1
大聖通天地之至理，極生民之能事，體妙繫於神機，作範留於器象。 然則制物成法，故冥賾可尋； 推變有因，而化生以驗。 昔黃帝採竹昆崘之陰，聽鳳岐陽之下，斷自然之物，寫自然之音。 音既協矣，黃鍾以立； 數既生矣，氣亦徵之。 於是乎備數、和聲、審度、嘉量、權衡之用，皆出於茲矣。 三古所共行，百王不能易。 漢孝武置協律之官，元帝時京房明六十律，事為密矣。 王莽世，徵天下通鍾律之士，劉歆總而條奏之，最為該博，故班固取以為志。 後漢待詔嚴嵩頗為知律，至其子宣不傳，遂罷。 魏世杜夔亦以通樂制律，晉中書監荀勗持夔律校練八音，以謂後漢至魏尺長古尺四分有餘。 又得古玉律，勗以新律命之，謂其應合，遂改晉調，而散騎侍郎阮咸譏其聲高。 永嘉以後，中原喪亂，考正鍾律，所未聞焉。 其存於夷裔，聲器而已。
The great sage grasps Heaven and Earth's deepest principles, brings human endeavor to its fullest extent, ties the subtle to divine design, and fixes enduring models in concrete forms and instruments. By fashioning things and setting norms, even hidden depths become traceable; and when change is pursued systematically, causes emerge and the processes of generation and transformation can be confirmed. Long ago the Yellow Emperor took bamboo from Kunlun's northern slopes, listened to the phoenix at Qiyang, cut what nature offered, and reproduced nature's own tones. Once the pitches were in harmony, the Yellow Bell was fixed as the standard; and once numbers arose from this, vital energy likewise registered their effect. From this came complete numeration, harmonized music, calibrated length, standard capacity, and weights and balances in full. Shared across the three antiquities, no king in a hundred reigns could overturn it. Han Emperor Wu created the office of pitch harmonization; under Emperor Yuan, Jing Fang elaborated the sixty-pitch system, and the subject grew exceedingly refined. Under Wang Mang, experts in bells and pitch were summoned empire-wide; Liu Xin summarized their work in a memorial of unmatched breadth, which Ban Gu adopted for his treatise. Later Han's expectant official Yan Song knew pitch theory well, but his son Xuan failed to pass it on, and the tradition lapsed. In Wei, Du Kui mastered music and set pitch standards; Jin's Xun Xu used Du Kui's pipes to tune the eight sounds and concluded that from Later Han through Wei the standard foot ran more than four tenths longer than the ancient foot. He also acquired ancient jade pitch pipes, renamed them according to his new standards, declared them a match, and changed Jin's tuning—though Palace Attendant Ruan Xian criticized the pitch as too high. After Yongjia, with the heartland in turmoil, no one spoke of correcting bells and pitch standards. All that endured among frontier peoples were the physical instruments themselves.
2
魏氏平諸僭偽，頗獲古樂。 高祖慮其永爽，太和中詔中書監高閭修正音律，久未能定。 閭出為相州刺史，十八年，閭表曰：「書稱『同律度量衡』，論云『謹權量，審法度』。 此四者乃是王者之要務，生民之所由。 四者何先？ 以律為首。 豈不以取法之始，求天地之氣故也。 孔子曰：『移風易俗，莫尚於樂。』 然則樂之所感，其致遠矣。 今調音制樂，非律無以克和，然則律者樂之本也。 臣前被敕理樂，與皇宗博士孫惠蔚、太樂祭酒公孫崇等考周官、國語及後漢律曆志，案京房法作準以定律，吹律以調絲，案律寸以孔竹，八音之別，事以粗舉。 書既三奏，備在前文。 臣年垂七十，日就衰頹，恐一朝先犬馬，竟無絲髮之益，使律法長絕，遺恨沒世，是以慺慺惓惓，不敢忘怠。 近在鄴見崇，臣先以其聰敏精勤，有挈瓶之智，雖非經國之才，頗長推考之術，故臣舉以教樂，令依臣先共所論樂事，自作鍾磬志議二卷，器數為備，可謂世不乏賢。 今崇徒教樂童書學而已，不恭樂事，臣恐音律一曠，精賞實難，習業差怠，轉乖本意。 今請使崇參知律呂鍾磬之事，觸類而長之，成益必深。 求持臣先所奏三表勘後漢律曆志，陛下親覽，以求厥衷，俱然易了。 又著作郎韓顯宗博聞強識，頗有史才，粗解音律，亦求令時往參知。 臣雖在外官，竊慕古人舉善之義，愚意所及，不能自己，雖則越分，志在補益，願不以言廢人。」 詔許之。
After the Wei conquered the rival states, they recovered a considerable store of ancient music. Emperor Gaozu feared permanent loss of the tradition; in Taihe he ordered Gao Lü to restore pitch and music, yet years passed without resolution. When Lü became governor of Xiangzhou, in year eighteen he submitted: "The Documents demand unified pitch, length, capacity, and balance; the Analects require scrupulous weights and careful standards. These four are essential duties of kingship and the basis on which the people rely. Which of the four should come first? Pitch standards come first. Surely because at the very start of establishing norms one must seek the vital breath of Heaven and Earth. Confucius said: 'To transform customs and change habits, nothing surpasses music.' Thus the influence of music extends very far indeed. To tune sounds and compose music today, harmony cannot be achieved without pitch standards; pitch standards are therefore the foundation of music. I was previously ordered to oversee music; with Imperial Clan academician Sun Huiwei, Director of Imperial Music Gongsun Chong, and others I studied the Offices of Zhou, the Discourses of the States, and the Later Han treatise on pitch and calendars. Following Jing Fang's method I constructed a pitch pipe to set the standards, used blown pipes to tune the strings, and bored bamboo to pitch-standard dimensions—the eight categories of sound have been sketched in outline. The memorial was presented three times; the full account appears in the preceding text. I am nearly seventy and growing weaker by the day; I fear I may die before contributing even the smallest service, leaving pitch law extinct forever and carrying regret to my grave—therefore I press on earnestly and dare not slacken. Recently at Ye I met Chong; I had judged him intelligent, diligent, and capable in small matters—though not a statesman, he excels at systematic inquiry. I recommended him to teach music and had him follow our earlier discussions, producing on his own two fascicles on bells and chimes with instruments and numbers fully specified; the age is not without able men. Now Chong only teaches music students literacy and nothing more, neglecting music itself; I fear that once pitch and music are abandoned, true mastery will be impossible, training will slacken, and the original purpose will be lost. I ask that Chong be assigned to pitch pipes, bells, and chimes; extending his work by analogy will yield substantial benefit. I ask that Your Majesty compare my three previously submitted tables with the Later Han treatise on pitch and calendars; a personal review will make the true meaning clear at once. Editorial Director Han Xianzong is erudite and retentive, talented in historiography, and has some grasp of pitch and music; I also ask that he be sent periodically to assist. Though I serve outside the capital, I admire the ancients' practice of recommending talent; I cannot restrain what I believe worth saying; though I overstep my rank, my aim is the public good—I pray you will not dismiss these men on account of my words." The edict granted his request.
3
景明四年，并州獲古銅權，詔付崇以為鍾律之準。 永平中，崇更造新尺，以一黍之長，累為寸法。 尋太常卿劉芳受詔修樂，以秬黍中者一黍之廣即為一分，而中尉元匡以一黍之廣度黍二縫，以取一分。 三家紛競，久不能決。 太和十九年，高祖詔，以一黍之廣，用成分體，九十黍之長，以定銅尺。 有司奏從前詔，而芳尺同高祖所制，故遂典修金石。 迄武定末，未有諳律者。
In Jingming year 4, Bingzhou recovered an ancient bronze weight; an edict assigned it to Chong as the standard for bells and pitch. During Yongping, Chong made a new foot-rule by stacking the length of single millet grains to define the inch. Soon Director of Ceremonies Liu Fang was ordered to revise music, defining one fen as the width of a single medium black millet grain; Commandant of the Guards Yuan Kuang, by contrast, used one grain's width to span two grain-gaps and thus obtain one fen. The three schools disputed fiercely, and for years no decision could be reached. In Taihe year 19, Emperor Gaozu decreed that the width of one millet grain should define the fractional units and ninety millet grains in length should fix the bronze foot-rule. The responsible officials reported compliance with the earlier decree; Fang's foot-rule matched the emperor's standard, so he was placed in charge of bells and stone chimes. Through the end of the Wuding era, no one was truly versed in pitch standards.
4
正光曆沿革
Evolution of the Zhengguang Calendar
5
曆者數之用，探靈測化，窮微極幽之術也。 所以上齊七政，下授萬方。 自軒轅以還，迄於三代，推元革統，厥事不一也。 秦世漢興，曆同顓頊，百有餘年，始行三統。 後漢孝章世改從四分，光和中易以乾象，魏文時用韓翊所定，至明帝行楊偉景初，終於晉朝，無所改作。 司天測象，今古共情，啟端歸餘，為法不等，協日正時，俱有得失。 太祖天興初，命太史令晁崇修渾儀以觀星象，仍用景初曆。 歲年積久，頗以為疏。 世祖平涼土，得趙𣤆所修玄始曆，後謂為密，以代景初。 真君中，司徒崔浩為五寅元曆，未及施行，浩誅，遂寢。 高祖太和中，詔祕書鍾律郎上谷張明豫為太史令，修綜曆事，未成，明豫物故。 遷洛，仍歲南討，而宮車晏駕。
The calendar applies number to probe the numinous and measure change—it is the art of exhausting the subtle and penetrating the deepest mysteries. Above it aligns the seven luminaries; below it instructs the myriad regions. From the Yellow Emperor through the Three Dynasties, each era established its own origin and changed its succession—the practice was never uniform. Under Qin and at Han's founding the calendar followed Zhuanxu; only after more than a century did the Triple Concordance system come into use. Later Han Emperor Zhang adopted the Quarter-Remainder calendar; in Guanghe it was replaced by the Supernal Icon; Wei Emperor Wen used Han Yi's system; Ming Emperor employed Yang Wei's Jingchu calendar, and through Jin no further changes were made. Celestial observers ancient and modern share the same concern; methods for opening cycles and handling remainders differ, and each system for harmonizing the sun and correcting time has its strengths and flaws. At the start of Tianxing, the Founding Emperor ordered Director of Astronomy Chao Chong to repair the armillary sphere for observing the heavens, continuing to use the Jingchu calendar. Over the years it came to be regarded as increasingly inaccurate. Emperor Shizu pacified the Liang region and obtained Zhao Fei's Xuanshi calendar; it was later judged more precise and replaced the Jingchu system. During Zhenjun, Minister of Works Cui Hao devised a calendar using the Five Yin as origin; before it could be adopted, Hao was executed and the project was abandoned. In Taihe, Emperor Gaozu appointed Zhang Mingyu of Shanggu, a secretary pitch-standards officer, as Director of Astronomy to compile the calendar; the work was unfinished when Mingyu died. After the move to Luoyang, southern campaigns continued year after year, and the emperor died.
6
世宗景明中，詔太樂令公孫崇、太樂令趙樊生等同共考驗。 正始四年冬，崇表曰：「臣頃自太樂，詳理金石，及在祕省，考步三光，稽覽古今，研其得失。 然四序遷流，五行變易，帝王相踵，必奉初元，改正朔，殊徽號、服色，觀于時變，以應天道。 故易，湯武革命，治曆明時。 是以三五迭隆，曆數各異。 伏惟皇魏紹天明命，家有率土，戎軒仍動，未遑曆事，因前魏景初曆，術數差違，不協晷度。 世祖應期，輯寧諸夏，乃命故司徒、東郡公崔浩錯綜其數。 浩博涉淵通，更修曆術，兼著五行論。 是時故司空、咸陽公高允該覽羣籍，贊明五緯，并述洪範。 然浩等考察未及周密，高宗踐祚，乃用敦煌趙𣤆甲寅之曆，然其星度，稍為差遠。 臣輒鳩集異同，研其損益，更造新曆。 以甲寅為元，考其盈縮，晷象周密，又從約省。 起自景明，因名景明曆。 然天道盈虛，豈曰必協，要須參候是非，乃可施用。 太史令辛寶貴職司玄象，頗閑祕數； 祕書監鄭道昭才學優贍，識覽該密； 長兼國子博士高僧裕乃故司空允之孫，世綜文業； 尚書祠部郎中宗景博涉經史； 前兼尚書郎中崔彬微曉法術：請此數人在祕省參候。 而伺察晷度，要在冬夏二至前後各五日，然後乃可取驗。 臣區區之誠，冀效萬分之一。」 詔曰：「測度晷象，考步宜審，可令太常卿芳率太學、四門博士等依所啟者，悉集詳察。」
In Jingming, Emperor Shizong ordered Gongsun Chong, Zhao Fansheng, and others jointly to examine and verify the calendar. In winter of Zhengshi year 4, Chong submitted: "I have recently left the Imperial Music Office, where I examined bells and stones in detail; in the Secretariat I calculated the motions of the sun, moon, and stars, reviewed ancient and modern systems, and studied their merits and flaws. The four seasons succeed one another and the five phases shift; each new emperor must establish a new origin, reform the calendar and new year, and change emblems, titles, and colors—observing the times to align with Heaven's Way. Thus the Changes says that Tang and Wu effected revolution and regulated the calendar to clarify the seasons. Hence the Three Sovereigns and Five Emperors rose in turn, each with a different calendar system. Great Wei has received Heaven's mandate and rules all within the seas; campaigns continue unceasingly and calendar reform has been deferred; the former Wei's Jingchu calendar has grown inaccurate and no longer matches gnomon observations. Emperor Shizu, responding to his age, pacified the realm and ordered the late Minister of Works, Duke of Dongjun Cui Hao, to rework the calendrical calculations. Hao was erudite and penetrating; he revised the calendrical methods and also wrote a treatise on the five phases. At that time the late Minister of Works, Duke of Xianyang Gao Yun, had mastered the classics, assisted in clarifying the five planetary paths, and also expounded the Great Plan. Yet Hao's investigation was incomplete; when Emperor Gaozong ascended the throne, Zhao Fei of Dunhuang's calendar with the jiayin origin was adopted, but its stellar positions were somewhat off. I have gathered divergent views, studied what should be revised, and devised a new calendar. With jiayin as the origin, I have examined its excess and deficit; gnomon observations are thorough, and the system is also simplified. Beginning in the Jingming era, it is therefore named the Jingming calendar. Yet Heaven's Way has its cycles of fullness and emptiness—how can we assume perfect accord? We must verify it through observation before adopting it. Director of Astronomy Xin Baogui, whose office oversees the heavens, is well versed in arcane calculations; Secretariat Director Zheng Daozhao is talented and learned, with comprehensive and penetrating knowledge; Senior Concurrent Academician of the Imperial University Gao Sengyu, grandson of the late Minister of Works Gao Yun, comes from a family long devoted to scholarship; Zong Jing of the Sacrifices Bureau in the Masters of Writing is broadly versed in the classics and histories; former Concurrent Secretary Cui Bin has some knowledge of computational methods: I ask that these men assist in verification within the Secretariat. Gnomon observation requires the five days before and after each solstice; only then can verification be obtained. In my humble way I hope to contribute even the smallest measure of service." The edict replied: "Gnomon measurement and calendrical calculation require care; let Director of Ceremonies Liu Fang lead Imperial University and Four Gates academicians and others as proposed, all to assemble for detailed review."
