon also has twelve zodiac animals, which are largely the same as China's twelve earthly branches, which is the second piece of evidence. Based on this, he concluded that the study of the eight trigrams originated from Babylon and not from China itself.

或因花氏之說，更加推求，謂伏羲畫卦，以備萬物之象。宇宙偉大之象，無不列舉，如天地水火風雷山澤，以配八卦。而海為天地間最大之一象，獨付闕如。而舉澤以為山之對，則亦一疑問也。巴比倫介歐亞之間，四面皆大陸，距海最遠。其間惟裡海死海，為瀦水最大之區，故稱之為澤，亦足證花氏之說不盡無因。花氏更稱巴比倫古代之王，有號福巨者，與伏羲二字音亦相近。當即為始畫八卦之人，亦可謂讀書得間矣。

Some followed Huazhi'an's view and went further to explore, saying that Fuxi's drawing of the eight trigrams was intended to include all the images of things in the universe, listing all great phenomena, such as heaven, earth, water, fire, wind, thunder, mountain, and marsh, assigning them to the eight trigrams. However, the sea, which is the largest phenomenon between heaven and earth, is missing from the eight trigrams, and there is no place for the sea in the trigrams. Instead, "marsh" is used to oppose "mountain," which raises a question. Babylonia is located between Europe and Asia, surrounded by land on all sides, and is farthest from the sea. Within its territory, the largest bodies of water are the Caspian Sea and the Dead Sea, which are considered "marshes." This shows that Huazhi'an's view is not entirely without basis. Huazhi'an also said that in ancient Babylonia, there was a king known as "Fujü," whose name is phonetically similar to "Fuxi," and should be the one who first drew the eight trigrams. This can be considered reading between the lines and understanding the meaning from the text.

但我國上古之史，雖無可稽考，然自伏羲而後，代有傳人。一畫開天，即文字所造始。儷皮為禮，已姓氏之足徵。在中國之佚聞古蹟，無有可為伏羲來自遠方之證者。況伏羲之陵，猶在中州，至今無恙。其果否為伏羲埋骨之所，雖無從徵實，但有一事足以參證，有決非人力所能為者，則古聖揲以求卦之蓍草是也。

But although the history of ancient China cannot be verified, there have been continuous records since the time of Fuxi. The "one line opening the heavens" refers to the beginning of writing; using pairs of deer skins as gifts already indicates the existence of surnames that can be verified. In China's legends and remains, there is no evidence that Fuxi came from afar. Moreover, Fuxi's tomb is still intact in the central plains. Whether or not it is indeed where Fuxi was buried cannot be confirmed for certain. However, there is one thing that can be used as evidence, and it is certainly not man-made. That is the yarrow stalks used by ancient sages for divination.

孔子曰：昔者聖人之作《易》也，幽贊於神明而生蓍。至今蓍草所產之地，厥惟伏羲文王周公孔子之墓，而他處無有也。（惟近今所生之蓍不及古時之長，余嘗採蓍於孔林，最長者乃不滿今尺三十寸，以合周尺僅四尺餘。所謂六尺及盈丈者，詢之孔氏，雲久未得見矣。）夫文周與孔子之墓，固確為聖穴，決無可疑。則伏羲之陵而有蓍草，亦斷然足證其非妄矣。其非巴比倫產，可不辨自明。

Confucius said in the "Shuo Gua Zhuan" that in ancient times the sages composed the Book of Changes, receiving divine assistance and producing the yarrow stalks. Up to the present day, the only places where yarrow stalks grow are the tombs of Fuxi, King Wen, the Duke of Zhou, and Confucius. Elsewhere, there are none. (Recently grown yarrow stalks are not as long as in ancient times. I once gathered yarrow stalks in the Confucius family graveyard, and the longest ones were no more than thirty inches long today, which is only four feet and more in Zhou measurements. Ancient texts mention six feet or even up to a zhang, but when I asked the descendants of Confucius, they said such stalks had long since disappeared.) The tombs of King Wen, the Duke of Zhou, and Confucius are indeed the resting places of sages, without any doubt; therefore, the fact that yarrow stalks grow on the tomb of Fuxi is sufficient to confirm that this tomb is not fabricated. If this is the case, then Fuxi is not from Babylon, and this can be clearly established without argument.

