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應天乾元儀天曆
The Yingtian, Qianyuan, and Yitian Calendars.
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古者，帝王之治天下，以律曆為先； 儒者之通天人，至律曆而止。 曆以數始，數自律生，故律曆既正，寒暑以節，歲功以成，民事以序，庶績以凝，萬事根本由茲立焉。 古人自入小學，知樂知數，已曉其原。 後世老師宿儒猶或弗習律曆，而律曆之家未必知道，各師其師，岐而二之。 雖有巧思，豈能究造化之統會，以識天人之蘊奧哉！ 是以審律造曆，更易不常，卒無一定之說。 治效之不古若，亦此之由，而世豈察及是乎！
In ancient times, rulers placed the regulation of pitch and the calendar foremost in governing the realm. Even for Confucian scholars who sought to unite Heaven and man, the limit of their learning was pitch and calendar. The calendar begins with number, and number arises from pitch; once pitch and calendar were set right, the seasons kept their rhythm, harvests succeeded, civil life fell into order, and every undertaking gained a foundation. The ancients learned music and reckoning in elementary school and already grasped these principles at the source. In later ages many senior scholars never studied pitch and calendar at all, while specialists in those arts often knew the techniques but not the larger Way — each following his own master until the two fields split apart. However clever they might be, how could they grasp the full unity of nature or plumb the deep mystery of Heaven and man! Hence pitch was revised and calendars replaced again and again, until no stable doctrine remained. That government fell short of antiquity owed much to this as well — yet who in the world ever noticed?
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宋初，承五代之季王朴制律曆、作律準，以宣其聲，太祖以雅樂聲高，詔有司考正。 和峴等以影表銅臬暨羊頭秬黍累尺制律，而度量權衡因以取正。 然累代尺度與望臬殊，黍有鉅細，縱橫容積，諸儒異議，卒無成說。 至崇寧中，徽宗任蔡京，信方士「聲為律、身為度」之說，始大盭乎古矣。
Early in the Song, inheriting the late Five Dynasties, Wang Pu devised pitch pipes and a calendar and made pitch standards to sound the tones; because Taizu found court music too sharp, he ordered officials to rectify it. He Yan and his colleagues derived pitch pipes from the shadow table, bronze gnomon, and black millet from Sheep Head Mountain stacked to a foot, and on that basis standardized measures, capacity, and weights. Yet scales differed from dynasty to dynasty and from the gnomon standard, millet grains varied in size, and scholars disputed length, breadth, and volume — in the end no consensus was reached. In the Chongning period Huizong entrusted Cai Jing, who believed the Daoist doctrine that "sound is pitch and the body is measure," and antiquity was overturned on a grand scale.
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顯德欽天曆亦朴所制也，宋初用之。 建隆二年，以推驗稍疏，詔王處訥等別造新曆。 四年，曆成，賜名應天，未幾，氣候漸差。 太平興國四年，行乾元曆，未幾，氣候又差。 繼作者曰儀天，曰崇天，曰明天，曰奉元，曰觀天，曰紀元，迨靖康丙午，百六十餘年，而八改曆。 南渡之後，曰統元，曰乾道，曰淳熙，曰會元，曰統天，曰開禧，曰會天，曰成天，至德祐丙子，又百五十年，復八改曆。 使其初而立法吻合天道，則千歲日至可坐而致，奚必數數更法，以求幸合玄象哉！ 蓋必有任其責者矣。
The Xiande Qintian calendar was also Wang Pu's work, and the early Song continued to use it. In Jianlong 2 (961), because its predictions had grown somewhat loose, the court ordered Wang Chunu and others to compile a new calendar. Four years later the calendar was finished and named Yingtian; before long its seasonal predictions began to drift. In Taiping Xingguo 4 (979) the Qianyuan calendar was adopted; again, before long the seasons fell out of alignment. Successors produced the Yitian, Chongtian, Mingtian, Fengyuan, Guantian, and Jiyuan calendars — by Jingkang bingwu (1126), in a little over 160 years, the calendar had been changed eight times. After the court crossed south came the Tongyuan, Qiandao, Chunxi, Huiyuan, Tongtian, Kaixi, Huitian, and Chengtian calendars — by Deyou bingzi (1296), in another 150 years, eight more revisions had been made. Had the first laws truly matched Heaven's way, the solstice a millennium hence could have been predicted without rising from one's seat — why revise the methods again and again merely to chase the heavens?! Surely someone must bear responsibility for this.
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雖然，天步惟艱，古今通患，天運日行，左右既分，不能無忒。 謂七十九年差一度，雖視古差密，亦僅得其概耳。 又況黃、赤道度有斜正闊狹之殊，日月運行有盈縮、朏朒、表裡之異。 測北極者，率以千里差三度有奇，晷景稱是。 古今測驗，止於岳臺，而岳臺豈必天地之中？ 餘杭則東南，相距二千餘里，華夏幅員東西萬里，發斂晷刻豈能盡諧？ 又造曆者追求曆元，踰越曠古，抑不知二帝授時齊政之法，畢殫於是否乎？ 是亦儒者所當討論之大者，諉曰星翁曆生之責可哉？ 至於儀象推測之具，雖亦數改，若熙寧沈括之議、宣和璣衡之制，其詳密精緻有出於淳風、令瓚之表者，蓋亦未始乏人也。 今其遺法具在方冊，惟奉元、會天二法不存。 舊史以乾元、儀天附應天，今亦以乾道、淳熙、會元附統元，開禧、成天附統天。 大抵數異術同，因仍增損，以追合乾象，俱無以大相過，備載其法，俾來者有考焉。
Yet the motion of Heaven is hard to track — a difficulty common to every age; the sky moves day by day, and once east and west are distinguished, some discrepancy is inevitable. The claim that precession amounts to one degree in seventy-nine years, though tighter than older estimates, still captures only a rough approximation. Moreover, ecliptic and equatorial degrees differ in obliquity and in apparent breadth, and the sun and moon vary in speed, phase, and apparent position. Observers of the north celestial pole commonly reckon a change of slightly more than three degrees for every thousand li, and gnomon shadows bear this out. All measurements, ancient and modern, were taken at the Yue Terrace — yet was the Yue Terrace truly the center of the world? Yuhang lies far to the southeast, more than two thousand li away, while China stretches ten thousand li from east to west — how could gnomon readings at every solstice and equinox agree everywhere? Calendar makers also chase ever more remote epoch origins — yet have they exhausted the method by which the sage emperors granted the seasons and aligned government with them? These are great questions Confucian scholars ought to debate — can they be dismissed as the concern of astrologers and calendar clerks alone? As for observational and computational instruments, though these too were revised repeatedly, Shen Kuo's proposals under Xining and the armillary sphere of the Xuanhe reign were in detail and precision finer than the work of Chunfeng and Ling Zan — the age was never without able men. Their surviving methods are preserved in the archives today, except that the Fengyuan and Huitian systems are lost. The earlier history appended Qianyuan and Yitian to Yingtian; here Qiandao, Chunxi, and Huiyuan are appended to Tongyuan, and Kaixi and Chengtian to Tongtian. In general the figures differ but the methods are the same, each revision inheriting and adjusting the last to chase the heavens — none stands far above the rest; all are recorded here so that posterity may study them.
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昔黃帝作律呂，以調陰陽之聲，以候天地之氣。 堯則欽若曆象，以授人時，以成歲功，用能綜三才之道，極萬物之情，以成其政化者也。 至司馬遷、班固敘其指要，著之簡策。 自漢至隋，歷代祖述，益加詳悉。 暨唐貞觀迄周顯德，五代隆替，踰三百年，博達之士頗亦詳緝廢墜，而律志皆闕。 宋初混一○內，能士畢舉，國經王制，悉復古道。 漢志有備數、和聲、審度、嘉量、權衡之目，後代因之，今亦用次序以志於篇：
Long ago the Yellow Emperor devised the pitch pipes to harmonize yin and yang in sound and to read the breath of Heaven and Earth. Yao reverently aligned the calendar and the heavens to grant the seasons to the people and complete the year's work, thereby mastering the way of Heaven, earth, and man and bringing all things into his civilizing rule. Sima Qian and Ban Gu later summarized these principles and committed them to writing. From Han through Sui each dynasty followed and elaborated the tradition with growing detail. From Tang Zhenguan to Later Zhou Xiande, through more than three centuries of the Five Dynasties, learned men did much to recover what had been lost — yet treatises on pitch were absent from the histories. When the Song unified the realm, able men were appointed everywhere, and state institutions were restored to the ancient pattern. The Han Treatise listed providing number, harmonizing sound, examining measure, standard capacity, and weights and balances; later histories followed that order, and this chapter does the same:
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曰備數。 周禮，保氏教國子以六藝，其六曰九數，謂方田、粟米、差分、少廣、商功、均輸、方程、贏朒、旁要，是為九章。 其後又有海島、孫子、五曹、張丘建、夏侯陽、周髀、綴術、緝古等法相因而起，歷代傳習，謂之小學。 唐試右千牛衛胄曹參軍陳從運著得一算經，其術以因折而成，取損益之道，且變而通之，皆合於數。 復有徐仁美者，作增成玄一法，設九十三問，以立新術，大則測於天地，細則極於微妙，雖粗述其事，亦適用於時。 古者命官屬於太史，漢、魏之世，皆在史官。 隋氏始置算學博士於國庠，唐增其員，宋因而不改。
Providing Number. The Rites of Zhou records that the Guardian taught the crown princes six arts, the sixth being the nine branches of reckoning: field measurement, grain, proportional parts, lesser width, engineering, fair transport, simultaneous equations, excess and deficit, and right triangles — the Nine Chapters. Later came the Sea Island, Master Sun, Five Officials, Zhang Qiujian, Xiahou Yang, Zhou Bi, Continuation of Methods, and Restoration of Antiquity texts, handed down through the ages as elementary mathematics. In Tang, Chen Congyun, probationary clerk of the Right Thousand-Ox Guard, wrote the Canon of Unified Calculation, whose methods build by successive adjustment through increase and decrease until all accord with number. Xu Renmei also devised the method of augmented formation and dark unity with ninety-three problems establishing new techniques — ranging from measurements of heaven and earth down to the most subtle quantities; though only roughly described, it served the needs of the age. In antiquity these offices belonged to the Grand Astrologer; under Han and Wei they remained with the historiographical staff. The Sui first appointed Doctors of Computational Learning at the National University; Tang increased their number, and Song left the arrangement unchanged.
