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律曆中
Treatises on Measures and the Calendar, Part Two.
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夫曆者，紀陰陽之通變，極往數以知來，可以迎日授時，先天成務者也。 然則懸象著明，莫大於二曜，氣序環複，無信於四時。 日月相推而明生矣，寒暑迭進而歲成焉，遂能成天地之文，極乾坤之變。 天數五，地數五，五位相乘而各有合。 天數二十有五，地數三十，凡天地之數五十有五，所以成變化而行鬼神也。 乾之策二百一十有六，坤之策一百四十有四，凡三百六十，以當期之日也。 至乃陰陽迭用。 剛柔相摩，四象既陳，八卦成列，此乃造文之元始，創曆之厥初者歟？ 洎乎炎帝分八節，軒轅建五部，少昊以鳳鳥司曆，顓頊以南正司天，陶唐則分命和仲，夏後乃備陳《鴻范》，湯武革命，咸率舊章。 然文質既殊，正朔斯革，故天子置日官，諸侯有日禦，以和萬國，以協三辰。 至於寒暑晦明之徵，陰陽生殺之數，啟閉升降之紀，消息盈虛之節，皆應躔次而不淫，遂得該浹生靈，堪輿天地，開物成務，致遠鉤深。 周德既衰，史官廢職，疇人分散，禨祥莫理。 秦兼天下，頗推五勝，自以獲水德之瑞，以十月為正。 漢氏初興，多所未暇，百有餘載，猶行秦曆。 至於孝武，改用夏正。 時有古曆六家，學者疑其紕繆，劉向父子，咸加討論，班固因之，采以為志。 光武中興，未能詳考。 逮于永平之末，乃複改行四分，七十餘年，儀式方備。 其後覆命劉洪、蔡邕，共修律曆，司馬彪用之以續《班史》。 當塗受命，亦有史官，韓翊創之于前，楊偉繼之於後，咸遵劉洪之術，未及洪之深妙。 中、左兩晉，迭有增損。 至於西涼，亦為蔀法，事蹟糾紛，未能詳記。 宋氏元嘉，何承天造曆，迄于齊末，相仍用之。 梁武初興，因循齊舊，天監中年，方改行宋祖沖之《甲子元曆》。 陳武受禪，亦無創改。 後齊文宣，用宋景業曆。 西魏入關，行李業興曆。 逮于周武帝，乃有甄鸞造《甲寅元曆》，遂參用推步焉。 大象之初，太史上士馬顯，又上《丙寅元曆》，便即行用。 迄於開皇四年，乃改用張賓曆，十七年，複行張胄玄曆，至於義寧。 今采梁天監以來五代損益之要，以著於篇雲。
The calendar records the ceaseless transformations of yin and yang, extrapolates from past reckonings to anticipate the future, aligns human affairs with the sun's course, and thus enables one to act ahead of Heaven itself. Among the celestial signs that hang in the heavens, none shines more brilliantly than the sun and moon; among the patterns by which vital force moves through the year, none is more reliable than the four seasons. The sun and moon drive one another in their courses and light comes into being; cold and heat succeed each other until the year is complete. In this way the pattern of Heaven and Earth is fulfilled and the full range of cosmic change is realized. Heaven has five numbers and Earth has five numbers; when these five positions are multiplied together, each pair combines in its proper way. Heaven's numbers sum to twenty-five and Earth's to thirty, giving fifty-five in all—the number by which transformation is achieved and the workings of spirits are carried out. The tally-sticks for qian total two hundred and sixteen, and those for kun one hundred and forty-four—three hundred and sixty altogether, matching the days of a full cycle. In this way yin and yang alternate in their operation. Hard and soft interact, the four emblems take their places, and the eight trigrams stand in order—is this not the primal source of written pattern and the very beginning of calendrical reckoning? From the Flame Emperor, who divided the eight seasonal nodes, through the Yellow Emperor, who established the five bureaus; from Shaohao, who entrusted the calendar to the phoenix bird, and Zhuanxu, who set the southern director to oversee Heaven; from Yao, who appointed He and Zhong, to the Xia, who fully codified the Great Plan—through the revolutions of Tang and Wu, all followed established calendrical precedent. Yet as ritual forms changed from age to age, so did the calendar's new year and first month; hence the Son of Heaven appointed officials of the sun, and feudal lords maintained their own calendrical officers, to harmonize the myriad states and keep the three celestial markers in accord. The signs of cold and heat, light and dark, the reckoning of yin and yang's generative and destructive phases, the cycles of opening and closing and of rising and falling, and the rhythms of waxing and waning—all matched the celestial stations without deviation. Thus the calendar could embrace all living things, span Heaven and Earth, open the way for human enterprise, and reach to the farthest depths. When Zhou's virtue waned, the historiographers neglected their duties, calendrical experts dispersed, and no one any longer attended to omens and portents. When Qin conquered the realm, it embraced the theory of cyclical conquest among the five phases, claimed the auspicious token of the Water Virtue, and made the tenth month the year's beginning. In the Han dynasty's early years, many reforms went unattended, and for more than a century the Qin calendar remained in use. Under Emperor Wu, the dynasty adopted the Xia calendar's first month as the new year. Six ancient calendrical schools were then in circulation, and scholars questioned their inaccuracies. Liu Xiang and his son examined them thoroughly, and Ban Gu drew on their work for his treatise. When Emperor Guangwu restored the Han, no thorough calendrical review was undertaken. Not until the end of the Yongping reign was the Quarter-Remainder calendar reinstated; only after more than seventy years were its procedures fully established. Later Liu Hong and Cai Yong were again commissioned to revise the standards of pitch and calendar, and Sima Biao incorporated their work into his continuation of Ban Gu's history. When the Cao Wei dynasty took power, it too maintained calendrical officers: Han Yi devised a system first, and Yang Wei followed, both adopting Liu Hong's methods but falling short of his depth and refinement. Under both the Western and Eastern Jin, the calendar underwent repeated revisions. Western Liang likewise employed an obscuration-cycle calendar, but the records are too confused to recount in detail. In the Song dynasty's Yuanjia era, He Chengtian devised a new calendar that remained in continuous use until the end of Qi. When Emperor Wu of Liang came to power, he initially retained Qi's calendar; only in mid-Tianjian did he adopt Zu Chongzhi's Jiazi Origin Calendar from the Liu Song. When Emperor Wu of Chen accepted the throne, he introduced no calendrical innovations. Later, Emperor Wenxuan of Northern Qi adopted the calendar of Song Jingye. When Western Wei entered the Guanzhong region, Li Yexing devised a new calendar. Under Emperor Wu of Northern Zhou, Zhen Luan created the Jiayin Origin Calendar, which was then adopted for calendrical computation. At the start of the Daxiang era, Senior Grand Astrologer Ma Xian submitted the Bingyin Origin Calendar, which was adopted at once. fourth year of In of Kaihuang the calendar of Zhang Bin was adopted; in 17th year of Kaihuang that of Zhang Zhouxuan was restored, and it remained in use until the Yining era. What follows selects the chief calendrical revisions of the five dynasties since Liang's Tianjian era and records them in this chapter.
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梁初因齊，用宋《元嘉曆》。 天監三年下詔定曆，員外散騎侍郎祖暅奏曰：「臣先在晉已來，世居此職。 仰尋黃帝至今十二代，曆元不同，周天、斗分，疏密亦異，當代用之，各垂一法。 宋大明中，臣先人考古法，以為正曆，垂之於後，事皆符驗，不可改張。」 八年，暅又上疏論之。 詔使太史令將匠道秀等，候新舊二曆氣朔、交會及七曜行度，起八年十一月，訖九年七月，新曆密，舊曆疏。 暅乃奏稱：「史官今所用何承天曆，稍與天乖，緯緒參差，不可承案。 被詔付靈台，與新曆對課疏密，前期百日，並又再申。 始自去冬，終於今朔，得失之效，並已月別啟聞。 夫七曜運行，理數深妙，一失其源，則歲積彌爽。 所上脫可施用，宜在來正。」 至九年正月，用祖沖之所造《甲子元曆》頒朔。 至大同十年，制詔更造新曆，以甲子為元，六百一十九為章歲，一千五百三十六為日法，一百八十三年冬至差一度，月朔以遲疾定其小餘，有三大二小。 未及施用而遭侯景亂，遂寢。
Early Liang, following Qi, continued to use the Song dynasty's Yuanjia Calendar. third year of In of Tianjian an edict ordered the calendar to be fixed. Supernumerary Gentleman Attendant Zu Geng memorialized: "My family has held this office for generations, going back to the Jin dynasty. Tracing from the Yellow Emperor down through twelve dynasties, each age has used a different calendar origin, with varying values for the celestial circuit and the dipper fraction; every dynasty in turn has handed down its own system. In the Liu Song's Daming era, my ancestor examined the ancient method and established the correct calendar, transmitting it to posterity. Its predictions have always been verified, and it should not be changed." eighth year of In of Tianjian, Zu Geng submitted another memorial on the matter. An edict ordered Grand Astrologer Jiang Daoxiu and others to compare the old and new calendars by observing seasonal nodes, new moons, conjunctions, and the motions of the seven luminaries from the eleventh month of eighth year of Tianjian through the seventh month of ninth year of Tianjian. The new calendar proved accurate; the old one did not. Zu Geng then reported: "The He Chengtian calendar currently used by the historiographers already diverges from the heavens; its underlying principles are inconsistent and cannot be maintained. By imperial order it was sent to the Observational Platform to be tested against the new calendar. Predictions were made a hundred days in advance and verified twice over. From last winter through the current new moon, the results have been reported month by month. The motions of the seven luminaries rest on principles of extraordinary subtlety; lose the underlying reckoning even once, and the error compounds with every passing year. The calendar submitted may be adopted and should take effect from the coming new year." In the first month of ninth year of Tianjian, Zu Chongzhi's Jiazi Origin Calendar was adopted and the new year's calendar was promulgated. tenth year of By of Datong an imperial decree ordered a new calendar: jiazi as the origin, 619 as the cycle year, 1,536 as the day divisor, one degree of precession at the winter solstice over 183 years, and new-moon small remainders determined by the moon's anomalistic motion, yielding three long months and two short months. Before it could be implemented, Hou Jing's rebellion broke out, and the project was abandoned.
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陳氏因梁，亦用祖沖之曆，更無所創改。 後齊文宣受禪，命散騎侍郎宋景業葉圖讖，造《天保曆》。 景業奏：依《握誠圖》及《元命包》，言齊受錄之期，當魏終之紀，得乘三十五以為蔀，應六百七十六以為章。」 文宣大悅，乃施用之。 期曆統曰：「上元甲子，至天保元年庚午，積十一萬五百六算外，章歲六百七十六，度法二萬三千六百六十，斗分五千七百八十七，曆餘十六萬二千二百六十一。」 至後主武平七年，董峻、鄭元偉立議非之曰：「宋景業移閏於天正，退命於冬至交會之際，承二大之後，三月之交，妄減平分。 臣案，景業學非探賾，識殊深解，有心改作，多依舊章，唯寫子換母，頗有變革，妄誕穿鑿，不會真理。 乃使日之所在，差至八度，節氣後天，閏先一月。 朔望虧食，既未能知其表裡，遲疾之曆步，又不可以傍通。 妄設平分，虛退冬至，虛退則日數減于周年，平分妄設，故加時差於異日。 五星見伏，有違二旬，遲疾逆留，或乖兩宿。 軌褵之術，妄刻水旱。 今上《甲寅元曆》，並以六百五十七為章，二萬二千三百三十八為蔀，五千四百六十一為斗分，甲寅歲甲子日為元紀。」 又有廣平人劉孝孫、張孟賓二人，同知曆事。 孟賓受業于張子信，並棄舊事，更制新法。 又有趙道嚴，准晷影之長短，定日行之進退，更造盈縮，以求虧食之期。 劉孝孫以百一十九為章，八千四十七為紀，九百六十六為歲餘，甲子為上元，命日度起虛中。 張孟賓以六百一十九為章，四萬八1千九百為紀，九百四十八為日法，萬四千九百四十五為斗分。 元紀共命，法略旨遠。 日月五星，並從斗十一起。 盈縮轉度，陰陽分至，與漏刻相符，共日影俱合，循轉無窮。 上拒春秋，下盡天統，日月虧食及五星所在，以二人新法考之，無有不合。 其年，訖幹敬禮及曆家豫刻日食疏密。 六月戊申朔，太陽虧，劉孝孫言食于卯時，張孟賓言食于甲時，鄭元偉、董峻言食于辰時，宋景業言食于巳時。 至日食，乃於卯甲之間，其言皆不能中。 爭論未定，遂屬國亡。
Chen, following Liang, also used Zu Chongzhi's calendar without further modification. When Emperor Wenxuan of Northern Qi took the throne, he ordered Gentleman Attendant Song Jingye, drawing on prognostic charts, to create the Tianbao Calendar. Jingye reported that according to the Chart of Grasping Sincerity and the Wrapping of the Primordial Mandate, Qi would receive the mandate at the end of Wei's era; thirty-five multiplied yields the obscuration cycle, and 676 the cycle year." Emperor Wenxuan was delighted and ordered the calendar into use. The calendar summary reads: "From the upper origin in jiazi to first year of Tianbao (gengwu), the accumulated count is 110,506 beyond the epoch; cycle year 676, degree divisor 23,660, dipper fraction 5,787, calendar remainder 162,261." seventh year of In of Wuping under the last ruler, Dong Jun and Zheng Yuanwei objected: "Song Jingye shifted the intercalary month to the celestial new year, deferred the epoch to the winter solstice conjunction, placed it after two long months, and at the third month's conjunction arbitrarily reduced the equal division. We find that Jingye's learning did not reach to the hidden depths, and his understanding was shallow. Though he intended reform, he mostly followed old formulas, merely swapping one constant for another—a forced and arbitrary contrivance that misses true principle. The result was solar positions off by as much as eight degrees, seasonal nodes lagging behind the heavens, and intercalary months arriving a month too early. For new and full moons and eclipses, he could not determine their limits; nor could his anomalistic reckoning be cross-checked by any alternative method. He arbitrarily imposed equal divisions and falsely shifted the winter solstice; this reduced the day count below a full year, and his arbitrary equal division caused the hour of occurrence to fall on the wrong day. The five planets' appearances and disappearances were off by as much as twenty days; their direct, retrograde, and stationary motions sometimes missed by two lunar lodges. His orbital methods arbitrarily predicted floods and droughts. We now submit the Jiayin Origin Calendar, with 657 as the cycle year, 22,338 as the obscuration cycle, 5,461 as the dipper fraction, and the jiazi day of the jiayin year as the epoch." Two men from Guangping, Liu Xiaosun and Zhang Mengbin, also shared responsibility for calendrical matters. Mengbin had studied under Zhang Zixin; both men abandoned the old system and devised new methods. Zhao Daoyan measured gnomon shadows to determine the sun's advance and retreat, and reconstructed the solar anomaly tables to calculate eclipse dates. Liu Xiaosun used 119 as the cycle year, 8,047 as the era, 966 as the year remainder, jiazi as the upper origin, and set the solar degree to begin at the middle of the Xu lodge. Zhang Mengbin used 619 as the cycle year, 48,900 as the era, 948 as the day divisor, and 14,945 as the dipper fraction. Their origin and era shared a single epoch; the method was compact in form but far-reaching in design. The sun, moon, and five planets all began from the eleventh degree of the Dipper lodge. Solar anomaly tables, rotating degrees, and the yin-yang divisions at the solstices matched the clepsydra graduations; gnomon shadows agreed as well, cycling without end. From the Spring and Autumn period down through the Tian Tong calendar, when solar and lunar eclipses and the positions of the five planets were checked against their new methods, every case matched. That year Gan Jingli and the calendrical specialists competed to predict the solar eclipse in advance. On the wushen day new moon of the sixth month, a solar eclipse occurred. Liu Xiaosun predicted it for the mao hour, Zhang Mengbin for the jia hour, Zheng Yuanwei and Dong Jun for the chen hour, and Song Jingye for the si hour. When the eclipse came, it fell between the mao and jia hours; none of them was correct. The dispute was never resolved before the state fell.
