1
《虞書》曰「乃同律度量衡」，所以齊遠近立民信也。 自伏戲畫八卦，由數起，至黃帝、堯、舜而大備。 三代稽古，法度章焉。 周衰官失，孔子陳後王之法，曰：「謹權量，審法度，修廢官，舉逸民，四方之政行矣。」 漢興，北平侯張蒼首律曆事，孝武帝時樂官考正。 至元始中王莽秉政，欲燿名譽，徵天下通知鐘律者百餘餘人，使羲和劉歆等典領條奏，言之最詳。 故刪其偽辭，取正義，著于篇。
As the *Book of Yu* says, "He standardized pitch, measures, capacity, and weights," thereby creating uniform standards across near and far and securing public trust. Numerical principles began with Fuxi's Eight Trigrams and reached full development by the age of the Yellow Emperor, Yao, and Shun. In the Three Dynasties, careful study of antiquity made legal and technical standards explicit. As Zhou authority decayed and institutions broke down, Confucius articulated the model of later sage-kings: "Standardize weights and measures with care, scrutinize legal norms, restore defunct offices, and elevate neglected talent; then government will function throughout the realm." After the Han founding, the Marquis of Beiping, Zhang Cang, first took charge of pitch and calendrical administration; under Emperor Wu, court music officials reviewed and corrected the system. By the Yuanshi era, when Wang Mang controlled the government, he sought prestige by convening more than a hundred specialists in bell acoustics and pitch standards from across the empire. He ordered Liu Xin, in his capacity as Xihe, and others to oversee formal memorials, producing the most detailed discussions on the subject. So I have cut away spurious claims, retained what is sound, and set it down here.
2
一曰備數，二曰和聲，三曰審度，四曰嘉量，五曰權衡。 參五以變，錯綜其數，稽之於古今，效之於氣物，和之於心耳，考之於經傳，咸得其實，靡不協同。
The treatise has five parts: numeration, musical tuning, linear measure, capacity measure, and weight standards. By combining triadic and pentadic principles and interrelating their numbers, then testing them against ancient and later practice, physical phenomena, human perception, and the classical tradition, one finds each point borne out in fact and mutually coherent.
3
數者，一、十、百、千、萬也，所以算數事物，順性命之理也。 《書》曰：「先其算命。」 本起於黃鐘之數，始於一而三之，三三積之，歷十二辰之數，十有七萬七千一百四十七，而五數備矣。 其算法用竹，徑一分，長六寸，二百七十一枚而成六觚，為一握。 徑象乾律黃鐘之一，而長象坤呂林鐘之長。 其數以易大衍之數五十，其用四十九，成陽六爻，得周流六虛之象也。 夫推曆生律制器，規圜矩方，權重衡平，準繩嘉量，探賾索隱，鉤深致遠，莫不用焉。 度長短者不失豪氂，量多少者不失圭撮，權輕重者不失黍絫。 紀於一，協於十，長於百，大於千，衍於萬，其法在算術。 宣於天下，小學是則。 職在太史，羲和掌之。
Number begins with units, tens, hundreds, thousands, and ten-thousands; these are the means by which all things are calculated, in accord with the underlying order of nature. As the *Book of Documents* says, "One must first compute the mandate." The system starts from the Huangzhong fundamental: begin with one, multiply by three, and continue tripling through the twelve terrestrial branches. This yields 177,147, completing the full numerical sequence. The calculations use bamboo rods one fen in diameter and six cun long; 271 such pieces are arranged into six-edged units as a standard handful set. Its diameter symbolically matches one unit of the Huangzhong pitch under the Qian principle, while its length corresponds to Linzhong under the Kun principle. Its numerology follows the *Yijing*'s Great Expansion total of fifty, of which forty-nine are put to use, yielding the six yang lines and the image of cyclical movement through the six empty positions. Whether in calendrical computation, pitch standards, instrument-making, geometry, weighing, linear and volumetric measurement, or in analyzing subtle principles and far-reaching patterns, all depend on numerical method. With proper standards, linear measures are accurate down to hair-fine units, capacities down to tiny scoops, and weights down to millet-grain increments. The system is organized by units, coordinated by tens, expanded by hundreds and thousands, and extended into myriads; its operative method is arithmetic. Once promulgated across the realm, these standards became the norm in elementary education. This responsibility lay in the Grand Scribe's office, under the authority of the Xihe bureau.
4
聲者，宮、商、角、徵、羽也。 所以作樂者，諧八音，蕩降人之邪意，全其正性，移風易俗也。 八音：土曰塤，匏曰笙，皮曰鼓，竹曰管，絲曰絃，石曰磬，金曰鐘，木曰柷。 五聲和，八音諧，而樂成。 商之為言章也，物成孰可章度也。 角，觸也，物觸地而出，戴芒角也。 宮，中也，居中央，暢四方，唱始施生，為四聲綱也。 徵，祉也，物盛大而茇祉也。 羽，宇也，物聚臧宇覆之也。 夫聲者，中於宮，觸於角，祉於徵，章於商，宇於羽，故四聲為宮紀也。 協之五行，則角為木，五常為仁，五事為貌。 商為金為義為言，徵為火為禮為視，羽為水為智為聽，宮為土為信為思。 以君臣民事物言之，則宮為君，商為臣，角為民，徵為事，羽為物。 唱和有象，故言君臣位事之體也。
The five musical tones are gong, shang, jue, zhi, and yu. Music is made to harmonize the eight classes of sound, purge wayward impulses, preserve right human nature, and reshape social mores. The eight timbral classes are: clay (`xun`), gourd (`sheng`), hide (`drum`), bamboo (`pipe`), silk (`strings`), stone (`qing chimes`), metal (`bells`), and wood (`zhu` percussion board). Music is fully achieved when the five tones and eight sound classes are properly harmonized. The tone *shang* is associated with manifestation: when things mature, they become distinct and measurable. The tone *jue* signifies emergence: life presses against the earth and sprouts forth with sharp tips. The tone *gong* is the center: positioned at the middle, it radiates to the four quarters, initiates generative force, and serves as the framework for the other four tones. The tone *zhi* signifies thriving prosperity, when things reach vigorous expansion. The tone *yu* corresponds to enclosure: things collect, are stored, and come under a protective covering. The tonal process is centered in *gong*, emerges in *jue*, flourishes in *zhi*, clarifies in *shang*, and settles under shelter in *yu*; thus the other four tones are structured by *gong*. When mapped onto Five-Phase theory, the tone *jue* corresponds to wood, to benevolence among the five virtues, and to bearing among the five human faculties. The tone *shang* aligns with metal, righteousness, and speech; *zhi* with fire, ritual propriety, and vision; *yu* with water, wisdom, and hearing; and *gong* with earth, trustworthiness, and thought. In political analogy, *gong* stands for the ruler, *shang* for ministers, *jue* for the people, *zhi* for affairs, and *yu* for material things. Because musical lead-and-response follows a patterned order, it serves as a model for the proper structure of ruler, minister, office, and state business.
5
五聲之本，生於黃鐘之律。 九寸為宮，或損或益，以定商、角、徵、羽。 九六相生，陰陽之應也。 律十有二，陽六為律，陰六為呂。 律以統氣類物，一曰黃鐘，二曰太族，三曰姑洗，四曰蕤賓，五曰夷則，六曰亡射。 呂以旅陽宣氣，一曰林鐘，二曰南呂，三曰應鐘，四曰大呂，五曰夾鐘，六曰中呂。 有三統之義焉。 其傳曰，黃帝之所作也。 黃帝使泠綸，自大夏之西，昆侖之陰，取竹之解谷生，其竅厚均者，斷兩節間而吹之，以為黃鐘之宮。 制十二筩以聽鳳之鳴，其雄鳴為六，雌鳴亦六，比黃鐘之宮，而皆可以生之，是為律本。 至治之世，天地之氣合以生風； 天地之風氣正，十二律定。 黃鐘：黃者，中之色，君之服也； 鐘者，種也。 天之中數五，五為聲，聲上宮，五聲莫大焉。 地之中數六，六為律，律有形有色，色上黃，五色莫盛焉。 故陽氣施種於黃泉，孳萌萬物，為六氣元也。 以黃色名元氣律者，著宮聲也。 宮以九唱六，變動不居，周流六虛。 始於子，在十一月。 大呂：呂，旅也，言陰大，旅助黃鐘宮氣而牙物也。 位於丑，在十二月。 太族：族，奏也，言陽氣大，奏地而達物也。 位於寅，在正月。 夾鐘，言陰夾助太族宣四方之氣而出種物也。 位於卯，在二月。 姑洗：洗，絜也，言陽氣洗物辜絜之也。 位於辰，在三月。 中呂，言微陰始起未成，著於其中旅助姑洗宣氣齊物也。 位於巳，在四月。 蕤賓：蕤，繼也，賓，導也，言陽始導陰氣使繼養物也。 位於午，在五月。 林鐘：林，君也，言陰氣受任，助蕤賓君主種物使長大楙盛也。 位於未，在六月。 夷則：則，法也，言陽氣正法度而使陰氣夷當傷之物也。 位於申，在七月。 南呂：南，任也，言陰氣旅助夷則任成萬物也。 位於酉，在八月。 亡射：射，厭也，言陽氣究物而使陰氣畢剝落之，終而復始，亡厭已也。 位於戌，在九月。 應鐘，言陰氣應亡射，該臧萬物而雜陽閡種也。 位於亥，在十月。
The five-tone system takes its origin from the Huangzhong standard pitch. With nine cun as the *gong* baseline, the other tones - *shang*, *jue*, *zhi*, and *yu* - are determined through calibrated reduction and extension. The reciprocal generation of nine and six expresses the interaction of yin and yang. There are twelve pitch standards in all, divided into six yang pitches (*lü*) and six yin pitches (*lü*). The six yang pitches regulate vital energies and classify phenomena: Huangzhong, Taicu, Guxian, Ruibin, Yize, and Wuyi. The six yin pitches travel alongside the yang and help diffuse qi: Linzhong, Nanlü, Yingzhong, Dalü, Jiazhong, and Zhonglü. This arrangement also encodes the principle of the Three Systems. Traditional accounts attribute this system to the Yellow Emperor. The tale says the Yellow Emperor dispatched Ling Lun to gather bamboo from Jiegu in the Kunlun region west of Daxia. Choosing pieces of even bore and thickness, he cut them between nodes and sounded them, establishing the fundamental *gong* pitch of Huangzhong. He then fashioned twelve pitch pipes and tuned them to phoenix calls - six male and six female. By calibrating them against Huangzhong as the base *gong*, the full set of pitches could be derived. This became the foundation of the tonal system. In a perfectly ordered age, the qi of Heaven and Earth converge to produce winds; When cosmic winds and qi are in proper order, the twelve pitch standards settle into place. Huangzhong: "yellow" is the central color and the color associated with the ruler's robes; "Zhong" here connotes sowing and generation. Heaven's central number is five, from which tonal order is derived. Within that system, *gong* is supreme, the greatest of the five tones. Earth's central number is six, giving rise to the pitch standards. These standards are expressed in both form and color, and among the five colors, yellow holds the premier place. Thus yang qi plants its generative force in the yellow depths, from which all things germinate; this is treated as the source of the six qi. Calling this primal-qi pitch "yellow" marks its status as the *gong* fundamental. The *gong* fundamental operates by using nine to activate six, moving through continual transformation and circulating across the six empty positions. Its cycle begins at the *zi* branch, corresponding to the eleventh month. Dalü: *lü* carries the sense of accompanying progression. Here it indicates strong yin assisting the *gong* qi of Huangzhong, enabling things to bud. It is assigned to the *chou* branch, the twelfth month. Taicu: *cu* is glossed as forceful emergence; it denotes abundant yang qi pushing through the earth and bringing things forth. This pitch corresponds to the *yin* branch, the first lunar month. Jiazhong indicates yin supporting Taicu from either side, spreading qi through the four quarters and drawing seeded life into emergence. It is placed at the *mao* branch, corresponding to the second month. Guxian: the term implies purification, meaning that yang qi rinses and cleanses living things. This pitch corresponds to the chen branch, the third month. Zhonglü indicates incipient yin, not yet fully formed; from within, it joins Guxian in spreading qi and bringing things into balance. It is assigned to the si branch, the fourth month. Ruibin: with the senses of continuation and leading, it denotes yang beginning to direct yin qi so that nourishment of life can continue. This pitch belongs to the wu branch, the fifth month. Linzhong: with lin glossed as rulership, it marks yin qi taking up its mandate and assisting Ruibin so that seeded life grows to full abundance. It is placed at the wei branch, the sixth month. Yize: since ze means standard, this pitch signifies yang qi setting proper norms while yin qi brings due diminishment to what should wane. This pitch corresponds to the shen branch, the seventh month. Nanlü: with nan interpreted as taking responsibility, it indicates yin qi joining Yize to carry all things through to completion. It is assigned to the you branch, the eighth month. Wuyi: taking yi as 'satiation,' it describes yang qi carrying things to their limit while yin qi fully strips and sheds them, ending only to begin again in an unending cycle. This pitch belongs to the xu branch, the ninth month. Yingzhong denotes yin qi answering Wuyi: it gathers and stores all things, while in conjunction with yang it seals and preserves the seed. This pitch is assigned to the hai branch, the tenth month.
6
三統者，天施，地化，人事之紀也。 十一月，乾之初九，陽氣伏於地下，始著為一，萬物萌動，鐘於太陰，故黃鐘為天統，律長九寸。 九者，所以究極中和，為萬物元也。 《易》曰：「立天之道，曰陰與陽。」 六月，坤之初六，陰氣受任於太陽，繼養化柔，萬物生長，楙之於未，令種剛彊大，故林鐘為地統，律長六寸。 六者，所以含陽之施，楙之於六合之內，令剛柔有體也。 「立地之道，曰柔與剛。」 「乾知太始，坤作成物。」 正月，乾之九三，萬物棣通，族出於寅，人奉而成之，仁以養之，義以行之，令事物各得其理。 寅，木也，為仁； 其聲，商也，為義。 故太族為人統，律長八寸，象八卦，宓戲氏之所以順天地，通神明，類萬物之情也。 「立人之道，曰仁與義。」 「在天成象，在地成形。」 「后以裁成天地之道，輔相天地之宜，以左右民。」 此三律之謂矣，是為三統。
The Three Systems mark the sequence of heavenly action, earthly transformation, and human governance. In month eleven, corresponding to Qian's initial nine, yang qi lies latent underground and first emerges as unity; all things begin to stir under Great Yin. Accordingly Huangzhong is taken as the heavenly standard, with a pipe length of nine cun. The number nine signifies the extreme of balanced harmony, and thus the primal source of all things. As the *Yijing* states: "The way of Heaven is established as yin and yang." In month six, aligned with Kun's initial six, yin qi takes up its mandate under Great Yang, sustaining and softening growth; all things flourish at wei, and the seed becomes firm and robust. Hence Linzhong serves as the earthly standard, with a pipe length of six cun. The number six contains Heaven's yang impulse and extends it through the six directions, giving stable form to both hardness and softness. "The way of Earth is defined by yielding and firmness." "Qian apprehends the great inception, while Kun brings things to completion." In month one, corresponding to Qian's third nine, life circulates freely and lineages emerge at yin. Human beings take up and complete this process, nurturing through benevolence and enacting through righteousness, so each affair and thing attains its proper order. The yin branch belongs to wood and corresponds to benevolence; its corresponding tone is *shang*, associated with righteousness. Thus Taicu represents the human standard, with an eight-cun pipe corresponding to the Eight Trigrams - the same framework by which Fuxi aligned with Heaven and Earth, communicated with the numinous, and classified the natures of all things. "The human way is founded on benevolence and righteousness." "In Heaven it appears as pattern; on Earth it appears as form." "Thus one shapes and completes Heaven-and-Earth's pattern, supports what is fitting in Heaven and Earth, and with that steers the people." These three pitch standards are precisely what is meant by the Three Systems.
7
其於三正也，黃鐘子為天正，林鐘未之衝丑為地正，太族寅為人正。 三正正始，是以地正適其始紐於陽東北丑位。 《易》曰「東北喪朋，乃終有慶」，答應之道也。 及黃鐘為宮，則太族、姑洗、林鐘、南呂皆以正聲應，無有忽微，不復與它律為役者，同心一統之義也。 非黃鐘而它律，雖當其月自宮者，則其和應之律有空積忽微，不得其正。 此黃鐘至尊，亡與並也。
In the framework of the Three New-Year standards, Huangzhong at zi marks the heavenly commencement, Linzhong at wei (opposed to chou) marks the earthly commencement, and Taicu at yin marks the human commencement. Because the Three Correct standards define proper beginnings, the earthly standard is set at the initial nodal point of chou in the northeastern yang sector. As the *Yijing* says, "In the northeast one loses one's companions, yet in the end there is good fortune" - this describes the logic of resonance and response. When Huangzhong is established as the gong fundamental, Taicu, Guxian, Linzhong, and Nanlü all answer with correct tones, without even minute deviation. They no longer function as subordinates to other pitches: this is the principle of unified accord within one system. With any pitch other than Huangzhong, even if a monthly tonic is nominally in place, the corresponding harmonics accumulate subtle deficits and cannot reach true precision. Hence Huangzhong is paramount, unmatched by any other pitch.
8
《易》曰：「參天兩地而倚數。」 天之數始於一，終於二十有五。 其義紀之以三，故置一得三，又二十五分之六，凡二十五置，終天之數，得八十一，以天地五位之合終於十者乘之，為八百一十分，應曆一統千五百三十九歲之章數，黃鐘之實也。 繇此之義，起十二律之周徑。 地之數始於二，終於三十。 其義紀之以兩，故置一得二，凡三十置，終地之數，得六十，以地中數六乘之，為三百六十分，當期之日，林鐘之實。 人者，繼天順地，序氣成物，統八卦，調八風，理八政，正八節，諧八音，舞八佾，監八方，被八荒，以終天地之功，故八八六十四。 其義極天地之變，以天地五位之合終於十者乘之，為六百四十分，以應六十四卦，大族之實也。 《書》曰：「天功人其代之。」 天兼地，人則天，故以五位之合乘焉，「唯天為大，唯堯則之」之象也。 地以中數乘者，陰道理內，在中餽之象也。 三統相通，故黃鐘、林鐘、太族律長皆全寸而亡餘分也。
As the *Yijing* says, "Take three for Heaven and two for Earth, and numbers are established on that basis." The heavenly numerical sequence starts from one and culminates at twenty-five. Its logic is organized by triplication: one becomes three, with an added ratio of six twenty-fifths. Carrying this through twenty-five placements to complete the heavenly number gives eighty-one; multiplied by ten (the completion of the five positional correspondences of Heaven and Earth), it yields 810. This matches the cycle chapter-count for one calendrical system of 1,539 years, namely the Huangzhong constant. On this basis, the dimensional ratios of the twelve pitch-pipes are derived. The terrestrial numerical sequence begins at two and ends at thirty. Its logic is organized by pairing: one gives two. Carrying this through thirty placements completes the earthly number at sixty; multiplied by Earth's central six, it yields 360 divisions, corresponding to the days of the annual cycle - the Linzhong constant. Humanity carries forward Heaven while conforming to Earth: ordering qi to complete things, coordinating the Eight Trigrams, eight winds, eight administrative domains, and eight seasonal divisions, harmonizing eight sounds, performing the eight-row dance, supervising the eight directions, and extending to the far reaches. Thus the human completion of Heaven-and-Earth's work is expressed as eight times eight, or sixty-four. Its logic extends to all Heaven-and-Earth transformations: multiplied by ten (the completion of the five positional correspondences), it gives 640, matching the sixty-four hexagrams. This is the Taicu constant. As the *Book of Documents* says, "Human beings carry out Heaven's work on its behalf." Because Heaven embraces Earth and human order takes Heaven as its norm, the calculation uses the five-position union as multiplier. This reflects the saying, "Only Heaven is truly great, and Yao alone took it as his model." Using Earth's central number as multiplier reflects the inward ordering of yin, symbolized as nourishment within the center. Since the Three Systems are mutually linked, Huangzhong, Linzhong, and Taicu all have whole-cun pipe lengths without fractional excess.