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曰和聲。 周禮，典同掌六律六同之和，凡為樂器，以十有二律為之數度。 古之聖人推律以製器，因器以宣聲，和聲以成音，比音而為樂。 然則律呂之用，其樂之本歟！ 以其相生損益，數極精微，非聰明博達，則罕能詳究。 故歷代而下，其法或存或闕，前史言之備矣。 周顯德中，王樸始依周法，以秬黍校正尺度，長九寸，虛徑三分，為黃鐘之管，作律準，以宣其聲。 宋乾德中，太祖以雅樂聲高，詔有司重加考正。 時判太常寺和峴上言曰：「古聖設法，先立尺寸，作為律呂，三分損益，上下相生，取合真音，謂之形器。 但以尺寸長短非書可傳，故累秬黍求為準的，後代試之，或不符會。 西京銅望臬可校古法，即今司天臺影表銅臬下石尺是也。 及以樸所定尺比校，短於石尺四分，則聲樂之高，蓋由於此。 況影表測於天地，則管律可以準繩。」 上乃令依古法，以造新尺并黃鐘九寸之管，命工人校其聲，果下於樸所定管一律。 又內出上黨羊頭山秬黍，累尺校律，亦相符合。 遂下尚書省集官詳定，眾議僉同。 由是重造十二律管，自此雅音和暢。
Harmonizing Sound. The Rites of Zhou assigns the Director of Harmonies to regulate the six pitch standards and six unisons; all instruments take their dimensions from the twelve pitch pipes. The ancient sages derived pitch to fashion instruments, used instruments to produce sound, harmonized sounds into tones, and combined tones into music. Thus pitch pipes are the very foundation of music! Because their cycles of generation and diminution involve numbers of the utmost refinement, only the most penetrating minds could master them fully. Down the generations the methods were sometimes preserved and sometimes lost — earlier histories record this at length. In Later Zhou Xiande, Wang Pu first followed Zhou methods, using black millet to set the scale: a nine-inch pipe with a hollow diameter of three fen for the yellow bell, and pitch standards to sound it. In Song Qiande (963–968), Taizu found court music too sharp and ordered officials to re-examine it. The acting director of the Court of Imperial Sacrifices, He Yan, memorialized: "The ancient sages first established dimensions and from them derived the pitch pipes, diminishing and augmenting by thirds to generate the scale until true pitch was reached — these are the physical instruments. Because exact lengths cannot be conveyed in writing, black millet was stacked to provide a standard, yet later trials often failed to match. The bronze gnomon of the Western Capital can verify the ancient standard — the stone foot beneath the shadow table at the Directorate of Astronomy today. Pu's foot proved four fen shorter than the stone foot — which explains why the music sounded too high. Since the shadow table is measured against heaven and earth, the pitch pipes can be set by that standard." The emperor then ordered a new foot and a nine-inch yellow bell pipe made by the ancient method; when craftsmen tested the sound, it proved a full pitch lower than Pu's pipe. The palace also supplied black millet from Sheep Head Mountain in Shangdang; stacked to a foot, it matched the pitch pipes as well. The matter was referred to the Department of State Affairs for a joint review, and all officials agreed. The twelve pitch pipes were then remade, and from that time court music sounded in proper harmony.
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曰審度者，本起於黃鐘之律，以秬黍中者度之，九十黍為黃鐘之長，而分、寸、尺、丈、引之制生焉。 宋既平定四方，凡新邦悉頒度量於其境，其偽俗尺度踰於法制者去之。 乾德中，又禁民間造者。 由是尺度之制盡復古焉。
Examining Measure: the standard derives from the yellow bell pitch pipe; the medium grain of black millet defines it — ninety grains equal the length of the yellow bell, from which the units of fen, inch, foot, rod, and cord are derived. After the Song pacified the realm, every newly subdued territory received official measures, and local scales that exceeded the legal standard were abolished. In Qiande private manufacture was also banned. Thus the system of linear measure was fully restored to the ancient standard.
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曰嘉量。 周禮，○氏為量。 漢志云，物有多少受以量，本起於黃鐘之管容秬黍千二百，而龠、合、升、斗、斛五量之法備矣。 太祖受禪，詔有司精考古式，作為嘉量，以頒天下。 其後定西蜀，平嶺南，復江表，泉、浙納土，并、汾歸命，凡四方斗、斛不中式者皆去之。 嘉量之器，悉復昇平之制焉。
Excellent Capacity. The Rites of Zhou assigns the Granary Master to make the measures. The Han Treatise states that things vary in amount and are measured accordingly; the standard began with the yellow bell pipe holding twelve hundred grains of black millet, establishing the five capacities from yue through hu. When Taizu received the abdication, he ordered officials to study ancient models carefully and cast standard capacity measures to distribute throughout the realm. After Western Shu was subdued, Lingnan pacified, the lower Yangzi recovered, Quanzhou and Zhejiang submitted, and Bing and Fen surrendered, every nonstandard bushel and picul in the realm was abolished. The standard capacity measures were thus restored to the unified system of the age.
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曰權衡之用，所以平物一民、知輕重也。 權有五，曰銖、兩、斤、鈞、石，前史言之詳矣。 建隆元年八月，詔有司按前代舊式作新權衡，以頒天下，禁私造者。 及平荊湖，即頒量、衡於其境。 淳化三年三月三日，詔曰：「書云：『協時、月，正日，同律、度、量、衡。』 所以建國經而立民極也。 國家萬邦咸乂，九賦是均，顧出納於有司，繫權衡之定式。 如聞秬黍之制，或差毫釐，錘鈞為姦，害及黎庶。 宜令詳定稱法，著為通規。」 事下有司，監內藏庫、崇儀使劉承珪言：「太府寺舊銅式自一錢至十斤，凡五十一，輕重無準。 外府歲受黃金，必自毫釐計之，式自錢始，則傷於重。」 遂尋究本末，別製法物。 至景德中，承珪重加參定，而權衡之制益為精備。 其法蓋取漢志子穀秬黍為則，廣十黍以為寸，從其大樂之尺， 〈（秬黍，黑黍也。 樂尺，自黃鐘之管而生也。 謂以秬黍中者為分寸、輕重之制。）〉 就成二術， 〈（二術謂以尺、黍而求氂、絫。）〉 因度尺而求氂， 〈（度者，丈、尺之總名焉。 因樂尺之源，起於黍而成於寸，析寸為分，析分為氂，析氂為毫，析毫為絲，析絲為忽。 十忽為絲，十絲為毫，十毫為氂，十氂為分。）〉 自積黍而取絫。 〈（從積黍而取絫，則十黍為絫，十絫為銖，二十四銖為兩。 錘皆以銅為之。）〉 以氂、絫造一錢半及一兩等二稱，各懸三毫，以星準之。 等一錢半者，以取一稱之法。 其衡合樂尺一尺二寸，重一錢，錘重六分，盤重五分。 初毫星準半錢，至稍總一錢半，析成十五分，分列十氂； 〈（第一毫下等半錢，當五十氂，若十五斤稱等五斤也。）〉 中毫至稍一錢，析成十分，分列十氂； 末毫至稍半錢，析成五分，分列十氂。 等一兩者，亦為一稱之則。 其衡合樂分尺一尺四寸，重一錢半，錘重六錢，盤重四錢。 初毫至稍，布二十四銖，下別出一星，等五絫； 〈（每銖之下，復出一星，等五絫，則四十八星等二百四十絫，計二千四百絫為十兩。）〉 中毫至稍五錢，布十二銖，列五星，星等二絫； 〈（布十二銖為五錢之數，則一銖等十絫，都等一百二十絫為半兩。）〉 末毫至稍六銖，銖列十星，星等絫。 〈（每星等一絫，都等六十絫為二錢半。）〉 以御書真、草、行三體淳化錢，較定實重二銖四絫為一錢者，以二千四百得十有五斤為一稱之則。 其法，初以積黍為準，然後以分而推忽，為定數之端。 故自忽、絲、毫、氂、黍、絫、銖各定一錢之則。 〈（謂皆定一錢之則，然後制取等稱也。）〉 忽萬為分， 〈（以一萬忽為一分之則，以十萬忽定為一錢之則。 忽者，吐絲為忽； 分者，始微而著，言可分別也。）〉 絲則千， 〈（一千絲為一分，以一萬絲定為一錢之則。）〉 毫則百， 〈（一百毫為一分，以一千毫定為一錢之則。 毫者，毫毛也。 自忽、絲、毫三者皆斷驥尾為之。）〉 氂則十， 〈（一十氂為一分，以一百氂定為一錢之則。 氂者，氂牛尾毛也，曳赤金成絲為之也。）〉 轉以十倍倍之，則為一錢。 〈（轉以十倍，謂自一萬忽至十萬忽之類定為則也。）〉 黍以二千四百枚為一兩， 〈（一龠容千二百黍為十二銖，則以二千四百黍定為一兩之則。 兩者，以二龠為兩。）〉 絫以二百四十， 〈（謂以二百四十絫定為一兩之則。）〉 銖以二十四， 〈（轉相因成絫為銖，則以二百四十絫定成二十四銖為一兩之則。 銖者，言殊異。）〉 遂成其稱。 稱合黍數，則一錢半者，計三百六十黍之重。 列為五分，則每分計二十四黍。 又每分析為一十氂，則每氂計二黍十分黍之四。 〈（以十氂分二十四黍，則每氂先得二黍。 都分成四十分，則一絫又得四分，是每氂得二黍十分黍之四。）〉 每四毫一絲六忽有差為一黍，則氂、絫之數極矣。 一兩者，合二十四銖為二千四百黍之重。 每百黍為銖，二百四十黍為絫，二銖四絫為錢，二絫四黍為分。 一絫二黍重五氂，六黍重二氂五毫，三黍重一氂二毫五絲，則黍、絫之數成矣。 其則，用銅而鏤文，以識其輕重。 新法既成，詔以新式留禁中，取太府舊稱四十、舊式六十，以新式校之，乃見舊式所謂一斤而輕者有十，謂五斤而重者有一。 式既若是，權衡可知矣。 又比用大稱如百斤者，皆懸鈞於架，植環於衡，鐶或偃，手或抑按，則輕重之際，殊為懸絕。 至是，更鑄新式，悉由黍、絫而齊其斤、石，不可得而增損也。 又令每用大稱，必懸以絲繩。 既置其物，則卻立以視，不可得而抑按。 復鑄銅式，以御書淳化三體錢二千四百暨新式三十有三、銅牌二十授於太府。 又置新式於內府、外府，復頒於四方大都，凡十有一副。 先是，守藏吏受天下歲貢金帛，而太府權衡舊式失準，得因之為姦，故諸道主者坐逋負而破產者甚眾。 又守藏更代，校計爭訟，動必數載。 至是，新制既定，奸弊無所指，中外以為便。 〈（度、量、權、衡皆太府掌造，以給內外官司及民間之用。 凡遇改元，即差變法，各以年號印而識之。 其印面有方印、長印、八角印，明制度而防偽濫也。）〉
Weights and balances serve to standardize goods, unite the people, and distinguish heavy from light. There are five weight units — zhu, liang, jin, jun, and shi — which earlier histories describe in full. In the eighth month of Jianlong 1 (960), officials were ordered to follow former dynasties' models in casting new weights and balances for distribution throughout the realm and to forbid private manufacture. When Jinghu was pacified, the new measures and weights were immediately distributed there. On the third day of the third month of Chunhua 3 (992), an edict cited the Book: "Harmonize the seasons and months, rectify the days, and unify pitch, measure, capacity, and balance. This is how a state establishes its foundations and sets the standard for the people. Though the realm is at peace and taxes are fairly levied, all receipts and disbursements depend on officials and ultimately on fixed weights and balances. We hear that even the black-millet standard may err by a hair or a fraction, allowing fraud with steelyards and scales that harms the common people. Let weighing methods be examined in detail and established as a general regulation." The matter was referred to the responsible offices; Liu Chenggui, commissioner of the Inner Storehouse, reported: "The Court of the Imperial Treasury's old bronze standards, from one cash to ten jin — fifty-one in all — had no consistent weight. The outer treasuries receive gold yearly and must reckon to the finest fraction; standards beginning at one cash are too coarse." He then traced the matter to its roots and cast new standard weights. By the Jingde era Chenggui revised them again, and the system of weights and balances became still more precise. The method took the Han Treatise's black millet as the standard — the width of ten grains as one inch, following the musical foot — (Black millet is black broomcorn millet. The musical foot is derived from the yellow bell pipe. That is, the medium grain of black millet defines the standards of length and weight.) And thereby produced two methods, (The two methods derive li and lei from the foot and millet.) And by the measure-foot derived li, (Measure is the general term for rod and foot. From the musical foot, beginning with millet and completing in