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臣案九章五紀之旨，三統四分之說，咸以節宣發斂，考詳晷緯，布政授時，以為皇極者也。 而乾維難測，斗憲易差，盈縮之期致舛，咎徵之道斯應。 甯止蛇或乘龍，水能沴火，因亦玉羊掩曜，金雞喪精。 王化關以盛衰，有國由其隆替，曆之時義，于斯為重。 自炎漢已還，迄于有魏，運經四代，事涉千年，日禦天官，不乏於世，命元班朔，互有沿改。 驗近則疊璧應辰，經遠則連珠失次，義難循舊，其在茲乎？ 大周受圖膺錄，牢籠萬古，時夏乘殷，斟酌前代，曆變壬子，元用甲寅。 高祖武皇帝索隱探賾，盡性窮理，以為此曆雖行，未臻其妙，爰降詔旨，博訪時賢，並敕太史上士馬顯等，更事刊定，務得其宜。 然術藝之士，各封異見，凡所上曆，合有八家，精粗踳駁，未能盡善。 去年冬，孝宣皇帝乃詔臣等，監考疏密，更令同造。 謹案史曹舊簿及諸家法數，棄短取長，共定今術。 開元發統，肇自丙寅，至於兩曜虧食，五星伏見，參校積時，最為精密。 庶鐵炭輕重，無失寒燠之宜，灰箭飛浮，不爽陰陽之度。 上元丙寅至大象元年己亥，積四萬一千五百五十四算上。 日法五萬三千五百六十三，亦名蔀會法。 章歲四百四十八，斗分三千一百六十七，蔀法一萬二千九百九十二。 章中為章會法。 日法五萬三千五百六十三，曆餘二萬九千六百九十三，會日百七十三，會餘一萬六千六百一十九，冬至日在斗十二度。 小周餘、盈縮積，其曆術別推入蔀會，分用陽率四百九十九，陰率九。 每十二月下各有日月蝕轉分，推步加減之，乃為定蝕大小餘，而求加時之正。
The Nine Chapters, Five Records, Triple Concordance, and Quarter-Remainder calendars all share one purpose: to regulate the seasons, measure gnomon shadows and celestial coordinates, govern the realm, and grant the proper seasons—the very pivot of imperial rule. Yet the celestial axis is hard to measure, the dipper standard easily drifts, solar anomaly periods go awry, and calamities follow in their wake. It is not only that snakes sometimes usurp the dragon's place and water overcomes fire; jade sheep dim their radiance and golden roosters lose their brilliance as well. Imperial fortunes rise and fall with the calendar; a dynasty's ascent and decline depend upon it. The calendar's timely significance could hardly be greater. From the Han through Wei, across four dynasties and a millennium, calendrical officers were never absent from the court, yet the calendar origin and new-moon promulgation were revised again and again. Tested at close range, predictions may align; traced over centuries, they fall out of sequence. How can one simply follow precedent? The need for reform is clear. Great Zhou received the mandate, embracing all antiquity; drawing on Xia and Yin, it weighed the achievements of former dynasties. Its calendar used renzi as the cycle and jiayin as the origin. Emperor Gaozu, probing hidden principles to the utmost, judged that although this calendar was in use, it had not reached perfection. He issued an edict seeking advice from the finest scholars of the age and ordered Senior Grand Astrologer Ma Xian and others to revise it until it was right. Yet calendrical experts held divergent views; eight schools in all submitted calendars, varying widely in quality, and none was fully satisfactory. Last winter Emperor Xiaoxuan ordered us to supervise comparative testing and jointly devise a new calendar. We have examined the historiography bureau's old records and the numerical methods of all schools, discarded weaknesses and selected strengths, and jointly established the present system. The epoch begins from bingyin; for eclipses of the sun and moon and the appearances and disappearances of the five planets, accumulated verification shows it to be the most precise. Thus the iron-and-charcoal test will not miss the proper balance of cold and warmth, and the clepsydra's floating and sinking markers will not deviate from the measure of yin and yang. From the upper origin in bingyin to first year of Daxiang (jihai), the accumulated count is 41,554. Day divisor: 53,563, also called the obscuration-conjunction divisor. Cycle year: 448; dipper fraction: 3,167; obscuration divisor: 12,992. The cycle medial serves as the cycle-conjunction divisor. Day divisor: 53,563; calendar remainder: 29,693; conjunction period: 173 days; conjunction remainder: 16,619; winter solstice sun at the twelfth degree of the Dipper lodge. Small circuit remainders and solar anomaly accumulations are computed separately for entry into obscuration and conjunction, using a yang rate of 499 and a yin rate of 9. Below each of the twelve months are rotating eclipse fractions; by adding and subtracting these in computation, one obtains the fixed large and small remainders of the eclipse and determines the correct hour of occurrence.
6
其術施行。 時高祖作輔，方行禪代之事，欲以符命曜於天下。 道士張賓，揣知上意，自雲玄相，洞曉星曆，因盛言有代謝之徵，又稱上儀錶非人臣相。 由是大被知遇，恆在幕府。 及受禪之初，擢賓為華州刺史，使與儀同劉暉、驃騎將軍董琳、索盧縣公祐、前太史上士馬顯、太學博士鄭元偉、前保章上士任悅、開府掾張撤、前蕩邊將軍張膺之、校書郎衡洪建、太史監候粟相、太史司曆郭翟、劉宜、兼算學博士張乾敘、門下參人王君瑞、荀隆伯等，議造新曆，仍令太常卿盧賁監之。 賓等依何承天法，微加增損，四年二月撰成奏上。 高祖下詔曰：「張賓等存心算數，通洽古今，每有陳聞，多所啟沃。 畢功表奏，具已披覽。 使後月複育，不出前晦之宵，前月之餘，罕留後朔之旦。 減朓就朒，懸殊舊准。 月行表裡，厥途乃異，日交弗食，由循陽道。 驗時轉算，不越纖毫，逖德前修，斯秘未啟。 有一於此，實為精密，宜頒天下，依法施用。」
This method was adopted. At that time Gaozu was chief minister, preparing the transfer of the throne, and wished to display tokens of the mandate to the world. The Daoist Zhang Bin, sensing the ruler's intent, claimed mystical insight and mastery of astronomy and calendrics. He spoke at length of signs that the mandate was changing and declared that Gaozu's bearing was not that of a mere subject. He thereby won great favor and remained constantly at Gaozu's headquarters. At the start of his reign, Gaozu promoted Zhang Bin to Inspector of Huazhou and commissioned him together with Liu Hui, Dong Lin, Gong You, the former Senior Grand Astrologer Ma Xian, Erudite Zheng Yuanwei, Ren Yue, Zhang Che, Zhang Yingzhi, Heng Hongjian, Su Xiang, Guo Di, Liu Yi, Zhang Qianxu, Wang Junrui, Xun Longbo, and others to devise a new calendar, with Minister of Ceremonies Lu Fen supervising the work. Zhang Bin and his colleagues based their work on He Chengtian's method with minor modifications. In the second month of fourth year of Kaihuang they completed the calendar and submitted it. Gaozu issued an edict: "Zhang Bin and his colleagues are devoted to computation, thoroughly versed in ancient and modern learning, and their reports have repeatedly offered valuable counsel. Your completed memorial has been received and fully reviewed. The following month begins its cycle without spilling past the last night of the month; the prior month's remainder rarely carries over to the next new-moon dawn. Adjusting from gibbous to crescent phase, it departs sharply from the old standard. The moon's path has inner and outer aspects with differing courses; when the sun crosses the node yet no eclipse occurs, it is because the crossing follows the yang path. Verified by timely calculation, the error does not exceed a hair's breadth—a secret that earlier masters had not yet unlocked. In each of these respects it is truly precise. It should be promulgated throughout the realm and put into use according to law."
7
張賓所造曆法，其要：
The essential parameters of Zhang Bin's calendar:
8
以上元甲子已來，至開皇四年歲在甲辰，積四百一十二萬九千一，算上。
From the upper origin in jiazi to fourth year of Kaihuang (jiachen year), the accumulated count is 4,129,001.
9
蔀法，一十萬二千九百六十。
Obscuration divisor: 102,960.
10
章歲，四百二十九。
Cycle year: 429.
11
章月，五千三百六。
Cycle month: 5,306.
12
通月，五百三十七萬二千二百九。
Common month: 5,372,209.
13
日法，一十八萬一千九百二十。
Day divisor: 181,920.
14
斗分，二萬五千六十三。
Dipper fraction: 25,063.
15
會月，一千二百九十七。
Conjunction month: 1,297.
16
會率，二百二十一。
Conjunction rate: 221.
17
會數，一百一十半。
Conjunction number: 110½.
18
會分，一十一億八千七百二十五萬八千一百八十九。
Conjunction fraction: 1,187,258,189.
19
會日法，四千二十萬四千三百二十。
Conjunction day divisor: 40,204,320.
20
會日，百七十三。
Conjunction period: 173 days.
21
餘，五萬六千一百四十三。
Remainder: 56,143.
22
小分，一百一十。
Small fraction: 110.
23
交法，五億一千二百一十萬四千八百。
Crossing divisor: 512,104,800.
24
交分法，二千八百一十五。
Crossing fraction divisor: 2,815.
25
陰陽曆，一十三。
Yin-yang calendar: 13.
26
餘，十一萬二百六十三。
Remainder: 110,263.
27
小分，二千三百二十八。
Small fraction: 2,328.
28
朔差，二。
New-moon difference: 2.
29
餘，五萬七千九百二十一。
Remainder: 57,921.
30
小分，九百七十四。
Small fraction: 974.
31
蝕限，一十二。
Eclipse limit: 12.
32
餘，八萬一千三百三。
Remainder: 81,303.
33
小分，四百三十三半。
Small fraction: 433½.
34
定差，四萬四千五百四十八。
Fixed difference: 44,548.
35
周日，二十七。
Circuit day: 27.
36
餘，一十萬八百五十九。 亦名少大法
Remainder: 100,859. Also called the lesser great divisor.
37
木精曰歲星，合率四千一百六萬三千八百八十九。
The essence of Wood is Jupiter; its conjunction rate is 41,063,889.
38
火精曰熒惑，合率八千二十九萬七千九百二十六。
The essence of Fire is Mars; its conjunction rate is 80,297,926.
39
土精曰鎮星，合率三千八百九十二萬五千四百一十三。
The essence of Earth is Saturn; its conjunction rate is 38,925,413.
40
金精曰太白，合率六千一十一萬九千六百五十五。
The essence of Metal is Venus; its conjunction rate is 60,119,655.
41
水精曰辰星，合率一千一百九十三萬一千一百二十五。
The essence of Water is Mercury; its conjunction rate is 11,931,125.
42
張賓所創之曆既行，劉孝孫與冀州秀才劉焯，並稱其失，言學無師法，刻食不中，所駁凡有六條：其一雲，何承天不知分閏之有失，而用十九年之七閏。 其二雲，賓等不解宿度之差改，而冬至之日守常度。 其三雲，連珠合璧，七曜須同，乃以五星別元。 其四雲，賓等唯知日氣餘分恰盡而為立元之法，不知日月不合，不成朔旦冬至。 其五雲，賓等但守立元定法，不須明有進退。 其六雲，賓等唯識轉加大餘二十九以為朔，不解取日月合會准以為定。 此六事微妙，歷數大綱，聖賢之通術，而暉未曉此，實管窺之謂也。 若乃驗影定氣，何氏所優，賓等推測，去之彌遠。 合朔順天，何氏所劣，賓等依據，循彼迷蹤。 蓋是失其菁華，得其糠粃者也。 又雲，魏明帝時，有尚書郎楊偉，修《景初曆》，乃上表立義，駁難前非，雲：「加時後天，食不在朔。」 然觀楊偉之意，故以食朔為真，未能詳之而制其法。 至宋元嘉中，何承天著曆，其上表雲：「月行不定，或有遲疾，合朔月食，不在朔望，亦非曆之意也。」 然承天本意，欲立合朔之術，遭皮延宗飾非致難，故事不得行。 至後魏獻帝時，有龍宜弟複修延興之曆，又上表雲：「日食不在朔，而習之不廢，據《春秋》書食，乃天之驗朔也。」 此三人者，前代善曆，皆有其意，未正其書。 但歷數所重，唯在朔氣。 朔為朝會之首，氣為生長之端，朔有告餼之文，氣有郊迎之典，故孔子命曆而定朔旦冬至，以為將來之範。 今孝孫曆法，並按明文，以月行遲疾定其合朔，欲今食必在朔，不在晦、二之日也。 縱使頻月一小、三大，得天之統。 大抵其法有三，今列之雲。
Once Zhang Bin's calendar was adopted, Liu Xiaosun and the Ji Province scholar Liu Chuo both challenged its errors, declaring that its methods lacked proper foundation and its eclipse predictions missed the mark. They raised six objections. First: He Chengtian did not recognize the error in distributing intercalary months and used the seven-intercalation-in-nineteen-years rule. Second: Zhang Bin and his colleagues did not account for the shifting discrepancy in lodge degrees and kept the winter solstice at a fixed position. Third: when the heavenly markers align, the seven luminaries must share a single origin, yet they assigned separate origins to the five planets. Fourth: Zhang Bin and his colleagues knew only that when the solar qi remainder is exactly exhausted one may set the origin, but did not grasp that without sun-moon conjunction, new-moon dawn and winter solstice cannot be fixed. Fifth: Zhang Bin and his colleagues merely clung to a fixed origin method and did not account for advance and retreat. Sixth: Zhang Bin and his colleagues knew only to add the rotating large remainder of twenty-nine to obtain the new moon, but did not understand using sun-moon conjunction as the basis for fixing it. These six points are subtle matters—the great framework of calendrical science and the common method of sages—yet Liu Hui did not grasp them. This was indeed viewing the heavens through a bamboo tube. In verifying shadows to fix the seasonal nodes, He's method was superior; Zhang Bin's deductions strayed ever further. In aligning new moons with Heaven, He's method was weaker; Zhang Bin followed and traced that mistaken path. In short, they discarded the essence and kept the chaff. They also noted that under Emperor Ming of Wei, Secretariat Gentleman Yang Wei revised the Jingchu Calendar and submitted a memorial refuting earlier errors: "The hour of occurrence lags behind Heaven, and the eclipse does not fall on the new moon." Yang Wei's intent was to treat eclipse-on-new-moon as the standard, but he could not fully explain it or formulate the method. In the Liu Song's Yuanjia era, He Chengtian compiled a calendar whose submitted memorial stated: "The moon's motion is not uniform; it may be slow or fast. Conjunction and lunar eclipse do not always fall on new or full moon—this too departs from the calendar's intent." He Chengtian's original intent was to establish a conjunction method, but Pi Yanzong's obstruction prevented it from being implemented. Under Emperor Xian of Later Wei, Long Yidi again revised the Yanxing calendar and submitted a memorial: "Solar eclipses do not fall on the new moon, yet the practice persists; according to the Spring and Autumn Annals' eclipse records, Heaven verifies the new moon." These three were skilled calendrists of earlier ages; each had the right insight but failed to correct their written methods. Yet calendrical reckoning values above all the new moon and the seasonal nodes. The new moon heads the court assembly; the seasonal node marks the start of growth. The new moon has its rite of announcing provisions; the node has its suburban greeting ceremony. Confucius therefore fixed the calendar and established new-moon dawn and winter solstice as the model for posterity. Liu Xiaosun's calendar follows the explicit texts, using the moon's anomalistic motion to fix conjunction so that eclipses fall on the new moon, not on the last or second day of the month. Even if months frequently alternate one short and three long, this accords with Heaven's unity. The method has three parts in all, listed below.