9
天之中數五，地之中數六，而二者為合。 六為虛，五為聲，周流於六虛。 虛者，爻律夫陰陽，登降運行，列為十二，而律呂和矣。 太極元氣，函三為一。 極，中也。 元，始也。 行於十二辰，始動於子。 參之於丑，得三。 又參之於寅，得九。 又參之於卯，得二十七。 又參之於辰，得八十一。 又參之於巳，得二百四十三。 又參之於午，得七百二十九。 又參之於未，得二千一百八十七。 又參之於申，得六千五百六十一。 又參之於酉，得萬九千六百八十三。 又參之於戌，得五萬九千四十九。 又參之於亥，得十七萬七千一百四十七。 此陰陽合德，氣鐘於子，化生萬物者也。 故孳萌於子，紐牙於丑，引達於寅，冒茆於卯，振美於辰，已盛於巳，咢布於午，昧薆於未，申堅於申，留孰於酉，畢入於戌，該閡於亥。 出甲於甲，奮軋於乙，明炳於丙，大盛於丁，豐楙於戊，理紀於己，斂更於庚，悉新於辛，懷任於壬，陳揆於癸。 故陰陽之施化，萬物之終始，既類旅於律呂，又經歷於日辰，而變化之情可見矣。
Heaven's central five and Earth's central six combine as a paired system. The number six designates the empty positions, while five denotes tone, and tonal operation circulates through those six positions. Within these six void positions, line-structure and pitch regulate yin and yang; through ascent, descent, and cyclical movement they are arranged as twelve, and the lü-lü system is harmonized. The Great Ultimate's primal qi contains a triad within unity. "Ultimate" here denotes centrality. "Primal" means the beginning. Its cycle runs through the twelve earthly branches, beginning at zi. Applying the tripling operation at chou yields three. Tripling once more at yin yields nine. A further tripling at mao produces twenty-seven. A further tripling at chen yields eighty-one. Tripling at si again produces 243. Another tripling at wu gives 729. Tripling at wei further yields 2,187. At shen, the next tripling gives 6,561. Tripling again at you yields 19,683. At xu, the next tripling produces 59,049. A further tripling at hai yields 177,147. This sequence represents yin and yang united in efficacy, qi concentrating at zi, and from there transforming into the generation of all things. So life first germinates at zi, puts down buds and roots at chou, stretches outward at yin, covers itself with shoots at mao, flourishes at chen, reaches fullness at si, spreads at wu, turns obscure at wei, hardens at shen, ripens and lingers at you, withdraws at xu, and is sealed in at hai. Accordingly it issues forth at jia, surges at yi, shines at bing, flourishes fully at ding, becomes rich at wu, enters ordered regulation at ji, gathers and shifts at geng, renews completely at xin, receives and bears at ren, and presents settled pattern at gui. Thus the operations of yin-yang and the cycles of beginning and ending in all things are mapped both through the pitch system and through calendrical branches, making the pattern of transformation intelligible.
10
玉衡杓建，天之綱也； 日月初纏，星之紀也。 綱紀之交，以原始造設，合樂用焉。 律呂唱和，以育生成化，歌奏用焉。 指顧取象，然後陰陽萬物靡不條鬯該成。 故以成之數忖該之積，如法為一寸，則黃鐘之長也。 參分損一，下生林鐘。 參分林鐘益一，上生太族。 參分太族損一，下生南呂。 參分南呂益一，上生姑洗。 參分姑洗損一，下生應鐘。 參分應鐘益一，上生蕤賓。 參分蕤賓損一，下生大呂。 參分大呂益一，上生夷則。 參分夷則損一，下生夾鐘。 參分夾鐘益一，上生亡射。 參分亡射損一，下生中呂。 陰陽相生，自黃鐘始而左旋，八八為伍。 其法皆用銅。 職在大樂，太常掌之。
The directional setting of the Jade Balance's handle forms Heaven's governing framework; and the initial paths of sun and moon provide the starry calendar's ordering sequence. At the intersection of these frameworks, one can trace first principles and found institutions, from which ritual music takes practical form. The lü-lü system works by call-and-response, giving rise to generation and transformation; sung and instrumental performance both depend on it. Once one takes these correspondences as formal models, yin and yang and all things can be ordered, unfolded, and comprehensively completed. So if the completed numerical total is reduced according to rule to one cun, that value gives the Huangzhong length. Apply the sanfen method and remove one part: the descending generation yields Linzhong. From Linzhong, apply sanfen and add one part: the ascending generation yields Taicu. From Taicu, subtract one third by the sanfen rule to generate Nanlü downward. From Nanlü, add one third by sanfen to generate Guxian upward. From Guxian, subtract one third to generate Yingzhong in descending order. From Yingzhong, add one third to generate Ruibin upward. From Ruibin, subtract one third to generate Dalü downward. From Dalü, add one third and generate Yize upward. From Yize, subtract one third to generate Jiazhong downward. From Jiazhong, add one third to generate Wuyi upward. From Wuyi, subtract one third to generate Zhonglü downward. Yin and yang mutually generate in sequence, beginning from Huangzhong and turning leftward in cyclical order, organized in an eight-by-eight structure. All of these instruments are made of bronze. This fell under the Grand Music office, administered by the Minister of Ceremonies.
11
度者，分、寸、尺、丈、引也，所以度長短也。 本起黃鐘之長。 以子穀秬黍中者，一黍之廣，度之九十分，黃鐘之長。 一為一分，十分為寸，十寸為尺，十尺為丈，十丈為引，而五度審矣。 其法用銅，高一寸，廣二寸，長一丈，而分寸尺丈存焉。 用竹為引，高一分，廣六分，長十丈，其方法矩，高廣之數，陰陽之象也。 分者，自三微而成著，可分別也。 寸者，忖也。 尺者，卺也。 丈者，張也。 引者，信也。 夫度者，別於分，忖於寸，卺於尺，張於丈，信於引。 引者，信天下也。 職在內官，廷尉掌之。
Linear measure consists of fen, cun, chi, zhang, and yin, by which length is determined. The system is grounded in the Huangzhong standard length. Using mid-sized black millet as the reference, the width of one grain is treated as a unit, and ninety such units define the Huangzhong length. The scale proceeds as follows: 1 fen, 10 fen = 1 cun, 10 cun = 1 chi, 10 chi = 1 zhang, and 10 zhang = 1 yin; in this way the five length units are established. The bronze rule is one cun high, two cun wide, and one zhang long, with fen, cun, chi, and zhang divisions inscribed. The yin rod is made of bamboo, one fen high, six fen wide, and ten zhang long; its rectilinear form and dimensions are taken as symbolic expressions of yin and yang. Fen signifies differentiation: beginning from tiny subtleties, distinctions become clear. Cun is glossed as measuring by estimation. Chi is explained as a unit for taking hold. Zhang is glossed as extension. Yin is understood as reliable verification. So the sequence of measurement is: differentiation in fen, gauging in cun, grasping in chi, extension in zhang, and final verification in yin. The yin unit is what secures common trust in measurement across the realm. This responsibility fell under the inner administration, under the Commandant of Justice.
12
量者，龠、合、升、斗、斛也，所以量多少也。 本起於黃鐘之龠，用度數審其容，以子穀秬黍中者千有二百實其龠，以井水準其概。 合龠為合，十合為升，十升為斗，十斗為斛，而五量嘉矣。 其法用銅，方尺而圜其外，旁有庣焉。 其上為斛，其下為斗。 左耳為升，右耳為合龠。 其狀似爵，以縻爵祿。 上三下二，參天兩地，圜而函方，左一右二，陰陽之象也。 其圜象規，其重二鈞，備氣物之數，合萬有一千五百二十。 聲中黃鐘，始於黃鐘而反覆焉，君制器之象也。 龠者，黃鐘律之實也，躍微動氣而生物也。 合者，合龠之量也。 升者，登合之量也。 斗者，聚升之量也。 斛者，角斗平多少之量也。 夫量者，躍於龠，合於合，登於升，聚於斗，角於斛也。 職在太倉，大司農掌之。
Capacity measurement consists of yue, he, sheng, dou, and hu, by which quantity is determined. The standard begins from the Huangzhong yue. Its volume is tested numerically: one yue is filled by 1,200 mid-grade black millet grains, and well water serves as the leveling reference. The conversion is: 2 yue = 1 he, 10 he = 1 sheng, 10 sheng = 1 dou, and 10 dou = 1 hu; in this way the five capacity units are standardized. The vessel is cast in bronze, with a one-chi square interior and a rounded exterior, with side projections. Its upper part corresponds to the hu unit, and its lower part to dou. The left handle corresponds to sheng, while the right corresponds to he and yue. Its shape imitates the ritual jue cup, linking measurement to the order of office and emolument. With three above and two below, it follows the Heaven-three Earth-two ratio; round exterior and square interior, with one left and two right, it is designed as an image of yin-yang structure. Its round contour corresponds to the compass model, and at a weight of two jun it completes the qi-material number set, totaling 11,520. Its tone is keyed to Huangzhong, beginning from Huangzhong and returning to it in repetition; this represents the ruler's normative pattern in instrument-making. The yue unit is the concrete expression of the Huangzhong pitch standard: from subtle stirring of qi, generation begins. He is the composite unit formed from yue. Sheng is the next unit built up from he. Dou is the aggregate unit composed from sheng. Hu is the unit used to calibrate and equalize larger dou-level quantities. The capacity scale progresses from yue to he, rises to sheng, aggregates as dou, and reaches its full form at hu. Administration of this standard was assigned to the Grand Granary under the Grand Minister of Agriculture.
13
衡權者，衡，平也，權，重也，衡所以任權而均物平輕重也。 其道如底，以見準之正，繩之直，左旋見規，右折見矩。 其在天也，佐助旋機，斟酌建指，以齊七政，故曰玉衡。 論語云：「
In weighing, *heng* denotes balancing and *quan* denotes weighted mass; the balance carries weights to equalize objects and determine relative heaviness. Its principle is foundational: it manifests level by the standard, straightness by the line, circularity by left rotation, and squareness by right-angled turn. In celestial terms, it supports the turning mechanism and sets the directional indicator so the seven luminary motions are harmonized; hence the name Jade Balance. The *Analects* says:
14
立則見其參於前也，在車則見其倚於衡也。」 又曰：「齊之以禮。」 此衡在前居南方之義也。
"Standing, one sees it aligned before oneself; riding in a carriage, one sees it resting against the balance beam." It also says, "Bring them into order through ritual." This is why the balance is placed in front, signifying the southern orientation.
15
權者，銖、兩、斤、鈞、石也，所以稱物平施，知輕重也。 本起於黃鐘之重。 一龠容千二百黍，重十二銖，兩之為兩。 二十四銖為兩。 十六兩為斤。 三十斤為鈞。 四鈞為石。 忖為十八，易十有八變之象也。 五權之制，以義立之，以物鈞之，其餘小大之差，以輕重為宜。 圜而環之，令之肉倍好者，周旋無端，終而復始，無窮已也。 銖者，物繇忽微始，至於成著，可殊異也。 兩者，兩黃鐘律之重也。 二十四銖而成兩者，二十四氣之象也。 斤者，明也，三百八十四銖，易二篇之爻，陰陽變動之象也。 十六兩成斤者，四時乘四方之象也。 鈞者，均也，陽施其氣，陰化其物，皆得其成就平均也。 權與物均，重萬一千五百二十銖，當萬物之象也。 四百八十兩者，六旬行八節之象也。 三十斤成鈞者，一月之象也。 石者，大也，權之大者也。 始於銖，兩於兩，明於斤，均於鈞，終於石，物終石大也。 四鈞為石者，四時之象也。 重百二十斤者，十二月之象也。 終於十二辰而復於子，黃鐘之象也。 千九百二十兩者，陰陽之數也。 三百八十四爻，五行之象也。 四萬六千八十銖者，萬一千五百二十物歷四時之象也。 而歲功成就，五權謹矣。
The weight system consists of zhu, liang, jin, jun, and shi, by which objects are weighed and their relative heaviness determined. Its standard is grounded in the Huangzhong weight value. A single yue containing 1,200 millet grains weighs 12 zhu; doubled, that becomes one liang. 24 zhu are equivalent to one liang. 16 liang are equivalent to one jin. 30 jin are equivalent to one jun. 4 jun are equivalent to one shi. The measure is set at eighteen units, corresponding to the *Yijing* image of eighteen transformations. The five-grade weight system is founded on normative principle and calibrated against actual objects; residual differences of scale are adjusted according to relative heaviness. Its ring form is circular, with body thickness twice the opening, symbolizing endless rotation: completion returning to beginning without limit. The zhu unit signifies how things emerge from minute beginnings into manifest form, where distinctions become clear. A liang represents a doubled Huangzhong standard weight. The equation 24 zhu = 1 liang is taken as an analogue of the twenty-four seasonal qi divisions. Jin is associated with clarification; at 384 zhu, it corresponds to the line count of the two *Yijing* books and symbolizes yin-yang transformation. The ratio 16 liang per jin represents the four seasons times the four directions. Jun signifies equilibrium: yang exerts qi, yin transforms things, and both arrive at balanced completion. When weights are brought into full equilibrium with objects, the figure is 11,520 zhu, symbolically matching the myriad things. The value of 480 liang is interpreted as an analogue of six ten-day cycles moving through eight seasonal nodes. The ratio of 30 jin to one jun symbolizes a month's cycle. Shi denotes largeness and is the largest unit in the weight scale. The sequence starts with zhu, doubles into liang, clarifies at jin, equalizes at jun, and culminates at shi, since shi is the largest terminal unit. The ratio 4 jun = 1 shi symbolizes the four seasons. A total weight of 120 jin represents the twelve-month cycle. The full turn through twelve branches and return to zi is taken as the Huangzhong pattern. The figure 1,920 liang is interpreted as a yin-yang number. The total of 384 lines is treated as an analogue of Five-Phase operation. The value 46,080 zhu symbolizes 11,520 entities cycling through all four seasons. In this way annual operations are completed and the five weight standards are made exact.
16
權與物鈞而生衡，衡運生規，規圜生矩，矩方生繩，繩直生準，準正則平衡而鈞權矣。 是為五則。 規者，所以規圜器械，令得其類也。 矩者，所以矩方器械，令不失其形也。 規矩相須，陰陽位序，圜方乃成。 準者，所以揆平取正也。 繩者，上下端直，經緯四通也。 準繩連體，衡權合德，百工繇焉，以定法式，輔弼執玉，以翼天子。 《詩》云：「尹氏大師，秉國之鈞，四方是維，天子是毗，俾民不迷。」 咸有五象，其義一也。 以陰陽言之，大陰者，北方。 北，伏也，陽氣伏於下，於時為冬。 冬，終也，物終臧，乃可稱。 水潤下。 知者謀，謀者重，故為權也。 大陽者，南方。 南，任也，陽氣任養物，於時為夏。 夏，假也，物假大，乃宣平。 火炎上。 禮者齊，齊者平，故為衡也。 少陰者，西方。 西，遷也，陰氣遷落物，於時為秋。 秋，胆也，物呙斂，乃成孰。 金從革，改更也。 義者成，成者方，故為矩也。 少陽者，東方。 東，動也，陽氣動物，於時為春。 春，蠢也，物蠢生，乃動運。 木曲直。 仁者生，生者圜，故為規也。 中央者，陰陽之內，四方之中，經緯通達，乃能端直，於時為四季。 土稼嗇蕃息。 信者誠，誠者直，故為繩也。 五則揆物，有輕重圜方平直陰陽之義，四方四時之體，五常五行之象。 厥法有品，各順其方而應其行。 職在大行，鴻臚掌之。
Equalized weights and objects produce balance; operating the balance yields the compass, from compass comes the circle, from circle the square, from square the line, and from straight line the level. When the level is correct, balance and weighing are both correct. This is what is meant by the Five Standards. The compass governs circular implements so each takes its proper form. The square-rule governs rectilinear implements so their form does not deviate. Compass and square are mutually dependent; once yin and yang are correctly sequenced, circular and square forms can be fully realized. The level is used to assess flatness and establish true alignment. The line establishes vertical straightness from top to bottom, while longitudinal and transverse directions connect through all four quarters. Level and line function as one system, and balance and weight embody the same principle. From these, all crafts establish technical standards; with such norms, ministers bearing court insignia can properly support the emperor. As the *Book of Odes* says: "The Yin clan's Grand Master held the state's balancing power, sustaining the four quarters and assisting the Son of Heaven, so that the people did not lose their way." All are expressed through five symbols, but their underlying principle is single. Within yin-yang correspondences, Great Yin is assigned to the north. North signifies latency, where yang qi is hidden beneath; seasonally, this is winter. Winter marks completion: things come to an end and are stored away, making assessment possible. Water's nature is to moisten and descend. Wisdom consists in deliberation, and deliberation has weight; hence it corresponds to weighting. Great Yang is assigned to the south. South signifies taking charge: yang qi assumes the task of nourishing life, corresponding to summer. Summer signifies expansion, when things reach great growth and unfold toward full expression. Fire's tendency is to rise. Ritual establishes ordered alignment, and ordered alignment produces equilibrium; thus it corresponds to balancing. Lesser Yin corresponds to the west. West signifies transition, where yin qi moves things toward decline; seasonally, this is autumn. Autumn signifies contraction and hardening, as things draw inward and come to maturity. Metal's nature is transformation and reformation. Righteousness brings things to completion, and completion takes square form; thus it corresponds to the square. Lesser Yang corresponds to the east. East signifies movement, where yang qi sets life in motion; seasonally, this is spring. Spring means awakening, when life stirs and begins active circulation. Wood's property is flexible straightness - bending and returning straight. Benevolence is life-generating, and generation takes circular form; hence it corresponds to the compass. The center lies within the yin-yang field and amid the four directions; only when longitudinal and transverse lines are fully connected can true uprightness be established. Seasonally, this corresponds to the four interseasonal nodes. Earth's function is to receive cultivation, store, and foster proliferation. Trust is grounded in sincerity, and sincerity in straightness; thus it corresponds to the line. The Five Standards evaluate things through paired principles - weight and lightness, round and square, level and straight, yin and yang - while embodying the structure of four directions, four seasons, and the correspondences of the five virtues and five phases. Their methods are classed by rank, each aligned to its proper directional position and phase-correspondence. This function was assigned to the Office of Grand Processions, administered by the Director of Guests.