或者當伏羲之時，西北之人物殷繁（其時東南皆水陸地不多），治化流被於歐亞兩州之交，故巴比倫得有伏羲之學說（逮洪水為災，地形改變，流沙阻隔，西道遂不復通，故禹域西限流沙），未可知也。至八卦之象無海，則有說焉。夫江河海洋，皆後起之名辭。伏羲時文字未興，烏得有此析類之名？卦象水火山澤，皆以對舉為文。海固無可對也，故以澤對山。洪荒之世，世界一澤國耳。舉目所見惟山與澤，則亦以山澤象之耳。《周易》為中古之書，取象較廣。坎為大川，大川亦即海也。焉得以數千年後之名詞，而致疑於上古之世哉！

Perhaps during the time of Fuxi, the area to the northwest was densely populated, while the southeast was mostly water and had little land. The teachings spread to the border of Eurasia, so Babylon preserved Fuxi's doctrines. Later, due to floods and changes in the terrain, shifting sands blocked the western route, hence the western boundary of the territory set by Yu the Great was marked by shifting sands. This may be possible. As for the absence of the sea in the imagery of the Eight Trigrams, there is another reason. The terms river, river, sea, and ocean are later names. Writing had not yet emerged in Fuxi's time, so how could there be such specific classifications? Moreover, in the trigrams, water, fire, mountain, and marsh are paired. Since the sea has nothing to pair with, it is paired with marsh. In the age of chaos and wilderness, the whole world was merely a marshland, and all that could be seen were mountains and marshes, so they used mountain and marsh as symbols. The Zhou Yi is a book from the middle ancient times, and its imagery is broader. The trigram Kan represents a great river, which is also the sea. How can we use terms that appeared only thousands of years later to question the ancient times?

西人物質之化分，譯之為化學者，乃近世紀所發明者也。不謂地隔三萬里，時閱七千年，而吾《易》之象數，能與之一一吻合無毫釐之差。嗚呼！是所謂範圍天地而不過，曲成萬物而不遺者，豈空談性理所能悉其奧旨哉！張氏之銳《易象闡微》，取氣之分劑性質，以卦位爻數乘除推衍，無不妙合。尤奇者，陽三而陰二，足證古聖參天兩地之數，固俟之萬世而不惑者也。

Westerners study matter by breaking it down, and they call this study "chemistry," a discipline that has only recently emerged in the last one or two hundred years. It is amazing that, despite being three thousand miles apart and seven thousand years apart in time, the images and numbers in our Book of Changes can correspond with it one by one, without any error at all. How regrettable! The "Great Treatise" says, "It encompasses all transformations in heaven and earth without exceeding, and skillfully completes all things without omission." How could such profound meanings be understood by those who merely talk about the nature of the mind and principles? Zhang Zhiui's work, "Explanations of the Images of the Book of Changes," uses the proportions and properties of various kinds of qi (i.e., atomic weights) and applies them through hexagram positions and line numbers for calculations, and they all perfectly match. Especially remarkable is the use of the number three for yang and two for yin, which clearly proves that the ancient sages' principle of "three for heaven and two for earth" is indeed something that will never be doubted in future generations.