7
延昌四年冬，侍中、國子祭酒領著作郎崔光表曰：「易稱『君子以治曆明時』； 書云『曆象日月星辰』，『廼同律度量衡』； 孔子陳後王之法，曰『謹權量，審法度』； 春秋舉『先王之正時也，履端於始』，又言『天子有日官』。 是以昔在軒轅，容成作曆； 逮乎帝唐，羲和察影。 皆所以審農時而重民事也。 太和十一年，臣自博士遷著作，忝司載述，時舊鍾律郎張明豫推步曆法，治己丑元，草創未備。 及遷中京，轉為太史令，未幾喪亡，所造致廢。 臣中修史，景明初奏求奉車都尉、領太史令趙樊生，著作佐郎張洪，給事中、領太樂令公孫崇等造曆，功未及訖，而樊生又喪，洪出除涇州長史，唯崇獨專其任。 暨永平初，云已略舉。 時洪府解停京，又奏令重修前事，更取太史令趙勝、太廟令龐靈扶、明豫子龍祥共集祕書，與崇等詳驗，推建密曆。 然天道幽遠，測步理深，候觀遷延，歲月滋久，而崇及勝前後並喪。 洪所造曆為甲午、甲戌二元，又除豫州司馬。 靈扶亦除蒲陰令。 洪至豫州，續造甲子、己亥二元。 唯龍祥在京，獨修前事，以皇魏運水德，為甲子元。 兼校書郎李業興本雖不預，亦和造曆，為戊子元。 三家之術並未申用。 故貞靜處士李謐私立曆法，言合紀次，求就其兄瑒追取，與洪等所造，遞相參考，以知精粗。 臣以仰測晷度，實難審正，又求更取諸能算術兼解經義者前司徒司馬高綽、駙馬都尉盧道虔、前冀州鎮東長史祖瑩、前并州秀才王延業、謁者僕射常景等日集祕書，與史官同檢疏密； 并朝貴十五日一臨，推驗得失，擇其善者奏聞施用。 限至歲終。 但世代推移，軌憲時改，上元今古，考準或異，故三代課步，始卒各別。 臣職預其事，而朽惰已甚，既謝運籌之能，彌愧意算之藝，由是多歷年世，茲業弗成，公私負責，俯仰慚靦。」 靈太后令曰：「可如所請。」
In winter of Yanxi year 4, Palace Attendant Cui Guang, Chancellor of the Imperial University and concurrent Editorial Director, submitted: "The Changes says, 'The noble man regulates the calendar to clarify the seasons'; the Documents say, 'the calendar emblems are the sun, moon, stars, and constellations,' and 'thus unify pitch, length, capacity, and balance'; Confucius set forth methods for later kings, saying, 'be scrupulous with weights and measures, examine standards and models'; the Spring and Autumn cites, 'the former kings corrected the seasons, beginning at the year's start,' and also says, 'the Son of Heaven has officers of the sun.' Thus in antiquity under the Yellow Emperor, Rong Cheng devised the calendar; down to Emperor Yao of Tang, Xi and He observed the shadows. All were meant to determine the agricultural seasons and honor the people's welfare. In Taihe year 11 I moved from academician to the editorial office, charged with historiography; at that time former pitch-standards officer Zhang Mingyu was calculating the calendar with the jichou origin, but the draft remained incomplete. When the capital moved to Luoyang he became Director of Astronomy; he soon died and his work was abandoned. While I was revising the history, at the start of Jingming I requested that Palace Horse Commandant Zhao Fansheng, concurrent Director of Astronomy, editorial assistant Zhang Hong, and Gongsun Chong, concurrent Director of Imperial Music, compile the calendar; the work was unfinished when Fansheng died, Hong was posted to Jingzhou, and Chong alone carried on. By the start of Yongping it was said to be roughly complete. When Hong's prefectural term ended and he remained in the capital, I again requested revision of the earlier work, appointing Zhao Sheng as Director of Astronomy, Pang Lingfu as Director of the Imperial Ancestral Temple, and Mingyu's son Longxiang to join Chong and others in the Secretariat for detailed verification toward a precise calendar. Yet Heaven's Way is remote, measurement profound, and observation dragged on for years—meanwhile Chong and Sheng died in succession. Hong's calendar used the jiawu and jiaxu origins; he was then appointed military aide of Yuzhou. Lingfu was also appointed magistrate of Puyin. After Hong reached Yuzhou, he continued work on calendars with the jiazi and jihai origins. Only Longxiang remained in the capital, continuing the earlier work alone, taking Great Wei's Water Virtue fortune and adopting the jiazi origin. Concurrent proofreader Li Yexing, though not originally involved, also collaborated on a calendar with the wuzi origin. None of the three schools' methods was adopted. The retired gentleman Li Mi privately devised a calendar, claiming it matched the chronological order; he sought to recover it from his brother Yang's estate and cross-reference it with Hong's and others' work to judge their relative accuracy. I find that gnomon measurement from below is hard to verify with certainty; I further ask that skilled calculators who also understand the classics—former Secretariat Marshal Gao Chuo, Commandant of Horse Lu Daoqian, former eastern Ji chief secretary Zu Ying, former Bingzhou graduate Wang Yanye, and Master of Ceremonies Chang Jing—assemble daily in the Secretariat with the historiographers to examine each system's strengths and weaknesses; and that court nobles review the work every fifteen days, test each system's merits, select the best, and memorialize for adoption. The deadline is the end of the year. But as generations pass, calendrical norms change with the times; high origins and verification standards differ between past and present, so the computational steps of the Three Dynasties vary from start to finish. Though my office has been involved, age and indolence have taken their toll; I no longer have the skill for strategic reckoning, and I am all the more ashamed of my deficiencies in computational arts. Years have piled up while the work remains unfinished—a debt owed both to the state and in private—and I am ashamed wherever I turn. The Empress Dowager Ling ordered: "Let it be as requested."
8
延昌四年冬，太傅、清河王懌，司空、尚書令、任城王澄，散騎常侍、尚書僕射元暉，侍中、領軍、江陽王繼奏：「天道至遠，非人情可量； 曆數幽微，豈以意輒度。 而議者紛紜，競起端緒，爭指虛遠，難可求衷，自非建標準影，無以驗其真偽。 頃永平中雖有考察之利，而不累歲窮究，遂不知影之至否，差失少多。 臣等參詳，謂宜今年至日，更立表木，明伺晷度，三載之中，足知當否。 令是非有歸，爭者息競，然後採其長者，更議所從。」
In the winter of Yanchang year 4, Grand Tutor Prince Yi of Qinghe, Minister of Works and Secretariat Director Prince Cheng of Rencheng, Regular Attendant and Vice Director Yuan Hui, and Palace Attendant and Director of Palace Guards Prince Ji of Jiangyang submitted: "Heaven's Way is exceedingly remote—not something human feeling can gauge; calendrical reckonings are subtle and recondite—how can one simply guess at them by intuition? Yet debaters swarm with competing lines of argument, disputing over abstractions with no way to reach a fair judgment; unless one sets up a gnomon and measures shadows, there is no way to verify what is true or false. Recently, during Yongping, investigation did yield some benefit, but because the inquiry was not pursued year after year to the end, no one knew whether the shadow readings were correct or how large the errors were. We have reviewed the matter and believe that at this year's winter solstice a new gnomon post should be erected and gnomon shadow degrees carefully observed; within three years that would suffice to determine what is correct. Then right and wrong will have a clear basis, disputants will cease their quarrels, and afterward the strongest method can be chosen and a fresh decision made on which system to adopt."
9
神龜初，光復表曰：「春秋載『天子有日官，諸侯有日御』，又曰『履端于始』，『歸餘于終』，皆所以推二氣，考五運，成六位，定七曜，審八卦，立三才，正四序，以授百官于朝，萬民于野。 陰陽剛柔，仁義之道，罔不畢備。 由是先代重之，垂於典籍。 及史遷、班固、司馬彪著立書志，所論備矣。 謹案曆之作也，始自黃帝，辛卯為元，迄于大魏，甲寅紀首，十有餘代，歷祀數千，軌憲不等，遠近殊術。 其消息盈虛，覘步疏密，莫得而識焉。 去延昌四年冬，中堅將軍、屯騎校尉張洪，故太史令張明豫息盪寇將軍龍祥，校書郎李業興等三家並上新曆，各求申用。 臣學缺章程，藝謝籌運，而竊職觀閣，謬忝厥司，奏請廣訪諸儒，更取通數兼通經義者及太史，並集祕書，與史官同驗疏密，并請宰輔羣官臨檢得失，至於歲終，密者施用。 奉詔聽可。 時太傅、太尉公、清河王臣懌等以天道至遠，非卒可量，請立表候影，期之三載，乃採其長者，更議所從。 又蒙敕許。 於是洪等與前鎮東府長史祖瑩等研窮其事，爾來三年，再歷寒暑，積勤構思，大功獲成。 謹案洪等三人前上之曆，并駙馬都尉盧道虔、前太極採材軍主衞洪顯、殄寇將軍太史令胡榮及雍州沙門統道融、司州河南人樊仲遵、定州鉅鹿人張僧豫所上，總合九家，共成一曆，元起壬子，律始黃鍾，考古合今，謂為最密。 昔漢武帝元封中治曆，改年為太初，即名太初曆； 魏文帝景初中治曆，即名景初曆。 伏惟陛下道唯先天，功𨗿稽古，休符告徵，靈蔡炳瑞。 壬子北方，水之正位； 龜為水畜，實符魏德； 修母子應，義當麟趾。 請定名為神龜曆。 今封以上呈，乞付有司重加考議。 事可施用，并藏祕府，附於典志。」 肅宗以曆就，大赦改元，因名正光曆，班於天下。 其九家共修，以龍祥、業興為主。
At the start of the Shengui era, Guang again submitted a memorial: "The Spring and Autumn Annals records that 'the Son of Heaven has a sun officer, and the feudal lords have sun stewards,' and also says 'take the beginning as the start' and 'return the surplus at the end'—all of this serves to extrapolate the two qi, examine the five phases, complete the six positions, fix the seven luminaries, scrutinize the eight trigrams, establish the three powers, and set the four seasons in order, so as to grant this to the hundred officials at court and the myriad people in the fields. Yin and yang, firmness and yielding, and the Way of humaneness and righteousness—nothing is left incomplete. For this reason former ages prized it and preserved it in the canonical records. When Sima Qian, Ban Gu, and Sima Biao composed their bibliographic treatises, their discussions were exhaustive. A careful review shows that calendar-making began with the Yellow Emperor, with the xinmao origin; down to Great Wei, with the jiayin cycle head—more than ten dynasties and thousands of years of rites, with unequal standards and methods that differed widely in their assumptions. Their waxing and waning, fullness and emptiness, and the tightness or looseness of their computational steps could not be fully understood. As of the winter of Yanchang year 4, Major General and Commandant of Mounted Escort Zhang Hong; Longxiang, General Who Quells Bandits and son of the former Director of Astronomy Zhang Mingyu; and proofreader Li Yexing—three parties together submitted new calendars, each requesting official adoption. My own learning lacks proper method, and my skill falls short of computational planning, yet I hold this office unworthily in the archive; I therefore request broad consultation with Confucian scholars, and the further selection of those skilled in numerology who also understand the classics, together with the Directors of Astronomy, all to assemble in the Secretariat and jointly test each system's precision with the historiographers; I also ask the chief ministers and officials to inspect the results in person, and by year's end whichever system proves tightest shall be adopted. The edict granted approval. At that time Grand Tutor and Grand Commandant Prince Yi of Qinghe and others, holding that Heaven's Way is too remote to be measured quickly, requested that a gnomon be erected to observe shadows, with a three-year term, after which the best method would be chosen and a fresh decision made on which system to follow. This too received the imperial assent. Thereupon Hong and the others, together with the former chief secretary of the Eastern Staff, Zu Ying, and others, pursued the matter to its depths; three years passed and two rounds of seasons elapsed before diligent labor and careful design at last brought the great work to completion. A careful review shows that the calendars previously submitted by Hong and his three colleagues, together with those of Commander of Horse Lu Daoqian; former Supreme Ultimate talent-selection army master Wei Hongxian; Director of Astronomy and General Who Exterminates Bandits Hu Rong; the monk leader Daorong of Yongzhou; Fan Zhongzun of Henan in Sizhou; and Zhang Sengyu of Julu in Dingzhou—in all nine parties—were combined into one calendar with the origin at renzi and pitch standards beginning at the Yellow Bell; comparing antiquity with the present, we judge it the most precise. Formerly, in the yuanfeng era of Han Emperor Wu, the calendar was revised and the era name changed to Taichu—it was therefore called the Taichu calendar; in the jingchu era of Wei Emperor Wen the calendar was revised—it was therefore called the Jingchu calendar. We humbly consider that Your Majesty's Way precedes Heaven itself, your achievements reach back to antiquity, blessed omens proclaim their signs, and the sacred tortoise and divining stalks blaze with auspicious portents. Renzi is the northern direction, the proper position of water; the tortoise is the creature of water and truly matches the Wei's Water Virtue; adjusting the resonance of mother and child pitch pipes, as ritually fitting as the qilin's footprint. We request that it be formally named the Shengui calendar. We now seal and submit it for Your Majesty's review, humbly requesting that the relevant offices conduct further examination and deliberation. If approved for use, let it also be stored in the secret archive and appended to the canonical treatises. Emperor Suzong, because the calendar was complete, issued a general amnesty and changed the era name; the calendar was named the Zhengguang calendar and promulgated throughout the realm. Of the nine parties who jointly compiled it, Longxiang and Yexing served as the principal authors.
10
正光曆
The Zhengguang Calendar
11
壬子元以來，至今大魏正光三年歲在壬寅，積十六萬七千七百五十，算外； 壬子歲入甲申紀以來，至今孝昌二年歲在丙午，積四萬六千五百五十四，算外。 從壬子元以來，至今大魏孝昌三年歲次丁未，積十六萬七千七百五十六，算上； 壬子歲入甲申紀以來，[1]至今大魏孝昌三年歲次丁未，積四萬六千五百五十六，算上。
From the renzi origin to the present, Great Wei Zhengguang year 3 (renyin)—the accumulated count is 167,750, exclusive; From the renzi year entering the jiashen era cycle to the present, Xiaochang year 2 (bingwu)—the accumulated count is 46,554, exclusive. From the renzi origin to the present, Great Wei Xiaochang year 3 (dingwei)—the accumulated count is 167,756, inclusive; From the renzi year entering the jiashen era cycle to the present,[1] Great Wei Xiaochang year 3 (dingwei)—the accumulated count is 46,556, inclusive.
