1
自天地剖判，日月運行，剛柔相摩，寒暑交謝，分之以氣序，紀之以星辰，弦望有盈缺，明晦有修短。 古先哲王則之成化，迎日推策，各有司存，以天下之至王，盡生民之能事，先天而天弗違，後天而奉天時。 及卯金受命，年曆屢改，當塗啟運，日官變業，分路揚鑣，異門馳騖，回互靡定，交錯不等。 豈是人情淺深，苟相違異？ 蓋亦天道盈縮，欲止不能。
Ever since Heaven and Earth were divided, the sun and moon have run their courses, yin and yang have ground against each other, and cold and heat have given way in turn. The seasons of qi mark their divisions, the stars mark their reckonings, the crescent and full moon know fullness and lack, and light and dark know long and short. The sage kings of old patterned themselves on this to civilize the world. They greeted the sun and advanced the calendar, each with his proper office, and as supreme rulers under Heaven they brought the people's capacities to fulfillment—acting ahead of Heaven without defying it, and acting after Heaven while honoring its seasons. When Han received the Mandate, calendars were changed again and again; when Wei rose to power, astronomers altered their methods. Schools went their separate ways, rival traditions raced along, and systems overlapped without ever settling into agreement. Was this merely a matter of scholars' temperaments clashing for no good reason? Rather, it reflected the ebb and flow of Heaven's way, which cannot be halted even when one wishes to.
2
正光之曆既行於世，發元壬子，置差令朔。 [1]測影清臺，懸炭之期或爽； 候氣重室，布灰之應少差。 伏惟陛下當璧膺符，大橫恊兆，乘機虎變，撫運龍飛，苞括九隅，牢籠萬宇，四海來王，百靈受職。 大丞相、勃海王降神挺生，固天縱德，負圖作宰，知機成務，撥亂反正，決江疏河，效顯勤王，勳彰濟世。 功成治定，禮樂惟新，以履端歸餘，術數未盡，乃命兼散騎常侍執讀臣李業興，大丞相府東閤祭酒、夷安縣開國公臣王春，大丞相府戶曹參軍臣和貴興等，委其刊正。 但回舍有疾徐，推步有疏密，不可以一方知，難得以一途揆。 大丞相主簿臣孫搴，驃騎將軍、左光祿大夫臣曄，前給事黃門侍郎臣季景，勃海王世子開府諮議參軍事、定州大中正臣崔暹，業興息國子學生、屯留縣開國子臣子述等，並令參豫，定其是非。
After the Zhengguang calendar had come into use, with its era beginning at the renzi year, discrepancies were set so that new moons would fall correctly. [1] Shadow measurements at the Clear Terrace, and the seasons marked by suspended charcoal, sometimes failed to agree; and in the sealed chamber for observing the qi of the seasons, the ash-line responses were slightly off. We humbly observe that Your Majesty has received Heaven's mandate, that great omens unite in your favor, that you seize the moment with tiger-like vigor and ascend with dragon-like fortune, encompassing all corners of the earth and holding the universe in your grasp, while the four seas pay homage and the hundred spirits take their posts. The Grand Chancellor, Prince of Bohai, was a man of Heaven-sent birth and Heaven-given virtue. Bearing the cosmic chart, he governed as chief minister, seized the right moment to complete great tasks, restored order from chaos, opened rivers and canals, and won renown both for loyal service to the throne and for deeds that saved the world. Once success was won and order restored, rites and music were renewed; yet the calendar's year-beginnings and intercalary remainders were still not fully settled in the computations. He therefore charged Li Yexing, Reader and Concurrent Regular Attendant of Scattered Cavalry, Wang Chun, Libationer of the Eastern Pavilion of the Grand Chancellor's office and Duke of Yi'an, He Guixing, staff member of the Household Section, and others with revising the calendar. Yet orbital circuits have faster and slower phases, and stepwise calculations have tighter and looser intervals; no single method can capture everything, nor can one path alone judge them all. Sun Qian, chief clerk to the Grand Chancellor; Ye, General of Agile Cavalry and Left Grand Master of Splendid Happiness; Ji Jing, former Attendant of the Yellow Gate; Cui Xian, staff adviser to the Heir of Bohai and chief rectifier of Ding Province; Zi Shu, Yexing's son, a student at the Imperial University and Baron of Tunliu; and others were all ordered to join the deliberations and settle disputed points.
3
臣等職司其憂，猶恐未盡。 竊以蒙戎為飾，必藉眾腋之華； 輪奐成宇，寧止一枝之用。 必集名勝，更共修理。 左光祿大夫臣盧道約，大司農卿、彭城侯臣李諧，左光祿大夫、東雍州大中正臣裴獻伯，散騎常侍、西兗州大中正臣溫子昇，太尉府長史臣陸操，尚書右丞、城陽縣開國子臣盧元明，中書侍郎臣李同軌，前中書侍郎臣邢子明，中書侍郎臣宇文忠之，前司空府長史、建康伯臣元仲悛，大丞相法曹參軍臣杜弼，尚書左中兵郎中、定陽伯臣李溥濟，尚書起部郎中臣辛術，尚書祠部郎中臣元長和，前青州驃騎府司馬、安定子臣胡世榮，太史令、盧鄉縣開國男臣趙洪慶，太史令臣胡法通，應詔左右臣張喆，員外司馬督臣曹魏祖，太史丞郭慶，太史博士臣胡仲和等，或器標民譽，或術兼世業，並能顯微闡幽，表同錄異，詳考古今，共成此曆。 甲為日始，子實天正，命曆置元，宜從此起。 運屬興和，以年號為目，豈獨太初表於漢代，景初冠於魏曆而已。 謹以封呈，乞付有司，依術施用。
We who hold this office remain anxious that our work may still fall short. We submit that a fur-trimmed robe needs many patches of fine fur to shine; and a great hall cannot be built from a single rafter alone. We must therefore gather eminent scholars to revise the work together. Lu Daoyue, Left Grand Master of Splendid Happiness; Li Xie, Minister of Grand Agriculture and Marquis of Pengcheng; Pei Xianbo, Left Grand Master and chief rectifier of Eastern Yong; Wen Zisheng, Regular Attendant of Scattered Cavalry and chief rectifier of Western Yan; Lu Cao, chief clerk of the Grand Commandant's office; Lu Yuanming, right assistant of the Masters of Writing and Baron of Chengyang; Li Tonggui, Secretariat Gentleman; Xing Ziming, former Secretariat Gentleman; Yuwen Zhongzhi, Secretariat Gentleman; Yuan Zhongjun, former chief clerk of the Minister of Works and Baron of Jiankang; Du Bi, legal-section staff member to the Grand Chancellor; Li Puji, director of the Left Central Troops Bureau and Baron of Dingyang; Xin Shu, Construction Bureau director; Yuan Changhe, Sacrificial Affairs Bureau director; Hu Shirong, former chief of staff of Qing Province's Agile Cavalry office and Viscount of Anding; Zhao Hongqing, Director of the Grand Astrologer and Baron of Luxiang; Hu Fatong, Director of the Grand Astrologer; Zhang Zhe, attendant at imperial command; Cao Weizu, supernumerary chief of staff; Guo Qing, assistant director of the Grand Astrologer; Hu Zhonghe, erudite of the Grand Astrologer; and others—some renowned among the people, some heirs to hereditary expertise—all able to bring out fine points and clarify obscurities, record agreements and differences, scrutinize past and present, and together complete this calendar. Jia marks the start of the day-cycle, and zi is the true celestial new year; the calendar's era beginning should be established from this point. As fortune now belongs to the Xinghe era, the reign title should head the calendar—just as Taichu headed Han reckoning and Jingchu crowned Wei's calendar. We respectfully submit this in sealed form and ask that the appropriate offices be ordered to put its methods into practice.
4
詔以新曆示齊獻武王田曹參軍信都芳，芳關通曆術，駁業興曰：「今年十二月二十日，新曆歲星在營室十三度，[2]順，疾； 天上歲星在營室十一度。 今月二十日，新曆鎮星在角十一度，留； 天上鎮星在亢四度，留。 今月二十日，新曆太白在斗二十五度，晨見，逆行； 天上太白在斗二十一度，逆行。 便為差殊。」
The throne ordered the new calendar shown to Xin Du Fang, a staff member in the Fields Section serving Prince Xianwu of Qi. Well versed in calendrical science, Fang challenged Yexing: "On the twentieth day of the twelfth month this year, the new calendar puts Jupiter in Encampment at thirteen degrees, [2] moving direct and fast; yet in the sky Jupiter is in Encampment at eleven degrees. On the twentieth of this month, the new calendar places Saturn in Horn at eleven degrees, stationary; but in the sky Saturn is in Neck at four degrees, stationary. On the twentieth of this month, the new calendar places Venus in Dipper at twenty-five degrees, visible in the morning, retrograde; yet in the sky Venus is in Dipper at twenty-one degrees, retrograde. These are clear discrepancies."
5
業興對曰：
Yexing replied:
6
歲星行天，伺候以來八九餘年，恒不及二度。 今新曆加二度。 至於夕伏晨見，纖毫無爽。 今日仰看，如覺二度，及其出沒，還應如術。 鎮星，自造壬子元以來，歲常不及，故加壬子七度，亦知猶不及五度，適欲并加，恐出沒頓校十度、十日，將來永用，不合處多。 太白之行，頓疾頓遲，取其會歸而已。 近十二月二十日，晨見東方，新舊二曆推之，分寸不異。 行星三日，頓校四度。 如此之事，無年不有，至其伏見，還依術法。
Since we began observing Jupiter's course eight or nine years ago, it has consistently lagged by less than two degrees. The new calendar therefore adds two degrees. For evening disappearance and morning appearance, there is not the slightest error. Viewed in the sky today it may look like two degrees off, but at its risings and settings it will still match the computation. Saturn, since the renzi era was established, has year after year lagged behind observation; seven degrees were therefore added to the renzi reckoning, yet it still falls about five degrees short. To add the full amount at once would throw risings and settings off by as much as ten degrees and ten days, and in long use the errors would multiply. Venus alternates between swift and slow motion; the calendar need only capture its convergent return. Around the twentieth of the twelfth month, at its morning appearance in the east, old and new calendars agree to the finest fraction. In three days a planet's position can shift by as much as four degrees. Such discrepancies occur every year; when planets hide or appear, they still follow the computed rules.
7
又芳唯嫌十二月二十日星有前却。 業興推步已來，三十餘載，上算千載之日月星辰有見經史者，與涼州趙𢾺、劉義隆廷尉卿何承天、劉駿南徐州從事史祖沖之參校，業興甲子元曆長於三曆一倍。 考洛京已來四十餘歲，五星出沒，歲星、鎮星、太白，業興曆首尾恒中，及有差處，不過一日二日、一度兩度； 三曆之失，動校十日十度。 熒惑一星，伏見體自無常，或不應度。 祖沖之曆多甲子曆十日六度，何承天曆不及三十日二十九度； 今曆還與壬子同，不有加增。 辰星一星，沒多見少，及其見時，與曆無舛，今此亦依壬子元不改。 太白、辰星，唯起夕合為異。 業興以天道高遠，測步難精，五行伏留，推考不易，人目仰窺，未能盡密，但取其見伏大歸，略其中間小謬，如此曆便可行。 若專據所見之驗，不取出沒之效，則曆數之道其幾廢矣。 夫造曆者，節之與朔貫穿於千年之間，閏餘斗分推之於毫釐之內。 必使盈縮得衷，間限數合，周日小分不殊錙銖，陽曆陰曆纖芥無爽，損益之數驗之交會，日所居度考之月蝕，上推下減，先定眾條，然後曆元可求，猶甲子難值。 又雖值甲子，復有差分，如此踳駁，參錯不等。 今曆發元甲子，七率同遵，合璧連珠，其言不失。 法理分明，情謂為可。 如芳所言，信亦不謬。 但一合之裏星度不驗者，至若合終必還。 依術，鎮星前年十二月二十日見差五度，今日差三度； 太白前差四度，今全無差。 以此準之，見伏之驗，尋效可知，將來永用，大體無失。
Fang's complaint, moreover, concerns only the forward or backward discrepancy of the stars on the twentieth of the twelfth month. For more than thirty years Yexing has worked the calendar, checking a millennium of sun, moon, and planetary records in the classics against the systems of Zhao Fei of Liangzhou, He Tiancheng of Liu Song, and Zu Chongzhi—and his jiazi-era calendar outperforms all three by a wide margin. Over the forty-odd years since Luoyang was capital, for the risings and settings of the five planets—Jupiter, Saturn, and Venus—Yexing's calendar has consistently matched observation; where it errs, the gap is at most a day or two, a degree or two; whereas the three rival calendars regularly miss by ten days and ten degrees. Mars alone is inherently irregular in its appearances and disappearances and sometimes fails to match the predicted degrees. Zu Chongzhi's system runs ten days and six degrees ahead of the jiazi calendar, while He Tiancheng's falls twenty-nine days and twenty-nine degrees behind; the present calendar still follows the renzi reckoning without further adjustment. Mercury hides often and appears rarely; when it does appear, it matches the calendar, and here too we keep the renzi era unchanged. For Venus and Mercury, only evening rising and conjunction present special difficulty. Yexing argues that Heaven's way is vast and observation imperfect; the five planets' stations and retrogradations are hard to compute exactly, and the naked eye cannot catch every detail. One need only capture the broad pattern of their appearances and disappearances and overlook minor mid-course errors—then the calendar is fit for use. If one insisted only on momentary sightings and ignored long-term risings and settings, calendrical science would be all but ruined. Calendar-makers must thread solar terms and new moons across a thousand years and push intercalary fractions to within the finest measure. Waxing and waning must balance, intervals and constants must align, circuit-day fractions must agree to the smallest unit, solar and lunar reckonings must match exactly, adjustment formulas must check at conjunctions, the sun's position must be tested against eclipses—only after fixing all these rules can the era beginning be found, and even then a jiazi year is hard to land on. Even when one reaches a jiazi year, fractional discrepancies remain—systems contradict and fail to align. The present calendar begins at jiazi, follows seven unified rates, and fits together like joined disks and strung pearls—its claims hold firm. Its principles are clear, and we believe it will serve. Fang's observations are not mistaken either. Yet where degrees fail to match within a single conjunction cycle, they will realign by the cycle's end. By computation, Saturn's discrepancy on the twentieth of the twelfth month two years ago was five degrees; today it is three; Venus formerly differed by four degrees; now there is none at all. Judging by this trend, appearances and disappearances will soon prove the method; for long-term use, the calendar will hold in the main.