《易》以陽為氣而陰為質，乾坤氣質之總綱，六子乃氣質之分類。故氣質之分晰，當於六子求之。輕氣（亦曰水素）比較各氣重量為最輕，於《易》為震象，以乾通坤初，出於震之卦氣也。震初一陽始復，其幾甚微，故其氣甚輕。（地山謙，坤艮皆重也。乃曰謙輕者，亦以三四五爻之互震也。）震為動為陽，陽之氣輕清而上升，故坤變為震。動而氣升，則乾即成巽而為風。蓋乾陽之坤，熱漲上騰，而乾之本體反虛。他處空氣入而補之，因動盪而為風也。

The Book of Changes takes yang as qi and yin as substance. The two trigrams Qian (Heaven) and Kun (Earth) are the general principles of qi and substance. The six trigrams Zhen (Thunder), Kan (Water), Gen (Mountain), Xun (Wind), Li (Fire), and Dui (Lake) are the specific classifications of qi and substance. Therefore, to distinguish between qi and substance, one should look to these six trigrams. Light qi (also called water element, which is today's hydrogen) is the lightest among all qi. In the Book of Changes, it is associated with the trigram Zhen. This is because it is the first line of Qian, the initial yang line that emerges from Zhen, representing the qi that flows from Heaven to Earth. The first line of Zhen is a single yang line that has just begun to arise, which is extremely subtle, hence its qi is very light. (The hexagram "Modesty" (Earth over Mountain) has both Kun and Gen, which are heavy, yet it is called "Modesty" and "Light," because the third, fourth, and fifth lines together produce the trigram Zhen.) Zhen represents movement and yang. Yang qi is light and rises, so Kun transforms into Zhen. When there is movement, the qi rises, and Qian transforms into Xun, becoming wind. This is because the yang of Qian enters the yin of Kun, expands due to heat, and rises. As a result, the original body of Qian becomes empty, and air from elsewhere flows in to fill the space. Because of this movement and turbulence, wind is produced.

凡卦爻本體，不能互成他卦者，當為獨立之原點。其可以互成他卦者，即為化合之合點。如坎（六畫卦）五陽以上無互體，為獨立養氣（亦曰酸素）原點。下互艮為淡氣，此即養氣當與淡氣相合之象也。若五陽下降至四，則亦互震體，而變為雷雨作解。此即輕氣上升，與天空養氣化合而成雨之象也。又如震初陽無互體，為輕氣獨立原點。四陽互艮互坎，此亦與淡氣相和，及化合成雨之象也。

Any hexagram line that cannot generate other hexagrams is considered an independent origin, just like chemical elements; if it can generate other hexagrams, it is a point of combination. For example, the hexagram of Kan (Water), with its nine in the fifth line, has no further interrelated hexagram above it, making it an independent origin of yang energy (also called acid element, i.e., oxygen); below it, it generates Gen (Mountain), which represents neutral energy (i.e., nitrogen), exactly reflecting the combination of yang energy with neutral energy. If the nine in the fifth line descends to the fourth position, it then generates Zhen (Thunder), transforming the hexagram into Jie (Release), which represents light energy rising and combining with yang energy in the sky to form rain. Similarly, the initial line of Zhen (Thunder) has no interrelated hexagram, making it an independent origin of light energy; the nine in the fourth line generates Gen and Kan, reflecting the combination with neutral energy and the formation of rain.

坎五之陽由乾二來也，以陽三（爻體陽三陰二）與五位相加為八，以二乘八得數十六。震初之陽，由乾四來也。以陽三與初位相減為二，以二除四，仍得二二：十六二一：八，故輕氣之分劑數（亦曰原子量）為一，養氣之分劑數為八，一以乘得數，一以除得數也。

The Yang of the fifth line in the Kan hexagram comes from the Yang of the second line in the Qian hexagram. Using the Yang number three (Yang takes three, Yin takes two) and adding it to the fifth position, we get eight. Then multiplying eight by two gives sixteen. The Yang of the first line in the Zhen hexagram comes from the Yang of the fourth line in the Qian hexagram. Using the Yang number three and subtracting it from the first position, we get two. Then dividing four by two still gives two. The ratio of two to sixteen reduces to one to eight, so the numerical value for the light element (also called atomic weight) is set to one, and the numerical value for the nourishing element is set to eight: one is derived from multiplication, and the other from division.