17
《書》曰：「予欲聞六律、五聲、八音、七始詠，以出內五言，女聽。」 予者，帝舜也。 言以律呂和五聲，施之八音，合之成樂。 七者，天地四時人之始也。 順以歌詠五常之言，聽之則順乎天地，序乎四時，應人倫，本陰陽，原情性，風之以德，感之以樂，莫不同乎一。 唯聖人為能同天下之意，故帝舜欲聞之也。 今廣延群儒，博謀講道，修明舊典，同律，審度，嘉量，平衡，鈞權，正準，直繩，立于五則，備數和聲，以利兆民，貞天下於一，同海內之歸。 凡律度量衡用銅者，名自名也，所以同天下，齊風俗也。 銅為物之至精，不為燥溼寒暑變其節，不為風雨暴露改其形，介然有常，有似於士君子之行，是以用銅也。 用竹為引者，事之宜也。
As the *Book of Documents* says: "I want to hear the six pitch standards, five tones, eight timbral classes, and the chants of the seven beginnings, so that the five forms of utterance may be properly issued and received. Listen well." The pronoun "I" here refers to Emperor Shun. The passage means that one should harmonize the five tones through the lü-lü system, apply them across the eight timbral classes, and integrate them into complete music. The "seven" refers to the primal origins of Heaven, Earth, the four seasons, and human order. Accordingly, song and chant should express the language of the five constants. Properly heard, they align with Heaven and Earth, follow the seasons, answer to human relations, root in yin-yang principle, and accord with human feeling and nature; through moral influence and musical transformation, all are brought into unity. Only a sage can unify the intentions of the whole realm, which is why Emperor Shun wished to hear such music. Now we have broadly convened scholars, sought wide-ranging counsel, and deliberated on principle, restoring and clarifying the old canons: unifying pitch, scrutinizing measures, standardizing capacities, balancing scales, equalizing weights, correcting levels, and straightening lines, all grounded in the Five Standards. By perfecting number and tonal harmony, we aim to benefit the people, secure unity in the realm, and bring all under heaven to a common order. For pitch standards, linear measures, capacities, and weights made in bronze, each designation matches a fixed standard; by this, the realm is unified and customs standardized. Bronze is a highly refined material: neither dryness nor moisture, neither cold nor heat alters its measure, and wind, rain, or exposure do not distort its shape. Its stable constancy resembles the conduct of the noble person, and for this reason it is used. Using bamboo for the yin rod is a matter of practical suitability.
18
曆數之起上矣。 傳述顓頊命南正重司天，火正黎司地，其後三苗亂德，二官咸廢，而閏餘乖次，孟陬殄滅，攝提失方。 堯復育重、黎之後，使纂其業，故《書》曰：「乃命羲、和，欽若昊天，曆象日月星辰，敬授民時。」 「歲三百有六旬有六日，以閏月定四時成歲，允釐百官，眾功皆美。」 其後以授舜曰：「咨爾舜，天之曆數在爾躬。」 「舜亦以命禹。」 至周武王訪箕子，箕子言大法九章，而五紀明曆法。 故自殷周，皆創業改制，咸正曆紀，服色從之，順其時氣，以應天道。 三代既沒，五伯之末史官喪紀，疇人子弟分散，或在夷狄，故其所記，有黃帝、顓頊、夏、殷、周及魯曆。 戰國擾攘，秦兼天下，未皇暇也，亦頗推五勝，而自以為獲水德，乃以十月為正，色上黑。
The beginnings of calendrical computation reach back to the highest antiquity. According to transmitted accounts, Zhuanxu appointed Zhong, the Southern Corrector, to oversee celestial affairs, and Li, the Fire Corrector, to oversee terrestrial affairs. Later, when the Three Miao disrupted moral order, both offices collapsed; intercalary reckoning went out of sequence, first-month alignment disappeared, and the Sheti cycle lost its orientation. Yao then restored the line of Zhong and Li and ordered their descendants to continue that work. Hence the *Book of Documents* says: "He charged Xi and He to revere High Heaven, calculate the patterns of sun, moon, stars, and constellations, and respectfully promulgate seasonal time for the people." "The year contains 366 days; with intercalary months, the four seasons are fixed and the annual cycle is completed. Thus the hundred offices are properly ordered and all undertakings succeed." He then transmitted this charge to Shun, saying: "Ah, Shun, Heaven's calendrical mandate now resides in your person." "Shun, in turn, entrusted the same charge to Yu." When King Wu of Zhou sought counsel from Jizi, Jizi presented the Nine Articles of the Great Plan, whose fifth section explicates calendrical governance. Accordingly, from Shang to Zhou, each founding regime that restructured institutions also corrected the calendrical cycle, with official color codes adjusted to match seasonal qi and align with Heaven's order. After the Three Dynasties fell, and by the late age of the Five Hegemons, historical offices had lost their archival continuity; calendrical families dispersed, some even beyond the central states. What remained were records of the calendars of the Yellow Emperor, Zhuanxu, Xia, Yin, Zhou, and Lu. Amid Warring States chaos, Qin unified all-under-heaven and had little leisure for full reform. Still, it adopted the Five-Phase conquest doctrine, claiming Water Virtue, and therefore set month ten as the year's beginning and elevated black as the dynastic color.
19
漢興，方綱紀大基，庶事草創，襲秦正朔。 以北平侯張蒼言，用顓頊曆，比於六曆，疏闊中最為微近。 然正朔服色，未睹其真，而朔晦月見，弦望滿虧，多非是。
At the Han founding, while state structures were still being built from the ground up, the regime initially retained Qin's calendar basis. On Zhang Cang's recommendation, Han adopted the Zhuanxu calendar, which - among the six transmitted systems - was judged rough in scope but closest in fine approximation. Even so, true correctness in New-Year reckoning and dynastic color had not been achieved, and many predictions of lunar phases - new, dark, rising, quarter, full, and waning - were inaccurate.
20
至武帝元封七年，漢興百二歲矣，大中大夫公孫卿、壺遂、太史令司馬遷等言「曆紀壞廢，宜改正朔」。 是時御史大夫兒寬明經術，上乃詔寬曰：「與博士共議，今宜何以為正朔？ 服色何上？」 寬與博士賜等議，皆曰：「帝王必改正朔，易服色，所以明受命於天也。 創業變改，制不相復，推傳序文，則今夏時也。 臣等聞學褊陋，不能明。 陛下躬聖發憤，昭配天地，臣愚以為三統之制，後聖復前聖者，二代在前也。 今二代之統絕而不序矣，唯陛下發聖德，宣考天地四時之極，則順陰陽以定大明之制，為萬世則。」 於是乃詔御史曰：「乃者有司言曆未定，廣延宣問，以考星度，未能讎也。 蓋聞古者黃帝合而不死，名察發斂，定清濁，起五部，建氣物分數。 然則上矣。 書缺樂弛，朕甚難之。 依違以惟，未能修明。 其以七年為元年。」 遂詔卿、遂、遷與侍郎尊、大典星射姓等議造漢曆。 乃定東西，立晷儀，下漏刻，以追二十八宿相距於四方，舉終以定朔晦分至，躔離弦望。 乃以前曆上元泰初四千六百一十七歲，至於元封七年，復得閼逢攝提格之歲，中冬十一月甲子朔旦冬至，日月在建星，太歲在子，已得太初本星度新正。 姓等奏不能為算，願募治曆者，更造密度，各自增減，以造漢太初曆。 乃選治曆鄧平及長樂司馬可、酒泉候宜君、侍郎尊及與民間治曆者，凡二十餘人，方士唐都、巴郡落下閎與焉。 都分天部，而閎運算轉曆。 其法以律起曆，曰：「律容一龠，積八十一寸，則一日之分也。 與長相終。 律長九寸，百七十一分而終復。 三復而得甲子。 夫律陰陽九六，爻象所從出也。 故黃鐘紀元氣之謂律。 律，法也，莫不取法焉。」 與鄧平所治同。 於是皆觀新星度、日月行，更以算推，如閎、平法。 法，一月之日二十九日八十一分日之四十三。 先藉半日，名曰陽曆； 不藉，名曰陰曆。 所謂陽曆者，先朔月生； 陰曆者，朔而後月乃生。 平曰：「陽曆朔皆先旦月生，以朝諸侯王群臣便。」 乃詔遷用鄧平所造八十一分律曆，罷廢尤疏遠者十七家，復使校曆律昏明。 宦者淳于陵渠復覆太初曆晦朔弦望，皆最密，日月如合璧，五星如連珠。 陵渠奏狀，遂用鄧平曆，以平為太史丞。
By the seventh year of Yuanfeng under Emperor Wu, 102 years had elapsed since Han's founding. Gongsun Qing, Hu Sui, Sima Qian, and others memorialized that the calendrical system had decayed and should be corrected, beginning with the proper first month. At the time, the Censor-in-Chief Ni Kuan was renowned for classical learning, so the emperor ordered him: "Consult the court Erudites: by what standard should we now set the year's first month? And which dynastic color should hold primacy?" Ni Kuan and the Erudites, including Ci, deliberated and replied: "Sovereign founders must revise calendrical commencement and change state colors to show clearly that they have received Heaven's mandate. At dynastic founding, institutional systems are transformed rather than simply repeated; judged from transmitted succession principles, the present should follow the Xia standard. Our learning is limited and inadequate, and we cannot claim full clarity on this point. Your Majesty's personal sageliness and determined effort are worthy of Heaven and Earth. In my humble view, under the Three-Order model, when one sage age succeeds another, two dynasties intervene between them. Now those two dynastic lines are interrupted and no longer properly ordered. If Your Majesty manifests sagely virtue and fully examines the ultimate principles of Heaven, Earth, and the four seasons, then by following yin and yang you can establish a great luminous system to serve as a norm for all later ages." The emperor then ordered the Censorate: "The ministries have reported that the calendar remains unsettled. We have widely solicited inquiry and consultation, examining stellar positions, but we still have not been able to reconcile the calculations. I have heard that in high antiquity the Yellow Emperor brought these domains into coherent order: he distinguished categories, observed cycles of emergence and withdrawal, defined clear and turbid, set out the five divisions, and established numerical ratios for qi and material forms. That is truly the highest precedent. Now that records are incomplete and ritual music has lapsed, I find the problem exceedingly difficult. Despite repeated deliberation and caution, we still have not been able to restore full clarity. Accordingly, let year seven be reset as year one." He then ordered Gongsun Qing, Hu Sui, Sima Qian, Attendant Gentleman Zun, Grand Astrologer Shexing, and others to deliberate and draft a new Han calendar. They established east-west orientation, erected gnomons and observing instruments, and set water clocks, so they could track the spacing of the twenty-eight lunar lodges across the four quarters and thereby determine new and dark moons, solstices and equinoxes, orbital position, and quarter/full phases. Using the old calendar, they back-calculated 4,617 years to the Taichu high epoch, and by Yuanfeng 7 they recovered a benchmark year in the sexagenary cycle: in midwinter, month eleven, a jiazi new-moon dawn coincided with the winter solstice, with sun and moon at the Jian stars and Jupiter at zi. On that basis they established the Taichu system's base stellar degrees and new opening date. Shexing and his colleagues reported that they could not complete the calculations and requested recruitment of calendrical specialists to produce tighter numerical measures, adjusting where needed, so as to compile the Han Taichu calendar. They therefore selected the calendrical expert Deng Ping, Sima Ke of Changle, Hou Yijun of Jiuquan, Attendant Gentleman Zun, and over twenty private calendar specialists; the technical masters Tang Du and Luo Xiahong of Ba commandery also joined. Tang Du was responsible for dividing the celestial sectors, while Luo Xiahong ran the core computations and cyclical calendrical conversions. Their method derived calendrical numbers from pitch standards: "A pitch vessel contains one yue; accumulated to 81 cun, this yields the day's fractional divisions. This is paired with the terminal value in the length sequence. With a pitch-pipe length of nine cun, the cycle returns to its starting point after 171 divisions. After three full recurrences, the cycle arrives at jiazi. The yin-yang pitch standards, structured by nine and six, are the source from which line-symbols are derived. Hence Huangzhong, as the register of primal qi, is what is meant by a lü standard. "Lü means normative law: all things take it as their model." This agreed with Deng Ping's calculations. They then reexamined new stellar-degree data and solar-lunar motion, recalculating according to the methods associated with Luo Xiahong and Deng Ping. In that method, a synodic month is 29 days plus 43/81 of a day. When a half-day offset is applied in advance, it is called the yang calendar; without that offset, it is called the yin calendar. In the so-called yang calendar, the moon is treated as appearing before the calendrical new-moon point; in the yin calendar, lunar appearance is reckoned only after the new-moon point. Deng Ping said: "Under the yang calendar, at each new moon the moon is reckoned to have appeared before dawn, which makes court morning audiences more convenient for kings and ministers." The court then ordered Sima Qian to implement Deng Ping's 81-part pitch-calendar method, discarding seventeen systems deemed especially imprecise, and to continue verification of calendar and pitch standards through dawn-and-dusk observations. The eunuch Chunyu Lingqu rechecked the Taichu calendar's new-moon, dark-moon, quarter, and full-phase calculations and found them the most exact: sun and moon aligned like paired jade discs, and the five planets like a string of pearls. After Lingqu submitted his report, the court adopted Deng Ping's calendar and appointed Deng Ping as Assistant Grand Historian.
21
後二十七年，元鳳三年，太史令張壽王上書言：「曆者天地之大紀，上帝所為。 傳黃帝調律曆，漢元年以來用之。 今陰陽不調，宜更曆之過也。」 詔下主曆使者鮮于妄人詰問，壽王不服。 妄人請與治曆大司農中丞麻光等二十餘人雜候日月晦朔弦望、八節二十四氣，鈞校諸曆用狀。 奏可。 詔與丞相、御史、大將軍、右將軍史各一人雜候上林清臺，課諸曆疏密，凡十一家。 以元鳳三年十一月朔旦冬至，盡五年十二月，各有第。 壽王課疏遠。 案漢元年不用黃帝調曆，壽王非漢曆，逆天道，非所宜言，大不敬。 有詔勿劾。 復候，盡六年。 太初曆第一，即墨徐萬且、長安徐禹治太初曆亦第一。 壽王及待詔李信治黃帝調曆，課皆疏闊，又言黃帝至元鳳三年六千餘歲。 丞相屬寶、長安單安國、安陵桮育治終始，言黃帝以來三千六百二十九歲，不與壽王合。 壽王又移帝王錄，舜、禹年歲不合人年。 壽王言化益為天子代禹，驪山女亦為天子，在殷周間，皆不合經術。 壽王曆乃太史官殷曆也。 壽王猥曰安得五家曆，又妄言太初曆虧四分日之三，去小餘七百五分，以故陰陽不調，謂之亂世。 劾壽王吏八百石，古之大夫，服儒衣，誦不詳之辭，作祅言欲亂制度，不道。 奏可。 壽王候課，比三年下，終不服。 再劾死，更赦勿劾，遂不更言，誹謗益甚，竟以下吏。 故曆本之驗在於天，自漢曆初起，盡元鳳六年，三十六歲，而是非堅定。
Twenty-seven years later, in the third year of Yuanfeng, the Grand Astrologer Zhang Shouwang submitted a memorial: "The calendar is the fundamental order of Heaven and Earth, established by the Supreme Deity." It was said that the Yellow Emperor set the pitch standards and calendar. In Han year one, the court compared the Zhuanxu calendar with the six systems: it was loose and imperfect, though still the closest fit. Even so, the official first month and ritual colors were not yet settled. "Now yin and yang are out of balance; the calendar is at fault and should be revised." The throne ordered Xianyu Wangren, the official responsible for calendrical matters, to examine the issue. His reply was judged best, including in water-clock calculations. Wangren asked to help reform the calendar. He was summoned for questioning, and his proposals were circulated alongside Zhang Shouwang’s. Wangren argued: "The sages who created calendars in antiquity based them on heavenly qi and aligned them with the Dipper’s pole." The throne approved the proposal. An edict ordered joint observations of stars and calendrical data by the chancellor, censor-in-chief, commander-in-chief, general of the right, court historians, and erudites. They determined that in Yuanfeng 3, on the first day of the eleventh month, the winter solstice occurred at dawn, with sun and moon in the Jian asterism. Shouwang’s calculations proved rough and inaccurate. An inquiry showed that the Han court had not used the Yellow Emperor-derived tuned calendar in its first year. An edict ordered that no impeachment be brought. Observations were resumed and continued for a full six years. In the first year under the Taichu calendar, sun and moon were in conjunction, and the five planets aligned like a string of pearls. Shouwang and Li Xin, an expectant appointee, worked on the Yellow Emperor-style tuned calendar. Bao from the chancellor’s office, Shan Anguo of Chang’an, and Bei Yu of Anling handled calculations of cyclical beginnings and endings. Shouwang also revised the imperial chronologies, assigning reign lengths to Shun and Yu that did not accord with normal human lifespans. Shouwang claimed that Hua Yi succeeded Yu as Son of Heaven, and even that a woman from Mount Li ruled as Son of Heaven between Shang and Zhou; such claims matched no recognized classical learning. Shouwang’s system was simply the old Yin calendar kept by the Office of the Grand Astrologer. Shouwang recklessly spoke of “recovering the five-family calendars,” and falsely argued that the Taichu system was short by calendrical fractions, which supposedly threw yin and yang out of balance and marked the age as chaotic. The impeachment charged that Shouwang, an 800-shi official of grandee standing, donned Confucian dress while reciting dubious claims and spreading ominous talk to unsettle state institutions, conduct contrary to proper principle. The throne approved the impeachment memorial. After three years of comparative observations, Shouwang’s results consistently ranked lower, but he still refused to concede. He was impeached again with capital punishment recommended, then pardoned once more under orders to stop further impeachment. He ceased open argument, but his slander intensified, and he was eventually turned over to the courts. Thus calendrical principles are ultimately tested against the heavens. From the start of Han calendrical reform to Yuanfeng 6, thirty-six years passed, and the dispute over right and wrong was decisively settled.
22
至孝成世，劉向總六曆，列是非，作五紀論。 向子歆究其微眇，作三統曆及譜以說春秋，推法密要，故述焉。
By Emperor Cheng’s time, Liu Xiang had collated the six calendrical systems, sorted out their merits and faults, and wrote the Treatise on the Five Epochs. Liu Xiang’s son Liu Xin examined the finest technical details, produced the Santong Calendar and accompanying tables to interpret the *Spring and Autumn Annals*, and developed compact, crucial computational methods; therefore they are set forth here.
23
夫曆春秋者，天時也，列人事而目以天時。 傳曰：「民受天地之中以生，所謂命也。 是故有禮誼動作威儀之則以定命也，能者養以之福，不能者敗以取禍。」 故列十二公二百四十二年之事，以陰陽之中制其禮。 故春為陽中，萬物以生； 秋為陰中，萬物以成。 是以事舉其中，禮取其和，曆數以閏正天地之中，以作事厚生，皆所以定命也。 易金火相革之卦曰「湯武革命，順乎天而應乎人」，又曰「治曆明時」，所以和人道也。
In the *Spring and Autumn* system, chronology is grounded in celestial time: human events are listed and marked according to heavenly timing. The *Commentary* says: "Human beings receive life from the balanced center of Heaven and Earth; this is what is called one’s mandate. Therefore norms of ritual, propriety, conduct, and bearing are established to stabilize that mandate. The capable cultivate it and gain blessing; the incapable ruin it and bring misfortune on themselves." Accordingly, it records the affairs of the twelve dukes across 242 years, using the balance of yin and yang as the standard for ordering ritual. Spring is the season centered in yang, when all things come into being; autumn is centered in yin, when all things come to completion. Hence policy must hold to central balance and ritual must seek harmony. Intercalary adjustments in calendrical calculation correct the median of Heaven and Earth, enabling governance and the nurturing of life; all this serves to stabilize the Mandate. The *Book of Changes* says in the Revolution hexagram: "Tang and Wu transformed the Mandate, following Heaven and answering the people," and also, "set the calendar in order and clarify the seasons." This is how the human order is brought into harmony.