12
章歲，五百五。 〈古十九年七閏，閏餘盡為章。 積至多年，月盡之日，月見東方，日蝕先晦，輒復變曆，以同天象。 二百年多一日，三百年多一日半，晦朔失。 故先儒及緯文皆言「三百年斗曆改憲」。 候天減閏，五百五年減閏餘一，九千五百九十五年減一閏月，則從僖公五年至今，日蝕不失晦與二日，合朔者多。 閏餘成月，餘盡為章。〉
Rule years: 505. 〈In antiquity, nineteen years with seven intercalations—the intercalary remainder entirely constituted the rule. Over many accumulated years, on the day the month ran out the moon would appear in the east and solar eclipses would precede the new moon; then the calendar would be revised again to match the heavenly signs. Over two hundred years the count ran one day long; over three hundred years, a day and a half—new and full moons fell out of alignment. Hence earlier Confucian scholars and apocryphal texts all say, "Every three hundred years the Dipper calendar must change its standard." By observing Heaven and reducing intercalations—subtracting one from the intercalary remainder every 505 years and one intercalary month every 9,595 years—from Duke Xi's fifth year to the present, solar eclipses rarely miss the last or second day of the month, and new-moon conjunctions match more closely. When the intercalary remainder completes a month, the remainder entirely forms the rule.〉
13
章閏，一百八十六。 〈五百五年閏月之數，其中減舊十九分之一。〉
Rule intercalations: 186. 〈The number of intercalary months in 505 years, reduced by one-nineteenth from the old value.〉
14
章月，六千二百四十六。 〈五百五年所有月之數并閏月。〉
Rule months: 6,246. 〈The total number of months in 505 years, including intercalary months.〉
15
蔀法，六千六十。 〈十二章為一蔀，至此年小餘成日，為度法。〉
Obscuration divisor: 6,060. 〈Twelve rules form one obscuration; when in that year the small remainder completes a day, it becomes the degree divisor.〉
16
斗分，一千四百七十七。 〈四分度法得一千五百一十五，為古法。 今減三十八者，從僖公五年以來減七日有奇，謂為最近。 一百一十三歲減□日，減之太深，是以三十餘年改徙四子也。〉
Dipper fraction: 1,477. 〈The quarter-degree divisor yields 1,515—that is the old method. The present reduction of thirty-eight reflects a reduction of seven and a fraction days since Duke Xi's fifth year—we deem this the closest fit. Reducing [one] day every 113 years cut too deeply, which is why in little more than thirty years they changed and shifted four origins.〉
17
紀法，六萬六百。 〈十蔀成紀，大餘十也。〉
Era divisor: 60,600. 〈Ten obscurations form an era; the big remainder is ten.〉
18
統法，十二萬一千二百。 〈二紀成統，大餘二十。〉
Concordance divisor: 121,200. 〈Two eras form a concordance; the big remainder is twenty.〉
19
元法，三十六萬三千六百。 〈三統成元，大餘盡。〉
Origin divisor: 363,600. 〈Three concordances form an origin; the big remainder is exhausted.〉
20
日法，七萬四千九百五十二。 〈十二乘章月為日法。 章月，一年之閏分。〉
Day divisor: 74,952. 〈Twelve times the rule months gives the day divisor. Rule months are one year's intercalary fraction.〉
21
周天分，二百二十一萬三千三百七十七。 〈以度法通三百六十五度，內斗分。〉
Circuits-of-Heaven fraction: 2,213,377. 〈Using the degree divisor to convert 365 degrees, plus the Dipper fraction.〉
22
氣法，二十四。 〈歲中十二，一年一十二次。 [2]次有初中，分為二十四。〉
Qi divisor: 24. 〈Twelve mid-year qi in a year—twelve lunar stations in a year. [2] Each station has a beginning and middle, divided into twenty-four.〉
23
經月，大餘二十九，小餘三萬九千七百六十九。 〈日法除周天分得之。 日法者一蔀之月數，周天分者一蔀之日數。 以蔀月除蔀日，[3]得一月二十九及餘，是周天分即為月通。〉
Canonical month: big remainder 29, small remainder 39,769. 〈Obtained by dividing the Circuits-of-Heaven fraction by the day divisor. The day divisor is the number of months in one obscuration; the Circuits-of-Heaven fraction is the number of days in one obscuration. Dividing obscuration days by obscuration months, [3] one obtains twenty-nine months and a remainder—the Circuits-of-Heaven fraction is thus the month commonality.〉
24
會數，百七十三，餘二萬三千二百八。 〈五月二十三分月之二十為一會。 以二十三乘五月，內二十，得一百三十五，以乘周天分。 以二十三乘日法除之，得一百七十三及餘。〉
Conjunction number: 173, remainder 23,208. 〈Twenty twenty-thirds of a month times five months form one conjunction. Multiplying five months by twenty-three and adding twenty gives 135; multiply the Circuits-of-Heaven fraction by this. Multiplying the day divisor by twenty-three and dividing, one obtains 173 and a remainder.〉
25
會通，一千二百九十八萬九千九百四。 〈以日法乘會數，內會餘。〉
Conjunction commonality: 1,298,999,904. 〈Multiplying the conjunction number by the day divisor and adding the conjunction remainder.〉
26
周日，二十七，餘四萬一千五百六十二。 〈以月一日行除周天，得二十七日及餘。〉
Circuit day: 27, remainder 41,562. 〈Dividing the Circuits of Heaven by the moon's daily motion yields twenty-seven days and a remainder.〉
27
通周，二百六萬五千二百六十六。 〈日法乘周日二十七，內周餘。〉
Common circuit: 2,065,266. 〈Multiplying circuit day 27 by the day divisor and adding the circuit remainder.〉
28
小周，六千七百五十一。 〈月一日行十三度，乘章歲，內章閏也。〉
Lesser circuit: 6,751. 〈The moon's daily motion of thirteen degrees, multiplied by rule years, plus rule intercalations.〉
29
月周，八萬一千一十二。 〈以十二乘小周即得，與度同。〉
Lunar circuit: 81,012. 〈Multiplying the lesser circuit by twelve gives this—it equals degrees.〉
30
推月朔術第一
Method I: Calculating New Moons
31
推積月術曰：置入紀年，算外，以章月乘之，如章歲為積月，不盡為閏餘。 閏餘滿三百一十九以上，其歲有閏。
Method for calculating accumulated months: set the years since era entry, count exclusive, multiply by rule months, divide by rule years to obtain accumulated months; the remainder is the intercalary remainder. If the intercalary remainder reaches 319 or more, that year has an intercalary month.
32
推朔積日術曰：以通數乘積月，為朔積分，分滿日法為積日，不盡為小餘。 六旬去積日，不盡為大餘。 命以紀，算外，則所求年天正十一月朔日。
Method for calculating accumulated new-moon days: multiply accumulated months by the commonality number to obtain accumulated new-moon parts; when the parts fill the day divisor they become accumulated days; the remainder is the small remainder. Remove six decades from the accumulated days; the remainder is the big remainder. Name it against the era cycle, count exclusive, and that gives the new moon of the eleventh month of the civil year sought.
33
推上下弦望術曰：加朔大餘七、小餘二萬八千六百八十、小分一，小分滿四，從小餘； 小餘滿日法，從大餘一； 大餘滿六十，去之，即上弦日。 又加，得望； 又加，得下弦； 又加，得後月朔。
Method for calculating first quarter, full moon, and last quarter: add to the new moon's big remainder 7, small remainder 28,680, and small parts 1; when small parts fill four, carry to the small remainder; when the small remainder fills the day divisor, carry one to the big remainder; when the big remainder fills sixty, remove it—that is the first-quarter day. Add again to obtain the full moon; add again to obtain the last quarter; add again to obtain the next month's new moon.
34
推二十四氣術第二
Method II: Calculating the Twenty-Four Qi
35
推二十四氣術曰：置入紀年以來，算外，以餘數乘之為實。 以蔀法除之，所得為積沒，不盡為小餘。 以六旬去積沒，不盡為大餘。 命以紀，算外，所求年天正十一月冬至日。 求次氣，加大餘十五、小餘一千三百二十四、小分一，小分滿氣法二十四，從小餘一； 小餘滿蔀法，從大餘一； 大餘滿六十，去之，命如上，即次氣日。
Method for calculating the twenty-four qi: set the years since era entry, count exclusive, multiply by the remainder number for the dividend. Divide by the obscuration divisor; the quotient is accumulated extinction, and the remainder is the small remainder. Remove six decades from accumulated extinction; the remainder is the big remainder. Name it against the era cycle, count exclusive, and that gives the winter solstice of the eleventh month of the civil year sought. For the next qi, add 15 to the big remainder, 1,324 to the small remainder, and 1 to the small parts; when small parts reach 24, carry 1 to the small remainder; when the small remainder fills the obscuration divisor, carry 1 to the big remainder; when the big remainder reaches 60, drop a full cycle and name as before—that is the day of the next qi.
36
推閏術曰：以閏餘減章歲五百五，餘以歲中十二乘之。 滿章閏一百八十六，得一月； 餘半法已上，亦得一月。 數從天正十一月起，算外，閏月月也。 閏有進退，以無中氣為正。
Method for calculating intercalation: subtract the intercalary remainder from rule years 505, then multiply the remainder by 12 year medians. when the product fills rule intercalations 186, obtain one month; if the remainder is at or above half the divisor, likewise count one month. Count from the eleventh month of the civil year, count exclusive, and that is the intercalary month. Intercalations may shift forward or back, but the governing rule is that the leap month contains no major solar term.
37
冬至十一月中小寒十二月節大寒十二月中立春正月節
Winter Solstice — 11th month, major qi; Minor Cold — 12th month, minor qi; Major Cold — 12th month, major qi; Start of Spring — 1st month, minor qi.
38
雨水正月中驚蟄二月節春分二月中清明三明節
Rain Water — 1st month, major qi; Awakening of Insects — 2nd month, minor qi; Spring Equinox — 2nd month, major qi; Clear and Bright — third-month minor qi.
39
穀雨三月中立夏四月節小滿四月中芒種五月節
Grain Rain — 3rd month, major qi; Start of Summer — 4th month, minor qi; Minor Fullness — 4th month, major qi; Grain in Ear — 5th month, minor qi.
40
夏至五月中小暑六月節大暑六月中立秋七月節
Summer Solstice — 5th month, major qi; Minor Heat — 6th month, minor qi; Major Heat — 6th month, major qi; Start of Autumn — 7th month, minor qi.
41
處暑七月中白露八月節秋分八月中寒露九月節
End of Heat — 7th month, major qi; White Dew — 8th month, minor qi; Autumn Equinox — 8th month, major qi; Cold Dew — 9th month, minor qi.
42
霜降九月中立冬十月節小雪十月中大雪十一月節
Frost Descent — 9th month, major qi; Start of Winter — 10th month, minor qi; Minor Snow — 10th month, major qi; Major Snow — 11th month, minor qi.
43
推交會術第三[4]
Method III: Calculating Nodal Conjunction[4]
44
推合朔交會月蝕去交度術曰：置入紀朔積分，又以交會差分并之，今用甲申紀，差分七百四十一萬八千七百八十四也。 以會通去之，所得為積交，餘不盡者，以日法除之，所得為度餘，即所求年天正十一月朔却去交度及餘。
Method for calculating nodal distance at syzygy: set the accumulated new-moon fraction since era entry and add the conjunction–node difference parts; under the present Jiashen Cycle the difference parts are 7,418,784. Reduce modulo the conjunction communication constant for accumulated crossing; divide the remainder by the day divisor for the degree remainder—the degrees and parts by which the eleventh-month new moon precedes the node in the year sought.
45
求次月去交度術曰：加度二十九日、度餘三萬九千七百六十九，除如上，則次月去交度及分。
Method for the next month's nodal distance: add 29 days and degree remainder 39,769, reduce as above, and that gives the next month's distance from the node in degrees and parts.
46
求望去交度術曰：加度十四日、度餘五萬七千三百六十半，度餘滿日法從度，滿會數去之，亦除其餘； 餘若不足減者，減度一，加會虛，則望去交度及分。 朔望去交度分如朔望合數十四度、度餘五萬七千三百六十半已下，入交限數一百五十八度、度餘四萬七百九十九半以上者，朔則交會，望則月蝕。
Method for full-moon nodal distance: add 14 days and 57,360½ degree parts; carry overflow at the day divisor and reduce modulo the conjunction number as before. if the remainder cannot be reduced, borrow one degree and add the conjunction deficit to obtain full-moon nodal distance in degrees and parts. When syzygy nodal distance falls within the entry limit—no greater than the 14-day 57,360½ conjunction interval and no less than 158° 40,799½—new moon yields nodal conjunction and full moon yields lunar eclipse.
47
甲子紀 〈合朔日月如合璧，交中。〉
Jiazi Cycle 〈At new moon the sun and moon align like paired jade disks, with the crossing at center.〉
48
甲戌紀 〈合朔月在日道裏。〉 交會差四十九度度餘三萬六千七百四十四
Jiaxu Cycle 〈At new moon the moon lies inside the ecliptic.〉 Conjunction–node difference: 49 degrees, degree remainder 36,744.
49
甲申紀 〈合朔月在日道裏。〉 交會差九十八度度餘七萬三千四百八十八
Jiashen Cycle 〈At new moon the moon lies inside the ecliptic.〉 Conjunction–node difference: 98 degrees, degree remainder 73,488.
50
甲午紀 〈合朔月在日道裏。〉 交會差一百四十八度度餘三萬五千二百八十[5]
Jiawu Cycle 〈At new moon the moon lies inside the ecliptic.〉 Conjunction–node difference: 148 degrees, degree remainder 35,280[5].
51
甲辰紀 〈合朔月在日道裏。〉 交會差二十四度度餘四萬八千八百一十六
Jiachen Cycle 〈At new moon the moon lies inside the ecliptic.〉 Conjunction–node difference: 24 degrees, degree remainder 48,816.
52
甲寅紀 〈合朔月在日道裏。〉 交會差七十四度度餘一萬六百八
Jiayin Cycle 〈At new moon the moon lies inside the ecliptic.〉 Conjunction–node difference: 74 degrees, degree remainder 10,608.
53
求交道所在月：以十一月朔却去交度及餘，減會數及餘，餘若不足減者，減一度，加日法，[6]乃減之。 乃以十一月朔小餘加之，滿日法，除去之，從日一，餘為日餘。 命起往年十一月，如曆月大小除之，不滿月者為入月，算外，交道日。 交在望前者，其月朔則交會，望則月蝕； 交在望後者亦其月月蝕，後月朔則交會。 交正在望者，其月月蝕既，前後朔皆交會； 交正在朔者，日蝕既，前後望皆月蝕。
To find the month of nodal crossing: subtract the conjunction number and remainder from the eleventh-month new moon's nodal distance; if the remainder is insufficient, borrow one degree and add the day divisor,[6] then subtract. Then add the eleventh-month new moon's small remainder; carry at the day divisor for whole days, and the remainder is the day remainder. Count from the prior year's eleventh month, divide by the calendar's long and short months; the partial month is the entered month, and counting exclusive gives the day of nodal crossing. If the crossing precedes full moon, that month has nodal conjunction at new moon and lunar eclipse at full moon; if the crossing follows full moon, that month has a lunar eclipse, and the following month's new moon yields nodal conjunction. if the crossing falls exactly at full moon, that month has a total lunar eclipse, with nodal conjunction at the new moons on either side; if the crossing falls exactly at new moon, there is a total solar eclipse, with lunar eclipses at the full moons on either side.
54
求後交月及日：以會數及餘加前入月日及餘，餘滿日法，從日一，如曆月大小除之，命起前蝕月，得後交月及餘。
To find the next crossing month and day: add the conjunction number and remainder to the prior entered month and day, carry at the day divisor, divide by calendar months from the prior eclipse month, and obtain the next crossing month and remainder.
55
推月在日道表裏術曰：置入紀朔積分，又以紀交會差分加之， 〈今用甲申紀[7]交會差分七百四十一萬八千七百八十四。〉 倍會通，去之。 餘不滿會通者：紀首裏者，則天正十一月合朔，月在日道裏； 紀首表者，則月在表。 若滿會通者：紀首表者，則月在裏； 紀首裏者，則月在表。 黃道南為表，北為裏。 其滿會通者去之，餘如日法而一，即往年天正十一月朔却交度及餘。 以却去交度及餘減會數及會餘，會餘若不足減者，減一度，加日法乃減。 餘為前去度及餘。 [8]又以十一月朔小餘加之，滿日法從度一。 命起十一月，如曆月大小除之，不滿月者為入月日及餘，算外，交道日。 〈若十一月朔月在日道裏者，此交為出外，後交為入內； 十一月朔在表者，此交為入內，後交為出外。 一出一入，常法也。〉 其交在朔後望前者，朔，月在日道表裏與十一月同，望則反矣。 若交在望後朔前者，望與十一月同，後月朔則異矣。 〈若先交會後月蝕者，朔，月在日道裏； 望，在表。 朔在表，則望在裏。 其先月蝕後交會者，望在表則朔在裏也。〉 望在裏則朔在表矣。
Method for determining whether the moon lies north or south of the ecliptic: set the accumulated new-moon fraction since era entry and add the cycle's conjunction–node difference parts, 〈Under the present Jiashen Cycle[7], the conjunction–node difference parts are 7,418,784.〉 Reduce modulo twice the conjunction communication constant. If the remainder is less than the conjunction communication constant: when the cycle opens inside, the eleventh-month new-moon lies inside the ecliptic; when the cycle opens outside, the moon lies outside the ecliptic. If the remainder fills the conjunction communication constant: when the cycle opens outside, the moon lies inside; when the cycle opens inside, the moon lies outside the ecliptic. South of the ecliptic is outside; north is inside. Drop full conjunction communication constants and divide the remainder by the day divisor to obtain the prior year's eleventh-month new-moon nodal distance in degrees and parts. Subtract nodal distance from the conjunction number and remainder; if the conjunction remainder is insufficient, borrow one degree and add the day divisor before subtracting. The remainder is the forward nodal distance in degrees and parts. [8]Then add the eleventh-month new moon's small remainder; carry one degree when the day divisor is filled. Count from the eleventh month, divide by calendar months; the partial month gives the entered month, day, and remainder, and counting exclusive yields the day of nodal crossing. 〈If the eleventh-month new moon lies inside the ecliptic, this crossing exits outward and the next enters inward; if the eleventh-month new moon lies outside the ecliptic, this crossing enters inward and the next exits outward. One exit and one entry—this is the standing rule.〉 If the crossing falls between new moon and full moon, the moon at new moon matches the eleventh month's inside/outside polarity, but at full moon the polarity reverses. If the crossing falls between full moon and the next new moon, full moon matches the eleventh month, but the following new moon differs. 〈When nodal conjunction precedes lunar eclipse, at new moon the moon lies inside the ecliptic; at full moon it lies outside. If new moon is outside, full moon is inside. When lunar eclipse precedes nodal conjunction, outside at full moon means inside at new moon.〉 Inside at full moon means outside at new moon.