43
第一，勘日食證恆在朔。
First: verify that solar eclipses always fall on the new moon.
44
引《詩》雲：「十月之交，朔日辛卯，日有食之。」 今以甲子元曆術推算，符合不差。 《春秋經》書日食三十五。 二十七日食，經書有朔，推與甲子元曆不差。 八食，經書並無朔字。 《左氏傳》雲：「不書朔，官失之也。」 《公羊傳》雲：「不言朔者，食二日也。」 《谷梁傳》雲：「不言朔者，食晦也。」 今以甲子元曆推算，俱是朔日。 丘明受經夫子，于理尤詳，《公羊》、《谷梁》皆臆說也。
It cites the Odes: "At the crossing of the tenth month, on the xinmao day of the new moon, the sun was eclipsed." Calculated by the Jiazi Origin Calendar, the match is exact. The Spring and Autumn Annals records thirty-five solar eclipses. Of twenty-seven eclipses where the classic records a new moon, calculation with the Jiazi Origin Calendar matches exactly. Eight eclipses are recorded without the new-moon character. The Zuo Commentary says: "Failure to record the new moon was an error of the officials." The Gongyang Commentary says: "Failure to state the new moon means the eclipse was on the second day." The Guliang Commentary says: "Failure to state the new moon means the eclipse was on the last day." Calculated by the Jiazi Origin Calendar, all eight fall on new-moon days. Zuo Qiuming received the classic directly from Confucius and is far more reliable; the Gongyang and Guliang commentaries are speculative interpretations.
45
《春秋左氏》隱公三年二月己巳，日有食之。 推合己巳朔
The Zuo Commentary, Duke Yin third year, second month, jisi day: a solar eclipse. Calculation confirms conjunction on the jisi day new moon.
46
莊公十八年春三月，日有食之。 推合壬子朔
Duke Zhuang, year 18, spring, third month: a solar eclipse. Calculation confirms conjunction on the renzi day new moon.
47
僖公十二年三月庚午，日有食之。 推合庚午朔
Duke Xi, twelfth year, third month, gengwu day: a solar eclipse. Calculation confirms conjunction on the gengwu day new moon.
48
十五年夏五月，日有食之。 推合癸未朔
fifteenth year of Year, summer, fifth month: a solar eclipse. Calculation confirms conjunction on the guiwei day new moon.
49
襄公十五年秋八月丁巳，日有食之。 推合丁巳朔
Duke Xiang, year 15, autumn, eighth month, dingsi day: a solar eclipse. Calculation confirms conjunction on the dingsi day new moon.
50
前、後漢及魏、晉四代所記日食，朔、晦及先晦，都合一百八十一，今以甲子元曆術推之，併合朔日而食。
The 181 solar eclipses recorded by the Former and Later Han, Wei, and Jin—whether dated to new moon, last day, or day before last—all calculate as new-moon eclipses under the Jiazi Origin Calendar.
51
前漢合有四十五食。 三食並先晦一日，三十二食並皆晦日，十食並是朔日
The Former Han recorded forty-five eclipses in all. Three fell one day before the last day; thirty-two on the last day; ten on the new-moon day.
52
後漢合有七十四食。 三十七食並皆晦日，三十七食並皆朔日
The Later Han recorded seventy-four eclipses in all. Thirty-seven fell on the last day; thirty-seven on the new-moon day.
53
魏合有十四食。 四食並皆晦日，十食並皆朔日
Wei recorded fourteen eclipses in all. Four fell on the last day; ten on the new-moon day.
54
晉合有四十八食。 二十五食並皆晦日，二十三食並皆朔日
Jin recorded forty-eight eclipses in all. Twenty-five fell on the last day; twenty-three on the new-moon day.
55
第二，勘度差變驗。 《尚書》雲：「日短星昴，以正仲冬。」 即是唐堯之時，冬至之日，日在危宿，合昏之時，昴正午。 案《竹書紀年》，堯元年丙子。 今以甲子元曆術推算得合堯時冬至之日，合昏之時，昴星正午。 《漢書》武帝太初元年丁丑歲，落下閎等考定太初曆冬至之日，日在牽牛初。 今以甲子元曆術算，即得斗末牛初矣。 晉時有薑岌，又以月食驗於日度，知冬至之日日在斗十七度。 宋文帝元嘉十年癸酉歲，何承天考驗乾度，亦知冬至之日日在斗十七度。 雖言冬至後上三日，前後通融，只合在斗十七度。 但堯年漢日，所在既殊，唯晉及宋，所在未改，故知其度，理有變差。 至今大隋甲辰之歲，考定歷數象，以稽天道，知冬至之日日在斗十三度。
Second: verify the changing discrepancy in celestial degrees. The Documents says: "The days are short and the star Mao culminates, marking mid-winter." This was in the time of Emperor Yao of Tang: on the winter solstice the sun was in the Wei lodge, and at dusk Mao was on the meridian. According to the Bamboo Annals, Yao's first year was bingzi. Calculated by the Jiazi Origin Calendar method, the winter solstice in Yao's time and Mao on the meridian at dusk match exactly. The History of Han records that in the first year of Taichu under Emperor Wu (dingchou), Luoxia Hong and others fixed the Taichu Calendar winter solstice, with the sun at the beginning of the Qianniu lodge. Calculated by the Jiazi Origin Calendar method, one obtains the end of the Dipper lodge and beginning of the Ox lodge. In Jin, Jiang Ji verified solar longitude by lunar eclipse and found the winter solstice sun at the seventeenth degree of the Dipper lodge. In the tenth year of Yuanjia under Emperor Wen of Song (guiyou), He Chengtian verified celestial longitude and likewise found the winter solstice sun at the seventeenth degree of the Dipper lodge. Although the winter solstice was reported three days late, reconciling the accounts, it should still fall at the seventeenth degree of the Dipper lodge. Yao's epoch and Han's day differ in solar position; only Jin and Song remained unchanged, showing that the degree has naturally shifted over time. Down to the jiachen year of Great Sui, verifying calendrical reckoning against the Way of Heaven, we know the winter solstice sun is at the thirteenth degree of the Dipper lodge.
56
第三，勘氣影長驗。
Third: verify by seasonal nodes and gnomon shadow length.
57
《春秋緯命曆序》雲：「魯僖公五年正月壬子朔旦冬至。」 今以甲子元曆術推算，得合不差。 《宋書》元嘉十年，何承天以土圭測影，知冬至已差三日。 詔使付外考驗，起元嘉十三年為始，畢元嘉二十年，八年之中，冬至之日恆與影長之日差校三日。 今以甲子元曆術推算，但是冬至之日恆與影長之符合不差。 詳之如左：
The Spring and Autumn Apocrypha, Calendar Sequence Command, states: "Duke Xi of Lu, fifth year, first month, renzi day, new-moon dawn, winter solstice." Calculated by the Jiazi Origin Calendar, the match is exact. The History of Song records that in tenth year of Yuanjia, He Chengtian measured shadows with an earth gnomon and found the winter solstice had already drifted by three days. An edict ordered external verification from thirteenth year of Yuanjia through 20th year of Yuanjia; over eight years, winter solstice consistently differed from the longest-shadow day by three days. Calculated by the Jiazi Origin Calendar, every winter solstice matches the longest-shadow day exactly. The details are as follows:
58
十三年丙子，
Thirteenth year (bingzi),
59
天正十八日曆注冬至，
Celestial first month, day 18: calendar notes winter solstice;
60
十五日影長，
fifteenth year of Day: longest shadow;
61
即是今曆冬至日。
This matches the present calendar's winter solstice.
62
十四年丁丑，
Fourteenth year (dingchou),
63
天正二十九日曆注冬至，
Celestial first month, day 29: calendar notes winter solstice;
64
二十六日影長，
26th year of Day: longest shadow;
65
即是今曆冬至日。
This matches the present calendar's winter solstice.
66
十五年戊寅，
Fifteenth year (wuyin),
67
天正十一日曆注冬至，
Celestial first month, day 11: calendar notes winter solstice;
68
陰，無影可驗，
Overcast—no shadow could be measured;
69
今曆八日冬至。
Present calendar: winter solstice on day 8.
70
十六年己卯，
Sixteenth year (jimao),
71
天正二十一日曆注冬至，
Celestial first month, day 21: calendar notes winter solstice;
72
十八日影長，
18th year of Day: longest shadow;
73
即是今曆冬至日。
This matches the present calendar's winter solstice.
74
十七年庚辰，
Seventeenth year (gengchen),
75
天正二日曆注冬至，
Celestial first month, day 2: calendar notes winter solstice;
76
十月二十九日影長，
29th year of Day: longest shadow;
77
即是今曆冬至日。
This matches the present calendar's winter solstice.
78
十八年辛巳，
Eighteenth year (xinsi),
79
天正十三日曆注冬至，
Celestial first month, day 13: calendar notes winter solstice;
80
十日影長，
tenth year of Day: longest shadow;
81
即是今曆冬至日。
This matches the present calendar's winter solstice.
82
十九年壬午，
Nineteenth year (renwu),
83
天正二十九日曆注冬至，
Celestial first month, day 29: calendar notes winter solstice;
84
陰，無影可驗，
Overcast—no shadow could be measured;
85
今曆二十二日冬至。
Present calendar: winter solstice on day 22.
86
二十年癸未，
Twentieth year (guiwei),
87
天正六日曆注冬至，
Celestial first month, day 6: calendar notes winter solstice;
88
三日影長，
third year of Day: longest shadow;
89
即是今曆冬至日。
This matches the present calendar's winter solstice.