8
芳又云，以去年十二月中算新曆，其鎮星以十二月二十日在角十一度留，天上在亢四度留，是新曆差天五度； 太白、歲星並各有差。 校於壬子舊曆，鎮星差天五度，太白歲星亦各有差，是舊曆差天為多，新曆差天為少。 凡造曆者，皆須積年累日，依法候天，知其疏密，然後審其近者，用作曆術。 不可一月兩月之間，能正是非。 若如熒惑行天七百七十九日，一遲、一疾、一留、一逆、一順、一伏、一見之法，七頭一終； 太白行天五百八十三日，晨夕之法，七頭一終； 歲星行天三百九十八日，七頭一終； 鎮星行天三百七十八日，七頭一終； 辰星行天一百一十五日，晨夕之法，七頭一終。 造曆者必須測知七頭，然後作術。 得七頭者造曆為近，不得頭者其曆甚疏，皆非一二日能知是非。 自五帝三代以來及秦、漢、魏、晉，造曆者皆積年久測，術乃可觀。 其倉卒造者，當時或近，不可久行。 若三四年作者，初雖近天，多載恐失。 今甲子新曆，業興潛構積年，雖有少差，校於壬子元曆，近天者多。 若久而驗天，十年二十年間，比壬子元曆，三星行天，其差為密。
Fang also argued that, reckoning from last December, the new calendar placed Saturn stationary in Horn at eleven degrees on the twentieth, while the sky showed it stationary in Neck at four degrees—a five-degree gap from Heaven; Venus and Jupiter showed discrepancies as well. Compared with the old renzi calendar, which also missed Saturn by five degrees and Venus and Jupiter by their own margins, the old calendar diverged more from Heaven and the new less. Every calendar-maker must observe Heaven year after year, learn where methods are loose or tight, and then adopt the approach that comes closest. One cannot settle such questions in a month or two. Mars, for instance, completes its cycle in 779 days through seven phases—slow, fast, stationary, retrograde, direct, hidden, and visible—one full round; Venus in 583 days, with its morning-and-evening pattern, seven phases to a cycle; Jupiter in 398 days, seven phases to a cycle; Saturn in 378 days, seven phases to a cycle; Mercury in 115 days, with its morning-and-evening pattern, seven phases to a cycle. Calendar-makers must observe all seven phases before they can build their methods. Those who master the seven phases produce accurate calendars; those who do not produce crude ones—and neither can be judged in a day or two. From the Five Emperors through Qin, Han, Wei, and Jin, every sound calendar rested on years of observation. Hastily made calendars may look accurate at first but cannot last. Even three- or four-year efforts, though close at first, tend to drift over time. Yexing's new jiazi calendar was built on years of hidden labor; though small errors remain, it matches Heaven more closely than the renzi calendar. Tested over ten or twenty years against the renzi calendar, the three planets' motions will prove far tighter.
9
獻武王上言之，詔付外施行。
Prince Xianwu reported upward, and an edict ordered the calendar put into use.
10
興和曆
The Xinghe Calendar
11
甲子之歲入甲戌紀已來，積十二萬四千一百三十六，算上。
From the jiazi year entering the jiaxu cycle to the present: 124,136 years, inclusive.
12
上元甲子以來，至大魏興和二年歲在庚申，積二十九萬三千九百九十七，算上。
From the High Origin at jiazi to the second year of Xinghe in Great Wei, a gengshen year: 293,997 years, inclusive.
13
甲子之歲入甲戌紀至今庚申，積十二萬五千三百九十七，算上。
From the jiazi year entering the jiaxu cycle to the present gengshen year: 125,397 years, inclusive.
14
元法，一百一萬一千六百。 〈三統之數。〉
Origin divisor: 1,011,600. 〈The number for three sequences.〉
15
統法，三十三萬七千二百。 〈二紀之數。〉
Sequence divisor: 337,200. 〈The number for two cycles.〉
16
紀法，十六萬八千六百。 〈十蔀成紀，[3]日數至十。〉
Cycle divisor: 168,600. 〈Ten obscurations form one cycle; [3] the day-count reaches ten.〉
17
蔀法，一萬六千八百六十。 〈三十乘章歲，得日月餘皆盡之年數。〉
Obscuration divisor: 16,860. 〈Multiply the rule-years by thirty to obtain the years in which solar and lunar remainders are fully cleared.〉
18
度法，一萬六千八百六十。 〈三十乘章歲，得此數。〉
Degree divisor: 16,860. 〈Multiply the rule-years by thirty to obtain this number.〉
19
日法，二十萬八千五百三十。 〈三十乘章月，得此數。〉
Day divisor: 208,530. 〈Multiply the rule-months by thirty to obtain this number.〉
20
氣時法，一千四百五。 〈十二分度法，[4]得一時之數。〉
Qi-time divisor: 1,405. 〈Divide the degree divisor by twelve; [4] this yields the count for one double-hour.〉
21
章歲，五百六十二。 〈二十九章、十一年減閏餘一，一萬六百七十八年減一閏月。〉 [5]
Rule-years: 562. 〈Twenty-nine sequences over eleven years reduce the intercalary remainder by one; over 10,678 years one intercalary month is dropped.〉 See the fifth editorial note.
22
章閏，二百七。 〈五百六十二年之閏月數。〉 [6]
Rule intercalations: 207. 〈The intercalary months in 562 years.〉 See editorial note [6].
23
章月，六千九百五十一。 〈五百六十二年之月數并閏。〉
Rule-months: 6,951. 〈The months in 562 years, intercalary months included.〉
24
章中，六千七百四十四。 〈五百六十二年月除閏月數。〉
Rule mid-climates: 6,744. 〈The ordinary months in 562 years, intercalary months excluded.〉
25
周天，六百一十五萬八千一十七。 〈度法通度，內斗分之數。〉
Circuit of heaven: 6,158,017. 〈Multiply the degree divisor through the degrees and include the dipper fraction.〉
26
通數，六百一十五萬八千一十七。 〈日法通二十九日，內經月餘之數。〉
Communication number: 6,158,017. 〈The day divisor extended through twenty-nine days, including the canonical lunar remainder.〉
27
沒分，六百一十五萬八千一十七。 〈餘數通經沒六十九，內分五萬七千二百四十四[7]得此數。〉
Submergence fraction: 6,158,017. 〈Apply the remainder number through the canonical sixty-nine-day submergence and add the fraction 57,244 [7] to obtain this figure.〉
28
餘數，八萬八千四百一十七。 〈度法通一年下五，內斗分之數。〉
Remainder number: 88,417. 〈The degree divisor run through a year minus five, including the dipper fraction.〉
29
沒法，八萬八千四百一十七。 〈一年之內成甲之外分數。〉
Submergence divisor: 88,417. 〈The fractional part of a year beyond a complete sexagenary day-count.〉
30
斗分，四千一百一十七。 〈從斗量周天至此，不成度之分。〉
Dipper fraction: 4,117. 〈The fractional part when the circuit of heaven is measured from the Dipper station and does not complete a whole degree.〉
31
虛分，九萬七千八百八十三。 〈經月二十九日外少此，不滿三十日。〉
Void fraction: 97,883. 〈The shortfall beyond a canonical twenty-nine-day month; it does not reach thirty days.〉
32
小分法，二十四。 〈二十四氣除周天分之數也。〉
Minor-part divisor: 24. 〈The divisor for splitting the circuit-of-heaven fraction among the twenty-four solar terms.〉
33
歲中，十二。 〈十二月之中氣。〉
Mid-year qi: 12. 〈The twelve mid-climate terms of the year.〉
34
會數，一百七十三。 〈月一出一入黃道之日數，五月、二十三分月之二十也。〉 [8]
Conjunction number: 173. 〈The days for the moon's one departure from and return to the ecliptic—five months plus twenty-thirds of a month.〉 See editorial note [8].
35
會餘，六萬七千一百一十七。 〈百七十三日[9]外不成日之分。〉
Conjunction remainder: 67,117. 〈The fractional remainder beyond 173 whole days [9].〉
36
會通，三千六百一十四萬二千八百七。 〈以日法通百七十三，[10]內會餘之數。〉
Conjunction communication: 36,142,807. 〈Extend 173 through the day divisor [10] and include the conjunction remainder.〉
37
會虛，十四萬一千四百一十三。 〈會餘之外不成度之數。〉
Conjunction void: 141,413. 〈The portion beyond the conjunction remainder that does not complete a degree.〉
38
周日，二十七。 〈周天用一日月行數除。〉 [11]
Circuit day: 27. 〈Divide the circuit of heaven by the moon's daily motion count.〉 See editorial note [11].
39
周餘，十一萬五千六百三十一。 〈周天用日外及本處之分數。〉
Circuit remainder: 115,631. 〈The fractional part beyond whole days in the circuit of heaven, including the fraction at the starting position.〉
40
通周，五百七十四萬五千九百四十一。 〈日法通二十七，內分。〉
Communication circuit: 5,745,941. 〈Extend twenty-seven through the day divisor, including the fractional part.〉
41
周虛，九萬二千八百九十九。 〈周餘[12]外不成日之數。〉
Circuit void: 92,899. 〈The portion beyond the circuit remainder [12] that does not complete a day.〉
42
小周，七千五百一十三。 〈月一日行之數。〉
Minor circuit: 7,513. 〈The distance the moon travels in one day.〉
43
月周，二十二萬五千三百九十。 〈通小周，內度數。〉
Lunar circuit: 225,390. 〈Extend the minor circuit and include the degree count.〉
44
朔望合數，十四。 〈半經月日數。〉
New-moon and full-moon conjunction number: 14. 〈Half the day-count of a canonical month.〉
45
度餘，十五萬九千五百八十八半。 〈半經月日餘。〉
Degree remainder: 159,588½. 〈Half the fractional remainder of a canonical month.〉
46
入交限數，一百五十八度。 〈月出入黃道減半月之數。〉
Entry-crossing limit: 158 degrees. 〈Half a lunation minus the moon's arc of entry to and exit from the ecliptic.〉
47
度餘，十一萬六千五十八半。 〈減半月小餘之外。〉 [13]
Degree remainder: 116,058½. 〈The remainder after subtracting half a month's minor fraction.〉 See editorial note [13].
48
推月朔弦望術第一
Method I: Calculating New Moons, First and Last Quarters, and Full Moons
49
推積月術曰：置入紀以來盡所求年，減一，以章月乘之，章歲如一，所得為積月，不盡為閏餘。 閏餘三百五十五以上，其年有閏。 餘五百一十五以上，進退在天正十一月前後，以冬至定之。
To find accumulated months: take all years from cycle entry through the target year, subtract one; multiply by rule-months (6,951) and divide by rule-years (562); the quotient is accumulated months, the remainder the leap fraction. A leap month occurs when the intercalary remainder reaches 355 or higher. When the remainder is 515 or more, adjust placement before or after the civil eleventh month according to where Winter Solstice falls.
50
推積日術曰：以通數乘積月，為朔積分，日法如一，為積日，不盡為小餘。 以六旬去積日，不盡為大餘。 命大餘以紀， 〈今命以甲戌紀。〉 算外，即所求年天正十一月朔日。
To find accumulated days: multiply accumulated months by the communication number (6,158,017), divide by the day divisor (208,530); the quotient is accumulated days, the residue the minor remainder. Reduce accumulated days modulo sixty; the remainder is the large remainder. Index the large remainder against the cycle, 〈At present count from the jiaxu cycle.〉 then step one past the count: the new moon of the eleventh civil month in the target year.