24
周道既衰，幽王既喪，天子不能班朔，魯曆不正，以閏餘一之歲為蔀首。 故春秋刺「十一月乙亥朔，日有食之」。 於是辰在申，而司曆以為在建戌，史書建亥。 哀十二年，亦以建申流火之月為建亥，而怪蟄蟲之不伏也。 自文公閏月不告朔，至此百有餘年，莫能正曆數。 故子貢欲去其餼羊，孔子愛其禮，而著其法於春秋。 經曰：「冬十月朔，日有食之。」 傳曰：「不書日，官失之也。 天子有日官，諸侯有日御，日官居卿以厎日，禮也。 日御不失日以授百官於朝。」 言告朔也。 元典曆始曰元。 傳曰：「元，善之長也。」 共養三德為善。 又曰：「元，體之長也。」 合三體而為之原，故曰元。 於春三月，每月書王，元之三統也。 三統合於一元，故因元一而九三之以為法，十一三之以為實。 實如法得一。 黃鐘初九，律之首，陽之變也。 因而六之，以九為法，得林鐘初六，呂之首，陰之變也。 皆參天兩地之法也。 上生六而倍之，下生六而損之，皆以九為法。 九六，陰陽夫婦子母之道也。 律娶妻而呂生子，天地之情也。 六律六呂，而十二辰立矣。 五聲清濁，而十日行矣。 傳曰「天六地五」，數之常也。 天有六氣，降生五味。 夫五六者，天地之中合，而民所受以生也。 故日有六甲，辰有五子，十一而天地之道畢，言終而復始。 太極中央元氣，故為黃鐘，其實一龠，以其長自乘，故八十一為日法，所以生權衡度量，禮樂之所繇出也。 經元一以統始，易太極之首也。 春秋二以目歲，易兩儀之中也。 於春每月書王，易三極之統也。 於四時雖亡事必書時月，易四象之節也。 時月以建分至啟閉之分，易八卦之位也。 象事成敗，易吉凶之效也。 朝聘會盟，易大業之本也。 故易與春秋，天人之道也。 傳曰：「龜，象也。 筮，數也。 物生而後有象，象而後有滋，滋而後有數。」
Once the Zhou order decayed and King You was lost, the Son of Heaven could no longer issue the monthly calendar. Lu’s calendar went astray, using a flawed intercalary-remainder year as its epoch point. Accordingly, the *Spring and Autumn Annals* marks this as a fault: "On yihai, first day of the eleventh month, there was a solar eclipse." The asterism should then have been in shen, but calendrical officials treated it as jian-xu, while the historical record entered it as jian-hai. In Duke Ai’s twelfth year, they likewise misidentified the jian-shen “flowing fire” month as jian-hai, then wondered why the hibernating insects had not yet gone dormant. From Duke Wen’s failure to proclaim the new moon in an intercalary month to this point, over a century elapsed with no one able to rectify the calendrical reckoning. This is why Zigong wanted to abolish the offering sheep, while Confucius valued the rite and preserved its rule in the *Spring and Autumn Annals*. The Annals state: "In winter, first day of the tenth month, there was a solar eclipse." The Commentary says: "When the exact day is not recorded, it means the officials failed in their duty. The Son of Heaven employs day officials, and the feudal lords have day attendants. Day officials hold ministerial rank to determine dates; that is the proper rite. Day attendants must not miss the date; they report it to the assembled officials at court." This refers to the proclamation of the first day of the lunar month. In the foundational calendrical canon, the starting point is called *yuan*, “origin.” The Commentary says, "Origin is the foremost principle of goodness." Goodness consists in jointly cultivating the three virtues. It also says, "Origin is the leading principle of embodied existence." Because it unites the three cosmic bodies into a single source, it is called *yuan*. Across spring’s three months, each month is marked with “king”; this reflects origin’s threefold system. The three systems converge into one origin. So, from one *yuan*, one sets 9 x 3 as the divisor and 11 x 3 as the dividend. Dividing the dividend by that rule yields one. Huangzhong is “initial nine,” the first of the regular pitch-pipes and a transformation of yang. From there, multiplying by six and dividing by nine gives Linzhong as “initial six,” first of the complementary tones and a transformation of yin. All this follows the ratio principle of “three for Heaven, two for Earth.” In upward generation one multiplies by six and doubles; in downward generation one multiplies by six and reduces. In all cases, nine is the divisor. Nine and six encode the yin-yang pattern of husband and wife, mother and child. The regular pitch “takes a wife,” and the complementary pitch “bears a child”; this mirrors the generative relation of Heaven and Earth. With six regular pitches and six complementary pitches, the twelve earthly branches are set in place. Once the five tones are ordered into clear and turbid registers, the ten-day cycle can operate. The Commentary says, "Heaven is six, Earth is five"; this is the fixed numerical pattern. Heaven possesses six vital influences, which descend and produce the five flavors. Five and six form Heaven and Earth in balanced union, and that is what people receive as the basis of life. Hence the day cycle has six jia and the branch cycle has five zi: together they make eleven, completing the pattern of Heaven and Earth, which ends only to begin again. At the center of the Great Ultimate is primal qi, represented by Huangzhong. Its standard capacity is one yue, and its length squared gives eighty-one as the calendrical day rule. From this derive weights and measures, and from these in turn come ritual and music. The Classic sets origin at one to govern all beginnings; this corresponds to the Great Ultimate at the start of the *Changes*. The *Spring and Autumn* uses a dual structure to mark the year, matching the Two Modes in the *Changes*. Writing “king” in each spring month reflects the *Changes* principle of the three ultimate powers brought under one governance. Across all four seasons, even uneventful periods still record season and month; that follows the *Changes* pattern of the Four Images and their divisions. Seasonal and monthly records mark the divisions of solstices, equinoxes, opening, and closing; this mirrors the positional scheme of the Eight Trigrams. By representing events as success or failure, it enacts the *Changes* logic of auspicious and inauspicious outcomes. Court audiences, embassies, assemblies, and alliances form the basis of major political action, as in the *Changes*. Thus the *Changes* and the *Spring and Autumn* together articulate the Way linking Heaven and humanity. The Commentary says: "The turtle shell signifies image. Milfoil divination is fundamentally a matter of number. Things come into being, then take form; form develops, and only then do numbers arise."
25
是故元始有象一也，春秋二也，三統三也，四時四也，合而為十，成五體。 以五乘十，大衍之數也，而道據其一，其餘四十九，所當用也，故蓍以為數。 以象兩兩之，又以象三三之，又以象四四之，又歸奇象閏十九及所據一加之，因以再扐兩之，是為月法之實。 如日法得一，則一月之日數也，而三辰之會交矣，是以能生吉凶。 故《易》曰：「天一地二，天三地四，天五地六，天七地八，天九地十。 天數五，地數五，五位相得而各有合。 天數二十有五，地數三十，凡天地之數五十有五，此所以成變化而行鬼神也。」 并終數為十九，易窮則變，故為閏法。 參天九，兩地十，是為會數。 參天數二十五，兩地數三十，是為朔望之會。 以會數乘之，則周於朔旦冬至，是為會月。 九會而復元，黃鐘初九之數也。 經於四時，雖亡事必書時月。 時所以記啟閉也，月所以紀分至也。 啟閉者，節也。 分至者，中也。 節不必在其月，故時中必在正數之月。 故傳曰：「先王之正時也，履端於始，舉正於中，歸餘於終。 履端於始，序則不愆； 舉正於中，民則不惑； 歸餘於終，事則不誖。」 此聖王之重閏也。 以五位乘會數，而朔旦冬至，是為章月。 四分月法，以其一乘章月，是為中法。 參閏法為周至，以乘月法，以減中法而約之，則六扐之數，為一月之閏法，其餘七分，此中朔相求之術也。 朔不得中，是謂閏月，言陰陽雖交，不得中不生。 故日法乘閏法，是為統歲。 三統，是為元歲。 元歲之閏，陰陽災，三統閏法。 易九厄曰：初入元，百六，陽九； 次三百七十四，陰九； 次四百八十，陽九； 次七百二十，陰七； 次七百二十，陽七； 次六百，陰五； 次六百，陽五； 次四百八十，陰三； 次四百八十，陽三。 凡四千六百一十七歲，與一元終。 經歲四千五百六十，災歲五十七。 是以春秋曰：「舉正於中。」 又曰：「閏月不告朔，非禮也。 閏以正時，時以作事，事以厚生，生民之道於是乎在矣。 不告閏朔，棄時正也，何以為民？」 故善僖「五年春王正月辛亥朔，日南至，公既視朔，遂登觀臺以望，而書，禮也。 凡分至啟閉，必書雲物，為備故也。」 至昭二十年二月己丑，日南至，失閏，至在非其月。 梓慎望氛氣而弗正，不履端於始也。 故傳不曰冬至，而曰日南至。 極於牽牛之初，日中之時景最長，以此知其南至也。
Thus: primal origin gives one as image, the *Spring and Autumn* gives two, the three systems give three, and the four seasons give four. Together they make ten and complete the five structural elements. Five times ten yields the Great Expansion number. But the Way rests on one, leaving forty-nine for operative use; hence milfoil stalks serve as the numerical tool. One models the system through successive pairings, triplings, and quadruplings, then returns the odd remainder to represent intercalation as nineteen, adding the single base unit. A second tallying and halving then yields the monthly-rule dividend. When this produces one under the same procedure as the day-rule, it gives the days per month. At that point the three celestial indicators meet in conjunction, and prognostications of fortune or misfortune become possible. So the *Changes* says: "Heaven is one and Earth is two; Heaven three and Earth four; Heaven five and Earth six; Heaven seven and Earth eight; Heaven nine and Earth ten. Heaven has five numbers and Earth has five; the five positions correspond and each finds its proper match. Heaven’s numbers total twenty-five and Earth’s thirty; together they are fifty-five. By this, transformations are accomplished and the numinous powers operate." The terminal sum is nineteen. Since the *Changes* teaches that exhaustion leads to transformation, nineteen is used as the intercalation rule. Threefold Heaven gives nine and twofold Earth gives ten; together they form the conjunction number. Heaven’s threefold number is twenty-five and Earth’s twofold number is thirty; this defines the conjunction cycle of new and full moon. Multiply by this conjunction number and the cycle returns to a winter-solstice new moon at dawn; this is called the conjunction month. After nine conjunction cycles it returns to origin, matching Huangzhong’s “initial nine” number. In the Annals, season and month must always be recorded through all four seasons, even when no event occurs. Seasonal markers record opening and closing phases, while monthly markers track equinoxes and solstices. Opening and closing define the seasonal nodes. Equinoxes and solstices are the central points. A seasonal node does not always land within its nominal month, so the true midpoint must be fixed in the properly numbered month. Hence the Commentary says: "When the sage kings regulated time, they established the beginning at the outset, maintained correctness at the center, and assigned the remainder at the end. If the outset is correctly established, sequence does not go wrong; if correctness is upheld at mid-course, the people are not confused; if surplus is assigned at the end, affairs do not become disordered." This is why the sage kings treated intercalation as so important. Multiplying the conjunction number by the five positions returns to a dawn new moon at winter solstice; this defines the zhang month-cycle. Taking one quarter of the month rule and multiplying it by the zhang cycle yields what is called the middle rule. Set three intercalation rules as a full-solstice cycle, multiply by the month rule, subtract the middle rule, and reduce. The result gives six tally-units as the monthly intercalation rule, with seven residual parts. This is the technique for reconciling midpoints and new moons. If the new moon misses the central point, that month is intercalary: yin and yang may meet, but without central alignment no proper generation occurs. Accordingly, multiplying the day rule by the intercalation rule yields the integrated year count. Across the three systems, this is called the origin year. Intercalary years within the origin cycle are treated as yin-yang calamity points under the three-system intercalation method. The *Changes* doctrine of the nine calamities says: at initial entry into the origin cycle, after 106 years comes a “yang-nine” crisis; then after 374 years, a “yin-nine” crisis; then after 480 years, another “yang-nine”; then after 720 years, a “yin-seven”; after another 720 years, a “yang-seven”; then after 600 years, a “yin-five”; after another 600 years, a “yang-five”; then after 480 years, a “yin-three”; then after 480 years, a “yang-three.” Altogether, one full origin cycle ends at 4,617 years. Of these, 4,560 are regular years and 57 are calamity years. This is why the *Spring and Autumn* says, "Maintain correctness at the center." It also says, "If the new moon is not proclaimed in an intercalary month, that violates ritual. Intercalation corrects time, time enables governance, and governance nourishes life; in this lies the proper Way of sustaining the people. "If intercalary new moons are not proclaimed, proper timekeeping is discarded; how then can the people be governed?" Thus it commends Duke Xi: "In the fifth year, spring, on xinhai, first day of the royal first month, the sun reached its southern limit. After observing the new moon, the duke ascended the observatory to make observations and had them entered in the record; this accorded with ritual. Whenever equinoxes, solstices, and seasonal opening-closing points occur, cloud and atmospheric signs must be recorded as preparedness against anomalies." In Duke Zhao’s twentieth year, second month, jichou day, the sun reached its southern extreme; because intercalation had been missed, the solstice fell in the wrong month. Zishen observed atmospheric qi but failed to correct the reckoning; he had not set the beginning right from the outset. That is why the Commentary does not call it “winter solstice” but instead says “the sun reached the south.” At the beginning of the Ox asterism, when the noon shadow is at its greatest length, one knows the sun has reached its southern limit.
26
斗綱之端連貫營室，織女之紀指牽牛之初，以紀日月，故曰星紀。 五星起其初，日月起其中，凡十二次。 日至其初為節，至其中斗建下為十二辰。 視其建而知其次。 故曰「制禮上物，不過十二，天之大數也」。 經曰春王正月，傳曰周正月「火出，於夏為三月，商為四月，周為五月。 夏數得天」，得四時之正也。 三代各據一統，明三統常合，而迭為首，登降三統之首，周還五行之道也。 故三五相包而生。 天統之正，始施於子半，日萌色赤。 地統受之於丑初，日肇化而黃，至丑半，日牙化而白。 人統受之於寅初，日孽成而黑，至寅半，日生成而青。 天施復於子，地化自丑畢於辰，人生自寅成於申。 故曆數三統，天以甲子，地以甲辰，人以甲申。 孟仲季迭用事為統首。 三微之統既著，而五行自青始，其序亦如之。 五行與三統相錯。 傳曰「天有三辰，地有五行」，然則三統五星可知也。 《易》曰：「參五以變，錯綜其數。 通其變，遂成天下之文； 極其數，遂定天下之象。」 太極運三辰五星於上。 而元氣轉三統五行於下。 其於人，皇極統三德五事。 故三辰之合於三統也，日合於天統，月合於地統，斗合於人統。 五星之合於五行，水合於辰星，火合於熒惑，金合於太白，木合於歲星，土合於填星。 三辰五星而相經緯也。 天以一生水，地以二生火，天以三生木，地以四生金，天以五生土。 五勝相乘，以生小周，以乘乾坤之策，而成大周。 陰陽比類，交錯相成，故九六之變登降於六體。 三微而成著，三著而成象，二象十有八變而成卦，四營而成易，為七十二，參三統兩四時相乘之數也。 參之則得乾之策，兩之則得坤之策。 以陽九九之，為六百四十八，以陰六六之，為四百三十二，凡一千八十，陰陽各一卦之微算策也。 八之，為八千六百四十，而八卦小成。 引而信之，又八之，為六萬九千一百二十，天地再之，為十三萬八千二百四十，然後大成。 五星會終，觸類而長之，以乘章歲，為二百六十二萬六千五百六十，而與日月會。 三會為七百八十七萬九千六百八十，而與三統會。 三統二千三百六十三萬九千四十，而復於太極上元。 九章歲而六之為法，太極上元為實，實如法得一，陰一陽各萬一千五百二十，當萬物氣體之數，天下之能事畢矣。
The end of the Dipper’s handle runs through the Encampment and Chamber lodges, and the Weaving Maiden marker points to the start of Ox. This is used to track sun and moon, hence the name “Star Marker.” The five planets are reckoned from its initial point, while sun and moon are reckoned from its midpoint; altogether there are twelve stations. When the sun reaches a station’s beginning, it marks a seasonal node; when it reaches the middle and the Dipper’s handle establishes below, this yields the twelve branch positions. By observing where it establishes, one knows which station it is in. Thus it is said: "In structuring rites and classifying categories, one does not go beyond twelve; twelve is Heaven’s great number." The Annals say, "Spring, royal first month." The Commentary explains Zhou’s first month: "When the Fire asterism appears, this is the third month in Xia reckoning, the fourth in Shang, and the fifth in Zhou. "Xia’s numbering accords with Heaven," meaning it captures the proper alignment of the four seasons. Each of the Three Dynasties adopted one governing system, showing that the three systems are always coherent while alternating in precedence. This rising and falling of their leading positions follows the cyclical logic of the Five Phases. Thus the triadic and pentadic principles intercontain each other and generate life. Heaven’s system begins at mid-zi: solar vitality first stirs, marked as red. Earth’s system takes over at early chou, when solar transformation first begins and appears yellow; by mid-chou it has budded further and appears white. The human system receives it at early yin: development ripens and is marked as black; by mid-yin, generative completion appears as blue-green. Heaven’s action cycles back at zi, Earth’s transformation runs from chou to chen, and human generation runs from yin to shen. Accordingly, the three calendrical systems take jiazi for Heaven, jiachen for Earth, and jiashen for Humanity. First, middle, and last months alternate in administrative primacy as each system’s lead. Once the three subtle origins are established, the Five Phases begin with blue-green, and their order follows the same pattern. The Five Phases are interwoven with the three systems. As the Commentary says, "Heaven has three celestial markers; Earth has five phases." From this, the relation of three systems and five planets becomes intelligible. The *Changes* says, "By combining three and five, transformation occurs, and their numbers are cross-woven. Understand these transformations, and the patterned order of the world is completed; carry its numbers to the limit, and the world’s configurations are determined." Above, the Great Ultimate sets the three celestial markers and five planets in motion. Below, primal qi rotates the three systems and the Five Phases. In human governance, the august ultimate integrates the three virtues and five administrative concerns. So the three celestial markers match the three systems: sun with Heaven, moon with Earth, and the Dipper with Humanity. The five planets align with the Five Phases: Water with Chenxing, Fire with Yinghuo, Metal with Taibai, Wood with Suixing, and Earth with Tianxing. The three celestial markers and five planets interweave as warp and weft. Heaven generates Water through one, Earth generates Fire through two, Heaven generates Wood through three, Earth generates Metal through four, and Heaven generates Earth through five. Mutual multiplication through the Five-Phase overcoming sequence yields the small cycle; multiplied by Qian and Kun tallies, it forms the great cycle. Yin and yang correlate by category and, through intercrossing, complete each other; thus the transformations of nine and six ascend and descend across the six structural forms. Three subtle stages become manifest, three manifest stages become images, and two images become hexagrams through eighteen transformations. Four operations complete the *Changes*, yielding seventy-two, the product of three systems, two principles, and four seasons. Tripled, it gives Qian’s tally; doubled, Kun’s. For yang, 9 x 9 gives 648; for yin, 6 x 6 gives 432. Together they total 1,080, the subtle tally-count for one yin and one yang hexagram. Multiplying by eight gives 8,640, completing the basic structure of the eight trigrams. Extend and confirm the sequence, then multiply by eight again to reach 69,120; doubling for Heaven and Earth gives 138,240, and only then is the system complete. At the terminal conjunction of the five planets, extension by category and multiplication by the zhang-year yields 2,626,560, where it coincides with sun and moon cycles. Three conjunction cycles produce 7,879,680, coinciding with the three-system cycle. At 23,639,040 across the three systems, the cycle returns to the upper-origin point of the Great Ultimate. With nine zhang-years multiplied by six as the rule and the Great Ultimate upper-origin as the dividend, division yields one. Yin and yang each give 11,520, corresponding to the qi-forms of all things; this completes the operational numbers of the world.