90
于時新曆初頒，賓有寵于高祖，劉暉附會之，被升為太史令。 二人協議，共短孝孫，言其非毀天曆，率意迂怪，焯又妄相扶證，惑亂時人。 孝孫、焯等，竟以他事斥罷。 後賓死，孝孫為掖縣丞，委官入京，又上，前後為劉暉所詰，事寢不行。 仍留孝孫直太史，累年不調，寓宿觀台。 乃抱其書，弟子輿櫬，來詣闕下，伏而慟哭。 執法拘以奏之，高祖異焉，以問國子祭酒何妥。 妥言其善，即日擢授大都督，遣與賓曆比校短長。 先是信都人張胄玄，以算術直太史，久未知名。 至是與孝孫共短賓曆，異論鋒起，久之不定。 至十四年七月，上令參問日食事。 楊素等奏：「太史凡奏日食二十有五，唯一晦三朔，依克而食，尚不得其時，又不知所起，他皆無驗。 胄玄所克，前後妙衷，時起分數，合如符契。 孝孫所克，驗亦過半。」 於是高祖引孝孫、胄玄等，親自勞徠。 孝孫因請先斬劉暉，乃可定曆。 高祖不懌，又罷之。 俄而孝孫卒，楊素、牛弘等傷惜之，又薦胄玄。 上召見之，胄玄因言日長影短之事，高祖大悅，賞賜甚厚，令與參定新術。 劉焯聞胄玄進用，又增損孝孫曆法，更名《七曜新術》，以奏之。 與胄玄之法，頗相乖爽，袁充與胄玄害之。 焯又罷。 至十七年，胄玄曆成，奏之。 上付楊素等校其短長。 劉暉與國子助教王頍等執舊曆術，迭相駁難，與司曆劉宜援據古史影等，駁胄玄雲：
The new calendar had just been promulgated. Zhang Bin enjoyed Gaozu's favor, and Liu Hui attached himself to Bin and was promoted to Grand Astrologer. The two men colluded to attack Liu Xiaosun, accusing him of slandering the official calendar and making willfully eccentric claims. Liu Chuo falsely backed them, sowing confusion among scholars of the day. Liu Xiaosun, Liu Chuo, and their allies were ultimately dismissed on other pretexts. After Zhang Bin's death, Liu Xiaosun, then Assistant Magistrate of Ye County, resigned and went to the capital to submit his calendar again. Liu Hui repeatedly blocked him, and the proposal went nowhere. Liu Xiaosun remained attached to the Grand Astrologer's office as a direct appointee, unrewarded for years, lodging at the Observational Platform. He then brought his writings; his disciples carried his coffin to the palace gates, where he prostrated himself and wept. Law officers detained him and reported to the throne. Gaozu was moved and consulted Director of the Imperial Academy He Tuo. He Tuo endorsed the work, and Liu Xiaosun was immediately promoted to Grand Commander and ordered to compare his calendar with Zhang Bin's. Earlier, Zhang Zhouxuan of Xindu, skilled in computation, had served under the Grand Astrologer in obscurity for years. Now he joined Liu Xiaosun in attacking Zhang Bin's calendar. Conflicting opinions multiplied and remained unresolved for a long time. In the seventh month of fourteenth year of Kaihuang, the emperor ordered an inquiry into solar eclipse predictions. Yang Su and others reported: "The Grand Astrologer's office submitted twenty-five eclipse predictions. Only four—one on the last day and three on new moons—even roughly matched the event, and those missed the correct hour and origin. The rest failed entirely. Zhang Zhouxuan's predictions were consistently accurate; the times and fractional parts matched like tally and seal. Liu Xiaosun's predictions were verified in more than half the cases." Gaozu then summoned Liu Xiaosun, Zhang Zhouxuan, and the others and personally received them. Liu Xiaosun then demanded that Liu Hui be executed before the calendar could be fixed. Gaozu was displeased and dismissed him again. Liu Xiaosun soon died. Yang Su, Niu Hong, and others mourned him and again recommended Zhang Zhouxuan. The emperor summoned Zhang Zhouxuan, who spoke on the lengthening of days and shortening of shadows. Gaozu was greatly pleased, richly rewarded him, and ordered him to help fix the new calendar. When Liu Chuo learned of Zhang Zhouxuan's promotion, he revised Liu Xiaosun's calendar, renamed it the New Method of the Seven Luminaries, and submitted it. It diverged significantly from Zhang Zhouxuan's method. Yuan Chong and Zhang Zhouxuan worked against him. Liu Chuo was dismissed again. By 17th year of Kaihuang, Zhang Zhouxuan's calendar was complete and submitted. The emperor entrusted it to Yang Su and others to evaluate its merits. Liu Hui and Academy Assistant Instructor Wang Yi defended the old calendar, trading refutations with Zhang Zhouxuan. Calendar Officer Liu Yi drew on ancient histories and shadow records to challenge him, stating:
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《命曆序》僖公五年天正壬子朔旦日至，《左氏傳》僖公五年正月辛亥朔日南至。 張賓曆，天正壬子朔冬至，合《命曆序》，差《傳》一日。 張胄玄曆，天正壬子朔，合《命曆序》，差《傳》一日； 三日甲寅冬至，差《命曆序》二日，差《傳》三日。 成公十二年，《命曆序》天正辛卯朔旦日至。 張賓曆，天正辛卯朔冬至，合《命曆序》。 張胄玄曆，天正辛卯朔，合《命曆序》； 二日壬辰冬至，差《命曆序》一日。 昭公二十年，《春秋左氏傳》二月己丑朔日南至，准《命曆序》庚寅朔旦日至。 張賓曆，天正庚寅朔冬至，併合《命曆序》，差《傳》一日。 張胄玄曆，天正庚寅朔，合《命曆序》，差《傳》一日； 二日辛卯冬至，差《命曆序》一日，差《傳》二日。 宜案《命曆序》及《春秋左氏傳》，並閏餘盡之歲，皆須朔旦冬至。 若依《命曆序》勘《春秋》三十七食，合處至多； 若依《左傳》，合者至少，是以知《傳》為錯。 今張胄玄信情置閏，《命曆序》及《傳》氣朔並差。 又宋元嘉冬至影有七，張賓曆合者五，差者二，亦在前一日。 張胄玄曆合者三，差者四，在後一日。 元嘉十二年十一月甲寅朔，十五日戊辰冬至，日影長。 張賓曆合戊辰冬至，張胄玄曆己巳冬至，差後一日。 十三年十一月己酉朔，二十六日甲戌冬至，日影長。 張賓曆癸酉冬至，差前一日，張胄玄曆合甲戌冬至。 十五年十一月丁卯朔，十八日甲申冬至，日影長。 二曆併合甲申冬至。 十六年十一月辛酉朔，二十九日己丑冬至，日影長。 張賓曆合己丑冬至，張胄玄曆庚寅冬至，差後一日。 十七年十一月乙酉朔，十日甲午冬至，日影長。 張賓曆合甲午冬至，張胄玄曆乙未冬至，差後一日。 十八年十一月己卯朔，二十一日己亥冬至，日影長。 張賓曆合己亥冬至，張胄玄曆庚子冬至，差後一日。 十九年十一月癸卯朔，三日乙巳冬至，影長。 張賓曆甲辰冬至，差前一日，張胄玄曆合乙巳冬至。
The Calendar Sequence Command records Duke Xi's fifth year, celestial first month, renzi day, new-moon dawn, winter solstice. The Zuo Commentary records Duke Xi's fifth year, first month, xinhai day, new moon, winter solstice. Zhang Bin's calendar: celestial first month, renzi day new moon, winter solstice—matches the Calendar Sequence Command, differs from the Zuo Commentary by one day. Zhang Zhouxuan's calendar: celestial first month, renzi day new moon—matches the Calendar Sequence Command, differs from the Zuo Commentary by one day; Third day, jiayin, winter solstice—off by two days from the Calendar Sequence Command, three from the Zuo Commentary. Duke Cheng, year 12: the Calendar Sequence Command records celestial first month, xinmao day new-moon dawn, winter solstice. Zhang Bin's calendar: celestial first month, xinmao day new moon, winter solstice—matches the Calendar Sequence Command. Zhang Zhouxuan's calendar: celestial first month, xinmao day new moon—matches the Calendar Sequence Command; Second day, renchen, winter solstice—off by one day from the Calendar Sequence Command. Duke Zhao, year 20: the Zuo Commentary records second month, jichou day new moon, winter solstice; the Calendar Sequence Command records gengyin day new-moon dawn, winter solstice. Zhang Bin's calendar: celestial first month, gengyin day new moon, winter solstice—matches the Calendar Sequence Command, differs from the Zuo Commentary by one day. Zhang Zhouxuan's calendar: celestial first month, gengyin day new moon—matches the Calendar Sequence Command, differs from the Zuo Commentary by one day; Second day, xinmao, winter solstice—off by one day from the Calendar Sequence Command, two from the Zuo Commentary. Liu Yi argues that according to the Calendar Sequence Command and the Zuo Commentary, in every year when the intercalary remainder is exhausted, new-moon dawn and winter solstice must coincide. Checking the Spring and Autumn Annals' thirty-seven eclipses against the Calendar Sequence Command yields numerous matches; checking against the Zuo Commentary yields very few—proving the Commentary is in error. Zhang Zhouxuan arbitrarily sets intercalation by personal judgment; the seasonal nodes and new moons diverge from both the Calendar Sequence Command and the Zuo Commentary. Of seven Yuanjia-era winter-solstice shadow records, Zhang Bin's calendar matched five and missed two—both one day early. Zhang Zhouxuan's calendar matched three and missed four—all one day late. twelfth year of Yuanjia, eleventh month, jiayin day new moon; fifteenth day, wuchen, winter solstice; longest shadow. Zhang Bin's calendar: wuchen winter solstice. Zhang Zhouxuan's calendar: jisi winter solstice—one day late. thirteenth year of Yuanjia, eleventh month, jiyou day new moon; twenty-sixth day, jiaxu, winter solstice; longest shadow. Zhang Bin's calendar: guiyou winter solstice—one day early. Zhang Zhouxuan's calendar: jiaxu winter solstice—correct. fifteenth year of Yuanjia, eleventh month, dingmao day new moon; eighteenth day, jiashen, winter solstice; longest shadow. Both calendars agree on jiashen winter solstice. 16th year of Yuanjia, eleventh month, xinyou day new moon; twenty-ninth day, jichou, winter solstice; longest shadow. Zhang Bin's calendar: jichou winter solstice. Zhang Zhouxuan's calendar: gengyin winter solstice—one day late. 17th year of Yuanjia, eleventh month, yiyou day new moon; tenth day, jiawu, winter solstice; longest shadow. Zhang Bin's calendar: jiawu winter solstice. Zhang Zhouxuan's calendar: yi-wu day winter solstice—one day late. 18th year of Yuanjia, eleventh month, jimao day new moon; twenty-first day, jihai, winter solstice; longest shadow. Zhang Bin's calendar: jihai winter solstice. Zhang Zhouxuan's calendar: gengzi winter solstice—one day late. 19th year of Yuanjia, eleventh month, guimao day new moon; third day, yisi, winter solstice; longest shadow. Zhang Bin's calendar: jiachen winter solstice—one day early. Zhang Zhouxuan's calendar: yisi winter solstice—correct.
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又周從天和元年丙戌至開皇十五年乙卯，合得冬夏至日影一十四。 張賓曆合得者十，差者四，三差前一日，一差後一日。 張胄玄曆合者五，差者九，八差後一日，一差前一日。 天和二年十一月戊戌朔，三日庚子冬至，日影長。 張賓曆合庚子冬至，張胄玄曆辛丑冬至，差後一日。 三年十一月壬辰朔，十四日乙巳冬至，日影長。 張賓曆合乙巳冬至，張胄玄曆丙午冬至，差後一日。 建德元年十一月己亥朔，二十九日丁卯冬至，日影長。 張賓曆丙寅冬至，差前一日，張胄玄曆合丁卯冬至。 二年五月丙寅朔，三日戊辰夏至，日影短。 張賓曆己巳夏至，差後一日，張胄玄曆庚午夏至，差後二日。 三年十一月戊午朔，二十日丁丑冬至，日影長。 張賓曆合丁丑冬至，張胄玄曆戊寅冬至，差後一日。 六年十一月庚午朔，二十三日壬辰冬至，日影長。 張賓曆合壬辰冬至，張胄玄曆癸巳冬至，差後一日。 宣政元年十一月甲午朔，五日戊戌冬至，日影長。 兩曆併合戊戌冬至。 開皇四年十一月己未朔，十一日己巳冬至，日影長。 張賓曆合己巳冬至，張胄玄曆庚午冬至，差後一日。 五年十一月甲寅朔，二十二日乙亥冬至，日影長。 張賓曆甲戌冬至，差前一日，張胄玄曆合庚辰冬至。 七年五月乙亥朔，九日癸未夏至，日影短。 張賓曆壬午夏至，差前一日，張胄玄曆合癸未夏至。 十一月壬申朔，十四日乙酉冬至，日影長。 張賓曆合乙酉冬至，張胄玄曆丙戌冬至，差後一日。 十一年十一月己卯朔，二十八日丙午冬至，日影長。 張賓曆合丙午冬至，張胄玄曆丁未冬至，差後一日。 十四年十一月辛酉朔旦冬至。 張賓曆合十一月辛酉朔旦冬至，張胄玄曆十一月辛酉朔，二日壬戌冬至，差後一日。 建德四年四月大、乙酉朔，三十日甲寅，月晨見東方。 張賓曆四月大、乙酉朔，三十日甲寅，月晨見東方，張胄玄曆四月小、乙酉朔，五月大，甲寅朔，月晨見東方。 宜案影極長為冬至，影極短為夏至，二至自古史分可勘者二十四，其二十一有影，三有至日無影。 見行曆合一十八，差者六。 旅騎尉張胄玄曆合者八，差者一十六，二差後二日，一十四差後一日。 又開皇四年，在洛州測冬至影，與京師二處，進退絲毫不差。 周天和已來案驗並在後。 更檢得建德四年，晦朔東見； 張胄玄曆，五月朔日，月晨見東方。 今十七年，張賓曆閏七月，張胄玄曆閏五月。 又審至以定閏，胄玄曆至既不當，故知置閏必乖。 見行曆四月、五月頻大，張胄玄曆九月、十月頻大，為胄玄朔弱，頻大在後晨，故朔日殘月晨見東方。
From Northern Zhou's first year of Tianhe (bingxu) through Sui's fifteenth year of Kaihuang (yimao), fourteen winter and summer solstice shadow records are available. Zhang Bin's calendar matched ten and missed four—three one day early, one day late. Zhang Zhouxuan's calendar matched five and missed nine—eight one day late, one day early. second year of Tianhe, eleventh month, wuxu day new moon; third day, gengzi, winter solstice; longest shadow. Zhang Bin's calendar: gengzi winter solstice. Zhang Zhouxuan's calendar: xinchou winter solstice—one day late. third year of Tianhe, eleventh month, renchen day new moon; fourteenth day, yisi, winter solstice; longest shadow. Zhang Bin's calendar: yisi winter solstice. Zhang Zhouxuan's calendar: bingwu winter solstice—one day late. first year of Jiande, eleventh month, jihai day new moon; twenty-ninth day, dingmao, winter solstice; longest shadow. Zhang Bin's calendar: bingyin winter solstice—one day early. Zhang Zhouxuan's calendar: dingmao winter solstice—correct. second year of Jiande, fifth month, bingyin day new moon; third day, wuchen, summer solstice; shortest shadow. Zhang Bin's calendar: jisi summer solstice—one day late. Zhang Zhouxuan's calendar: gengwu summer solstice—two days late. third year of Year, eleventh month, wu-wu day new moon;, day 20, ding-chou, Winter Solstice, longest shadow. Zhang Bin's calendar matches ding-chou winter solstice; Zhang Zhouxuan's calendarwu-yin winter solstice; —one day late. sixth year of Year, eleventh month, geng-wu day new moon;, day 23, ren-chen, Winter Solstice, longest shadow. Zhang Bin's calendar matches ren-chen winter solstice; Zhang Zhouxuan's calendargui-si winter solstice; —one day late. Xuanzheng, first year, eleventh month, jia-wu day new moon;, day 5, wu-xu, winter solstice, longest shadow. Both calendars agree on wuxu winter solstice. fourth year of Kaihuang of Year, eleventh month, ji-wei day new moon;, day 11, ji-si, Winter Solstice, longest shadow. Zhang Bin's calendar matches ji-zi winter solstice; Zhang Zhouxuan's calendargeng-zi winter solstice; —one day late. fifth year of Year, eleventh month, jia-yin day new moon;, day 22, yi-hai, Winter Solstice, longest shadow. Zhang Bin's calendar: jiaxu winter solstice—one day early. Zhang Zhouxuan's calendar: gengchen winter solstice. seventh year of Year, fifth month, yi-hai day new moon;, day 9, gui-wei, Summer Solstice, shortest shadow. In Zhang Bin's calendar, renwu was the summer solstice, one day too early; Zhang Zhouxuan's calendar correctly placed the summer solstice on guiwei. Eleventh month, ren-shen day new moon;, day 14, yi-you, Winter Solstice, longest shadow. Zhang Bin's calendar matches yi-you winter solstice; Zhang Zhouxuan's calendarbing-xu winter solstice; —one day late. eleventh year of Year, eleventh month, ji-mao day new moon;, day 28, bing-wu, Winter Solstice, longest shadow. Zhang Bin's calendar matches bing-wu winter solstice; Zhang Zhouxuan's calendarding-wei winter solstice; —one day late. fourteenth year of Kaihuang, eleventh month, xinyou day new-moon dawn, winter solstice. Zhang Bin's calendar: eleventh month, xinyou day new-moon dawn, winter solstice. Zhang Zhouxuan's calendar: xinyou day new moon; renchen day 2, winter solstice—one day late. fourth year of Jiande, fourth month (long), yiyou day new moon; thirtieth day, jiayin—the moon was visible in the morning eastern sky. Zhang Bin's calendar: fourth month (long), yiyou new moon; day 30, jiayin—morning moon in the east. Zhang Zhouxuan's calendar: fourth month (short), yiyou new moon; fifth month (long), jiayin new moon—morning moon in the east. Liu Yi argues that the longest shadow marks winter solstice and the shortest marks summer solstice. Of twenty-four solstices verifiable from ancient records, twenty-one have shadow data and three record the solstice day without shadow measurements. The calendar then in use matched eighteen cases; six did not. Zhang Zhouxuan's calendar matched eight cases and missed sixteen—two by two days, fourteen by one day. fourth year of In of Kaihuang, winter solstice shadows measured at Luoyang and the capital agreed to the finest precision at both locations. From Northern Zhou's Tianhe era onward, all verified cases fall one day late. Further review found fourth year of Jiande: on the last day of the month at new moon, the moon was sighted in the morning east; Zhang Zhouxuan's calendar: fifth-month new-moon day—the moon visible in the morning east. Examining 17th year of Kaihuang: Zhang Bin's calendar places the intercalary month in the seventh month; Zhang Zhouxuan's in the fifth. Since intercalation should be determined by the solstice, and Zhang Zhouxuan's solstice is wrong, his intercalary placement must be incorrect. The official calendar frequently assigns long fourth and fifth months; Zhang Zhouxuan's frequently assigns long ninth and tenth months. His new-moon reckoning is weak, producing long months in later morning hours—hence the waning moon appears in the eastern morning sky on new-moon day.