51
求次月朔術曰：加大餘二十九、小餘十一萬六百四十七，滿除如上，命以紀，算外，即次月朔日。 其小餘滿虛分九萬七千八百八十三者，其月大； 減者，其月小。
For the following new moon add 29 to the large remainder and 110,647 to the small remainder, carrying overflows as before; index against the cycle and step past the count for the next conjunction date. If the minor remainder reaches the void fraction (97,883), the month is long (thirty days); otherwise it is short (twenty-nine days).
52
求上下弦望術曰：加朔大餘七、小餘七萬九千七百九十四、小分一。 小分滿四，從小餘； 小餘滿日法，從大餘； 大餘滿六十，去之，命以紀，算外，[14]即上弦日。 又加，得望、下弦、後月朔。
To find the upper quarter, full moon, and lower quarter: add 7 to the large remainder, 79,794 to the small remainder, and 1 minor part. Carry at four minor parts into the small remainder; carry at the day divisor into the large remainder; reduce the large remainder modulo sixty, index against the cycle, and step past the count [14] for the first-quarter date. Repeat the same addition to reach full moon, last quarter, and the following new moon.
53
推二十四氣閏術第二
Method II: Calculating the Twenty-Four Solar Terms and Intercalary Months
54
推二十四氣術曰：置入紀以來盡所求年，減一，以餘數乘之，蔀法如一，為積沒，不盡為小餘。 以六旬去積沒，不盡為大餘，命以紀，算外，即所求年天正十一月冬至日。
For the twenty-four solar terms: take all years from cycle entry through the target year, subtract one; multiply by the remainder number (88,417) and divide by the obscuration divisor (16,860); the quotient is accumulated submergence, the residue the minor remainder. Reduce modulo sixty for the large remainder, index against the cycle, and step past the count for the Winter Solstice of the eleventh civil month in the target year.
55
求次氣術術曰：加大餘十五、小餘三千六百八十四、小分一，小分滿小分法二十四，從小餘； 小餘滿蔀法，從大餘一，命如上，[15]算外，即次氣日。
For the next solar term add 15 to the large remainder, 3,684 to the small remainder, and 1 minor part; carry at twenty-four minor parts into the small remainder; carry at the obscuration divisor into the large remainder, index as before, and step past the count [15] for the next term's date.
56
推閏術曰：以閏餘減章歲，餘以歲中十二乘之，滿章閏二百七得一，月餘半法以上亦得一月，數起天正十一月，算外，即閏月。 閏月有進退，[16]以無中氣定之。
For the leap month: subtract the leap fraction from rule-years (562); multiply the result by twelve and divide by rule intercalations (207); add another month if the monthly remainder reaches half the divisor; count forward from the civil eleventh month and step past the count to name the intercalary month. Placement of the leap month may shift; [16] fix it by which month lacks a mid-climate.
57
推閏又法術曰：以歲中乘閏餘，如章閏得一，[17]盈章中六千七百四十四，數起冬至，算外，中氣終閏月也。 盈中氣在朔若二日，即前月閏。
Alternate leap-month procedure: multiply the leap fraction by twelve and divide by rule intercalations (207); [17] when the product exceeds rule mid-climates (6,744), count from Winter Solstice and step past the count—where a mid-climate ends, place the intercalary month. When the overflow mid-climate falls on the conjunction day or the day after, the previous month is the leap month.
58
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 (mid-eleventh month) · Minor Cold (twelfth month, first node) · Great Cold (mid-twelfth month) · Beginning of Spring (first month, first node)
59
雨水正月中驁蟄二月節春分二月中清明三月節
Rain Water (mid-first month) · Awakening of Insects (second month, first node) · Spring Equinox (mid-second month) · Clear and Bright (third month, first node)
60
穀雨三月中立夏四月節小滿四月中芒種五月節
Grain Rain (mid-third month) · Beginning of Summer (fourth month, first node) · Small Full (mid-fourth month) · Grain in Ear (fifth month, first node)
61
夏至五月中小暑六月節大暑六月中立秋七月節
Summer Solstice (mid-fifth month) · Minor Heat (sixth month, first node) · Great Heat (mid-sixth month) · Beginning of Autumn (seventh month, first node)
62
處暑七月中白露八月節秋分八月中寒露九月節
Abiding Heat (mid-seventh month) · White Dew (eighth month, first node) · Autumn Equinox (mid-eighth month) · Cold Dew (ninth month, first node)
63
霜降九月中立冬十月節小雪十月中大雪十一月節
Frost Descent (mid-ninth month) · Beginning of Winter (tenth month, first node) · Minor Snow (mid-tenth month) · Major Snow (eleventh month, first node)
64
推合朔却去度表裏術第三
Method III: Syzygy and Ecliptic Latitude (Outer and Inner)
65
推合朔却去交度術曰：置入紀以來朔積分，又以所入紀交會差分并之， 〈甲戌紀交會差分二千六百五十二萬二千六百四十九。〉 以會通去之，所得為積交； 不盡者，以日法約之，為度，不盡者為度餘，即所求年天正十一月朔却去交度及度餘。 甲子紀 〈紀首合朔，日月合璧，交中。〉
To find how far a syzygy stands from the lunar node: take the accumulated new-moon fraction from cycle entry and add the conjunction-difference fraction for the current cycle, 〈For the jiaxu cycle the conjunction-difference fraction is 26,522,649.〉 divide by conjunction communication (36,142,807); the quotient is accumulated crossing; the remainder, divided by the day divisor, gives degrees and a degree remainder—the node distance for the eleventh-month new moon of the target year. Jiazi cycle 〈At the cycle's opening syzygy, sun and moon are conjoined on the node.〉
66
甲子紀 〈紀首合朔，日月合璧，交中。〉
Jiazi cycle 〈At the cycle's opening syzygy, sun and moon are conjoined on the node.〉
67
甲戌紀 〈紀首合朔，月在日道表。〉 交會差一百二十七度度餘三萬九千三百三十九[18]
Jiaxu cycle 〈At the cycle's opening syzygy, the moon lies north of the ecliptic.〉 Conjunction offset: 127°; degree remainder: 39,339; see note 18.
68
甲申紀 〈紀首合朔，月在日道裏。〉 交會差八十一度度餘一萬一千五百六十一
Jiashen cycle 〈At the cycle's opening syzygy, the moon lies south of the ecliptic.〉 Conjunction offset: 81°; degree remainder: 11,561
69
甲午紀 〈紀首合朔，月在日道裏。〉 交會差三十四度度餘十九萬二千三百一十三
Jiawu cycle 〈At the cycle's opening syzygy, the moon lies south of the ecliptic.〉 Conjunction offset: 34°; degree remainder: 192,313
70
甲辰紀 〈紀首合朔，月在日道表。〉 交會差一百六十二度度餘二萬三千一百二十二
Jiachen cycle 〈At the cycle's opening syzygy, the moon lies north of the ecliptic.〉 Conjunction offset: 162°; degree remainder: 23,122
71
甲寅紀 〈紀首合朔，月在日道表。〉 交會差一百一十五度度餘二十萬三千八百七十四
Jiayin cycle 〈At the cycle's opening syzygy, the moon lies north of the ecliptic.〉 Conjunction offset: 115°; degree remainder: 203,874
72
求次月却去交度術曰：加度二十九、度餘十一萬六百四十七，度餘滿日法從度，度滿會數去之，亦除其會餘，即次月朔却去交度及度餘。
For the next month: add 29° and remainder 110,647; carry remainder into degrees at the day divisor, drop full conjunction cycles from degrees, and trim the conjunction remainder—the result is the nodal distance for the following new moon.
73
求望却去交度術曰：加度十四、度餘十五萬九千五百八十八半，滿除如上，即望却去交度及度餘。
For the full moon add 14° and remainder 159,588½ and apply the same carry-and-reduce rules to obtain its nodal distance.
74
推月在日道表裏術曰：置入紀以來朔積分，又以紀交會差分并之，倍會通去之，餘以會通減之，得一減者，為月在日道裏； 無所得者，為月在日道表。
To place the moon north or south of the ecliptic: take accumulated new-moon parts from cycle entry, add the cycle's conjunction-difference fraction, divide by twice the conjunction communication, and subtract the remainder from one full conjunction communication—a single subtraction means the moon is south of the ecliptic; no subtraction means the moon is north of the ecliptic.
75
求次月表裏術曰：加次月度及度餘，加表滿會數及會數餘，則在裏； 加裏滿會數及會餘，則在表。
For the following month add the tabulated degree increment; when starting from the outer track, filling the conjunction number and its remainder switches the moon to the inner track; from the inner track, the same overflow places it back on the outer track.
76
推交道所在日術曰：以十一月朔却去交度及餘減會數及會餘，會餘若不足減者，減一度，加日法乃減之； 又以十一月朔小餘加之，滿日法從度，餘為度餘，即是天正十一月朔前去交度及餘。 如曆月大小除之，起天正月十一月，不滿月者為入月，算外，交道所在日。 又以歲中乘入月小餘，日法除之，所得命以子，算外，[19]即交道所在辰。 其交在望前者，其月朔則交會，[20]望則月蝕。 交在望後者，其月月蝕，後朔交會。 交正在望者，其月月蝕既，前後朔交會。 交正朔者，日蝕既，前後月望皆月蝕。
To find the day of nodal passage: subtract the November new moon's nodal distance from the conjunction number and remainder; if the remainder is too small, borrow one degree and add the day divisor before subtracting; add the November syzygy's fine fraction; carry at the day divisor into degrees—the remainder gives the nodal arc elapsed before civil-year November conjunction. Divide by successive month lengths from civil month 11; the fractional month is months entered; step past the count to name the day of nodal crossing. Multiply the entered month's fine parts by twelve and divide by the day divisor; count the quotient from zi and step past the count [19] for the double-hour of the crossing. If the node precedes full moon, that month has a solar eclipse at conjunction [20] and a lunar eclipse at full moon. If the node follows full moon, the month has a lunar eclipse first and solar eclipse at the next syzygy. When the node aligns with full moon, a total lunar eclipse occurs, with solar eclipses at the adjacent new moons. With the node on conjunction day, a total solar eclipse; the full moons before and after both yield lunar eclipses.
77
求後交月及日術曰：以會數及會餘加前入月算及餘，餘滿日法從日，日如曆月大小除之，起前交月，算外，即後交月及日。 以次放之。
For the next crossing: add the conjunction number and remainder to the prior month count and fraction; carry at the day divisor, divide by month lengths from the previous crossing month, and step past the count for the next crossing date. Repeat the same steps for each subsequent crossing.
78
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，合交中者，蝕之既。 其月蝕在日之衝，起角亦如之。
To find the eclipse quadrant: on the outer track, if syzygy precedes the node, the bite begins at the southeast limb; if the node precedes syzygy, the bite begins at the southwest limb. On the inner track, syzygy before the node starts the bite at the northwest limb; a central crossing yields a total eclipse. For lunar eclipses at opposition, apply the same quadrant rule.
79
推蝕分多少術曰：其朔望去交度及度餘如入交限數一百五十八度、度餘十一萬六千五十八半以上者，以減會數及會數餘，餘為不蝕度。 若朔望去交度如朔望合數十四度、度餘十五萬九千五百八十八半以下者，即是不蝕度。 [21]皆以減十五，餘為蝕分。 [22]朔望去交度盡者，蝕之既。
For eclipse magnitude: if the syzygy's nodal distance reaches the entering limit (158°; remainder 116,058½) or more, subtract from the conjunction number and remainder—the remainder is the non-eclipse arc. If the nodal distance is at or below the syzygy conjunction number (14°; remainder 159,588½), there is no eclipse. [21] Subtract each case from 15; the remainder is the eclipse magnitude in fifteenths. [22] When the nodal distance is zero, the eclipse is total.
80
推合朔月蝕入遲疾曆盈縮術第四
Method IV: Syzygy, Lunar Eclipse, and the Lunar Anomaly Table (Surplus and Deficit)
81
推合朔入遲疾曆術曰：置入紀以來朔積分，又以所入紀遲疾差分并之， 〈甲戌紀遲差分二百三十五萬二千一百九十一。〉 [23]以通周去之，所得為積周； [24]不盡者，以日法約之，為日，不盡者為日餘。 命日，算外，即所求年天正月十一月合朔入曆日。
To place a syzygy in the lunar anomaly table: take accumulated new-moon parts from cycle entry and add the slow-fast difference fraction for the current cycle, 〈For the jiaxu cycle the slow-fast difference fraction is 2,352,191.〉 [23] divide by communication cycles; the quotient is accumulated cycles; [24] the remainder, divided by the day divisor, gives whole days and a day remainder. Index the day and step past the count for the anomaly-table day of the target year's civil November conjunction.