淡氣（亦曰窒素）者，愛力甚小。而其性安靜無為，其作用但在節止養氣之太過，故知此氣必為艮之卦氣（艮靜而止）也。如艮九三上九兩陽爻，皆為淡氣。上九之陽由乾三來，九三之陽由乾上來。以陽三與上位六相加則為三，三以陽三加入三位，則為二三。

The affinity of the subtle energy (also called inert gas, i.e., nitrogen) is very weak, and its nature is quiet and inactive. Its function is only to restrain and regulate the vital energy so that it does not become excessive. From this we can infer that this kind of energy must belong to the Qian trigram (Gen), since Gen's nature is quiet and capable of stopping. For example, in the Gen hexagram, the ninth line in the third position and the ninth line at the top are both yang lines that belong to the subtle energy. The yang line at the top comes from the ninth line in the third position of the Qian trigram, and the yang line in the third position comes from the top line of the Qian trigram. Adding the yang number three with the upper position six gives the divisor three used later; adding the yang number three with the number three gives the multiplier two used later.

艮下體之卦位，其和數為六（初二三之和）。上體之卦位，其和數為十五（四五六之和）。六與十五之和數為二十一，以三除之得七，後以二乘七得十四（以三除者陽，加於上位得三，陽數也。以二乘者陽，加於三位得二陽數也），為淡氣一分劑數。輕養二氣之乘除，以反對為用，淡氣則自為乘除。由此推之，可見卦氣每生一物，必變化其方式。方式不同，故形質有異也。

The third line of the lower trigram in the "Gen" hexagram adds up to six (the sum of the first, second, and third lines). The third line of the upper trigram adds up to fifteen (the sum of the fourth, fifth, and sixth lines). Adding six and fifteen gives twenty-one. Dividing twenty-one by three results in seven. Then multiplying seven by two gives fourteen. The reason for using three as the divisor is that the yang number three is added to the upper position, making it three, which is a yang number. The reason for using two as the multiplier is that the yang number three is added to the third position, resulting in two, which is also a yang number. The number fourteen is thus a component number of the "dàn qì" (thin qi). The multiplication and division of "qīng qì" (light qi) and "yǎng qì" (nourishing qi) are used in opposition to each other, while "dàn qì" uses multiplication and division within itself. From this, it can be seen that whenever a hexagram generates an object, it must change its calculation method. Since the calculation methods differ, the forms and substances produced are also different.

輕淡養為三少陽氣類原點。此三氣者，常混合瀰漫於空中者也。試以三類氣質合成卦象如□，即蒙卦也。上艮為淡氣，下坎為養氣。九二六三六四互震為輕氣，九二至上九共歷五位。陽自初至二進一位，自初至上進五位。養氣即生，則應消去淡氣一位。故養氣居空氣五分之一，淡氣居空氣五分之四，而輕氣體積則甚小。輕養淡三氣混合瀰漫而成蒙象。蒙象者，其地球表面之蒙氣乎？

The three types of qi, light qi, thin qi, and nourishing qi, originate from the category of the three yangs. These three types of qi often mix together and spread throughout the air. If we combine these three types of qi into a single hexagram (the original hexagram is now missing), it forms the Hexagram of Enlightenment. The upper trigram is Gen (Mountain), representing thin qi, and the lower trigram is Kan (Water), representing nourishing qi. The lines nine in the second, six in the third, and six in the fourth interact to form Zhen (Thunder), representing light qi. Counting from nine in the second to nine at the top, there are five positions. Yang moves from the initial position to the second position, advancing by one position, and from the initial position to the top position, advancing by five positions. When nourishing qi is generated, it should decrease by one position from thin qi. Therefore, nourishing qi occupies one fifth of the air, thin qi occupies four fifths of the air, and the volume of light qi is extremely small. The mixture of light, nourishing, and thin qi forms the hexagram of Enlightenment. The term 'Mont' probably refers to the layer of misty qi that envelops the Earth's surface.