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宜又案開皇四年十二月十五日癸卯，依曆月行在鬼三度，時加酉，月在卯上，食十五分之九，虧起西北。 今伺候，一更一籌起食東北角，十五分之十，至四籌還生，至二更一籌複滿。 五年六月三十日，依曆太陽虧，日在七星六度，加時在午少強上，食十五分之一半強，虧起西南角。 今伺候，日乃在午後六刻上始食，虧起西北角，十五分之六，至未後一刻還生，至五刻複滿。 六年六月十五日，依曆太陰虧，加時酉，在卯上，食十五分之九半弱，虧起西南，當其時陰雲不見月。 至辰巳，雲裡見月，已食三分之二，虧從東北，既還雲合。 至巳午間稍生，至午後，雲裡暫見，已複滿。 十月三十日丁丑，依曆太陽虧，日在斗九度，時加在辰少弱上，食十五分之九強，虧起東北角。 今候所見，日出山一丈，辰二刻始食，虧起正西，食三分之二，辰後二刻始生，入巳時三刻上複滿。 十年三月十六日癸卯，依曆月行在氐七度，時加戌，月在辰太半上，食十五分之七半強，虧起東北。 今候，月初出卯南，帶半食，出至辰初三分，可食二分許，漸生，辰未已複滿。 見行曆九月十六日庚子，月行在胃四度，時加醜，月在未半強上，食十分之三半強，虧起正東。 今伺候，月以午後二刻，食起正東，須臾如南，至未正上，食南畔五分之四，漸生，入申一刻半複滿。 十二年七月十五日己未，依曆月行在室七度，時加戌，月在辰太強上，食十五分之十二半弱，虧起西北。 今伺候，一更三籌起西北上，食准三分之二強，與曆注同。 十三年七月十六日，依曆月在申半強上，食十五分之半弱，虧起西南。 十五日夜，從四更候月，五更一籌起東北上，食半強，入雲不見。 十四年七月一日，依歷時加巳弱上，食十五分之十二半強。 至未後三刻，日乃食，虧起西北，食半許，入雲不見，食頃暫見，猶未複生，因即雲鄣。 十五年十一月十六日庚午，依曆月行在井十七度，時加亥，月在巳半上，食十五分之九半強，虧西北。 其夜一更四籌後，月在辰上起食，虧東南，至二更三籌，月在巳上，食三分之二許，漸生，至三更一籌，月在丙上，複滿。 十六年十一月十六日乙丑，依曆月行在井十七度，時加醜，月在未太弱上，食十五分之十二半弱，虧起東南。 十五日夜伺候，至三更一籌，月在丙上，雲裡見，已食十五分之三許，虧起正東，至丁上，食既，後從東南生，至四更三籌，月在未末，複滿。 而胄玄不能盡中。
Liu Yi also cites fourth year of Kaihuang, twelfth month, day 15 (guimao): per the calendar the moon was at 3° Ghost; hour you; moon above mao; eclipse magnitude 9/15; obscuration began in the northwest. Observed: at the first watch, first tally, obscuration began at the northeast limb, magnitude 10/15; restored by the fourth tally; fully restored by the second watch, first tally. fifth year of Kaihuang, sixth month, day 30: per the calendar, solar eclipse; sun at 6° Seven Stars; hour of occurrence mid-wu; magnitude 1.5/15; obscuration began at the southwest limb. Observed: eclipse began after the sixth quarter-hour of wu; obscuration from the northwest, magnitude 6/15; restoration began after the first quarter of wei; full restoration by the fifth quarter. sixth year of Kaihuang, sixth month, day 15: per the calendar, lunar eclipse; hour you; moon above mao; magnitude 9.5/15; obscuration from the southwest; clouds obscured the moon at the time. By chen-si hours the moon appeared through clouds, already two-thirds eclipsed from the northeast; after totality clouds closed again. Restoration began around si-wu; by afternoon a break in the clouds showed the moon fully restored. Tenth month, day 30 (dingchou): per the calendar, solar eclipse; sun at 9° Dipper; hour chen slightly weak; magnitude 9/15; obscuration from the northeast. Observed: sun one zhang above the horizon at sunrise; eclipse began at chen 2; obscuration from the west, two-thirds magnitude; restoration after chen 2; full restoration by si 3. tenth year of Kaihuang, third month, day 16 (guimao): per the calendar moon at 7° Di; hour xu; moon well past chen; magnitude 7.5/15; obscuration from the northeast. Observed: at first rising south of mao the moon was half eclipsed; by early chen about two-thirds; gradual restoration; fully restored before wei. Official calendar, ninth month, day 16 (gengzi): moon at 4° Stomach; hour chou; moon above wei; magnitude 3.5/10; obscuration from the east. Observed: eclipse began due east at the second quarter after wu; briefly turning south; at exact wei, four-fifths of the southern limb eclipsed; gradual restoration; full by shen 1.5. twelfth year of Kaihuang, seventh month, day 15 (jiwei): per the calendar moon at 7° Room; hour xu; moon above chen; magnitude 12.5/15; obscuration from the northwest. Observed: at the first watch, third tally, obscuration began northwest at about two-thirds magnitude—matching the calendar prediction. thirteenth year of Kaihuang, seventh month, day 16: per the calendar moon above shen; magnitude 0.5/15; obscuration from the southwest. On the night of the 15th, watching from the fourth watch: at the fifth watch, first tally, obscuration began northeast at half magnitude; then lost in clouds. fourteenth year of Kaihuang, seventh month, day 1: per the calendar, hour si weak; magnitude 12.5/15. By wei 3 the sun was eclipsed from the northwest at about half magnitude; lost in clouds; briefly visible during totality but not yet restored; then obscured again. fifteenth year of Kaihuang, eleventh month, day 16 (gengwu): per the calendar moon at 17° Well; hour hai; moon above si; magnitude 9.5/15; obscuration northwest. That night after the first watch, fourth tally: moon above chen, eclipse began southeast; by second watch, third tally, moon above si, about two-thirds; gradual restoration; by third watch, first tally, moon above bing, fully restored. 16th year of Kaihuang, eleventh month, day 16 (yichou): per the calendar moon at 17° Well; hour chou; moon above wei; magnitude 12.5/15; obscuration from the southeast. Observed on the night of the 15th: by third watch, first tally, moon above bing seen through clouds, already about 3/15 eclipsed from the east; totality at ding; restoration from the southeast; by fourth watch, third tally, moon at end of wei, fully restored. Yet Zhang Zhouxuan could not fully hit the mark.
94
迭相駁難，高祖惑焉，逾時不決。 會通事舍人顏慜楚上書雲：「漢落下閎改《顓頊曆》作《太初曆》，雲後八百歲，此曆差一日。」 語在胄玄傳。 高祖欲神其事，遂下詔曰：「朕應運受圖，君臨萬宇，思欲興複聖教，恢弘令典，上順天道，下授人時，搜揚海內，廣延術士。 旅騎尉張胄玄，理思沉敏，術藝宏深，懷道白首，來上曆法。 令與太史舊曆，並加勘審。 仰觀玄象，參驗璿璣，胄玄歷數與七曜符合，太史所行，乃多疏舛，群官博議，咸以胄玄為密。 太史令劉暉，司曆郭翟、劉宜，驍騎尉任悅，往經修造，致此乖謬。 通直散騎常侍、領太史令庾季才，太史丞邢俊，司曆郭遠，曆博士蘇粲，曆助教傅俊、成珍等，既是職司，須審疏密。 遂虛行此曆，無所發明。 論暉等情狀，已合科罪，方共飾非護短，不從正法。 季才等附下罔上，義實難容。」 於是暉等四人，元造詐者，並除名； 季才等六人，容隱奸慝，俱解見任。 胄玄所造曆法，付有司施行。 擢拜胄玄為員外散騎侍郎，領太史令。 胄玄進袁充，互相引重，各擅一能，更為延譽。 胄玄言充曆妙極前賢，充言胄玄曆術冠於今古。 胄玄學祖沖之，兼傳其師法。 自茲厥後，克食頗中。 其開皇十七年所行曆術，命冬至起虛五度。 後稍覺其疏，至大業四年劉焯卒後，乃敢改法，命起虛七度，諸法率更有增損，朔終義寧。 今錄戊辰年所定曆術著之於此雲。
The parties traded refutations until Gaozu was perplexed and no decision was reached for a long time. Then Palace Attendant Yan Minting submitted a memorial: "Under the Han, Luoxia Hong revised the Zhuanxu Calendar into the Taichu Calendar, declaring that after eight hundred years it would err by one day." The account appears in Zhang Zhouxuan's biography. Gaozu wished to invest the matter with portentous significance and issued an edict: "Having received the mandate, We rule the realm, seeking to revive sage teaching and expand the statutes—aligning with Heaven above and granting the seasons to the people below, searching far and wide for masters of calendrical science. Cavalry Commandant Zhang Zhouxuan, deep of mind and vast in skill, devoted to the Way into old age, submitted his calendar method. He was ordered to compare his calendar jointly with the Grand Astrologer's existing system. Observing the heavens and verifying against the armillary sphere, Zhang Zhouxuan's reckoning matched the seven luminaries, while the Grand Astrologer's system was largely erroneous. The assembled officials judged Zhang Zhouxuan's method the more precise. Grand Astrologer Liu Hui, Calendar Officers Guo Di and Liu Yi, and Cavalry Commandant Ren Yue—who had previously devised the calendar—were responsible for these errors. Direct Palace Attendant and Grand Astrologer Yu Jicai, Assistant Grand Astrologer Xing Jun, Calendar Officer Guo Yuan, and Erudites Su Can, Fu Jun, and Cheng Zhen—as calendrical officers—should have reviewed the system's accuracy. Yet they allowed this faulty calendar to remain in use without objection. Liu Hui and his colleagues already deserved punishment; instead they embellished errors and shielded faults, defying proper procedure. Yu Jicai and his colleagues deceived their superiors—conduct that could not be tolerated." Liu Hui and three others, the original authors of the faulty calendar, were stripped of rank; Yu Jicai and five others, who had concealed the fraud, were dismissed from office. Zhang Zhouxuan's calendar was entrusted to the relevant offices for implementation. Zhang Zhouxuan was promoted to Supernumerary Gentleman Attendant and appointed Grand Astrologer. Zhang Zhouxuan recommended Yuan Chong; they promoted each other, each claiming mastery in one area, and further burnished each other's reputations. Zhang Zhouxuan declared Yuan Chong's calendar the finest since antiquity; Yuan Chong declared Zhang Zhouxuan's calendrical art unmatched in all history. Zhang Zhouxuan studied Zu Chongzhi and also transmitted his master's methods. From this point forward, eclipse predictions largely hit the mark. The calendar adopted in 17th year of Kaihuang set the winter solstice at the fifth degree of the Xu lodge. Later its imprecision became apparent; by fourth year of Daye, after Liu Chuo's death, Zhang Zhouxuan revised the method to set the winter solstice at the seventh degree of Xu, with further adjustments to various parameters, continuing until the Yining era. What follows records the calendar method fixed in the wuchen year.
95
自甲子元至大業四年戊辰，百四十二萬七千六百四十四年，算外。
From the jiazi origin to fourth year of Daye (wuchen): 1,427,644 years beyond the epoch.
96
章歲，四百一十。
Cycle year: 410.
97
章閏，百五十一。
Cycle intercalation: 51.
98
章月，五千七十一。
Cycle month: 5,071.
99
日法，千一百四十四。
Day divisor: 144.
100
月法，三萬三千七百八十三。
Month divisor: 33,783.
101
辰法，二百八十六。
Chronogram divisor: 286.
102
歲分，一千五百五十七萬二千九百六十三。
Year fraction: 15572963.
103
度法，四萬二千六百四十。
Degree divisor: 42,640.
104
沒分，五百一十九萬一千三百一十一
Submergence fraction: 5,191,311.
105
沒法，七萬四千五百二十一。
Submergence divisor, 21.
106
周天分，一千五百五十七萬四千四百六十六。
Circuit-of-heaven fraction: 15574466.
107
斗分，一萬八百六十六。
Dipper fraction: 10866.
108
氣法，四十六萬九千四十。
Qi divisor: 469,040.
109
氣時法，一萬六百六十。
Qi-time divisor: 10660.
110
周日，二十七。
Circuit day: 27.
111
日餘，一千四百一十三。
Day remainder: 1413.
112
周通，七萬二百九。
Circuit common: 70209.
113
周法，二千五百四十八。
Circuit divisor: 2548.
114
推積月術：
Method for computing accumulated months:
115
置入元已來至所求年，以章月乘之，如章歲得一，為積月，餘為閏餘。 閏餘三百九十七巳上，若冬至不在其月，加積月一
From the origin to the year sought, multiply by the cycle month and divide by the cycle year to obtain accumulated months; the remainder is the intercalary remainder. If the intercalary remainder is 397 or above and winter solstice does not fall in that month, add one to the accumulated months.
116
推月朔弦望術：
Method for computing new moon, first quarter, full moon, and last quarter:
117
以月法乘積月，如法得一，為積日，餘為小餘。 以六十去積日，餘為大餘，命以甲子算外，為所求年天正月朔日。 天正月者，建子月也，今為去年十一月。 凡朔小餘五百四十七巳上，其月大。
Multiply accumulated months by the month divisor; divide by the divisor to obtain one, yielding accumulated days; the remainder is the small remainder. Remove multiples of sixty from accumulated days; the remainder is the large remainder; count from jiazi beyond the tally—this is the new-moon day of the celestial first month of the year sought. The celestial first month establishes zi—now taken as the eleventh month of the prior year. If the new-moon small remainder is 547 or above, the month is long.