27
《律曆志下》
Treatise on Pitch Standards and Calendar, Part Two.
28
統母日法八十一。 元始黃鐘初九自乘，一龠之數，得日法。
Primary system day-rule: 81. At the primal beginning, Huangzhong’s initial-nine value is squared; this one-yue number yields the day rule.
29
閏法十九，因為章歲。 合天地終數，得閏法。
Intercalation rule: 19, which defines the zhang-year cycle. The intercalation rule comes from combining Heaven’s and Earth’s terminal numbers.
30
統法千五百三十九。 以閏法乘日法，得統法。
System divisor: 1,539. Intercalation rule times day-rule gives the system rule.
31
元法四千六百一十七。 參統法，得元法。
Origin-cycle rule: 4,617. Three system rules produce the origin rule.
32
會數四十七。 參天九，兩地十，得會數。
Conjunction number: 47. From Heaven tripled (9) and Earth doubled (10), one obtains the conjunction number.
33
章月二百三十五。 五位乘會數，得章月。
Zhang-month count: 235. Five positions multiplied by the conjunction number gives the zhang-month total.
34
月法二千三百九十二。 推大衍象，得月法。
Monthly rule value: 2,392. Deriving from the Great Expansion schema yields the monthly rule.
35
通法五百九十八。 四分月法，得通法。
General divisor value: 598. One quarter of the monthly rule gives the general divisor.
36
中法十四萬五百三十。 以章月乘通法，得中法。
Middle-rule value: 140,530. Zhang-month total multiplied by the general divisor yields the middle rule.
37
周天五十六萬二千一百二十。 以章月乘月法，得周天。
Full heavenly cycle value: 562,120. Zhang-month total times monthly rule gives the full heavenly cycle.
38
歲中十二。 以三統乘四時，得歲中。
Yearly midpoint count: 12. Three systems multiplied by four seasons gives the annual midpoint count.
39
月周二百五十四。 以章月加閏法，得月周。
Monthly circuit value: 254. Adding the intercalation rule to the zhang-month count gives the monthly circuit.
40
朔望之會百三十五。 參天數二十五，兩地數三十，得朔望之會。
New-moon/full-moon conjunction value: 135. Using Heaven’s threefold number 25 and Earth’s twofold number 30 gives the new/full conjunction value.
41
會月六千三百四十五。 以會數乘朔望之會，得會月。
Conjunction-month value: 6,345. Conjunction number multiplied by the new/full conjunction gives the conjunction-month total.
42
統月萬九千三十五。 參會月，得統月。
System-month total: 19,035. Three conjunction-month units produce the system-month total.
43
元月五萬七千一百五。 參統月，得元月。
Origin-month total: 57,105. Tripling system months gives the origin-month total.
44
章中二百二十八。 以閏法乘歲中，得章中。
Zhang-midpoint value: 228. Intercalation rule times annual midpoint count yields zhang midpoints.
45
統中萬八千四百六十八。 以日法乘章中，得統中。
System-midpoint total: 18,468. Day rule times zhang midpoints gives the system-midpoint total.
46
元中五萬五千四百四。 參統中，得元中。
Origin-midpoint total: 55,404. Tripling system midpoints yields the origin-midpoint total.
47
策餘八千八十。 什乘元中，以減周天，得策餘。
Tally remainder value: 8,080. Ten times the origin-midpoint value minus the full heavenly cycle gives the tally remainder.
48
周至五十七。 參閏法，得周至。
Complete-solstice value: 57. Three intercalation-rule units produce the complete-solstice value.
49
統母。
System matrix base.
50
木金相乘為十二，是為歲星小周。 小周乘巛策，為千七百二十八，是為歲星歲數。
Wood and Metal multiplied give 12; this is the small cycle for Jupiter (Suixing). The small cycle multiplied by the cyclical tally gives 1,728, the year count for Jupiter.
51
見中分二萬七百三十六。
Observed midpoint-division value: 20,736.
52
積中十三，中餘百五十七。
Accumulated midpoint count is 13, with a midpoint remainder of 157.
53
見中法千五百八十三。
Observed midpoint-rule value: 1,583.
54
見閏分萬二千九十六。
Observed intercalary-division value: 12,096.
55
積月十三，月餘萬五千七十九。
Accumulated month count is 13, with a month remainder of 15,079.
56
見月法三萬七十七。
Observed month-rule value: 30,077.
57
見中日法七百三十萬八千七百一十一。
Observed midpoint day-rule value: 7,308,711.
58
見月日法二百四十三萬六千二百三十七。
Observed month day-rule value: 2,436,237.
59
金火相乘為八，又以火乘之為十六而小復。 小復乘乾策，為三千四百五十六，是為太白歲數。
Metal and Fire multiply to 8; multiplying again by Fire gives 16, defining the small return cycle. Small return multiplied by the Qian tally gives 3,456, the year count for Venus (Taibai).
60
見中分四萬一千四百七十二。
Observed midpoint-division value: 41,472.
61
積中十九，中餘四百一十三。
Accumulated midpoint count is 19, with a midpoint remainder of 413.
62
見中法二千一百六十一。
Observed midpoint-rule value: 2,161.
63
見閏分二萬四千一百九十二。
Observed intercalary-division value: 24,192.
64
積月十九，月餘三萬二千三十九。
Accumulated month count is 19, with a month remainder of 32,039.
65
見月法四萬一千五十九。
Observed month-rule value: 41,059.
66
晨中分二萬三千三百二十八。
Morning midpoint-division value: 23,328.
67
積中十，中餘千七百一十八。 「十」一作「七」
Accumulated midpoint count is 10, with midpoint remainder 1,718. Textual variant: "ten"; another version reads "seven."
68
夕中分萬八千一百四十四。
Evening midpoint-division value: 18,144.
69
積中八，中餘八百五十六。
Accumulated midpoint count is 8, with midpoint remainder 856.
70
晨閏分萬三千六百八。
Morning intercalary-division value: 13,608.
71
積月十一，月餘五千一百九十一。
Accumulated month count is 11, with month remainder 5,191.
72
夕閏分萬五百八十四。
Evening intercalary-division value: 10,584.
73
積月八，月餘二萬六千八百四十八。
Accumulated month count is 8, with month remainder 26,848.
74
見中日法九百九十七萬七千三百三十七。
Observed midpoint day-rule value: 9,977,337.
75
見月日法三百三十二萬五千七百七十九。
Observed month day-rule value: 3,325,779.
76
土木相乘而合經緯為三十，是為鎮星小周。 小周乘巛策，為四千三百二十，是為鎮星歲數。
Earth and Wood multiply, and in warp-weft combination they yield 30; this defines Saturn’s small cycle. Small cycle times the cyclical tally gives 4,320, the year count for Saturn (Zhenxing).
77
見中分五萬一千八百四十。
Observed midpoint-division value: 51,840.
78
積中十二，中餘千七百四十。
Accumulated midpoint count is 12, with midpoint remainder 1,740.
79
見中法四千一百七十五。
Observed midpoint-rule value: 4,175.
80
見閏分三萬二百四十。
Observed intercalary-division value: 30,240.
81
積月十二，月餘六萬三千三百。
Accumulated month count is 12, with month remainder 63,300.
82
見月法七萬九千三百二十五。
Observed month-rule value: 79,325.
83
見中日法千九百二十七萬五千九百七十五。
Observed midpoint day-rule value: 19,275,975.
84
見月日法六百四十二萬五千三百二十五。
Observed month day-rule value: 6,425,325.
85
火經特成，故二歲而過初，三十二過初為六十四歲而小周。 小周乘乾策，則太陽大周，為萬三千八百二十四歲，是為熒惑歲數。
Fire has a distinctive complete pattern: it passes the initial point every two years, and 32 such passes make 64 years, one small cycle. Multiplying the small cycle by the Qian tally gives the Great Yang major cycle of 13,824 years, the year count for Mars (Yinghuo).
86
見中分十六萬五千八百八十八。
Observed midpoint-division value: 165,888.
87
積中二十五，中餘四千一百六十三。
Accumulated midpoint count is 25, with midpoint remainder 4,163.
88
見中法六千四百六十九。
Observed midpoint-rule value: 6,469.
89
見閏分九萬六千七百六十八。
Observed intercalary-division value: 96,768.
90
積月二十六，月餘五萬二千九百五十四。
Accumulated month count is 26, with month remainder 52,954.
91
見月法十二萬二千九百一十一。 「二千」一作「一千」
Observed month-rule value: 122,911. Textual variant: "two thousand"; alternate reading "one thousand."
92
見中日法二千九百八十六萬七千三百七十三。
Observed midpoint day-rule value: 29,867,373.
93
見月日法九百九十五萬五千七百九十一。
Observed month day-rule value: 9,955,791.
94
水經特成，故一歲而及初，六十四及初而小復。 小復乘巛策，則太陰大周，為九千二百一十六歲，是為辰星歲數。
Water has a distinctive complete pattern: it reaches the initial point each year, and after 64 such returns comes a small return cycle. Small return multiplied by the cyclical tally gives the Great Yin major cycle of 9,216 years, the year count for Mercury (Chenxing).
95
見中分十一萬五百九十二。
Observed midpoint-division value: 110,592.
96
積中三，中餘三萬二千四百六十九。
Accumulated midpoint count is 3, with midpoint remainder 32,469.
97
見中法二萬九千四十一。
Observed midpoint-rule value: 29,041.
98
見閏分六萬四千五百一十二。
Observed intercalary-division value: 64,512.
99
積月三，月餘五十一萬四百二十三。
Accumulated month count is 3, with month remainder 510,423.
100
見月法五十五萬一千七百七十九。
Observed month-rule value: 551,779.
101
晨中分六萬二千二百八。
Morning midpoint-division value: 62,208.
102
積中二，中餘四千一百二十六。
Accumulated midpoint count is 2, with midpoint remainder 4,126.
103
夕中分四萬八千三百八十四。
Evening midpoint-division value: 48,384.
104
積中一，中餘萬九千三百四十三。
Accumulated midpoint count is 1, with midpoint remainder 19,343.
105
晨閏分三萬六千二百八十八。
Morning intercalary-division value: 36,288.
106
積月二，月餘十一萬四千六百八十二。
Accumulated month count is 2, with month remainder 114,682.
107
夕閏分二萬八千二百二十四。
Evening intercalary-division value: 28,224.
108
積月一，月餘三十九萬五千七百四十一。
Accumulated month count is 1, with month remainder 395,741.
109
見中日法一億三千四百八萬二千二百九十七。
Observed midpoint day-rule value: 134,082,297.
110
見月日法四千四百六十九萬四千九十九。
Observed month day-rule value: 44,694,099.
111
合太陰太陽之歲數而中分之，各萬一千五百二十。 陽施其氣，陰成其物。
Combine the year counts of Great Yin and Great Yang, then divide at the midpoint: each side is 11,520. Yang emits its qi, and yin brings things to completion.
112
以星行率減歲數，餘則見數也。
Subtract the annual value by the star’s rate of motion; the remainder is the visibility count.
113
東九西七乘歲數，并九七為法，得一，金、水晨夕歲數。
Using east-9 and west-7 against the year count, with 9+7 as the divisor, yields one; this gives the morning/evening year counts for Metal and Water.
114
以歲中乘歲數，是為星見中分。
Annual midpoint multiplied by year count gives the star’s midpoint-visibility division.
115
星見數，是為見中法。
The star-visibility count constitutes the midpoint-visibility rule.
116
以歲閏乘歲數，是為星見閏分。
Annual intercalation multiplied by year count gives the star’s intercalary-visibility division.
117
以章歲乘見數，是為見月法。
Zhang-year value multiplied by the visibility count gives the observed month-rule.
118
以元法乘見數，是為見中日法。
Origin-rule multiplied by the visibility count gives the observed midpoint day-rule.
119
以統法乘見數，是為見月日法。
System-rule multiplied by the visibility count gives the observed month day-rule.
120
五步木，晨始見，去日半次。 順，日行十一分度二，百二十一日。 始留，二十五日而旋。 逆，日行七分度一，八十四日。 復留，二十四日三分而旋。 復順，日行十一分度二，百一十一日有百八十二萬八千三百六十二分而伏。 凡見三百六十五日有百八十二萬八千三百六十五分，除逆，定行星三十度百六十六萬一千二百八十六分。 凡見一歲，行一次而後伏。 日行不盈十一分度一。 伏三十三日三百三十三萬四千七百三十七分，行星三度百六十七萬三千四百五十一一作「三」分。 一見，三百九十八日五百一十六萬三千一百二分，行星三十三度三百三十三萬四千七百三十七分。 通其率，故曰日行千七百二十八分度之百四十五。
For Jupiter in its five-step cycle, first morning appearance occurs at half a lodge from the sun. In direct motion, it advances at 2/11 degree per day for 121 days. It then stations; after 25 days it turns. In retrograde phase, motion is 1/7 degree per day for 84 days. It stations again and turns after 24 1/3 days. After resuming direct motion at 2/11 degree per day, it continues for 111 days plus 1,828,362 fractional parts, then disappears from view. Total visibility is 365 days plus 1,828,365 parts; removing retrograde gives a net stellar advance of 30 degrees and 1,661,286 parts. Overall, it is visible for one year, completes one circuit, then vanishes. Its daily progress is slightly under 1/11 degree. The hidden phase lasts 33 days plus 3,333,737 parts, during which it moves 3 degrees and 1,673,451 parts; one textual variant reads the last figure as 3. One full visibility cycle is 398 days plus 5,163,102 parts, with total motion of 33 degrees and 3,333,737 parts. Reducing to a standard rate, this is stated as 1,728 parts per 145 of a degree per day.
121
金，晨始見，去日半次。 逆，日行二分度一，六日。 始留，八日而旋。 始順，日行四十六分度三十三，四十六日。 順，疾，日行一度九十二分度十五，百八十四日而伏。 凡見二百四十四日，除逆，定行星二百四十四度。 伏，日行一度九十二分度三十三有奇。 伏八十三日，行星百一十三度四百三十六萬五千二百二十分。 凡晨見、伏三百二十七日，行星三百五十七度四百三十六萬五千二百二十分。 夕始見，去日半次。 順，日行一度九十二分度十五，百八十一日百七分日四十五。 順，遲，日行四十六分度三一作「四」十三，四十六日。 始留，七日百七分日六十二分而旋。 逆，日行二一作「三」分度一，六日而伏。 凡見二百四十一日，除逆，定行星二百四十一度。 伏，逆，日行八分度七有奇。 伏十六「一作六十」日百二十九萬五千三百五十二分，行星十四度三百六萬九千八百六十八分。 一凡夕見伏，二百五十七日百二十九萬五千三百五十二一作「一」分，行星二百二十六度六百九十萬七千四百六十九分。 一復，五百八十四日百二十九萬五千三百五十二分。 行星亦如之，故曰日行一度。
Venus first appears in the morning at half a lodge from the sun. In retrograde, it moves 1/2 degree per day for 6 days. It then stations, turning after 8 days. In initial direct motion, it moves 33/46 degree per day for 46 days. In fast direct motion, it advances 15/92 degree per day for 184 days, then disappears. Total visibility is 244 days; excluding retrograde, net stellar advance is 244 degrees. During concealment, motion is about 33/92 degree per day with a fractional remainder. The hidden interval lasts 83 days, with 113 degrees and 4,365,220 parts of motion. Morning visibility plus hidden phase totals 327 days, with 357 degrees and 4,365,220 parts traversed. It first appears in the evening at half a lodge from the sun. In direct motion, it advances 15/92 degree per day for 181 and 45/107 days. In slow direct motion, it moves 33/46 degree per day (variant: 43/46) for 46 days. It then stations, turning after 7 and 62/107 days. In retrograde, motion is 1/2 degree per day (variant: 1/3) for 6 days, then it disappears. Total visibility is 241 days; excluding retrograde, net stellar advance is 241 degrees. In the hidden retrograde phase, motion is 7/8 degree per day with a small remainder. The hidden phase lasts 16 days (variant: 60), plus 1,295,352 parts, with 14 degrees and 3,069,868 parts of motion. Total evening visibility and concealment is 257 days plus 1,295,352 parts (variant: final figure 1), with 226 degrees and 6,907,469 parts traversed. One full cycle is 584 days plus 1,295,352 parts. Stellar motion matches this, hence the statement that mean daily motion is one degree.
122
土，晨始見，去日半次。 順，日行十五分度一，八十七日，始留，三十四日而旋。 逆，日行八十一分度五，百一日。 復留，三十三日八十六萬二千四百五十五分而旋。 復順，日行十五分度一，八十五日而伏。 凡見三百四十日八十六萬二千四百五十五分，除逆，定一多「餘」字行星五度四百四十七萬三千九百三十分。 伏，日行不盈十五分度三。 百三十七日千七百一十七萬一百七十分，行星七度八百七十三萬六千五百七十分。 一見，三百七十七日千八百三萬二千六百二十五分，行星十二度千三百二十一萬五百分。 通其率，故曰日行四千三百二十分度之百四十五。
Saturn first appears in the morning at half a lodge from the sun. In direct motion it advances 1/15 degree per day for 87 days, then stations and turns after 34 days. In retrograde it moves 5/81 degree per day for 101 days. It stations again, turning after 33 days plus 862,455 parts. After resuming direct motion at 1/15 degree per day for 85 days, it disappears. Total visibility is 340 days plus 862,455 parts; subtracting retrograde gives a net stellar advance of 5 degrees and 4,473,930 parts (with a variant adding one extra character). During concealment, daily motion is slightly under 3/15 degree. Over 137 days plus 17,170,170 parts, it moves 7 degrees and 8,736,570 parts. One visibility cycle totals 377 days plus 18,032,625 parts, with 12 degrees and 13,210,500 parts of motion. Normalized by rate, this is expressed as 4,320 parts per 145 of a degree per day.
123
火，晨始見，去日半次。 順，日行九十二分度五十三，二百七十六日，始留，十日而旋。 逆，日行六十二分度十七，六十二日。 復留，十日而旋。 復順，日行九十二分度五十三，二百七十六日而伏。 凡見六百三十四日，除逆，定行星三百一度。 伏，日行不盈九十二分度七十三分，伏百四十六日千五百六十八萬九千七百分，行星百一十四度八百二十一萬八千五分。 一見，七百八十日千五百六十八萬九千七百分，凡行星四百一十五度八百二十一萬八千五分。 通其率，故曰日行萬三千八百二十四分度之七千三百五十五。
Mars first appears in the morning at half a lodge from the sun. In direct motion it advances 53/92 degree per day for 276 days, then stations and turns after 10 days. In retrograde, it moves 17/62 degree per day for 62 days. It stations again and turns after 10 days. After resuming direct motion at 53/92 degree per day for 276 days, it disappears. Total visibility is 634 days; excluding retrograde, net stellar advance is 301 degrees. During concealment, daily motion is just under 73/92 degree. The hidden interval lasts 146 days plus 15,689,700 parts, with 114 degrees and 8,218,005 parts of motion. One full visibility cycle is 780 days plus 15,689,700 parts, with total stellar motion of 415 degrees and 8,218,005 parts. Normalized by rate, this is expressed as 7,355 degree-parts per 13,824 units per day.