82
求次月入曆日術曰：加一日、日餘二十萬三千五百四十六，日蝕滿日法從日，[25]日滿周日及周餘去之，命如上，算外，即次月入曆日。
For the following month add 1 day and remainder 203,546; carry remainder into days at the day divisor; [25] drop full circuit weeks; index as before to obtain the next entry day.
83
月行遲疾度及分[26]損益率盈縮并率盈縮積分
Table heading: lunar speed (degrees and parts) [26], adjustment rates, combined excess/deficit, accumulated excess/deficit
84
一日十四度 〈四百二分〉 益七百五十七盈初
Day 1: 14° 〈402 parts〉 +757 benefit; initial surplus
85
二日十四度 〈三百三十四分〉 [27]益六百八十九盈七百五十七[28]盈積分二萬一千一十一
Day 2: 14° 〈334 parts〉 [27] +689 benefit; running surplus 757 [28]; surplus integral 21,011
86
三日十四度 〈二百六十一分〉 [29]益六百一十六[30]盈一千四百四十六[31]盈積分四萬一百三十五
Day 3: 14° 〈261 parts〉 [29] +616 benefit [30]; running surplus 1,446 [31]; surplus integral 40,135
87
四日十四度 〈一百九十分〉 益五百四十五盈二千六十二盈積分五萬七千二百三十二
Day 4: 14° 〈190 parts〉 +545 benefit; running surplus 2,062; surplus integral 57,232
88
五日十四度 〈一百一十一分〉 益四百六十六盈二千六百七盈積分七萬二千三百六十
Day 5: 14° 〈111 parts〉 +466 benefit; running surplus 2,607; surplus integral 72,360
89
六日十三度 〈五百二十三分〉 益三百一十五盈三千七十三盈積分八萬五千二百九十四
Day 6: 13° 〈523 parts〉 +315 benefit; running surplus 3,073; surplus integral 85,294
90
七日十三度 〈二百九十六分〉 益八十九盈三千三百八十八盈積分九萬四千三十七
Day 7: 13° 〈296 parts〉 +89 benefit; running surplus 3,388; surplus integral 94,037
91
八日十三度 〈六十八分〉 損一百三十九盈三千四百七十七盈積分九萬六千五百七
Day 8: 13° 〈68 parts〉 −139 damage; running surplus 3,477; surplus integral 96,507
92
九日十二度 〈四百六十八分〉 [32]〈〉 損二百八十三盈三千三百三十八盈積分九萬二千六百四十九
Day 9: 12° 〈468 parts〉 See editorial note [32]. −283 damage; running surplus 3,338; surplus integral 92,649
93
十日十二度 〈三百七十九分〉 損三百九十盈三千五十五盈積分八萬四千七百九十四
Day 10: 12° 〈379 parts〉 −390 damage; running surplus 3,055; surplus integral 84,794
94
十一日十二度 〈二百六十七分〉 [33]〈〉 損五百二盈二千六百六十五盈積分七萬三千九百六十九
Day 11: 12° 〈267 parts〉 See editorial note [33]. −502 damage; running surplus 2,665; surplus integral 73,969
95
十二日十二度 〈一百五十一分〉 損六百一十八盈二千一百六十三盈積分六萬三十六
Day 12: 12° 〈151 parts〉 −618 damage; running surplus 2,163; surplus integral 60,036
96
十三日十二度 〈四十分〉 損七百二十九盈一千五百四十五盈積分四萬二千八百八十三
Day 13: 12° 〈40 parts〉 −729 damage; running surplus 1,545; surplus integral 42,883
97
十四日十一度 〈五百一十五分〉 損八百一十六盈八百一十六盈積分二萬二千六百四十九
Day 14: 11° 〈515 parts〉 −816 damage; shrinkage 816; shrinkage integral 22,649
98
十五日十二度 〈三十八分〉 益七百三十一縮初
Day 15: 12° 〈38 parts〉 +731 benefit; initial shrinkage
99
十六日十二度 〈一百二十三分〉 益六百四十六[34]縮七百三十一縮積分二萬二百九十
Day 16: 12° 〈123 parts〉 +646 benefit [34]; running shrinkage 731; shrinkage integral 20,290
100
十七日十二度 〈二百一十一分〉 益五百五十八縮一千三百七十七縮積分三萬八千二百二十
Day 17: 12° 〈211 parts〉 +558 benefit; running shrinkage 1,377; shrinkage integral 38,220
101
十八日十二度 〈三百二十四分〉 [35]〈〉 益四百四十五縮一千九百三十五縮積分五萬三千七百
Day 18: 12° 〈324 parts〉 See editorial note [35]. +445 benefit; running shrinkage 1,935; shrinkage integral 53,700
102
十九日十二度 〈四百三十五分〉 益三百三十四縮二千三百八十縮積分六萬六千五十九
Day 19: 12° 〈435 parts〉 +334 benefit; running shrinkage 2,380; shrinkage integral 66,059
103
二十日十二度 〈五百五十五分〉 益二百一十四縮二千七百一十四縮積分七萬五千三百二十九
Day 20: 12° 〈555 parts〉 +214 benefit; running shrinkage 2,714; shrinkage integral 75,329
104
二十一日十三度 〈一百二十八分〉 益七十九縮二千九百二十八縮積分八萬一千二百六十九
Day 21: 13° 〈128 parts〉 +79 benefit; running shrinkage 2,928; shrinkage integral 81,269
105
二十二日十三度[36] 〈二百七十分〉 損六十三縮三千七縮積分八萬三千四百六十三
Day 22: 13°; see note 36. 〈270 parts〉 −63 damage; running shrinkage 3,007; shrinkage integral 83,463
106
二十三日十三度 〈四百三十二分〉 損二百二十五縮二千九百四十四縮積分八萬一千七百一十三
Day 23: 13° 〈432 parts〉 −225 damage; running shrinkage 2,944; shrinkage integral 81,713
107
二十四日十四度 〈三十三分〉 損三百八十八縮二千七百一十九縮積分七萬五千四百六十八
Day 24: 14° 〈33 parts〉 −388 damage; running shrinkage 2,719; shrinkage integral 75,468
108
二十五日十四度 〈一百九十四分〉 損五百四十九縮二千三百三十一縮積分六萬四千六百九十九
Day 25: 14° 〈194 parts〉 −549 damage; running shrinkage 2,331; shrinkage integral 64,699
109
二十六日十四度 〈三百一十九分〉 損六百七十四縮一千七百八十三縮積分四萬九千四百六十一
Day 26: 14° 〈319 parts〉 −674 damage; running shrinkage 1,783; shrinkage integral 49,461
110
二十七日十四度 〈三百四十六分〉 [37]〈〉 損七百一縮一千一百八縮積分三萬七百五十四
Day 27: 14° 〈346 parts〉 See editorial note [37]. −701 damage; running shrinkage 1,118; shrinkage integral 30,754
111
周日十四度 〈三百七十九分〉 損七百三十四縮四百七縮積分一萬一千二百九十七
Circuit day: 14° 〈379 parts〉 −734 damage; running shrinkage 407; shrinkage integral 11,297
112
推合朔交會月蝕定大小餘[38]術曰：以入曆日餘乘所入曆下損益率，以小周七千五百一十三除之，所得損益盈縮積分為定積分。 積分盈者，[39]以減本朔望小餘； 縮者，加之。 加之滿日法者，交會加時在後日； 減之，不足減者，減一日，加日法乃減之，交會加時在前日。 月蝕者，隨定大小餘[40]為定日加時。
To fix syzygy, nodal, and eclipse remainders [38]: multiply the ephemeris-entry day remainder by the tabulated benefit/damage rate and divide by Lesser Zhou 7,513—the surplus or deficit integral is the fixed integral. If the integral is in surplus [39], subtract from the base syzygy small remainder; if in shrinkage, add. If addition fills the day divisor, the nodal event falls on the next day; on subtraction, borrow one day and add the day divisor if needed—the nodal event falls on the previous day. For lunar eclipse, take the fixed great and small remainders [40] as the true time of day.
113
推加時術曰：以歲中乘定小餘，日法除之，所得命以子，算外。 朔望加時有餘不盡者，四之，如法得一為少，二為半，三為太。 半又有餘者，三之，如法得一為強，半法以上排成一，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，以之并少為半弱，以之并半為太弱，以之并太為一辰弱。 隨所在辰而命之，即其強弱。 日之衝為破，月常在破下蝕。 推日月合朔弦望度術第五[41]推日度術曰：置入紀以來朔積日，以日度法一萬六千八百六十乘之，滿周天去之，餘以日度法約之為度，不盡為餘。 [42]命起牛前十二度，宿次除之，不滿宿者，算外，即所求年天正十一月朔夜半日所在度及分。
To find the hour of occurrence: multiply the fixed small remainder by 12 (year medial) and divide by the day divisor; index from zi and step past the count. For a fractional hour at syzygy, quadruple the remainder: 1/4 of the divisor is shao, 2/4 is ban, 3/4 is tai. If the half-degree still has remainder, triple it: one third of the divisor is qiang; at half the divisor or more round up, otherwise discard. Add qiang to shao for shao-qiang, to ban for ban-qiang, to tai for tai-qiang. Two qiang yield shao-ruo; added to shao gives ban-ruo, to ban gives tai-ruo, to tai gives one full lodge-ruo. Name the lodge reached—that is the strength grade. The sun’s opposition point is po; eclipses occur when the moon stands there. Method V: Syzygy and Lunar Phase Longitudes [41]. Solar longitude: set accumulated new-moon days from cycle entry, multiply by solar-degree divisor 16,860, remove full circuits of heaven, and reduce the remainder to degrees and a remainder. [42] Index from 12° before Ox, walk through the lodges, and step past the count—the remainder is the sun’s midnight longitude at civil-year November new moon.
114
推日度又法，術曰：置周天三百六十五度、斗分四千一百一十七，以冬至去朔日數減一，以減周天度，冬至小餘減斗分； 斗分不足減者，減一度，加日度法乃減之。 命起如上，算外，即所求年天正十一月朔夜半日所在度及分。
Alternate solar-longitude method: set 365° plus Dipper fraction 4,117, subtract (days from winter solstice to new moon − 1) from the circuit, and subtract the winter-solstice small remainder from the Dipper fraction; if the Dipper fraction is too small, borrow 1° and add the solar-degree divisor, then subtract. Index as before and step past the count for the sun’s midnight longitude at civil-year November new moon.
115
求次月次日日所在度[43]術曰：月大者加度三十，月小者加度二十九，次日者加度一，宿次除之，逕斗除其分。 [44]
For later days [43]: add 30° in a long month, 29° in a short month, 1° for the next day; walk the lodges and deduct the Dipper fraction. See editorial note [44].
116
推合朔日月共度術曰：以章歲五百六十二乘朔小餘，以章月六千九百五十一除之，所得為大分，不盡為小分。 以加夜半日度分，分滿日度法從度，命如上，算外，即所求年天正十一月合朔日月共度。
Shared syzygy longitude: multiply the new-moon small remainder by rule years 562 and divide by rule months 6,951 for great and small fractions. Add to midnight solar longitude, carry at the solar-degree divisor, index the lodges, and step past the count for November syzygy longitude.
117
推合朔日月共度又法，術曰：加度二十九、大分八千九百四十五、小分六千九百一十九，小分滿章月從大分； 大分滿日度法從度，宿次除之，逕斗去其分，算外，即次月合朔日月共度。
Alternate syzygy longitude: add 29°, great fraction 8,945, and small fraction 6,919; carry small to great at rule months; carry great to degrees at the solar-degree divisor, walk the lodges, deduct the Dipper fraction, and step past the count for the next month’s syzygy longitude.
118
推月度術曰：置入紀以來朔積日，以月周二十二萬五千三百九十乘之，[45]滿周天去之。 餘以日度法約之為度，餘為度分，命起牛前十二度，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。
Lunar longitude: set accumulated new-moon days from cycle entry, multiply by lunar circuit 225,390, [45] and remove full circuits of heaven. Reduce to degrees and fractional parts, index from 12° before Ox, walk the lodges, and step past the count for the moon’s midnight longitude at November new moon.
119
推月度又法，術曰：以小周乘朔小餘為實，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，餘為小分。 所得以減合朔度及度分，算外，即所求年天正十一月朔夜半月所在度及分。
Alternate lunar longitude: Lesser Zhou × new-moon small remainder over rule years × day divisor gives degrees; the remainder divides by rule months for great fraction, with the residue as small fraction. Subtract from syzygy longitude and step past the count for the moon’s midnight position at November new moon.
120
求次月月度術[46]曰：月小，加度二十二、分七千三百七十三，月大，加度三十五、分一萬三千五百八十三，分滿日度法從度，宿次除之，不滿宿者，算外，即月次月所在度。
Next month’s moon [46]: add 22° and fraction 7,373 in a short month or 35° and fraction 13,583 in a long month, carry at the solar-degree divisor, walk the lodges, and step past the count.