118
加大餘七，小餘四百三十七太； 凡四分一為少，二為半，三為太。 小餘滿日法去之，從大餘； 滿六十去之，命如前，為上弦日。 又加，得望、下弦、後月朔。 朔餘滿五百三十七，其月大，減者小餘。
Add seven to the large remainder and 437 plus three-quarters to the small remainder. One-quarter is "less," two-quarters is "half," three-quarters is "greater". When the small remainder fills the day divisor, remove it and carry to the large remainder. When it fills sixty, remove it; count as before—this is the first-quarter day. Add again to obtain full moon, last quarter, and the next month's new moon. If the new-moon remainder fills 537, the month is long; if reduced, use the small remainder.
119
推二十四氣術：
Method for computing the twenty-four qi:
120
以月法乘閏餘，又以章歲乘朔小餘，加之，如氣法得一，為日，命朔算外，為冬至日。 不盡者，以十一約之，為日分。
Multiply the intercalary remainder by the month divisor, then multiply the new-moon small remainder by the cycle years and add it. Divide by the qi divisor to obtain days, then count from the new moon, excluding the starting count, to get the winter solstice day. Divide remainder by eleven for day fractions.
121
求次氣：加日十五，日分九千三百一十五，小分一； 小分滿八從日分一，日分滿度法從日一； 如月大小去之，日不滿月，算外，為次氣日。 其月無中氣者，為閏。
：15, day fraction, small fraction1. Small fraction 8 carries to day fraction, day fraction full carries to days. Remove month lengths according to whether the months are large or small; when the remaining days do not fill a month, count beyond the tally to obtain the day of the next qi. Month without mid-season node is intercalary.
122
求朔望入氣盈縮術：
Method for finding solar anomaly at new and full moon upon entering qi:
123
以入氣日算乘損益率，如十五得一，餘八已上，從一； 以損益盈縮數為定盈縮。 其入氣日十五算者，如十六得一，餘半法已上亦從一，以下皆准此。
Multiply days since entering qi by the decrease-increase rate; divide by 15 to get one; if the remainder is 8 or above, carry 1; . Apply decrease or increase to the anomaly value to obtain the fixed anomaly. For an entering-qi day counted as fifteen, divide as though sixteen makes one; if the remainder is half the divisor or more, also count it as one. The cases below follow this rule.
124
推土王術：
Method for computing the Earth phase:
125
加分至日二十七，日分一萬六千七百六十七，小分九； 小分滿四十從日分一，滿去如前，即分至後土始王日。
27, day fraction, small fraction9. When small fractions reach 40, carry 1 to day fractions; when full, remove as before—this gives the day Earth begins to reign after the solstice.
126
推沒日術：
Method for computing submergence days:
127
其氣有小分者，以八乘日分，內小分，又以十五乘之，以減沒分； 無小分者，以百二十乘日分，以減之； 滿沒法為日，不盡為日分，以其氣去朔日加之，去、命如前。
When the qi has a small fraction, multiply the day fraction by eight, include the small fraction, then multiply by fifteen and subtract this from the submergence fraction. When there is no small fraction, multiply the day fraction by 120 and subtract it. When full counts as days; remainder is day fraction; add days from new moon since entering qi; remove and count as before.
128
求次沒：加日六十九，日分四萬九千三百七十二； 日分滿沒法，從日，去、命如前。
：, day fraction. When day fraction fills submergence divisor, carry to days; remove and count as before.
129
推入遲疾曆術：
Method for entering the slow-fast calendar:
130
以周通去朔積日，餘以周法乘之，滿周通又去之，餘滿周法得一日，餘為日餘，即所求年天正朔算外夜半入曆日及餘。
Remove the accumulated new-moon days by the full circuit. Multiply the remainder by the circuit divisor, remove another full circuit when it fills, and count one day whenever the remainder fills the circuit divisor. The remainder is the day remainder: this gives the calendar day and remainder, after the count, for midnight at the first new moon of the requested year.
131
求次月：大月加二日，小月加一日，日餘皆千一百三十五，滿周日及日餘去之。
To find the next month: add 2 days for a long month, 1 day for a short month; day remainders all 1,135; when full remove circuit day and day remainder.
132
求次日：加一，滿、去如前。
To find the next day: add 1; when full, remove as before.
133
求朔望加時入曆術：
Method for finding the hour of new/full moon entry into the calendar:
134
以四十九乘朔小餘，滿二十二得一為日餘，不盡為小分，以加夜半入曆日及餘分。
Multiply the new-moon small remainder by forty-nine; every full twenty-two makes one day remainder, and what does not fill twenty-two is the small fraction. Add this to the midnight entering-calendar day and remainder.
135
求次月：加日一，餘二千四百八十六，小分二十一，滿、去如前，即次月入曆日及餘。
To find the next month: add 1 day and remainder 2,486, small fraction 21; when full remove as before—this gives next month calendar entry day and remainder.
136
求望：加日十四日，餘千九百四十九，小分二十一半，滿、去如前，為望入曆日及餘。
To find full moon: add 14 days, remainder 1,949, small fraction 21½; when full remove as before—this gives full-moon calendar entry day and remainder.
137
推朔望加時定日及小餘術：
Method for computing fixed day and small remainder at new/full moon hour:
138
以入曆日餘乘所入曆日損益率，以損益盈縮積分，如差法而一，為定積分。 如差法乃與入氣定盈縮，皆以盈減、縮加本朔望小餘； 不足減者，加日法乃減之，加時在往日； 加之，滿日法者去之，則在來日； 餘為定小餘。 無食者不須氣盈縮。
Multiply the remainder of the entered-calendar day by that day's decrease-or-increase rate, use it to decrease or increase the accumulated excess-or-deficit fraction, divide by the difference divisor, and this gives the fixed accumulated fraction. Divide by difference divisor together with entering-qi fixed anomaly; all apply expansion-subtract and contraction-add to base new/full moon small remainder. When insufficient to subtract, add the day divisor and then subtract—the hour falls on the prior day. When adding, if it fills the day divisor remove it—the hour falls on the following day. Remainder is small remainder. When no eclipse occurs, solar anomaly correction is not required.
139
角十二度亢九度氐十五度房五度心五度尾十八度箕十一度
twelfth year of Horn°, ninth year of Neck°, fifteenth year of Root°, fifth year of Room°, fifth year of Heart°, 18th year of Tail°, Winnowing eleventh year of Basket°.
140
東方七宿七十五度
Eastern seven lodges: 75°.
141
斗二十六度牛八度女十二度虛十度危十七度室十六度壁九度
26th year of Dipper°, eighth year of Ox°, twelfth year of Girl°, tenth year of Emptiness°, 17th year of Rooftop°, 16th year of Encampment°, ninth year of Wall°.
142
北方七宿九十八度
Northern seven lodges: 98°.
143
奎十六度婁十二度胃十四度昴十一度畢十六度觜二度參九度
16th year of Legs°, twelfth year of Bond°, fourteenth year of Stomach°, Hairy eleventh year of Head°, 16th year of Net°, second year of Turtle of Beak°, ninth year of Three of Stars°.
144
西方七宿八十度
Western seven lodges: 90°.
145
井三十三度鬼四度柳十五度星七度張十八度翼十八度軫十七度
33th year of Well°, fourth year of Ghost°, fifteenth year of Willow°, seventh year of Star°, Extended 18th year of Net°, 18th year of Wings°, 17th year of Axletree°.
146
南方七宿百一十二度
Southern seven lodges: 112°.
147
推日度術：
Method for computing the solar degree:
148
置入元至所求年，以歲分乘之，為通實，滿周天分去之，餘如度法而一，為積度，不盡為度分。 命度以虛七度宿次去之，經斗去其分，度不滿宿，算外，即所求年天正冬至日所在度及分。 以冬至去朔日以減分度數，分不足減者，減度一，加度法，乃減之，命如前，即天正朔前夜半日所在度及分。 須求朔共度者，用去定用日數減之，俟後所須。
Set down the years from the epoch to the requested year and multiply by the year fraction to make the total. Remove full celestial circuits; divide the remainder by the degree divisor to obtain accumulated degrees, with the remainder as the degree fraction. beyond tally, Winter Solstice. Subtract the days from winter solstice to the new moon from the degree and fraction. If the fraction is insufficient to subtract, subtract one degree and add the degree divisor before subtracting. Count as above to obtain the solar degree and fraction at midnight before the first new moon. When the new-moon shared degree is needed, subtract the fixed day count used; apply as later required.
149
求次月：大月加度三十，小月加度二十九，宿次去之，經斗去其分。
To find the next month: add 30° for a long month, 29° for a short month; remove by lodge sequence; when passing the Dipper remove its fraction.
150
求次日：加度一，去、命如前。
To find the next day: add 1°; remove and count as before.
151
求朔望加時日所在度術：
Method for finding the sun degree at new/full moon hour:
152
各以定小餘乘章歲，滿十一為度分，以加其前夜半度分，滿之去如前。 凡朔加時日月同度
For each fixed small remainder, multiply by the cycle years; every full eleven makes a degree fraction. Add this to the preceding midnight degree fraction, removing full units as above. At new-moon hour, the sun and moon share the same degree.
153
求轉分：以千四十約度分，不盡為小分。
：, small fraction.
154
求望加時月所在度術：
Method for finding the moon's degree at the time of full moon:
155
置望加時日所在度及分，加度一百八十二，轉分二十五，小分七百五十三； 小分滿千四十從轉分一，轉分滿四十一從度； 去、命如前，經斗去轉分十，小分四百六十六。
25, small fraction. When small fractions reach 1,040, carry 1 to rotating fraction; when rotating fractions reach 41, carry 1 to degrees; Remove and count as before; when passing Dipper remove 10 rotation fractions and 466 small fractions.
156
求月行遲疾日轉定分術：
Method for finding fixed daily rotation fraction of lunar slow-fast motion:
157
以夜半入曆日餘乘轉差，滿周法得一為變差，以進加、退減日轉分為定分。
Multiply midnight calendar-entry day remainder by rotation difference; divide by circuit divisor to obtain one—this is variation difference; advance-add and retreat-subtract to daily rotation fraction for fixed fraction.
158
推朔望夜半月定度術：
Method for computing fixed midnight lunar degree at new and full moon:
159
以定小餘乘所入曆日轉定分，滿日法得一為分，分滿四十一為度，各以減加時月所在度，即各其前夜半定度。
Multiply the fixed small remainder by the adjusted rotation fraction of the entered-calendar day. Every full day divisor makes one fraction, and every full forty-one fractions makes one degree. Subtract each from the moon's degree at the hour to obtain the fixed degree for the preceding midnight.
160
求次日：以日轉定分加轉分，滿四十一從度，去、命如前； 朔日不用前加。
To find the next day: add the fixed daily rotation fraction to the rotation fraction; when 41 carry to degrees; remove and count as before. On new-moon day, the prior addition is not applied.
161
推五星術：
Method for computing the five planets:
162
木數，千七百萬八千三百三十二四分
Wood number (Jupiter): 7008332 and 4 parts.
163
火數，三千三百二十五萬六千二十六。
Fire number (Mars): 33256026.
164
土數，千六百一十二萬一千七百六十七。
Earth number (Saturn): 6121767.
165
金數，二千四百八十九萬八千四百一十七。
Metal number (Venus): 24898417.
166
水數，四百九十四萬一千九十八。
Water number (Mercury): 4941098.,,
167
木終日，三百九十八，日分，三萬七千六百一十二四分。
Jupiter's terminal period is 398 days and 37,612 and one-quarter day fractions.
168
火終日，七百七十九，日分，三萬九千四百六十六。
Mars's terminal period is 779 days and 39,466 day fractions.
169
土終日，三百七十八，日分，三千八百四十七。
Saturn's terminal period is 378 days and 3,847 day fractions.
170
金終日，五百八十三，日分，三萬九千二百九十七。 晨見伏，三百二十七日，分同； 夕見伏，二百五十六日。
Venus's terminal period is 583 days and 39,297 day fractions. Morning appearance and disappearance: 327 days, fractions the same. Evening appearance and disappearance: 256 days。.,,
171
水終日，百一十五，日分，三萬七千四百九十八。 晨見伏，六十三日，分同； 夕見伏，五十二日。
Mercury's terminal period is 115 days and 37,498 day fractions. Morning appearance and disappearance: 63 days, fractions the same. Evening appearance and disappearance: 52 days。.
172
求星見術：
Method for finding the planets' appearances:
173
置通實，各以數去之，餘以減數，其餘如度法得一為日，不盡為日分，即所求年天正冬至後晨平見日及分。 其金、水，以夕見伏日去之，得者餘為夕平見日及分。
degree divisor, day fraction, Winter Solsticemean appearance. For Venus and Mercury, subtract evening appearance-disappearance days; the remainder is evening mean-appearance day and fraction.
174
求平見見月日：置冬至去朔日數及分，各以冬至後日數及分加之，分滿度法從日，起天正月，依大小去之，不滿月者為去朔日，命日算外，即星見所在月日及分。
To find mean appearance:Winter Solsticeremove parts, each Winter Solstice parts thereof , parts when full degree divisor carry to , celestial first month, remove , not when full as remove , beyond the tally, then that which at parts.
175
求後見：各以終日及分加之，滿去如前。 其金、水各以晨夕加之，滿去如前，加晨得夕，加夕得晨。
：, when full remove as before. For Venus and Mercury, add morning or evening respectively; when full, remove as before—adding morning yields evening, adding evening yields morning.
176
木：平見在春分前者，以三千三百四十乘去大寒後十日數，以加平見分，滿法去之，以為定見日及分。 立秋後者，以四千二百乘去寒露日，加之，滿同前。 春分至清明均加四日，後至立夏五日，以後至芒種加六日，均至立秋。 小雪前者，以七千四百乘去寒露日數，以減平見日分； 冬至後者，以八千三百乘去大寒後十日數，以減之； 小雪至冬至均減八日，為定日數。 初見伏去日各十四度。
Jupiter: when its mean appearance is before Spring Equinox, multiply the days elapsed since ten days after Great Cold by 3,340 and add this to the mean appearance fraction. When it fills the divisor, remove it as above to obtain the fixed appearance day and fraction. For those after Start of Autumn, multiply the days until Cold Dew by 4,200 and add this; when it fills the divisor, proceed as before. From spring equinox through Pure Brightness add 4 days uniformly; then through Start of Summer add 5; then through Grain in Ear add 6, uniformly through Start of Autumn. Lesser Snow, Cold Dew, mean appearance day fraction. For those after Winter Solstice, multiply the days elapsed since ten days after Great Cold by 8,300 and subtract this. From Lesser Snow through Winter Solstice uniformly subtract eight days, yielding the fixed day count. At first appearance and disappearance, each is fourteen degrees from the sun.