56
推交會起角術曰：其月在外道，先會後交者，虧從東南角起； 先交後會者，虧從西南角起。 其月在內道，先會後交者，虧從西北角起； 先交後會者，虧從西北角起。 合交中者，蝕之既。 其月蝕在日之衝，起角亦如之。 凡日月蝕，去交十五為限，十以下是蝕也，十以上，虧蝕微少，光影相接而已。
Method for the eclipse starting corner: when the moon is on the outer path and conjunction precedes crossing, obscuration begins at the southeast limb; when crossing precedes conjunction, obscuration begins at the southwest limb. When the moon is on the inner path and conjunction precedes crossing, obscuration begins at the northwest limb; when crossing precedes conjunction, obscuration begins at the northwest limb. When syzygy and nodal crossing coincide at center, the eclipse is total. At lunar eclipse the moon stands opposite the sun, and the starting corner follows the same rule. For all solar and lunar eclipses, 15 degrees from the node is the limit; within 10 degrees there is true eclipse; beyond 10, obscuration is slight and only light and shadow graze.
57
推蝕分多少術曰：置入交限十五度，以朔望去交日數減之，餘則蝕分。
Method for eclipse magnitude: set the crossing limit at 15 degrees, subtract syzygy nodal distance in days, and the remainder is the eclipse fraction.
58
推合朔入曆遲疾盈縮第四
Method IV: New-Moon Entry into the Slow-Fast and Surplus-Shrinkage Tables
59
推合朔入曆遲疾術曰：置入紀以來朔日積分，又以紀遲疾差分并之。 〈今用甲申紀，遲疾差分一百八十一萬九千七百九十二。〉 [9]以通周如一為積周，不盡者以日法約之，為日，不盡為日餘。 命日算外，即所求年天正十一月合朔入曆日。
Method for new-moon entry into the anomaly table: set the accumulated new-moon fraction since era entry and add the cycle's slow-fast difference parts. 〈Under the present Jiashen Cycle, the slow-fast difference parts are 1,819,792.〉 [9]Divide by the communication cycle for accumulated cycles; divide the remainder by the day divisor for whole days and the day remainder. Count days exclusive, and that gives the anomaly-table day for the eleventh-month new moon of the civil year sought.
60
甲子紀遲疾差二十四日日餘六萬三千五百六十八
Jiazi Cycle: slow-fast difference 24 days, day remainder 63,568.
61
甲戌紀遲疾差二十四日日餘四萬二千二百五十六
Jiaxu Cycle: slow-fast difference 24 days, day remainder 42,256.
62
甲申紀遲疾差二十四日日餘二萬九百四十四
Jiashen Cycle: slow-fast difference 24 days, day remainder 20,944.
63
甲午紀遲疾差二十三日日餘七萬四千五百八十四[10]
Jiawu Cycle: slow-fast difference 23 days, day remainder 74,584[10].
64
甲辰紀遲疾差二十三日日餘五萬三千二百七十二
Jiachen Cycle: slow-fast difference 23 days, day remainder 53,272.
65
甲寅紀遲疾差二十三日日餘三萬一千九百六十
Jiayin Cycle: slow-fast difference 23 days, day remainder 31,960.
66
求次月入曆日術曰：加一日、日餘七萬三千一百五十九，日餘滿日法從日，日滿二十七去之，亦除餘如周日餘； 日餘若不足，減一日，加周虛。 日滿二十七而餘不滿周日日餘者，為入曆值，周日法滿去之，為入曆一日。
Method for the next month's anomaly day: add 1 day and 73,159 day parts; carry at the day divisor, drop full 27-day cycles, and treat the remainder like the circuit-day remainder; if the day remainder is insufficient, borrow one day and add the cycle deficit. When days reach 27 but the remainder falls short of the circuit-day remainder, that is the table entry value; drop full circuit-day divisors to obtain one day entered into the table.
67
求望入曆術曰：加十四日、日餘五萬七千三百六十半。 又加，得後月曆日。
Method for full-moon table entry: add 14 days and 57,360½ day parts. Add again to obtain the following month's table day.
68
月行遲疾度及分損益率盈縮并盈縮積分
Lunar speed table: degrees and parts, excess-deficit rates, combined surplus-shrinkage, accumulated surplus-shrinkage.
69
一日十四度 〈三百六十一分〉 [11]〈〉 益六百八十盈初
Day 1: 14 degrees. 〈361 parts.〉 Editorial note eleven in the lunar motion table. Benefit 680; initial surplus.
70
二日十四度 〈三百分〉 益六百一十九盈六百八十盈積分七千五百五十
Day 2: 14 degrees. 〈300 parts.〉 Benefit 619; surplus 680; surplus accumulated parts 7,550.
71
三日十四度 〈二百三十六分〉 [12]〈〉 益五百五十五盈一千二百九十九盈積分一萬四千四百二十二
Day 3: 14 degrees. 〈236 parts.〉 Editorial note twelve in the lunar motion table. Benefit 555; surplus 1,299; surplus accumulated parts 14,422.
72
四日十四度 〈一百七十一分〉 益四百九十盈一千八百五十四盈積分二萬五百八十四
Day 4: 14 degrees. 〈171 parts.〉 Benefit 490; surplus 1,854; surplus accumulated parts 20,584.
73
五日十四度 〈九十九分〉 益四百一十八盈二千三百四十四盈積分二萬六千二十四
Day 5: 14 degrees. 〈99 parts.〉 Benefit 418; surplus 2,344; surplus accumulated parts 26,024.
74
六日十三度 〈四百七十一分〉 益二百八十五盈二千七百六十二盈積分三萬六百六十五
Day 6: 13 degrees. 〈471 parts.〉 Benefit 285; surplus 2,762; surplus accumulated parts 30,665.
75
七日十三度 〈二百六十六分〉 益八十盈三千四十七盈積分三萬三千八百二十九[13]
Day 7: 13 degrees. 〈266 parts.〉 Benefit 80; surplus 3,047; surplus accumulated parts 33,829[13].
76
八日十三度 〈六十一分〉 損一百二十五盈三千一百二十七盈積分三萬四千七百一十七
Day 8: 13 degrees. 〈61 parts.〉 Damage 125; surplus 3,127; surplus accumulated parts 34,717.
77
九日十二度 〈四百三十九分〉 [14]〈〉 損二百五十二盈三千二盈積分三萬三千三百二十九
Day 9: 12 degrees. 〈439 parts.〉 Editorial note fourteen in the lunar motion table. Damage 252; surplus 3,002; surplus accumulated parts 33,329.
78
十日十二度 〈三百三十八分〉 損三百五十三盈二千七百五十盈積分三萬五百三十一
Day 10: 12 degrees. 〈338 parts.〉 Damage 353; surplus 2,750; surplus accumulated parts 30,531.
79
十一日十二度 〈二百三十七分〉 損四百五十四[15]盈二千三百九十七盈積分二萬六千六百一十二
Day 11: 12 degrees. 〈237 parts.〉 Damage 454[15]; surplus 2,397; surplus accumulated parts 26,612.
80
十二日十二度 〈一百三十六分〉 損五百五十五盈一千九百四十三[16]盈積分二萬一千五百七十二
Day 12: 12 degrees. 〈136 parts.〉 Damage 555; surplus 1,943[16]; surplus accumulated parts 21,572.
81
十三日十二度 〈三十五分〉 損六百五十六盈一千三百八十八盈積分一萬五千四百一十
Day 13: 12 degrees. 〈35 parts.〉 Damage 656; surplus 1,388; surplus accumulated parts 15,410.
82
十四日十一度 〈四百六十四分〉 損七百三十二[17]盈七百三十二盈積分八千一百二十七
Day 14: 11 degrees. 〈464 parts.〉 Damage 732[17]; surplus 732; surplus accumulated parts 8,127.
83
十五日十二度 〈三十六分〉 益六百五十五縮初
Day 15: 12 degrees. 〈36 parts.〉 Benefit 655; initial shrinkage.
84
十六日十二度 〈一百九分〉 [18]〈〉 益五百八十二縮六百五十五縮積分七千二百七十二[19]
Day 16: 12 degrees. 〈109 parts.〉 Editorial note eighteen in the lunar motion table. Benefit 582; shrinkage 655; shrinkage accumulated parts 7,272[19].
85
十七日十二度 〈一百八十九分〉 益五百二縮一千二百三十七縮積分一萬三千七百三十四
Day 17: 12 degrees. 〈189 parts.〉 Benefit 502; shrinkage 1,237; shrinkage accumulated parts 13,734.
86
十八日十二度 〈二百九十分〉 益四百一縮一千七百三十七縮積分一萬九千三百九[20]
Day 18: 12 degrees. 〈290 parts.〉 Benefit 401; shrinkage 1,737; shrinkage accumulated parts 19,309[20].
87
十九日十二度 〈三百九十二分〉 [21]益二百九十九縮二千一百四十縮積分二萬三千七百五十九
Day 19: 12 degrees. 〈392 parts.〉 [21] Benefit 299; shrinkage 2,140; shrinkage accumulated parts 23,759.
88
二十日十二度 〈四百九十六分〉 益一百九十五縮二千四百三十九縮積分二萬七千七十九
Day 20: 12 degrees. 〈496 parts.〉 Benefit 195; shrinkage 2,439; shrinkage accumulated parts 27,079.
89
二十一日十三度 〈一百一十八分〉 益六十八縮二千六百三十四縮積分二萬九千二百四十四[22]
Day 21: 13 degrees. 〈118 parts.〉 Benefit 68; shrinkage 2,634; shrinkage accumulated parts 29,244[22].
90
二十二日十三度 〈二百四十三分〉 [23]〈〉 損五十七縮二千七百二縮積分二萬九千九百九十九
Day 22: 13 degrees. 〈243 parts.〉 Editorial note twenty-three in the lunar motion table. Damage 57; shrinkage 2,702; shrinkage accumulated parts 29,999.
91
二十三日十三度 〈三百八十八分〉 損二百二縮二千六百四十五縮積分二萬九千三百六十六
Day 23: 13 degrees. 〈388 parts.〉 Damage 202; shrinkage 2,645; shrinkage accumulated parts 29,366.
92
二十四日十四度 〈二十九分〉 損三百四十八縮二千四百四十三縮積分二萬七千一百二十三
Day 24: 14 degrees. 〈29 parts.〉 Damage 348; shrinkage 2,443; shrinkage accumulated parts 27,123.
93
二十五日十四度 〈一百七十四分〉 損四百九十三縮二千九十五縮積分二萬三千二百五十九
Day 25: 14 degrees. 〈174 parts.〉 Damage 493; shrinkage 2,095; shrinkage accumulated parts 23,259.
94
二十六日十四度 〈二百八十七分〉 [24]〈〉 損六百六縮一千六百二縮積分一萬七千七百八十六
Day 26: 14 degrees. 〈287 parts.〉 Editorial note twenty-four in the lunar motion table. Damage 606; shrinkage 1,602; shrinkage accumulated parts 17,786.
95
二十七日十四度 〈三百一十二分〉 [25]〈〉 損六百三十一縮九百九十六縮積分一萬一千五十八[26]
Day 27: 14 degrees. 〈312 parts.〉 Editorial note twenty-five in the lunar motion table. Damage 631; shrinkage 996; shrinkage accumulated parts 11,058[26].
96
周日十四度 〈三百三十九分小分九千六百八十四〉 損六百五十 〈小分九千六百八十四分〉 縮三百六十五縮積分四百五十二
Circuit day: 14 degrees. 〈339 parts; small parts 9,684.〉 Damage 650. 〈9,684 small parts.〉 Shrinkage 365; shrinkage accumulated parts 452.
97
推合朔交會月蝕定大小餘術曰：以入曆日餘乘所入曆下損益率，以小周六千七百五十一除之，所得以損益盈縮積分，為定積分。 [27]值盈者，以減本朔望小餘； 值縮者，加之。 加之滿日法者，[28]交會加時在後日。 減之，不足減者，減上一日，加下日法乃減之，交會加時在前日。 月蝕者，隨定大小餘為定日加時。
Method for fixing conjunction, nodal, and eclipse remainders: multiply the anomalistic day remainder by the tabulated excess-deficit rate, divide by 67,051, and apply the result to the surplus-shrinkage accumulated parts to obtain the fixed accumulated parts. [27]When the result is surplus, subtract it from the base syzygy small remainder; When the result is deficit, add it to the base syzygy small remainder. If the addition fills the day divisor, [28] the nodal-crossing hour falls on the next day. Subtract it; if the remainder is insufficient, borrow one day from the upper count, add the day divisor, then subtract—the nodal-crossing hour falls on the previous day. For lunar eclipse, take the fixed large and small remainders as the fixed day and hour.
98
推加時術曰：以時法六千二百四十六除定小餘，所得命以子起，算外。 朔望加時有餘不盡者，四之，如法得一為少，[29]二為半，三為太半。 又有餘者，三之，如法得一為強，半法以上排成之，不滿半法棄之。 以強并少為少強，并半為半強，并太為太強，得二強者為少弱，以之并少為半強，[30]以之并半為太弱。 以之并太為一弱，隨所在辰命之，則其強弱。 日之衝為破，月常在破下蝕。
Method for calculating the hour-mark: divide the fixed small remainder by the hour divisor, 6,246; name the result starting from zi (midnight), count exclusive. When the syzygy hour-mark leaves a remainder, quadruple it; one per divisor is "minor," [29] two is "half," three is "major half." If a remainder still remains, triple it; one per divisor is "strong"; at or above half the divisor, round up; below half the divisor, discard. Combine "strong" with "minor" to get "minor-strong," with "half" for "half-strong," with "major" for "major-strong"; two "strongs" make "minor-weak"; combine that with "minor" for "half-strong," [30] with "half" for "major-weak." Combine that with "major" for "one-weak"; name it against the double-hour in which it falls, and that gives its strength grade. The sun's opposition point is po (break); the moon is always eclipsed below it.
99
入曆值周日者術曰：以周日日餘乘損率，[31]以周日度小分并。 又以入曆日餘乘之為實，以小周乘周日日餘為法，實如法得一，以減縮積積分。 有餘者，以加本朔望小餘，小餘滿日法從大餘一，是為蝕後日。 推加時如上法。
Method when anomalistic entry falls on circuit day: multiply the circuit-day remainder by the damage rate, [31] and add the circuit-day degree small parts. Multiply again by the anomalistic day remainder for the dividend; multiply the lesser cycle by the circuit-day remainder for the divisor; take one per divisor from the dividend and subtract from the shrinkage accumulated parts. If a remainder remains, add it to the base syzygy small remainder; when the small remainder fills the day divisor, carry 1 to the big remainder—this is the day after the eclipse. Calculate the hour-mark by the method above.