the inch: the inch is divided into fen, fen into li, li into hao, hao into si, and si into hu. Ten hu make one si, ten si one hao, ten hao one li, and ten li one fen.) From accumulated millet grains, one derives lei. (Deriving lei from accumulated millet: ten grains make one lei, ten lei one zhu, and twenty-four zhu one liang. All weight pieces are cast in copper.) Using li and lei, he made two scales calibrated at one and a half qian and one liang, each fitted with three hao markers aligned by star graduations. The scale set at one and a half qian provided the standard for a single-balance method. Its beam measured one chi two cun on the musical foot; it weighed one qian, with a six-fen weight and a five-fen pan. From the first hao marker, star-aligned at half a qian, to the tip totaling one and a half qian, it was divided into fifteen fen, each subdivided into ten li; (The first hao below equals half a qian, or fifty li—like the five-jin mark on a fifteen-jin steelyard.) From the middle hao to the tip at one qian, it was divided into ten fen, each into ten li; From the last hao to the tip at half a qian, it was divided into five fen, each into ten li. The scale set at one liang likewise served as the standard for a single balance. Its beam measured one chi four cun on the musical fractional foot; it weighed one and a half qian, with a six-qian weight and a four-qian pan. From the first hao to the tip, twenty-four zhu were laid out; below each, an extra star mark equaled five lei; (Below each zhu another star equals five lei; forty-eight stars thus equal two hundred forty lei, and two thousand four hundred lei total ten liang.) From the middle hao to the tip at five qian, twelve zhu were laid out with five stars, each star equaling two lei; (Twelve zhu laid out for five qian means one zhu equals ten lei, or one hundred twenty lei for half a liang.) From the last hao to the tip at six zhu, each zhu bore ten stars, each star equaling one lei. (Each star equals one lei; sixty lei in all make two and a half qian.) Calibrating against Chunhua coins in regular, cursive, and running imperial scripts, he fixed the true weight at two zhu four lei per qian; two thousand four hundred grains thus yielded fifteen jin as the standard for the larger scale. The method began with accumulated millet as the standard, then derived hu through fen as the foundation of fixed quantities. Thus hu, si, hao, li, millet, lei, and zhu each received a fixed standard at one qian. (That is, each unit was fixed at one qian before the graduated scales were manufactured.) Ten thousand hu make one fen; (Ten thousand hu define one fen; one hundred thousand hu define one qian. Hu is spun silk; fen is the first perceptible division, the point at which quantities become distinguishable.) A thousand si make one fen; (One thousand si make one fen; ten thousand si define one qian.) A hundred hao make one fen; (One hundred hao make one fen; one thousand hao define one qian. Hao is fine down hair. Hu, si, and hao were all made from cut strands of swift-horse tail.) Ten li make one fen; (Ten li make one fen; one hundred li define one qian. Li is yak-tail hair, drawn through red gold to form filaments.) Each unit increases tenfold until one qian is reached. (The tenfold progression fixes standards from ten thousand hu up to one hundred thousand hu.) Millet uses two thousand four hundred grains per liang; (One yue holds twelve hundred grains for twelve zhu; two thousand four hundred grains define one liang. A liang is two yue combined.) Lei uses two hundred forty per liang; (Two hundred forty lei define one liang.) Zhu uses twenty-four per liang; (Lei convert to zhu in succession; two hundred forty lei yield twenty-four zhu per liang. Zhu means that which is distinct.) The scales were thereby completed. Aligning the scale with millet counts, the one-and-a-half-qian scale weighed three hundred sixty grains. Divided into five fen, each fen equaled twenty-four grains. Each fen was further divided into ten li, each li equaling two grains and four-tenths of a grain. (Dividing twenty-four grains among ten li gives two grains per li initially. Divided into forty parts, one lei adds four-tenths more—two grains and four-tenths per li.) Four hao, one si, and six hu together approximate one grain—the finest limits of li and lei. One liang combined twenty-four zhu, weighing two thousand four hundred grains. One hundred grains made one zhu, two hundred forty one lei, two zhu four lei one qian, and two lei four grains one fen. One lei plus two grains weighed five li; six grains two li five hao; three grains one li two hao five si—completing the millet and lei reckoning. The standards were cast in copper with inscriptions marking each weight. When the new system was complete, an edict kept the new weights in the inner palace; forty old scales and sixty old patterns from the Imperial Storehouse were tested against the new standard. Ten of those marked one jin proved light; one marked five jin proved heavy. With standards so defective, the state of weights and balances was plain. Large scales of up to a hundred jin had hooks on frames and rings on the beam; if the ring tilted or the hand pressed down, readings could swing wildly. New patterns were cast so jin and shi derived entirely from millet and lei, allowing no arbitrary adjustment. It was further ordered that large scales must hang from silk cords. Once the load was set, the weigher stood back to read it—no hand could press on the beam. Copper standards were cast again; two thousand four hundred Chunhua coins in three imperial scripts, thirty-three new patterns, and twenty copper plaques were delivered to the Imperial Storehouse. New sets were placed in the inner and outer treasuries and distributed to major capitals in all directions—eleven sets in all. Previously, treasury keepers received annual tribute in gold and silk; the Imperial Storehouse's uncalibrated weights enabled fraud, and many regional officials were ruined for shortfalls. Successions of keepers brought accounting disputes that often dragged on for years. Once the new system was fixed, abuses had no loophole; throughout the court and provinces it was welcomed. (Length, volume, weights, and balances were all manufactured by the Imperial Storehouse for government and public use. At each era change, new standards were issued, each stamped with the reign title. Square, elongated, and octagonal seals marked the regulations and deterred counterfeiting.)
12
宋初，用周顯德欽天曆，建隆二年五月，以其曆推驗稍疏，乃詔司天少監王處訥等別造曆法。 四年四月，新法成，賜號應天曆。 太平興國間，有上言應天曆氣候漸差，詔處訥等重加詳定。 六年，表上新曆，詔付本監集官詳定。 會冬官正吳昭素、徐瑩、董昭吉等各獻新曆，處訥所上曆遂不行。 詔以昭素、瑩、昭吉所獻新曆，遣內臣沈元應集本監官屬、學生參校測驗，考其疏密。 秋官正史端等言：「昭吉曆差。 昭素、瑩二曆以建隆癸亥以來二十四年氣朔驗之，頗為切準。 復對驗二曆，唯昭素曆氣朔稍均，可以行用。」 又詔衛尉少卿元象宗與元應等，再集明曆術吳昭素、劉內真、苗守信、徐瑩、王熙元、董昭吉、魏序及在監官屬史端等精加詳定。 象宗等言：「昭素曆法考驗無差，可以施之永久。」 遂賜號為乾元曆。 應天、乾元二曆皆御製序焉。
Early in the Song, the Zhou Xiande Qintian Calendar was in use. In the fifth month of the second year of Jianlong, because its predictions proved somewhat loose, the court ordered Vice-Director of the Astronomy Bureau Wang Chunei and others to devise a new calendar. In the fourth month of the fourth year, the new system was finished and named the Yingtian Calendar. During Taiping Xingguo, a memorial reported that the Yingtian Calendar's seasonal terms were drifting; Chunei and his colleagues were ordered to revise it thoroughly. In the sixth year they presented a new calendar; the edict referred it to the bureau for collective review. Meanwhile Attendants of the Winter Office Wu Zhaosu, Xu Ying, Dong Zhaoji, and others each submitted new calendars, and Chunei's version was not adopted. The edict ordered that the calendars submitted by Zhaosu, Ying, and Zhaoji be tested by the eunuch Shen Yuanying with bureau officials and students to gauge their accuracy. Attendant Shi Duan of the Autumn Office reported: "Zhaoji's calendar is deficient. Zhaosu's and Ying's calendars, tested against the qi and new moons of the twenty-four years since the guihai year of Jianlong, matched fairly well. On further comparison, only Zhaosu's calendar had the more even qi and new-moon alignment and could be adopted." The court then ordered Vice Chamberlain Yuan Xiangzong and Yuanying to convene again the calendrical experts Wu Zhaosu, Liu Neizhen, Miao Shouxin, Xu Ying, Wang Xiyuan, Dong Zhaoji, Wei Xu, and bureau officers including Shi Duan for a thorough review. Xiangzong and his colleagues reported: "Zhaosu's calendar passed examination without error and may be employed permanently." It was then named the Qianyuan Calendar. Both the Yingtian and Qianyuan calendars received prefaces composed by the emperor.