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火：平見在雨水前，以二萬六千八百八十乘去大寒日數； 在立夏後，以萬三千四百四十乘去立秋日數，以加見日分，滿去如前； 雨水至立夏，均加二十九日。 小雪前，以萬一千五百八十乘去處暑日數； 冬至後，以三萬四千三百八十乘去大寒日數，滿去如前，以減之； 小雪至冬至，均減二十五日。 初見伏去日各十七度。
Mars: mean appearance at Rain Water, Great Cold. After Start of Summer, multiply the number of days until Start of Autumn by 13,440 and add this to the appearance-day fraction; when it fills the divisor, remove it as before. Rain Water through Start of Summer, uniformly add 29day. Lesser Snow, End of Heat. After Winter Solstice, multiply the number of days elapsed since Great Cold by 34,380; when it fills the divisor, remove it as before and subtract it. Lesser Snow through Winter Solstice, uniformly subtract 25day. At first appearance and disappearance, each is seventeen degrees from the sun.
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土：平見在處暑前，以萬二千三百七十乘去大暑日數； 白露後，以八千三百四十乘去霜降日數，以加見日分，滿去如前； 處暑至白露均加九日。 小寒前，以四千九百八十乘去霜降日數，小寒至立春均減九日，立春後減八日，啟蟄後去七，氣別去一，至穀雨去三，夏至後十日去一，至大暑去盡。 初見伏去日各十七度。
Saturn: mean appearance at End of Heat, Great Heat. After White Dew, multiply the number of days until Frost Descent by 8,340 and add this to the appearance-day fraction; when it fills the divisor, remove it as before. From End of Heat through White Dew add 9 days uniformly. Before Lesser Cold, multiply the days elapsed since Frost Descent by 4,980. From Lesser Cold to Start of Spring, subtract nine days throughout; after Start of Spring subtract eight, after Waking of Insects subtract seven, then reduce the subtraction by one at each qi until Grain Rain, when three are subtracted. After Summer Solstice, subtract one for every ten days, until the subtraction is exhausted at Great Heat. At first appearance and disappearance, each is seventeen degrees from the sun.
179
金：晨平見，在立春前者，以四千一百二十乘去小寒日數小滿後，以四千一百二十乘去夏至日數，以加見日分，滿去如前立春至小滿均加三日。 立秋前，以四千一百二十乘去小暑日數，小雪後以四千一百二十乘去冬至日數，滿去如前，以減之，立秋至小雪均減三日。 夕平見，在啟蟄前，以六千三百九十乘去小雪日數。 清明後，以六千二百九十乘去芒種日數，滿去如前，以減之，啟蟄至清明均減九日。 處暑前，以六千二百九十乘去夏至日數； 寒露後，以六千二百九十乘去大雪日數； 以加之，處暑至寒露均加九日。 初見伏去日各十一度。
Venus: when its morning mean appearance is before Start of Spring, multiply the number of days since Lesser Cold by 4,120; after Lesser Fullness, multiply the number of days until Summer Solstice by 4,120. Add this to the appearance-day fraction, and when it fills the divisor remove it as before. From Start of Spring to Lesser Fullness, uniformly add three days. Start of Autumn, Lesser Heat, Lesser Snow Winter Solstice, when full remove as before and subtract, Start of Autumn Lesser Snow. For evening mean appearance before Waking of Insects, multiply the days until Lesser Snow by 6,390. After Pure Brightness, multiply the number of days until Grain in Ear by 6,290; when it fills the divisor, remove it as before and subtract it. From Waking of Insects to Pure Brightness, uniformly subtract nine days. End of Heat, Summer Solstice. After Cold Dew, multiply the number of days until Great Snow by 6,290. When adding: from End of Heat through Cold Dew uniformly add 9 days. At first appearance and disappearance, each is eleven degrees from the sun.
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水：晨平見，在雨水後、立夏前者，應見不見。 啟蟄至雨水，去日十八度外、四十六度內，晨有木、火、土、金一星已上者，見； 無者不見。 立夏至小滿，去日度如前，晨有木、火、土、金一星已上者，見； 無者亦不見。 從霜降至小雪加一日，冬至至小寒減四日，立春至雨水減三日。 冬至前，一去三，二去二，三去一。 夕平見，在處暑後、霜降前者，應見不見。 立秋至處暑，夕有星，去日如前者，見； 無者亦不見。 霜降至立冬，夕有星，去日如前者，見； 無者亦不見。 從穀雨至夏至，減二日。 初見伏去日各十七度。
Mercury: morning mean appearance before Start of Summer and after Rain Water—should appear but does not. From Awakening of Insects through Rain Water, 18° to 46° from the sun—in the morning, if Jupiter, Mars, Saturn, or Venus is present, it appears. When absent, it is not visible. From Start of Summer through Lesser Fullness, solar distance as before—in the morning, if one or more of Jupiter, Mars, Saturn, or Venus is present, it appears. When absent, it is likewise not visible. From Frost's Descent through Lesser Snow add 1 day; from winter solstice through Lesser Cold subtract 4; from Start of Spring through Rain Water subtract 3. Winter Solstice, 3, 2, 1. Evening mean appearance after End of Heat and before Frost Descent—should appear but does not. From Start of Autumn to End of Heat, if a star appears in the evening at the same solar distance as above, it is visible. When absent, it is likewise not visible. From Frost Descent to Start of Winter, if a star appears in the evening at the same solar distance as above, it is visible. When absent, it is likewise not visible. Carry to Grain Rain through Summer Solstice, subtract2day. At first appearance and disappearance, each is seventeen degrees from the sun.
181
行五星法：
Method for computing the five planets' motions:
182
置星定見之前夜半日所在宿度算及分，各以定見日分加其分，滿度法從度。 又以星初見去日度數，晨減、夕加之，滿去如前，即星初見所在度及分。
Set the star fixed-appearance prior midnight solar lodge degree count and fraction; add the fixed-appearance day fraction to each fraction; when full carry from the degree divisor to degrees. Take the star's first-appearance distance from the sun; subtract for morning, add for evening; when full, remove as before—this gives the star's first-appearance degree and fraction.
183
求次日：各加一日所行度及分，有小分者，各日數為母，小分滿其母去從分，分滿度法從度。 其行有益疾遲者，副置一日行分，各以其分疾益遲損之。 留者因前，退則減之，伏不注度，順行出斗去其分，退行入斗先加分。 訖，皆以千四十約分，為大分，以四十一為母。
To find the following day, add the degree and fraction traveled in one day. When there is a small fraction, use the number of days as the denominator; when the small fraction fills its denominator, carry it to the fraction, and when the fraction fills the degree divisor, carry it to degrees. Its Fast motion Slow motion, 1 daily motion parts, each its parts Fast motion Slow motion thereof . Stationary, then subtract thereof , disappearance not °, direct remove its parts, parts. When done, all reduce fractions by 1,040 to obtain large fractions, with 41 as the denominator.
184
木：初見，順，日行萬六百一十八分，日益遲六十分，一百一十四日行十九度、萬三千八百三十二分而留。 二十六日乃退，日六千一百一分，八十四日退十二度、八百四分。 又留二十五日、三萬七千六百一十二分、小分四，乃順。 初日行三千八百三十七分，日益疾六十分，百一十四日行十九度、萬三千七百一十八分而伏。
Jupiter ：first appearance, direct , daily motion 618, daily decreasing slow 60, 114daily motion 19 and 3,832then stationary. 26then retrograde , 6,101, 84retrograde 12 and 804. Also stationary 25 days, 37,612 fractions, small fraction 4—then direct motion. Initial daily motion 3,837 fractions, increasing by 60 per day; in 114 days it travels 19° 13,718 fractions, then disappears.
185
土：初見，順，日行三千八百一十四分，八十三日行七度、萬八千八十二分而留。 三十八日乃退，日二千五百六十三分，百日退六度、四百六十分。 又留三十七日、三千八百四十七分乃順，日三千八百一十三分，八十三日行七度萬七千九百九十九分，如初乃伏。
Saturn ：first appearance, direct , daily motion 3,814, 83daily motion 7 and 8,082then stationary. 38then retrograde , 2,563, retrograde 6 and 460. Also stationary 37 days, 3,847 fractions, then direct at 3,813 fractions daily; in 83 days travels 7° 17,999 fractions—as initially—then disappears.
186
火：初見已後各如其法：
Mars: after first appearance, each phase follows its method:
187
見在雨水前，以見去小寒日數，小滿後，以去大暑日數； 三約之，所得減日為定日； 雨水至小滿，均去二十日為定日。 已前皆前疾日數及度數。 各計冬至後日數，依損益之，為定日數及度數。 以度法乘定度，如定日得一，即平行一日分，不盡為小分。 大寒至立秋差行，餘平行。 處暑至白露，皆去定日，定度六。 白露至寒露，初日行半度，四十日行二十度，餘日及餘度續同前。 置日數減一，以三十乘之，加平行一日分，為初日分。 差行者，日益遲六十分，各盡其日度而遲。 初日行二萬六百分，日益遲百分，六十日行二十四度、三萬五千六百四十分其前疾去度六者，此遲初日加四千二百六十四分，六十日行三十度，分同。 而留。 十三日前去日者，分、日於二留，奇縱後留。 乃退，日萬二千八十二分，六十日退十七度、四十分。 又留，十二日三萬九千四百六十六分。 又順，遲，初日行萬四千七百分，日益疾百分，六十日行二十四度，分同前，此遲在立秋至秋分加一日，行分四千二百六十四，六十日行四十度，分同前。 而後疾。
If the appearance falls before Rain Water, use the number of days from the appearance to Lesser Cold; after Lesser Fullness, use the number of days until Great Heat. Divide by 3; subtract the result from days to obtain fixed days. From Rain Water to Lesser Fullness, uniformly remove twenty days to make the fixed day. All preceding entries give the day and degree counts of the prior fast-motion phase. For each, count the days after Winter Solstice and increase or decrease according to them, making the fixed number of days and degrees. Degree divisor, 1, namely uniform motion 1day fraction, small fraction. Great Cold Start of Autumn, remainder uniform motion. From End of Heat to White Dew, subtract the fixed day in each case; the fixed degree is six. From White Dew to Cold Dew, the initial daily motion is half a degree; in forty days it travels twenty degrees, and the remaining days and degrees continue according to the preceding rule. 1,, uniform motion 1day fraction, day fraction, slow motion60fraction, slow motion. The first day's motion is 20,600 fractions, decreasing by 100 each day. In sixty days it travels 24 degrees and 35,640 fractions. In the preceding fast section, when six degrees are removed, add 4,264 fractions to this slow section's first day, so that in sixty days it travels 30 degrees, with the same fraction. Then stationary. When thirteen days before leaving the sun, distribute fraction and days between two stationary phases; odd remainders go to the later stationary phase. Then retrograde , 2,082, 60retrograde 17 and 40. Also stationary , 139466 parts.,, It then moves direct and slow; the first day's motion is 14,700 fractions, increasing by 100 each day, and in sixty days it travels 24 degrees with the same fraction as before. When this slow motion falls from Start of Autumn to Autumn Equinox, add one day and 4,264 fractions of motion, so that in sixty days it travels 40 degrees, with the same fraction as before. Then later fast motion.
188
後遲加六度者，此後疾去度為定度，已前皆後疾日數及度數。 其在立夏至，小暑，日行半度，盡六十日，行三十度。 小暑至立秋，盡四十日，行二十度。 計餘日及度，從前法。 前法皆平行。 求行分亦如前。 各盡其日度而伏。
When later slow motion adds 6°, subtract from later fast motion for fixed degrees; all prior entries give later-fast day and degree counts. When it falls from Start of Summer to Lesser Heat, it moves half a degree per day; through sixty days it travels 30 degrees. From Lesser Heat to Start of Autumn, through forty days it travels 20 degrees. Compute remaining days and degrees according to the prior method. All preceding phases use uniform motion. To find motion fractions, proceed as before. Each completes its allotted days and degrees and disappears.
189
金：晨初見，乃退，日半度，十日退五度而留。 九日乃順，遲，差行，先遲日益五百分，四十日行三十度。 小暑前以去芒種日數，十日減一度； 立冬後以去大雪日數，十日減一度； 小暑至立冬，均減三度為定度。 大雪至芒種不加減。 求初日，以三十乘度法，四十得一為平分。 又以三十九乘二百五十，以減平分為初日行分。 平行，日一度，十五日行十五度。 小寒後十日，益日度各一，至雨水二十一日，行二十一度。 均至春分後十日減一，至小滿，複十五日行十五度。 其後六日減一，至處暑，日及度皆盡。 至霜降後，四日益一，至冬至複十五日行十五度疾，百七十日行二百四度。 前順遲減度者，計減數益此度為定度。 求一日行度分者，以百七十日日一度以減定度，餘乘度法，如百七十得一，為一日平行度分。 晨伏東方。 夕初見，順，疾，百七十日行二百四度。 夏至前，以見去小滿日數，六日加一度； 小暑後，以去立秋日數，六日加一度，夏至至小暑均加五度，為定度。 白露至清明，差行，先疾日益遲百分。 清明至白露，平行，求一日平行同，晨疾求差行，以五十乘百六十九，加之，為初日行度分。 平行，日一度，十五日行十五度。 冬至後十日減日度各一，至啟蟄九日行九度。 均至夏至後五日益一，至大暑複十五日行十五度。 均至立秋後六日益一，至寒露二十五日行二十五度。 後六日減一，至大雪複十五日行十五度，均至冬至。 順，遲，差行，先疾，日益五百分，四十日行三十度。 前加度者，此依數減之，求初日行分。 如晨遲，唯減者為加之。 又留，九日乃退，日半度，十日退五度，而夕伏西方。
Venus: at its first morning appearance, it retrogrades half a degree per day; in ten days it retreats five degrees and then becomes stationary. After nine days it moves direct and slow, with variable motion: first slow, then increasing by 500 fractions each day; in forty days it travels 30 degrees. Before Lesser Heat, counting from Grain in Ear, reduce 1° every 10 days; After Start of Winter, counting from Great Snow, reduce 1° every 10 days; From Lesser Heat to Start of Winter, uniformly subtract three degrees to make the fixed degree. From Great Snow through Grain in Ear, no addition or reduction. To find the initial day: multiply the degree divisor by 30; divide by 40 to obtain one equal fraction. Also 39multiply 250, subtract parts as daily motion parts. In uniform motion it travels one degree per day, so in fifteen days it travels 15 degrees. After ten days past Lesser Cold, add one to the days and degrees each day; by the twenty-first day at Rain Water, it travels 21 degrees. Uniformly through 10 days after spring equinox reduce by 1; through Lesser Fullness, again 15 days travel 15°. After that, subtract one every six days; by End of Heat, both days and degrees are exhausted. After Frost Descent, increase by one degree every four days; by Winter Solstice, it again travels fifteen degrees in fifteen days at rapid speed, and in 170 days it travels 204 degrees. When prior direct-slow reduces degrees, compute the reduction and add to this degree for fixed degree. To find the degree fraction for one day of motion, subtract 170 days of one degree per day from the fixed degree, multiply the remainder by the degree divisor, and divide by 170 to obtain the degree fraction of one day's mean motion. Morning disappearance in the east. At its first evening appearance, it moves direct and fast; in 170 days it travels 204 degrees. Before Summer Solstice, use the number of days from the appearance to Lesser Fullness and add one degree for every six days. After Lesser Heat, use the number of days until Start of Autumn and add one degree for every six days; from Summer Solstice to Lesser Heat, uniformly add five degrees to make the fixed degree. From White Dew to Pure Brightness, its motion is variable: first fast, then each day slower by 100 fen. From Pure Brightness to White Dew, the method for finding one day of mean motion is the same. To find the variable fast morning motion, multiply fifty by 169 and add it to make the first day's degree fraction. In uniform motion it travels one degree per day, so in fifteen days it travels 15 degrees. 10 days after winter solstice reduce days and degrees by 1 each; through Awakening of Insects, 9 days travel 9°. Uniformly through 5 days after summer solstice increase by 1; through Great Heat, again 15 days travel 15°. Uniformly through 6 days after Start of Autumn increase by 1; through Cold Dew, 25 days travel 25°. 1, Great Snow15 daily motion, Winter Solstice.,,,,, daily motion.,, daily motion. As with morning slow motion—only where it says subtract, add instead. Also stationary , 9 then retrograde , °, 15°, and evening disappearance.