100
推日月合朔弦望度術第五
Method V: Calculating Sun and Moon Degrees at Conjunction, Quarters, and Full Moon
101
推日度術曰：置入紀朔積日，以日度法乘之，滿周天去之，餘滿日度法為度，不盡為餘。 命度起牛前十二度， 〈牛前十二度，在斗十五度也。〉 宿次除之，不滿宿者，算外，即天正十一月朔夜半日所在度。
Method for calculating solar longitude: set the accumulated new-moon days since era entry, multiply by the solar-degree divisor, remove circuit heaven when filled; the quotient in solar-degree divisors is degrees, the remainder is fractional parts. Name the degrees starting twelve degrees before Ox, 〈Twelve degrees before Ox—that is fifteen degrees into Dipper.〉 Remove by lodge sequence; count exclusive at the partial lodge, and that gives the sun's degree at midnight on the new moon of the eleventh month of the civil year.
102
推日度又法，術曰：置周天三百六十五度、斗分一千四百七十七，以冬至去朔日數減一，餘以減周天度，冬至小餘減斗分，不足減者，減度一，加日度法，乃減之。 命起如上，即所求年天正十一月朔日夜半日所在度。
Alternate method for solar longitude: set circuit heaven at 365 degrees and Dipper fraction 1,477; subtract one from the days from winter solstice to new moon, subtract that from circuit-heaven degrees, and subtract the Dipper fraction from the winter-solstice small remainder; if insufficient, borrow one degree, add the solar-degree divisor, then subtract. Name starting as above, and that gives the sun's degree at midnight on the new moon of the eleventh month of the year sought.
103
求次月日所在度術曰：月大加三十度，月小加二十九度，求次日加一度，宿次除之，逕斗去其分一千四百七十七。
Method for the next month's solar degree: add 30 degrees in a long month, 29 in a short month; add 1 degree for the next day; remove by lodge sequence; when passing Dipper, subtract its extra fraction of 1,477.
104
推合朔日月共度術曰：以章歲乘朔小餘，以章月除之，所得為大分，不盡為小分，[32]以加夜半日度分，分滿日度法從度，命起如前，即所求年天正十一月合朔日月共度。
Method for the shared sun-moon degree at conjunction: multiply the new-moon small remainder by rule years and divide by rule months; the quotient is big parts, the remainder small parts; [32] add these to the midnight solar degree parts, carrying to degrees when parts fill the solar-degree divisor; name as before, and that gives the shared sun-moon degree at the eleventh-month conjunction of the year sought.
105
求次月合朔共度術曰：加度二十九、大分三千二百一十五、小分二千四百五十五，小分滿章月從大分，大分滿日度法從度，宿次除之，逕斗除其分，則次月合朔日月共度。
Method for the next month's shared conjunction degree: add 29 degrees, 3,215 big parts, and 2,455 small parts; carry from small parts to big parts when they fill rule months, and from big parts to degrees when they fill the solar-degree divisor; remove by lodge sequence and subtract Dipper's extra fraction—the result is the next month's shared sun-moon degree at conjunction.
106
推月度術曰：置入紀朔積日，以月周八萬一千一十二乘之，滿周天去之，餘以日度法約之為度，不盡為度分。 命度起牛前十二度，宿次除之，不滿宿者，算外，即所求年天正十一月朔夜半月所在度及分。
Method for lunar longitude: set the accumulated new-moon days since era entry, multiply by the lunar circuit 81,012, remove circuit heaven when filled, and reduce the remainder by the solar-degree divisor for degrees; the remainder is degree parts. Name degrees starting twelve degrees before Ox, remove by lodge sequence, count exclusive at the partial lodge, and that gives the moon's degree and parts at midnight on the new moon of the eleventh month of the year sought.
107
推月度又一法，術曰：以小周乘朔小餘為實，以章歲乘日法為法，實如法得一為度； 不滿法者，以章月除之，為大分，不盡為小分。 [33]所得以減合朔度及分，餘即所求年天正十一月朔夜半月所在度及分。
Alternate method for lunar longitude: multiply the new-moon small remainder by the lesser cycle for the dividend and rule years by the day divisor for the divisor; one per divisor from the dividend gives degrees; What does not fill the divisor, divide by rule months for big parts; the remainder is small parts. [33]Subtract the result from the conjunction degree and parts; the remainder is the moon's degree and parts at midnight on the new moon of the eleventh month of the year sought.
108
求次月度術曰：小月加度二十二、分二千六百五十一，大月加度三十五、分四千八百八十三，分滿日度法從度，宿次除之，不滿宿者，算外，[34]次月所在度。
Method for the next month's lunar degree: add 22 degrees and 2,651 parts in a short month, 35 degrees and 4,883 parts in a long month; carry to degrees when parts fill the solar-degree divisor; remove by lodge sequence and count exclusive at the partial lodge—[34] that is the next month's degree.
109
求次日月行度術曰：加度十三、分二千二百三十二，分滿日度法從度，宿次除之，逕斗去其分。
Method for the next day's lunar travel: add 13 degrees and 2,232 parts; carry to degrees when parts fill the solar-degree divisor; remove by lodge sequence and subtract Dipper's extra fraction.
110
求弦望日所在度術曰：加合朔度七、大分二千三百一十八、小分五千二百九十八、微分一，微分滿四從小分，小分滿章月從大分，大分滿日度法從度，命如上，則上弦日所在度。 又加得望、上弦、次月合朔。 [35]
Method for solar longitude at quarter and full moon: add to the conjunction degree 7 degrees, 2,318 big parts, 5,298 small parts, and 1 minute part; carry from minute parts to small parts when they fill four, from small parts to big parts when they fill rule months, and from big parts to degrees when they fill the solar-degree divisor; name as above, and that gives the sun's degree at first quarter. Repeat the addition to obtain full moon, last quarter, and the next month's conjunction. Editorial note thirty-five in the degree-calculation procedures.
111
斗二十六度牛八度女十二度虛十度
Dipper: 26 degrees; Ox: 8 degrees; Woman: 12 degrees; Void: 10 degrees.
112
危十七度室十六度壁九度
Rooftop: 17 degrees; House: 16 degrees; Wall: 9 degrees.
113
北方玄武七宿：九十八度一千四百七十七分
Northern Black Tortoise—seven lodges: 98 degrees, 1,477 parts.
114
奎十六度婁十二度胃十四度昴十一度
Legs: 16 degrees; Bond: 12 degrees; Stomach: 14 degrees; Hairy Head: 11 degrees.
115
畢十六度觜二度參九度
Net: 16 degrees; Turtle Beak: 2 degrees; Three Stars: 9 degrees.
116
西方白虎七宿：八十度
Western White Tiger—seven lodges: 80 degrees.
117
井三十三度鬼四度柳十五度星七度
Well: 33 degrees; Ghost: 4 degrees; Willow: 15 degrees; Star: 7 degrees.
118
張十八度翼十八度軫十七度
Extended Net: 18 degrees; Wings: 18 degrees; Chariot Base: 17 degrees.
119
南方朱鳥七宿：一百一十二度
Southern Vermilion Bird—seven lodges: 112 degrees.
120
角十二度亢九度氐一十五度房五度
Horn: 12 degrees; Neck: 9 degrees; Root: 15 degrees; Room: 5 degrees.
121
心五度尾十八度箕十一度
Heart: 5 degrees; Tail: 18 degrees; Winnowing Basket: 11 degrees.
122
東方倉龍七宿：七十五度
Eastern Azure Dragon—seven lodges: 75 degrees.
123
周天三百六十五度、六千六十分度之一千四百七十七。 通分，得二百二十一萬三千七百七十七，名曰周天分。
Circuit heaven: 365 degrees and 1,477 parts in 6,060 parts of a degree. Convert to a common denominator to obtain 2,213,777, called circuit-heaven parts.
124
五行沒滅易卦氣候上朔術第六
Method VI: Calculating the Five Phases, Extinction and Submersion Days, Hexagrams, Climate Pentads, and Upper New Moon
125
推五行用事日：水、火、木、金、土各王七十三日、小餘二百九十五、小分九、微分三，春木、夏火、秋金、冬水四立即其用事。 始求土者，置立春大小餘及分，以木王七十三日、小餘二百九十五、小分九、微分三加之，微分滿五從小分一，小分滿氣法二十四從小餘一，小餘滿蔀法從大餘一，大餘滿六十去之，命以紀，得季春土王日。 又加土王十八日、小餘一千五百八十八、小分二十、微分二，滿從命如上，即得立夏日。 求次如法。 又一法求土王用事日：各置四立大小餘及分，各減大餘十八、小餘一千五百八十八、小分二十、微分二，命以紀，算外，即四立土王日。 若大餘不足減者，加六十而後減之； 小餘不足減者，減取大餘一，加蔀法乃減之。
Calculating the days when the five phases hold sway: Water, Fire, Wood, Metal, and Earth each reign for 73 days, 295 small remainder, 9 small parts, and 3 minute parts; Wood in spring, Fire in summer, Metal in autumn, and Water in winter take domain at the four establishment days. To find Earth first: set the Establishment of Spring remainders and parts, add Wood's reign of 73 days, 295 small remainder, 9 small parts, and 3 minute parts; carry from minute parts to small parts when they fill five, from small parts to small remainder when they fill the qi divisor 24, and from small remainder to big remainder when they fill the obscuration divisor; remove sixty-day cycles from the big remainder and name against the era cycle to obtain the late-spring Earth-reign day. Then add Earth's reign of 18 days, 1,588 small remainder, 20 small parts, and 2 minute parts; carry and name as above to obtain the Establishment of Summer day. Find each subsequent day by the same method. Alternate method for the Earth-reign domain days: set the remainders and parts for each of the four establishment days, subtract 18 from the big remainder, 1,588 from the small remainder, 20 small parts, and 2 minute parts; name against the era cycle, count exclusive, and that gives the Earth-reign day at each establishment. If the big remainder is insufficient, add 60 and then subtract; if the small remainder is insufficient, borrow one from the big remainder, add the obscuration divisor, then subtract.
126
推沒滅術曰：因冬至積沒有小餘者，加積一，以沒分乘之，如沒法而一，為積日，不盡為沒餘。 以六旬去積日，餘為沒日，命以紀，算外，即所求年天正十一月冬至後沒日。
Method for extinction and submersion days: from the winter solstice, if the accumulated submersion has a small remainder, add 1 to the accumulation; multiply by submersion parts and divide by the submersion divisor for accumulated days; the remainder is the submersion remainder. Remove six-day decades from the accumulated days; name the remainder against the era cycle, count exclusive, and that gives the first submersion day after the winter solstice of the eleventh month of the year sought.
127
求次沒術曰：加沒日六十九、沒餘二萬七百六十四，沒餘滿沒法三萬一千七百七從沒日一，沒日滿六十去之，命以紀，算外，即次沒日。 [36]一歲常有五沒或六沒，小餘盡者為滅日。
Method for the next submersion day: add 69 submersion days and 20,764 to the submersion remainder; carry 1 to submersion days when the remainder fills the submersion divisor 31,707; remove sixty-day cycles from submersion days; name against the era cycle, count exclusive, and that gives the next submersion day. [36]A year usually has five or six submersion days; when the small remainder is exhausted, that day is an extinction day.
128
又，以冬至去朔日加沒日，冬至小餘滿蔀法從沒日，命日起天正十一月，如曆月大小除之，不足，除者入月算，命以朔，算外，即冬至後沒日。 求次沒：加沒沒日六十九，沒餘三千九百五十九、沒分二萬四千六百九十七，分滿沒法從沒餘，沒餘滿蔀法從沒日，[37]命起前沒月，曆月大小除之，[38]即後沒日及餘。
Also: add the submersion days to the days from winter solstice to new moon, carrying to submersion days when the winter-solstice small remainder fills the obscuration divisor; count from the eleventh month of the civil year and divide by long and short months as in the calendar; what does not fill a month enters the month count; name from the new moon, count exclusive, and that gives the submersion day after the winter solstice. Find the next submersion: add 69 submersion days, 3,959 to the submersion remainder, and 24,697 submersion parts; carry from parts to submersion remainder when they fill the submersion divisor, and from submersion remainder to submersion days when it fills the obscuration divisor; [37] count from the month of the prior submersion and divide by long and short months—[38] that gives the next submersion day and remainder.
129
推四正卦術曰：因冬至大小餘，即坎卦用事日； 春分，即震卦用事日； 夏至，即離卦用事日； 秋分，即兌卦用事日。
Method for the four standstill hexagrams: from the winter solstice remainders, that is the day when the Kan hexagram holds sway; the spring equinox is the day when the Zhen hexagram holds sway; the summer solstice is the day when the Li hexagram holds sway; the autumn equinox is the day when the Dui hexagram holds sway.
130
求中孚卦：加冬至小餘五千五百三十、小分九、微分一，微分滿五從小分，小分滿氣法從小餘，小餘滿蔀法從大餘，命以紀，算外，即中孚卦用事日。 其解加震，咸加離，賁加兌，亦如中孚加坎。
Find the Zhongfu hexagram: add 5,530 to the winter-solstice small remainder, 9 small parts, and 1 minute part; carry from minute parts to small parts when they fill five, from small parts to small remainder when they fill the qi divisor, and from small remainder to big remainder when it fills the obscuration divisor; name against the era cycle, count exclusive, and that gives the day when Zhongfu holds sway. For Jie add to Zhen, for Xian add to Li, for Bi add to Dui—by the same rule as Zhongfu added to Kan.
131
求次卦：加坎大餘六、小餘五百二十九、小分十四、微分四，微分滿五從小分，小分滿氣法從小餘，小餘滿蔀法從大餘，命以紀，算外，即復卦用事日。 大壯加震，姤加離，觀加兌，如中孚加坎。
Find the next hexagram: add 6 to Kan's big remainder, 529 to the small remainder, 14 small parts, and 4 minute parts; carry as above; name against the era cycle, count exclusive, and that gives the day when Fu holds sway. Dazhuang adds to Zhen, Gou adds to Li, Guan adds to Dui—by the same rule as Zhongfu added to Kan.
132
十一月，未濟、蹇、頤、中孚、復； 十二月，屯、謙、睽、升、臨； 正月，小過、蒙、益、漸、泰； 二月，需、隨、晉、解、大壯； 三月，訟、豫、蠱、革、夬； 四月，旅、師、比、小畜、乾； 五月，大有、家人、井、咸、始； 六月，鼎、豐、渙、履、遯； 七月，恒、節、同人、損、否； 八月，巽、萃、大畜、賁、觀； 九月，歸妹、无妄、明夷、困、剝； 十月，艮、既濟、噬嗑、大過、坤。
Eleventh month: Weiji, Jian, Yi, Zhongfu, Fu; Twelfth month: Tun, Qian, Kui, Sheng, Lin; First month: Xiaoguo, Meng, Yi, Jian, Tai; Second month: Xu, Sui, Jin, Xie, Dazhuang; Third month: Song, Yu, Gu, Ge, Guai; Fourth month: Lü, Shi, Bi, Xiaoxu, Qian; Fifth month: Dayou, Jiaren, Jing, Xian, Shi; Sixth month: Ding, Feng, Huan, Lü, Dun; Seventh month: Heng, Jie, Tongren, Sun, Pi; Eighth month: Xun, Cui, Daxu, Bi, Guan; Ninth month: Guimei, Wuwang, Mingyi, Kun, Bo; Tenth month: Gen, Jiji, Shike, Daguo, Kun.