121
求次日月度術[47]曰：加度十三、分六千二百一十，分滿日度法從度，除如上，算外，即月次日所在度。
Next day’s moon [47]: add 13° and fraction 6,210, carry, walk the lodges, and step past the count.
122
求弦望日所在度術曰：加合朔度七、大分六千四百五十一、小分三千四百六十七、[48]微分二，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，算外，即上弦日所在度。 又加，得望、[49]下弦、後月合朔。
First-quarter sun: add 7°, great fraction 6,451, small fraction 3,467, and minute fraction 2 to syzygy longitude with the usual carries, then index the lodges for first-quarter solar longitude. Repeat the same increment for full moon, last quarter, and the next syzygy.
123
求弦望月所在度術曰：加合朔度九十八、大分一萬一千六百九十五、小分五千二百二十五、微分一，滿除如上，算外，即上弦日月所在度。 又加，得望、下弦、後月合朔。
First-quarter moon: add 98°, great fraction 11,695, small fraction 5,225, and minute fraction 1 to syzygy longitude with the usual carries for first-quarter lunar longitude. Repeat for full moon, last quarter, and the next syzygy.
124
斗二十六度牛八度女十二度虛十度
Dipper 26°; Ox 8°; Maid 12°; Void 10°
125
危十七度室十六度壁九度
Rooftop 17°; Encampment 16°; Wall 9°
126
北方玄武七宿：九十八度分四千一百一十七
Northern Dark Warrior (seven lodges): 98°; fraction 4,117
127
奎十六度婁十二度胃十四度昴十一度
Legs 16°; Bond 12°; Stomach 14°; Hairy Head 11°
128
畢十六度觜二度參九度
Net 16°; Turtle Beak 2°; Three Stars 9°
129
西方白虎七宿：八十度
Western White Tiger (seven lodges): 80°
130
井三十三度鬼四度柳十五度星七度
Well 33°; Ghost 4°; Willow 15°; Star 7°
131
張十八度翼十八度軫十七度
Spread 18°; Wings 18°; Chariot Shaft 17°
132
南方朱鳥七宿：一百一十二度
Southern Vermilion Bird (seven lodges): 112°
133
角十二度亢九度氐十五度房五度
Horn 12°; Neck 9°; Root 15°; Chamber 5°
134
心五度尾十八度箕十一度
Heart 5°; Tail 18°; Winnowing Basket 11°
135
東方倉龍七宿：七十五度
Eastern Azure Dragon (seven lodges): 75°
136
周天三百六十五度、一萬六千八百六十分度之四千一百一十七，通之，得六百一十五萬八千一十七，名曰周天。
Circuit of heaven: 365° plus 4,117/16,860 per degree—combined total 6,158,017, called the circuit of heaven.
137
推土王滅沒卦候上朔術第六
Method VI: Earth-King Days, Extinction and Submersion, Hexagrams, Pitch-Pipes, and Upper Syzygy
138
推土王日術曰：置四立大小餘，各減其大餘十八、小餘四千四百二十、小分十八、微分二，大餘不足減者，加六十乃減之； 小餘不足減者，減一日，加蔀法乃減之； 小分不足減者，減小餘一，加小分法二十四乃減之； 微分不足減者，減小分一，加五，然後皆減之。 命以紀，算外，即四立前土王日。
Earth-king days: take the four establishment remainders and subtract great remainder 18, small remainder 4,420, small fraction 18, and minute fraction 2; borrow 60 for the great remainder if needed; borrow one day and add the obscuration divisor for the small remainder if needed; borrow one small remainder and add 24 small-fraction parts if needed; borrow one small fraction and add five minute parts, then complete the subtraction. Index by cycle and step past the count for the earth-king day before each establishment.
139
推土王又法，術曰：加冬至大餘二十七、小餘六千六百三十一、小分六、微分三，微分滿五從小分，小分滿小分法從小餘，小餘滿蔀法從大餘一，命以紀，算外，即季冬土王日。
Alternate earth-king method: add to winter solstice great remainder 27, small remainder 6,631, small fraction 6, and minute fraction 3 with the usual carries, index by cycle, and step past the count for the late-winter earth-king day.
140
求次季土王日術曰：加大餘九十一、小餘五千二百四十四、小分六，小分滿小分法從小餘，小餘滿蔀法從大餘，大餘滿六十去之，命以紀，算外，即次季土王日。
Next season’s earth-king day: add great remainder 91, small remainder 5,244, and small fraction 6 with the usual carries, drop full sexagenary cycles, index, and step past the count.
141
推滅沒術曰：因冬至積沒有小餘者，加積沒一，以沒分乘之，以沒法八萬八千四百一十七除之，所得為積日，不盡為沒餘。 六旬去積日，不盡為沒日，命以紀，算外，即所求天正十一月冬至後沒日。
Extinction and submersion: if winter solstice leaves a submersion small remainder, increment accumulated submersion, multiply by the submersion fraction, and divide by submersion divisor 88,417 for accumulated days and submersion remainder. Remove sixties from accumulated days; the remainder is the submersion day—index by cycle and step past the count for the first submersion after November winter solstice.
142
求次滅術曰：加沒日六十九、沒餘五萬七千二百四十四，沒餘滿沒法從沒日，沒日滿六十去之，命以紀，算外，即次沒日。 餘盡者為滅。
Next submersion: add day 69 and remainder 57,244 with carry at the submersion divisor, drop full sexagenary cycles, index, and step past the count. A zero remainder marks extinction.
143
求次沒術曰：加沒日六十九、沒餘一萬九百一十五、沒分六萬二千二百八十五，沒分滿沒法從沒餘，沒餘滿蔀法從沒日。 命起前沒月，曆月大小除之，不滿月者，即後沒日及沒餘、沒分。 命曰如上，算外，即次沒日。
Next extinction: add day 69, remainder 19,915, and fraction 62,285 with carries at the submersion and obscuration divisors. From the prior submersion month, walk long and short months; the fractional month yields the next submersion day, remainder, and fraction. Index as before and step past the count for the next submersion day.
144
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。 中孚因坎卦。
Four cardinal hexagrams: winter solstice remainders give the Kan dominion day; spring equinox gives Zhen; summer solstice gives Li; autumn equinox gives Dui. Zhong Fu follows the Kan hexagram.
145
求次卦術曰：加坎卦大餘六，小餘一千四百七十三、小分十四、微分四，微分滿五從小分，小分滿小分法從小餘，小餘滿蔀法從大餘，大餘滿六十去之，命以紀，算外，即復卦用事日。
Next hexagram: add to Kan great remainder 6, small remainder 1,473, small fraction 14, and minute fraction 4 with the usual carries, drop full sexagenary cycles, index, and step past the count for the Fu dominion day.
146
十一月，未濟、蹇、頤、中孚、復。
Eleventh month: Wei Ji, Jian, Yi, Zhong Fu, and Fu.
147
十二月，屯、謙、睽、升、臨。
Twelfth month: Zhun, Qian, Kui, Sheng, and Lin.
148
正月，小過、蒙、益、漸、泰。
First month: Xiao Guo, Meng, Yi, Jian, and Tai.
149
二月，需、隨、晉、解、大壯。
Second month: Xu, Sui, Jin, Xie, and Da Zhuang.
150
三月，豫、訟、蠱、革、夬。
Third month: Yu, Song, Gu, Ge, and Guai.
151
四月，旅、師、比、小畜、乾。
Fourth month: Lü, Shi, Bi, Xiao Chu, and Qian.
152
五月，大有、家人、井、咸、姤。
Fifth month: Da You, Jia Ren, Jing, Xian, and Gou.
153
六月，鼎、豐、渙、履、遯。
Sixth month: Ding, Feng, Huan, Lü, and Dun.
154
七月，恒、節、同人、損、否。
Seventh month: Heng, Jie, Tong Ren, Sun, and Pi.
155
八月，巽、萃、大畜、賁、觀。
Eighth month: Xun, Cui, Da Chu, Bi, and Guan.
156
九月，歸妹、无妄、明夷、困、剝。
Ninth month: Gui Mei, Wu Wang, Ming Yi, Kun, and Bo.
157
十月，艮、既濟、噬嗑、大過、坤。
Tenth month: Gen, Ji Ji, Shi He, Da Guo, and Kun.
158
四正為方伯，中孚為三公，復為天子，屯為諸侯，謙為大夫，睽為九卿，升還從三公，周而復始。
The four cardinal hexagrams map to regional lords; Zhong Fu to the Three Excellencies; Fu to the emperor; Zhun to feudal lords; Qian to grandees; Kui to the Nine Ministers; Sheng again follows the Three Excellencies in a repeating cycle.
159
九三應上九，清淨、微溫、陽風； 九三應上六，降赤、决溫、陰雨。 [50]六三應上六，日澤、寒、陰雨； [51]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。
When nine-three meets nine on top: clear air, mild warmth, and yang wind; When nine-three meets six on top: red haze, sudden warmth, and yin rain. [50] When six-three meets six on top: marshy sun, cold, and yin rain; [51] When six-three meets nine on top: dusty haze, sharp cold, and yang wind. Hexagrams with yang in the upper line yield yang wind; those with yin above yield yin rain.
160
推七十二候術曰：因冬至大小餘即虎始交日，加大餘五、小餘一千二百二十八、微分一，微分滿三從小分，小分滿小分法從小餘，小餘滿蔀法從大餘，大餘滿六十去之，命以紀，算外，依次候日。
Seventy-two hou: from winter solstice remainders locate the “tigers begin mating” day; add great remainder 5, small remainder 1,228, and minute fraction 1 with the usual carries, drop full sexagenary cycles, index, and step past the count for each successive phenological hou.
161
冬至虎始交芸始生荔挺生
Winter Solstice — tigers begin mating; winter cress sprouts; bricolea shoots forth.
162
小寒蚯蚓結麋角解水泉動
Minor Cold — earthworms coil up; elk shed antlers; springs begin to move.
163
大寒雁北向鵲始巢雉始雊
Major Cold — geese turn north; magpies start nests; pheasants first call.
164
立春鷄始乳東風解凍蟄蟲始振
Start of Spring — hens brood; the east wind thaws ice; dormant insects awaken.
165
雨水魚上負冰獺祭魚鴻雁來
Rain Water — fish swim beneath ice; otters “sacrifice” fish; wild geese return.
166
驚蟄始雨水桃始華倉庚鳴
Awakening of Insects — rains begin; peach blossoms open; orioles sing.
167
春分鷹化為鳩玄鳥至雷始發聲
Spring Equinox — hawks become turtledoves; swallows arrive; thunder first rolls.
168
清明電始見蟄蟲咸動蟄蟲啟戶
Pure Brightness — lightning appears; dormant insects stir; they open their burrows.
169
穀雨桐始華田鼠化為鴽虹始見
Grain Rain — paulownia flowers; field mice turn to quail; rainbows first show.
170
立夏萍始生戴勝降桑螻蟈鳴
Start of Summer — duckweed appears; hoopoes alight on mulberry; frogs begin calling.
171
小滿蚯蚓出王瓜生苦菜秀
Minor Fullness — earthworms surface; snake-gourd grows; bitter herbs flourish.
172
芒種靡草死小暑至螗蜋生
Grain in Ear — tender grasses die; minor heat begins; mantises hatch.
173
夏至鵙始鳴反舌無聲鹿角解
Summer Solstice — shrikes call; mockingbirds go quiet; deer shed antlers.
174
小暑蟬始鳴半夏生木槿榮
Minor Heat — cicadas chirp; pinellia grows; hibiscus flowers.
175
大暑溫風至蟋蟀居壁鷹乃學習
Major Heat — warm winds blow; crickets move to walls; young hawks learn to fly.
176
立秋腐草化為螢土潤溽暑涼風至
Start of Autumn — rotting grass turns to fireflies; earth grows damp and sultry; cool winds come.
177
處暑白露降寒蟬鳴鷹祭鳥
Limit of Heat — white dew descends; cold cicadas sing; hawks offer captured birds.
178
白露天地始肅暴風至鴻雁來
White Dew — heaven and earth turn stern; gales blow; wild geese arrive.
179
秋分玄鳥歸羣鳥養羞雷始收聲
Autumn Equinox — swallows depart; birds store food; thunder falls silent.
180
寒露蟄蟲附戶殺氣浸盛陽氣日衰
Cold Dew — insects shelter by doors; killing qi strengthens; yang qi wanes day by day.
181
霜降水始涸鴻雁來賓雀入大水化為蛤
Frost Descent — waters shrink; visiting geese arrive; sparrows dive into the floods and become clams.
182
立冬菊有黃華豺祭獸水始冰
Start of Winter — chrysanthemums bloom yellow; jackals offer prey; water starts to freeze.