綠氣（亦曰鹽素）能漂白物色，其臭甚烈，能傷人，此巽之象也。《說卦傳》“巽為白，又為臭”，又曰“巽也者言萬物之絜齊也”，皆綠氣之性能功用也。如巽□象，下體陽爻，積與位之和為十一（陽三加二位為五，陽三加三位為六，合之為十一也）。上體陽爻積與位之和為十七（陽三加五位為八，陰三加上位為九，合之為十七也），下體陰爻積與位之和為三（陰二加初位為三），上體陰爻積與位之和為六（陽二加四位為六），以三減六得三，以初加四得五。三五相加，再與十一十七相和，即等三十六，為綠氣之一分劑數（綠氣之一分劑數或作三五一五）。

Green gas (also called salt, which is today's chlorine) can bleach the color of objects, has a very strong smell, and can injure people when inhaled. This is exactly the image of Xun. The "Say on the Hexagrams" says, "Xun represents white," and also says "Xun represents smell," and further says, "Xun is the one who speaks of the purity and order of all things." "Purity and order" refers to the function and ability of green gas. According to the image of the Hexagram Xun (the original hexagram diagram is now missing), the sum of the number of yang lines and their positions in the lower trigram is eleven (three yang lines plus two positions equal five, three yang lines plus three positions equal six, altogether eleven); the sum of the number of yang lines and their positions in the upper trigram is seventeen (three yang lines plus five positions equal eight, three yang lines plus six positions equal nine, altogether seventeen); the sum of the number of yin lines and their positions in the lower trigram is three (two yin lines plus the first position equal three); the sum of the number of yin lines and their positions in the upper trigram is six (two yin lines plus the fourth position equal six). Subtracting three from six gives three, adding the first position and the fourth position gives five, and three plus five equals eight. Adding this eight to eleven and seventeen gives exactly thirty-six, which is a number representing green gas (there is also a version of the number for green gas as thirty-five point fifteen, which is slightly different from this).

炭氣（亦曰炭素）為有形質之氣，故化學家謂之非金類，而煤炭所含者為最多。若在氣質內，則常與養氣化合，所謂炭養二是也。蓋炭氣之原點屬離，離陰卦也。陰常附陽（離麗也，必附麗於他物而始見。離為火，火必附於物始燃），是為有形質之物。故炭氣與綠氣，不能如輕淡養三氣，能卓然獨立於空氣界也。

Carbon gas (also called carbon, which is a tangible gas) is classified by chemists as a non-metal, and it is most abundant in coal. When carbon gas exists in a gaseous state, it often combines with vital gas, known as "carbon and vital," with one part carbon combining with two parts vital. The origin of carbon gas is associated with the hexagram Li, which is a yin trigram. Yin often depends on yang (the term "Li" means "adhering," meaning that it must adhere to something else to be visible. Li represents fire, which also needs to be attached to an object in order to burn). Therefore, carbon gas is a tangible substance. Because of this, carbon gas and green gas cannot exist independently in the air like the light, thin, and vital gases.

□離卦二陰自坤五而來，下體陽爻積與位之和為十（陽三加初位為四，陽三加三位為六，合為十也）。上體陽爻積與位之和為十六（陽三加四位為七，陽三加上位為九，合為十六）。以上體陽五位數，除下體陽爻積與位之和，得數二。以下體陰二位數，除上體陽爻積與位之和得數八。二與八之較六，即為炭氣之一分劑數（綠氣一分劑數由和而得，炭氣一分劑數由較而得方式亦不同也）。