133
四正為方伯，中孚為三公，復為天子，屯為諸侯，謙為大夫，暌為九卿，升還從三公，周而復始。 九三應上九，清淨微溫陽風； 九三應上六，絳赤決溫陰雨。 六三應上六，白濁微寒陰雨； 六三應上九，麴塵決寒陽風。 諸卦上有陽爻者，陽風； 上有陰爻者，陰雨。
The four standstill hexagrams correspond to regional lords; Zhongfu to the Three Dukes; Fu to the Son of Heaven; Tun to feudal lords; Qian to grandees; Kui to the nine ministers; Sheng returns in the train of the Three Dukes, and the cycle begins anew. When line nine-three responds to upper nine: clear skies, slight warmth, and a yang wind; When line nine-three responds to upper six: deep red skies, decidedly warm, with yin rain. When line six-three responds to upper six: white and murky, slightly cold, with yin rain; When line six-three responds to upper nine: yeast-colored dust, decidedly cold, with a yang wind. When a hexagram has a yang line on top, expect a yang wind; when a yin line is on top, expect yin rain.
134
推七十二候術曰：因冬至大小餘，即虎始交日，加大餘五、小餘四百四十一、小分八、微分一，微分滿三從小分，小分滿氣法從小餘，小餘滿蔀法從大餘，[39]命以紀，算外，所候日。
Method for the seventy-two climate pentads: from the winter solstice remainders, that is the day when tigers begin mating; add 5 to the big remainder, 441 to the small remainder, 8 small parts, and 1 minute part; carry as above; [39] name against the era cycle, count exclusive, and that gives the pentad day.
135
冬至虎始交芸始生荔挺出
Winter Solstice: tigers begin mating; cress begins to sprout; iris sprouts emerge.
136
小寒蚯蚓結麋角解水泉動
Lesser Cold: earthworms coil; elk antlers shed; spring waters stir.
137
大寒雁北向鵲始巢雉始雊
Greater Cold: wild geese turn north; magpies begin nesting; pheasants first crow.
138
立春鷄始乳東風解凍蟄蟲始振
Establishment of Spring: hens begin brooding; the east wind melts the ice; hibernating insects first stir.
139
雨水魚上冰獺祭魚鴻雁來
Rain Water: fish rise beneath the ice; otters offer fish; wild swans arrive.
140
驚蟄始雨水桃始華倉庚鳴
Awakening of Insects: rain water begins; peach first blooms; orioles call.
141
春分鷹化鳩玄鳥至雷始發聲
Spring Equinox: hawks turn to cuckoos; dark swallows arrive; thunder first sounds.
142
清明電始見蟄蟲咸動蟄蟲啟戶
Clear and Bright: lightning first appears; hibernating insects all stir; hibernating insects open their doors.
143
穀雨桐始花田鼠化為鴽[40]虹始見
Grain Rain: paulownia first blooms; field mice transform into quail[40]; the rainbow first appears.
144
立夏萍始生戴勝降於桑螻蟈鳴
Establishment of Summer: duckweed first sprouts; hoopoes alight on mulberry trees; tree frogs call.
145
小滿蚯蚓出王瓜生苦菜秀
Lesser Fullness of Grain: earthworms emerge; snake gourds grow; sow thistles flourish.
146
芒種靡草死小暑至螳蜋生
Grain in Beard: withering grasses die; Lesser Heat arrives; mantises hatch.
147
夏至鵙始鳴反舌無聲鹿角解
Summer Solstice: shrikes first call; mockingbirds fall silent; deer antlers shed.
148
小暑蟬始鳴半夏生木槿榮
Lesser Heat: cicadas first chirp; pinellia sprouts; hibiscus blooms.
149
大暑溫風至蟋蟀居壁鷹乃學習
Greater Heat: warm winds arrive; crickets retreat to walls; hawks begin training their young.
150
立秋腐草化螢土潤溽暑涼風至
Establishment of Autumn: rotting grass becomes fireflies; the earth moistens in humid heat; cool winds arrive.
151
處暑白露降寒蟬鳴鷹祭鳥
End of Heat: white dew descends; cold cicadas chirp; hawks sacrifice birds.
152
白露天地始肅暴風至鴻雁來
White Dew: heaven and earth begin to chill; gales arrive; wild swans come.
153
秋分玄鳥歸羣鳥養羞雷始收聲
Autumn Equinox: dark swallows return; flocks of birds store provisions; thunder first falls silent.
154
寒露蟄蟲附戶殺氣浸盛陽氣始衰
Cold Dew: hibernating creatures cling to doors; killing qi gradually swells; yang qi begins to wane.
155
霜降水始涸鴻雁來賓雀入大水化為蛤
Frost Descent: waters first dry up; wild swans arrive as guests; sparrows entering great waters transform into clams.
156
立冬菊有黃華豺祭獸水始冰
Establishment of Winter: chrysanthemums show yellow blooms; jackals sacrifice game; water first freezes.
157
小雪地始凍雉入大水化為蜃虹藏不見
Lesser Snow: earth first freezes; pheasants entering great waters transform into oysters; rainbows hide from view.
158
大雪冰益壯地始坼鶡旦不鳴
Greater Snow: ice grows stronger; earth first splits; crossbill-hawks at dawn cease calling.
159
術曰：因冬至虎始交後，五日一候。
The method says: from after the winter solstice pentad when tigers begin mating, there is one climate pentad every five days.
160
推上朔法：置入紀年減一，加八，以六律乘之，以六十去之，[41]餘為大餘，以甲子算外，上朔日。
Method for calculating the upper new moon: set era-entry years minus one, add eight, multiply by the six pitch pipes, and remove by sixty; [41] the remainder is the big remainder—count exclusive from jiazi, and that gives the upper new moon day.
161
推五星六通術第七
Method VII: Calculating the Five Planets and the Six Passages
162
上元壬子以來至春秋隱公元年己未，積十六萬六千五百七，算外； 至今大魏熙平二年，歲次丁酉，積十六萬七千七百四十五，算外。
From the renzi upper origin to the first year of Duke Yin of Spring and Autumn, year jiwei, the accumulated count is 166,507, exclusive; To the present, Great Wei Xiping year 2, cyclical year dingyou, the accumulated count is 167,745, exclusive.
163
木精曰歲星，其數二百四十一萬六千六百六十。
The essence of Wood is the Year Star; its number is 2,416,660.
164
火精曰熒惑，[42]其數四百七十二萬五千八百四十八。
The essence of Fire is Dazzling Delusion; [42] its number is 4,725,848.
165
土精曰鎮星，其數二百二十九萬一千二十一。 [43]
The essence of Earth is Quelling Star; its number is 2,291,021. Editorial note forty-three in the five-planet procedure.
166
金精曰太白，其數三百五十三萬八千一百三十一。
The essence of Metal is Great White; its number is 3,538,131.
167
水精曰辰星，其數七十萬二千一百八十二。
The essence of Water is Chronogram Star; its number is 702,182.
168
推五星：置上元以來盡所求年，減一，以周天二百二十一萬三千三百七十七乘之，名為六通之實。 以蔀法除之，所得為冬至積日，不盡為小餘。 以六旬去積日，不盡為大餘，命以甲子，算外，即冬至日。 以章歲五百五除冬至小餘，所得命子，算外，即律氣加時。
To calculate the five planets: set the years from the upper origin through the year sought, subtract one, multiply by circuit heaven, 2,213,377, and call the result the six-passage dividend. Divide by the obscuration divisor; the quotient is accumulated winter-solstice days, the remainder the small remainder. Remove six decades from the accumulated days; the remainder is the big remainder—name from jiazi, count exclusive, and that gives the winter solstice day. Divide the winter-solstice small remainder by rule years, 505; name the result from zi (midnight), count exclusive, and that gives the pitch-qi hour-mark.
169
五星各以其數為法，除六通實，所得為積合，不盡為合餘。 以合餘減法，餘為入歲度分，以日度約之，所得即所求天正十一月冬至後晨夕合度算及餘。 其金、水，以一合日數及合餘減合度算及餘，得一者為夕見，無所得為晨見。 若度餘不足減，減合度算一，加日度法乃減之。 命起牛前十二度，宿次除之，不滿宿者，算外，即天正十一月冬至後晨夕合度及餘。
Each planet uses its own number as divisor to divide the six-passage dividend; the quotient is accumulated conjunction, the remainder conjunction remainder. Subtract the conjunction remainder from the divisor; the remainder is the degree-fraction entered into the year—reduce by the solar-degree divisor, and the result is the dawn-and-dusk conjunction degree count and remainder after the eleventh-month winter solstice of the year sought. For Venus and Mercury, subtract the single-conjunction day number and conjunction remainder from the conjunction degree count and remainder; a quotient of one means evening appearance, none means morning appearance. If the degree remainder is insufficient, borrow one from the conjunction degree count, add the solar-degree divisor, then subtract. Name starting twelve degrees before Ox, remove by lodge sequence, count exclusive at the partial lodge, and that gives the dawn-and-dusk conjunction degree and remainder after the eleventh-month winter solstice.
170
求星合月及日：置冬至朔日數減一，以加合度算，以冬至小餘加度餘，度餘滿日度法去之，加度一，合度算變成合日算，餘為日餘。 命起天正十一月，如曆月大小除之，不滿月者，算外，星合月及日，有閏計之。
To find the planet-conjunction month and day: set the winter-solstice new-moon day number minus one and add it to the conjunction degree count; add the winter-solstice small remainder to the degree remainder; when the degree remainder fills the solar-degree divisor, remove it and carry one degree—the conjunction degree count becomes the conjunction day count, and the remainder is the day remainder. Name starting from the eleventh month of the civil year, divide by long and short months as in the calendar, count exclusive at the partial month, and that gives the planet-conjunction month and day—account for intercalation when present.
171
求後合月及日：以合終日數及餘如前入月算及餘，餘滿日度從日，曆月大小除之，起前合月，算外，即後合月及日。 其金、水，以一合日數及餘，加晨得夕，加夕得晨。
To find the next conjunction month and day: add the conjunction terminal day number and remainder to the entered month count and remainder as before; carry to days when the remainder fills the solar-degree divisor; divide by long and short months; start from the prior conjunction month, count exclusive, and that gives the next conjunction month and day. For Venus and Mercury, add the single-conjunction day number and remainder—adding to morning appearance yields evening appearance, and adding to evening yields morning.
172
求後合度：以行星度及餘加前合度算及餘，餘滿日度從度，命起前合度，宿次除之，不滿宿者，算外，即後合度及餘。 逕斗去其分一千四百七十七。
To find the next conjunction degree: add the planetary degree and remainder to the prior conjunction degree count and remainder; carry to degrees when the remainder fills the solar-degree divisor; name from the prior conjunction degree, remove by lodge sequence, count exclusive at the partial lodge, and that gives the next conjunction degree and remainder. When passing Dipper, subtract its extra fraction of 1,477.
173
歲星：合終日數三百九十八，合終日餘四千七百八十，行星三十三度，度餘三千三百三，周虛一千二百八十。
Year Star: conjunction terminal days 398, terminal day remainder 4,780, planetary motion 33 degrees, degree remainder 3,303, circuit void 1,280.
174
歲星：晨與日合，在日後伏，十六日、餘二千三百九十，行星二度、餘四千六百八十一半。 去日十三度半，晨見東方，順，疾，日行五十七分之十一，五十七日行十一度。 順，遲，日行九分，五十七日行九度而留。 不行，二十七日而旋。 逆，日行七分之一，八十四日退十二度。 [44]復留二十七日。 復順，遲，日行九分，五十七日行九度。 復疾，日行十一分，五十七日行十一度，在日前，夕伏西方。 順，遲，十六日、日餘二千三百九十，行星二度、餘四千六百八十一半，與日合。 凡一見三百六十六日，行星二十八度； 在日前後伏三十二日、餘四千七百八十，[45]行星五度、度餘三千三百三，復終於晨見。
Year Star: conjoined with the sun at dawn, it disappears behind the sun; after sixteen days plus remainder 2,390 and two degrees plus remainder 4,681½ of motion. Thirteen and a half degrees from the sun, it reappears in the east at dawn; prograde and swift: eleven parts per fifty-seven each day, eleven degrees in fifty-seven days. Prograde and slow: nine parts per day, nine degrees in fifty-seven days, then it stops. It remains motionless for twenty-seven days, then reverses. Retrograde: one part in seven per day, twelve degrees back in eighty-four days. [44] It halts again for twenty-seven days. Prograde and slow again: nine parts per day, nine degrees in fifty-seven days. Swift again at eleven parts per day, eleven degrees in fifty-seven days; standing ahead of the sun, it sets in the west at dusk. Prograde and slow for sixteen days plus day remainder 2,390 and two degrees plus remainder 4,681½, then it conjoins with the sun. In all, one appearance lasts 366 days and 28 degrees of motion; Hidden before and after the sun for thirty-two days plus remainder 4,780, [45] moving five degrees plus degree remainder 3,303, then completing the cycle at morning appearance.
175
熒惑：合終日數七百七十九，合終日餘五千一百八，[46]周虛九百五十二，行星四十九度，度餘二千一百五十四。
Dazzling Delusion: conjunction terminal days 779, terminal day remainder 5,108, [46] circuit void 952, planetary motion 49 degrees, degree remainder 2,154.
176
熒惑：晨與日合，在日後伏，七十一日、餘五千五百八十四，行星五十五度、餘四千八百四十五半。 去日十六度，晨見東方，順，疾，日行二十三分之十四，一百八十四日行一百一十二度。 順，遲，日行二十三分之十二，九十二日行四十八度而留。 不行，十一日而旋。 逆，日行六十二分之十七，六十二日退十七度。 [47]復留十一日。 復順，疾，日行十四分，一百八十四日行一百一十二度。 在日前，夕伏西方，順，七十一日、餘五千五百八十四，行星五十五度、度餘四千八百四十五半，而與日合。 凡一見六百三十六日，行星三百三度； 在日前後伏一百四十三日、餘五千一百八，行星一百一十一度、餘三千六百三十一，[48]過周四十九度、度餘二千一百五十四，復終於晨見。
Dazzling Delusion: conjoined with the sun at dawn, it disappears behind the sun; after seventy-one days plus remainder 5,584 and fifty-five degrees plus remainder 4,845½ of motion. Sixteen degrees from the sun, it reappears in the east at dawn; prograde and swift: fourteen parts per twenty-three each day, 112 degrees in 184 days. Prograde and slow: twelve parts per twenty-three each day, forty-eight degrees in ninety-two days, then it stops. It remains motionless for eleven days, then reverses. Retrograde: seventeen parts per sixty-two each day, seventeen degrees back in sixty-two days. [47] It halts again for eleven days. Prograde and swift again: fourteen parts per day, 112 degrees in 184 days. Standing ahead of the sun, it sets in the west at dusk; prograde for seventy-one days plus remainder 5,584 and fifty-five degrees plus degree remainder 4,845½, then it conjoins with the sun. In all, one appearance lasts 636 days and 303 degrees of motion; Hidden before and after the sun for 143 days plus remainder 5,108, moving 111 degrees plus remainder 3,631, [48] passing the circuit week by 49 degrees plus degree remainder 2,154, then completing the cycle at morning appearance.
177
鎮星：合終日數三百七十八日、餘三百四十一，行星十二度、餘四千九百二十四，周虛五千七百一十九。
Quelling Star: conjunction terminal days 378, terminal day remainder 341, planetary motion 12 degrees, degree remainder 4,924, circuit void 5,719.
178
鎮星：晨與日合，在日後伏，十八日、日餘一百七十半，行星二度、餘二千四百六十二，去日十五度半，晨見東方。 順，日行十二分之一，八十四日行七度而留。 不行，三十六日而旋。 逆，日行十七分之一，[49]一百二日退六度。 復留三十六日。 復順，日行十二分之一，八十四日行七度，在日前，伏西方。 順，十八日、日餘一百七十半，行星二度、餘二千四百六十二，而與日合。 凡見三百四十二日，行星八度； 在日前後，伏三十六日、日餘三百四十一，行星四度、度餘四千九百二十四，復終於晨見。
Quelling Star: conjoined with the sun at dawn, it disappears behind the sun; after eighteen days plus day remainder 170½ and two degrees plus remainder 2,462, it reappears in the east at dawn, fifteen and a half degrees from the sun. Prograde: one part in twelve per day, seven degrees in eighty-four days, then it stops. It remains motionless for thirty-six days, then reverses. Retrograde: one part in seventeen per day; [49] six degrees back in 102 days. It halts again for thirty-six days. Prograde again at one part in twelve per day, seven degrees in eighty-four days; standing ahead of the sun, it disappears in the west. Prograde for eighteen days plus day remainder 170½ and two degrees plus remainder 2,462, then it conjoins with the sun. In all, visibility lasts 342 days and 8 degrees of motion; Hidden before and after the sun for thirty-six days plus day remainder 341, moving four degrees plus degree remainder 4,924, then completing the cycle at morning appearance.