13
真宗嗣位，命判司天監史序等考驗前法，研覈舊文，取其樞要，編為新曆。 至咸平四年三月，曆成來上，賜號儀天曆。 凡天道運行，皆有常度，曆象之術，古今所同。 蓋變法以從天，隨時而推數，故法有疏密，數有繁簡，雖條例稍殊，而綱目一也。 今以三曆參相考校，以應天為本，乾元、儀天附而注之，法同者不復重出，法殊者備列於後。
When Zhenzong succeeded to the throne, he ordered Shi Xu, director of the Astronomy Bureau, and others to test the earlier methods, study the old texts, extract their essentials, and compile a new calendar. By the third month of the fourth year of Xianping, the completed calendar was presented and named the Yitian Calendar. The motions of heaven all follow constant measures; the arts of calendrics and astronomy have been the same in all ages. Methods are revised to accord with heaven and numbers are derived as the seasons require; hence procedures may be coarse or refined and computation more or less elaborate, yet though the articles differ slightly, the framework is one. The three calendars are here compared and collated, with Yingtian as the base and Qianyuan and Yitian added as annotations; identical procedures are not repeated, divergent ones are fully set out below.
14
建隆應天曆
The Jianlong Yingtian Calendar
15
演紀上元木星甲子，距建隆三年壬戌，歲積四百八十二萬五千五百五十八。 〈（乾元上元甲子距太平興國六年辛巳，積三千五十四萬三千九百七十七。 儀天自上元土星甲子至咸平四年辛丑，積七十一萬六千四百九十七。）〉
From the superior origin of jiazi with Jupiter, counting to the renxu year, third year of Jianlong, the accumulated years are 4,825,558. (For Qianyuan, from the superior origin jiazi to the xinsi year, sixth year of Taiping Xingguo, the accumulation is 30,540,377. For Yitian, from the superior origin jiazi with Saturn to the xinchou year, fourth year of Xianping, the accumulation is 716,497.)
16
步氣朔
Procedure for the qi terms and new moons
17
元法：一萬二。 〈（乾元元率九百四十。 儀天宗法一萬一百。 又總謂之日法。）〉
Origin factor: 12,000. (Qianyuan origin rate: 940. Yitian lineage factor: 10,100. Both are also called the day factor.)
18
歲盈：二十六萬九千三百六十五。 〈（乾元歲周二十一萬四千七百六十四。 儀天歲周三十六萬八千八百九十七。 儀天有周天三百六十五、餘二千四百七十，約餘二千四百四十五； 歲餘五萬二千九百七十、餘二千四百七十。 應天、乾元無此法，後皆倣此。）〉
Tropical year excess: 269,365. (Qianyuan year circuit: 214,764. Yitian year circuit: 368,897. Yitian sets the full circuit of heaven at 365, remainder 2,470, approximated remainder 2,445; and year remainder at 52,970, remainder 2,470. Yingtian and Qianyuan have no such entry; what follows this pattern.)
19
月率：五萬九千七十三。 〈（乾元不置此法。 儀天合率二十九萬八千二百五十九。 又儀天有歲閏一萬九千八百六十二，月閏九千一百一十五、秒六。）〉
Lunar rate: 59,073. (Qianyuan has no such entry. Yitian conjunction rate: 298,259. Yitian also has year intercalation 19,862 and month intercalation 9,115, 6 seconds.)
20
會日：二十九、小餘五千三百七。 〈（乾元朔策二十九、小餘一千五百六十。 儀天會日二十九、小餘五千三百五十七。）〉
Conjunction day: 29; minor remainder 5,307. (Qianyuan new-moon interval: 29; minor remainder 1,560. Yitian conjunction day: 29; minor remainder 5,357.)
21
弦策：七、小餘三千八百二十七、秒六。 〈（乾元小餘一千一百二十五。 儀天小餘三千八百六十四、秒二十七。 策並同。）〉
Crescent interval: 7; minor remainder 3,827; 6 seconds. (Qianyuan minor remainder: 1,125. Yitian minor remainder: 3,864; 27 seconds. The interval constants are the same.)
22
望策：十四、小餘七千六百五十四、秒一十二。 〈（乾元小餘二千二百五十七。 儀天小餘七千七百二十七、秒一十八。 策並同。）〉
Full moon interval: 14; minor remainder 7,654; 12 seconds. (Qianyuan minor remainder: 2,257. Yitian minor remainder: 7,727; 18 seconds. The interval constants are the same.)
23
氣策：十五、小餘二千一百八十五、秒二十四。 〈（乾元小餘六百四十二半。 儀天小餘二千二百七、秒三。 策並同。 又儀天有氣盈四千四百一十四、秒六。）〉
Qi interval: 15; minor remainder 2,185; 24 seconds. (Qianyuan minor remainder: 642½. Yitian minor remainder: 2,207; 3 seconds. The interval constants are the same. Yitian also has qi excess 4,414, 6 seconds.)
24
朔虛分：四千六百九十五。 〈（乾元一千三百八十。 儀天四千七百四十一。）〉
New-moon void fraction: 4,695. (Qianyuan: 1,380. Yitian: 4,741.)
25
沒限：七千八百一十六、秒九。 〈（乾元二千二百九十七半。 儀天七千八百九十二。 又儀天有紀實六十萬六千。）〉
Extinction limit: 7,816; 9 seconds. (Qianyuan: 2,297½. Yitian: 7,892. Yitian also has era solid 606,000.)
26
秒法：二十四。 〈（乾元一百。 儀天秒母三十六。）〉 紀法：六十。 〈（二曆同。）〉 推元積： 〈（乾元、儀天皆謂之求歲積分。）〉 置所求年，以歲盈展之為元積。
Second divisor: 24. (Qianyuan: 100. Yitian second mother: 36.) Era divisor: 60. (The two agree.) To derive the origin accumulation: (Qianyuan and Yitian both term this "seeking the year-accumulation fraction.") Place the year sought and expand it by the year excess to obtain the origin accumulation.
27
求天正所盈之日及分并冬至大小餘：以八十四萬一百六十八去元積，不盡者，半而進位，以元法收為所盈日，不滿為小餘。 日滿六十去之，不滿者，命從甲子，算外，即冬至日辰、大小餘也。 〈（乾元以歲周乘積年為歲積分，以七萬五百六十去之，不盡，以五因，滿元率收為日，不滿為餘日。 儀天以歲周乘積年，進一位，為歲積分； 盈宗法而一為積日，不滿為餘日。 去命並同應天。）〉
To find the days and fractional parts of solar excess at the celestial first month and the major and minor remainders of the winter solstice: remove 840,168 from the origin accumulation; of the remainder, halve and round up, then collect by the origin factor into days of excess—the remainder is the minor remainder. Remove full cycles of sixty days; of the remainder, name from jiazi outward—that gives the day, sequence, and major and minor remainders of the winter solstice. (For Qianyuan, multiply the accumulated years by the year circuit to get the year-accumulation fraction; remove 70,560; of the remainder, multiply by five and, when full, collect by the origin rate into days—the remainder being remainder-days. For Yitian, multiply accumulated years by the year circuit and advance one digit to obtain the year-accumulation fraction; divide by the lineage factor for accumulated days, the remainder being remainder-days. Removal and naming follow Yingtian.)
28
求次氣：以天正冬至大、小餘遍加諸常數，盈六十去之，不盈者，命如前，即得諸氣日辰、大小餘秒也。 〈（乾元置中氣大、小餘，以氣策加之，命如前，即次氣日辰也。 儀天置冬至大、小餘，加氣策及餘秒，秒盈秒母從小餘，盈紀法去之，皆命如前法，各得次氣常日辰及餘秒。）〉
To find the successive qi: to the major and minor remainders of the winter solstice at heaven's first month, add each constant in turn; remove full sixties; name as before—the qi days, sequences, and major, minor, and second remainders follow. (For Qianyuan, set the major and minor remainders of the mid-qi, add the qi interval, name as before—that yields the next qi day and sequence. For Yitian, set the major and minor remainders of the winter solstice, add the qi interval and remainder seconds; when seconds fill the second mother, carry to the minor remainder; remove full era divisors; name by the same method—each successive qi regular day, sequence, and remainder seconds follow.)
29
求天正十一月朔中日： 〈（乾元謂之經朔。 儀天謂之天正合朔。）〉 以月率去元積，不盡者，為天正十一月通餘； 以通餘減七十三萬六百三十五，餘，半而進位，以元法收為日，不滿為分，即得所求天正十一月朔中日及餘秒。 〈（乾元以一萬七千三百六十四去歲積分，不盡為朔餘； 以歲積分為朔積分，又倍五萬二千九百二十，除之，餘以五因，滿元率為日，不滿為分。 儀天以合率去歲積分，不盡為閏餘； 滿宗法為閏日，不滿為餘，以閏日及餘減天正冬至大、小餘，為天正合朔大、小餘； 去命如前，即得合朔日辰、大小餘。）〉
To find the mid-day of the new moon of the eleventh month at heaven's first: (Qianyuan calls this "canonical new moon." Yitian calls it the new moon of heaven's first.) Remove the origin accumulation by the lunar rate; the remainder is the communication remainder for the eleventh month at heaven's first; Subtract that from 730,635; halve the remainder and round up; collect by the origin factor into days, the remainder being parts—the mid-day of the sought new moon of the eleventh month at heaven's first and its remainder seconds follow. (For Qianyuan, remove the year-accumulation fraction by 17,364; the remainder is the new-moon remainder; take the year-accumulation fraction as the new-moon accumulation; double 52,920 and divide; multiply the remainder by five; when full, collect by the origin rate into days, the remainder being parts. For Yitian, remove the year-accumulation fraction by the conjunction rate; the remainder is the intercalation remainder; divide by the lineage factor for intercalation days, the remainder being remainder; subtract these from the major and minor remainders of the winter solstice at heaven's first to obtain the major and minor remainders of the new moon at heaven's first; Remove and name as before—the day, sequence, and major and minor remainders of the conjunction new moon follow.)
30
求次朔望中日： 〈（乾元謂之求弦望經朔。 儀天謂之求次朔。）〉 置朔中日，累加弦策餘秒，即得弦、望及次朔中日。 〈（乾元以弦策加經朔大、小餘，即得次朔經日； 以弦策及餘秒加經朔，得上弦； 再加，得望； 三之，得下弦。）〉
To find the mid-days of the successive new and full moons: (Qianyuan calls this "seeking the crescent, full moon, and canonical new moons." Yitian calls it "seeking the successive new moons.") Set the mid-day of the new moon and add the crescent interval and remainder seconds in turn—the mid-days of the crescent, full moon, and next new moon follow. (For Qianyuan, add the crescent interval to the major and minor remainders of the canonical new moon to obtain the next canonical new-moon day; add the crescent interval and remainder seconds to the canonical new moon for the first crescent; add again for the full moon; thrice it for the last crescent.)