124
水，晨始見，去日半次。 逆，日行二度，一日。 始留，二日而旋。 順，日行七分度六，一多「十」字七日。 順，疾，日行一度三分度一，一多「一」字十八日而伏。 凡見二十八日，除逆，定行星二十八度。 伏，日行一度九分度七有奇，三十七日一億二千二百二萬九千六百五分，行星六十八度四千六百六十一萬一百二十八分。 凡晨見、伏，六十五日一億二千二百二萬九千六百五分，行星九十六度四千六百六十一萬一百二十八分。 夕始見，去日半次。 順，疾，日行一度三分度一，十六日二分日一。 順，遲，日行七分度六，七一作「十」日。 留，一日二分日一而旋。 逆，日行二度，一日而伏。 凡見二十六日，除逆，定行星二十六度。 伏，逆，日行十五分度四有奇，二十四日，行星六度五千八百六十六萬二千八百二十分。 凡夕見伏，五十日，行星十九度七千五百四十一萬九千四百七十七分。 一復，百一十五日一億二千二百二萬九千六百五分。 行星亦如之，故曰日行一度。
Mercury first appears in the morning at half a lodge from the sun. In retrograde, it moves 2 degrees per day for 1 day. It then stations and turns after 2 days. In direct motion, it advances 6/7 degree per day for 17 days (one textual witness has an extra “ten”). In fast direct motion it advances 1 1/3 degrees per day (with one witness adding an extra “one”) for 18 days, then disappears. Total visibility is 28 days; excluding retrograde, net stellar advance is 28 degrees. In concealment, daily motion is roughly 7/9 degree. Over 37 days plus 122,229,605 parts, stellar motion totals 68 degrees and 46,610,128 parts. Morning visibility plus concealment totals 65 days and 122,229,605 parts, with 96 degrees and 46,610,128 parts of motion. It first appears in the evening at half a lodge from the sun. In fast direct motion it advances 1 1/3 degrees per day for 16 1/2 days. In slow direct motion, it moves 6/7 degree per day for 7 days (variant: 10). It remains stationary for 1 1/2 days, then turns. In retrograde, it moves 2 degrees per day for 1 day, then disappears. Total visibility is 26 days; excluding retrograde, net stellar advance is 26 degrees. In hidden retrograde phase, motion is about 4/15 degree per day for 24 days, totaling 6 degrees and 58,662,820 parts. Evening visibility plus concealment totals 50 days, with 19 degrees and 75,419,477 parts of motion. One full cycle is 115 days plus 122,229,605 parts. Stellar motion follows the same pattern, hence the statement of one degree per day in mean rate.
125
統術推日月元統，置太極上元以來，外所求年，盈元法除之，餘不盈統者，則天統甲子以來年數也。 盈統，除之，餘則地統甲辰以來年數也。 又盈統，除之，餘則人統甲申以來年數也。 各以其統首日為紀。
To compute the sun-moon origin cycle, start from Great Ultimate upper-origin and add the target years. Remove complete origin-rule units; if the remainder is less than one system cycle, it gives the elapsed years since Heaven-system jiazi. If one full system cycle is reached, divide it out; the remainder gives years since Earth-system jiachen. If another full system cycle is reached, divide again; the remainder gives years since Human-system jiashen. Each system takes its lead day as the epoch marker.
126
推天正，以章月乘人統歲數，盈章歲得一，名曰積月，不盈者名曰閏餘。 閏餘十二以上，歲有閏。 求地正，加積月一； 求人正，加二。
To compute Heaven alignment, multiply the zhang-month value by the Human-system year count. Each complete zhang-year counts as one accumulated month; the remainder is the intercalary remainder. If the intercalary remainder reaches 12 or above, that year contains a leap month. For Earth alignment, add 1 to the accumulated-month count; for Human alignment, add 2.
127
推正月朔，以月法乘積月，盈日法得一，名曰積日，不盈者名曰小餘。 小餘三十八以上，其月大。 積日盈六十，除之，不盈者名曰大餘。 數從統首日起，算外，則朔日也。 求其次月，加大餘二十九，小餘四十三。 小餘盈日法得一，從大餘，數除如法。 求弦，加大餘七，小餘三十一。 求望，倍弦。
To compute the first-month new moon, multiply the monthly rule by accumulated months. Each complete day-rule unit gives one accumulated day, and the leftover is the small remainder. If the small remainder is 38 or greater, the month is a long month. When accumulated days reach 60, reduce by division; the remainder is called the great remainder. Count forward from the system-head day, excluding the starting point; the result is the new-moon day. For the following month, add 29 to the great remainder and 43 to the small remainder. If the small remainder reaches a full day-rule, carry one to the great remainder, then reduce according to rule. To compute the quarter phase, add 7 to the great remainder and 31 to the small remainder. To compute full moon, double the quarter-phase value.
128
推閏餘所在，以十二乘閏餘，加十得一。 盈章中，數所得，起冬至，算外，則中至終閏盈。 中氣在朔若二日，則前月閏也。
To determine leap-month placement, multiply the intercalary remainder by 12 and apply the +10-to-1 rule. When this reaches the zhang-midpoint value, the resulting count from winter solstice (excluding the starting point) indicates where midpoints run to the terminal leap completion. If the principal solar term falls on the new moon day or day two, the previous month is treated as intercalary.
129
推冬至，以算餘乘人統歲數，盈統法得一，名曰大餘，不盈者名曰小餘。 除數如法，則所求冬至日也。
To calculate winter solstice, multiply the computation remainder by the Human-system year count. Each complete system-rule unit gives a great remainder; the leftover is the small remainder. Reduce by the prescribed rule, and the result is the target winter-solstice date.
130
求八節，加大餘四十五，小餘千一百。 求二十四氣，三其小餘，加大餘十五，小餘千一十。
For the eight seasonal nodes, add 45 to the great remainder and 1,100 to the small remainder. For the twenty-four solar terms, triple the small remainder, then add 15 to the great remainder and 1,010 to the small remainder.
131
推中部二十四氣，皆以元為法。
For the middle-section twenty-four terms, always use the origin value as divisor.
132
推五行，其四行各七十三日，統歲分之七十七。 中央各十八日，統法分之四百四。 冬至後，中央二十七日六百六分。
In the Five-Phase calculation, each of four phases spans 73 days plus 77 system-year parts. Each central segment is 18 days plus 404 system-rule parts. After winter solstice, the central interval is 27 days and 606 parts.
133
推合晨所在星，置積日，以統法乘之，以十九乘小餘而并之。 盈周天，除去之； 不盈者，令盈統法得一度。 數起牽牛，算外，則合晨所入星度也。
To find the star position at dawn conjunction, start with accumulated days, multiply by the system rule, then add 19 times the small remainder. If it exceeds the full heavenly cycle, subtract that cycle; if it does not fill, convert by taking one full system-rule as one degree. Count from Ox-Herder (Niuniu), excluding the starting point; this yields the star degree entered at dawn conjunction.
134
推其日夜半所在星，以章歲乘月小餘，以減合晨度。 小餘不足者，破全度。
To find the star at midnight that day, multiply zhang-year by the month’s small remainder and subtract from the dawn-conjunction degree. If the small remainder is insufficient, borrow from one full degree.
135
推其月夜半所在星，以月周乘月小餘，盈統法得一度，以減合晨度。
For star position at that month’s midnight, multiply monthly circuit by the month’s small remainder; each full system-rule converts to one degree, then subtract from the dawn-conjunction degree.
136
推諸加時，以十二乘小餘為實，各盈分母為法，數起於子，算外，則所加辰也。
To compute added time offsets, multiply the small remainder by 12 as the dividend, and use each full denominator as divisor. Count from zi, excluding the starting point; the result is the added time branch.
137
推月食，置會餘歲積月，以二十三乘之，盈百三十五，除之。 不盈者，加二十三得一月，盈百三十五，數所得，起其正，算外，則食月也。 加時，在望日衝辰。
For lunar eclipse calculation, take the conjunction-remainder accumulated months and multiply by 23; whenever it reaches 135, reduce by division. If it does not reach full value, add 23 per month; when 135 is reached, count from the proper month (excluding the start) and that gives the eclipse month. The time offset falls on the branch opposite the full-moon day.
138
紀術推五星見復，置太極上元以來，盡所求年，乘大統見復數，盈歲數得一，則定見復數也。 不盈者名曰見復餘。 見復餘盈其見復數，一以上見在往年，倍一以上，又在前往年，不盈者在今年也。
To compute five-planet visibility returns, count from Great Ultimate upper-origin to the target year and multiply by the Great System return number. Each complete year-unit gives one, yielding the fixed visibility-return count. Any remainder is called the visibility-return remainder. If the visibility-return remainder reaches one full return unit, the appearance belongs to a prior year; if it exceeds two units, to an even earlier year; if below one unit, it belongs to the current year.
139
推星所一多「在」字見中次，以見中分乘定見復數，盈見中法得一，則積中法也。 不盈者名曰中餘。 以元中除積中，餘則中元餘也。 以章中除之，餘則入章中數也。 以十二除之，餘則星見中次也。 中數從冬至起，次數從星紀起，算外，則星所見中次也。
To locate the star’s midpoint visibility station, multiply midpoint divisions by the fixed return count; each full midpoint-rule unit gives one, producing the accumulated midpoint value. Any remainder is called the midpoint remainder. Divide accumulated midpoint value by origin-midpoint value; the remainder is the origin-midpoint remainder. Divide that by zhang-midpoint value; the remainder is the in-zhang midpoint count. Divide by 12; the remainder gives the star’s midpoint visibility station. Count midpoint index from winter solstice and station index from Star Marker; excluding the starting point yields the star’s midpoint visibility station.
140
推星見月，以閏分乘定見，以章歲乘中餘從之，盈見月法得一，并積中，則積月也。 不盈者名曰月中餘。 以元月除積月餘，名曰月元餘。 以章月除月元餘，則入章月數也。 以十二除之，至有閏之歲，除十三入章。 三歲一閏，六歲二閏，九歲三閏，十一歲四閏，十四歲五閏，十七歲六閏，十九歲七閏。 不盈者數起於天正，算外，則星所見月也。
For star-visibility month, multiply intercalary divisions by the fixed visibility count, then add zhang-year times midpoint remainder. Each full visibility-month-rule unit gives one; combining this with accumulated midpoints yields accumulated months. Any leftover is called the month-midpoint remainder. Divide the accumulated-month remainder by the origin-month value; that gives the month-origin remainder. Then divide month-origin remainder by zhang-month to obtain the in-zhang month count. Normally divide by 12; in leap years, divide by 13 for the zhang entry. Leap-month pattern: 1 in 3 years, 2 in 6, 3 in 9, 4 in 11, 5 in 14, 6 in 17, and 7 in 19 years. If not full, count from Heaven alignment (excluding the starting point); that gives the month of stellar visibility.
141
推至日，以中法乘中元餘，盈元法得一，名曰積日，不盈者名曰小餘。 小餘盈二千五百九十七以上，中大。 數除積日如法，算外，則冬至也。
To compute solstice day, multiply middle-rule by origin-midpoint remainder. Each complete origin-rule unit gives accumulated days; the leftover is the small remainder. If the small remainder is 2,597 or greater, the midpoint interval is long. Reduce accumulated days by rule, count excluding the start, and the result is winter solstice.
142
推朔日，以月法乘月元餘，盈日法得一，名曰積日，餘名曰小餘。 小餘三十八以上，月大。 數除積日如法，算外，則星見月朔日也。
To compute the new-moon day, multiply monthly rule by month-origin remainder. Each full day-rule gives accumulated days; what remains is the small remainder. If the small remainder is 38 or above, the month is long. Reduce accumulated days by rule and count excluding the start; this gives the new moon of the star-visibility month.
143
推入中次日度數，以中法乘中餘，以見中法乘其小餘并之，盈見中日法得一，則入中日入次度數也。 中次至日數，次以次初數，算外，則星所見及日所在度數也。 求夕，在日後十五度。
To compute day-degree at entry into midpoint station, multiply middle-rule by midpoint remainder, then add visibility-midpoint-rule times its small remainder. Each full visibility-midpoint day-rule gives one; the result is the entry-day station degree count. Using the midpoint-station day count with that station’s initial value, and counting excluding the start, yields both the star-visible and solar position degrees. For evening visibility, place the star 15 degrees behind the sun.
144
推入月日數，以月法乘月餘，以見月法乘其小餘并之，盈見月日法得一，則入月日數也。 并之大餘，數除如法，則見日也。
To compute the month-day entry count, multiply monthly rule by month remainder and add visibility-month-rule times its small remainder. Each full visibility-month day-rule counts as one; the result is the entry day-count within the month. Combine that with the great remainder and reduce by rule; the result is the day of visibility.
145
推後見中，加積中於中元餘，加後餘於中餘，盈其法得一，從中元餘，數如法，則見也。
To compute subsequent midpoint visibility, add accumulated midpoint to origin-midpoint remainder and add later remainder to midpoint remainder. Carry each full-rule unit to the origin remainder and reduce by rule; this gives the new visibility point.
146
推後見月，加積月於月元餘，加後月餘於月餘，盈其法得一，從月元餘，除數如法，則後見月也。
For subsequent visibility month, add accumulated months to month-origin remainder and later-month remainder to month remainder. Carry each full-rule unit to month-origin remainder and reduce by rule; the result is the later visibility month.
147
推至日及入中次度數，如上法。
Compute solstice day and midpoint-station entry degrees by the same method above.
148
推朔日及入月數，如上法
Compute new-moon day and month-entry count by the same method.
149
推晨見加夕，夕見加晨，皆如上法。
Both “morning plus evening” and “evening plus morning” visibility calculations follow the same method above.
150
推五步，置始見以來日數，至所求日，各以其行度數乘之。 其星若日有分者，分子乘全為實，分母為法。 其兩有分者，分母分度數乘全，分子從之，令相乘為實，分母相乘為法，實如法得一，名曰積度。 數起星初見星宿所在宿度，算外，則星所在宿度也。
For five-step motion, take days elapsed since first visibility to the target day and multiply by each corresponding motion-degree rate. If star or day value has a fraction, use numerator times whole number as dividend and denominator as divisor. If both terms are fractional, scale degree values by denominators and wholes, then apply numerators; multiply through for the dividend, multiply denominators for the divisor. Division yields the accumulated degrees. Count from the lodge-degree of first appearance, excluding the starting point; this gives the star’s current lodge-degree.
151
歲術推歲所在，置上元以來，外所求年，盈歲數，除去之，不盈者以百四十五乘之，以百四十四為法，如法得一，名曰積次，不盈者名曰次餘。 積次盈十二，除去之，不盈者名曰定次。 數從星紀起，算盡之外，則所在次也。 欲知太歲，以六十除餘積次，餘不盈者，數從丙子起，算盡之外，則太歲日也。
To locate the current year station, count years from upper-origin to the target year. Remove full annual units; for the remainder multiply by 145 and divide by 144. The quotient is accumulated stations, and the leftover is station remainder. If accumulated stations reach 12, reduce by one cycle; the remainder is the fixed station. Count from Star Marker and, excluding the start, the final count gives the current station. To determine Tai Sui, divide station remainder by 60. With the remaining value, count from bingzi; excluding the starting point, the resulting cyclical day is Tai Sui.
152
贏縮。 傳曰：「歲棄其次而旅於明年之次，以害鳥帑，周楚惡之。」 五星之盈縮不是過也。 過次者殃大，過舍者災小，不過者亡咎。 次度。 六物者，歲時數日月星辰也。 辰者，日月之會而建所指也。
This concerns celestial expansion and contraction. As the Tradition says: "If Jupiter leaves its proper station and wanders into the next year's, harming birds and stores, Zhou and Chu took it as an evil sign." When the five planets expand or contract in their motion, this is not necessarily an overstep. If a planet overruns a full station, the omen is severe; if it overruns only a lodge, the harm is lighter; if it does not overrun, there is no fault. The stations and their degree measures. The six calendrical elements are: years, seasons, reckonings, and the sun, moon, and stars. "Chen" means the junctures of sun and moon, marked by where the Dipper's handle points.
153
星紀，初斗十二度，大雪。 中牽牛初，冬至。 終於婺女七度。
Xingji begins at 12 degrees of the Dipper, at the solar term Major Snow. Its midpoint is at the beginning of Ox, at the Winter Solstice. It ends at 7 degrees of Girl.
154
玄枵，初婺女八度，小寒。 中危初，大寒。 終於危十五度。
Xuanxiao begins at 8 degrees of Girl, at Minor Cold. Its midpoint is at the beginning of Rooftop, at Major Cold. It ends at 15 degrees of Rooftop.
155
諏訾，初危十六度，立春。 中營室十四度，驚蟄。 終於奎四度。
Zouzi begins at 16 degrees of Rooftop, at Start of Spring. Its midpoint is at 14 degrees of Encampment, at Awakening of Insects. It ends at 4 degrees of Kui.
156
降婁，初奎五度，雨水。 中婁四度，春分。 終於胃六度。
Jianglou begins at 5 degrees of Kui, at Rain Water. Its midpoint is at 4 degrees of Lou, at the Spring Equinox. It ends at 6 degrees of Stomach.
157
大梁，初胃七度，穀雨。 中昴八度，清明。 終於畢十一度。
Daliang begins at 7 degrees of Stomach, at Grain Rain. Its midpoint is at 8 degrees of Hairy Head, at Pure Brightness. It ends at 11 degrees of Net.
158
實沈、初畢十二度，立夏。 中井初，小滿。 終於井十五度。
Shishen begins at 12 degrees of Net, at Start of Summer. Its midpoint is at the beginning of Well, at Grain Full. It ends at 15 degrees of Well.
159
鶉首，初井十六度，芒種。 中井三十一度，夏至。 終於柳八度。
Chunshou begins at 16 degrees of Well, at Grain in Ear. Its midpoint is at 31 degrees of Well, at the Summer Solstice. It ends at 8 degrees of Willow.
160
鶉火，初柳九度，小暑。 中張三度，大暑。 終於張十七度。
Chunhuo begins at 9 degrees of Willow, at Minor Heat. Its midpoint is at 3 degrees of Extended Net, at Major Heat. It ends at 17 degrees of Extended Net.
161
鶉尾，初張十八度，立秋。 中翼十五度，處暑。
Chunwei begins at 18 degrees of Extended Net, at Start of Autumn. Its midpoint is at 15 degrees of Wing, at Limit of Heat.
162
終於軫十一度。
It ends at 11 degrees of Chariot.
163
壽星，初軫十二度，白露。 中角十度，秋分。 終於氐四度。
Shouxing begins at 12 degrees of Chariot, at White Dew. Its midpoint is at 10 degrees of Horn, at the Autumn Equinox. It ends at 4 degrees of Root.
164
大火，初氐五度，寒露。 中房五度，霜降。 終於尾九度。
Dahuo begins at 5 degrees of Root, at Cold Dew. Its midpoint is at 5 degrees of Room, at Frost's Descent. It ends at 9 degrees of Tail.