183
小雪地始凍雉入大水為蜃虹藏不見
Minor Snow — earth freezes; pheasants enter the waters as oysters; rainbows vanish.
184
大雪冰益壯地始坼鶡旦不鳴[52]
Major Snow — ice hardens; earth cracks; crossbills fall silent at dawn; see note 52.
185
推上朔術曰：置入紀以來盡所求年，減一，以六律乘之，以六旬去之，不盡者命以甲子，算外，[53]即上朔日。
Upper new moon: set years since cycle entry minus one, multiply by six lü, remove sixties, index from jiazi, [53] and step past the count for the upper new-moon day.
186
推五星見伏術第七
Treatise 7: computing the five planets' apparitions and occultations.
187
上元甲子以來至今大魏興和二年，歲在庚申，積二十九萬三千九百九十七算。
From Upper Origin jiazi to the present Great Wei Xinghe 2 (year gengshen), the accumulated count is 293,997.
188
木精曰歲星，其數六百七十二萬三千八百八十八。
Wood's essence is Jupiter; its divisor is 6,723,888.
189
火精曰熒惑，其數一千三百一十四萬九千八十三。
Fire's essence is Mars; its divisor is 13,149,083.
190
土精曰鎮星，其數六百三十七萬四千六十一。
Earth's essence is Saturn; its divisor is 6,374,061.
191
金精曰太白，其數九百八十四萬三千八百八十二。
Metal's essence is Venus; its divisor is 9,843,882.
192
水精曰辰星，其數一百九十五萬三千七百一十六。 [54]
Water's essence is Mercury; its divisor is 1,953,716. See editorial note [54].
193
推五星術曰：置上元以來盡所求年，減一，以周天乘之，為五星之實； 各以其數為法，除之，所得為積合，不盡為合餘。 以合餘減法，餘為入歲度分。 以日度法約之，所得即所求年天正十一月冬至後晨夕合度算及度餘。 其金、水，以一合日數及合餘減合度算及度餘，得一者為晨，無所得者為夕； 若度餘不足減者，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，即所求年天正十一月冬至後晨夕合度及度餘。
Five planets: set years since Upper Origin minus one, multiply by the circuit of heaven for the planetary dividend; divide each by its own constant for accumulated conjunction and conjunction remainder. Subtract the remainder from the divisor to obtain the degree-fraction entered into the year. Reduce by the solar-degree divisor for the dawn/dusk conjunction count and remainder after November winter solstice of the year sought. For Venus and Mercury, subtract one synodic day-count and remainder from the conjunction tally; a positive result is morning, none is evening; borrow one from the conjunction tally and add the solar-degree divisor if the degree remainder is too small. Index from twelve degrees before Ox, walk the lodges, and step past the count for the dawn/dusk conjunction longitude and remainder after November winter solstice.
194
徑推五星術曰：置上元以來盡所求年，減一，如法算之。 合度餘滿日度法，加合度算一，合度算滿合終日數去之。 亦以合終日餘減合度餘，[55]若不足減者，減合度算一，加周虛。 [56]所得即所求年天正十一月冬至後晨夕合度算及度餘。 其求金水及命度，[57]皆如上法。
Shortcut for the five planets: years since Upper Origin minus one, reckoned directly by each divisor. Carry one to the conjunction tally when degree remainder fills the solar-degree divisor; drop full synodic cycles from the tally. Subtract synodic day-remainder from degree remainder, [55] borrowing one tally and adding circuit void if needed. [56] The result is the dawn/dusk conjunction tally and remainder after November winter solstice of the year sought. Venus, Mercury, and lodge indexing all use the same method; see note 57.
195
求星合月及日術曰：置冬至去朔日數，減一，加合度算。 冬至小餘以加合度餘，合度餘滿日度法去之，加合度算一。 合度算變成合日算，合度餘為日餘，命日起天正十一月，如曆月大小除之，不滿月者，算外，即星合月及日。 有閏以閏計之。
Conjunction month and day: add winter-solstice-to-new-moon days minus one to the conjunction tally. Add winter solstice small remainder to degree remainder, carrying to the tally when it fills the solar-degree divisor. Convert tally to day-count and remainder to day-fraction, walk months from November, and step past the count for the conjunction month and day. Include intercalary months in the reckoning.
196
求後合月及日術曰：以合終日數及合終日餘加前入月算及餘，餘滿日度法從日，[58]如曆月大小除之，起前合月，算外，即後合月及日。 其金、水以一合日數及合日餘加之，[59]加夕得晨，加晨得夕也。
Next conjunction: add synodic days and remainder to the prior month tally, carry at the solar-degree divisor, [58] walk long and short months from the prior conjunction, and step past the count. For Venus and Mercury, add one synodic day and remainder; [59] evening plus the increment gives morning, morning gives evening.
197
求後合度術曰：以行星度餘加前合度及度餘，度餘滿日度法從度，命起前合度，宿次除之，不滿宿者，算外，即後合度餘。 逕斗除其分，其分四千一百一十七。
Later conjunction longitude: add planetary motion to prior degree and remainder, carry, index from the prior conjunction, walk lodges, and step past the count for the remainder. Deduct the Dipper shortcut fraction of 4,117.
198
歲星：合終日數三百九十八，合終日餘一萬二千六百八，周虛三千二百五十二，行星三十三度，度餘八千四百九十一。 [60]
Jupiter: synodic days 398, day-remainder 12,608, circuit void 3,252, motion 33° with remainder 8,491. See editorial note [60].
199
歲星：晨與日合，在日後伏，十六日、日餘六千八百四，行星二度、度餘一萬三千一百七十五。 晨見東方，順，疾，日行五十八分之十一，五十八日行十一度。 順，遲，日行九分，五十八日行九度而留。 不行，二十五日而旋。 逆，日行七分之一，八十四日退十二度。 復留，二十五日。 復順，遲，日行九分，五十八日行九度。 復順，疾，日行十一分，五十八日行十一度。 在日前，夕伏西方，順，十六日、日餘六千八百四，行星二度、度餘一萬三千一百七十六，而與日合。
Jupiter: morning conjunction hides behind the sun for sixteen days plus remainder 6,804 while advancing 2° with remainder 13,175. Dawn apparition in the east: fast direct motion at 11/58° per day, 11° in fifty-eight days. Slow direct motion at nine parts per day for fifty-eight days to nine degrees, then stations. It halts twenty-five days, then turns retrograde. Retrograde at 1/7° per day, retreating 12° in eighty-four days. It stations again for twenty-five days. Again slow direct at nine parts per day for fifty-eight days and nine degrees. Again fast direct at eleven parts per day for fifty-eight days and eleven degrees. West of the sun it evening-hides for sixteen days plus remainder 6,804, advancing 2° with remainder 13,176, then re-conjoins.
200
熒惑：合終日數七百七十九，合終日餘一萬五千一百四十三，周虛一千七百一十七，行星四十九度，度餘六千九百九。
Mars: synodic days 779, day-remainder 15,143, circuit void 1,717, motion 49° with remainder 699.
201
熒惑：晨與日合，在日後伏，七十一日、日餘一萬六千一，行星五十五度、度餘一萬三千九百四十三。 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，日行十二分，九十二日行四十八度而留。 [61]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。 復留，十一日。 復順，遲，日行十二分，九十二日行四十八度。 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、日餘一萬六千二，行星五十五度、[62]度餘一萬三千九百四十三，而與日合。
Mars: morning conjunction hides for seventy-one days plus remainder 16,001 while advancing 55° with remainder 13,943. Dawn apparition in the east: fast direct at 14/23° per day, 112° in 184 days. Slow direct at twelve parts per day for ninety-two days to 48°, then stations. [61] It halts eleven days, then turns retrograde. Retrograde at 17/62° per day, retreating 17° in sixty-two days. It stations again for eleven days. Again slow direct at twelve parts per day for ninety-two days and 48°. Again fast direct at fourteen parts per day for 184 days and 112°. West of the sun it evening-hides for seventy-one days plus remainder 16,002, advancing 55° [62] with remainder 13,943, then re-conjoins.
202
鎮星：合終日數三百七十八，合終日餘九百八十一，周虛一萬五千八百七十九，行星十二度，度餘一萬三千七百二十四。
Saturn: synodic days 378, day-remainder 981, circuit void 15,879, motion 12° with remainder 13,724.
203
鎮星：晨與日合，在日後伏，十八日、日餘四百九十，行星二度、度餘六千八百六十二。 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，日行十七分之一，一百二日退六度。 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，夕伏西方，順，十八日、日餘四百九十一，行星二度、度餘六千八百六十二而與日合。
Saturn: morning conjunction hides for eighteen days plus remainder 490 while advancing 2° with remainder 6,862. Dawn apparition in the east: direct at 1/12° per day, 7° in eighty-four days, then stations. It halts thirty-six days, then turns retrograde. Retrograde at 1/17° per day, retreating 6° in 102 days. It stations again for thirty-six days. Again direct at 1/12° per day for eighty-four days and seven degrees. West of the sun it evening-hides for eighteen days plus remainder 491, advancing 2° with remainder 6,862, then re-conjoins.
204
太白：合終日數五百八十三，合終日餘一萬四千五百二，周虛二千三百五十八，行星二百九十一度， 〈亦曰一合日數。〉 度餘一萬五千六百八十一。 [63] 〈亦曰一合日數。〉
Venus: synodic days 583, day-remainder 14,502, circuit void 2,358, motion 291°, 〈Also called the synodic day-count.〉 degree-remainder 15,681. See editorial note [63]. 〈Also called the synodic day-count.〉
205
太白：夕與日合，在日前伏，四十一日、日餘一萬五千六百八十一，行星五十一度、度餘一萬五千六百八十一。 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。 [64]順，遲，日行一度十三分之二，九十一日行一百五度。 [65]順，大遲，日行十五分之十一，[66]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，伏六日，退四度，而與日晨合。
Venus: evening conjunction hides before the sun for forty-one days plus remainder 15,681 while advancing 51° with remainder 15,681. Evening apparition in the west: fast direct at 1° 3/13 per day, 112° in ninety-one days. [64] Slow direct at 1° 2/13 per day for ninety-one days and 105°. [65] Very slow direct at 11/15° per day; [66] 33° in forty-five days, then stations. It halts eight days, then turns retrograde. Retrograde at 2/3° per day, retreating 6° in nine days. West of the sun it evening-hides six days while retreating 4°, then morning-conjoins.
206
太白：晨與日合，在日後，伏六日，退四度。 晨見東方，逆，日行三分之二，九日退六度而留。 不行，八日。 順，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。 順，大疾，日行一度十三分之三，[67]九十一日行一百一十二度。 在日後，晨伏東方，順，四十一日、日餘一萬五千六百八十一，行星五十一度、度餘一萬五千六百八十一，而與日夕合。
Venus: morning conjunction hides behind the sun six days while retreating 4°. Dawn apparition in the east: retrograde at 2/3° per day, retreating 6° in nine days, then stations. It halts eight days. Direct at 11/15° per day for forty-five days and 33°. Fast direct at 1° 2/13 per day for ninety-one days and 105°. Very fast direct at 1° 3/13 per day; [67] 112° in ninety-one days. East of the sun it dawn-hides for forty-one days plus remainder 15,681, advancing 51° with remainder 15,681, then evening-conjoins.
207
辰星：合終日數一百一十五，合終日餘一萬四千八百一十六，[68]周虛二千四十四，行星五十七度， 〈亦曰一合日數。〉 度餘一萬五千八百三十八。 [69] 〈亦曰一合日數。〉
Mercury: synodic days 115, day-remainder 14,816, [68] circuit void 2,044, motion 57°, 〈Also called the synodic day-count.〉 degree-remainder 15,838. See editorial note [69]. 〈Also called the synodic day-count.〉
208
辰星：夕與日合，在日前伏，十七日、日餘一萬五千八百三十八，行星三十四度、度餘一萬五千八百三十八。 [70]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。 不行，四日。 在日前，夕伏西方，逆，十一日退六度，而與日晨合。
Mercury: evening conjunction hides before the sun for seventeen days plus remainder 15,838 while advancing 34° with remainder 15,838. [70] Evening apparition in the west: fast direct at 1° 1/3 per day, 24° in eighteen days. Slow direct at 5/7° per day for seven days to 5°, then stations. It halts four days. West of the sun it evening-hides retrograde eleven days while retreating 6°, then morning-conjoins.
209
辰星：晨與日合，在日後伏，十一日，退六度。 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、日餘一萬五千八百三十八，行星三十四度、度餘一萬五千八百三十八，[71]而與日夕合。
Mercury: morning conjunction hides behind the sun eleven days while retreating 6°. Dawn apparition in the east: it stations and halts four days. Slow direct at 5/7° per day for seven days and 5°. Fast direct at 1° 1/3 per day for eighteen days and 24°. East of the sun it dawn-hides for seventeen days plus remainder 15,838, advancing 34° with remainder 15,838, [71] then evening-conjoins.