In the hexagram Li (Fire), the two yin lines are both derived from the fifth line of the hexagram Kun (Earth). The sum of the number of yang lines and their positions in the lower trigram is ten (three for yang plus the initial position gives four, three for yang plus the third position gives six, totaling ten). The sum of the number of yang lines and their positions in the upper trigram is sixteen (three for yang plus the fourth position gives seven, three for yang plus the top position gives nine, totaling sixteen). Using the fifth position occupied by the yang lines in the upper trigram to divide the sum of the lower trigram, which is ten, the result is two. Using the second position occupied by the yin lines in the lower trigram to divide the sum of the upper trigram, which is sixteen, the result is eight. Subtracting eight from two gives a difference of six, which is the numerical component of the charcoal qi. The numerical component of the green qi is obtained by addition, while that of the charcoal qi is obtained by subtraction. The calculation methods differ, hence their forms and qualities are also different.

六子之卦，震坎艮巽離，皆有其自成之原點。惟兌獨無，蓋陽數大，陰數小，陽順生，陰逆生，陽始於震，陰始於兌。震之原點為一，則兌始生之陰數，必更小於一，退入零位，幾不可見，必不能成為獨立之原點矣。求之於化學氣類中，惟喜氣於兌象為近。兌為悅，悅喜意也（喜氣嗅之令人狂笑）。考喜氣為淡養所合成，西名為NO。此亦足證兌無自生之原點矣。

Among the six sons' hexagrams, Zhen, Kan, Gen, Xun, and Li each have their own independent origin, but Dui does not. This is because the yang number is large and the yin number is small, with yang generating in a forward direction and yin generating in a reverse direction: yang begins with Zhen, while yin begins with Dui. If the origin of Zhen is counted as one, then the initial yin number of Dui must be smaller than one, retreating to zero, which is almost imperceptible and therefore cannot serve as an independent origin. In terms of chemical qi types, only the qi of joy is most similar to Dui. Dui represents joy, which means happiness (the scent of joy can cause laughter). Investigating the nature of joy, it is composed of a light qi and a nurturing qi, known in Western terms as NO. Since it is formed by the combination of two qis, it is not a fundamental substance itself, which precisely demonstrates that Dui does not have an independent origin.

喜氣之分劑數為二十二，即八與十四所合之數（養氣一分劑數為八，淡氣一分劑數為十四）。淡氣為艮象，養氣為坎象。艮之上體三爻積與位之和為二十二（陰二加四位為六，陰二加五位為七，陽三加上位為九，合之為二十二也），坎之上體三爻積與位之和亦二十二（陰二加四位為六，陽三加五位為八，陰二加上位亦八，合之亦為二十二也）。以艮之上體，合兌之下體，則為□損卦。以坎之上體，合兌之下體，則為□節卦。

The numerical value of the joyful qi is twenty-two, which is exactly the sum of the numerical values of the nurturing qi (eight) and the mild qi (fourteen). This confirms that it is indeed composed of two qis. The mild qi belongs to the image of Gen, and the nurturing qi belongs to the image of Kan, so the numbers must be taken from the hexagrams Gen and Kan. The sum of the yao numbers and their positions in the upper trigram of Gen is twenty-two (the sum of two yin lines added to the fourth position gives six, two yin lines added to the fifth position gives seven, and three yang lines added to the upper position gives nine, totaling twenty-two). Similarly, the sum of the yao numbers and their positions in the upper trigram of Kan is also twenty-two (two yin lines added to the fourth position gives six, three yang lines added to the fifth position gives eight, and two yin lines added to the upper position also gives eight, totaling twenty-two). When the upper trigram of Gen is combined with the lower trigram of Dui, it forms the hexagram Shen; when the upper trigram of Kan is combined with the lower trigram of Dui, it forms the hexagram Jie (the original hexagram figures are now missing). These two hexagrams serve as the basis for the subsequent calculations.