179
太白，金：再合終日數五百八十三日、日餘五千一百五十一，周虛九百九，行星二百九十一度、 〈亦曰一合日數。〉 度餘五千六百五半。 〈亦曰一合日餘。〉
Great White (Venus): double-conjunction terminal days 583, day remainder 5,151, circuit void 909, planetary motion 291 degrees, 〈Also called the single-conjunction day number.〉 degree remainder 5,605½. 〈Also called the single-conjunction day remainder.〉
180
太白：晨與日合，在日後伏，六日，退四度，去日十度，晨見東方。 逆，日行三分之二，九日退六度。 留，不行八日。 順，遲，日行十五分之十一，四十五日行三十三度。 順，疾，日行一度、十三分之二，九十一日行一百五度。 大疾，日行一度、十三分之三，九十一日行一百一十二度，在日後，晨伏東方。 順，四十一日、餘五千六百五半，行星五十一度、度餘五千六百五半，而與日合。 凡見東方二百四十四日，行星二百四十四度，[50]在日後伏，四十一日、餘五千六百五半，行星五十一度、餘五千六百五半，[51]而與日合。 〈見西方亦如之。〉 夕與日合，在前，伏四十一日、餘五千六百五半，行星五十一度、餘五千六百五半，去日十度，夕見西方。 順，疾，日行一度、十三分之三，九十一日行一百一十二度。 順，遲，日行一度、十三分之二，九十一日行一百五度。 順，遲，日行十五分之十一，四十五日行三十三度而留。 不行，八日而旋。 逆，日行三分之二，九日退六度，在日前，夕伏西方。 六日退四度，而與日合。 凡再見四百八十八日，行星四百八十八度； [52]在日前後，伏八十三日、餘五千一百五十一，行星一百三度、[53]度餘五千一百五十一，過周二百一十八度、度餘三千六百七十四，復終於晨見。
Great White: conjoined with the sun at dawn, it disappears behind the sun; after six days and four degrees of retrograde motion, it reappears in the east at dawn, ten degrees from the sun. Retrograde: two parts in three per day, six degrees back in nine days. It stops and remains motionless for eight days. Prograde and slow: eleven parts per fifteen each day, thirty-three degrees in forty-five days. Prograde and swift: one degree plus two-thirteenths per day, 105 degrees in ninety-one days. Very swift: one degree plus three-thirteenths per day, 112 degrees in ninety-one days; standing behind the sun, it disappears in the east at dawn. Prograde for forty-one days plus remainder 5,605½ and fifty-one degrees plus degree remainder 5,605½, then it conjoins with the sun. In all, eastern visibility lasts 244 days and 244 degrees of motion; [50] hidden behind the sun for forty-one days plus remainder 5,605½ and fifty-one degrees plus remainder 5,605½, [51] then it conjoins with the sun. 〈Appearance in the west follows the same pattern.〉 Conjoined with the sun at dusk, it disappears ahead of the sun for forty-one days plus remainder 5,605½ and fifty-one degrees plus remainder 5,605½; ten degrees from the sun, it reappears in the west at dusk. Prograde and swift: one degree plus three-thirteenths per day, 112 degrees in ninety-one days. Prograde and slow: one degree plus two-thirteenths per day, 105 degrees in ninety-one days. Prograde and slow: eleven parts per fifteen each day, thirty-three degrees in forty-five days, then it stops. It remains motionless for eight days, then reverses. Retrograde: two parts in three per day, six degrees back in nine days; standing ahead of the sun, it sets in the west at dusk. After six days and four degrees of retrograde motion, it conjoins with the sun. In all, two appearances total 488 days and 488 degrees of motion; [52] Hidden before and after the sun for eighty-three days plus remainder 5,151, moving 103 degrees plus [53] degree remainder 5,151, passing the circuit week by 218 degrees plus degree remainder 3,674, then completing the cycle at morning appearance.
181
水星：辰星再合終日數一百一十五、餘五千二百八十二，行星五十七度、 〈亦曰一合日數。〉 餘五千六百七十一， 〈亦曰一合日餘。〉 周虛七百七十八。
Mercury (Chronogram Star): double-conjunction terminal days 115, remainder 5,282, planetary motion 57 degrees, 〈Also called the single-conjunction day number.〉 remainder 5,671, 〈Also called the single-conjunction day remainder.〉 Circuit void 778.
182
辰星：與日合，在日後，伏十一日，退六度，去日十七度，晨見東方而留。 不行，四日。 順，遲，日行七分之五，七日行五度。 順，疾，日行一度、三分之一，十八日行二十四度，在日後，晨伏東方。 順，十七日、餘五千六百七十一，行星三十四度、[54]餘五千六百六十一，而與日合。 凡見東方二十九日，行星二十九度，[55]在日後，伏二十八日、餘五千六百七十一，行星三十四度、餘五千六百七十一，[56]而與日合。 見西方亦然。
Mercury (Chronogram Star): conjoined with the sun, it disappears behind the sun for eleven days and six degrees of retrograde motion; seventeen degrees from the sun, it reappears in the east at dawn and stops. It remains motionless for four days. Prograde and slow: five parts in seven per day, five degrees in seven days. Prograde and swift: one degree plus one-third per day, twenty-four degrees in eighteen days; standing behind the sun, it disappears in the east at dawn. Prograde for seventeen days plus remainder 5,671 and thirty-four degrees plus [54] remainder 5,661, then it conjoins with the sun. In all, eastern visibility lasts twenty-nine days and twenty-nine degrees of motion; [55] hidden behind the sun for twenty-eight days plus remainder 5,671 and thirty-four degrees plus remainder 5,671, [56] then it conjoins with the sun. Western visibility follows the same pattern.
183
辰星：夕與日合，在日前，伏十七日、餘五千六百七十一，行星三十四度、餘五千六百七十一，去日十七度，夕見西方。 順，疾，日行一度、三分之一，十八日行二十四度。 順，遲，日行七分之五，七日行五度，而留四日，在日前，夕伏西方。 逆，十一日退六度，而晨與日合。 凡再見五十八日，行星四十六度； [57]在日前後，伏五十七日、餘五千二百八十二，行星六十九度、[58]餘五千二百八十二，復終於晨見。
Mercury (Chronogram Star): conjoined with the sun at dusk, it disappears ahead of the sun for seventeen days plus remainder 5,671 and thirty-four degrees plus remainder 5,671; seventeen degrees from the sun, it reappears in the west at dusk. Prograde and swift: one degree plus one-third per day, twenty-four degrees in eighteen days. Prograde and slow: five parts in seven per day, five degrees in seven days; it stops for four days; standing ahead of the sun, it sets in the west at dusk. Retrograde for eleven days and six degrees back, then it conjoins with the sun at dawn. In all, two appearances total fifty-eight days and forty-six degrees of motion; [57] Hidden before and after the sun for fifty-seven days plus remainder 5,282, moving sixty-nine degrees plus [58] remainder 5,282, then completing the cycle at morning appearance.
184
斗一至牛五，星紀，丑。 牛五至危五，玄枵，子。
From one degree of Dipper to five degrees of Ox: Star Chronogram; earthly branch chou. From five degrees of Ox to five degrees of Rooftop: Xuanxiao; earthly branch zi.
185
危五至壁三，陬訾，亥。 壁三至婁八，降婁，戌。
From five degrees of Rooftop to three degrees of Wall: Zouzi; earthly branch hai. From three degrees of Wall to eight degrees of Bond: Jianglou; earthly branch xu.
186
婁八至畢二，大梁，酉。 畢二至井五，實沈，申。
From eight degrees of Bond to two degrees of Net: Great Bridge; earthly branch you. From two degrees of Net to five degrees of Well: Real Sink; earthly branch shen.
187
井五至鬼三，鶉首，未。 鬼三至張七，鶉火，午。
From five degrees of Well to three degrees of Ghost: Quail Head; earthly branch wei. From three degrees of Ghost to seven degrees of Extended Net: Quail Fire; earthly branch wu.
188
張七至軫一，鶉尾，巳。 軫一至亢三，壽星，辰。
From seven degrees of Extended Net to one degree of Chariot Crossbar: Quail Tail; earthly branch si. From one degree of Chariot Crossbar to three degrees of Neck: Longevity Star; earthly branch chen.
189
亢三至心四，大火，卯。 心四至斗一，析木，寅。
From three degrees of Neck to four degrees of Heart: Great Fire; earthly branch mao. From four degrees of Heart to one degree of Dipper: Split Wood; earthly branch yin.
190
校勘記
Collation Notes
191
歲中十二一年一十二次諸本「十二一年」作「十三年」。 錢氏考異卷三０云：「『歲中十三，年一十二次』，當云『歲中十二，一年一十二次』，誤併『二一』為『三』字。」 錢說是，今改正。
Twelve mid-year qi, one year—twelve stations: all editions read "thirteen years" for "twelve years in one." Qian's Collation Notes, vol. 30, say: "'Mid-year thirteen, years one twelve stations' should read 'Mid-year twelve, one year twelve stations'—'two-one' was wrongly merged into 'three.' Qian's view is correct; emended accordingly.
192
以蔀月除蔀日諸本「蔀月」作「用月」，「蔀日」作「眾日」。 按上文云「日法者一蔀之月數」當簡稱「蔀月」； 「周天分者一蔀之日數」，當簡稱「蔀日」。 「用月」、「眾日」顯誤，今改正。
Dividing obscuration days by obscuration months: all editions read "use months" for "obscuration months" and "crowd days" for "obscuration days." The text above says "Day divisor is the obscuration month count"—this should be abbreviated "obscuration months"; "Circuit-heaven parts are the obscuration day count"—this should be abbreviated "obscuration days." "Use months" and "crowd days" are clearly wrong; emended.
193
推交會術第三諸本皆無此標題。 錢氏考異卷三０云：「案正光術凡七篇，推月朔術第一，推二十四氣術第二，推合朔入曆遲疾盈縮第四，……獨不見第三。 蓋推合朔交會去交度以下，本別為一篇，當云推交會術第三，列於此行之前。 轉寫遺脫爾。」 錢說是，今據補。
Method III for Calculating Nodal Crossings: all editions lack this heading entirely. Qian's Collation Notes, vol. 30, say: "The Zhengguang procedure has seven sections: Method I for new moons, Method II for the twenty-four qi, Method IV for syzygy anomalistic entry and surplus-shrinkage, and the rest—Method III alone is missing. Presumably from "degrees from syzygy nodal crossing node" onward was originally a separate section—it should read "Method III for Calculating Nodal Crossings" and stand before this line. Lost in transcription." Qian's view is correct; supplied accordingly.
194
度餘三萬五千二百八十諸本「八十」作「二十八」。 按於甲申紀交會差九十八度、度餘七萬三千四百八十八，加一紀交會差四十九度、度餘三萬六千七百四十四，度餘滿日法進度一，得甲午紀交會差一百四十八度、度餘三萬五千二百八十。 原文「二十八」誤，今改正。
Degree remainder 35,280: all editions read "twenty-eight" for "eighty." From the Jiashen-cycle nodal difference of ninety-eight degrees plus degree remainder 73,488, add one cycle's nodal difference of forty-nine degrees plus degree remainder 36,744; when the degree remainder fills the day divisor, carry one degree—yielding the Jiawu-cycle nodal difference of 148 degrees plus degree remainder 35,280. The original "twenty-eight" is wrong; emended.
195
加日法諸本「日」作「入」。 按滿日法成一度，今退一度，則於度餘當加日法。 「入」字顯誤，今改正。
Add day divisor: all editions read "enter" for "day." When the day divisor fills to make one degree, and one borrows one degree, the degree remainder should have the day divisor added. The character "enter" is clearly wrong; emended.
196
甲申紀諸本「紀」作「如」。 按上文云「以紀交會差分加之」，今用者當為甲申紀交會差分，「如」顯為「紀」之誤，今改正。
Jiashen cycle: all editions read "as if" for "cycle." The text above says "add the cycle nodal-difference parts"; what is used here should be the Jiashen-cycle nodal-difference parts—"as if" is clearly an error for "cycle"; emended.
197
餘為前去度及餘諸本「及」作「乃」。 依文義改。
Remainder is prior departure degree and remainder: all editions read "then" for "and." Changed according to the sense of the text.
198
遲疾差分一百八十一萬九千七百九十二諸本「八十一萬九千」作「八十二萬千」。 按以甲申紀遲疾差二十四日乘日法七萬四千九百五十二，并日餘二萬九百四十四，得遲疾差分一百八十一萬九千七百九十二。 原數顯誤，今改正。
Slow-swift difference parts 1,819,792: all editions read "820,000-odd" for "819,000-odd." Using the Jiashen-cycle slow-swift difference of twenty-four days multiplied by day divisor 74,952, plus day remainder 20,944, yields slow-swift difference parts 1,819,792. The original number is clearly wrong; emended.
199
日餘七萬四千五百八十四諸本「七萬」作「十萬」。 按各紀間遲疾差之差率為日餘二萬一千三百十二。 從甲申紀遲疾差二十四日、日餘二萬九百四十四減此差率，日餘不足減，退一日，加日法再減，即得甲午紀遲疾差二十三日、日餘七萬四千五百八十四。 原文「十萬」誤，今改正。
Day remainder 74,584: all editions read "100,000" for "70,000." The inter-era slow-swift difference rate is day remainder 21,312. From the Jiashen-cycle slow-swift difference of twenty-four days plus day remainder 20,944 subtract this difference rate; when the day remainder is insufficient, borrow one day, add the day divisor, and subtract again—yielding the Jiawu-cycle slow-swift difference of twenty-three days plus day remainder 74,584. The original "100,000" is wrong; emended.
200
三百六十一分諸本「三百」均誤作「二百」。 按本表各項數字譌誤甚多，今依下述方法加以校算： (一) 月行遲疾度及分：以月日平行分六千七百五十一加減本日損益率 (損減、益加) ，以章歲五日五約之為度，餘為分。 (二) 損益率：以各日月實行度分化成分，減以月日平行六千七百五十一 (即小周) ，即得。 (三) 盈縮并：將本日前各日損益率累加得之。 (四) 盈縮積分：盈縮并乘日法七萬四千九百五十二，約以小周六千七百五十一，即得。 以下至校勘記[二六]，逕指出校正原文譌誤之所在，其具體演算不再贅述。
361 parts: all editions wrongly read "two hundred" for "three hundred." The figures in this table contain many errors; they are now recalculated by the following methods: (1) Lunar slow-swift degrees and parts: add or subtract the current day's damage-benefit rate to or from the lunar daily mean motion, 6,751, (subtract for damage, add for benefit) ; reduce by rule years, 505, to obtain degrees; the remainder is parts. (2) Damage-benefit rate: convert each day's actual lunar motion degrees and parts to parts, subtract the lunar daily mean, 6,751, (that is, the lesser cycle) ; and thus obtain it. (3) Surplus-shrinkage aggregate: accumulate the damage-benefit rates of all days before the current day. (4) Surplus-shrinkage accumulated parts: multiply the surplus-shrinkage aggregate by day divisor 74,952, reduce by lesser cycle 6,751, and thus obtain it. From here through Collation Note [26], errors in the original text are indicated directly; the detailed calculations are not repeated.
201
二百三十六分諸本「三十」訛「四十」，今據校算改正。
236 parts: all editions wrongly read "forty" for "thirty"; emended by recalculation.