210
五星曆步術曰：以術法伏日度及餘加星合日度及餘，餘滿日度法一萬六千八百六十得一，從全命之如前，[72]得星見日度及餘。 以星行分母乘見度分，日度法如一得一分，不盡半法以上亦得一，以加所行分，分滿其母得一度。 逆順母不同，以當行之母乘故分，故母如一，為當行分。 留者承前，逆則減之，伏不書度，[73]除斗分，以行母為率，分有損益，前後相御。 [74]
Five-star step method: add the tabulated hidden-day degree and remainder to the conjunction day-degree and remainder; carry one full unit when the sum fills the day-degree divisor 16,860, index as before, [72] for the star's appearance day and fractional degree. Multiply the appearance degree fraction by the motion denominator, divide by the day-degree divisor rounding up at half, add to the daily motion fraction, and carry a degree when the fraction fills its denominator. When retrograde and direct use different denominators, convert the prior fraction by the ratio of denominators to obtain the motion fraction for the current phase. Stationary phases carry forward the prior value, retrograde phases subtract, hidden phases omit degrees; [73] clear dipper fractions, use the motion denominator as the rate, and when fractional parts shift, successive phases adjust one against another. See editorial note [74].
211
求五星行所在度術曰：以行分子乘行日數，分母除之，所得即星行所在度。
To find a planet's longitude: multiply the motion numerator by the days in the phase and divide by the denominator.
212
校勘記
Collation Notes
213
置差令朔「令朔」不可解，疑「令」乃「合」之訛。
Set the difference to make new moon: "make new moon" ("ling shuo") is unintelligible; "ling" is likely a corruption of "he" (conjoin).
214
新曆歲星在營室十三度諸本「新曆」下脫「歲星」兩字，依下文鎮星、太白例補。
New calendar, Jupiter in Encampment-and-Room 13°: all editions omit "sui xing" after "xin li"; supplied per the Saturn and Venus passages below.
215
十蔀成紀諸本「十」作「千」。 按蔀法一萬六千八百六十，十蔀成十六萬八千六百，正為一紀之數。 「千」顯誤，今改正。
Ten obscurations form a cycle: all editions read "qian" for "shi". Check: obscuration divisor 16,860; ten obscurations yield 168,600—the full cycle count. "Qian" is clearly wrong; corrected here.
216
十二分度法諸本「十」作「小」。 按以十二除度法一萬六千八百六十，得一千四百五，乃為一氣時法之數。 「小」係形近致誤，今改正。
Twelve-part degree divisor: all editions read "xiao" for "shi". Check: 16,860 ÷ 12 = 1,405, the one-qi double-hour divisor. "Xiao" is a near-graph error; corrected here.
217
二十九章十一年減閏餘一一萬六百七十八年減一閏月諸本「閏餘一，一」作「閏餘二」，「六百」作「一百」，「減」下有「右」字。 按興和曆一章歲等於古法 (十九年一章) 之二十九章加十一年，而閏餘減一； 積十九章歲，即一萬六百七十八年則減一閏月。 傳本將「閏餘一，一萬」之兩「一」字誤并成「二」字，「六百」誤為「一百」，又衍「右」字，今俱改正。
Twenty-nine sequences plus eleven years reduce the intercalary remainder by 1; after 10,678 years one intercalary month is dropped. Editions read "run yu er" for "run yu yi,yi", "yi bai" for "liu bai", and add "you" under "jian". Note: in the Xinghe calendar one rule-year equals the ancient method (19 years per rule) —twenty-nine such rules plus eleven years, with the intercalary remainder reduced by 1; and nineteen rule-years of years (10,678 years) drop one intercalary month. The transmitted text merged the two "yi" of "run yu yi,yi wan" into "er", wrote "yi bai" for "liu bai", and added "you"; all corrected.
218
閏月數諸本「閏」訛「間」，今依文義改。
Intercalary month count: all editions read "jian" for "run"; emended by sense.
219
內分五萬七千二百四十四諸本「二百四十四」作「一百八十四」。 按以沒法八萬八千四百一十七除沒分六百一十五萬八千一十七，得經沒六十九、分五萬二百四十四。 數字顯誤，今改正。
Inner fraction 57,244: editions read "yi bai ba shi si" for "er bai si shi si". Check: 6,158,017 ÷ 88,417 = 69 canonical submergences, remainder 50,244. The figures are clearly wrong; corrected here.
220
五月二十三分月之二十也諸本「五月」作「周髀六」。 按會數為五月二十三分月之二十的整日數，「周髀六」三字當衍，又脫「五月」兩字。 今改正。
Fifth month, 20/23 of a month: all editions read "Zhou Bi liu" for "wu yue". Check: the conjunction number is the whole-day count for fifth month, 20/23 month; "Zhou Bi liu" is intrusive and "wu yue" is missing. Corrected here.
221
百七十三日諸本「三」作「二」。 按此數即上會數，「二」字訛，今改正。
173 days: all editions read "er" for "san". Check: this is the upper conjunction count above; "er" is wrong; corrected.
222
百七十三諸本「三」作「二」。 按此數亦即上會數，「二」字訛，今改正。
173: all editions read "er" for "san". Check: this too is the upper conjunction count; "er" is wrong; corrected.
223
周天用一日月行數除諸本「日」上脫「一」字，「數」下脫「除」字。 按周日為小周 (即一日月行數) 除周天所得之整數。 原文脫漏，今補。
Circumference of heaven ÷ one-day moon-motion number: editions drop "yi" before "ri" and "chu" after "shu". Check: circuit-day is the lesser circuit (i.e., one-day moon-motion number) the integer from dividing the circumference of heaven. Lacuna in the original; supplied here.
224
周餘諸本「周」訛「用」，今依文義改正。
Circuit remainder: all editions read "yong" for "zhou"; emended by sense.
225
減半月小餘之外按入交限數為會通減通數之半，除以日法，即得一百五十八度。 其餘數為十一萬六千五十八半。 此注或由錯簡，致與正文迥無關係，依術當作「一百五十八度外不成度之數」。
Beyond subtracting half the month's small remainder. Check: entering-crossing limit = conjunction-communicant − half communicant; ÷ day divisor = 158°. The remainder is 116,058½. This gloss may be a displaced passage unrelated to the main text; procedurally it should read "the fractional degrees outside 158°."
226
命以紀算外諸本「算」下脫「外」字，今依文義補。
Count from the era, outside the tally: editions omit "wai" after "suan"; supplied by sense.
227
命如上諸本「上」訛「止」，今依文義改正。
Count as above: all editions read "zhi" for "shang"; emended by sense.
228
閏月有進退諸本「退」訛「即」，今依文義改正。
Intercalary months may advance or retreat: all editions read "ji" for "tui"; emended by sense.
229
如章閏得一諸本「如」訛「加」，今依文義改正。
As rule-intercalation yields one: all editions read "jia" for "ru"; emended by sense.
230
度餘三萬九千三百三十九諸本「三十九」作「四十九」。 按甲戌紀交會差二千六百五十二萬二千六百四十九除以日法二十萬八千五百三十，得交會差一百二十七度、度餘三萬九千三百三十九。 「四十」乃「三十」之訛，今改正。
Degree remainder 39,339: editions read "si shi jiu" for "san shi jiu". Check: jiaxu-era crossing-conjunction difference 26,522,649 ÷ day divisor 208,530 = 127° with remainder 39,339. "Si shi" is a corruption of "san shi"; corrected here.
231
算外諸本「算」下脫「外」字，今依文義補。
Outside the tally: editions omit "wai" after "suan"; supplied by sense.
232
其月朔則交會諸本「會」訛「道」，今依文義改正。
That month's new moon then crosses in conjunction: all editions read "dao" for "hui"; emended by sense.
233
即是不蝕度諸本「蝕」訛「餘」，今依文義改正。
That is the non-eclipse degree: all editions read "yu" for "shi"; emended by sense.
234
餘為蝕分諸本「為」下衍「餘」字，今依文義刪。
The remainder is the eclipse fraction: editions add "yu" after "wei"; deleted by sense.
235
甲戌紀遲差分二百三十五萬二千一百九十一諸本「二百」作「二日」，「二千」作「三千」。 按以一紀月數二百八萬五千三百乘通數六百一十五萬八千一十七，以通周五百七十四萬五千九百四十一去之，得甲戌紀遲差分二百三十五萬二千一百九十一。 知「日」乃「百」之訛，「三千」乃「二千」之訛，今改正。
Jiaxu-era slow-motion difference fraction 2,352,191: editions read "er ri" for "er bai" and "san qian" for "er qian". Check: 2,085,300 × 6,158,017, less communicant circuit 5,745,941, yields jiaxu-era slow-motion difference 2,352,191. "Ri" corrupts "bai" and "san qian" corrupts "er qian"; corrected here.
236
所得為積周諸本「為積周」訛「日餘周」，今依正光曆例改正。
The result is accumulated circuits: editions read "ri yu zhou" for "wei ji zhou"; emended per the Zhengguang calendar.
237
日蝕滿日法從日諸本「日法從日」誤倒作「日從日法」，今依文義乙正。
Solar eclipse: when the divisor is full, carry to days—editions reverse "ri fa cong ri" as "ri cong ri fa"; transposed by sense.
238
月行遲疾度及分諸本「分」訛「合」，今依文義改正。
Moon slow-and-fast motion in degrees and parts: all editions read "he" for "fen"; emended by sense.
239
十四度 〈三百三十四分〉 諸本「三十四分」作「三十十四分」，衍一「十」字。 按本表各項數字，今依下列方法加以校算： (一) 月行遲疾度及分：以月日平行分七千五百一十三 (即小周) 加減本日損益率 (損減、益加) ，以章歲五百六十二約之為度，餘為分。 (二) 損益率：以各日月實行度分化為分，減以月日平行分七千五百一十三即得。 (三) 盈縮并率：將本日前各日損益率累加得之。 (四) 盈縮積分：盈縮并率乘日法二十萬八千五百三十，約以小周七千五百一十三，即得。 以下至校勘記[三七]，逕指出傳本譌誤所在，校改理由不再贅述。
14° 〈334 parts〉 Editions read "san shi si fen" for "san shi si fen", with an extra "shi". Each entry in this table has been rechecked by the following procedures: (1) Moon slow-and-fast degrees and parts: use parallel moon-sun motion fraction 7,513 (i.e., the lesser circuit) Add or subtract the present day's gain-loss rate (on a decrease entry, subtract; on an increase entry, add) , divide by rule-year 562 to get whole degrees; the remainder counts as parts. (2) Gain-loss rate: turn each day's true solar-lunar motion in degree-parts into fractional parts, then subtract parallel moon-sun motion fraction 7,513. (3) Excess-deficit combined rate: sum the daily gain-loss rates from the start of the period through the day before. (4) Excess-deficit accumulated parts: multiply the combined rate by day divisor 208,530, then reduce by lesser circuit 7,513. From here through Collation Note [37], each entry only marks where transmitted texts err; the rationale for each fix is not repeated.
240
盈七百五十七諸本「十」下脫「七」字，今據校算補。
Excess 757: editions omit "qi" below "shi"; supplied per recalculation.
241
二百六十一分諸本「二百」下衍「四十」兩字，今據校算刪。
261 parts: editions add "si shi" below "er bai"; deleted per recalculation.
242
益六百一十六諸本「十六」訛「十七」，今據校算改正。
Increase 616: editions read "shiqi" for "shiliu"; corrected per recalculation.
243
盈一千四百四十六諸本「四十六」訛「三十六」，今據校算改正。
Excess 1,446: editions read "sanshiliu" for "sishiliu"; corrected per recalculation.
244
四百六十八分諸本「六十八分」訛「八十六分」，今據校算乙正。
468 parts: editions read "bashi liu fen" for "liu shi ba fen"; transposed per recalculation.
245
二百六十七分諸本「二百」訛「三百」，今據校算改正。
267 parts: editions read "sanbai" for "erbai"; corrected per recalculation.
246
益六百四十六諸本「四十六」訛「三十六」，今據校算改正。
Increase 646: editions read "sanshiliu" for "sishiliu"; corrected per recalculation.
247
三百二十四分諸本「三百」訛「二百」，今據校算改正。
324 parts: editions read "erbai" for "sanbai"; corrected per recalculation.
248
十三度諸本「十三」訛「十二」，今據校算改正。
13 degrees: editions read "shi'er" for "shisan"; corrected per recalculation.
249
三百四十六分諸本「四十六」訛「三十六」，今據校算改正。
346 parts: editions read "sanshiliu" for "sishiliu"; corrected per recalculation.
250
定大小餘諸本「餘」訛「蝕」，今依文義改正。
Determine major and minor remainders: editions read "shi" for "yu"; emended by sense.
251
積分盈者諸本「盈」訛「盛」，今依文義改正。
Where accumulated parts are in excess: editions read "sheng" for "ying"; emended by sense.