31
求望中月：置朔中月，加半交，盈交正去之，餘為望中月。 〈（二曆不立此法。）〉
To find the mid-month of the full moon: place the mid-month of the new moon, add half the crossing; when the crossing is full, remove by the crossing correction—the remainder is the mid-month of the full moon. (Neither Qianyuan nor Yitian uses this procedure.)
32
求朔弦望入氣：置朔、望中日，各以盈縮準去，不盡者，為入氣日及分。 〈（二曆不立此法。）〉
To find when the new moon, crescent, and full moon enter a qi: place the mid-days of the new and full moons; from each subtract the excess-deficit standard—the remainder is the day and parts of qi-entry. (Neither calendar establishes this method.)
33
推沒日：置有沒之氣小餘， 〈（其小餘七千八百一十六、秒九以上者求之也。）〉 返減元法，餘以八因之，一千九十二、秒一十九半除為沒日，命起氣初，即得沒日辰。 其秒不足者，退一分，加二十四秒，然後除之，四分之三以上者進。 〈（乾元置有沒之氣小餘，在二千二百九十七半以上者，以十五乘之，用減四萬四千七百四十二半，餘以六百四十二半除為沒日。 儀天以秒母通常氣小餘及秒，而從之以減歲周，餘滿五千二百九十七為沒日，去命如前。）〉
To compute extinction days: place the minor remainder of the qi bearing extinction, (This applies when the minor remainder is 7,816 and 9 seconds or more.) Subtract back through the origin factor; multiply the remainder by eight; divide by 1,092 and 19½ seconds for extinction days; name from the start of the qi—that gives the extinction day and its sequence. Where seconds fall short, drop one step and add twenty-four seconds before dividing; round up at three-fourths or more. (For Qianyuan, set the minor remainder of qi with extinction; when it is 2,297½ or more, multiply by fifteen, subtract from 44,742½, and divide the remainder by 642½ for extinction days. For Yitian, adjust the minor remainder and seconds of the regular qi by the second mother, follow to subtract the year circuit—when the remainder reaches 5,297 it is an extinction day; remove and name as before.)
34
推滅日：以冬至大、小餘，遍加朔日中為上位，有分為下位，在四千六百九十五以下者，為有滅之分也。 置有滅之分，進位，以一千五百六十五除為滅日，以滅日加上位，命從甲子，算外，即得月內滅日。 〈（乾元置有滅之經朔小餘，在一千一百八十以下者，以八因之，滿三百六十八除為滅日。 儀天經朔小餘在朔虛法以下者，三因，進位，以朔虛分除為滅日。）〉
To compute quenching days: take the major and minor remainders of the winter solstice and add them throughout to the mid-day of the new moon as the upper position, with fractional parts as the lower; values at 4,695 or below count as quenching portions. Place the quenching portions, round up, and divide by 1,565 for quenching days; add the quenching days to the upper position, name from jiazi outward—that yields the quenching days within the month. (For Qianyuan, set the minor remainder of the canonical new moon bearing quenching; when it is 1,180 or less, multiply by eight and, when full, divide by 368 for quenching days. When Yitian's canonical new-moon minor remainder is below the new-moon void divisor, multiply by three, round up, and divide by the new-moon void parts for quenching days.)
35
求發斂
To find emittance and restraint
36
候策：五、小餘七百二十八、秒二，母二十四。 〈（乾元候數五、小餘一百一十四、秒十二，秒母七十二。 儀天候率五、小餘七百三十五、秒二十五，秒母三十六。）〉
Pentad interval: 5; minor remainder 728; 2 seconds; mother 24. (Qianyuan pentad number: 5; minor remainder 114; 12 seconds; second mother 72. Yitian pentad rate: 5; minor remainder 735; 25 seconds; second mother 36.)
37
卦策：六、小餘八百七十四、秒六。 〈（乾元卦位六、小餘二百五十七，秒母六十。 儀天卦率六、小餘八百八十三、秒二十。）〉
Hexagram interval: 6; minor remainder 874; 6 seconds. (Qianyuan hexagram position: 6; minor remainder 257; second mother 60. Yitian hexagram rate: 6; minor remainder 883; 20 seconds.)
38
土王策：十二、小餘一千七百四十八、秒一十二。 〈（乾元策三、小餘一百二十八半，秒母一百一十。 儀天土王率三、小餘四百四十、秒五，秒母同上。）〉
Earth-king interval: 12; minor remainder 1,748; 12 seconds. (Qianyuan interval: 3; minor remainder 128½; second mother 110. Yitian earth-king rate: 3; minor remainder 440; 5 seconds; second mother as above.)
39
辰數：八百三十三半。 〈（乾元辰法二百四十五，辰率千五百二十。）〉 刻法：一百。 〈（乾元一百四十七。 儀天刻三百。）〉
Double-hour number: 833½. (Qianyuan double-hour divisor: 245; double-hour rate: 1,520.) Quarter-hour divisor: 100. (Qianyuan: 147. Yitian quarters: 300.)
40
求七十二候：各因諸氣大、小餘秒命之，即初候日也； 各以候策加之，得次候日； 又加之，得末候日。 〈（二曆同法。）〉
To find the seventy-two pentads: from each qi's major and minor remainders and seconds derive the day—that is the first-pentad day; add the pentad interval to obtain the next-pentad day; add again for the last-pentad day. (The two calendars use the same method.)
41
求六十四卦：各置諸中氣大、小餘秒命之，即公卦用事日； 以卦策加之，得次卦用事日； 又加之，得終卦用事日。 十有二節之初，皆諸侯外卦用事日。 〈（二曆同法。）〉
To find the sixty-four hexagrams: set the major and minor remainders and seconds of each mid-qi and name them—that is the duke-hexagram governing day; add the hexagram interval to obtain the next hexagram governing day; add again for the final hexagram governing day. At the opening of each of the twelve nodes, all are the outer-hexagram governing days of the feudal lords. (The two calendars use the same method.)
42
求五行用事：各因四立大、小餘秒命之，即春木、夏火、秋金、冬水首用事日； 以土王策加四季之節大、小餘秒，命從甲子，算外。 即其月土王用事日。 〈（乾元以土王策減四季中氣大、小餘。 儀天以土王率加四季大、小餘。）〉
To find the governing days of the five phases: from the major and minor remainders and seconds of the four establishments derive the days—spring wood, summer fire, autumn metal, and winter water each have their first governing day; add the earth-king interval to the major and minor remainders and seconds of the seasonal nodes, name from jiazi outward. That yields the earth-king governing day of the month. (For Qianyuan, subtract the earth-king interval from the major and minor remainders of the four seasons' mid-qi. For Yitian, add the earth-king rate to the major and minor remainders of the four seasons.)
43
求二十四氣加時辰刻： 〈（乾元謂之辰刻。 儀天謂之求時。）〉 各置小餘，以辰數除之為時數，不滿，百收為刻分，命起子正，算外，即所在。 〈（乾元時數同，其不盡，以五因之，以刻法除為刻分。 儀天以三因小餘，以辰率除之為時數，不盡者，滿刻率除為刻，餘為分。）〉
To find the hour and quarter added to each of the twenty-four qi: (Qianyuan calls this "double-hour quarters." Yitian calls it "seeking the hour.") Place the minor remainder and divide by the double-hour number for the hour count; collect the remainder by hundreds into quarter-parts; name from midnight upright outward—that is the position. (For Qianyuan the hour count is the same; of the remainder, multiply by five and divide by the quarter-hour divisor for quarter-parts. For Yitian, multiply the minor remainder by three and divide by the double-hour rate for the hour count; of the remainder, divide by the quarter rate for quarters, the remainder being parts.)
44
常數月中節四正卦初候中候末候始卦中卦末卦冬至十一月中坎初六蚯蚓結麋角解水泉動公中孚辟復侯屯內小寒十二月節坎九二鴈北鄉鵲始巢雉始雊侯屯外大夫謙卿睽大寒十二月中坎六三始乳鷙鳥厲疾水澤腹堅公升辟臨侯小過內立春正月節坎六四東風解凍蟄蟲始振魚上冰侯小過外大夫蒙卿益雨水正月中坎九五獺祭魚鴻鴈來草木萌動公漸辟泰侯需內驚蟄二月節坎上六桃始華倉庚鳴鷹化為鳩侯需外大夫隨卿晉春分二月中震初九玄鳥至雷乃發聲始電公解辟大壯侯豫內清明三月節震六二桐始華田鼠化鴽虹始見侯豫外大夫訟卿蠱穀雨三月中震六三萍始生鳴鳩拂羽戴勝降桑公革辟夬侯旅內立夏四月節震九四螻蟈鳴蚯蚓出王瓜生侯旅外大夫師卿比小滿四月中震六五苦菜秀靡草死小暑至公小畜辟乾侯大有內芒種五月節震上六螗螂生鵙始鳴反舌無聲侯大有外大夫家人卿井夏至五月中離初九鹿角解蜩始鳴半夏生公咸辟姤侯鼎內小暑六月節離六二溫風至蟋蟀居壁鷹乃學習侯鼎外大夫豐卿渙大暑六月中離九三腐草為螢土潤溽暑大雨時行公履辟遯侯恆內立秋七月節離九四涼風至白露降寒蟬鳴侯恆外大夫節卿同人處暑七月中離六五鷹乃祭鳥天地始肅禾乃登公損辟否侯巽內白露八月節離上九鴻鴈來玄鳥歸鳥養羞侯巽外大夫萃卿大畜秋分八月中兌初九雷乃收聲蟄蟲壞戶水始涸公賁辟觀侯歸妹內寒露九月節兌九二鴻鴈來賓雀入水為蛤菊有黃花侯歸妹外大夫無妄卿明夷霜降九月中兌六三豺乃祭獸草木黃落蟄蟲咸俯公困辟剝侯艮內立冬十月節兌九四水始冰地始凍雉入大水為蜃侯艮外大夫既濟卿噬嗑小雪十月中兌九三虹藏不見天氣上騰地氣下降閉塞成冬公大過辟坤侯未濟內大雪十一月節兌上六鶡鳥不鳴虎始交荔挺出侯未濟外大夫蹇卿頤二曆同
Yingtian constants table: for each of the twenty-four qi, the mid-month node and four upright hexagram line, three pentads, and opening, middle, and closing hexagrams with noble ranks (duke, marquis, feudal lord, grandee, minister), from Winter Solstice through Greater Snow; the two calendars agree.
45
求日躔
To find the sun's lodge motion
46
天總：七十三萬六百五十八、秒六十四。 〈（乾元軌率二十一萬四千七十七、秒七千五百一十、小分七十。 儀天乾元數三百六十八萬九千八十八、秒九十九。）〉
Heaven total: 730,658; 64 seconds. (Qianyuan orbit rate: 214,077; 7,510 seconds; 70 minor parts. Yitian Qianyuan number: 3,689,088; 99 seconds.)