190
水：晨初見，留六日。 順，遲，日行萬六百六十分，四日行一度。 大寒至雨水不須此遲行。 平行，日一度，十日行十度。 大寒後二日，去日度各一，盡二十日，日及度俱盡。 疾，日行一度三萬八千三百七十六分，十日行十九度，前無遲行者，減此分萬二千七百九十二分，十日行十六度。 晨伏東方。 夕初見，順，疾，日行一度三萬八千三百七十六分，十日行十九度。 小暑至白露減萬二千七百九十二分，十日行十六度。 平行，日一度，十日行十度。 大暑後二日，去日度各一，盡二十日，日及度俱盡。 遲，日行萬六百六十分，四日行一度。 疾減萬二千七百九十二分者，不須此遲。 行又留六日，夕伏西方。
Mercury: at its first morning appearance, it remains stationary for six days. Direct, Slow motion, daily motion 660 parts, 4 daily motion 1°. From Great Cold through Rain Water, this slow-motion phase is not required. In uniform motion it travels one degree per day, so in ten days it travels 10 degrees. After two days past Great Cold, remove one from the days and degrees each day; after twenty days, both days and degrees are exhausted. Fast motion, daily motion 1°38376 parts, 10 daily motion 19°, prior none Slow motion, subtract this parts2792 parts, 10 daily motion 16°. Morning disappearance in the east. Evening first appearance, direct, Fast motion, daily motion 1°38376 parts, 10 daily motion 19°. From Lesser Heat through White Dew reduce 12,792 fractions; in 10 days travel 16°. In uniform motion it travels one degree per day, so in ten days it travels 10 degrees. After two days past Great Heat, remove one from the days and degrees each day; after twenty days, both days and degrees are exhausted. Slow motion, daily motion 660 parts, 4 daily motion 1°. When fast motion reduces 12,792 fractions, this slow phase is not required. Also stationary 6, evening disappearance.
191
推交會術：
Method for computing crossings and conjunctions:
192
會通，千六十四萬六千七百二十九。
Conjunction common: 646729.
193
朔差，九十萬七千五十七。
New-moon difference: 907057.
194
望差，四十五萬三千五百二十八半。
Full-moon difference: 453528 and a half.
195
單數，五百三十二萬三千三百六十四半。
Single number: 5323364 and a half.
196
時法，三萬二千六百四。
Hour divisor: 32604.
197
望數，五百七十七萬六千八百九十三。
Full-moon number: 5776893.
198
外限，四百八十六萬九千八百三十六。
Outer limit: 4869836.
199
內限，千一十九萬三千二百半。
Inner limit: 193200 and a half.
200
中限，五百六十四萬九千四百四半。
Middle limit: 5649404 and a half.
201
次限，千三十二萬六百八十九。
Secondary limit: 320689.
202
推入交法：
Method for computing entry into crossing:
203
以會通去積月，餘以朔望差乘之，滿會通又去之，餘為所求年天正朔入交餘。
Remove accumulated months by the conjunction circuit. Multiply the remainder by the new/full-moon difference and remove another conjunction circuit when it fills; the remainder is the crossing remainder for the first new moon of the requested year.
204
求望，望數加之，滿、去如前。
To find full moon: add the full-moon number; when full, remove as before.
205
求次月，以朔差加之，滿、去如前。
To find the next month: add the new-moon difference; when full, remove as before.
206
推交道內外及先後去交術：
Method for computing crossing inner/outer path and prior/later node distance:
207
其朔望在啟蟄前，以一千三百八十乘去小寒日數； 在穀雨後，以乘去芒種日數，為氣差以加之，啟蟄至穀雨均加六萬三千六百； 滿會通去之，餘為定餘。 其小寒至春分，立夏至芒種，朔值盈二時已下，皆半氣差而加之； 二時已上，皆不加。 朔入交餘如望差、望數已下，中限已上，有星伏，木、土去見十日外，火去見四十日外，金、晨伏去見二十二日外。 有一星者不加氣差。 朔望在白露前者，以九百乘去小暑日數； 在立冬後者，以千七百七十乘去大雪日數，以減之； 白露至立冬均減五萬五千，不足減者，加會通乃減之，餘為定餘。 朔入交餘如外限、內限已上，單數次限已下有星伏，如前者，不減氣差。 定餘不滿單數者，為在外； 滿去之，餘在內。 其餘如望差已下、外限已上，望則月食； 在內者，朔則日食。 其餘如望差已下者，即為去先交餘； 如外限已上者，以減單數，餘為去後交餘。 如時法得一，然為去交時數。
When new or full moon is before Awakening of Insects, multiply 1,380 by days since Lesser Cold. After Grain Rain, multiply by the number of days until Grain in Ear to make the qi difference and add it; from Waking of Insects to Grain Rain, uniformly add 63,600. When full, remove the conjunction common; the remainder is the fixed remainder. From Lesser Cold to Spring Equinox and from Start of Summer to Grain in Ear, when the new moon falls at two hours or less of excess, halve the qi difference and add it. At two hours or above, do not add. When new-moon crossing remainder is below full-moon difference and full-moon number, above middle limit, with star disappearance: Jupiter and tenth year of Saturn+ days from appearance, 40th year of Mars+, Venus morning disappearance 22+. When only one luminary is involved, do not add the qi difference. White Dew, Lesser Heat. For those after Start of Winter, multiply the days until Great Snow by 1,770 and subtract this. From White Dew to Start of Winter, uniformly subtract 55,000; if there is not enough to subtract, add the conjunction circuit and then subtract. The remainder is the fixed remainder. When new-moon crossing remainder is at outer/inner limit or above, with star disappearance below single-number secondary limit as before—do not subtract qi difference. When the fixed remainder is less than the single number, it is outside. When full, remove it; the remainder is inside. Remainder below full-moon difference and above outer limit—at full moon there is lunar eclipse. When inside the node, a solar eclipse occurs at new moon. The remainder below the full-moon difference is the prior-crossing remainder removed.,, remainder is. Divide by the hour divisor to obtain one—this is the crossing hour count removed.
208
推月食加時術：
Method for computing lunar eclipse hour:
209
置食定日小餘，三之，如辰法得一辰，命以子算外，即所在辰。 不盡為時餘，四之，如法，無所得為辰初，一為少，二為半，三為太。 又不盡者，三之，如法，得一為強，以並少為少強，並半為半強，並太為太強； 得二強者為少弱，並少為半弱，並半為太弱，並太為辰末。 此加時謂食時月在沖也。
Set down the fixed-day small remainder for the eclipse and triple it; every full chronogram divisor makes one double-hour. Count from zi, excluding the starting count, to obtain the double-hour. The remainder is the hour remainder; multiply by 4; divide by the divisor—zero gives chronogram start, 1 is "less," 2 is "half," 3 is "greater". If still remainder, multiply by 3; divide by the divisor—one gives "strong"; combined with "less" gives "less-strong," with "half" gives "half-strong," with "greater" gives "greater-strong". Two "strong" gives "less-weak"; combined with "less" gives "half-weak," with "half" gives "greater-weak," with "greater" gives chronogram end. This hour of occurrence means the moon at opposition during the eclipse.
210
推日食加時術：
Method for computing solar eclipse hour:
211
置食定日小餘，秋三月，內道，去交八時已上，加二十四，十二時以加四十八； 春三月，內道，去交七時已上，加二十四。 乃以三乘之，如辰法得一辰，以命子算外，即所在辰。 不盡為時餘。 副置時餘，仲辰不滿半辰，減半辰，已上去半辰； 季辰者直加半辰； 孟辰者減辰法，餘加半辰為差率。
Set down the fixed-day small remainder for the eclipse. In the three autumn months, on the inner path, if the distance from the crossing is eight hours or more, add 24; if twelve hours, add 48. In the third month of spring, inner path, node distance seven hours or above—add 24.,, Then multiply it by three; every full chronogram divisor makes one double-hour. Count from zi, excluding the starting count, to obtain the double-hour. The remainder is the time remainder. Set aside the hour remainder. If the middle double-hour remainder is less than half a double-hour, subtract half a double-hour; if it is half or more, remove half a double-hour. For seasonal chronograms, add half a chronogram directly; For primary chronograms subtract the chronogram divisor; add half a chronogram to the remainder for the difference rate.
212
又，置去交時數，三已下加三，六已下加二，九已下加一，九已上依數，十二已上從十二； 以乘差率，如十四得一為時差。 子半至卯半、午半至酉半，以加時餘； 卯半至午半、酉半至子半，以減時餘。 加之，滿辰法去之，進一辰，減之若不足，退一辰，餘為定時餘。 乃如月食法，子午卯酉為仲，辰戌醜未為季，寅申巳亥為孟。 日出前入後各二時外，不注日食。 三乘氣時法得一，命子算外為時。
Also set down the number of hours from the crossing: add three if it is three or less, add two if it is six or less, add one if it is nine or less, use the number itself if it is nine or more, and follow twelve if it is twelve or more. Multiply by the difference rate; divide by 14 to obtain one—this is the hour difference. Half half and half , Multiply by add remainder. Half half and half , Multiply by subtract remainder.,,,,, remainder is. As with the lunar eclipse method: zi-wu-mao-you are mid-chronograms; chen-xu-chou-wei are seasonal chronograms; yin-shen-si-hai are primary chronograms. Solar eclipses more than two hours before sunrise or after sunset are not recorded. 1, beyond tally.
213
求外道日食法：
Method for finding solar eclipses on the outer path:
214
去交一時內者，食。 夏去交二時內，加時在南方三辰者，食。 若去分至十二時內，去交六時內者，亦食。 若去春分三日內，後交二時內，秋分三日內，先交二時內者，亦食。 先交二時內，值盈二時外，及後交二時內，值縮二時外，亦食。 諸去交三時內，星伏如前者，食。
When node distance is within one hour, there is an eclipse. In summer, node distance within two hours, hour in the southern three chronograms—eclipse. If within twelve hours of an equinox or solstice, node distance within six hours—also eclipse. If it is within three days of Spring Equinox with the later crossing within two hours, or within three days of Autumn Equinox with the earlier crossing within two hours, there is also an eclipse. Prior crossing within two hours with expansion beyond two hours, or later crossing within two hours with contraction beyond two hours—also eclipse. Node distance3, star disappearance, eclipse.
215
求內道日不食法：
Method for finding when the sun is not eclipsed on the inner path:
216
加時南方三辰，五月朔先交十三時外，六月朔後交十三時外，不食。 啟蟄至穀雨，先交十三時外，值縮加時在未以西者，不食。 處暑至霜降，後交十三時外，值盈加時在巳以東者，不食。
Hour of occurrence3chronogram, 513, 613, not eclipse. Awakening of Insects through Grain Rain, 13, hour of occurrence, not eclipse. From End of Heat to Frost Descent, when the later crossing is outside thirteen hours and the excess-added hour is east of si, there is no eclipse.
217
求月食分：
To find the lunar eclipse fraction:
218
春後交、秋先交、冬後交，皆去不食餘一時，不足去者，食既。 餘以三萬二百三十五為法，得一為不食分。 不盡者，半法已上為半強，已下為半弱，以減十五，餘為食分。
Crossing and crossing and crossing , remove eclipse remainder 1, remove , eclipse. 30235, 1eclipse fraction.,,, 15, remainder is.
219
推日食分術：
Method for computing solar eclipse magnitude:
220
在秋分前者，以去夏至日數乘二千，以減去交餘，餘為不食餘； 不足減者，反減十八萬四千，餘為不食餘。 亦減望差為定法。 其後交值縮，並不減望差，直以望差為定法。 在啟蟄後者，以去夏至日數乘千五百以減之； 秋分至啟蟄，均減十八萬四千，不足減者，如前； 大寒至小滿，去後交五時外，皆去不食餘一時。 時差減者，先交減之，後交加之，不足減者食既； 值加，先交加之，後交減之。 不足減者食。
For those before Autumn Equinox, multiply the number of days since Summer Solstice by 2,000 and subtract this from the crossing remainder; the remainder is the non-eclipse remainder. remainder is. Also subtract the opposition difference to make the fixed divisor. When a later crossing value contracts, do not subtract the opposition difference; simply use the opposition difference as the fixed divisor. At Awakening of Insects, Summer Solstice500. From Autumn Equinox to Waking of Insects, uniformly subtract 184,000; if there is not enough to subtract, proceed as before. From Great Cold to Lesser Fullness, when the later crossing is more than five hours away, remove one hour from the non-eclipse remainder. eclipse already. Add , crossing add , crossing subtract. When the subtraction cannot be completed, an eclipse occurs.
221
求所起：內道西北，虧東北； 外道西南，虧東南。 十三分以上，正左起。 虧皆據甚時，月則行上起。
To find the initial point: inner path northwest, obscuration northeast. Outer path southwest, obscuration southeast. At thirteen parts or above, begin at due left. Obscuration is always reckoned from maximum; for the moon it begins from the upper limb.
222
求日出入所在術：
Method for finding sunrise and sunset locations:
223
以所入氣辰刻及分，與後氣辰刻及分相減，餘乘入氣日算，如十五得一，以損益所入氣，依刻及分為定刻。
Subtract the entering qi's hour, quarter, and fractional parts from those of the following qi; multiply the remainder by the entering-qi day count, divide by fifteen to obtain one, and decrease or increase the entering qi accordingly to obtain the fixed quarter and parts.