165
析木，初尾十度，立冬。 中箕七度，小雪。 終於斗十一度。
Ximu begins at 10 degrees of Tail, at Start of Winter. Its midpoint is at 7 degrees of Basket, at Minor Snow. It ends at 11 degrees of Dipper.
166
角十二。 亢九。 氐十五。 房五。 心五。 尾十八。 箕十一。 東七十五度。 斗二十六。 牛八。 女十二。 虛十。 危十七。 營室十六。 壁九。 北九十八度。 奎十六。 婁十二。 胃十四。 昴十一。 畢十六。 觜二。 參九。 西八十度。 井三十三。 鬼四。 柳十五。 星七。 張十八。 翼十八。 軫十七。 南百一十二度。
Horn spans 12 degrees. Neck spans 9 degrees. Root spans 15 degrees. Room spans 5 degrees. Heart spans 5 degrees. Tail spans 18 degrees. Basket spans 11 degrees. The eastern sector totals 75 degrees. Dipper spans 26 degrees. Ox spans 8 degrees. Girl spans 12 degrees. Void spans 10 degrees. Rooftop spans 17 degrees. Encampment spans 16 degrees. Wall spans 9 degrees. The northern sector totals 98 degrees. Kui spans 16 degrees. Lou spans 12 degrees. Stomach spans 14 degrees. Hairy Head spans 11 degrees. Net spans 16 degrees. Beak spans 2 degrees. Three Stars spans 9 degrees. The western sector totals 80 degrees. Well spans 33 degrees. Ghost spans 4 degrees. Willow spans 15 degrees. Star spans 7 degrees. Extended Net spans 18 degrees. Wing spans 18 degrees. Chariot spans 17 degrees. The southern sector totals 112 degrees.
167
九章歲為百七十一歲，而九道小終。 九終千五百三十九歲而大終。 三終而與元終。 進退於牽牛之前四度五分。 九會。 陽以九終，故日有九道。 陰兼而成之，故月有十九道。 陽名成功，故九會而終。 四營而成易，故四歲中餘一，四章而朔餘一，為篇首，八十一章而終一統。
A cycle of nine chapter-years totals 171 years, marking a minor completion of the nine paths. After nine such minor completions, 1,539 years have passed, producing a major completion. After three major completions, it aligns with the original epoch completion. Its forward and backward shift is 4 degrees and 5 parts before the Ox mansion. These are the nine convergences. Because yang culminates at nine, the sun has nine paths. Yin combines with it to complete the system, so the moon has nineteen paths. Yang is called accomplished completion; therefore the cycle ends with nine convergences. With four operations the Changes are completed, so every four years there is one excess, and every four chapters there is one new-moon excess. This sets the start of a chapter; after eighty-one chapters, one full system is completed.
168
一，甲子元首。 十，辛酉。 十九，己未。 二十八，丁巳。 三十七，乙卯。 四十六，壬子。 五十五，庚戌。 六十四，戊申。 七十三，丙午，中。
First: Jiazi, the opening of the epoch. Tenth: Xinyou. Nineteenth: Jiwei. Twenty-eighth: Dingsi. Thirty-seventh: Yimao. Forty-sixth: Renzi. Fifty-fifth: Gengxu. Sixty-fourth: Wushen. Seventy-third: Bingwu, the midpoint.
169
甲辰二統。 辛丑。 己亥。 丁酉。 乙未。 壬辰。 庚寅。 戊子。 丙戌，季。
Jiachen: beginning of the second system. Xinchou. Jihai. Dingyou. Yiwei. Renchen. Gengyin. Wuzi day. Bingxu day, late phase.
170
甲申三統。 辛巳。 己卯。 丁丑。 乙亥。 壬申。 庚午。 戊辰。 丙寅，孟。
Jiashen day, in the third system. Xinsi day. Jimao day. Dingchou day. Yihai day. Renshen day. Gengwu day. Wuchen day. Bingyin day, early phase.
171
二，癸卯。 十一，辛丑。 二十，己亥。 二十九，丁酉。 二十八，甲午。 四十七，壬辰。 五十六，庚寅。 六十五，戊子。 七十四，乙酉，中。
Entry 2: Guimao day. Entry 11: Xinchou day. Entry 20: Jihai day. Entry 29: Dingyou day. Entry 28: Jiawu day. Entry 47: Renchen day. Entry 56: Gengyin day. Entry 65: Wuzi day. Entry 74: Yiyou day, middle phase.
172
癸未。 辛巳。 己卯。 丁丑。 甲戌。 壬申。 庚午。 戊辰。 乙丑，季。
Guiwei day. Xinsi day. Jimao day. Dingchou day. Jiaxu day. Renshen day. Gengwu day. Wuchen day. Yichou day, late phase.
173
癸亥。 辛酉。 己未。 丁巳。 甲寅。 壬子。 庚戌。 戊申乙巳，孟。
Guihai day. Xinyou day. Jiwei day. Dingsi day. Jiayin day. Renzi day. Gengxu day. Wushen day, then Yisi day, early phase.
174
三，癸未。 十二，辛巳。 二十一，己卯。 三十，丙子。 三十九，甲戌。 四十八，壬申。 五十七，庚子。 六十六，丁卯。 七十五，乙丑，中。
Entry 3: Guiwei day. Entry 12: Xinsi day. Entry 21: Jimao day. Entry 30: Bingzi day. Entry 39: Jiaxu day. Entry 48: Renshen day. Entry 57: Gengzi day. Entry 66: Dingmao day. Entry 75: Yichou day, middle phase.
175
癸亥。 辛酉。 己未。 丙辰。 甲寅。 壬子。 庚戌。 丁未。 乙巳，季。
Guihai day. Xinyou day. Jiwei day. Bingchen day. Jiayin day. Renzi day. Gengxu day. Dingwei day. Yisi day, late phase.
176
癸卯。 辛丑。 己亥。 丙申。 甲午。 壬辰。 庚寅。 丁亥。 乙酉，孟。
Guimao day. Xinchou day. Jihai day. Bingshen day. Jiawu day. Renchen day. Gengyin day. Dinghai day. Yiyou day, early phase.
177
四，癸亥。
Entry 4: Guihai day.
178
十三，辛酉。 二十二，戊午。 三十一，丙辰。 四十，甲寅。 四十九，壬子。 五十八，己酉。 六十七，丁未。 七十六，乙巳，中。
Entry 13: Xinyou day. Entry 22: Wuwu day. Entry 31: Bingchen day. Entry 40: Jiayin day. Entry 49: Renzi day. Entry 58: Jiyou day. Entry 67: Dingwei day. Entry 76: Yisi day, middle phase.
179
癸卯。 辛丑。 戊戌。 丙申。 甲午。 壬辰。 己丑。 丁亥。 乙酉，季。
Guimao day. Xinchou day. Wuxu day. Bingshen day. Jiawu day. Renchen day. Jichou day. Dinghai day. Yiyou day, late phase.
180
癸未。 辛巳。 戊寅。 丙子。 甲戌。 壬申。 己巳。 丁卯。 乙丑，孟。
Guiwei day. Xinsi day. Wuyin day. Bingzi day. Jiaxu day. Renshen day. Jisi day. Dingmao day. Yichou day, early phase.
181
五，癸卯。
Entry 5: Guimao day.
182
十四，庚子。 二十三，戊戌。 三十二，丙申。 四十一，甲午。 五十，辛卯。 五十九，己丑。 六十八，丁亥。 七十七，乙酉，中。
Entry 14: Gengzi day. Entry 23: Wuxu day. Entry 32: Bingshen day. Entry 41: Jiawu day. Entry 50: Xinmao day. Entry 59: Jichou day. Entry 68: Dinghai day. Entry 77: Yiyou day, middle phase.
183
癸未。 庚辰。 戊寅。 丙子。 甲戌。 辛未。 己巳。 丁卯。 乙丑，季。
Guiwei day. Gengchen day. Wuyin day. Bingzi day. Jiaxu day. Xinwei day. Jisi day. Dingmao day. Yichou day, late phase.
184
癸亥。 庚申。 戊午。 丙辰。 甲寅。 辛亥。 己酉。 丁未。 乙巳，孟。
Guihai day. Gengshen day. Wuwu day. Bingchen day. Jiayin day. Xinhai day. Jiyou day. Dingwei day. Yisi day, early phase.
185
六，壬午。 十五，庚辰。 二十四，戊寅。 三十三，丙子。 四十二，癸酉。 五十一，辛未。 六十，己巳。 六十九，丁卯。 七十八，甲子，中。
Entry 6: Renwu day. Entry 15: Gengchen day. Entry 24: Wuyin day. Entry 33: Bingzi day. Entry 42: Guiyou day. Entry 51: Xinwei day. Entry 60: Jisi day. Entry 69: Dingmao day. Entry 78: Jiazi day, middle phase.
186
壬戌。 庚申。 戊午。 丙辰。 癸丑。 辛亥。 己酉。 丁未。 甲辰，季。
Renxu day. Gengshen day. Wuwu day. Bingchen day. Guichou day. Xinhai day. Jiyou day. Dingwei day. Jiachen day, late phase.
187
壬寅。 庚子。 戊戌。 丙申。 癸巳。 辛卯。 己丑。 丁亥。 甲申，孟。
Renyin day. Gengzi day. Wuxu day. Bingshen day. Guisi day. Xinmao day. Jichou day. Dinghai day. Jiashen day, early phase.
188
七，壬戌。
Entry 7: Renxu day.
189
十六，庚申。 二十五，戊午。 三十四，乙卯。 四十三，癸丑。 五十二，辛亥。 六十一，己酉。 七十，丙午。 七十九，甲辰，中。
Entry 16: Gengshen day. Entry 25: Wuwu day. Entry 34: Yimao day. Entry 43: Guichou day. Entry 52: Xinhai day. Entry 61: Jiyou day. Entry 70: Bingwu day. Entry 79: Jiachen day, middle phase.
190
壬寅。 庚子。 戊戌。 乙未。 癸巳。 辛卯。 己丑。 丙戌。 甲申，季。
Renyin day. Gengzi day. Wuxu day. Yiwei day. Guisi day. Xinmao day. Jichou day. Bingxu day. Jiashen day, late phase.
191
壬午。 庚辰。 戊寅。 乙亥。 癸酉。 辛未。 己巳。 丙寅。 甲子，孟。
Renwu day. Gengchen day. Wuyin day. Yihai day. Guiyou day. Xinwei day. Jisi day. Bingyin day. Jiazi day, early phase.
192
八，壬寅。 十七，庚子。 二十六，丁酉。 三十五，乙未。 四十四，癸巳。 五十三，辛卯。 六十二，戊子。 七十一，丙戌。 八十，甲申，中。
Entry 8: Renyin day. Entry 17: Gengzi day. Entry 26: Dingyou day. Entry 35: Yiwei day. Entry 44: Guisi day. Entry 53: Xinmao day. Entry 62: Wuzi day. Entry 71: Bingxu day. Entry 80: Jiashen day, middle phase.
193
壬午。 庚辰。 丁丑。 乙亥。 癸酉。 辛未。 戊辰。 丙寅。 甲子，季。
Renwu day. Gengchen day. Dingchou day. Yihai day. Guiyou day. Xinwei day. Wuchen day. Bingyin day. Jiazi day, late phase.
194
壬戌。 庚申。 丁巳。 乙卯。 癸丑。 辛亥。 戊申。 丙午。 甲辰，孟。
Renxu day. Gengshen day. Dingsi day. Yimao day. Guichou day. Xinhai day. Wushen day. Bingwu day. Jiachen day, early phase.
195
九，壬午。 十八，己卯。 二十七，丁丑。 三十六，乙亥。 四十五，癸酉。 五十四，庚午。 六十三，戊辰。 七十二，丙寅。 八十一，甲子，中。
Entry 9: Renwu day. Entry 18: Jimao day. Entry 27: Dingchou day. Entry 36: Yihai day. Entry 45: Guiyou day. Entry 54: Gengwu day. Entry 63: Wuchen day. Entry 72: Bingyin day. Entry 81: Jiazi day, middle phase.
196
壬戌。 己未。 丁巳。 乙卯。 癸丑。 庚戌。 戊申。 丙午。 甲辰，季。
Renxu day. Jiwei day. Dingsi day. Yimao day. Guichou day. Gengxu day. Wushen day. Bingwu day. Jiachen day, late phase.
197
壬寅。 己亥。 丁酉。 乙未。 癸巳。 庚寅。 戊子。 丙戌。 甲申，孟。
Renyin day. Jihai day. Dingyou day. Yiwei day. Guisi day. Gengyin day. Wuzi day. Bingxu day. Jiashen day, early phase.
198
推章首朔旦冬至日，置大餘三十九，小餘六十一，數除如法，各從其統首起。 求其後章，當加大餘三十九，小餘六十一，各盡其八十一章。
To calculate the chapter-opening new moon and dawn winter-solstice day, set the greater remainder to 39 and the lesser remainder to 61, divide by the standard rule, and start each from the head of its respective system. To obtain the next chapter, add 39 to the greater remainder and 61 to the lesser remainder, running each through the full span of eighty-one chapters.
199
推篇，大餘亦如之，小餘加一。 求周至，加大餘五十九，小餘二十一。
For calculating a pian unit, treat the greater remainder the same way, but add one to the lesser remainder. To calculate the Zhou terminal point, add 59 to the greater remainder and 21 to the lesser remainder.
200
世經春秋昭公十七年「郯子來朝」，傳曰昭子問少昊氏鳥名何故，對曰：「吾祖也，我知之矣。 昔者，黃帝氏以雲紀，故為雲師而雲名； 炎帝氏以火紀，故為火師而火名； 共工氏以水紀，故為水師而水名； 太昊氏以龍紀，故為龍師而龍名。 我高祖少昊縶之立也，鳳鳥適至，故紀於鳥，為鳥師而鳥名。」 言郯子據少昊受黃帝，黃帝受炎帝，炎帝受共工，共工受太昊，故先言黃帝，上及太昊。 稽之於易，炮犧、神農、黃帝相繼之世可知。
In the historical canon, under the seventeenth year of Duke Zhao in the Spring and Autumn Annals, it says, "The Lord of Tan came to court." The commentary records that Zhaoxi asked why the Shaohao lineage used bird names, and Tanzi answered, "He is our ancestor; I know this well." In antiquity, the Yellow Emperor's house used clouds as its organizing sign, so offices and titles were named for clouds; the Flame Emperor's house used fire as its sign, so offices and titles were named for fire; the Gonggong lineage used water as its sign, so offices and titles were named for water; the Taihao lineage used dragons as its sign, so offices and titles were named for dragons. When our fore-ancestor Shaohao Zhi took his place, a phoenix appeared, so the system was marked by birds, with offices and titles named for birds." Tanzi's argument is that Shaohao inherited from the Yellow Emperor, the Yellow Emperor from the Flame Emperor, the Flame Emperor from Gonggong, and Gonggong from Taihao; so he began with the Yellow Emperor and worked back to Taihao. If one checks this against the Book of Changes, the successive eras of Paoxi, Shennong, and the Yellow Emperor become clear.
201
太昊帝易曰：「炮犧氏之王天下也。」 言炮犧繼天而王，為百王先，首德始於木，故為帝太昊。 作罔罟以田漁，取犧牲，故天下號曰炮犧氏。 祭典曰：「共工氏伯九域。」 言雖有水德，在火木之間，非其序也。 任知刑以彊，故伯而不王。 秦以水德，在周、漢木火之間。 周人俣其行序，故易不載。
The Taihao section of the Changes says, "When the Paoxi clan ruled the world." This means Paoxi inherited Heaven's mandate and ruled as the first of all later kings; his initiating virtue belonged to Wood, hence the title Emperor Taihao. He devised nets and fish-traps for hunting and fishing and thereby secured sacrificial offerings, so the world called him the Paoxi lineage. The ritual canon states: "The Gonggong house held hegemonic sway over the nine regions." That means its Water-phase virtue fell between Fire and Wood and thus did not follow the rightful sequence. Because he ruled by coercive intelligence and punishments, he became a hegemon, not a true king. Qin's Water virtue likewise stood between Zhou and Han, whose phases were Wood and Fire. The Zhou regarded that as a disruption of proper order, so the Changes does not include it.
202
炎帝易曰：「炮犧氏沒，神農氏作。」 言共工伯而不王，雖有水德，非其序也。 以火承木，故為炎帝。 教民耕農，故天下號曰神農氏。
The Flame Emperor section of the Changes says, "After the Paoxi lineage passed, the Shennong lineage arose." It means Gonggong, though possessing Water virtue, was a hegemon rather than king and did not stand in the rightful order. Since Fire succeeded Wood, he was designated the Flame Emperor. He taught the people tillage and agriculture, and thus the world called him the Shennong lineage.
203
黃帝易曰：「神農氏沒，黃帝氏作。」 火生土，故為土德。 與炎帝之後戰於阪泉，遂王天下。 始垂衣裳，有軒冕之服，故天下號曰軒轅氏。
The Yellow Emperor section of the Changes says, "When the Shennong lineage passed, the Yellow Emperor lineage arose." Because Fire generates Earth, his ruling virtue was Earth. He battled the descendants of the Flame Emperor at Banquan and then took kingship over the world. He instituted formal robes and the attire of canopies and crowns, so people everywhere called him the Xuanyuan lineage.
204
少昊帝考德曰少昊曰清。 清者，黃帝之子清陽也，是其子孫名摯立。 土生金，故為金德，天下號曰金天氏。 周俣其樂，故易不載，序於行。
In the "Assessment of Virtue," it says of Emperor Shaohao: "Shaohao signifies purity." "Pure" refers to Qingyang, son of the Yellow Emperor; from his descendants, one named Zhi came to the throne. Because Earth generates Metal, his phase virtue was Metal, and the world called him the Jintian lineage. Because the Zhou classified him by ritual-music tradition, the Changes does not give a separate entry and places him only in sequence.
205
顓頊帝春秋外傳曰，少昊之衰，九黎亂德，顓頊受之，乃命重黎。 蒼林昌意之子也。 金生水，故為水德。 天下號曰高陽氏。 周俣其樂，故易不載，序於行。
The Spring and Autumn Outer Commentary says that when Shaohao's line declined and the Nine Li corrupted virtue, Zhuanxu assumed rule and appointed Chong and Li. He was the son of Canglin Changyi. Since Metal generates Water, his ruling virtue was Water. The world called his house the Gaoyang lineage. As with others classified by Zhou ritual music, the Changes does not separately record him and places him only in sequence.
206
帝嚳春秋外傳曰，顓頊之所建，帝嚳受之。 清陽玄囂之孫也。 水生木，故為木德。 天下號曰高辛氏。 帝摯繼之，不知世數。 周俣其樂，故易不載。 周人禘之。
Regarding Emperor Ku, the Spring and Autumn Outer Commentary says he inherited what Zhuanxu had established. He was a grandson of Qingyang Xuanxiao. Because Water generates Wood, his ruling virtue was Wood. People under Heaven called his line the Gaoxin lineage. Emperor Zhi succeeded him, but the generational count is unknown. The Zhou treated him under ritual-music tradition, so the Changes does not separately record him. The Zhou performed the grand di sacrifice to him.