252
隨定大小餘諸本「小」下衍「蝕」字，今依文義刪。
Follow to determine major and minor remainders: editions add "shi" below "xiao"; deleted by sense.
253
推日月合朔弦望度術第五諸本「度」下脫「術」字，今依本術前後各標題例補。
Procedure 5, computing solar-lunar conjunction, new moon, quarter, and full-moon degrees: editions omit "shu" below "du"; supplied per adjacent section titles.
254
不盡為餘諸本「餘」上脫「不盡為」三字，今依文義補。
What does not exhaust becomes remainder: editions omit "bu jin wei" before "yu"; supplied by sense.
255
求次月次日日所在度諸本「次月次日」作「日次月次」。 按依術文，先求次月日所在度，次求次日日所在度，傳本誤倒，今乙正。
Find the degree where the next month and next day stand: editions read "ri ci yue ci" for "ci yue ci ri." Note: per the procedure text, one first finds the next month's daily degree, then the next day's solar degree; transmitted texts reverse this; transposed.
256
逕斗除其分諸本「逕」訛「遙」，今依文義改正。
Pass through Dipper and drop the fractional parts: editions read "yao" for "jing"; emended by sense.
257
以月周二十二萬五千三百九十乘之諸本「周」上脫「月」字，今依文義補。
Multiply by lunar circuit 225,390: editions omit "yue" before "zhou"; supplied by sense.
258
求次月月度術諸本「次月」訛「月次」，今據上正光曆例乙正。
Procedure to find the next month's lunar degree: editions read "yue ci" for "ci yue"; transposed per the Zhengguang calendar example above.
259
求次日月度術諸本「次日月度」訛「月次日度」，今據上正光曆乙正。
Procedure to find the next day's lunar degree: editions read "yue ci ri du" for "ci ri yue du"; transposed per the Zhengguang calendar above.
260
小分三千四百六十七諸本「七」作「一」。 按以四除一經月度數：度二十九、大分八千九百四十五、小分六千九百一十九，得上弦度數：度七、大分六千四百五十一、小分三千四百六十七、微分二。 小分尾數「一」字乃「七」之訛，今改正。
Minor parts 3,467: editions read "yi" for "qi." Check: dividing one lunation's degrees 29, major parts 8,945, minor parts 6,919 by 4 yields first quarter 7°, 6,451 major parts, 3,467 minor parts, 2 micro-parts. The terminal minor part "yi" is a corruption of "qi"; corrected here.
261
又加得望諸本「加」作「如」。 殿本考證云「如字當作加」，說是，今改正。
Then add again to obtain full moon: editions read "ru" for "jia." The Palace Edition textual note says "ru" should read "jia"; that view is accepted; corrected here.
262
降赤決溫陰雨「降赤」，本志上正光曆作「絳赤」，疑作「絳赤」是。
Crimson fall, decisive warmth, yin rain: the Zhengguang calendar above reads "jiang chi" (deep crimson); "jiang chi" is likely correct.
263
日澤寒陰雨「日澤、寒、陰雨」，本志上正光曆作「白濁、微寒、陰雨」，疑此「日澤」乃「白濁」之訛，又脫「微」字。
Sun marsh, cold, yin rain: Zhengguang has white turbidity, slight cold, yin rain; "ri ze" here is likely a corruption of "bai zhuo," and "wei" is missing.
264
鶡旦不鳴諸本「鳴」上無「不」字。 按本志上正光曆作「鶡旦不鳴」。 據禮月令作「鶡旦不鳴」，呂氏春秋仲冬作「鶡鴠不鳴」，淮南子時則篇作「鳱鴠不鳴」，則以有「不」字為是。 今據補。
Crossbill silent at dawn: editions lack "bu" above "ming." Note: the Zhengguang calendar in this treatise reads "he dan bu ming." The Rites' Monthly Ordinances has "he dan bu ming"; Lüshi chunqiu mid-winter has "he dan bu ming"; Huainanzi Seasonal Regulations has "guan dan bu ming"—so the negative particle is warranted. Supplied here accordingly.
265
算外諸本「外」訛「上」，今據正光曆推上朔法改正。
Outside the tally: editions read "shang" for "wai"; corrected per Zhengguang new-moon procedure.
266
其數一百九十五萬三千七百一十六諸本「十六」作「十七」。 按辰星合終日數一百一十五乘以度法一萬六千八百六十，并以合終日餘一萬四千八百一十六 (此數末位尾數「六」原文誤作「八」，參閱校勘記[六八]) ，得一百九十五萬三千七百一十六。 尾數「十七」乃「十六」之訛，今改正。
The number 1,953,716: editions read "shiqi" for "shiliu." Check: Mercury conjunction-cycle days 115 × degree divisor 16,860, plus conjunction-cycle day remainder 14,816 (The terminal digit "liu" in this figure was written as "ba" in the source; see Collation Note [68].) , yielding 1,953,716. Terminal digits "shiqi" are a corruption of "shiliu"; corrected here.
267
減合度餘諸本「度」下脫「餘」字，今依文義補。
Subtract conjunction degree remainder: editions omit "yu" below "du"; supplied by sense.
268
加周虛諸本於「加周虛」下衍「積年盡」三字，今依文義刪。
Add circuit void: editions add "ji nian jin" below "jia zhou xu"; deleted by sense.
269
其求金水及命度諸本「水」上奪「金」字。 按上推五星術於求得冬至後晨夕合度算及度餘後，述求金、水冬至後晨夕合度及度餘術。 則此處不當單求水一星，今補。
Finding Metal, Water, and fate-degree: editions lack "jin" above "shui." Note: after the five-star procedure obtains winter-solstice morning and evening conjunction tallies and remainders, it gives procedures for Metal and Water winter-solstice morning and evening conjunctions and remainders. This passage should not treat Water alone; "jin" is supplied here.
270
餘滿日度法從日諸本「從」訛「後」，「從日」下又衍「一日」兩字，今依文義刪改。
When remainder fills day-degree divisor, carry to days: editions read "hou" for "cong," and add "yi ri" below "cong ri"; deleted and emended by sense.
271
以一合日數及合日餘加之諸本「以一合日數及合日數」誤倒作「以合日數及一合日餘」，今依文義乙正。
Add one-conjunction day count and conjunction day remainder: editions reverse this as "conjunction day count and one-conjunction day remainder"; transposed by sense.
272
度餘八千四百九十一諸本「八千」作「九千」。 按歲星數六百七十二萬二千八百八十八減以周天六百一十五萬八千一十七，餘除以度法一萬六千八百六十，得行星三十三度、度餘八千四百九十一。 知「九千」乃「八千」之訛，今改正。
Degree remainder 8,491: editions read "jiu qian" for "ba qian." Check: Jupiter number 6,722,888 − circuit of heaven 6,158,017; remainder ÷ degree divisor 16,860 = planetary motion 33° with remainder 8,491. "Jiu qian" is a corruption of "ba qian"; corrected here.
273
九十二日行四十八度而留諸本「二」作「一」。 按熒惑日行二十三分之十二分，則九十二日行四十八度，不當為九十一日，今改正。
92 days, 48 degrees, then station: editions read "yi" for "er." Check: Mars moves 12/23 of a degree per day, so 92 days yield 48°, not 91 days; corrected.
274
行星五十五度諸本「十」下無「五」字。 按熒惑在日前後，伏行星度當相等，上文在日後伏作「行星五十五度」； 依其伏、行、留、逆之總行度計，當以此數為是，傳本脫「五」字，今據補。
Planetary motion 55°: editions lack "wu" below "shi." Check: Mars before and after the sun should have equal hidden motion; the passage above, hidden after the sun, reads 55°; by total hidden, direct, stationary, and retrograde motion this figure is correct; transmitted texts omit "wu"; supplied here.
275
度餘一萬五千六百八十一諸本「度餘一萬」誤倒作「度一餘萬」，今依文義乙正。
Degree remainder 15,681: editions read "du yi yu wan" for "du yu yi wan"; transposed by sense.
276
九十一日行一百一十二度諸本「一百」作「二百」。 按太白日行一度十三分之三，乘以九十一，則得一百一十二度，「二」字訛，今改正。
91 days, 112 degrees: editions read "er bai" for "yi bai." Check: Venus at 1° 3/13 per day × 91 = 112°; "er" is wrong; corrected.
277
九十一日行一百五度諸本「五」上脫「一百」。 按太白日行一度十三分之二，乘以九十一日，應行一百五度； 下太白在日后，以同速運行九十一日，正作「一百五度」。 今據補。
91 days, 105 degrees: editions omit "yi bai" above "wu." Check: Venus at 1° 2/13 per day × 91 days should yield 105°; below, Venus after the sun at the same speed for 91 days correctly reads 105°. Supplied here accordingly.
278
順大遲日行十五分之十一諸本「遲」作「疾」。 按太白日行十五分之十一為最遲，不當為大疾，今改正。
Prograde, great slow, 11/15 per day: editions read "ji" for "chi." Check: Venus at 11/15 is slowest motion; it cannot read great fast; corrected.
279
日行一度十三分之三諸本下「三」字作「二」。 按以九十一除一百一十二度，得日行一度十三分之三。 「二」乃「三」之訛，今改正。
Daily motion 1° 3/13: editions read "er" for "san" below. Check: 112° ÷ 91 = 1° 3/13 per day. "Er" is a corruption of "san"; corrected here.
280
合終日餘一萬四千八百一十六諸本「十六」作「十八」。 按以度法一萬六千八百六十除辰星數一百九十五萬三千七百一十七，得合終日數一百一十五，合終日餘一萬四千八百一十六； 而以周虛二千四十四減度法一萬六千八百六十，所得日餘亦同，足證末位尾數當為「六」。 今改正。
Conjunction-cycle day remainder 14,816: editions read "shiba" for "shiliu." Check: Mercury number 1,953,717 ÷ degree divisor 16,860 = conjunction-cycle days 115, day remainder 14,816; subtracting circuit void 2,044 from degree divisor 16,860 yields the same day remainder—confirming the terminal digit should be "liu." Corrected here.
281
度餘一萬五千八百三十八諸本「三十八」作「四十八」。 按以二除辰星合終日數一百一十五、日餘一萬四千八百一十六，得行星度五十七、度餘一萬五千八百三十八。 知「四十八」乃「三十八」之訛，今改正。
Degree remainder 15,838: editions read "siba" for "sanba." Check: Mercury conjunction-cycle days 115 and day remainder 14,816 ÷ 2 = planetary motion 57° with remainder 15,838. "Siba" is a corruption of "sanba"; corrected here.
282
日餘一萬五千八百三十八行星三十四度度餘一萬五千八百三十八諸本日餘末兩位數「三十八」作「四十八」，其下並脫「行星三十四度、度餘一萬五千八百三十八」數字。 按辰星在日前后之伏行留逆互為對稱。 下文在日後晨伏東方項下作：「十七日、日餘一萬五千八百三十八」，其下有「行星三十四度、度餘一萬五千八百三十八」十七字。 其在日前後伏行留逆之總日數各為五十七日、日餘一萬五千八百三十八，總行度各為五十七度、度餘一萬五千八百三十八，知此處脫文。 今改補。
Day remainder 15,838; planetary motion 34°; degree remainder 15,838: editions read "siba" for the last two digits of day remainder, and omit "planetary motion 34 degrees, degree remainder 15,838" below. Check: Mercury's hidden, direct, stationary, and retrograde phases before and after the sun are symmetric. Below, under morning hidden east after the sun: "17 days, day remainder 15,838," with "planetary motion 34 degrees, degree remainder 15,838" following. Total days and remainders for hidden, direct, stationary, and retrograde motion before and after the sun are each 57 days with remainder 15,838, and total degrees each 57° with remainder 15,838—showing text is missing here. Emended and supplied here.
283
度餘一萬五千八百三十八諸本「三十八」訛「四十八」，今改正。 詳上條校記。
Degree remainder 15,838: editions read "siba" for "sanba"; corrected here. See the preceding collation note.
284
從全命之如前諸本「全」誤作「令」，今依文義改正。
Follow the full fate as before: editions read "ling" for "quan"; emended by sense.
285
伏不書度諸本「書」作「盡」。 按李銳四分術注謂「伏不書度」云「謂不以所在宿度書於術」。 李說是，於此亦然，「盡」當作「書」，今據改。
In hiding, do not record degrees: editions read "jin" for "shu." Note: Li Rui's Quarter-remainder commentary on "in hiding, do not record degrees" says this means not writing the lodge-degree of its position into the procedure. Li is right; here too "jin" should read "shu"; emended accordingly.
286
前後相御諸本「相御」下衍「十四」兩字，今據晉書卷一七律曆志中載景初曆五星曆步術文刪。
Before and after mutually govern: editions add "shi si" below "xiang yu"; deleted per Jin shu 17, Treatise on Music and Calendars, Jingchu calendar five-star step procedure.