兌下體三爻積與位之和為十四（陽三加初位為四，陽三加二位為五，陰二加三位為五，合之為十四也），以下體兌之十四，減上體艮之二十二，即得坎養之八。以坎養之八，復減上體坎之二十二，即得艮淡之十四。故八與十四之和，即喜氣之分劑數。其數之妙合，極參伍錯綜之致，而無不吻合，誠不可思議矣。

The third line of the lower trigram兑 (Dui) has a total of fourteen when the number of lines and their positions are added together (a yang line three plus the first position gives four, a yang line three plus the second position gives five, and a yin line two plus the third position gives five, totaling fourteen). Using fourteen from the lower trigram兑 to subtract from the upper trigram艮 (Gen) which is twenty-two, we get exactly eight, which corresponds to the nurturing energy of坎 (Kan). Then using the eight from坎 to subtract from the upper trigram坎 again, we get fourteen, which corresponds to the subtle energy of艮. By subtracting each pair, the results exactly match the respective energies. Therefore, adding eight and fourteen gives twenty-two, which is the number of the joyful energy. The elegance of these numbers fitting together is so intricate and complex that it is truly beyond comprehension.

以上輕（震象）養（坎象）淡（艮象）綠（巽象）炭（離象）喜（兌象）六氣，即乾坤二元所化生，震坎艮三少陽，巽離兌三少陰之六元象也。然三少陽所成之原質有三類，而三少陰所成原質只有兩類。可見陽三陰二之定則，離造物者亦不能越此範圍（五行之數亦天三地二，天地之生數亦天三地二）也。此外之原質，若碘，若砩，若金類，若非金類，為數甚夥。苟能各按其天然分劑之數與其品性，依卦象變化之條理，以方程求等之法細繹之，當無不有一定之公式，可斷言矣。

The six kinds of qi, namely light qi (Zhen image), nourishing qi (Kan image), mild qi (Gen image), green qi (Xun image), charcoal qi (Li image), and joyful qi (Dui image), are derived from the two fundamental principles of Heaven and Earth. These correspond to the three lesser Yang trigrams, Zhen, Kan, and Gen, and the three lesser Yin trigrams, Xun, Li, and Dui, making up the six fundamental elements. Each type of qi corresponds to one trigram, and there is a perfect match. However, the original substances formed by the three lesser Yang trigrams are categorized into three types: light, nourishing, and mild. In contrast, the original substances formed by the three lesser Yin trigrams are only categorized into two types: green and charcoal. This shows that the principle of "Yang three, Yin two" is a fixed rule that even the Creator cannot transcend (the Five Phases also follow the principle of Heaven three and Earth two, and the generating numbers of Heaven and Earth are also Heaven three and Earth two). As for other original substances, such as iodine, fluorite, metals, and non-metals, there are many of them. As long as each is classified according to its natural division and properties, following the logic of trigram transformations, and carefully deduced using equations, there must be a definite formula for each. This can be confidently asserted, and it is not only these six kinds of qi that conform to the principles of images and numbers.

《化學鑑原》云：輕氣與諸質化合，其性大似金類（與鋅或銅化合，顯此性尤大），故疑為氣質金類，如汞為流質金類是也。輕氣與汞若汞與鉑，雖視無金光，而叩之不堅，亦不足為必非金類之據。汞加熱可化為明氣質，亦與金類為例，是可見輕氣亦或為金類也。

The Chemical Origin says: Light gas combines with various substances in a manner similar to metals, especially when it combines with zinc or copper. Therefore, it is suspected to be a gaseous metal, just as mercury is a liquid metal. Although light gas and mercury, as well as mercury and platinum, do not appear to have a metallic luster and are not hard when struck, this is not sufficient to conclude that they are definitely not metals. Mercury can be turned into a transparent gas when heated, which is similar to metals. Therefore, it is possible that light gas is also a type of metal.