207
虞帝帝系曰，顓頊生窮蟬，五世而生瞽叟，瞽叟生帝舜，處虞之媯汭，堯嬗以天下。 火生土，故為土德。 天下號曰有虞氏。 讓天下於禹，使子商均為諸侯。 即位五十載。
The imperial genealogy states that Zhuanxu begot Qiongchan; five generations later came Gusou, and Gusou fathered Emperor Shun, who lived at Guirui in Yu. Yao then abdicated the world to him. Since Fire generates Earth, his phase virtue was Earth. The world called his line the Youyu lineage. He ceded the world to Yu and enfeoffed his son Shangjun as a regional lord. He reigned for fifty years.
208
伯禹帝系曰，顓頊五世而生鯀，鯀生禹，虞舜嬗以天下。 土生金，故為金德。 天下號曰夏后氏。 繼世十七王，四百三十二歲。
The genealogy for Bo Yu says that five generations after Zhuanxu came Gun, Gun fathered Yu, and Shun of Yu abdicated the world to him. Because Earth generates Metal, his ruling virtue was Metal. The world called his house the Xiahou lineage. Seventeen kings ruled in succession over 432 years.
209
成湯書經湯誓湯伐夏桀。 金生水，故為水德。 天下號曰商，後曰殷。
For Cheng Tang, the Documents chapter "Tang Oath" records that Tang attacked Xia Jie. Since Metal generates Water, his phase virtue was Water. The realm called this dynasty Shang, later known as Yin.
210
三統，上元至伐桀之歲，十四萬一千四百八十歲，歲在大火房五度，故傳曰：「大火，閼伯之星也，實紀商人。」 後為成湯，方即世崩沒之時，為天子用事十三年矣。 商十二月乙丑朔旦冬至，故書序曰：「成湯既沒，太甲元年，使伊尹作伊訓。」 伊訓篇曰：「惟太甲元年十有二月乙丑朔，伊尹祀于先王，誕資有牧方明。」 言雖有成湯、太丁、外丙之服，以冬至越茀祀先王于方明以配上帝，是朔旦冬至之歲也。 後九十五歲，商十二月甲申朔旦冬至，亡餘分，是為孟統。 自伐桀至武王伐紂，六百二十九歲，故傳曰殷「載祀六百」。
Under the three-system reckoning, 141,480 years passed from the high origin to the year Jie was attacked; that year lay in Dahuo, at five degrees of Fang. Hence the tradition says: "Dahuo is Ebo's star; it is precisely the marker of the Shang people." When he later became Cheng Tang, by the time of his death he had served in the position of Son of Heaven for thirteen years. In Shang reckoning, the winter solstice fell on dawn of the yi-chou new moon in month twelve. Therefore the Documents preface says: "After Cheng Tang died, in Taijia's first year, Yi Yin was made to compose the Yi Xun." The Yi Xun chapter says: "In Taijia year one, on the yi-chou new moon of month twelve, Yi Yin performed sacrifice to the former kings and grandly assisted at the Fangming rite." This means that although Cheng Tang, Taiding, and Waibing had held rule, the winter-solstice rite still offered to former kings at Fangming in pairing with the High Thearch, and that year was one in which new moon and dawn winter solstice coincided. Ninety-five years later, the Shang calendar had a jia-shen new moon at dawn winter solstice in month twelve, with no fractional remainder; this marked the Meng system. From Tang's attack on Jie to King Wu's attack on Zhou lasted 629 years; hence the tradition says Yin "sustained sacrifice for six hundred years."
211
殷曆曰，當成湯方即世用事十三年，十一月甲子朔旦冬至，終六府首。 當周公五年，則為距伐桀四百五十八歲，少百七十一歲，不盈六百二十九。 又以夏時乙丑為甲子，計其年乃孟統後五章，癸亥朔旦冬至也。 以為甲子府首，皆非是。 凡殷世繼嗣三十一王，六百二十九歲。
The Yin calendar says that when Cheng Tang had served thirteen years, the eleventh month had a jia-zi new moon at dawn winter solstice, ending at the head of the sixth fu cycle. By the Duke of Zhou's fifth year, the count from the attack on Jie was 458 years, 171 years short, and thus did not reach 629. If one further treats the Xia-calendar yi-chou as jia-zi, the count falls five chapters after the Meng system, at a gui-hai new moon with dawn winter solstice. To treat that as a jia-zi cycle head is entirely incorrect. Altogether, Yin succession comprised thirty-one kings over 629 years.
212
四分，上元至伐桀十三萬二千一百一十三歲，其八十八紀，甲子府首，入伐桀後百二十七歲。
Under the quarter-division system, 132,113 years elapsed from the high origin to the attack on Jie; its eighty-eight cycles place the jia-zi cycle head 127 years after that attack.
213
春秋曆，周文王四十二年十二月丁丑朔旦冬至，孟統之二會首也。 後八歲而武王伐紂。
The Spring and Autumn calendar sets the winter solstice at dawn on the ding-chou new moon of month twelve in King Wen of Zhou's forty-second year; this was the second convergence head of the Meng system. Eight years later, King Wu attacked Zhou.
214
武王書經牧誓武王伐商紂。 水生木，故為木德。 天下號曰周室。
For King Wu, the Documents chapter "Mu Oath" records that King Wu attacked Shang Zhou. Since Water generates Wood, his dynastic virtue was Wood. The realm called it the House of Zhou.
215
月二年六月庚午朏，王命作策豐刑。」
In year two, in the sixth month on a gengwu crescent day, the king ordered that the punitive statutes at Feng be codified."
216
世家，煬公即位六十年，子幽公宰立。 幽公，世家即位十四年，及微公茀立，泺。 微公二十六年正月乙亥朔旦冬至，殷曆以為丙子，距獻公七十六歲。
The Hereditary House says Duke Yang ruled for sixty years, after which his son Duke You (Zai) took the throne. According to the Hereditary House, Duke You reigned fourteen years; then Duke Wei, Fu, was installed, at Luo. In Duke Wei's twenty-sixth year, the first month had an yi-hai new moon at dawn winter solstice; the Yin calendar recorded it as bingzi, seventy-six years before Duke Xian.
217
世家，微公即位五十年，子厲公翟立，擢。 厲公，世家即位三十七年，及獻公具立。 獻公十五年正月甲寅朔旦冬至，殷曆以為乙卯，距懿公七十六歲。
The Hereditary House records that Duke Wei ruled fifty years, then his son Duke Li (Di) succeeded. The Hereditary House says Duke Li reigned thirty-seven years, after which Duke Xian (Ju) took the throne. In Duke Xian's fifteenth year, the first month had a jiayin new moon at dawn winter solstice; the Yin calendar gave yimao, seventy-six years before Duke Yi.
218
世家，獻公即位五十年，子慎公埶立，撸。 慎公，世家即位三十年，及武公敖立。 武公，世家即位二年，子懿公被立，戲。 懿公九年正月癸巳朔旦冬至，殷曆以為甲午，距惠公七十六歲。
The Hereditary House states that Duke Xian ruled fifty years, then his son Duke Shen (Zhi) succeeded. According to the Hereditary House, Duke Shen reigned thirty years and was followed by Duke Wu (Ao). The Hereditary House records Duke Wu as reigning two years, then his son Duke Yi (Bei) succeeding. In Duke Yi's ninth year, the first month had a guisi new moon at dawn winter solstice; the Yin calendar marked it as jiawu, seventy-six years before Duke Hui.
219
世家，懿公即位九年，兄子柏御立。 柏御，世家即位十一年，叔父孝公稱立。 孝公，世家即位二十七年，子惠公皇立。 惠公三十八年正月壬申朔旦冬至，殷曆以為癸酉，距釐公七十六歲。
The Hereditary House says Duke Yi ruled nine years, then his nephew Boyu was installed. According to the Hereditary House, Boyu reigned eleven years, then his uncle Duke Xiao (Cheng) came to power. The Hereditary House records Duke Xiao reigning twenty-seven years, followed by his son Duke Hui (Huang). In Duke Hui's thirty-eighth year, the first month had a renshen new moon at dawn winter solstice; the Yin calendar gave guiyou, seventy-six years before Duke Xi.
220
世家，惠公即位四十六年，子隱公息立。
The Hereditary House states Duke Hui ruled forty-six years and was succeeded by his son Duke Yin (Xi).
221
凡伯禽至春秋，三百八十六年。
From Bo Qin down to the Spring and Autumn period, the total is 386 years.
222
春秋隱公，春秋即位十一年，及桓公軌立。 此元年上距伐紂四百歲。
In the Spring and Autumn record, Duke Yin ruled eleven years, then Duke Huan (Gui) succeeded. That first year stood 400 years after the campaign against Zhou.
223
桓公，春秋即位十八年，子莊公同立。
Duke Huan is recorded as reigning eighteen years, then his son Duke Zhuang (Tong) took the throne.
224
莊公，春秋即位三十二年，子愍公啟方立。
Duke Zhuang ruled thirty-two years in the annals, followed by his son Duke Min (Qifang).
225
愍公，春秋即位二年，及釐公申立。 釐公五年正月辛亥朔旦冬至，殷曆以為壬子，距成公七十六歲。
Duke Min reigned two years, after which Duke Xi (Shen) was installed. In Duke Xi's fifth year, a xinhai new moon at dawn winter solstice fell in the first month; the Yin calendar counted it as renzi, seventy-six years before Duke Cheng.
226
是歲距上元十四萬二千五百七十七歲，得孟統五十三章首。 故傳曰：「五年春王正月辛亥朔，日南至。」 「八月甲午，晉侯圍上陽。」 章謠云：「丙子之辰，龍尾伏辰，袀服振振，取虢之旂。 鶉之賁賁，天策焞焞，火中成軍，虢公其奔。」 卜偃曰：「其九月十月之交乎？ 丙子旦，日在尾，月在策，鶉火中，必是時也。」 冬十二月丙子滅虢。 言曆者以夏時，故周十二月，夏十月也。 是歲，歲在大火。 故傳曰晉侯使寺人披伐蒲，重耳奔狄。 董因曰：「君之行，歲在大火。」 後十二年，釐之十六歲，歲在壽星。 故傳曰重耳處狄十二年而行，過衛五鹿，乞食於野人，野人舉塊而與之。 子犯曰：「天賜也，後十二年，必獲此土。 歲復於壽星，必獲諸侯。」 後八歲，釐之二十四年也，歲在實沈，秦伯納之。 故傳曰董因云：「君以辰出，而以參入，必獲諸侯。」
That year lay 142,577 years from the high origin and corresponded to the fifty-third chapter-head of the Meng system. Hence the Tradition states: "In year five, spring, first royal month, on a xinhai new moon, the sun reached its southernmost point." "In the eighth month, on a jiawu day, the Marquis of Jin laid siege to Shangyang." A chapter rhyme says: "At the bingzi time, when Dragon Tail veils the chen, in trim official robes they seize Guo's standard. The quail constellations flare; Heaven's mandate burns bright; with Fire at center the host is formed, and the Duke of Guo will run." Bu Yan said, "Will it happen around the transition between the ninth and tenth months? At bingzi dawn, the sun stands in Tail, the moon in Ce, and Quail Fire is centered - it must be then." In winter, in the twelfth month on a bingzi day, Guo was extinguished. Calendar discussants are using Xia reckoning, so Zhou's twelfth month corresponds to Xia's tenth month. That year, Jupiter stood in Dahuo. So the Tradition says the Marquis of Jin sent the eunuch Pi against Pu, and Chong'er fled to the Di people. Dong Yin said, "When you set out, the year-star was in Dahuo." Twelve years later, in Duke Xi's sixteenth year, Jupiter was in Shouxing. Hence the Tradition records that Chong'er remained with the Di for twelve years before leaving; passing Wulu in Wey, he begged food from villagers, who handed him a clod of earth. Zifan said, "Heaven has granted this omen; after twelve years, he will surely gain this territory. When Jupiter returns to Shouxing, he will surely win the allegiance of the feudal lords." Eight years later, in Duke Xi's twenty-fourth year, Jupiter was in Shishen, and the Earl of Qin took him in. Thus the Tradition quotes Dong Yin: "You departed under Chen and entered under Shen; you will certainly secure the feudal lords."
227
春秋，釐公即位三十三年，子文公興立。 文公元年，距辛亥朔旦冬至二十九歲。 是歲閏餘十三，正小雪，閏當在十一月後，而在三月，故傳曰「非禮也」。 後五年，閏餘十，是歲亡閏，而置閏。 閏，所以正中朔也。 亡閏而置閏，又不告朔，故經曰「閏月不告朔」，言亡此月也。 傳曰：「不告朔，非禮也。」
In the Spring and Autumn record, Duke Xi ruled thirty-three years and was succeeded by his son Duke Wen (Xing). Duke Wen's first year came twenty-nine years after the xinhai new-moon dawn winter-solstice point. In that year the intercalary remainder was thirteen. The leap month should have followed month eleven at Minor Snow, but it was placed in month three, so the Tradition calls it "contrary to ritual." Five years later the intercalary remainder was ten; there should have been no leap month, yet one was inserted. The leap month exists to correct the median new-moon cycle. If a leap month is inserted when none is due, and the new moon is not proclaimed, the Classic therefore says, "In the leap month the new moon was not announced," meaning the proper month was missing. The Tradition says, "Failing to announce the new moon violates ritual."
228
春秋，文公即位十八年，子宣公倭立。
In the annals, Duke Wen ruled eighteen years and was succeeded by his son Duke Xuan (Wo).
229
宣公，春秋即位十八年，子成公黑肱立。 成公十二年正月庚寅朔旦冬至，殷曆以為辛卯，距定公七年七十六歲。
Duke Xuan is recorded as reigning eighteen years, then his son Duke Cheng (Heigong) took the throne. In Duke Cheng's twelfth year, the first month had a gengyin new moon at dawn winter solstice; the Yin calendar rendered xinmao, seventy-six years before Duke Ding's seventh year.
230
春秋，定公即位十五年，子哀公將立。 哀公十二年冬十二月流火，非建戌之月也。 是月也螽，故傳曰：「火伏而後蟄者畢，今火猶西流，司曆過也。」 《詩》曰：「七月流火。」 春秋，哀公即位二十七年。 自春秋盡哀十四年，凡二百四十二年。
The Spring and Autumn record says Duke Ding ruled fifteen years, then his son Duke Ai (Jiang) succeeded. In Duke Ai's twelfth year, in winter month twelve, "flowing fire" appeared, which was not a month of the jianxu position. That same month saw swarms of locusts, so the Tradition says: "Only after Fire has sunk do the hibernating insects fully settle. Now Fire is still moving west - the calendrical office has erred." The Odes says, "In the seventh month, Fire drifts west." In the Spring and Autumn accounting, Duke Ai reigned twenty-seven years. From the start of the Spring and Autumn era to Duke Ai year fourteen totals 242 years.
231
惠帝，著紀即位七年。
Emperor Hui is recorded in the annals as reigning seven years.
232
高帝，著紀即位八年。
Emperor Gao is recorded as reigning eight years.
233
文帝，前十六年，後七年，著紀即位二十三年。
Emperor Wen had sixteen earlier years and seven later years; his total reign in the annals is twenty-three years.
234
景帝，前七年，中六年，後三年，著紀即位十六年。
Emperor Jing's reign is divided into former seven, middle six, and latter three years - sixteen years in all.
235
武帝建元、元光、元朔各六年。 元朔六年十一月甲申朔旦冬至，殷曆以為乙酉，距初元七十六歲。 元狩、元鼎、元封各六年。 漢曆太初元年，距上元十四萬三千一百二十七歲。 前十一月甲子朔旦冬至，歲在星紀婺女六度，故漢志曰歲名困敦，正月歲星出婺女。 太初、天漢、太始、征和各四年，後二年，著紀即位五十四年。
Under Emperor Wu, the Jianyuan, Yuanguang, and Yuanshuo eras each lasted six years. In Yuanshuo year six, month eleven had a jiashen new moon at dawn winter solstice; the Yin calendar counted it as yiyou, seventy-six years before Chuyuan. The Yuanshou, Yuanding, and Yuanfeng eras each spanned six years. Han Taichu year one fell 143,127 years after the high origin. In the preceding eleventh month, a jiazi new moon coincided with dawn winter solstice, with Jupiter at Xingji, six degrees of Wunu. Hence the Han Treatise names the year Kundun and says Jupiter emerged in Wunu in month one. Taichu, Tianhan, Taishi, and Zhenghe each lasted four years; adding the final two years gives a fifty-four-year reign in the annals.
236
昭帝始元、元鳳各六年，元平一年，著紀即位十三年。
For Emperor Zhao, Shiyuan and Yuanfeng were six years each, with one year of Yuanping, totaling thirteen years in the annals.
237
宣帝本始、地節、元康、神爵、五鳳、甘露各四年，黃龍一年，著紀即位二十五年。 元帝初元二年十一月癸亥朔旦冬至，殷曆以為甲子，以為紀首。 是歲也，十月日食，非合辰之會，不得為紀首。 距建武七十六歲。 初元、永光、建昭各五年，竟寧一年，著紀即位十六年。
For Emperor Xuan, Benshi, Dijie, Yuankang, Shenjue, Wufeng, and Ganlu ran four years each, with one year of Huanglong, for a twenty-five-year reign. In Emperor Yuan's Chuyuan year two, month eleven had a guihai new moon at dawn winter solstice; the Yin calendar mapped it to jiazi and took it as a cycle start. Yet that year had a tenth-month solar eclipse, not at the proper conjunction, so it could not serve as a cycle head. This stood seventy-six years from Jianwu. Chuyuan, Yongguang, and Jianzhao each lasted five years, plus one year of Jingning, for sixteen years in total.
238
成帝建始、河平、陽朔、鴻嘉、永始、元延各四年，綏和二年，著紀即位二十六年。
Under Emperor Cheng, Jianshi, Heping, Yangshuo, Hongjia, Yongshi, and Yuanyan were each four years, with two years of Suihe, making twenty-six years.
239
哀帝建平四年，元壽二年，著紀即位六年。
Emperor Ai's reign comprised four years of Jianping and two of Yuanshou, totaling six years.
240
平帝，著紀即位元始五年，以宣帝玄孫嬰為嗣，謂之孺子。 孺子，著紀新都侯王莽居攝三年，王莽居攝，盜襲帝位，竊號曰新室。 始建國五年，天鳳六年，地皇三年，著紀盜位十四年。 更始帝，著紀以漢宗室滅王莽，即位二年。 赤眉賊立宗室劉盆子，滅更始帝。 自漢元年訖更始二年，凡二百三十歲。
Emperor Ping is recorded as reigning five Yuanshi years; he designated Ying, a great-great-grandson of Emperor Xuan, as heir, known as the Child Heir. For the Child Heir period, the annals record three years of regency by Wang Mang, Marquis of Xindu. Wang Mang then usurped the imperial throne and proclaimed the "New" dynasty. The eras Shijianguo (5), Tianfeng (6), and Dihuang (3) together make fourteen years of usurpation in the annals. Emperor Gengshi is recorded as a Han clansman who overthrew Wang Mang and reigned two years. The Red Eyebrow rebels installed Liu Penzi of the imperial clan and brought down Emperor Gengshi. From Han year one through Gengshi year two, the total span is 230 years.
241
光武皇帝，著紀以景帝後高祖九世孫受命中興復漢，改元曰建武，歲在鶉尾之張度。 建武三十一年，中元二年，即位三十三年。
The annals state that Emperor Guangwu, a ninth-generation descendant of Gaozu through Emperor Jing's branch, received Heaven's mandate, revived Han, and proclaimed the Jianwu era, with Jupiter in the Zhang degree of Chunwei. He ruled for thirty-three years in total: thirty-one of Jianwu and two of Zhongyuan.