47
天度：三百六十五、小餘二千五百六十三，微八十八。 〈（乾元周天三百六十五度、小餘二千五百六十三。 儀天乾則三百六十五度、小餘二千五百八十八、秒九十九。 應天諸法皆在天總數中。 乾元、儀天各立其法。 乾元周天策一百七萬三千八百五十三、秒七千五百五十三半，會周一萬七千三百六十四，會餘二十一萬四千七百六十四，天中一百八十二、六千二百八十一半。 儀天歲差一百一十八、秒九十九，一象度九十一、餘三千一百四十二、秒五十，盈初縮末限分八十九萬七千六百九十九、秒五十，限日八十八、餘八千八百九十九、秒五十，縮初盈末限分九十四萬六千七百八十五、秒十五，限日九十三、餘七千四百八十五、秒五十，盈縮積二萬四千五百四十三，進退率一千八百三十六，秒母一百。）〉
Heaven degrees: 365; minor remainder 2,563; 88 micro-parts. (Qianyuan full circuit of heaven: 365 degrees; minor remainder 2,563. Yitian Qian rule: 365 degrees; minor remainder 2,588; 99 seconds. All Yingtian methods lie within the heaven-total figure. Qianyuan and Yitian each set out their own methods. Qianyuan: full-heaven interval 1,073,853; 7,553½ seconds; conjunction circuit 17,364; conjunction remainder 214,764; heaven mid 182,6281½. Yitian year difference 118, 99 seconds; one image degree 91, remainder 3,142, 50 seconds; excess-opening deficit-closing limit 899,769.50; limit days 88, remainder 8,899.50; deficit-opening excess-closing limit 946,785.15; limit days 93, remainder 7,485.50; excess-deficit accumulation 24,543; advance-retreat rate 1,836; second mother 100.)
48
常氣盈縮準常數定日損益準先後積冬至十四五千四十五秒十五十五二千一百八十五秒十五十四五千四十五秒十五損六十四後二十小寒一十九一千二百八十六三十四千三百七十一十四六千二百三十六秒十五損六十九先五百二十九大寒四十三八千七百五秒二十一四十五六千五百五十六秒二十一十四七千四百二十五秒十五損七十六先九百七十五立春五十八七千三百二十半六十八千七百四十二半十四八千六百一十六秒十五損八十二先一千三百三十五雨水七十三七千三百六十三七十六九百二十六十五四十二秒十五損八十九先一千六百六驚蟄八十八八千八百三十四太九十一三千一百一十一太十五一千四百七十秒十五損九十七先一千七百七十一春分一百四一千三百三十三九一百六五千二百九十七秒九十五二千八百九十九秒十五益九十七先一千八百一十九清明一百十九六千六十一空一百二十一七千四百八十三空十五四千三百二十八秒十五益八十九先一千七百八十穀雨一百三十五一千八百一十五十五一百三十六六千六百六十八秒十五十五五千七百五十七秒十五益八十三先一千六百五立夏一百五十八千七百六十五六一百五十二一千八百五十二秒六十五六千九百四十七秒十五益七十八先一千三百五十小滿一百六十六六千八百九十七二十一一百六十七四千三十一秒二十十五八千一百三十六秒十五益七十二先九百九十五芒種一百八十二六千二百二十三半一百八十二六千二百三十三半十五九千三百七十二秒十五益六十六先五百四十一夏至一百九十八五千五百四十九三一百九十七八千四百九秒三十五九千三百二十七秒十五損六十五先五小暑二百十四三千六百八十三十八二百十三五百九十二太十五八千一百三十六秒十五損七十二後五百四十九大暑二百三十六百二十九九二百二十八二千七百七十八秒九十五八千一百三十六秒十五損七十七後九百八十五立秋二百四十五六千三百八十六空二百四十三四千九百六十四空十五五千七百五十六秒十五損八十三後一千三百四十六處暑二百六十一七百一十二十五二百五十八七千二百四十九秒十五十五四千三百二十八秒十五損八十九後一千六百一十一白露二百七十六三千六百一十二六二百五十八七千一百四十九秒十五十五四千三百二十八秒十五損九十七後一千七百八十秋分二百九十一五千八十三二十一二百八十九七千五百十八秒五十一十五益九十七後一千八百三十一寒露三百六五千一百二十六十二二百四三千七百四半十五四十二秒十五益八十九後一千七百八十六霜降二百二十一三千四百四十一三三百一十九五千八百九十秒三十四八千六百一十六秒三益八十二後一千六百二十一立冬三百三十六一千六百六十四十六三百三十四八千七十五太十四七千四百二十五秒十五益七十五後一千三百五十七小雪三百五十七千四百十九三百五十三百九十九秒十五十四六千二百三十六秒十五益七十後九百八十八大雪三百六十五二千四百四十五三百六十五二千四百四十五十四五千四十五秒十五益六十四後五百五十
Yingtian ready table (regular qi, excess-deficit standard, regular number, fixed day, increase-decrease standard, before-after accumulation): Regular qi; surplus-deficit standard; regular number; fixed day; augments-diminishes standard; prior-later accumulation - Winter Solstice, the cited text the cited text the cited text diminishes the cited text later the cited text; Lesser Cold, the cited text the cited text diminishes the cited text prior the cited text; Greater Cold, the cited text the cited text diminishes the cited text prior the cited text; Start of Spring, the cited text the cited text diminishes the cited text prior the cited text; Rain Water, the cited text the cited text diminishes the cited text prior the cited text; Awakening of Insects, the cited text surplus the cited text surplus the cited text diminishes the cited text prior the cited text; Spring Equinox, the cited text the cited text augments the cited text prior the cited text; Clear Bright, the cited text empty the cited text empty the cited text augments the cited text prior the cited text; Grain Rain, the cited text the cited text augments the cited text prior the cited text; Start of Summer, the cited text the cited text augments the cited text prior the cited text; Lesser Fullness, the cited text the cited text augments the cited text prior the cited text; Grain in Ear, the cited text the cited text augments the cited text prior the cited text; Summer Solstice, the cited text the cited text diminishes the cited text prior the cited text; Lesser Heat, the cited text the cited text surplus the cited text diminishes the cited text later the cited text; Greater Heat, the cited text the cited text diminishes the cited text later the cited text; Start of Autumn, the cited text empty the cited text empty the cited text diminishes the cited text later the cited text; End of Heat, the cited text the cited text diminishes the cited text later the cited text; White Dew, the cited text the cited text diminishes the cited text later the cited text; Autumn Equinox, the cited text the cited text augments the cited text later the cited text; Cold Dew, the cited text the cited text augments the cited text later the cited text; Frost Descent, the cited text the cited text augments the cited text later the cited text; Start of Winter, the cited text surplus the cited text augments the cited text later the cited text; Lesser Snow, the cited text the cited text augments the cited text later the cited text; Greater Snow, the cited text the cited text augments the cited text later the cited text
49
乾元二十四氣日躔陰陽度
Qianyuan: twenty-four qi solar lodge motion yin-yang degrees
50
陰陽分陰陽度損益率陰陽差冬至陽分二千二百七十六卷陽度空益一百七十陽差空小寒陽分一千七百八十四卷陽初度二千二百七十六卷益一百三十三卷陽差一百七十大寒陽分一千三百四十四卷陽一度一千一百二十益一百一陽差三百三立春陽分九百五十六陽一度二千四百六十四卷益七十一陽差四百四雨水陽分五百八十一陽二度四百八十益四十三陽差四百七十五卷驚蟄陽分二百九十三卷陽二度一千六十一益十四陽差五百一十八卷春分陽分一百九十四卷陽二度一千二百五十五卷損十四陽差五百三十二清明陽分五百八十一陽二度一千六十一損四十三陽差五百一十八卷穀雨陽分九百五十六卷陽二度四百八十損七十一陽差四百七十五卷立夏陽分一千三百四十四卷陽一度二千四百六十四卷損一百一陽差四百四小滿陽分一千七百八十四卷陽一度一千一百二十損一百三十三陽差三百三芒種陽分二千二百七十六卷陽初度二千二百七十六卷損一百七十陽差一百七十夏至陰分二千二百七十六卷陰度空益一百七十陰差空小暑陰分一千七百八十四卷陰度二千二百七十六卷益一百三十三卷陰差一百七十大暑陰分一千三百四十四卷陰一度一千一百二十益一百一陰差三百三立秋陰分九百五十六陰一度二千四百六十四卷益七十一陰差四百四處暑陰分五百八十一陰二度四百八十益四十三陰差四百七十五卷白露陰分一百九十四卷陰二度一千六十一益十四陰差五百一十八卷秋分陰分一百九十四卷陰二度一千二百五十五卷損十四陰差五百二十一寒露陰分五百八十一陰二度一千六十一損四十三陰差五百一十八卷霜降陰分九百五十六卷陰二度四百八十損七十一陰差四百七十五卷立冬陰分一千三百四十四卷陰一度二千四百六十四卷損百一陰差四百四小雪陰分一千七百八十四卷陰一度一千一百二十損一百三十三陰差三百三大雪陰分二千二百七十六卷陰初度二千二百七十六卷損一百七十陰差一百七十
Qianyuan ready table: Yin-yang parts, yin-yang degrees, increase-decrease rates, and yin-yang differences for the twenty-four qi—from Winter Solstice through Greater Snow—tabulating yang and yin portions and degrees, rates of increase and decrease, and accumulated yin-yang differences for solar lodge motion.
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〈（應天、乾元二曆，以常氣求其陰陽差，故有二十四氣立成。 儀天以盈縮定分、四限直求二十四氣陰陽差，乃更不制二十四氣差法。）〉
(Yingtian and Qianyuan derive yin-yang difference from regular qi, hence their ready tables for the twenty-four qi. Yitian obtains the twenty-four qi yin-yang difference directly from fixed excess-deficit parts and the four limits, and therefore no longer sets out a separate twenty-four-qi difference method.)