又云：輕氣雖為萬物中至輕至稀之品，而分劑數為最小者，然依其分劑數而計其愛力，則甚大也。如一分重之輕氣，能與綠氣三十六分之重化合，能與溴八十分之重化合，與碘一百二十五分之重化合。極少之數，亦足支配此各質而變化其性。且所成之質，最難分析。化學以為極少之分劑，而生極大之愛力性，為甚奇也。

It is also said that light gas, though the lightest and rarest of all things with the smallest proportion, has a very great affinity when calculated by its proportion. For example, one part of light gas can combine with thirty-six parts of green gas, eighty parts of bromine, and one hundred and twenty-five parts of iodine—the combined weight being many times heavier than itself. A very small amount is sufficient to control these substances and change their properties, and the resulting compounds are extremely difficult to decompose. Chemists find it strange that such a small proportion can produce such a great affinity.

按化學家疑輕氣為金類，而不敢下確定之斷語。又奇其分劑數小而愛力極大，此由不明輕氣所以化生之原理也。若以《易》象證之，則所疑者皆渙然冰釋矣。蓋輕氣為震象，震者乾始交坤，其象恆通乾（震為乾之長子）。《說卦傳》“震其究為健”，健即乾也。乾為金，震者金氣之初發動者也。知輕氣之為震，則知輕氣為氣類之金可無疑矣。

Note: Chemists suspect that hydrogen is a metal and are hesitant to make a definite conclusion, also finding it strange that its atomic weight is small while its affinity is very great. This is because they do not understand the principle by which hydrogen is generated. If we use the imagery of the Book of Changes to verify this, all their doubts will be clearly resolved. Hydrogen belongs to the image of Zhen (Thunder), which is formed when Qian (Heaven) first interacts with Kun (Earth). The image of Zhen always connects with Qian (Zhen is the eldest son of Qian). The "Commentary on the Hexagrams" says, "Zhen, when pursued to its conclusion, becomes Qian," meaning that Zhen, when examined thoroughly, ultimately returns to Qian, "Qian" meaning strength. Qian represents metal, so Zhen represents the initial movement of metal energy. If we know that hydrogen belongs to Zhen, then hydrogen is a gaseous metal, and there is nothing to doubt.

一陽初復，故分劑數最小，所謂復小而辨於物是也。其感動力反甚大者，以乾健而震動也。震為物之始生，此天地之仁氣也。且又為最尊最大之乾卦最肖之長子，然則能以至小之分劑數化合最大之分劑數，而顯最大之愛力者，又何足怪乎！

A single yang line has just begun to revive, so its dividing number is the smallest. The Great Treatise says, "Recovery is small yet distinct in things," which exactly expresses this meaning. Its power of influence is actually great because the Qian trigram is strong and vigorous, and Zhen trigram is the master of movement. Zhen is the place where all things begin to grow, which is the benevolence of heaven and earth; moreover, it is the eldest son most similar to the greatest Qian trigram. Therefore, it can combine the largest dividing number with the smallest one, showing the greatest affinity, so what is strange about it?

以上諸說，不但足與物理相發明，尤足以證明象之切切實實，非模糊影響而意為擬議者也。向來說《易》者，以空談性理為高，能精研象數者已不可多得。間有談象數者，又莫明象數之原理，於是東牽西扯，曲折附會以求合，而不知去《易》之道愈遠，而象數反為說《易》之累矣。故顯明象數，必知物理。離物理以言象數，亦與離象數而談性理者，敝正相等耳。

The above statements are not only sufficient to complement physics, but also sufficient to prove that the images in the Book of Changes are concrete realities, not vague or subjective inventions. In the past, those who interpreted the Book of Changes often regarded discussions on moral principles and human nature as refined, while those who studied the principles of images and numbers were already rare. Occasionally, those who discussed images and numbers did not understand their underlying principles, so they twisted and forced interpretations to fit, yet they did not realize that this only took them further away from the true path of the Book of Changes. Thus, to explain images and numbers clearly, one must first understand physics. To discuss images and numbers without reference to physics is just as flawed as discussing moral principles without reference to images and numbers; both approaches are equally faulty, and neither is superior to the other.