202
盈積分三萬三千八百二十九諸本「三萬」訛「二萬」，今據校算改正。
Surplus accumulated parts 33,829: all editions wrongly read "twenty thousand" for "thirty thousand"; emended by recalculation.
203
十二度四百三十九分諸本「十二度」訛「十三度」，「三十九分」訛「一十九分」，今據校算改正。
Twelve degrees 439 parts: all editions wrongly read "thirteen degrees" for "twelve degrees" and "nineteen parts" for "thirty-nine parts"; emended by recalculation.
204
損四百五十四諸本「四百」訛「五百」，今據校算改正。
Damage 454: all editions wrongly read "five hundred" for "four hundred"; emended by recalculation.
205
[一六]盈一千九百四十三諸本「三」訛「二」，今據校算改正。
[the cited text] Surplus 1,943: all editions wrongly read the cited text for the cited text; emended by recalculation.
206
損七百三十二諸本「二」訛「一」，今據校算改正。
Damage 732: all editions wrongly read "one" for "two"; emended by recalculation.
207
一百九分諸本「一」訛「二」，今據校算改正。
109 parts: all editions wrongly read "two" for "one"; emended by recalculation.
208
縮積分七千二百七十二諸本「二百」訛「一百」，今據校算改正。
Shrinkage accumulated parts 7,272: all editions wrongly read "one hundred" for "two hundred"; emended by recalculation.
209
縮積分一萬九千三百九諸本「三百九」訛「三百七」，今據校算改正。
Shrinkage accumulated parts 19,309: all editions wrongly read "307" for "309"; emended by recalculation.
210
三百九十二分諸本「三百」訛「二百」，今據校算改正。
392 parts: all editions wrongly read "two hundred" for "three hundred"; emended by recalculation.
211
縮積分二萬九千二百四十四諸本「二百」訛「一百」，今據校算改正。
Shrinkage accumulated parts 29,244: all editions wrongly read "one hundred" for "two hundred"; emended by recalculation.
212
二百四十三分諸本「四十三」訛「三十三」，今據校算改正。
243 parts: all editions wrongly read "thirty-three" for "forty-three"; emended by recalculation.
213
二百八十七分諸本「二百」訛「三百」，今據校算改正。
287 parts: all editions wrongly read "three hundred" for "two hundred"; emended by recalculation.
214
三百一十二分諸本「一十二」訛「一十一」，今據校算改正。
312 parts: all editions wrongly read "eleven" for "twelve"; emended by recalculation.
215
縮積分一萬一千五十八諸本原均闕「一千」兩字，今據校算補。
Shrinkage accumulated parts 11,058: all editions originally lacked the two characters "one thousand"; supplied by recalculation.
216
為定積分諸本「為定積分」上有「加之」兩字。 按上既云「所得以損益盈縮積分」，則下不當再云「加之」。 本曆推合朔交會月蝕定大小餘術襲自景初諸曆，術文也基本相同，此句見宋書卷一二、卷一三曆志，景初、元嘉皆作「以損益盈縮積分為定積分」。 此處「加之」兩字顯衍，今刪。
Fixed accumulated parts: all editions have the two characters "add to" above "fixed accumulated parts." The text above already says "take what is obtained to damage or benefit the surplus-shrinkage accumulated parts," so below it should not again say "add to." This calendar's method for fixing syzygy, nodal, and eclipse remainders derives from the Jingchu and related calendars; the procedure text is essentially identical. This line appears in Song Shu calendar treatises, vols. 12–13; both Jingchu and Yuanjia read "take the surplus-shrinkage accumulated parts damaged or benefited as the fixed accumulated parts." The two characters "add to" here are clearly redundant; deleted.
217
加之滿日法者諸本脫「加之」兩字，今參照景初、元嘉術補。
When adding fills the day divisor: all editions omit the two characters "add to"; supplied with reference to the Jingchu and Yuanjia procedures.
218
如法得一為少諸本「如」作「加」。 錢氏考異卷三０云：「加當作如。」 錢說是，今據改。
As one per divisor gives minor: all editions read "add" for "as." Qian's Collation Notes, vol. 30, say: "Add should read as." Qian's view is correct; changed accordingly.
219
以之并少為半強諸本「之」作「定」。 錢氏考異卷三０云：「定當作之。」 錢說是，今據改。
Combine it with minor for half-strong: all editions read "fixed" for "it." Qian's Collation Notes, vol. 30, say: "Fixed should read it." Qian's view is correct; changed accordingly.
220
以周日日餘乘損率諸本「日餘」作「月餘」，顯誤，今改正。
Multiply circuit-day remainder by damage rate: all editions read "month remainder" for "day remainder"—clearly wrong; emended.
221
不盡為小分諸本脫「為」字，今依文義補。
What is not exhausted is the small part: all editions omit the character "is"; supplied according to the sense of the text.
222
不盡為小分諸本脫「分」字，今依文義補。
What is not exhausted is the small part: all editions omit the character "part"; supplied according to the sense of the text.
223
不滿宿者算外諸本脫「外」字，今依文義補。
What does not fill a lodge, count exclusive: all editions omit the character "exclusive"; supplied according to the sense of the text.
224
次月合朔諸本脫「次」字，今依文義補。
On “next month’s new-moon conjunction”: all editions (the cited text) omit the character the cited text; it is supplied herein on grounds of textual sense.
225
即次沒日諸本「日」作「月」，今依文義改。
On “that is, the next extinction day”: in all editions (the cited text) the cited text appears as the cited text; the reading is emended herein according to textual sense.
226
沒餘滿蔀法從沒日諸本脫「沒餘」及「法」字，今依文義補。
On “when the extinction remainder fills the obscuration divisor, carry from the extinction day”: all editions (the cited text) omit the cited text and the cited text; both are supplied herein on grounds of textual sense.
227
曆月大小除之諸本「月」作「日」，今依文義改。
On “divide by the calendrical month’s great and small portions”: in all editions (the cited text) the cited text appears as the cited text; the reading is emended herein according to textual sense.
228
小餘滿蔀法從大餘諸本脫「法」字，今依文義補。
On “when the minor remainder fills the obscuration divisor, carry from the major remainder”: all editions (the cited text) omit the cited text; it is supplied herein on grounds of textual sense.
229
田鼠化為鴽諸本脫「化」字。 按禮月令：「田鼠化為鴽」，本志下興和曆推七十二候也作「田鼠化為鴽」。 今據補。
On “field mice transform into quails”: all editions (the cited text) omit the character the cited text. Note: the Monthly Ordinances in the Rites gives “field mice transform into quails,” and the Xinghe calendar’s derivation of the seventy-two seasonal terms later in this treatise uses the same wording. The character is restored on that authority.
230
以六十去之諸本「六十」作「六千」。 殿本考證云：「千字疑十字之訛。 又下卷作『以六旬去之』。」 按考證說是，今據改。
On “remove it by sixty”: in all editions (the cited text) the cited text appears as the cited text. the cited text observes: “The character the cited text is likely a corruption of the cited text. The lower fascicle likewise reads, ‘remove it by six ten-day weeks.’” The verification is accepted; the text is emended accordingly.
231
火精曰熒惑諸本「熒惑」下有「星」字。 按據各史五星命名及後興和曆例，熒惑下「星」字衍，今刪。
On “the essence of fire is called the Sparkling Deluder”: all editions (the cited text) add the character the cited text after the cited text. On the authority of the five-planet nomenclature in the various histories and the Xinghe calendar’s precedents, the character the cited text after the cited text is redundant and is deleted.
232
其數二百二十九萬一千二十一諸本「二百」作「三百」。 按以鎮星合終日數三百七十八日乘蔀法六千六十，并以餘三百四十一，得二百二十九萬一千二十一。 原文「三百」誤，今改正。
On “its number is 2,291,021”: in all editions (the cited text) the cited text appears as the cited text. Calculation: the Quelling Star’s total conjunction of 378 days, multiplied by the obscuration divisor 6,060 and combined with the remainder 341, yields 2,291,021. The original reading “three hundred” is in error and is corrected herein.
233
退十二度諸本「十二」作「十三」。 按日行七分之一，以七約八十四，得十二度。 原文「三」誤，今改正。
On “retreat twelve degrees”: in all editions (the cited text) the cited text appears as the cited text. At one-seventh of a degree per day, 84 divided by 7 yields twelve degrees. The original reading is in error—thirteen should be twelve—and is corrected herein.
234
餘四千七百八十諸本「八」下脫「十」字。 按歲星在日後伏十六日、餘二千三百九十，在日前伏十六日、餘二千三百九十，則在日前後共伏三十二日、餘四千七百八十。 今據補。
On “remainder 4,780”: all editions (the cited text) omit the character the cited text after the cited text. The Year Star occults sixteen days with remainder 2,390 after the sun, and sixteen days with remainder 2,390 before the sun; together that is thirty-two days of occultation with remainder 4,780. The missing character is restored on that authority.
235
合終日餘五千一百八諸本「一百」作「一十」。 按熒惑數四百七十二萬五千八百四十八約以蔀法，得合終日數七百七十九，合終日餘五千一百八。 原文「十」字誤，今改正。
On “total conjunction day-remainder 5,108”: in all editions (the cited text) the cited text appears as the cited text. Dividing the Sparkling Deluder’s figure of 4,725,848 by the obscuration divisor gives 779 total conjunction days and a day-remainder of 5,108. The original character the cited text is in error and is corrected herein.
236
六十二日退十七度諸本「十」下脫「七」字。 按日行六十二分之十七，則六十二日應行十七度。 今據補。
On “in sixty-two days retreat seventeen degrees”: all editions (the cited text) omit the character the cited text after the cited text. At seventeen sixty-second parts per day, sixty-two days should yield seventeen degrees of motion. The missing character is restored on that authority.
237
餘三千六百三十一諸本「三十」作「四十」。 按熒惑在日後，伏，行星五十五度、餘四千八百四十五半； 在日前，伏，行星五十五度、餘四千八百四十五半，則日前後伏時共行星一百一十一度、餘三千六百三十一。 原文「四十」誤，今改正。
On “remainder 3,631”: in all editions (the cited text) the cited text appears as the cited text. After the sun the Sparkling Deluder occults with fifty-five degrees of stellar motion and remainder 4,845 and a half; and before the sun it occults with fifty-five degrees of stellar motion and remainder 4,845 and a half; taken together, occultation before and after the sun yields one hundred eleven degrees of stellar motion with remainder 3,631. The original reading “forty” is in error and is corrected herein.
238
日行十七分之一諸本「行」上脫「日」字，今依文義補。
On “daily motion one-seventeenth of a degree”: all editions (the cited text) omit the cited text before the cited text; it is supplied herein on grounds of textual sense.
239
行星二百四十四度諸本「十」下脫「四」字。 按依太白於東方所見各日行星度，順加、逆減，共計行星度當為二百四十四。 今據補。
On “stellar motion two hundred forty-four degrees”: all editions (the cited text) omit the character the cited text after the cited text. Summing the Great White’s daily degrees of motion when seen in the east—adding in direct motion and subtracting in retrograde—the total stellar motion should be two hundred forty-four degrees. The missing character is restored on that authority.
240
在日後伏四十一日餘五千六百五半行星五十一度餘五千六百五半按太白在日後先伏六日，退四度； 次伏四十一日、餘五千六百五半，順，行星五十一度、度餘五千六百五半，共伏四十七日、餘五千六百五半，行星四十七度、度餘五千六百五半。 「四十一日」當作「四十七日」，「五十一度」當作「四十七度」。 原文未將先伏之六日、退四度計算在內，以致錯誤； 下前後伏總計數亦同。
“After the sun, occultation of forty-one days with remainder 5,605 and a half, and stellar motion of fifty-one degrees with degree-remainder 5,605 and a half.” Note: after the sun the Great White first occults six days and retreats four degrees; then occults forty-one days with remainder 5,605 and a half while advancing fifty-one degrees in direct motion with the same degree-remainder; in total that is forty-seven days of occultation with remainder 5,605 and a half and forty-seven degrees of stellar motion with the same degree-remainder. “Forty-one days” should read “forty-seven days,” and “fifty-one degrees” should read “forty-seven degrees.” The original text did not account for the initial six-day occultation and four-degree retreat, which led to the error; the aggregate figures for occultation before and after the sun below are affected in the same way.
241
凡再見四百八十八日行星四百八十八度諸本兩「八十」下俱脫「八」字。 按太白見東方二百四十四日，行星二百四十四度； 見西方亦二百四十四日，行星二百四十四度，凡再見共四百八十八日，行星四百八十八度。 今據補。
On “in all, two appearances totaling four hundred eighty-eight days and four hundred eighty-eight degrees of stellar motion”: in all editions (the cited text) both occurrences of the cited text lack the character the cited text. The Great White is seen in the east for two hundred forty-four days with two hundred forty-four degrees of stellar motion; and in the west likewise for two hundred forty-four days with two hundred forty-four degrees; two appearances together thus yield four hundred eighty-eight days and four hundred eighty-eight degrees of motion. The missing characters are restored on that authority.
242
在日前後伏八十三日餘五千一百五十一行星一百三度按「八十三日」當作「九十五日」，「一百三度」當作「九十五度」，原文計算錯誤。 詳見校勘記[五一]。
“Occultation before and after the sun of eighty-three days with remainder 5,151, and stellar motion of one hundred three degrees.” Note: “eighty-three days” should read “ninety-five days,” and “one hundred three degrees” should read “ninety-five degrees”; the original calculation is in error. For full discussion see Collation Note [the cited text].
243
行星三十四度諸本「三十」作「四十」。 按下辰星在日前伏時行星度數作「三十四」，參照伏時總行星度，「四十四」乃「三十四」之誤，今改正。
On “stellar motion thirty-four degrees”: in all editions (the cited text) the cited text appears as the cited text. Below, the Chronogram Star’s degrees of motion during occultation before the sun are given as thirty-four; measured against the total degrees during occultation, “forty-four” is a mistake for “thirty-four” and is corrected herein.
244
行星二十九度諸本「九」作「二」。 按辰星晨見東方，先行五度，次行二十四度，共行星二十九度。 原文顯誤，今改正。
On “stellar motion twenty-nine degrees”: in all editions (the cited text) the cited text appears as the cited text. When the Chronogram Star appears in the morning east, it first advances five degrees and then twenty-four, for a total stellar motion of twenty-nine degrees. The original reading is plainly in error and is corrected herein.
245
行星三十四度餘五千六百七十一按辰星在日後，先伏十一日，退六度； 次伏十七日、餘五千六百七十一，順，行星三十四度、餘五千六百七十一，共伏二十八日、餘五千六百七十一，行星二十八度、餘五千六百七十一。 志未將先伏退行度數減去，致總日數未誤，而總行星度數「二十八」誤作「三十四」； 下前后伏總行星度數亦同以致誤。
“Stellar motion of thirty-four degrees with remainder 5,671.” Note: after the sun the Chronogram Star first occults eleven days and retreats six degrees; then occults seventeen days with remainder 5,671 while advancing thirty-four degrees in direct motion with the same remainder; in total that is twenty-eight days of occultation with remainder 5,671 and twenty-eight degrees of stellar motion with the same remainder. The treatise did not subtract the retrograde degrees from the initial occultation; the total days are unaffected, but the aggregate stellar motion wrongly reads thirty-four instead of twenty-eight; the combined stellar-motion totals for occultation before and after the sun below are wrong for the same reason.
246
行星四十六度按辰星見東方，共行星二十九度，見西方，亦行星二十九度，再見共行星五十八度。 志誤。
“Stellar motion forty-six degrees.” Note: the Chronogram Star moves twenty-nine degrees when seen in the east and twenty-nine when seen in the west; two appearances together yield fifty-eight degrees. The treatise is mistaken.
247
行星六十九度按「六十九」當作「五十七」，詳見校勘記[五六]。
“Stellar motion sixty-nine degrees.” Note: “sixty-nine” should read “fifty-seven”; for full discussion see Collation Note [the cited text].