52
求日躔損益盈縮度： 〈（乾元謂之求每日陰陽差。 儀天謂之求入盈縮分先後定數。）〉 各置定日及分，以冬至常數相減，百收，通為分，自雨水後十六為法，自霜降後十五為法。 除分為氣中率，二相減，為合差； 半之，加減率為初、末率。 〈（後多者，減為初、加為末； 後少者，加為初、減為末。）〉 又法，以除合差，為日差； 〈（後少者，日損初率； 後多者，日益初率。）〉 為每日日躔損益率； 累積其數，為盈縮度分。 〈（乾元各置氣數，以一百二十乘之，以一千八百二十六除之，所得為平行率； 相減，為合差； 初、末並如應天。 儀天以宗法乘盈縮積，以其限分除之，為限率分； 倍之，為未限平率； 日分乘之，亦以限分除之，為日差； 半之，加減初、末限平率，在初者減初加末，在末者減末加初，為末定率； 乃以日差累加減限初定率，初限以減、末限以加，為每日盈縮定分； 各隨其限盈加縮減其下先後數，為每日先後定數； 冬至後積盈為先，在縮減之； 夏至後，積縮為後，在盈減之。 其進退率、昇平積準此求之，即各得其限每日進退率、昇平積也。）〉
To find the sun's lodge motion increase-decrease and excess-deficit degrees: (Qianyuan calls this "seeking the daily yin-yang difference." Yitian calls it "seeking the entering excess-deficit parts and before-after fixed numbers.") Place the fixed day and parts, subtract mutually from the winter solstice regular number, collect by hundreds and combine as parts; from Rain Water onward use 16 as divisor, from Frost Descent onward use 15. divide the parts for the qi mid-rate; subtract the two to get the combined difference; halve it and add or subtract from the rate for the opening and closing rates. (When the later rate is greater, subtract for the opening and add for the closing; when the later is lesser, add for the opening and subtract for the closing.) By another method, divide the combined difference to obtain the day difference; (When the later is lesser, decrease the opening rate daily; when the later is greater, increase the opening rate daily.) that gives the daily sun's lodge motion increase-decrease rate; accumulate the numbers for the excess-deficit degree parts. (For Qianyuan, set each qi number, multiply by 120, divide by 1,826—the quotient is the parallel rate; subtract one from the other to get the combined difference; the opening and closing rates follow Yingtian likewise. For Yitian, multiply the excess-deficit accumulation by the lineage factor and divide by the limit parts to obtain limit-rate parts; double this for the pre-limit mean rate; multiply by day-parts and again divide by limit-parts to get the day difference; halve it and add or subtract from the opening and closing limit mean rates—in the opening limit subtract the opening and add the closing, in the closing limit subtract the closing and add the opening—for the closing fixed rate; then use the day difference to add or subtract cumulatively from the limit opening fixed rate—decrease in the opening limit and increase in the closing limit—for the daily excess-deficit fixed parts; in each limit, add for excess and subtract for deficit from the before-after numbers below, yielding the daily before-after fixed numbers; after the winter solstice, accumulated excess counts as earlier and is reduced where there is deficit; after the summer solstice, accumulated deficit counts as later and is reduced where there is excess. advance-retreat rates and ascent-level accumulations are found by the same procedure, each limit yielding its daily advance-retreat rate and ascent-level accumulation.)
53
求日躔先後定數： 〈（乾元謂之求入氣、求弦望氣入、求日躔陰陽差。）〉 各以朔、弦、望入氣日及減本氣定日及分秒通之，下以損益率展，以元法為分，損減益加次氣下先後積為定數。 〈（乾元以其月氣節減經朔大、小餘，即得入氣日及分； 又以弦策累加天正朔日入氣大、小餘，滿氣策去之，即得弦、望經朔入氣日及分； 以其日損益率乘入氣日餘分，所得，用損益其日陰陽差為定數。 儀天法見上。 又儀天有求四正節定日，去冬、夏二至盈縮之中，先後皆空，以常為定； 其春、秋二分盈縮之極，以一百乘盈縮積，滿宗法為日，先減後加，去命如前，各得定日。 若求朔、弦、望盈縮限日，以天正閏日及餘減縮末限日及分，餘為天正十一月經朔加時入限日及餘； 以弦策累加之，即得弦、望及後朔初、末限日； 各置入限日及餘，以其日進退率乘之，如宗法而一，所得，以進退其日下昇平積，即各為定數。）〉
To find the sun's lodge motion before-after fixed numbers: (Qianyuan calls these "seeking qi-entry," "seeking crescent- and full-moon qi-entry," and "seeking the sun's lodge motion yin-yang difference.") For each new moon, crescent, and full moon, combine the qi-entry day with the fixed day and fractional parts and seconds of the parent qi; spread downward by the increase-decrease rate with the origin method as divisor; decrease where there is subtract and increase where there is add to the before-after accumulation under the next qi for the fixed numbers. (For Qianyuan, subtract the canonical new-moon major and minor remainders from that month's qi node to obtain the qi-entry day and parts; then add the crescent interval cumulatively to the heaven's-first new-moon qi-entry major and minor remainders, removing full qi intervals, to obtain the canonical new-moon qi-entry days and parts for crescent and full moon; multiply the qi-entry day remainder by that day's increase-decrease rate and apply the product to increase or decrease that day's yin-yang difference for the fixed number. The Yitian method appears above. Yitian also has a method to find fixed days for the four cardinal nodes: at the winter and summer solstices the before-after values are all void, taken as the regular fixed value; at the spring and autumn equinoxes, the extremes of excess and deficit: multiply the excess-deficit accumulation by 100; when it fills the lineage factor, that many days—first subtract then add, remove and name as before—to obtain each fixed day. to find the excess-deficit limit days for new moon, crescent, and full moon: subtract the deficit-closing limit day and parts from the heaven's-first intercalation day and remainder; the remainder is the limit-entry day and remainder at canonical new-moon hour-addition for the eleventh month of heaven's first; add the crescent interval cumulatively to obtain the limit days for crescent, full moon, and later new moons at opening and closing limits; place each limit-entry day and remainder, multiply by that day's advance-retreat rate and divide by the lineage factor, and apply the quotient to advance or retreat the ascent-level accumulation under that day—the fixed numbers follow.)
54
赤道宿度斗：二十六。 牛：八。 女：十二。 虛：十。 〈（及分。）〉 危：十七。 室：十六。 壁：九。 〈（二曆同。）〉
Equatorial lodge degrees—Dipper: 26. Ox: 8. Girl: 12. Void: 10. (and fractional parts.) Rooftop: 17. Room: 16. Wall: 9. (The two calendars agree.)
55
北方七宿九十八度。 虛分二千五百六十三，秒一十九。 〈（乾元七千五百三十五、秒二十五。 儀天二千五百八十八、秒九十九。）〉
Northern Seven Lodges: 98 degrees. Void: 2,563 parts, 19 seconds. (Qianyuan: 7,535, 25 seconds. Yitian: 2,588, 99 seconds.)
56
奎：十六。 婁：十二。 胃：十四。 昴：十一。 畢：十七。 觜：一。 參：十。 西方七宿八十一度。 〈（二曆同。）〉
Straddler: 16. Bond: 12. Stomach: 14. Hairy Head: 11. Net: 17. Turtle Beak: 1. Three Stars: 10. Western Seven Lodges: 81 degrees. (The two calendars agree.)
57
井：三十三。 鬼：三。 柳：十五。 星：七。 張：十八。 翌：十八。 軫：十七。 南方七宿一百一十一度。 〈（二曆同）〉
Well: 33. Ghost: 3. Willow: 15. Star: 7. Extended Net: 18. Wings: 18. Chariot Crossbar: 17. Southern Seven Lodges: 111 degrees. (The two calendars agree)
58
角：十二。 亢：九。 氐：十五。 房：五。 心：五。 尾：十八。 箕：十一。 東方七宿七十五度。 〈（二曆同。）〉
Horn: 12. Neck: 9. Root: 15. Chamber: 5. Heart: 5. Tail: 18. Winnowing Basket: 11. Eastern Seven Lodges: 75 degrees. (The two calendars agree.)
59
〈（又儀天云：「前皆赤道度，自古以來，累依天儀測定，用為常準。 赤道者，天中紘帶，儀極攸憑，以格黃道也。」）〉
(Yitian also states: "The foregoing are all equatorial degrees; since antiquity they have been fixed by successive measurements with the celestial armillary and serve as the regular standard. The equator is the girdle band at heaven's center, the pole of the armillary sphere by which the ecliptic is framed.")
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求赤道變黃道度： 〈（乾元謂之求黃道度。 儀天謂之推黃道度。）〉 準二至赤道日躔宿次。 前後五度為限，初限十二，每限減半，終九限減盡。 距二立之宿，減一度少強，又從盡起限，每限增半，九限終於十二。 距二分之宿，皆乘限度，身外除一，餘滿百為度分，命曰黃赤道差。 二至前後各九限，以差為減； 二分前後各九限，以差為加。 各加減赤道度為黃道度，有餘分就近收為太、半、少之數。 〈（乾元初率九，每限減一，末率一。 儀天初數一百七，每限減一十，末率二十七，其餘限數加減並同應天。）〉
To convert equatorial degrees to ecliptic degrees: (Qianyuan calls this "seeking ecliptic degrees." Yitian calls it "deducing ecliptic degrees.") Take the solar lodge at the two solstices on the equator as the standard. Within five degrees on either side, begin at 12 and halve each successive band until after nine bands the reduction is complete. From the lodges of the two Establishment qi, subtract one degree slightly strong; then begin again from zero, adding half per band through nine bands until the tally is 12. From the lodges of the two equinoxes, multiply all by the limit value, divide outside the body by 1; of the remainder, what fills 100 becomes degree-parts—called the ecliptic-equatorial difference. Within nine bands on either side of each solstice, subtract the difference; Within nine bands on either side of each equinox, add the difference. Add or subtract from equatorial degrees to obtain ecliptic degrees; round remaining parts to the nearest major, half, or minor fraction. (Qianyuan: initial rate 9, decrease by 1 per band, closing rate 1. Yitian: initial number 107, decrease by 10 per band, closing rate 27; the remaining limit additions and subtractions all follow Yingtian.)
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黃道宿度
Ecliptic lodge degrees
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斗：二十三度半。 牛：七度半。 〈（二曆同。）〉 女：十一度太。 〈（二曆並十一度半。）〉 虛：十度少強。 〈（二千五百六十三、秒十九。 乾元無分。 儀天六十三分，九十九秒。）〉 危：十七度少。 〈（乾元同。 儀天十七度太。）〉 室：十六度太。 壁：十度。 〈（乾元九度太。 儀天同。）〉
Dipper: 23.5 degrees. Ox: 7.5 degrees. (The two calendars agree.) Girl: 11 degrees major. (Both calendars give 11.5 degrees.) Void: 10 degrees, slightly strong. (2,563 parts, 19 seconds. Qianyuan has no fractional parts. Yitian: 63 parts, 99 seconds.) Rooftop: 17 degrees minor. (Qianyuan agrees. Yitian: 17 degrees major.) Room: 16 degrees major. Wall: 10 degrees. (Qianyuan: 9 degrees major. Yitian agrees.)
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北方七宿九十七度二千五百六十三、秒十九。 〈（乾元九十六度半、儀天九十七度半、六十三、秒九十九。）〉
Northern Seven Lodges: 97 degrees, 2,563 parts, 19 seconds. (Qianyuan: 96.5 degrees, Yitian: 97.5 degrees, 63 parts, 99 seconds.)
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奎：十七度半。 〈（二曆同。）〉 婁：一十二度太。 〈（乾元十三度。 儀天同。）〉 胃：十四度少。 〈（二曆並十四度太。）〉 昴：十一度。 〈（二曆同。）〉 畢：十六度半。 〈（乾元同。 儀天十六度少。）〉 觜：一度。 參：九度少。 〈（二曆並同。）〉 西方七宿八十二度少。 〈（乾元八十三度。 儀天八十二度半。）〉
Straddler: 17.5 degrees. (The two calendars agree.) Bond: 12 degrees major. (Qianyuan: 13 degrees. Yitian agrees.) Stomach: 14 degrees minor. (Both calendars give 14 degrees major.) Hairy Head: 11 degrees. (The two calendars agree.) Net: 16.5 degrees. (Qianyuan agrees. Yitian: 16 degrees minor.) Turtle Beak: 1 degree. Three Stars: 9 degrees minor. (Both calendars agree.) Western Seven Lodges: 82 degrees minor. (Qianyuan: 83 degrees. Yitian: 82.5 degrees.)
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井：三十度。 鬼：二度太。 〈（二曆並同。）〉 柳：十四度半。 〈（乾元、儀天十四度少。）〉 星：七度。 〈（乾元、儀天並六度太。）〉 張：十八度少。 〈（乾元同。 儀天十八度太。）〉 翼：十九度少。 〈（乾元十九度。 儀天同。）〉 軫：十八度太。 〈（二曆同。）〉 南方七宿一百一十度半。 〈（乾元一百九度太。 儀天同。）〉 角：十三度。 亢：九度半。 〈（二曆並同。）〉 氐：十二度少。 〈（乾元、儀天並十五度半。）〉 房：五度。 〈（二曆同。）〉 心：五度。 〈（乾元同。 儀天四度太。）〉 尾：十七度少。 〈（乾元同。 儀天十七度。）〉 箕：十度 〈（乾元十度太。 儀天十度。）〉 東方七宿七十五度少。 〈（乾元七十六度。 儀天七十四度太。）〉
Well: 30 degrees. Ghost: 2 degrees major. (Both calendars agree.) Willow: 14.5 degrees. (Qianyuan and Yitian: 14 degrees minor.) Star: 7 degrees. (Qianyuan and Yitian both give 6 degrees major.) Extended Net: 18 degrees minor. (Qianyuan agrees. Yitian: 18 degrees major.) Wings: 19 degrees minor. (Qianyuan: 19 degrees. Yitian agrees.) Chariot Crossbar: 18 degrees major. (The two calendars agree.) Southern Seven Lodges: 110.5 degrees. (Qianyuan: 109 degrees major. Yitian agrees.) Horn: 13 degrees. Neck: 9.5 degrees. (Both calendars agree.) Root: 12 degrees minor. (Qianyuan and Yitian both give 15.5 degrees.) Chamber: 5 degrees. (The two calendars agree.) Heart: 5 degrees. (Qianyuan agrees. Yitian: 4 degrees major.) Tail: 17 degrees minor. (Qianyuan agrees. Yitian: 17 degrees.) Winnowing Basket: 10 degrees (Qianyuan: 10 degrees major. Yitian: 10 degrees.) Eastern Seven Lodges: 75 degrees minor. (Qianyuan: 76 degrees. Yitian: 74 degrees major.)
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求赤道日度： 〈（儀天謂之推日度。）〉 以天總除元積，為總數； 不盡，半而進位，又以一百收總數從之，以元法收為度，不滿為分秒，命起赤道虛宿四度分。 〈（乾元以軌率去歲積分，餘以五因之，滿軌率收為度，不滿，退除為分，餘同。 儀天以乾數去歲積分，宗法收為度，命起盧宿二度，餘同應天。 又以一象度及餘秒累加之，滿赤道宿度即去之，各得四正，即初日加時赤道日度也。）〉 求黃道日度：置冬至赤道日躔宿度，以所入限數乘之，所得，身外除一，滿百為度，不滿為分，用減赤道日度，為冬至加時黃道日度及分。 〈（乾元、儀天亦如其法。 乾元即以八十四，儀天以一百一除為度，餘同應天。）〉
To find equatorial solar degrees: (Yitian calls this "deducing solar degrees.") Divide the origin accumulation by the heavenly total to obtain the aggregate number; Of the remainder, halve and round up; then collect the aggregate by 100 and add it; collect into degrees by the origin factor—the remainder being parts and seconds—and name from Void lodge at 4 degrees on the equator. (For Qianyuan, remove the year-accumulation fraction by the orbital rate; multiply the remainder by five; when full, collect by the orbital rate into degrees—if not full, reduce to parts—the rest as above. For Yitian, remove the year-accumulation fraction by the Qian factor and collect into degrees by the lineage factor, naming from Hut lodge at 2 degrees; the rest follows Yingtian. Then accumulate by one image-degree and remainder seconds; when full, remove by equatorial lodge degree to obtain each of the four uprights—the equatorial solar degree at the initial day's added hour.) To find the ecliptic solar degree: set the winter solstice equatorial solar position, multiply by the applicable limit number, divide the product (discarding one place outside), take hundreds as degrees and the remainder as parts, subtract from the equatorial degree — yielding the ecliptic solar degree and parts at the winter solstice hour. (The Qianyuan and Yitian calendars follow the same method. Qianyuan divides by 84 and Yitian by 101 to obtain degrees; otherwise the procedure matches Yingtian.)
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求朔望常日月： 〈（乾元謂之求黃道平朔日度。）〉 置朔、望日躔先後定數，進一位，倍之，身外除之，以元法收為度分，先加後減朔望中日、月，為朔望中常日、月度分； 用加冬至黃道之宿，命如前，即得朔望常日、月所在。 〈（乾元置會週一萬七千三百六十，以距十一月後來月數乘之，所得，減去朔餘，加會餘而半之，以二百九十四收為度，不盡，退除為分。 儀天法在後。 乾元又有求黃道加時朔日度，置平朔日，以日躔陽加陰減之，又以冬至黃道日度加而命之，即其朔加時黃道日度及分也。 若求望日度者，以半朔策加之，即得望日度及分也。 用陽度，即依本術。）〉
To find the mean solar and lunar positions at new and full moon: (The Qianyuan calendar calls this finding the mean ecliptic solar degree at new moon.) Set the new- and full-moon solar before-after fixed numbers, advance one place and double, divide accordingly, and reduce by the origin factor to degrees and parts; add or subtract from the mid-month sun and moon to obtain the mean solar and lunar positions at new and full moon; Add this to the winter solstice ecliptic lodge and count forward as before to locate the mean sun and moon at new and full moon. (In Qianyuan, set the conjunction circuit at 17,360, multiply by months elapsed since the eleventh month, subtract the new-moon remainder, add half the conjunction remainder, and reduce by 294 for degrees; the remainder becomes parts. The Yitian method is given below. Qianyuan also provides the hour-added ecliptic solar degree at new moon: take the mean new-moon day, apply solar motion (adding in yang and subtracting in yin), add the winter solstice ecliptic degree, and count forward — yielding the ecliptic degree and parts at the new-moon hour. For the full-moon solar degree, add half the synodic interval to obtain the degree and parts at full moon. When using yang degrees, follow the base procedure.)
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每日加時黃道日度： 〈（乾元謂之每日行分。）〉 以定朔、望日所在相減，餘以距後日數除之，為平行分； 二行分相減，為合差； 半之，加減平行分，為初行分； 〈（後平行多，減為初； 後平行少，加為初。）〉 以距後日數除合差，為日差； 後少者損，後多者益，為每日行分； 累加朔、望日，即得所求。 〈（乾元同。 儀天不立此法。 又儀天有求次正定日加時黃道日度，置歲差，以限數乘之，退一位，滿一百一為差秒及小分，再析之，乃以加一象度，所得，累加冬至黃道日，滿黃道宿次去之，各得四正，即加時黃道日度也。 若求四正定日夜半黃道日度，置其定日小餘副之，以其日盈縮分乘之，滿宗法而一，盈加縮減其副，乃以減其日加時，即為夜半黃道日度。 又有求每日夜半日度，因四正初日夜半度，累加一策，以其日盈縮分盈加縮減，滿黃道宿次去之，即得每日夜半日度。 又有求定朔、弦、望加時日度，置定朔、望小餘副之，以其日盈縮分乘之，以宗法收之為分，盈加縮減其副，以加其日夜半度，各得其時加日躔所次。 如朔、望有進退者，此術不用。）〉
Daily hour-added ecliptic solar degree: (Qianyuan calls this the daily motion in parts.) Subtract the fixed new- and full-moon positions; divide the remainder by the days elapsed to obtain the mean daily motion; subtract the two motion rates to get the combined difference; halve it and add or subtract from the mean motion to obtain the initial motion; (If the later mean motion is greater, subtract to obtain the initial; if the later mean motion is less, add to obtain the initial.) Divide the combined difference by the days elapsed to obtain the daily increment; diminish when the later rate is less and augment when it is greater — yielding the daily motion in parts; accumulate from the new- or full-moon day to obtain the desired result. (The same in Qianyuan. Yitian does not include this procedure. Yitian also gives the hour-added ecliptic degree on each fixed upright day: take the annual precession, multiply by the limit number, reduce one place, and divide by 101 for seconds and minor parts; subdivide again, add one image degree, accumulate from the winter solstice ecliptic position, and drop full lodges — obtaining the four uprights as hour-added ecliptic degrees. For midnight ecliptic degrees on the four upright days: take the fixed day's minor remainder as auxiliary, multiply by that day's excess-deficit, divide by the lineage factor, adjust the auxiliary for excess or deficit, subtract from the hour-added degree — yielding the midnight ecliptic position. To find each day's midnight solar degree: begin from the midnight degree at the first night of each upright, add one day's interval, adjust by that day's excess-deficit, and drop full ecliptic lodges. There is also the hour-added solar degree at fixed new moon, first quarter, and full moon: use the fixed minor remainder as auxiliary, multiply by the day's excess-deficit, reduce by the lineage factor, adjust the auxiliary, add to the midnight degree — locating the sun at the desired hour. When new or full moon requires advance or retreat, this method is not used.)